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

ssrsuur dzikzh ),,()exp(),,(

Wigner distribution function (WDF)

Second order characterization

• Beam matrix P (+)

0U

0WWt

t

U

UM

MWP t

Second order characterization

• Gauss Schell model (GSM) beams (+)

)(det k

12 2

g

t

tggg

tIII

σ

RRR

0σσσ

0σσσ

First order optical systems

ABCD matrix:

DC

BAS

S symplectic, JtSJS

0I

I0J

Propagation

tinout SPSP

invariant,

at

4effM

1)(tr 22

1det422

24eff

2MUW

P

k Tkt

kM

4effMta

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

P matrix:

P =µ

W MM t U

¶

=

0

BBB@

hx2i hxyi hxui hxvihxyi hy2i hyui hyvihxui hyui hu2i huvihxvi hyvi huvi hv2i

1

CCCA

GSM beam:

¾I =µ

¾I x ¾I xy

¾I xy ¾I y

¶

; ¾g =µ

¾gx ¾gxy¾gxy ¾gy

¶

; R =µ

Rx Rxy

Rxy Ry

¶

; ¿

ABCD system:

S =

0

BBB@

Axx Axy Bxx Bxy

Ayx Ayy Byx Byy

Cxx Cxy Dxx Dxy

Cyx Cyy Dyx Dyy

1

CCCA

The problem• Decoupled beam:

(trivial or) no crossed terms

BBB@

CCCA

P matrix:

P =µ

W MM t U

¶

=

0

BBB@

hx2i 0 hxui 00 hy2i 0 hyvi

hxui 0 hu2i 00 hyvi 0 hv2i

1

CCCA

GSM beam:

¾I =µ

¾I x 00 ¾I y

¶

; ¾g =µ

¾gx 00 ¾gy

¶

; R =µ

Rx 00 Ry

¶

; ¿ = 0

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

Class Subclasses Symmetry propertiesIS ST Rotationally-symmetric

(a = 0, SA ASA Orthogonally-symmetricor RSA

I = 0) GA RGA Non-orthogonalSA ASA Orthogonally-symmetric

IA RSA(a > 0, NRGA PST Rotationally-symmetric

or GA (pseudo- PSA PASA Orthogonally-symmetricI > 0) -type) PRSA

RGA Non-orthogonal

*

*

• IS beams: Pd rotationally symmetric• IA beams: Pd 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

BBB@

CCCA

P matrix:

P =µ

W MM t U

¶

=

0

BBB@

hx2i 0 hxui 00 hy2i 0 hyvi

hxui 0 hu2i 00 hyvi 0 hv2i

1

CCCA

GSM beam:

¾I =µ

¾I x 00 ¾I y

¶

; ¾g =µ

¾gx 00 ¾gy

¶

; R =µ

Rx 00 Ry

¶

; ¿ = 0

The proof: optical systems

Free space RSA thin lens

tLL

LL

CC

IC

0IS

yfxyfxyfxf

/1/1

/1/1

L'LL 21 F'FF 21

0

z

z

I0

IISF

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

,, gI σσ

axes diagonalcommon ,

0

gI σσ

τ

L / beam is decoupled lens R

does not affect

conditions:

2. L (single lens)

• P matrix

0)tr(δ

0)tr(δ

2

1

WJU MJM

WJM

01

10J

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

00δ1

3. F L

Propagate conditions 1, 2 in free space

0

0

z)(tr

z)(tr

2δ

zδδ

(z)δ

(z)δ 2

2

21

2

1

MJU

MJU

0δδ )( tr 4

0z )( tr 2

δz

221

2

MJUMJU

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. L1 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

L1 L2 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

0)()(

/0

21

LL

L

zz

z

0LzYES

NO

YESYES

YES

NONO

)0(

0)(1

Lz

P/GSM

Use L

Use LFL

Use FLNO (zL>0)

P PRSA

Use F Use LConverts 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 Pd P’

Engineering: starting point to handle GA (rotating or non rotating beams)