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

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68

 

 

FRESNEL AND FRAUNHOFER APPROXIMATIONS

where xb and xe are given by

 

r

 

 

 

 

 

 

 

 

 

 

 

 

 

D

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

xb ¼

 

 

 

 

 

 

 

þ x0

 

 

lz

2

 

 

 

 

 

r

 

 

 

 

 

 

 

 

 

 

xe ¼

2

 

D

þ x0

 

 

 

lz 2

 

 

 

Eq. (5.2-15) can be written as

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

xe

 

 

 

 

1

 

 

xb

 

 

 

p 2

 

 

p 2

ð5:2-17Þ

Ix ¼ p2

ð ej2v

dv p2

ð ej2v

dv

 

 

0

 

 

 

 

 

 

 

 

 

0

 

 

The Fresnel integrals C(z) and S(z) are defined by

 

 

 

 

 

z

 

 

 

 

 

 

 

v2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CðzÞ ¼ ð0

cos

p

 

dv

 

ð5:2-18Þ

2

 

 

 

 

 

z

 

 

 

 

 

 

 

v2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SðzÞ ¼ ð0

sin

p

dv

 

ð5:2-19Þ

 

2

 

In terms of C(z) and S(z), Eq. (5.2-17) is written as

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ð5:2-20Þ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ix ¼ p2 f½CðxeÞ CðxbÞ& þ j½SðxeÞ SðxbÞ&g

The integral above also occurs with respect to y in Eq. (5.2-15). Let it be denoted by Iy, which can be written as

 

 

ye

 

 

 

 

 

 

 

 

 

1

p 2

 

 

 

 

 

 

 

 

 

Iy ¼

p2

ð ej2v dv

 

 

 

 

 

 

 

 

 

 

1

yb

 

 

 

 

 

 

 

 

ð5:2-21Þ

 

 

 

 

 

 

 

 

 

 

 

 

¼ p2 f½Cðye CðybÞ& þ j½Sðye SðybÞ&g

where

 

r

 

 

 

 

 

 

D

 

 

yb ¼

2

 

 

þ y0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lz

2

 

 

 

r

D

 

 

 

 

 

Ye ¼

 

2

 

 

y0

 

 

lz

2

DIFFRACTION IN THE FRESNEL REGION

69

Equation (5.2-15) for Uðx0; y0; zÞ can now be written as

 

 

 

 

ejkz

 

 

 

 

Uðx0; y0; zÞ ¼

 

 

jxjy

ð5:2-22Þ

 

2j

The intensity Iðx0; y0; zÞ ¼ jUðx0; y0; zÞj2 can be written as

 

Iðx0; y0; zÞ ¼

1

f½CðxeÞ CðxbÞ&2 þ ½SðxeÞ SðxbÞ&2g

 

 

 

 

 

4

 

 

f½CðyeÞ CðybÞ&2 þ ½SðyeÞ SðybÞ&2g

ð5:2-23Þ

The Fresnel number NF is defined by

 

 

 

 

NF ¼

D2

ð5:2-24Þ

 

 

 

4lz

 

For constant D and l, NF decreases as z increases.

The Fresnel diffraction intensity from a square aperture is visualized in Figures 5.2–5.5. As z gets less, the intensity starts resembling the shape of the input square aperture. This is further justified in Example 5.5. Figure 5.2 shows the image of the square aperture used. Figure 5.3 is the Fresnel intensity diffraction pattern from the square aperture. Figure 5.4 is the same intensity diffraction pattern as a 3-D plot.

Figure 5.2. The input square aperture for diffraction studies.

70

FRESNEL AND FRAUNHOFER APPROXIMATIONS

Figure 5.3. The intensity diffraction pattern from a square aperture in the Fresnel region.

100

80

60

40

20

0

0.5

0

0.5

 

 

0

–0.5

–0.5

–1 –1

Figure 5.4. The same intensity diffraction pattern from the square aperture in the Fresnel region as a 3-D plot.

EXAMPLE 5.5 Using the Fresnel approximation, show that the intensity of the diffracted beam from a rectangular aperture looks like the rectangular aperture for small z.

Solution: The intensity pattern from the rectangular aperture is given by Eq. (5.2-22) where Dx and Dy are used instead of D along the x- and y-directions, respectively.

DIFFRACTION IN THE FRESNEL REGION

 

 

 

 

 

 

 

71

The variables for very small z can be written as

 

 

 

 

 

 

r

 

 

 

 

1

 

 

 

 

 

 

2

 

D

! (

 

 

x0 > Dx=2

xb ¼

 

 

lz

 

2

þ x0

1x0 < Dx=2

r

 

1

 

< Dx=2

 

 

2 D

 

 

 

 

x0

xe ¼

 

lz 2

x0

! ( 1 x0 > Dx=2

 

r

 

 

 

 

1

 

 

 

 

 

 

 

 

2 D

! (

1 y0 > Dy=2

yb ¼

 

 

lz

 

2

þ y0

 

y0 < Dy=2

r

 

1

 

< Dy=2

 

 

2

D

 

 

 

 

y0

ye ¼

 

lz

 

2

y0

! ( 1 y0 > Dy=2

The asymptotic limits of cosine and sine integrals are given by

Cð1Þ ¼ Sð1Þ ¼ 0:5; Cð 1Þ ¼ Sð 1Þ ¼ 0:5

Consequently, Eq. (5.2-22) becomes

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x0

rect

y0

 

 

 

 

Iðx0; y0; zÞ ’ rect

 

 

 

 

 

 

D

 

D

 

for small z.

EXAMPLE 5.6 Show under what conditions the angular spectrum method reduces to the Fresnel approximation.

Solution: The transfer function in the angular spectrum method is given by

 

 

 

(

p

q

HA

ð

fx; fy

Þ ¼

ejz k2 4p2ðfx2þfy2Þ

fx2 þ fy2 < l1

 

 

0

otherwise

The transfer function in the Fresnel approximation is given by

HFðfx; fyÞ ¼ ejkze jplzðfx2þfy2Þ

If

q

4p2ð fx2 þ fy2Þ

 

2

2

2

2

 

k

 

4p

ð fx

þ fy

Þ k

2k

that corresponds to keeping the first two terms of the Taylor expansion HAð fx; fyÞ equals HFð fx; fyÞ. The approximation above is valid when jlfxj, jlfyj 1. In conclusion, the Fresnel approximation and the angular spectrum method are approximately equivalent for small angles of diffraction.

72

FRESNEL AND FRAUNHOFER APPROXIMATIONS

5.3FFT IMPLEMENTATION OF FRESNEL DIFFRACTION

Fresnel diffraction can be implemented either as convolution given by Eq. (5.2-10) or by directly using the DFT after discretizing Eq. (5.2-13). The convolution implementation is similar to the implementation of the method of the angular spectrum of plane waves. The second approach is discussed below.

Let Uðm1; m2; 0Þ ¼ Uð xm1; ym2; 0Þ be the discretized u(x,y,0). Thus, x and y are discretized as xm1 and xm2, respectively. Similarly, x0 and y0 are discretized as x0n1 and x0n2, respectively. We write

U0ðm1; m2; 0Þ ¼ Uðm1; m2; 0Þej

k

½ð xm1Þ2þð ym2Þ2&

ð5:3-1Þ

2z

In order to use the FFT, the Fourier kernel in Eq. (5.2-13) must be expressed as

e j2lpzðx0xþy0yÞ ¼ e j2lpzð x x0n1m1þ y y0n2m2Þ ¼ e j2p

n m

1

 

n m

2

 

 

1

 

þ

2

ð5:3-2Þ

N1

 

N2

 

where N1 N2 is the size of the FFT to be used. Hence,

 

 

 

 

 

 

 

2p

2p

 

 

n1m1

 

 

 

 

 

 

 

 

x0x ¼

 

x x0n1m1 ¼ 2p

 

 

 

 

 

 

 

 

 

lz

lz

N1

 

 

 

 

 

 

This yields

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x x0 ¼

lz

 

 

 

 

 

 

 

 

 

ð5:3-3Þ

 

 

 

N1

 

 

 

 

 

 

 

 

Similarly, we have

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

y y0 ¼

lz

 

 

 

 

 

 

 

 

 

ð5:3-4Þ

 

 

 

N2

 

 

 

 

 

 

 

 

In other words, small x or y gives large x0 or y0 and vice versa, respectively. Equations (5.3-3) and (5.3-4) can be compared to Eq. (4.4-12) for the angular spectrum method. For fx ¼ x0=lz and fy ¼ y0=lz, these equations are the same. However, now the output is directly given by Eq. (5.2-3), there is no inverse transform to be computed, and hence the input and output windows can be of different sizes.

 

ð

 

 

Þ

 

2

2

 

In order to use the FFT, U0

 

m1

; m2; 0

 

is shifted by

N1

;

N2

, as discussed

previously in Section 4.4. Suppose the result is U00ðm1; m2; 0Þ.

 

 

 

The discretized output wave Uðn1; n2; zÞ ¼ Uð x0n1; y0n2; zÞ is expressed as

 

ejkz

 

k

2

2

 

X X

n1m1

 

n2m2

 

 

 

 

N1

1

N2

1

 

 

Uðn1; n2; zÞ ¼

jlz

ej

2z

½ð x0n1

Þ

þð y0n2Þ

& x y m1

¼

0

m2

¼

0 U00ðm1; m2; 0Þe j2p

N1

þ

N2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ð5:3-5Þ

Uðn1; n2; zÞ is shifted again by

N1

;

N2

in order to correspond to the real wave.

 

2

2

 

PARAXIAL WAVE EQUATION

73

EXAMPLE 5.7 Consider Example 4.4. With the same definitions used in that example, the pseudo program for Fresnel diffraction can be written as

u1 ¼ u exp j k ðx2 þ y2Þ

2z u1p ¼fftshiftðu1Þ

ejkz

u3 ¼ jlz ej2kzðx20þy20Þ u2

U¼fftshiftðu3Þ

5.4PARAXIAL WAVE EQUATION

The Fresnel approximation is actually a solution of the paraxial wave equation derived below. Consider the Helmholtz equation . If the field is assumed to be propagating mainly along the z-direction, it can be expressed as

Uðx; y; zÞ ¼ U0ðx; y; zÞejkz

ð5:4-1Þ

where U0ðx; y; zÞ is assumed to be a slowly varying function of z. Substituting Eq. (5.4-1) in the Helmholtz equation yields

r2U0

d

ð5:4-2Þ

þ 2jk dz U0 ¼ 0

As U0ðx; y; zÞ is a slowly varying function of z, Eq. (5.4-2) can be approximated as

d2U0 þ d2U0 þ 2jk d U0 ¼ 0 dx2 dy2 dz

Equation (5.4-3) is called the paraxial wave equation. Consider

1 jkðx2þy2Þ

U1ðx; yÞ ¼ jlz e 2z

ð5:4-3Þ

ð5:4-4Þ

It is easy to show that U1 is a solution of Eq. (5.4-3). As Eq. (5.4-3) is linear and shift invariant, U1ðx x1; y y1Þ for arbitrary x1 and y1 is also a solution. Superimposing

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