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ELG5106 Fourier Optics
Trevor Hall
[email protected]
Fourier Optics
DIFFRACTION
2
Propagation between Planes in Free Space
x2
y2
x1
y1
k
x3

x3=0
2

k  0
2
x3=z
3
Plane Wave Expansion I

2

 k 2   0 ,   a exp ik.x  k12  k 22  k32  k 2
 for a forward going wave (implicit exp  it  dependence )
k3 k1 , k 2   k 2  k12  k 22

k3 k1 , k 2   i k  k  k
2


2
2
2
1

k  k   k
k  k   k
2
1
,
,
2
1
2
2
2
2
2
2
by superposit ion this generalise s to :
 x  
 
  ak , k expik x
1
2
1 1
 k 2 x2  k3 k1 , k 2 x3 dk1dk 2
  
Evanescent wave
4
Plane Wave Expansion II
Noting that at x3  0 this reduces to an inverse Fourier tr ansform
within a multiplica tive constant :
 
  ak , k expik x
 x1 , x2 , x3  0 
1
2
1 1
 k 2 x2 dk1dk 2
  
and setting :
u x1 , x2     x1 , x2 , x3  0 , v y1 , y2     x1  y1 , x2  y2 , x3  z 
then
v y1 , y2  
 
uˆ k , k  exp ik y


2 
1
1
2
2
1 1
 k 2 y2  k3 k1 , k 2 z dk1dk 2
  
where
uˆ k1 , k 2  
 
  ux , x exp ik x
1
2
1 1
 k 2 x2 dk1dk 2
  
5
Plane Wave Expansion III
Explicity
v y1 , y2  
   
u x , x  exp ik  y




2 
1
1
2
2
1
1
 x1   k 2  y2  x2   k3 k1 , k 2 z dx1dx2 dk1dk 2
    

v y1 , y2  
 
  h y
1
 x1 , y2  x2 u  x1 , x2 dx1dx2
  
h y1 , y2  
 
exp ik y


2 
1
1 1
2
 k 2 y2  k3 k1 , k 2 z  dk1dk 2
  
hˆk1 , k 2   exp ik 3 k1 , k 2 z
Spatial Frequency Response
Impulse Response /Point Spread Function
Linear Shift Invariant System
6
Propagation as a filter
u
v

ĥ
2

 k2   0
unimodular phase
function
k2
k
0
exponential decay
1
k1
7
Why is the angular spectrum of plane waves expansion rarely used?
h y1 , y2  
 
exp ik y


2 
1
1 1
2
 k 2 y2  k3 k1 , k 2 z  dk1dk 2
  
may be rewriten
k2
h y1 , y2  
2 2
  k1 y1 k 2 y2 k3 k1 , k 2     k1   k 2 
exp ikz  k z  k z  k   d  k d  k 
 
or
k2
h y1 , y2  
2 2
 
  y1
y2

exp i  p z  q z  m  dpdq
m  1 p2  q2
,
 i p2  q2 1 ,
p2  q2  1
p2  q2  1
  kz
8
Oscillatory Integrals
• We are left with the consideration of integrals of the form:

I   a p  exp i   p dp

a,   C 
,  

• If
 0 then the integrand is highly oscillatory and
p
I 0 ,  
• If   p*   0 then there is a contribution from the
p
integrand in the neighbourhood of the
stationary point p*
9
Stationary Phase Condition
 y1   y1 
  p , q   p    q    m p , q  ; m  1  p 2  q 2
 z  z


 y1  p
 y2  q
 0   0 ;
 0   0
p
p
 z m
 z  m
y1
p
m
p k1

m k3
,
q k2

m k3
z
The stationary phase condition corresponds to a ray from
source point to observation point ( recall shift invariance)
10
Paraxial Approximation I
In a paraxial system rays are inclined at small angles to the
optical axis. One may then make the paraxial approximation:
1 2 1 2
m  1 p  q  1 p  q
2
2

y2
y1
  p , q   p  q  m p , q 
z
z
y2 1 2 1 2
y1
 1 p  q  p  q
z 2 z 2
z
2
2
2
2
2
y2 
y1  1 
1  y1  1  y2  1 
 1        p     q  
z 
z  2
2 z  2 z  2
11
2
Paraxial Approximation II
k2
h y1 , y2  
2 2
k2

2 2
k2

2 2
 
  expi   p, q  dpdq
  
2
2
  


  1  y  2 1  y  2  

y1  
y2   
1
2
   exp  i  p     q     dpdq  exp i 1        
z 
z   
  2  z  2  z   

 

  1  y  2 1  y  2  
 

2
2
1
2
   exp  i p  q dpdq  exp i 1        
  2  z  2  z   


 

  1  y  2 1  y  2  
k  2 
1
2

 exp i 1        
2 
2   i 
  2  z  2  z   
  1  y  2 1  y  2  
 1 ik

 
exp ikz  exp ikz 1   1    2   
 2 z

  2  z  2  z   
2
2
12
Fresnel Diffraction
v y1 , y2  
 
  h y
1
 x1 , y2  x2 u x1 , x2 dx1dx2
  
 
  1  y  x  2 1  y  x  2  

 1 ik
exp ikz    u x1 , x2  exp ikz 1   1 1    2 2   dx1dx2
 
 
 2 z
  2  z  2  z   
  1  y  2 1  y  2 
1 ik
exp ikz1   1    1  

2 z
  2  z  2  z  
 
  1  x  2 1  x  2  

 ik
   u x1 , x2  exp ikz 1   1    2    exp  x1 y1  x2 y2  dx1dx2

 z
  2  z  2  z   
  

Up to a multiplicative quadratic phase factor (that is often neglected), the field at
the observation plane is given by the Fourier transform of the field at the source
plane multiplied by a quadratic phase factor.
13
Fraunhoffer Diffraction
If the source filed u has compact support (is zero outside some bounded
aperture) and z is sufficiently large the variation of the quadratic phase factor
over the support of u becomes negligible. The leading phase factor is also
often neglected either because the region of interest in the observation plane
subtends a sufficiently small angle with respect to the origin at the source plane
or because it is the intensity only that is observed. The diffracted field
distribution is then given by a Fourier transform of the field distribution in the
source plane.
 
 ik

v y1 , y2     u  x1 , x2  exp  x1 y1  x2 y2  dx1dx2
 z

  
14
Notes
• The oscillatory integral representation of the impulse response of
this optical system can be evaluated asymptotically without
recourse to the paraxial approximation using the method of
stationary phase.
• The magnitude but not the phase of the leading multiplicative phase
factors of the Fresnel and Faunhoffer diffraction integrals may be
evaluated by appealing to energy conservation – the integral over
the source and observation planes of the field intensity must be
equal.
• The choice of outgoing plane waves in the plane wave spectrum
ensures that all three diffraction integrals (plane wave expansion,
Fresnel & Fraunhoffer formulae satisfy the Sommerfeld radiation
condition at infinity.