decoupling laser beams with the minimal number of optical elements julio serna december 14, 2000

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)