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

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138 FOURIER TRANSFORMS AND IMAGING WITH COHERENT OPTICAL SYSTEMS

Thus, the wave field at the focal plane of the lens is the 2-D Fourier transform of the wave field at z ¼ f if the effect of P(x,y) is neglected.

The limitation imposed by the finite lens aperture is called vignetting and can be avoided by choosing d1 small. To include the effect of the finite lens aperture, geometrical optics approximation can be used [Goodman]. With this approach, the initial wave field is approximated such that the finite pupil function at the lens equals

Pðx1 þ ðd1=f Þxf þ ðd1=f Þyf Þ. Then, Aðfx; fy; d1Þ with fx ¼ xf =lf and fy ¼ yf =lf is computed with respect to this pupil function. The wave field at z ¼ f becomes

Uðxf ; yf ; f Þ ¼ ej2f ð1 f Þðxf þyf Þ

ðð

Uðx; y; d1ÞP x þ

f1 xf ; y þ y1 yf e j l f ðxxf þyyf Þdxdy

 

k d1 2 2

1

 

d

d

2p

 

 

1

 

 

 

ð9:3-7Þ

 

 

 

 

 

 

9.3.3Wave Field to the Right of the Lens

The third possibility is to consider the initial wave field behind the lens, a distance d from the focal plane of the lens. This is the case when the lens is illuminated with a plane perpendicular wave field, and an object transparency is located at z ¼ f d as shown in Figure 9.4.

By geometrical optics, the amplitude of the spherical wave at this plane is proportional to f=d and can be neglected as constant. If ðx2; y2; f dÞ indicates a point on this plane, the equivalent pupil function on the object plane is Pðx2ðf =dÞ; y2ðf =dÞÞ due to the converging spherical wave. If Uðx2; y2; f dÞ is the wave field due to the object transparency, the total wave field at the object plane is given by

 

 

k

2

2

 

f

 

f

 

 

U0

ðx2; y2; f dÞ ¼ e j

2d

ðx2

þy2

ÞP x2

 

; y2

 

Uðx2; y2; f dÞ

ð9:3-8Þ

d

d

Let A0ðfx; fy; f dÞ be the angular spectrum of U0ðx2; y2; f dÞ. Provided that the Fresnel approximation is valid for the distance d, the angular spectrum of the wave field at the focal plane is given by

Aðxf ; yf ; f Þ ¼ A0ð fx; fy; f dÞe j

k

ðxf2þyf2Þ

ð9:3-9Þ

2d

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 9.4. The geometry for the input to the right of the lens.

IMAGE FORMATION AS 2-D LINEAR FILTERING

139

 

 

 

 

 

 

 

 

 

 

 

 

Figure 9.5. The geometry used for image formation.

where constant terms are neglected fx and fy are xf =ld and yf =ld. With this geometry, the scale of the Fourier transform can be varied by adjusting d. The size of the transform is made smaller by decreasing d. In optical spatial filtering, this is a useful feature because the size of the Fourier transform must be the same as the size of the spatial filter located at the focal plane [Goodman].

9.4IMAGE FORMATION AS 2-D LINEAR FILTERING

Lenses are generally known as imaging devices. Imaging by a positive, aberrationfree thin lens in the presence of monochromatic waves is discussed below.

Consider Figure 9.5 in which transparent object is placed on plane z ¼ d1, and illuminated by a wave so that the wave field is Uðx1; y1; d1Þ on this plane.

The wave field Uðx0; y0; d0Þ on the plane z ¼ d0 is to be determined. Then, under what conditions an image is formed is discussed.

Since wave propagation is a linear phenomenon, the fields at ðx0; y0; d0Þ and ðx; y; d1Þ can always be related by a superposition integral:

Uðx0; y0; d0Þ ¼

ðð

hðx0; y0; x1; y1ÞUðx1; y1; d1Þdx1dy1

ð9:4-1Þ

 

1

 

 

 

1

 

 

where hðx0; y0; x1; y1Þ is the impulse response of the system. In order to find h, Uðx; y; d1Þ will be assumed to be a delta function at ðx1; y1; d1Þ. This is physically equivalent to a spherical wave originating at this point.

We will assume that the lens at z ¼ 0 has positive focal length f. All the constant terms due to wave propagation will be neglected. The field incident on the lens within the paraxial approximation is given by

 

k

 

2

2

 

Uðx; y; 0Þ ¼ e j

 

ððx x1

Þ þðy y1

Þ Þ

ð9:4-2Þ

2d1

The field after the lens is given by

U0ðx; y; 0Þ ¼ Uðx; y; 0ÞPðx; yÞe j

k

ðx2þy2Þ

ð9:4-3Þ

2f

140 FOURIER TRANSFORMS AND IMAGING WITH COHERENT OPTICAL SYSTEMS

The Fresnel diffraction formula, Eq (5.2-13), yields

Uðx0; y0; d0Þ ¼ hðx0; y0; x1; y1Þ

¼

ðð

U0ðx; y; 0Þej 2dk0ððx0 xÞ2þðy0 yÞ2Þ dxdy

ð9:4-4Þ

 

1

 

 

 

 

 

1

 

 

 

 

In practical imaging applications, the final image is detected by a detector system that is sensitive to intensity only. Consequently, the phase terms of the form ejy can be neglected if they are independent of the integral in Eq. (9.4-4). Then, Eq. (9.4-4) can be simplified to

hðx0; y0; x1; y1Þ ¼

ðð

Pðx; yÞe j 2 ðx

þy

Þe jk d0

þd1

xþ d0

þd1 y dxdy

ð9:4-5Þ

 

1

kD 2

2

 

x0

 

x1

 

y0

 

y1

 

 

1

where

1

1

1

ð9:4-6Þ

D ¼

 

þ

 

 

 

d1

d0

f

Consider the case when D is equal to zero. In this case, Eq. (8.4-5) becomes

hðx0; y0; x1; y1Þ ¼

ðð

Pðx; yÞe jl2dp0 ½ðx0þMx1Þxþðy0þMy1Þy&dxdy

 

1

 

 

 

 

1

 

 

 

where

M ¼ d0 d1

If P(x,y) is neglected, Eq. (8.4-7) is the same as

ðð1

hðx0; y0; x1; y1Þ ¼ e j2p½ðx1þxM0Þx0þðy1þyM0Þy0&dx0dy0

1

where x0 ¼ x=ld0 and y0 ¼ y=ld0, and a constant term is dropped. Equation (9.4-9) is the same as

 

x0

 

 

y0

hðx0; y0; x1; y1Þ ¼ d x1 þ

 

; y1

þ

 

 

M

M

Using this result in Eq. (9.4-1) gives

Uðx0; y0; d0Þ ¼ U x0 ; y0 ; d1 M M

ð9:4-7Þ

ð9:4-8Þ

ð9:4-9Þ

ð9:4-10Þ

ð9:4-11Þ

IMAGE FORMATION AS 2-D LINEAR FILTERING

141

We conclude that Uðx0; y0; d0Þ is the magnified and inverted image of the field at z ¼ d1, and M is the magnification when D given by Eq. (9.4-6) equals zero. Then, d0 and d1 are related by

1

1

1

 

 

¼

 

 

 

ð9:4-12Þ

d0

f

d1

This relation is known as the lens law.

9.4.1The Effect of Finite Lens Aperture

Above the effect of the finite size pupil function P(x,y) was neglected. Letting

 

 

x10 ¼ Mx1

 

ð9:4-13Þ

 

 

y10 ¼ My1

 

ð9:4-14Þ

Equation (9.4-7) can be written as

 

 

 

 

 

 

 

1

 

 

 

 

 

hðx0; y0; x10 ; y10 Þ ¼ ðð Pðld0x; ld0yÞe j2p½ðx0 x10 Þxþðy0 y10 Þy&dxdy

ð

9:4-15

Þ

 

 

1

 

 

 

 

 

¼ hðx0 x10 ; y0 y10 Þ

 

 

 

 

 

Then, Eq. (8.4-1) can be written as

 

 

 

 

 

Uðx0; y0

; d0

Þ ¼ ðð hðx0 x10 ; y0 y10

ÞUðx10

; y10 ; d1Þdx10 dy10

ð9:4-16Þ

 

 

1

 

 

 

 

 

 

 

1

 

 

 

 

 

where a constant term is again dropped. Equation (9.4-16) is a 2-D convolution:

Uðx0; y0; d0Þ ¼ hðx0; y0Þ U

x0

 

y0

 

 

 

;

 

; d1

ð9:4-17Þ

M

M

where Uðð x0=MÞ; ð y0=MÞ; d1Þ is the ideal image, and

 

 

hðx0; y0Þ ¼

ðð

Pðld0x; ld0yÞe j2pðx0xþy0yÞdxdy

ð9:4-18Þ

 

1

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

The impulse response is observed to be the 2-D FT of the scaled pupil function of the lens. The final image is the convolution of the perfect image with the system

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