# Frequency analysis of optical imaging system

```Frequency analysis of optical
imaging system
Dinesh Ganotra
Imaging System
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Lens Design Software
Effective Focal Length
Max. Field Angle
Stop Surface Number
Afocal EFL
Back Focal Length
Zoom Surface
Working Distance
Wavelength (Primary)
No. of Zoom Positions
Telephoto Ratio
Refractive Index (Primary)
Overall Physical Length
Abbe Number
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Entrance Pupil Diameter
No. of Glass Elements
Max. Parax Image Height
Overall Glass Length
Lateral Magnification
No. of Optical Surfaces
Angular Magnification
No. of Physical Surfaces
Zoom Ratio
No. of Cemented Groups
Numerical Aperture
Number of Examples
Imaging system

v

zo
u
zi
Geometrical optics Diffraction optics








U
u
,
v

h
u
,
v
;,
U
d
d
i
o,

U i u, v 
hu,v;,
: image amplitude
: amplitude at image coordinates
u, v 
in response to a point source object at
 , 
Amplitude point spread function
Amplitude Point Spread Function





 

A 
2










h
u
,
v
;
,

P
x
,
y
exp

j
u

M
x

v

M
y
d



z
z
i
i




P  x, y 
: Pupil function : unity inside and zero outside the projection aperture.
Superposition integral
MTF
PSF
Reduced coordinates
~
  M
~  M

;
Ideal image

~~


1
~
~

U
, 
g ,  U
o


M
M
M






Amplitude PSF in reduced coordinates















A 
2
~
~
~
~




h
u

,
v


P
x
,
y
exp

j
u

x

v

y
d



z
z
i
i












U
u
,
v

h
u
,
v
;,
U
d
d
i
o,








~~~
~
~
~

U
u
,
v

h
u

,
v

U
,
d
d
i
g

A
hu, v  
z i



2




P
x
,
y
exp

j
ux

vy

dxdy

z i



Diffraction limited system
• regard the image as being a convolution of the
image predicted by geometrical optics with an
impulse response that is the Fraunhofer
diffraction pattern of the exit pupil.
Spatial coherence









~
~
~
~
~
~


U
u
,
v
;
t

h
u

,
v

U
,
;
t

d
d
i
g

where  is the time delay associated with propagation from
in general ,  is a function of the coordinates involved.
~,~ 
to
u, v 
Intensity


I
u
,v

U
u
,v
;
t
i
i
2
 
  



~
~
~
~




U

,

;
t


U

,

;
t


~
~
~~
~~

~
~


I
u
,
v

d
d
d
d
h
u

,
v

h
u

,
v

i
1
1
2
2
1 1
2 2
g11

1 g2 2
2
Drop time delays


















~
~
~
~
~
~

~
~
~
~
~
~


I
u
,
v

d
d
d
d
h
u

,
v

h
u

,
v

J
,
;
,
i
1
1
2
2
1
1
2
2
g
1
1
2
2

where

 










~
~
~
~

~
~
~
~
J
,
;
,

U
,
;
t
U
,
;
t
g
1
1
2
2
g
1
1 g
2
2
known as
mutual intensity.
Take time-varying phasor at the origin
as reference
For a perfectly coherent illumination






~
~
~
~




U
,
U
0
,
0
;
t
U
,
U
0
,
0
;
t
~
~
g
1
1
g
g
2
2
g
~
~
U
,1
;
t
U
,2
;
t
g 1
g 2
2
2




U
0
,
0
;
t
U
0
,
0
;
t
g
g


Thus



 










~
~
~
~

~
~
~
~
J
,
;
,

U
,
U
,
g
1122
g
11g2
2
Coherent object illumination is linear
in complex amplitude

 

~
~ ~ ~ ~2
2
~
I i u, v    h u   , v   U g  , d d  U i u, v 
Frequency response
• Coherent illumination
• Incoherent illumination
Coherent illumination
• Define
Gg  f X , f Y  
Gi  f X , f Y  

 U u, vexp
j 2  f X u  f Y v dudv
 U u, vexp
j 2  f X u  f Y v dudv
g



i

Amplitude
H  f X , fY  

transfer function
 hu, v exp
j 2  f X u  f Y v dudv

Fourier transform of PSF
Coherent imaging …







~~~
~
~
~

U
u
,
v

h
u

,
v

U
,
d
d
i
g

Taking Fourier transform on both the sides and using convolution theorem
Gi  f X , f Y   H  f X , f Y U g  f X , f Y

Substituting h(u,v)
H  f X , fY  

 hu, v exp
j 2  f X u  f Y v dudv


A
H  f X , fY   
z i



2




P
x
,
y
exp

j
ux

vy

dxdyexp j 2  f X u  f Y v dudv


z
i




H  f X , f Y   Pzi f X , zi f y 
Take
Az i  1
and ignore negative signs
 x 
 y 
Px, y   rect
rect

 2w 
 2w 
 z i f X 
 z i f Y 
H  f X , f Y   rect
rect



2
w
2
w




Cut off frequency
f0 
w
z i
Example  = 10-4 cm w=1 cm zi = 10cm give cut off frequency
of 100 cycles / mm
Incoherent illumination























~
~
~
~
~
~
~

~
~
~
~
~
~
~
J
,
;
,

U
,
;
t
U
,
;
t

I
,

,

g
1
1
2
2 g
1
1
g
2
2
g
1
1
1
2
1
2







2 ~
~
~
~
~
~

I
u
,
v

h
u

,
v

I
,
d
d
i
g

Convolution of intensity impulse response with ideal image intensity
Define

Gg  f X , fY

 I u, v  exp
g
j 2  f X u  f Y v dudv


 I u, v dudv
g


Gi  f X , f Y

 I u, v  exp
i
j 2  f X u  f Y v dudv


 I u, v dudv
i


H  f X , fY


hu , v  exp j 2  f X u  f Y v dudv
2




hu , v  dudv
2







2 ~
~
~
~
~
~

I
u
,
v

h
u

,
v

I
,
d
d
i
g

Take FT on both sides and use
convolution theorem
Gi  f X , f Y   H  f X , f Y Gg  f X , f Y
Optical transfer function

Relationship between OTF and amplitude
transfer function
Amplitude Transfer Function
H  f X , fY  

 hu, v exp
j 2  f X u  f Y v dudv


H  f X , fY

fX
fY   
fX
fY 

H
p

,
q

H
p

,
q




dpdq

2
2
2
2







H  p, q  dpdq
2

Optical Transfer Function
OTF is normalized autocorrelation function of amplitude transfer function
Why frequency analysis?
```