Optical Sensing Techniques and Signal Processing-3

8/6/2019 Optical Sensing Techniques and Signal Processing-3

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

Dr. Gao-Wei Chang1

Chap 4 Fresnel and FraunhoferDiffraction


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Dr. Gao-Wei Chang2

Content

4.1 Background

4.2 The Fresnel approximation

4.3 The Fraunhofer approximation

4.4 Examples of Fraunhofer diffraction patterns


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Dr. Gao-Wei Chang3

max2223 ])()[(4/ L\PT "" yxz 2/)( 22 L\ "" kz

),( L\


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Dr. Gao-Wei Chang4

4.1 Background

These approximations, which are commonly made in many fields

that deal with wave propagation, will be referred to as Fresnel and

Fraunhofer approximations.

In accordance with our view of the wave propagation phenomenon

as a system, we shall attempt to find approximations that are valid

for a wide class of input field distributions.


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Dr. Gao-Wei Chang5

4.1.1 The intensity of a wave field

Poyntings thm.

HESXXX

v!

E

VES

1

2

1

)2

1(

20

20

!

!X

XX

2EIS w

X

When calculation a diffraction pattern, we will general regard the intensity

of the pattern as the quantity we are seeking.


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Dr. Gao-Wei Chang6

scos)(

1

)( 0110

01

dr

e

PUjPU

jkr

U!

4.1.2 The Huygens-Fresnel principle in rectangular coordinates

Before we introducing a series of approximations to the Huygens-Fresnel principle, it will be helping to first state the principle in

more explicit from for the case of rectangular coordinates.

As shown in Fig. 4.1, the diffracting aperture is assumed to lie in the

plane, and is illuminated in the positivezdirection.

According to Eq. (3-41), the Huygens-Fresnel principle can be

stated as

.toropointing

ectortheandnor alout ardebet een thangletheishere

10

01

PP

rn X

U

(1)


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Dr. Gao-Wei Chang7

Fig. 4.1 Diffraction geometry

y

y

L

1P

0P

\


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Dr. Gao-Wei Chang8

byexactlygiveniscostermThe U

01

cosr

z !

and therefore the Huygens-Fresnel principle can be rewritten

L\L\ ddr

eU

j

zx,yU

jkr

012

01

),()( !

byexactlyentancethewhere 01r

)()( 22201 y-x-zr !

(2)

(3)


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Dr. Gao-Wei Chang9

.01 r ""

There have been only two approximations in reaching this expression.

1.One is the approximation inherent in the scalar theory

apert retheromengthsmany wavelis

istancenobservatiothat theass mptiontheisseconThe.


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Dr. Gao-Wei Chang10

4.2 Fresnel Diffraction

Recall, the mathematical formulation of the Huygens-Fresnel , the

first Rayleigh- Sommerfeld sol.

The Fresnel diffraction means the Fresnel approximation to

diffraction between two parallel planes. We can obtain the

approximated result.

!

z

n

jkr

o dsarr

epU

jpU ).,cos()(

1)( 01

01

1

01

P

? A

g

g

! L\L\

P

L\ddeU

zj

eyxU

yxz

kj

jkz 22 )()(2),(),( (1)


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Dr. Gao-Wei Chang11

z

x

y

\

L

? A222

"" L\ yxz

Kj

e (Why?)

(wave propagation)

wave propagation z

Aperture PlaneObservation Plane

Corresponding to

The quadratic-phase exponential with positive phase

i.e, ,for z>? A22 )()(2

L\ yxz

kj

e


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Dr. Gao-Wei Chang12

N

ote: The distance from the observation point to an aperture point

Using the binominal expansion, we obtain the approximation to

? A2

1

22

21

222

01

)()(1

)()(

!

!

z

y

z

xz

yxzr

L\

L\

=b

? AL\

L\

!

!

yxz

z

z

y

z

x

zr


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Dr. Gao-Wei Chang13

as the term

is sufficiently small.

The first Rayleigh Sommerfeld sol for diffraction between two

parallel planes is then approximated by

22 )()(

z

y

z

x L\

L\L\P

L\

ddzr

eU

jyxU

yxz

zjk

7

!2

])()(2

1[

01

22

),(1

),(


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Dr. Gao-Wei Chang14

( ) , the r01 in denominator of the

integrand is supposed to be well approximated by the first term only

in the binomial expansion, i.e,

In addition, the aperture points and the observation points areconfined to the ( , ) plane and the (x,y) plane ,respectively. )

Thus, we see

),cos( r

z

ar n !3

zr !01

\ L

? A

! L\L\

P

L\ddeU

zj

eyxU

yxz

kj

jkz 22 )()(2),(),(


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Dr. Gao-Wei Chang15

Furthermore, Eq(1) can be rewritten as

(2a)

where the convolution kernel is

(2b)

Obviously, we may regard the phenomenon of wave propagation asthe behavior of a linear system.

g

g L\L\L\ ddyxhUyxU ),(),(),(

)](2

exp[),( 22 yxz

jk

zj

eyxh

jkz

!P


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Dr. Gao-Wei Chang16

Another form of Eq.(1) is found if the term

is factored outside the integral signs, it yields

)( yxz

kj

e

g

g

4

! L\L\P

L\P

L\

ddeeUezj

e

yxU

yxz

jz

kjyx

z

kj

jkz)(

2)(

2)(

2

]),([),(

2222

(3)

which we recognize (aside from the multiplicative factors) to be the

Fourier transform of the complex field just to the right of the aperture

and a quadratic phase exponential.


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Dr. Gao-Wei Chang17

We refer to both forms of the result Eqs. (1) and (3), as the Fresneldiffraction integral . When this approximation is valid, the observer

is said to be in the region of Fresnel diffraction or equivalently in

the near field of the aperture.

Note:

In Eq(1),the quadratic phase exponential in the integrand

? A22 )()(2 L\ yxzkje


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Dr. Gao-Wei Chang18

do not always have positive phase for z> .Its sign depends on the

direction of wave propagation. (e.g, diverging of converging

spherical waves)

In the next subsection ,we deal with the problem of positive or

negative phase for the quadratic phase exponent.


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Dr. Gao-Wei Chang19

4.2.1 Positive vs. Negative Phases

Since we treat wave propagation as the behavior of a linear system

as described in chap.3 of Goodman), it is important to descries the

direction of wave propagation.

As a example of description of wave propagation direction, if we

move in space in such a way as to intercept portions of a wavefield

(of wavefronts ) that were emitted earlier in time.


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Dr. Gao-Wei Chang20

),2( tzzf c

),( tzzf c

),( tzf

cz

cz2

z

z

z

tzftf !

ct

),()( cc ttzfttf !

ct2

ct

t

t

t

In the above two illustrations, we assume the wave speed v=zc/tc

where zc and tc are both fixed real numbers.


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Dr. Gao-Wei Chang21

In the case of spherical waves,

r

k

r

k

Diverging spherical wave Converging spherical wave


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Dr. Gao-Wei Chang22

Consider the wave func. r

e rkj

,where rar r!

and r > andP

Tkk

akak !!

If rk aa ,then

rkjrkj er

er

!11

(Positive phase)

implies a diverging spherical wave.

Or ifrk

aa !


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Dr. Gao-Wei Chang23

rkjrkj

erer

!

11

i lies a c er i s erical a e.

(Negative phase)

Note

For spherical wave ,we say they are diverging or converging ones

instead or saying that they are emitted earlier in time or later in

time.


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Dr. Gao-Wei Chang24

The term standing for the time dependence of a traveling

wave implies that we have chosen our phasors to rotate in the clockwisedirection.

Earlier in time

Positive phasecttvj

e T

vtjttvjee c

TT 2)(2

vtje

T2

Specifically, for a time interval tc > , we see the following relations,


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Dr. Gao-Wei Chang25

Therefore, we have the following seasonings

Earlier in time Positive phase

(e.g., diverging spherical waves)

Later in time Negative phase

(e.g., converging spherical waves)

Note

Earlier in time means the general statement that if we move in space in

such a way as to intercept wavefronts (or portions of a wave-field ) that

were emitted earlier in time.


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Dr. Gao-Wei Chang26

0"cU

za

ya

Propagation direction

Spatial distribution of

wavefronts

To describe the direction of wave propagation for plane waves, we cannot

use the term diverging or converging .Instead .we employ the generalstatement ,for the following situations.


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Dr. Gao-Wei Chang27

The phasor of a plane wave,yj

eTE2

, (whereE

multiplied by the time dependence gives

(222 cttvjvtjyj eee

!TTTE , where cc y

vE

1!

We may say that ,if we move in the positive y direction , the argument of

the exponential increases in a positive sense, and thus we are moving to a

portion of the wave that was emitted earlier in time.

> )


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Dr. Gao-Wei Chang28

0cU

Propagation direction

In a similar fashion , we may deal with the situation for or cUE


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Dr. Gao-Wei Chang29

Note

Show that the Huygens-Fresnel principle can be expressed by

dsarr

epU

jpU n

jkr

),cos()()(XX

!P

Recall the wave field at observation point P

dsn

GU

n

uGpU

x

x

x

x! )(

4)(

T( )


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Dr. Gao-Wei Chang30

For the first RayleighSommerfeld solution ,the Green func.

1

~

1~

11

r

e

r

eG

rjkjkr

!

Note we put the subscript -, i.e, G- to signify this kind of Green

func.

Substituting Eq(2) into Eq.(1) gives

(2)


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Dr. Gao-Wei Chang31

( )

dsn

p k xx! )(1)( 0 T (4)

or

where the Green func. proposed by Kirchhoff

01

01

r

eG

jkr

k !

dsn

G

UpU xx

!

)(4

1

)( 0 T


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Dr. Gao-Wei Chang32

The term in the integrand of Eq.(4)

010101

2

0101

0101

01

01

0101

01

01

)1)(o (

)1(1

)o (

)(

)(

re

rjar

rej er

ar

a

r

e

r

a

aGn

G

j r

n

j rj rn

n

j r

r

nKK

!

!

x

x!

!x

x


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Dr. Gao-Wei Chang33

as1

12

rK ""

PT or P""r

!x

x

n

GK ),cos(2

1

1

1

n

j r

r

r

ej

P

T

Finally, substituting Eq.(5) into Eq.(4) yields

dsarrpjp n

jkr

),cos()(

1

)( 010110

01 XX

! P

(5)


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Dr. Gao-Wei Chang34

4.2.2 Accuracy of Fresnel Approximation

Recall Fresnel diffraction integral

? A

g

g

! L\L\

P

L\ddeU

zj

eyxU

yxz

kj

jkz 22

2,,

observation point (fixed)Aperture point (varying with)

Parabolic wavelet

(4.14)

We compare it with the exact formula

g

g

d! L\L\P

ddnarr

eU

jyxU

jkrXX

10

01

cos,1

,01

Spherical wavelet

01r

z

where

!

z

y

z

xzr L\

(or )

! .

2222

018

1

2

11

z

y

z

x

z

y

z

xzr

L\L\


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Dr. Gao-Wei Chang35

since the binomial expansion

.! 2218

1

2

111 bbb

where22

!z

y

z

x L\

The max.approx.error (i.e.,( )max)

bb2

111 2

1

222

2

8

1

8

1

!z

y

z

x L\

and the corresponding error of the exponential

8bjkz

e

is maximized at the phase (or approximately 1 radian)T


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Dr. Gao-Wei Chang36

A sufficient condition for accuracy would be

a

z

y

z

xz L\

P


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Dr. Gao-Wei Chang37

6

222

3

1050.4

210114.3

vv

v

z or6 v! m0.4z

za

This sufficient condition implies that the distance z must be

relatively much larger than

? AmaL\P

T yx


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Dr. Gao-Wei Chang38

Since the binomial expansion

HOTbbbb !! 21

18

1

2

111 22

1

. (high order term)

where22

!

z

y

z

xb

L\

we can see that the sufficient condition leads to a sufficient small

value of b

However, this condition is not necessary. In the following, we will

give the next comment that accuracy can be expected for much

smaller values of z (i.e., the observation point (x , y) can be located

at a relatively much shorter distance to an arbitrary aperture point

on the (,) plane)


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Dr. Gao-Wei Chang39

We basically malcr use of the argument that for the convolutionintegral of Eq.(4-14), if the major contribution to the integral comes

from points (,) for whichx andy, then the values of

the HOTs of the expansion become sufficiently small.(That is, as

(,) is close to (x , y)

!z

y

z

xb

L\gives a relatively small value

Consequently, can be well approximated by . ) 2

1

1b

b2

1

1


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Dr. Gao-Wei Chang40

In addition it is found that the convolution integral of Eq.(4-14),

? A g

g

! LLP

L

P

T

ddeUzj

eyxU

yxz

jjkz22

,,

g

g

! L\L\P

P

L

P

\T

ddeUzj

eyxU

z

y

z

xjjkz

22

,,or

7 L\L\

PT ddeU

zj

e YXjjkz

22

,

where and ,z

x

X P

\

! z

y

Y P

L

!


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Dr. Gao-Wei Chang41

can be governed by the convolution integral of the function

with a second function (i.e., U(,)) that is smooth and

slowly varying for the rang 2 < X < 2 and 2 < Y < 2. Obviously,

outside this range, the convolution integral does not yield a

significant addition.

22 YXje T

( Note

For one dimensional case

12

!g

gdXe XjT is governed by

dXe XjT

we can see that

! g

g

dXdYe YXj

T

is well approximated by

dXdYeYXjT


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Dr. Gao-Wei Chang42

Finally, it appears that the majority of the contribution to the

convolution integral for the range - < X < and - < Y < or the aperture area comes from that for a square in the (,)

plane with width and centered on the point = x,= y

(i.e., the range 2


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Dr. Gao-Wei Chang43

From another point of view, since the Fresnel diffraction

integral

? A

7

! L\L\

L\

T

ddeUzj

eyxU

yxz

jjkz 22

,,

? A

L\L\

P

L\PT

ddeUzj

e yxz

jjkz 22

,

Corresponding square area

yields a good approximation to the exact formula

7! dsarre

PU

j

PUn

j rXX

,cos1

01

01

10

01

Pwhere

z

y

z

xzr

L\


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Dr. Gao-Wei Chang44

we may say that for the Fresnel approximation (for the aperture area

or the corresponding square area) to give accurate results, it is not

necessary that the HOTs of the expansion be small, only that they do

not change the value of the Fresnel diffraction integral significantly.

NoteFrom Goodmans treatment (P. 9 7 ), we see that

X

X

XjdXe

2T

can well approximate

g

gdXe Xj

2Tor

7dXe Xj

2T

Where the width of the diffracting aperture is larger than the

length of the region 2 < X < 2


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Dr. Gao-Wei Chang45

For the scaled quadratic-phase exponential of Eqs.(4-14) and

Eq.(4-1 ), the corresponding conclusion is that the majority of the

contribution to the convolution integral comes from a square in the

(,) plane, with width and centered on the point (=

x ,= y)zP4

In effect,1. When this square lie entirely within the open portion of the

aperture, the field observed at distance z is, to a good

approximation, what it would be if the aperture were not

present. (This is corresponding to the light region)


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Dr. Gao-Wei Chang46

2. When the square lies entirely behind the obstruction of the

aperture, then the observation point lies in a region that is, to agood approximation, dark due to the shadow of the aperture.

3. When the square bridges the open and obstructed parts of the

aperture, then the observed field is in the transition (or gray)

region between light and dark.

For the case of a one-dimensional rectangular slit, boundaries

among the regions mentioned above can be shown to be

parabolas, as illustrated in the following figure.


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Dr. Gao-Wei Chang47

zP4

zP2

zP2


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Dr. Gao-Wei Chang48

Thus, the upper (or lower) boundary between the transition

(or gray) region and the light region can be expressed by

zwx P4! (or ) zwx P4!

The light region

W x , x

W + x , x

zP2zP2


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Dr. Gao-Wei Chang49

4.2.3 The Fresnel approximation and the Angular Spectrum

In this subsection, we will see that the Fourier transform of theFresnel diffraction impression response identical to the transfer func.

of the wave propagation phenomenon in the angular spectrum

method of analysis, under the condition of small angles.

From Eqs.(4-15)and (4-1 ), We have

g

g! L\L^EL^ ddyhUyxU )()()(

Where the convolution kernel (or impulse response) is

ee yxk

jk

jyxh

)()(

x

x

!


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Dr. Gao-Wei Chang50

The FT of the Fresnel diffraction impulse response becomes

g

g

!! dxdy

jyxhF eeffH

yxjkj

jk

yxF

fyfxyx )()(z

z

z),()],([

T

The integral term

dxdyee yxjj fyfxyx g

g )(

2)(z

22

T

T

can be rewritten a

g

g

dpdqee

qpfyfx jj )(

z

))z((- )z(P

TP

P

TP

where

fx

zxp P! fy

zyq P!and


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Dr. Gao-Wei Chang51

( eca se t e e e ts

(( ]([ fzxzffx xzj

xzxzj x PP P

T

PP

T

!where f

xzxp P!

)()(222

])(2[ fzyzffy yz

j

yzy

z

j

yPP

P

TP

P

T!

where fy

zyq P!

as a result,

eeffzfyzfx

zjjkz

yxFH

)( )()(

),(PP

P

T

dpdq

zj eqp

zj )(1P

T

P=1

P

q


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Dr. Gao-Wei Chang52

soeeH

y

xzjjkz

yxF

)( 22),(

!

TP

On the other hand, the transfer function of the wave propagation

phenomenon in the angular spectrum method of analysis is expressed by

!

otherwise,

, ,)--()-((jk

yxa

yxyx

eH

under the condition of small angles (as noted below the term)

e yxjkz )()( PP

can be approximated by

ee

efyfx

fyfx

zjj z

j z

)(

)2

1

2

11(

22

)( 2)( 2

TP

PP

(becauseP

T!k )


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Dr. Gao-Wei Chang53

(Note: because

)()(122

2

1

1

z

y

z

xr zo

L\

!

For Fresnel approximation, the sufficient condition ma be

][22

4 maxL\

P

T

"" yxz

The obliquity factorco ra on then approache

That i co ran XX!U is small angle


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Dr. Gao-Wei Chang54

Which is the transfer function of the wave propagation phenomenon

in the angular spectrum method of analysis under the condition of

small angles.

a ffHffH yxyx !

Therefore, we have shown that the FT of the Fresnel diffraction

impulse response


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Dr. Gao-Wei Chang55

4.2.4 Fresnel Diffraction between Confocal Spherical surfaces.

\

ro1

ro1

L

ro1


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Dr. Gao-Wei Chang56

)2

2

2

21(

)2

1

2

11(

2

22

2

22

22

1 )()(

zz

z

zz

r

yxz

zo

L

L\

L\

!

$

as L\yx are all very close to zero, (i.e, the paraxial condition)

z

y

z

xzro

L\ $

1

Recall the Rayleigh Sommerfeld sol, (for the paraxial condition

!

!

L\L\P

L\L\P

L\P ddU

zj

ddUj

yxU

ee

arre

yxz

kjj

noo

jkro

)(2

),(

),cos(),(),(


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Dr. Gao-Wei Chang57

as a result, for the paraxial region,

This Fresnel diffraction eq. expresses the field ,L\U

observed on the right hand spherical cap as the FT of the filed

U(x,y) on the left-hand spherical cap.

Comparison of the result with Eq(4-17),the Fresnel diffraction

integral (including Fourier-transform-like operation)

! L\L\

PL\

PT

ddUzj

yxU eeyx

zj

jkz)(

),(),(

(including the paraxial representation of spherical phase)


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Dr. Gao-Wei Chang58

g

! L\L\

P

L\P

TL\

dd

zj

yx eeee yx

zj

z

kj

z

kj

jkzyx )(

2)(

2)(

2 ]),([),(2222

quadratic phase parabolic phase

Note: Recall


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Dr. Gao-Wei Chang59

The two quadratic phase factors in Eq(4-17)are in fact simply

paraxial representations of spherical phase surfaces, (since the

Rayleigh Sommerfeld sol. can be applied only to the planar screens),

and it is therefore reasonable that moving to the spheres has

eliminated them.

For the diffraction between two spherical caps, it is not really validto use the Rayleigh-Sommerfeld result as the basis for the

calculation (only for the diffraction between two parallel planes).

However, the Kirchhoff analysis remains valid, and its predictions

are the same as those of the Rayleigh-Sommerfeld approachprovided paraxial conditions hold.


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Dr. Gao-Wei Chang60

4.3 The Fraunhofer approximation

From Eq(4-17), We see

g

g

! L\L\

P

L\P

TL\

ddUz

yxU eeee yx

zzz

zyx )(

2)(

2)(

2 ]),([),(2222

If the exponent

22)](

2[

max

z

k

We have

a

a

L\

L\P

T

""

""

zor

z

(4-17)


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Dr. Gao-Wei Chang61

The observed filed strength U(x,y) can be found directly from a FT

of the aperture function itself (because )e zk

j )(2

22

L\

1

0

ej

That is, Eq.(4-17)with the Fraunhofer approximation becomes

g

g

! \\P

\T

ddUzjyxU eee fyfx

yx

jz

kjjkz

)(2)(

2

),(),(

22

(Aside from the multiplicative phase factors, this expression is simply

the FT of the aperture distribution)

where z

y

andz ff yx PP !!x

(4-2 )

(4-25)


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Dr. Gao-Wei Chang62

Note

Recall the different forms of Fresnel diffraction integral

! )14-4........(..........),(),(

][ )()(L\L\

P

L\

PT

ddUzj

yxU ee yx

zj

jkz

)15-4.........(....................),(),(),(

g

g ! ddyxhUyxU

where the Fresnel diffraction impulse response

ee yx

z

kj

jkz

zjyxh

!

P

(4-1 )

and that of Eq(4-17)


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Dr. Gao-Wei Chang64

4.4 Examples of Fraunhofer diffraction patterns

4.4.1 Rectangular Aperture If the aperture is illuminated by a unit-amplitude, normally incident,

monochromatic plane wave, then the field distribution across the

aperture is equal to the transmittance function .Thus using Eq.(4-25),

the Fraunhofer diffraction pattern is seen to be

zY

zXyfxf

yxz

kj

jkz

UFzj

ee

yxU

PP

L\P

//

)(2

)},({),(

22

!!

! \


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Dr. Gao-Wei Chang65

4.4.2 Circular Aperture

Suggests that the Fourier transform of Eq.(4-25) be rewritten as a

Fourier-Bessel transform. Thus if Kis the radius coordinate in the

observation plane, we have

zrp

jkz

qUz

kjzj

eUP

FP /

2

)( )}({)2

exp(!

!


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Dr. Gao-Wei Chang66

4.4.3 Thin Sinusoidal Amplitude Grating

In practice, diffracting objects can be far more complex. In accord

with our earlier definition (3- ),the amplitude transmittance of a

screen is defined as the ratio of the complex field amplitude

immediately behind the screen to the complex amplitude incident on

the screen . Until now ,our examples have involved only

transmittance functions of the form

ape tu etheut

ape tu ethein

tA0

1

),( L\


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Dr. Gao-Wei Chang67

Spatial patterns of phase shift can be introduced by means of

transparent plates of varying thickness, thus extending the realizable

values oftA to all points within or on the unit circle in the complexplane.

As an example of this more general type of diffracting screen,

consider a thin sinusoidal amplitude grating defined by the

amplitude transmittance function

!

w

rectw

rectfm

tA

222cos

22

1 L\\TL\ (4-33)

where for simplicity we have assumed that the grating structure isbounded by a square aperture of width 2w. The parameter m represents

the peak-to-peak change of amplitude transmittance across the screen

andf0

is the spatial frequency of the grating.


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4.4.4 Thin sinusoidal phase grating

or x)(\Binary phase grating

)2

()2

()()]2(sin

2[ 0

w

rect

w

recte,yU

fm

j !

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Optical Sensing Techniques and Signal Processing-3