1. Synopsis of Yang/Wang’s Analysis and Optimization on Single-Zone Binary Flat-Top Beam Shaper Blake Anderton Mon, Dec. 8 2008

2. The BIG problem … Overview: gain medium illuminated as shown. Very expensive laser gain medium f Fourier-conjugate planes

3. The BIG problem … Common input: circularly symmetric Gaussian Very expensive laser gain medium f Gaussian input Gaussian output

4. The BIG problem … Result: Strange and not-so-wonderful things happen. Very expensive laser gain medium f Gaussian input Gaussian output Too much light at center (scoring, crack) (also not good) Too little light at edge (lost opportunity) (not good)

5. But what if… …you modify the input beam’s phase to produce a uniform (“flat-top”, “circ”) pattern at gain medium? Very expensive laser gain medium Gaussian input UNIFORM output beam (!) Benefits of uniform beam at gain medium no scoring (too much light) no missed gain opportunities at edges (too little light) ? Phase plate (e i  scaling factor)

6. The “ideal” … Phase plate produces “Bessinc” entering lens Output: “perfect” circ in Fourier plane Gaussian input ? Phase plate “Bessinc”“Circ”

7. The “ideal” … & why it costs too much Phase plate produces “Bessinc” entering lens Output: “perfect” circ in Fourier plane Requires: continuous-phase plate (high precision etching) Gaussian input $$$$$$ CONTINUOUS-phase plate “Bessinc”“Circ”

8. How to mimic a Bessinc Compare plate input and ideal output –Input: Gaussian –Output: Bessinc Key difference: y-values (pos/neg) BEFORE phase plate AFTER phase plate (ideal) ? GaussBessinc

9. How to mimic a Bessinc Where do first negative Bessinc values occur (radially)? We designate that ring’s width by w. w w

10. How to mimic a Bessinc Where do first negative Bessinc values occur (radially)? We designate that ring’s width by w. Add remaining bands with approx. same width w w w w w w w

11. How to mimic a Bessinc Make Gaussian values negative within these bands. Result is negative where Bessinc is negative. w w w w w w

12. How to mimic a Bessinc Net result (right): –Not quite Bessinc –better than Gaussian. With this going into the lens, how do we build a phase plate to make this? ? Goes here

13. How do we make this phase plate? We seek to specify  in e i . What values must  take?

14. We seek to specify  in e i . What values must  take? –Everything btwn 0 and 2  How do we make this phase plate?

15. We seek to specify  in e i . What values must  take? –Everything btwn 0 and 2  –Just 0 or  (inside/outside the bands). How do we make this phase plate?

16. We seek to specify  in e i . What values must  take? –Everything btwn 0 and 2  –Just 0 or  (insdie/outside the bands). How to produce these  ’s? –Etch a substrate (index n) height profile h(r) such that = 0 or    How do we make this phase plate?

17. How many bands do we really need? At right: effect of 1 st, 2 nd bands Bands 2+ have minimal effect. Effect of 1 st band (BIG) Effect of 2 nd band (less than big) What if phase plate ONLY had 1 band (no 2 nd, 3 rd, …)?

18. Plot: comparing 1-, 2-, 3-band plates. Result: Not too different! Bottom line: only 1 band is needed. Don’t need Don’t need Don’t need Don’t need Don’t need Don’t need How many bands do we really need?

19. How far are we from ideal? 3 Figures of Merit (FOMs) How do you compare the Fourier-plane field produced by a binary-phase plate to the “ideal” circ? With 3 figures-of-merit: –Uniformity: “ringing” in central zone (less ringing is better) Plot of typical Fourier-plane intensity produced by single-banded binary-phase plate Central zone is defined as having intensity ≥ 90% peak

20. How far are we from ideal? 3 Figures of Merit (FOMs) How do you compare the Fourier-plane field produced by a binary-phase plate to the “ideal” circ? With 3 figures-of-merit: –Uniformity: “ringing” in central zone (less ringing is better) –Steepness: slope of central-zone boundary (steeper is better) Plot of typical Fourier-plane intensity produced by single-banded binary-phase plate Central zone is defined as having intensity ≥ 90% peak

21. How far are we from ideal? 3 Figures of Merit (FOMs) How do you compare the Fourier-plane field produced by a binary-phase plate to the “ideal” circ? With 3 figures-of-merit: –Uniformity: “ringing” in central zone (less ringing is better) –Steepness: slope of central-zone boundary (steeper is better) –Efficiency: energy spread beyond central-zone (less loss is better) Plot of typical Fourier-plane intensity produced by single-banded binary-phase plate Central zone is defined as having intensity ≥ 90% peak

22. Check with experiment System characteristics – = 633 nm (wavelength) –    = 420  m (Gaussian e -1 radius and inner-radius r 0 of single phase-plate zone) – f = 200 mm (lens focal length) – d = 184 mm (obs. plane location) –  = 0.8768  (phase on a h = 0.52  m, n = 1.534 single-zone, binary-phase plate) Value of d was changed to compensate the departure in etch depth h (designed for 0.47 um, actually got 0.52 um)

23. Check with experiment Comparing predicted/actual results: –Predicted Uniformity: U = 2.2% Steepness: K = 0.61 Efficiency:  = 75% –Actual Uniformity: U < 3% Steepness: K = 0.59 Efficiency:  = 72.3% Experi- mental results: Intensity along x- direction Experi- mental results: Intensity along y- direction Good agreement!

24. Summary Driving problem: non-uniform beam at gain medium. Costly ideal: continuous-phase plates. Affordable alternative: binary-phase modification of a Gaussian’s similarity to “ideal” (circ-producing) Bessinc Manufacturing: relating phase level  to etch depth h in a substrate of index n Simplifying: marginal benefits of more than one zone Figures-of-merit: uniformity, steepness, efficiency Verification: through experiment

25. Conclusion Single-zone, binary-phase plates provide an affordable, mechanically-feasible option for producing a uniform field in an optical system’s focal plane.

26. Relevance to Optomechanics Provides mechanically-feasible “phase grating” implementation. Exemplifies a system with mechanical compensation capability (d,  ). Yang/Wang also include tolerancing examples: zone width, etch depth sensitivities at +/- 10% of design values. Exemplifies cost benefits of designing for manufacturability (1-zone vs. 2+).

27. Thank you!

28. How far are we from ideal? 3 Figures of Merit (FOMs) How do you compare the Fourier-plane field produced by a binary-phase plate to the “ideal” circ? With 3 figures-of-merit: Plot of typical Fourier-plane intensity produced by single-banded binary-phase plate Central zone is defined as having intensity ≥ 90% peak

29. Selecting observation distance d and phase  for optimum FOMs Now, best d and  depend on FOMs: U, K, . –They’ll differ from (d = f,  ). If we set d = f, we need  If we set , we need d = 0.81f  d = f 0.7925  d =0.81 f  Best FOMs

30. Check with experiment System characteristics – = 633 nm (wavelength) –    = 420  m (Gaussian e -1 radius and inner-radius r 0 of single phase-plate zone) – f = 200 mm (lens focal length) – d = 184 mm (obs. plane location) –  = 0.8768  (phase on a h = 0.52  m, n = 1.534 single-zone, binary-phase plate) Value of d was changed to compensate the departure in etch depth h (designed for 0.47 um, actually got 0.52 um)