frequency analysis of optical imaging systems

1

Frequency Analysis of Optical Imaging Systems6.1 Generalized treatment of optical imaging systems

6.1.1 A generalized model6.1.2 Polychromic illumination: the coherent and incoherent cases

6.2 Frequency response for diffraction-limited coherent imaging6.2.1 The amplitude transfer function6.2.2 Examples of amplitude transfer functions

6.3 Frequency response for diffraction-limited incoherent imaging6.3.1 The optical transfer function6.3.2 General properties of the OTF6.3.3 The OTF of an aberration-free system6.3.4 Examples of diffraction-limited OTFs

6.4 Aberrations and their effects on frequency response6.4.1 The generalized pupil function6.4.2 Effects of aberrations on the amplitude transfer function6.4.3 Effects of aberrations on the OTF6.4.4 Examples of a simple aberration: a focusing error6.4.5 Apodization and its effects on Frequency response

2

Frequency Analysis of Optical Imaging Systems• Frequency analysis and linear systems theory are relatively new to

optics but they have a very fundamental place in the theory of imaging systems

• Introduction of Fourier analysis to optical systems– Earnest Abbe (1840-1905) and Loard Rayleigh (1842-1919) Laid

the foundations of the Fourier optics – P. M.Duffieux in France in1930s published a book on Fourier

optics in 1946 translated to English “The Fourier transform and its applications to optics” Wiley 1983

– Otto Schade in US in 1948 employed methods of linear system theory in analysis and improvement of TV camera lenses.

– H.H. Hopkins in UK used transfer function methods for the assessment of the quality of optical imaging systems

• This chapter: – Coherent imaging systems important in microscopy, holography– Incoherent imaging systems wider applications everything else

3

6.1 Generalized treatment of optical imaging systems• Treatment of more general lens systems• Treatment of

– Quasi-monochromatic systems• Spatially coherent • Spatially incoherent

z

η

ξ

y

x

V

u

Entrance pupil

Black box ImageObjectz0 zi

Exit pupil y

x

4

6.1.1 A generalized model• System: some positive and negative lenses with various distances between

them possibly thick.• Assumption: the system ultimately produces a real image in space.• If a system is producing a virtual image, it can be converted to a real image by a

lens. • All imaging elements are lumped into a single “black box” (aggregate system).• The aggregate is defined by its “terminal properties” at the planes containing the

entrance and exit pupils. These are images of the physically limiting apertures (source of the diffraction effects) within the system.

• We assume the passage of light between the entrance and exit pupils is properly described by geometrical optics

• Diffraction-limited imaging system: if a diverging spherical wave of a point source is converted by the system to a new perfectly spherical wave that converges towards the image point on the image plane whose location on the image plane is related to location of the object on the object plane by a simple scaling relation that stays the same for all the points on the image field of interest.

• Terminal property of a diffraction limited system: converting a diverging spherical wave at the entrance pupil to a converging spherical wave at the exit pupil.

• For any optical system this property will be satisfied over a limited region of the object plane.

• Aberration: if the wavefront for a point source on the image plane at the exit pupil departs significantly from an ideal spherical wave, then the system is said to have aberration. Aberrations lead to defects in the spatial-frequency response of the imaging system.

5

6.1.2 Effects of diffraction on the image I

Two equivalent views: Image resolution is limitted by 1) the finite entrance pupil seen from the object space2) the finite exit pupil seen from the image spacesince two pupils are images of each other.

z

η

ξ

y

x

V

u

Black box ImageObjectz0 zi

y

x

Geometrical optics explainsthis part

Diffraction affects the image

Diffraction affects the image

6

The Abbe theory of image formation

Focal plane Image

Certain portions of diffracted components are interrupted by finite entrance pupil.Higher orders

Uninterrupted components generated by high frequency components of the object amplitude transmittance.

Object: a diffraction grating

( )Ernst Abbe (1873): diffraction affects the result from the entrance pupil.

Loard Rayleigh 1896 : diffraction affects the result from the exit pupil.

We will adopt Raylegh's view.

7

( ) ( )( )

( )0-, , ; , ,

Amplitude distributionAmplitude at image coordinates u,V transmitted by the objectin response to a point-source at ( , )

Image amplitude superposition integral:

iU u V h u V U d d

ξ η

ξ η ξ η ξ η∞

∞= � � ��

When there is no aberration arises from a spherical wave converging from the exit pupil toward the ideal image point at and The light amplitude about the ideal image point is the Fraunhofer

h

u M V Mξ η= =

( ) ( )( ) ( )

( )

2

-, ; , ,

1,

0

diffractionpattern of the exit pupil, centered on image coordinates and

inside the aperture is the pupil fu

outside the aperture

ij u M x V M y

z

i

u M V M

Ah u V P x y e dxdy

z

P x y

π ξ ηλ

ξ η

ξ ηλ

− − + −∞

∞

= =

=

�= ��

� �

,

nction and

is the distance from the exit pupil to the image plane are the coordinates in the plane of exit pupil

The quadratic phase factors in the object and image planes are ignored.

iz

x y

6.1.2 Effects of diffraction on the image II

8

( ) ( )

,

, ,

To achieve space-invariance in the imaging oeration, we need to remove effects of magnification and inversion from the equation for by the following transformation:

i

h

M M

Ah u V P x y e

z

ξ ξ η η

ξ ηλ

= =

− − =

� �

� �( ) ( )

( )

( ) ( )

2

-

0

1, ,

| |

, ,

impluse response that is the Fraunhofer diff

Ideal image: the geometrical optics prediction of the image for a perfect system

ij u x V y

z

g

i

dxdy

U UM M M

U u V h u V

π ξ ηλ

ξ ηξ η

ξ η

� �− − + −∞ � �

∞

= � �

�

= − −

� ��

� ��

� � ( )

( ) ( )( )2

-

,

, ,

image predicted by raction geometrical optics

pattern of the exit pupil.

Where

So in general for a diffraction-limited system we can rega

i

g

j ux Vyz

i

U d d

Ah u V P x y e dxdy

z

πλ

ξ η ξ η

λ

∞

−∞

− +∞

∞=

� �

� �

� ��

rd the image as being a convolution of the image predicted by geometrical optics with an impluse response

that is the Fraunhofer diffraction pattern of the exit pupil.

6.1.2 Effects of diffraction on the image III

9

6.2.3 Polychromic illumination: the coherent and incoherent cases (heuristic approach)There is no perfectly monochromatic source in nature or in the lab. The time variations of illumination amplitude and phase have statistical nature.

These fluctuations influence behavior of the imaging system. Here we consider the effect of polychromacity. Monochromatic case: the amplitude and phase of the field presented by a complexphasor that is function of the space coordinates.Polychromatic narrowband case: the amplitude and phase of the field presented by

a complex time-varying phasor that is function of the space coordinates and time.

Only statistical techniques can produce a satisfactory explanation of the image produced by such sources. Consider two types of Polychromatic illumination (narrow band):1) Spatially coherent sources: the phasor amplitudes of the field at all object points

vary in unison. Obtained whenever light is originated from a single point or lasers.

In coherent illumination the various impulse responses in the image plane vary in unison and must be added in complex amplitude basis. In other words a coherent imaging system is linear in complex amplitude and we can use the results of the

monochromatic analysis understanding that the is a time-invariant phasU or that depends on the relative phases of the light.2) Spatially incoherent sources: the phasor amplitudes of the field at all object points

vary in totally uncorrelated fashin. Obtained from diffused or extended sources.

For incoherent illumination the uncorrelated impulse responses must be added ona power or intensity basis. Incoherent imaging system is linear in intensity.

10

6.2.3 Polychromic illumination: the coherent and incoherent cases (rigorous approach)

( )( )

2

,

, ( , )Time varying phasor representation of

Polychromatic wave

where is the center frequency of the optical wave.For a narrowband system, the amplitude impulse response dose not

j t

u P t

u P t U P t e πν

ν

±± =

��

change dramatically for the various frequencies. So we can expres the time-varying phasors representing the image as convolution of a wavelength-independent impulse response with the time-varying pha

( ) ( )( ) ( )

( )

( ) ( )

2

-

0

, ,

1, ,

| |

, ; ,Time varying phasor Wavelength-indepenrepresentation of the image

sor representation of the object.

ij u x V y

z

i

g

i

Ah u V P x y e dxdy

z

U UM M M

U u V t h u V

π ξ ηλξ η

λ

ξ ηξ η

ξ η

� �− − + −∞ � �

∞− − =

= � �

�

= − −

� ��

� �

� ��

� ��

( )

( ) ( )

, ;

, ,

dent Time varying phasor impulse response representation of the

object in reduced coordinates

is the time delay associated with propagation from to

gU t d d

u V

ξ η τ ξ η

τ ξ η

∞

−∞−� � � ��

��

� �

11

( ) 2, ; ,

(~ 50 )

To calculate the image intensity we must time average the instantaneous

intensity represented by due to the fact that the detector integration

time is extremely long compared to theiU u V t

ps

( ) ( )

( ) ( ) ( )

2

*1 1 2 2 1 1 2 2

1

(~ 1/100 ~ 100 ) ( 1 )

, , ;

, , ,

,

reciprocal of the bandwidth even for narrowband optical sources .

Therefore

i it

i

g

GHz fs nm

I u V U u V t

I u V d d d d h u V h u V

U

λ

ξ η ξ η ξ η ξ η

ξ η

∞ ∞

−∞ −∞

= ∆ =

=

= − − − −

×

� � � ��

� �( ) ( )

( ) ( )

*1 1 2 2 2

1 1 2 2

; , ;

, ,

For a fixed image point, is nonzero only for small region about the ideal image

point. So and are very close to each other and the difference

between the time de

gt U t

h

τ ξ η τ

ξ η ξ η

− −� �

� ��

( ) ( ) ( )( )

( ) ( ) ( )

*1 1 2 2 1 1 2 2

1 1 2 2

*1 1 2 2 1 1 2 2

, , ,

, ; ,

, ; , , ; , ;

1 2lays and is negligible under the naorrowband condition.

Where is

i

g

g g g

I u V d d d d h u V h u V

J

J U t U t

τ τ


ξ η ξ η


∞ ∞

−∞ −∞= − − − −

×

=

� � � ��

� ��

� � � �� the "mutual intensity" and

it is a measure of the sptial coherence of the light at the two object points.


12


( ) ( ) ( )1 1 1 1

0,0;, ; ,

Tim

Complex Constant

For a perfectly coherent illumination, the time-varying phasor amplitudes across the object plane vary only by a complex constant. So we can write:

gg g

U tU t Uξ η ξ η=� ��

�� ( )( ) ( ) ( )

( )

( ) ( ) ( )( )

2 2 2 21/ 2 1/ 22 2

*1 1 2 2 1 1 2 2

0,0;, ; ,

0,0; 0,0;

, ; , , ,

, ,

e varying phasoramplitude at the origin

Complex Constant

and

the new "mutual intensity"

gg g

g g

g g g

i

U tU t U

U t U t

J U U

I u V h u V

ξ η ξ η


ξ

=

=

= − −

��

� ��

� � � ��

� �( )2

-

1

( , )

( ,

When the object illumination is perfectly incoherent, the phasor amplitudes acros the object vary in statistically independent fashion. This property is represented by:

g

g

U d d

U

η ξ η ξ η

ξ η

∞

∞� � � ��

� � ( )*1 2 2 1 1 1 2 1 2

2

; ) ( , ; ) ( , ) ,

Where is a real constant.For incoherent illumination the image intensity is found as a convolution of the

intensity impulse response with the ideal i

g gt U t I

h

ξ η κ ξ η δ ξ ξ η η

κ

= − −� � � ��

( ) ( ) 2

-, , ( , )

mage intensity g

i g

I

I u V h u V I d dκ ξ η ξ η ξ η∞

∞= − −� � � � ��

13

6.2 Frequency response for diffraction-limited coherent imagingA coherent imaging system is linear in complex amplitude.Therefore the intensity mapping will be nonlinear.The frequency analysis should be applied to the linear amplitude mapping

14

6.2.1 The amplitude transfer functionAnalysis of the coherent systems has yielded a space-invariant form of amplitude mapping. The amplitude at the image plane are convolution of the geometrical image amplitudes with a space-invariant im

( ) ( ) ( )( ){ } ( ){ } ( ){ }

, , ,

, , ,

pulse response.

We will apply transfer-function concept to this picture. Defining the frequency spectra of input and output and aplitude transfer fu

i g

i g

U u V h u V U d d

U u V h u V U u V

ξ η ξ η ξ η∞

−∞= − −

=

� � � � ��

� � �

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

2

2

2

, ,

, ,

, ,

, , ,

nction:

Applying the convolution theorem

This is effect of diffraction-limited i

X Y

X Y

X Y

j f u f Vg X Y g

j f u f Vi X Y i

j f u f VX Y

i X Y X Y g X Y

G f f U u V e dudV

G f f U u V e dudV

H f f h u V e dudV

G f f H f f G f f

π

π

π

∞ − +

−∞∞ − +

−∞∞ − +

−∞

=

=

=

=

� �

� �

� �

maging in the frequency domaine.

15

6.2.1 The amplitude transfer function and physical characteristics of the system

( ) ( ) ( )2, ,

Note is defined as the Fourier transform of the amplitude point-spread function. We can aslo express H as the Fraunhofer diffraction pattern and is Fourier transform

X Yj f u f VX YH f f h u V e dudV

H

π∞ − +

−∞= � �

( ) ( )( ) ( )

2

, , ,

1

of the pupil function (eq 6-5).

If we set and ignore the negative signs since for all of the interestingapplications pupil fun

ij ux Vy

zX Y i i X i Y

i

i

AH f f P x y e dxdy A z P z f z f

z

A z

πλ λ λ λ

λ

λ

− +∞

−∞

� �� = = − −� ��

=

� ��

( ) ( ), ,

ctions are symmetric in and .

If the pupil function is unity within a region and zero elsewhere, then there exists a finite frequency band for which the diffraction-limitted

imag

X Y i X i Y

x y

H f f P z f z fλ λ=

ing system passes all of the frequency components whithin that band

without distortion. This result is for an aberration-free system.

16

6.2.2 Examples of amplitude transfer functions

( )

( ) ( )( )

2

,2 2

, ,

,

Frequency response of diffraction-limited coherent imaging systems: 1) A square with width of

Using the amplitude transfer function is:X Y i X i Y

X Y

w

x yP x y rect rect

w w

H f f P z f z f

H f f rect

λ λ

λ

= � � � � � �

=

=

( )

( )( ) ( )

0 0

0

2 2

2 2

2 22

2 2 2 2

,

,

The cutoff frequency is

2) A circular aperture of radius

i X i Y X Y

i

i X i Y X YX Y

z f z f f frect rect rect

w w f f

wf

z

w x y

x yP x y circ

w

z f z f f fH f f circ circ

w

λ

λ

λ λ

= � � � ��

=

= +

+� �=� � �

+ +� �= =� ��

2 2 2

0

0

/

The cutoff frequency is

X Y

i

i

f fcirc

w z f

wf

z

λ

λ

+� � � �=� � � � � �

=

17

6.3 Frequency response for diffraction-limited incoherent imaging

( ) ( ), ,

The relationship between the pupil and amplitude transfer function in

coherent case:

Goal: to find the releationship between the amplitude transfer function and the system pupils fo

X Y i X i YH f f P z f z fλ λ=

r incoherent illumination. This may not be as direct a relationship as that of the coherent case.Systems: only diffraction-limitted.

18

6.3.1 The optical transfer function

( ) ( ) ( )2

, , ,

Intensity convolu

ConstantIdeal image intensityIntensity impulse response

Incoherent (illumination) imaging systems are linear in intensity and obey the:

i gI u V h u V I d dκ ξ η ξ η ξ η∞

−∞= − −� � � � ��

��

( )( ) ( )

( )

( )

2,,

,

,

tion integral

Zero frequency value of the spectra

We define the normalized frequency spectra of and as:

X Y

g

g i

j f u f Vg

g X Y

g

I

i

i X Y

I I

I u V e dudVf f

I u V dudV

If f

π∞ − +

−∞∞

−∞

=

=

� �

� �

��

��

�

�

�

�( ) ( )

( )

( )

2,

,

,


Maximum valule of the Furier transform of any real and nonnegative function

Such as occures at the oringin. T

X Y

i

j f u f V

i

I

g i

u V e dudV

I u V dudV

I I

π∞ − +

−∞∞

−∞

� �

� ��

hat maximum value is chosen as the

normalization constant. This constant is related to the background light and contrast of the image.

19


( )( ) ( )

( )

( ) ( )

2 2

2

,,

,

, ,


We define the normalized transfer function of the system

Apply the convolution theorem to

X Yj f u f V

X Y

h

i

h u V e dudVf f

h u V dudV

I u V h u V

π

κ ξ η

∞ − +

−∞∞

−∞

−

=

= − −

� �

� ��

� �

�

( )( ) ( ) ( )

( )( )

2

,

, , ,

,

,

We get:

Based on an international agreement:

is known as Optical Transfer Function (OTF) of the system.

is known as Modulation Transfer Function (M

g

i X Y X Y g X Y

X Y

X Y

I d d

f f f f f f

f f

f f

ξ η ξ η∞

∞

=� � � ��

� ��

�

�

( ), ,

TF) of the system

that specifies the complex weighting factor applied by the system to the

frequency component at relative to the weighting factor applied

to the zero-frequency component.X Yf f

20


( ) { } ( ){ }

( )

2

2, ,

,

Normalized

Incoherent

Coherent impulse response

The relationship between the OTF and MTF through

Using the Rayleigh's (Parseval's) theorem and autocorrela

X Y X Y

h

hf f h f f

h u V dudV∞

−∞

= → =� �

��

��

��

( ) ( )

( ){ } ( ) ( )

( )( ) ( )

( )

2 2

2 *

*

2

, ,

, , ,

', ' ' , ' ' ',

', ' ' '

'- / 2, '- / 2

tion theorem

With a change of variables: we get a symmetrical

X Y X Y

X Y

x Y

X Y

X Y

h x y dxdy H f f df df

h x y H H f f d d

H p q H p f q f dp dqf f

H p q dp dq

p p f q q f

ξ η ξ η ξ η

∞ ∞

−∞ −∞

∞

−∞

∞

−∞∞

−∞

=

= − −

− −=

= =

� � � �

� �

� �

� �

�

�

( )( )

*, ,2 2 2 2,

,

Autocorrelation function of the amplitude transfer function of the coherent system

OTFIncoherent system

expression of:

X Y X Y

X Y

f f f fH p q H p q dpdq

f fH p q

∞

−∞

+ + − −� � � � � �=

� �

��

��

2

.

Normalization

This equation is the primary link between the coherent and incoherent

systems and it is valied for systems with and without aberrations

dpdq∞

−∞� ��

21

6.3.2 General properties of the OTF I

( )( )

( )

*

2

, ,2 2 2 2,

,

1) 0,0 1

Some simple and elegant properties of the OTF:OTF is a normalized autocorrelation function.

(For pr

X Y X Y

X Y


f fH p q dpdq

∞

−∞

∞

−∞

+ + − −� � � � � �=

=

� �

� ��

�

( ) ( )

( ) ( )

*

0, 0)

2) , ,

3) , 0,0

oof substitute

(For proof use "Fourier transform of a real

function has Hermitian symmetry")

(For p

X Y

X Y X Y

X Y

f f

f f f f

f f

= =

− − =

≤

� �

� �

( ) ( ) ( )2 2 2

*

, , , ,

roof use the Schwarz's inequality)

If and are any two complex valued functions of then

The equality holds ony if where is a complex constant.

X p q Y p q p q

XYdpdq X dpdq Y dpdq

Y KX K

≤

=� � � � � �

22

6.3.2 General properties of the OTF II( ) ( )

( )

*

2*

2 2*

2

, , , ,2 2 2 2

, ,2 2 2 2

, ,2 2 2 2

,

Now let and

Then we find

X Y X Y

X Y X Y

X Y X Y

f f f fX p q H p q Y p q H p q


f f f fH p q dpdq H p q dpdq

H p q d

∞

−∞

∞ ∞

−∞ −∞

= + + = − −� � � � � �

+ + − −� � � � � �

≤ + + − −� � � � � �

=

� �

� � � �

( )

( ) ( )( ) ( )

2

*

2

, ,2 2 2 2

1,

, 1 0,0 1

, 0,0

therefore

using the property (1) we get

Absolute intensity of the image background is never the s

X Y X Y

X Y

X Y

pdq


H p q dpdq

f f

f f

∞

−∞

∞

−∞

∞

−∞

� ��

+ + − −� � � � � � ≤

≤ =

≤

� �

� �

� ��

� �

am as the absolute intensity of the object background. The normalization has removed the information about the absolute intensity levels.

23

6.3.3 The OTF of an aberration-free system I

( ) ( ), ,

Up to this point we have not made any assumptions about the aberrationof the system. Now we consider the diffraction-llimitted incoherent system:

We have for coherent systems

For an X Y i X i YH f f P z f z fλ λ=

( )( )

( ) 2

,

, ,2 2 2 2,

,

, ,

incoherent system, it follows from the equation for OTF with a change of variables as:

We used

X i X Y i Y

i X i Y i X i Y

X Y

f z f f z f

z f z f z f z fP x y P x y dxdy

f fP x y dxdy

P x y P x

λ λλ λ λ λ∞

−∞

∞

−∞

= =

+ + − −� � � � � �=

=

� �

� ��

( ) ( ), since is equal to zero or unity.y P x y

24

( )/ 2, / 2

The numinator of the OTF of a diffraction-limitted incoherent system represents the area of overlap of two displaced pupil functions:

1) One centered at

2) Second centered at dimetricai X i Yz f z fλ λ

( )

( )

/ 2, / 2

,

lly opposite point

The denuminator normalizes the area of overlap by the total area of the pupil. area of overlap

Thus total area

Figure shows a geometrical optics interpretatio

i X i Y

X Y

z f z f

f f

λ λ− −

=�

n of the OTF of a diffraction-limitted incoherent system. This interpretation implies that OTF is always real and nonnegative.

6.3.3 The OTF of an aberration-free system II

Area over which value of the both of the aperture functions is one

x

y

x

y

λzi|fX|

λzi|fY|

P(x,y)

25

6.3.3 The OTF of an aberration-free system III

( )( )

, ,2 2 2 2,

,

Computational appraoch to calculation of the OTF of a complicated diffraction-limitted system:

1) Inverse Fourier transform

i X i Y i X i Y

X Y


f fP x y dxdy

λ λ λ λ∞

−∞

∞

−∞

+ + − −� � � � � �=

� �

� ��

( )( )

,

, ,

the reflected pupil function or

Fourier transform the pupil function to find the amplitude

point-spread function.2) Take the squared magnitude of the amplitude point-spread functionto fin

P x y

P x y

− −

d the Intensity point-spread function.3) Take the Fourier transform of the Intensity point-spread function to find the unnormalized OTF.4) Normalize the OTF to unity at the origin.

26

6.3.3 The OTF of an aberration-free system IV

( )( )

( )

, ,2 2 2 2,

,

,Question: how a sinusoidal componet at a particular frequency pair is generated?

Answere: one way is by interference o

i X i Y i X i Y

X Y

X Y


f fP x y dxdy

f f

λ λ λ λ∞

−∞

∞

−∞

+ + − −� � � � � �=

� �

� ��

( )( )

| |, | | .

| |, | |

f light in the image plane from two separate patches

on the exit pupil of the system with separation

There is more than one pair of patches that are separated by .

Weight

i X i Y

i X i Y

z f z f

z f z f

λ λλ λ

of each frequency component is determined by the number of ways the corresponding

separation can be fit into the pupil. The number of ways a particular separation can be fit into the exit pupil is proportional to the

area of overlap of pupils separated by this spacing.

λz||fX|

λz||fY|d

D=zi

1

sin

tan /

/

/

1/ /

Separation of the patchesthat light is coming from to

produce fringes of freq.

i

i

m m X i

X X i

X i

X

d m

y z

y z m d

y y z d

f d z

d f z

f

θ λθ

λλ λ

λ λλ

+

==

≈

− = =

= =

=

y

z

27

( )( ) ( )

( ) ( )

1) 2 .

,

/ 2, / 2 / 2, / 2 .

2,

OTF of a system with square pupil of width We need to calculate

the area of overlap between two pupils separated by centered at

and

i X i Y

i X i y i X i y

i XX Y

w

z f z f

z f z f z f z f

w z fA f f

λ λ

λ λ λ λ

λ

− −

−=

( )

( )( ) ( )

2

2 22

0

4

2 2 2 2, 2 2

0

,

otherwise

Normalizing this area with the total area of we get:

,

i Y X Yi i

i X i YX Y

X Y i i

w ww z f f f

z z

w

w z f w z f w wf f

f f w w z z

λλ λ

λ λλ λ

� − ≤ ≤��

− −≤ ≤

=�

( )

( )

0 0

0

0

2 21 1

, 2 2

0

,2

otherwise

,

otherwise

Where

X YX Y

X Y i i

i

XX Y

f f w wf f

f f f f z z

wf

z

ff f

f

λ λ

λ

��

� − − ≤ ≤�� = � � �

��

=

= Λ� �

�

�

�0

0

2

22Cutoff frequency:

In this case OTF extends twice the cutoff frequency of the coherent system. Compare Fig. 6.7 and 6.3

Y

i

ff

wf

zλ

Λ� � �

=

6.3.4 Examples of diffraction-limited OTFs

λzi|fX|

x

y

λzi|fY|

λzi|fX|

λzi|fY|

fX/2f0

fY/2f

H

0 1-1

28

( )2) 2 .

/ 2, / 2 .

OTF of a system with circular pupil of diameter We need to calculate

the area of overlap between two pupils centered at

It is not as simple as the square case. We know the OTF wi X i Y

w

z f z fλ λ

( ) ( )2 c

2

ill be circularly symmetric. So we calculate along the x axis and then rotate that around the center of it. The shaded area in (a) is four times the area in (b). First get B B

Area A B wθ ππ

� �+ = =� ��

�

( ) ( )

( )

( ) ( ) ( )( ) ( )

( )

12

22

212 2

2 2

os / 2

2

12 2 2

cos / 2 14 2 2 2 2

,0

For general radial distance in the frequency plane

i X

i X i X

i X i X i X

X

z f ww

z f z fArea A w

z f w z f z fw w

area A B area Af

w w

λπ

π

λ λ

λ λ λππ

π πρ

ρ

−

−

� ��

= −� � � � � �

� � − −� � � � � �� + − � �� = =�

�

2

10 0

0 0 0

2cos 1- 2

2 2 2

0

,where

otherwise

In this case again OTF extends twice the cutoff frequency of the coheren

i

wz

ρ ρ ρ ρ ρ ρπ ρ ρ ρ λ

−� � � � � �− ≤ =� � � � �� = � � ��

t system. Compare Fig. 6.9 and 6.3

6.3.4 Examples of diffraction-limited OTFs

fX/2f0

fY/2f0

λzi|fX|

y (a)

A Bθw

λzi|fX|/2

(b)

29

6.4 Aberrations and their effects on frequency response• For diffraction-limited systems we assumed that a point

source object yields at the exit pupil a perfect spherical wavefront converging towards the ideal geometrical image point.

• For systems with aberrations the exit pupil wavefront departs from a perfect sphere.

• We do not treat the aberration subject completely. • We will concentrate on general effects of aberration on

the frequency response of a system. • We will show these effects with a simple example.

30

6.4.1 The generalized pupil function IFor a diffraction limited system, amplitude point-spread function is

the Fraunhofer diffraction pattern of the exit pupil centered on the

ideal image point.

To include effects of the aberrations in th

( ),

is picture we assume:1) the pupil is still illuminated by a perfect spherical wave2) there is a phase mask in the aperture deforming the wavefront that leaves the pupil.

3) represents effect of tkW x y ( )( )

( ) ( )

,

2 / ,

, ,

he phase error at point where

and is an effective path length error or

the aberration function .4) The complex amplitude transmittanc of the imaginary phase shift

plane is: j

x y

k W x y

x y P x y e

π λ=

=�( )

( )

,

,

the complex function is the generalized pupil function.

kW x y

x y�

31

6.4.1 The generalized pupil function II( ) ( ) ( ),, ,The generalized pupil function defined as:

Now with the above assumptions a) The amplitude transmittance function of an aberrated coherent system is

the Fraunhofer diffraction pat

jkW x yx y P x y e=�

( ), .tern of the generalized pupil function

b) The intensity impulse response of an aberrated incoherent system is the squared magnitude of the amplitude impulse response.

c) Aberration function is def

x y�

ined with respect to a Gaussian reference sphere.

d) Gaussian reference sphere is the ideal spherical surface centered at the ideal image point and passing through the point where the optical axis pierces the exit pupil.e) The actual wavefront is also defined to intercept the optical axis in the exit pupil.f) The error caused by aberration function can be negative or positive.

Ideal image point

zi

Gaussian reference sphere

Actual wavefrontW(x,y)

Exit pupil

Negative W(x,y)

Positive W(x,y)

32

6.4.2 Effects of aberrations on the amplitude transfer function

( ) ( )( )2

-, ,

For a diffraction-limited coherent system:

1) impulse response is Fourier transform of the pupil function.

2) The amplitude transfer function is the Fourier transform

ij ux Vy

z

i

Ah u V P x y e dxdy

z

πλ

λ− +∞

∞= � �

( ) ( ) ( )

( ) ( )

2, ,

, ,

of the

amplitude impulse response

3) Therefore the amplitude transfer function was found to be proportional to

the scaled pupil function

For diffrac

X Yj f u f VX Y

X Y i X i Y

H f f h u V e dudV

H f f P z f z f

π

λ λ

∞ − +

−∞=

=

� �

( ) ( ) ( )

( ) ( ) ( ) ( )

,

,

, ,

, ,

tion limited coherent systems with aberration:

a) The generalized pupil function is:

b) The amplitude transfer function is written as:

c) The b

i X i Y

jkW x y

jkW z f z fX Y i X i Y

x y P x y e

H f f z f z f P x, y eλ λλ λ

=

= =

�

�

( ),and-limitation of the that was imposed by the finite exit pupil

has not changed in presence of the aberrations.

d) Aberrations only introduce phase distortions within the passband.

X YH f f

33

6.4.3 Effects of aberrations on the OTF (incoherent systems) I

( )( )

( )

, ,2 2 2 2,

,

,

OTF of an aberration-free incoherent diffraction-limited system:

We define the function as the area of overlap of

i X i Y i X i Y

X Y

X Y


f fP x y dxdy

f f

P x

λ λ λ λ∞

−∞

∞

−∞

+ + − −� � � � � �=

+

� �

� ��

�

( ) ( )

( )

( )

, ,2 2 2 2

,

0,0

, ,2 2 2 2

,

,

Wavefron

OTF

and

then

In presence of aberrations:

X Y

z f z fz f z fi X i Xi Y i Yjk W x y W x y

i X i Y i X i Y

f fX Y

X Y

z f z f z f z fy P x y

dxdyf f

dxdy

ef f

λ λλ λ

λ λ λ λ

� � + + − − −� � � ��

+ − −� � � � � �

=

=

� �

� �

��

�

�

�

�( )

( )

( )

,

0,0

,

t errors caused by aberrations

Aberrations will never increase the or the modulus of the OTF.

X Yf f

X Y

dxdy

dxdy

MTF f f=

� �

� �

��

�

�

�

34

6.4.3 Effects of aberrations on the OTF (incoherent systems) II

( ) ( ), ,2 2

With aberrations Without aberrations

Important conclusions in presence of aberrations:

1)

implies that aberrations can never increase contrast of any spatial frequency component of the im

X Y X Yf f f f≤� �

age.2) The absolute cutoff frequency of the system remains thae same however, severe aberrations can reduce high frequency portions of OTF to an extent that the effective cutoff is much lower than the diffraction-limited cutoff.3) For certian frequency bands OTF may have negative or even complex value. when OTF is negative, the image components at that

frequency can undergo a contrast reversal; i.e. intensity peaks become intentity nulls and vice versa.

35

6.4.4 Examples of a simple aberration: a focusing errorMost aberrations are mathematically very challenging subjecst. The simplest example is a focusing error for an square aperture system:

The center of curvature of a spherical wavefront converging toward

,

s image of a point-source object is either to the left or right of the image plane.For simplicity we assume this point is still on the optical axis.

Ideal phase distribution across the exit pupil: i xφ ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

2 2

2 2

2 2 2 2

,

, , ,

1 1 1, ,

2

Actual phase distribution across the exit pupil:

where

The path length error or aberration function:

i

aa

a i

a i

a i a i

y x yz

x y x yz

z z

kW x y x y x y

kW x y x y x y W x yz z z z

πλ

πφλ

φ φ

π πλ λ

= − +

= − +

≠= −

= − + + + → = − −� �

�( )2 2

2

1 12

Quadratic dependenceon the space variables on the exit pupil

For a square aperture of width the maximum path-length difference

at the edge of the aperture along the or axis is: m

x y

w

x y Wz

+

= −

��

( )

2

2 2

2

1

,

a i

m

wz

x yW x y W

w

−� �

�

+=

36

6.4.4 Examples of a simple aberration: a focusing error

( )

( ) ( )

( )

( )

, ,2 2 2 2

2

2 2

2 2

2

,

0,0

,

,

,

Now substitute the path-length difference in the OTF:

z f z fz f z fi X i Xi Y i Yjk W x y W x y

X Y

z fW z fi Xm i Yjk x yw

m

f fX Y

X Y

x yW x y W

w

e dxdyf f

dxdy

ef f

λ λλ λ

λ λ

� � + + − − −� � � ��

+ + +� ��

�

+=

=

=

� �

� ��

�

�

�( )

( )

( )

22 2

2 2 2

,

0,0

0 0 0 0 0 0

8 8, sin 1 sin 1

2 2 2 2 2 2

see page 144 eq 6.31

z f z fi X i Yx y

X Yf f

X Ym mX Y X YX Y

dxdy

dxdy

f fW Wf f f ff f c c

f f f f f f

λ λ

λ λ

� � � �− − + −� ��

� � � = Λ Λ − −� ��

� � � � � � � ��

� �

� ��

�

�

( )/

0

/ 2

We plot this OTF for various values of

gives the diffration-limited no aberration OTF.

For sign of the OTF is reversed meaning a contrast reversal.

m

m

m

W

W

W

λ

λ

��

=>

37

Cutofffrequency

Wm>λ/2

Wm<λ/2

OTF with focusing error in a system with square aperture

38

Spatial frequency increases moving to center

Local contrast or MTF changes as a result of change in spatial frequency

Focused and misfocused images of a spoke target

The position of the fringes is determined by the phase associated with OTF at each frequency

39

6.4.4 Examples of a simple aberration: a focusing error

( )0 0 0 0 0 0

/ 2

8 8, sin 1 sin 1

2 2 2 2 2 2

Consider the form of OTFwhen the focusing error is very sever

In this case value o

m

X Ym mX Y X YX Y

W

f fW Wf f f ff f c c

f f f f f f

λ

λ λ

>>

� � � � = Λ Λ − −� � � ��

� � � � � � � � � ��

( )

0 0 0 0

0 0 0

0

1 1 1 12 2 2 2

12 2 2

8, sin sin

2

f is negligable over large frequencies and only it is nonzero for

small so <<1 and and

for <<1

X X X Y

XX Y

m XX Y

f f f f

f f f f

ff ff f f

W ff f c c

fλ

− ≈ − ≈

Λ ≈ Λ ≈� � � � � �

� � = � ��

� � ��

�

�0

82

This is precisely the OTF predicted by the geometrical optics saying that the point-spread function of the system is going to be the geometrical projection of the exit pupil int

m YW ffλ

� � � ��

o the image plane.

40

Geometrical optics prediction of the point-spread function of a system with square aperture and sever focusing error

zozi

Fourier transform of such a PSF function gives the OTF of the system in presence of severe aberrations, Fourier transform of the geometrical PSF is a good approximation for the OTF of the system and diffraction plays a negligible role is shape of the image

frequency analysis of optical imaging systems

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