1. Optical Sciences CenterThe University of Arizona ERROR ANALYSIS FOR CGH OPTICAL TESTING Yu-Chun Chang and James Burge Optical Science Center University of Arizona

2. Optical Sciences CenterThe University of Arizona Applications of CGH in Optical Testing Optical interferometry measures shape differences between a reference and the test piece; Test pieces with complex surface profiles require reference surfaces with matched shapes or null lenses; Using CGHs to produce reference wavefronts eliminates the need of making expensive reference surfaces or null optics.

3. Optical Sciences CenterThe University of Arizona CGHs in Optical Interferometry

4. Optical Sciences CenterThe University of Arizona CGHs in Optical Interferometry Quality of the wavefront generated by CGHs affects the accuracy of interferometric measurements; Abilities to predict and analyze these phase errors are essential.

5. Optical Sciences CenterThe University of Arizona CGH Fabrication Errors Traditional fabrication method is done through automated plotting and photographic reduction; Modern technique uses direct laser/electron beam writing; Fabrication uncertainties are mostly responsible for the degradation of the quality of CGHs;

6. Optical Sciences CenterThe University of Arizona Sources of Errors CGH Fabrication A CGH may simply be treated as a set of complicated interference fringe patterns written onto a substrate material; CGH substrate figure errors; CGH pattern errors includes; –fringe position errors; –fringe duty-cycle errors; –fringe etching depth errors.

7. Optical Sciences CenterThe University of Arizona Substrate Figure Errors Typical CGH substrate errors are low spatial frequency surface figure errors; Produce low spatial frequency wavefront aberrations in the diffracted wavefront. CGH substrate Transmitted wavefront Reflected wavefront Incident wavefront ss  s (n = index of refraction ) (n-1)  s

8. Optical Sciences CenterThe University of Arizona Pattern Distortion The hologram used at m th order adds m waves per line; CGH pattern distortions produce wavefront phase error:

9. Optical Sciences CenterThe University of Arizona Binary Linear Grating Model Binary linear grating models are used to study grating duty-cycle and etching depth errors; Scalar diffraction theory is used for wavefront phase and amplitude calculations; Both phase gratings and chrome-on- glass gratings are studied; Analytical results are achieved.

10. Optical Sciences CenterThe University of Arizona Binary Linear Grating Model Output wavefront from a binary linear grating (normally incident plane wavefront): b A1eiA1ei A0A0 S x where A 0 and A 1 are amplitude functions and  is phase depth

11. Optical Sciences CenterThe University of Arizona Binary Linear Grating Model Diffraction wavefront function at Fraunhofer plane: where.

12. Optical Sciences CenterThe University of Arizona Binary Linear Grating Model Diffraction efficiency functions: Wavefront phase functions:

13. Optical Sciences CenterThe University of Arizona Diffraction Efficiency for Zero (m=0) Diffraction Orders (Phase Grating)

14. Optical Sciences CenterThe University of Arizona Diffraction Efficiency for Non-zero Diffraction Orders (Phase Grating)

15. Optical Sciences CenterThe University of Arizona Diffraction Wavefront Phase as a Function of Duty-cycle and Phase Depth phase grating at m=0

16. Optical Sciences CenterThe University of Arizona Wavefront Phase vs. Etching Depth for Non-zero Order Beams Duty-cycle: 0% - 100% m=1 Duty-cycle: 50% - 100% Duty-cycle: 0% - 50% m=2

17. Optical Sciences CenterThe University of Arizona Wavefront Phase vs. Duty-cycle for Non-zero Order Beams m=4 m=5m=6 m=1 m=2m=3

18. Optical Sciences CenterThe University of Arizona Phase Grating Sample

19. Optical Sciences CenterThe University of Arizona Chrome-on-glass Grating (Top view) Duty-cycle = 40% Spacing = 50 um Duty-cycle = 50% Spacing = 50 um 20um gap D = 40%D = 50%

20. Optical Sciences CenterThe University of Arizona Interferograms Obtained at Different Diffraction Orders (for chrome-on-glass grating) m=0

21. Optical Sciences CenterThe University of Arizona Wavefront Phase Sensitivity Functions Wavefront phase sensitivities to grating duty-cycle and phase depth.            cosAA2AA AAA 0 D :,...2,1m 10 2 0 2 1 10 2 10m 0m 00, for sinc(mD)=0 otherwise

22. Optical Sciences CenterThe University of Arizona Wavefront Phase Sensitivity Functions Wavefront phase sensitivity functions provide an easy solution for CGH fabrication errors analysis; Applications of wavefront phase sensitivity functions in optical testing are given.

23. Optical Sciences CenterThe University of Arizona CGH Errors Analysis Using Wavefront Sensitivity Functions Fizeau interferometer Phase type CGH Asphere Test Piece Spherical reference (Sample Phase CGH)

24. Optical Sciences CenterThe University of Arizona Sources of Errors Wavefront errors come from: –Surface figure * (n-1) –Pattern distortion/spacing –Etch depth variation * sensitivity from diffraction analysis –duty cycle variation * sensitivity from diffraction analysis RSS all terms give test error due to CGH (Sample Phase CGH)

25. Optical Sciences CenterThe University of Arizona Wavefront Phase Sensitivities to Grating Phase Depth Errors (Phase Grating at Zero-order Diffraction)

26. Optical Sciences CenterThe University of Arizona CGH Errors Analysis Using Wavefront Sensitivity Functions (Sample Phase CGH)

27. Optical Sciences CenterThe University of Arizona CGH Errors Analysis Using Wavefront Sensitivity Function (Sample Chrome CGH)

28. Optical Sciences CenterThe University of Arizona CGH Errors Analysis Using Wavefront Sensitivity Function (Sample Chrome CGH)

29. Optical Sciences CenterThe University of Arizona CGH Errors Analysis Using Wavefront Sensitivity Functions (Sample Chrome CGH)

30. Optical Sciences CenterThe University of Arizona Wavefront Phase Sensitivities Functions (Chrome-on-Glass Grating at Zero-order Diffraction) Duty-cycle ErrorsEtching Depth Errors

31. Optical Sciences CenterThe University of Arizona Fizeau interferometer Spherical reference Test plate (Sample Chrome CGH ) CGH Errors Analysis Using Wavefront Sensitivity Function

32. Optical Sciences CenterThe University of Arizona Conclusions Wavefront phase deviations due to CGH fabrication errors are studied; Analytical solutions are obtained and verified with experimental results; Applications of wavefront sensitivity functions in optical testing are demonstrated; Wavefront sensitivity functions provide a direct and intuitive method for CGH error analysis and error budgeting.