Chapter 6 Image Formation Theory

In this chapter, we will discuss the mathematical formalism of image formation

process and relevant image resolution issue.

6.1 Intensity Impulse Response

First, let us consider an imaging system as depicted below. The object is

illuminated by an incoherent light source.

Since Maxwell’s equations, which govern the propagation of light in free

space, are linear and the optical system is stationary, the following principles can

be invoked to facilitate the description of image formation:

∎ Linearity: Assuming 1 2( ') ( ')f x and f x are two sources lying on the object plane

and the optical system images 1 1( ') ( )f x into g x , 2 2( ') ( )f x into g x , respectively, then

the system will produce 1 2( ) ( )a g x b g x for the linearly combined source

1 2( ') ( ')a f x b f x .

∎ Stationarity: If the system maps 1 1( ') ( )f x g x , then it will also map

1 0 1 0( ' ') ( )f x x g x x .

Based on the above properties, we can further draw the following conclusions:

(i) Let ( )s x be the image for a point source on the object plane, i.e., ( ') ( )x s x ,

an arbitrarily distributed intensity distribution on the object plane can be

expressed as ( ') ( ") ( ' ") "obI x f x x x dx .

(ii) From the linearity and stationarity of the system, the image distribution

formed on the image plane will be ( ) ( ') ( ') 'imI x f x s x x dx .

Here ( )s x is called as the point spread function (psf), which is the intensity

impulse response, or the point diffraction pattern of the optical system.

6.1.1 The Expression of Intensity Impulse Response

Refer to the figure

.

and recall that the effect of a thin lens is to impose a phase factor

2exp[ (2 )]j k f on the incident wave with a transmission factor ( )A . Based on

the Huygens-Fresnel principle, the optical field on the image plane ( , )x x y can

be expressed as

2

1 2

2

1( ) ( ') ( ) '

j kjkr j k sf

im ob

e eU x K U x A e d dx

r s

.

By using the paraxial approximation at the far-field

2

2

2 2 2 2

2 2 2 2

( ' ) ( ' ) ' 2 ' | | ( ' )2 '

2

'

'

( ) ( ) 2 | | ( )

Rr x x

x

R

x

R

R x R

s x RR

x R x

,

the optical field distribution at the image plane can be simplified to be

2

1 2

1 1 1( ) '

( )'

('

1

2')

( ) ( ') ( ) ''

x xj

jk R R

i

kR

j

R

k

m b

R f

o

ReU x K U x A e e d dx

RR

.

With the paraxial approximation of ' 'x z and x z f , we achieve

2

1 2

1 1 1( )

2 '

'( )

'( ) ( ') ( ) '

j k

z

x xjk

zz f

bz

im oU x K U x A e e d dx

,

where1 1 1

( ) 0'z z f

on the image plane. And finally we have

2 1

'2 ( )( )

'( ) ( ') ( ) ' ( ) ( )'

x xjjk

z z

x

zim ob obU x K U x A e dx d K U A e d

z

,

where 2 ( ) '

'( ) ( ') ''

j xz

ob obU U x e dxz

.

For a point source of ( ') ( ')obu x x , 2 ( ) '

'( ) ( ') ' 1'

j xz

obU x e dxz

, we then

obtain 2

2 ( )

( ) ( )j

zim

x

U x K A e d

.

By defining 2

2 ( )* 2( ) ( ) ( ) ( )

x

zim i

j

ms x U x U x K A e d

, which indicates that

the intensity impulse response s(x) can be determined directly from the aperture

transmission function A(), which is also called pupil function.

6.1.2 Image Formation in Terms of the Intensity Impulse Response

To facilitate further discussion, let us start with an 1-D case with a

rectangular aperture transmission function ( )A

.

Based on the previous result,2 2 2

2 ( )

( ) 4 ( )a j

z

a

x axs x e d a sinc

f

, indicating

that when a bar object of '

( ')0 '

o

ob

I x bI x

x b

is imaged, then its image will be

2 2

22

'( ) ( ') ( ') ' 4 ( ) '

24 (2 ) (2 ) { }

2

x b

im ob ox b

o o

axI x s x x I x dx a I sinc dx

f

f x b x b a Ca I I

a C C

S S S S,

where C denotes the spot size of the intensity impulse response

#2 22

fC f

a , and

2

0( ) ( )

2

x ux sinc du S .

Let us define 2bC

(the ratio of object dimension and image spot size)

.

A series of images taking by an imaging apparatus with different #f (i.e.,

different C ) are shown below

.

6.1.3 Resolution in Terms of the Intensity Impulse Response

Now consider the image of two points ( ') [ ( ' ) ( ' )]ob oI x I x b x b in 1-D case

We calculate the image distribution

2 2

2 2 2

( ')( ) 4 [ ( ' ) ( ' )] ( ) '

( ) ( )4 ( ) ( )

im o

o

ka x xI x a I x b x b sinc dx

z

ka x b ka x ba I sinc sinc

z z

,

where 2 2

2 ( )2( ) 4 ( )

xa j

z

a

kaxs x e d a sinc

z

.

Rayleigh proposed a resolution criterion for an incoherent image by choosing a

separation 2 'rb of the two sinc2 functions such that the central maximum of one

coincides with the first minimum of the other. The resulting image intensity

distribution has a small minimum at the center of the two-point images

,

with #2 '

2r

zb f if z f

a

.

For partially coherent imaging, we can choose the separation of the two sinc2

functions such that

2

02( ) 0

im xI xx

. Note

2 2( ) ( )( ) ( ) ( )im

ka x b ka x bI x Sinc Sinc

z z

and ( ) ( )Sinc x Sinc x . The

resulting criterion (i.e., the Sparrow criterion) then becomes

2.6062 '

2s

zb

a

.

For 2-D case, 2 2

1( ) 4 [2 ( ) ( )]kax kax

s x a Jz z

; where a is the aperture radius of

the optical system. The corresponding resolution criteria become

2.9762 ' 1.22 , 2 '

2 2or r s

z zb b

a a

.

6.2 Image Formation Described by the Intensity Transfer Function

Image formation described by the Intensity Transfer Function is usually more

suitable for the imaging analysis of complicated objects. Consider an object with a

cosinusoidal intensity distribution as ( ') 1 cos(2 ')ob oI x A x , where A describes

the modulation amplitude, and o the spatial frequency of the object. In view that

s(x) is a real function, the image distribution formed on the image plane becomes

2 ' 2 ' 2 ( ')

2

2 2

2

( ) ( ') ( ') '

[1 ( )] ( ) '2

[ ( ) ( ) ( )] ( )2 2

(0) [ ( ) ( ) ]2

(0) Re[ ( ) ]

( )(0) 1

o o

o o

o

im ob

j x j x j x x

j x

o o

j x j x

o o

j x

o

o

I x I x s x x dx

Ae e s e d dx

A As e d

As s e s e

s A s e

ss A

cos(2 ( )(0)

o oxs

.

Visibility of the periodic image pattern can be displayed as

( )( )

(0)

o

o

sV A A H

s

,

where ( )oH is the optical transfer function.

Consider a general object, which the object intensity distribution can be described

by 2( ) ( ) j x

ob obI x I e d , with ( )obI being the object spectrum.

Then

2

2 ' 2 '( ')

2

( ) ( )

( ') ( ') ' [ ( ) ][ ( ') '] '

( ) ( )

j x

im im

j x j x x

ob ob

j x

ob

I x I e d

I x s x x dx I e d s e d dx

I s e d

.

Thus, ( ) ( ) ( )im obI I s . Note

22

( ) ( )x

jzs x A e d

2 2 '* 2

*

( ) [ ( ) ][ ( ') ']

( ') ( ' ) '

x xj j

j xz zs A e d A e d e dx

A A z d

,

implying that the un-normalized optical transfer function is given by the

unnormalized autocorrelation of the aperture function with its complex conjugate.

6.2.2 Image Formation in Terms of the Optical Transfer Function

To facilitate further discussion on the optical transfer function, let us consider

the following simple aperture function

1

( )

0

a

A

a

.

*( ) ( ') ( ' ) 's A A z d = the measure of the overlapping area as the

aperture is slided across itself by z : ( ) 2 2 [1 ]2

a z

a

zs d a z a

a

.

By normalizing ( )s , we deduce the optical transfer function to be

#

| ( ) | | | 2 1( ) (1 )

(0) 2

s z aH as

s a f f

.

By defining #

2

zz

a ,

#( ) (1 )H z .

Some examples are given below for further illustration:

∎ Image of a Ronchi Ruling

Consider an image of a Ronchi ruling with incoherent light, the ruling is

described by

1 ' 4

( ') 0 4 ' 3 4

1 3 4 '

ob

x

I x x

x

and ( ' ) ( ')ob obI x I x .

Then,2( ) ( ) ( ') ( ') 'j x

im im obI x I e d I x s x x dx , where ( ) ( ) ( )im obI I H .

For the object with Ronchi ruling

2 ' 22 2( ) ( ') ' [ ] [ ] ( )

4 4 4 4

j x j n

ob ob

n n

nI I x e dx sinc e sinc

and #( ) (1 )H z for

#1 ( )z .

A. If #

1 1

z , only dc terms of the ruling passes through the optical system

described by ( )H , resulting in no intensity variation.

B. If #

1 1 3

z , the resultant image consists of the central order of the sinc

function.

C. #

4 1

z

D. #

6 1

z

The imaging properties were summarized in the following figure:

6.3 Image Formation with Coherent Light

Consider the case that two fields 1( , )U x t and 2 ( , )U x t are superposed on the space-

time point (x, t): 1 2( , ) ( , ) ( , )U x t U x t U x t .

The total intensity becomes *

1 2 12( ) ( , ) ( , ) ( ) ( ) 2Re ( )I x U x t U x t I x I x x ,

where U satisfies the wave equation for a quasi-monochromatic light with

( ) 2 2( , ) ( ) ( )j x j t j tU x t A x e e A x e .

6.3.1 The Imaging Problem with Coherent Light

Note ( , ) ( ', ') ( ') 'im obU x t U x t K x x dx

, where ( ')K x x is the complex

amplitude in the image plane due to a point object, i.e., an amplitude impulse

response of the optical system. The light intensity distribution ( )imI x can be formed

on the image plane

* * *

'( ) ( , ) ( , ) ( ', ') ( ", ') ( ') ( ") ' "im im im ob ob tI x U x t U x t U x t U x t K x x K x x dx dx .

∎ for an incoherent source *

'( ', ') ( ", ') ( ') ( ' ")ob ob t obU x t U x t I x x x ,

* *

2

( ) ( , ) ( , ) ( ') ( ') ( ') '

( ') ( ') ' ( ') ( ') '

im im im t ob

ob ob

I x U x t U x t I x K x x K x x dx

I x K x x dx I x s x x dx

.

∎ for a coherent source,

2

* *( )

(

( ') ( ')

') ( ') '

'( ") ( ") "obm oi b

ob

U x K xUI x K x x dxx

U x K x x dx

x dx

.

6.3.2 Amplitude Impulse Response

Referring to the optical imaging process shown above,

2

2 1

1 2

1 2 1 2

2 1

'

2

.1 1 1

'( ) '( )

( , ) ' ( ) ( ')

' ( ) ( ')

xjks jkr jkf

im ob

paraxial approxi

and yxz z f jkx jky

z z z z

ob

e eU x y K d x d U A x e

s r

K d x d U A x e

.

Let ( ) ( , )obU , therefore

2 2

2

2 [ '( ) '( )]

( , ) ' ' ( ', ')

x yj x y

z z

imU x y K dx dy A x y e

.

In terms of polar coordinates,

2 ( ) cos( )2

0 0

12

00

( , )

22 ( )

22 ( )

2

rja

zim

a

U r K d e d

raJ

r zK J d K araz

z

.

6.3.3 Coherent Imaging Resolution

Consider an imaging process of two point sources ( ) ( )[ ( ) ( )]ob oU U b b

in an optical system

*

1 2 1 2

*

1 2 1 1 2 2

0 1 2 1 1 2 2

( , ) ( ) ( )

( ) ( ) [ ( ) ( )][ ( ) ( )]

( , )[ ( ) ( )][ ( ) ( )]

ob ob ob

o o

U U

U U b b b b

I b b b b

.

Then *1 1 2 2

1 2 1 2 1 2

2 1 2 1

( , ) ( , ) ( ) ( )im ob

x xx x K K d d

z z z z

, and

2 2 *

1 2 0( ) ( ) ( ') ( ') 2Re ( ' ') ) ), ( (im imI x x x I K x b K x b K x Kb xb bb ,

where 2

1

'z

b b b mz

.

∎ 1-D system

2

( ) 2kax

K x a sincz

.

2 2 2

0

2 2 2 2

( ') ( ) ( ') ( ')( ) 4 [ ] [ ] 2Re ( , )im

ka x b ka x b ka x b ka x bI x a I sinc sinc b b sinc sinc

z z z z

.

(a) Incoherent Limit with ( , ) 0b b

2 2 2

0

2 2

( ') ( ')( ) 4 [ ] [ ]im

ka x b ka x bI x a I sinc sinc

z z

.

Rayleigh Criterion : 2 2

( ') ( ')0,

ka x b ka x b

z z

22 'rb

z

ka .

Sparrow Criterion : 2

2

0

( )0im

x

I x

x

22.6062 's

z

kab .

(b) Coherent Limit with ( , ) 1b b

2 2 2

0

2 2 2 2

( ') ( ') ( ') ( ')( ) 4 [ ] [ ] 2 [ ] [ ]im

ka x b ka x b ka x b ka x bI x a I sinc sinc sinc sinc

z z z z

.

Sparrow Criterion : 2

2

0

( )0im

x

I x

x

24.1642 's

z

kab .

∎ 2-D system

1

22 2

1

2

2

2

( )

ka rJ

z ka rK x a a

ka r z

z

.

4 2 2

0 1 1 1 1

2 2 2 2

( ') ( ') ( ') ( ')( ) [ ] [ ] 2Re ( , ) [ ] [ ]im

ka r b ka r b ka r b ka r bI r a I b b

z z z z

.

(a) Incoherent Limit with ( , ) 0b b

Rayleigh Criterion : 23.8322 'r

z

kab .

Sparrow Criterion : 22.9762 's

z

kab .

(b) Coherent Limit with ( , ) 1b b

Sparrow Criterion : 24.62 's

z

kab .

6.4 Coherent Imaging: An Example

Consider an edge object such that

0 0( )

0 0ob

II

.

∎ Incoherent Illumination of the Edge

*

2 1 2 1

2 0 0 20

02 1 2

2

( ) ( ) ( ) ( )

1 cos( )

[ ( )] ( )2

im ob

x xI x I K K d

z z z z

kax

I I zx kaxI sinc ka d Si

z z z kax

z

.

∎ Coherent Illumination of the Edge with 1 2( , ) 1

2

*

1 2 1 2

2 1 2 1 2 1

2

2

0

2

( ) ( , ) ( ) ( ) ( ) ( )

1 1( )

2

im ob ob

x x xI x K K d d I K d

z z z z z z

kaxI Si

z

,

where 0

sin( )

x uSi x du

u . For your reference, the image distribution ( )imI x as a

function of 2x ka z is shown in the following:

The photograph showing the edge patterns with incoherent (a) and coherent

imaging (b) are shown below. The corresponding traces across the edge are

presented on (c) and (d).

(a) (b) (c) and (d)