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

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INVERSION OF THE ANGULAR SPECTRUM REPRESENTATION

85

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 6.1. Geometry for inverse diffraction.

where zr and z0 are the input and output z variables, and z0r ¼ z0 zr.

The Fraunhofer diffraction is governed by Eq. (5.4-2). Its inversion is given by

 

je jkz0r

1 1

 

k

2

2

 

2p

 

Uðx; y; zrÞ ¼

 

ð ð

Uðx0; y0; z0Þe j

 

ðx0

þy0

Þe j

 

ðx0xþy0yÞdx0dy0 ð6:2-2Þ

 

2z0r

lz0r

lz0r

 

 

1 1

 

 

 

 

 

 

 

6.3 INVERSION OF THE ANGULAR SPECTRUM REPRESENTATION

Below the method is first discussed in 2-D. Then, it is generalized to 3-D. Equation (4.3-12) in 2-D can be written as

 

1

 

 

Uðx0; z0Þ ¼

1

ð6:3-1Þ

ð

Aðfx; zrÞejz0r pk2 4p2fx2 e j2pfxx0 dfx

As this is a Fourier integral, recovery of Aðfx; zrÞ involves the computation of the Fourier transform of Uðx0; z0Þ:

 

1

Uðx0; z0Þe j2pfxx0 dx0

 

1

ð6:3-2Þ

Aðfx; zrÞ ¼ e jz0r pk2 4p2fx2

ð

Uðx; zrÞ is the inverse Fourier transform of Aðfx; zrÞ:

Uðx; zrÞ ¼

1

2 1

Uðx0; z0

Þe j2pfxx0 dx03e jz0r pk2

4p2fx2 e j2pfxxdfx

ð6:3-3Þ

 

ð

ð

 

 

 

 

1

41

 

5

 

 

With F[.] and F 1[.] indicating the forward and inverse Fourier transforms, Eq. (6.3-3) can be expressed as

h

pi

ð6:3-4Þ

Uðx; zrÞ ¼ F 1 F½Uðx0; z0Þ&e jz0r

k2 4p2fx2
1
p
When 2pfx > k, the propagator in the last integral becomes ekz0r 4p2fx2 k2 , and the integral in Eq. (6.4-2) diverges. This is due to the inclusion of evanescent waves. In practice, it can be shown that the results are sufficiently accurate when the limits of
1ð

86

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

INVERSE DIFFRACTION

In 3-D, the corresponding equation is given by

 

 

 

 

U

ð

x

; ;

rÞ ¼

F

 

h

½

U

ð

x

0;

y

0;

z

0Þ&

i

ð :

3-5

Þ

 

y z

 

 

1

F

 

 

 

e jz0r pk2 4p2ðfx2þfy2Þ

6

 

Note that zr can be varied above to reconstruct the wave field at different depths. In this way, a 3-D wave field reconstruction is obtained.

If measurements are made with nonmonochromatic waves, resulting in a wave field Uðx0; y0; z0; tÞ, where t indicates the time dependence, a single time frequency component Uðx0; y0; z0; f Þ can be chosen by computing

 

1

 

 

 

 

Uðx0; y0; z0; f Þ ¼

ð

Uðx0; y0; z0; tÞe j2pftdt

ð6:3-6Þ

1

 

 

 

 

Then, the equations discussed above can be used with k given by

 

k ¼

 

2p

¼ 2p

f

ð6:3-7Þ

 

l

v

 

where v is the phase velocity. The technique discussed above was used with ultrasonic and seismic image reconstruction [Boyer, 1971; Boyer et al., 1970; Ljunggren, 1980]. It is especially useful when z0r is small so that other approximations to the diffraction integral, such as Fresnel and Fraunhofer approximations discussed in Chapter 5, cannot be used.

6.4 ANALYSIS

Interchanging orders of integration in Eq. (6.4-3) results in

1ð

Uðx; zrÞ ¼ Uðx0; z0ÞBðx; x0Þdx0

ð6:4-1Þ

1

where

p

Bðx; x0Þ ¼

e jz0r k2 4p2fx2 e j2pfxðx x0Þdfx

ð6:4-2Þ

ANALYSIS

87

integration are restricted such that k > 2pfx [Lalor, 1968]. Then, fx is restricted to the range

1

ð6:4-3Þ

jfxj l

Suppose that the exact wave field at z ¼ zr is T(x,zr). How does U(x,zr) as computed above compare to T(x,zr)? In order to answer this question, we can first determine Uðx0; z0Þ in terms of T(x,zr) by forward diffraction. This is given by

 

 

Uðx0; z0Þ ¼

 

ð

 

 

ð

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ð6:4-4Þ

 

 

 

M 2

1 Tðx; zrÞe j2pfxxdx3ejz0r pk2 4p2fx2 ej2pfxx0 dfx

 

 

 

 

 

 

 

 

 

 

M

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

41

 

 

 

 

 

 

5

 

 

 

 

 

 

 

 

where jfxj M is used. The upper bound for M is 1/l.

 

 

 

 

 

 

Substituting this result in Eq. (6.4-3) and allowing fx to the range fx Q results in

 

 

Uðx; zrÞ ¼

ð

 

 

ð

2

ð

 

 

 

 

 

 

 

 

 

9

 

 

 

 

Q

 

8 M

1 Tðx; zrÞe j2pfx0xdx3ejz0r pk2 4p2fx02 ej2pfxx0 dfx0

 

 

 

 

 

 

 

 

Q

<

 

M

 

 

 

 

 

 

 

 

5

 

 

=

 

 

 

 

 

 

 

 

 

:

 

 

 

 

 

 

 

x

 

 

 

 

;

 

 

 

 

 

 

 

 

 

 

41

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

e jz0r pk2 4p2fx2 ej2pfxxdf

 

 

 

 

 

 

 

 

 

 

 

 

Interchanging orders of integration, this can be written as [Van Rooy, 1971]

 

 

 

ð

ð

 

dfx0e j2pfx0xejz0r pk2

4p2fx0

2

 

 

 

 

 

 

 

 

 

Uðx; zrÞ ¼ 1

2 M

 

 

 

 

 

 

 

 

 

 

 

 

 

1

4

M

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 Q

dfxe j2pfxxe jz0r pk2 4p2fx2

 

1

dx0ej2px0ðfx0 fxÞ 13T

 

x; zr

dx

 

 

 

 

 

 

 

 

ð

 

 

 

 

 

 

 

2

ð

3

 

ð

 

Þ

 

 

 

 

 

 

B Q

 

 

 

 

 

 

4

1

C7

 

 

 

 

 

 

 

 

 

 

@

 

 

 

 

 

 

 

 

 

 

 

 

5A5

 

 

 

 

 

As

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dðfx fx0Þ ¼

ð

 

ej2pxðfx0 fxÞdx

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

Eq. (6.4-4) becomes

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

U x; zr

1 2 Me j2pfx0x0 ejz0r pk2 4p2fx02 dfx0 Qd fx fx0 ej2pfxxe jz0r pk2 4p2fx2 dfx3T x; zr dx

ð

 

16 M

 

 

 

 

 

 

 

Q

 

 

 

 

 

 

7

ð

 

Þ

 

Þ ¼ð

ð

 

 

 

 

 

 

 

 

 

 

ð

ð Þ

 

 

5

 

 

 

 

4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

88

INVERSE DIFFRACTION

Depending on the values of M and Q, there are two possible cases that are discussed below.

Case I: Q M

In this case, Eq. (6.4-5) reduces to

Uðx; zrÞ ¼ 1

2 M

ej2pfx0ðx x0Þdfx03Tðx; zrÞdx

ð

ð

 

 

 

 

 

1

4 M

 

 

5

 

¼ Tðx; zrÞ sin c

2Mx

 

 

l

 

¼ Tðx; zrÞ

 

 

 

 

 

because T(x,zr) is bandlimited.

 

 

 

 

 

 

Case II: Q < M

 

 

 

 

 

 

In this case, Eq. (6.4-12) reduces to

 

 

 

 

 

Uðx; zrÞ ¼ Tðx; zrÞ sin c

Qx

 

2

l

ð6:4-5Þ

ð6:4-6Þ

This means lowpass filtering of T(x,zr) because the Fourier transform of the sin c function is the rectangular function. In other words, frequencies above Q=l are filtered out. The reconstructed image may have smoothed edges and loss of detail as a consequence.

As Q is restricted to the values Q 1=l in practice, the resolution achievable in the image is l. This is in agreement with other classical results that indicate that linear imaging systems cannot resolve distances less than a wavelength.

EXAMPLE 6.1 Determine

U(x,zr)

when

Uðx0; z0Þ is measured in

an aperture

jx0j R.

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution: In this case, Eq. (6.4-3) becomes

 

 

 

ð

x; zr

Þ ¼

ð

8e jkz0r pk2 4p2fx2

ð

dx0e j2pfxx0

 

 

U

 

Q

R

 

 

 

 

 

 

Q

<

 

 

 

R

 

dxTðx; zrÞe j2pfx0x3319

 

 

 

 

 

 

02

 

dfx0ej2pfx0x0 2

1

 

dfx

 

 

 

 

:

M

 

 

 

ð

=

j2pfxx

 

 

 

 

 

@4

ð

 

4

 

 

 

 

 

 

 

 

 

 

55A

 

 

 

 

 

 

 

 

 

 

 

 

 

;

 

 

M 1

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