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?