frequency analysis of optical imaging system dinesh ganotra
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Frequency analysis of optical imaging system
Dinesh Ganotra
Imaging System
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
• 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
u
zo zi
Point Spread
Geometrical optics Diffraction optics
ddUvuhvuU oi ,,;,,
vuU i , : image amplitude
,;,vuh : amplitude at image coordinates vu,in response to a point source object at ,
Amplitude point spread function
Amplitude Point Spread Function
dxdyyMvxMuz
jyxPz
Avuh
ii
2
exp,,;,
yxP , : Pupil function : unity inside and zero outside the projection aperture.
Superposition integral
MTF
PSF
Reduced coordinates
M~
;
M~
MMU
MU og
~,
~1~,
~Ideal image
Amplitude PSF in reduced coordinates
dxdyyvxuz
jyxPz
Avuh
ii
~~2
exp,~,~
~~~,~~,
~, ddUvuhvuU gi
ddUvuhvuU oi ,,;,,
dxdyvyuxz
jyxPz
Avuh
ii
2
exp,,
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
~~;~,~~,
~;, ddtUvuhtvuU gi
where is the time delay associated with propagation from ~,~
to vu,
in general , is a function of the coordinates involved.
Intensity
2;,, tvuUvuI ii
222111
22112211
;~,~
;~,~
~,~~,
~~~~~,
tUtU
vuhvuhddddvuI
gg
i
Drop time delays
221122112211~,
~;~,
~~,~~,
~~~~~, gi JvuhvuhddddvuI
where tUtUJ ggg ;~,~
;~,~~,
~;~,
~22112211
known as mutual intensity.
Take time-varying phasor at the origin as reference
For a perfectly coherent illumination
211
11
;0,0
;0,0~,~
;~,~
tU
tUUtU
g
ggg
222
22
;0,0
;0,0~,~
;~,~
tU
tUUtU
g
ggg
Thus 22112211
~,~~,
~~,~;~,
~ ggg UUJ
Coherent object illumination is linear in complex amplitude
22
,~~~,~~,
~, vuUddUvuhvuI igi
Frequency response• Coherent illumination• Incoherent illumination
Coherent illumination• Define
dudvvfufjvuUffG YXgYXg
2exp,,
dudvvfufjvuUffG YXiYXi
2exp,,
dudvvfufjvuhffH YXYX
2exp,,
Amplitude transfer function
Fourier transform of PSF
Coherent imaging … ~~~,
~~,~
, ddUvuhvuU gi
YXgYXYXi ffUffHffG ,,,
Taking Fourier transform on both the sides and using convolution theorem
Substituting h(u,v) dudvvfufjvuhffH YXYX
2exp,,
dudvvfufjdxdyvyuxz
jyxPz
AffH YX
iiYX
2exp
2exp,,
yiXiYX fzfzPffH ,, Take 1izA
and ignore negative signs
w
yrect
w
xrectyxP
22,
w
fzrect
w
fzrectffH YiXi
YX 22,
Cut off frequency iz
wf
0
Example = 10-4 cm w=1 cm zi = 10cm give cut off frequency of 100 cycles / mm
Incoherent illumination 21211122112211
~~,~~~,
~;~,
~;~,
~~,~;~,
~ gggg ItUtUJ
~~~,~~,
~,
2ddIvuhvuI gi
Convolution of intensity impulse response with ideal image intensity
dudvvuI
dudvvfufjvuI
ffG
g
YXg
YXg
,
2exp,
,
dudvvuI
dudvvfufjvuI
ffG
i
YXi
YXi
,
2exp,
,
dudvvuh
dudvvfufjvuh
ffHYX
YX
2
2
,
2exp,
,
Define
Take FT on both sides and use convolution theorem
~~~,~~,
~,
2ddIvuhvuI gi
YXgYXYXi ffGffHffG ,,,
Optical transfer function
Relationship between OTF and amplitude transfer function
dpdqqpH
dpdqf
qf
pHf
qf
pH
ffH
YXYX
YX
2
,
2,
22,
2,
dudvvfufjvuhffH YXYX
2exp,,
Amplitude Transfer Function Point Spread Function
Optical Transfer FunctionOTF is normalized autocorrelation function of amplitude transfer function
Why frequency analysis?