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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

Chap 5 Wave-optics Analysis of Coherent Systems

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

Outline

• 5.1 A thin lens as a phase transformation

• 5.2 Lens as a Fourier transform

• 5.3 Image formation ： monochromatic illumination

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

5.1 A thin lens as a phase transformation

x

y

),( yx

0

Figure 5.1The thickness function. (a) Front view, (b) side view

(a) (b)

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

The total phase delay suffered by the wave at coordinates

in passing through the lens may be written

Where n is the refractive index of the lens material.

is the phase delay introduced by the lens.

is the phase delay introduced by

the remaining region of free space

between the two planes.

),( yx

)],([),(),( 0 yxkyxknyx

),( yxkn

)],([ 0 yxk

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

Equivalently the lens may be represented by a

multiplicative phase transformation of the form

The complex field across a plane immediately

behind the lens is then related to the complex field

incident on a plane immediately in front of the lens by

)],()1(exp[]exp[ 0 yxnjkjktl

),(' yxU l

),( yxU l

),(),(),(' yxUyxtyxU lll

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

222

11 yxRR

01

1R),( yx

02

),( yx2R

03

222

22 yxRR

(a) (b)

(c)Figure 5.2Calculation of the thickness function. (a) Geometry for , (b) geometry for , and (c) geometry for

01 0203

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

Total thickness function as the sum of three individual

thickness functions

The thickness function is given by

),(),(),(),( 321 yxyxyxyx

),(1 yx

)11(

)(),(

2

1

22

101

222

11011

R

yxR

yxRRyx

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

The second component of the thickness function comes

from a region of glasses of constant thickness .

The third component is given by

02

)11(

)(),(

2

2

22

203

222

22033

R

yxR

yxRRyx

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

The total thickness is seen to be

where

)11()11(),(2

2

22

22

1

22

10 R

yxR

R

yxRyx

0302010

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

We consider only values of x and y sufficiently small to

allow the following approximations to be accurate

With the help of these approximation, the thickness

function becomes

2

2

22

2

2

22

2

1

22

2

1

22

211

211

R

yx

R

yx

R

yx

R

yx

)11

(2

),(21

22

0 RR

yxyx

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

Substitution into

yields the following

approximation to the lens transformations ：

Note ： For a thin lens

)],()1(exp[]exp[ 0 yxnjkjktl

)11

(2

),(21

22

0 RR

yxyx

)]11

(2

)1(exp[]exp[),(21

22

0 RR

yxnjkjknyxtl

00

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

The physical properties of the lens (that is, n, and )

can be combined in a single number f called the focal

length, which is defined by )11

)(1(1

21 RRn

f

1R 2R

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

Neglecting the constant phase factor, which we shall drop

hereafter, the phase transformation may now be written

This equation will serve as our basic representation of the

effects of a thin lens on an incident disturbance.

)](2

exp[),( 22 yxf

kjyxtl

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

5.2 Lens as a fourier transform

),(' yxU l

),( vuU

),( U

),( yxU l

Eq (4-17) Eq (4-17)

),( vu

Lens as a phasor transformer

)(2'

22

),(),(yx

f

kj

ll eyxUyxU

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

Pupil function ：

otherwise ,0

aperture lens theinside ,1),( yxP

)](2

exp[),(),(),( 22' yxf

kjyxPyxUyxU ll

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

0

(x,y))115(),(),( yxAtyxU Al

),(' yxU l

Object Image),( ),( vu

Transparency

),( vuU f

fbfl lt),( U Z

lbffZ

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

dxdyeyxUfj

evuU

vyuxf

j

l

vuf

kj

f

)(2)(

2

),(),(

22

is similar to

where is called Fraunhofer (or far-field) diffraction

formula (or FT of )

ddeUzj

eeyxU

yxz

jyx

z

kj

jkz )(2)(

2

),(),(

22

Quadratic phase

),( U

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

Recall Fresnel diffraction formula

ddeeUezj

eyxU

yxz

jz

kjyx

z

kjjkz )(

2)(

2)(

2 ]),([),(2222

dxdyeeyxUe

fj

evuU

vyuxf

jyxf

kj

l

vuf

kjjkf

f

)(2

)(2'

)(2 ]),([),(

2222

f is constant.It is neglected.

),(),( yxPyxU l If Lens is enough big,then P(x,y) is

equal 1.

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

is neglected.

dxdyeyxUfj

evuU

vyuxf

j

l

vuf

kj

f

)(2)(

2

),(),(

22

Object

),( yxP

Fraunhofer diffraction formula → )],([),( yxUFvuU lf

focal length

Fraunhofer diffraction patten

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

Lens as a Fourier transformer ：

dxdyeyxUfj

evuU

yvxuf

j

l

vuf

kj

f

)(2)(

2

),(),(

22

neglected)],([ yxUF l

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

5.3 Image formation ： monochromatic illumination

),(' yxU l

),( U

),( yxU l

),;,(or ),(),( vuhvuhvuLi

1Z

),(),(0 U

Object

2Z

Image

Linear system

v)(u,U i

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

By the shifting property, we see

),(

),(),;,(

vuh

MvMukvuh

ddvuhUvuU i ),,,(),(),( 0

Output Intput Impulse response

Space invariant ),( vuh

Ideally),(0 MvMukU

Perfect image

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

),(),( vuvuh

),( yxP),()],([ vuhyxPF

(Actually, it will be derived later )],([),( yxPFvuh

Lens coordinate

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

To find the impulse response (or called point spread

function) h, let and substituting it to

gives

),(),(0 U

ddeUzj

eyxU

yxz

kjjkz 22 )()(2),(),(

]})()[(2

exp{1

),( 22

11

yxz

jk

zjyxU l

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

After passage through the lens (focal length f), the filed

distribution becomes

)](2

exp[),(),(),( 22' yxf

jkyxPyxUyxU ll

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

Finally, using the Fresnel diffraction equation

to account for

propagation over distance , we have

ddeUzj

eyxU

yxz

kjjkz 22 )()(2),(),(

2z

dxdyyvxuz

jkyxU

zjvuh l ]})()[(

2exp{),(

1),;,( 22

2

'

2

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

Where constant phase factors have been dropped.

Combining ，

and

and

again neglecting a pure phase factor, yields the formidable

result

]})()[(2

exp{1

),( 22

11

yxz

jk

zjyxU l

)](2

exp[),(),(),( 22' yxf

jkyxPyxUyxU ll

dxdyyvxuz

jkyxU

zjvuh l ]})()[(

2exp{),(

1),;,( 22

2

'

2

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

Let the magnification (Ref. geometrical optics), we

see

dxdyyz

v

zx

z

u

zjk

yxfzz

jkyxP

z

jkvu

z

jk

zzvuh

]})()[(exp{

)])(111

(2

exp[),(

)][(2

exp[)][(2

exp[1

),;,(

2121

22

21

22

1

22

2212

1Const.

0

Classical lens law of geometrical opticals

1

1

2

Z

ZM

dxdyyMvxMuz

jyxPzz

vuh ]})()[(2

exp{),(1

),;,(221

2

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

As it currently stands, the impulse response is that of a

linear space-variant system, so the object and image are

related by a superposition integral but not by a convolution

integral. This space-variant attribute is a direct result of the

magnification and image inversion that occur in the

imaging operation. To reduce the object-image relation to a

convolution equation, we must normalize object-plane

variables be introduced ：

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

in which case the impulse response of

reduces to

M~ M~

dxdyyMvxMuz

jyxPzz

vuh ]})()[(2

exp{),(1

),;,(221

2

dxdyyvxuz

jyxPzz

vuh ]})~()~

[(2

exp{),(1

)~,~

;,(221

2

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

which depends only on the differences of coordinates

A final set of coordinate normalizations simplifies the

results even further. Let

)~,~

( vu

2

~z

xx

2

~z

yy

h

Mh

1~

ydxdyvxujyzxzPvuh ~~)]~~(2exp[)~,~(),(~

22

or

Impulse response or point spread function. (PSF) )],([),( yxPFvuh

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

Then the object-image relationship becomes

or

Where

is the geometrical-optics prediction of the image.

~~)]

~,

~(

1[)~,

~(

~),( 0 dd

MMU

MvuhvuU i

),(),(~

),( vuUvuhvuU gi (convolution form)

)~

,~

(1

),( 0 MMU

MvuU g

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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

There are two main conclusions from the analysis and

discussion above.

• The ideal image produced by a diffraction-limited optica

l system (i.e. a system that is free from aberrations) is a s

caled and inverted version of the object.

• The effect of diffraction is to convolve that ideal image

with the Fraunhofer diffraction pattern of the lens pupil.