Freeform, φ-polynomial Optical Surfaces: Optical Design, Fabrication and Assembly. Kyle Fuerschbach. Submitted in Partial Fulfillment of the

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1 Freeform, φ-polynomial Optical Surfaces: Optical Design, Fabrication and Assembly by Kyle Fuerschbach Submitted in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Supervised by Professor Jannick Rolland The Institute of Optics Arts, Science and Engineering Edmund A. Haim School of Engineering and Applied Sciences University of Rochester Rochester, New York 1

2 ii Biographical Sketch Kyle Fuerschbach is originally from Albuquerque, NM. After graduating high school, he attended the University of Arizona in Tucson, AZ and graduated summa cum laude with a Bachelor of Science degree in Optical Sciences and Engineering. He began doctoral studies at The Institute of Optics at the University of Rochester in 8. During his tenure he was awarded the Robert L. and Mary L. Sproull University Fellowship in 8, the Frank J. Horton Research Fellowship from 8-1, and the Michael Kidger Memorial Scholarship in Optical Design in 11. He has also served as an elected representative for the University of Rochester s student chapter of SPIE. He pursued his research in optical design and fabrication of optical systems with freeform optics under the direction of Professor Jannick Rolland and co-direction of Dr. Kevin Thompson. The following peer reviewed publications and patents were a result of work conducted during doctoral study: K. Fuerschbach, K. P. Thompson, and J. P. Rolland, "Assembly of an off-axis optical system employing three φ-polynomial, Zernike mirrors," Optics Letters (Accepted to appear April 1). K. Fuerschbach, K. P. Thompson, and J. P. Rolland, "Interferometric measurement of a concave, phi-polynomial, Zernike mirror," Optics Letters 39, 18-1 (1). J. P. Rolland and K. Fuerschbach, "Nonsymmetric optical system and design method for nonsymmetric optical system," US8171 B (13). K. Fuerschbach, J. P. Rolland, and K. P. Thompson, "Extending Nodal Aberration Theory to include mount-induced aberrations with application to freeform surfaces," Opt. Express, (1). K. Fuerschbach, J. P. Rolland, and K. P. Thompson, "A new family of optical systems employing phi-polynomial surfaces," Opt. Express 19, (11). S. Vo, K. Fuerschbach, K. P. Thompson, M. A. Alonso, and J. P. Rolland, "Airy beams: a geometric optics perspective," J. Opt. Soc. Am. A 7, (1).

3 iii Acknowledgments I would like to thank Professor Jannick Rolland and Dr. Kevin Thompson for their support and guidance during the Ph.D. process. The original idea to work on a three mirror freeform design came about when we were deciding what I could present at the 1 International Optical Design Conference. At the time, I didn t know it would eventually become part of my thesis, but through their direction and my hard work, we were able to explore many avenues in freeform optical surfaces that were all prompted by the first pamplemousse design. I would like to thank John Miller at the university machine shop and Gregg Davis and Alan Hedges at II-VI Infrared for providing me with fabrication support. These men helped translate my crazy ideas into tangible, working pieces of hardware that have been critical to the success of my research. I would like to thank all my labmates and officemates: Dr. Cristina Canavesi, Robert Gray, Jinxin Huang, Jianing Yao, Eric Schiesser, Jacob Reimers, and Aaron Bauer. Specifically, I would like to thank Aaron Bauer for answering all my questions throughout the years. He was always willing to help me through a problem or read something I had written. Thanks also to all the students who helped me with my research in the lab: Eddie Lavilla, Jean Inard-Charvin, Johan Thivollet, and Isaac Trumper. Without them, I d still be in the lab working to get my experiments finished. Thanks to Elizabeth for her support during my academic career. She made many personal and professional sacrifices along the way and they have not gone unnoticed.

4 iv Also, thanks to my parents, Phil and Marcie, for helping me get to this point. Without their guidance, I may have never studied optics. Lastly, I would like to thank my support, the Frank J. Horton Research Fellowship, the II-VI Foundation, and the National Science Foundation (EECS-1179) as well as Zygo for their partnership in optical testing, Synopsys Inc. for the student license of CODE V, and Photon Engineering for the student license of FRED.

5 v Abstract Freeform optical surfaces are creating exciting new opportunities in optics for design, fabrication, metrology, and assembly. While the term freeform is currently being applied over a broad range of surface shapes, in our research on imaging with freeform optical surfaces, a freeform is a surface whose sag varies not only with the radial component but also with the azimuthal component, φ, also known as a φ-polynomial optical surface. Interestingly, these surfaces are readily fabricated with techniques like single point diamond turning; however, challenges remain in their optimization during optical design and characterization after fabrication. In this dissertation, we propose a more effective optical design approach based in nodal aberration theory that considers the aberrations induced by a φ-polynomial optical surface up to sixth order. Specifically, when a φ-polynomial overlay is placed on a surface away from the aperture stop, there is both a field constant and field dependent contribution to the net aberration field. These findings are validated through the design, implementation, and wavefront measurement of an aberration generating Schmidt telescope that employs a custom fabricated φ-polynomial plate. The measured wavefront behavior is in good agreement with the theoretical predictions of nodal aberration theory throughout the field of view. The design methods are also applied to a specific example: a wide field, fast focal ratio, long wave infrared, unobscured reflective imager. The system employs three, tilted φ-polynomial surfaces to provide diffraction limited performance throughout the field of view. The surfaces were fabricated with diamond turning and a novel metrology

6 vi approach based on an inteferometric null is proposed for characterizing the figure error of the fabricated surfaces. A mechanical design is also presented for the housing structure that simplifies the system assembly. The as-built optical system maintains diffraction limited performance throughout the field of view. The work conducted in this dissertation provides a foundation for the efficient design of optical systems employing freeform surfaces and demonstrates that a system based on freeform surfaces is realizable in the long wave infrared and may be extended to shorter wavelength regimes.

7 vii Contributors and Funding Sources This work was supervised by a dissertation committee consisting of Professors Jannick Rolland (advisor) and Miguel Alonso of The Institute of Optics, Dr. Kevin Thompson of Synopsys, and Professor Victor Genberg of The Department of Mechanical Engineering. The original matlab code to plot the Full Field Displays in Chapter 3 and Chapter was developed by Dr. Christina Dunn. The fabrication of the components and experiments in Chapter were assisted by Isaac Trumper (undergraduate research assistant) and in part by Edward Lavilla (summer research assistant). The experiments in Chapter were assisted in part by Johan Thivollet (graduate research assistant). The mirror surfaces and optical housing in Chapter 7 were manufactured by II-VI Infrared. All other work conducted for the dissertation was completed by the student independently. Graduate study was supported by the Frank J. Horton Research Fellowship from the Laboratory for Laser Energetics, the II-VI foundation, and the National Science Foundation (EECS-1179).

8 viii Table of Contents Biographical Sketch... ii Acknowledgments... iii Abstract... v Contributors and Funding Sources... vii Table of Contents... viii List of Figures... xiii List of Tables... xxix List of Acronyms... xxxi Chapter 1. Introduction Off-Axis Reflective Systems Offset Aperture and/or Biased Field Tilted Optical Surfaces Freeform Optical Surfaces Motivation Dissertation Outline... 1 Chapter. Aberration Fields for Tilted and Decentered Optical Systems with Rotationally Symmetric Components Aberration Field Centers... 1

9 ix. Wave Aberration Expansion in a Perturbed Optical System Full Field Aberration Display... Chapter 3. Aberration Fields in Optical Systems with φ-polynomial Optical Surfaces Formulating Nodal Aberration Theory for Freeform, ϕ-polynomial Surfaces away from the Aperture Stop The Aberration Fields of ϕ-polynomial Surface Overlays Zernike Astigmatism Zernike Coma Zernike Trefoil (Elliptical Coma) Zernike Oblique Spherical Aberration Zernike Fifth Order Aperture Coma APPLICATION: The Astigmatic Aberration Field Induced by Three Point Mount-Induced Trefoil Surface Deformation on a Mirror of a Reflective Telescope Astigmatic Reflective Telescope Configuration ( W ) in the Presence of a Three Point Mount-Induced Surface Deformation on the Secondary Mirror Anastigmatic Reflective Telescope Configuration ( W = ) in the Presence of a Three Point Mount-Induced Surface Deformation on the Secondary Mirror...

10 x Validation of the Nodal Properties of a Reflective Telescope with Three Point Mount-Induced Figure Error on the Secondary Mirror Extending Nodal Aberration Theory to Include Decentered Freeform ϕ-polynomial Surfaces away from the Aperture Stop... 9 Chapter. Experimental Validation of Nodal Aberration Theory for φ-polynomial Optical Surfaces Design of an Aberration Generating Schmidt Telescope Fabrication of the Aspheric Corrector/Nonsymmetric Plate Experimental Setup of the Aberration Generating Schmidt Telescope Experimental Results The Generated Field Conugate, Field Linear Astigmatic Field Rotation of the Aberration Generating Plate Lateral Displacement of the Aberration Generating Plate... 9 Chapter 5. Design of a Freeform Unobscured Reflective Imager Employing φ-polynomial Optical Surfaces The New Method of Optical Design The Starting Form The Unobscured Form Creating Field Constant Aberration Correction... 98

11 xi 5.3. Creating Field Dependent Aberration Correction The Final Form Mirror Surface Figures Chapter. Interferometric Null Configurations for Measuring φ-polynomial Optical Surfaces Concave Surface Metrology First Order Design Optimization of the Interferometric Null System Experimental Setup of Interferometric Null System Experimental Results Convex Surface Metrology Chapter 7. Assembly of an Optical System with φ-polynomial Optical Surfaces Mechanical Design Sensitivity Analysis Stray Light Analysis As-built Optical System As-built Optical Performance Conclusion and Future Work

12 xii Appendix A. Vector Multiplication and Its Vector Properties and Identities List of References... 1

13 xiii List of Figures Figure 1-1. Demonstration of how an on-axis optical system is made unobscured by offsetting the aperture, biasing the field, or a combination of both.... Figure 1-. Single point diamond turning surface roughness evolution through time. Each color represents a lateral measurement of a part from a specific time period. (Adapted from Schaefer [5])... 1 Figure 1-3. Optical design space defined by the light collection (F/number), area collection (FOV), and packaging for various surface representations Figure -1. Coordinate system for aberration theory of a perturbed optical system where both the pupil and field coordinate are represented as vectors Figure -. Representation of the new effective field vector.... Figure -3. Node locations for third order astigmatism in a perturbed optical system. There are two points in the field where the aberration can be zero Figure -. Full field display (FFD) showing (a) third order field quadratic in a centered system and (b) in perturbed optical system that yields binodal astigmatism Figure 3-1. (a) When the aspheric corrector plate of a Schmidt telescope is displaced longitudinally from the aperture stop, the beam for any off-axis field point will displace along the corrector plate. The displacement depends on the paraxial quantities for the marginal ray height, y, chief ray height, y, chief ray angle, u, and the distance between the stop and plate, t. (b) Alternatively, the beam displacement on the corrector plate can be thought of as a field dependent

14 xiv decenter of the aspheric corrector, h, that modifies the mapping of the normalized pupil coordinate from ρ to ρ ' Figure 3-. Generation of coma and astigmatism as the aspheric corrector plate in a Schmidt telescope is moved longitudinally (along the optical axis) from the physical aperture stop located at the center of curvature of the spherical primary mirror for various positions (a-d). For each field point in the FFD, the plot symbol conveys the magnitude and orientation of the aberration. (e) Plots of the magnitude of coma and astigmatism generated as the aspheric plate is moved longitudinally for two field points, (, ) (blue square) and (, ) (red triangle) Figure 3-3. Fringe Zernike polynomial set up to 5 th order ( th order in wavefront). The set includes Z 1 (piston), Z /3 (tilt), Z (defocus), Z 5/ (astigmatism), Z 7/8 (coma), Z 9 (spherical aberration), Z 1/11 (elliptical coma or trefoil), Z 1/13 (oblique spherical aberration or secondary astigmatism), Z 1/15 (fifth order aperture coma or secondary coma), and Z 1 (fifth order spherical aberration or secondary spherical aberration). The φ-polynomials to be explored include Z 5/, Z 7/8, Z 1/11, Z 1/13, and Z 1/ Figure 3-. Surface map describing the freeform Zernike overlay for astigmatism on an optical surface over the full aperture. The error is quantified by its magnitude z and its orientation FF 5/ FFξ5/ that is measured clockwise with respect to the ŷ axis. P and V denote where the surface error is a peak rather than a valley... 39

15 xv Figure 3-5. The characteristic field dependence of field constant astigmatism that is generated by a Zernike astigmatism overlay on an optical surface in an optical system. This induced aberration is independent of stop position Figure 3-. The characteristic field dependence of (a) field constant coma, (b) field asymmetric, field linear astigmatism, and (c) field linear, field curvature that is generated by a Zernike coma overlay on an optical surface away from the stop surface.... Figure 3-7. The characteristic field dependence of (a) field constant elliptical coma, (b) field conugate, field linear astigmatism, which is generated by a Zernike elliptical coma overlay on an optical surface away from the stop surface Figure 3-8. The characteristic field dependence of (a) field constant oblique spherical aberration, (b) field asymmetric, field linear trefoil, (c) field conugate, field linear coma, (d) field constant, field quadratic astigmatism, and (e) field quadratic, field curvature that is generated by a Zernike oblique spherical aberration overlay on an optical surface away from the stop surface Figure 3-9. The characteristic field dependence of (a) field constant, fifth order aperture coma, (b) field linear medial oblique spherical aberration, (c) field asymmetric, field linear oblique spherical aberration, (d) field quadratic trefoil, (e) field quadratic coma, (f) field asymmetric, field cubed astigmatism, and (g) field cubic, field curvature that is generated by a Zernike fifth order aperture coma overlay on an optical surface away from the stop surface

16 xvi Figure 3-1. (a) The nodal behavior for an optical system with conventional third order field quadratic astigmatism and Zernike trefoil at a surface away from the stop, e.g., a two mirror telescope with a three point mount-induced error on the secondary mirror, is displayed in a reduced field coordinate, Π, where the node located by ( MNTERR x ) has an orientation angle of MNTERRξ 1/11 and a magnitude that is proportional to MNTERRC 333, SM. The two related nodes on the circle are then advanced by 1º and º for this special case. (b) When the nodal solutions are re-mapped to the conventional field coordinate, H, the node located by ( MNTERR x ) has an orientation angle of MNTERRξ 1/11 and a magnitude that is 3 proportional to C... 3 MNTERR 333, SM Figure A measurement or simulation of the mount-induced error on the secondary mirror yields the magnitude and orientation of MNTERRC 333, SM... 3 Figure 3-1. (a) Layout for a F/8, 3 mm Ritchey-Chrétien telescope and (b) a Full Field Display (FFD) of the RMS WFE of the optical system at.33 µm over a ±. FOV. Each circle represents the magnitude of the RMS WFE at a particular location in the FOV.... Figure Displays of the magnitude and orientation of Fringe Zernike astigmatism (Z 5/ ) and Fringe Zernike trefoil, elliptical coma, (Z 1/11 ) throughout the FOV for (a) a Ritchey-Chrétien telescope in its nominal state and (b) the telescope when.5λ of three point mount-induced error oriented at has been added to the

17 xvii secondary mirror. It is important to recognize that these displays of data are FFDs that are based on a Zernike polynomial fit to real ray trace OPD data evaluated on a grid of points in the FOV. For each field point, the plot symbol conveys the magnitude and orientation of the Zernike coefficients pairs, Z 5/ on the left and Z 1/11 on the right.... Figure 3-1. (a) Layout for a JWST-like telescope geometry and (b) a Full Field Display (FFD) of the RMS WFE of the optical system at 1. µm over a ±. FOV. The system utilizes a field bias (outlined in red) to create an accessible focal plane.. 8 Figure Displays of the magnitude and orientation of Fringe Zernike astigmatism (Z 5/ ) and Fringe Zernike trefoil, elliptical coma, (Z 1/11 ) throughout the FOV for (a) a JWST-like telescope in its nominal state and (b) the telescope when.5λ of three point mount-induced error oriented at has been added to the secondary mirror Figure 3-1. Displays of the magnitude and orientation of Fringe Zernike astigmatism (Z 5/ ) and Fringe Zernike trefoil, elliptical coma, (Z 1/11 ) throughout the FOV for a JWST-like telescope with.5λ of three point mount-induced error oriented at on the off-axis tertiary mirror Figure -1: Testing configuration for the Schmidt telescope to demonstrate the field dependent aberration behavior of a freeform optical surface. A freeform, Zernike plate can purposely be placed at or away from the stop surface to induce field dependent aberrations. The aberration field behavior of the telescope is measured

18 xviii interferometrically by acquiring the double pass wavefront over a twodimensional FOV with a scanning mirror Figure -. Layout of the nominal Schmidt telescope configuration. The aspheric and Zernike trefoil plate are both fabricated in NBK7 substrates and the primary mirror is a commercially available 15. mm, F/1 concave, spherical mirror Figure -3. Simulated interferogram at a wavelength 3.8 nm of the 3 µm trefoil deformation added on one surface of the 1 mm, NBK7 plate to be added into the optical path of the nominal Schmidt telescope Figure -. (a) The predicted astigmatism (Z 5/ ) and (b) elliptical coma (Z 1/11 ) FFDs over a square, 5 degree full FOV for the Schmidt telescope system with the Zernike trefoil plate oriented at and located 1mm away from the stop surface. The Zernike trefoil plate generates both field constant elliptical coma and field conugate, field linear astigmatism Figure -5. The predicted magnitude of the (a) astigmatism (Z 5/ ) and (b) elliptical coma (Z 1/11 ) as a function of the Zernike trefoil plate position relative to the stop surface for the ( Hx 1, Hy ) = = field point of Schmidt telescope configuration Figure -. First order layout demonstrating how the retro-reflector must be designed to ensure that the pupil of the Schmidt telescope is conugate to the pupil of the concave mirror that sends the wavefront back towards the interferometer Figure -7. (a) Measured surface departure of the aspheric corrector plate for the Schmidt telescope and (b) residual error when the nominal optical design surface is

19 xix subtracted from the measured surface. The error is about.5λ PV or.λ RMS at the testing wavelength of 3.8 nm Figure -8. (a) Measured surface departure of the Zernike trefoil plate and (b) residual error when the nominal optical design surface is subtracted from the measured surface. The error is about.3λ PV or.5λ RMS at the testing wavelength of 3.8 nm Figure -9. Experimental setup of the Schmidt telescope system. The scanning mirror and retro-reflector are motorized so that the FOV can be scanned over a twodimensional grid of points. The trefoil plate is also motorized so that effect of plate position on magnitude of generated aberration field can be studied Figure -1. (a) Measured interferograms after baseline subtraction for a 3x3 grid of field points spanning a square, 5 degree diagonal FOV for the Schmidt telescope system with the Zernike trefoil plate oriented at and displaced roughly 1 mm longitudinally away from the stop surface and (b) the 3x3 grid of wavefronts with the field constant elliptical coma removed, revealing the generated field conugate, field linear astigmatism induced by the trefoil plate Figure -11. The measured Zernike astigmatism (Z 5/ ) FFD after baseline subtraction, left, and theoretical Zernike astigmatism (Z 5/ ) FFD predicted by NAT, right, over a 9x9 grid spanning a square, 5 full FOV for the Schmidt telescope system with the Zernike trefoil plate oriented at and located (a) 1.81 mm, (b) mm, and (c) mm away from the stop surface

20 xx Figure -1. Plot of the mean magnitude of the Zernike trefoil and astigmatism after baseline subtraction for two field points, ( Hx 1, Hy ) circle and ( Hx 1, Hy ) = = represented by the blue = = represented by the red star, for five measured plate positions. The error bars on the data points represent plus or minus one standard deviation from the mean value over the ten measurements acquired at each plate position. In black, the magnitude of the Zernike trefoil and astigmatism based on the theoretical predictions of NAT is plotted as a function of plate position Figure -13. The (a) measured Zernike astigmatism (Z 5/ ) FFD after baseline subtraction and (b) theoretical Zernike astigmatism Z 5/ FFD predicted by NAT over a 9x9 grid spanning a square, 5 degree full FOV for the Schmidt telescope system with the Zernike trefoil plate oriented at 5 and located roughly 1 mm away from the stop surface Figure -1. The (a) measured Zernike astigmatism (Z 5/ ) FFD after baseline subtraction and (b) theoretical Zernike astigmatism (Z 5/ ) FFD predicted by NAT over a 9x9 grid spanning a square, 5 degree full FOV for the Schmidt telescope system with the Zernike trefoil plate oriented at, located roughly 1 mm away from the stop surface, and displaced laterally 1 mm in the x-direction and -1 mm in the y- direction Figure 5-1. (a) Layout of U.S. Patent 5,39,7 consisting of three off-axis sections of rotationally symmetric mirrors and a fourth fold mirror (mirror 3). The optical system had, at the time of its design, the unique property of providing the largest

21 xxi planar, circular input aperture in the smallest overall spherical volume for a gimbaled application. (b) The new optical design based on tilted φ-polynomial surfaces to be coupled to an uncooled microbolometer Figure 5-. (a) Layout for a fully obscured solution for a F/1.9, 1 full FOV LWIR imager. The system utilizes three conic mirror surfaces. (b) A FFD of the RMS WFE of the optical system. Each circle represents the magnitude of the RMS wavefront at a particular location in the FOV. The system exhibits a RMS WFE of < λ/5 over 1 full FOV Figure 5-3. The lens layout, Zernike coma (Z 7/8 ) and astigmatism (Z 5/ ) FFDs for a ± FOV for the (a) on-axis optical system, (b) halfway tilted, 5% obscured system, and (c) fully tilted, 1% unobscured system. The region in red shows the field of interest, a 1 diagonal FOV Figure 5-. The lower order spherical (Z 9 ), coma (Z 7/8 ), astigmatism (Z 5/ ), higher order, elliptical coma (Z 1/11 ), oblique spherical aberration (Z 1/13 ), and fifth order aperture coma (Z 1/15 ) Zernike aberration contributions and RMS WFE FFDs over a ±5 degree FOV for the fully unobscured, on-axis solution. It can be seen that the system is dominated by field constant coma and astigmatism which are the largest contributors to the RMS WFE of ~1λ Figure 5-5. The lower order spherical (Z 9 ), coma (Z 7/8 ), astigmatism (Z 5/ ), higher order, elliptical coma (Z 1/11 ), oblique spherical aberration (Z 1/13 ), and fifth order aperture coma (Z 1/15 ) Zernike aberration contributions and RMS WFE FFDs over

22 xxii a ±5 degree FOV for the optimized system where Zernike astigmatism and coma were used as variables on the secondary (stop) surface. When the system is optimized, the field constant contribution to astigmatism and coma are greatly reduced improving the RMS WFE from ~1λ to ~.75λ Figure 5-. The lower order spherical (Z 9 ), coma (Z 7/8 ), astigmatism (Z 5/ ), higher order, elliptical coma (Z 1/11 ), oblique spherical aberration (Z 1/13 ), and fifth order aperture coma (Z 1/15 ) Zernike aberration and RMS WFE FFDs over a ±5 degree FOV for the optimized system where Zernike coma is added as an additional variable to the primary surface. The RMS WFE has been reduced from ~.75λ to ~.15λ Figure 5-7. The lower order spherical (Z 9 ), coma (Z 7/8 ), astigmatism (Z 5/ ), higher order, elliptical coma (Z 1/11 ), oblique spherical aberration (Z 1/13 ), and fifth order aperture coma (Z 1/15 ) Zernike aberration contributions and RMS WFE FFDs over a ±5 degree FOV for the optimized system where Zernike coma is added as an additional variable to the tertiary surface. The RMS WFE has been reduced from ~.75λ to ~.18λ Figure 5-8. The lower order spherical (Z 9 ), coma (Z 7/8 ), astigmatism (Z 5/ ), higher order, elliptical coma (Z 1/11 ), oblique spherical aberration (Z 1/13 ), and fifth order aperture coma (Z 1/15 ) Zernike aberration contributions and RMS WFE FFDs over a ±5 degree FOV for the optimized system where the mirror conic constants are added as additional variables in addition to Zernike elliptical coma, oblique

23 xxiii spherical aberration, fifth order aperture coma on the secondary surface. The RMS WFE has been reduced from ~.18λ to ~.5λ Figure 5-9. The lower order spherical (Z 9 ), coma (Z 7/8 ), astigmatism (Z 5/ ), higher order, elliptical coma (Z 1/11 ), oblique spherical aberration (Z 1/13 ), and fifth order aperture coma (Z 1/15 ) Zernike aberration contributions and RMS WFE FFDs over a ±5 degree FOV for the optimized system where Zernike astigmatism, elliptical coma, and oblique spherical aberration are added as additional variables to the tertiary surface. The RMS WFE has been reduced from ~.5λ to ~.1λ Figure 5-1. (a) Layout of LWIR imaging system optimized with φ-polynomial surfaces and (b) the RMS WFE of the final, optimized system, which is < λ/1 (.1λ) over a 1 diagonal full FOV Figure (a) Sag of the primary mirror surface various Zernike components removed from the base sag, (b) sag of the secondary mirror surface various Zernike components removed from the base sag, and (c) sag of the tertiary mirror surface mirror surface various Zernike components removed from the base sag. When the piston, power, and astigmatism are removed from the base sags of the three mirrors, the asymmetry induced from the coma being added into the surface is observed Figure -1. (a) Sag of the secondary mirror surface with the piston, power, and tilt Zernike components removed revealing the astigmatic contribution of the surface,

24 xxiv (b) sag with the astigmatic component additionally removed, and (c) sag with the spherical component additionally removed Figure -. First order layout of the Offner null to compensate spherical aberration. The rays in red show the illumination path for the testing wavefront whereas the rays in blue show the imaging path for the pupils of the Offner null Figure -3. First order layout of the comatic and higher order null. A collimating lens is uses to couple the wavefront to an actuated, deformable membrane mirror. The rays in red show the illumination path for the testing wavefront whereas the rays in blue show the imaging path for the pupils of the comatic null Figure -. Layout of the optimized interferometric null for the concave, secondary mirror to be coupled to a conventional Fizeau interferometer with a transmission flat. The interferometric null is composed of three nulling subsystems: an Offner null to null spherical aberration, a tilted geometry to null astigmatism, and a retroreflecting DM to null coma and any higher order aberration terms Figure -5. Simulation of the double pass wavefront exiting the concave interferometric null (a) before and (b) after the deformable null has been applied at a testing wavelength of 3.8 nm Figure -. (a) Layout of the setup to create the comatic and higher order null on the DM surface. The setup uses a Shack-Hartmann wavefront sensor to run a closed loop optimization to set the shape of the DM. The DM is also interrogated with a Fizeau interferometer. (b) The setup realized in the laboratory

25 xxv Figure -7. (a) DM comatic null surface measured by the interferometer and (b) the residual after the theoretical shape has been subtracted. The residual has a PV error of µm PV Figure -8. Custom designed kinematic indexing mount for counter rotating the test mirror during alignment of the interferometric null. The plates are machined in 3 stainless steel and employ three hardened C stainless steel 7/1 spheres Figure -9. The interferometric null configuration realized in the laboratory. A rotation stage with a rail affixed is used to create the tilted geometry. The secondary mirror is measured using a Zygo Fizeau-type interferometer Figure -1. (a) Initial surface error map of the test mirror with power and (b) with the power removed. The PV error of the surface residual before and after the power is removed is 3.81 µm and.5 µm, respectively. (c) Final surface error map of the test mirror after the software null has been subtracted (c) before and (d) after the power has been removed. In this case, the PV error is 3.3 µm before and 1.1 µm after the power has been removed Figure -11. (a) Sag of the primary mirror surface with the piston, power, and tilt Zernike components removed, (b) sag with the astigmatic component additionally removed, and (c) sag with the spherical component additionally removed. With the piston, power, tilt, astigmatism, and spherical components removed, the asymmetry induced from the coma being added into the surface can be seen... 17

26 xxvi Figure -1. Layout of the optimized interferometric null for the convex, Primary mirror to be coupled to a conventional Fizeau interferometer with a transmission flat. The interferometric null is composed of three nulling subsystems: an afocal Offner null to null spherical aberration, a tilted geometry to null astigmatism and coma, and a retro-reflecting DM to null any higher order aberration terms Figure -13 Simulation of the double pass wavefront exiting the convex interferometric null (a) before and (b) after the deformable null has been applied at a testing wavelength of 3.8 nm Figure 7-1. (a) Layout of the housing structure of the three mirror freeform optical system and (b) exploded view of the tertiary mirror subassembly consisting of the optical mirror surface, adaptor plate, and steel dowel pins for alignment Figure 7-. The tertiary mirror subassembly and values that determine its alignment, namely, the pin hole position tolerances and their relative spacings Figure 7-3. Cumulative probability as a function of as-built RMS WFE for the three mirror optical system over nine field points assuming only passive alignment. 1 Figure 7-. The astigmatism (Z 5/ ) and coma (Z 7/8 ) Zernike aberration FFDs over an 8 x full FOV for the (a) nominal system and with.1 α tilt of the (b) primary, (c) secondary, and (d) tertiary mirror surfaces Figure 7-5. Cumulative probability as a function of as-built RMS WFE for the three mirror optical system over nine field points assuming active alignment where secondary mirror tilt and focal plane tilt are used as compensators.... 1

27 xxvii Figure 7-. The computed elevation log(pst) for the baseline optical housing with the walls of the housing material assumed to be machined aluminum, resulting in a near specular surface with 8% reflectance Figure 7-7. The computed elevation log(pst) for the optical system with blackened walls in blue and the computed elevation log(pst) for the baseline optical housing in gray. An improvement is observed when the walls of the housing are blackened versus left machined aluminum Figure 7-8. Cutaway of the optical system (a) without a baffle and (b) with a baffle and its solid angle to the environment from the focal plane shown in red for each case. With the baffle added to the housing, the solid angle to the environment goes to zero Figure 7-9. The computed elevation log(pst) for the optical system with blackened walls as well as baffling near the image plane in red and the computed elevation log(pst) for the optical housing with blackened walls in light blue. A two order of magnitude improvement is observed in the regions of large stray light when baffling is added near the image plane Figure 7-1. The computed elevation log(pst) for the optical system with blackened walls, baffling near the image plane, and baffling at the primary mirror in green and the computed elevation log(pst) for the optical housing with blackened walls and baffling near the image plane in light red. A two order of magnitude

28 xxviii improvement is observed for large positive elevation angles where scattering is the dominant contributor to stray light Figure As-built subassemblies for the (a) primary, (b) secondary, and (c) tertiary mirrors of the three mirror system that are to be mated to the optical housing. Each subassembly mates to one face of the optical housing and rests on three raised, diamond turned pads Figure 7-1. Assembled three mirror optical system. The system consists of a housing structure and three mirror subassemblies that are mated to the faces of the housing Figure Experimental setup for measuring the full field performance of the as-built optical system Figure 7-1. Measured wavefronts for a 3x3 grid of field points spanning an 8 mm x mm FOV for the directly assembled three mirror optical system. The RMS WFE in microns displayed within the wavefront for each field Figure Measured wavefronts for a 3x3 grid of field points spanning an 8 mm x mm FOV for the directly assembled three mirror optical system with the secondary mirror tilted roughly 1 arc minute with a 3 µm shim. The RMS WFE in microns is displayed within the wavefront for each field Figure 7-1. Sample LWIR image from the optical system Figure A-1. Concept of vector multiplication.... 1

29 xxix List of Tables Table -1. Names of the aberration terms from the wavefront expansion up to fifth order Table -. Summary of the first sixteen Fringe Zernike polynomials and their relation to the standard Zernike set.... Table -3. Field dependence of the Zernike coefficients in terms of the wave aberration coefficients. (Adapted from Gray et al. [35])... 7 Table 3-1. Field aberration terms that are generated from the longitudinal shift of an aspheric plate from the stop surface in a Schmidt telescope Table 3-. Image degrading aberration terms that are generated by a Zernike coma overlay and how the terms link to existing concepts of NAT... 5 Table 3-3. Image degrading aberration terms that are generated by a Zernike elliptical coma overlay and how the terms link to existing concepts of NAT... 7 Table 3-. Image degrading aberration terms that are generated by a Zernike oblique spherical aberration overlay and how the terms link to existing concepts of NAT... 5 Table 3-5. Image degrading aberration terms that are generated by a Zernike fifth order aperture coma overlay and how the terms link to existing concepts of NAT... 5 Table -1. Design specifications for the nominal aberration generating Schmidt telescope

30 xxx Table 7-1. Summary of the initial sensitivity analysis of the three mirror optical system. For each tolerance, the change in RMS WFE from nominal is computed and the RSS is compiled to provide the as-built RMS WFE. The RMS WFE is terms of waves at the central operating wavelength of 1 µm Table 7-. Summary of the quantities used to derive the tolerances for the Monte Carlo sensitivity analysis. The pin hole tolerances are used to derive the mirror x/y decenter and mirror clocking angle

31 xxxi List of Acronyms CGH COTS DM DOF(s) FFD FOV(s) JWST LWIR MRF NAT OAR OPD PST PV RMS RSS TMA WALRUS WFE Computer Generated Hologram Commercial Off The Shelf Deformable Mirror Degree(s) of Freedom Full Field Display Field(s) of View James Webb Space Telescope Long Wave InfraRed MagnetoRheological Finishing Nodal Aberration Theory Optical Axis Ray Optical Path Difference Point Source Transmittance Peak to Valley Root Mean Square Root Sum Square Three Mirror Anastigmat Wide Angle Large Reflective System Wavefront Error

32

33 Chapter 1. Introduction 1 In the introductory part of this dissertation, a brief history of off-axis reflective systems and freeform optical surfaces is presented. Our motivation for this research is then presented and the dissertation is outlined. 1.1 Off-Axis Reflective Systems Reflective telescopes are commonly used for astronomical and earth based surveying because they provide large apertures for light collection; however, most classical telescope forms, i.e. Newtonian, Cassegrain, Gregorian, and Ritchey-Chrétien, have an obscured aperture that will affect the overall image quality from diffraction of the obscuration and its spider supports. The obscuration may also cause stray light in infrared applications because the warm mechanical structure from the obscuration exists in the beam path. Handling the obscuring aperture and creating an accessible image plane becomes even more difficult when trying to design a system to correct the three primary aberrations, i.e. spherical aberration, coma, and astigmatism, where three mirror surfaces are required [1-3]. One way to avoid an obscured configuration is to operate off-axis creating an unobscured form. Historically, there are two principal ways to operate off-axis. The first is to take a nominally rotationally symmetric reflective form and either offset the aperture, bias the field, or a combination of both []. In this configuration each optical surface is a section of a larger parent surface where each parent surface lies on a common optical axis. The other way to operate off-axis is to tilt the optical surfaces themselves to create an unobscured form [5, ]. In this fashion, each optical surface is not arranged along a common optical axis. In some unobscured configurations that tilt the optical

34 surfaces, a nonsymmetric surface is employed to restore the optical performance after the surfaces have been tilted [] Offset Aperture and/or Biased Field A reflective telescope that is made unobscured by operating off-axis in aperture, in field, or both, only uses part of a larger, rotationally symmetric optical system. As an example of this concept, Figure 1-1 shows an F/5, inverse telephoto made unobscured by these techniques. For the biased field system, the aberration performance does not change because the incoming beam has only been tilted with respect to the optical axis. When the aperture is offset, the stop surface has been decentered with respect to the optical axis; therefore, the aberration performance of the optical system will change. Leveraging a combination of both field bias and aperture offset is often required to find an optimal 9:5:3 unobscured solution with minimal impact on the aberration performance. On-Axis 1:1:5 9:57:3 Biased Field Offset Aperture 39. MM New lens from CVMACRO:cvnewlens.seq Scale:. 13-Mar-1 1::1 Biased Field and Offset Aperture 39. MM 38. MM New lens from CVMACRO:cvnewlens.seq Scale:. 13-Mar-1 New lens from CVMACRO:cvnewlens.seq Scale:.5 13-Mar MM Figure 1-1. Demonstration of how an on-axis optical system is made unobscured by offsetting the aperture, biasing the field, or a combination of both. New lens from CVMACRO:cvnewlens.seq Scale:.5 13-Mar-1

35 3 One of the earliest three mirror forms that used aperture offset and field bias was introduced by Cook [7, 8] who took a three mirror anastigmat (TMA) form by Korsch [9] off-axis in aperture and field. Each mirror is an off-axis conic section with the primary and secondary elements forming a Cassegrainian pair that creates an intermediate image. The image is then re-imaged with a tertiary element of approximately unit magnification. The design has the degrees of freedom (DOFs) to provide an aplanatic, anastigmatic, and flat image plane. This type of system is useful for infrared applications because it has an accessible exit pupil, but is best suited for a narrow, strip field of view (FOV). Another set of unobscured forms with large FOVs were created in the late 197s by using combinations of systems that are designed with symmetry principles in mind. These principles are: stop at the center of curvature of a sphere (i.e. Schmidt telescope), concentricity about the obect/image or pupil (i.e. Schwarzschild obective), and confocal (Mersenne) parabolas [1]. From these principles, several wide field obscured forms can be derived from which an unobscured form is obtained. The basic form found by Baker [11] combines confocal parabolas with a Schmidt telescope to form an aplanatic, anastigmatic telescope obective. In this configuration, the secondary element, which is located at the center of curvature of the tertiary, is aspherized to provide spherical aberration correction. Two other variations of the form presented by Baker are the reflective triplet [1], which is on-axis in aperture but off-axis in field, and the wide angle large reflective system (WALRUS) [13], which is an inverse Baker design. These designs yield larger FOVs than the TMA and maintain similar performance but sacrifice the accessible exit pupil.

36 A still larger FOV is achieved in a three mirror unobscured form by using a principle proposed by Brueggemann in which the stop surface is placed at one of the mathematical foci of a conic mirror to remove astigmatism [1]. Following this principle Egdall formed a three mirror obective called the three mirror long using a hyperboloidal primary and an ellipsoidal tertiary [15]. A nearly flat secondary is placed at the common conic focus and serves as the stop of the optical system to correct astigmatism. By aspherizing this element, spherical aberration can also be corrected creating an aplanatic, anastigmatic obective. The shortcomings of this configuration are its length, usually around four focal lengths [1], and its mirrors, which will have a larger diameter than the entrance pupil of the system when using a large FOV Tilted Optical Surfaces Another approach to create an unobscured system is to tilt the optical surfaces directly. For this method there are several approaches on how to create an unobscured optical system from tilted components. One approach takes a well corrected system that is nominally rotationally symmetric and adusts the tilts of the optical surfaces to create an unobscured form []. The performance of the optical system after applying the tilt is restored by adusting the system parameters or by adding additional DOFs to the optical surfaces by changing their surface shape. One of the first examples of a system designed with this method was introduced by Kutter where he took a two mirror telescope and tilted the mirrors in a configuration that removes the obscuration while keeping the astigmatism and coma at a minimum [17]. Leonard used similar principles to obtain a three mirror telescope he called the Yolo that used two conic elements and one anamorphic conic (different curvatures in orthogonal

37 5 directions) [18]. The system is slow at roughly F/1 but provides good performance over a diameter field. Buchroeder [5] developed a modified Seidel aberration theory to understand the behavior of tilted component systems. In his theory the net aberration fields are still the superposition of the individual surface aberration field contributions; however, each contribution will have its own center defined by its decentration or tilt. Shack [19] then developed an expression for the wave aberration expansion that used the concepts of Buchroeder. In the new expansion of the wave aberration, the aberration types can have multiple points in the field where they may go to zero and these zeros are called nodes. The theory of Shack, often called vector or nodal aberration theory (NAT), was developed through fifth order by Thompson [-5] and was applied to the tolerancing of optical systems. Rogers applied NAT as a design technique for three mirror telescope obectives [-8]. In his method, two tilted optical components are combined to yield a system with linear coma and constant astigmatism. Next, a third optical element is added with some cylindrical power to eliminate axial astigmatism. Lastly, the elements are aspherized to correct the residual coma and spherical aberration. With this method systems of similar performance to the Yolo are obtained in a different packaging geometry. The Yolo and the systems proposed by Rogers are slow (greater than F/1), have small FOVs, and do not utilize freeform surfaces to improve performance; rather, they are special configurations where the net aberration fields are arranged to be near zero. In another approach, the optical system is designed from the outset in an unobscured form. Systems designed in this manner require a method to set up the initial system parameters but give the designer the freedom to control the geometry, i.e. volume, while

38 selecting an initial design. One method to generate systems composed of three spherical mirrors has been proposed by Howard [9]. In this method, the imaging properties about a central ray are Taylor expanded. The coefficients of this expansion represent the first order imaging properties and are used to constrain the system parameters like the distances between elements, curvatures of the mirrors, and their tilts. Only solutions that yield no first order blur are considered and these solutions are found using a systematic search or a global optimization technique. Such a method allows the designer to explore a larger design space more rapidly but does not guarantee a practical solution with useful performance. Another three mirror, tilted component system has been proposed by Nakano [3] in which the geometry is derived to maximize the compactness as well as the input aperture. Setting the optical path configuration fixes the mirror positions and then Cartesian surfaces are used to correct spherical aberration and minimize astigmatism. Coma is minimized by adding higher than second order deformation to the surfaces. The system achieves a compact geometry operating over a x square FOV at F/.. 1. Freeform Optical Surfaces In the systems described above, the symmetry of the optical system is broken out of necessity, either to avoid an obscuration or to meet the size and/or weight constraints of the optical system. However, in general, unless special configurations are exploited, the performance of the optical system degrades when the system symmetry is broken. As a result, the surfaces of the optical system can be freeform to help recover from the performance degradation. We define freeform surfaces as nonsymmetric surfaces that include coma and potentially higher orders to their surface departure and go beyond

39 7 anamorphic. One of the first examples of an optical system that utilized a freeform optical surface is the Polaroid SX-7 [31]. The commercial product was designed to be collapsible and the need for flatness of the overall package prompted Baker, the lead optical designer, to use mirrors rather than a penta-prism for the viewfinder. The constraints on the system geometry forced the use of two freeform lenses that are described by up to an eighth order power series in both the x and y directions of the optical surface. Around the same time, Tatian [3-3] began studying nonsymmetric surfaces for the design of unobscured reflective systems. The surface representation dubbed the unusual optical surface is described by a section of an aspheric surface with bilateral symmetry in both the x and y directions where within the local origin of the section may exist up to a tenth order power series in both the x and y directions. With this surface description, Tatian was able to achieve roughly a 3X improvement in the root mean square (RMS) wavefront error (WFE) of a three mirror WALRUS design with unusual surfaces versus the same design with only aspheric surfaces. Shafer also applied a nonsymmetric optical surface to the design of unobscured systems. In his approach, he proposed a two-axis aspheric surface that is the summation of two aspheres that are shifted relative to one another and may be anamorphically stretched. In the region that these two aspheres overlap, lower order aberration contributions like coma and astigmatism are generated. With this approach, special optical configurations like a two-axis asphere at a pupil location can be exploited to yield an unobscured two mirror optical system that is corrected for all third order aberrations. Shafer mentions that these surfaces could be described and optimized with a

40 8 two-dimensional polynomial set over the entire surface, but the computational power required to do so at the time was prohibitive. More recently, now that computational power is no longer nearly as restrictive, two-dimensional polynomial sets to describe an optical surface have started to appear. As mentioned in Section 1.1., Nakano [3] used an orthogonal polynomial set called the Zernike polynomial set (described in detail in Chapter ) to describe an optical surface. The Zernike set is expressed in polar coordinates and is desirable as it directly relates to the wavefront aberrations proposed by Hopkins [35]. A related two-dimensional orthogonal polynomial set has been proposed by Forbes [3] to describe freeform surfaces. Forbes set is also based on Jacobi polynomials but arranged and normalized so that the slope of the optical surface can be minimized. Also, rigid body terms like defocus and tilt have been eliminated from the description. Since both the set proposed by Forbes and the Zernike polynomial set are orthogonal, they can be used interchangeably to describe one another. The surface representations described above consider the global surface shape so that the variables describing the surface affect the entire surface. A more localized optical representation based on a bicubic spline has been proposed for nonsymmetric optical systems by Vogl et al. [37] and implemented further by Stacy [38] for the design of an unobscured optical system. For a spline surface, the optical surface is sampled by a grid of points. At each point, the surface deformation at that point becomes a variable that can be optimized. The values between these mesh points are interpolated by a cubic polynomial. A benefit of the spline surface is that the deformations at each point are only partially correlated to surrounding points. Stacy applied the spline surface to a mirror near the focal plane of a four mirror telescope to improve the field performance of the

41 9 system. The final surface shape exhibited strong oscillations that do cause image degradation. Spline surfaces are computationally intensive because many variables are required to describe them. Another approach at local shape control was proposed by Cakmakci et al. [39] where the optical surface is written as a sum of basis functions, in this case, a two-dimensional Gaussian. In this approach, the surface is sampled by a grid of points where at each point, the Gaussian shape can be varied. This surface description was applied to a single mirror head-worn display. In a local approach the key is to ensure that the performance metric of the optical system is appropriately sampled throughout the FOV [, 1]. For this reason, a global or hybrid surface representation may be more effective for a sparsely sampled field that is often the case during optimization in optical design. 1.3 Motivation The concept of a freeform optical surface is not new and was recognized early on as a promising tool for the design of the nonsymmetric optical systems; however, unless the surfaces can be manufactured, they are little more than an academic exercise. For example, in 197, Gelles when studying unobscured two mirror systems wrote that progress in surface generation will undoubtedly permit the use of exotic types of surfaces in the future []. Until recently, the fabrication capabilities did not exist to manufacture these types of optical surfaces in a cost effective manner. One of these recent advances has been in diamond turning technology where servos have been integrated into the axes geometry in either a fast tool servo or slow slide servo configuration [3, ]. This integration allows for surfaces that are nonsymmetric to be routinely manufactured. Moreover, the residual surface roughness after diamond turning

42 1 has been reduced so that post-polishing is no longer required, further reducing the cost of manufacturing [5]. To demonstrate this progress, Figure 1- shows the improvement of optical surface finish with single point diamond turning as a function of time. As a result of this progress, freeform optical surfaces may be specified for application in the long wave infrared (LWIR) with the technology continuing to push towards shorter wavelength regimes as the residual surface roughness continues to get smaller. Actual Measured Data ~198 ~198 ~199 ~1998 ~ Figure 1-. Single point diamond turning surface roughness evolution through time. Each color represents a lateral measurement of a part from a specific time period. (Adapted from Schaefer [5]) The other component to the fabrication of freeform surfaces is the form error that results from the manufacturing process and how the final surface figure is quantified. The metrology component of the manufacturing chain is the current limitation and cost driver for freeform optics manufacturing as there are very few metrology techniques available. One method available is profilometry where a probe, either contact or non-contact, is scanned along the optical surface and the vertical displacement is recorded [, 7]. This method can be very accurate but it acquires the overall measurement on a point by point basis. Therefore, the measurement process is time intensive, which relates directly to cost

43 11 on the manufacturing floor. Another method is based on the use of a computer generated hologram (CGH) that acts as a nulling component in an interferometric arrangement [8]. The quality of the measurement obtained with the CGH depends strongly on the fabrication of the CGH and the arrangement in which it is placed in the interferometer [9]. Moreover, each CGH is unique to one specific surface and can be cost prohibitive for multiple surfaces [5]. Another potential method is to arrange optical elements (i.e. lenses or mirrors) in a null configuration. These methods exist for measuring off-axis sections of conics and aspherics [51] but have not been developed for freeform surfaces. In addition to fabrication, one of the challenges with freeform optical surfaces is the excess of variables introduced during optimization. If a global surface representation like the Zernike polynomial set is used, the optical designer has access to an impractical number of variables per surface during optimization. In a more localized approach, the number of coefficients grows rapidly as the sampling is increased on the optical surface. In 1978, Shafer recognized this point and to motivate his two-axis asphere approach over a set of polynomials, he wrote, a Zernike set of aspheric coefficients would be able to describe these surfaces and could be used to design systems. That, however, would be a very cumbersome way to proceed, and would probably have a poor convergence rate during optimization [5]. Even with modern day computational power, where the time per optimization cycle is minimal, a more efficient approach for choosing which surfaces would benefit from a freeform surface and which variables to optimize on the surface is desirable.

44 1 With the challenges described above for the design and fabrication of a freeform surface, there has to be some direct benefit that cannot be achieved without a freeform surface to ustify their use in an optical system. To describe this benefit, consider the specifications of an optical system. Any optical system will be required to meet some sort of image quality metric with a certain light collection capability like F/number and with a certain area coverage like FOV. Another more esoteric constraint may be the packaging of the optical system. For example, the weight or size of the optical system might be constrained for certain applications. These three items, F/number, FOV, and packaging, define the design space for optical design. The extent of the design space that may be covered by a particular surface representation is demonstrated in Figure 1-3. The most restrictive optical design shape is the sphere. If the package is to be made smaller with the same performance, thus widening the optical design space, conics or aspheric surfaces are usually employed. Examples here are the use of conics in astronomical applications [1, 53] and the use of aspheres for mobile phone optics [5]. If non-inline geometries are considered like a tilted or decentered optical system, the aberration correction capability is limited with conic or aspheric surfaces. Innovative packaging geometries are the strength of freeform surfaces as they provide the necessary DOFs to operate in this space, thus, increasing the optical design space.

45 13 Packaging Conics/Aspheres Freeform FOV Spheres F/# Figure 1-3. Optical design space defined by the light collection (F/number), area collection (FOV), and packaging for various surface representations. In this dissertation, our research is focused on exploring these innovative package geometries that are enabled by freeform surfaces. We propose a method based in NAT for describing the aberration field behavior of a freeform surface, specifically, φ-polynomial (Zernike based) surfaces. With an analytical theory, the selection of variables during optimization becomes structured and is no longer purely a brute-force approach. In addition, we explore the state of the art in freeform manufacturing through the development of a specific optical system. This system allows for each step in the manufacturing chain of freeform optical surfaces to be studied and identify what links are missing. In the case of metrology for freeform surfaces, we propose a new technique; in particular, a new null based interferometric method for measuring freeform surfaces. An end goal of the research is to demonstrate that a high performing optical system can be designed, fabricated, and assembled with freeform optical surfaces. The principles described in this work extend to a wide variety of applications.

46 1 1. Dissertation Outline The dissertation is organized as follows: Chapter discusses NAT in the context of a perturbed optical system with rotationally symmetric components. The misalignment induced aberration fields are reviewed through fifth order with the concept of the aberration field center. Also, the concept of the full field display, a visualization tool for studying the aberration behavior of a nonsymmetric optical system, is described. Chapter 3 presents a method for integrating freeform optical surfaces, specifically φ-polynomial (Zernike) optical surfaces, into NAT. Using this method, the aberration fields generated by a Zernike overlay away from the stop surface are derived up to sixth order and linked to preexisting concepts of NAT. This theory is then applied to a specific example, three-point mount induced error for both two and three mirror telescopes. Chapter experimentally validates the extension of NAT to freeform optical surfaces by measuring the aberration behavior of a specially designed Schmidt telescope. The Schmidt telescope is composed of two corrector plates, one to remove third order spherical aberration, and the other to induce an aberration field known as field linear, field conugate astigmatism. The generated aberration field is studied under several conditions including both axial and lateral displacement and rotation of the aberration generating plate. Chapter 5 presents the design of an unobscured three mirror imager that utilizes three, tilted φ-polynomial optical surfaces. The design shows how the concepts derived in NAT for freeform surfaces can be used to effectively choose variables for optimization. These

47 15 strategies target either field constant or field dependent aberration correction and utilize the full field display as an analysis technique. Chapter demonstrates a new interferometric nulling technique for the measurement of φ-polynomial optical surfaces. In this method, several adaptable subsystems are combined that each null an aberration type present in the departure of the mirror surface. This method is used to design configurations for measuring both convex and concave optical surfaces. An experimental measurement of an as-fabricated concave, φ-polynomial optical surface is also demonstrated. Chapter 7 demonstrates the design and assembly of an optical housing for the optical system described in Chapter 5. The mechanical housing and its sensitivity to manufacturing error is studied as well as its susceptibility to stray light. Finally, the as-built system is presented along with its as-built optical performance.

48 1 Chapter. Aberration Fields for Tilted and Decentered Optical Systems with Rotationally Symmetric Components The wavefront expansion and surface contributions to the individual aberrations that describe the imaging properties of an optical system have historically assumed the optical system is rotationally symmetric [55]. In this case, the third order aberrations are the sum of the individual surface contributions. For the unobscured reflective systems that were described in Chapter 1, the symmetry has been broken by either offsetting the aperture or tilting the optical components. As a result, a new foundation needs to be established that can handle the imaging behavior of nonsymmetric optical systems..1 Aberration Field Centers The extension of aberration theory to nonsymmetric optical systems was approached by Buchroeder [5] in which he proposed that the aberration fields of any optical system are composed of the contributions of rotationally symmetric surfaces that may be aspheric where each surface contributes rotationally symmetric aberration fields; however, the center of the aberration fields will be offset and defined by the surface s decentration or tilt. As a result, the net aberration fields are still the summation of the shifted, individual surface contributions. The shift of the aberration fields is relative to the center of the Gaussian image plane that is located by the optical axis ray (OAR). The OAR corresponds to the ray that connects the center of the obect, to be chosen arbitrarily, with the center of the aperture stop in the system [, ]. The intersection of the OAR with the image plane defines the field center for each individual surface s offset. Buchroeder sph introduced a vector, σ, to quantify the shift of the aberration field contributions for a sph spherical surface. Specifically, σ represents a vector that lies in the plane of the image and points to the intersection of a line that connects the center of curvature and the center

49 of a local entrance or exit pupil (image of the aperture stop in the local space) of surface 17 with the image plane. For an aspheric cap on surface, there is an additional asph σ parameter that is defined by the intersection of a line that connects the vertex of the asphere (relative to the OAR) and the local pupil of surface with the image plane.. Wave Aberration Expansion in a Perturbed Optical System In a centered, rotationally symmetric system, the common way to express the aberrations of the system is through the use of the scalar wave expansion of Hopkins [5], which is represented in the form where k l m = ( ) cos ( ), (.1) W W H ρ ϕ p n m klm k = p+ m, l = n+ m, (.) W is the total wave aberration and is the sum of all the individual surface contributions, H is the normalized field coordinate, ρ is the normalized pupil coordinate, and φ is the azimuthal coordinate in the pupil. The scalar expansion assumes rotational symmetry so only terms containing powers of H, ρ, and H ρcos( ϕ) are valid. If Eq. (.1) is expanded through sixth order (fifth order in transverse ray aberration), W takes the form where cos( ) cos( ) S ( ) ( ) ( ) cos ( ) cos( ) ( ) S cos ( ) ( φ) W = W ρ + W Hρ φ + W ρ + W Hρ φ + W H ρ W H ρ cos φ + W H ρcos φ + W ρ + W H ρ cos φ W H ρ + W H ρ φ + W H ρ φ 3 3 S W H ρ cos φ + W H ρ + W H ρ φ + W H ρcos, (.3) W = W. (.) klm klm The total wave aberration in Eq. (.) is a summation over all the intrinsic surface contributions that are derived with paraxial quantities. For the fifth order aberrations, the

50 18 surface contributions consist of both intrinsic and induced contributions. The induced contributions at a surface depend on a sum of the third order image and pupil aberrations at the previous surface, though, they are still calculated from paraxial quantities [57, 58]. A set of naming conventions commonly used in optical design to refer to the terms of the wavefront expansion in Eq. (.3) is presented in Table -1. Note the name for each term refers to the transverse ray aberration at the image plane so the pupil order is one order lower than the wavefront order. Table -1. Names of the aberration terms from the wavefront expansion up to fifth order. k H l ρ cos m ( φ ) Coeff. Transverse Ray Aberration Name W Defocus 1 1 W11 Tilt th Order Wave Aberration Type W 3 rd order spherical aberration W rd order coma W S 3 rd order sagittal focal surface W 3 rd order astigmatism W rd order distortion th Order Wave Aberration Type W 5 th order spherical aberration W th order field linear coma W S 5 th order sagittal focal surface for oblique spherical aberration W 5 th order oblique spherical aberration W th order field cubed coma W th order elliptical coma W S 5 th order sagittal focal surface W 5 th order astigmatism W th order distortion Now, to describe the aberrations of a nonsymmetric optical system, Shack [19] proposed allowing both H and ρ to have independent orientation angles, θ and φ, so

51 that the vector H represents the normalized position in the two-dimensional field and ρ represents the normalized position in the two-dimensional pupil as shown in Figure Hy Hx H ρx ρ φ ρy θ z Image Pupil Figure -1. Coordinate system for aberration theory of a perturbed optical system where both the pupil and field coordinate are represented as vectors. Updating the scalar expansion of Hopkins to the vector form proposed by Shack, the representation in Eq. (.1) takes the following form W W HH H (.5) = p n m p n m ( ) ( ) ( ρ ρ) ( ρ). klm With Buchroeder s insight that each surface has its own aberration field defined by σ, the aberration offset is included in the expansion, so Eq. (.5) is modified to the final form p n m W ( W ) klm ( H σ ) ( H σ ) ( ρ ρ) = ( H σ ) ρ, (.) p n m where a new effective field component has been defined for surface that accounts for that surface s aberration offset. This new effective field vector is written as H = H σ, A (.7) and is illustrated at the image plane in Figure -. Depending on the surface of interest, there may be an effective field vector for the spherical part of the surface and the aspheric cap resulting in two contributions at the surface.

52 H y H H A H x Figure -. Representation of the new effective field vector. If Eq. (.), which now has the effective field vector in the expansion, is expanded through fifth order following Thompson [59], W takes the form W W W H W 11 M W H W H H W H W H H H W W H 1 W H H M H 3 3 M 1 W H W H H W H H H W W 331M H H H W 511 H, H H (.8) where through the use of several vector identities described in Appendix A 1 WM WS W, 1 WM WS W, 3 W331M W331 W333, 1 WM WS W, (.9)

53 1 have been defined. The identities used to arrive at the terms in Eq. (.9) and the final wavefront expansion in Eq. (.8) have made use of an operation known as vector multiplication. This operation is essential to represent the aberration fields of nonsymmetric optical systems and can be thought of as an extension to the mathematics of complex numbers. The properties of vector multiplication are described in more detail in Thompson [, 59] and are summarized in Appendix A. Also, it is important to re-emphasize that the aberration surface contributions observed in Eq. (.9) are not affected by a perturbation because they are functions of paraxial quantities. Though Eq. (.8) is complete, it does not provide any insight into the behavior of the aberrations in a nonsymmetric optical system. This behavior is revealed by performing a summation over the surfaces to get the total aberration effect. To simplify the notation, Thompson [59] proposed using substitutions for the summations as follows A B B W klm klm W klm klm 3 3 klm klm ( σ σ ) klm klm W C = W σ σ σ C D = = = ( ) klm klm = = W W σ σ σ ( σ σ ) klm klm D = W σ σ σ ( ) klm klm Eklm = Wklm ( σ ) σ σ, (.1) creating a set of image plane perturbation vectors that are only created when the symmetry of the optical system is broken. With the perturbation vectors defined, the wavefront expansion in Eq. (.8) is expanded further and simplified following

54 Thompson [59] to reveal how the field behavior of rotationally symmetric aberration fields are modified when the symmetry of the optical system is broken, which is given as, W W ( ρρ) W11 ( H ρ) W ( ρρ) ( W131H A131 ) ρ = ( ρρ) WM ( HH) ( HAM ) B + + M ( ρ ρ) 1 + WH HA B ρ + * + W311 ( HH) H ( HA311 ) H+ B311H ( HH) A311 + B311H C 311 ρ 3 + W ( ρρ) ( W ) ( ) 151H A151 ρ + ρρ WM ( HH) ( HAM ) B + + M ( ρ ρ) 1 ( W H HA B ) ρ + + ( ρ ρ) W331M ( HH ) H ( HA 331M ) H+ B331MH ( HH ) A 331M + ρ ( ρ ρ ) * + B331MH C 331M W333H 3H A333 3HB333 C ρ WM ( HH )( HH ) ( HH )( HA M ) + BM ( HH ) + ( ρ ρ ) ( H BM ) ( HCM ) D + + M 1 W ( HH ) H ( HH ) HA + 3( HH ) B ( HA ) H + ρ 3 * CH + 3BH 3HC D + (.11) W511 ( HH )( HH ) H ( HH )( HA 511 ) H+ B511 ( HH ) H + + ( HB511 ) H ( HC 511 ) H+ 3D511H ( HH )( HH ) A 511 ρ. * * 3 * * + ( HH ) B511H ( HHC ) 511 HC511 C511H + D511H E511 In Eq. (.11), for each rotationally symmetric aberration type, several additional components of the aberration are induced when the symmetry is broken. The perturbation induced aberration components have a field dependence of lower order than their parent rotationally symmetric aberration and the induced aberration components change how the aberration behaves throughout the field. More specifically, the zero location of the field dependent aberrations will change [3-5]. For a centered optical system, the field dependent aberrations are always zero on-axis and increase from that zero point

55 3 depending on their inherent field dependence. When the optical system is perturbed, the zero location is altered by the perturbation vectors and may result in the aberration going to zero at more than one field location. The order of the field dependence of the rotationally symmetric aberration determines how many zeros (or nodes) may exist when the system is perturbed. If the field component of the various aberration types is solved for its zeros, the nodes can be analytically predicted. As an example, consider the case of third order astigmatism where the wavefront aberration in a nonsymmetric optical system is 1 W = WH HA + B ρ. (.1) To compute the nodes of this aberration, the field component is set to zero, as follows W H HA + B =. (.13) With the concept of vector multiplication, the quadratic formula is used as if the vectors were scalar quantities and the nodes locations are solved for as where A ± i W B A H = W, (.1) ± i defines a rotation of the vector by ±9. A graphical interpretation of these node locations for third order astigmatism is shown in Figure -3. H y i A W W B A W + i W B A W H x Figure -3. Node locations for third order astigmatism in a perturbed optical system. There are two points in the field where the aberration can be zero.

56 In addition to the general case, there are several special cases to be addressed for this aberration as well. The first case is when the system is corrected for third order astigmatism, that is, W =. In this case, the wavefront aberration becomes If both A and B 1 W = HA + B ρ. (.15) are non-zero, the aberration takes the form of linear astigmatism (i.e. astigmatism that depends linearly with field) and a node will exist at B H =. (.1) A Lastly, if W = and A = then the wavefront aberration becomes 1 W = B ρ, (.17) where the astigmatism is constant throughout the field in both magnitude and orientation and governed by B. Similar methodology is applied for the other aberrations to reveal their characteristic behavior throughout the field and to analytically determine their node locations for different tilt and decenter perturbations as detailed in [3-5]..3 Full Field Aberration Display In a centered optical system with rotationally symmetric components, the aberrations need only be assessed in one field direction (historically, the +y-field). With the aberrations known in this field direction, they are also known in every other field direction since the aberrations will also be rotationally symmetric. In most optical design software packages, a common way to assess the aberration performance is through a transverse ray aberration plot. In this plot, the transverse ray aberration is computed as a function of pupil position for various field heights along one field direction. This plot is useful for rotationally symmetric systems, but if the optical system symmetry is broken,

57 5 the transverse ray aberration plot is no longer useful because the aberrations are no longer known for every field direction. A more useful plot would be one that computes the aberrations over a two-dimensional field. One such plot developed by Thompson [] is the full field display (FFD). This plot computes the aberrations over a two-dimensional grid of field points and then displays those aberrations using symbols to represent the magnitude and orientation of the aberration at a particular field point. The method for computing the aberrations is based on an orthogonal polynomial fit to the wavefront at the exit pupil. The orthogonal polynomial set used is known as the Fringe Zernike polynomial set [1], a modified form of the standard Zernike polynomial set [1, ]. These sets have several benefits over other polynomials sets including the fact that they are orthogonal and complete over a unit radius circular pupil, they represent balanced aberrations, and they can be equated to the aberrations of the Hopkins wavefront expansion. More specifically, the standard Zernike polynomial set is given by Z ± m n m ( ρφ) R ( ρ) ( φ ) ( φ ) cos m for + m, = n, sin m for m m where m is a positive integer (or zero) and R ( ρ ) is the radial component given by n (.18) R ( ρ) = ( 1 )( n s)! ρ. (.19) ( n m) m n s n s= n+ m n m s! s! s! The Fringe Zernike set was developed by John Loomis [3] and is based on the standard Zernike polynomial set but has a specific ordering that is more aligned with that of aberration theory. The first 1 Fringe Zernike terms, their relationship to the standard Zernike set, and their naming convention are summarized in Table -.

58 Table -. Summary of the first sixteen Fringe Zernike polynomials and their relation to the standard Zernike set. Fringe Standard ± m Z ( ρφ, ) Zn ( ρφ, ) Z 1 Z Z Z 1 1 Z 3 Z 1-1 Z Z ρ 1 Z 5 Z Z Z - Z 7 Z 3 1 Z 8 Z 3-1 Zernike Polynomial 1 Piston ρcos( φ ) Tilt ρsin ( φ ) ( φ ) ( φ ) ρ cos ρ sin ( 3 ρ 3 ρ) cos ( φ) ( 3 ρ 3 ρ) sin ( φ) Defocus Name Pri. Astigmatism Pri. Coma Z 9 Z ρ ρ + 1 Pri. Spherical Z 1 3 Z 3 ρ 3 cos( 3φ ) Trefoil Z 11-3 Z 3 ρ 3 sin ( φ ) (Elliptical Coma) Z 1 Z ( ρ 3 ρ ) ( φ) Sec. Astigmatism Z 13 - Z ρ 3 ρ sin φ (Oblique Spherical) Z 1 Z 5 1 Z 15 Z 5-1 ( ) ( ) ( 1 ρ 5 1 ρ ρ) cos ( φ) ( 1 ρ 5 1 ρ ρ) sin ( φ) Sec. Coma Z 1 Z ρ 3ρ + 1ρ 1 Sec. Spherical The relationship of the Fringe Zernike set to the Hopkins wavefront expansion was presented by Gray et al. [35] and the resulting relationships are displayed in Table -3. In Table -3 it is seen that the Zernike terms do not directly relate to the wave aberration coefficients. Some low order Zernike coefficients, like Z 5/ and Z 7/8, are composed of both third and fifth order wave aberration types. Therefore, when evaluating a FFD, it is important to understand that the display does not isolate a single aberration type but will display the dominant aberration characteristics. The higher order Zernike terms appear to be directly related to a wave aberration type, though, it is only because the wavefront

59 expansion is up to fifth order. If seventh order components are considered, additional factors will exist for these terms as well. Table -3. Field dependence of the Zernike coefficients in terms of the wave aberration coefficients. (Adapted from Gray et al. [35]) Fringe Z ( ρφ, ) Z 1 Z /3 Z Z 5/ Z 7/8 Z 9 Z 1/11 Z 1/13 Z 1/15 1 Z 1 W Wavefront Expansion Coefficient Function Through 5 th order W + W + W + WMH + WMH + WMH ( ) cos θ W11 + W131 + W151 + W311H + W331M H + W511H H 3 3 sin ( θ ) W + W + W + WMH + WMH + WMH ( ) cos θ W + W + WH H 8 sin ( θ ) 1 1 ( ) cos θ W131 + W151 + W331M H H sin ( θ ) W + W + WM H 1 cos( 3θ ) 3 W333 H sin ( 3θ ) 1 cos( θ ) W H 8 sin ( θ ) 1 cos( θ ) W151 H 1 sin ( θ ) 7 As an example of the FFD, Figure - (a-b) shows the Zernike pair for astigmatism (Z 5/ ) for the case when an optical system is centered so only field quadratic astigmatism is present, as shown in Figure - (a), and when the optical system is perturbed to create a binodal response, as shown in Figure - (b). With the FFD, the two nodes are readily visible. Note that the Zernike pair, Z 5/, is plotted together so both the magnitude of the

60 8 entire aberration and its orientation, in this case, at the image plane, can be visualized with the FFD. (a) (b) Figure -. Full field display (FFD) showing (a) third order field quadratic in a centered system and (b) in perturbed optical system that yields binodal astigmatism.

61 Chapter 3. Aberration Fields in Optical Systems with φ-polynomial Optical Surfaces 9 Nodal aberration theory (NAT) describes the aberration fields of optical systems when the constraint of rotational symmetry is not imposed. Historically the theory, discovered by Shack [19] and developed by Thompson [], has been limited to optical imaging systems made of rotationally symmetric components, or offset aperture portions thereof, that are tilted and/or decentered. Recently, the special case of an astigmatic optical surface located at the aperture stop (or pupil) was introduced into NAT by Schmid et al. [] and analyzed for the case of a primary mirror in a two mirror telescope. At the stop surface, the beam footprint is the same for all field points, so all field angles receive the same contribution from the astigmatic surface. The net astigmatic field dependence, as predicted by NAT, and as validated by real ray tracing, takes on characteristic nodal features that allow the presence and magnitude/orientation of astigmatic figure error to be readily distinguished from the presence and magnitude/orientation of any misalignment of the secondary mirror. In this chapter, a path based in NAT is presented for developing an analytic theory for the aberration fields of nonsymmetric optical systems with freeform surfaces. With this extension to NAT, the zeros (or nodes) of the aberration contributions, which are distributed throughout the FOV, can be anticipated analytically and targeted directly for the correction or control of the aberrations in an optical system with freeform surfaces. We consider an optical surface defined by a conic plus a ϕ-polynomial overlay, where the sag of the overlay depends on the radial component, ρ, as well as the azimuthal component, φ, within the aperture of the surface. Significantly, the freeform overlay can be placed anywhere within the optical imaging system. Under these more general

62 3 conditions, it will be shown that the aberration contributions of the freeform surface contribute both field constant and field dependent terms to the net aberration field of the optical system. These aberration terms are derived for a specific ϕ-polynomial set, the Zernike polynomial set up to sixth order. For each term in this subset, the aberration behavior throughout the field is examined. Unexpectedly, we find that the impact of integrating ϕ-polynomial freeform surfaces into NAT does not introduce new forms of field dependence; rather, the freeform parameters link directly with the terms presented for the generally multinodal field dependence of the sixth order wavefront aberrations derived for tilted and decentered rotationally symmetric surfaces as reviewed in Chapter. As an example of the types of analyses that can now be carried out with NAT, the impact of three point mount-induced error (trefoil) on the field dependence of astigmatism is presented here. 3.1 Formulating Nodal Aberration Theory for Freeform, ϕ-polynomial Surfaces away from the Aperture Stop To analytically characterize the impact of a ϕ-polynomial optical surface away from the stop on the net aberration field, first consider a classical Schmidt telescope configuration. The telescope is composed of a rotationally symmetric third order (fourth order in wavefront) aspheric corrector plate in coincidence with a mechanical aperture that is the stop of the optical system, located at the center of curvature of a spherical mirror. In such a configuration, the net aberration contribution of the aspheric corrector plate, W Corrector, Stop, is described by the overall third order spherical aberration it induces, given by W = W ρ ρ Corrector, Stop ( ASPH ) ( ) (3.1)

63 ( ASPH ) where W denotes the spherical aberration wave aberration contribution from the aspheric corrector plate and ρ is a normalized two-dimensional pupil vector that denotes a location in the pupil of the Schmidt telescope. Nominally, the Schmidt telescope is corrected for third order spherical aberration by the corrector plate and for third order coma and astigmatism by locating the stop at the center of curvature of the spherical mirror, leaving only field curvature as the limiting third order aberration. The case where an aspheric corrector plate located at the stop or pupil of an optical system is decentered from the optical axis was previously treated in the context of NAT by Thompson [5] and was more recently revisited by Wang et al. []. If the aspheric plate is instead shifted axially (i.e. longitudinally along the optical axis) relative to a physical aperture stop, as shown in Figure 3-1 (a), the beam for any off-axis field point will begin to displace across the aspheric plate. The amount of relative beam displacement, h, is given by y ut h H = H, y y 31 (3.) where y is the paraxial marginal ray height on the aspheric plate, y is the paraxial chief ray height on the aspheric plate, u is the paraxial chief ray angle, t is the distance between the aspheric corrector plate and the mechanical aperture that is the optical system stop, and H is the normalized two-dimensional field vector that locates the field point of interest in the image plane (i.e. H 1). Conceptually, the beam displacement on the corrector plate when it is shifted away from the stop can be thought of as a field dependent decenter of the aspheric corrector when it is located at the aperture stop as shown in Figure 3-1 (b) where the mapping of

64 3 the normalized pupil coordinate is modified from ρ to ρ '. Therefore, the net aberration contribution of the aspheric corrector described by Eq. (3.1) must be modified to account for this effect. By replacing ρ with ρ ' + h and expanding the pupil dependence leads to a modified aberration contribution, W, Corrector Not Stop, that is given by ( ASPH ) WCorrector, Not Stop W = ( ρ' + h) ( ρ' + h) ( ASPH ) = W ( ρ ρ) + ( h ρ)( ρ ρ) + ( h h)( ρ ρ) + ( h ρ ) + ( h h)( h ρ) + ( h h), (3.3) where it is recognized that the measurement is done in the shifted pupil coordinate and the primes have been dropped from the final expression of Eq. (3.3). As can be seen from Eq. (3.3), the original spherical aberration contribution from the aspheric plate generates lower order field dependent aberration components as the plate is shifted away from the stop. Note that the operation of vector multiplication, introduced in [], is being used in this expansion. The aberration terms that are generated by this expansion are the conventional third order field aberration terms summarized in Table 3-1, which could be anticipated since the field aberrations are the product of spherical aberration in the presence of a stop shift from the center of curvature.

65 1:5:15 33 Stop Corrector Plate y uh yh h ' t Spherical Mirror (a) (b) 31.5 MM Figure 3-1. (a) When the aspheric corrector plate of a Schmidt telescope is displaced longitudinally from the aperture stop, the beam for any off-axis field point will displace along the corrector plate. The displacement depends on the paraxial quantities for the marginal ray height, y, chief ray height, y, chief ray angle, u, and the distance between the stop and plate, t. (b) Alternatively, the beam displacement on the corrector plate can Flat-field Schmidt Scale:.8 ORA 1-Jun-1 be thought of as a field dependent decenter of the aspheric corrector, h, that modifies the mapping of the normalized pupil coordinate from to '. Table 3-1. Field aberration terms that are generated from the longitudinal shift of an aspheric plate from the stop surface in a Schmidt telescope. Terms in Eq. (3.3) ASPH W ASPH W 3 rd Order Vector Aberration using Eq. (3.1) and (3.3) ASPH ASPH W H W h y y ASPH ASPH W H H W h h y y ASPH ASPH y W H W h y 3 ASPH ASPH y W H H H W h h h y ASPH ASPH y W H H W h h y 3 rd Order Naming Convention Spherical Aberration Coma Field Curvature Astigmatism Distortion Piston Figure 3- (a-d) demonstrates the generation of astigmatism and coma for an example F/1. Schmidt telescope analyzed using a FFD over a ± FOV. The aberration

66 3 components of the displays are calculated based on real ray optical data using either a generalized Coddington close skew ray trace for astigmatism [7] or a Fringe Zernike polynomial fit to the wavefront optical path difference (OPD) data in the exit pupil for coma and any higher order aberration terms. In Figure 3- (e), the magnitude of the generated coma and astigmatism is evaluated at two specific field points for several longitudinal positions of the fourth order aspheric corrector plate. From Figure 3- it can be seen that as the plate moves longitudinally away from the aperture stop along the optical axis, third order field linear coma is generated linearly with the distance from the aperture stop. In addition, third order field quadratic astigmatism is generated quadratically with distance from the aperture stop, matching the predictions described in Table 3-1. These observed dependencies parallel observations made by Burch [8] when he introduced his see-saw diagram concept and by Rakich [9] when he used the see-saw diagram to simplify the third order analysis of optical systems. What has been recognized for the first time in the context of NAT is that this method for generating the aberration terms displayed in Eq. (3.3) is not restricted to rotationally symmetric corrector plates and it can be applied, with interpretation, to the general class of ϕ-polynomial surfaces. This approach is a pathway for melding freeform optical surfaces into NAT. More significantly, the outcome is that freeform surfaces in the ϕ-polynomial family fit directly into the existing discoveries for the characteristic aberration fields of a perturbed (i.e. tilted or decentered) optical system through sixth order that are developed in [, 3-5].

67 1:17: 1:17:3 8:3: 8:3: Flat-field Schmidt Flat-field Schmidt Flat-field Schmidt Flat-field Schmidt 5. MM ORA 9-May-1 5. MM ORA 9-May-1 8:35: 8:35:55 5. MM ORA 3-May-1 5. MM ORA 3-May-1 8:3:8 8:3:1 Y Field Angle in Obect Space - degrees Y Ob. Field (deg.) Y Field Angle in Obect Space - degrees Y Ob. Field (deg.) Y Field Angle in Obect Space - degrees Y Ob. Field (deg.) Y Field Angle in Obect Space - degrees Y Ob. Field (deg.) FRINGE ZERNIKE PAIR Z7 AND Z8 vs FIELD ANGLE IN OBJECT SPACE Flat-field Schmidt Minimum =.198e-1 Maximum = 58.8 Average = 33.9 Std Dev = ORA - - Flat-field Schmidt -Minimum =.331e-11 Maximum =.58 Average =.3 Std Dev =.851 ORA 3-May-1 15waves ( 587. nm) - - Flat-field Schmidt -Minimum =.911e-13 Maximum = Average = Std Dev =.17 ORA 3-May-1 15waves ( 587. nm) - - Flat-field Schmidt -Minimum =.895e-1 Maximum =.11 Average =.7 Std Dev = 8.3 ORA 3-May-1 15waves ( 587. nm) Coma (waves) Coma - X Field -Angle in Obect Space - degrees (a) 9:17: Flat-field Schmidt 1-Jan-1 FRINGE ZERNIKE PAIR Z7 AND Z8 vs FIELD ANGLE IN OBJECT SPACE - X Field -Angle in Obect Space - degrees (b) 1-Jan-1 FRINGE ZERNIKE PAIR Z7 AND Z8 vs FIELD ANGLE IN OBJECT SPACE - X Field -Angle in Obect Space - degrees (c) FRINGE ZERNIKE PAIR Z7 AND Z8 vs FIELD ANGLE IN OBJECT SPACE - X Field -Angle in Obect Space - degrees (d) Y Field Angle in Obect Space - degrees May-1 15waves ( 587. nm) 8:35:39 8:35:51 8:3: 8:3:17 Y Field Angle in Obect Space - degrees Y Ob. Field (deg.) - - Minimum = Maximum =.5797 Average =.598 Std Dev = Field -Angle Obect degrees X in Space - Flat-field Schmidt Minimum = Maximum =.88e- Average =.1e- Std Dev =.5357e-5 ORA 3-May mm Y Field Angle in Obect Space - degrees Y Ob. Field (deg.) Minimum = Maximum =.5797 Average =.598 Std Dev = Field -Angle Obect degrees X in Space - Flat-field Schmidt Minimum = Maximum =.55 Average =.1751 Std Dev =.138 Y Field Angle in Obect Space - degrees Y Ob. Field (deg.) ORA Minimum = Maximum =.5797 Average =.598 Std Dev = Field -Angle Obect degrees X in Space - Flat-field Schmidt Minimum = Maximum =.81 Average =.8755 Std Dev =.5578 ORA 3-May mm Y Field Angle in Obect Space - degrees Y Ob. Field (deg.) Minimum = Maximum =.5797 Average =.598 Std Dev = Field -Angle Obect degrees X in Space - ASTIGMATIC LINE IMAGE vs FIELD ANGLE IN OBJECT SPACE Flat-field Schmidt Minimum = Maximum =.558 Average =.1998 Std Dev =.178 ORA X Field Angle in Obect Space - degrees - FRINGE ZERNIKE PAIR Z7 AND Z8 vs - FIELD ANGLE IN OBJECT SPACE X Ob. ORA Field (deg.) 15λ (.587µm) 9:17: Y Field Angle in Obect Space - degrees - Flat-field Schmidt waves ( 587. nm) - - X Field Angle in Obect Space - degrees - FRINGE ZERNIKE PAIR Z7 AND Z8 vs - FIELD ANGLE IN OBJECT SPACE X Ob. ORA Field (deg.) 15λ (.587µm) waves ( 587. nm) - vs - - vs - Astigmatism X Ob. Field (deg.) ASTIGMATIC LINE IMAGE vs FIELD ANGLE IN OBJECT SPACE X Ob. Field (deg.) ASTIGMATIC LINE IMAGE vs FIELD ANGLE IN OBJECT SPACE 3-May mm X Ob. ORA Field 1-Jan-1 (deg.) waves ( λ (.587µm) nm) X Ob. Field (deg.) 9:17: 9:17: Y Field Angle in Obect Space - degrees - Flat-field Schmidt Y Field Angle in Obect Space - degrees - Flat-field Schmidt - - X Field Angle in Obect Space - degrees FRINGE ZERNIKE PAIR Z7 AND Z8 FIELD ANGLE IN OBJECT SPACE - - X Field Angle in Obect Space - degrees FRINGE ZERNIKE PAIR Z7 AND Z8 FIELD ANGLE IN OBJECT SPACE X Ob. ORA Field 1-Jan-1 (deg.) 15λ (.587µm) 5 Astigmatism (waves) 3 1 waves ( 587. nm) ASTIGMATIC LINE IMAGE vs FIELD ANGLE IN OBJECT SPACE X Ob. Field (deg.) 3-May mm Plate Postion from Stop (mm) (e) 1 18 Plate Postion from Stop (mm) Figure 3-. Generation of coma and astigmatism as the aspheric corrector plate in a Schmidt telescope is moved longitudinally (along the optical axis) from the physical aperture stop located at the center of curvature of the spherical primary mirror for various positions (a-d). For each field point in the FFD, the plot symbol conveys the magnitude and orientation of the aberration. (e) Plots of the magnitude of coma and astigmatism generated as the aspheric plate is moved longitudinally for two field points, (, ) (blue square) and (, ) (red triangle).

68 3 3. The Aberration Fields of ϕ-polynomial Surface Overlays The set of ϕ-polynomial overlays to be placed on an optical surface is the Fringe Zernike polynomial set that is presented in Figure 3-3 up to sixth order. This set differs from other Zernike polynomial sets in its arrangement of terms where they are ordered by wavefront expansion order, with the third order aberration components appearing before the fifth order components. Out of the sixteen terms displayed in Figure 3-3, twelve are nonsymmetric, ϕ-polynomial types and of the twelve nonsymmetric terms, ten will blur the image if they are placed on a surface of an optical system. Moreover, these ten terms form five pairs to be explored, namely, they are Zernike astigmatism (Z 5/ ), Zernike coma (Z 7/8 ), Zernike trefoil or elliptical coma (Z 1/11 ), Zernike secondary astigmatism or oblique spherical aberration (Z 1/13 ), and Zernike secondary coma or fifth order aperture coma (Z 1/15 ). In Section 3.1, when describing the aspheric corrector plate of the Schmidt telescope, it was found that the aberration contribution from the corrector is field constant when the plate is located at the aperture stop and develops a field dependent contribution as the surface is shifted longitudinally away from the aperture stop. For the aspheric corrector plate of the Schmidt telescope, the field constant aberration that results is third order spherical aberration. By analogy, if a plate placed at the stop is deformed by one of the Zernike terms described above, it will also introduce a field constant aberration. By utilizing the vector pupil dependence of the Zernike overlay terms, the induced field constant aberration is predicted by NAT and it can be added to the total aberration field.

69 37 Z1 Z3 Z Z Z Z5 Z11 Z8 Z7 Z1 Z13 Z9 Z1 Z15 Z1 Z1 Figure 3-3. Fringe Zernike polynomial set up to 5 th order ( th order in wavefront). The set includes Z 1 (piston), Z /3 (tilt), Z (defocus), Z 5/ (astigmatism), Z 7/8 (coma), Z 9 (spherical aberration), Z 1/11 (elliptical coma or trefoil), Z 1/13 (oblique spherical aberration or secondary astigmatism), Z 1/15 (fifth order aperture coma or secondary coma), and Z 1 (fifth order spherical aberration or secondary spherical aberration). The φ-polynomials to be explored include Z 5/, Z 7/8, Z 1/11, Z 1/13, and Z 1/ Zernike Astigmatism In order of increasing radial dependence, the first freeform overlay term to consider is astigmatism. In optical metrology terminology, Zernike astigmatism (Fringe polynomial terms Z5 and Z ) is given by ( φ) ( φ) Z5 z5 ρ cos =, Z z ρ sin (3.) where z5 and z are the coefficient values for the astigmatism term, ρ is the normalized radial coordinate, and φ represents the azimuthal angle on the surface. In optical testing, the Fringe Zernike set is described in a right-handed coordinate system with φ measured

70 38 counter-clockwise from the ˆx axis. The magnitude, z, and orientation, Test FF 5/ FFξ 5/, of the freeform Zernike astigmatism overlay is then calculated from the coefficients by z = z + z (3.5) FF 5/ 5 ξ Test FF 5/ z = z5 1 tan 1, where the superscript Test denotes the optical testing coordinate system. (3.) Zernike astigmatism can be introduced in the vector multiplication environment of NAT with the following observation, which is the basis for NAT, ( φ) ( φ) ( φ) ( φ) sin sin if ρ = ρ then ρ = ρ cos cos,, (3.7) where consistent with commercial optical design raytrace programs, a right-handed coordinate system is employed with φ measured clockwise from the ŷ axis. To implement a coordinate system for the overlay term that is consistent with its generated aberration field within the context of the real ray based environment of NAT, the orientation in Eq. (3.) must be modified. A new orientation, FFξ 5/, is defined and is displayed in Figure 3- and given by FF ξ 5/ π z = z5 1 tan 1. (3.8)

71 λ (P) y pupil coordinate V FF ξ 5/ P P V x pupil coordinate - -1.λ (V) Figure 3-. Surface map describing the freeform Zernike overlay for astigmatism on an optical surface over the full aperture. The error is quantified by its magnitude z and FF 5/ its orientation FFξ5/ that is measured clockwise with respect to the ŷ axis. P and V denote where the surface error is a peak rather than a valley. From the vector pupil dependence in Eq. (3.7), it is deduced that the astigmatism overlay will induce field constant astigmatism that is predicted by NAT when the optical surface is placed at the aperture stop. Based on this observation, it is added to the total aberration field as where B FF 1 W, Stop = ( FF B ρ ), (3.9) is a two-dimensional vector that describes the magnitude and orientation of the astigmatic overlay, which is related to the overall Zernike astigmatism by FF B n 1 z exp i ξ, ( ) ( ) FF 5/ FF 5/ where n is the index of refraction of the substrate medium. (3.1) If a surface with a Zernike astigmatism overlay is now placed away from the stop, the beam footprint for an off-axis field angle will begin to displace across the surface resulting in the emergence of a number of field dependent terms. Replacing ρ with ρ ' + h in Eq. (3.9), expanding the pupil dependence, and simplifying leads to a specific

72 set of additive terms for the wavefront expansion when a surface is located away from the stop, 1 W ( ), Not Stop = FF B ρ ' h + 1 = FF B ρ + FF B hρ + FF B h, (3.11) where, as in Section 3.1, the primes on the pupil coordinate have been dropped from the final expression. To map the impact of these additive terms on the overall field dependent wave aberration expansion of an optical system, the pupil dependence needs to be converted into existing aberration types. To this end, an additional vector operation, introduced in [], is used, (3.1) * A BC = AB C, * where B is a conugate vector with the standard properties of a conugate variable in the mathematics of complex numbers * B = Bexp i = Bxˆ+ Byˆ. ( β ) x y (3.13) By applying the vector identity of Eq. (3.1), Eq. (3.11) takes the form 1 * W, Not Stop = FF B ρ + FF B h ρ + FF B h. (3.1) In Eq. (3.1) two additional field dependent aberration terms are generated in addition to the anticipated field constant astigmatism term. The second and third terms, however, are a tilt and piston that do not affect the image quality but affect the mapping and phase. Here we are focusing on the image quality; therefore, these terms will not be directly addressed for this or any subsequent Zernike overlay terms. In this case, the only image degrading aberration is field constant astigmatism that is independent of where the Zernike astigmatism overlay is located with respect to the stop. A magnitude and

73 1 orientation plot for the field constant astigmatism contribution is illustrated in Figure 3-5. Since the aberration has no dependence on the field vector, the magnitude and orientation are the same everywhere throughout the field. 1.5 Y Field X Field Figure 3-5. The characteristic field dependence of field constant astigmatism that is generated by a Zernike astigmatism overlay on an optical surface in an optical system. This induced aberration is independent of stop position. The field constant astigmatism contribution can be added to the existing concepts of NAT by re-defining the field constant astigmatic term of NAT, B, as B B B N = ALIGN + FF, = 1, (3.15) where the summation and index has been introduced to generalize the result to include a multi-element optical system where the Zernike astigmatism overlay is on the th optical surface and B ALIGN misalignment. From the definition of defines the existing astigmatic component that may result from a B in Eq. (3.15), the conventional strategies of NAT can be applied to solve for the nodal properties of the astigmatic aberration field when a Zernike astigmatism overlay is placed on an optical surface of a multi-element optical system. 3.. Zernike Coma The next freeform overlay term in order of pupil dependence is Zernike coma (Fringe polynomial terms Z7 and Z 8 ) that, in optical metrology terminology, is written as

74 ( ) ( ) 3 Z7 z7 3ρ cos φ ρcos φ =, Z 3 8 z8 3ρ sin ( φ) ρsin ( φ) (3.1) where z7 and z 8 are the coefficient values for the coma term. Within this term there is 3 cubic aperture ( ρ ) coma term and a linear aperture ( ρ ) tilt term. The tilt term is inherently built into Zernike coma to minimize the RMS WFE of the aberration polynomial over the aperture, a property of the Zernike polynomial set. In order to generate coma that can be introduced in the vector multiplication environment of NAT, an adusted Zernike coma is used that combines both Zernike coma and Zernike tilt and is written as ( φ) ( φ) Ad Ad 3 Z 7 Z7 + Z z 7 3ρ cos = =. Ad Z Z Ad Z + 3 z 8 3ρ sin Similar to Zernike astigmatism, the magnitude, z Ad, and orientation, Ad FF 7/8 FFξ 7/8 (3.17), of the freeform, adusted Zernike coma overlay term is then calculated from the coefficients by ( ) ( ) Ad Ad Ad 7/8 7 8 = + (3.18) FF z z z ξ Ad FF 7/8 π z = Ad 1 8 tan, Ad z7 (3.19) where the orientation in Eq. (3.19) creates an orientation consistent within the real ray based environment of NAT. The overlay term in Eq. (3.17) can be linked to the vector multiplication environment of NAT, with the following observation, ( φ) ( φ) sin if ρ = ρ, then ρ ρ ρ = ρ, cos sin ( φ) 3 ( ) (3.) cos( φ) where a right-handed coordinate system is employed with φ measured clockwise from the ŷ axis. From the vector pupil dependence in Eq. (3.), it is deduced that the

75 overlay will induce field constant coma when located at the stop surface and is added to the total aberration field as W A ρ ρρ 3 ( FF )( ) (3.1) 131, Stop = 131, where A FF 131 is a two-dimensional vector that describes the magnitude and orientation of the Zernike coma overlay, which is related to the overall Zernike coma by Now replacing ρ with ρ ' + h FF A 3 n 1 z exp i ξ. Ad Ad ( ) ( FF ) 131 FF 7/8 7/8 (3.) in Eq. (3.1), expanding the pupil dependence, and simplifying leads to a specific set of additive terms for the wavefront expansion when a surface with a Zernike coma overlay is located away from the stop, W 131, Not Stop = A131 + h + h + h FF ( ρ' ) ( ρ' ) ( ρ' ) ( FF A ρ)( ρρ ) + ( )( ) 131 FF A hρ FF A h ρρ 131 * + ( h h)( FF A ρ) + A h ρ + ( )( ) FF A h h h 131 =, (3.3) where, as in Section 3.1, the primes on the pupil coordinate have been dropped from the final expression. As can be seen from Eq. (3.3), five additional field dependent aberration terms are generated in addition to the anticipated field constant coma term. The first, field constant term is added into NAT by re-defining the field constant coma term in NAT, A 131, as N A A A = 131 ALIGN 131 FF 131, = 1, (3.) where A ALIGN 131 is any comatic contribution from misalignment. The second term is recognized to be an astigmatic term based on the ρ aperture dependence. When Eq. (3.) is used to replace h in the astigmatic term of Eq. (3.3), it becomes y (3.5) FF A131 h ρ = FF A131, H ρ. y

76 Equation (3.5) is a form of field asymmetric, field linear astigmatism that was first seen in the derivation for the nodal structure of third order (fourth order in wavefront) astigmatism by Thompson [, 1]. This contribution is added to the field linear astigmatism contribution of NAT, A, as where A ALIGN N y A = ALIGN A FF A131,, (3.) = 1 y defines the existing astigmatic component that may result from a misalignment. The third term is recognized to be a field curvature term based on the ( ρρ ) aperture dependence and when h is replaced in it, it takes the form y ( FF A131 h)( ρρ ) = ( FF A131H)( ρρ ). (3.7) y Equation (3.7) is now recognized as a form of field curvature, seen in the derivation for the nodal structure of third order field curvature by Thompson [], that yields a tilted focal surface relative to the Gaussian image plane and is added to the field linear, field curvature contribution of NAT, A M, as where A ALIGN M N y A = 131,. M ALIGN A M FF A (3.8) = 1 y defines the existing field linear, field curvature component that may result from a misalignment. The process of linking the aberration terms generated by a Zernike coma overlay to existing concepts of NAT is summarized in Table 3- where the aberration terms from Eq. (3.3) are displayed in column one with h replaced by its form using Eq. (3.), column two displays the NAT analog term that has the same field and pupil behavior as the generated terms in column one, and column three displays how

77 the NAT analog term is re-defined to include both the misalignment and freeform overlay components. Table 3-. Image degrading aberration terms that are generated by a Zernike coma overlay and how the terms link to existing concepts of NAT Aberration Terms for a Zernike Coma Overlay NAT Analog ( FF A131, ρ)( ρρ ) ( A 131 ρ)( ρρ) y y FF y y A 131, H Addition of overlay term into NAT N A131 A131 A131, = ALIGN FF = 1 ρ ( ) N 1 AH ρ A = ALIGN A FF A131, = 1 y ( FF A131, H)( ρρ ) ( A H)( ρρ) M y N y A = A M A M 131, = 1 y ALIGN FF 5 The magnitude and orientation plots of the aberration terms generated by a Zernike coma overlay, summarized in Table 3-, are depicted throughout the field in Figure 3- (a-c). In Figure 3- (a), the field constant comatic contribution from a Zernike coma overlay is displayed. The magnitude and orientation are the same everywhere throughout the field and are governed by the vector describing the overlay term, A FF 131. In Figure 3- (b), the astigmatic contribution from a Zernike coma overlay away from the stop is displayed. As can be seen from the line images, the aberration is asymmetric with field while increasing linearly from a single node. Lastly, in Figure 3- (c), the field curvature contribution is displayed. This form of field curvature increases linearly with field in the direction of the vector describing the overlay term, A FF 131.

78 Y Field Y Field Y Field X Field X Field X Field (a) (b) (c) Figure 3-. The characteristic field dependence of (a) field constant coma, (b) field asymmetric, field linear astigmatism, and (c) field linear, field curvature that is generated by a Zernike coma overlay on an optical surface away from the stop surface Zernike Trefoil (Elliptical Coma) The next freeform overlay term that has the same pupil dependence as coma but a higher order azimuthal dependence is Zernike trefoil (Fringe polynomial terms Z1 and Z 11 ) that, in optical metrology terminology, is written as Z Z 1 3 z1 cos 3, z sin (3.9) where z1 and z 11 are the coefficient values for the trefoil term. The magnitude, z, FF 1/11 and orientation, FF 1/11, of the Zernike trefoil is then calculated from the coefficients by FF z z z (3.3) FF 1/ /11 z 3 z1 1 tan (3.31) The overlay term in Eq. (3.9) can be linked to the vector multiplication environment of NAT, with the following observation sin sin 3 if cos cos 3 3 3, then, (3.3) where a right-handed coordinate system is employed with measured clockwise from the ŷ axis. From the vector pupil dependence in Eq. (3.3), it is deduced that the trefoil

79 deformation will induce field constant, elliptical coma when located at the stop surface and is added to the total aberration field as where 3 C FF W333, Stop = ( FF C333 ρ ), (3.33) is a two-dimensional vector that describes the magnitude and orientation of field constant elliptical coma, which is related to the overall Zernike trefoil by Now replacing ρ with ρ ' + h FF C n 1 z exp i3 ξ. ( ) ( ) FF 1/11 FF 1/11 (3.3) in Eq. (3.33), expanding the pupil dependence, and simplifying leads to a specific set of additive terms for the wavefront expansion when a surface with a Zernike trefoil overlay is located away from the stop, 1 3 W ( ) 3 333, Not Stop = FF C333 ρ ' h * 1 FF C333ρ + 3 FF C333 h ρ =, 3 * FF C333 h ρ + FF C333 h (3.35) where, as in Section 3.1, the primes on the pupil coordinate have been dropped from the final expression. In Eq. (3.35), three additional field dependent aberration terms are generated in addition to the anticipated field constant elliptical coma (trefoil) term. Following the method outlined for the Zernike coma overlay, Table 3-3 displays the image degrading aberration terms generated by the Zernike trefoil overlay with h replaced in each term and shows how each term links to existing concepts of NAT. Table 3-3. Image degrading aberration terms that are generated by a Zernike elliptical coma overlay and how the terms link to existing concepts of NAT Aberration Terms for a Zernike Trefoil Overlay C ρ NAT Analog 1 C ρ , FF y y FF C H ρ 3 * 333, 1 ( ) 3 * C H ρ Addition of overlay term into NAT N C C C = 333 ALIGN 333 FF 333, = 1 C C 3 y C N = ALIGN FF 333, = 1 y

80 8 In Table 3-3, it can be seen that the field constant elliptical coma term pairs with C 3 ALIGN 333 which is a fifth order (sixth order in wavefront) misalignment induced aberration 3 component. Normally, since C ALIGN 333 is a cubic vector, this contribution is small and dominated by lower order misalignment contributions. However, with the use of freeform overlays, particularly any overlay of equal or higher order than Zernike trefoil, the fifth 3 order aberration space and their misalignment induced aberration components like C 333 can be roughly equal to or greater than the third order misalignment induced aberration components of NAT. The second term from Eq. (3.35) is seen to be an astigmatic term based on the ρ aperture dependence and it is a form of field linear astigmatism that was first seen in the derivation for the nodal structure of field quartic fifth order astigmatism by Thompson [5], and reported in Table 3-3 (second row, second column). This linear astigmatism term has not previously been isolated as an observable field dependence and it represents the first time any aberration with conugate field dependence has been linked to an observable quantity [7]. In Chapter, this aberration and its characteristic field behavior are experimentally validated through the design and implementation of an aberration generating telescope. The magnitude and orientation plots of the aberration terms generated by a Zernike trefoil overlay, summarized in Table 3-3, are depicted throughout the field in Figure 3-7 (a-b). In Figure 3-7 (a), the field constant elliptical coma contribution from a Zernike trefoil overlay is displayed. The magnitude and orientation are the same everywhere throughout the field and are governed by the vector describing the overlay term, 3 C FF 333. In Figure 3-7 (b), the astigmatic contribution from a Zernike trefoil overlay

81 9 away from the stop is displayed. The aberration is of the same order as field asymmetric, field linear astigmatism but it depends on the conugate vector so it takes on a different orientation throughout the field. This form of astigmatic field dependence was reported in the literature by Stacy [71], but its analytical origin has remained unexplained until now Y Field Y Field X Field (a) X Field (b) Figure 3-7. The characteristic field dependence of (a) field constant elliptical coma, (b) field conugate, field linear astigmatism, which is generated by a Zernike elliptical coma overlay on an optical surface away from the stop surface. 3.. Zernike Oblique Spherical Aberration Moving to the next pupil order, the next freeform overlay term is Zernike oblique spherical (Fringe polynomial terms Z1 and Z 13 ) that, in optical metrology terminology, is written as Z Z 1 13, z1 cos 3 cos z13 sin 3 sin (3.3) where z1 and z 13 are the coefficient values for the oblique spherical term. Within this term there is a quartic aperture ( ) oblique spherical aberration term and a quadratic aperture ( ) astigmatism term. Similar to the case of Zernike coma, there is an included astigmatic term to minimize the RMS WFE of the oblique spherical aberration term. In order to generate oblique spherical aberration that can be introduced in the vector multiplication environment of NAT, an adusted Zernike oblique spherical aberration is

82 5 used that combines both Zernike oblique spherical aberration and Zernike astigmatism and is written as Ad The magnitude, 1/13 ( φ) Ad Ad Z 1 Z1 + 3Z5 z1 ρ cos = =. Ad Z Z Ad Z z13 ρ sin ( φ) (3.37) + z, and orientation, Ad FF FFξ 1/13, of the freeform, adusted Zernike oblique spherical aberration overlay term is then calculated from the coefficients by ( ) ( ) Ad Ad Ad 1/ = + (3.38) FF z z z Ad Ad π 1 z13 FFξ1/13 = tan. Ad (3.39) z1 The overlay term in Eq. (3.37) can be linked to the vector multiplication environment of NAT, with the following observation ( φ) ( φ) ( φ) ( ) (3.) ( φ) sin sin if ρ= ρ, then ρρ ρ = ρ, cos cos where a right-handed coordinate system is employed with φ measured clockwise from the ŷ axis. From the vector pupil dependence in Eq.(3.), it is deduced that the oblique spherical overlay will induce field constant, oblique spherical aberration when located at the stop surface and is added to the total aberration field as where B FF 1 W, Stop = B ρ ρρ ( FF )( ), (3.1) is a two-dimensional vector that describes the magnitude and orientation of field constant oblique spherical aberration, which relates to adusted Zernike oblique spherical aberration by FF B 8 n 1 z exp i ξ. Ad Ad ( ) ( FF ) FF 1/13 1/13 (3.)

83 Now replacing ρ with ρ ' + h in Eq. (3.1), expanding the pupil dependence, and 51 simplifying leads to a specific set of additive terms for the wavefront expansion when a surface with a Zernike oblique spherical aberration overlay is located away from the stop, 1 W, Not Stop FF B ( ρ' h) = ( ρ' h) ( ρ' h) * 3 ( FF B ρ )( ρρ ) + 3( FF B h ρ)( ρρ ) + FF B hρ 1 + 3( h h)( FF B ρ ) + 3( )( ) FF B h ρρ = *, + ( FF B h )( hρ ) + ( h h) ( FF B h ρ ) + ( h h)( FF B h ) (3.3) where, as in Section 3.1, the primes on the pupil coordinate have been dropped from the final expression. As can be seen from Eq. (3.3), seven additional field dependent aberration terms are generated in addition to the anticipated field constant oblique spherical aberration term. Table 3- displays the image degrading aberration terms generated by the Zernike oblique spherical aberration overlay with h replaced in each term and shows how each term links to existing concepts of NAT. In order of decreasing pupil dependence, the first field dependent term is identified as an elliptical coma 3 aberration based on the ρ dependence, where, the elliptical coma is linear throughout the field. The second term is identified as a comatic aberration based on the ρρ ρdependence. The aberration field is linear with conugate field dependence and ( ) belongs with the misalignment induced aberrations of field cubed coma. The third term is a fifth order astigmatic aberration based on the ρ dependence where the aberration is quadratic with field from the ( HH ) the orientation only depends on the vector B component; however, since this quantity is a scalar, FF, and, as a result, the orientation is constant throughout the field. The final term is a fifth order field curvature aberration

84 5 based on the ( ρρ ) dependence that yields a saddle shaped focal surface relative to the Gaussian image plane. Table 3-. Image degrading aberration terms that are generated by a Zernike oblique spherical aberration overlay and how the terms link to existing concepts of NAT Aberration Terms for a Zernike Oblique Spherical Aberration Overlay 1 NAT Analog 1 ( B ρ )( ρρ ) ( )( ), FF 1 y y FF B H ρ 3, B ρ ρρ 1 ( 3 3 B ) 333H ρ Addition of overlay term into NAT B B B N = ALIGN + FF, = 1 B B y B N 333 = ALIGN FF, 3 = 1 y 3 y y * ( FF B )( ), H ρ ρρ ( B * 331 M H ρ)( ρρ) N 3 y B = B + B M M = 1 y 331 ALIGN 331 FF, 3 y y 3 y y 1 ( HH )( FF B ), ρ 3 ( HH )( B ρ ) ( FF B )( ), H ρρ ( B H )( ρρ ) M N y = ALIGN + FF, = 1 y B B B N 3 y = M ALIGN + M FF = 1 y B B B The magnitude and orientation plots of the aberration terms generated by a Zernike oblique spherical aberration overlay, summarized in Table 3-, are depicted throughout the field in Figure 3-8 (a-e). In Figure 3-8 (a), the field constant oblique spherical aberration contribution from a Zernike oblique spherical aberration overlay is displayed. The magnitude and orientation are the same everywhere throughout the field and are governed by the vector describing the overlay term, B FF. In Figure 3-8 (b), the elliptical coma contribution from a Zernike oblique spherical aberration overlay away from the stop is displayed. The field behavior of this elliptical coma term is of the same form as field asymmetric, field linear astigmatism. Figure 3-8 (c) displays the field cubed comatic contribution where the conugate field dependence of the aberration yields a unique

85 53 orientation when compared to conventional third order field linear coma. The fifth order astigmatic contribution, Figure 3-8 (d), exhibits a field constant orientation while the magnitude of the aberration varies quadratically with the field vector. Lastly, Figure 3-8 (e), displays a fifth order field curvature contribution that equates to a saddle shaped focal plane as the aberration curves up in one direction and down in the other Y Field Y Field Y Field X Field X Field X Field (a) (b) (c) Y Field Y Field X Field (d) X Field (e) Figure 3-8. The characteristic field dependence of (a) field constant oblique spherical aberration, (b) field asymmetric, field linear trefoil, (c) field conugate, field linear coma, (d) field constant, field quadratic astigmatism, and (e) field quadratic, field curvature that is generated by a Zernike oblique spherical aberration overlay on an optical surface away from the stop surface Zernike Fifth Order Aperture Coma The next pupil order and last freeform overlay term is Zernike fifth order aperture coma (Fringe polynomial terms Z1 and Z 15 ) that, in optical metrology terminology, is written as Z Z z1 1 cos 1 cos 3 cos, 5 3 z15 1 sin 1 sin 3 sin (3.)

86 5 where z 1 and z 15 are the coefficient values for the fifth order coma term. Within this term 5 3 there is a quintic aperture ( ρ ) coma term, a cubic aperture ( ρ ) coma term, and a linear aperture ( ρ ) tilt term to minimize the RMS WFE of the fifth order aperture coma term. To generate a fifth order aperture coma that can be introduced in the vector multiplication environment of NAT, an adusted Zernike fifth order coma is used that combines Zernike fifth order aperture coma, Zernike coma, and Zernike tilt and is written as Ad The magnitude, 1/15 Ad Ad 5 Z 1 Z1 + Z7 + 5Z z1 1ρ cos = =. Ad Z Z8 5Z Ad Z 3 z15 1ρ sin ( φ) (3.5) + + z, and orientation, Ad FF FFξ 1/15, of the freeform, adusted Zernike fifth order aperture coma overlay term is then calculated from the coefficients by ( ) ( ) Ad Ad Ad 1/ FF z z z ( φ) = + (3.) Ad Ad π 1 z15 FFξ1/15 = tan. Ad (3.7) z1 The overlay term in Eq. (3.37) can be linked to the vector multiplication environment of NAT, with the following observation ( φ) ( φ) sin ( φ) 5 ( ) (3.8) cos( φ) sin if ρ = ρ, then ρ ρ ρ = ρ, cos where a right-handed coordinate system is employed with φ measured clockwise from the ŷ axis. From the vector pupil dependence in Eq.(3.), it is deduced that the fifth order aperture coma overlay will induce field constant, fifth order aperture coma when located at the stop surface and is added to the total aberration field as W ( )( ) 151, Stop = FF A151 ρ ρρ, (3.9) where A FF 151 is a two-dimensional vector describing the magnitude and orientation of field constant, fifth order aperture coma, which relates to adusted Zernike fifth order coma by

87 Now replacing ρ with ρ ' + h FF A 1 n 1 z exp i ξ. Ad Ad ( ) ( FF ) 151 FF 1/15 1/15 55 (3.5) in Eq. (3.1), expanding the pupil dependence, and simplifying leads to a specific set of additive terms for the wavefront expansion when a surface with a Zernike fifth order aperture coma overlay is located away from the stop, W 151, Not Stop = A151 + h + h + h = + FF ( ρ' ) ( ρ' ) ( ρ' ) 3 ( FF A ρ)( ρρ ) + 3( )( ) ( ) 151 FF A h ρρ FF A h ρ 151 ( FF A151 h) h 3( h h) FF A ρ( ρρ ) + ( FF A151 h) h + ( h h) FF A151 h ρ ( FF A15 hρ )( ρρ ) + ( h h)( )( ) 1 FF A h ρρ ( h h) FF A151 + ( FF A151 h)( h h) h ρ + ( h h) ( FF A151 h), (3.51) where, as in Section 3.1, the primes on the pupil coordinate have been dropped from the final expression. As can be seen from Eq. (3.3), eleven additional field dependent aberration terms are generated in addition to the anticipated field constant fifth order aperture coma term. Table 3-5 displays the image degrading aberration terms generated by the Zernike fifth order aperture coma overlay with h replaced in each term and shows how each term links to existing concepts of NAT. In order of decreasing pupil dependence, the first field dependent term is identified as medial oblique spherical aberration and it equates to a tilted medial surface for oblique spherical aberration relative to the Gaussian image plane. The second term is identified as oblique spherical aberration based on the ( ρρ ) ρdependence where the aberration is linear throughout the 3 field. The third term is an elliptical coma aberration based on the ρ dependence where the aberration is quadratic with field. The fourth and fifth terms are both identified as a form of coma based on the ( ) ρρ ρdependence and are found as a misalignment induced

88 5 aberration of fifth order, field cubed coma. Likewise, the six and seventh terms are a form astigmatism based on the ρ dependence and are found as a misalignment induced aberration of fifth order, astigmatism. Lastly, the eighth term is a fifth order field curvature term that manipulates the focal surface relative to the Gaussian image plane. Table 3-5. Image degrading aberration terms that are generated by a Zernike fifth order aperture coma overlay and how the terms link to existing concepts of NAT Aberration Terms for a Zernike Fifth Order NAT Analog Aperture Coma Overlay ( )( ) FF A151, ρ ρρ ( A )( ) 151 ρ ρρ y 3 y y y ( H A )( ) 151, ρρ ( H A )( ) ρρ M 1 ( FF A151, Hρ )( ρρ ) ( HA ρ )( ρρ) Addition of overlay term into NAT N A151 A151 A151, = ALIGN FF = 1 N 3 y A = A M A M 151, = 1 y ALIGN FF N y A = A A151, = 1 y ALIGN FF y y FF y y y 3 y 3 y y 3 y y 3 y y 3 A H ρ 151, 1 3 ( 3A 333H ρ ) ( FF A151, H)( Hρ)( ρρ ) ( HA331 )( Hρ)( ρρ ) M ( HH )( FF A151, ρ)( ρρ ) ( HH )( A331 M ρ)( ρρ ) 1 ( FF A151, H)( H ρ ) ( HA )( H ρ ) 1 ( HH )( FF A151, H ρ ) ( H H )( HA ρ ) ( HH )( FF A151, H)( ρρ ) ( HH )( A H)( ρρ ) M N y 333 = ALIGN 333 FF 151, 3 = 1 y A A A N y 331 = M ALIGN M FF 151, = 1 y A A A 3 N y = ALIGN FF 151, = 1 y A A A 3 N 3 y = M ALIGN M FF 151, = 1 y A A A

89 The magnitude and orientation plots of the aberration terms generated by a Zernike fifth order aperture coma overlay, summarized in Table 3-5, are depicted throughout the field in Figure 3-9 (a-g). In Figure 3-9 (a), the field constant fifth order aperture coma contribution from a Zernike fifth order aperture coma overlay is displayed. The magnitude and orientation are the same everywhere throughout the field and are governed by the vector describing the overlay term, A FF 151. In Figure 3-9 (b), the medial oblique spherical aberration contribution from a Zernike fifth order aperture coma overlay away from the stop is displayed. The field behavior of this term resembles that of the field curvature term generated by a Zernike coma overlay. Figure 3-9 (c) displays the oblique spherical aberration contribution where the field behavior resembles that of field asymmetric, field linear astigmatism. The elliptical coma contribution, Figure 3-9 (d), is field quadratic and depending on the vector describing the overlay term, A FF 151, the aberration orientation may appear rotationally symmetric as is depicted in Figure 3-9 (d). In Figure 3-9 (e), the two field quadratic coma contributions are displayed together resulting in a net field asymmetric aberration. Similarly, Figure 3-9 (f) displays together the two field cubed, fifth order astigmatism contributions, resulting in a net aberration that is field asymmetric. Lastly, Figure 3-9 (g), displays a fifth order field curvature contribution that equates to a cubic shaped focal plane. 57

90 Y Field Y Field Y Field X Field X Field X Field (a) (b) (c) Y Field Y Field Y Field X Field X Field X Field (d) (e) (f) 1.5 Y Field X Field (g) Figure 3-9. The characteristic field dependence of (a) field constant, fifth order aperture coma, (b) field linear medial oblique spherical aberration, (c) field asymmetric, field linear oblique spherical aberration, (d) field quadratic trefoil, (e) field quadratic coma, (f) field asymmetric, field cubed astigmatism, and (g) field cubic, field curvature that is generated by a Zernike fifth order aperture coma overlay on an optical surface away from the stop surface. 3.3 APPLICATION: The Astigmatic Aberration Field Induced by Three Point Mount-Induced Trefoil Surface Deformation on a Mirror of a Reflective Telescope With a theoretical framework in place for understating the aberration behavior of a freeform overlay, deformations that exist in an as-built configuration of a telescope can

91 be studied. The deformation of particular interest here is the self weight deflection of an optic located away from the aperture stop being held at three points, a kinematically stable condition. An error of this nature is usually measured interferometrically by measuring the optic in its on-axis, null configuration while in its in-use mounting configuration; or, the error can be simulated by the use of finite element methods [7]. In either the measured or simulated case, the deformation is quantified based on the values of its Fringe Zernike coefficients. The predominant surface error that arises with this mount configuration is trefoil, in optical testing terminology, Fringe polynomial terms Z1 and Z 11. In Section 3., the field aberration influence of a trefoil surface overlay was described. The next step is to apply these results to various reflective telescope forms to observe the impact of the mount-induced trefoil deformation from the nominal telescope configuration. Depending on the telescope optical configuration, the third order aberrations, i.e. spherical aberration, coma, and astigmatism, may or may not be corrected. For the case of a two mirror telescope, the system is corrected for third order spherical aberration and may be corrected for third order coma depending on the conic distribution of the mirrors. Whether or not coma is corrected, third order astigmatism remains uncorrected. If a third mirror is added, the telescope system may also be corrected for third order astigmatism. In either the two or three mirror case, when the secondary mirror is deformed by a three point mount, it will generate a field dependent astigmatic contribution, assuming the secondary mirror is not the stop surface. Under these conditions, the astigmatic response of the telescope is of interest because it reveals information into the as-built state of the 59

92 telescope. In the case described above, the astigmatic response, takes the nodal form W AST, of the telescope 1 3 y SM 3 * WAST = WH + MNTERRC333, SM H ρ, (3.5) ysm where the subscript SM signifies that the mount-induced trefoil deformation is on the secondary mirror surface and depending on whether the telescope is anastigmatic, W may or may not be equal to zero. To emphasize, Eq. (3.5) presents the magnitude and orientation of the astigmatic Fringe Zernike coefficients (Z 5/ ) that would be measured if an interferogram was collected at the field point H in the FOV of the perturbed telescope. The perturbation, in this case, is a three point kinematic mount deformation on the secondary mirror, 3 characterized by C MNTERR 333, SM Zernike trefoil (Z 1/11 ) following Eq. (3.3)., and is directly related to the measured values of the Fringe To exploit the strength of NAT for developing insight into the relationships between alignment, fabrication, uncorrected aberration fields, and now mount-induced errors, the next step is to understand the nodal response of the astigmatism to these deviations from a nominal design depending on whether the system is corrected for third order astigmatism Astigmatic Reflective Telescope Configuration ( W ) in the Presence of a Three Point Mount-Induced Surface Deformation on the Secondary Mirror In order to determine the possible nodal geometry for the case where residual third order astigmatism exists, the term inside the brackets of Eq. (3.5) is set equal to zero, as represented in Eq. (3.53),

93 1 1 3 y SM 3 * WH + MNTERRC333, SM H =. (3.53) ysm The first step in solving the vector formulation represented in Eq. (3.53) is to establish a path for arranging H and H * in a form that can be solved, ideally using previously developed techniques. This step is accomplished by multiplying both sides of Eq. (3.53) by unity in the form of ˆ 1 ˆ HH H H i + = =, H H * * (3.5) where a vector multiplication relation presented in [] has been applied. Since Eq. (3.5) is a unit, vector formulation, it does not affect the magnitude or orientation of either vector in Eq. (3.53). Multiplying the identity in Eq. (3.5) through Eq. (3.53) yields y 3 SM 3 * * WH + MNTERRC333, SM HH H =. (3.55) ysm H Again, making use of the identity in Eq. (3.5), Eq. (3.55) takes the form 3 1 H 3 y SM 3 * W MNTERRC + 333, SM H =. (3.5) H y SM It can now be seen based on the powers of H that there is a quadranodal astigmatic response in the FOV to a mount-induced trefoil deformation on the secondary mirror with the term in the brackets of Eq. (3.5) exhibiting equilateral trinodal behavior with a fourth zero located on-axis at H =. In order to find the nodal response, the term inside the bracket of Eq. (3.5) is rearranged, and set to zero, taking the form H 3 y 3 SM 3 + MNTERRC333, SM =. (3.57) W ysm H 3 The first term of Eq. (3.57) is substituted with a new reduced field vector Π written in complex notation as

94 Π = = = H H 3 i3θ H H e i 3 θ 3 i He H e 3 θ, (3.58) where the new vector represented in Eq. (3.58) has the same orientation, θ, as H but with a magnitude equal to the cube root of H. In this new form, Eq. (3.57) takes the form 3 3 y SM 3 Π + MNTERRC333, SM =. (3.59) W ysm Following the method proposed by Thompson and detailed in [3] for solving the nodes of a cubic vector equation, that has been applied to the case of elliptical coma and fifth order astigmatism [3, 5] in tilted and decentered systems, the node locations for a trinodal form are governed by two vectors, x and x, which, in this case are equal, and given by y SM 3 3 MNTERR x = MNTERR x = ( MNTERRC333, SM ) W ysm. (3.) In terms of these cubic equation solution vectors x MNTERR and x MNTERR, which are best kept independent for later generalizations, the three node locations referenced to the intersection of the OAR with the image plane are, for this case, equidistant from the on-axis node with W AST = at, 3, 3. (3.1) ( MNTERR x ) ( MNTERR x ) + i ( MNTERR x ) ( MNTERR x ) i ( MNTERR x ) The four field points at which astigmatism is found to be zero are illustrated in Figure 3-1 (a) where the solutions are plotted in the Π reduced field coordinate. In Figure 3-1 (b), the four nodal solutions have been re-mapped into the conventional H field coordinate. The solution vectors follow a notation introduced in [3, 5] for characterizing the cubic nodal behavior of elliptical coma and fifth order astigmatism. In this case, the vectors are proportional to MNTERRC 333, SM, which is directly computed from a

95 measurement or simulation of the mount-induced trefoil deformation on the secondary mirror, as visualized in Figure y H y MNTERR 1/11 MNTERR x C MNTERR 333, SM MNTERR 1/11 MNTERR C 3 333, SM i x 3 MNTERR MNTERR x i x 3 MNTERR x H x (a) (b) Figure 3-1. (a) The nodal behavior for an optical system with conventional third order field quadratic astigmatism and Zernike trefoil at a surface away from the stop, e.g., a two mirror telescope with a three point mount-induced error on the secondary mirror, is displayed in a reduced field coordinate,, where the node located by MNTERR x has an orientation angle of MNTERR 1/11 and a magnitude that is proportional to MNTERR C 333, SM. The two related nodes on the circle are then advanced by 1º and º for this special case. (b) When the nodal solutions are re-mapped to the conventional field coordinate, H, the node located by MNTERR x has an orientation angle of MNTERR 1/11 and a magnitude that is proportional to MNTERR C , SM y WAVEFRONT ABERRATION Cassegrain Ritchey-Chretien y λ (.33µm) Waves 1. MNTERR 1/11 MNTERR 3 MNTERR 1/11 C 333, SM MNTERR 1/11 x MNTERR C 3 333, SM i 3x x x x.5.5 i 3x (a). Field = (.,.) Degrees Wavelength = 3.8 nm Defocusing =. mm (b) Figure A measurement or simulation of the mount-induced error on the secondary mirror yields the magnitude and orientation of MNTERR C 333, SM.

96 3.3. Anastigmatic Reflective Telescope Configuration ( W = ) in the Presence of a Three Point Mount-Induced Surface Deformation on the Secondary Mirror For the case where the telescope configuration is corrected for third order astigmatism, the first term inside the brackets of Eq. (3.5) is set to zero yielding 3 y SM 3 * WAST = MNTERRC333, SM H ρ. (3.) ysm In Eq. (3.) it can be seen that the only astigmatic contribution is now from the mount-induced perturbation on the secondary mirror. In this case, the nodal solution is trivial where if the term inside the brackets of Eq. (3.) is set to zero, the only solution is located on-axis at H =. For both the astigmatic and anastigmatic cases presented above, the astigmatism takes on a unique distribution throughout the FOV when there is a mount-induced error on the secondary mirror. These unique distributions are significant because by measuring only the Fringe Zernike pair (Z 5/ ) and reconstructing the nodal geometry from these measurements, it can be determined whether the as-built telescope is dominated by mount error versus other errors like alignment or residual figure error Validation of the Nodal Properties of a Reflective Telescope with Three Point Mount-Induced Figure Error on the Secondary Mirror Astigmatic Reflective Telescope Configuration ( W ) in the Presence of a Three Point Mount-Induced Surface Deformation on the Secondary Mirror As a validation of the predicted nodal behavior summarized in Figure 3-11 (a) for the case of a two mirror telescope with a mount-induced perturbation on the secondary mirror, an F/8, 3 mm Ritchey-Chrétien telescope, displayed in Figure 3-1 (a), has been simulated in commercially available lens design software, in this case, CODE V. The aberration performance throughout the FOV in terms of a total measure of image

97 5 quality, the RMS WFE is displayed in Figure 3-1 (b). The RMS WFE increases as a function of FOV because of the uncorrected field quadratic astigmatism. When it comes to assembling and aligning an optical system of this type, it is becoming increasingly common to measure the system interferometrically and use information that is available about significant characteristic aberrations through a polynomial fit to the wavefront OPD. Figure 3-13 (a-b) displays separately the Fringe Zernike astigmatism (Z 5/ ) and Fringe Zernike trefoil (Z 1/11 ) that would be measured at selected, discrete points in the FOV. As can be seen from Figure 3-13 (a), the system suffers from third order astigmatism. The higher order aberrations, like elliptical coma, are near zero, which is expected for a system with a modest F/number and FOV. When a.5λ, orientation, trefoil mount error is added to the secondary mirror, the aberration displays are modified as shown in Figure 3-13 (b). The astigmatic contribution has developed a quadranodal behavior and there is now a field constant contribution to the elliptical coma. The astigmatic behavior matches the general case shown in Figure 3-11 (a) where the orientation angle, MNTERR 1/11, has been set to zero. A quantitative evaluation of the zeros in the display for astigmatism from Figure 3-13 (b) confirms the predictions made by NAT described in Section The displays are based on real ray data and the zero locations for the astigmatic contribution are independent of NAT so they are an excellent validation of the theoretical developments presented in Section and 3.3..

98 -1 1:: X Real Ray Image Height - mm 1::1 1:: telescope_19 KPT 13-May-1 1. FRINGE ZERNIKE PAIR Z5 AND Z vs REAL RAY IMAGE HEIGHT Minimum =.158e-9 Maximum =.379 Average =.175 Std Dev =.157 1λ 1waves (.33µm) ( 3.8 nm) telescope_19 Scale:.33 KPT 5-Jun MM telescope_19 Scale:.33 KPT 5-Jun-1 (a) 75. mm 75. MM RMS WAVEFRONT ERROR vs REAL RAY IMAGE HEIGHT telescope_19 Minimum =.1575 Figure 3-1. (a) Layout for a F/8, 3 mm Ritchey-Chrétien telescope and Maximum (b) = a.333 Full Field Average =.888 Std Dev =.37 KPT 5-Jun-1 1waves ( 3.8 nm) Each circle represents the magnitude of the RMS WFE at a particular location in the 1:8:8 Y Field Angle (deg) Y Real Ray Image Height - mm X Real. Image Height mm. 1 Ray (b) X Field Angle (deg) (b) Display (FFD) of the RMS WFE of the optical system at.33 µm over a ±. FOV. FOV. 1 1:: Y Real Ray Image Height - mm KPT Y Field Angle (deg) 1:: Y Field Angle (deg) Y Real Ray Image Height - mm telescope_ X Real Ray Image Height - mm PAIR FRINGE ZERNIKE REAL X Real RAY Z5 AND Z Image HEIGHT.Height mm 1. vs IMAGE Ray FRINGE ZERNIKE PAIR Z5 AND Z 13-May-1 1waves ( 3.8 vs nm) REAL RAY IMAGE HEIGHT telescope_19 Minimum =.158e-9 1 Maximum =.379. Average =.175 Std Dev =.157 KPT 13-May-1 1waves ( 3.8 nm) 5.1 Y Real Ray Image Height - mm Z 5/ Z 1/11 Minimum =.158e-9 X Maximum Field =.379 Angle (deg) Average =.175 Std Dev =.157 1λ (.33µm) Y Field Angle (deg) 1:3: Y Field Angle (deg) Y Real Ray Image Height - mm (a) X Real Image.Height mm 1. Ray telescope_19 1. FRINGE ZERNIKE PAIR Z1 AND Z11 vs REAL RAY IMAGE HEIGHT Minimum =.81e-15 Maximum =.133 Average =.37e- Std Dev =.31e- KPT 13-May-1 1waves ( 3.8 nm) 5.1 Y Real Ray Image Height - mm X Field Angle (deg) FRINGE ZERNIKE PAIR Z5 AND Z FRINGE ZERNIKE PAIR Z1 AND Z11 vs vs REAL RAY IMAGE HEIGHT REAL RAY IMAGE HEIGHT telescope_19 telescope_19 Minimum =.3e-7 Minimum = Figure Displays of the magnitude Maximum =.5958 and orientation of Fringe Maximum = Zernike astigmatism Average =.1871 Average =.9835 Std Dev =.187 Std Dev =.989e- 1:1: X Real Image.Height mm 1. Ray X Field Angle (deg) 1:11: (b) X Real Image.Height mm 1. Ray X Field Angle (deg) (Z 5/ ) and Fringe Zernike trefoil, elliptical coma, (Z KPT 13-May-1 1waves ( 3.8 nm) KPT 13-May-1 1/11 ) throughout the FOV for (a) a 1waves ( 3.8 nm) Ritchey-Chrétien telescope in its nominal state and (b) the telescope when.5λ of three point mount-induced error oriented at has been added to the secondary mirror. It is important to recognize that these displays of data are FFDs that are based on a Zernike polynomial fit to real ray trace OPD data evaluated on a grid of points in the FOV. For each field point, the plot symbol conveys the magnitude and orientation of the Zernike coefficients pairs, Z 5/ on the left and Z 1/11 on the right.

99 Anastigmatic Reflective Telescope Configuration ( W = ) in the Presence of a Three Point Mount-Induced Surface Deformation on the Secondary Mirror In the case of an anastigmatic telescope with a mount-induced perturbation on the secondary mirror, the nodal behavior is simplified as discussed in Section 3.3. where the node is on-axis at H =. As a validation for this prediction, a relevant TMA geometry based on the James Webb Space Telescope (JWST) [73] has been simulated and analyzed for a trefoil perturbation on the secondary mirror. The optical system operates at F/ with a. m entrance pupil diameter and is shown in Figure 3-1 (a). In order to yield an accessible focal plane, the FOV is biased so that an off-axis portion of the tertiary mirror is utilized. The RMS WFE of the system is displayed in Figure 3-1 (b) over a ±. FOV and the portion of the field that is utilized for the biased system is bounded by the red rectangle. In the center of the on-axis FOV, the RMS WFE is well behaved because the third order aberrations are well corrected. The performance does increase at the edge of the FOV due to higher order aberration contributions. Following a similar approach to that outlined in Section , the individual aberration contributions that make up the total RMS WFE can be evaluated over the FOV. Figure 3-15 displays separately the Fringe Zernike astigmatism (Z 5/ ) and Fringe Zernike trefoil (Z 1/11 ) that would be measured at selected, discrete points in the FOV for the JWST-like system. As can be seen from Figure 3-15 (a), the system is anastigmatic and the elliptical coma is near zero throughout the FOV. If a.5λ, orientation, trefoil error is added to the secondary mirror, the aberration displays are modified as shown in Figure 3-15 (b). The astigmatic contribution has developed field linear, field conugate astigmatism with a single node centered on-axis. The node lies outside the usable FOV for the field biased telescope. As with the previous case, there is also a field constant 7

100 8 contribution to the elliptical coma. Both contributions match the theoretical developments presented in Section and Y Field Angle (deg) (a) 15. mm 15. MM λ (1.µm) X Field Angle (deg) (b) Figure 3-1. (a) Layout for a JWST-like telescope geometry and (b) a Full Field Display (FFD) of the RMS WFE of the optical system at 1. µm over a ±. FOV. The system utilizes a field bias (outlined in red) to create an accessible focal plane.. Z 5/. Z 1/11 Y Field Angle (deg) Y Field Angle (deg) X Field Angle (deg) 1λ (1.µm) -. (a) Field Angle. Obect.1 - degrees. X in Space X Field Angle (deg).. Y Field Angle (deg) Y Field Angle (deg) X Field Angle (deg) -. (b) X Field Angle (deg) Figure Displays of the magnitude and orientation of Fringe Zernike astigmatism (Z 5/ ) and Fringe Zernike trefoil, elliptical coma, (Z 1/11 ) throughout the FOV for (a) a JWST-like telescope in its nominal state and (b) the telescope when.5λ of three point mount-induced error oriented at has been added to the secondary mirror.

101 3. Extending Nodal Aberration Theory to Include Decentered Freeform ϕ-polynomial Surfaces away from the Aperture Stop In the case of the JWST-like geometry in Figure 3-1 (a), the tertiary mirror is an off-axis section of a larger rotationally symmetric surface. If a trefoil deformation is to be applied to the tertiary mirror, the error must be centered with respect to the off-axis portion of the surface, not the larger parent surface. Therefore, an additional parameter must be defined that accounts for a shift of the nonsymmetric deformation from the reference axis that is defined to be the OAR []. Following the method used in [] for the decenter of an aspheric cap of an optical surface, the nonsymmetric deformation is treated as a zero-power thin plate. When the nonsymmetric deformation is shifted, there is a freeform 9 sigma vector ( ) σ FF that is expressed as * where ( ) δ v FF σ = ( ) * ( δ vff ) FF y, (3.3) is the distance between the OAR and the freeform departure vertex. For the case of a freeform, φ-polynomial surface, the freeform vertex corresponds to the origin of the unit circle that bounds the polynomial set. To compute the overall aberration field from the shifted freeform deformation, a new effective aberration field height ( FF ) H is defined, following the notation of [], as ( HFF ) = H ( σ FF ). (3.) The astigmatic response of a telescope with a mount-induced trefoil deformation can now be modified to account for the new effective aberration field height. Updating Eq. (3.5) with the effective field height ( FF ) th optical surface, W AST takes the form H and generalizing the perturbation to be on the

102 7 1 3 y 3 * WAST = WH + MNTERRC333, ( HFF ) ρ. y (3.5) The nodal solution for the astigmatic response represented in Eq. (3.5) is best found numerically and may be quadranodal but degenerates to special cases where only three or two nodes exist. For the anastigmatic case where the third order astigmatism is zero, Eq. (3.5) simplifies to where there is a single node located at ( ) 3 y 3 * WAST = MNTERRC333, ( HFF ) ρ, y (3.) H = σ FF. As a validation of these predictions, the JWST-like system evaluated in Section is reevaluated where the.5λ, orientation, trefoil error is now added to the off-axis section of the tertiary mirror. In this case, the aberration displays are modified as shown in Figure 3-1. The astigmatic contribution has developed field linear, field conugate astigmatism with a single node now centered off-axis. The node has moved off-axis because the trefoil deformation is no longer located along the OAR and now lies in the center of the field biased FOV. It is also interesting to note that for this configuration, the induced astigmatic contribution is larger than the induced field constant contribution to the elliptical coma. At the tertiary mirror, the beam footprints for each field are widely spread about the optical surface; as a result, the field dependent contribution has a larger net effect than the field constant contribution.

103 71. Z 5/. Z 1/11 Y Field Angle (deg) Y Field Angle (deg) Field Angle. Obect.1 - degrees. X in Space X Field Angle (deg) -. 1λ (1.µm) Field Angle. Obect Space.1 - degrees. X in X Field Angle (deg) Figure 3-1. Displays of the magnitude and orientation of Fringe Zernike astigmatism (Z 5/ ) and Fringe Zernike trefoil, elliptical coma, (Z 1/11 ) throughout the FOV for a JWSTlike telescope with.5λ of three point mount-induced error oriented at on the off-axis tertiary mirror.

104 7 Chapter. Experimental Validation of Nodal Aberration Theory for φ-polynomial Optical Surfaces Chapter 3 presented a theoretical foundation for the general, unrestricted aberration theory for optical systems that employ φ-polynomial surfaces. In this chapter, this theoretical foundation is verified experimentally by the design and implementation of an aberration generating telescope. Within the basic telescope framework, a surface with a φ-polynomial departure is placed in the optical path. When the surface is displaced axially from the aperture stop of the optical system, aberrations of lower radial order than the nonsymmetric departure of the surface are generated. The particular nonsymmetric departure to be studied in this chapter is elliptical coma or Zernike trefoil. As will be verified, when the trefoil surface is displaced from the stop surface, field conugate, field linear astigmatism is generated throughout the FOV. It will be shown that the aberration is centered in the image plane about a point that depends on the lateral offset of the Zernike trefoil vertex from the OAR of the telescope. Moreover, it will be verified that the magnitude of the astigmatic aberration is generated linearly with relative axial distance from the stop surface as predicted theoretically in Chapter 3..1 Design of an Aberration Generating Schmidt Telescope In its simplest form, a Schmidt telescope is composed of a spherical mirror, stop, and aspheric corrector plate. The stop lies at the center of curvature of the spherical mirror so that the system is corrected for third order coma and astigmatism. By placing a sixth order aspheric corrector plate at the stop surface, the system is also corrected for third and fifth order spherical aberration. In this configuration, the telescope is corrected for all third order aberrations except field curvature and, as such, makes an excellent baseline optical system for introducing controlled amounts of individual, isolated aberration types

105 73 to study their field behavior. In this particular case, depicted in Figure -1, the aberrations are generated by inserting an additional plate with Fringe Zernike trefoil (or elliptical coma) polished directly into the surface, yielding a Zernike, freeform surface. When the plate is shifted away from the stop surface, aberrations of lower radial order than trefoil, which is cubic, are generated. The aberration fields induced by the plate are observed by evaluating the wavefront with a 1 mm aperture, Zygo Fizeau-type He-Ne laser interferometer. The wavefront is reflected back to the interferometer by the use of a re-imaging retro-reflecting component placed near the image plane. In order to evaluate the wavefront across a two-dimensional FOV, a scanning mirror is introduced into the path between the output of the interferometer and the entrance aperture of the Schmidt telescope. When the FOV is scanned, the retro-reflector must follow the beam to track the image displacement, including field curvature, created by the off-axis field angle. Spherical Primary Retro-reflector Zernike Plate Corrector Plate Interferometer Field of view generator Figure -1: Testing configuration for the Schmidt telescope to demonstrate the field dependent aberration behavior of a freeform optical surface. A freeform, Zernike plate can purposely be placed at or away from the stop surface to induce field dependent aberrations. The aberration field behavior of the telescope is measured interferometrically by acquiring the double pass wavefront over a two-dimensional FOV with a scanning mirror.

106 7 The specifications for the nominal Schmidt system are summarized in Table -1 and are based on the constraints of pre-existing optical/mechanical components. The focal length is constrained by the selection of the primary mirror. In this case, an existing 15. mm diameter, 15. mm focal length spherical mirror is selected to maintain a small package for the telescope. The pupil size is constrained by the 1 mm aperture of the interferometer. Because the FOV scanning mirror does not lie in a pupil plane, the beam will displace on the focal plane of the interferometer. As a result, a 7 mm entrance pupil diameter is selected so that the entire FOV can be measured by the interferometer without vignetting. The scanning of the retro-reflector limits the achievable FOV. The stages that move the retro-reflector are limited to ±.5 mm of motion so the maximum measureable diagonal full FOV is roughly 5. Lastly, the departure of the Fringe Zernike trefoil plate is constrained to 3 µm so a discernible amount of aberration is induced by the plate but ensures that the surface is still manufacturable with available resources. The nominal Schmidt design without the Fringe Zernike trefoil deformation on the plate is shown in Figure -. The aspheric corrector plate is concave and has about 37 µm of departure from planar or 9.5 µm of departure from the best fit sphere of -197 mm. The overall system is diffraction limited throughout the FOV (relative to a curved focal plane) with a maximum RMS WFE of.13λ at 3.8 nm.

107 75 Table -1. Design specifications for the nominal aberration generating Schmidt telescope. Parameter Target Value Schmidt telescope Spherical, primary mirror Type th and th order NBK7 corrector NBK7, 3 µm Fringe Zernike trefoil plate at 5 mm normalization radius Diagonal Full FOV (deg.) 5 F/#. Focal Length (mm) 15. Entrance pupil diameter (mm) 7 Wavelength (nm) RMS Wavefront Error on a curved image plane, R=-151 mm λ = 3.8 nm) [nominal design] Image Quality [nominal design] 1:57:1 3.8 (He-Ne) On axis <.7.7 FOV <.7 1. FOV <.7 Zero 3 rd and 5 th order spherical, 3 rd order coma, 3 rd order astigmatism 7. mm Dia. 8 mm thick, NBK7 window 15. mm Dia. 3.8 mm ROC Spherical Mirror 1 mm Dia. 1 mm thick, NBK7 window 1.7 MM Figure F/, -. Schmidt Layout of the nominal Schmidt telescope configuration. Scale:. The KHF aspheric -Jan-1 and Zernike trefoil plate are both fabricated in NBK7 substrates and the primary mirror is a commercially available 15. mm, F/1 concave, spherical mirror.

108 7 When the 3 µm trefoil deformation, as shown in Figure -3, is added to the plate, the aberration behavior of the optical system is altered. In Figure - (a-b), the FFDs for astigmatism (Z 5/ ) and elliptical coma (Z 1/11 ) are simulated across a square, 5 diagonal FOV for the trefoil plate oriented at and located 1 mm away from the stop surface. With this visualization, a line symbol is used at each field point to represent the magnitude and orientation of the aberration. In the presence of the trefoil plate, the wavefront of the optical system now exhibits field conugate, field linear astigmatism, shown in Figure - (a), and field constant elliptical coma, shown in Figure - (b), as predicted by Eq. (3.35). Moreover, if the magnitude of the Z 5/ and Z 1/11 contributions to the wavefront for the ( Hx 1, Hy ) = = field point is tracked as a function of the plate position as shown in Figure -5, it can be seen that the magnitude of the astigmatism increases linearly with plate position and the elliptical coma term remains roughly constant as a function of plate position, as expected from the equations in Table 3-3. Any discrepancy in the trend of the aberration magnitudes as a function of the plate position is attributable to the residual higher order aberrations present in the nominal Schmidt telescope design. 3.8 nm 1..5 Figure -3. Simulated interferogram at a wavelength 3.8 nm of the 3 µm trefoil deformation added on one surface of the 1 mm, NBK7 plate to be added into the optical path of the nominal Schmidt telescope..

109 77 Y Field Angle in Obect Space - degrees Y Field Angle in Obect Space - degrees Y Field Angle in Obect Space - degrees X Field X Field Angle Angle in Obect in Obect Space Space - degrees - degrees Z 5/ Z 1/11 Y Field Angle in Obect Space - degrees Y Field Angle in Obect Space - degrees Y Field Angle in Obect Space - degrees X Field X Field Angle Angle in Obect in Obect Space Space - degrees - degrees - - 9:59:3 9:59:3 KHF KHF F/, F/, Schmidt Schmidt 9:59:3 1-May-13 1-May-13 F/, Schmidt FRINGE FRINGE ZERNIKE ZERNIKE PAIR Z5 PAIR AND Z5 ZAND Z vs vs X Field FIELD Angle FIELD ANGLE in ANGLE IN Obect OBJECT IN OBJECT Space SPACE SPACE - degrees Minimum Minimum =.717e- =.717e- F/, F/, Schmidt Schmidt Maximum Maximum =.1885 =.1885 Average Average =.15 =.15 Std Dev Std = Dev.899 =.899 FRINGE ZERNIKE PAIR Z5 AND Z 1waves 1waves ( 3.8 ( 3.8 nm) KHF nm) KHF vs FIELD ANGLE IN OBJECT SPACE (a) 9:59:19 9:59:19 9:59:19 1-May-13 1-May-13 F/, Schmidt FRINGE FRINGE ZERNIKE ZERNIKE PAIR Z1 PAIR AND Z1 Z11 AND Z11 vs vs X Field FIELD Angle FIELD ANGLE ANGLE in IN Obect OBJECT IN OBJECT SPACE Space SPACE - degrees Minimum Minimum =.8881 =.8881 Maximum Maximum =.8937 =.8937 Average Average =.895 =.895 Std Dev Std = Dev.1319 =.1319 FRINGE 1waves ZERNIKE 1waves ( PAIR 3.8 ( Z1 3.8 nm) AND nm) Z11 vs FIELD ANGLE IN OBJECT SPACE Minimum =.717e- Minimum =.8881 Maximum =.1885 Maximum =.8937 Average =.15 Average =.895 Figure -. (a) The predicted Std Dev astigmatism =.899 (Z 5/ ) and (b) elliptical coma Std (ZDev 1/11 =.1319 ) FFDs over KHF 1-May-13 1waves ( 3.8 nm) a square, 5 degree full FOV for the Schmidt KHF telescope 1-May-13 system with the Zernike 1waves ( trefoil 3.8 nm) plate oriented at and located 1mm away from the stop surface. The Zernike trefoil plate generates both field constant elliptical coma and field conugate, field linear astigmatism. (b) Z5/ Z1/11 Waves (@ 3.8 nm) Trefoil Plate Position Relative to Stop (mm) Figure -5. The predicted magnitude of the (a) astigmatism (Z 5/ ) and (b) elliptical coma (Z 1/11 ) as a function of the Zernike trefoil plate position relative to the stop surface for the Hx 1, Hy field point of Schmidt telescope configuration.

110 78 In order to couple the Schmidt telescope to the interferometer, the beam must be retro-reflected at or near the image plane. When the beam is retro-reflected, it is important that the beam traverses the same path heading back to the interferometer. In this case, the beam must strike normal to the retro-reflector so that the light reflects back on itself and to minimize re-trace errors, the pupil of the telescope must be conugate to the retro-reflector. Several retro-reflector configurations can be designed to meet these criteria, however, some of them require the fabrication of custom optical components. In order to employ commercial, off-the-shelf (COTS) components, a retro-reflector that utilizes a concave mirror and plano-convex field lens is selected. A first order layout of this retro-reflector is shown in Figure - where the illumination light (red rays) is retro-reflected by the concave mirror and the field lens ensures the concave mirror and aperture stop of the Schmidt telescope are conugate to one another by imaging the image of the aperture stop as seen through the primary mirror onto the concave mirror (blue rays). More specifically, knowing that the aperture stop is mm away from the primary mirror with a radius of curvature 3.8 mm, the image of the aperture stop is found to be mm in front of the primary mirror or roughly collocated with the aperture stop. Next, based on readily available COTS components, the concave, retro mirror is selected to be a 1.7 mm diameter, 19 mm radius of curvature mirror. With the concave mirror defined, the field lens focal length is calculated to be 1.83 mm. Based on readily available COTS components, the most similar lens that can be found is an NBK7, plano-convex, 9 mm diameter, 18 mm focal length lens.

111 79 From Primary Mirror Field Lens Concave Mirror Image of Stop as seen through Primary R mirror z stop Figure -. First order layout demonstrating how the retro-reflector must be designed to ensure that the pupil of the Schmidt telescope is conugate to the pupil of the concave mirror that sends the wavefront back towards the interferometer.. Fabrication of the Aspheric Corrector/Nonsymmetric Plate With a completed optical design in place, the next step is to procure and fabricate the optical components. The spherical mirror is an existing COTS component; however, the aspheric and trefoil plates are nonstandard optical components that require custom fabrication. Moreover, conventional full aperture lapping techniques cannot be employed to fabricate these components without special tooling; these components require the use of a sub-aperture polishing process. One accessible sub-aperture polishing process at the University of Rochester in the Robert E. Hopkins Center is magnetorheological finishing (MRF). In this process a magnetic, abrasive impregnated fluid is pumped over a polishing wheel. At the apex of the wheel there is a magnetic field. When the fluid encounters the magnetic field, the fluid hardens. An optical surface is set into this hardened region, creating a small polishing zone. The removal of the material is determined by the dwell of the optic in the fluid, and through computer control, a multitude of shape profiles can be polished into the surface. The MRF machine located at the University of Rochester is the QED Q-XE that is capable of polishing both

112 8 rotationally symmetric and nonsymmetric surfaces up to 1 mm in diameter and is well suited for polishing the aspheric and trefoil plates. The NBK7 substrates for fabrication are COTS pre-polished flats to λ/ or better. The aspheric plate substrate is 7. mm in diameter and 8 mm thick whereas the trefoil plate substrate is 1 mm in diameter and 1 mm thick. For each plate only one surface is to be polished by the MRF machine. In order to create the dwell maps for polishing, the removal within the polishing zone must be known. To determine this removal, four spots are polished into a sacrificial 5 mm NBK7 substrate and characterized by a Zygo laser interferometer with a transmission reference flat attached. Once the removal of the polishing zone has been characterized, the machine computes from the initial surface shape a dwell map to create the desired final surface shape. Because a large amount of material needs to be removed for both plates and the removal rate of MRF is generally small, the polishing is split into multiple runs. During each run only part of the overall departure is polished into the surface. With this method, the polishing runtimes are shorter, providing better stability of the polishing parameters during fabrication. After each polishing run, the plates are measured with an interferometer in reflection or transmission if the departure of the surface from planar is too great. The measurement after each polishing run serves as the input for the subsequent polishing cycle. The final surface shape of the third and fifth order aspheric correcting plate and its residual from the theoretical design are shown in Figure -7 (a-b) over a 7 mm clear aperture. As depicted in Figure -7 (a), roughly. µm peak-to-valley (PV) of departure has been polished into the substrate material over the course of fourteen polishing runs that were each removing µm PV. The residual error after polishing, Figure -7 (b), is

113 81 about.5λ PV or.λ RMS at the testing wavelength of 3.8 nm. There is some coma present in the surface but most of the residual is higher order and present at the edges where the slopes are largest. A small center feature as well as a spoking pattern (a mid-spatial frequency error) is observed. Both features are a residual from the subaperture MRF polishing process that is difficult to correct once polished into the part. However, since the measured errors are higher order and not low order astigmatism, it will not impact or prevent any features to be observed during the measurement of the assembled Schmidt optical system. (a) (b) Figure -7. (a) Measured surface departure of the aspheric corrector plate for the Schmidt telescope and (b) residual error when the nominal optical design surface is subtracted from the measured surface. The error is about.5λ PV or.λ RMS at the testing wavelength of 3.8 nm. For the trefoil plate, the final shape and its residual from theoretical are shown in Figure -8 (a-b) over a 98 mm clear aperture. The surface departure of the final surface, Figure -8 (a), is 5.75 µm PV and has been polished in over the course of seven runs that each removed about 1. µm of material. The residual for this surface, Figure -8 (b), is.3λ PV or.5λ RMS. Similar to the case with the asphere, the residual is primarily higher order with a center artifact and a mid-spatial frequency spoking pattern. Some

114 8 fringe features are observed in the residual and are most likely caused by interference from reflections of the back surface. Similar to the case of the asphere, since the residual features are primarily higher order, they will not impact the low order astigmatism that is to be measured by the Schmidt telescope. (a) (b) Figure -8. (a) Measured surface departure of the Zernike trefoil plate and (b) residual error when the nominal optical design surface is subtracted from the measured surface. The error is about.3λ PV or.5λ RMS at the testing wavelength of 3.8 nm..3 Experimental Setup of the Aberration Generating Schmidt Telescope The assembled Schmidt telescope is displayed in Figure -9. The test wavefront from the interferometer is reflected off the motorized, FOV generating fold mirror nominally at 9 where it enters the telescope. The telescope is composed of the aperture stop, aspheric plate, trefoil plate, primary mirror, and retro-reflector. The mount that holds the aspheric plate also serves as the aperture stop of the optical system. The trefoil plate is motorized so that the position of the plate relative to the stop can be varied. Moreover, the trefoil plate and aspheric plate are assembled so that they can be moved as close to each other as possible. The illumination light is focused by the primary mirror and reflected back through the system by the motorized retro-reflector. In this configuration, a

115 83 wide variety of FOVs can be directed into the telescope and the retro-reflector is re-positioned in x, y, and z to send the wavefront back towards the interferometer without defocus or tilt. The entire optical system is computer controlled by a custom written LabVIEW program so that the FOV can be scanned over a grid of points, acquiring the double pass wavefront at each point, for multiple plate positions. The software uses a lookup table for the actuator positions of the FOV mirror and retro-reflector. These lookup tables are created by using the relationship between image displacement and FOV since the focal length of the primary mirror is known. The lookup table for the focus position of the retro-reflector is determined from its x and y position and the radius of curvature of the image plane for the nominal Schmidt telescope. During initial alignment of the telescope, the Zernike trefoil plate is replaced with a λ/1, flat NBK7 plate of the same center thickness as the trefoil plate. This plate provides the correct optical path length between optical elements but does not impart any additional aberrations into the telescope. With this plate in place, the aberration free, on-axis field angle of the telescope is found. This angle lies parallel to the axis that connects the center of curvature of primary mirror and the aperture stop. To find this point, the aperture stop and aspheric plate combination are longitudinally displaced from the center of curvature of the primary mirror. When the aperture stop and aspheric plate are displaced from the center of curvature, third order coma and astigmatism are generated. For a small displacement of the aperture stop and aspheric plate, the generated comatic contribution is largest. By observing the generated third order coma and inherent field curvature in the double pass wavefront with the interferometer as a function of input field angle, the field angle at which both aberrations go to zero is found. This field angle

116 8 corresponds to the aberration free, on-axis field angle. Once the on-axis field is found, the wavefront of an off-axis field angle is observed with the interferometer. The aspheric plate is now translated longitudinally until the coma in the wavefront is zero at the interrogated field point as well as throughout the entire FOV. This step ensures that the aperture stop and aspheric plate are now again at the center of curvature of the primary mirror. With the telescope aligned and the on-axis field angle determined, the Zernike plate is placed into the optical path, replacing the surrogate NBK7 plate. The Zernike plate is aligned so that the generated field conugate, linear astigmatism for a plate position away from the stop is roughly zero on-axis. In addition, the orientation of the Zernike plate is determined by evaluating the Z 1 contribution to Zernike trefoil and adusting the orientation of the plate until this term is zero. This equates to an orientation of for the Zernike plate. With the plate aligned, the experiment proceeds. The LabVIEW program cycles through five longitudinal Zernike plate positions. The first plate position is roughly 1 mm away from the stop surface and the other plate positions are equally spaced at roughly mm increments. At each plate position, the program acquires ten wavefront measurements with a one second difference between measurements over a 9x9 grid throughout the FOV. Averaging multiple measurements over time helps reduce the effects of vibration and air turbulence on the overall measurement since the interferometric cavity is long and there are many mounted optical components that lie in the optical path. In addition to acquiring a set of interferograms over a 9x9 grid of field points with the Zernike plate in the Schmidt telescope, a baseline measurement is also acquired with the

117 85 flat plate in the telescope over the same field grid. This baseline measurement is subtracted from the Zernike plate measurement set as it accounts for any residual misalignment induced aberrations present in the telescope configuration as well as the surface figure error of the folding mirror that generates the FOV for the telescope. This baseline subtraction ensures that the analyzed wavefront only reflects the aberrations induced by the trefoil plate. Zygo Interferometer FOV Mirror Trefoil Plate Aspheric Plate Retro-Reflector Primary Mirror Figure -9. Experimental setup of the Schmidt telescope system. The scanning mirror and retro-reflector are motorized so that the FOV can be scanned over a two-dimensional grid of points. The trefoil plate is also motorized so that effect of plate position on magnitude of generated aberration field can be studied.. Experimental Results..1 The Generated Field Conugate, Field Linear Astigmatic Field As an example of the measurement process, a 3x3 grid of wavefronts spanning a square, 5 diagonal FOV is shown in Figure -1 (a). The data shown is the difference at each field point between the baseline measurement with the flat plate and the actual measurement with the Zernike plate oriented at and displaced approximately 1 mm

118 8 longitudinally away from the stop surface. The obscuration present in the wavefront is from the retro-reflector mounted on a half inch optical post. In evaluating the structure of the wavefront, there is a field constant elliptical coma contribution to the wavefront. In Figure -1 (b) the field constant elliptical coma is subtracted from the wavefront revealing the generated astigmatic contribution. The astigmatic contribution is recognized as field conugate, field linear astigmatism by evaluating the orientation and magnitude of the wavefront throughout the FOV. (a) (b) Figure -1. (a) Measured interferograms after baseline subtraction for a 3x3 grid of field points spanning a square, 5 degree diagonal FOV for the Schmidt telescope system with the Zernike trefoil plate oriented at and displaced roughly 1 mm longitudinally away from the stop surface and (b) the 3x3 grid of wavefronts with the field constant elliptical coma removed, revealing the generated field conugate, field linear astigmatism induced by the trefoil plate. Another way to visualize the generated astigmatic field is to plot the astigmatism, Z 5/, FFD throughout the FOV. In Figure -11 (a-c), left, the measured astigmatism with the baseline subtracted are plotted over a 9x9 grid of field points spanning a square, 5 diagonal FOV at three different plate positions: 1.81 mm, mm, and mm. As a point of comparison, the theoretical aberration fields predicted by NAT are displayed as

119 87 well in Figure -11 (a-c), right. In evaluating Figure -11 (a-c), there is good agreement in both magnitude and orientation between the experimental and theoretical predictions of NAT for all three plate positions. To create a more quantitative comparison, Figure -1 compares the magnitude of the measured versus theoretical Zernike trefoil and Zernike astigmatism for two field points, ( Hx = 1, Hy = ) and ( Hx 1, Hy ) = =, as a function of plate position. As demonstrated earlier in Figure -5, the magnitude of the Zernike trefoil remains constant as a function of plate position and the magnitude of the generated Zernike astigmatism increases linearly with plate position. For the five measured plate positions, the measured and theoretical data agree within the uncertainty in the measurement for both field points analyzed. Any error in the measurement and deviation from theoretical is attributed to air turbulence and vibration in the measurement that has the greatest effect for small plate offsets where the magnitude of the generated astigmatism is small.

120 88.5 Experimental Measured.5 Theoretical Y Ob. Field (deg) Y Ob Field (deg) X Ob. Field (deg) 1λ (λ=.3.8nm) -.5 (a) X Ob Field (deg) Y Ob. Field (deg) Y Ob Field (deg) X Ob. Field (deg) -.5 (b) X Ob Field (deg) Y Ob. Field (deg) Y Ob Field (deg) X Ob. Field (deg) -.5 (c) X Ob Field (deg) Figure -11. The measured Zernike astigmatism (Z 5/ ) FFD after baseline subtraction, left, and theoretical Zernike astigmatism (Z 5/ ) FFD predicted by NAT, right, over a 9x9 grid spanning a square, 5 full FOV for the Schmidt telescope system with the Zernike trefoil plate oriented at and located (a) 1.81 mm, (b) mm, and (c) mm away from the stop surface.

121 89 Magnitude (waves at 3.8 nm) Zernike Trefoil Zernike Astigmatism H=(-1,) H=( 1,) Theoretical Plate Position (mm) Figure -1. Plot of the mean magnitude of the Zernike trefoil and astigmatism after H = 1, H = represented by the blue circle baseline subtraction for two field points, ( x y ) and ( Hx 1, Hy ) = = represented by the red star, for five measured plate positions. The error bars on the data points represent plus or minus one standard deviation from the mean value over the ten measurements acquired at each plate position. In black, the magnitude of the Zernike trefoil and astigmatism based on the theoretical predictions of NAT is plotted as a function of plate position... Rotation of the Aberration Generating Plate In Section..1, the Zernike plate in the Schmidt telescope configuration was aligned on-axis and oriented at. If the Zernike plate is now rotated, the orientation of the astigmatic line images throughout the FOV will also rotate. To demonstrate this effect, the Zernike plate in the Schmidt telescope is rotated 5 and the experiment is repeated to acquire the wavefront throughout the FOV. Figure -13 (a) displays the measured results for the rotated Zernike plate at 5 when the plate is roughly 1 mm away from the stop surface and Figure -13 (b) displays the theoretical predictions from NAT. Similar to the results shown earlier for a plate on-axis and oriented at, there is very good agreement between the measured astigmatic field and the theoretical simulations. In both cases, the zero of the field conugate, field linear astigmatism stays nearly on-axis and the line

122 9 images rotate with the orientation of the Zernike plate. If the astigmatic field for the 5 oriented Zernike plate shown in Figure -13 is compared to the oriented Zernike plate, shown in Figure -11 (c), it can be seen that the field structure is the same except that the entire aberration field is rotated by Y Ob. Field (deg.) Y Ob. Field (deg.) X Ob. Field (deg.) X Ob. Field (deg.) (a) 1λ (λ=.3.8nm) (b) Figure -13. The (a) measured Zernike astigmatism (Z 5/ ) FFD after baseline subtraction and (b) theoretical Zernike astigmatism Z 5/ FFD predicted by NAT over a 9x9 grid spanning a square, 5 degree full FOV for the Schmidt telescope system with the Zernike trefoil plate oriented at 5 and located roughly 1 mm away from the stop surface...3 Lateral Displacement of the Aberration Generating Plate The Zernike trefoil plate has been initially aligned so that the vertex of the Zernike deformation is coincident with the on-axis field point. If the Zernike plate is now shifted laterally, the generated astigmatic aberration field will shift. The shift of the aberration field is predicted by NAT as outlined in Chapter 3 with the introduction of a freeform sigma vector that modifies the astigmatic contribution. For the Schmidt telescope configuration, the Zernike plate is displaced +1 mm in the x-direction and -1 mm in the y-direction. Based on the telescope configuration with the Zernike plate approximately 1 mm away from the stop surface, the freeform sigma vector is computed as

123 σ FF * δ v FF + mm + = = =, y 3.1mm 1mm.31 where the freeform sigma vector defines a new effective field height H FF defined as, 91 (.1) H = H σ FF FF. (.) Since the only astigmatic aberration generated is field conugate, field linear astigmatism, NAT predicts a single node at H = σ FF. To verify this prediction, the experiment to acquire the wavefront throughout the FOV proceeds as described above with, in this case, a laterally shifted Zernike plate. Figure -1 (a) displays the measured results for the shifted Zernike plate and Figure -1 (b) displays the theoretical predictions from NAT. Similar to the results shown earlier for a plate on-axis, there is very good agreement between the measured astigmatic field and the theoretical simulations predicted by NAT. In both cases, the zero of the field conugate, field linear astigmatism has moved off-axis with the zero approximately at +.57 H = deg..57 (.3)

124 Y Ob. Field (deg.) Y Ob. Field (deg.) X Ob. Field (deg.) X Ob. Field (deg.) (a) 1λ (λ=.3.8nm) (b) Figure -1. The (a) measured Zernike astigmatism (Z 5/ ) FFD after baseline subtraction and (b) theoretical Zernike astigmatism (Z 5/ ) FFD predicted by NAT over a 9x9 grid spanning a square, 5 degree full FOV for the Schmidt telescope system with the Zernike trefoil plate oriented at, located roughly 1 mm away from the stop surface, and displaced laterally 1 mm in the x-direction and -1 mm in the y-direction.

125 Chapter 5. Design of a Freeform Unobscured Reflective Imager Employing φ-polynomial Optical Surfaces 93 Historically, optical designers had a reputation for designing optical systems that exceed the industry capabilities for fabrication and/or assembly. In general, these systems were intrinsically rotationally symmetric using spheres, aspheres, or off-axis segments of a rotationally symmetric surface (other than the occasional use of cylindrical or toric surfaces for special case anamorphic systems). Recently, the optical fabrication industry changed this paradigm by implementing a capability to fabricate diamond turned, optical quality surfaces in the LWIR that are not rotationally symmetric. In particular, it is now possible to fabricate an optical surface that is defined as a conic plus the lower order terms of a Zernike polynomial (less than Fringe term 1). In this chapter, an optical system design that is composed of tilted Zernike polynomial mirrors is optimized to create a compact, LWIR optical system that will couple to a 3x5 pixel, 5 µm pixel pitch, uncooled microbolometer detector. The optimization strategies that are employed during the optical design of this nonsymmetric system use the principles of NAT applied to Zernike polynomial optical surfaces, as described in Chapter 3. With this new understanding of freeform optical surfaces and their intrinsic aberration fields, it is now possible to apply a NAT based optical analysis approach to optimization. 5.1 The New Method of Optical Design In the 19s, the first optical designs that involved three or more mirrors in an unobscured configuration started to be declassified and began to appear in limited distribution government reports [7]. Motivated by the advance in LWIR detectors and the accompanying need for stray light control, a number of systems were designed as

126 9 concept designs for missile defense. While many of these systems appear to lack rotational symmetry, detailed analysis reveals that any successful design with a significant FOV was, in fact, based on a rotationally symmetric design with an offset aperture, a biased field, or both. Analysis shows that this fact could be anticipated, as many systems that depart from rotational symmetry immediately display on-axis coma, where the axis for a nonsymmetric system is defined by the OAR []. While there are special configurations that eliminate axial coma, there are very few practical forms that do not reduce to a rotational symmetric form. In 199, an optical system designed by Rodgers was patented that had the property of providing the largest planar, circular input aperture in the smallest overall spherical volume [75]. A design attempting to meet similar constraints can be found in 5 by Nakano [3]. The particular form embodied in the patent of [75] is shown in Figure 5-1 (a). This optical design is a 9:1 afocal relay that operates over a 3 full FOV using four mirrors. It provides a real, accessible exit pupil that is often a requirement in earlier infrared systems requiring cooled detectors. In use, it is coupled with a fast F/number refractive component in a dewar near the detector. It is based on using off-axis sections of rotationally symmetric conic mirrors that are folded into the spherical volume by using one fold mirror (mirror 3).

127 95 5. MM Zernike Polynomial FULL SCALE KHF 17-Sep-11 (a) (b) 5. MM Figure 5-1. (a) Layout of U.S. Patent 5,39,7 consisting of three off-axis sections of rotationally symmetric mirrors and a fourth fold mirror (mirror 3). The optical system had, at the time of its design, the unique property of providing the largest planar, circular input aperture in the smallest overall spherical volume for a gimbaled application. (b) The new optical design based on tilted φ-polynomial surfaces to be coupled to an uncooled microbolometer. As is often the case, many applications would exploit a larger FOV if it were available with usable performance. In addition, if an optical form could be developed at a fast enough F/number, it becomes feasible to transition to an uncooled detector thereby abandoning the need for the reimaging configuration, the external exit pupil, and the refractive component in the dewar. Using the new paradigm of tilted freeform, φ-polynomial optical surfaces, a three mirror, F/1.9 form with a 1 diagonal full FOV has been developed using the methods of NAT for the optimization. The nominal optical design is shown in Figure 5-1 (b) and has an overall RMS WFE of less than λ/1 at a wavelength of 1 µm over a 1 full FOV where the overall RMS WFE is computed as the average plus one standard deviation RMS WFE for all field points. The remainder of this chapter will detail how this solution was developed using the tools and concepts of NAT applied to tilted φ-polynomial surfaces. 5. The Starting Form The first step in the new design process is to design a well corrected rotationally symmetric optical form without regard for the fact that no light can pass through the

128 9 system because of the obscuration by the mirrors. This step corrects the spherical aberration, coma, and astigmatism and creates a basic configuration with conic mirrors to minimize the use of the Zernike terms, which can challenge the testing program. Figure 5- (a-b) shows the result of this step for a system with aggressive goals for the F/number and FOV. The primary and tertiary mirrors are oblate ellipsoids whereas the secondary mirror is hyperbolic and is also the stop surface. The system is well corrected throughout the FOV where the overall average RMS WFE over the 1 full FOV, as displayed in Figure 5- (b), is less than λ/5 (.λ). The next step is to make this fictitious starting point design unobscured. Typically, the solution to creating an unobscured design from an obscured form is to go off-axis in aperture and/or bias the input field []. It is difficult to do so with this geometry because the primary mirror is smaller than the secondary and tertiary mirrors. With the knowledge that there is a path to removing axial coma by using the new design DOF, machining coma directly onto the surface, the new strategy is to simply tilt the surfaces until the light clears the mirrors. 1:1:5 1:1:5 k Sec =-.5 k Ter =.3 KHF 9-Jun-11 11:5:1 - Zernike Polynomial k Pri =.95 11:5:1 Y Field KHF - - Y Field Angle in Obect Space - degrees X Field Angle in Obect Space - degrees - Minimum =.877 Maximum =.388 Average =.313 Std Dev = Jun-11 RMS WAVEFRONT ERROR vs FIELD ANGLE IN OBJECT SPACE Minimum =.877 Maximum =.388 Average =.313 Std Dev =..5waves (1. nm).5waves (1. nm) 1.71 MM - Zernike Polynomial Scale: 1.7 KHF 3-Jun-11 Zernike Polynomial Scale: 1.7 KHF 3-Jun-11 (a) 1.71 MM X Field Angle in Obect Space - degrees Figure 5-. (a) Layout for a fully obscured solution for a F/1.9, 1 full FOV LWIR imager. The system utilizes three conic mirror surfaces. (b) A FFD of the RMS WFE of the optical system. Each circle represents the magnitude of the RMS wavefront at a particular location in the FOV. The system exhibits a RMS WFE of < λ/5 over 1 full FOV. (b)

129 The Unobscured Form Tilting the on-axis solution breaks the rotational symmetry of the system and changes where the aberration field zeros (nodes) are located for each aberration type. The shift of the aberration fields drastically degrades the overall performance of the system. A strategy for tracking the evolution of the nodal structure as the unobscured design form unfolds is to oversize the FOV to many times the intended FOV. As an example of this strategy, Figure 5-3 (a-c) shows the design form at %, 5% and 1% unobscured accompanied by an evaluation of Zernike coma (Z 7/8 ) and Zernike astigmatism (Z 5/ ) across a ± field (note there is a 1X scale change between Figure 5-3 (a) and Figure 5-3 (b-c) so the nodal behavior can be seen for each tilt position). As can be seen from Figure 5-3 (a), the on-axis solution is well corrected for astigmatism and coma within the 1 diagonal full FOV (sub-region in red) and the nodes (blue star and green dot) are centered on the optical axis (zero field). As the system is tilted halfway to an unobscured solution, shown in Figure 5-3 (b), the node for coma has moved immediately beyond the field being evaluated resulting in what is a field constant coma. For this intermediate tilt, one of the two astigmatic nodes remains within the extended analysis field, moving linearly with tilt. When the system is tilted to a fully unobscured solution, shown in Figure 5-3 (c), field constant coma is increased while the astigmatic node also moves out of the 8X oversized analysis field leaving the appearance of a field constant astigmatism. The first significant observation regarding formulating a strategy for correction is that in the unobscured configuration the nodes have moved so far out in the field that the astigmatism and coma contributions within the region of interest, a 1 full FOV, are nearly constant.

130 :37:35 1:37:35 1:37:35 1:5: 1:5: Zernike Polynomial 5. MM 1:37:35 1:5: Zernike Polynomial 1::7 1:5: 1::7 1::7 Zernike Polynomial Zernike Polynomial Zernike Polynomial 1::7 Zernike Polynomial Zernike Zernike Polynomial Polynomial Y Field Angle in Obect Space - degrees 5. MM - 5. KHF MM 8-Jun-11 1:5: Z Coma Z Coma X Field Angle in Obect Space X Field - - degrees Angle in Obect Space - - degrees FRINGE ZERNIKE PAIR - Z5 AND Z FRINGE ZERNIKE PAIR - Z5 AND Z vs vs FIELD ANGLE IN OBJECT SPACEFIELD ANGLE IN OBJECT SPACE Zernike Polynomial Zernike Polynomial - FRINGE - ZERNIKE Minimum PAIR =.19 - Z5 AND Z - Minimum = -.19 Maximum vs = 8.53 Maximum = 8.53 X Field Angle in Obect Space - degrees Average = X Field Angle in Obect Space - degrees - - X Field FIELD Angle ANGLE in Obect IN OBJECT Space SPACE - degrees Average = X Field Angle in Obect Space - degrees - - Std Dev = 1.31 Std Dev = 1.31 Zernike Polynomial - - Minimum = KHF 8-Jun-11KHF Maximum 8-Jun-11 = λ 3waves (1µm) (1. nm).5λ 3waves (1µm) (1. nm) X Field Angle X in Field Obect Space (deg.) - degrees Average = X Field Angle in Obect Space - degrees X Field (deg.) Std Dev = 1.31 FRINGE ZERNIKE PAIR Z7 AND Z8 FRINGE ZERNIKE PAIR Z5 AND Z 5. KHF MM 8-Jun-11 3waves vs vs FIELD ANGLE IN OBJECT SPACE (a) (a).5λ (1µm) (1. (1µm) nm) FIELD ANGLE IN OBJECT SPACE Zernike Polynomial Zernike Polynomial KHF 8-Jun-11 FRINGE ZERNIKE PAIR Z7 AND Z8 Minimum =.11e-5 FRINGE ZERNIKE PAIR Z5 AND Z Minimum =.1957e-5 vs Maximum =.1889 vs Maximum =.81 (a) FIELD ANGLE IN OBJECT SPACE Average =.77 FIELD ANGLE IN OBJECT SPACE Average =.8757 Std Dev =.913 Std Dev =.838 Zernike Polynomial Minimum =.11e-5 Minimum =.1957e-5 KHF Maximum 8-Jun-11 = waves (1. nm) KHF Maximum 8-Jun-11 =.81.5waves (1. nm) Average =.77 Average =.8757 Y Field Angle in Obect Space - degrees 1::1 Y Field Angle in Obect Space - degre Y Field Angle in Obect Space - degrees Zernike Polynomial Zernike - FRINGE - ZERNIKE Polynomial Minimum PAIR - Z5 = AND.19 Z - Minimum = -.19 Maximum vs = 8.53 Maximum = 8.53 X Field Angle in Obect Space X Field - degrees Angle in Obect Space - degrees X Field Angle in Obect Space X Field - degrees Angle in Obect Space - degrees - - FIELD ANGLE IN Average OBJECT = SPACE Average = Std Dev = 1.31 Std Dev = 1.31 Zernike Polynomial - - Minimum = KHF 8-Jun-11KHF Maximum 8-Jun-11 = λ 3waves (1µm) (1. nm) 3waves (1. nm) X Field Angle 3λ (1µm) X in Field Obect Space (deg.) - degrees Average = X Field Angle in Obect Space - degrees X Field (deg.) Std Dev = 1.31 FRINGE ZERNIKE PAIR Z7 AND Z8 FRINGE ZERNIKE PAIR Z7 AND Z8 FRINGE ZERNIKE PAIR Z5 AND Z FRINGE ZERNIKE PAIR Z5 AND Z 5. KHF MM 8-Jun-11 vs.5λ 3waves vs (1. nm) vs vs FIELD ANGLE IN OBJECT SPACEFIELD ANGLE IN OBJECT SPACE FIELD ANGLE IN OBJECT SPACEFIELD ANGLE IN OBJECT SPACE (b) (b) 3λ (1µm) (1µm) Zernike Polynomial Zernike FRINGE ZERNIKE Polynomial Zernike Polynomial Zernike Polynomial Minimum PAIR Z7 = AND 3.7 Z8 Minimum = 3.7 FRINGE ZERNIKE Minimum PAIR Z5 =.978 AND Z Minimum =.978 KHF 8-Jun-11 Maximum vs = Maximum = Maximum vs = 1.38 Maximum = 1.38 (b) Average =.879 Average =.879 Average = (a) FIELD ANGLE IN OBJECT SPACE FIELD ANGLE IN OBJECT SPACE 9.98 Average = 9.98 Std Dev =.591 Std Dev =.591 Std Dev = 3.91 Std Dev = 3.91 Zernike Polynomial Minimum Minimum =.11e-5 = 3.7 Minimum = =.1957e Jun-11KHF Maximum 8-Jun-11 = waves (1. nm) KHF 3waves 8-Jun-11 (1. nm) KHF Maximum 8-Jun-11 = waves (1. nm) 3waves (1. nm) Average = = Average = Std Dev =.591 Std Dev = Zernike Polynomial KHF 8-Jun-11 KHF 1::1 Y Field Angle in Obect Space - degrees Y Field Angle in Obect Space - degrees X Field Angle in Obect Space - - degrees - FRINGE ZERNIKE PAIR - Z5 AND Z FRINGE ZERNIKE PAIR Z5 AND Z - vs - vs FIELD ANGLE IN OBJECT SPACEFIELD ANGLE IN OBJECT SPACE - Zernike Polynomial Zernike - FRINGE - ZERNIKE Polynomial Minimum PAIR - Z5 = AND.19 Z - Minimum = -.19 Maximum vs = 8.53 Maximum = 8.53 X Field Angle in Obect Space X Field - degrees Angle in Obect Space - degrees - - FIELD ANGLE IN Average OBJECT = SPACE Average X Field = Angle in Obect Space X Field - degrees Angle in Obect Space - degrees - - Std Dev = 1.31 Std Dev = 1.31 Zernike Polynomial - - Minimum = KHF 8-Jun-11 KHF Maximum 8-Jun-11 = 8.53 X Field Angle in Obect Space - degrees 3λ 3waves (1µm) (1. nm) 3λ 3waves (1µm) (1. nm) X Field Angle in Obect Space - degrees X Field (deg.) Average = X Field (deg.) Std Dev = 1.31 FRINGE ZERNIKE PAIR Z7 AND Z8 FRINGE ZERNIKE PAIR Z7 AND Z8 5. MM 5. MM FRINGE ZERNIKE PAIR Z5 AND Z FRINGE ZERNIKE PAIR Z5 AND Z KHF 8-Jun-11 vs 3waves (1. nm) vs vs vs FIELD ANGLE IN OBJECT SPACEFIELD ANGLE IN OBJECT SPACE FIELD ANGLE IN OBJECT SPACEFIELD ANGLE IN OBJECT SPACE (c) (c) 3λ (1µm) (1µm) Zernike Polynomial KHF 8-Jun-11 Zernike Polynomial KHF 8-Jun-11 Zernike FRINGE ZERNIKE Polynomial PAIR Z7 AND Z8 Minimum = Minimum Zernike = Polynomial Zernike FRINGE Polynomial 5. MM ZERNIKE PAIR Z5 AND Z Minimum = Minimum = MM vs vs Maximum = Maximum = Maximum = Maximum = FIELD ANGLE IN OBJECT SPACE FIELD (c) Average = Average = ANGLE IN OBJECT SPACE (b) Average = 9.83 Average = 9.83 Std Dev = 5.38 Std Dev = 5.38 KHF 8-Jun-11 Polynomial Zernike Polynomial Std Dev = Std Dev = Minimum = 3.7 Minimum = = KHF 8-Jun-11 KHF 8-Jun-11KHF Maximum 8-Jun-11 = waves (1. nm) KHF 3waves 8-Jun-11 (1. nm) Maximum KHF = 8-Jun Figure 5-3. The lens layout, Zernike coma (Z 3waves (1. nm) 3waves (1. nm) Average = Average = Std 7/8 ) and astigmatism (Z Dev = / ) FFDs for a ± Std Dev = KHF 8-Jun-11 3waves (1. nm) KHF 8-Jun-11 3waves (1. nm) FOV for the (a) on-axis optical system, (b) halfway tilted, 5% obscured system, and (c) fully tilted, 1% unobscured system. The region in red shows the field of interest, a 1 diagonal FOV. 1::7 8-Jun-11 Y Field Angle in Obect Space - degrees KHF Zernike Polynomial 8-Jun-11 1:37:35 1:37:35 1:5: 1:5: 1::7 Y Field Angle in Obect Space - degrees KHF Y Field Angle in Obect Space - degrees KHF Y Field Angle in Obect Space - degrees ::1 8-Jun-11 1::1 1::1 1::1 Y Field Angle in Obect Space - degrees Y Field Angle in Obect Space 1:37:35 - degrees Y Field Angle in Obect Space 1:5: - degrees Y Field Angle in Obect Space 1::7 - degrees Y Field Angle in Obect Space - degrees Y Field Angle in Obect Space - degrees - Z Coma ::1 1::1 1::1 Y Field Angle in Obect Space - degrees Y Field Angle in Obect Space - degrees Z Coma 1:5: Y Field Angle in Obect Space - degrees X Field Angle in Obect Space - - degrees Std Dev =.913 Y Field Angle in Obect Space - degrees - - X Field Angle in Obect Space X Field - degrees Angle in Obect Space - degrees Y Field Angle in Obect Space - degrees 1:37:3 1:37:3 1:5: FRINGE ZERNIKE PAIR - Z5 AND Z FRINGE ZERNIKE PAIR - Z5 AND Z vs vs FIELD ANGLE IN OBJECT SPACEFIELD ANGLE IN OBJECT SPACE X Field Angle in Obect Space X Field - - degrees Angle in Obect Space - - degrees 1:: Y Field Angle in Obect Space - degrees KHF.5waves (1. nm) Y Field Angle in Obect Space - degrees X Field Angle in - Obect Space - degrees Std Dev =.913.5waves 3waves (1. nm) KHF 1:: 1:5: Y Field Angle in Obect Space - degrees Y Field Angle in Obect Space - degrees - X Field Angle in Obect Space - degrees Creating Field Constant Aberration Correction 8-Jun-11 8-Jun-11 1:37:3 1:5: 1:: Z Astig Z Astig Z Astig Y Field Angle in Obect Space - degrees - - Y Field Angle in Obect Space - degrees - - Y Field Angle in Obect Space - degrees - Std Dev =.838.5waves (1. nm) 3waves.5waves (1. nm) - X Field Angle in Obect Space - - degrees With a baseline unobscured system - established, the FRINGE ZERNIKE next PAIR - Z5 AND Zstep is to use the new DOFs, vs FIELD ANGLE IN OBJECT SPACE Average = Std Dev = 1.31 KHF 8-Jun-11 3λ 3waves (1µm) (1. nm) efficiently and effectively, to create a usable performance over the 1 diagonal full FOV FRINGE ZERNIKE PAIR Z7 AND Z8 5. MM FRINGE ZERNIKE PAIR Z5 AND Z vs vs FIELD ANGLE IN OBJECT SPACE FIELD ANGLE IN OBJECT SPACE (c) KHF 8-Jun-11 Zernike Polynomial Minimum = Zernike Polynomial Minimum = Maximum = Maximum = Average = Average = 9.83 Std Dev = 5.38 Std Dev = and at an F/number that allows the use of an uncooled microbolometer (less than F/). Zernike Polynomial 1::7 Y Field (deg.) Y Field (deg.) Y Field (deg.) Y Field Angle in Obect Space - degrees KHF 8-Jun-11 1:: Y Field Angle in Obect Space - degrees 3waves (1. nm) KHF 8-Jun-11 Z Astig Zernike Polynomial - - Minimum = Maximum = 8.53 X Field Angle in Obect Space - degrees X Field Angle in Obect Space - degrees 1::1 Y Field (deg.) Y Field (deg.) Y Field (deg.) 3waves (1. nm) Now that the nodal evolution has been established, it is more effective to return to an analysis only over the target FOV. Figure 5- shows that when the field performance is

131 99 evaluated over a smaller field, ±5, the field constant behavior is clearly observed for both coma and astigmatism as well as for the higher order aberration contributions, such as elliptical coma (Z 1/11 ) and oblique spherical aberration (Z 1/13 ). It is worth noting that Figure 5- shows that the spherical aberration (Z 9 ) is nearly unchanged even for this highly tilted system. An evaluation of the RMS WFE is also added (far lower right) to determine when adequate correction is achieved. For this starting point, the RMS WFE is ~1λ and is predominately due to the astigmatism and coma contributions both of which are, significantly, field constant. 13:53: Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial KHF X Field (deg.) - - Field - Angle Obect Space degrees X in - 3-Sep-13 13:53: Z Spher FRINGE ZERNIKE COEFFICIENT Z9 vs FIELD ANGLE IN OBJECT SPACE Y Field Angle in Obect Space - degrees Y Field (deg.) Minimum = -. Maximum = Average = Std Dev = :5:59 1waves (1. nm) Zernike Polynomial Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial KHF Z Obl.Spher. - - X Field (deg.) X Field (deg.) - - Field - Angle Obect Space degrees X in - 3-Sep Field - Angle Obect Space degrees X in - 1:37:1 FRINGE ZERNIKE PAIR Z1 AND Z13 vs KHF FIELD ANGLE IN OBJECT SPACE Minimum =.85 Maximum =.5357 Average =.3919 Std Dev =.887 Y Field Angle in Obect Space - degrees :53: Z Coma FRINGE ZERNIKE PAIR Z7 AND Z8 vs FIELD ANGLE IN OBJECT SPACE Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial Minimum = Maximum = 1.99 Average = Std Dev = :5:59 1waves (1. nm) Zernike Polynomial Y Field Angle in Obect Space - degrees Y Field Angle in Obect Space - degrees Zernike Polynomial KHF X Field Angle in Obect Space - degrees 3-Sep-13 13:53:1 - - Field - Angle Obect Space degrees X Z - 5th Coma in - X Field (deg.) FRINGE ZERNIKE PAIR Z5 AND Z - - vs FIELD ANGLE - IN OBJECT SPACE 3-Sep-13 Minimum = Maximum =.173 Average =.335 Std Dev =.8811 FRINGE ZERNIKE PAIR Z1 AND Z15.5waves (1. vs nm) 1.λ µm) FIELD ANGLE IN OBJECT SPACE Minimum =.1717 Maximum =. Average =.1959 Std Dev =.157 Zernike Polynomial Minimum = 31.1 X Field Angle Maximum = 3. in Obect Space - degrees Average = 3.98 Std Dev = :53: 1waves (1. nm) KHF 3-Sep-13 1waves (1. nm) KHF 3-Sep-13 1waves (1. nm) KHF 3-Sep-13 1waves (1. nm) elliptical coma (Z 1/11 ), oblique spherical aberration (Z 1/13 ), and fifth order aperture coma Y Field (deg.) FRINGE - ZERNIKE - COEFFICIENT Z9 - - X Field - Angle KHF vs in Obect Space 3-Sep-13 - degrees FIELD ANGLE IN OBJECT SPACE X Field (deg.) Zernike Polynomial 13:53:1 Z Astig Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial Y Field Angle in Obect Space - degrees Y Field (deg.) KHF RMS WFE 3-Sep-13 RMS WAVEFRONT ERROR vs FIELD ANGLE IN OBJECT SPACE Minimum = 11.9 Maximum = Average = Std Dev = X Field (deg.) waves (1. nm) X Field Angle in Obect Space - degrees RMS WAVEFRONT ERROR vs FIELD ANGLE IN OBJECT SPACE - - Field - Angle Obect Space degrees X in - X Field (deg.) Minimum = 11.9 Maximum = Average = Std Dev =.9 Z Ellip. Coma Figure 5-. The lower order spherical (Z 9 ), coma (Z 7/8 ), astigmatism (Z 5/ ), higher order, (Z 1/15 ) Zernike aberration contributions and RMS WFE FFDs over a ±5 degree FOV for the fully unobscured, on-axis solution. It can be seen that the system is dominated by field constant coma and astigmatism which are the largest contributors to the RMS WFE of ~1λ. FRINGE ZERNIKE PAIR Z1 AND Z11 vs FIELD ANGLE IN OBJECT SPACE Minimum =.993 Maximum = 1.38 Average = 1.13 Std Dev = waves (1. nm) It is possible to correct the field constant aberrations shown in Figure 5- by using the fact that the stop location for this optical system is the secondary mirror. In Chapter 3, it was shown that when a Zernike polynomial overlay is placed at the stop location, a field constant aberration is induced. In this design case, Zernike coma and astigmatism are

132 1 added as variables to the secondary conic surface, so they will introduce, when optimized, the opposite amount of field constant coma and astigmatism present from tilting the optical system to create an unobscured form. The effect of optimizing the optical system with these variables is shown in Figure 5-5 where the field constant coma and astigmatism have been removed. The RMS WFE has gone from ~1λ for the tilted system without φ-polynomials to ~.75λ for the tilted system with Zernike coma and astigmatism on the secondary surface (note that there is a 1X scale change from Figure 5- to Figure 5-5 to show the residual terms in further detail). 13:57:1 Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial KHF X Field (deg.) - - Field - Angle Obect Space degrees X in - 3-Sep-13 13:57: Z Spher FRINGE ZERNIKE COEFFICIENT Z9 vs FIELD ANGLE IN OBJECT SPACE Y Field Angle in Obect Space - degrees Y Field (deg.) Minimum =.311 Maximum =.18 Average =.778 Std Dev = waves (1. nm) Zernike Polynomial 13:57:1 Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial KHF Z Obl.Spher. - - X Field (deg.) X Field (deg.) - - Field - Angle Obect Space degrees X in - 3-Sep Field - Angle Obect Space degrees X in - 1:37:1 FRINGE ZERNIKE PAIR Z1 AND Z13 vs KHF FIELD ANGLE IN OBJECT SPACE Minimum =.51 Maximum =.1889 Average =.1137 Std Dev = Y Field Angle in Obect Space - degrees :57: Z Coma FRINGE ZERNIKE PAIR Z7 AND Z8 vs FIELD ANGLE IN OBJECT SPACE Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial Minimum =.518 Maximum =.73 Average =.5737 Std Dev = waves (1. nm) Zernike Polynomial 13:57:1 Y Field Angle in Obect Space - degrees Y Field Angle in Obect Space - degrees FRINGE ZERNIKE PAIR Z5 AND Z vs - - FIELD ANGLE IN -OBJECT SPACE Zernike Polynomial Zernike Polynomial Minimum =.5353 X Field Maximum Angle = in Obect Space - degrees Average =.7771 Std Dev = 1.97 KHF X Field Angle in Obect Space - degrees 3-Sep-13 13:57: - - Field - Angle Obect Space degrees X Z 5th Coma - in - X Field (deg.) 3-Sep-13 Minimum = Maximum =.173 Average =.335 Std Dev =.8811 FRINGE ZERNIKE PAIR Z1 AND Z15.5waves (1. vs nm) 1.λ µm) FIELD ANGLE IN OBJECT SPACE Minimum =.38 Maximum =.955 Average =.7818 Std Dev =.117 1waves (1. nm) 13:57: Y Field Angle in Obect Space - degrees KHF Sep-13 FRINGE ZERNIKE PAIR Z1 AND Z11 vs FIELD ANGLE IN OBJECT SPACE KHF 3-Sep-13 1waves (1. nm) KHF 3-Sep-13 1waves (1. nm) KHF 3-Sep-13 1waves (1. nm) elliptical coma (Z 1/11 ), oblique spherical aberration (Z 1/13 ), and fifth order aperture coma Y Field (deg.) Zernike Polynomial FRINGE - ZERNIKE - COEFFICIENT Z9 - - X Field - KHF 3-Sep-13 vs Angle in Obect Space - degrees FIELD ANGLE IN OBJECT SPACE X Field (deg.) 13:57: Z Astig Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial Y Field (deg.) RMS WFE RMS WAVEFRONT ERROR vs FIELD ANGLE IN OBJECT SPACE Minimum = Maximum = 1.7 Average =.7179 Std Dev = waves (1. nm) - - X Field - Angle in Obect Space - degrees X Field (deg.) RMS WAVEFRONT ERROR vs FIELD ANGLE IN OBJECT SPACE - - Field - Angle Obect Space degrees X in - X Field (deg.) Minimum = Maximum = 1.7 Average =.7179 Std Dev =.519 Z Ellip. Coma Figure 5-5. The lower order spherical (Z 9 ), coma (Z 7/8 ), astigmatism (Z 5/ ), higher order, (Z 1/15 ) Zernike aberration contributions and RMS WFE FFDs over a ±5 degree FOV for the optimized system where Zernike astigmatism and coma were used as variables on the secondary (stop) surface. When the system is optimized, the field constant contribution to astigmatism and coma are greatly reduced improving the RMS WFE from ~1λ to ~.75λ. Minimum =.73 Maximum = 1.75 Average =.5787 Std Dev =.785 1waves (1. nm) 5.3. Creating Field Dependent Aberration Correction By studying the residual behavior of the optical system after optimization of Zernike coma and astigmatism on the secondary surface, it can be seen from the displays, shown

133 11 in Figure 5-5, that the dominant aberration contribution is Zernike astigmatism and it is the largest contributor to the RMS WFE of ~.75λ. Moreover, the astigmatism has taken the form of field linear, field asymmetric astigmatism. In Chapter 3, it was discovered that a Zernike coma overlay displaced axially away from the stop surface will introduce field linear, field asymmetric astigmatism as well as field linear, field curvature. Using this result, Zernike coma is placed on an optical surface away from the stop location, that is, the primary or tertiary surface, and optimized to reduce (and in some cases eliminate) the residual field linear, field asymmetric astigmatism. However, in order to effectively use this added variable, the tilt angle of the focal plane must also be varied to compensate the induced field linear, field curvature component also introduced by the Zernike coma overlay. Since both the primary and tertiary surfaces lie away from the stop surface, a Zernike coma overlay can be placed on either surface. If the equation for the generated field linear, field asymmetric astigmatism from a Zernike coma overlay away from the stop is investigated, Eq. (3.5), it can be seen that the magnitude of the aberration depends linearly on the ratio of the chief to marginal ray on the optical surface. For the current configuration, the ratio at the primary surface is roughly.17 whereas the ratio is.8 on the tertiary surface. Based on this first order analysis, it appears that the primary surface will be a much more effective variable for removing the residual field linear, field asymmetric astigmatism since less comatic departure will be required to create the equivalent induced astigmatic aberration. The effectiveness of the Zernike coma overlay on the primary mirror surface is demonstrated in Figure 5- where the relevant aberration contributions after optimization

134 1 are shown. As can be seen from Figure 5-, which is on the same scale as Figure 5-5, the astigmatism contribution has been reduced and the RMS WFE has been improved by another factor of X going from.75λ to roughly.15λ. The astigmatism contribution has not been eliminated completely by the use of the Zernike coma overlay because while the Zernike coma overlay is used to correct the residual third order, field linear, field asymmetric astigmatism, it also induces higher order aberration components on subsequent surfaces that will impact the overall RMS WFE. The optimized solution is a balance of these two effects. 13:58: Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial KHF X Field (deg.) - - Field - Angle Obect Space degrees X in - 3-Sep-13 13:58:3 Z Spher FRINGE ZERNIKE COEFFICIENT Z9 vs FIELD ANGLE IN OBJECT SPACE Y Field Angle in Obect Space - degrees Y Field (deg.) Minimum =.7859 Maximum =.88 Average =.815 Std Dev = waves (1. nm) Zernike Polynomial 13:58: Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial KHF Z Obl.Spher. - - X Field (deg.) X Field (deg.) - - Field - Angle Obect Space degrees X in - 3-Sep Field - Angle Obect Space degrees X in - 1:37:1 FRINGE ZERNIKE PAIR Z1 AND Z13 vs KHF FIELD ANGLE IN OBJECT SPACE Minimum =.889 Maximum =.531 Average =.8583 Std Dev =.183 Y Field Angle in Obect Space - degrees :58:3 Z Coma FRINGE ZERNIKE PAIR Z7 AND Z8 vs FIELD ANGLE IN OBJECT SPACE Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial Minimum =.3373 Maximum =.58 Average =.315 Std Dev = waves (1. nm) Zernike Polynomial 13:58: Y Field Angle in Obect Space - degrees Y Field Angle in Obect Space - degrees FRINGE ZERNIKE PAIR Z5 AND Z vs - - FIELD ANGLE IN -OBJECT SPACE Zernike Polynomial Zernike Polynomial Minimum =.73 X Field Maximum Angle =.13 in Obect Space - degrees Average =.1883 Std Dev =.9157 KHF X Field Angle in Obect Space - degrees 3-Sep-13 13:58: - - Field - Angle Obect Space degrees X Z 5th Coma - in - X Field (deg.) 3-Sep-13 Minimum = Maximum =.173 Average =.335 Std Dev =.8811 FRINGE ZERNIKE PAIR Z1 AND Z15.5waves (1. vs nm) 1.λ µm) FIELD ANGLE IN OBJECT SPACE Minimum =.337 Maximum =.397 Average =.713 Std Dev = waves (1. nm) 13:58:3 Y Field Angle in Obect Space - degrees KHF Sep-13 FRINGE ZERNIKE PAIR Z1 AND Z11 vs FIELD ANGLE IN OBJECT SPACE KHF 3-Sep-13 1waves (1. nm) KHF 3-Sep-13 1waves (1. nm) KHF 3-Sep-13 1waves (1. nm) elliptical coma (Z 1/11 ), oblique spherical aberration (Z 1/13 ), and fifth order aperture coma Y Field (deg.) Zernike Polynomial FRINGE - ZERNIKE - COEFFICIENT Z9 - - X Field - KHF 3-Sep-13 vs Angle in Obect Space - degrees FIELD ANGLE IN OBJECT SPACE X Field (deg.) 13:58: Z Astig Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial Y Field (deg.) RMS WFE RMS WAVEFRONT ERROR vs FIELD ANGLE IN OBJECT SPACE Minimum =.7 Maximum =.159 Average =.155 Std Dev = waves (1. nm) - - X Field - Angle in Obect Space - degrees X Field (deg.) RMS WAVEFRONT ERROR vs FIELD ANGLE IN OBJECT SPACE - - Field - Angle Obect Space degrees X in - X Field (deg.) Minimum =.7 Maximum =.159 Average =.155 Std Dev =.175 Z Ellip. Coma Figure 5-. The lower order spherical (Z 9 ), coma (Z 7/8 ), astigmatism (Z 5/ ), higher order, (Z 1/15 ) Zernike aberration and RMS WFE FFDs over a ±5 degree FOV for the optimized system where Zernike coma is added as an additional variable to the primary surface. The RMS WFE has been reduced from ~.75λ to ~.15λ. Minimum =.88 Maximum =.895 Average =.397 Std Dev = waves (1. nm) If the Zernike coma overlay is instead added onto the tertiary mirror surface and optimized, there is a similar improvement to the RMS WFE where the average performance is around.18λ but the relevant aberration contributions after optimization, shown in Figure 5-7, are much different. In this configuration the residual astigmatism is

135 13 no longer field linear, field asymmetric and now resembles field conugate, field linear astigmatism, a fifth order aberration. Moreover, the residual coma and elliptical coma contributions are smaller than the case where the Zernike coma overlay is applied to the primary surface; however, the Zernike spherical and oblique spherical aberration contributions are larger. The difference in these higher order aberration components is a result of induced aberrations that stem from the arrangement of Zernike overlay terms on the mirror surfaces. While the overlay terms help correct residual aberrations in the design, they may, depending on their location in the optical system, induce higher order aberrations since the beam shape and ray angles on subsequent surfaces will change. 13:59:51 Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial KHF X Field (deg.) - - Field - Angle Obect Space degrees X in - 3-Sep-13 13:59:51 Z Spher FRINGE ZERNIKE COEFFICIENT Z9 vs FIELD ANGLE IN OBJECT SPACE Y Field Angle in Obect Space - degrees Y Field (deg.) Minimum = -.11 Maximum = Average = -.18 Std Dev = waves (1. nm) Zernike Polynomial 13:59:51 Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial KHF Z Obl.Spher. - - X Field (deg.) X Field (deg.) - - Field - Angle Obect Space degrees X in - 3-Sep Field - Angle Obect Space degrees X in - 1:37:1 FRINGE ZERNIKE PAIR Z1 AND Z13 vs KHF FIELD ANGLE IN OBJECT SPACE Minimum =.18 Maximum =.158 Average =.155 Std Dev =.119 Y Field Angle in Obect Space - degrees :59:51 Z Coma FRINGE ZERNIKE PAIR Z7 AND Z8 vs FIELD ANGLE IN OBJECT SPACE Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial Minimum =.133 Maximum =.399 Average =.1397 Std Dev = waves (1. nm) Zernike Polynomial 13:59:5 Y Field Angle in Obect Space - degrees Y Field Angle in Obect Space - degrees FRINGE ZERNIKE PAIR Z5 AND Z vs - - FIELD ANGLE IN -OBJECT SPACE Zernike Polynomial Zernike Polynomial Minimum =.5711 X Field Maximum Angle =.853 in Obect Space - degrees Average =.83 Std Dev =.13 KHF X Field Angle in Obect Space - degrees 3-Sep-13 13:59:5 - - Field - Angle Obect Space degrees X Z 5th Coma - in - X Field (deg.) 3-Sep-13 Minimum = Maximum =.173 Average =.335 Std Dev =.8811 FRINGE ZERNIKE PAIR Z1 AND Z15.5waves (1. vs nm) 1.λ µm) FIELD ANGLE IN OBJECT SPACE Minimum =.117 Maximum =.338 Average =.1188 Std Dev =.88 1waves (1. nm) 13:59:51 Y Field Angle in Obect Space - degrees KHF Sep-13 FRINGE ZERNIKE PAIR Z1 AND Z11 vs FIELD ANGLE IN OBJECT SPACE KHF 3-Sep-13 1waves (1. nm) KHF 3-Sep-13 1waves (1. nm) KHF 3-Sep-13 1waves (1. nm) elliptical coma (Z 1/11 ), oblique spherical aberration (Z 1/13 ), and fifth order aperture coma Y Field (deg.) Zernike Polynomial FRINGE - ZERNIKE - COEFFICIENT Z9 - - X Field - KHF 3-Sep-13 vs Angle in Obect Space - degrees FIELD ANGLE IN OBJECT SPACE X Field (deg.) 13:59:5 Z Astig Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial Y Field (deg.) RMS WFE RMS WAVEFRONT ERROR vs FIELD ANGLE IN OBJECT SPACE Minimum =.177 Maximum =.1 Average =.181 Std Dev = waves (1. nm) - - X Field - Angle in Obect Space - degrees X Field (deg.) RMS WAVEFRONT ERROR vs FIELD ANGLE IN OBJECT SPACE - - Field - Angle Obect Space degrees X in - X Field (deg.) Minimum =.177 Maximum =.1 Average =.181 Std Dev =.185 Z Ellip. Coma Figure 5-7. The lower order spherical (Z 9 ), coma (Z 7/8 ), astigmatism (Z 5/ ), higher order, (Z 1/15 ) Zernike aberration contributions and RMS WFE FFDs over a ±5 degree FOV for the optimized system where Zernike coma is added as an additional variable to the tertiary surface. The RMS WFE has been reduced from ~.75λ to ~.18λ. Minimum =.899 Maximum =.588 Average =.597 Std Dev =.359 1waves (1. nm)

136 1 5. The Final Form With the successful creation of a nearly compliant unobscured form, the remaining optimization proceeds with additional use of Zernike coefficients for either field constant or field dependent correction. Continuing from the configuration with Zernike astigmatism and coma on the secondary mirror and Zernike coma on the tertiary surface, Figure 5-7 shows that the optical system is now limited by field constant aberrations, namely, field constant oblique spherical aberration that shows up in the Z 9 and Z 1/13 FFDs, field constant elliptical coma that shows up in the Z 1/11 FFD, and field constant fifth order aperture coma that shows up in the Z 1/15 FFD. These field constant aberrations are reduced by adding the conic constants of the surfaces as additional variables as well as adding variables for elliptical coma (Z 11 ), oblique spherical aberration (Z 1 ), and fifth order aperture coma (Z 15 ) at the secondary mirror. When the system is optimized with these additional variables, the field constant aberrations are decreased as shown in Figure 5-8 and the RMS WFE has improved from.18λ to.5λ (note that there is a X scale change from Figure 5-7 to Figure 5-8 to show the residual terms in further detail).

137 15 1:8:8 Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial KHF X Field (deg.) - - Field - Angle Obect Space degrees X in - 3-Sep-13 1:8:8 Z Spher FRINGE ZERNIKE COEFFICIENT Z9 vs FIELD ANGLE IN OBJECT SPACE Y Field Angle in Obect Space - degrees Y Field (deg.) Minimum = -.51 Maximum =.317 Average =.77 Std Dev = :8:8.5waves (1. nm) Zernike Polynomial Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial KHF Z Obl.Spher. - - X Field (deg.) X Field (deg.) - - Field - Angle Obect Space degrees X in - 3-Sep Field - Angle Obect Space degrees X in - 1:37:1 FRINGE ZERNIKE PAIR Z1 AND Z13 vs KHF FIELD ANGLE IN OBJECT SPACE Minimum =.155 Maximum =.151 Average =.51 Std Dev =.9353 Y Field Angle in Obect Space - degrees :8:8 Z Coma FRINGE ZERNIKE PAIR Z7 AND Z8 vs FIELD ANGLE IN OBJECT SPACE Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial Minimum =.789 Maximum =.19 Average =.8879 Std Dev = :8:8.5waves (1. nm) Zernike Polynomial Y Field Y Angle Field Angle in in Obect Obect Space Space - degrees - degrees FRINGE ZERNIKE PAIR Z5 AND Z FRINGE ZERNIKE PAIR Z1 AND Z11 vs vs - -FIELD ANGLE IN - OBJECT SPACE FIELD ANGLE IN OBJECT SPACE Zernike Polynomial Zernike Polynomial Minimum =.178 Minimum =.97 X Field Maximum Angle =.51 in Obect Space - degrees Maximum =.7587 Average =.78 Average =.981 Std Dev =.1199 Std Dev =.13 KHF X Field Angle in Obect Space - degrees 3-Sep-13 1:8:9 - - Field - Angle Obect Space degrees X Z 5th Coma - in - X Field (deg.) 3-Sep-13 Minimum = Maximum =.173 Average =.335 Std Dev =.8811 FRINGE ZERNIKE PAIR Z1 AND Z15.5waves (1. vs nm).5λ µm) FIELD ANGLE IN OBJECT SPACE Minimum =.595e- Maximum = Average =.3835 Std Dev =. 1:8:8.5waves (1. nm) KHF 3-Sep-13.5waves (1. nm) KHF 3-Sep-13.5waves (1. nm) KHF 3-Sep-13.5waves (1. nm) elliptical coma (Z 1/11 ), oblique spherical aberration (Z 1/13 ), and fifth order aperture coma Y Field (deg.) Zernike Polynomial FRINGE - ZERNIKE - COEFFICIENT Z9 - - X Field - KHF 3-Sep-13 vs Angle in Obect Space - degrees FIELD ANGLE IN OBJECT SPACE X Field (deg.) 1:8:9 Z Astig Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial Y Field Angle in Obect Space - degrees Y Field (deg.) KHF RMS WFE 3-Sep-13 RMS WAVEFRONT ERROR vs FIELD ANGLE IN OBJECT SPACE Minimum =.719 Maximum =.111 Average =.9 Std Dev = X Field - Angle in Obect Space - degrees X Field (deg.) waves (1. nm) RMS WAVEFRONT ERROR vs FIELD ANGLE IN OBJECT SPACE - - Field - Angle Obect Space degrees X in - X Field (deg.) Minimum =.719 Maximum =.111 Average =.9 Std Dev =.1999 Z Ellip. Coma Figure 5-8. The lower order spherical (Z 9 ), coma (Z 7/8 ), astigmatism (Z 5/ ), higher order, (Z 1/15 ) Zernike aberration contributions and RMS WFE FFDs over a ±5 degree FOV for the optimized system where the mirror conic constants are added as additional variables in addition to Zernike elliptical coma, oblique spherical aberration, fifth order aperture coma on the secondary surface. The RMS WFE has been reduced from ~.18λ to ~.5λ..5waves (1. nm) After optimization to remove the higher order field constant aberrations, the dominant residual aberrations are now fifth order, field conugate, field linear astigmatism and a fifth order comatic contribution that resembles field conugate, field linear coma. In Chapter 3, it was discovered that a Zernike trefoil overlay displaced axially away from the stop surface will introduce field conugate, field linear astigmatism so it is added as an additional variable on the tertiary surface. To compensate the comatic contribution, a Zernike oblique spherical overlay is added at the tertiary surface as it primarily induces field conugate, field linear coma when placed away from the stop. Because a Zernike oblique spherical overlay has a Zernike astigmatism component built into its term, it also helps to add Zernike astigmatism to the tertiary surface as an additional independent variable so the oblique spherical aberration term can be independently controlled relative

138 to the astigmatism. Figure 5-9 shows the resulting aberration contributions after optimization with these three additional variables. There has been a drastic improvement in the RMS WFE going from.5λ to.1λ. 1:37:1 Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial KHF Sep-13 1:37:1 Z Spher FRINGE ZERNIKE COEFFICIENT Z9 vs FIELD ANGLE IN OBJECT SPACE Y Field Angle in Obect Space - degrees Minimum = Maximum =.173 Average =.335 Std Dev = :37:1.5waves (1. nm) Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial KHF Sep-13 FRINGE ZERNIKE PAIR Z1 AND Z13 vs KHF FIELD ANGLE IN OBJECT SPACE FRINGE ZERNIKE PAIR Z7 AND Z8 vs FIELD ANGLE IN OBJECT SPACE Minimum =.159 Maximum =.9 Average =.18 Std Dev =.915 1:37:.5waves (1. nm) Y Field Angle in Obect Space - degrees Y Field Angle in Obect Space - degrees FRINGE ZERNIKE PAIR Z5 AND Z FRINGE ZERNIKE PAIR Z1 AND Z11 vs vs - -FIELD ANGLE IN -OBJECT SPACE FIELD ANGLE IN OBJECT SPACE Zernike Polynomial Zernike Polynomial Minimum =.398 Minimum =.1177 X Field Maximum Angle =.111 in Obect Space - degrees Maximum =.3538 Average =.33 Average =.111 Std Dev =.13 Std Dev =.7717 KHF 3-Sep-13 1:37:1.5waves (1. nm) Zernike Polynomial Zernike Polynomial Zernike Polynomial Minimum =.195 Minimum =.1119 Minimum =.8 Maximum =.8797 Maximum =.83 Maximum =.131 Average =.1 Average =.3851 Average =.1715 Std Dev =.97 Std Dev =.138 Std Dev =.1 Figure 5-9. The lower order spherical (Z KHF 3-Sep-13.5waves (1. nm) KHF 3-Sep-13 9 ), coma (Z.5waves 7/8 ), astigmatism (Z (1. nm) KHF 3-Sep-13 5/ ), higher order,.5waves (1. nm) 1:37:1 Z Coma Y Field Angle in Obect Space - degrees Minimum = Maximum =.173 Average =.335 Std Dev =.8811 FRINGE ZERNIKE PAIR Z1 AND Z15.5waves (1. vs nm).5λ µm) FIELD ANGLE IN OBJECT SPACE 1:37: Z Astig - - X Field - Angle in Obect Space - degrees X Field Angle in Obect Space - degrees X Field Angle in Obect Space - degrees X Field (deg.) X Field (deg.) X Field (deg.) Z Obl.Spher. Z 5th Coma - Y Field (deg.) X Field - Angle in Obect Space - degrees X Field (deg.) 1:37:1 Y Field Angle in Obect Space - degrees Y Field (deg.) Zernike Polynomial Sep-13 Y Field (deg.) X Field Angle in Obect Space - degrees FRINGE ZERNIKE COEFFICIENT Z9 - - X Field - Angle KHF in Obect Space 3-Sep-13 vs - degrees FIELD ANGLE IN OBJECT SPACE X Field (deg.) 1:37: - - Zernike Polynomial Y Field Angle in Obect Space - degrees Y Field (deg.) - - Y Field Angle in Obect Space - degrees Y Field (deg.) RMS WFE KHF Sep-13 RMS WAVEFRONT ERROR vs FIELD ANGLE IN OBJECT SPACE Minimum =.8 Maximum =.131 Average =.1715 Std Dev = X Field - Angle in Obect Space - degrees X Field (deg.) RMS WAVEFRONT ERROR vs FIELD ANGLE IN OBJECT SPACE Z Ellip. Coma X Field Angle in Obect Space - degrees X Field (deg.).5waves (1. nm) elliptical coma (Z 1/11 ), oblique spherical aberration (Z 1/13 ), and fifth order aperture coma (Z 1/15 ) Zernike aberration contributions and RMS WFE FFDs over a ±5 degree FOV for the optimized system where Zernike astigmatism, elliptical coma, and oblique spherical aberration are added as additional variables to the tertiary surface. The RMS WFE has been reduced from ~.5λ to ~.1λ..5waves (1. nm) Ultimately, further optimization leads to the system shown in Figure 5-1 (a) where the overall average RMS WFE over the 1 full FOV, as displayed in Figure 5-1 (b), is less than λ/1 (.1λ), well within the diffraction limit (.7λ). In this final optimization, the Zernike distribution on the three mirror surfaces is manipulated as it helps to alter the induced aberration components. Since the induced aberration behavior is not currently predicted by NAT, the optimizer is useful for distributing the Zernike contributions about the mirror surfaces; however, the method presented here is useful for determining which variables will be effective for reducing the intrinsic aberration components. As a point of comparison for the final system performance, if the field and

139 17 F/number of the unobscured, conic only solution presented in Figure 5-3 (c) are reduced to produce a diffraction limited system, the field must be reduced to a 3 diagonal full FOV and the system speed must be reduced to F/. Thus with the φ-polynomial surface, there is a substantial advance in usable FOV and light collection capability in this design space. 1:5: Zernike Polynomial KHF -Sep-11 1:5: Y Field Angle Y Field Angle in Obect Space - degrees - Zernike Polynomial KHF X Field Angle in Obect Space - degrees - -Sep-11 FIELD ANGLE IN OBJECT SPACE Minimum =.951 Maximum =.1538 Average =.857 Std Dev = RMS WAVEFRONT ERROR vs FIELD ANGLE IN OBJECT SPACE Minimum =.951 Maximum =.1538 Average =.857 Std Dev = waves (1. nm).1waves (1. nm) - 5. MM X Field Angle in Obect Space - degrees (a) Figure 5-1. (a) Layout of LWIR imaging system optimized with φ-polynomial surfaces and (b) the RMS WFE of the final, optimized system, which is < λ/1 (.1λ) over a 1 diagonal full FOV. 5.5 Mirror Surface Figures The sags of three mirrors for the final design are displayed in Figure 5-11 (a-c) where they are evaluated with different Zernike components removed from the base sag. In Figure 5-11 (a), the sags are evaluated with the piston, tilt, and power Zernike contributions removed so that the dominant astigmatic contribution present in the surfaces can be seen. When the astigmatism is also removed from the surface sags, Figure 5-11 (b), the remaining sag components are observed. An asymmetry is now seen in the sags that results from the comatic departure present in the surfaces. If the spherical aberration is now removed, Figure 5-11 (c), the comatic departure on the surfaces is more readily visible. The primary mirror surface has the smallest amount of Zernike departure, which is on the order of 5 µm and is primarily composed of higher order coma. The (b)

140 18 secondary mirror surface has roughly 1 µm of Zernike departure where most of the departure is composed of astigmatism. Since the secondary mirror is the stop surface, it makes sense that the surface is primarily astigmatic. The tertiary has the most Zernike departure out of the three mirror surfaces, which is on the order of 7 µm. Similar to the secondary mirror, the tertiary surface is largely composed of astigmatism with the next most dominant contribution being coma. Sag of Primary minus Piston/Power/Tilt in µm Sag of Secondary minus Piston/Power/Tilt in µm Sag of Primary minus Piston/Power/Tilt/Astig. in µm Sag of Secondary minus Piston/Power/Tilt/Astig. in µm Sag of Primary minus Piston/Power/Tilt/Astig./Spher. in µm Sag of Secondary minus Piston/Power/Tilt/Astig./Spher. in µm Sag of Tertiary minus Piston/Power/Tilt in µm (a) (b) Sag of Tertiary minus Piston/Power/Tilt/Astig. in µm Sag of Tertiary minus Piston/Power/Tilt/Astig./Spher. in µm (c) Figure (a) Sag of the primary mirror surface various Zernike components removed from the base sag, (b) sag of the secondary mirror surface various Zernike components removed from the base sag, and (c) sag of the tertiary mirror surface mirror surface various Zernike components removed from the base sag. When the piston, power, and astigmatism are removed from the base sags of the three mirrors, the asymmetry induced from the coma being added into the surface is observed.

141 Chapter. 19 Interferometric Null Configurations for Measuring φ-polynomial Optical Surfaces As seen in the Chapter 5, a φ-polynomial surface will usually have some amount of spherical aberration, astigmatism, coma, and some higher order aberration terms placed into the surface departure. As a result, a conventional interferometer that is designed for measuring spherical surfaces has insufficient dynamic range to measure the as-fabricated surface because the departure between the spherical reference wavefront and the test wavefront reflected off the surface of the mirror is too great. However, if the test wavefront is manipulated to null or partially null each aberration type present in the mirror, the departure between the test and measurement wavefronts can be minimized and brought within the dynamic range of the interferometer. This chapter presents a method for measuring concave and convex φ-polynomial surfaces by utilizing a series of adaptive subsystems that each null a particular aberration type present in the departure of the freeform surface..1 Concave Surface Metrology As a demonstration of a realizable null configuration for a concave optical surface, the secondary mirror of the optical system designed in Chapter 5 will be used as an example. This mirror was diamond turned by II-VI Infrared, as were the other two mirror surfaces, in a copper substrate with a gold protective coating. For reference, the sag of the secondary mirror surface of the three mirror design is shown in Figure -1 (a-c) where it is evaluated with different Zernike components removed from the base sag. In Figure -1 (a), the sag is evaluated with the piston, power, and tilt Zernike contributions removed so that the dominant astigmatic contribution present in the surface can be seen. When the astigmatism is removed from the surface sag, shown in Figure -1 (b), the

142 11 remaining sag components are observed. An asymmetry is seen in the sag that results from the comatic departure present in the surface. If the spherical aberration, the next most dominant contribution, is removed from the surface sag, shown in Figure -1 (c), the comatic departure of the surface is more readily visible. Sag of Secondary minus Piston/Power/Tilt in µm Sag of Secondary minus Piston/Power/Tilt/Astig. in µm Sag of Secondary minus Piston/Power/Tilt/Astig./Spher. in µm (a) (b) (c) Figure -1. (a) Sag of the secondary mirror surface with the piston, power, and tilt Zernike components removed revealing the astigmatic contribution of the surface, (b) sag with the astigmatic component additionally removed, and (c) sag with the spherical component additionally removed. The goal of the interferometric null is to systematically subtract the spherical aberration, astigmatism, and coma present in the secondary mirror surface. The first step in designing the null configuration is to select either a planar or spherical reference wavefront out of the interferometer. For this particular design, a planar wavefront, translating to a flat reference surface at the output of the interferometer, is chosen because the alignment of the null to the interferometer will be less critical since it can lie anywhere within the aperture of the interferometer/transmission flat. From the output of the interferometer, the aberration terms can be nulled in multiple configurations. For this design, the spherical aberration component is first nulled by the use of a refractive Offner null [7]. Next, the astigmatic component is removed by tilting the test surface. Lastly, the residual comatic and higher order terms are nulled by adding their opposite departure on an adaptive mirror that also acts as retro-reflector to send the light back towards the

143 111 measurement interferometer [77]. Together these three components form a configuration that enables the optical surface to be measured with a conventional interferometer. In the section below, the first order design of the null components is described and finally the entire optimized, null system is presented..1.1 First Order Design Spherical Null One of the common methods for creating a spherical null is to implement a refractive Offner null lens consisting of two refractive elements as shown in Figure -. The first element focuses the planar wavefront from the interferometer in such a manner that the exiting wavefront has the same amount of spherical aberration present in the test mirror so that the beam incident the mirror is normal to the surface. The second element is a field lens placed at or near the focus of the first null lens and it images the pupil of the null lens to the pupil of the test mirror as shown by the blue dashed ray in Figure -. Conugating the pupil between the null lens and the test surface ensures that higher order aberrations are not generated as the beam propagates through the null system. Null Lens y null Field Lens Test Mirror f null z mirror y mirror Figure -. First order layout of the Offner null to compensate spherical aberration. The rays in red show the illumination path for the testing wavefront whereas the rays in blue show the imaging path for the pupils of the Offner null. The focal lengths of the two lenses depend on the first order parameters of the testing configuration. For this mirror the diameter of the pupil located at the null lens, y null selected to be 5 mm so that the lens is not overly difficult to fabricate. In addition, the, is

144 11 region of interest on the mirror, y mirror, is 7 mm and to keep the overall length of the null system small, the distance between the focus of the null lens and the test mirror, z mirror, is chosen to be at or near the radius of curvature of the mirror under test, which for this system has been chosen to be 37.5 mm. Based on these parameters, the focal length of the null lens, f null, is computed using the magnification as, f null z mirror = ynull, (.1) ymirror and yields for the parameters described above, a null lens focal length of 3. mm. From the null lens focal length, the focal length of the field lens is computed from the thin lens equation as, f field 1 1 = zmirror f and yields a field lens focal length of 13. mm. null 1, (.) Now that the first order parameters of the Offner null have been computed, the next step is to select the curvatures for the null lens that will yield the correct amount of spherical aberration to create a null for the mirror under test. Since the null lens and mirror are conugate to one another, the Fringe Zernike spherical contribution on the Mirror mirror surface, z 9, is related to the required null lens spherical wave aberration, by taking the opposite transmitted wavefront aberration of the mirror calculated as, Null W, W = z ( 1 1). (.3) Null Mirror 9 where the refractive index of the mirror is assumed to be -1. For the.5 µm of Fringe Zernike spherical present in the test mirror, 3 µm of spherical wave aberration must be created by the null lens. From the required wave aberration of the null lens, the equation

145 for spherical aberration of a thin lens at the stop surface [55] is rearranged to compute the required shape factor, β Null β Null where 1 Null Null 1 Null Null Null 1, for the null lens and is given by, ( n ) 3 f 3 n( n ) R + R 1 1 Null Null n n = = + W + R R n+ y Null n 1 n+ n+ R and R Null 1/, 113 (.) are the front and back radius of curvature of the null lens and n is the refractive index of the lens. For ease of fabrication, NBK7 glass is chosen, which has a refractive index of at interferometer testing wavelength of 3.8nm. Substituting all known parameters in Eq. (.) leads to a shape factor of roughly.81, which indicates a strong meniscus for the null lens. Finally using the relation of the curvature of the null lens to the focal length and shape factor, the front and back radius of curvature is computed as, f ( n 1 Null ) R1/ =, (.5) ( β ± 1) where after substitution, the two radii come to.9 mm and mm, respectively. With all the first order parameters of the Offner null calculated, the next step is to calculate the required tilt angle of the test mirror to null astigmatism Astigmatic Null When a spherical mirror is operated off-axis at or near the center of curvature, the dominant aberration is third order astigmatism. The astigmatism is minimized by adding a toroidal shape to the mirror. The principal radii of curvature, R x and R y, that determine the ideal toroidal mirror to minimize astigmatism are found by the Coddington equations [78] that are expressed for a mirror as, and 1 1 T' T R i + = (.) cos( ) t

146 cos( i) + =, (.7) S' S R where T and S are the distances from the tangential (T) and sagittal (S) astigmatic focal surfaces of the obect to the mirror, T and S are the distances from the mirror to the astigmatic focal surfaces of the image, and i is the angle of the mirror with respect to the optical axis in the YZ plane. If the obect and image are to be anastigmatic, T must be equivalent to S as well as T must also be equal to S. In this special case, Eq. (.) and (.7) reduce to, R x x ( i) = (.8) cos, where Rx < Ry for there to be a valid mirror angle, i. For the case where Ry < Rx, the mirror angle, i, must be re-defined in the XZ plane, so that Eq. (.8) becomes, R R y y ( i) = (.9) cos. For the mirror of interest, a toroidal shape has been intentionally polished into the surface by prescribing some combination of primary and secondary Zernike astigmatism. Following Eq. (.8), in order to null the toroidal shape, the mirror must be tilted at an angle where the angle depends on the principal radii of curvature. The principal radii of curvature are derived from the Zernike terms present in the mirror surface by following a method proposed by Schwiegerling et al. [79]. In this method, the sag of the optical surface with the Fringe Zernike overlay is approximated as parabolic, so that the sag along the x-direction is written as, R x ρ ρ ρ ρ ρ ρ Rx RB RN RN RN RN sagx = = + z z + z 3 z, and the sag in the y-direction is written as, (.1)

147 ρ ρ ρ ρ ρ ρ Ry RB RN RN RN RN sag y = = + z z z + 3 z, 115 (.11) where R B is the base radius of curvature of the mirror, ρ is the radial coordinate, RN is the normalizing radius of Fringe Zernike overlay, and z,8,5,1 are the Fringe Zernike coefficients for power, spherical aberration, astigmatism, and oblique spherical aberration. By simplifying and manipulating Eq. (.1) and (.11), computed as, R x and R y are R x/ y RN =. R N + z z8 ± z5 z1 RB (.1) After substituting the prescription parameters of the mirror in Eq. (.1), R x and Ry are calculated to be -39. mm and mm leading to a tilt angle from Eq. (.8) of More specifically, if the mirror under test is tilted at then the astigmatic contribution from the mirror will be nulled. However, an obstacle to overcome with a tilted geometry is that the reflected wavefront from the mirror will no longer be reflected back on itself and requires the use of an additional element to return the test wavefront back to the interferometer. In this case, a deformable mirror (DM) that is nominally flat but can be deformed into a wide variety of shapes is employed. It provides retro-reflection without inversion and also nulls the residual coma and any higher order aberrations present in the test wavefront Comatic and Higher Order Null In order to use the quasi-flat DM, the wavefront reflected off the test mirror must be collimated with the use of a collimating lens. In addition, the DM with a clear aperture of 15 mm must also be conugate to the test mirror. This configuration is diagramed in Figure -3 where the illumination path from the interferometer is shown in red and the

148 11 imaging path between the test mirror and DM is shown in blue. As can be seen from Figure -3, the collimating lens is performing two first order imaging functions. y mirror Test Mirror z' mirror Collimating Lens f DM z DM y DM Deformable Mirror Figure -3. First order layout of the comatic and higher order null. A collimating lens is uses to couple the wavefront to an actuated, deformable membrane mirror. The rays in red show the illumination path for the testing wavefront whereas the rays in blue show the imaging path for the pupils of the comatic null. With the constraints laid out above, it is possible to derive the first order parameters for this section of the interferometric null. The focal length of the collimating lens is found using the magnification between the test mirror and DM, calculated as, where z ' mirror f DM 1 y DM 1 y DM = z' mirror = +, ymirror zmirror RB ymirror (.13) is the image distance of the test mirror that is related to the obect distance of the mirror z mirror by the thin lens imaging equation. Based on the values for the distances and sizes of the optics, the focal length of the collimating lens is found to be roughly 81.7 mm. The other parameter that needs to be calculated is the distance between the collimating lens and the DM, z ' DM, ensuring both the test mirror and DM are conugate to one another. Using the thin lens equation, this distance is calculated as, z ' DM = + = +, fdm zdm fdm z' mirror + fdm 1 (.1) and yields for the parameters above a distance of 99. mm. The comatic and higher order departure that needs to be applied to the DM is negative two times the departure present on the test mirror since the test wavefront is reflected off the mirror twice. Also, because of the imaging condition between the test mirror and DM, the Zernike contribution on the

149 117 DM must be rotated by 18. With all paraxial parameters for the inteferometric null now established, the null configuration can now be optimized using commercially available lens design software to provide a final thick lens solution..1. Optimization of the Interferometric Null System The paraxial solution described in the section above creates a starting point for further optimization. The end goal for optimization is to produce a double pass, thick lens solution that provides a null or quasi-null wavefront exiting the interferometric system. Using CODE V, user defined constraints are written for nulling the Fringe Zernike spherical aberration, astigmatism, coma, and any higher order aberration terms while maintaining conugates between nulling components. As for the parameters allowed to vary during optimization, the radii of the null lens are roughly set by the first and third order constraints but are allowed to vary to account for variations from the lens thickness; however, the focal length is kept fixed to its first order value. For the field lens, the lens has been chosen to be NBK7. Since the size of the beam footprint at the field lens is small, it introduces little spherical aberration, so for ease of fabrication, positioning, and alignment, its shape is chosen to be bi-convex. The focal length is allowed to vary from its paraxial value because the field lens parameters can be used to minimize higher order spherical aberration. The collimating lens near the DM is chosen to be a high index material, SF, so it introduces less spherical aberration. For ease of fabrication and to introduce as little spherical aberration as possible, the shape is chosen to be plano-convex as it is near the shape factor for minimum spherical aberration. The focal length is kept roughly the same as its first order value to ensure that the size of the beam on the DM does not exceed 15 mm. The mirror tilt about the y-axis is allowed to vary from its

150 118 paraxial value and the Fringe Zernike tilt (Z 3 ), coma (Z 8 ), trefoil (Z 11 ), oblique spherical aberration (Z 1 ), and higher order coma (Z 1 ) are allowed to vary at the DM surface. The final, optimized system is shown in the XZ plane in Figure -. As can be seen from the figure, an aspect ratio of at least 7:1 is selected for each lens to aid in manufacturability. The overall package of the interferometric null is roughly mm x 5 mm. The theoretical wavefront exiting the interferometric null is shown in Figure -5 (a) before the DM is active and in Figure -5 (b) after the comatic and higher order null has been applied. In Figure -5 (a), the astigmatism and spherical aberration have been nulled from the wavefront but there is still a departure of 38λ PV at the testing wavelength of 3.8 nm in the double pass wavefront. After the DM has been applied, the residual present is on the order of λ PV or.λ RMS. At the operating wavelength of around 1 µm, the residual in the double pass null wavefront corresponds to.5λ PV and.3λ RMS. The residual is non-zero because of the tilt angle required to null the astigmatic part of the surface. With a tilted geometry, the pupils cannot be perfectly conugate to one another since a tilted obect must be imaged to a tilted image per the Scheimpflug principle [8]. Moreover, the beam incident on the test mirror is slightly elliptical and will alter the Zernike composition of the wavefront. The residual in the exiting wavefront can be compensated either in hardware or software by using the DM to subtract the residual or simulating a software null in CODE V to subtract from the measured data.

151 119 Deformable Mirror Collimating Lens Output of Interferometer Offner Null Mirror Under Test 75. MM Figure -. Layout of the optimized interferometric null for the concave, secondary mirror to be coupled to a conventional Fizeau interferometer with a transmission flat. The interferometric null is composed of three nulling subsystems: an Offner null to null spherical aberration, a tilted geometry to null astigmatism, and a retro-reflecting DM to null coma and any higher order aberration terms. 11.λ 11.λ.5λ.5λ (a).λ (b).λ Figure -5. Simulation of the double pass wavefront exiting the concave interferometric null (a) before and (b) after the deformable null has been applied at a testing wavelength of 3.8 nm..1.3 Experimental Setup of Interferometric Null System The first step in assembling the interferometric null system is to create the comatic and higher order null on the DM surface. The DM selected for this proect is mirao 5-e, a fifty two actuator reflective membrane mirror, from Imagine Eyes. On the underside of the membrane surface, small magnets are affixed at each actuator site. The actuation of the surface is achieved through variation of the voltage in a small coil that creates a magnetic field at the actuator site. The magnetic field influences the magnet either

152 1 pushing or pulling the membrane surface depending on the applied voltage. This type of DM is capable of achieving large deformations of the surface and is well suited for creating the comatic null for the interferometric null system. The system for setting this shape is shown in Figure - (a) where a 3:1 afocal telescope relays collimated light from a 3.8 nm Zygo laser interferometer through a cube beamsplitter and onto the DM. The wavefront then reflects off the surface of the mirror and half the light is directed back to the interferometer and the other half is directed through a :1 afocal telescope that images the DM surface onto a.8 x 3. mm Shack-Hartmann wavefront sensor. Using the wavefront sensor to interrogate the DM surface, it is operated in a closed loop configuration where the influence functions of the actuators on the DM are known a priori and they are iteratively adusted in software to converge to a desired shape. The optimized Fringe Zernike coefficients of the comatic null from the lens design are the target for the closed loop optimization. The laser interferometer is used as an additional aid to measure the shape of the comatic null as the DM is adusted to its final form. The assembled optical system for the DM calibration is shown in Figure - (b). Any aberrations induced from the afocal telescopes can be subtracted from the measured wavefront by first replacing the DM with a flat reference mirror of high quality and using this measurement as a baseline.

153 11 Deformable Mirror Shack-Hartmann Wavefront Sensor 3:1 Afocal Relay :1 Afocal Relay F Afocal Relay (:1) F Afocal Relay (3:1) Zygo DynaFiz Interferometer Deformable Mirror Wavefront Sensor (a) (b) Figure -. (a) Layout of the setup to create the comatic and higher order null on the DM surface. The setup uses a Shack-Hartmann wavefront sensor to run a closed loop optimization to set the shape of the DM. The DM is also interrogated with a Fizeau interferometer. (b) The setup realized in the laboratory. The shape of the comatic null measured by the interferometer is shown in Figure -7 (a). The dynamic range of the DM is capable of creating this large departure null with a surface PV of roughly 1 µm. However, when the theoretical shape is subtracted from the actual comatic null, there is a large residual as displayed in Figure -7 (b). The residual is on the order of µm PV and is mostly composed of higher order deformations that result from the local deformation at or near the actuator sites. The voltages of the actuators are near their maximum for this surface shape so some residual is to be expected. Since the deformable surface has been measured, it can be applied as a hitmap in CODE V and the residual wavefront can be simulated to create a software null in CODE V to subtract from the measured data.

154 µm +1.35µm (a) µm (b) -.588µm Figure -7. (a) DM comatic null surface measured by the interferometer and (b) the residual after the theoretical shape has been subtracted. The residual has a PV error of µm PV. With the correct shape set on the DM, the rest of the interferometric null is assembled. The three lens components, the Offner null lens, field lens, and collimating lens, have been fabricated by Optimax Systems. Each lens is coated with a V-coating that ensures back reflections to the interferometer are minimized. The optics are mounted with commercially available optomechanical components that provide four DOF movement (x/y decenter and tip/tilt). Moreover, each optical component sits on a kinematic base that provides stable six DOF positioning. With kinematic couplings, each element can be removed for the alignment of subsequent components and after alignment, the element can be replaced repeatably. The tilted geometry for the test mirror, collimating lens, and DM is created by using a precision rotation stage with a rail attached to the tabletop of the stage. Since the mirror is to be tilted at α, the optical axis or rail must be rotated by α. The mirror, whose vertex lies at the axis of rotation of the rotation stage, also rotates by α and must be counter-rotated by α. The counter-rotation of the test mirror is made possible by the use of a custom kinematic base known as a Kelvin clamp. The two part base, shown in

155 13 Figure -8, has a bottom plate with three conical cups milled about a radius separated by 1. In these cups sit three spheres. On the top plate, sets of 1 spaced vee grooves are milled into the plate. Each sphere of the bottom plate sits in one vee groove, constraining two DOFs. In total, all six DOFs are uniquely determined. Each set of grooves defines one index (rotation) of the top plate. For the test mirror measurement there are three vee sets milled into the plate:, α, and α. Using the specialized base, the mirror under test can first be aligned perpendicular to the optical axis in the zero index position. When the stage is rotated by α, the base can be indexed to the α position, bisecting the rotation angle of the stage. Since the stage is kinematic, the positioning will be repeatable for multiple iterations of positioning. Figure -8. Custom designed kinematic indexing mount for counter rotating the test mirror during alignment of the interferometric null. The plates are machined in 3 stainless steel and employ three hardened C stainless steel 7/1 spheres. The assembled and aligned interferometric null configuration is displayed in Figure -9. The interferometric null is coupled to a phase shifting Zygo DynaFiz interferometer with a reference transmission flat. The optical axis of the null (shown in red) is defined by a line that interests the vertex of the test mirror and is normal to the transmission flat of the interferometer. Because the vertex of the mirror is not readily accessible, a precision external target that couples to the mechanical alignment features

156 1 of the test mirror is used to locate the vertex. Using this target and the transmission flat, an alignment telescope is aligned to these datums and it defines the optical axis for subsequent alignment of the other optical components. The alignment telescope sits behind the test mirror in the interferometric null. During alignment the test mirror is removed to provide an unobstructed view of the other components. In this fashion the Offner null is easily aligned. In order to align the collimating lens and DM, the rotation stage and rail are first aligned to the optical axis defined by the alignment telescope. In this case, the Offner null components are removed. Once the components have been aligned, the rotation stage is set to its angle and the Offner null components are replaced. Deformable Mirror Secondary Mirror Offner Null Zygo DynaFiz Interferometer Figure -9. The interferometric null configuration realized in the laboratory. A rotation stage with a rail affixed is used to create the tilted geometry. The secondary mirror is measured using a Zygo Fizeau-type interferometer..1. Experimental Results With the interferometric null aligned, the Zygo interferometer is used to acquire an interferogram of the optical surface. One important consideration for the raw interferogram acquired by the interferometer is the scaling factor, or the relationship between the fringe pattern on the wavefront and surface error on the mirror. A typical

157 Fizeau measurement is double pass, resulting in one fringe being equivalent to λ surface 15 error. For this test configuration, the test wavefront reflects off the tilted mirror twice, so, in this case, one fringe on the wavefront is estimated as λ cos( α ) (.15) on the surface, where α is the angle of incidence on the mirror with respect to the optical axis. The cosine term is included to account for the proection of surface height from the tilted plane back to a normal condition. Moreover, the raw interferogram acquired by the interferometer is rotated 18 from the actual surface of the mirror since the light passes through an intermediate focus in the Offner null. Taking these items into consideration, the initial surface error of the test mirror surface is shown in Figure -1 (a-b) where the surface error maps are presented in microns. In Figure -1 (a), the surface error is presented with the residual power present in the surface. The PV error is 3.81 µm and.819 µm RMS. When the dominant power is subtracted from the measurement, Figure -1 (b), the PV error goes to.5 µm and.35 µm RMS. With the power subtracted, the less dominant features of the residual can be discerned and these errors resemble the residual of the comatic null from its theoretical state presented in Figure -1 (b). In order to observe the errors of the mirror surface and not the errors of the comatic null, a software null is created in CODE V. The software null simulates the wavefront at the exit pupil and it includes the effects of the residual aberrations present in the testing setup and incorporates a hitmap of the comatic null. The surface error maps after subtracting the software null from the measured data are depicted in Figure -1 (c-d). When the power is present in the surface error, Figure -1 (c), the PV error is now

158 1 3.3 µm and.798 µm RMS. After the power is subtracted, Figure -1 (d), the PV error is reduced to 1.1 µm and.15 µm RMS. At a wavelength of 1 µm, the center operating wavelength of the optical system, the PV error is.11λ and.1λ RMS. Therefore, for an LWIR application, the surface is almost a tenth wave. In evaluating the features in the surface error, it can be seen that, while small, the error is mostly astigmatism that may be a residual from the mounting process during fabrication µm +1.3µm (a) -1.9µm +1.99µm (b) -.81µm +.58µm (c) -1.31µm (d) -.559µm Figure -1. (a) Initial surface error map of the test mirror with power and (b) with the power removed. The PV error of the surface residual before and after the power is removed is 3.81 µm and.5 µm, respectively. (c) Final surface error map of the test mirror after the software null has been subtracted (c) before and (d) after the power has been removed. In this case, the PV error is 3.3 µm before and 1.1 µm after the power has been removed.

159 17. Convex Surface Metrology As a demonstration of a realizable null configuration for a convex optical surface, the primary mirror of the optical system designed in Chapter 5 will be used as an example. For reference the sag of the primary mirror surface of the three mirror design is shown in Figure -11 (a-c) where it is evaluated with different Zernike components removed from the base sag. In Figure -11 (a), the sag is evaluated with the piston, power, and tilt Zernike contributions removed. For this particular surface, the dominant aberration component is coma. This fact can more readily be seen by additionally subtracting the astigmatism from the surface sag, as shown in Figure -11 (b), and also the spherical aberration, as shown in Figure -11 (c). When these two aberration components are additionally removed, there is little change in the surface sag residual and based on the characteristic asymmetric behavior, the residual is recognized as coma. Sag of Primary minus Piston/Power/Tilt in µm Sag of Primary minus Piston/Power/Tilt/Astig. in µm Sag of Primary minus Piston/Power/Tilt/Astig./Spher. in µm (a) (b) (c) Figure -11. (a) Sag of the primary mirror surface with the piston, power, and tilt Zernike components removed, (b) sag with the astigmatic component additionally removed, and (c) sag with the spherical component additionally removed. With the piston, power, tilt, astigmatism, and spherical components removed, the asymmetry induced from the coma being added into the surface can be seen. Similar to the concave null configuration, the aberration components will be nulled with a series of subsystems starting with a planar wavefront, translating to a flat reference surface at the output of the interferometer. From the output of the interferometer, the aberration terms can be nulled in multiple configurations. For this design, the spherical

160 18 aberration component is first nulled by the use of an afocal, refractive Offner null, consisting of two refractive elements, one of which is a null lens that introduces the opposite amount of spherical aberration present in the mirror under test and the other is a field lens that collimates the beam and conugates the null lens to the mirror under test. Next, the astigmatic component is removed by operating the mirror off-axis, or tilting the mirror. Unlike for the case of a concave test mirror where the wavefront converges to a point, the beam exiting the tilted, convex mirror will diverge so an additional auxiliary optic is needed to focus the wavefront. In this case, a large concave mirror is used similar to a Hindle sphere [81]. This additional mirror adds another DOF so that its tilt is used to remove the residual comatic contribution present in the wavefront. Finally, similar to the concave null configuration, the residual higher order terms are nulled by adding their opposite departure on a DM that also acts as a reimaging retro-reflector to send the light back towards the measurement interferometer. In order to couple the wavefront to the quasi-flat DM, the wavefront is collimated with the use of an additional lens. Together these three components form a configuration that allows the optical surface to be measured with a conventional interferometer. The sizing of the various optical components constrains the layout of the interferometric null. The region of interest on the test mirror is 5 mm in diameter while the output from the Zygo interferometer is 11. mm. To keep the components of the refractive null readily commercially available, a 1:1 afocal Offner null is selected so only 5 mm of the 11. mm aperture is used. After the wavefront passes through the null lens and reflects off the test mirror, its beam size will grow rapidly as the wavefront is diverging. As a result, the distance between the test mirror and auxiliary mirror should be

161 19 minimized to keep the beam footprint on the auxiliary mirror small. The auxiliary mirror clear aperture and its beam footprint are constrained to 15 mm diameter maximum, which is the largest mirror size that is readily commercially available at relatively fast focal ratios. The focal ratio must be fast to keep the length of the null small. An F/1, 15 mm COTS sphere is used to meet these constraints. The distance between the auxiliary sphere and the mirror under test also impacts how the wavefront traverses through the entire null configuration. The beam exiting the afocal Offner null must pass by the auxiliary sphere and the sphere should not obscure the incoming beam. Similarly, after the beam has reflected off the auxiliary sphere, it passes by the mirror under test and the mirror should not obscure the beam either. The distance between the test and auxiliary mirror is set so that when the two mirrors are tilted to null both astigmatism and coma, they do not obscure any part of the beam. Lastly, the DM has a 15 mm clear aperture so the collimating lens after the auxiliary sphere must be arranged to meet this constraint. With all these constraints in mind, a solution is optimized in CODE V with user defined constraints to null the Fringe Zernike spherical aberration, astigmatism, coma, and any higher order aberration terms while ensuring that the clear aperture limitations are met, the beam is not obscured, and the imaging conugates between components are maintained. The final, optimized system is shown in the YZ plane in Figure -1. The overall package of the interferometric null is roughly 7 mm x 35 mm. All the lens components are plano-convex making them readily commercially available. The theoretical interferogram exiting the interferometric null is shown in Figure -13 (a) before the DM is active and in Figure -13 (b) after the higher order null has been

162 13 applied. In Figure -13 (a), the spherical aberration, astigmatism, and coma have been nulled from the wavefront but there is still about 5λ PV of departure present in the double pass wavefront at the testing wavelength of 3.8 nm where most of the departure resembles that of Zernike trefoil. The Zernike trefoil that is present in the wavefront is not all from the amount present in the mirror surface. The fast auxiliary sphere is tilted at a fairly large angle so the beam footprint has become elliptical. The elliptical beam on the auxiliary sphere results in the generation of elliptical coma also known as Zernike trefoil. This residual that is a result of the testing configuration is subtracted at the DM null. After the higher order null has been applied, the residual is on the order of λ PV or 1.λ RMS. At the operating wavelength of around 1 µm, the residual in the double pass null wavefront corresponds to.5λ PV and.1λ RMS. The residual is non-zero for several reasons. The ideal shape factor for the null lens in the spherical null is near plano-convex but not perfectly plano-convex. To aid in commercial availability, the lens was forced to be plano-convex at the cost of some residual spherical aberration. Moreover, the beam incident on the test mirror is slightly elliptical and will alter the Zernike composition of the wavefront. The residual in the exiting wavefront can be compensated either in hardware or software by using the DM to subtract the residual or by simulating a software null in the lens design software to subtract from the measured data.

163 131 Auxiliary Sphere Mirror Under Test Output of Interferometer Afocal Offner Null Deformable Mirror Figure -1. Layout of the optimized interferometric null for the convex, Primary mirror to be coupled to a conventional Fizeau interferometer with a transmission flat. The interferometric null is composed of three nulling subsystems: an afocal Offner null to null spherical aberration, a tilted geometry to null astigmatism and coma, and a retro-reflecting DM to null any higher order aberration terms. a e 1 1.λ 1.λ.5λ.5λ (a).λ (b).λ Figure -13 Simulation of the double pass wavefront exiting the convex interferometric null (a) before and (b) after the deformable null has been applied at a testing wavelength of 3.8 nm.

164 13 Chapter 7. Assembly of an Optical System with φ-polynomial Optical Surfaces The optical system described in Chapter 5 opens a new space for optical design where a freeform overlay may be utilized on an optical surface to enable a non-inline, tilted geometry of the overall optical system. New fabrication and assembly challenges arise when building an optical system of this type because conventional methods of fabrication must be abandoned to enable these new optical design forms. Chapter showed how interferometric metrology can be configured to measure this new class of optical surfaces. When it comes to assembling an optical system of this type, the mounting and fiducialization of the optical surfaces becomes critical. In particular, the optical surfaces must be oriented in a particular manner with respect to the optical housing and constrained in all six DOFs because of their nonsymmetric shape. In this chapter, the mechanical design of the optical design in Chapter 5 is presented and the sensitivity of the design to assembly alignment residuals is evaluated. In addition to a sensitivity analysis, the mounting structure is evaluated for stray light and the problems are mitigated through baffles and surface preparation. Lastly, the assembled optical system and its optical performance are presented. 7.1 Mechanical Design The housing structure of the three mirror system is displayed in Figure 7-1 (a) and was developed in collaboration with II-VI Infrared. It is constructed from an aluminum block with the faces of the block machined to the required tilt angle for each mirror. The mirrors are designed to be back surface mounted so an adaptor plate, as shown in Figure 7-1 (b), is used to couple the mirror to its corresponding face. Steel dowel pins are used to position the mirror correctly within the mechanical housing. These dowel pins

165 133 provide a good mechanical datum to the optical surface because during the fabrication process they register the optical surface to a tooling plate with a reference flat that is trued to the diamond turning machine. In total there are two mechanical connections for each mirror subassembly. The first connection is between the optical surface and the adaptor plate and the second connection is between the adaptor plate and the housing face. At each connection three diamond turned raised pads are used as the mounting interface to provide a quasi-kinematic condition when the two surfaces are mated together. In total, the pin connections constrain the x decenter, y decenter, and clocking angle of the optical surface with the pads and screws providing preload thus constraining any in-plane movement. Paths are bored through the housing and are sized to ensure the light passes through the housing without vignetting. Mirror Dowel Pins Adaptor Plate Raised Pads (a) (b) Figure 7-1. (a) Layout of the housing structure of the three mirror freeform optical system and (b) exploded view of the tertiary mirror subassembly consisting of the optical mirror surface, adaptor plate, and steel dowel pins for alignment.

166 Sensitivity Analysis As is the case with any piece of hardware, there is some tolerance on how well the mirrors can be positioned in the housing relative to their nominal value. The key is to ensure that within the manufacturing tolerances, the as-built optical system remains diffraction limited. In addition to hardware tolerances, the assembly method of the optical system may impact the manufacturing tolerances. If the optical system is to be passively aligned, that is, no adustments are made with the exception of focus, the manufacturing tolerances will have to be tighter. If the system is to be actively aligned, that is, a compensator is used to restore the optical performance during assembly, the assessment of the performance during alignment is important and the mechanical complexity of the housing will have to increase because a DOF must now be made adustable. In this section, both approaches are explored Passive Alignment In a passive alignment approach, the three mirrors must be constrained in their x/y translation, tip/tilt, and clocking angle. There are also two vertex spacings that must be held between the three mirrors. The spacing between the tertiary mirror and focal plane is used as a focus adustment after assembly. The focus compensation is performed by shimming the detector in 1.5 µm steps and determining through an optical assessment the shim that provides the best performance. The detector must also be held in tip and tilt relative to the housing and there is a separate tip and tilt tolerance for the focal plane relative to the mounting fixtures on the detector. In total, there are 1 positioning tolerances to consider for this assembly. To check the sensitivity of the optical housing to manufacturing errors, each tolerance is perturbed by its expected error value and the change in the RMS WFE after focus

167 135 compensation is recorded for several field points. Once each tolerance and its resulting change in performance is computed, the total change in performance for each field is computed as the root sum square (RSS) of all the tolerances. The results of this analysis are displayed in Table 7-1 where the change in RMS WFE is displayed for each tolerance at two field points: on-axis (, ) and the most sensitive field (, 3 ). For this analysis the x decenter, y decenter, and despace of the mirrors is assumed to be ±5 µm, the tip and tilt of the mirrors and detector (α and β tilt) is assumed to be ±1 arc min or ±.17, and the clocking angle (γ rotation) is assumed to be larger at ±.1. From the mechanical drawing of the detector, the focal plane tip and tilt tolerances are calculated to be roughly ±.5. With all the tolerances considered, the as-built RMS WFE is found to be roughly.5λ for the (, ) field and.λ for the (, 3 ) field, both of which are within the diffraction limit of.7λ. Looking at the tolerances on a term by term basis, the primary contributors to the overall loss in performance are the tilt tolerances on the secondary mirror, tertiary mirror, and focal plane. Therefore, the initial tolerances selected are sufficient for meeting the performance specification; however, they do not consider how the optical components will actually be mated together. Since the components are to be assembled with pin connections, an alternative analysis would be to model the sensitivity of the connections directly. The position of the dowel pin holes affects the x decenter, y decenter, and clocking angle of the mirror surfaces.

168 13 Table 7-1. Summary of the initial sensitivity analysis of the three mirror optical system. For each tolerance, the change in RMS WFE from nominal is computed and the RSS is compiled to provide the as-built RMS WFE. The RMS WFE is terms of waves at the central operating wavelength of 1 µm. Tolerance Δ RMS WFE (waves) Field: (, ) Δ RMS WFE (waves) Field: (,3 ) Pri. Mirror x decenter ±5 µm.. y decenter ±5 µm..5 α tilt ± β tilt ± γ rotation ±.1.. Pri.-Sec. despace ±5 µm.1.1 Sec. Mirror x decenter ±5 µm.3. y decenter ±5 µm.1. α tilt ± β tilt ± γ rotation ±.1.3. Sec.-Ter. despace ±5 µm.. Ter. Mirror x decenter ±5 µm.. y decenter ±5 µm.3.3 α tilt ± β tilt ± γ rotation ± Detector α tilt ±.17.. β tilt ±.17.. Focal Plane α tilt ± β tilt ±.5..3 RSS.5.5 Nominal.11.1 Predicted As-Built.5. For each set of dowel pins, the diametrical true position of the two dowel pin holes must be considered with respect to a reference datum. Specifically, the diametrical true position defines a region in which the dowel pin hole must lie. Since it is a diametrical

169 137 tolerance zone, as the x decenter of the dowel pin hole increases, the y decenter must decrease accordingly. Furthermore, if the top dowel pin hole is not collinear with the bottom dowel pin hole, the mirror will be rotated. For the analysis, an initial diametrical true position tolerance, φ, is selected for the position of the top and bottom dowel pin holes. From this tolerance a random x decenter of the hole is selected from a position within the tolerance zone, computed as x T/ B dec φ φ = + ( RAND), (7.1) T/ B where x is the x decenter of either the top or bottom hole and RAND is a normally dec distributed random number between and 1. From the x decenter, the maximum possible / y decenter, y T B max dec, is calculated as y T/ B φ / max dec =. T B ( xdec ) From Eq. (7.) a random y decenter of either the top or bottom hole is computed as (7.) = + ( ). (7.3) y y RAND y T/ B T/ B T/ B dec max dec max dec With the x and y decenter computed for both the top and bottom dowel pin holes, the total x and y decenter of the set is computed as and x dec T B xdec + xdec =, (7.) Lastly, the clocking angle error of the set, γ, is computed as y dec T B ydec + ydec =. (7.5) x x γ = d pin T B 1 tan dec dec, (7.)

170 138 where d pin is the spacing between the two pins. From Eq. (7.) it can be seen that if the pin spacing is increased, the clocking angle error of the set will decrease for the same diametrical true position tolerance. For each mirror there are two pin connections that yield four sets of dowel pin holes. Therefore, the total mirror x decenter, y decenter, and clocking angle error is the summation of the decenter and clocking angle tolerances of the four sets. These totals are computed as and x = x + x + x + x (7.7) mirror pin 1 pin pin 3 pin dec dec dec dec dec, y = y + y + y + y (7.8) mirror pin 1 pin pin 3 pin dec dec dec dec dec, γ = γ + γ + γ + γ (7.9) mirror pin 1 pin pin 3 pin dec dec dec dec. As an example, the quantities required for deriving an overall random x decenter, y decenter, and clocking angle error are shown in Figure 7- for the tertiary mirror and are tabulated for all the mirror surfaces in Table 7-. The diametrical tolerance zone is different depending on the mating interface. For example, the housing has the loosest tolerance on the position of the dowel pin holes because it is difficult to machine a hole into a tilted plane with a high level of accuracy. Also, while the tolerance on the position of the dowel pin holes are the same for each mirror, the pin spacings, d pin, are different because the primary mirror is a different physical size than secondary and tertiary. As a result, the clocking angle of primary mirror will be larger than the secondary and tertiary mirrors because the pin spacing is the smallest for this mirror.

171 139 A φ plate/housing =5 µm A B φ mirror/plate =5 µm A B d mirror/plate = mm B d plate/housing = 1 mm φ mirror =1.5 µm A B Figure 7-. The tertiary mirror subassembly and values that determine its alignment, namely, the dowel pin hole position tolerances and their relative spacings. Table 7-. Summary of the quantities used to derive the tolerances for the Monte Carlo sensitivity analysis. The dowel pin hole position tolerances are used to derive the mirror x/y decenter and mirror clocking angle. Item Value Dowel Pin Hole Position Tolerance (All Mirrors) Mirror 1.5 µm Adaptor Plate to Mirror 5. µm Adaptor Plate to Housing 5. µm Housing to Adaptor Plate.5 µm Dowel Pin Hole Spacing (Pri.) Mirror to Adaptor Plate 35 mm Adaptor Plate to Housing mm Dowel Pin Hole Spacing (Sec. and Ter.) Mirror to Adaptor Plate mm Adaptor Plate to Housing 1 mm From these tolerance ranges and the other tolerances mentioned in Table 7-1, a Monte Carlo sensitivity analysis is performed. In this analysis, the DOFs of the system are perturbed and the optical performance is computed. From multiple trials the probability of the as-built optical system meeting a given performance metric may be determined. Figure 7-3 displays the results of a simulation with 5 trials where the cumulative probability of the as-built optical system meeting a given RMS WFE at the operating wavelength of 1 µm is displayed for nine field points across the FOV. As to be expected from the prior sensitivity analysis, the edge of field is most heavily impacted and the

172 1 change in performance is most likely dominated by the large focal plane tilt tolerance. If a cumulative 95% is selected as the passing metric, all the field points lie below an as-built RMS WFE of.λ. These results are in good agreement with the prior analysis where the as-built performance was around.λ. From these two analyses, it can be concluded that the optical system can be machined and assembled with standard machine shop tolerances and the final system will remain diffraction limited, less than.7λ, throughout the FOV. 1% Cumulative Percentage 8% % % % (+,+ ) (-,+ ) (+,+ ) (-3,+ ) (+3,+ ) (+,+3 ) (+,-3 ) (-,+3 ) (-,-3 ) % As-Built RMS Wavefront Error (waves at 1µm) Figure 7-3. Cumulative probability as a function of as-built RMS WFE for the three mirror optical system over nine field points assuming only passive alignment Active Alignment The two analyses performed in section assumed that the optical system is to be passively aligned where the as-built optical performance is determined by the as-built manufacturing tolerances and a shim for focus compensation. If now some additional DOFs are allowed to vary during the assembly process, it may be possible to improve the as-built optical performance. Active alignment becomes more likely as the optical design form is considered for shorter wavebands.

173 11 The previously shown sensitivity analyses use the overall RMS WFE as the performance metric to assess the as-built optical performance; however, it does not provide any information on which aberration contributions are limiting the performance and how they vary throughout the FOV. If the DOFs are perturbed a known amount and now the Zernike aberration contributions are monitored throughout the FOV using the FFD, additional insight can be gathered on alignment strategies as well as which DOFs are going to be best for compensators. From the results of the sensitivity analysis in Table 7-1, it is seen that for the optical system, in general, there is a greater loss in performance when the mirror components are tilted versus decentered, so an effective compensator will be the mirror tilt. Figure 7- (a-d) displays the FFDs for Zernike astigmatism (Z 5/ ) and Zernike coma (Z 7/8 ) for the nominal optical system, shown in Figure 7- (a), and for the case where the mirrors are individually tilted +.1 in the YZ plane, shown in Figure 7- (b-d). When each mirror is tilted, the primary aberration component induced is field constant astigmatism. Some field constant coma is induced as well; though, its magnitude is about ten times less than that of the astigmatism. From this result, several conclusions can be drawn. First, since field constant astigmatism is primarily induced when the telescope is misaligned, during active alignment only one field point needs to be monitored to get a good representation of how the other field points are behaving. Second, for the same tilt of the three mirrors, the secondary is most sensitive to the perturbation. Consequently, this mirror may make the most effective compensator because a small perturbation of the mirror will have a large net effect on the overall system performance, thus requiring less mechanical movement of the compensator, assuming the compensator has enough mechanical resolution. The

174 1 tilt of the focal plane is also a key compensator. It reduces the focus variation across the FOV that will not be compensated by tilting the secondary mirror which primarily compensates astigmatism. Having selected the two most effective compensators, the Monte Carlo simulation is re-analyzed with the addition of the secondary mirror tilt and focal plane tilt as compensators. Figure 7-5 displays the results of a simulation with 5 trials where the cumulative probability of the as-built optical system meeting a given RMS WFE at the operating wavelength of 1 µm is displayed for nine field points across the FOV. In comparing these results to those of Figure 7-3, it can be seen that when these compensators are allowed to vary, the variance of the RMS WFE has decreased and now all the field points lie below an as-built RMS WFE of.5λ if the cumulative 95% point is used as the passing metric. In this case, the as-built optical performance is near nominal so there is little degradation in performance when assembly tolerances are considered with the secondary mirror tilt and focal plane tilt being used as compensators. In this case, the tolerances on the components could be relaxed if any of the tolerances were challenging to meet during fabrication or driving the cost of the components. The sensitivity analyses shown above have not considered the irregularity of the optical surfaces. In Chapter where the one of the optical surfaces was measured interferometrically, the predominant residual error in the surface was found to be astigmatism. As a result, the system performance is going to be degraded by a field constant astigmatic aberration as explained in Chapter 3. In this case, active alignment can be used to restore the optical functionality of the optical system. More specifically, the secondary mirror tilt is used to introduce the opposite amount of field constant astigmatism that results from the three fabricated optical surfaces. Ultimately, this

175 15:15:7 15:15:7 15:15:7 15:58:9 15:15:7 1:3:1 15:15:7 1:: property makes the entire optical system robust to both misalignment and fabrication induced errors. Y Field 15:15:7 Angle Y in Field Obect Angle Space in Obect - degrees Space - degrees Y Field 15:15:7 Angle Y Field in Obect Angle Space in Obect - degrees Space - Y degrees Field 15:15:7 Angle Y Field in Obect Angle Space in Obect - degrees Space - degrees Y Field Angle Y Field in Obect Angle Space in Obect - degrees Space - degrees KHF 1-Nov-13 Zernike Polynomial Zernike Polynomial 15:15:7 KHF Y Field Angle in Obect Space - degrees Y Ob. Field (deg.) Y Field Angle in Obect Space - degrees Y Ob. Field (deg.) Zernike Polynomial X Field Angle in Obect Space - degrees KHF 1-Nov-13 1-Nov X Ob. Field (deg.) Minimum =.9777 Zernike Polynomial Maximum =.19 Average FRINGE =.1191 ZERNIKE PAIR Z5 AND Z Std Dev =.5vs FRINGE ZERNIKE PAIR Z5 AND Z FIELD ANGLE IN OBJECT SPACE vs.5waves (1. nm) KHF 1-Nov-13 FIELD ANGLE IN OBJECT SPACE Minimum =.9777 Zernike Zernike Polynomial Polynomial Maximum =.19 Minimum =.1597 Average =.1191 Maximum =.1519 Std Dev =.5 Average =.1987 Std Dev = waves (1. KHF nm) KHF 1-Nov-13 1-Nov-13 1::1.5waves (1. nm) 15:15:7 Y Field Angle Y Field in Obect Angle Space in Obect - degrees Space - degrees Y Field 15:15:7 Angle Y Field in Obect Angle Space in Obect - degrees Space - degrees Y Field 15:15:7 Angle Y Field in Obect Angle Space in Obect - degrees Space - degrees Y Field Angle Y Field in Obect Angle Space in Obect - degrees Space - degrees Minimum =.9777 X Field X Angle Field in FRINGE Angle Obect ZERNIKE in Space Obect PAIR - Space degrees Z5 AND - Zdegrees Maximum =.19 X Field X Angle Field in FRINGE Angle Obect ZERNIKE in Space Obect PAIR - Space degrees Z5 AND - Zdegrees vs Average =.1191 vs FIELD ANGLE IN OBJECT SPACE Std Dev =.5 FIELD ANGLE IN OBJECT SPACE KHF - 1-Nov-13 Zernike Zernike Polynomial Polynomial KHF Y Field Angle in Obect Space - degrees Y Ob. Field (deg.) Zernike Polynomial Minimum = Minimum =.9777 Maximum =.19 Maximum =.19 FRINGE Average ZERNIKE FRINGE = PAIR.1191 ZERNIKE Z5 AND PAIR Z Z5 AND Z FRINGE Average ZERNIKE FRINGE = PAIR.1191 ZERNIKE Z7 AND PAIR Z8 Z5 AND Z Std Dev = vs.5 vs Std Dev = vs.5 vs - FIELD ANGLE FIELD IN OBJECT ANGLE IN SPACE OBJECT SPACE FIELD ANGLE FIELD IN OBJECT ANGLE IN SPACE OBJECT SPACE - - Zernike Zernike Polynomial 1-Nov-13.5waves (1. nm) Polynomial KHF Zernike Minimum =.9777 Zernike Polynomial 1-Nov-13.5waves (1. nm) Polynomial - Minimum =.9777 Minimum = Minimum.115 = Maximum = - Maximum.19= Maximum = Maximum =.19 Average = Average.1191=.1191 Average = Average.3117=.1191 X Field Angle in Obect Space - degrees Std Dev = Std.5 Dev =.5 Std Dev = Std.171 Dev = KHF 1-Nov-13-1-Nov-13.5waves.5waves (1. nm) (1. KHF nm) - 1-Nov-13.5waves (1. nm) KHF - 1-Nov-13.5waves (1. nm) X Field Angle in Obect Space - degrees X Field Angle in Obect Space - degrees X Ob. Field (deg.) FRINGE ZERNIKE PAIR Z5 AND Z X Ob. Field (deg.) (b) vs FIELD ANGLE IN OBJECT SPACE KHF Y Field Angle in Obect Space - degrees Zernike Polynomial KHF Y Ob. Field (deg.) - X Field Angle X Field in Angle Obect in Space Obect - degrees Space - degrees FRINGE ZERNIKE PAIR Z5 AND Z vs FIELD ANGLE IN OBJECT SPACE - - X Field Angle in Obect Space - degrees 15:15:7 Y Field Angle in Obect Space - degrees KHF 1-Nov-13 1-Nov X Ob. Field (deg.) Zernike Polynomial KHF Z Astig - - X Ob. Field (deg.) X Field Angle in Obect Space - degrees :15:7 1::1 15:59:5 15:15:7 15:15:7 - Y Field Angle in Obect Space - degrees Y Ob. Field (deg.) X Field X Angle Field in Angle Obect in Space Obect - Space degrees - degrees FRINGE ZERNIKE PAIR Z5 AND Z X Field X Angle Field in Angle Obect in Space Obect - Space degrees - degrees FRINGE ZERNIKE PAIR Z5 AND Z vs vs FIELD ANGLE IN OBJECT SPACE FIELD ANGLE IN OBJECT SPACE Zernike Polynomial Minimum = Zernike Polynomial - Minimum =.9777 Maximum =.19 Maximum =.19 FRINGE ZERNIKE PAIR Z5 AND Z FRINGE ZERNIKE PAIR Z7 AND Z8 Average FRINGE =.1191 ZERNIKE PAIR Z5 AND Z Average FRINGE =.1191 ZERNIKE PAIR Z5 AND Z vs vs Std Dev =.5 vs Std Dev =.5vs FIELD ANGLE FIELD IN OBJECT ANGLE IN SPACE OBJECT SPACE FIELD ANGLE FIELD IN OBJECT ANGLE IN SPACE OBJECT SPACE - - KHF Zernike Zernike Polynomial 1-Nov-13 Polynomial.5waves (1. nm) KHF Zernike Polynomial Minimum Minimum =.3753 Zernike 1-Nov-13.5waves (1. nm) Polynomial =.9777 Minimum Minimum =.17 = Maximum = Maximum.3955=.19 Maximum = Maximum.85=.19 Average = Average.33898=.1191 Average = Average.198= Std Dev = Std.89 Dev =.5 Std Dev = Std.1515 Dev = KHF KHF 1-Nov-13-1-Nov-13.5waves.5waves (1. nm) (1. KHF nm) - KHF 1-Nov-13-1-Nov-13.5waves.5waves (1. nm) (1. nm) :15:7 1:: Y Field Angle in Obect Space - degrees Y Field Angle in Obect Space - degrees Y Ob. Field (deg.) Minimum 1-Nov-13 Zernike.5waves Polynomial (1. nm) =.9777 Minimum = Maximum =.19 Maximum =.19.5λ (1µm) FRINGE Average ZERNIKE FRINGE = PAIR.1191 ZERNIKE Z5 AND PAIR Z Z5 AND Z FRINGE Average ZERNIKE FRINGE = PAIR.1191 ZERNIKE Z7 AND PAIR Z8Z5 AND Z Std Dev = vs.5 vs Std Dev = vs.5vs FIELD ANGLE FIELD IN OBJECT ANGLE IN SPACE OBJECT SPACE FIELD ANGLE FIELD IN OBJECT ANGLE IN SPACE OBJECT SPACE.5waves (1. nm) KHF - 1-Nov-13.5waves (1. nm) Zernike Zernike Polynomial Polynomial Minimum = Minimum.35 = Minimum = Minimum.115 =.9777 Maximum = Maximum.73=.19 - Maximum = Maximum =.19 Average = Average.9=.1191 Average = Average.3117=.1191 Std Dev = Std.18 Dev =.5 Std Dev = Std.171 Dev =.5 -.5waves.5waves (1. nm) (1. KHF nm) - KHF 1-Nov-13-1-Nov-13.5waves.5waves (1. nm) (1. nm) 15:15:7 (a) Y Field Angle in Obect Space - degrees Y Ob. Field (deg.) Y Ob. Field (deg.) Zernike Polynomial (c) (d) X Field Angle X Field in Angle Obect in Space Obect - degrees Space - degrees FRINGE ZERNIKE PAIR Z5 AND Z vs FIELD ANGLE IN OBJECT SPACE - Z Coma - - X Ob. Field (deg.) X Field Angle in Obect Space - degrees - - X Ob. Field (deg.) X Field Angle in Obect Space - degrees - - X Ob. Field (deg.) X Field Angle in Obect Space - degrees Figure X The Field - astigmatism FRINGE Angle ZERNIKE in Obect PAIR Space Z5 (ZAND 5/ - Z ) degrees and coma (Z 7/8 ) Zernike X Field X Angle Field aberration in FRINGE Angle Obect ZERNIKE in Space Obect FFDs PAIR - Space Z5 degrees AND over - Zdegrees an X Field Angle in Obect Space vs - degrees vs 8 x full FOV for FIELD the ANGLE (a) IN OBJECT nominal SPACE system and with.1 α tilt of FIELD the ANGLE (b) IN OBJECT primary, SPACE (c) secondary, and (d) tertiary mirror surfaces. Minimum =.9777 Maximum =.19 FRINGE Average ZERNIKE FRINGE =.1191 PAIR ZERNIKE Z7 AND PAIR Z8Z5 AND Z Std Dev =.5vs FIELD ANGLE FIELD IN ANGLE OBJECT IN SPACE OBJECT SPACE.5waves (1. nm) Minimum Minimum =.385 =.9777 Maximum = Maximum.3387 =.19 Average = Average.81 =.1191 Std Dev = Std Dev =.5.5waves.5waves (1. (1. nm) nm) 13

176 1 1% Cumulative Percentage 8% % % % (+,+ ) (-,+ ) (+,+ ) (-3,+ ) (+3,+ ) (+,+3 ) (+,-3 ) (-,+3 ) (-,-3 ) % As-Built RMS Wavefront Error (waves at 1µm) Figure 7-5. Cumulative probability as a function of as-built RMS WFE for the three mirror optical system over nine field points assuming active alignment where secondary mirror tilt and focal plane tilt are used as compensators Stray Light Analysis In addition to considering the manufacturing tolerances of the mechanical design that relate to the alignment of the optical system, another consideration for the design is its susceptibility to stray light. Therefore, an additional component to the mechanical design is to limit the light outside the FOV from reaching the detector through the use of baffling and surface preparation. Similar to the case of the sensitivity analysis, a figure of merit is established that measures how well the mechanical structure is reecting unwanted radiation. For this analysis, the figure of merit is the point source transmittance (PST) [8]. The PST computes the ratio of the average detector irradiance to the incident source irradiance as a function of input angle of the source. In an ideal system, the PST would be one everywhere within the FOV and zero elsewhere. However, in reality, some radiation from a source outside the intended FOV will reach the detector either through direct paths to the focal plane or through multiple bounces or scattering off the mechanical and optical surfaces. For this optical system, since the system is

177 15 nonsymmetric, the PST will also be nonsymmetric. Furthermore, the optical system is tilted only in the XZ plane so the primary contributor to the PST is the elevation angle of the source. Using FRED, a non-sequential raytrace program from Photon Engineering, the elevation PST of the optical system and base mechanical design presented in Figure 7-1 is computed. In the software, rays from a 51x51, 1 mm diameter, 1 µm source at the input aperture are traced in 1 increments over a 18 elevation. At each input angle, the PST is computed. Figure 7- shows the log(pst) as a function of input elevation angle for the base optical system where the optical surfaces are assumed to be perfect reflectors and the walls of the optical housing are assumed to be near specular with 8% reflectivity, which is a good representation for the specular component of machined aluminum in the LWIR [83]. In Figure 7-, it can be seen that the stray light reection of the base optical system is poor. There is a large region of stray light from roughly to 5 that results from a direct path to the focal plane from the input aperture. A similar region of stray light is observed from -5 to - as the light in this region reaches the focal plane by reflecting off one of the input faces of housing thus creating a mirror image of the region between and 5. These problem regions should be mitigated to improve the overall signal-to-noise ratio of the optical system, if possible.

178 1 - Log1(PST) Input Angle (deg.) Figure 7-. The computed elevation log(pst) for the baseline optical housing with the walls of the housing material assumed to be machined aluminum, resulting in a near specular surface with 8% reflectance. As a first step to mitigate the stray light, the walls of the optical housing are made less specular by blackening the walls with a suitable paint. In this fashion, the walls of the housing now become scattering surfaces. In FRED, the surface preparation is modeled as a flat black paint with a reflectance coefficient of.1. Moreover, importance sampling is added that preferentially traces rays that are scattered towards the focal plane, primary mirror, and tertiary mirror, which are the most direct ray paths to the focal plane. When this surface preparation is added to the walls of the optical housing and the PST is re-computed, the elevation PST is improved as shown in Figure 7-7 where the new PST is shown in blue and the previous PST with near specular walls is shown in gray. With the walls of the housing less specular, the amount of radiation reaching the focal plane is lessened. The large region of stray light that resulted from a reflection off the input face is no longer present because the surface now scatters the incoming light. The main

179 contributor to the stray light is now the direct path to the focal plane from the input aperture of the optical system Log1(PST) Input Angle (deg.) Figure 7-7. The computed elevation log(pst) for the optical system with blackened walls in blue and the computed elevation log(pst) for the baseline optical housing in gray. An improvement is observed when the walls of the housing are blackened versus left machined aluminum. To lessen the direct path to the focal plane from the input aperture of the optical housing, a baffle is added near the image plane that blocks most of the input radiation from the source. The effect of the baffle near the image plane is observed in Figure 7-8 (a-b), where Figure 7-8 (a) shows a cutaway of the optical system without the baffle and rays are drawn from the focal plane to the limiting mechanical structure to demonstrate the solid angle of the outside environment that can be seen by the focal plane. When the baffle, which is a hemispherical aluminum mask, is added as shown in Figure 7-8 (b), the solid angle of the environment seen by the focal plane is zero with only a direct view to the input aperture of the housing. An additional baffle is added on the other side of the image plane to block a region of stray light observed in Figure 7-7 around - that is

180 18 caused by radiation from the source reflecting off the tertiary and primary and reaching the focal plane. Image Plane Image Plane Solid Angle (a) (b) Figure 7-8. Cutaway of the optical system (a) without a baffle and (b) with a baffle and its solid angle to the environment from the focal plane shown in red for each case. With the baffle added to the housing, the solid angle to the environment goes to zero. The effect of adding these baffles to the optical housing is observed by re-computing the PST as shown in Figure 7-9 where the PST with the baffles is shown in red and the PST without the baffles is shown in light blue. In comparing the two PST plots, it can be seen that the magnitude of the stray light around - and between 5 and has decreased by about two orders of magnitude. These regions of stray light have not completely vanished because some scattered radiation still reaches the focal plane through these ray paths. As a final step to decrease some of the light that reaches the focal plane through scattering, an additional baffle is added at the primary mirror. This baffle is a cylindrical sleeve that fits around the clear aperture of the primary mirror as observed in Figure 7-8 (b). Its effect is observed by re-computing the PST, as shown in Figure 7-1, where the PST with the primary baffle is shown in green and the PST without the baffle is shown in light red. There is a slight decrease to the stray light around - and a two

181 19 order of magnitude decrease in the scattered stray light for elevation angles greater than when compared to the previous PST plot. Overall, through the use of several baffles and a black surface preparation on the walls of the optical housing, the optical surface is much better suited for the reection of stray light that will improve the signal-to-noise ratio of the optical system when operated. - Log1(PST) Input Angle (deg.) Figure 7-9. The computed elevation log(pst) for the optical system with blackened walls as well as baffling near the image plane in red and the computed elevation log(pst) for the optical housing with blackened walls in light blue. A two order of magnitude improvement is observed in the regions of large stray light when baffling is added near the image plane.

182 15 - Log1(PST) Input Angle (deg.) Figure 7-1. The computed elevation log(pst) for the optical system with blackened walls, baffling near the image plane, and baffling at the primary mirror in green and the computed elevation log(pst) for the optical housing with blackened walls and baffling near the image plane in light red. A two order of magnitude improvement is observed for large positive elevation angles where scattering is the dominant contributor to stray light. 7. As-built Optical System Working with II-VI Infrared, the optical housing has been manufactured offsite and directly assembled at the University of Rochester. In Figure 7-11 (a-c) the subassemblies of the three mirrors are shown for the primary, secondary, and tertiary. Within each subassembly, the three diamond turned raised pads and dowel pins can be seen that mate the subassembly to the optical housing. The secondary mirror subassembly, shown in Figure 7-11 (b), differs from the other two subassemblies as it includes the aperture stop of the optical system. The elliptical knife edge rests above the secondary mirror and ensures the correct ray bundle enters the optical system.

183 151 (a) (b) (c) Figure As-built subassemblies for the (a) primary, (b) secondary, and (c) tertiary mirrors of the three mirror system that are to be mated to the optical housing. Each subassembly mates to one face of the optical housing and rests on three raised, diamond turned pads. The as-built optical system with the three subassemblies mated to the housing is shown in Figure 7-1. With the use of the slip fit steel dowel pins, the subassemblies readily mate to the faces of the optical housing. To minimize mounting distortion of the mirror components, the screws are tightened ust enough to ensure that the subassemblies are secure to the housing as well as the mirrors secure to the adaptor plates. 1/- tapped holes are machined into both sides of the optical housing so that the housing can be secured to other mechanical components. The layout of the hole pattern is 1 on center and is designed to be perpendicular to the input face of the optical system. Figure 7-1. Assembled three mirror optical system. The system consists of a housing structure and three mirror subassemblies that are mated to the faces of the housing.

184 As-built Optical Performance The as-built full field performance of the assembled optical system is measured interferometrically with a Zygo 3.8 nm wavelength DynaFiz laser interferometer in a double pass configuration as shown in Figure The interferometer is affixed with an F/1.5 transmission sphere, providing a spherical wavefront output that overfills the F/1.9 optical system. With this interferometer configuration, the optical system is oriented backwards, that is, the output face of the optical system faces the interferometer. The point source focus of the interferometer is located at the image plane of the optical system by adusting the position of the optical system which is mounted on a z-axis translation stage. When the point source is at the correct image plane location, the beam exiting the optical system is collimated. A 15 mm diameter, λ/ high quality flat mirror is inserted at the input of the optical system so the wavefront is retro-reflected back towards the interferometer. The mirror must be oversized relative to the 3 mm entrance pupil diameter of the optical system because as the point source is scanned along the image plane surface, the angle of the exiting beam will change and at the retro-reflector, the beam will displace along the mirror surface. The retro-reflector mirror mount has variable tip and tilt and the optical system is mounted on both an x-axis and y-axis translation stage so various field points on the focal plane surface can be measured with the interferometer.

185 153 Figure Experimental setup for measuring the full field performance of the as-built optical system. Before any measurements are made with the optical system in the interferometric configuration, the system must be aligned to the interferometer. Specifically, the output face of the optical system must be aligned normal to the output of the interferometer. Without this alignment step, the measurements acquired throughout the FOV will not be relative to the correct image plane and the field curvature present in the measured wavefronts will be incorrect. To perform this alignment, the transmission sphere is first replaced with a transmission flat and the flat is aligned to the interferometer. Using the reflection from the output face of the optical housing as a guide, the tip and tilt of the optical system is adusted until it is nulled relative to the interferometer. With the alignment complete, the transmission sphere is replaced and aligned relative to the interferometer. The next step is to find the image plane location that corresponds to the on-axis field point of the telescope. This point is found by placing a reference optical flat on the input

186 15 face of optical housing, which is designed to be perpendicular to the on-axis field angle. The wavefront reflected back towards the interferometer from the reference flat is used to adust the x, y, and z position of the optical housing until the wavefront is nulled in both tilt and defocus. With the on-axis field point found, the rest of the FOV is measured. In total, a 3x3 grid of field points is measured over the 8 mm x mm image plane. At each field point, the retro-reflecting mirror must be re-positioned in tip and tilt to null the tilt present in the resulting interferogram. No adustments are made to the focus of the optical system during the measurement process to ensure that the field curvature of the optical system is appropriately measured relative to the nominally designed image plane. The measured 3x3 grid of wavefronts for the directly assembled system are shown below in Figure 7-1 where the RMS WFE at each field point is displayed inside the wavefront in microns. As can be seen in the structure of the wavefronts, the as-built system does suffer from field constant astigmatism oriented at ; however, the magnitude of the aberration is small and the RMS WFE throughout the FOV is less than.λ at 1 µm, below the diffraction limit of.7λ. Based on the analysis performed in Section , the residual field constant astigmatism could be a result of either misalignment, figure error, or both. The as-built measured data of the optical housing from II-VI Inc. can be analyzed to determine if the residual aberration in the optical system is the result of a misalignment. At II-VI using a coordinate measuring machine, the angular errors of the optical housing faces were measured. For the three mirror faces, the magnitude of angle error was found to be.5 arcsec for the primary mirror, 13 arcsec for the secondary mirror, and 1 arcsec for the tertiary mirror. Based on these magnitudes, which are very small, it is concluded that the observed field constant

187 155 astigmatism is not the result of a manufacturing error. On the other hand, the surface characterization of the mirror surface in Chapter did reveal astigmatism as the predominant surface error and would suggest that figure error of the mirror surfaces is the main contributing factor to the residual field constant astigmatism observed in the overall system. If the optical system is to be pushed to a shorter wavelength regime, further alignment will be required that compensates for the residual field constant astigmatism. 3.5 µm.8 µm.559 µm Y Image Plane Height (mm).395 µm.3 µm µm Wavefront Error (µm) µm. µm.9 µm - X Image Plane Height (mm) Figure 7-1. Measured wavefronts for a 3x3 grid of field points spanning an 8 mm x mm FOV for the directly assembled three mirror optical system. The RMS WFE in microns displayed within the wavefront for each field. The field constant astigmatism present in the directly assembled optical system that results from figure error of the as-fabricated surfaces can be removed by tilting the secondary mirror as discussed in Section The required tilt of the secondary mirror is dictated by the amount of field constant astigmatism present in the optical system. Using the on-axis field point as the reference point, the Zernike astigmatism (Z 5/ ) is measured for the as-built system and compared to its nominal value. The measured Z 5

188 15 and Z astigmatism for the as-built system is -.85 µm and µm compared to.7 µm and. µm for the nominal system. This difference is simulated in a commercial lens design software package, in this case CODE V, by adding -1.5 µm of Z 5 astigmatism and µm of Z astigmatism to the entrance pupil. Next, in CODE V, the secondary mirror tilt is re-optimized to remove the residual field constant astigmatism. Only the tilt in the YZ plane is allowed to vary because the XZ plane tilt is difficult to implement in the actual as-built system where shims will have to be used. From the simulation the optimum tilt is found to be and it improves the overall performance so that it is near nominal with a maximum RMS WFE of.λ at a wavelength of 1 µm. However, there is a tradeoff for this improvement as the boresight of the optical system does change and the image shifts down 7 µm in the y-direction. For a camera with 5 µm pixels, the secondary tilt results in a boresight error of three pixels. Based on the optical housing geometry, the shim required to be placed underneath the raised pad where the secondary mirror subassembly mounts to the optical housing is roughly 3 µm. As an example of this implementation, a 3 µm shim at the secondary mirror has been implemented. Figure 7-15 shows the resulting measured wavefronts for the 3x3 grid of field points with the corresponding RMS WFE at each field point displayed inside the wavefront in microns. The maximum RMS WFE has improved by a factor of two to.3λ at a wavelength of 1 µm. Further improvement is still possible if the shim size is increased to 3 µm and the mirror is shimmed out of plane to remove the residual field constant astigmatism oriented at 5. However, even at this stage in the alignment, the optical system is well within the diffraction limit of.7λ at a wavelength of 1 µm and would perform well if operated at 1 µm. As an example to demonstrate the

189 157 image quality of the assembled optical system, Figure 7-1 shows a sample image of the optical system affixed with the 8 mm x mm, 5 µm pixel pitch uncooled microbolometer detector µm.13 µm.95 µm Y Image Plane Height (mm).1 µm.157 µm µm Wavefront Error (µm) µm.19 µm. µm - X Image Plane Height (mm) Figure Measured wavefronts for a 3x3 grid of field points spanning an 8 mm x mm FOV for the directly assembled three mirror optical system with the secondary mirror tilted roughly 1 arc minute with a 3 µm shim. The RMS WFE in microns is displayed within the wavefront for each field. Figure 7-1. Sample LWIR image from the optical system