Numerical propagation of light beams in refracting / diffracting devices

$: Numerical propagation of light beams in refracting / diffracting devices$
J.-Y. Vinet 1

Numerical propagationof light beams in

refracting/diffracting devicesJean-Yves VINET

Observatoire de la Côte d’Azur(Nice, France)

KASHIWA-October-2011

Summary

•Needs for optical simulations

•General principles of numerical propagation : several methods

• Some examples :• Fourier Transform• Hankel Transform• Modal• Monte-Carlo

•Advantages/drawbacks 2

J.-Y. VinetKASHIWA-October-2011

J.-Y. Vinet

Needs for Optical simulations in GW interferometer design

KASHIWA-October-2011

1) Sensitivity of a GW interferometer is strongly dependent on the quality of the Fabry-Perot cavities-Efficiency of power recycling-Power in sidebands2) Quality of Fabry-Perot’s depends on the quality of the mirrors3) Mirrors are not perfect ! Requirements are needed for manufacturers4) Heated mirrors change of internal/external properties

KASHIWA-October-2011 4J.-Y. Vinet

General principles of Propagation Methods

Expand optical field on a family of functions of which propagation is well known

•Plane waves•Bessel waves•Gaussian modes (eg. HG or LG)•Photons

5J.-Y. VinetKASHIWA-October-2011

Propagation by Fourier Transform : General principles


Paraxial diffraction theory

Maxwell+single frequency Helmholtz :2 2 2 2 2/ ( , , , ) 0x y z c E x y z

Slowly varying envelope : ( , , , ) exp ( , , , ) and zE x y z ikz F x y z F kF

2 2 2 ( , , , ) 0x y zik F x y z

Paraxial diffraction equation ( math. analogous to Heat-Fourier and to Schrödinger eq. ) :

2 22 ( , , , ) 0zik p q F p q z

2

( , ) ( , ) ipx iqy

R

f p q f x y e e dxdy2D Fourier Tr. :

1 0

2 2

1 0exp ( )( , , , ) ( , , , )4

p qi z zF p q z F p q z

propagator

2kc


Propagation by Fourier Transform

0( , , )A x y z 0( , , )A x y z z

0( , , )A p q z0( , , )A p q z z2 2

exp2

p qi zk

FT FT-1

Diffraction over z

Use of Discrete Fourier Transform (in practice : FFT)

x-window1 2 3 N

x=0 x=W

/x W N

p-windowNegative frequenciesPositive frequencies

p=0

N/2

max /p p N W

2 /p W


Mode of a Fabry-Perot cavityA E

B E’

2 2

exp 2 ( , ) ( 1, 2)2a a a

a

x yR r ik f x y a

2 2

exp 2 ( , ) ( 1,2)2a a

aa

x yr ik f y aR x

exp ( , )a a aT t ikh x y

Curvature radius Measured roughness(Lyon’s surface charts)

Optical thickness

Mirroroperatorsin xy plane

1 1 2( )L LP P ERE AT R

L XP X -1 PF .FPropagation

Implicit equation :

propagator

1M 2M

L


1 1n nE T A E C

Solution by simple relaxation scheme :

With initial guess : 10 00

1 21tE TEMrr

Large number of iterations if large finesse and/or large defects

Accelerated convergence (a la Aitken):1 1 1 1 1( )n n n n nE E T A E C

1 2M P M P C

With optimal choice of ,n n at each iteration

See e.g. : Saha, JOSA A, Vol 14, No 9, 1997

of simple relaxation'nE


Black fringe :( ideal TEM00) – (TEM00 reflected by a Virgo cavity) 10-8 W/W

30 cm

2 x perfect 35cm mirors


Propagation by Bessel Transform : General principles

Suitable for axisymmetrical problems

1( , ) exp ( ) ( , )2

f p q dxdy i px qy f x y

1( , ) exp ( ) ( , )2

f p q dxdy i px qy f x y

cos , sin , cos , sinx r y r p q

2 2( , )f x y f r x y

Assume (axial symmetry) :

then

01( ) exp cos( () ) ( )( )

2f rdrd i r f r J r f r rdr

Fourier Transform :

Bessel Transform


0( ) ( ) ( )f r J r f d

Inverse B transform :

Assume negligible for ( )f r r a

Let , 1,..., be the zeros of 1( )J r

Sturm-Liouville theorem : the 0( ) ( / )r J r a are a complete, orthogonal family on 0,a

220

0

( ) ( ) , ( )2

a ar r rdr p p J So that

1

( ) ( )ff r rp

0

( ) ( )a

f f r r rdr with


Example of a sampling grid with 20 nodes

/i i Nx a 1 N

The first 20 zeros of

0. a

1( )J x

x

1( )J x


1

( ) ( )ff r rp

02 21 1 0

2( ) ( )( ) N

ff f r r J fp a J

0(( ) )

2 201

2with

( )N

JH

a Jf H f

Reciprocal F transform :

20

(( ) )

12 2

0

2with

( )N

N

a JH

Jf H f

Direct and inverse Bessel transforms are done with explicit matrices

/ Nr a

( ) with f fa

Direct F transform :


Propagator in the Fourier space over distance z :

2 2( , , ) exp4zP p q z i p q

In the Fourier-Bessel space :

a

2 2 2p q

After sampling :

22( ) exp

4zP z ia

Diffraction step by a simple matrix product :

( ) ( )

1 1

( ) (z) with ( ) ( )z z P H P HP z z

To be computed once


Example : propagation of a TEM00 over 3000 m

Initial wave

Diffraction theory

Bessel propagated


Representation of mirrorsAxially symmetrical defects : diagonal operator

24exp ( ) with 2 c N

r aiM r f r rR

Pure paraboliccontribution defects

Reflected field :

' M


A

B

EL

Intracavity field :

Example : reflectance of a Fabry-Perot cavity

21 1 2e ( ) ( )ikLE t A M P L M P L E

C Matrix operator

1M 2M

121e ikLE Id t A

CIntracavity field by matrix inversion :

B R A Reflected field by matrix product :

1† 21 1 2 1e ikLR M t PM P Id t

CWith the reflectance operator


Modal propagation : general principlesThe set of all complex functions of integrable square modulus has the structure of a Hilbert space, with a scalar product

( , )x y

2

*, ( , ) ( , )dxdy x y x y

An example of a basis of such a HS is the Hermite-Gauss family of optical modes

2 2 2 2

2

2( , ) 2 2 exp expnm nm n mx y x y x yx y H H iw w w R

So that any optical amplitude can be expanded in a series of HG modes

,

( , ) ( , )nm nmm n

A x y A x y


Propagation of a HG mode of parameter (waist) :2 2

2 2

2 1( , , ) exp ( )( ) 2 ! ! ( ) 2 ( )

2 2

lm lmm n

l m

P r rx y z ik iG zw z m n w z R z

x yH Hw w

20 /b w

20( ) 1 ( / )w z w z b

2

( ) bR z zz

( ) ( 1)arctan( / )lmG z l m z b

0w

Rayleigh parameter :

Beam width :

Curvature radius of the wavefront :

Gouy phase


2

( ) bR z zz

z

20( ) 1 ( / )w z w z b

0w

0z

Diffraction of a Gaussian beam


HG01 HG22

HG05 HG55

Representation of mirrors by their matrix elements

,abcd ab cdM M

,abcd ab cdM M

Modal expansion widely used by Andreas Freise’s « Finesse » package

Propagation of light in complex structures by Monte-Carlo photons

Principle : send random pointlike particles from identified sources

Rough mirrorsurface

« Main beam »

Scattered light

Reflection of a photon

k

n

'k

' 2( . )k k k n n

Refraction of a photonk n

'k

2 21' . 1 ( . )k k k n N k n nN

Diffusion of a photon

Rough surface

'Random variablewith a PD that mimicsthe BRDF of the material

Diffraction of photons ?


Probability Densityof emission point

Probability Densityof direction

source

target

2( , )lmdP x ydS

2

2

1 ( , )lmdPd

2 2( , ) sin cos , sin sin

( , )

lm

lm

p q p q

Example 1 : Propagation of a beam


Example 1 : propagation of a TEM00 over 3000 mw0=2cm

Initial wave : MC

Analytical initial TEM

MC propagated

Diffraction theory

Monte-Carlo methods

Radial coord. [m]


Example 2 : Management of diffraction by obstacles

: Centered random deviate of standard deviation* arctan4 x

x

screen

Emissionof photons

target

. ,p x p k


Example 2 : diffraction by an edge

Histogram : Monte-Carlo

Diffraction theory(Fresnel Integral)

Screen at 5m

transverse distances [m]

Conclusion*FFT propagation : general purpose codes (DarkF), suitable even for short spatial wavelength defects of mirrors

•Propagation by Bessel transform : suitable for axisymmetricalproblems (eg. heating by axisymmetrical beams)

•Propagation by modal expansion : ideal for nearly perfect instruments, small misalignments, small ROC errors, etc….•Photons : mandatory for propagation of scattered light

in complex structures (vacuum tanks, etc…)

Numerical propagation of light beams in refracting / diffracting devices

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