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

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

Outline Introduction The problem The proof Consequences and conclusions

Laser beam characterization Wigner distribution function (WDF)

Second order characterization Beam matrix P ( + )

Second order characterization Gauss Schell model (GSM) beams ( + )

First order optical systems ABCD matrix: S symplectic,

Propagation

The problem (ST beam)

The problem (ASA beam)

The problem (GA beam)

The problem (PST beam?)

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

The problem

Decoupled beam: (trivial or) no crossed terms

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

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

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

What do we know 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 Decoupling

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

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

The proof: optical systems Free space RSA thin lens

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

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

2. L (single lens) P matrix  L / beam is decoupled

2. L (single lens) $L / beam is decoupled Note:   last element L: end in waist possible  L covers all IS beams, and more

3. F L Propagate conditions  1,  2 in free space

3. F L Beams not decoupled via F, FL: 1.PST, PASA, PRSA   (z) = 0 constant   1 (z)  0  go to LFL 2.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!)

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

4a. L F L, NRGA beams 1.L 1 to have tr M = 0 (waist) 2.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 L 1 L 2 F L = L F L

4b. L F L, RGA with  (z)  0 1.GA  PST, PASA, PRSA: L is enough, since  (z)  0 2.Go to 4a L’ L F L = L” F L

5. F L F L 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

YES NO YES NO P/GSM Use L Use LFL Use FL NO (z L >0) P  PRSA Use F Use L Converts into PRSAModifies  1 / 

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  P d  P’ ® Engineering: starting point to handle GA (rotating or non rotating beams)