ERROR ANALYSIS FOR CGH OPTICAL TESTING

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

2. 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. CGHs in Optical Interferometry

4. 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. 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. 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. CGH substrate Incident wavefront (n = index of refraction) Transmitted wavefront Reflected wavefront (n-1)ds ds 2ds Substrate Figure Errors • Typical CGH substrate errors are low spatial frequency surface figure errors; • Produce low spatial frequency wavefront aberrations in the diffracted wavefront.

8. Pattern Distortion • The hologram used at mth order adds m waves per line; • CGH pattern distortions produce wavefront phase error:

9. 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. S b x A1ei A0 Binary Linear Grating Model • Output wavefront from a binary linear grating (normally incident plane wavefront): where A0 and A1 are amplitude functions and f is phase depth

11. Binary Linear Grating Model • Diffraction wavefront function at Fraunhofer plane: where .

12. Binary Linear Grating Model • Diffraction efficiency functions: • Wavefront phase functions:

13. Diffraction Efficiency for Zero (m=0) Diffraction Orders (Phase Grating)

14. Diffraction Efficiency for Non-zero Diffraction Orders (Phase Grating)

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

16. m=1 m=2 Duty-cycle: 0% - 50% Duty-cycle: 0% - 100% Duty-cycle: 50% - 100% Wavefront Phase vs. Etching Depthfor Non-zero Order Beams

17. Wavefront Phase vs. Duty-cycle for Non-zero Order Beams m=2 m=3 m=1 m=5 m=6 m=4

18. Phase Grating Sample

19. D = 40% D = 50% 20um gap Duty-cycle = 50% Spacing = 50 um Duty-cycle = 40% Spacing = 50 um Chrome-on-glass Grating(Top view)

20. Interferograms Obtained at Different Diffraction Orders (for chrome-on-glass grating) m=0

21. = ± ± m 1 , 2 ,... : 00 , for sinc(mD)=0 ¶ Y ¹ m 0 = 0 otherwise ¶ D ¶ Y - f 2 A A A cos = ¹ m 0 1 0 1 ¶ f + - f 2 2 A A 2 A A cos 1 0 0 1 Wavefront Phase Sensitivity Functions • Wavefront phase sensitivities to grating duty-cycle and phase depth.

22. 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. Spherical reference Phase type CGH Asphere Test Piece Fizeau interferometer CGH Errors Analysis Using Wavefront Sensitivity Functions (Sample Phase CGH)

24. Sources of Errors (Sample Phase CGH) • 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

25. Wavefront Phase Sensitivities to Grating Phase Depth Errors (Phase Grating at Zero-order Diffraction)

26. CGH Errors Analysis Using Wavefront Sensitivity Functions (Sample Phase CGH)

27. CGH Errors Analysis Using Wavefront Sensitivity Function (Sample Chrome CGH)

28. CGH Errors Analysis Using Wavefront Sensitivity Function (Sample Chrome CGH)

29. CGH Errors Analysis Using Wavefront Sensitivity Functions (Sample Chrome CGH)

30. Wavefront Phase Sensitivities Functions (Chrome-on-Glass Grating at Zero-order Diffraction) Duty-cycle Errors Etching Depth Errors

31. Spherical reference Test plate Fizeau interferometer CGH Errors Analysis Using Wavefront Sensitivity Function (Sample Chrome CGH )

32. 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.