Decoupling laser beams with the minimal number of optical elements

1. Decoupling laser beams with the minimal number of optical elements Julio Serna December 14, 2000

2. In collaboration with: Decoupling laser beams with the minimal number of optical elements George Nemeş

3. Outline • Introduction • The problem • The proof • Consequences and conclusions

4. Laser beam characterization Wigner distribution function (WDF)

5. Second order characterization • Beam matrix P (+l)

6. Second order characterization • Gauss Schell model (GSM) beams (+l)

7. First order optical systems ABCDmatrix: Ssymplectic,

8. Propagation

9. The problem (ST beam)

10. The problem (ASA beam)

11. The problem (GA beam)

12. The problem (PST beam?)

13. Cylindrical lens fx=184 mm Cylindrical lens fx=184 mm PST beam & cyl. lens

14. The problem

15. The problem • Decoupled beam: (trivial or) no crossed terms

16. The problem Question: Which is the minimum number of optical systems F, L needed to decouple a (any) laser beam? Answer:F L F L

17. Why the question? • Laser beam properties can be changed using optical systems • Nice mathematical properties. Further insight into P/GSM, S • I like it

18. What do we know Optical systems • Any optical system can be synthesized using a finite number of F and L • Shudarshan et al. (2D/3D) OA85 • Nemes (constructive method) LBOC93

19. What do we know Decoupling • Any beam can be decoupled using ABCD systems • Shudarshan et al. (general proof, no method) PR85 • Nemes (constructive method) LBOC93 • Anan’ev el al. (constructive method) OSp94 • Williamson (pure math) AJM36

20. * * * to decouple • IS beams:  Pd rotationally symmetric • IA beams:  Pd rotationally symmetric rounded beams/non-rotating beams/ blade like beams/angular momentum... What do we know? Beam classification

21. The proof: beam conditions • Decoupled beam conditions • P: M symmetrical, W, M, U same axes • GSM: I, g, R same axes,  = 0

22. Free space RSA thin lens The proof: optical systems

23. 1.F(free space) • Impossible: F does not change ST, ASA or RSA property • Consequences: • no use alone • no point in having F at the end

24. L / beam is decoupled lens  R  does not affect conditions: 2.L(single lens) • GSM

25. L / beam is decoupled 2.L(single lens) • Pmatrix

26. 2.L(single lens) • L / beam is decoupled Note:   last elementL: end in waist possible  Lcovers all IS beams, and more

27. 3.FL Propagate conditions 1, 2 in free space

28. 3. FL Beams not decoupled via F, FL: • PST, PASA, PRSA  (z) = 0 constant  1(z)  0  go to LFL • What if 1(z) =  = 0 but 2(z)  0?  go to LFL? Not enough 1(z) =  = 0 invariant under L  go to FLFL (at least!)

29. 4. L FL • Left beams: (z)  0 • Aproach: find a particular solution • NRGA (pseudo-symmetrical, twisted phase) beams • RGA (twisted irradiance) + (z)  0

30. 4a. L FL, NRGA beams • L1 to have tr M = 0 (waist) • Use a “de”twisting system • Simon et al. (matrix) JOSAA93 • Beijersbergen et al.OC93 • Friberg et al. josaa94 • Zawadzki (general case) SPIE95 L F L L1 L2 F L = L F L

31. 4b. L FL, RGA with (z)  0 • GA  PST, PASA, PRSA: L is enough, since (z)  0 • Go to 4a L’ L F L = L” F L

32. 5. F L FL • Leftovers from F L: beams with 1(0) = (z) = 0 2  0 Solution: free space F ( is not invariant under F)  then go to L F L

33. P/GSM YES YES Use L NO (zL>0) Use FL NO YES YES Use LFL P PRSA NO NO Use L Use F Modifies1/ Converts into PRSA

34. Consequences and conclusions • To decouple any beam we need FLFL or less • The output beam can be at its waist • We can use the result to “move around” PP’ solved via PPdP’ • Engineering: starting point to handle GA (rotating or non rotating beams)