Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 1 Chap 4 Fresnel and Fraunhofer Diffraction

$Page 1: Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 1 Chap 4 Fresnel and Fraunhofer Diffraction$
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

Dr. Gao-Wei Chang

Chap 4 Fresnel and Fraunhofer Diffraction

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Dr. Gao-Wei Chang

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 Chang

Full wave equation

z~λ

Rayleigh-Sommerfield &Fersnel-Kirchhoff

z>>λ

Fresnel(near field)

max2223 ])()[(4/ yxz

Fraunhoffer(far field)

2/)( 22 kz光線追跡

BeampropBPM CAD

GsolverDOE CAD

ZEMAXCode VOSLOASAP

(x,y)),( λ = 850 nmλ = 1550 nm

850 nm1550 nm

966 um791 um

4.6 mm2.5 mm

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Dr. Gao-Wei Chang

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 Chang

• 4.1.1 The intensity of a wave field

• Poynting’s thm.

HES

μεEε

VEεS

1

2

1

)2

1(

20

20

2EIS

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 Chang

scos)(1

)(01

1001

dr

ePU

jλPU

jkr

Σ

• 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 positive z direction.

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

. to from pointing

vector theand ˆ normal outward ebetween th angle theis where

1 0

01

PP

r n

(1)

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Dr. Gao-Wei Chang

Fig. 4.1 Diffraction geometry

X

Z

y

1P

0P

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Dr. Gao-Wei Chang

byexactly given is cos termThe

01cos

r

zθ

and therefore the Huygens-Fresnel principle can be rewritten

ddr

eU

jλ

zx,yU

jkr

Σ 012

01

),()(

byexactly given is distance thewhere 01 r

)()( 22201 y-ηx-ξzr

(2)

(3)

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Dr. Gao-Wei Chang

.01 λ r

There have been only two approximations in reaching this expression.

1.One is the approximation inherent in the scalar theory

aperture, thefrom engthsmany wavel is

distancen observatio that theassumption theis second The 2.

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Dr. Gao-Wei Chang

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

011

01

ddeU

zj

eyxU

yxz

kjjkz 22 )()(2),(),( (1)

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Dr. Gao-Wei Chang

z

x

y

22

2"" yx

z

Kj

e (Why?) (wave propagation)

wave propagation z

Aperture PlaneObservation Plane

Corresponding to

The quadratic-phase exponential with positive phasei.e, ,for z>0 22 )()(

2 yx

z

kj

e

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Dr. Gao-Wei Chang

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

Using the binominal expansion, we obtain the approximation to

2

1

22

21

22201

)()(1

)()(

z

y

z

xz

yxzr

=b

22

2201

)()(2

1

)(2

1)(

2

11

yxz

z

z

y

z

xzr

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Dr. Gao-Wei Chang

• 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

ddzr

eU

jyxU

yxz

zjk

2

])()(2

1[

01

22

),(1

),(

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Dr. Gao-Wei Chang

• ( ) , 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 are confined to the ( , ) plane and the (x,y) plane ,respectively. )

• Thus, we see

0101 ),cos(

r

zar n

zr 01

ddeU

zj

eyxU

yxz

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

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Dr. Gao-Wei Chang

• Furthermore, Eq(1) can be rewritten as

(2a)

• where the convolution kernel is

(2b)

•

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

ddyxhUyxU ),(),(),(

)](2

exp[),( 22 yxz

jk

zj

eyxh

jkz

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Dr. Gao-Wei Chang

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

is factored outside the integral signs, it yields

)(2

22 yxz

kj

e

ddeeUe

zj

eyxU

yxz

jz

kjyx

z

kjjkz )(

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 Chang

•

We refer to both forms of the result Eqs. (1) and (3), as the Fresnel diffraction 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

22 )()(2

yxz

kj

e

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Dr. Gao-Wei Chang

do not always have positive phase for z>0 .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 Chang

• 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 Chang

),2( tzzf c

),( tzzf c

),( tzf

cz

cz2

z

z

z

),()( tzftf

ct

)2,()2( 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 Chang

• In the case of spherical waves,

r

k

r

k

Diverging spherical wave Converging spherical wave

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Dr. Gao-Wei Chang

Consider the wave func. r

e rkj

,where rar r

and r >0 and2

kk akak

If rk aa ,then

rkjrkj er

er

11

(Positive phase)

implies a diverging spherical wave.

Or if rk aa

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Dr. Gao-Wei Chang

rkjrkj er

er

11

implies a converging spherical wave.

(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 Chang

The term standing for the time dependence of a traveling wave implies that we have chosen our phasors to rotate in the clockwise direction.

“Earlier in time ”

Positive phase)(2 cttvje

vtjttvj ee c 2)(2

vtje 2

Specifically, for a time interval tc >0, we see the following relations,

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Dr. Gao-Wei Chang

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 Chang

0cza

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 general statement ,for the following situations.

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Dr. Gao-Wei Chang

The phasor of a plane wave, yje 2 , (where

multiplied by the time dependence gives

)(222 cttvjvtjyj eee , where cc yv

t 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.

>0)

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Dr. Gao-Wei Chang

0c

Propagation direction

In a similar fashion , we may deal with the situation for 0 or 0 c

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Dr. Gao-Wei Chang

Note：

Show that the Huygens-Fresnel principle can be expressed by

dsarr

epU

jpU n

jkr),cos()(

1)( 01

0110

01 <pf>

Recall the wave field at observation point P0

dsn

GU

n

uGpU

)(4

1)( 0 (1)

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Dr. Gao-Wei Chang

For the first Rayleigh –Sommerfeld solution ,the Green func.

01

~

01~

0101

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 Chang

(3)

dsn

GUpU k

)(2

1)( 0 (4)

or

where the Green func. proposed by Kirchhoff

01

01

r

eG

jkr

k

dsn

GUpU

)(4

1)( 0

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Dr. Gao-Wei Chang

The term in the integrand of Eq.(4)

010101

2

0101

0101

01

01

0101

01

01

)1

)(,cos(

)1(1

),cos(

)(

)(

r

e

rjkar

rejker

ar

ar

e

ra

aGn

G

jkr

n

jkrjkrn

n

jkr

r

nKK

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Dr. Gao-Wei Chang

as 01

12

rK

or 01r

n

GK ),cos(2

0101

01

n

jKrar

r

ej

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

dsarr

epU

jpU n

jkr),cos()(

1)( 01

0110

01

(5)

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Dr. Gao-Wei Chang

4.2.2 Accuracy of Fresnel Approximation

Recall Fresnel diffraction integral

ddeU

zj

eyxU

yxz

kjjkz 22

2,,

observation point (fixed)Aperture point (varying withΣ)

Parabolic wavelet

…(4.14)

We compare it with the exact formula

ddnar

r

eU

jyxU

jkr 10

01

cos,1

,01

Spherical wavelet

01r

z

where 2

122

01 1

z

y

z

xzr

(or )

2222

01 8

1

2

11

z

y

z

x

z

y

z

xzr

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Dr. Gao-Wei Chang

since the binomial expansion

221

8

1

2

111 bbb

where

22

z

y

z

xb

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

bb

2

111 2

1

2222

8

1

8

1

z

y

z

xb

and the corresponding error of the exponential

2

8

1bjkz

eis maximized at the phase (or approximately 1 radian) 2

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Dr. Gao-Wei Chang

A sufficient condition for accuracy would be

max

222

8

12

z

y

z

xz

<<1

max2223

4

yxz

For example

(ξ,η)

(x,y)1cm

(x,y)

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Dr. Gao-Wei Chang

6

222

3

1050.4

210114.3

z or 3m28.6 m 0.4z

(x,y)(x,y)

(ξ,η) is variable

ξ

η

z

x

y

observation point (fixed)

za

This sufficient condition implies that the distance z must be relatively much larger than

max222

4

yx

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Dr. Gao-Wei Chang

Since the binomial expansion

HOTbbbb 2

11

8

1

2

111 22

1 (high order term)

where 22

z

y

z

xb

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 (ξ,η) plane)

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Dr. Gao-Wei Chang

We basically malcr use of the argument that for the convolution integral of Eq.(4-14), if the major contribution to the integral comes from points (ξ,η) for which ξ≒x and η≒y, then the values of the HOTs of the expansion become sufficiently small.(That is, as (ξ,η) is close to (x , y)

22

z

y

z

xb

gives a relatively small value

Consequently, can be well approximated by . )

21

1 b b2

11

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Dr. Gao-Wei Chang

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

ddeUzj

eyxU

yxz

jjkz 22

,,

ddeUzj

eyxU

z

y

z

xjjkz

22

,,or

ddeU

zj

e YXjjkz 22

,

where and , z

xX

z

yY

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Dr. Gao-Wei Chang

can be governed by the convolution integral of the function with a second function (i.e., U(ξ,η)) 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

( Note For one dimensional case

12

dXe Xj is governed by

2

2

2

dXe Xj

we can see that 122

dXdYe YXj

is well approximated by

2

2

2

2

22

dXdYe YXj

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Dr. Gao-Wei Chang

Finally, it appears that the majority of the contribution to the convolution integral for the range -∞ < X < ∞ and -∞ < Y < ∞ or the aperture area Σ comes from that for a square in the (ξ,η) plane with width and centered on the point ξ= x,η= y (i.e., the range –2 < <2 and –2< <2 or

< and < )

z4

z

x

z

y

x z2yz2

As a result within the square area, the expansion

221

8

1

2

111 bbb

as well approximated, since

22

z

y

z

xb

is small enough.

(x,y)(x,y)ξ

η

z

x

y

zaz4

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Dr. Gao-Wei Chang

From another point of view, since the Fresnel diffraction integral

ddeUzj

eyxU

yxz

jjkz 22

,,

ddeUzj

e yxz

jjkz 22

,Corresponding square area

yields a good approximation to the exact formula

dsarr

ePU

jPU n

jkr ,cos

101

0110

01

where 2

122

01 1

z

y

z

xzr

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Dr. Gao-Wei Chang

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

Note ： From Goodman’s treatment (P.69 70), we see that

X

X

Xj dXe2

can well approximate

dXe Xj 2

or dXe Xj 2

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 Chang

For the scaled quadratic-phase exponential of Eqs.(4-14) and Eq.(4-16), the corresponding conclusion is that the majority of the contribution to the convolution integral comes from a square in the (ξ,η) plane, with width and centered on the point (ξ= x ,η= y)

z4

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 Chang

2. When the square lies entirely behind the obstruction of the aperture, then the observation point lies in a region that is, to a good 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 Chang

z4

z2

x

y

z

Aperture stop

Incidentwavefront

z2

x

Dark

Dark

Light

Transition(Gray)

Consider the rectangular slit

with the width 2w

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Dr. Gao-Wei Chang

Thus, the upper (or lower) boundary between the transition (or gray) region and the light region can be expressed by

zwx 42 (or ) zwx 42

The light region

W – x , x 0 ≧ ≧W + x ≧ , x ＜ 0

z2z2

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Dr. Gao-Wei Chang

• 4.2.3 The Fresnel approximation and the Angular Spectrum

• In this subsection, we will see that the Fourier transform of the Fresnel 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 (4-16), We have

ddyhUyxU ),(),(),(

Where the convolution kernel (or impulse response) is

ee yxz

kjkz

zjyxh

)( 22),(

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Dr. Gao-Wei Chang

• The FT of the Fresnel diffraction impulse response becomes

dxdy

jyxhF eeffH

yxjk

jjk

yxFf yf xyx )(2)(

z2

z22

z),()],([

The integral term

dxdyeeyxjj f yf xyx

)(2)(

z22

can be rewritten a

dpdqeeqpf yf x j

zj )(

z))z((- 2222)z( 22

where

fx

zxp fy

zyq and

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Dr. Gao-Wei Chang

• (because the exponents

)()(222

])(2[ fzxzffx xz

jxzx

zj

x

where fx

zxp

)()(222

])(2[ fzyzffy yz

jyzy

z

jy

where fy

zyq

as a result,

eeff zf yzf xz

jjkz

yxFH)( )( 22)( 22

),(

dpdq

zj eqp

zj )( 221

=1

P

q

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Dr. Gao-Wei Chang

• so eeffH

f yf xzjjkz

yxF)( 22

),(

On the other hand, the transfer function of the wave propagation phenomenon in the angular spectrum method of analysis is expressed by

otherwise , 0

),(

111z 2222

λff,)-λλ-()-(λ(jk

yxa

yxyxeffH

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

ef yf xjkz )()(

221

can be approximated by

eee

f yf x

f yf x

zjjkz

jkz

)(

)2

1

2

11(

22

)( 2)( 2

(because

2

k )

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Dr. Gao-Wei Chang

• (Note: because

])()(1[22

2

1

1z

yz

xr z

o

For Fresnel approximation, the sufficient condition ma be

])()([ 224 max

3

yxz

The obliquity factor ),cos(1ra on then approaches 1

That is, ),(cos 01-1 ran

is small angle

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Dr. Gao-Wei Chang

•

• Which is the transfer function of the wave propagation phenomenon in the angular spectrum method of analysis under the condition of small angles.

),(),(aF ffHffH yxyx

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

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Dr. Gao-Wei Chang

4.2.4 Fresnel Diffraction between Confocal Spherical surfaces.

ro1

ro1

x

y

Paraxial region

ro1

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Dr. Gao-Wei Chang

)2

2

2

21(

)2

1

2

11(

2

22

2

22

22

1 )()(

zy

zzx

z

y

z

xr

yxz

zo

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

z

y

z

xzr o

1

Recall the Rayleigh Sommerfeld sol, (for the paraxial condition

ddUzj

ddUj

yxU

ee

arre

yxzk

jj

noo

jk ro

)(2

11

),(1

),cos(),(1

),(1

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Dr. Gao-Wei Chang

• as a result, for the paraxial region,

This Fresnel diffraction eq. expresses the field ),( 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)

ddUzj

yxU ee yxz

jjkz

)(2

),(),((including the paraxial representation of spherical phas

e)

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Dr. Gao-Wei Chang

ddU

zjyxU eeee yx

zj

zk

jz

kj

jkzyx )(

2)(

2)(

2 ]),([),(2222

quadratic phase parabolic phase

Note: Recall

sphere

Parabola

Paraxial region

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Dr. Gao-Wei Chang

• 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 valid to 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 approach provided paraxial conditions hold.

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Dr. Gao-Wei Chang

4.3 The Fraunhofer approximation

• From Eq(4-17), We see

ddU

zjyxU eeee yx

zj

zk

jz

kj

jkzyx )(

2)(

2)(

2 ]),([),(2222

If the exponent

122 )](2

[max

z

k

We have

)][(

)][(22

22

max

max

2

k

zor

z

(4-17)

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Dr. Gao-Wei Chang

• The observed filed strength U(x,y) can be found directly from a FT of the aperture function itself (because ) e z

kj )(2

22

10 e

j

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

ddUzj

yxU eee f yf x

yxjz

kjjkz

)(2)(

2),(),(

22

(Aside from the multiplicative phase factors, this expression is simply the FT of the aperture distribution)

where z

yand

zff

yx

x(4-26)

(4-25)

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Dr. Gao-Wei Chang

• Note：• Recall the different forms of Fresnel diffraction integral

)14-4........(..........),(),(][ )( 2)( 2

ddUzj

yxU ee yxz

jjkz

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

ddyxhUyxU

where the Fresnel diffraction impulse response

ee yxz

kj

jkz

zjyxh

)(2

22),(

(4-16)

and that of Eq(4-17)

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Dr. Gao-Wei Chang

• Comparison of Eqs(4-15)and (4-16) with Eqs.(4-25)and (4-26) tell us that there is no transfer function for the Fraunhofer (or far-field) diffraction since Eqs(4-25) and (4-26) do not include impulse response.

• Nonetheless, since Fraunhofer diffraction is only a special case of Fresnel diffraction, the transfer function Eq(4-21) remains valid throughout both the Fresnel and the Fraunhofer regimes. That is, it is always possible to calculate diffracted field in the Fraunhofer region by retaining the full accuracy of the Fresnel approximation.

Treating the wave propagation phenomenon as a linear system

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Dr. Gao-Wei Chang

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

kjjkz

UFzj

eeyxU

//

)(2

)},({),(

22

Wx-Wx

1

2Wx

rect(x)

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Dr. Gao-Wei Chang

• 4.4.2 Circular Aperture

Suggests that the Fourier transform of Eq.(4-25) be rewritten as a Fourier-Bessel transform. Thus if is the radius coordinate in the observation plane, we have

zrp

jkz

qUz

kj

zj

eU

/

2

)( )}({)2

exp(

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Dr. Gao-Wei Chang

• 4.4.3 Thin Sinusoidal Amplitude Grating

• In practice, diffracting objects can be far more complex. In accord with our earlier definition (3-68),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

aperturetheout

aperturetheint A

0

1),(

Binary transmission (amplitude grating)

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Dr. Gao-Wei Chang

• Spatial patterns of phase shift can be introduced by means of transparent plates of varying thickness, thus extending the realizable values of tA to all points within or on the unit circle in the complex plane.

• As an example of this more general type of diffracting screen, consider a thin sinusoidal amplitude grating defined by the amplitude transmittance function

wrect

wrectf

mt A 22

2cos22

1, 0

(4-33)

where for simplicity we have assumed that the grating structure is bounded by a square aperture of width 2w. The parameter m represents the peak-to-peak change of amplitude transmittance across the screen,and f0 is the spatial frequency of the grating.

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Dr. Gao-Wei Chang

• 4.4.4 Thin sinusoidal phase grating

or x) (Binary phase grating

)2

()2

()()]2(sin

2[ 0

w

ηrect

w

ξrecteξ,yU

ξπfm

j

Documents

Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Dr. Gao-Wei Chang 1 Chap 4 Fresnel and Fraunhofer Diffraction