Download Numerical simulation tools for optics

Numerical propagation
of light beams in
refracting/diffracting devices
Jean-Yves VINET
Observatoire de la Côte d’Azur
(Nice, France)
KASHIWA-October-2011
J.-Y. Vinet
1
Summary
•Needs for optical simulations
•General principles of numerical propagation :
several methods
•Some examples :
•Fourier Transform
•Hankel Transform
•Modal
•Monte-Carlo
•Advantages/drawbacks
KASHIWA-October-2011
2
J.-Y. Vinet
Needs for Optical simulations
in GW interferometer design
1) Sensitivity of a GW interferometer is strongly
dependent on the quality of the Fabry-Perot cavities
-Efficiency of power recycling
-Power in sidebands
2) Quality of Fabry-Perot’s depends on the quality of the
mirrors
3) Mirrors are not perfect ! Requirements are needed for
manufacturers
4) Heated mirrors change of internal/external properties
KASHIWA-October-2011
J.-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
KASHIWA-October2011
J.-Y. Vinet
4
Propagation by Fourier Transform :
General principles
KASHIWA-October-2011
J.-Y. Vinet
5
Paraxial diffraction theory
Maxwell+single frequency  Helmholtz :
Slowly varying envelope :
 2x   2y   2z   2 / c 2  E ( , x, y, z )  0
E(, x, y, z)  exp ikz  F (, x, y, z) and  z F  kF
 2
k 
c

Paraxial diffraction equation ( math. analogous to Heat-Fourier and to Schrödinger eq. ) :
 2x   2y  2ik  z  F ( , x, y, z )  0
2D Fourier Tr. : f ( p, q) 

f ( x, y )eipx eiqy dxdy
R2
 2ik  z  p 2  q 2  F ( , p, q, z )  0


p2  q2
F (, p, q, z1 )  exp  i
( z1  z0 )  F (, p, q, z0 )
4


propagator
KASHIWA-October-2011
J.-Y. Vinet
6
6
Propagation by Fourier Transform
Diffraction over z
A( x, y, z0 )
A( x, y, z0  z )
FT-1
FT
 p2  q2 
exp  i
z 
2
k


A( p, q, z0 )
A( p, q, z0  z)
Use of Discrete Fourier Transform (in practice : FFT)
x=0
x=W
x-window
1
p=0
2
3
 x W / N
N
p  pmax  N / W
p-window
Positive frequencies
KASHIWA-October-2011
N/2
J.-Y. Vinet
 p  2 / W Negative frequencies
7
Mode of a Fabry-Perot cavity
E
A
L
B
E’
M1
Implicit equation :
M2
E  T1  A  ( R1  PL  R2  PL )  E
   xx 2 y y 2


Ra  ra exp
2ik2ik 
) 2) 
R 
r
exp
 f (
x, yf) a ( x(,a y
 1,
2





   2 a
2
Mirror
operators
in
xy plane
a
2
a
a
Curvature radius
Ta  ta exp ikha ( x, y)
Optical thickness
Propagation
KASHIWA-October-2011
(a  1, 2)
a
Measured roughness
(Lyon’s surface charts)
propagator
PL  X  F -1 P.F  X  
J.-Y. Vinet
8
Solution by simple relaxation scheme :
En  T1 A  C  En1
With initial guess :
E0 
C  M1  P  M 2  P
t1
TEM 00
1  r1r2
Large number of iterations if large finesse and/or large defects
Accelerated convergence (a la Aitken):
En   n1En1  n1 (T1 A  C  En1 )
En' of simple relaxation
With optimal choice of
 n , n
at each iteration
See e.g. : Saha, JOSA A, Vol 14, No 9, 1997
KASHIWA-October-2011
J.-Y. Vinet
9
Black fringe :( ideal TEM00) – (TEM00 reflected by a Virgo cavity) 10-8 W/W
2 x perfect 35cm mirors
30 cm
KASHIWA-October-2011
J.-Y. Vinet
10
Propagation by Bessel Transform :
General principles
Suitable for axisymmetrical problems
Fourier Transform :
1
f ( p, q ) 
dxdy exp i ( px  qy )  f ( x, y )

2
Assume (axial symmetry) :

f ( x, y )  f r  x 2  y 2

then
x  r cos  , y  r sin  ,
f ( ) 
p   cos  , q   sin 
1
rdrd exp i  r cos(   ) f (r )   J 0 ( r ) f (r )rdr

2
Bessel Transform
KASHIWA-October-2011
J.-Y. Vinet
11
Inverse B transform :
f (r )   J 0 ( r ) f (  )  d 
Assume
Let
f (r )
r a
negligible for
 ,   1,..., 
be the zeros of
J1 (r )
 (r )  J 0 (  r / a)
are a complete, orthogonal family on 0, a 
Sturm-Liouville theorem : the
a
  (r ) (r )rdr  p  ,
0
So that

f (r )  
 1
KASHIWA-October-2011
f
 (r )
p
a2 2
p 
J 0 (  )
2
a
with
f   f (r ) (r )rdr
0
J.-Y. Vinet
12
J1 ( x )
The first 20 zeros of J1 ( x)
x
Example of a sampling grid with 20 nodes
0.
1
xi   i a /  N
KASHIWA-October-2011
J.-Y. Vinet
a
N
13

f (r )  
 1
r    a /  N
f
 (r )
p

f   f (r )  
 1

  
f
2
 (r )   2 2
J0 
p
 1 a J 0 (  )
 N
Reciprocal F transform :

f   H( ) f 
 1
with H( )
Direct F transform :

   
2J0 


 2 2 N 
a J 0 (  )
   
2a J 0 


 N 

 N2 J 02 (  )
2
()
f   H
f
 1

 f

()
with H
f  f (  ) with  

a
Direct and inverse Bessel transforms are done with explicit matrices
KASHIWA-October-2011
J.-Y. Vinet
14
Propagator in the Fourier space over distance z :
 z 2
2 
P( p, q, z )  exp  i
p  q 

 4

p2  q2   2
In the Fourier-Bessel space :
 
After sampling :

a
 z 2 
P (z )  exp  i

2 
 4 a

Diffraction step by a simple matrix product :


 1
 1
()
 ( z  z )   P   (z) with P (z)   H( ) P (z) H
To be computed once
KASHIWA-October-2011
J.-Y. Vinet
15
Example : propagation of a TEM00 over 3000 m
Initial wave
Diffraction theory
Bessel propagated
KASHIWA-October-2011
J.-Y. Vinet
16
Representation of mirrors
Axially symmetrical defects : diagonal operator
 4i
M   r exp 
 
 r2

 f (r )  

 2 Rc

Pure parabolic
contribution
 a
with r 
N
defects
Reflected field :
 '  M 
KASHIWA-October-2011
J.-Y. Vinet
17
Example : reflectance of a Fabry-Perot cavity
A
E
L
B
Intracavity field :
M1
M2
E  t1 A  e2ikL M 1P( L) M 2 P( L) E
C Matrix operator
1
Intracavity field by matrix inversion :
E   Id  e
Reflected field by matrix product :
B  R A
With the reflectance operator
R  M  t1 PM 2 P  Id  e
KASHIWA-October-2011

J.-Y. Vinet
†
1
C  t1 A
2ikL
1
C  t1
2 ikL

18
Modal propagation : general principles
The set of all complex functions  ( x, y ) of integrable square modulus
has the structure of a Hilbert space, with a scalar product
,  
*
dxdy

( x, y)( x, y)

2
An example of a basis of such a HS is the Hermite-Gauss family of optical modes
 x2  y 2 
 2 x 2  y 2 
x
y


 nm ( x, y)   nm H n  2  H m  2  exp 
exp i

2
w
w
w 
R 



 
So that any optical amplitude can be expanded in a series of HG modes
A( x, y )   Anm  nm ( x, y )
m,n
KASHIWA-October-2011
J.-Y. Vinet
19
Propagation of a HG mode of parameter (waist) w0 :
2P
 lm ( x, y, z ) 
 w( z )2


1
r2
r2
exp  
 ik
 iGlm ( z )  
mn
2
2 m !n !
2 R( z )
 w( z )

x
y


H l  2  H m  2 
w
w


Rayleigh parameter :
Beam width :
b   w02 / 
w( z )  w0 1  ( z / b) 2
Curvature radius of the wavefront :
Gouy phase
KASHIWA-October-2011
b2
R( z )  z 
z
Glm ( z )  (l  m  1) arctan( z / b)
J.-Y. Vinet
20
Diffraction of a Gaussian beam
w( z )  w0 1  ( z / b)
2
w0
z 0
KASHIWA-October-2011
J.-Y. Vinet
z
b2
R( z )  z 
z
21
HG01
HG22
HG55
HG05
KASHIWA-October-2011
J.-Y. Vinet
22
Representation of mirrors by their matrix elements
M abcd  ab , M cd
M abcd  ab , M cd
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
Scattered light
« Main beam »
Rough mirror
surface
Reflection of a photon
k
n
k'
k '  k  2(k .n ) n
Refraction of a photon
k
n
1
k'
N
k'


 k  k .n  N 2  1  (k .n ) 2 n 


Diffusion of a photon

'
Rough surface
Random variable
with a PD that mimics
the BRDF of the material
Diffraction of photons ?
Example 1 : Propagation of a beam
target
source
Probability Density
of direction
2
dP
1
 2 lm ( ,  )
d 
Probability Density
of emission point
dP
2
 lm ( x, y )
dS
KASHIWA-October-2011
lm ( p, q) p 
2

sin  cos  , q 
2

sin  sin 
 lm ( ,  )
J.-Y. Vinet
29
Monte-Carlo methods
Example 1 : propagation of a TEM00 over 3000 m
w0=2cm
Initial wave : MC
Analytical initial TEM
MC propagated
Diffraction theory
Radial coord. [m]
KASHIWA-October-2011
J.-Y. Vinet
30
Example 2 : Management of diffraction
by obstacles
Emission
of photons

x
target
screen
p.x  , p  k

: Centered random deviate of standard deviation
KASHIWA-October-2011
J.-Y. Vinet
  
 4x 
 *  arctan 
31
Example 2 : diffraction by an edge
Screen at 5m
Histogram : Monte-Carlo
Diffraction theory
(Fresnel Integral)
transverse distances [m]
KASHIWA-October-2011
J.-Y. Vinet
32
Conclusion
*FFT propagation : general purpose codes (DarkF),
suitable even for short spatial wavelength defects
of mirrors
•Propagation by Bessel transform : suitable for axisymmetrical
problems (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…)