Numerical propagation
of light beams in
refracting/diffracting devices
Jean-Yves VINET
Observatoire de la Côte d’Azur
(Nice, France)
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J.-Y. Vinet
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Summary
•Needs for optical simulations
•General principles of numerical propagation :
several methods
•Some examples :
•Fourier Transform
•Hankel Transform
•Modal
•Monte-Carlo
•Advantages/drawbacks
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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
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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
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Propagation by Fourier Transform :
General principles
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Paraxial diffraction theory
Maxwell+single frequency  Helmholtz :   2x   2y   2z   2 / c 2  E ( , x , y , z )  0

Slowly varying envelope :

E ( , x , y , z )  exp  ikz  F (  , x , y , z )
k 

c

and
 z F   kF
2

Paraxial diffraction equation ( math. analogous to Heat-Fourier and to Schrödinger eq. ) :
  2x   2y  2 ik  z  F ( , x , y , z )  0


2D Fourier Tr. :
f ( p, q) 

R
f ( x, y )e
ip x
e
iq y
d xd y
2
 2 ik  z  p 2  q 2  F ( , p , q , z )  0


2
2


p q
F ( , p , q , z1 )  exp   i 
( z1  z 0 )  F ( , p , q , z 0 )
4



propagator
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Propagation by Fourier Transform
Diffraction over  z
A( x, y, z0 )
A( x, y , z0  z )
FT-1
FT
2
2


p q
exp   i
z 
2k


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
p  p max
x W /N
 N /W
N
p-window
Positive frequencies
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 p  2 / W
Negative frequencies
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Mode of a Fabry-Perot cavity
E
A
L
B
E’
M1
Implicit equation :
M2
E  T1  A  ( R1  PL  R 2  PL )  E
   x 2  y 2


x  y
R a  ra exp
2 ik
R 
r 
exp
 2 ik
  2   f (x , yf) a ( x(,a y 1,) 2)

 2


a



2
Mirror
operators
in
xy plane
a
2
a
a
( a  1, 2)
a
Curvature radius
Measured roughness
(Lyon’s surface charts)
T a  t a ex p  ikh a ( x , y ) 
propagator
Optical thickness
Propagation
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PL  X  F  P .F  X  
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Solution by simple relaxation scheme :
E n  T1 A  C  E n 1
With initial guess :
E0 
C  M1  P  M 2  P
t1
1  r1 r2
T E M 00
Large number of iterations if large finesse and/or large defects
Accelerated convergence (a la Aitken):
E n   n 1 E n 1   n 1 (T1 A  C  E n 1 )
'
E n of simple relaxation
With optimal choice of
n, n
at each iteration
See e.g. : Saha, JOSA A, Vol 14, No 9, 1997
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Black fringe :( ideal TEM00) – (TEM00 reflected by a Virgo cavity) 10-8 W/W
2 x perfect 35cm mirors
30 cm
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Propagation by Bessel Transform :
General principles
Suitable for axisymmetrical problems
Fourier Transform :
f ( p, q) 
1
2
 dxdy exp  i ( px  qy )  f ( x , y )
Assume (axial symmetry) :

f ( x, y )  f r 
x  y
2
2

then
x  r cos  , y  r sin  ,
f ( ) 
1
2
p   cos  , q   sin 
 rdrd  exp  i  r cos(   )  f ( r )   J
0
(  r ) f ( r ) rdr
Bessel Transform
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Inverse B transform :
f (r ) 
J
0
( r ) f (  )  d 
Assume f ( r ) negligible for r  a
Let
  , 
 1, ...,   be the zeros of J 1 ( 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
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f
p
 ( r )
p 
a
2
2
J 0 (  )
2
a
with
f 

f ( r )  ( r ) rdr
0
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J1 ( x)
The first 20 zeros of J 1 ( x )
x
Example of a sampling grid with 20 nodes
0.
1
xi   i a /  N
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a
N
13

f (r ) 

 1
f
p
r    a /  N
 ( r )

f   f ( r ) 

 1
f
p

  ( r ) 

 1
Reciprocal F transform :

f 

()
H  f 
w ith
()
H 
 1
Direct F transform :

f 

   
J0 
 f
2
2
a J 0 (  )
 N 
2
   
2J0 


 N 

2
2
a J 0 (  )
   
2a J 0 


 N 

2
2
 N J 0 (  )
2
()
H  f 
w ith
( )
H 
 1
f   f (   ) w ith   

a
Direct and inverse Bessel transforms are done with explicit matrices
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Propagator in the Fourier space over distance z :
 z
2
2 
P ( p , q , z )  exp   i
p  q 

4


p q  
2
In the Fourier-Bessel space :
 
After sampling :
2
2

a
z 2 

P (  z )  exp   i
 
2
 4 a

Diffraction step by a simple matrix product :

  (z  z) 
 P    (z)

w ith
P  (  z ) 
 1
 H   P (  z ) H 
()
()
 1
To be computed once
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Example : propagation of a TEM00 over 3000 m
Initial wave
Diffraction theory
Bessel propagated
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Representation of mirrors
Axially symmetrical defects : diagonal operator
M
 4 i  r2

 r exp 
 f ( r )  


   2 Rc
Pure parabolic
contribution
w ith
r 
 a
N
defects
Reflected field :
 '  M   
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Example : reflectance of a Fabry-Perot cavity
A
E
L
B
Intracavity field :
M1
E  t1 A  e
M2
2 ikL
M 1P ( L )M 2 P ( L ) E
C Matrix operator
Intracavity field by matrix inversion :
E   Id  e
Reflected field by matrix product :
B  R  A 
With the reflectance operator
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
R  M
J.-Y. Vinet
†
1
2 ikL
1
C  t1 A
 t1 P M 2 P  Id  e
2 ikL
1
C  t1

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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
, 

dx dy  ( x , y )  ( x , y )
*
2
An example of a basis of such a HS is the Hermite-Gauss family of optical modes

 nm ( x , y )   nm H n 

x 

2 Hm 
w

 x2  y2
y
2  exp  
2
w
w


 2 x 2  y 2 
 exp  i

R

 

So that any optical amplitude can be expanded in a series of HG modes
A( x, y ) 

Anm  nm ( x , y )
m ,n
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Propagation of a HG mode of parameter (waist)
 lm ( x , y , z ) 

H l 

2P
 w( z)
x 

2 Hm 
w

2
2


r
r
exp  
 ik
 iG lm ( z )  
mn
2
2
m !n !
2R(z)
 w(z)

1
2
y 
2 
w
Rayleigh parameter :
Beam width :
w0 :
b   w0 / 
2
w ( z )  w0 1  ( z / b )
Curvature radius of the wavefront :
R(z)  z 
2
b
2
z
Gouy phase
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G lm ( z )  ( l  m  1) arctan( z / b )
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Diffraction of a Gaussian beam
w ( z )  w0 1  ( z / b )
2
w0
z
z  0
R(z)  z 
b
2
z
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HG01
HG22
HG55
HG05
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Representation of mirrors by their matrix elements
M a b cd   a b , M  cd
M a b cd   a b , 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 '
k  k .n 
N 
k '
N  1  ( k .n )
2
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
dP
d
Probability Density
of emission point
dP
dS
  lm ( x , y )
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 lm ( p , q ) p 


2

1

2
 lm ( ,  )
sin  cos  , q 
2
2

sin  sin 
 lm ( ,  )
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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]
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Example 2 : Management of diffraction
by obstacles
Emission
of photons

x
target
screen
 p . x  , p  k

: Centered random deviate of standard deviation
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  
  arctan 

 4  x 
*
31
Example 2 : diffraction by an edge
Screen at 5m
Histogram : Monte-Carlo
Diffraction theory
(Fresnel Integral)
transverse distances [m]
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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…)