MISALIGNMENT INDUCED NODAL ABERRATION FIELDS AND THEIR USE IN THE ALIGNMENT OF ASTRONOMICAL TELESCOPES

Transcription

1 MISALIGNMENT INDUCED NODAL ABERRATION FIELDS AND THEIR USE IN THE ALIGNMENT OF ASTRONOMICAL TELESCOPES by TOBIAS SCHMID Dipl. Ing. University of Applied Sciences Regensburg Germany 004 M.S. University of Central Florida USA 006 A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the College of Optics and Photonics at the University of Central Florida Orlando Florida Summer Term 010 Major Professor: Jannick P. Rolland

2 010 Tobias Schmid ii

3 ABSTRACT Following the foundation of aberration theory for rotationally symmetric optical systems established by Seidel Schwarzschild Burch Conrady Buchdahl and in its most useful form H. H. Hopkins Shack Buchroeder Thompson and Rogers developed a vectorial form of the wave aberration theory that enables addressing optical systems without symmetry. In this research a vectorial theory is utilized and extended for the alignment of two- and three-mirror astronomical telescopes including the effects of pointing changes and astigmatic figure errors. Importantly it is demonstrated that the vectorial form of aberration theory also referred to as nodal aberration theory not only provides valuable insights but also facilitates a quantitative description of the aberrations in optical systems without symmetry. Specifically nodal aberration theory has been utilized to establish key insights into the aberration field response of astronomical telescopes to misalignments. Important nodal properties have been derived and discussed and the theoretical predictions have been validated with optical design software. It has been demonstrated that the removal of on-axis coma in some of the most common astronomical telescopes in use today directly leads to a constraint for one of the nodes for astigmatism to be located at the field center which is exactly true for Cassegrain or Gregorian telescopes and approximately true for Ritchey-Chrétien (or aplanatic Gregorian) telescopes. These observations led to important conclusions concerning the alignment of astronomical telescopes. First the correction of these telescopes on-axis for zero coma removes all misalignment induced aberrations only on-axis. Secondly given that the image quality at the field center remains stigmatic in the presence of misalignments for these telescopes non-zero astigmatism measured at the field-center directly reveals astigmatic mirror figure errors. iii

4 Importantly the effects of misalignments and astigmatic figure error can be clearly distinguished if present in combination even in the presence of significant boresight errors. Having the possibility to clearly distinguish between misalignment and astigmatic mirror figure error provides an important prerequisite for the optimal operation of active/adaptive optics systems that are becoming standard in observatory class telescopes. Subsequent work on TMA telescopes revealed that even though TMAs are limited by fifth order aberrations in their nominal alignment state third order nodal aberration theory provides accurate image quality predictions for misalignments and astigmatic figure (third order) effects in these optical systems. It has been demonstrated for the first time that analytical expressions can be devised that describe the characteristic misalignment induced aberration fields of any TMA telescope leading to two main image quality degrading aberrations fieldconstant coma and field-linear field-asymmetric astigmatism. These new insights can be strategically leveraged in the development of alignment strategies for TMAs. The final part of this work analyzed how third and fifth order nodal aberration fields can be utilized in the alignment of wide-angle telescopes with the specific example of the Large Synoptic Survey Telescope (LSST). In cooperation with the National Optical Astronomy Observatory (NOAO) an alignment strategy has been developed for the LSST (without camera) to expedite the commissioning of the telescope providing for the first time analytical expressions for the computation of misalignment parameters in three-mirror telescopes taking into account fabrication tolerances for the alignment of the tertiary mirror on the primary mirror substrate. Even though the discussion has been focused primarily on alignment strategies of iv

5 astronomical telescopes the methods and algorithms developed in this work can be equally applied to any imaging system. v

6 To my wife Noel and son Carsten. vi

7 ACKNOWLEDGMENTS I would like to thank Dr. Roland Schiek a former professor during my studies in electrical engineering who motivated and supervised my first research projects in optics together with Dr. George Stegeman. I started my graduate degree in optics at CREOL The College of Optics and Photonics in 005 and after initial work in the development of sources for EUV Lithography I focused my dissertation research on the aberration theory of optical systems without symmetry. Many thanks to Optical Research Associates Pasadena California for providing a student license for CODE V and LightTools and also for an opportunity to do an internship in the engineering group in Fall 008. Special thanks for numerous discussions especially with Dr. John Tamkin Dr. John Rogers John Isenberg Matthew Rimmer Dr. Mary Kate Crawford and Dave Hasenauer for engineering work. I also would like to thank Brian McGrath Aaron Kruglyak Dr. Mike Hayford Dr. Bill Cassarly Mike Zollers for valuable discussions concerning software testing. Special thanks to my advisor Prof. Jannick Rolland who motivated and enabled my research and helped me improve my writing skills and to Dr. Kevin Thompson (Optical Research Associates) for the extensive theoretical and historical input. I also would like to extend special thanks to Dr. Andrew Rakich for numerous discussions on telescope alignment and specifically on many discussions involving practical considerations. Much thank also to Dr. Jacques Sebag William Gressler and Douglas Neill from LSST Corporation/NOAO for the joint work on an alignment strategy for the Large Synoptic Survey Telescope. vii

8 I also would like to thank my current and previous labmates/officemates: Dr. Ozan Cakmakci Florian Fournier Kyle Fuerschbach Sophie Vo Dr. Kye-Sung Lee Panomsak Meemon Dr. Supraja Murali Ilhan Kaya Bob Gray and Cristina Canavesi. viii

9 TABLE OF CONTENTS LIST OF FIGURES... XII LIST OF TABLES... XX CHAPTER 1: INTRODUCTION Historical Perspective of Reflective Astronomical Telescopes Alignment Methods for Astronomical Telescopes Motivation Research Summary Dissertation Outline CHAPTER : NODAL ABERRATION THEORY FOR OPTICAL SYSTEMS WITHOUT SYMMETRY Concepts and Definitions Local Coordinate System Paraxial Raytrace Definition of Surface Local Object/Image and Entrance/Exit-Pupil Planes # Definition of an Equivalent Tilt Parameter for a Spherical Surface: β Locating the Centers of the Local Objects/Images (Q Q ) and Entrance/ Exit-Pupils (EE ) Parameterizing the OAR The Vector that Locates the Center of the Aberration Field Symmetry for a Tilted and Decentered Surface ( sph) Shifted Aberration Field Center Associated with the Spherical Base Curve σ Shifted Aberration Field Center Associated with the Aspheric Departure from the ( asph) Spherical Base Curve σ Real-Ray Equivalent Approach to Find Individual Surface Aberration Field Centers 34.5 Nodal Properties of Third Order Aberrations Concept of Shifted Aberration Field Centers and Effective Field Height Third Order Spherical Aberration in Optical Systems Without Symmetry Third Order Coma in Optical Systems Without Symmetry Third Order Astigmatism in Optical Systems Without Symmetry Third Order Medial Focal Surface in Optical Systems Without Symmetry CHAPTER 3: MISALIGNMENT ABERRATION FIELDS IN ASTRONOMICAL TELESCOPES Definition of Structural and Systems Parameters Wave Aberration Coefficients for Two-Mirror Telescopes Effects Induced By Transverse Misalignments Misalignment Induced First Order Effects/Boresight Error Locating the Aberration Field Centers in Misaligned Two-Mirror Telescopes Third Order Nodal Aberrations in Misaligned Two-Mirror Telescopes and Computation of Misalignment Parameters Pivot Points in Two-Mirror Telescopes Rotations of a Mirror about its Center of Curvature Rotations of a Mirror about the Coma-Free Pivot Point ix

10 3.5 A Unique Astigmatic Nodal Property in Misaligned Cassegrain Gregorian and Ritchey-Chrétien Telescopes With Misalignment Coma Removed Nodal Constraints of a Coma-Aligned Cassegrain or Gregorian Telescope with the Aperture Stop Located on the Primary Mirror Nodal Constraints of a Coma-Aligned Ritchey-Chrétien or Aplanatic Gregorian Telescope Assuming the Aperture Stop is Located on the Primary Mirror CHAPTER 4: MISALIGNMENT ABERRATION FIELDS IN THREE-MIRROR ANASTIGMATIC ASTRONOMICAL TELESCOPES Abberation Fields of Telescope Configurations Throughout the 0 th Century Application of Nodal Aberration Theory to Derive the Dominant Characteristic Response of any TMA Telescope to Misalignments Interpretation of the Dominant Characteristic Misaligned TMA Aberration Field Response Functions Application James-Webb-like Space Telescope Utilizing Full-Field-Displays to Explore the Effects of Misalignments on the Aberration Fields Developing Expressions for Misalignment Parameters based on Nodal Aberration Theory CHAPTER 5: ABERRATION FIELD RE-CONSTRUCTION BASED ON SPARSE WAVEFRONT MEASUREMENTS Prediction of Point Spread Functions for Misaligned Astronomical Telescopes Discussion of the Minimum Field Sampling to Fully Re-Construct the Aberration Fields Third Order Spherical Aberration Third Order Coma Third Order Astigmatism Effects of Uncertainties in the Wavefront Measurement on the Accuracy in the Prediction of Misalignment Parameters CHAPTER 6: DISTINGUISHING PRIMARY MIRROR FIGURE ERROR FROM MISALIGNMENT INDUCED ASTIGMATI IN ASTRONOMICAL TELESCOPES Consequences of Actively Controlled Primary Mirrors in the Context of Telescope Alignment Formulating Nodal Aberration Theory of a Misaligned Ritchey-Chrétien Telescope with Astigmatic Figure Error on the Primary Mirror Binodal Response of the Astigmatic Field Dependence of a Ritchey-Chrétien Telescope with Primary Mirror Astigmatic Figure Error Binodal Response of the Astigmatic Field Dependence of a Ritchey-Chrétien Telescope with Misalignments and Astigmatic Figure Error Conditions Imposed as a Result of Aligning a Ritchey-Chrétien Telescope to Remove Field-Constant Coma Conditions Resulting from Primary Mirror Astigmatic Figure Error and Misalignments in a Ritchey-Chrétien Telescope x

11 6.5 Validation of the Nodal Properties of a Ritchey-Chrétien Telescope with Misalignments and Astigmatic Figure Error Example: Computation of Misalignment Parameters in a Misaligned Ritchey-Chrétien Telescope with the Aperture Stop at the Primary Mirror Including Primary Astigmatic Figure Error CHAPTER 7: USING A MULTINODAL THEORY OF ABERRATIONS IN THE ALIGNMENT OF THREE-MIRROR WIDE-ANGLE ASTRONOMICAL TELESCOPES WITH THE EXAMPLE OF THE LARGE SYNOPTIC SURVEY TELESCOPE Optical Layout of the Large Synoptic Survey Telescope (LSST) First Order Calculations/Aberration Field Centers Computing the Surface Contributions to the Wave Aberrations Development of an Alignment Strategy Assuming Ideal Mirror Shapes... 8 CHAPTER 8: CONCLUSION AND FUTURE WORK APPENDIX A : DERIVATIONS FOR THE LOCAL COORDINATE SYSTEM PARAXIAL RAYTRACE APPENDIX B : CONCEPT OF VECTOR MULTIPLICATION APPENDIX C : FIRST ORDER QUANTITIES OF TWO-MIRROR TELESCOPES APPENDIX D : THIRD ORDER QUANTITIES OF TWO-MIRROR TELESCOPES APPENDIX E : DERIVATION OF THE COMA-FREE PIVOT POINT IN TWO-MIRROR TELESCOPES WITH THE APERTURE STOP LOCATED ON THE PRIMARY MIRROR... 6 APPENDIX F : DERIVATION OF THE AMOUNT OF SECONDARY MIRROR TIP- TILT TO COMPENSATE FIELD-CONSTANT COMA IN A RITCHEY- CHRÉTIEN TELESCOPE WITH THE APERTURE STOP LOCATED ON THE PRIMARY MIRROR LIST OF REFERENCES xi

12 LIST OF FIGURES Figure -1: Illustration of the concept of local object and pupil for each surface in an optical system # Figure -: Definition of the equivalent tilt parameter β 0 for a spherical surface which defines the location of the perturbed center of curvature relative to the mechanical coordinate axis (MCA) # # Figure -3: Illustrating object/image displacement δ Q and ( δq )' (and pupil displacement # # δε and ( δε )' by substituting the entrance/exit-pupil points (EE ) for the object/ image points (QQ ))... 9 Figure -4: Perturbed optical system with OAR (top) and paraxial marginal and chief-ray of nominal optical system (bottom) Figure -5: Illustrating important OAR parameters for a tilted and decentered spherical surface Figure -6: Concept of the shifted aberration field center for the spherical base curve evaluated in the local object space Figure -7: (a) Concept of the shifted aberration field center for the contribution of the aspheric departure from the spherical surface and (b) paraxial raytrace of the nominal optical system in local space Figure -8: Perturbed optical system with OAR parameters for a given surface j Figure -9: Concept of aberration field center in the image plane and effective field height Figure -10: Pupil dependence of third order spherical aberration Figure -11: Pupil dependence of third order coma Figure -1: Magnitude (left) and orientation (right) of third order coma across the field for (a) a rotationally symmetric optical system and (b) an optical system without symmetry Figure -13: Two wave aberration contributions for coma across the field with different magnitudes and aberration field centers and projection of the intersection line of the two surface contributions for coma onto the x-y - field Figure -14: Orientation of the two surface aberration contributions for coma showing the individual aberration field centers and the coma node location Figure -15: Aberration field vector for third order coma showing the node location for third order coma Figure -16: Pupil dependence of third order astigmatism for the (a) tangential (b) medial and (c) sagittal focus Figure -17: (a) Magnitude and (b) orientation of third order astigmatism across the field for a rotationally symmetric optical system and (c) magnitude and (d) orientation of third order astigmatism across the field for an optical system without symmetry Figure -18: Two wave aberration contributions for astigmatism across the field with different magnitudes but (a) equal signs for the wave aberration contributions and (b) unequal signs for the wave aberration contributions xii

13 Figure -19: Orientation of the two surface aberration contributions for astigmatism showing the individual aberration field centers and the two astigmatic node locations for (a) equal signs of the two surface contributions and (b) unequal signs of the two surface contributions. Note the insets are magnified illustrations of the line orientation at the astigmatic nodes Figure -0: Aberration field vectors for third order astigmatism showing the two astigmatic node locations Figure -1: Pupil dependence of the medial astigmatic component Figure -: Shift of the medial focal surface with respect to the design image plane Figure -3: Magnitude of the wave aberration for the medial focal surface across the field for (a) a rotationally symmetric optical system and (b) an optical system without symmetry.. 55 Figure -4: Aberration field vector for the third order medial focal surface showing the transverse displacement of its vertex. In addition the surface is shifted longitudinally by δ z0 = 8( f #) W M 0 b M M Figure 3-1: Visualization of the sign conventions for the ray intersection height the ray inclination angle the angle of incidence and axial spacings Figure 3-: Design forms for two-mirror astronomical telescopes with the stop at the primary mirror showing (a) the Cassegrain type and (b) the Gregorian type Figure 3-3: (a) Marginal-ray (axial field point) and OAR for an aligned Ritchey-Chrétien telescope with the stop on the primary mirror and (b) marginal-ray and OAR in the presence of secondary mirror misalignments with the stop on the primary mirror. Note that the OAR no longer intersects the image plane at the center of the AR which itself is assumed to be aligned with the primary mirror axis Figure 3-4: (a) Schematic layout of a two-mirror telescope with the aperture stop located on the primary mirror and (b) OAR raytrace starting at the aspheric vertex of the primary mirror Figure 3-5: Schematic layout of a two-mirror telescope with the aperture stop located on the secondary mirror showing the (a) varying pupil footprint on the primary mirror with field angle and (b) OAR raytrace showing the intercept at the secondary mirror (aspheric) vertex Figure 3-6: FFDs showing the magnitude and orientation of FRINGE Zernike polynomials Z7/Z8 related to third order coma for a Cassegrain telescope (a) without misalignments and (b) in the presence of secondary mirror misalignments Figure 3-7: FFDs showing the magnitude and orientation of FRINGE Zernike polynomials Z5/Z6 related to third order astigmatism for a Cassegrain telescope (a) without misalignments and (b) in the presence of secondary mirror misalignments Figure 3-8: Schematic diagram visualizing the node positions for third order coma and astigmatism in Cassegrain/Gregorian telescopes including the boresight error with respect to the center of the AR Figure 3-9: FFDs showing the magnitude and orientation of FRINGE Zernike polynomials Z7/Z8 related to third order coma for a Ritchey-Chrétien telescope (aplanatic) (a) without misalignments and (b) in the presence of secondary mirror misalignments xiii

14 Figure 3-10: FFDs showing the magnitude and orientation of FRINGE Zernike polynomials Z5/Z6 related to third order astigmatism for a Ritchey-Chrétien telescope (aplanatic) (a) without misalignments and (b) in the presence of secondary mirror misalignments Figure 3-11: Rotation of the secondary mirror about its center of curvature Figure 3-1: Coma-free pivot point for the secondary mirror of a two-mirror telescope with the aperture stop located on the primary mirror Figure 3-13: Coma-free pivot point for the primary mirror of a two-mirror telescope with the aperture stop located on the secondary mirror Figure 3-14: Dependence of the location of the coma-free pivot point in Cassegrain telescopes on the secondary mirror magnification and axial obstruction ratio expressed as distance from the secondary mirror vertex normalized by the system focal length Figure 3-15: Distance of the coma-free pivot point from the secondary mirror vertex normalized with the system focal length dependent on secondary mirror magnification and axial obstruction ratio Figure 3-16: Nominal system (third order) coma for various distributions for the conic constants to achieve a particular position of the coma-free pivot point Figure 3-17: (a) Secondary mirror conic constant as a function of secondary mirror magnification and axial obstruction ratio to obtain zero decentering sensitivity for coma and (b) primary conic constant to provide in combination with the secondary mirror conic constant stigmatic imaging on-axis (zero spherical aberration) Figure 3-18: FFDs for the case of a Cassegrain telescope showing the FRINGE Zernike coefficients (a) Z7/Z8 and (b) Z5/6 assuming arbitrary secondary mirror misalignments.109 Figure 3-19: FFDs for the case of a Cassegrain telescope showing FRINGE Zernike coefficients (a) Z7/Z8 and (b) Z5/6 after aligning the secondary mirror for zero on-axis coma Figure 3-0: Surface-by-surface wave aberration coefficients for (a) coma and (b) astigmatism of a typical Ritchey-Chrétien telescope including the separation of the contribution from the spherical base curve of a conic/aspheric mirror and the contribution from the conic/aspheric departure and (c) aberration field center vectors for the spherical base curve and conic/ aspheric contributions of the secondary mirror ( ) of a misaligned Ritchey-Chrétien telescope. The opposing alignment of sph ( asph) σ and σ is a key constraint imposed by the correction of misalignment induced coma Figure 3-1: Example of (a) field-quadratic astigmatism (magnitude) in case of an aligned Ritchey-Chrétien telescope (b) misalignment induced binodal astigmatism (third order) of a Ritchey-Chrétien telescope aligned for zero field-constant third order coma Figure 3-: (a) Visualizing the ratio of Equation (3-105) and Equation (3-106) for secondary mirror magnifications m =..10 axial obstruction ratio γ = and (b-d) astigmatic node and node midpoint positions for various secondary mirror magnifications assuming identical secondary mirror perturbations as given in Figure xiv

15 Figure 4-1: Field of view of a TMA telescope exhibiting (a) zero coma in the nominal alignment state (b) constant coma in the presence of residual misalignments (c) zero astigmatism in the nominal alignment state and (d) field-asymmetric field-linear third order astigmatism in the presence of residual misalignments Figure 4-: Schematic layout of the JWST like telescope Figure 4-3: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for the nominal JWST like telescope Figure 4-4: (a) Field curves for the JWST like telescope and (b) astigmatic line images indicating the orientation reversal beyond the ring zone Figure 4-5: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for a primary mirror decenter of YDE PM = 0. mm Figure 4-6: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for a primary mirror tip-tilt of ADE PM = deg Figure 4-7: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for a secondary mirror decenter of YDE = 0. mm Figure 4-8: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for a secondary mirror tip-tilt of ADE = deg Figure 4-9: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for a tertiary mirror decenter of YDE TM = 0.5 mm Figure 4-10: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for a tertiary mirror tip-tilt of ADE TM = 0.05 deg Figure 4-11: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for the initially misaligned JWST-like configuration given by YDE PM = mm ADE PM = deg YDE = mm and ADE = deg Figure 4-1: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for the JWST-like configuration after decentering the primary mirror to compensate misalignment induced coma given by YDE PM = mm ADE PM = deg YDE = mm and ADE = deg Figure 4-13: Variation of constant coma with secondary tilt angle and pivot length Figure 4-14: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for the JWST-like configuration after rotating the secondary mirror about its coma-free pivot point to compensate field-linear astigmatism resulting in YDE PM = mm ADE PM = deg YDE = mm and ADE = deg Figure 4-15: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for the JWST-like configuration after decentering the primary and secondary mirror to align the medial focal surface given by YDE PM = 0.95 mm ADE PM = deg YDE = mm and ADE = deg Figure 4-16: FDDs for the James-Webb-like space telescope showing (a) field-constant coma and (b) field-linear field-asymmetric astigmatism and (c) magnitude of FRINGE Zernike coefficients Z7/8 and (d) magnitude of the FRINGE Zernike coefficients Z5/ xv

16 Figure 5-1: Correspondence of the wavefront in the exit-pupil of the telescope to the point spread function in the image plane Figure 5-: (a) Pupil transmission function chosen for the example of a Ritchey-Chrétien telescope and (b) phase function for a particular field point under consideration Figure 5-3: (a) Point spread functions calculated based on Equations (5-1) - (5-6) compared to (b) point spread functions computed in CODE V Figure 5-4: (a) Point spread function for the field point ( ) arcmin obtained through focus based on nodal aberration theory and (b) corresponding point spread functions computed in CODE V Figure 5-5: (a) Predicted defocused off-axis point spread function mainly affected by misalignment induced coma and astigmatism and (b) simulated point spread function in CODE V Figure 5-6: Despace induced third order spherical aberration showing (a) the point spread function and (b) FFD for the FRINGE Zernike coefficient Z9 related to spherical aberration Figure 5-7: Point spread functions for the on-axis and several off-axis field points for (a) an aligned Cassegrain telescope showing nominal coma and (b) a misaligned Cassegrain telescope showing on-axis coma and a displaced coma node Figure 5-8: Point spread functions for the on-axis and several off-axis field points for (a) an aligned Ritchey-Chrétien telescope showing zero nominal coma and (b) a misaligned Ritchey-Chrétien telescope showing dominant constant coma across the field of view Figure 5-9: Point spread functions for the on-axis and several off-axis (edge of the field) field points for (a) an aligned Ritchey-Chrétien telescope showing nominal astigmatism (b) a misaligned Ritchey-Chrétien telescope showing binodal astigmatism and (c) FFD showing FRINGE Zernike coefficients Z5/6 visualizing astigmatism across the field of view Figure 5-10: Point spread functions for the on-axis and several off-axis (intermediate) field points for (a) an aligned TMA telescope showing zero nominal astigmatism and (b) a misaligned TMA telescope showing dominant field-asymmetric field-linear third order astigmatism. (c) FFD showing the FRINGE Zernike coefficients Z5/Z6 across the field Figure 5-11: Spread of the re-constructed astigmatic node locations caused by the uncertainty in the wavefront measurements ε = ± 0.0 shown for (a) 4 wavefront measurements (b) 9 wavefront measurements and (c) 16 wavefront measurements Figure 5-1: RMS deviation of secondary mirror decenters and tilts with respect to the mean value dependent on the number of measured field points and the magnitude of the uncertainty in the wavefront measurements. (ab) Reconstructed x- and y-decenter values dependent on the number of measured field points and (cd) re-constructed x-zand y-z tip-tilt values dependent on the number of measured field points Figure 6-1: Aspheric corrector plate with aperture stop positions (a) centered on the optical axis indicating spherical aberration (bcd) shifted aperture stop causing field-constant astigmatism xvi

17 Figure 6-: Coordinate system definition in the exit-pupil showing (a) the definition for the FRINGE Zernike polynomials and (b) the coordinate system orientation utilized in nodal aberration theory Figure 6-3: (a) Interferogram of the primary mirror astigmatic figure error (b) binodal astigmatism caused by an astigmatic figure error on the primary mirror in the case of a fully aligned Ritchey-Chrétien telescope and (c) the magnitude of astigmatism corresponding to (b) Figure 6-4: (a) Wave aberration contributions for coma (top) and astigmatism (bottom) showing the spherical base curve and conic/aspheric surface contributions (b) secondary mirror aberration field centers (spherical and aspheric) before (denoted by * ) and after removing misalignment induced coma Figure 6-5: (a) Aberration field center of symmetry for the spherical and aspheric aberration field contributions of the secondary mirror after aligning the telescope for zero fieldconstant coma. (b) Astigmatic primary mirror figure error for a Ritchey-Chrétien telescope Figure 6-6: The vector a that locates the center of biplanar symmetry of the binodal astigmatic field for a Ritchey-Chrétien telescope with secondary misalignment and primary mirror astigmatic figure error. (a) Contribution from secondary mirror misalignments under the condition that field-constant coma has been removed and (b) the contribution from astigmatic figure error on the primary mirror as derived in Section and 3 has no a component Figure 6-7: The vector b that points from the endpoint of a to the nodal points for binodal astigmatism shown here for a Ritchey-Chrétien telescope with secondary misalignment and primary mirror astigmatic figure error consisting of (a) ( ALI ) b denoting the contribution caused by secondary mirror misalignments (b) b ( FIG) determined by interferogram data combined with the knowledge of the total astigmatism in the nominal optical system resulting in (c) the overall vector b and b the final astigmatic node locating vector when combined with a Figure 6-8: The characteristic node geometry for (a) astigmatism in the absence of misalignments or astigmatic figure errors (b) misalignment induced binodal astigmatism (c) astigmatic figure error induced binodal astigmatism and (d) both contributions (b) and (c) combined Figure 6-9: Magnitude of astigmatism of (a) the nominal Ritchey-Chrétien telescope (b) binodal astigmatism in the presence of misalignments after alignment to remove constant coma (c) an aligned telescope with only astigmatic figure error and (d) combined misalignments (b) and astigmatic figure error (c) Figure 6-10: Magnitude of astigmatism of (a) the nominal Ritchey-Chrétien telescope (b) binodal astigmatism in the presence of misalignments after alignment to remove constant coma (c) the aligned telescope including astigmatic figure error and (d) combined misalignments (b) and astigmatic figure error (c) to collapse the astigmatic nodes xvii

18 Figure 6-11: Plot of the RMS wavefront error across the field for (a) a misaligned telescope with added astigmatic figure error to obtain coincident astigmatic nodes and (b) same as (b) but with adjusted focal plane tilt Figure 6-1: Real-Ray based verification of the prediction of astigmatic nodal positions by a generalized Coddington raytrace illustrating the (a) astigmatic aberration field of the nominal system (b) node positions for secondary mirror misalignments only in a configuration that is corrected for zero coma (c) node positions corresponding to the astigmatic figure error illustrated in Figure 6-8c (d) net display including the secondary misalignments and astigmatic figure error and (e) combined secondary mirror misalignments and primary mirror astigmatic figure chosen to obtain coincident astigmatic nodes Figure 6-13: (a) FFD for astigmatism and (b) Point Spread Functions simulated on-axis and at a ring of field points at 5.9 arcmin in the field for the nominal alignment state of the Ritchey-Chrétien telescope Figure 6-14: Simulated point spread functions demonstrating constant coma across the FOV Figure 6-15: Optical performance of the telescope after eliminating misalignment induced coma showing (a) FFD for Zernike polynomial coefficients Z5/6 and (b) simulated point spread functions and (c) re-constructed aberration field for astigmatism with overlaid field points for the wavefront data Figure 6-16: Accuracy of secondary mirror misalignment parameter prediction with offset of the adapter axis with respect to the primary mirror optical axis for (a) secondary mirror decenter in the y-z -plane and (b) secondary mirror tilt in the y-z -plane Figure 6-17: Optical performance of the telescope after compensation of secondary mirror misalignments showing the remaining image quality degradation caused by astigmatic figure error of the primary mirror (a) FFD for Zernike polynomial coefficients Z5/6 and (b) simulated point spread functions Figure 7-1: Layout of the LSST (a) including the camera and (b) without the camera utilizing only a small portion of the total field of view Figure 7-: FFDs displaying FRINGE Zernike coefficients for the nominal LSST without the camera (a) Z5/6 related to third order astigmatism (b) Z7/8 related to third order coma and (c) Z14/15 related to fifth order coma Figure 7-3: FFDs displaying FRINGE Zernike coefficients for the perturbed case of the LSST without the camera assuming a secondary mirror decenter of YDE = 1 mm (a) Z5/6 related to third order astigmatism (b) Z7/8 related to third order coma and (c) Z14/15 related to fifth order coma Figure 7-4: FFDs displaying FRINGE Zernike coefficients for the perturbed case of the LSST without the camera assuming a secondary mirror tip-tilt of ADE = 0.03 deg (a) Z5/6 related to third order astigmatism (b) Z7/8 related to third order coma and (c) Z14/15 related to fifth order coma Figure 7-5: FFDs displaying FRINGE Zernike coefficients for the perturbed case of the LSST without the camera assuming a tertiary mirror decenter of YDE TM = 1 mm (a) Z5/6 related to third order astigmatism (b) Z7/8 related to third order coma and (c) Z14/15 related to fifth order coma xviii

19 Figure 7-6: FFDs displaying FRINGE Zernike coefficients for the perturbed case of the LSST without the camera assuming a tertiary mirror tip-tilt of ADE TM = 0.03 deg (a) Z5/6 related to third order astigmatism (b) Z7/8 related to third order coma and (c) Z14/15 related to fifth order coma Figure 7-7: Schematic layout of the LSST without camera module defining the notation for secondary mirror misalignments and fabrication errors (fixed misalignments of the tertiary mirror with respect to the primary mirror). Note that the spheres correspond to the centers of curvature and vertices of the mirrors. As shown the primary and tertiary mirrors are made from one monolithic substrate Figure 7-8: Schematic nodal behavior (left) and FFD (right) for (a) third order coma (b) fifth order coma and (c) and third order astigmatism Figure 7-9: Schematic drawing of the LSST showing a fabrication induced (a) tip-tilt of the tertiary mirror axis with respect to the primary mirror optical axis and (b) decenter of the tertiary mirror axis with respect to the primary mirror optical axis Figure 7-10: Schematic drawing of the LSST showing a pure (a) tip-tilt of the secondary mirror and (b) decenter of the secondary mirror Figure 7-11: (a) Node sensitivity for third order coma and (b) node midpoint sensitivity for third order astigmatism and (c) node sensitivity for fifth order coma Figure 7-1: Schematic drawing of the field shift and aberration field vectors for third and fifth order coma and third order astigmatism Figure 7-13: Schematic drawing of field shift and aberration field vectors for third and fifth order coma and third order astigmatism with a slightly misplaced detector Figure 7-14: Schematic drawing of the relative aberration field vectors referenced from the fifth order coma node to the third order coma node and the midpoint between the astigmatic nodes. Since only relative quantities are utilized a displaced detector does not affect the accuracy in the prediction of misalignment parameters Figure 7-15: Full-Field-Displays for Zernike coefficients (a) Z5/6 (b) Z7/8 and (c) Z14/15 for the case of secondary mirror misalignments given by XDE = mm YDE = mm ADE = deg and BDE = deg Figure B-1: Concept of vector multiplication xix

20 LIST OF TABLES Table 3-1: Wave aberration coefficients for spherical aberration (third order) of a two-mirror telescope with the aperture stop located on the primary mirror Table 3-: Wave aberration coefficients for coma (third order) of a two-mirror telescope with the aperture stop located on the primary mirror Table 3-3: Wave aberration coefficients for astigmatism (third order) of a two-mirror telescope with the aperture stop located on the primary mirror Table 3-4: Wave aberration coefficients for the medial focal surface (third order) of a twomirror telescope with the aperture stop located on the primary mirror Table 3-5: Wave aberration coefficients for distortion (third order) of a two-mirror telescope with the aperture stop located on the primary mirror Table 4-1: The two important cases for zero third order astigmatism in a misaligned TMA. The value of H determines the location in the field of view where the third order misalignment induced astigmatism is zero. The value of xx B is proportional to the residual astigmatic figure error in the three mirrors that comprise the TMA telescope Table 4-: Wave aberration contributions for coma and astigmatism of the nominal JWSTlike telescope computed at a field angle of 0.13 deg at a wavelength of 1 μm Table 6-1: Decenter and tilt perturbations of the initially misaligned Ritchey-Chrétien telescope Table 6-: Decenter and tip-tilt misalignments after compensating misalignment induced coma Table 7-1: Definition of symbols utilized in Equations (7-1) - (7-1) Table 7-: Wave aberration coefficients for third order coma fifth order coma and third order astigmatism Table C-1: First order quantities of two-mirror telescopes Table D-1: Seidel coefficients for the spherical base sphere contributions (left column) and contributions originating from an aspheric departure (right column) (Shack 005) Table D-: Conversion of Seidel coefficients (spherical and aspheric) to wave aberrations Table D-3: Wave aberration coefficients for spherical aberration (third order) of a twomirror telescope with the aperture stop located on the secondary mirror Table D-4: Wave aberration coefficients for coma (third order) of a two-mirror telescope with the aperture stop located on the secondary mirror Table D-5: Wave aberration coefficients for astigmatism (third order) of a two-mirror telescope with the aperture stop located on the secondary mirror Table D-6: Wave aberration coefficients for the medial focal surface (third order) of a twomirror telescope with the aperture stop located on the secondary mirror Table D-7: Wave aberration coefficients for distortion (third order) of a two-mirror telescope with the aperture stop located on the secondary mirror Table D-8: Optical prescription of the Ritchey-Chrétien telescope (EPD = 3000 mm)

21 Table D-9: Wave aberrations (for a nominal wavelength of 63.8 nm) computed based on the analytical equations provided in Table Table 3-5 assuming a location of the aperture stop on the primary mirror Table D-10: Wave aberrations (for a nominal wavelength of 63.8 nm) computed in CODE V assuming a location of the aperture stop on the primary mirror Table D-11: Wave aberrations (for a nominal wavelength of 63.8 nm) computed based on the analytical equations provided in Table D-3 - Table D-7 assuming a location of the aperture stop on the secondary mirror Table D-1: Wave aberrations (for a nominal wavelength of 63.8 nm) computed in CODE V assuming a location of the aperture stop on the secondary mirror xxi

22 CHAPTER 1: INTRODUCTION Chapter One begins with a historical perspective of reflective astronomical telescopes to provide a fundamental understanding of past and future developments. Following this history alignment methods for astronomical telescopes are reviewed. Finally our motivation for the research and a research summary is presented Historical Perspective of Reflective Astronomical Telescopes The history of reflective telescopes goes back as far as 1609 to the time of Galileo who already had an understanding of basic imaging properties. It was this insight which led him to conclude already in the very early stages of refractive telescopes that the convex lens objective could be replaced by a correspondingly shaped concave mirror (Wilson 1996). One of the very first reflective telescopes was constructed by Zucchi in 1616 (King 1979) where he utilized a concave mirror out of bronze in combination with a negative powered Galilean eyepiece. Because only one mirror was used the telescope had to be used as a frontview telescope a form which later became well known through William Herschel. To achieve clearance between the observers head behind the eyepiece and the concave mirror the latter had to be tilted consequently causing lower order field aberrations. In 1630 the French mathematician Descartes invented analytical geometry which provided the theoretical framework for a comprehensive understanding of mirror shapes to obtain perfect axial imagery for infinite and finite conjugates. He recognized the importance of 1

23 conic sections (curves of nd degree) and Cartesian ovals (curves of fourth degree) and its potential in obtaining stigmatic imagery of a point (Wilson 1996). The results of analytic geometry were also well known to Mersenne who was in close contact with Descartes. Mersenne has to be credited with the theoretical invention of the compound two-mirror telescope. Being heavily discouraged by other contemporaries especially Descartes Mersenne actually never tried to build one of the telescopes which he proposed. But given the manufacturing capabilities at the time the required strong aspheric secondary mirrors would have likely been infeasible from a fabrication point of view. The potential of Mersenne s ideas including the possibility of having a strong telephoto effect in a two-mirror telescope remained unrecognized until 1663 when Gregory proposed what is known as the Gregorian telescope today. Given the limited manufacturing techniques at the time the challenges in manufacturing the steep aspheric forms for the secondary mirror could not be overcome for several decades (King 1979). Similarly the theoretical invention of the Cassegrain form by Cassegrain in 167 could not be realized in practice for ~100 years mainly because of the complexity in manufacturing the convex hyperbolic secondary mirror (Wilson 1996). Newton recognized the difficulty in the manufacturing of the proposed two-mirror telescopes utilizing two powered surfaces at the time and focused his interest on a two-mirror design which had a plane secondary mirror. In 1668 he finished his first telescope which had an aperture size of 1.3 inches (primary mirror) and a system focal ratio of f/5 (Wilson 1996 Watson 005). Given by the small tilted plane mirror the primary mirror obscuration could be kept at a minimum while greatly facilitating its assembly compared to telescopes with two powered

24 mirrors. Keeping the aperture size of the primary mirror small enabled Newton to use a spherical primary mirror because of the small departure compared to a paraboloid at the corresponding f-numbers. In 171 the Hardley brothers John Henry and George succeeded in the manufacturing of parabolic primary mirrors. This was the foundation for increasingly larger mirrors in the subsequent 150 years which enabled the construction of Newtonian telescopes of ever increasing size (Wilson 1996). It was not until the developments made by the optician Short advanced the precision in the manufacturing of aspheric concave mirrors that telescope designs utilizing two powered mirrors became practical (~1734). Most of the two-mirror telescopes built by James Short at the time were of the Gregorian or Newtonian type because of the reduced complexity in the manufacturing of concave mirrors compared to convex Cassegrain secondary mirrors (Wilson 1996). While small aperture Gregorian telescopes became feasible with the work of Short essentially all large aperture telescopes remained of the Herschel type especially because of less stringent requirements on the mechanical structures to mount the optics which would have been required for alignment purposes otherwise (~177). William Herschel was successful in completing a 0-foot (f/1) Herschel telescope in 1784 corresponding to a primary mirror aperture size of 18.8 inches. Shortly after Herschel already started the construction of a 40-foot focus (f/10) telescope with an aperture size of 4 feet (1789). The high copper concentration required to guarantee the mechanical stability of the large mirrors caused rapid reflectivity degradation and the considerable weight of the mirrors caused mechanical problems which 3

25 became the limiting factor preventing the construction of even larger telescopes at the time (Todd 1901 Wilson 1996). In 1835 Grubb succeeded in manufacturing the first (convex) Cassegrain secondary mirror which led to the 15 inch Newtonian/Cassegrain telescope for the Armagh Observatory in Northern Ireland (King 1979). In ~1845 James Nasmyth a Scottish engineer and inventor developed what is known today as the Nasmyth telescope (King 1979). The Nasmyth telescope consists of a Cassegrain telescope with an additional flat mirror mounted at 45 degrees to deflect the light along the altitude axis which does not affect the aberrations of the telescope but has significant mechanical advantages. The light exits from a hole in the middle of the altitude bearing and consequently rotations about the altitude axis do not require the repositioning of instrumentation it merely requires a derotation of the field which can be accomplished with an adapter-rotator. The advantages of the Nasmyth configuration are particularly attractive with the use of heavy instrumentation as spectrographs since no re-positioning of the instruments is necessary which could compromise the mechanical balance of the telescope or lead to varying stresses on the instruments otherwise. This configuration of a Cassegrain telescope with a Nasmyth focus is used in many of the Very Large Telescopes (VLT s) today. Despite the overwhelming progress in the production of mirror blanks and in the grinding/polishing of the mirrors to the desired mirror figure a serious challenge was the material of the mirrors itself. Until ~1850 the primary mirrors were fabricated using speculum metal an alloy consisting of ~60-70% Cu and 30-40% Sn. While the initial reflectivity was 4

26 ~60% the effort of having to re-polish the surface frequently to compensate for the reflectivity degradation over time was considerable (Wilson 1996). Based on a process to deposit thin silver films on a glass substrate developed by Liebig Foucault and von Steinheil produced silver coated mirrors with higher and more long-lived reflectivity reducing the need for re-polishing the optics (1857). Having the capability of providing silver coatings to increase the mirror reflectivity while also reducing premature reflectivity degradation was a crucial prerequisite for the success of reflective telescopes from then on (Watson 005). Between 1850 and ~1900 a notable improvement in manufacturing techniques combined with the development of testing methods like the Knife-Edge Test by Focault were enabling factors in the development of primary mirrors with ever decreasing f/#. One example of these fast primary reflectors was the f/5.8 Crossley reflector (36 inch) remounted at the Lick Observatory in 1900 (Wilson 1996). After ~70 years since the invention of analytic geometry by Decartes Schwarzschild extended the aberration theory to include field aberrations in reflecting telescopes (1905). With the aim of obtaining a field of several degrees he developed an aplanatic and anastigmatic telescope form both which suffered from large obscurations and were less practical because of their long telescope tubes compared to the classical two-mirror telescopes. Nevertheless his theoretical contributions were of fundamental importance for future telescope designs (Schwarzschild 1905a 1905b). Motivated by the goal of nebular photography and the required decrease in the primary focal ratios Ritchey successfully developed a 3 1/ inch reflector working at f/3.9 which was 5

27 faster than any mirror fabricated before. The reflector was used in a Newton configuration for photography and in the Cassegrain configuration for spectroscopy. Ritchey experimented with stopping down the primary mirror aperture and recognized the cubic dependence of third order coma with aperture height which eventually led him to the discovery of the aplanatic Ritchey- Chrétien form. In 1908 Ritchey completed the 60 inch Reflector on Mt. Wilson taking advantage of several new concepts for the testing of the optics as well as the astatic support concept previously introduced by Lassell. The 60 inch telescope was equipped with Newton Cassegrain and Coude focus making it consequently one of the most versatile telescopes at the time. The 60 inch telescope together with the 100 inch Mt. Wilson telescope (also referred to as the Hooker telescope) completed in 1917 facilitated some of the most remarkable discoveries in the 0th century (Wilson 1996). In the following years after the completion of the 100 inch telescope several somewhat smaller telescopes followed with only minor advances compared to Ritchey s previous work with one exception namely the use of low expansion glass (e.g. Pyrex) for the primary mirrors therefore annihilating some of the thermal deficiencies in previously used plate glass (Wilson 1996). In 196 Couder revisited previous work by Schwarzschild and proposed a modified version of the Scharzschild telescope by allowing non-zero field curvature. This two-mirror design had the main advantage of a considerably reduced axial obstruction ratio compared to Schwarzschild s original design. One drawback was the long length of the telescope and the poor stray light characteristics which led to the rare use of this two-mirror configuration even though 6

28 it had the best imaging potential across a wide field based on its state of correction (Couder 196). Inspired by his previous observations of the coma dependence on field and aperture size Ritchey completed the first aplanatic (corrected for third order coma across the full field) telescope with an aperture size of 0.5 m in 197 with further aplanatic Ritchey-Chrétien telescopes of increased aperture sizes in subsequent years (King 1979). A special two-mirror telescope design form comprising a spherical secondary mirror of the Cassegrain type appears to have been developed in the 1930 s by Horace Dall of Luton England. The advantages of this form also referred to as Dall-Kirkham is clearly the considerably reduced complexity of the spherical secondary mirror at the cost of severe field limitations set by field-linear coma (Manly 1991). Starting in ~1934 Hale was trying to build an even larger telescope (00 inch) on Mt. Palomar following a proposal by Pease to use a light weighted blank for the primary mirror reducing the weight by ~x. Also revolutionary was the introduction of aluminum for the reflective coatings enhancing the durability of the coating. Even though Hale died in 1939 and further delays were caused by the Second World War the 00 inch Palomar telescope was eventually completed in late 1947 (King 1979). In the post era of the Second World War a 6 m aperture telescope was built in the USSR at Mt. Pashukov Zelenchuck having a non-light weighted parabolic primary mirror working at f/4. Even though the telescope suffered from problems of thermal inertia caused by the huge mirror mass combined with dome seeing effects and limited seeing conditions for the particular site the telescope was ground breaking concerning the mechanics. Instead of using an equatorial 7

29 mount as preferred in the West at that time the telescope was equipped with an Altitude/Azimuth (Alt-Az) mount providing advantages of gravitational symmetry (Wilson 1996). Based on the increasingly used photo multipliers/image-intensifiers tubes at the time instead of the conventionally used photographic plates Bowen s proposal concerning desired focal ratios in 1967 led to a whole generation of telescopes of the Bowen type. Based on his analysis he proposed the use of focal ratios of f/3 f/8 f/30 at the prime Cassegrain and Coude foci respectively (Wilson 1996). Many two-mirror telescopes have been built after 1980 in aperture sizes of ~4 m across the world with some even up to an aperture size (diameter) of 10 m (1990 s). While significant developments were made in numerous areas for example electronic detectors wavefront sensing guiding active and adaptive optics in the segmentation and positioning of the mirrors etc. many of the fundamental concepts found in very large telescopes already completed or under construction were established based on previous discoveries. With major developments in the manufacturing of reflective coatings the development of three and more mirror telescopes became feasible and started in the 1930 s-1940 s with pioneering work done by Paul 1935 and Baker 1969 motivated by the goal to minimize field aberrations. Numerous optical designs for three and more mirror concepts have evolved since then with extensive contributions from Buchroeder 1969 Cook Korsch 1991 Kutter 1953 Robb 1978 Shafer Wilson 1996 and others too numerous to be included here. A good summary of three and more mirror telescopes can be found in Wilson 1996 and Gross et al

30 1.. Alignment Methods for Astronomical Telescopes Rimmer 1990 described a computer aided optical alignment method which utilizes measured interferograms combined with a lens model of the optical system. Using differential raytracing the lens model is perturbed to approximate the measured interferograms thereby providing an estimate of the misalignment parameters. Since the wavefront corresponding to a particular field point can exhibit good correction while different field points exhibit large aberrations Rimmer suggested the use of several field points for each of which an interferogram has to be provided. Lundgren et al presented an alignment strategy based upon ray aberration measurements and reverse optimization and applied it to three-mirror on-axis telescopes. Hartmann measurements at multiple field points and several focal planes were performed and by misaligning the nominal system in ACCOS the ray aberration data was reproduced with a leastsquares iterative process. Knowing the tilts and decenters which showed the same ray aberrations provided estimates for alignment (opposite tilts and decenters). McLeod 1996 reported a technique which minimizes coma and astigmatism across the field applicable to two-mirror telescopes. The alignment process described is a two stage process. In the first stage coma was eliminated by tilting or decentering the secondary mirror to center the obscuration shadow in an on-axis star image. In the second stage a technique based on off-axis measurements was used which allowed for the minimization of alignment induced astigmatism. The required tilts to make the primary and secondary mirror optical axis collinear were determined by a least-squares fit using position angles and axis ratios obtained from defocused star images (at several field points). 9

31 Chapman et al developed a method for compensation selection and alignment of optical systems based on SVD (singular value decomposition). The main idea of their method was to find compensators which influence the system aberration field in the most independent or orthogonal way. The described method allows for the smallest number of compensators to achieve a defined optical performance. Consequently the more stringent the performance requirements are (e.g. RMS wavefront error) the more compensators have to be utilized. Chapman and Sweeney introduced the aberration vector that contains all considered Zernike coefficients evaluated at several different field points. They showed that the best alignment state corresponds to the minimum of the norm of the aberration vector which depends on configurational parameters (tilts decenters element spacings). Noethe et al. 000 derived analytical expressions for the field dependence of astigmatism in the case of decentered two-mirror telescopes based on previous expressions given by Wilson The derived expressions describe the field dependence of astigmatism dependent on all characteristic design parameters based on earlier work of McLeod 1996 and Schroeder Noethe et al. derived special forms from the general results which enabled the treatment of afocal telescopes. The theory was applied in the alignment effort of the European Southern Observatory s (ESO) Very Large Telescope (VLT) and results obtained from numerical simulations showed good agreement with experimental data. Rakich et al. 008 reported the use of field aberrations for the alignment of the Large Binocular Telescope (LBT). Explicitly he used expressions for misalignment induced aberrations derived based on an extension of the plate diagram (Burch 194). The derived expressions were utilized for the realignment of the prime focus cameras using out of focus star 10

32 images. Additionally he reported a strategy for the alignment of the LBT in the bent Gregorian- Focus mode. Similarly as for the case of the prime focus camera astigmatism at the edge of the field has been utilized for the alignment now obtained by Shack-Hartmann wavefront measurements Motivation In the past most classical telescope forms have been traditionally aligned based on the removal of on-axis coma. In the case of slow f-numbers (~f/15 or slower) and small field of view Cassegrain telescopes (without active optics) alignment strategies based upon the removal of onaxis coma can be sufficient in some cases. Even though Cassegrain telescopes are typically seen as less critical for alignment compared to configurations with better correction state (e.g. aplanatic or anastigmatic) as shown through this work even for Cassegrain telescopes some configurations exist where the sole removal of on-axis coma leads to notable image quality degradation caused by misalignment induced astigmatism. In most large telescopes today with steadily increasing aperture size and more demanding image quality requirements active optics systems are planned or are already utilized to continually adjust the primary mirror figure during operation to compensate for low spatial frequency figure errors. The required wavefront measurements for active optics corrections are typically done outside the science field of view to maximize the available field of view for observing and the wavefront inside the science field of view is computed based on the measured wavefront data. Because of the binodal field dependence of astigmatism in the presence of misalignments image degradation at large field angles (outside the science field of view) cannot 11

33 be neglected since misalignment induced astigmatism can result in incorrect predictions for active optics adjustments if not taken into account properly. With the trend to utilize faster focal ratios and the move to aplanatic or even anastigmatic design forms the requirements for alignment have become more and more demanding. In many cases the surfaces require a notable departure from their spherical base curve. The aspheric departure defines an aspheric vertex and surfaces have to be centered and aligned for the correct orientation of its optical axis 1 to fully leverage the improved performance of these telescopes. Consequently the alignment process typically has to encompass five degrees of freedom three decenters and two surface tilts assuming a rotationally symmetric surface compared to only three degrees of freedom in the case of a spherical surface. While the importance of using field aberrations to meet the increased requirements on the alignment accuracy has been recognized recently most existing expressions for misalignment induced aberrations cannot easily be extended to take into account details of particular optical systems for example field correctors or additional instruments. In fact extending expressions for misalignment induced aberrations as given by Wilson or Schroeder 1987 to threemirror telescopes causes major complications has not been reported in the literature yet and its practical value is commonly seen as questionable because of its complexity. The most promising approach reported in the literature with the capability to extend expressions for misalignment induced aberrations to three and more-mirror telescopes without major complications is an extension of the plate diagram proposed by Rakich 008. Even though an extension to multi-surface optical systems seems very feasible its use in the current form is 1 The optical axis of an aspheric surface is uniquely defined as the line connecting the center of curvature and the aspheric vertex. 1

34 limited to small off-axis angles (Simon 1978) and to third order aberrations. When analyzing optical systems with more than two-mirrors having the capability to analyze fifth and higher order aberrations becomes an important feature since higher than third order aberrations are typically not negligible particularly then when large fields are used. We propose an alternative approach for describing misalignment induced aberrations capable of overcoming the aforementioned shortcomings of other methods. Specifically we propose the utilization of a vectorized form of the traditionally used aberration theory also referred to as nodal aberration theory developed by Thompson Using nodal aberration theory to describe misalignment induced aberrations in astronomical telescopes is motivated by several significant and unique advantages compared to existent approaches on the subject which will be summarized as follows. As demonstrated in Thompson 005 the well known wave aberration expansion by Hopkins 1950 can be extended to optical systems without rotational symmetry based on a single new concept the concept of shifted aberration field centers which will be explained in detail in Chapter. Consequently nodal aberration theory provides a unified theory for all nominal and misalignment induced aberrations based on a single new concept. A crucial advantage of describing misalignment induced aberrations with nodal aberration theory is its applicability to optical systems with an arbitrary number of surfaces. In contrast to other methods reported in the literature nodal aberration theory provides an aberration model that yields to obtain insight into the aberration response to misalignments. Also the complexity is not notably increased while providing a surface by surface sensitivity to misalignments. 13

35 Thompson has developed the multinodal properties of all aberrations through fifth order based on the same concept of the shifted aberration field centers as in the case of third order aberrations. While fifth order nodal expressions become more complicated than expressions for third order aberrations considering the nodal properties remains an insightful strategy and provides valuable information when developing an alignment plan as shown for the particular case of the LSST in Chapter 7 (Schmid 010e 010f). Moving to a multinodal theory of aberrations becomes most important when considering both the design and alignment of the new emerging class of three-mirror anastigmatic telescopes which are corrected for spherical aberration coma and astigmatism to third order. Nodal aberration theory facilitates an understanding of these more complex higher performance telescope systems to misalignments. Given its potential to be applied to optical systems with an arbitrary number of surfaces and its capability of including fifth (and higher order) aberration fields nodal aberration theory is currently seen as the most promising candidate for the development of a comprehensive and extensible theory of misalignment induced aberrations and is utilized and extended in this research Research Summary A limitation of the previously used first order calculations for locating the aberration field centers has been the dependency on how tilts and decenters in optical design software are modeled. Specifically only one decenter/tilt type (of ~ 5 typically available in lens design software) was allowed in the Local Coordinate System Paraxial Raytrace as detailed in Thompson Schmid and Rolland 009. To overcome this limitation and to facilitate the 14

36 integration of the first order calculations into commercial optical design software a real-ray equivalent formulation to locate the aberration field centers has been developed and implemented in CODE V s macro language. Using the concept of aberration field centers a deterministic alignment strategy for twomirror astronomical telescopes has been developed. It has been shown how misalignments of the secondary (or primary) mirror can be found through the evaluation of coma and astigmatism across the field of view. Explicitly it has been shown that the location of the nodes (coma and astigmatism) provides all the information required for the extraction of misalignment parameters in the absence of figure errors (Schmid 010d). By analyzing two-mirror telescopes a crucial discovery has been made concerning the astigmatic node locations after the telescope has been collimated for zero coma. It has been found that one of the astigmatic nodes remains at (Cassegrain and Gregorian telescope) or near (Ritchey-Chrétien telescope) the field center (i.e. the location in the image plane corresponding to the axial field point). This finding is important since it allows two immediate conclusions (Schmid 010a). First measuring astigmatism exclusively on-axis does not provide any meaningful insight into the alignment state of the telescope since one astigmatic node (zero astigmatism) remains on-axis in the absence of coma regardless of existing misalignments. This property emphasizes that astigmatism has to be measured off-axis close to the edge of the field of view to reveal residual misalignments. Secondly the same property suggests that figure induced astigmatism should be measured on-axis since alignment induced astigmatism remains zero. It has been found that 15

37 astigmatic figure error (at a pupil) affects the astigmatic node separation but importantly it does not affect the location of the midpoint between the astigmatic nodes. Based on these findings an algorithm has been developed to compute secondary mirror misalignments and figure error parameters of the primary mirror of any two-mirror telescopes (Schmid 010c 010d). The analysis of a three-mirror anastigmatic telescope (James Webb-Space Telescope) revealed similar nodal behavior for astigmatism. While astigmatism is not binodal as in the aplanatic case in the presence of misalignments the sole astigmatic node remains typically at or close to the field center if the telescope has been collimated for zero coma. Derivations based on nodal aberration theory also confirmed theoretically that this finding is a typical property of anastigmatic telescopes which can be leveraged for the determination of astigmatic figure error as mentioned in the case of two-mirror telescopes. It also could be shown that misaligned anastigmatic telescopes have two main aberration responses to misalignments field-constant coma and field-asymmetric field-linear astigmatism. The final part of this research investigated the nodal properties of wide-angle telescopes with the specific example of the Large Synoptic Survey Telescope (LSST). With the objective to accelerate the commissioning phase of the telescope a deterministic alignment strategy has been developed in corporation with LSST cooperation. For the first time an alignment plan has been devised that is entirely independent of existing boresight errors therefore considerably facilitating a practical implementation of the approach. This has been accomplished by including higher order field aberrations and their nodal response to misalignments. 16

38 1.5. Dissertation Outline The dissertation is organized as follows: Chapter reviews the previous first order calculations to locate the aberration field centers and details the newly developed equivalent real-raytrace based approach. A summary of nodal properties of third order aberrations is provided to the extent required for the discussion of misalignment aberration fields in astronomical telescopes and additional graphical interpretations are given facilitating a more intuitive and conceptual understanding. Chapter 3 discusses the effects of misalignments on the aberration fields in the case of astronomical telescopes and demonstrates how the misalignment state of classical two-mirror telescopes can be computed. It is demonstrated how nodal aberration theory can be utilized to reveal fundamental properties of these telescopes in response to misalignments the understanding of which is crucial in the alignment process (and in the design) of these systems. Chapter 4 details misalignment aberration fields in three-mirror anastigmatic astronomical telescopes and shows that these telescopes have two main aberration characteristics in the presence of misalignments field-constant coma and field-linear field-asymmetric astigmatism. The theory describing the characteristic nodal behavior of all anastigmatic optical systems is discussed and applied to the example of a James-Webb-like Space Telescope. Chapter 5 demonstrates that the expressions developed for describing misalignment induced aberrations in Chapter 3 and 4 are also well suited when analyzing the optical performance of misaligned telescopes including diffraction effects. It is also demonstrated how one can reconstruct the aberration fields based on wavefront measurements at several field 17

39 points. The influence of finite measurement accuracy on the prediction of misalignments is discussed. Chapter 6 discusses the nodal behavior in the presence of primary mirror astigmatic figure errors and misalignments. It is demonstrated how the two effects can be distinguished an important prerequisite for facilitating adequate mirror alignment and mirror figure adjustments. Chapter 7 describes how a multinodal theory of aberrations can be used in the development of alignment strategies for multi-mirror wide-angle telescopes. The theory is applied to the Large Synoptic Survey Telescope (LSST) the largest survey telescope project in history with the aim of developing an alignment plan for the initial commissioning of the telescope without its camera. Chapter 8 presents conclusions and a summary of our contributions as well as future work. 18

40 CHAPTER : NODAL ABERRATION THEORY FOR OPTICAL SYSTEMS WITHOUT SYMMETRY In Section.1 fundamental concepts in the context of nodal aberration theory are summarized. The local coordinate system paraxial raytrace for the tracing of the optical axis ray is summarized in Section.. In Section.3 it is demonstrated how the aberration field centers can be computed based on the quantities of the optical axis ray and a paraxial raytrace of the nominal optical system. A real-ray equivalent formulation for the computation of the aberration field centers is demonstrated in Section.4 facilitating the implementation of the concepts in optical design software where several decenter types are commonly used. Section.5 concludes the chapter by integrating the concept of shifted aberration field centers in a vectorized form of the wave aberration expansion and an intuitive discussion of the nodal behavior of all third order image degrading aberrations is given. The discussion deliberately excludes distortion since it does not degrade image quality but changes its position and is typically accounted for by calibration methods for the optical systems considered in this work..1 Concepts and Definitions It is well known that in the theory of third order aberrations of rotationally symmetric optical systems (with rotationally symmetric optical surfaces) the total aberrations can be described as the summation over all individual surface-by-surface contributions. This is a special case in the aberration theory of n th -order since for higher than third order aberrations additional concepts 19

41 have to be introduced such as the concept of intrinsic and transferred aberrations. What is less known is that the field dependence of all of the third order image quality degrading aberrations of an individual spherical optical surface (i.e. spherical aberration coma astigmatism and field curvature) is rotationally symmetric about a common axis for each surface. For a spherical surface this common axis is the line that connects the center of curvature of the surface and the center of the local entrance-pupil (i.e. the image of the optical system aperture stop in the local object space) (Buchroeder 1976). In optical systems exhibiting rotational symmetry this line coincides with the optical axis 3 and the importance of this fact as a concept in itself may seem dispensable at first sight. In a tilted and decentered optical system either by design or caused by component misalignments the third order aberrations continue to be the sum of the individual surface contributions. However now these surface contributions must be added together accounting for the fact that each is characterized by the rotationally symmetric characteristic field dependence for that term that is rotationally symmetric about a different point in the Gaussian image plane for each tilted and decentered surface (Thompson 005). In the absence of system symmetry the concept of the local symmetry axis defined by the line that connects the center of curvature and the center of the local entrance-pupil for the particular surface becomes fundamentally important. For each spherical surface the point of symmetry in the Gaussian image plane can be determined in the local space defined by the particular surface its local object and entrance- Intrinsic aberration contributions are aberrations that are caused exclusively by a particular surface. This would be the case if an aberration free wavefront is incident on a surface. When considering aberrations of order higher than third (in transverse aberrations) the propagated and aberrated wavefront will generate additional aberration contributions on subsequent surfaces (beyond the intrinsic components) that are typically referred to as transferred or induced aberrations. 3 The optical axis defines the axis of rotational symmetry in rotationally symmetric optical systems and contains all aspheric vertices and centers of curvatures of all surfaces. 0

42 pupil. This is possible through proper normalization as discussed in sections to follow. The point in the Gaussian image plane about which the surface aberration fields are symmetric is found as the intersection point of the individual surface axis with the local object plane. For subsequent discussions of aberrations in optical systems without rotational symmetry the concept of the Optical Axis Ray (OAR) becomes important. In the case of an aligned rotationally symmetric optical system the OAR is defined as the ray that connects the center of the object with the center of the image and passes through the center of the aperture stop while at the same time intersects all centers of curvature and vertices of the individual surfaces and aspherics. In the specific case of optical systems exhibiting rotational symmetry the OAR coincides with the optical axis. In a misaligned or generally nonsymmetrical optical system made of otherwise rotationally symmetric optical surfaces the OAR is defined as the ray that connects the center of the object with the center of the circular physical aperture stop within the optical system. This could be a paraxial ray computed using the local coordinate system (LCS) paraxial equations of Buchroeder 1976 or a real-ray computed using Snell s law as in any commercial optical design software package (typically called a principal ray for the on-axis field point). It is worth noting that while the center of the circular physical aperture stop of the optical system is a completely defined point in space the center of the object field is arbitrary and can be selected at the designer s discretion. The intersection point of the OAR with the image surface determines the image plane displacement (if any) typically referred to as boresight or pointing error relative to the mechanical axis. This intersection point with the image plane also serves as the reference point from which the aberration field offsets of each individual surface contribution are defined. 1

43 The surface shapes in optical systems frequently comprise surfaces that exhibit an aspheric departure from their underlying spherical base curve typically expressed as a power series expansion. Prior to Chapter 6 where we specifically include astigmatic figure errors in the analysis we limit the discussion of optical systems to ones where the aspheric departures are rotationally symmetric or can be described as portion of a rotationally symmetric aspheric departure of a parent surface. A fundamental concept in nodal aberration theory (third order) is the decomposition of the surface wave aberration contributions into two separate contributions each one associated with the spherical base curve the other determined by the aspheric departure (if any) from the spherical base curve. For conceptual and practical purposes an aspheric surface can then be represented mathematically by two superimposed optical surfaces each with their own family of aberrations one for the spherical base curve surface contribution and one for the aspheric departure contribution. As with the spherical base curve surface contribution the aberration fields associated with the aspheric sag of a particular surface are all rotationally symmetric about the same point in the image plane. However significantly in a tilted and decentered optical system the point in the image plane that the aberration fields of the aspheric contribution of the surface are rotationally symmetric about is typically different from the point in the image plane about which the spherical surface components are rotationally symmetric. Treating these as the individual spherical surface aberration field contributions and the individual aspheric cap aberration field contributions provides the best working concept. For a spherical surface the relevant optical characteristic point is the location of the center of curvature of the surface. This is a physical uniquely defined point in space. It is useful to not only characterize a spherical optical surface by this point but to also associate a local optical

44 axis with the surface. A second point to be used in constructing this axis is the vertex of the surface. The vertex of an individual spherical optical surface like the center of the object field can be defined in many ways at the discretion of the optical designer. To facilitate the treatment of aspheric surfaces a convenient definition for the vertex of a spherical optical surface which will be adopted here is the intersection point of the OAR with the spherical surface. With this definition the decenter of an aspheric optical surface is the offset between the vertex defined by the OAR and the vertex defined by the center of symmetry for the rotationally symmetric aspheric departure from the spherical surface. For the aspheric surface departure from the spherical base curve sometimes also referred to as aspheric cap the physically relevant characteristic point is the surface vertex about which the departure from the spherical surface is rotationally symmetric. For the aspheric cap the location of the point of rotational symmetry for its aberration field contributions in the Gaussian image plane is found by a line intersection with the local object plane where the line is constructed using the aspheric vertex of the individual surface and the center of the local entrance-pupil of the surface. In general then when the aspheric vertex is displaced from the intersection point of the OAR with the surface the aberration field contribution originating from the aspheric departure of the surface is displaced in the image plane from the aberration contribution originating from the spherical base sphere.. Local Coordinate System Paraxial Raytrace To provide a concise reference and comparison point for the real-ray based methods that will be developed for the first time in Section.4 the LCS paraxial raytrace equations for optical 3

45 systems with tilted and decentered surfaces as developed by Buchroeder 1976 will be summarized. The purpose of Buchroeder s work was to parameterize the line that connects the center of the local entrance-pupil for the surface to the center of curvature of that surface within the approximations used in paraxial raytracing. The process involves primarily a series of surface local coordinate transformations. The intersection point of the line defined by the center of curvature of the surface and the center of the entrance-pupil with the local object space specifies the center of rotational symmetry (referred hereafter as the aberration field center) of the field dependence for all of the aberration fields associated with that spherical optical surface relative to the OAR (aspheric surfaces will be treated separately). In principle this derivation can be done in either object or image space but here the derivation in local object space has been chosen...1 Definition of Surface Local Object/Image and Entrance/Exit-Pupil Planes The calculation of the surface aberration field centers requires a raytrace of the OAR through an optical system that contains tilted and decentered optical surfaces elements and/or groups specifically to determine the relative positions of the images of the physical aperture stop to the centers of curvature of the surface. While in the standard paraxial approximation surface decenters and tilts are neglected in Buchroeder s approach the tilts and decenters are retained. The overall strategy is to create a local coordinate system for each optical surface and to retain the tilts and/or decenters in determining the local coordinate axes. Then a traditional paraxial raytrace is conducted in the tilted and/or decentered local coordinate space. Upon transferring to the next surface the tilts and decenters are again applied to locate the next surface local 4

46 coordinate system and so on. This can be accomplished by considering each surface as a partial optical imaging system and recalling that the OAR (or its extension to the other side of the surface) by definition intersects the center of the object/image and entrance/exit-pupil. This representation allows associating an object plane an entrance-pupil an exit-pupil and an image plane with each surface as illustrated in Figure -1. All of the intermediate planes can be interpreted as relayed images of the object and the entrance-pupil (where the entrance-pupil is the image of a real physical circular limiting aperture) for a specific surface. Specifically the image of the object located at plane j produced by surface j becomes the object for surface j+1. Equivalently the exit-pupil formed by the entrance-pupil j and surface j serves as the entrancepupil for imaging on surface j+1. exit-pupil for surface j-1 entrance-pupil for surface j exit-pupil for surface j entrance-pupil for surface j+1 n j-1 n j c = 1 r OAR F F +y object plane for surface j image plane for surface j-1 refracting or reflecting surface j image plane for surface j object plane for surface j+1 +z Figure -1: Illustration of the concept of local object and pupil for each surface in an optical system. #.. Definition of an Equivalent Tilt Parameter for a Spherical Surface: β 0 To begin note that the paraxial and aberrational optical properties of any tilted and/or decentered spherical optical surface can be fully described by the location of the center of curvature for that particular surface. In other words the vertex of a spherical surface is not a relevant optical 5

47 parameter (however it is relevant for an aspheric surface as discussed later in Section.3.). This property of spherical surfaces leads to the definition of an equivalent tilt parameter for a spherical surface as illustrated in Figure - which locates the center of curvature based on a # single parameter β 0 measured from the mechanical coordinate axis (MCA) which is coincident with the OAR in a rotationally symmetric optical system (i.e. one without tilts and/or decenters). The MCA is a mechanical reference axis that represents a fixed reference line from which the misalignments (e.g. decenters and tilts) to the optical surfaces denoted here by # (measured from MCA) and β (measured from MCA ) are applied. In Figure - MCA is a reference axis that is parallel to the MCA intersecting the decentered spherical surface. Hereafter all quantities referring to one of the mechanical axes (MCA MCA etc.) are denoted with a pound sign (# # etc.). cc pert is the point that locates the center of curvature after the application of a decenter followed by a tilt to the surface which is also parameterized by # δ c # measured from the MCA. The equivalent tilt angle β 0 # # # # 0 +c v = c c. is then defined as β β δ δ (-1) # Note that β 0 is a vector with projected components in the x-z - and the y-z -planes. δ v #..3 Locating the Centers of the Local Objects/Images (Q Q ) and Entrance/Exit-Pupils (EE ) The first necessary step is to locate the centers of the local objects/images and the local entrance/exit-pupils which supports the overall tracing of the OAR that connects the user- defined center of the object field to the corresponding center of the image field (located at Q and 6

48 (Refracting Surface) j +y pert +z nn j-1 I 1 nom n j cc pert # δ v # β 0 # β # δ c MCA'(= #') cc nom MCA(= #) 1 r = ( > 0 ) c # Figure -: Definition of the equivalent tilt parameter β 0 for a spherical surface which defines the location of the perturbed center of curvature relative to the mechanical coordinate axis (MCA). Q respectively) and the center of the entrance-pupil (E) to the center of the exit-pupil (E ) # including all intermediate images. The local displacements of the object/image Q δ /( δ Q ) # ' and pupils # δε / ( # )' δε referenced to the MCA can be obtained from the geometry shown in Figure -3. To facilitate the coordinate transfer note the solid and then dashed line in image space (blue) constructed to join the local center of the object field (Q) the displaced center of curvature (cc pert ) and the local center of the image field (Q ). To aid in following the coordinate transformation note that one can draw a reference axis MCA (also blue) that is parallel to the original MCA but passes through the center of curvature of the perturbed surface cc pert. Then the point Q (the center of the image in the local space of the surface) can be computed using #'' #'' traditional paraxial equations as illustrated in the drawing using the quantities ( yo ) and ( O ) #'' #'' where ( y O )' is obtained by multiplying ( O ) y ' y with the paraxial magnification of the surface. 7

49 One can show that the location of the center of the local object/image (Equation (-)) and the entrance/exit-pupil (Equation (-3)) for a particular surface are given by (APPENDIX (A.1)) # ( δq ) ( yo )' O # ( δe ) ' # y ( n) δ Q = + y ( ye )' Ж ' # y ( n) δ E = y E Ж β β # 0 # 0 (-) (-3) where ( δ Q # / y O ) and # ( )' ( y O ) δq / ' locate the center of the object field and the center of the image field respectively normalized by the paraxial field size y O and ( y O )' (see the lower half of Figure -4). Similarly the centers of entrance- and exit-pupils are located at # ( )'/ ( y E ) δ E # / ye and δ E ' respectively where these locations are normalized by the corresponding pupil size y E and ( y E )'. To simplify the notation the surface subscript j has been omitted. In Equation (-) and Equation (-3) each quantity other than the optical invariant would have a j subscript. This practice of omitting the surface subscript will be continued for the remainder of this section. 8

50 MCA' '(= #'') #'' y 0 Object Q # ( δ Q ) n j-1 (Refracting Surface) j pert n j # 0 cc pert MCA''(= #) l r l Image β δc # ( ) cc nom δ Q # ' Q #'' ( y 0 ) MCA'(= #') MCA(= #) ' # # Figure -3: Illustrating object/image displacement δ Q and ( δq )' (and pupil # # displacement δε and ( δε )' by substituting the entrance/exit-pupil points (EE ) for the object/image points (QQ ))...4 Parameterizing the OAR With Equation (-) that locates the transverse displacements of the points that define the centers of the object (Q) and image fields (Q ) and Equation (-3) that locates the transverse displacements of the centers of the local entrance-pupils (E) and exit-pupils (E ) the path of the OAR through the system is established. It is useful to parameterize the OAR by the height of the # OAR intercept at the surface referenced to the MCA denoted as y OAR and the OAR inclination angle relative to the MCA denoted as u # OAR. These quantities as illustrated in Figure -4 are given by (APPENDIX (A.)) # δq# δe# u OAR = u + u y ye O (-4) # δq# δe# y OAR = y + y. y y O E (-5) 9

51 Object (real) Center of Object Q # δ Q OAR # u OAR # y OAR (Refracting Surface) j pert # β 0 cc pert. Entrance-Pupil (virtual) Center of Pupil E # δ E MCA Q n j-1 y n j Marginal-Ray u cc nom y E y O l u y r i s y cc Chief-Ray E MCA Figure -4: Perturbed optical system with OAR (top) and paraxial marginal and chief-ray of nominal optical system (bottom). Considering Figure -5 it is useful to define the equivalent tilt angle #'' β that locates the perturbed center of curvature corresponding to a particular spherical surface relative to MCA. Using the geometry of the local coordinate systems #'' β is given by # Together with the inclination angle of the OAR u OAR β #'' = β # y # OAR c. (-6) 0 #'' β facilitates the calculation of the key quantity that is related to the location of the center of curvature of the surface the angle of incidence of the OAR on the local surface of interest * i given by * # #'' i =u β OAR. (-7) 30

52 The parameterization of the OAR also allows developing an expression for the * displacement of the center of curvature ( δ c ) with respect to the extension of the OAR past the refracting surface (dashed red line) given by δc = ri = r( u β ) = r( u + y c β ). (-8) * * # #'' # # # OAR OAR OAR 0 Object (real) Center of Object Q # δ Q OAR n j-1 # u OAR (Refracting Surface) j # y OAR * i # β 0 cc pert #'' β n j * δ c MCA'' Entrance Pupil (virtual) # δ E +y Center of Pupil E +z r MCA Figure -5: Illustrating important OAR parameters for a tilted and decentered spherical surface..3 The Vector that Locates the Center of the Aberration Field Symmetry for a Tilted and Decentered Surface.3.1 Shifted Aberration Field Center Associated with the Spherical Base Curve ( sph) σ The center of symmetry for the field dependence of the aberration fields for an individual spherical surface is given by the intersection of the line connecting the center of curvature (perturbed) with the center of the local entrance-pupil (located on the OAR) when extended to ( sph) the local object plane (and then normalized). Figure -6 can now be used to establish σ which is the vector whose base is at the OAR and whose end point locates the aberration field center of symmetry for the field dependence of the aberrations associated with the surface in the local object/image. 31

53 * From Figure -5 the displacement of the center of curvature δ c is measured from the extension of the OAR beyond the refracting surface of the OAR (dashed red line passing through E). The AFA is then the line connecting the point E with the center of curvature in the perturbed * configuration located by δ c extended to intersect the local object plane associated with the * surface (Figure -6). For small perturbations the ratio of δ c and ycc can be utilized to express ( sph) σ the normalized vector locating the aberration field center for the specific surface given by (APPENDIX (A.3)) which also can be expressed as u β + y c = y u + yc * δc 0 (-9) ( sph) OAR OAR σ = cc # # # σ ( sph) i i *. (-10) * σ sph. y O y O # δq Q Object (real) u Chief-Ray # y OAR s n j-1 # u OAR (Refracting Surface) j Surface normal of unperturbed system i y OAR cc pert. cc nom +z E Figure -6: Concept of the shifted aberration field center for the spherical base curve evaluated in the local object space. * i r n j * δ c y cc AFA +y # δe MCA Entrance Pupil (virtual) 3

54 .3. Shifted Aberration Field Center Associated with the Aspheric Departure from the ( asph) Spherical Base Curve σ In the case of an aspheric surface two separate sigma vectors have to be calculated one originating from the spherical base curve (as described in Section.3.1) and the second one originating from the aspheric departure from the spherical surface. The departure of the aspheric surface from a spherical surface can be treated as a zero-power thin optical plate following concepts in plate theory introduced by Burch 194. This contribution to the net aberration fields is within third order approximations not dependent on the angle of incidence but exclusively on the intersection height of the OAR with respect to the aspheric vertex. Based on the geometry provided in Figure -7 the aspheric sigma vector ( asph) σ is given by (APPENDIX (A.4)) σ ( asph) = * δv. y (-11) Object (real) Center of Object Q y O ( σ ) asph OAR (Refracting Surface) j # y OAR * δv = YSC y # OAR Entrance Pupil (virtual) Center of Object E # δ E +y +z δ Q # n j-1 YSC n j Q y O u u y (a) y cc (b) Marginal-Ray Chief-Ray y E E MCA MCA Figure -7: (a) Concept of the shifted aberration field center for the contribution of the aspheric departure from the spherical surface and (b) paraxial raytrace of the nominal optical system in local space. 33

55 .4 Real-Ray Equivalent Approach to Find Individual Surface Aberration Field Centers While Equation (-10) and Equation (-11) and the accompanying figures (Figure -6 Figure -7) are perfectly correct and provide valuable insight into properties of perturbed optical systems the LCS paraxial equations of Buchroeder are not practical to implement in a multi-user optical design and analysis environment. The difficulty is found in the numerous conceptual methods that are provided to the optical designer for entering tilts and decenters into the model for an optical system. Specifically of the approximately five types of perturbation models that are currently available only one of these models would yield the correct form required for computing the individual aberrations field centers. Remembering the properties of a vector cross-product a real-ray based equation for computing the σ -vector for each surface j can be obtained. Here it is assumed that the ray propagation in the unperturbed optical system is in the z-direction which is consistent with most optical design software. Referring to Figure -8 for each surface j let N j be a unit vector normal to the local object plane that we assume perpendicular to the z-axis in an unperturbed system. Thus N j has direction cosines [0 0-1] R j corresponds to a unit vector along the incident ray (OAR) using L M N as normalized direction cosines and S j denotes a unit vector along the local surface normal using SRL SRM SRN as normalized direction cosines. The cross-product of R j and S j is then given by R ( ) ˆ ( ) ˆ j Sj = M SRN N SRM x + N SRL L SRN y + ( L SRM M SRL )ˆ z. (-1) 34

56 The σ -vectors for each surface are contained in the local Gaussian image plane. To impose this condition the projection of this cross-product in the x-y local Gaussian image plane is formed. The projection is created by taking the cross-product with N j which is expressed as N ( ) ( ) ˆ ( ) ˆ 0 ˆ j Rj Sj = N SRL L SRN x+ N SRM M SRN y+ z (-13) The expression provided in Equation (-13) provides a convenient means for the computation of the σ ( sph) j -vector. Alternatively the cross-product can be expressed as N [( R S )] = sin( i )( N yˆ) = sin( i ) xˆ (-14) * * j jx z jx z jx z j jx z N [( R S )] = sin( i )( N xˆ) = sin( i ) yˆ (-15) * * j jy z jy z jy z j jy z defining a vector i * j with components along the x- and y- directions with values sin( i * j x z ) and -vector for each surface j may be expressed as a cross- sin( i * j y z ) respectively. Then the product given by σ ( sph) j σ * i [ ( )] ( sph) j N j Rj Sj j = =. i i j j (-16) We have now established a compact expression for σ that is conceptually intuitive ( sph ) j and independent of the implementation of the tilts and decenters of the individual surfaces. The aberration field center of an individual surface is determined by the angle between the OAR and the surface normal. Equation (-16) greatly facilitates the implementation in a commercial lens design environment. This real-ray based approach can be readily extended to incorporate aspheric surfaces. The σ ( asph) j -vector for the aspheric surface contribution has been stated in Equation (-11) with 35

57 its corresponding illustration in Figure -7. Since the coordinates of the surface intersection of the OAR relative to the vertex of the aspheric component of the surface can be obtained by one real raytrace of the perturbed optical system and the paraxial chief-ray incidence angle and chief-ray height of the unperturbed optical system are also readily available the overall approach is no longer decenter type dependent. E Entrance Pupil (virtual) n j-1 (Refracting Surface) j n j # δ E Object (real) Nx(RxS) # u OAR #'' β cc pert S * i -R # y OAR RxS r MCA OAR N # δ Q Q Figure -8: Perturbed optical system with OAR parameters for a given surface j..5 Nodal Properties of Third Order Aberrations This section summarizes how the concept of the shifted aberration field centers can be combined with a vectorized form of the wave aberration expansion (Section.5.1). Subsequent subsections discuss the nodal behavior of third order spherical aberration (Section.5.) third order coma (Section.5.3) third order astigmatism (Section.5.4) and the medial focal surface (Section.5.5). The equations describing the nodal behavior of the aberrations are discussed in 36

58 detail in Thompson 005 and are summarized in Section.5. Nevertheless the main objective of this section is to provide more of an intuitive understanding of the nodal behavior of the aberrations which will be capitalized on in chapters to follow..5.1 Concept of Shifted Aberration Field Centers and Effective Field Height The net image plane aberration of rotationally symmetric optical systems consists of the sum of all the individual surface contributions. This is concisely described by the wave aberration expansion published in its original form by Hopkins 1950 given by k l m W = ( W ) H ρ cos φ (-17) j p n m klm where k = p + m and l = n + m W klm denotes the particular aberration type H denotes the normalized field height in the image plane and ρ denotes the normalized position in the exitpupil. As suggested by Shack Equation (-17) can be expressed in vector form given by j ( ) W N N N klm j H H H j p n m p n m = ( W )( )( ρ ρ)( ρ ) (-18) where H denotes a normalized vector for the field height in the image plane and ρ is a normalized vector describing the position in the pupil. In optical systems exhibiting rotational symmetry the normalized field height H is referenced from the axial field point (i.e. the point where the optical axis intersects the image plane). Both Equation (-17) and Equation (-18) are completely equivalent in the case of rotational symmetry. In the absence of system symmetry the third order aberration fields associated with the individual surface j third order spherical aberration coma astigmatism field curvature and 37

59 distortion remain all rotationally symmetric in the field about the point denoted by the σ -vector with constant linear quadratic quadratic and cubic field dependence respectively. In fact the field dependence of each of the aberration field types is now characterized by the displaced aberration field height vector H = H σ (for the aberration contribution originating ( sph) ( sph) Aj j from the spherical base curve) and H ( asph) = H σ ( asph) (for the aberration contribution Aj originating from the aspheric departure). The concept of the shifted aberration field centers and the effective field height is visualized in Figure -9 for a misaligned aspheric surface which j shows the normalized vectors H = H σ ( sph) ( sph) Aj j H = H σ H ( asph) ( asph) Aj j σ ( sph) j and σ ( asph) j. Note the aberration field centers σ and ( sph ) j σ are referenced with respect to the ( asph ) j OAR intercept with the Gaussian image plane which is generally different from the mechanical axis of the system. It is critical to realize that for an individual aspheric surface each of the associate field dependent aberration components associated with the spherical base curve remain coincident about the point located by the σ ( sph) j - vector and that each term retains its original field dependence. Equivalently for the field dependent aberration components associated with the aspheric departure each of the associate field dependent aberration components remain coincident about the point located by the σ ( asph) j - vector and each term retains its original field dependence. To mathematically take the displacements of the individual surface aberration field centers σ and ( sph ) j σ into account when considering the net composite aberration field at ( asph ) j the image plane the wave aberration expansion of Equation (-18) is re-written substituting the 38

60 individual surface aberration field vectors H σ and ( sph ) j H σ for the original surface ( asph ) j independent aberration field dependence vector H giving after converting to a vector notation (Thompson 005) W ( sph asph) ( sph asph) ( sph asph) ( ) ( ) ( ) ( ) p n sph asph m W klm j j j ( ) ( j ) j p n m H H H = σ σ ρ ρ σ ρ (-19) which can be simplified by substituting H = H σ in Equation (-19) given ( sph asph) ( sph asph) Aj j by W ( sph asph) ( sph asph) ( sph asph) ( ) ( ) p n sph asph m W klm j Aj Aj ( ) Aj j p n m H H H = ρ ρ ρ (-0) where H and ( sph ) Aj ( asph) H Aj denote the effective aberration field heights for the spherical base curve and the aspheric departure of surface j respectively measured from the shifted center of the aberrations fields associated with the spherical and separately aspheric contributions of a surface. This form of the wave aberration expansion first as vector formulation of the wave aberration equation of Hopkins 1950 and then including the discoveries of Buchroeder 1976 was ( ) developed by Shack The vectors σ sph and σ j ( asph) j originating at the intersection point of the OAR with the image plane denote the new aberration field centers for an individual aspheric surface j. 39

61 Exit-pupil Plane ρ y φ ρ aberration field center for the aspheric departure for surface j Aspheric Surface j ρ x OAR * H y ( asph ) Aj ( asph) σ j Η ( sph) Η Η Aj MCA Image Plane H y H x * ( sph) σ j z H x aberration field center for the spherical base curve contribution for surface j Figure -9: Concept of aberration field center in the image plane and effective field height..5. Third Order Spherical Aberration in Optical Systems Without Symmetry The first wave aberration to third order is referred to as spherical aberration. It has a quartic pupil dependence and is constant with field given by W SA3 W ( ) 040 j ρ ρ (-1) j = where W 040 j designates the wave aberration term for third order spherical aberration at surface j. Since the absence of system symmetry (assuming rotationally symmetric surfaces) only affects the field dependence of aberrations spherical aberration is not affected by transverse decenters or tilts and remains constant across the field. Even though transverse decenters or tilts have no effect on spherical aberration the effect of longitudinal decenters (in the context of telescopes also referred to as despace ) on spherical aberration becomes important. 40

62 Spherical Aberration ρ y ρ x Figure -10: Pupil dependence of third order spherical aberration..5.3 Third Order Coma in Optical Systems Without Symmetry In optical systems with rotational symmetry the wave aberration term with linear field and cubic pupil dependence is denoted as third order coma. The pupil dependence for third order coma is visualized in Figure -11 where field point that is under analysis. ρ is always rotated to be aligned with the orientation of the y Third order Coma ρ y ρ x Figure -11: Pupil dependence of third order coma. 41

63 In optical systems without symmetry the field dependence of coma develops characteristic nodal properties. An additional coma term that is constant with field is typically present in addition to the linear field dependence in the rotationally symmetric case. The general expression for third order coma in the presence of decenters and tilts is given by W = W [( H σ ) ρ]( ρ ρ) COMA3 131 j j j (-) or can be re-written using the definition of the un-normalized aberration field vector A 131 given by with W = [( W H A ) ρ]( ρ ρ ) (-3) COMA ( sph) ( sph) ( asph) ( asph) ( W j j W j j ) 131 = j A σ σ (-4) where A131 corresponds to the amount of field-constant coma in the case of non-zero aberration field centers. In the case of an optical system exhibiting rotational symmetry with non-zero coma in the nominal state (i.e. W131 0 ) a normalized expression can be found by introducing the normalized aberration field vector a 131 given by with where W and ( sph) 131 j ( asph) 131 j W = W [( H a ) ρ]( ρ ρ) COMA a 131 = A W (-5) (-6) W designate the wave aberration terms for third order coma for surface j ( sph) ( asph) ( ) associated with the spherical base curve ( W ) and aspheric departure ( W ) σ sph and 131 j j j

64 σ ( asph) j describe the normalized aberration field centers for surface j associated with the spherical base curve and aspheric departure respectively and a 131 denotes the net normalized aberration field vector for third order coma which denotes the location of the field point exhibiting zero coma also referred to as the coma node. In the most general case of a non-aplanatic optical system without symmetry the coma node is displaced from the axial to an off-axis field point while the linear field dependence itself remains. An example to visualize the modified field dependence is given in Figure -1 (left) where the magnitude of coma is plotted for a rotationally symmetric optical system (a) and for an optical system without symmetry (b). Because coma grows linearly with field it can be represented as conical shape with it s vertex located at the aberration field center. Depending on the size of coma for the particular surface contribution the steepness of the conical shape varies; a surface contributing large coma causes a steep conical shape and vice versa. In a rotationally symmetric optical system coma is always centered with respect to the axial field point and consequently its orientation (inward or outward) is fully described with the sign of the surface contribution. In the absence of symmetry the coma node shifts in the field denoted by the field vector a 131 (as stated in Equation (-6)) and consequently the orientation of coma across the field becomes important. The orientation of coma across the field is visualized in Figure -1 (right) for the case of (a) rotational symmetry and (b) without system symmetry using Full-Field-Displays (FFDs) generated in CODE V. It is important to recognize that while FFDs were developed to illustrate the nodal aberration field behavior is it simply a display method. The data that is being 43

65 displayed is based on real ray-tracing with no knowledge of nodal aberration theory. As a result this is an excellent validation of the theoretical developments presented here. 4 H y Magnitude of Coma H x H y (a) H x H y Magnitude of Coma a 131 H x a 131 H y (b) H x Figure -1: Magnitude (left) and orientation (right) of third order coma across the field for (a) a rotationally symmetric optical system and (b) an optical system without symmetry. To more intuitively understand that there remains just one node for coma regardless of how many surfaces the system under consideration might have the following construction for the node location has been created visualizing how the coma node could be found for the 4 It should be kept in mind that Full-Field-Displays visualize the Zernike polynomial coefficients of the fit to the wavefront in the exit-pupil. Consequently the orientation of the displayed Zernike polynomial coefficients corresponds to the orientation in the exit-pupil rather than in the image plane. The location of the nodes however remains unchanged regardless of observing the field dependence in the exit-pupil or in the image plane. 44

66 example of a misaligned aspheric surface. For the particular example shown it is assumed that an aspheric surface has been tip-tilted about its vertex which remained at the intersection point of the OAR. Consequently the aspheric surface contribution (surface contribution I) is centered in the field while the spherical surface aberration contribution to coma originating from the base sphere is shifted to a new aberration field center (surface contribution II Figure -13). To find the resulting coma node two conditions have to be met. First the magnitudes of the two surface contributions have to be equal and secondly the orientation has to be opposite. The first condition for the location of a coma node is met at the field points corresponding to the oval described by the intersection line of the two cones which includes all the locations at which the two coma contributions have an identical magnitude (Figure -13). To find the corresponding field points fulfilling the condition of opposite orientation the oval has been projected onto the x-y - field plane and is overlaid onto the Full-Field-Display visualizing the orientation of coma at each field point (Figure -14) for the two surface contributions. Examining the orientation for coma it becomes clear that the orientation of the two coma contributions are opposite at only one field point and therefore canceling each other only at this particular point. Note that the coma node is located on the line joining the two aberration field centers. The outlined approach could be directly applied to find the coma node location for an arbitrary number of surface contributions by successively applying the outlined procedure for all surfaces. The notation in Figure -15 is used from here on to designate the coma node location. 45

67 Surface Contribution I Magnitude of Coma σ I = 0 H x σ II H y Surface Contribution II Figure -13: Two wave aberration contributions for coma across the field with different magnitudes and aberration field centers and projection of the intersection line of the two surface contributions for coma onto the x-y - field. Coma Node Location of Equal Magnitude Aberration Field Center II Aberration Field Center I Figure -14: Orientation of the two surface aberration contributions for coma showing the individual aberration field centers and the coma node location. H y Image Plane a 131 H x Figure -15: Aberration field vector for third order coma showing the node location for third order coma. 46

68 .5.4 Third Order Astigmatism in Optical Systems Without Symmetry In optical systems exhibiting rotational symmetry astigmatism is defined with quadratic field and pupil dependence. The pupil dependence of astigmatism across the exit-pupil is visualized in Figure -16 for three focal positions the tangential (a) medial (b) and sagittal focus (c). In the presence of misalignments astigmatism develops an unique new field dependence as first shown in Shack et al given by W 1 = W [( H σ ) ρ ] (-7) AST3 j j j which can be reduced if W 0 to W AST3 1 = W[( H a) + b] ρ (-8) with A 1 a = ( W jσ j) (-9) W W j and B 1 b a σ a (-30) ( W j j ) W = W j where W is the total wave aberration for astigmatism W j designates the wave aberration term for third order astigmatism for surface j a denotes a vector from the center of the field to the midpoint between the two astigmatic nodes and ± ib are two vectors pointing from the endpoint of the a vector to the two astigmatic nodes. 47

69 Astigmatism ρ y Astigmatism ρ y ρ x (a) ρ x (b) Astigmatism ρ y ρ x (c) Figure -16: Pupil dependence of third order astigmatism for the (a) tangential (b) medial and (c) sagittal focus. The squared vectors used in Equations (-7) - (-8) and Equation (-30) can be calculated using the concept of vector multiplication which is summarized in APPENDIX B and is explained in detail in Thompson 005. Similarly as shown for coma in Section.5.3 the sigma vector contributions for the spherical base curve and the aspheric departure have to be treated separately. Equation (-9) and Equation (-30) can be re-written given by and 1 a = σ + σ (-31) ( sph) ( sph) ( asph) ( asph) ( W j j W j j ) W j 48

70 b sph sph ( ) asph asph W j j W j j 1 = σ + σ a (-3) ( ) ( ) ( ) ( ) W j where W is the contribution to astigmatism from the base sphere of surface j and ( sph) j W is ( asph) j the contribution to astigmatism that comes from the aspheric cap which is the departure of the surface from the base curvature of surface j. When compared to the common field dependence of astigmatism in rotationally symmetric optical systems Equations (-7) - (-8) show that there are two additional astigmatic components in the presence of misalignments one linear with field the other one constant with field. This new field dependence causes astigmatism to exhibit two distinct nodes that are positions in the field where astigmatism is zero which is characteristic of every misaligned optical system nominally uncorrected for astigmatism. The changes in the field dependence of astigmatism are demonstrated in Figure -17 (left) by plotting the magnitude of astigmatism across the field where (a) shows the case of a rotationally symmetric optical system and (b) shows the most general case without symmetry which clearly exhibits the binodal behavior. To also visualize the orientation of astigmatism in both cases the corresponding line images are shown in Figure -17 (right). Note the FFDs shown in Figure -17 (right) were computed by utilizing a Coddington real-raytrace in CODE V and consequently the orientation of the line elements corresponds to the orientation of astigmatism in image space. To obtain a conceptual understanding why astigmatism in optical systems without symmetry (not corrected for astigmatism) is typically binodal a graphical construction of the node location is shown on an example with two surface contributions. Two surface contributions for astigmatism with equal (Figure -18(a)) and opposite (Figure -18(b)) sign are shown where 49

71 the aberration field center for the larger surface contribution has been shifted from the axial field point. Similarly as in the previous case of coma the first requirement for the formation of a node is equal magnitude (absolute value) of the two surface contributions. The intersection line of the two contributions is found and projected onto the x-y - field plane (Figure -18(ab)). H y Magnitude of Astigmatism H y H x H x (a) (b) H y + ib H y ib a Magnitude of Astigmatism H x H x (c) (d) Figure -17: (a) Magnitude and (b) orientation of third order astigmatism across the field for a rotationally symmetric optical system and (c) magnitude and (d) orientation of third order astigmatism across the field for an optical system without symmetry. Along the ovals in Figure -18(ab) the absolute values of both surface contributions for astigmatism are equal. The second requirement for the occurrence of a node is that the astigmatic 50

72 line images of the two surfaces have to be perpendicular with respect to each other. Examining the line orientation along the oval (denoting equal magnitude of astigmatism) there are two positions where the line images of the two surface contributions are perpendicular with respect to each other specifying the astigmatic node locations (Figure -19). Surface Contribution I σ I = 0 Magnitude of Astigmatism Surface Contribution II σ II 0 location of equal magnitude H x Surface Contribution II Surface Contribution I Magnitude of Astigmatism σ I = 0 location of equal magnitude σ II 0 H x H y (a) H y (b) Figure -18: Two wave aberration contributions for astigmatism across the field with different magnitudes but (a) equal signs for the wave aberration contributions and (b) unequal signs for the wave aberration contributions. Also note that for the specific case of equal signs of individual surface contributions the astigmatic nodes are located on a line that is perpendicular to the line connecting the two aberration field centers (Figure -19(a)) as also previously pointed out by Rogers Similarly in the case of opposite signs of the individual surface contributions (e.g. the case for the secondary mirror wave aberration contributions of a Ritchey-Chrétien telescope) the two line orientations are perpendicular to each other on a line connecting the two aberration field centers (Figure -19(b)). Knowing that third order astigmatism can only become binodal in the most general case the simplified node diagram in Figure -0 will be used from here on. 51

73 zero astigmatism equal magnitude zero astigmatism Aberration Field Center I Aberration Field Center II (a) zero astigmatism zero astigmatism zero astigmatism Aberration Field Center I Aberration Field Center II (b) equal magnitude Figure -19: Orientation of the two surface aberration contributions for astigmatism showing the individual aberration field centers and the two astigmatic node locations for (a) equal signs of the two surface contributions and (b) unequal signs of the two surface contributions. Note the insets are magnified illustrations of the line orientation at the astigmatic nodes. 5

74 H y ib Image Plane a + ib H x Figure -0: Aberration field vectors for third order astigmatism showing the two astigmatic node locations..5.5 Third Order Medial Focal Surface in Optical Systems Without Symmetry While astigmatism is commonly referenced with respect to the sagittal focal surface nodal aberration theory has been developed using the concept of the medial astigmatic surface as reference surface for astigmatism the focal surface of minimum RMS wavefront error. The quadratic pupil dependence for the medial astigmatic component is visualized in Figure -1. The purely quadratic field dependence in optical systems with symmetry changes in the presence of tilts and decenters and is given by with ( ) ( ) W0 = W0M H a0m H a 0M + b0 M (-33) 1 a = W σ (-34) 0M j 0Mj j W0M 53

75 Magnitude of the Medial Focal Surface ρ y ρ x Figure -1: Pupil dependence of the medial astigmatic component. b 1 ( ) σ σ a a (-35) 0M = W0Mj j j j 0M 0M W0M where W0 denotes defocus W 1 0 = W j 0 = W j 0 + W aberration for the third order medial focal surface is the system wave M Mj j j W describes the surface wave aberration 0Mj coefficient for the third order medial focal surface a 0M and b 0M correspond to a transverse and a longitudinal shift of the vertex of the medial focal surface respectively. The focal shift in image space is given by δ z = 8( f #) W b (-36) 0M 0M 0M where f # corresponds to the sytem working f-number of the optical system. The transverse and longitudinal displacements of the medial focal surface in the absence of system symmetry result in a tilted and defocused best focal surface as visualized in Figure -. 54

76 Gaussian Image Plane a 0M δ z 0M OAR Medial Focal Surface Design Image Plane Figure -: Shift of the medial focal surface with respect to the design image plane. The field dependence of the wave aberration term corresponding to the medial focal surface is visualized in Figure -3 for the case of rotational symmetry (a) and for the case without symmetry (b). Magnitude of the Medial Focal Surface H y H x Magnitude of the Medial Focal Surface H x (a) H y (b) Figure -3: Magnitude of the wave aberration for the medial focal surface across the field for (a) a rotationally symmetric optical system and (b) an optical system without symmetry. 55

77 Knowing that the shape of the medial focal surface remains quadratic in the absence of symmetry and that the only effects are a decentered medial focal surface and a focal plane shift with respect to the design image plane the simplified representation of the aberration field vector given in Figure -4 will be used from here on. H y Image Plane a 0M H x δ z = 8( f #) W b 0M 0M 0M Figure -4: Aberration field vector for the third order medial focal surface showing the transverse displacement of its vertex. In addition the surface is shifted longitudinally by δ z = 8( f #) W b. 0M 0M 0M 56

78 CHAPTER 3: MISALIGNMENT ABERRATION FIELDS IN ASTRONOMICAL TELESCOPES In Section.5 it has been demonstrated how the third order aberrations in optical systems without symmetry can be described based on the unchanged wave aberration coefficients and the concept of shifted aberration field centers. In the conceptual design phase of astronomical telescopes it is of great help for trade-off studies to have analytical equations that describe the complete solution space of a telescope configuration instead of a purely numerical model that only provides information for one specific configuration. Therefore Chapter 3 provides analytical expressions for both key quantities the wave aberration coefficients (third order) and the shifted aberration field centers for two-mirror astronomical telescopes. In Section 3.1 the chosen set of parameters for subsequent derivations is summarized. In Section 3. analytical expressions for the wave aberration surface contributions grouped into contributions from the conic/aspheric departure and spherical base curve are derived. The effects of lateral misalignments caused by surface decenters and tip-tilts are described in Section 3.3 for the important case of a two-mirror telescope with the aperture stop either on the primary or secondary mirror providing a model to describe the third order misalignment induced aberrations in those systems. Additionally expressions are provided that facilitate the computation of misalignment parameters based on measured nodal properties. In Section 3.4 the concept of pivot-points is introduced with application to two-mirror astronomical telescopes. Section 3.5 concludes the chapter with important nodal properties found as a result of the general 57

79 model for describing the misalignment induced aberrations which are of fundamental importance during the development of an alignment plan of those systems. 3.1 Definition of Structural and Systems Parameters The wave aberration coefficients depend on several parameters some of which describe the telescope on a system level others are specific to a particular design. Consequently an intuitive classification of these parameters into systems parameters and structural parameters can be made. It can be found that for the case of two-mirror telescopes there are only three systems parameters that are independent. Several such combinations are conceivable with the best choice of parameters dependent on the particular application. Here the entrance pupil diameter (EPD) the system focal length (f SYS ) and the (semi-) field angle in object space u PM are utilized in subsequent derivations. Prior to defining the characteristic parameters the sign conventions utilized throughout this work are summarized. In general the mechanical axis is assumed to be oriented along the z- axis and relative surface to surface distances are assumed positive after an even number of reflections (including none). On the contrary the surface to surface thickness is assumed negative after an odd number of reflections. The sign for a particular surface curvature is defined based on the location of the center of curvature with respect to the surface vertex. If the z- coordinate of the center of curvature is larger than the corresponding z-coordinate of the surface vertex the sign is assumed to be positive and vice versa. To treat the case of reflection we adhere to the commonly used convention that the refractive index n is assumed to be -1 after an odd number of reflections and +1 after an even number of reflections. 58

80 The signs for the ray height at an arbitrary position are defined by the location of the ray with respect to the mechanical axis which also defines an x-y coordinate system as shown in Figure 3-1. The ray inclination angle is defined to be positive for a ray segment if the ray height (measured from the mechanical axis) along the ray increases in the positive z-direction and vice versa. While the sign of the ray inclination angle is defined by the ray orientation with respect to the mechanical axis the sign for the angle of incidence and angle of reflection are referenced with respect to the surface normal at the ray-surface intersection. The angle of incidence and the angle of reflection always have opposite signs where the signs are defined as shown in Figure 3-1. y j ( > 0) u = 0 j i j ( < 0) i j ( > 0) u j ( > 0) +y +x y j ( > 0) u > 0 j u j ( < 0) +y i j ( > 0) +x i j ( < 0) u j < 0 cc j +z cc j +z d j ( < 0) d j ( > 0) c j ( < 0) c j ( < 0) Figure 3-1: Visualization of the sign conventions for the ray intersection height the ray inclination angle the angle of incidence and axial spacings. Concerning structural parameters two parameters that relate to the scale of the system are required to completely describe the first order properties of a two-mirror telescope as illustrated in Figure 3-. The ones chosen here are the mirror spacing (d 1 ) and the primary mirror focal length (f PM ). Furthermore two additional derived parameters have been defined one of them being the axial obstruction ratio (Wilson 1996) defined as 59

81 y γ = (3-1) y where y PM and y denote the marginal-ray height on the primary and secondary mirror respectively. The second derived quantity utilized here is the secondary mirror magnification given by PM f m = f SYS PM fpm L d 1 (3-) where f SYS = f f f f d PM PM 1 (3-3) where f corresponds to the system focal length L denotes the distance from the secondary SYS mirror to the Gaussian image plane and f denotes the secondary mirror focal length. The two common design forms for two-mirror astronomical telescopes are depicted in Figure 3- where Figure 3-(a) refers to the Cassegrain type (concave primary and convex secondary mirror) and Figure 3-(b) refers to the Gregorian type (concave primary and secondary mirror). All the expressions derived in the following sections apply equally to both configurations with the only difference being that in the Cassegrain telescope the system focal length f and SYS the axial obstruction ratio γ are assumed to be positive quantities. In the Gregorian type the system focal length f and the axial obstruction ratio SYS γ are assumed to be negative quantities which also agrees with the conventions chosen in common optical design software. 60

82 P c PM k PM y chief-ray marginal-ray y y c k y PM = EPD/ u PM F PM Stop z F SYS y IMG f < PM ( 0) d 1 (<0) L ( > 0) f SYS ( > 0) (a) c PM k PM y chief-ray marginal-ray P c k y PM = EPD/ F SYS F PM chief ray z y IMG u PM y y Stop d 1(<0) f ( 0) PM < L(>0) (b) f ( < 0) SYS Figure 3-: Design forms for two-mirror astronomical telescopes with the stop at the primary mirror showing (a) the Cassegrain type and (b) the Gregorian type. 3. Wave Aberration Coefficients for Two-Mirror Telescopes In Section.5.1 it has been shown how the scalar wave aberration expansion can be first rewritten in vectorized form and then merged with the concept of shifted aberration field centers. It is interesting to note that the surface wave aberration coefficients utilized in Equation (-0) remain unchanged compared to the case of rotational symmetry. While the wave aberration coefficients can be conveniently computed numerically based on paraxial raytrace data in optical design software particularly in the early stages of the design 61

83 of a telescope analytical expressions for the wave aberrations can be highly advantageous for several reasons. First they directly provide the dependencies on the structural system and derived parameters and important relationships concerning the sensitivities with respect to misalignments can be obtained. Secondly the surface wave aberration contributions can be separated into the contribution originating from the base sphere and its aspheric departure which provides valuable insights into design forms exhibiting special aberration properties which will be discussed in more detail in Section 3.4. The derivation of the Seidel terms for the base sphere and aspheric/conic aberration contributions requires several first order quantities which are summarized in APPENDIX C (Table C-1) and the utilized definitions for the Seidel terms are summarized in APPENDIX D (Table D-1). Substituting the paraxial quantities into the expressions for the Seidel terms and converting the expressions to wave aberrations (APPENDIX D Table D-) provides the expressions that will be utilized throughout this work. The third order wave aberration contributions spherical aberration coma astigmatism medial focal surface and (for completeness) distortion separated into spherical base curve and aspheric surface contributions for the case of a two-mirror telescope with the stop at the primary mirror are listed in Table 3-1 Table 3-5 respectively. Similar expressions for the wave aberrations in two mirror astronomical telescopes with the aperture stop located on the secondary mirror are given in APPENDIX D Table D-3 - Table D-7. To verify the validity of the derived expressions for the wave aberration coefficients the example of a typical Ritchey-Chrétien telescope is given and analytical results are compared to results obtained in CODE V (APPENDIX D Table D-9 - Table D-1). 6

84 Table 3-1: Wave aberration coefficients for spherical aberration (third order) of a two-mirror telescope with the aperture stop located on the primary mirror. SUR# W040 PM (sph) EPD m 4 3 (STOP) 51( fsys ) PM (asph) (STOP) 51( fsys ) k PM 3 EPD m (sph) (asph) 4 EPD m+ m ( 1) (1 ) γ ( f ) 51 SYS 4 3 k EPD (1 m) γ 51 3 ( fsys ) 3 Table 3-: Wave aberration coefficients for coma (third order) of a two-mirror telescope with the aperture stop located on the primary mirror. SUR# W131 PM (sph) 3 EPD m upm (STOP) 3( fsys ) PM (asph) (STOP) 0 3 (sph) ( 1) [ 1 ( 1) γ ] EPD m m m+ u 64 m ( fsys ) PM (asph) (1 ) ( 1) 3 3 k EPD m γ upm 64 ( fsys ) m 63

85 Table 3-3: Wave aberration coefficients for astigmatism (third order) of a twomirror telescope with the aperture stop located on the primary mirror. SUR# W PM (sph) EPD mupm (STOP) 8( fsys ) PM (asph) (STOP) 0 (sph) (1 )[ 1 ( 1) γ ] EPD m m m+ u 3 ( fsys ) m γ PM (asph) (1 ) ( 1) 3 k EPD m γ upm 3 ( fsys ) m γ Table 3-4: Wave aberration coefficients for the medial focal surface (third order) of a two-mirror telescope with the aperture stop located on the primary mirror. SUR# W0M PM (sph) EP Dm (STOP) 16( fsys ) u PM PM (asph) (STOP) 0 { } 3( fsys ) m γ (sph) EPD (1 m) ( γ 1) + m( γ 1) + m [ γ( + γ) 1] PM u (asph) k EPD (1 m) ( γ 1) u 3 PM 3 ( fs YS ) m γ 64

86 Table 3-5: Wave aberration coefficients for distortion (third order) of a twomirror telescope with the aperture stop located on the primary mirror. W 311 PM (sph) (STOP) PM (asph) (STOP) 0 0 (sph) [ ][ ] 3 EPD(1 m)( γ 1) 1 m ( m+ 1) γ 1 γ m(3+ γ) u PM 3 16m γ (asph) k EPD(1 m) ( γ -1) u 3 16m γ PM 3.3 Effects Induced By Transverse Misalignments In Section the effect of lateral misalignments on boresight error (also referred to as pointing change) is discussed. In Section 3.3. it is discussed in detail how the aberration field centers of two-mirror telescopes can be computed for the most common cases of the aperture stop located at the primary or secondary mirror. Cases where the aperture stop is located in front of the primary mirror as for example in a Schmidt telescope are not considered here but the aberration field centers could be readily obtained following the procedures described. Utilizing the expressions for the aberration field centers the behavior of coma and astigmatism in the presence of lateral misalignments is described for two-mirror telescopes and analytical expressions are derived that allow the re-construction of misalignment parameters based on the observed nodal properties. 65

87 3.3.1 Misalignment Induced First Order Effects/Boresight Error In optical systems exhibiting rotational symmetry (i.e. without misalignments) the field center at the image plane is typically defined by the intersection point of the optical axis ray with the image plane (Figure 3-3). The significance of this particular field point is two-fold. First at the field center all field dependent aberrations are zero given that the aberration field centers of all surfaces coincide at this point. Secondly aberrations considered at field points equidistant from the field center all have the same magnitude and the orientation at a particular field point (if any) is given by the line connecting the field point of interest and the field center with the orientation (if any) inward or outward depending on the sign of the particular aberration type. In this work the reference coordinate system (denoted as H x * and H y * in Figure 3-3) is defined to be perpendicular to the optical axis of the surface that acts as the aperture stop. In the most often encountered case of having the aperture stop coincide with the primary mirror misalignments of the secondary mirror are computed relative to the primary mirror with no loss of generality when the system aperture stop is physically registered to the center of the primary mirror. In the absence of system symmetry assuming misalignments of the secondary mirror with respect to the primary mirror the zero field point can but will typically not coincide with the center of the adapter-rotator 5 (AR) which is shown in Figure 3-3(b). Consequently dependent on particular misalignments the new field center (defined by the intersection point of the OAR with the Gaussian image plane) is located at an off-axis field point (the location of which may not be known in practice) compared to the original field coordinate system of the 5 In telescopes with an Alt-az mount the adapter-rotator forms the mechanical interface between the telescope and the scientific instruments and provides several key functions including field acquisition and guiding wave front sensing and instrument rotation. In practice all field points are referenced with respect to its center. 66

88 aligned system. From an aberration point of view there will be in general two aberration field centers for each mirror one associated with the spherical base curve referred to as σ ( sph) j and determined by the location of the center of curvature and one associated with the center of ( ) symmetry of the aspheric departure (if any) of the surface denoted as σ asph. Even though the reference field center required for the computation of the aberrations changes in the presence of misalignments (denoted as H x and H y in Figure 3-3(b)) in practice all measurements are made with respect to the center of the AR which can be significantly different depending on the magnitude of boresight errors. As shown here for the particular case of a two-mirror telescope with the aperture stop located on the primary mirror the shift of the field center in the presence of misalignments can be included in the analysis. This can be accomplished by expressing the normalized boresight error dependent on the aberration field center associated with the spherical base curve of the secondary mirror which is given by H AR ( sph) σ = L( upm ( 1 + dc 1 )) y IMG j (3-4) where u PM is the chief-ray angle of incidence at the primary mirror. The shift of the object space field center defined as the incoming ray that enters the telescope coaxial with the primary mirror axis with respect to the center of the adapter is visualized in Figure 3-3(b). 67

89 Marginal-Ray OAR MA Marginal-Ray misaligned MA OAR PM (STOP) PM (STOP) Adapter- Rotator (b) Adapter- Rotator (a) H y * AR H H y =H y * H y z H x = H x * σ j H * H H x * H x H Aj T Figure 3-3: (a) Marginal-ray (axial field point) and OAR for an aligned Ritchey- Chrétien telescope with the stop on the primary mirror and (b) marginal-ray and OAR in the presence of secondary mirror misalignments with the stop on the primary mirror. Note that the OAR no longer intersects the image plane at the center of the AR which itself is assumed to be aligned with the primary mirror axis Locating the Aberration Field Centers in Misaligned Two-Mirror Telescopes In Section.5.1 it has been demonstrated how the aberrations in misaligned optical systems can be described by integrating the concept of shifted aberration field centers in a vectorial form of the wave aberration expansion. The aim of this subsection is to compute the shifted aberration field centers for misaligned two-mirror telescopes where the focus will be on the two most important cases for the aperture stop location. Specifically we provide the analytical expressions for the aberration field center associated with the spherical base curve and separately with the aspheric departure for the case of the aperture stop being located on either of the mirrors (i.e. the primary or secondary mirror). In both cases the aperture stop is not allowed to decenter with 68

90 respect to the center of symmetry of the aspheric departure of the respective mirror. Equivalent expressions can be developed for arbitrary aperture stop positions but are generally less important in practice Stop at the Primary Mirror and Secondary Mirror Misaligned Most large astronomical telescopes in use today have the aperture stop located on the primary mirror. The reasoning for the particular choice is mainly caused by the large relative cost of the primary mirror aperture size. As visualized in Figure 3-4 having the stop located on the primary mirror all field points utilize the full aperture (and therefore the full light gathering power) of the primary mirror. As described in Section.4 (Thompson Schmid Cakmakci Rolland 009) the shifted aberration field centers are fully defined by one real-raytrace of the OAR and one paraxial raytrace. For analysis purposes the real-raytrace can be most conveniently performed in optical design software. In the system design phase however it is extremely useful to have expressions that show the dependencies on structural or system parameters. Since for aspheric surfaces no analytical solution for the ray-surface intersection exists it may initially seem impossible to determine such analytical expressions. Nevertheless when considering the typical telescope dimensions in practice (e.g. aperture sizes ~ m) and anticipated system perturbations after initial mechanical alignment (decenter misalignments ~ mm tip-tilts ~ arcmin) it becomes clear that the deviation of the OAR from the optical axis of the nominal optical system is small compared to the surface apertures. Consequently the small sag difference by considering only the spherical base curve of in general an aspheric surface for the computation of the OAR is negligible. This 69

91 approximation facilitates then the derivation of analytical expressions for the aberration field centers. For two-mirror telescopes one surface can always be chosen as reference coordinate system without loss of generality (when the stop is physically registered to the center of the aspheric departure) and misalignment perturbations of the other surface are expressed with respect to this reference frame. A particularly convenient choice for the reference coordinate system is to position the origin of the reference coordinate system at the center of the aperture stop and to align the z-axis with respect to the corresponding mirror axis where the aperture stop is located. Since in this subsection the aperture stop is assumed to be located on the primary mirror the origin of the reference coordinate system coincides with the primary mirror vertex and the z-axis of the coordinate system is coincident with the optical axis of the primary mirror. The consequence of this reference coordinate system definition is that the primary mirror can be considered as perfectly aligned leading to ( sph) 0 σ PM (3-5) 0 ( asph) 0 σ PM. 0 (3-6) OAR PM (STOP) (a) MM (b) MM Figure 3-4: (a) Schematic layout of a two-mirror telescope with the aperture stop located on the primary mirror and (b) OAR raytrace starting at the aspheric vertex of the primary mirror. 70

92 In the presence of secondary mirror misalignments the corresponding secondary mirror ( ) aberration field center displacement vectors σ sph ( ) and σ asph are generally non-zero. Re-calling the definition of the aberration field center σ ( sph) for the wave aberration contributions originating from the spherical base curve one has to find the angle of incidence of the OAR with respect to the decentered and tilted secondary mirror. Normalizing this angle of incidence with the paraxial chief-ray angle of incidence one can find a field normalized expression for the aberration field center for the wave aberration contributions originating from the spherical base curve of the secondary mirror given by σ ( sph) ( ) ( ) BDE XDE 1 BDE c XDE = ADE YDE upm (1 + c d1) ADE cyde (3-7) where XDE and YDE are the secondary mirror vertex decenters in the x-z and y-z plane respectively and BDE and ADE are the secondary mirror tip-tilts in the x-z and y-z plane respectively. Since the location of the aberration field center for the wave aberration contribution originating from the spherical base curve is defined by the location of the center of curvature of the surface both decenters and tip-tilts appear in Equation (3-7). The aberration field center for the wave aberration contributions originating from the aspheric departure of the secondary mirror σ ( asph) is determined by the displacement of the secondary mirror aspheric vertex with respect to the OAR intersection height at the secondary mirror normalized by the paraxial chief-ray height at the secondary mirror. Re-calling Equation (-11) an expression for the aberration field center for the wave aberration contributions originating from the aspheric departure can be obtained and is given by 71

93 ( XDE ) ( YDE ) ( asph) 1 XDE σ = (3-8) du 1 YDE PM where du corresponds to the chief-ray height at the secondary mirror. 1 PM The introduced boresight error (normalized with the chief-ray height in image space) caused by secondary mirror misalignments can be found by tracing the OAR to the image plane and is given (dependent on secondary mirror decenter and tip-tilt perturbations) by H AR IMG ( XDE BDE ) ( E ) L BDE + c XDE =. YDE AD y ADE IMG + + cyde (3-9) Even though Equation (3-7) and Equation (3-8) facilitate the computation of the aberration field centers for a specific telescope configuration more insight on a system level is obtained when re-writing the expressions dependent on systems (and derived) parameters utilizing first order expressions as summarized in APPENDIX C given by σ ( sph) ( fsys ) γ XDE BDE ( BDE XDE ) m(1 m) (1 m) = ( ADE YDE ) ( fsys )[ 1 m ( m+ 1) γ ] u PM γ ( fsys ) YDE + ADE (1 m ) (3-10) σ ( asph) ( XDE ) m XDE = ( YDE ) ( fsys ) ( γ 1) u YDE PM and correspondingly for the boresight error given by ( fsys ) γ XDE BDE ( ) ( BDES M XDE ) AR L( 1 m) 1 m H IMG =. ( ADE YDE ) ( fsys ) γ y IMG ( fsys ) γ YDE + ADE 1 m (3-11) (3-1) 7

94 Examining Equation (3-10) one recognizes that the tip-tilt contribution to the aberration field center for the spherical base curve becomes increasingly dominant with increasing system focal length f SYS. For observatory class telescopes with typical system focal lengths of several tens of meters the tip-tilt term is frequently larger than the decenter term assuming realistic misalignment residuals. It is also revealed that the decenter term in Equation (3-10) and Equation (3-11) becomes increasingly important with increasing (i.e. larger values for m ) secondary mirror magnifications Stop at the Secondary Mirror and Primary Mirror Misaligned Even though the full aperture of the primary is not utilized when placing the aperture stop at the secondary mirror there are several circumstances which may motivate this compromise. One example is when thin shell adaptive secondary mirrors are used with the prime example of the Large Binocular Telescope (LBT) a Gregorian telescope with adaptive secondary mirror. Several constraints require the placement of the aperture stop on the secondary mirror one of them being the limited aperture size of the adaptive secondary mirror caused by its extremely high aspect ratio = aperture size/thickness (910mm/1.5mm) of ~ 607 (Gallieni 000). An additional requirement is imposed by the function of the secondary mirror as adaptive optics system to perform wavefront correction (from tip-tilt up to high orders). If the adaptive optics system would be deployed not at a pupil undesired field dependencies in the wavefront correction would be introduced and a mixture of aberrations would be generated since the pupil footprint on the secondary mirror would be different for each field point. 73

95 Similarly as discussed in Section the optical axis of one mirror can be utilized as coordinate reference surface. In this case it is most convenient to utilize the secondary mirror as coordinate reference surface which is assumed to coincide with the aperture stop. Consequently XDE BDE YDE ADE 0 while the primary is considered to be misaligned with respect to the secondary mirror. Nevertheless in this case aberration contributions from the primary and secondary mirror have to be included in the analysis. The reason for a shifted aberration field of the secondary mirror is the deviated OAR caused by the misaligned primary mirror. Specifically the misaligned primary mirror will result in shifted aberration field centers for the spherical base curve and aspheric departure of the primary mirror. Additionally the deviated OAR will introduce a shifted aberration field center associated with the spherical base curve of the secondary mirror. The aberration field center associated with the aspheric departure of the secondary mirror remains zero since by definition the OAR intercept will be at the center of the aperture stop which is assumed to coincide with the secondary mirror aspheric vertex. A schematic drawing of the OAR for a two-mirror telescope with the aperture stop on the secondary mirror assuming primary mirror misalignments is shown in Figure 3-5(b). Expressing the OAR angle of incidence on the primary mirror normalized with the paraxial chief-ray angle of incidence on the primary mirror the aberration field center for the ( ) spherical base curve σ sph is given by PM σ ( XDEPM BDEPM ) ( ) OAR ( ) OAR ( ) BDE + c XDE H ( sph) 1 PM PM PM PM x PM = YDEPM ADEPM cpm yp M + u PM +ADEPM + cpm YDEPM HPM y (3-13) where the intersection height of the OAR on the primary mirror is given by 74

96 leading to H OAR PM dc 1 PM = 1 c d PM 1 XDE YDE PM PM 1 c PM 1 + c PM BDE ADE PM PM (3-14) σ ( sph) PM ( XDEPM BDEPM ) cpm = ( YDE ADE ) ( c d 1)( c y + u ) PM PM PM 1 PM PM PM XDE YDE PM PM 1 c PM 1 + c PM BDE ADE PM PM. (3-15) Similarly the aberration field center for the secondary mirror associated with its spherical base curve can be obtained by expressing the OAR angle of incidence and normalizing the angle by the paraxial chief-ray incidence angle given by σ ( XDEPM BDEPM ) ( YDE ADE ) ( sph) 1 = PM PM c ypm + cpm + upm (1 + c d1)... 1 c c XDE BDE + c H... 1 c cpm YDEPM + ADEPM + cpm cp M (1 + c d1) OAR PM PM PM PM PM x cpm (1 + c d1) OAR H PM y (3-16) leading to σ ( sph) ( XDEPM BDEPM ) cpm = ( YDE ADE ) ( c d ) y c ( c d ) ( ( c ) u ( c M d )) PM PM PM 1 PM PM 1 PM S 1 1 XDEPM BDEPM cpm YDEPM ADE + PM c PM (3-17) 75

97 (STOP) OAR PM (a) MM (b) MM Figure 3-5: Schematic layout of a two-mirror telescope with the aperture stop located on the secondary mirror showing the (a) varying pupil footprint on the primary mirror with field angle and (b) OAR raytrace showing the intercept at the secondary mirror (aspheric) vertex. For the aberration field center associated with the aspheric departure of the primary mirror an expression can be found by determining the distance of the OAR intersection point from the aspheric vertex normalized by the paraxial chief-ray height at the primary mirror given by ( DEPM ) ( YDE ) OAR X ( ) 1 XDEPM H asph PM x σ PM =. OAR PM y PM YDEPM H (3-18) PM y Note the form of Equation (3-18) (for the primary mirror) is similar to Equation (3-8) (for the secondary mirror) with the difference that we now have to subtract the non-zero intersection height of the OAR on the primary mirror surface. Substituting Equation (3-14) into Equation (3-18) the aberration field center associated with the aspheric departure of the primary mirror is again determined by the offset of the OAR from the aspheric vertex which is given by σ ( XDEPM BDEPM ) 1 ( YDE ADE ) y ( 1 c d ) XDE ( ) PM d1bde asph PM PM PM PM PM PM 1 YDEPM d1adepm =. + (3-19) While in the case of a perturbed secondary mirror with the primary mirror (stop) being the reference surface only surface decenters appeared in Equation (3-8) in Equation (3-19) both 76

98 primary mirror decenter and tip-tilts are present. The reason for this occurrence is the definition of the OAR to intersect the secondary mirror at its aspheric vertex. Consequently tip-tilts of the primary mirror cause a deviation of the OAR which automatically shifts the OAR in the primary mirror aperture to still intersect the center of the stop at the secondary mirror. The aberration field center for the aspheric departure of the secondary mirror is zero by definition since the aspheric vertex is centered at the aperture stop expressed as ( asph) 0 σ. (3-0) 0 The boresight error at the detector is determined by continuing the raytrace of the OAR to the image plane given by H AR IMG ( XDEPM BDEPM ) ( YDE ADE ) PM PM ( 1 Lc ) OAR H PM x = +... OAR y IMG H PM y OAR ( L+ d ( )) ( ) 1 c L BDEPM + cpm H PM x XDEPM 1... y ADE + c H OAR ( YDE ) IMG PM PM PM y PM. (3-1) Substituting the expression for the OAR intersection height at the primary mirror in Equation (3-1) with the expression given in Equation (3-14) an expression for the boresight error at the image plane is obtained given by H AR IMG ( XDEPM BDEPM ) L = ( E ADE ) ( 1 c d ) + BDE c XDE YD PM PM y ADE c YDE PM PM PM PM 1 IMG PM PM PM. (3-) Equation (3-) can be re-written in terms of structural derived and systems parameters and an alternative expression is given by 77

99 H AR IMG ( XDE BDE ) PM PM ( YDEPM ADE ) γ ( f ) ( fsys ) XDEPM + L m m = y YDEPM m ( f S ) PM SYS IMG SY BDE ADE PM PM. (3-3) An alternative set of expressions for the aberration field center of the primary (Equation (3-15) Equation (3-17)) and secondary mirror (Equation (3-19)) can be derived dependent on systems and derived parameters given by σ ( sph) PM σ ( fsys ) XDEPM + ( BDEPM XDEPM ) m = m ( ADEPM YDEPM ) ( ( fsy S ) upm ypm m) γ f YDEPM m σ ( sph) ( asph) PM ( fsys ) ( SYS ) ( γ 1) XDEPM BDE ( XDEPM BDEPM ) 1 = m ( YDEPM ADEPM ) ypmγ ( fsys )( γ 1) YDEPM + ADE m ( ) ( YDEPM ADEPM ) ( fsys ) + ( fsys ) ( fsys ) ( fsys ) BDE ADE XDEPM BDEPM m =... (1 mu ) ( m 1)( u y m) γ ( asph) 0 σ. 0 PM PM PM XDE YDE PM PM + m m PM PM BDE ADE PM PM PM PM (3-4) (3-5) (3-6) (3-7) 78

100 3.3.3 Third Order Nodal Aberrations in Misaligned Two-Mirror Telescopes and Computation of Misalignment Parameters The aberration fields of two-mirror telescopes with ideal rotationally symmetric mirrors with the correct spacing but in any state of misalignment with respect to tilts and decenters can be developed by combining the expressions for the wave aberration contributions given in Section 3. and the expressions for the shifted aberration field centers developed in Section Writing the aberration contributions in terms of aberration field vectors a general expression for describing the misalignment induced aberration fields in two-mirror nonaplanatic astronomical telescopes is obtained given by 1 WNOSYM = W [ ] ( H a131) ρ ( ρ ρ) + W ( )... H a + b ρ W 0M ( H a0m) ( H a0m) + b 0M ρ (3-8) or for the aplanatic case given by 1 WNOSYM = [ A ] ρ ( ρ ρ) + W ( )... H a + b ρ W 0M ( H a0m) ( H a0m) + b 0M ρ. (3-9) In Section it is shown how the nodal properties of coma and astigmatism can be described for the most common case in practice a two-mirror telescope with the aperture stop located on the primary mirror. It is also demonstrated how these nodal properties can be utilized in the re-construction of misalignment parameters and analytical expressions for the decenter and tip-tilt parameters are given assuming known nodal properties of coma and astigmatism. In Section it is discussed how the nodal properties of coma and astigmatism can be described in the presence of misalignments if the aperture stop is located on the secondary mirror and it is demonstrated how expressions for misalignment parameters can be devised. 79

101 Determining the State of Misalignment of Two-Mirror Telescopes with the Aperture Stop Located on the Primary Mirror and Secondary Mirror Misalignments Here we utilize the same conventions concerning the reference coordinate system as described in Section Since the aperture stop is defined to be coincident with the primary mirror the reference coordinate system is fully defined by the local coordinate system of the primary mirror and secondary mirror misalignments are determined relative to the primary mirror. Because in practice when measuring the aberrations of a telescope the field points are typically referenced with respect to the center of the AR as shown in Section expressions are given that include the boresight error with respect to the center of the AR. In two-mirror telescopes with a conic/aspheric secondary mirror coma increases linearly with field as for example in the case of a Cassegrain or Gregorian telescope or it is corrected in its nominal state as in the case of a Ritchey-Chrétien or aplanatic Gregorian telescope. The nodal properties for two-mirror telescopes exhibiting field-linear coma are most conveniently expressed utilizing a normalized description of the aberration field vectors. If the specific optical system exhibits rotational symmetry the aberration field vector a 131 = 0 coincides with the center of the AR. In the presence of misalignments additional field-constant coma is introduced which generally shifts the coma node to a point in the field described by the characteristic vector a 131. Assuming that the aperture stop of the telescope is located on the primary mirror the node location for third order coma with respect to the center of the AR can be expressed as a combination of boresight error and third order coma node displacement given by a a H (3-30) AR AR 131 =

102 with 1 a = σ + σ (3-31) ( sph) ( sph) ( asph) ( asph) ( W W ) W131 where AR H is a normalized vector and corresponds to the shifted field center as described in Equation (3-4) W131 is the total wave aberration contribution for coma W ( sph) 131 is the coma contribution from the base sphere of the secondary mirror and W ( asph) 131 is the coma contribution from the aspheric cap of the secondary mirror that is the departure of the surface from the base ( sph) curvature σ is the aberration field center attributed to the spherical base curve of the ( asph) secondary mirror and σ is the aberration field center attributed to the aspheric departure of the secondary mirror. Since the coordinate reference surface was defined to coincide with the primary mirror optical axis and secondary mirror misalignments are expressed relative to the primary mirror no primary mirror contributions appear in Equation (3-31). A graphical visualization of the third order coma aberration field of a misaligned Cassegrain or Gregorian telescope is shown in Figure 3-6 where FFDs for the FRINGE Zernike polynomials Z7/Z8 are shown centered at the OAR intercept with the image plane (i.e. without the AR H component). Similarly an expression for the midpoint between the two astigmatic nodes referenced from the center of the AR is given by a a H (3-3) AR AR = + with 1 a = σ + σ ( sph) ( sph) ( asph) ( asph) ( W W ) W (3-33) 81

103 where W is the total wave aberration contribution for astigmatism W ( sph) is the astigmatism contribution from the secondary mirror base sphere and W ( asph) is the astigmatism contribution from the aspheric cap of the secondary mirror. The nodal behavior of astigmatism in the nominal (Figure 3-7(a)) and misaligned (Figure 3-7(b)) state is visualized through FFDs showing the FRINGE Zernike polynomials Z5/6 with respect to the OAR intercept at the image plane. Note the FFDs shown in Figure 3-7 show the orientation of astigmatism in the exit-pupil rather in image space. This is equally valid for all FFDs showing the field-dependence of particular Zernike polynomials for example Z7/8 as shown in Figure 3-6. AR A drawing of the aberration field vectors for coma ( a 131 a 131 ) and astigmatism a a ) is given in Figure 3-8 to visualize the expressions given in Equation (3-30) - (3-33). AR ( AR AR Knowing the aberration field vectors for coma a 131 and astigmatism a with respect to the center of the AR from measurements of the wavefront at several field points (see Chapter 5 for more details) combined with the expression for the boresight component AR H (Equation (3-4)) two linear vector equations (Equation (3-30) and Equation (3-3)) are provided which can be ( ) solved for the aberration field centers σ sph and σ ( asph) for the secondary mirror. Knowing σ and ( sph ) σ ( asph) the secondary mirror decenter and tip-tilts can then be obtained based on the dependence of the aberration field centers on nominal systems and perturbation parameters given by XDE = u d σ (3-34) ( asph) PM 1 x YDE u d σ ( asph) = PM 1 y (3-35) 8

104 ( sph) ( asph) (( 1 ) σ 1 σ ) BDE = u 1 + c d c d (3-36) PM x x ( sph) ( asph) (( 1 ) σ 1 σ ) ADE = upm 1 + c d y c d y. (3-37) Equations (3-34) - (3-37) provide analytical expressions for misalignment parameters of the secondary mirror utilizing the measured nodal properties of coma and astigmatism while it is assumed in Equation (3-31) that the optical system is not aplanatic (i.e. W 131 0). In Ritchey-Chrétien and aplanatic Gregorian telescopes both third order spherical aberration and coma are corrected across the field. While in those telescopes the nodal behavior of astigmatism in the presence of secondary mirror misalignments is similar to the case of Cassegrain or Gregorian telescopes the response of coma to misalignments is different. Since no linear field component is present misalignments cause the appearance of field-constant coma with the orientation dependent on the particular decenter and tilt perturbations. To avoid the singularity caused by W131 0 in Equation (3-31) a non-normalized aberration field vector for coma can be defined given by A σ σ ( sph) ( sph) ( asph) ( asph) 131 = W131 + W131 (3-38) where the aberration field vector A 131 directly corresponds to the amount of third order coma anywhere in the field (in x- and y-direction). This nodal behavior is also visualized through FFDs computed in CODE V for the case of a misaligned Ritchey-Chrétien telescope (Figure 3-9(b)) compared to the nominal case shown in (Figure 3-9(a)). Similarly following the outline for the non-aplanatic case for the aplanatic (coma- AR corrected) case the aberration field vector for coma ( A ) and astigmatism ( 131 a ) with respect to 83

105 the center of the adapter can be utilized. The aberration field vectors can be determined from wavefront measurements at a sparse set of field points providing two linear vector equations (Equation (3-3) and Equation (3-38)) which can be solved for the aberration field centers ( asph) σ and σ of the secondary mirror. The decenter and tilt parameters of the secondary ( sph ) mirror can be computed as previously described in Equation (3-34) - (3-37) Y Field Angle in Object Space - degrees a 131 = 0 Y Field Angle in Object Space - degrees a X Field Angle in Object Space degrees (a) X Field Angle in Object Space degrees Figure 3-6: FFDs showing the magnitude and orientation of FRINGE Zernike polynomials Z7/Z8 related to third order coma for a Cassegrain telescope (a) without misalignments and (b) in the presence of secondary mirror misalignments. (b) 84

106 Y Field Angle in Object Space - degrees a = 0 Y Field Angle in Object Space - degrees a ib +b i X Field Angle in Object Space degrees (a) X Field Angle in Object Space degrees (b) Figure 3-7: FFDs showing the magnitude and orientation of FRINGE Zernike polynomials Z5/Z6 related to third order astigmatism for a Cassegrain telescope (a) without misalignments and (b) in the presence of secondary mirror misalignments. H y a = H + a AR AR H y * a 131 AR H a H x a = H + a AR AR AR center H x * Figure 3-8: Schematic diagram visualizing the node positions for third order coma and astigmatism in Cassegrain/Gregorian telescopes including the boresight error with respect to the center of the AR. 85

107 Y Field Angle in Object Space - degrees Corrected For 3 rd Order Coma Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (a) X Field Angle in Object Space degrees (b) Figure 3-9: FFDs showing the magnitude and orientation of FRINGE Zernike polynomials Z7/Z8 related to third order coma for a Ritchey-Chrétien telescope (aplanatic) (a) without misalignments and (b) in the presence of secondary mirror misalignments Y Field Angle in Object Space - degrees a = 0 Y Field Angle in Object Space - degrees a ib +b i X Field Angle in Object Space degrees (a) X Field Angle in Object Space degrees (b) Figure 3-10: FFDs showing the magnitude and orientation of FRINGE Zernike polynomials Z5/Z6 related to third order astigmatism for a Ritchey-Chrétien telescope (aplanatic) (a) without misalignments and (b) in the presence of secondary mirror misalignments. 86

108 Determining the State of Misalignment of Two-Mirror Telescopes with the Aperture Stop Located on the Secondary Mirror and Primary Mirror Perturbations For a two-mirror telescope with the aperture stop located on the secondary mirror (centered on the secondary mirror optical axis) three aberration field centers have to be considered as derived in Section Similarly as in Section the node location for third order coma (assuming a non-aplanatic two-mirror telescope) can be described as given by where and a a H (3-39) AR AR 131 = a = σ + σ + σ (3-40) ( sph) ( sph) ( asph) ( asph) ( sph) ( sph) ( W PM PM W PM PM W ) W131 H AR ( XDEPM BDEPM ) L ( YDE ADE ) ( 1 ) PM PM + BDEPM cpm XDEPM =. cpm d1 yimg ADEPM cpmydepm (3-41) Similarly the node midpoint for third order astigmatism can now be expressed as given by with a a H (3-4) AR AR = + 1 a = σ + σ + σ. (3-43) ( sph) ( sph) ( asph) ( asph) ( sph) ( sph) ( W PM PM W PM PM W ) W When attempting to compute the misalignment parameters utilizing the nodal properties of third order coma and astigmatism it may seem at first sight that there are three unknown aberration field centers to determine. Nevertheless comparing Equation (3-15) with Equation (3-17) it can be seen that the aberration field center associated with the secondary mirror spherical base curve can be expressed as a function of the primary mirror aberration field center associated with its spherical base curve given by 87

109 σ ( sph) ( PM PM ) ( ) ( PM PM + PM ) ( ) XDE BDE c y u = YDEPM ADEPM ypm cpm 1 c d c upm 1 c d σ ( ( + 1 ) + ( + 1) ) ( sph) PM ( XDEPM BDEPM ) ( YDE ADE ) PM PM.... (3-44) Consequently Equation (3-40) and Equation (3-43) can be expressed as where a 1 = ( W + ξw ) σ + W σ (3-45) ( sph) ( sph) ( sph) ( asph) ( asph) PM 131 PM 131 PM PM W 131 a 1 = ( W + ξw ) σ + W σ (3-46) ( sph) ( sph) ( sph) ( asph) ( asph) PM PM PM PM W ξ = ( cpm ypm + up M ) ( + ) + ( + c d ) y c 1 c d c u 1 PM PM 1 PM 1 (3-47) which shows that there are only two unknown parameters (primary mirror aberration field centers) to solve for. After substituting Equations (3-45) - (3-46) into Equations (3-39) and (3-4) the two linear vector equations can be solved for the aberration field centers of the primary mirror. Utilizing Equation (3-15) and Equation (3-19) the primary mirror aberration field centers can be related to primary mirror decenter and tip-tilts given by ( ) XDE = y σ d c y + u σ (3-48) ( asph) ( sph) PM PM PM x 1 PM PM PM PM x ( ) YDE = y σ d c y + u σ (3-49) ( asph) ( sph) PM PM PM y 1 PM PM PM PM y 88

110 ( y u ) ADE = c y σ + c + σ (3-50) ( asph) ( sph) PM PM PM PM y PM PM PM PM y BDE =+ c y c ( y u ) ( asph) ( sph) PM PM PMσPM x PM PM PM σpm x. + (3-51) Alternative expressions for the decenter and tip-tilts of the primary mirror dependent on systems and derived parameters are given by ( f ) ( γ 1) SYS ypm m ( ) ( ) XDEPM = up ( ) sph asph M PM x + ypm PM x m f σ σ (3-5) SYS ( f ) ( γ 1) SYS ypm m ( ) ( ) YDEPM = up ( ) sph asph M PM y + ypm PM y m f σ σ (3-53) SYS BDE PM ypm m ( asph) ypm m ( sph) = σ PM x upm σ PM x (3-54) ( fsys ) ( fsys ) ADE PM ypm m ( asph) ypm m ( sph) = σ PM y + upm σ PM y. (3-55) ( fsys ) ( fsys ) In the case of an aplanatic two-mirror telescope with the aperture stop located on the secondary mirror the only difference compared to the non-aplanatic case is the definition of the aberration field vector for coma. Here an un-normalized form of Equation (3-45) is utilized given by (( W ( sph) ( sph) ) ( sph) ( asph) ( asph) PM ξw PM W PM PM ) A = + σ + σ (3-56) where the vector A 131 describes the amount of coma in x- and y-direction across the field. After computing the aberration field centers for the primary mirror based on Equation (3-4) and Equation (3-56) misalignment parameters can be found utilizing Equations (3-48) - (3-51). 89

111 3.4 Pivot Points in Two-Mirror Telescopes An important concept when considering the alignment of initially randomly misaligned optical systems as well as in the design of optical systems without symmetry is the concept of pivot points. Several misalignment perturbations of an optical surface are possible in general. Surfaces can be decentered with respect to some reference; they can be tilted about their aspheric vertex or a combination of both. When tilting and decentering an optical surface the optical surface movement can be alternatively described as a rotation about a particular external pivot point. In the alignment of astronomical telescopes there are in particular two pivot points of special interest. One pivot point results in zero pointing changes when a combination of tiptilt/decenter is applied that results in a rotation of the surface about its center of curvature which will be discussed in Section the other maintains the field behavior of third order coma which will be discussed in Section Rotations of a Mirror about its Center of Curvature It is important to note that the first order properties of an optical system remain unchanged when rotating a surface about its center of curvature since the aspheric departure (if any) is neglected in the paraxial approximation. Considering the case of a two-mirror telescope with the aperture stop on the primary with a misaligned secondary mirror we could use Equation (3-9) and ( ) AR require H ( )( ) XDE BDE YDE ADE = 0 giving BDE + c XDE = + ADE + cyde = 0 0. (3-57) (3-58) 90

112 Based on Equation (3-57) - (3-58) or directly from Figure 3-11 it can be found that the ratio of the vertex decenter and the tip-tilt corresponds to the distance of the pivot point with respect to the aspheric vertex of the surface given by XDE BDE 1 = = r (3-59) c YDE ADE 1 = = r c (3-60) which corresponds to the center of curvature for the particular surface. When considering the effects of a surface rotation about the center of curvature on the third order aberrations the effect on the surface aberration contribution originating from the spherical base curve have to be considered separately from the effects on the surface aberration contribution originating from the aspheric departure. Considering Equation (3-7) it becomes evident that the aberration field center associated with the spherical base curve remains zero (assuming it has been zero to begin with) for rotations about its center of curvature given by σ ( sph) ( ) ( ) BDE XDE 1 BDE c XDE = = 0. ADE YDE upm (1 + c d1) ADE cyde (3-61) ( sph) An alternative intuitive conclusion that σ has to remain zero can be drawn already from the invariance of a spherical surface for rotations about the center of curvature. ( sph) On a side note if a non-zero σ is assumed as starting point rotations of the surface ( sph) about its current center of curvature leave σ unchanged/constant (consequently leaving also boresight unaffected) and 91

113 σ ( sph) ( BDE + BDE XDE + XDE ) ( ADE + ADE YDE + YDE ) 1 ( BDE + BDE ) c ( XDE + XDE ) ( ADE + ADE ) c ( YDE + YDE ) u (1 c d ) PM = + 1 (3-6) with BDE c XDE = 0 (3-63) ADE c YDE = 0 (3-64) which results in ( BDE + BDE XDE + XDE ) ( ADE + ADE YDE + YDE ) ( sph) σ = 1 BDE c XDE. u (1 c d ) ADE c YDE PM + 1 (3-65) The aberration field center associated with the aspheric departure however does depend on rotations of the surface about the center of curvature and is given by where BDE and ( cc ) ( cc) ( BDE ) ( cc) ( ) ( cc) ( asph) 1 r BDE σ = ( cc) (3-66) ADE d1upm r ADES M ( cc ) ADE are the rotation angles of the secondary mirror about its center of curvature. In practice there are some examples where a rotation of an optical surface about its center of curvature is utilized. One such example is for the alignment of astronomical telescopes. Since typically a fixed detector is utilized centered on some reference axis (typically the AR axis) the property of zero pointing change makes the rotation of a surface about its center of curvature useful. 9

114 center of curvature ( cc) ADE OAR chief- ray marginal-ray ( cc) ADE YDE OA primary mirror and detector OA : optical axis r Figure 3-11: Rotation of the secondary mirror about its center of curvature Rotations of a Mirror about the Coma-Free Pivot Point Another pivot point which becomes fundamentally important in the alignment of optical systems is the concept of the coma-free pivot point. As the name indicates a rotation of the particular surface about this pivot point leaves the (third order) coma aberration field unchanged. In case field-linear coma has been present as encountered for example in Cassegrain telescopes the field point of zero coma remains at the field center. For aplanatic optical systems (i.e. they are corrected for third order coma in the nominal state) coma remains zero across the FOV for rotations about the coma-free pivot point. The general condition for the locating of the coma-free pivot point for an optical system with N surfaces is given by N ( sph) ( sph) ( asph) ( asph) ( W j j W j j ) A = σ + σ = j= 1 which can be simplified in the case of two-mirror telescopes. (3-67) 93

115 Location of the Coma-Free Pivot Point for Two-Mirror Telescopes with the Aperture Stop Located on the Primary Mirror Assuming the most common case of a two-mirror telescope with aperture stop on the primary mirror the coma-free pivot point for the secondary mirror can be obtained by ( sph) ( sph) ( asph) ( asph) ( W W ) A131 = 131 σ σ =0 (3-68) which provides a very useful condition for the sigma vector components (assuming σ 0 σ 0) given by ( sph) ( asph) σ σ W ( sph) ( asph) ( x y) 131 = ( asph) ( sph) ( x y) W131. (3-69) Re-writing Equation (3-69) in structural parameters provides an expression for the distance from the secondary mirror aspheric vertex (in its nominal state) to its coma-free pivot point which is given by (APPENDIX E) L ( cfp) YDE d W = ADE = W W (3-70) ( sph) ( asph) c d1 131 where L denotes the location of the coma-free pivot point measured from the nominal vertex ( cfp ) position of the secondary mirror YDE corresponds to the decenter of the secondary mirror optical axis ADE corresponds to the tilt angle of the secondary mirror optical axis (in rad) y corresponds to the paraxial chief-ray height at the secondary mirror i corresponds to the angle of incidence of the paraxial chief-ray at the secondary mirror (in rad) and c denotes the curvature of the secondary mirror. The right hand side of Equation (3-70) only depends on aberration coefficients of the nominal optical system and structural parameters. 94

116 ( cfp) Any rotation of the secondary mirror about this pivot point (denoted as ADE in Figure 3-1) affects astigmatism and the medial focal surface but it does not affect coma to third order. This clearly demonstrates that having zero-coma at the field center is possible with an infinite amount of combinations of spherical and aspheric sigma vectors all fulfilling the condition given in Equation (3-67). While on-axis coma in a two-mirror telescope exhibiting rotational symmetry in its nominal state always implies misalignments the existence of a coma-free pivot point clearly shows that zero coma on-axis does not necessarily correspond to an aligned telescope. Importantly an alignment process with the aim to remove misalignment induced coma and astigmatism can be decoupled when utilizing the coma-free pivot point. Specifically one can tiptilt or decenter the secondary mirror to remove misalignment induced coma and subsequent rotations of the mirror about the coma-free pivot point facilitate the removal of misalignment induced astigmatism without affecting the coma aberration field. coma free pivot point ( cfp) ADE OAR chief-ray marginal-ray ( cfp) ADE YDE primary mirror and detector prime focus ( cfp) L Figure 3-1: Coma-free pivot point for the secondary mirror of a two-mirror telescope with the aperture stop located on the primary mirror. 95

117 3.4.. Location of the Coma-Free Pivot Point for Two-Mirror Telescopes with the Aperture Stop Located on the Secondary Mirror In the case of a two-mirror telescope with the aperture stop location on the secondary mirror for reasons as already discussed in Section there exists a coma-free pivot point for the primary mirror. One can determine the location of the coma-free pivot point for the primary mirror by requiring ( ) ( ) ( ) ( ) ( ) ( ) ( W sph sph asph asph sph sph PM PM W PM PM W ) A σ σ σ (3-71) 131 = =0. Comparing Equation (3-71) with Equation (3-68) one may notice that both the aberration field centers of the primary mirror and the aberration field center associated with the spherical base curve of the secondary mirror have to be included while the aberration field center for the secondary mirror aspheric departure is zero. Since by definition the OAR will reflect of the secondary mirror at the center of the aperture stop there will be no displacement of the aspheric vertex of the secondary mirror with respect to the OAR. An expression for the coma-free pivot point for the primary mirror (aperture stop located on the secondary mirror) referenced to the secondary mirror optical axis can be obtained by substituting Equation (3-15) Equation (3-17) and Equation (3-19) into Equation (3-71) giving L i W + W + W ( ) ( asph) ( sph) ( sph) 131 PM PM ( cfp) YDEPM ipm PM = = ADE i PM ( asph) cpm i ( sph) ( sph) W131 PM + W131 PM + cpmw131 ypm ipm. (3-7) The derived expressions given in Equation (3-70) and Equation (3-7) for the location of the coma-free pivot point of the primary and secondary mirror have been kept completely 96

118 general. Here expressions are given for the specific cases of a Cassegrain and Ritchey-Chrétien telescope with the aperture stop located at the primary mirror. ( cfp) ADE PM OAR chief ray marginal ray PM YDE PM prime focus ( cfp) ADE PM coma free pivot point ( ) cfp L PM Figure 3-13: Coma-free pivot point for the primary mirror of a two-mirror telescope with the aperture stop located on the secondary mirror. Special Case A: Cassegrain Telescope with the Aperture Stop Located on the Primary Mirror The location of the secondary mirror coma-free pivot point for a Cassegrain telescope can be obtained by substituting the expressions for the wave aberration coefficients in Equation (3-70). For the Cassegrain telescope the result for the coma-free pivot point takes on a particularly simple form given by L ( fsys ) γ L = = m m (3-73) ( cfp) CASSEGRAIN. Equation (3-73) shows that the location of the coma-free pivot point for the secondary mirror coincides with the prime focus. In the specific case of the Cassegrain telescope one can alternatively find the coma-free pivot point merely by a thought experiment. Given by the 97

119 parabolic primary mirror a stigmatic image is formed at the prime focus for the on-axis field point. If such a coma-free pivot point for the secondary mirror exists then coma at the field center has to remain zero which is only the case by pivoting the secondary mirror about the virtual object for the secondary mirror which is located at the prime focus. The dependence of the coma-free pivot point location with respect to its aspheric vertex is visualized for different secondary mirror magnifications and axial obstruction ratio in Figure 3-14 in normalized form with respect to the system focal length. (distance secondary mirror coma-free pivot to vertex) /f SYS m secondary mirror magnification γ axial obstruction ratio Figure 3-14: Dependence of the location of the coma-free pivot point in Cassegrain telescopes on the secondary mirror magnification and axial obstruction ratio expressed as distance from the secondary mirror vertex normalized by the system focal length. Special Case B: Ritchey-Chrétien Telescope with the Aperture Stop Located on the Primary Mirror Similarly as in Case A the wave aberration coefficients for coma have been substituted in Equation (3-70) assuming the aperture stop is located on the primary mirror. This substitution provides an expression for the location of the coma-free pivot point for the secondary mirror of a Ritchey-Chrétien telescope given by 98

120 L ( cfp) RITCHEY = ( fsys ) m ( 1) γγ ( 1). m m ( γ 1) γ (3-74) Equation (3-74) can be re-cast to better visualize the difference of the coma-free pivot point location compared to the Cassegrain telescope given by L ( fsys ) γ 1 1 = 1+ = L 1 + m m ( γ 1) γ m ( γ 1) γ ( cfp) ( cfp) RITCHEY CASSEGRAIN (3-75) with its visual interpretation given in Figure secondary mirror magnification m (distance secondary mirror coma-free pivot to vertex)/f SYS γ axial obstruction Figure 3-15: Distance of the coma-free pivot point from the secondary mirror vertex normalized with the system focal length dependent on secondary mirror magnification and axial obstruction ratio. Special Case C: Consequences of Distributing the Conic Constants to Obtain a Two- Mirror Coma Stabilized Optical System for a Particular Coma-Free Pivot Point Distance Assuming the Aperture Stop is Located on the Primary Mirror So far it has been demonstrated that for a particular conic constant of the secondary mirror a coma-free pivot point can be found. The question arises whether one can re-distribute the conic constants among the primary and secondary mirror to obtain a stigmatic image on axis (i.e. zero 99

121 spherical aberration) while choosing a desired location of the coma-free pivot point. Assuming the aperture stop is located on the primary mirror a set of conic constants can be derived fulfilling the condition for the coma-free pivot point (i.e. A 131 = 0) and at the same time leading to zero spherical aberration given by k PM ( fsys ) ( cfp) ( m 1) γ( L m+ γ) = 1 + (3-76) ( cfp) 3 L m ( fsys ) ( cfp) ( m+ 1) L (1 m) γ k (3-77) =. ( cfp) L (1 m) The resulting nominal coma of the telescope assuming conic constants as described in Equation (3-76) and Equation (3-77) can be expressed as given by ( SYS )( m 1)( 1) f γ γ W W m L m ( γ ) Cassegrain 131 = γ ( cfp) 1 (3-78) where 3 Cassegrain EPD upm W 131 =. (3-79) 3 ( f ) SYS The nominal coma according to Equation (3-78) normalized with the corresponding coma of the equivalent Cassegrain solution is visualized in Figure 3-16 for various locations of the coma-free pivot point and secondary mirror magnifications for typical secondary mirror axial obstruction ratio. It is shown that for each axial obstruction ratio and secondary mirror magnification there exists only one location for the coma-free pivot point where W 131 = 0 corresponding to the well known Ritchey-Chrétien telescope design. All other locations for the coma-free pivot point will result in non-zero coma of the nominal system. Note that while one can design a telescope configuration that has a short distance of the coma-free pivot point to the 100

122 surface vertex as one typically desires for infrared chopping the price to be paid is an increased amount of coma of the nominal system. γ = 0.15 γ = 0.5 γ = 0.35 W131/W131Cassegrain distance secondary mirror pivot to vertex/f SYS Ritchey-Chretien solution m secondary mirror magnification Figure 3-16: Nominal system (third order) coma for various distributions for the conic constants to achieve a particular position of the coma-free pivot point. Special Case D: Extreme Cases for the Location of the Coma-Free Pivot Point Assuming the Aperture Stop is Located on the Primary Mirror Considering Equation (3-70) several special cases arise when the numerator or denominator approach zero or infinity. In the case of the numerator approaching zero (i.e. W ) the secondary mirror has zero curvature. Consequently any secondary mirror ( sph) tilt only affects the pointing but not the aberrations of the optical system. Also of limited importance is the case when the coma-free pivot point approaches the secondary mirror vertex ( asph) for W which also can be observed in Figure The extreme case of the denominator of Equation (3-70) approaching zero (i.e. W ( asph) 131 c d1w ) is more interesting and reveals some rather unusual behavior. In this case the coma-free pivot point moves to infinity; a rotation of the secondary mirror about the 101

123 coma-free pivot point located at infinity corresponds to a pure decenter movement of the secondary mirror. Consequently if W + c d W the telescope has zero sensitivity ( asph) for coma induced by secondary decenter which especially under harsh environmental conditions (shock vibrations temperature transients etc.) might be desirable. Utilizing W ( asph) 131 c d1w and substituting the wave aberration expressions for coma as given in Table 3- an expression for the conic constant of the secondary mirror can be obtained given by k [ γ ] c d ( m+ 1) 1 m ( m+ 1) = 1 (1 + c d1)(1 m) ( γ 1). (3-80) Given the conic constant for the secondary mirror one can determine a corresponding primary mirror constant to obtain stigmatic on-axis imaging corresponding to zero spherical aberration. An expression for the primary mirror conic constant for W040 0 can be obtained based on expressions for spherical aberration provided in Table 3-1 given by k PM 3 3 (1 + c d1) m + 1 (1+ c d1) m+ m + ( + 3 c d1) m γ + ( m+ 1) (1 m) γ =. + 3 (1 c d1) m ( γ 1) (3-81) Expressing the structural design parameters in terms of secondary mirror magnification and axial obstruction ratio Equations (3-80) and Equations (3-81) can be drastically simplified resulting in an expression for the conic constants for the location of the coma-free pivot point at infinity and stigmatic on-axis imaging given by k m + 1 = m 1 (3-8) k PM = 1+. m γ (3-83) 10

124 To demonstrate the required mirror shapes as defined by Equation (3-8) - (3-83) the conic constants are visualized in Figure It is shown that the secondary mirror corresponds to an oblate ellipsoid (Figure 3-17(a)) consequently somewhat challenging to fabricate and test and the primary mirror has the shape of a prolate ellipsoid (Figure 3-17(b)). k k PM m secondary mirror magnification (a) γ m γ axial obstruction ratio secondary mirror magnification (b) axial obstruction ratio Figure 3-17: (a) Secondary mirror conic constant as a function of secondary mirror magnification and axial obstruction ratio to obtain zero decentering sensitivity for coma and (b) primary conic constant to provide in combination with the secondary mirror conic constant stigmatic imaging on-axis (zero spherical aberration). 3.5 A Unique Astigmatic Nodal Property in Misaligned Cassegrain Gregorian and Ritchey- Chrétien Telescopes With Misalignment Coma Removed We present the aberration field response of Cassegrain Gregorian and Ritchey-Chrétien telescopes with the aperture stop on the primary mirror to secondary mirror misalignments. More specifically we derive a general condition for the geometry of the binodal astigmatic aberration field for a telescope that has been aligned to remove field-constant coma using the methods as described in the previous sections. It has been observed that when the coma caused by secondary mirror misalignments is removed the astigmatic field is typically not symmetric 103

125 around the periphery but significantly it is always effectively zero on-axis. This observation is a manifestation of binodal astigmatism where one of the astigmatic nodes remains at or near the field center. Here we show how the condition to remove field-constant coma simultaneously creates a constraint whereby one of the astigmatic nodes must remain effectively on-axis. This result points to why the alignment of a Cassegrain Gregorian and Ritchey-Chrétien telescopes based on axial imagery is insufficient and demonstrates exactly the geometry of the remaining misalignment aberration field providing insights into more complete alignment approaches. Historically until relatively recently astronomers have been aligning large telescopes onsite on the basis of removing the appearance of image asymmetry on-axis or with the equally effective technique of centering the obscuration shadow in a slightly defocused on-axis image. More specifically the tilt or the decenter of the secondary mirror is adjusted until the on-axis image is coma-free. Since coma depends on aperture-cubed it is readily seen as asymmetry in an astronomical image. Alternatively coma can be measured with wavefront sensing techniques that are deployed on-site to identify the aberrations of the wavefront often in terms of Zernike coefficients which are readily converted to the wave aberration coefficients to be used here. Prior to the present decade most deployed professional astronomical telescopes have been twomirror designs and in the last 70 years many have been Ritchey-Chrétien telescopes. McLeod 1996 and others (Wilson 1999 Noethe 000 and Rakich 008 Schmid 010b ) have recognized that in fact an alignment method based exclusively on axial imagery is not sufficient to ensure alignment. They report methods for achieving full alignment in the case of rotationally symmetric mirrors whereby both the tilts and decenters of the secondary mirror are controlled to both eliminate on-axis degradation and also equalize the performance (third order 104

126 astigmatism) around the periphery of the field of view to be unchanged in magnitude and orientation compared to astigmatism of the aligned telescope. More specifically there is a fixed external pivot point away from the secondary mirror vertex but located on the optical axis of the primary mirror) which the secondary mirror can be rotated about that will affect the astigmatic field but will not result in the introduction of coma into the optical system. Other types of secondary mirror misalignment perturbations will induce field-constant coma thus including the presence of coma on-axis. Under the conditions where field-constant coma has been removed through secondary mirror alignment McLeod 1996 describes the variation in astigmatism around the periphery of the field of view. A recent paper by the authors (Schmid 010b) puts the findings of McLeod 1996 in the context of Nodal Aberration Theory and in doing so provides a common framework to plan the alignment of observatory-class two-mirror telescopes built dominantly during the 0 th century. Significantly Schmid 010b provides a foundation for the current generation of more than two-mirror telescopes such as the Large Synoptic Survey Telescope where in-fact an understanding of the nodal properties of higher order aberration fields is significant in the alignment process. It was as part of Schmid 010b that the unique nodal behavior now explained here (Schmid 010a) was discovered through an example. A discussion of field-astigmatism in misaligned two-mirror telescopes has also been given by Wilson 1999 and more recently in a detailed paper by Noethe and Guisard 000 who showed that for the specific case of a misaligned Cassegrain telescope that has been aligned to obtain zero misalignment induced coma the on-axis point is free of astigmatism. Also Noethe and Guisard postulated that the conclusions arrived at for Cassegrain telescopes should be approximately valid for Ritchey-Chrétien telescopes. 105

127 Using nodal aberration theory this work illustrates why the alignment of Cassegrain Gregorian and Ritchey-Chrétien telescopes (stop at primary mirror) to remove field-constant coma also results in the removal of axial astigmatism even though the astigmatic field varies around the periphery of the field. In this context the term field center or on-axis refers to the intersection point of a ray that is parallel to the primary mirror optical axis with the detector. In general in the presence of secondary mirror misalignments the intersection point differs from the primary mirror optical axis intersection point with the image plane. This result of having one astigmatic node located very close to the field center is an important special case of misalignment aberration fields of astronomical telescopes which has significant consequences in the development of alignment plans. Specifically this result illustrates that wavefront sensors placed on-axis will not reveal residual component misalignments Nodal Constraints of a Coma-Aligned Cassegrain or Gregorian Telescope with the Aperture Stop Located on the Primary Mirror In Cassegrain or Gregorian telescopes the conic constants are chosen that spherical aberration of each mirror is compensated independently. The dominant image degrading aberration that remains is third order coma. This choice of conic constants has important and far-reaching consequences on the nodal properties of coma and astigmatism in the presence of misalignments assuming an aperture stop location at the primary mirror. As will be shown a thorough understanding of this nodal behavior is of fundamental importance in the development of alignment strategies. 106

128 It has been observed that in the presence of secondary mirror misalignments one of the astigmatic nodes typically lies close to the field center (Figure 3-18(b)) for realistic misalignments. Furthermore it has been found that after the secondary mirror has been adjusted to compensate misalignment induced coma that the node that was already close to the field center has moved to exactly coincide with the field center (Figure 3-19(b)). To understand why this nodal behavior is in fact a general property of Cassegrain or Gregorian telescopes one has to consider the aberration field vectors for coma and astigmatism. Utilizing Equation (-30) it can be determined that having one astigmatic node at the field center (in the presence of misalignments) requires following condition on the aberration field vectors of astigmatism given by where B b = a (3-84) W = 0 1 a = σ + σ (3-85) ( sph) ( sph) ( asph) ( asph) ( W W ) W B σ σ (3-86) ( sph) ( sph) ( asph) ( asph) = W + W =0. While one could utilize Equation (3-84) in the remaining derivation it is found to be less tedious to utilize the condition stated in Equation (3-86). Applying the rules of vector multiplication in Equation (3-86) and substituting an expression for the ratio of the sigma vector components obtained by (W 131 0) resulting in 1 a = σ + σ =0 (3-87) ( sph) ( sph) ( asph) ( asph) ( W W ) W

129 σ σ W ( sph) ( asph) ( x y) 131 ( x y) = ( asph) ( sph) ( x y) W131 ( x y) (3-88) a general condition on the secondary wave aberration contributions for coma and astigmatism can be found given by ( asph) ( asph) 131 W = ( sph) ( sph) 131 W W. W (3-89) Substituting expressions for the wave aberration terms for Cassegrain (or Gregorian) telescopes into Equation (3-89) one obtains the left and right hand side of Equation (3-89) given by and ( asph) 131 (1 m) ( γ 1) ( sph) = 131 (1 (1 ) γ ) W + W m + m (3-90) W + = W m + m ( asph) (1 m) ( γ 1) ( asph) (1 (1 ) γ ). (3-91) Comparing Equation (3-90) and Equation (3-91) one finds that the required equality obtained in Equation (3-89) to have one astigmatic node coincident at the field center is fulfilled exactly for the case of a Cassegrain or Gregorian telescope which explains the nodal behavior observed. 108

130 0.06 waves (63.8 nm) wave (63.8 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees a +b i ib X Field Angle in Object (a) Space degrees X Field Angle in Object (b) Space degrees Figure 3-18: FFDs for the case of a Cassegrain telescope showing the FRINGE Zernike coefficients (a) Z7/Z8 and (b) Z5/6 assuming arbitrary secondary mirror misalignments waves (63.8 nm) wave (63.8 nm) Y Field Angle in Object Space - degrees a 131 = 0 Y Field Angle in Object Space - degrees a ib +b i X Field Angle in Object (a) Space degrees X Field Angle in Object (b) Space degrees Figure 3-19: FFDs for the case of a Cassegrain telescope showing FRINGE Zernike coefficients (a) Z7/Z8 and (b) Z5/6 after aligning the secondary mirror for zero on-axis coma. 109

131 3.5. Nodal Constraints of a Coma-Aligned Ritchey-Chrétien or Aplanatic Gregorian Telescope Assuming the Aperture Stop is Located on the Primary Mirror With the aim of providing a versatile functionality many observatories today deploy Ritchey- Chrétien telescopes allowing the imaging of a wider FOV because of aplanatism compared to classical Cassegrain telescopes. In Ritchey-Chrétien telescopes with the aperture stop located on the primary mirror misalignment induced third order coma can be expressed as given by (sph) ( sph ) (asph) ( asph) A131 = W131σ + W131 σ (3-9) W =W + W +W =0 (3-93) ( sph) ( sph) ( asph) PM Figure 3-0(a) shows the surface-by-surface distribution of individual mirror wave aberration coefficients for coma and astigmatism including a separation of the base sphere and aspheric/conic contributions for a typical Ritchey-Chrétien telescope (stop at the primary mirror) fully described in Table D-8. It is important to note that both coma contributions (spherical and conic/aspheric) of the secondary mirror are of the same sign. This is a general condition of a Ritchey-Chrétien telescope (with the stop at the primary mirror) as the terms are working to cancel the intrinsically larger contribution from the primary mirror (Figure 3-0(a)). In contrary considering the signs for the secondary mirror conic/aspheric and spherical contributions of astigmatism it is observed that the signs are opposite (Figure 3-0(b)) which will be important for the constraint on the astigmatic node locations discussed. Apart from the trivial case of ( sph) ( asph) perfect secondary mirror misalignments ( σ = σ = 0 ) the two σ -vectors must point into opposite directions when requiring zero misalignment induced constant coma (i.e. A = ) ( sph) ( asph) given the identical signs and similar magnitudes of W and W

132 Figure 3-0(c) illustrates the constraint on the aberration field center vectors when requiring zero misalignment induced constant coma. For a typical set of secondary mirror misalignments (listed in Figure 3-0) the aberration field center vectors σ and ( sph )* σ are ( asph )* visualized where the * refers to the initial misalignments before the compensation of misalignment induced coma. In this case a tip-tilt to the secondary mirror was applied to cancel constant coma leading to a new vector pointing to the center of the aberration field due to the base sphere of the secondary mirror denoted by σ ( sph). As shown in Figure 3-0(c) the applied tip-tilt to the secondary mirror only modified the σ ( sph) -vector which now points in exactly opposite direction to the aspheric aberration field center vector. The aspheric aberration field ( asph)* ( asph) center vector remains unaffected ( σ = σ ) since a tip-tilt of the secondary mirror does not change the location of the secondary mirror vertex. In general there are any number of combinations of tilt and decenter of the secondary mirror that effectively rotate the secondary mirror about a fixed point on the optical axis of the primary mirror that will result in the cancellation of constant coma. The nodal vector for astigmatism describing the distance between the astigmatic nodes b is defined (Shack 1980 Thompson 005) as b B a W (3-94) with for a Ritchey-Chrétien telescope with the aperture at the primary mirror B σ σ (3-95) ( sph) ( sph) ( asph) ( asph) = W + W 111

133 Coma [waves] W ( sph) 131PM W (a sph) 131PM W ( sph) 131 W COMA (a) (a sph) 131 Astigmatism [waves] W ( sph) PM W ASTIGMATI (a sph) PM W (b) ( sph) W (a sph) y-field [norm.] σ σ asph ( = ) ( asph) ( )* before coma compensation: XDE =.5 mm YDE = mm ADE = deg BDE = deg ( sph) ( sph) ( asph) σ σ = σ ( sph)* σ 0.5 x-field [norm.] after coma compensation: XDE =.5 mm YDE = mm ADE = deg BDE = deg 1.0 (c) Figure 3-0: Surface-by-surface wave aberration coefficients for (a) coma and (b) astigmatism of a typical Ritchey-Chrétien telescope including the separation of the contribution from the spherical base curve of a conic/aspheric mirror and the contribution from the conic/aspheric departure and (c) aberration field center vectors for the spherical base curve and conic/ aspheric contributions of the secondary mirror of a misaligned Ritchey-Chrétien telescope. The opposing ( ) alignment of sph ( ) σ and σ asph is a key constraint imposed by the correction of misalignment induced coma. A = 1 a σ + σ. (3-96) ( sph) ( sph) ( asph) ( asph) ( W W ) W W In investigating this model a general result was discovered that cannot be easily seen from the original work but is in-fact a useful general insight for the form of the misalignment 11

134 induced astigmatic field of a Ritchey-Chrétien telescope with the stop at the primary mirror. In the original work the nodes for astigmatism are generally found given by 1 ( ) W AST 3 = W H a + b = ρ 0 (3-97) ( ) H a = b (3-98) ( ) H a =± b (3-99) H = a ±i b. (3-100) Graphically as shown in (Shack 1980 Thompson 005) the operation of taking the square root of a negative vector b results in a rotation of the b -vector by ± 90 before it is added to a. This operation does not result in any general constraint on either of the astigmatic node locations. In the case of a Ritchey-Chrétien telescope that has been aligned to remove field-constant coma one can make an important general conclusion that does provide a constraint on one of the two astigmatic nodes. To recognize the intrinsic astigmatic nodal behavior of a coma-aligned Ritchey-Chrétien telescope one needs to look at the interaction of the components of A 131 a B knowing now that requiring zero field-constant coma ( A = ) results in the constraint and that the two σ -vectors for the secondary mirror aberration field centers must take on opposite directions. With this new knowledge consider the implications for b and more specifically the components of B given by Equation (3-95). As presented in APPENDIX A of Thompson 113

135 005 a squared vector is the same vector whether the component vector is considered positive or negative. From Figure 3-0 the components for the two wave aberration coefficients for the secondary mirror for astigmatism are of opposite sign which is in fact a general property of Ritchey-Chrétien telescopes. They are also of similar magnitude assuming a typical location of the image plane (behind the primary mirror) a secondary mirror magnification between ~ - 6 and an axial obstruction ratio within 0.15 to Therefore for any practical Ritchey-Chrétien telescope with the aperture stop on the primary mirror B << a and as a general result ( ) b a (3-101). Now revisiting Equation (3-98) with this new result for a coma-aligned telescope ( ) ( ) H a = b a a (3-10) H a ± a (3-103) H = a ± a 0 a. (3-104) This is a general result that describes the most common state of alignment for a Ritchey- Chrétien telescope that has been aligned to remove field-constant coma but has not been aligned based on the periphery of the field (Schmid 010a). As demonstrated aligning the telescope for zero field-constant coma removes all axial wavefront aberration but it does not put any constraints on image quality degradation at the edge of the field of view which is illustrated in Figure 3-1. Specifically Figure 3-1(a) compares the astigmatism in the case of rotational 114

136 symmetry with binodal astigmatism after aligning the telescope for zero field-constant coma Astigmatism [waves] Astigmatism [waves] (Figure 3-1(b)). a ( 00) y-field angle y-field angle x-field angle x-field angle (b) (a) Figure 3-1: Example of (a) field-quadratic astigmatism (magnitude) in case of an aligned Ritchey-Chrétien telescope (b) misalignment induced binodal astigmatism (third order) of a Ritchey-Chrétien telescope aligned for zero fieldconstant third order coma. An alternative approach to assess the nodal behavior of astigmatism in Ritchey-Chrétien telescopes with the aperture stop at the primary mirror after compensating misalignment induced coma is to utilize the general expression developed in Equation (3-89). Substituting the expressions for coma and astigmatism for the secondary mirror into Equation (3-89) one obtains for the Ritchey-Chrétien telescope (aperture stop at primary mirror) { 1 m [ 1 + (m + 1)m(γ 1) γ ] + γ } = + ( 1) 1 ( 1) γ m m m [ ] (3-105) (γ 1) { 1 m [ 1 + (m + 1)m(γ 1) γ ] + γ } =. (1 m) [1 m (m + 1)γ ] (3-106) ( asph ) W131 ( sph ) W 131 ( asph ) W ( sph ) W 115

137 Since these expressions are not identical in contrast to the classical Cassegrain telescope the ratio between Equation (3-105) and Equation (3-106) is visualized in Figure 3-. As one can see for very small secondary mirror magnifications the expressions given in Equation (3-105) and Equation (3-106) differ but are quite similar for common values of m ~ 4 or larger (i.e. the plotted ration in Figure 3-(a) is almost equal to 1). Even though Figure 3-(a) indicates that the nodal behavior in the case of large secondary mirror magnifications should be similar to the classical Cassegrain telescope since the wave aberration ratio given in Equation (3-105) and Equation (3-106) fulfill the condition stated in Equation (3-89) it is not clear to which extent the nodal behavior is different for small secondary mirror magnifications. To analyze the effect at small secondary mirror magnifications the astigmatic node positions for an observatory class Ritchey-Chrétien telescope (EPD = 3 m f SYS = 3 m) assuming secondary misalignments are shown in Figure 3-(b-d) for various choices of secondary mirror magnifications and axial obstruction ratio. As demonstrated the node displacement from the field center is indeed negligible in essentially all cases even at low secondary mirror magnifications (Figure 3-(b-d)). 116

138 W W W W ( asph) 131 ( sph) 131 ( asph) ( sph) m secondary mirror magnification (a) ib +b i a γ axial obstruction ratio ib +b i a (b) m = m = 3 ib m = 5 a +b i (c) (d) Figure 3-: (a) Visualizing the ratio of Equation (3-105) and Equation (3-106) for secondary mirror magnifications m =..10 axial obstruction ratio γ = and (b-d) astigmatic node and node midpoint positions for various secondary mirror magnifications assuming identical secondary mirror perturbations as given in Figure

139 CHAPTER 4: MISALIGNMENT ABERRATION FIELDS IN THREE- MIRROR ANASTIGMATIC ASTRONOMICAL TELESCOPES In Chapter 4 it is demonstrated how a complete model (to third order) for misalignment induced aberrations can be developed for three-mirror anastigmats (TMAs) as a direct extension from the same underlying concepts as presented for two-mirror telescopes as shown in Chapter 3. In Section 4.1 a brief summary of the most important astronomical telescope configurations with corresponding aberration fields are given for the previous century. In Section 4. it is demonstrated how the third order response of any TMA telescope to lateral decenters and tip-tilts can be derived based on the framework of nodal aberration theory. In Section 4.3 an interpretation of the aberration field response function is given and important consequences for alignment strategies are discussed. Section 4.4 concludes the chapter by applying the discussed theory to a James Webb Space Telescope (JWST) like TMA demonstrating how the obtained insight into the nodal properties can be leveraged in the development of alignment algorithms. 4.1 Abberation Fields of Telescope Configurations Throughout the 0 th Century Two-mirror telescopes have dominated large astronomical telescope forms for over 100 years beginning with the 100 telescope at the Lick Observatory (1890) and Mt. Wilson (1900) moving on to Mt. Palomar s 00 (199) and continuing to the Hubble Space Telescope (.3 m 1994). Even though the first concepts for three mirror telescopes capable of providing widefield imaging based on the absence of third order spherical aberration coma and astigmatism date back to Paul 1935 it has been only recently that these configurations are considered for 118

140 large observatory class telescopes. One of the biggest remaining disadvantages of telescopes utilizing three or more reflections is the reduced throughput compared to conventional twomirror two-reflection telescopes. As an example assuming a typical reflectivity of an aluminum coated mirror (aperture diameter size ~ 8 m) in the order of ~ 85% yielding ~7% reflectivity for two reflections. In comparison moving to three reflections would already decrease the overall reflectivity to ~ 61% but since most three-mirror telescope configurations are disadvantageous from stray light considerations and/or the focal plane position (Wilson 1996) four mirrors will be used in many cases consequently even further reducing the throughput to ~5%. Having recognized the severe imbalance of other developments in the context of astronomical instrumentation compared to mirror reflectivity lead to enhanced efforts in the development of advanced coatings and a reflectivity/reflection of ~95% seems feasible within the next decade (Wilson 1996). The next major space-borne observatory the James-Webb-Space Telescope will be a 6.6 m field-biased obscured three-mirror anastigmat (TMA) leveraging the absence of the three primary third order aberrations: spherical aberration coma and astigmatism. Assuming residual misalignments the performance over the used field of view of TMA telescopes is typically dominated by third order misalignment aberrations. Here it is shown that there are mainly two dominant third order misalignment aberrations that arise for any TMA telescope. One aberration field-constant third order coma is a well known misalignment aberration form commonly seen in two-mirror Ritchey-Chrétien telescopes. The second aberration fieldasymmetric field-linear third order astigmatism is a new and unique field dependence derived 119

141 here for the first time using nodal aberration theory. Knowledge of this new field dependence can greatly aid in developing TMA alignment plans. For three-mirror anastigmatic (TMA) telescopes the aberration field response to misalignments is also predictable irrespective of the details of the telescope as was the case for two-mirror telescopes. Here third order nodal aberration theory as developed in Thompson 005 predicts the response of TMA telescope systems to misalignments and can be used to demonstrate what aberration fields dominate during the alignment of TMA telescopes. Significantly while the appearance of third order coma on-axis is still a key characteristic of misaligned TMAs the second key and common characteristic is the appearance of third order astigmatism with a new unique aberration field dependence field-asymmetric field-linear third order astigmatism. Most importantly this form of astigmatism typically remains centered in the field. As a result if alignment is conducted using only on-axis measurements it is nearly guaranteed that the TMA will remain in an unacceptably misaligned condition. The alignment of a TMA telescope requires measurements at multiple field points some or all of which should be near the edge of the field format. For an astronomical TMA telescope we can concentrate on the third order aberrations that degrade the quality of the image when the telescope is in a misaligned state. These include third order coma and third order astigmatism. More specifically TMA telescopes by definition have a corrected third order coma and a corrected third order astigmatic field. In addition to the field aberrations there is a need to control axial spacings to avoid the introduction of mainly third order spherical aberration which is well known from the context of two-mirror telescopes and applies equally to TMA telescopes. 10

142 4. Application of Nodal Aberration Theory to Derive the Dominant Characteristic Response of any TMA Telescope to Misalignments Given that third order coma and third order astigmatism are the aberrations that will limit the performance of TMA-based astronomical telescopes Equation 4.3 of Thompson 005 can be used to initiate the derivation of the dominant aberrations of a misaligned TMA. Specifically the wave aberration expansion through third order for a misaligned TMA telescope (or any misaligned optical system for that matter) is given by ( ρ ρ) ( H ρ) ( ρ ρ) W = W + W + W j j ( H σ j) ρ ( ρ ρ)... + W 131 j +... j ( H σ ) ρ W ( H σ ) ( H σ ) ( ρ ρ)... + W j j j 0S jj j j ( H ) ( H ) ( H )... + W 311 j σ j σ j σ j ρ j (4-1) or in slightly different form by utilizing the concept of the medial focal surface given by ( ρ ρ) ( H ρ) ( ρ ρ) W = W + W + W j j ( H σ j) ρ ( ρ ρ)... + W 131 j +... j W ( ) + W ( ) ( ) ( ) +... H σ ρ H σ H σ ρ ρ j j j 0Mj j j j ( H ) ( H ) ( H )... + W 311 j σ j σ j σ j ρ j (4-) where all notations are described in Section.5.1. Concentrating first on the coma term (W 131 ) in Equation (4-1) given by it is instructive to re-write the summation as given by 11 ( ) ( ) W = W H σ ρ ρ ρ (4-3) COMA3 131 j j j

143 W = W W ( H ) ( σ ) ρ ( ρ ρ) COMA j 131 j j. j j (4-4) Here the first summation results in the contribution of the rotationally symmetric system which for a TMA telescope is zero given by W 0 j 131 j H = W131 H =. (4-5) The second summation is the sum of the surface contribution displacement vectors in the image plane σ j each weighted by the corresponding surface contribution to the wave aberration for third order coma W 131j where the subscript j refers to the surface number which typically will contain an aberration contribution originating from the spherical base curve and an aberration contribution caused by its aspheric departure. This summation results in a net unnormalized vector in the image plane given by ( sph) ( sph) ( asph) ( asph) ( W j j W j j ) A σ + σ j (4-6) which parameterizes the magnitude and the orientation of the coma aberration field. This leads immediately to the equation that describes the misalignment induced third order coma aberration in TMAs given by [ ] W COMA 3 = A131 ρ ( ρ ρ ). (4-7) This is a third order coma aberration that is constant over the field of view which is characteristic of any misaligned optical system with overall correction of third order coma. Following a similar derivation for third order astigmatism (W ) the term for astigmatism in Equation (4-) is utilized given by 1

144 1 WAS ( ) 3 T= W jh H W jσ j + W jσ j ρ. j j j (4-8) For a TMA telescope which is corrected for third order astigmatism the first term in Equation (4-8) is zero leading to which then gives W j H = W H = 0 (4-9) j 1 W ( W ) W AS 3 T= j j j j. H σ + σ ρ j j (4-10) Here it is important to recall that Equation (4-10) in particular involves the use of an uncommon operation referred to as vector multiplication which is summarized in APPENDIX B. Similarly as done for coma two unnormalized aberration field vectors for astigmatism are defined given by and W j j j A σ (4-11) W j j j B σ (4-1) which allows writing the characteristic field dependence for astigmatism in a misaligned TMA as given by 1 WAST = 3 +. HA B ρ (4-13) Unlike coma which in a misaligned TMA is constant over the field in magnitude and orientation the astigmatism for a misaligned TMA has both a linear with field component and a constant with field component. 13

145 Combining Equations (4-7) with Equation (4-13) the general aberration state for a misaligned TMA telescope is given by where the notation HA and 1 WNOSYM [ A ]( ) ρ ρ ρ + + HA B ρ (4-14) B denote vector multiplications. These two components are illustrated in Figure 4-1. A terminology for the astigmatic term which is isolated here as a dominant characteristic of misaligned TMA telescopes for the first time is field-asymmetric field-linear third order astigmatism. To better understand the orientation of field-asymmetric field-linear third order astigmatism one may assume for a moment that astigmatism is small but non-zero. Equation (4-8) can be re-written utilizing the definitions of the corresponding aberration field vectors given by W AST3 1 = W +. H HA B ρ (4-15) Expressing the zeroes of Equation (4-15) also referred to as astigmatic nodes one obtains A A B H = ±. (4-16) W W W Comparing the two terms under the square root one recognizes that the first term tends to be considerably larger mainly for two reasons. First the wave aberration coefficients contained in the A -vector are squared as well in contrast to B where only the σ j -vectors are squared (which tend to be small for realistic misalignment residuals) and secondly 1 1 (4-17) W W 14

146 since W has been assumed to be small. Consequently in many cases one may approximate Equation (4-16) given by A A A H ± = 0. (4-18) W W W Equation (4-18) demonstrates that one of the nodes is approximately at the field center and the other zero moves rapidly outside of the field of view as W becomes small. As observed in Figure 4-1(d) even in the case of W 0 the orientation of astigmatism is still dependent on the second node position which can from a purely conceptual point of view be considered as being located at infinity. 15

147 Y Field Angle in Object Space - degrees Corrected For 3 rd Order Coma Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (a) X Field Angle in Object Space degrees (b) Y Field Angle in Object Space - degrees Corrected For 3 rd Order Astigmatism Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (c) X Field Angle in Object Space degrees (d) Figure 4-1: Field of view of a TMA telescope exhibiting (a) zero coma in the nominal alignment state (b) constant coma in the presence of residual misalignments (c) zero astigmatism in the nominal alignment state and (d) fieldasymmetric field-linear third order astigmatism in the presence of residual misalignments. 16

148 4.3 Interpretation of the Dominant Characteristic Misaligned TMA Aberration Field Response Functions It is well known that a misaligned two-mirror telescope displays axial coma and for the case of a Ritchey-Chrétien design which is nominally corrected for third order coma this coma is in fact constant over the field of view. This coma is often corrected by a tip-tilt and/or decenter of the secondary mirror based on measurements of the wavefront or other performance criteria on-axis. In Section 3.4 it has been shown utilizing nodal aberration theory (in the context of two-mirror telescopes) that there exists a pivot point for a surface which has the special property of maintaining an unchanged coma aberration field when an optical surface is rotated with respect to this point. Even though the coma-aberration field remains unchanged for rotations about this point the aberration field for astigmatism will change and cause astigmatism to become binodal in general. The same concepts apply to TMA telescopes and because in addition TMA telescopes are in fact corrected for third order astigmatism in the aligned state a TMA telescope is substantially more sensitive to residual misalignment induced third order astigmatism than a corresponding Ritchey Chrétien two-mirror form. In general if the misalignment induced coma is corrected by a tip-tilt and/or decenter of one of the TMA mirrors this correction will result in zero aberration for the on-axis field point but it will not in all but the rarest of cases control misalignment induced astigmatism over the field. It is critical to realize that the location for the point of zero misalignment induced aberration for the field-asymmetric field-linear astigmatism will in the absence of astigmatic figure error on any of the mirrors typically reside close to the center of the field of view. This 17

149 property can be seen from Figure 4-1(d) or Equation (4-13) which shows that the solution for the field point with zero misalignment astigmatism is given by H =1/ ( B / A ) (4-19) or expressed in component form (APPENDIX B) given by H x 1 = A ( B ) A y( B ) x y x Ax + A y (4-0) H y 1 = A ( B ) + A y( B ) x x Ax + A y y. (4-1) Considering Equation (4-14) it is observed that the term B is a contribution to third order astigmatism that is constant over the field of view. The term B originating from component misalignments is in most cases small since it depends on the perturbations (which are small for all practical purposes) in a squared manner. On the other hand astigmatic figure error at a pupil leads to constant astigmatism which would result in cases are summarized in Table 4-1. B 0. The two possible Table 4-1: The two important cases for zero third order astigmatism in a misaligned TMA. The value of H determines the location in the field of view where the third order misalignment induced astigmatism is zero. The value of B is proportional to the residual astigmatic figure error in the three mirrors that comprise the TMA telescope. B 0 No figure error astigmatic node onaxis i.e. H= 0 No axial aberration B 0 Astigmatic figure error astigmatic node shifted wrt the field center i.e. H 0 Axial astigmatism 18

150 The most significant result is that during the assembly of TMA telescopes that are aligned to provide diffraction-limited performance on-axis based on on-axis measurements alone will not necessarily result in an aligned state of the telescope. Most likely there will be significant field-asymmetric field-linear third order astigmatism considerably degrading image quality. In particular knowing that there is a specific intrinsic field-asymmetric behavior is an important result for engineers attempting to interpret sparsely sampled off-axis performance measurements especially those based on Zernike component reductions of measured wavefront data over a limited set of measurement points in the field of view. Providing a quantitative understanding of this observation can be used to substantial advantage in the alignment of any TMA. A second important result is that if a TMA telescope is aligned to remove axial coma and under this condition if astigmatism is measured on-axis this astigmatism is almost exclusively caused by astigmatic mirror figure error and not by component misalignments. Importantly the on-axis astigmatism can only be removed by correcting the mirror figure error; it cannot be corrected or even significantly reduced through alignment. 4.4 Application James-Webb-like Space Telescope In Section FFDs are utilized to present the effects of misalignments on the aberration fields. It is demonstrated that perturbing the nominal state and evaluating the aberration response with FFDs can reveal valuable information that can be leveraged in the development of an alignment plan. In Section 4.4. it is demonstrated how nodal aberration theory can be utilized to reconstruct misalignment parameters or to compute the particular location of the coma-free pivot point for the secondary mirror. 19

151 4.4.1 Utilizing Full-Field-Displays to Explore the Effects of Misalignments on the Aberration Fields To demonstrate how the knowledge of the general nodal behavior of misalignment induced aberrations in TMA telescopes as discussed in Section 4. and 4.3 can be utilized in the development of alignment strategies the example of a telescope similar to the James-Webb- Space telescope is discussed which is shown schematically in Figure 4-. The optical performance for any presented alignment state is characterized and visualized through the use of FFDs for the FRINGE Zernike coefficients Z4 Z5/6 Z7/8 (related to the medial focal surface astigmatism and coma respectively) which for the given example experience the most significant changes in the presence of misalignment perturbations. As overall metric independent of the particular aberration type causing the image quality degradation FFDs for the RMS wavefront error are utilized. The nominal performance of the TMA over the field is shown in Figure 4-3. Note that for the actual telescope only the portion of the field within the overlaid rectangle is utilized but for the better understanding of the aberration fields a larger portion of the field is shown. PM F TM IMG JWST-Public-SPIE MM Figure 4-: Schematic layout of the JWST like telescope. 130

152 In Figure 4-4 the corresponding field curves are visualized next to the FFD for the astigmatic line images in the image plane (generated with a Coddington real-raytrace in CODE V) which reveals that the design is limited in field mainly by fifth order astigmatism in its nominal aligned state. Note that astigmatism is zero at the field center and at a ring-shaped zone in the field where third and fifth order astigmatism balance each other. To obtain an understanding of misalignment effects for each mirror decenter and tip-tilt perturbations are applied to each mirror and FFDs are computed for Z4 Z5/6 Z7/8 and RMS wavefront which is shown for the primary mirror in Figure 4-5 and Figure 4-6 for the secondary mirror in Figure 4-7 and Figure 4-8 and for the tertiary mirror in Figure 4-9 and Figure

153 0.3 1 wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees X Field Angle in Object Space degrees (a) (b) wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees X Field Angle in Object Space degrees (c) (d) Figure 4-3: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for the nominal JWST like telescope. 13

154 ANGLE(deg) X Y mm FOCUS(MILLIMETERS) Y Field Angle in Object Space - degrees (a) X Field Angle in Object Space degrees (b) Figure 4-4: (a) Field curves for the JWST like telescope and (b) astigmatic line images indicating the orientation reversal beyond the ring zone. While FFDs provide quantitative data for the Zernike polynomials over the field it is more the conceptual understanding of the effects that will be emphasized in this discussion. As derived in Section 4. the main aberration response is field-constant third order coma and fieldlinear field-asymmetric astigmatism. In summary decenter perturbations of the primary mirror mainly generate constant coma with only small changes in astigmatism and the medial focal surface. A tip-tilt perturbation of the primary will generate both sizeable amounts of field-constant coma and field-linear astigmatism. The secondary mirror behaves fairly similar in the presence of decenter and tilt perturbations. The tertiary mirror which will not be adjustable in the actual JWST has been found to be fairly insensitive to misalignment perturbations. Its main contribution to the aberrations in the presence of misalignments is field-linear astigmatism for large tertiary mirror tip-tilts. 133

155 0.3 1 wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (a) X Field Angle in Object Space degrees (b) 0.3 1waves (1000nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (c) X Field Angle in Object Space degrees (d) Figure 4-5: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for a primary mirror decenter of YDE PM = 0. mm. 134

156 0.3 1 wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (a) X Field Angle in Object Space degrees (b) wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (c) X Field Angle in Object Space degrees (d) Figure 4-6: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for a primary mirror tip-tilt of ADE PM = deg. 135

157 0.3 1 wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (a) X Field Angle in Object Space degrees (b) wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (c) X Field Angle in Object Space degrees (d) Figure 4-7: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for a secondary mirror decenter of YDE = 0. mm. 136

158 0.3 1 wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (a) X Field Angle in Object Space degrees (b) wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (c) X Field Angle in Object Space degrees (d) Figure 4-8: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for a secondary mirror tip-tilt of ADE = deg. 137

159 0.3 1 wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (a) X Field Angle in Object Space degrees (b) wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (c) X Field Angle in Object Space degrees (d) Figure 4-9: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for a tertiary mirror decenter of YDE TM = 0.5 mm. 138

160 0.3 1 wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (a) X Field Angle in Object Space degrees (b) wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (c) X Field Angle in Object Space degrees (d) Figure 4-10: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for a tertiary mirror tip-tilt of ADE TM = 0.05 deg. 139

161 Based on the aberration response obtained in Figure Figure 4-10 a possible alignment strategy is identified demonstrated on a misaligned configuration of the JWST like telescope assuming primary and secondary mirror perturbations (YDE PM = mm ADE PM = deg YDE = mm and ADE = deg with a fixed tertiary mirror which is also utilized as coordinate reference surface. The main image degrading aberration in the initial misaligned state is field-constant third order coma (Figure 4-11(c)) which is suggested to be removed with a primary mirror decenter leading to an optical performance as shown in Figure 4-1. The image quality at the field center is already dramatically improved but the periphery of the field is still severely impacted by field-linear field-asymmetric astigmatism. For the next step in the alignment process the coma-free pivot point for the secondary mirror has been determined numerically by damped least-square (DLS) optimization in CODE V (Figure 4-13) which was found to be located mm with respect to the secondary mirror vertex (i.e mm to the left of the secondary mirror vertex in Figure 4-). As visualized in Figure 4-13 rotations about this pivot point do not result in misalignment induced constant coma. 140

162 0.3 1 wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (a) X Field Angle in Object Space degrees (b) 0.3 1waves (1000nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (c) X Field Angle in Object Space degrees (d) Figure 4-11: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for the initially misaligned JWST-like configuration given by YDE PM = mm ADE PM = deg YDE = mm and ADE = deg. 141

163 0.3 1 wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (a) X Field Angle in Object Space degrees (b) wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (c) X Field Angle in Object Space degrees (d) Figure 4-1: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for the JWST-like configuration after decentering the primary mirror to compensate misalignment induced coma given by YDE PM = mm ADE PM = deg YDE = mm and ADE = deg. 14

164 zero coma pivot point distance wrt mirror vertex [mm] rotation angle about pivot point [deg] Figure 4-13: Variation of constant coma with secondary tilt angle and pivot length. A rotation of the secondary mirror about the coma-free pivot point is then applied to compensate the image degradation caused by field-linear astigmatism. FFDs demonstrating the optical performance after removing misalignment induced field-linear astigmatism are shown in Figure Even though the image quality is further improved residual variations in the FFD for the RMS wavefront error remain as shown in Figure This residual image degradation is caused by a shifted medial focal surface (i.e. the focal surface resulting in the smallest RMS wavefront error is displaced and tilted) caused by residual misalignments. Knowing from the sensitivity analysis that a tertiary mirror decenter essentially only affects the medial focal surface while being insensitive for coma and astigmatism it would be useful to decenter the tertiary mirror from its current position. Since in the actual JWST the tertiary mirror cannot be 143

165 0.3 1 wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (a) X Field Angle in Object Space degrees (b) wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (c) X Field Angle in Object Space degrees (d) Figure 4-14: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for the JWST-like configuration after rotating the secondary mirror about its coma-free pivot point to compensate field-linear astigmatism resulting in YDE PM = mm ADE PM = deg YDE = mm and ADE = deg. 144

166 adjusted one could achieve the same relative movement of the tertiary mirror with respect of the primary and secondary mirror by applying equal decenters to both primary and secondary mirror. This last step restores the performance of the telescope (Figure 4-15) to be essentially identical as provided by the nominal design (Figure 4-3). This analysis clearly demonstrates how an effective and also intuitive alignment plan can be developed based on a thorough understanding of the effects to the aberration fields in the presence of misalignments. It should be pointed out that such an approach is not limited by the size of the aberrations and consequently by the size of the misalignment perturbations present as long as the wavefront can be determined by some means. This feature poses a significant advantage compared to other optimization based approaches that can suffer from stagnation problems and poor convergence if the initial misalignment perturbations are large. 145

167 0.3 1 wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (a) X Field Angle in Object Space degrees (b) wave (1000 nm) wave (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space degrees (c) X Field Angle in Object Space degrees (d) Figure 4-15: FFDs for FRINGE Zernike coefficients (a) Z4 (b) Z5/6 (c) Z7/8 and (d) RMS wavefront error for the JWST-like configuration after decentering the primary and secondary mirror to align the medial focal surface given by YDE PM = 0.95 mm ADE PM = deg YDE = mm and ADE = deg. 146

168 4.4. Developing Expressions for Misalignment Parameters based on Nodal Aberration Theory In Section FFDs have been utilized to obtain a conceptual understanding of the effects of misalignments on the nodal aberration fields. While FFDs are a unique choice to visualize the aberration fields the results are purely numerical in nature; the displayed results are obtained by tracing a grid of real rays for one particular set of perturbation parameters. On the example of a JWST-like telescope we demonstrate that nodal aberration theory provides the foundation for the development of an analytical model describing the aberration fields in the absence of system symmetry. Here we assume eight perturbation parameters two x- and y-decenters of the primary and secondary mirror each and similarly two tip-tilts of the primary and secondary mirror respectively. As in the real JWST design the tertiary mirror is assumed without actuators for adjustment. As discussed in Section.5 and already utilized for the case of two-mirror telescopes expressions for third order coma and astigmatism can be formulated given by A ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 131 = W sph sph sph sph asph asph sph sph asph asph 131 PMσPM + W131 σ + W131 σ + W131 TMσTM + W131 TMσTM (4-) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A = W sph sph sph sph asph asph sph sph asph asph PMσPM + W σ + W σ + W TMσTM + W TMσ TM (4-3) B σ σ σ σ σ (4-4) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = W sph sph sph sph asph asph sph sph asph asph PM PM + W + W + W TM TM + W TM TM with wave aberration coefficients for coma and astigmatism as summarized in Table 4- separated into the surface contribution originating from the spherical base curve and its aspheric departure. 147

169 Table 4-: Wave aberration contributions for coma and astigmatism of the nominal JWST-like telescope computed at a field angle of 0.13 deg at a wavelength of 1 μm. W ( sph) 131 j 1 μm 0.13 deg] W ( asph) 131 j 1 μm 0.13 deg] W ( sph) j 1 μm 0.13 deg] W ( asph) j 1 μm 0.13 deg] PM (STOP) TM SUM Expressions for the shifted aberration field centers can be derived equivalently as outlined in Section 3.3. and are given by ( sph) 1 BDEPM σ PM = upm ADE (4-5) PM σ ( sph) BDEPM c XDE PM XDE = u + ADE u + c d YDE YDE ( 1 ) ( 1 ) PM PM PM 1 PM 1 BDE... + u c d ADE σ ( sph) TM PM + 1 BDEPM = +... upm + ADEPM c ( 1+ ctm d ) ( 1+ ( 1) + 1( 1+ ) ) 1 ( ctm d ) ( 1+ ( 1) + 1( 1+ ) ) PM + + c YDE TM d d c d ctm d upm PM YDE + + BDE c ADE TM d d c d ctm d u PM + ctm c d d c d c d u ( 1+ ( 1) + 1( 1+ ) ) TM TM PM XDE XDE YDE PM P M XDE... (4-6) (4-7) 148

170 σ = 0 (4-8) ( asph) 0 PM σ ( asph) 1 XDEPM XDE BDEPM = + du 1 YDE PM PM YDE upm + ADEPM (4-9) ( asph) BDEPM 1 XDEPM σ TM = upm ADEPM ( d+ d1(c d 1)) upm YDEPM d BDE dc XDE XDE + + ( d+ d1(c d 1)) upm ADE ( d+ d1(c d 1)) upm YDEPM YDE... PM. (4-30) While Equations (4-5) - (4-30) accurately describe the behavior of the aberrations in the presence of primary and secondary mirror misalignments another important aspect is the boresight or pointing error that results from the misalignments. Expressions for the OAR intersection point with the image plane for the misaligned optical system can be found dependent on perturbation parameters given by ( ( ) ( ( ( ) ))) Η = c d d + c d d + d c d d 1 XDE +... IMG x TM TM 3 4 PM yimg ( ( 4 3)( ) 1( ( 3 4) ( ( 4 3)( ) )))...+ d + d d 1+ ctm d + d ctm d d 1+ c d + d d 1+ ctm d BDEPM +... y IMG...+ c ( d4 d3 + d( 1+ ctm ( d4 d3) )) XDE +... y IMG... + ( d3 d4+ d( ctm ( d3 d4) 1 )) BDE y IMG ( ( ) ( ( ( ) ))) Η = c d d + c d d + d c d d 1 YDE +... IMG y TM TM 3 4 PM yimg ( ( 4 3)( ) 1( ( 3 4) ( ( 4 3)( ) )))... d + d d 1+ ctm d + d ctm d d 1+c d + d d 1+ ctm d ADEPM +... y IMG...+ c ( d4 d3+ d( 1+ ctm ( d4 d3) )) YDE +... y IMG... ( d3 d4+ d( ctm ( d3-d4) 1 )) ADE y IMG (4-31) (4-3) 149

171 where the image plane location is calculated with respect to the primary mirror optical axis. Combining (4-) - (4-3) provides eight equations for the eight unknown misalignment parameters which is re-written in numerical form obtained after substituting expressions for the shifted aberration field centers (Equation (4-5)-(4-30)) and the wave aberration coefficients (Table 4-) into the Equation (4-)-(4-4) given by A = BDE BDE XDE XDE (4-33) ( meas) 131 x PM PM A = ADE ADE YDE YDE (4-34) ( meas) 131 y PM PM A = BDE BDE XDE XDE (4-35) ( meas) x PM PM A = ADE ADE YDE YDE (4-36) ( meas) y PM PM B ( meas) x = { ADE BDE BDE XDE XDE 6 PM ( PM PM ) ADE ( BDE BDE XDE XDE )+... PM PM BDE YDE BDE YDE XDE YDE XDE } PM PM PM PM PM BDE YDE BDE YDE XDE YDE XDE YD PM PM YD (4-37) B ( meas) y = { ADEPM ADE BDEPM BDEPM BDE BDE BDEPM XDEPM BDE XDEPM XDEPM BDE XDE BDE XDE XDE XDE XDE } PM ADEYDEPM YDEPM ADEYDE YDEPMYDE YDE + ADEPM ( ADE YDEPM YDE ) PM (4-38) ( meas) H AD x = BDEPM BDE XDEPM XDE (4-39) ( meas) HAD y = ADEPM ADE YDEPM YDE (4-40) 150

172 where all the left-hand sides of Equations (4-33) - (3-40) would be known based on measurements. Assuming primary and secondary mirror misalignment parameters as given by YDE PM = 0.49 mm YDE = mm ADE PM = deg ADE = deg the aberration fields for coma and astigmatism are simulated as shown in Figure 4-16(ab) and reconstructed based on sparsely measured field points (Figure 4-16(cd)). The fit of the wavefront data for coma and astigmatism provided following aberration field vectors given by = waves A 131 (4-41) = 0.3 waves A (4-4) = waves B (4-43) ( meas) 0.00 = mm 6.41 ΔH (4-44) Substituting the measured aberration field vectors and the displacement of the field center from the center of the CCD into Equations (3-33) - (4-40) the misalignment parameters for the primary and secondary mirror are obtained given by XDE PM = mm YDE PM = 0.51 mm ADE PM = deg BDE PM = deg XDE = mm YDE = mm ADE = deg BDE = deg. Even though the chosen perturbations were applied in the y-z plane only the derived Equations (4-) - (4-3) apply equally for any non-coplanar misalignment perturbations of the primary and secondary mirror. 151

173 In Section it has been shown how the location of the coma-free pivot point can be determined by simulations in optical design software. Here we show based on nodal aberration theory how the specific location can be determined based on wave aberration coefficients for coma and based on nominal system data. Specifically the location of the coma-free pivot point is obtained by requiring ( sph) ( sph) ( asph) ( asph) ( sph) ( sph) ( asph) ( asph) A131 = W131 σ + W131 σ + W131 TMσTM + W131 TMσ TM =0 (4-45) and after substituting the expressions for the aberration field centers given in Equations (4-5) - (4-30) solving the resulting expression for YDE ADE. This ratio of secondary mirror decenter and tip-tilt specifies the location of the coma-free pivot point for the secondary mirror given by L YDE = = ( asph) ( sph) ( sph) W131 TM d W131 (1+ ctm d) W131 TM d d ( 1+c d ) 1+ c d 1+ c ( d d ) + c d (1+ c d ) 1+ + (1+ ) ( cfp) TM 1 1 TM ( asph) ( asph) ( sph) ( sph) ADE W 131 W131 TM d W131 (1+ ctm d) W131 TM + c d1 d1 d( 1+c d1) 1+ c d1 ctm ( d d1) c d1 ctm d (4-46) where the resulting position of the coma-free pivot point is obtained as L ( cfp) = mm which agrees well with the value obtained through DLS optimization in CODE V in Section

174 0.3 waves (1000 nm) 0.3 waves (1000 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space - degrees (a) X Field Angle in Object Space - degrees (b) Zernike Polynomials Z7/8 Zernike Polynomials Z5/6 (c) sample field points (d) Figure 4-16: FDDs for the James-Webb-like space telescope showing (a) fieldconstant coma and (b) field-linear field-asymmetric astigmatism and (c) magnitude of FRINGE Zernike coefficients Z7/8 and (d) magnitude of the FRINGE Zernike coefficients Z5/6. 153

175 CHAPTER 5: ABERRATION FIELD RE-CONSTRUCTION BASED ON SPARSE WAVEFRONT MEASUREMENTS In Section 5.1 the relationship between the wavefront in the exit-pupil and the obtained pointsspread functions in the presence of misalignments is discussed and results for the obtained pointspread functions predicted with nodal aberration theory are compared to purely numerical results obtained in CODE V. Important practical aspects like how many field points need to be measured for the re-construction of the complete aberration field for a particular aberration type are addressed in Section 5.. The chapter concludes by demonstrating how uncertainties during the wavefront measurements can be linked to uncertainties in the re-constructed misalignment parameters in (Section 5.3). 5.1 Prediction of Point Spread Functions for Misaligned Astronomical Telescopes All previous discussions have been exclusively concerned with how the aberrations behave in optical systems without symmetry based on geometrical optics. Some of the most significant assumptions made so far are that the wavefront is perpendicular to the rays and that the boundary limiting the rays defines the pupil size and shape but importantly does not affect the rays in their propagation. The diffraction theory for optical systems with symmetry is well established (Born and Wolf 1999). The most common approach for diffraction analysis is to compute the exit-pupil diffraction (Figure 5-1) which exhibits an excellent approximation in the context of 154

176 astronomical telescopes with a monolithic primary mirror given that the divergence angles are typically small which is the case for most two-mirror observatory class telescopes. Here we want to demonstrate how the developed expressions for the aberration field centers can be integrated in a diffraction analysis to compute the point spread functions of optical systems without symmetry. An expression for the diffracted electric field at a plane located at along the z-axis is given by the Fresnel Diffraction integral given by ikz π π π e i ( x + y ) i ( η + ξ ) i ( ηx1+ ξy1) λz λz λz E ( ηξ H) = e P( x1 y1 ) e e dx1dy1 iλz H (5-1) where ( ) P x y H describes the wavefront in the exit-pupil which is given by 1 1 W ( H ρ) P ( x1 y1 H ) = A( x y) [ cyl( x y w) cyl( x y γw) ] ex p π j λ (5-) where A( x y) [ cyl( x y w) cyl( x y γ w) ] denotes the transmittance function and W ( H ρ ) denotes the particular aberrations for a particular field point H. Utilizing the expressions obtained for the aberration field centers we can substitute the expression for the wavefront W ( H ρ) in Equation (5-) given by n W( H ρ) = ( W ) klm j ( H σ j ) ( H σ j ) ( ρ ρ) ( H σ j ) ρ (5-3) j p n m p m or re-written in terms of aberration field vectors given by 155

177 W ( H ρ) = W ( ρ ρ) + W ( ρ ρ) + ( W H A ) W ρ ( ρ ρ) H H A H + B ( ρ ρ) + 0M 0M 0M (5-4) 1 [ WH AH + B ] ρ. Since the intention is to compute the point spread functions either at or very close to the Gaussian image plane we can utilize the Fraunhofer approximation for the specific case given by ikz π π e i ( x + y ) i ( η x1+ ξ y1) λz λz E ( ηξ H) = e P( x1 y1 ) e dx1dy1. iλz H (5-5) To compute the PSF one can utilize the diffracted electric field obtained in Equation (5-5) and take the modulus squared given by * ( 0 0) = ( ηξ H) ( ηξ H) x0 y0 PSF x y E E. η= ξ= EFLλ EFLλ (5-6) To demonstrate qualitatively the achievable accuracy of the model to describe the point spread functions in misaligned optical systems an example case of a misaligned Ritchey- Chrétien telescope is considered. The pupil amplitude function is visualized in Figure 5-(a) and the phase function for a particular field point expressed based on Equation (5-4) is visualized in Figure 5-(b). The computed point spread functions for a ring of field points based on Equations (5-1) - (5-6) is demonstrated in Figure 5-3(a) compared to the point spread functions computed in CODE V (Figure 5-3(b)) and excellent agreement is found. Clearly noticeable is the circular diffraction spot in the center of the field and additionally at one off-axis point which is caused by the second astigmatic node. 156

178 E ( ηξ H ) ( ) P x y H 1 1 E * ( ηξ H) E( ηξ H) x0 y0 PSF x η= ξ= EFLλ EFLλ ( y ) MM Figure 5-1: Correspondence of the wavefront in the exit-pupil of the telescope to the point spread function in the image plane. y p [mm] (a) x p [mm] y p [mm] x p [mm] (b) Figure 5-: (a) Pupil transmission function chosen for the example of a Ritchey- Chrétien telescope and (b) phase function for a particular field point under consideration. 157

179 (0 6 ) (0 6 ) ( ) (4. 4. ) ( ) (4. 4. ) (-6 0 ) (0 0 ) (6 0 ) (-6 0 ) (0 0 ) (6 0 ) ( ) ( ) ( ) ( ) (0-6 ) (0-6 ) (a) (b) Figure 5-3: (a) Point spread functions calculated based on Equations (5-1) - (5-6) compared to (b) point spread functions computed in CODE V. The evaluation of point spread functions is well known from star testing optical systems where the point spread function is typically analyzed through focus. The expressions given in Equations (5-1) - (5-6) can directly be applied to analyze slightly defocused images where the shift of the image plane is related to the amount of defocus in the wavefront given by # ( ) δz 8 F W λ = 00 (5-7) with δ z corresponding to the movement of the image plane along the z-axis. Applied to the same example as shown in Figure 5-3 through-focus point spread functions are computed in Figure 5-4 for the field point ( ) arcmin. 158

180 W 00 =-0.866λ W 00 =-0.69λ W 00 = λ W 00 = λ W 00 =-0.173λ W 00 = 0 λ W 00 = 0.173λ W 00 = 0.346λ W 00 = 0.519λ W 00 = 0.69λ W 00 = 0.866λ (a) W 00 =-0.866λ W 00 =-0.69λ W 00 = λ W 00 = λ W 00 =-0.173λ W 00 = 0 λ W 00 = 0.173λ W 00 = 0.346λ W 00 = 0.519λ W 00 = 0.69λ W 00 = 0.866λ (b) Figure 5-4: (a) Point spread function for the field point ( ) arcmin obtained through focus based on nodal aberration theory and (b) corresponding point spread functions computed in CODE V. Since the point spread functions shown in Figure 5-3 or Figure 5-4 were computed for the special case where misalignment induced coma has been removed by tilting/decentering the 159

181 secondary mirror we demonstrate in Figure 5-5 that the predictions work equally well for generally misaligned optical systems also exhibiting other aberrations in the particular case a mixture of misalignment induced third order coma and nominal and alignment induced astigmatism. (a) (b) Figure 5-5: (a) Predicted defocused off-axis point spread function mainly affected by misalignment induced coma and astigmatism and (b) simulated point spread function in CODE V. 5. Discussion of the Minimum Field Sampling to Fully Re-Construct the Aberration Fields Even though one can analyze most metrics describing the performance of a particular optical system in optical design software across the field in a matter of minutes or less at a large number of sample points in practice for example wavefront measurements are typically limited to a few points in the field. Some of the reasons are motivated by cost but other reasons include technical difficulties that arise when a large number of measurements has to be acquired. Assuming ground-based observatories for a moment one needs to keep the exposure time long enough to 160

182 integrate out seeing effects when measuring the wavefront. Additionally some time is required to adjust the camera in the focal plane and to let the system stabilize after moving components etc. to mention just a few aspects that set a lower limit of the time/measurement. Particularly telescopes that include an active optics system have one optical element (typically the primary mirror) which has a large aspect ratio (i.e. the thickness is small compared to its diameter) to operate most effectively. On the other hand this means that on the time scale of the measurements the optical system can t be considered anymore as time invariant since for example the figure of the optics is changing on a time scale similar to that of the measurements. The resulting challenge then is to deduce the same information obtained by evaluating the wavefront at an arbitrary number of field points in optical design software by evaluating only a few wavefront measurements in the field. The following discussion points out what the minimum number of field points required is for particular aberration types relevant in the context of telescope alignment (i.e. spherical aberration coma and astigmatism) Third Order Spherical Aberration In the context of astronomical telescopes assuming a correct figuring and mounting of the individual mirrors third order spherical aberration is the most notable consequence of despace between the mirrors. The characteristic behavior of spherical aberration is its invariance with field. Consequently one measurement at an arbitrary field point fully characterizes the aberration field for spherical aberration which is also visualized in Figure

183 (a) (b) Figure 5-6: Despace induced third order spherical aberration showing (a) the point spread function and (b) FFD for the FRINGE Zernike coefficient Z9 related to spherical aberration. 5.. Third Order Coma Misalignment induced third order coma is typically the most prominent image degradation in the presence of misalignments. As shown in Section.5.3 the nodal behavior of third order coma depends on the correction state of the nominal system. In the case of a misaligned non-aplanatic optical system (e.g. a Cassegrain telescope) the coma node typically decenters in the field as visualized with simulated point spread functions for the on-axis field point and a ring of field points taken at a field radius of 6 arcmin for the aligned (Figure 5-7(a)) and misaligned case of a 16

184 Cassegrain telescope (Figure 5-7(b)). Mathematically this nodal behavior of non-aplanatic optical systems can be described given by W ( ) = W [ H a ρ]( ρ ρ). COMA (5-8) Since wavefront measurements typically provide the Zernike polynomial coefficients to the wavefront the correspondence between the Seidel and Zernike coefficients can be utilized in the computation of the unknowns. Note that the exact correspondence between the Seidel coefficients and Zernike polynomials is generally an infinite sum as discussed in detail in Tyson 198 but for most two- and even some three-mirror systems higher than third order aberrations are reasonably small assuming the typical small FOVs that the correspondence takes on the simple form as shown here. Nevertheless it should be kept in mind that such a description has to be modified for the application to wide-angle systems. Equation (5-8) can then be re-written in terms of Zernike polynomials given by ( x x) x( x y ) 3 Z = W H a ρ ρ + ρ ( y y) y( x y ) 3 Z = W H a ρ ρ + ρ (5-9) (5-10) When trying to characterize the coma aberration field by wavefront measurements the number of required field points depends on the knowledge of the nominal optical system. If no knowledge of the optical system is assumed the unknowns are one scalar (W 131 ) and one vector quantity (a 131 ) which characterizes the shifted coma node. For each wavefront measurement at a particular field point the wavefront is fit to Zernike polynomials and two equations (Equations (5-9) - (5-10)) are obtained consequently requiring wavefront measurements at a 163

185 minimum of two field points if W 131 is not known. If W 131 is known from an optical prescription one wavefront measurement is sufficient to fully characterize the coma aberration field. (a) (b) Figure 5-7: Point spread functions for the on-axis and several off-axis field points for (a) an aligned Cassegrain telescope showing nominal coma and (b) a misaligned Cassegrain telescope showing on-axis coma and a displaced coma node. In the special case of an aplanatic optical system W 131 = 0 a non-normalized expression for misalignment induced coma is utilized given by W = [ A ρ]( ρ ρ) COMA3 131 which can be re-cast in terms of Zernike polynomials given by (5-11) ( x) x( x y ) 3 Z = A ρ ρ + ρ (5-1) ( y) y( x y ) 3 Z = A ρ ρ + ρ. (5-13) In this case if the nominal optical system is free of coma (Figure 5-8(a)) in the presence of misalignments coma is constant across the field (Figure 5-8(b)) and one measurement is required to fully characterize the coma aberration field. 164

186 (a) (b) Figure 5-8: Point spread functions for the on-axis and several off-axis field points for (a) an aligned Ritchey-Chrétien telescope showing zero nominal coma and (b) a misaligned Ritchey-Chrétien telescope showing dominant constant coma across the field of view Third Order Astigmatism It has been shown in Chapter 3 that merely aligning an astronomical telescope for zero coma is generally not a sufficient condition to assure alignment if other metrics are not imposed additionally. The consequences of residual misalignments on astigmatism are shown in Figure 5-9 showing astigmatism in the aligned (Figure 5-9(a)) and misaligned (Figure 5-9(b)) state (assuming the optical system is not anastigmatic). Clearly visible in Figure 5-9(b) is the essentially unchanged point spread function at the field center and the region with decreased astigmatism (close to the second astigmatic node) while exhibiting larger astigmatism than in the nominal system on the other side of the field of view. It has been shown that binodal astigmatism can be described mathematically (assuming W 0) as given by 165

187 W AST3 1 = W[( H a) + b] ρ. (5-14) Converting Equation (5-14) into expressions utilizing Zernike polynomials gives ( ( x y) ( y x x y) x)( x y) 1 Z6 = W HH HA + HA + B ρρ (5-15) ( ( ) ( ) )( y x y y x x y y x ) 1 Z5 = W H H HA HA + B ρ ρ. (5-16) Assuming that no a priori knowledge about the optical system is available there are five unknowns to be determined and each equation has four unknowns. Consequently when solving Equations (5-15) - (5-16) separately for the unknowns a minimum number of four field points for wavefront measurements is required to re-construct the astigmatic aberration field. It can be shown that it is more efficient to solve Equations (5-15) - (5-16) together as expressed in matrix form (for the case of the minimum number of required sample field points) given by ( x y) ( x y) ( ) ( x y) ( x y) Z5 1 H 1H 1Hy 1Hx 1Hx 1H 0 1 y W Z5 H H Hy Hx Hx Hy 0 A Z5 3 Hx 3H y = 3Hy 3Hx 3Hx 3Hy 0 1 A Z H Hx Hy Hy Hx 1H B Hx Hy Hy Hx 1 0 Z6 H H B x y x y (5-17) where the indices in front of the field variables denote the particular field point. When solving Equations (5-15) - (5-16) combined wavefront measurements at three distinct field points are sufficient. Note that Equation (5-17) can be equally utilized in the presence of astigmatic figure error and misalignments even though the interpretation of the aberration field vectors has to follow accordingly. 166

188 (a) (b) 0.0 Y Field Angle in Object Space - degrees X Field Angle in Object Space - degrees (c) Figure 5-9: Point spread functions for the on-axis and several off-axis (edge of the field) field points for (a) an aligned Ritchey-Chrétien telescope showing nominal astigmatism (b) a misaligned Ritchey-Chrétien telescope showing binodal astigmatism and (c) FFD showing FRINGE Zernike coefficients Z5/6 visualizing astigmatism across the field of view. It has been demonstrated that for the new generation of TMAs the main aberration field response beyond constant coma is field-asymmetric field-linear third order astigmatism which 167

189 is visualized in terms of point spread functions for the on-axis and several off-axis field points in Figure 5-10(b) compared to the case without misalignments shown in Figure 5-10(a). When trying to re-construct the astigmatic aberration field Equations (5-15) - (5-16) simplify since W = 0 given by ( ( y x x y) x)( x y) 1 Z6 = HA + HA + B ρρ (5-18) ( ( y y x x) y)( y x ) 1 Z5 = HA HA + B ρ ρ. (5-19) Equations (5-18) - (5-19) contain four unknowns and for each field point two equations are obtained since the Zernike coefficients Z 5 and Z 6 will be known from the wavefront fit. Consequently wavefront measurements at a minimum of two field points reveal the complete astigmatic aberration field. 168

190 (a) (b) 0.0 Y Field Angle in Object Space - degrees X Field Angle in Object Space - degrees Figure 5-10: Point spread functions for the on-axis and several off-axis (intermediate) field points for (a) an aligned TMA telescope showing zero nominal astigmatism and (b) a misaligned TMA telescope showing dominant field-asymmetric field-linear third order astigmatism. (c) FFD showing the FRINGE Zernike coefficients Z5/Z6 across the field. (c) 169

191 5.3 Effects of Uncertainties in the Wavefront Measurement on the Accuracy in the Prediction of Misalignment Parameters The discussion of how to choose sufficient sampling for wavefront measurements at various points in the field to uniquely re-construct the specific aberration fields assumed zero uncertainty in the wavefront measurements. In reality the wavefront measurements will comprise various uncertainties with the magnitude and exact nature dependent on the specific approach chosen to measure the wavefront. To demonstrate how the effects on the misalignment parameters can be determined if a statistical model of the wavefront errors is available the case of a misaligned Ritchey-Chrétien telescope is considered assuming a simple error model in which the errors to ). the Zernike coefficients Z 5 and Z 6 are added relative to their magnitude (i.e. Z = Z ( ± ε ) Depending on the number of wavefront measurements utilized in the reconstruction of the astigmatic node locations the node spread is shown in Figure 5-11 (Schmid 008a). a +b i ib a +b i ib a ib +b i (a) (b) (c) Figure 5-11: Spread of the re-constructed astigmatic node locations caused by the uncertainty in the wavefront measurements ε = ± 0.0 shown for (a) 4 wavefront measurements (b) 9 wavefront measurements and (c) 16 wavefront measurements. 170

192 The uncertainty in the node positions results in an uncertainty in the re-constructed misalignment parameters. In practice one would determine how accurate the prediction of misalignment parameters would have to be for the particular application considering a tolerance analysis of the specific system and given the estimate of the uncertainty in the wavefront measurements one could choose an appropriate number of field points to be measured. For the case of ground-based telescopes a large number of field points (~>8) becomes impractical since particularly in large observatory class type telescopes parameters like mirror figure change on the time-scale required for the measurements. Consequently one could determine for a given number of measured field points how accurate the wavefront has to be measured to achieve a specific uncertainty in the re-constructed misalignment parameters. This could be accomplished by computing the uncertainties in the re-constructed misalignment parameters dependent on the number of wavefront measurements and magnitude of error terms as shown for the example case in Figure

193 ε ε ε ε = = = = ±0.05 ± ± ± ε = ±0.01 ε ε ε ε = = = = ±0.05 ± ± ± ε = ±0.01 x-decenter y-decenter (a) ε ε ε ε = = = = ±0.05 ± ± ± ε = ±0.01 (b) ε ε ε ε = = = = ±0.05 ± ± ± ε = ±0.01 y-z - Tip/Tilt x-z - Tip/Tilt (c) (d) Figure 5-1: RMS deviation of secondary mirror decenters and tilts with respect to the mean value dependent on the number of measured field points and the magnitude of the uncertainty in the wavefront measurements. (ab) Reconstructed x- and y-decenter values dependent on the number of measured field points and (cd) re-constructed x-z- and y-z tip-tilt values dependent on the number of measured field points. 17

194 CHAPTER 6: DISTINGUISHING PRIMARY MIRROR FIGURE ERROR FROM MISALIGNMENT INDUCED ASTIGMATI IN ASTRONOMICAL TELESCOPES In Chapter 6 the assumption of rotationally symmetric surfaces previously made is suspended. The effects of small (< ~ λ RMS) figure errors (astigmatism) on the nodal properties are discussed where it is assumed that the optical surface exhibiting figure errors is coincident with a pupil. In Section 6.1 the objectives of this work are discussed in the context of telescope alignment. The integration of astigmatic figure terms into the mathematical framework of nodal aberration theory is described in Section 6. and the effects on the nodal properties of an otherwise aligned astronomical telescope with the aperture stop located on the primary mirror are explained in Section 6.3. In Section 6.4 it is shown how the behavior of astronomical telescopes in the presence of misalignments and astigmatic figure errors on the primary mirror can be described mathematically and also it is demonstrated how an intuitive understanding of both effects in combination can be obtained. Important new nodal properties are described that reveal even without detailed analysis if only misalignments only astigmatic figure errors or both are present. Also the consequences of attempting a compensation of misalignment induced aberrations and figure induced aberrations are described in Section 6.4 and expressions are derived that will result in quasi field-quadratic astigmatism shifted in the field for a particular combination of misalignments and astigmatic figure error. The developed theoretical predictions are validated with CODE V in Section 6.5 where excellent agreement between the analytical predictions and numerical simulations is found. The chapter closes with an example of a 173

195 misaligned Ritchey-Chrétien telescope with astigmatic primary mirror figure and it is shown how misalignments and figure error can be separated. 6.1 Consequences of Actively Controlled Primary Mirrors in the Context of Telescope Alignment The trend to steadily increase the aperture size of the emerging generation of astronomical telescopes while maintaining good image quality has only been feasible with the introduction of active optics systems to control the particular mirror shapes. As described in detail by Noethe 00 for primary mirror aperture sizes larger than ~ m the stability of the mirror cannot be solely provided by increasing the mirror thickness. Also in the evening as the ambient temperature reduces the large thermal inertia caused by a large primary mirror mass would require long times for the mirror surface to reach thermal equilibrium with the surrounding air. Observations during this period would suffer from image quality degradation caused by local air turbulences in proximity to the mirror surface (with the prime example of the Russian 6 m telescope at the Zelenchuk Observatory in the Caucasus with a non-light weighted primary mirror (Wilson 1996)). To overcome these difficulties the trend has been to move to thin primary mirrors that are actively controlled. The shape of the mirror is adjusted based on information obtained through wavefront measurements from a star and/or through pre-calculated deformations of the mirror for varying zenith angles. An additional advantage of having the capability of adjusting the mirror figure are the more relaxed manufacturing tolerances. On the other hand this also means that a telescope with a thin actively controlled primary mirror can only perform well in combination with an active optics system. 174

196 Even though the mirror figure can be adjusted during operation the remaining challenge has been to determine which aberrations measured in for example the wavefront analyzer are caused by mirror misalignments and which aberration contributions are caused by mirror figure. Having the possibility of distinguishing misalignment and figure effects some of the consequences of compensating misalignments with mirror figure and vice versa can be avoided. For example compensating one with the other typically leads to a laterally decentered exit-pupil and an inclination of the exit beam with respect to the primary mirror optical axis (Wilson 1996) requiring not only focus but also tilt adjustments of the focal plane. As discussed by Wilson 1996 some of these consequences can be dealt with for example by utilizing pointing software but it seems desirable to prevent the cause for these additional steps to the extent possible to begin with. Here we describe how to integrate an astigmatic primary mirror figure error characterized by a Zernike polynomial description for the case of a monolithic mirror into nodal aberration theory. The analysis shows that understanding the nodal behavior of astigmatism provides an opportunity to deterministically isolate misalignment- and figure-induced performance degradation. By isolating these two causes of telescope performance degradation the misalignments of the secondary mirror and the residual astigmatic figure error in the primary mirror the dynamic range of the active optics system can be reserved for figure correction only consequently extending the performance envelope for the telescope. 175

197 6. Formulating Nodal Aberration Theory of a Misaligned Ritchey-Chrétien Telescope with Astigmatic Figure Error on the Primary Mirror In this work astigmatic figure errors on the primary mirror (coincident with the stop) of a Ritchey-Chrétien telescope will be introduced in nodal aberration theory which is until now limited to modeling optical systems with rotationally symmetric surfaces (or portions thereof). This set of conditions is more restrictive than necessary but by imposing them here they can provide a succinct development of the concept which can then be extended from this specific case along obvious paths. The introduction of primary mirror figure error will be accomplished by adding the corresponding low order astigmatic Zernike polynomial characterization of the surface error to the wave aberration expression extended to the vector form for optical systems that are not rotationally symmetric as discussed in Section.5. The key observation leading to the integration of mirror figure error in nodal aberration theory is to consider the equations of nodal aberration theory for the special case of an aspheric surface placed at the aperture stop. As developed in Appendix C of Thompson 1980 consider a third order (fourth order in wavefront) aspheric plate corrector as in a Schmidt telescope placed in the entrance/exit-pupil. When the aspheric plate is decentered by an amount ρ the resulting new additive vector wave aberration terms are W = W [( ρ + ρ) ( ρ + ρ)] [( ρρ ) 4( ρρ )( ρρ ) 4( ρ ρ)( ρρ ) = W + + ( ρ ρ ) 4( ρ ρ)( ρ ρ) ( ρ ρ) ] (6-1) where W 040 denotes the wave aberration for spherical aberration ρ is a normalized vector that denotes a location in the exit-pupil and ρ is the normalized offset of the rotationally symmetric conic/aspheric plate from the optical axis. When the aspheric contribution to the wavefront is 176

198 placed at the stop/pupil and then decentered relative to the stop/pupil the original contribution (exclusively) to spherical aberration generates lower order aberration components including astigmatism that are each themselves constant over the field of view since the beam footprints of all field points on the surface are identical. In fact for large decenters of a rotationally symmetric parent aspheric the dominant aberration is astigmatism which is the aberration to be developed exclusively here because it is typically the dominant residual figure error in large monolithic primary mirrors. Referring to the form for astigmatism in a misaligned optical system presented in Equation (6-1) it is clear that the contribution to astigmatism in Equation (6-1) comes from the fourth term; ( ) ρ ρ. Figure 6-1 shows an example of how a rotationally symmetric spherical aberration contribution evolves to a dominantly astigmatic wavefront term as the asphere is decentered relative to the aperture. This approach to introducing figure error then makes use of an existing portal into nodal aberration theory that enables the addition of optical surfaces that in general are not rotationally symmetric. Here we limit this development to small astigmatic deviations from the ideal conic/aspheric surface of the primary mirror of a Ritchey- Chrétien telescope. Having identified an access point within the context of nodal aberration theory of misaligned optical systems where the concept of a misfigured mirror (or more generally any non-rotationally symmetric optical surface placed at the stop or in a pupil) can be inserted consider first a common representation of a misfigured mirror an interferogram as the quantitative data source. An interferogram of the primary mirror that contains and displays the departure from the aspheric/conic surface of the nominal design for the surface as if it was tested on-axis in a null configuration is an effective and practical method for quantifying the figure 177

199 error of the primary mirror. The interferogram can be considered as an aspheric surface contribution of the form presented in Equation (6-1) in the context of this development. This interferometric data which is readily measured at in particular the on-axis field point is typically quantified based on the value of the FRINGE Zernike coefficients. surface surface stop stop (a) (b) surface surface stop stop (c) (d) Figure 6-1: Aspheric corrector plate with aperture stop positions (a) centered on the optical axis indicating spherical aberration (bcd) shifted aperture stop causing field-constant astigmatism. An astigmatic wavefront can be parameterized based on the measured Zernike polynomial coefficient values for terms C 5 and C 6. With these measured/simulated values the magnitude ( FIGERR) C 5/6 and orientation * ( FIGERR) ξ 5/6 caused by the residual state of the primary mirror figure is given by of the astigmatic error in the wavefront 178

200 ( ) ( ) ( FIGERR) C5/6 ( FIGERR) C5 ( FIGERR) C6 = + (6-) ξ * ( FIGERR) 5/6 1 = ArcTan ( ( FIGERR) C6) ( ( FIGERR) C5). (6-3) Referring to the FRINGE version of the Zernike polynomials the polynomial dependence for the low order astigmatic terms are given by (Wyant 199) ( FIGERR) Z5 ( FIGERR) C5ρ cos( φ) = (6-4) ( FIGERR) Z6 ( FIGERR) C6ρ sin( φ) = (6-5) where ρ is the radial variable describing the corresponding zone in the exit-pupil and φ describes the azimuthal frequency with sign conventions as shown in Figure 6-(a) where the azimuthal dependence is measured from the positive ρ x axis. The radial and azimuthal dependence as shown in Equations (6-4) - (6-5) is notably exactly the dependence obtained as the result of the vector multiplication which is the basis for nodal aberration theory i.e. ( φ ) ( φ ) sin ρ = ρ (6-6) cos with j ρ = ρe φ (6-7) where a different orientation of the ρ x axis and a different reference for the azimuthal dependence (from the ρ y axis) are assumed as shown in Figure 6-(b). Since the interferograms in this work are shown in the same coordinate system as used for the aberration field vectors in nodal aberration theory there is a sign reversal in Equation (6-3) which is given by 179

201 Zernike Polynomials ξ ( FIGERR) 5/6 1 = ArcTan ( ( FIGERR) C6) ( ( FIGERR) C5). Nodal Aberration Theory (6-8) ρ y ρ y ρ x (a) Exit-Pupil (b) ρ x Exit-Pupil Figure 6-: Coordinate system definition in the exit-pupil showing (a) the definition for the FRINGE Zernike polynomials and (b) the coordinate system orientation utilized in nodal aberration theory. Although only the astigmatic term is presented here clearly any Zernike polynomial term that is placed on a surface in a pupil or at the stop can be accommodated with this methodology. For surfaces away from the pupil or the stop the approach can be extended to take the specific pupil footprint for each field at the particular surface into account causing in general a field dependence of the figure error contributions. While the primary mirror is typically the most critical element in the context of figure errors mainly caused by its aperture size figure effects away from a pupil are gaining increased importance with the current trend to telescopes of increased secondary or tertiary mirror aperture sizes and resulting effects will be addressed in future publications. To integrate an astigmatic figure error at the pupil into the existing characterization of the misalignment induced aberration fields it is only necessary to label this contribution as a figure 180

202 error component (FIGERR) and add it directly to the existing astigmatic component that is independent of the field of view and has a squared vector behavior given by B B B (6-9) = ( MISALIGN ) + ( FIGERR) where ( ( ) C ) 56 j ( ( ) ξ56) B exp (6-10) ( FIGERR) FIGERR FIGERR (sph) ( sph ) (asph) ( asph) ( MISALIGN ) W + W. B σ σ (6-11) Figure 6-3(ab) illustrates the interrelationship of the interferogram data and the construction of the nodal vector component B ( FIGERR). Specifically the angle describing the orientation of the vector ( FIGERR) B in Figure 6-3(b) is twice the angle describing the orientation of the interferogram. Utilizing Equations (6-9) - (6-11) the total wave aberration of a misaligned Ritchey- Chrétien telescope with primary mirror astigmatic figure error aligned to remove field-constant coma and ignoring the field curvature term is given by W = W = 1 [ ] RC MISALIGN FIGERR W H a + b ρ (6-1) ( ) AST ( ) B b a (6-13) W B B B (6-14) = ( MISALIGN ) + ( FIGERR) a (6-15) W A A σ σ (6-16) (sph) ( sph ) (asph) ( asph) W + W 181

203 where a denotes the midpoint between the two astigmatic nodes b is related to the distance between the astigmatic nodes in Equation (6-10). ( ) MISALIGN B is given in Equation (6-11) and B ( FIGERR) is given pupil coordinate x Y Field Angle in Object Space - degrees i ( FIGERR) ξ56 pupil coordinate x (a) ( FIGERR) W B + i ( FIGERR) W ( FIGERR) B W B ( FIGERR) W X Field Angle in Object Space - degrees (b) B ( ) RC ALIGNED FIGERR W Astigmatism [waves] B ( ( ) C ) 56 j ( ( ) ξ56) exp ( FIGERR) FIGERR FIGERR ( ) ( ) ( ( FIGERR) C6) ArcTan ( ( FIGERR) C5) ( FIGERR) C56 = ( FIGERR) C5 + ( FIGERR) C6 ξ ( FIGERR) 56 = 1 x-field angle [deg] = 1 ( FIG) W H + B ρ (c) y-field angle [deg] Figure 6-3: (a) Interferogram of the primary mirror astigmatic figure error (b) binodal astigmatism caused by an astigmatic figure error on the primary mirror in the case of a fully aligned Ritchey-Chrétien telescope and (c) the magnitude of astigmatism corresponding to (b). 18

204 6.3 Binodal Response of the Astigmatic Field Dependence of a Ritchey-Chrétien Telescope with Primary Mirror Astigmatic Figure Error Before treating the more general case of the combination of misalignments with astigmatic figure error the astigmatic binodal response to astigmatic figure error applied to the primary mirror of a Ritchey-Chrétien telescope will be presented. In the case of an aligned Ritchey-Chrétien telescope a = B = 0 with astigmatic figure error ( MISALIGN ) ( MISALIGN ) FIGERR B 0 Equation (6-1) reduces to ( ) ( ) RC ALIGNED FIGERR W = 1 ( FIGERR). W H + B ρ (6-17) The locations of the astigmatic nodes are found by finding the zeros of Equation (6-17) i.e. the positions in the field H for which W = ( RC ALIGNED FIGERR ) 0. This condition leads to ( FIGERR) H = (6-18) W B with the binodal solution ( FIGERR) H = ± i (6-19) W B using the methods of vector multiplication described in APPENDIX B and where B is ( FIGERR) computed from the astigmatic figure error measurement of the coefficients ( FIGERR) C 5 and ( FIGERR) C 6 which are then used in Equations (6-) - (6-5). Equation (6-19) denotes two vectors pointing in opposing directions to each of the astigmatic nodes significantly originating from the field center ( a = 0). Consequently in the case of an astigmatic figure error in an otherwise aligned Ritchey-Chrétien telescope (which is 183

205 not corrected for field-quadratic astigmatism as an intrinsic property of the optical design form i.e. W 0 ) the two astigmatic nodes always exhibit symmetry with respect to the field center as visualized in Figure 6-3(bc). Note that for the case of a telescope design where the uncorrected astigmatism of the optical design is greater than zero W > 0 (which is a general property of Ritchey-Chrétien telescopes) the nodes emerge at ±90 to the direction of the B -vector which is determined from the interferogram data. If one would apply the same concepts to optical systems (e.g. telescopes that are not of a Ritchey-Chrétien form) where W < 0 the astigmatic nodes would emerge along the B -vector since in that case the minus sign in the denominator of Equation (6-19) would contribute an additional 90. An important characteristic of an astigmatic figure error at the aperture stop (or images thereof) is that it is fully characterized by B and does not contribute to the a -vector. 6.4 Binodal Response of the Astigmatic Field Dependence of a Ritchey-Chrétien Telescope with Misalignments and Astigmatic Figure Error While astigmatism in a Ritchey-Chrétien telescope becomes binodal in the presence of either misalignments or astigmatic figure errors or both in the case of astigmatic figure errors and misalignments a characteristic difference in the nodal geometry allows separating the two effects as will be presented in this section. 184

206 6.4.1 Conditions Imposed as a Result of Aligning a Ritchey-Chrétien Telescope to Remove Field-Constant Coma It has been shown by the authors that the operation of removing field-constant coma does result in placing some important conditions on one of the node locations for binodal astigmatism (Schmid 010a). Specifically it has been shown that one of the astigmatic nodes remains essentially at the field center caused by the secondary mirror coma contributions ( W W ( asph) 131 ( sph) 131 ) which are of equal sign and similar magnitude. Consequently after the telescope has been aligned for zero constant coma following condition is fulfilled W ( sph) ( sph) ( asph) ( asph) W131 =0 and σ σ (6-0) where the vectors σ ( sph) ( ) and σ asph are almost equal in length and point into opposite directions as shown in Figure 6-4(b). When calculating the quantity ( sph) sph ( asph) ( ) ( asph) ( MISALIGN ) W + W B σ σ it is found to be small for any realistic decenters and tilts since the squared vectors σ ( sph) ( ) and σ asph point into the same direction while ( sph) W and ( asph) W are opposite in sign and similar in magnitude. As result from Equation (6-13) b a and the astigmatic nodes are located at H = a ± ib = ( 0a ). ( MISALIGN ) 185

207 wave aberration wave aberration W W ( sph) 131PM ( sph) PM COMA ( asph) ASTIGMATI (a) ( sph) ( asph) (a) W W = ( sph) PM 0 ( asph) W W = PM 0 W W ( asph) 131 ( sph) σ ( sph)* σ y-field [norm.] σ (b) W ( asph) ( asph) σ ( sph) σ ( sph) ( asph) W x-field [norm.] Figure 6-4: (a) Wave aberration contributions for coma (top) and astigmatism (bottom) showing the spherical base curve and conic/aspheric surface contributions (b) secondary mirror aberration field centers (spherical and aspheric) before (denoted by * ) and after removing misalignment induced coma Conditions Resulting from Primary Mirror Astigmatic Figure Error and Misalignments in a Ritchey-Chrétien Telescope Figure 6-5(b) illustrates a characteristic interferogram dominated by astigmatic figure error which is the most common form of residual figure error on a monolithic primary mirror of a large astronomical telescope. In parallel with Figure 6-5(a) which is directly linked to the misalignments of the secondary mirror in a coma-corrected state the residual figure error in the primary mirror is most commonly characterized by a Zernike polynomial decomposition of the measured wavefront on-axis during commissioning or in some cases during or just prior to operation. As stated in Equation (6-1) the two vectors that control the node positions for the misalignment/figure error induced binodal astigmatic field dependence are a and b. The vector a shown in Figure 6-6(a) determines the center of symmetry (planar) of the binodal 186

208 astigmatic field. In this case it is caused exclusively by secondary mirror misalignments and is computed from Equation (6-15) - (6-16) using the wave aberration coefficients shown in Figure 6-4(a) and the σ ( sph asph) -vectors shown in Figure 6-4(b). As developed in Section 6. and 6.3 and a key to distinguishing between secondary mirror misalignments and astigmatic figure errors on the primary mirror the astigmatic figure errors do not have a contribution to the a - vector as illustrated in Figure 6-6(b). Figure 6-7 illustrates the other component contribution to the geometry of the nodes associated with binodal astigmatism b. This is the vector that decomposes to point from the center of biplanar symmetry located by a to the location of each of the nodes which are symmetric about the end point of a. Here the alignment component of b ( MISALIGN ) is shown in Figure 6-7(a) for a misaligned Ritchey-Chrétien telescope that has been partially aligned to achieve zero field-constant coma. The figure contribution ( FIGERR) = ( FIGERR) / W b B which is a quantitative representation of the primary mirror figure error is computed directly from the interferogram and the knowledge of W (from nominal system data) using Equation (6-10) and is visualized in Figure 6-7(b). The vectorial addition of the secondary mirror misalignment contribution and the primary mirror astigmatic figure error contribution b and ( MISALIGN ) ( FIGERR) b respectively is shown in Figure 6-7(c) (as an illustration of Equation (6-14)). By taking the square root of the final composite vector b the final vector b is obtained that determines the locations of the astigmatic nodes as shown in Figure 6-8(d). 187

209 Misalignment y-field [norm.] W ( sph) ( asph) 131 W131 Figure Error y ( FIGERR) ξ ( sph) σ ( ) asph σ x-field [norm.] x 0.5 (a) (b) 0.0 Figure 6-5: (a) Aberration field center of symmetry for the spherical and aspheric aberration field contributions of the secondary mirror after aligning the telescope for zero field-constant coma. (b) Astigmatic primary mirror figure error for a Ritchey-Chrétien telescope. Misalignment Figure Error y-field angle [deg] a y-field angle [deg] Fundamentally figure error has no equivalent a -vector x-field angle [deg] x-field angle [deg] (a) (b) Figure 6-6: The vector a that locates the center of biplanar symmetry of the binodal astigmatic field for a Ritchey-Chrétien telescope with secondary misalignment and primary mirror astigmatic figure error. (a) Contribution from secondary mirror misalignments under the condition that field-constant coma has been removed and (b) the contribution from astigmatic figure error on the primary mirror as derived in Section and 3 has no a component. 188

210 Misalignment Figure Error y-field angle [deg] y ( FIGERR ) ξ b MISALIGN ( ) x-field angle [deg] ( FIGERR) ξ b 56 FIGERR ( ) b ( FIGERR) x 0.5 y-field angle [deg] b ( FIGERR) b MISALIGN ( ) (a) Misalignment and Figure Error b b ( NET ) x-field angle [deg] (b) Figure 6-7: The vector b that points from the endpoint of a to the nodal points for binodal astigmatism shown here for a Ritchey-Chrétien telescope with secondary misalignment and primary mirror astigmatic figure error consisting of (a) ( ALI ) b denoting the contribution caused by secondary mirror misalignments (b) b ( FIG) determined by interferogram data combined with the knowledge of the total astigmatism in the nominal optical system resulting in (c) the overall vector b and b the final astigmatic node locating vector when combined with a (c) Figure 6-8 presents the most important graphical realization for the results of this work (Schmid 010c). In Figure 6-8(a) the nominal astigmatic aberration field is shown having the two astigmatic nodes coincide at the field center and in Figure 6-8(b) the general nodal symmetry for the case of misalignments only for the condition where axial (field-constant) coma has been removed is shown. The general feature to be recognized here is that one of the 189

211 astigmatic nodes is located at the field center which has been shown to be a direct consequence of removing misalignment induced field-constant coma (Schmid 010a). In comparison Figure 6-8(c) demonstrates the effects of astigmatic figure error on the third order astigmatic aberration field which demonstrates that the nodes are constrained to remain symmetric with respect to the field center (here the telescope is otherwise perfectly aligned). Figure 6-8(d) combines the two sources of degradation secondary misalignments and primary mirror astigmatic figure error. It can be observed that the binodal field dependence displays symmetry about the field point denoted by the vector a. Another property of the aberration field vectors a and ± ib becomes apparent when comparing Figure 6-8(b) with Figure 6-8(d). While in the absence of misalignment induced fieldconstant coma and astigmatic figure errors the direction of the node splitting always occurs along the orientation of the vector a (Figure 6-8(b)) this constraint is removed in the presence of astigmatic figure errors (Figure 6-8(d)). With these new insights into these fundamental nodal properties of astigmatism it becomes apparent that measuring non-zero astigmatism at the field center can only be caused by an astigmatic figure error since in the case of pure misalignments one astigmatic node will remain at the field center causing misalignment induced astigmatism to be effectively zero. Consequently measuring astigmatism at the field center would completely quantify an astigmatic figure error even in the presence of secondary mirror misalignments. Knowing the behavior of astigmatism as described mathematically in Equations (6-9) - (6-16) is most valuable since the astigmatic nodal positions are readily found from as few as three measurement points in the field of view of the wavefront in some metric that can be reduced to 190

212 Zernike coefficient terms. Given the nodal positions as illustrated in Figure 6-8 it can be determined if secondary mirror misalignment astigmatic figure or a combination of both are present simultaneously. If neither astigmatic node is effectively on-axis after the telescope has been aligned for zero field-constant coma and if the nodes are not symmetrically placed about the axis then both types of errors are present. Nominal Misalignment y-field angle [deg] y-field angle [deg] i b a ( ALI ) x-field angle [deg] + i b ( ALI ) (ALI) b x-field angle [deg] y-field angle [deg] i (a) Figure Error b ( FIGERR) ( ) b FIGERR x-field angle + i( FIGERR) b [deg] (b) Misalignment and Figure Error y-field angle [deg] ib + ib a b x-field angle [deg] (c) (d) Figure 6-8: The characteristic node geometry for (a) astigmatism in the absence of misalignments or astigmatic figure errors (b) misalignment induced binodal astigmatism (c) astigmatic figure error induced binodal astigmatism and (d) both contributions (b) and (c) combined. It has been shown that astigmatism can become binodal in the presence of misalignments or astigmatic figure errors. Consequently the important question arises whether it is possible to obtain a quadratic field dependence of astigmatism corresponding to two overlapping astigmatic 191

213 nodes in the presence of both astigmatic figure and tilts and/or decenters. A qualitative discussion of transforming binodal astigmatism into quasi field-quadratic astigmatism has previously been given in an excellent paper by Rogers 000 in the context of design strategies for tilted component systems. He showed that by adding a cylindrical corrector to an optical system exhibiting binodal astigmatism quasi field-quadratic astigmatism can be achieved centered at the midpoint between the astigmatic nodes if the appropriate amount of constant astigmatism generated by the cylindrical surface is chosen. Since for this discussion coma is corrected in the nominal state (Ritchey-Chrétien) and compensated by alignment before analyzing astigmatism the problematic of having the coma node coincident with the midpoint of the astigmatic nodes does not exist. For the two astigmatic nodes to overlap the absolute value b has to be identical to zero. Equation (6-13) provides two conditions for collapsing astigmatic nodes given by ( ) A A + C B B W = (6-1) x y ( FIG) 6 x y 0 ( ) A A + B B + C W = 0. (6-) x y y x ( FIG) 5 Solving Equations (6-1) - (6-) provides then the corresponding expressions for the Zernike coefficients ( FIG) C 5 and ( FIG) C 6 for collapsing astigmatic nodes given by A A 1 y x ( FIG) C5 = + B x B y W C = B B A A W ( FIG) 6 x y ( ) x y. (6-3) (6-4) 19

214 Astigmatism [waves] Astigmatism [waves] x-field angle [deg] (a) y-field angle [deg] x-field angle [deg] (b) y-field angle [deg] Astigmatism [waves] Astigmatism [waves] x-field angle [deg] (c) y-field angle [deg] x-field angle [deg] (d) y-field angle [deg] Figure 6-9: Magnitude of astigmatism of (a) the nominal Ritchey-Chrétien telescope (b) binodal astigmatism in the presence of misalignments after alignment to remove constant coma (c) an aligned telescope with only astigmatic figure error and (d) combined misalignments (b) and astigmatic figure error (c). Equations (6-1) - (6-) reveal that for the two astigmatic nodes to coincide a specific ratio of field-linear and field-constant astigmatism has to be given. Any mismatch of this required ratio can be compensated by introducing field-constant astigmatism by applying an astigmatic figure error at the pupil as described in Equations (6-3) - (6-4). To illustrate this result the identical example of a misaligned Ritchey-Chrétien has been utilized showing the misalignment induced binodal response in Figure 6-10(b). Utilizing Equation (6-3) - (6-4) the required astigmatic figure error has been computed to obtain collapsing astigmatic nodes which would cause in the presence of zero misalignments the 193

215 binodal response shown in Figure 6-10(c). The resultant combination of secondary mirror misalignments and astigmatic primary figure error is then shown in Figure 6-10(d). Note that the field dependence of astigmatism is essentially unchanged compared to Figure 6-10(a) which shows the field dependence of astigmatism in the nominal system with the difference that the two coincident astigmatic nodes are displaced from the field center. The field point at which the two nodes coincide is given by the aberration field vector a which is solely determined by secondary mirror misalignments and can t be modified by primary mirror figure adjustments. The consequences of utilizing mirror figure to collapse the two astigmatic nodes are shown in Figure 6-11(a) which shows the RMS wavefront error plotted across the FOV. It is noticeable that even though the telescope is aligned for zero coma and astigmatism is merely displaced in the field but behaves otherwise quadratic the RMS wavefront error is not symmetric with respect to the field point denoted by a. Further analysis shows that the residual misalignments cause the medial focal surface (i.e. the surface of minimum RMS wavefront error) to decenter. If other consequences like a displaced exit-pupil with respect to the telescope axis (defined here by the primary mirror optical axis) among other effects are not prohibitive for the particular application the performance of the telescope can be improved by tilting the detector (if this is physically possible) resulting in an RMS wavefront error as shown in Figure 6-11(b). 194

216 Astigmatism [waves] Astigmatism [waves] x-field angle (a) y-field angle x-field angle (b) y-field angle Astigmatism [waves] Astigmatism [waves] x-field angle (c) y-field angle x-field angle (d) y-field angle Figure 6-10: Magnitude of astigmatism of (a) the nominal Ritchey-Chrétien telescope (b) binodal astigmatism in the presence of misalignments after alignment to remove constant coma (c) the aligned telescope including astigmatic figure error and (d) combined misalignments (b) and astigmatic figure error (c) to collapse the astigmatic nodes. 6.5 Validation of the Nodal Properties of a Ritchey-Chrétien Telescope with Misalignments and Astigmatic Figure Error To validate the predicted nodal behavior of astigmatism as shown in Figure Figure 6-10 FFDs for the astigmatic line images for alignment errors alone figure errors alone and the simultaneous combination are shown in Figure 6-1(a-e) respectively. Comparing the node locations predicted by nodal aberration theory and real-raytrace data in all cases in the presence of misalignments only in the case of astigmatic primary mirror figure errors only and misalignments combined with astigmatic figure error excellent agreement has been found. 195

217 0. waves (63.8 nm) 0. waves (63.8 nm) Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees (a) X Field Angle in Object Space - degrees (b) X Field Angle in Object Space - degrees Figure 6-11: Plot of the RMS wavefront error across the field for (a) a misaligned telescope with added astigmatic figure error to obtain coincident astigmatic nodes and (b) same as (b) but with adjusted focal plane tilt. Specifically Figure 6-8(b) (generated with Equation (6-11) - (6-16)) and Figure 6-1(b) (Coddington raytrace in optical design software) show the astigmatic node locations in the presence of misalignments only. Similarly Figure 6-8(c) (generated with Equation (6-17)) corresponds to the case shown in Figure 6-1(c) (Coddington raytrace in optical design software) visualizing the case where only astigmatic figure errors are present and the node locations in the presence of combined misalignments and astigmatic figure error are shown in Figure 6-8(d) (generated with Equations (6-10) - (6-16) and verified with the FFD in Figure 6-1(d). The specific combination of misalignments and astigmatic figure error to collapse the astigmatic nodes is shown in Figure 6-10(d) with its real-ray equivalent Coddington raytrace shown in Figure 6-1(e). Even though not obvious from visually comparing Figure 6-1(c) and Figure 6-1(d) a quantitative comparison of the FRINGE Zernike coefficients at the field center (i.e. x-field = y-field = 0) reveals identical values for Z 5 and Z 6 confirming the predicted behavior with nodal theory. 196

218 mm mm Y Field Angle in Object Space - degrees X Field Angle in Object Space - degrees (a) Y Field Angle in Object Space - degrees X Field Angle in Object Space - degrees (b) mm mm Y Field Angle in Object Space - degrees Y Field Angle in Object Space - degrees X Field Angle in Object Space - degrees (c) 0. mm X Field Angle in Object Space - degrees (e) Y Field Angle in Object Space - degrees X Field Angle in Object Space - degrees (d) Figure 6-1: Real-Ray based verification of the prediction of astigmatic nodal positions by a generalized Coddington raytrace illustrating the (a) astigmatic aberration field of the nominal system (b) node positions for secondary mirror misalignments only in a configuration that is corrected for zero coma (c) node positions corresponding to the astigmatic figure error illustrated in Figure 6-8c (d) net display including the secondary misalignments and astigmatic figure error and (e) combined secondary mirror misalignments and primary mirror astigmatic figure chosen to obtain coincident astigmatic nodes. 197

219 6.6 Example: Computation of Misalignment Parameters in a Misaligned Ritchey-Chrétien Telescope with the Aperture Stop at the Primary Mirror Including Primary Astigmatic Figure Error The outlined approach to compute the secondary mirror misalignment parameters in the presence of astigmatic figure error of the primary mirror is demonstrated on the case of a misaligned Ritchey-Chrétien telescope (F/10.6 EPD = 3000 mm). An optical prescription of the telescope is given in Table D-8. Given by its correction state the performance of the nominal telescope is limited by field astigmatism which is illustrated by a FFD for Zernike polynomials coefficients Z5/6 (Figure 6-13(a)). Also shown are point spread functions computed at nine sample field points one in the center of the adapter and eight equally spaced field points located on a circle (5.9 arcmin) which is centered at the adapter (Figure 6-13(b)). The applied secondary mirror misalignments and primary mirror figure errors are summarized and listed in Table 6-1 and the aim of this analysis is to compute the applied misalignments and primary mirror figure error based on the nodal properties of third order coma and astigmatism. In practice axial spacings also have to be adjusted during the alignment of the telescope. Since despace manifests itself in increased spherical aberration which does not exhibit any field dependence and remains unchanged in the presence of transverse tilts and decenters of the secondary mirror taking measurements of spherical aberration at the center of the adapter is sufficient to directly compute the amount of despace. Consequently this aspect of the alignment process is not included in this example. In the initially misaligned telescope constant coma is the dominant aberration response which is visualized through point spread functions as shown in Figure Utilizing the 198

220 correspondence between the Zernike polynomials and the wave aberration coefficients given that higher order aberrations are negligible (i.e. W 3 Z + Z (Tyson 198)) the amount of coma in x- and y-field direction was found to be δ A = ( ) [waves@ 63.8 nm]. waves(63.8 nm) Y Real Ray Image Height - mm arcmin X Real Ray Image Height - mm (a) (b) Figure 6-13: (a) FFD for astigmatism and (b) Point Spread Functions simulated on-axis and at a ring of field points at 5.9 arcmin in the field for the nominal alignment state of the Ritchey-Chrétien telescope. Table 6-1: Decenter and tilt perturbations of the initially misaligned Ritchey- Chrétien telescope. ( fig ) ( fig ) XDE YDE ADE BDE C5 C 6 [mm] [mm] [arcmin] [arcmin] [nm] [nm] Applied decenter/tip-tilt misalignments of the and astigmatic PM figure error PM (STOP)

221 Knowing the amount of coma a perturbation of the secondary mirror can be calculated to cancel the amount of coma present. While in principle both decenters or tilts could be used to compensate misalignment induced coma in this case a tip-tilt of the secondary mirror about its vertex is chosen since in this case the vertex remains stationary. Consequently only the coma contribution from the spherical base curve of the secondary mirror is affected by a tip-tilt of the secondary mirror about its vertex. Expressions for secondary mirror tip-tilts are given by (APPENDIX F) δade u (1 + dc ) = δa (6-5) PM y ( sph) W131 δbde u (1 + dc ) = δa (6-6) PM x ( sph) W131 δ δ = arcmin to collimate the telescope for zero resulting in ( ADE BDE ) ( ) coma. The decenter and tip-tilt parameters after collimating the telescope for zero coma are summarized in Table 6-. As expected caused by the existence of a coma-free pivot point for the secondary mirror residual misalignments of the secondary remain leading to image quality degradation caused by misalignment and figure induced astigmatism which is shown in Figure

222 0.mm Figure 6-14: Simulated point spread functions demonstrating constant coma across the FOV. Table 6-: Decenter and tip-tilt misalignments after compensating misalignment induced coma. ( fig ) ( fig ) XDE YDE ADE BDE C 5 C 6 [mm] [mm] [arcmin] [arcmin] [nm] [nm] Decenter and tilt misalignments after compensating misalignment induced coma PM (STOP) Utilizing the wavefront data for the field points shown in Figure 6-13(b) Equations (5-15) - (5-16) have been utilized to re-construct the binodal astigmatic aberration field providing data for the aberration field vector a AR = ( ) visualized in Figure (normalized) as 6 AR Note the astigmatic aberration field is plotted across the detector in units of mm. The vector which is a normalized quantity can be converted to units of mm on the detector by multiplication with the paraxial chief-ray AR height at the detector i.e. a y IMG. 01 a

223 Solving Equations (3-3) (3-33) and (3-38) provides the aberration field centers for the secondary mirror and combined with Equations (3-34) - (3-37) the decenter and tilt parameters of the secondary mirror given by XDE.497 mm YDE.005 mm ADE = arcmin BDE = arcmin. A comparison of the re-constructed secondary mirror decenters and tip-tilts with Table 6- indicates excellent agreement. All equations derived in Section 3 assume that the primary mirror and adapter axis are coaxial. If this assumption is not met the accuracy of the predicted misalignment parameters for the secondary mirror will degrade since slightly different aberration field vectors are measured. To analyze the impact of possible misalignments of the primary mirror and adapter axis for the particular example the secondary mirror decenter and tip-tilt y-z - plane are computed for various offsets of the AR axis which is shown for secondary mirror decenters in Figure 6-16(a) and for secondary mirror tip-tilts in Figure 6-16(b). As shown the accuracy for the reconstructed misalignment parameters degrades linearly with primary mirror axis to image plane center offsets but given realistic offsets to expect in practice of ~ < 1 mm a few alignment iterations will be sufficient. Having aligned the secondary mirror with respect to the primary mirror residual astigmatism can be solely attributed to astigmatic primary mirror figure error assuming the secondary mirror is free of astigmatic figure error and the optical performance is illustrated in Figure 6-17 showing a FFD for the astigmatic line images (Figure 6-17(a)) and point spread functions across the field (Figure 6-17(b)). Clearly visible is the binodal field dependence for astigmatism centered with the midpoint between the nodes at the field center. 0

224 waves(63.8 nm) 100 Y Real Ray Image Height - mm arcmin X Real Ray Image Height - mm (a) (b) a AR y IMG (c) Figure 6-15: Optical performance of the telescope after eliminating misalignment induced coma showing (a) FFD for Zernike polynomial coefficients Z5/6 and (b) simulated point spread functions and (c) re-constructed aberration field for astigmatism with overlaid field points for the wavefront data. 03