Four-Mirror Freeform Design

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23:06:34 38.46 MM Four-Mirror, Tuned Scale: 0.65 02-Jul-17 Four-Mirror Freeform Design Jonathan C. Papa, Joseph M. Howard, and Jannick P. Rolland 1

NASA Technology Roadmap This work was supported by a NASA Space Technology Research Fellowship 2

Project Overview 23:04:56 Central objective: survey of the four-mirror freeform solution space that considers geometries that could be advantageous for system constraints, such as mass, volume, stray light control, or radiation shielding. 23:06:34 27.47 MM On Axis Version Scale: 0.91 02-Jul-17 Methods/techniques: use analytically designed starting points before adding/varying freeform terms to explore different design forms. 38.46 MM Four-Mirror, Tuned Scale: 0.65 02-Jul-17 3

Parts of the Design Process Selection of Suitable Starting Point Investigate multiple starting point designs. Optimization Techniques Investigate multiple optimization approaches. Tolerancing/Sensitivity Analysis Compare sensitivity across the different design forms. 4

Procedure Overview: Off-Axis Conics 23:04:56 with Aspheric Caps on Top 1. Choose first order layout, and model as Cartesian reflectors, also known as confocal conics (If a flat-field is desired, mirror powers need to be balanced in this step). 2. Solve for 4 th order aspheric terms on top of Cartesian reflector to correct third-order aberrations. 3. Choose tilts for the surfaces such that the system is unobscured and fieldasymmetric field-linear astigmatism is canceled. 23:04:56 27.47 MM On Axis Version Scale: 0.91 02-Jul-17 23:06:34 27.47 MM On Axis Version Scale: 0.91 02-Jul-17 38.46 MM Four-Mirror, Tuned Scale: 0.65 02-Jul-17 5

Key Points from Literature (Steps 1&2) Correcting Seidel third-order aberrations in a given first-order layout for 4 mirrors using aspheric deformations (aperture^4) is a linear system of equations [Korsch, 1973]. A system made of Cartesian reflectors (conics with stigmatic imaging at each individual surface for the base field point, also known as confocal conics) can only be corrected through third-order if the system has a magnification of 1X, or if the system is afocal [Korsch, 1991]. To make a focal system, with an infinite conjugate, that is corrected through third-order, you cannot use only Cartesian reflectors/confocal conics For example: a classic Cassegrain with a stigmatically imaging parabolic primary and stigmatically imaging hyperbolic secondary, cannot correct for coma. A Ritchey-Chretien has spherical aberration introduced at the primary and canceled at the secondary in a way that cancels coma). Korsch, Dietrich. "Closed-form Solutions for Imaging Systems, Corrected for Third-order Aberrations." Journal of the Optical Society of America 63.6 (1973): 667. Korsch, D. Reflective Optics. Boston: Academic, 1991. Print. 6

Solving for Aspheric Caps 1 1 1 1 4 y 1 y 1 4 y 2 y 2 4 y 3 y 3 4 y 4 4 y 1 y 1 4 y 1 y 1 2 3 4 y 2 y 2 4 y 2 y 2 2 3 4 y 3 y 3 4 y 3 y 3 2 3 y 4 4 y 4 y 4 4 y 4 y 4 2 3 W 040 1 W 040 2 W 040 3 W 040 4 = W 040 W 131 W 222 W 311 27.47 MM On Axis Version Scale: 0.91 02-Jul-17 Spherical Coma Astigmatism Distortion Related to Sag of 4 th order aspheric caps 7

Key Points from Literature (Step 3) It is possible to find tilted/decentered systems that exhibit aberrations of the ordinary kind (where two astigmatism nodes collapse to single node at center of field, while keeping the coma node at the center of field as well) [Rogers, 1986]. Tilted aspheric term (looks like spherical aberration) produces aberrations of the ordinary kind, meaning field-linear coma, and field-quadratic astigmatism [Rogers, 1986]. 1.0 1.0 Y Field Angle in Object Space - degrees 0.5 0.0-0.5 Y Field Angle in Object Space - degrees 0.5 0.0-0.5-1.0-1.0-1.0-0.5 0.0 0.5 1.0 X Field Angle in Object Space - degrees -1.0-0.5 0.0 0.5 1.0 X Field Angle in Object Space - degrees Rogers, John R. "Vector Aberration FRINGE ZERNIKE PAIR Theory Z5 AND Z6 And The Design Of Off-Axis Systems." 1985 International FRINGE ZERNIKE PAIR Z5 AND Z6 Lens Design vs vs Conference (1986). 12:36:54 Four-Mirror, Tuned 09-Jul-17 Minimum = 0.38216e-10 Maximum = 1.2371 Average = 0.48322 Std Dev = 0.31528 2.6waves (587.6 nm) Tech Days Nov. 15, 2017 09-Jul-17 12:33:46 Four-Mirror, Tuned Minimum = 0.22542e-11 Maximum = 0.64546 Average = 0.33534 Std Dev = 0.19634 1.4waves (587.6 nm) 8

1.0 Y Field Angle in Object Space - degrees 0.5 0.0-0.5-1.0-1.0-0.5 0.0 0.5 1.0 X Field Angle in Object Space - degrees As the base FRINGE ray ZERNIKE PAIR is Z5 AND bent Z6 at M1, M2 vs Four-Mirror, Tuned bends to keep Minimum ordinary = 0.22542e-11 Maximum = 0.64546 Average = 0.33534 Std Dev = 0.19634 astigmatism 09-Jul-17 field1.4waves (587.6 nm)

Key Points from Literature: Aberration Coefficients for Plane-Symmetric Systems Constraining a surface to be a stigmatically imaging off-axis conic eliminates [Sasian, 1994]: Spherical Aberration Field-Constant Coma Field-Constant Astigmatism Anamorphism Ignoring distortion and piston terms, the remaining image degrading aberrations are: Field-Linear Coma Field-Asymmetric Field-Linear Astigmatism (FAFL) Field-Quadratic Astigmatism Field Curvature The designer needs to eliminate FAFL Astigmatism to make the aberrations ordinary and correctable using aspheric caps. Sasian, Jose M. "How to Approach the Design of a Bilateral Symmetric Optical System."Optical Engineering 33.6 (1994): 2045.

11

Four-Mirror Example Stop Foci of conics (not coaxial) 12

Create On-Axis Equivalent, Third-Order Corrected 23:04:56 27.47 MM On Axis Version Scale: 0.91 02-Jul-17 M1 M2 M3 M4 Radius -743.522-186.402-448.082-1246.81 K -1-0.11681-0.0161-40.8171 A 6.46E-12-5.05E-09-7.06E-10-6.38E-09 Save Aspheric Caps For Later Step K is the conic constant that corresponds to a stigmatic imaging conic for the axial field point. On top of these stigmatic conics is a 4 th order A asphere coefficient that is used to correct the third-order field aberrations (field-linear coma, field-quadratic astigmatism) while leaving third-order spherical corrected. 13

22:50:31 Create Chain of Stigmatic Imaging Conics (Aspheric Caps Removed), with Tilts That Cancel FAFL Astigmatism According to Aberration Coefficients 1.0 Y Field Angle in Object Space - degrees 0.5 0.0-0.5-1.0 41.67 MM Four-Mirror, No Tuning Scale: 0.60 02-Jul-17 Astigmatism is not quite ordinary 00:17:38 FRINGE ZERNIKE PAIR Z5 AND Z6 (WAVES AT 587.6 NM) Four-Mirror, No Tuni ng 22:53:28 0.390 0.347 0.305 0.262 0.220 0.177 0.134 0.092 0.049 0.007 11-Jul-17 Four-Mirror, No Tuni ng 02-Jul-17-1.0-0.5 0.0 0.5 1.0 X Field Angle in Object Space - degrees Max 0.3535 waves Field Limits ( 0.000, -1.000) ( 0.000, 1.000) -0.036-1.00-0.80-0.60-0.40-0.20 0.00 0.20 0.40 0.60 0.80 1.00 FRINGE ZERNIKE PAIR Z5 AND Z6 vs Minimum = 0.12388e-7 Maximum = 0.35357 Average = 0.23142 Std Dev = 0.082739 0.73waves (587.6 nm) 14

22:55:34 Tune the Angle of the Last Mirror to Bring Astigmatism Nodes Together 1.0 Y Field Angle in Object Space - degrees 0.5 0.0-0.5-1.0 Four-Mirror, Tuned Scale: 0.61 00:18:47 02-Jul-17 M4 tilt adjusted to -8.174 from -9.298 Astigmatism has single node, closer to ordinary 40.98 MM FRINGE ZERNIKE PAIR Z5 AND Z6 (WAVES AT 587.6 NM) Four-Mirror, Tuned 0.0260 0.0232 0.0203 0.0175 22:55:09 0.0146 0.0118 0.0090 0.0061 0.0033 0.0004 Four-Mirror, Tuned -1.0-0.5 0.0 0.5 1.0 02-Jul-17 X Field Angle in Object Space - degrees Max 0.063194 waves 11-Jul-17 Field Limits ( 0.000, -1.000) ( 0.000, 1.000) -0.0024-1.00-0.80-0.60-0.40-0.20 0.00 0.20 0.40 0.60 0.80 1.00 FRINGE ZERNIKE PAIR Z5 AND Z6 vs Minimum = 0.65632e-8 Maximum = 0.063194 Average = 0.023468 Std Dev = 0.016078 0.13waves (587.6 nm) 15

23:04:56 Put Aspheric Caps Back On 23:06:34 27.47 MM -0.1375 waves On Axis Version Scale: 0.91 02-Jul-17-2.0949 waves 3.1585 waves -0.9260 waves 38.46 MM Four-Mirror, Tuned Scale: 0.65 02-Jul-17 16

Aberrations Before and After Caps: Spherical Aberration Before After 1.0 1.0 Y Field Angle in Object Space - degrees 0.5 0.0-0.5 Y Field Angle in Object Space - degrees 0.5 0.0-0.5-1.0-1.0-1.0-0.5 0.0 0.5 1.0-1.0-0.5 0.0 0.5 1.0 X Field Angle in Object Space - degrees Max 0.00071 waves X Field Angle in Object Space - degrees Max 0.015 waves 13:22:18 Four-Mirror, Tuned 03-Jul-17 FRINGE ZERNIKE COEFFICIENT Z9 vs Minimum = -0.71345e-3 Maximum = -0.38835e-9 Average = -0.36188e-3 Std Dev = 0.00021177 13:20:32 0.0015waves (587.6 nm) Four-Mirror, Asphere Caps 03-Jul-17 FRINGE ZERNIKE COEFFICIENT Z9 vs Minimum = -0.014755 Maximum = -0.44387e-3 Average = -0.0053689 Std Dev = 0.0037095 0.031waves (587.6 nm) 17

Aberrations Before and After Caps: Coma Before After 1.0 1.0 13:05 FRINGE ZERNIKE PAIR Z7 AND Z8 (WAVES AT 587.6 NM) Four-Mirror, Tuned 0.540 0.481 0.422 0.363 0.304 0.245 0.187 0.128 0.069 0.010 11-Jul-17 23:12:37 Y Field Angle in Object Space - degrees Four-Mirror, Tuned 0.5 0.0 Field Limits ( 0.000, -1.000) ( 0.000, 1.000) -0.5-1.0-1.0-0.5 0.0 0.5 1.0 02-Jul-17 X Field Angle in Object Space - degrees -0.049-1.00-0.80-0.60-0.40-0.20 0.00 0.20 0.40 0.60 0.80 1.00 Max 0.48257 waves FRINGE ZERNIKE PAIR Z7 AND Z8 vs Minimum = 0.57452e-7 Maximum = 0.48257 Average = 0.32428 Std Dev = 0.11675 1waves (587.6 nm) 23:22:29 Y Field Angle in Object Space - degrees Four-Mirror, Asphere Caps 0.5 0.0-0.5-1.0-1.0-0.5 0.0 0.5 1.0 02-Jul-17 X Field Angle in Object Space - degrees -1.00-0.80-0.60-0.40-0.20 0.00 0.20 0.40 0.60 0.80 1.00 Max 0.083627 waves 00:14:23 FRINGE ZERNIKE PAIR Z7 AND Z8 vs Minimum = 0.00018816 Maximum = 0.083627 Average = 0.023135 Std Dev = 0.019828 FRINGE ZERNIKE PAIR Z7 AND Z8 (WAVES AT 587.6 NM) Four-Mirror, Asphere Caps 0.0930 0.0829 0.0728 0.0626 0.0525 0.0424 0.0323 0.0222 0.0120 0.0019-0.0082 0.18waves (587.6 nm) 11-Jul-17 Field Limits ( 0.000, -1.000) ( 0.000, 1.000) 18

Aberrations Before and After Caps: Astigmatism (Coddington) Before After 1.0 1.0 :40 ASTIGMATIC LINE IMAGE (MILLIMETERS) Four-Mirror, Tuned 0.00210 0.00187 0.00164 0.00141 0.00118 0.00095 0.00073 0.00050 0.00027 0.00004 09-Jul-17 23:15:43 Y Field Angle in Object Space - degrees Four-Mirror, Tuned 0.5 0.0 Field Limits ( 0.000, -1.000) ( 0.000, 1.000) -0.5-1.0-0.00019-1.00-0.80-0.60-0.40-0.20 0.00 0.20 0.40 0.60 0.80 1.00-1.0-0.5 0.0 0.5 1.0 02-Jul-17 X Field Angle in Object Space - degrees Avg. Line 1.81 microns ASTIGMATIC LINE IMAGE vs Minimum = 0 Maximum = 0.0048247 Average = 0.0018075 Std Dev = 0.0012178 0.01mm 23:17:43 Y Field Angle in Object Space - degrees Four-Mirror, Asphere Caps 0.5 0.0-0.5-1.0 02-Jul-17 X Field Angle in Object Space - degrees Avg. Line 1.67 microns ASTIGMATIC LINE IMAGE vs Minimum = 0 Maximum = 0.0070252 Average = 0.0016681 Std Dev = 0.0015674 0.015mm 12:14:18 Four-Mirror, Asphere 0.00014-1.0-0.5 0.0 0.5 1.0-0.00071-1.00-0.80-0.60-0.40-0.20 0.00 0.20 0.40 0.60 0.80 1.00 ASTIGMATIC LINE IMAGE (MILLIMETERS) Caps 0.00780 0.00695 0.00610 0.00525 0.00440 0.00354 0.00269 0.00184 0.00099 09-Jul-17 Field Limits ( 0.000, -1.000) ( 0.000, 1.000) 19

RMS Wavefront Error Before and After Caps Before After 1.0 1.0 Y Field Angle in Object Space - degrees 0.5 0.0-0.5 Y Field Angle in Object Space - degrees 0.5 0.0-0.5-1.0-1.0 16:01:40 Four-Mirror, Tuned -1.0-0.5 0.0 0.5 1.0 09-Jul-17 X Field Angle in Object Space - degrees Max 0.17 waves RMS WAVEFRONT ERROR vs Minimum = 0.23046e-7 Maximum = 0.17389 Average = 0.11652 Std Dev = 0.04235 0.36waves (587.6 nm) 15:59:59 Four-Mirror, Asphere Caps -1.0-0.5 0.0 0.5 1.0 09-Jul-17 X Field Angle in Object Space - degrees Max 0.09 waves RMS WAVEFRONT ERROR vs Minimum = 0.014957 Maximum = 0.090362 Average = 0.030667 Std Dev = 0.016946 0.19waves (587.6 nm) 20

Approaching NASA Tech Challenge 12:10:21 200 mm Aperture 5 x 5 FOV F/4 119.05 MM Four-Mirror Freeform 200mm EPD Scale: 0.21 09-Nov-17 21

Four-Mirror Freeform Spot Diagram 16:18:45 FIELD POSITION -0.60,-0.60-1.50,-1.50 DG 0.60,-0.60 1.500,-1.50 DG 0.00,-0.60 0.000,-1.50 DG 0.80, 0.60 2.000,1.500 DG -0.80, 0.60-2.00,1.500 DG -1.00, 0.00-2.50,0.000 DG 1.00, 0.00 2.500,0.000 DG 0.00,-1.00 0.000,-2.50 DG 0.00, 1.00 0.000,2.500 DG -1.00,-1.00-2.50,-2.50 DG 1.00,-1.00 2.500,-2.50 DG -1.00, 1.00-2.50,2.500 DG 1.00, 1.00 2.500,2.500 DG 0.00, 0.00 0.000,0.000 DG DEFOCUSING 0.00000 Four-Mirror Freeform 200mm EPD.482E-01 MM RMS 09-Nov-2017 = 0.005024 100% = 0.013877 RMS = 0.005024 100% = 0.013877 RMS = 0.004164 100% = 0.010580 RMS = 0.004202 100% = 0.011846 RMS = 0.004202 100% = 0.011846 RMS = 0.005956 100% = 0.014401 RMS = 0.005956 100% = 0.014401 RMS = 0.005830 100% = 0.012755 RMS = 0.006757 100% = 0.015076 RMS = 0.005537 100% = 0.013123 RMS = 0.005537 100% = 0.013123 RMS = 0.004172 100% = 0.012032 RMS = 0.004172 100% = 0.012032 RMS = 0.003949 100% = 0.008402 12:10:25 Y Field Angle in Object Space - degrees Four-Mirror Freeform 200mm EPD 3 2 1 0-1 -2-3 -3-2 -1 0 1 2 3 X Field Angle in Object Space - degrees 09-Nov-17 RMS SPOT DIAMETER vs Minimum = 0.0030443 Maximum = 0.0073866 Average = 0.0051233 Std Dev = 0.0010977 0.018mm 22

Freeform Caps on Original Base Off-Axis Conic 23

Freeform Caps on Refit Base Off-Axis Conic 24

Summary Several analytical starting point design methods exist with various symmetries and states of correction. A combination of these methods can allow for unobscured starting points that are corrected for third order image degrading aberrations, which will be used to facilitate a survey of the four-mirror freeform solution space. 25

References Howard, J.M, and B. D. Stone, "Imaging with four spherical mirrors," Appl. Opt.39, 3232-3242 (2000) Rakich, A. "Four-mirror Anastigmats, Part 1: A Complete Solution Set for All-spherical Telescopic Systems." Optical Engineering Opt. Eng46.10 (2007b): 103001. Korsch, Dietrich. "Closed-form Solutions for Imaging Systems, Corrected for Third-order Aberrations." Journal of the Optical Society of America 63.6 (1973): 667. Korsch, D. Reflective Optics. Boston: Academic, 1991. Print. Rogers, John R. "Vector Aberration Theory And The Design Of Off-Axis Systems." 1985 International Lens Design Conference (1986). Sasian, Jose M. "How to Approach the Design of a Bilateral Symmetric Optical System."Optical Engineering 33.6 (1994): 2045. Elazhary T.T., P. Zhou, C. Zhao, and J. H. Burge, "Optical Design Method for Free Form Optics," in Renewable Energy and the Environment, OSA Technical Digest (online) (Optical Society of America, 2013), paper FM4B.4. Benitez, P., M. Nikolic, and J. C. Miñano. "Analytical Solution of an Afocal Two Freeform Mirror Design Problem," Optics Express 25.4 (2017): 4155. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, Theory of aberration fields for general optical systems with freeform surfaces, Opt. Express 22(22), 26585-26606 (2014). A. Bauer and J. P. Rolland, "Design of a freeform electronic viewfinder coupled to aberration fields of freeform optics," Opt. Express 23(22), 28141-28153 (2015). Seunghyuk Chang, "Linear astigmatism of confocal off-axis reflective imaging systems with N-conic mirrors and its elimination," J. Opt. Soc. Am. A32, 852-859 (2015) 26

Put Aspheric Caps Back On M1 M2 M3 M4 Radius -743.522-186.402-448.082-1246.81 K -1-0.11681-0.0161-40.8171 A 6.46E-12-5.05E-09-7.06E-10-6.38E-09 A Coefficients from On-Axis Equivalent from Before M1 M2 M3 M4 Asphere 6.46E-12 Asphere -5.05E-09 Asphere -7.06E-10 Asphere -6.38E-09 Existing Needed Sum Existing Needed Sum Existing Needed Sum Existing Needed Sum x^4 2.94E-10 6.46E-12 3.00E-10 8.10E-10-5.05E-09-4.24E-09-2.93E-11-7.06E-10-7.35E-10 2.66E-09-6.38E-09-3.72E-09 x^3*y 0.00E+00 0 0.00E+00 0.00E+00 0 0.00E+00 0.00E+00 0 0.00E+00 0.00E+00 0 0.00E+00 x^2*y^2 5.47E-10 1.29E-11 5.60E-10 2.97E-09-1.01E-08-7.13E-09 7.78E-12-1.41E-09-1.40E-09 4.99E-09-1.28E-08-7.77E-09 x*y^3 0.00E+00 0 0.00E+00 0.00E+00 0 0.00E+00 0.00E+00 0 0.00E+00 0.00E+00 0 0.00E+00 y^4 2.55E-10 6.46E-12 2.61E-10 2.14E-09-5.05E-09-2.91E-09 3.63E-11-7.06E-10-6.70E-10 2.34E-09-6.38E-09-4.04E-09-0.1375 waves -2.0949 waves 3.1585 waves -0.9260 waves Spherical Aberration Contribution of Caps 27

On Axis Version 0.99E-03 09-Jul-17 Field Limits ( 0.000, -1.000) ( 0.000, 1.000) ASTIGMATIC LINE IMAGE (MILLIMETERS) 0.88E-03 0.77E-03 0.67E-03 0.56E-03 0.45E-03 0.34E-03 0.23E-03 0.13E-03 0.18E-04 -.90E-04-1.00-0.80-0.60-0.40-0.20 0.00 0.20 0.40 0.60 0.80 1.00 28

Four-Mirror, No Tuni ng 09-Jul-17 Field Limits ( 0.000, -1.000) ( 0.000, 1.000) 0.0300 ASTIGMATIC LINE IMAGE (MILLIMETERS) 0.0267 0.0235 0.0202 0.0169 0.0137 0.0104 0.0071 0.0038 0.0006-0.0027-1.00-0.80-0.60-0.40-0.20 0.00 0.20 0.40 0.60 0.80 1.00 29

Four-Mirror, Tuned Field Limits 09-Jul-17 ( 0.000, -1.000) ( 0.000, 1.000) 0.00210 ASTIGMATIC LINE IMAGE (MILLIMETERS) 0.00187 0.00164 0.00141 0.00118 0.00095 0.00073 0.00050 0.00027 0.00004-0.00019-1.00-0.80-0.60-0.40-0.20 0.00 0.20 0.40 0.60 0.80 1.00 30

Four-Mirror, Asphere Caps 09-Jul-17 Field Limits ( 0.000, -1.000) ( 0.000, 1.000) 0.00780 ASTIGMATIC LINE IMAGE (MILLIMETERS) 0.00695 0.00610 0.00525 0.00440 0.00354 0.00269 0.00184 0.00099 0.00014-0.00071-1.00-0.80-0.60-0.40-0.20 0.00 0.20 0.40 0.60 0.80 1.00 31

Aberration Coefficients (Spherical Contribution) 32

Non-Spherical Contributions Choose these, so that these cancel the field constant contributions 33

Remaining Coefficients after Adding Non- Spherical Contributions from Off-Axis Conics Anamorphism conveniently cancels too 34

Screenshot of Geogebra Tool

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