Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf

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58

SCALAR DIFFRACTION THEORY

P0

r

01

 

 

 

r01

P0

A

 

 

 

 

 

 

 

 

 

n

P1

Figure 4.9. Rayleigh–Sommerfeld modeling of diffraction by a plane aperture.

Green’s function:

 

e

jkr01

 

e

jkr01

 

G2ðrÞ ¼

 

 

 

 

ð4:6-1Þ

 

 

 

 

r01

 

r01

where r01 is the distance from P0 to P1, P0 being the mirror image of P0 with respect

to the initial plane. This is shown in Figure 4.9.

The use of the second Green’s function leads us to the first Rayleigh–Sommerfeld diffraction formula given by

Uðx0; y0; zÞ ¼ j1l

ðð

Uðx; y; 0Þ r01 er01

dxdy

ð4:6-2Þ

 

 

1

 

z jkr01

 

 

 

 

1

 

 

 

 

 

 

Another derivation of this equation by using the convolution theorem is given in the next section.

Equation (4.6-2) shows that Uðx0; y0; zÞ may be interpreted as a linear superposition of diverging spherical waves, each of which emanates from a point

ðx; y; 0Þ and is weighted by 1 z Uðx; y; 0Þ. This mathematical statement is also

jl r01

known as the Huygens–Fresnel principle.

It is observed that Eq. (4.6-2 ) is a 2-D linear convolution with respect to x,y:

Uðx0; y0; zÞ ¼ Uðx; y; 0Þ hðx; y; zÞ

ð4:6-3Þ

where the impulse response hðx; y; zÞ is given by

 

 

 

 

 

 

 

 

 

jkz 1

 

2

2

1=2

 

 

 

1

 

e

þ

x

þy

 

 

 

 

 

 

 

 

z2

 

 

 

 

hðx; y; zÞ ¼ jlz

 

 

þ

 

 

z2

y2

 

 

ð4:6-4Þ

1

x2

þ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The transfer function Hð f1; f2; zÞ, the 2-D Fourier transform of hðx; y; zÞ, is given by

Hð fx; fy; zÞ ¼ e jz½k2 4p2ð fx2þfyÞ&1=2

ð4:6-5Þ

ANOTHER DERIVATION OF THE FIRST RAYLEIGH–SOMMERFELD

59

This is the same as the transfer function for the propagation of angular spectrum of plane waves discussed in Section 4.5. Thus, there is complete equivalence between the Rayleigh–Sommerfeld theory of diffraction and the method of the angular spectrum of plane waves.

4.6.1The Kirchhoff Approximation

Incorporating the Kirchhoff approximation into the first Rayleigh–Sommerfeld integral yields

Uðx; y; zÞ ¼ jl ððA

Uðx; y; 0Þ r01

r01

dxdy

ð4:6-6Þ

1

 

 

z

e jkr01

 

 

z=r01 is sometimes written as cosðn; r01Þ, indicating the cosine of the angle between the z-axis and r01 shown in Figure 4.9.

4.6.2The Second Rayleigh–Sommerfeld Diffraction Formula

Another valid Green’s function that can be used instead of G2ðrÞ is given by

 

 

 

 

e jkr01

 

 

e jkr010

 

 

 

 

 

 

 

 

 

G3ðrÞ ¼

 

 

 

þ

 

 

 

 

 

 

 

 

ð4:6-7Þ

 

 

U2

r01

 

r010

 

Þ

 

r01

 

 

 

ðx; y; zÞ ¼ 2p ððA

d

 

 

ðd;n ;

 

 

dxdy

ð4:6-8Þ

 

 

 

 

1

 

 

 

 

U

x y

0

 

e jkr01

 

 

 

dUðx;y;0Þ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

where

is the partial derivative of Uðx; y; 0Þ in the normal direction (z-direction).

dn

It can be shown that the Kirchoff solution discussed in Section 4.5 is the arithmetic average of the first and second Rayleigh–Sommerfeld solutions [Goodman, 2004].

4.7 ANOTHER DERIVATION OF THE FIRST RAYLEIGH–SOMMERFELD DIFFRACTION INTEGRAL

We assume that the field UðrÞ is due to sources in the half space z0 and that it is known in the plane z ¼ 0. We want to determine UðrÞ for z0. Uðx; y; 0Þ in terms of its angular spectrum is given by

Uðx; y; 0Þ ¼

ðð

Aðfx; fy; 0Þej2pðfxxþfyyÞdfxdfy

ð4:7-1Þ

 

1

 

 

 

1

 

 

In terms of the convolution theory, this equation can be interpreted as finding the output of a LSI system whose transfer function is unity. The impulse response

60 SCALAR DIFFRACTION THEORY

of the system is given by

ðð

 

ð4:7-2Þ

hðx; y; 0Þ ¼

ej2pðfxxþfyyÞdfxdfy

 

1

 

 

1

What if z ¼6 0? hðx; y; zÞ consistent with Eq. (4.7-1) and other derivations of

diffraction integrals as in Section 4.6 is given by

ð4:7-3Þ

hðx; y; zÞ ¼ 4p2

ðð

e jk rdkxdky

1

1

 

 

 

1

 

 

where k and r are given by Eqs. (3.2-28) and (3.2-30), respectively. Equation (4.7-3) can be written as

 

p

 

 

 

 

p

ðð

 

z

 

 

 

 

1

@

2

 

j

1 e jk

r

 

hðx; y; zÞ ¼

 

 

 

 

 

 

 

dkxdky&

ð4:7-4Þ

2 @z

2

 

1

k

 

 

 

 

 

4

 

 

 

 

 

 

 

The quantity inside the brackets can be shown to be the plane wave expansion of a spherical wave [Weyl]:

 

er

 

¼ 2jp

ðð ekz

r

dkxdky

ð4:7-5Þ

 

jkr

1

 

jk

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

Substituting this result in Eq. (4.7-4) yields

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

@

 

 

e jkr

 

 

 

hðx; y; zÞ ¼

 

 

 

 

 

 

 

 

 

ð4:7-6Þ

 

2p

@z

r

The above equation is the same as

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

hðx; y; zÞ ¼

 

1 z e jkr

 

ð4:7-7Þ

 

 

jl

 

r

 

r

 

 

The diffracted field Uðx; y; zÞ is interpreted as the convolution of Uðx; y; 0Þ with hðx; y; zÞ. Thus,

Uðx; y; zÞ ¼ jl

ðð

Uðx; y; 0Þ r01

 

r01

dxdy

ð4:7-8Þ

1

1

 

z

e jkr01

 

 

 

 

1

 

 

 

 

 

 

Using Eq. (4.8-6), this result is sometimes written as

1 1

 

d

 

jkr

 

 

 

Uðx; y; zÞ ¼

1

ð

ð

Uðx; y; 0Þ

 

e

01

dxdy

ð4:7-9Þ

2p

dz

r01

 

1

1

 

 

 

 

 

 

 

THE RAYLEIGH–SOMMERFELD DIFFRACTION INTEGRAL

61

4.8 THE RAYLEIGH–SOMMERFELD DIFFRACTION INTEGRAL FOR NONMONOCHROMATIC WAVES

In the case of nonmonochromatic waves, Uðr; tÞ is represented in terms of its Fourier transform Uf ðr; f Þ ¼ Uf ðx; y; z; f Þ as in Eq. (4.2-5). By substituting f 0 ¼ f , this equation can also be written as

 

1

 

 

Uðx; y; z; tÞ ¼

ð

Uf ðx; y; z; f 0Þe j2pf 0tdf 0

ð4:8-1Þ

 

1

 

 

Uf ðx; y; z; f 0Þ satisfies the Rayleigh–Sommerfeld integral at an aperture:

Uf ðx; y; z;

f 0Þ ¼ jl ððA

Uf ðx; y; 0;

 

f 0Þ r01

r01

dxdy

ð4:8-2Þ

 

 

 

 

1

 

 

 

 

 

 

z

e jkr01

 

 

Substituting this result in Eq. (4.8-1) yields

 

 

 

 

 

 

 

 

Uðx; y; z; tÞ ¼

ð

 

l

ðð

 

 

01

01

dxdy&e

 

ð4:8-3Þ

1

2 j1

Uf ðx; y; 0; f 0

Þ rz

erjkr01

j2pf 0tdf 0

 

1

4

 

A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Note that

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lf ¼ c

 

 

 

 

 

 

 

ð4:8-4Þ

where c is the phase velocity of the wave. Using this relation and exchanging orders of integration in Eq. (4.9-3) results in

Uðx; y; z; tÞ ¼ ðð 2pcr012

2 ð

 

j2pf 0Uf ðx; y; 0;

f 0Þe j2pf 0ðt c Þdf 03dxdy ð4:8-5Þ

 

 

 

z

 

1

 

 

 

 

 

 

r01

 

 

A

4

1

 

 

 

 

 

 

 

5

As

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

d

 

 

d

ð

 

f 0Þe

j2pf 0tdf 0

 

 

 

 

Uðx; y; 0; tÞ ¼

 

 

Uf ðx; y; 0;

 

 

 

dt

dt

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

¼

ð

 

j2pf 0Uf ðx; y; 0;

f 0Þe j2pf 0tdf 0

1

62

 

 

 

 

 

 

 

 

 

SCALAR DIFFRACTION THEORY

Eq. (4.8-5) can also be written as

 

 

 

 

 

 

 

 

 

 

 

 

ð

Þ ¼

 

ðð

 

 

 

h

ð

 

 

 

ð

Þ

 

2pc

r012

dt

 

c

U x; y; z; t

 

z

A

 

 

 

i

4:8-6

 

 

 

1 d

U x; y; 0; t

r01 dxdy

 

Thus, the wave at ðx; y; zÞ, z0, is related to the time derivative of the wave at ðx; y; 0Þ with a time delay of r01=c, which is the time for the wave to propagate from P1 to P0.

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