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

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FREQUENCY RESPONSE OF A DIFFRACTION-LIMITED IMAGING SYSTEM

167

where Hð fx; fyÞ is the 2-D FT of the impulse response:

Hð fx; fyÞ ¼

ðð

hðx; yÞe j2pð fxxþfyyÞdxdy ¼ Pðld0 fx; ld0 fyÞ

ð10:8:1-2Þ

1

1

Hð fx; fyÞ is known as the coherent transfer function.

It is observed that a coherent imaging system is equivalent to an ideal low-pass filter, which passes all frequencies within the pupil function’s ‘‘1’’ zone and cuts off all frequencies outside this zone.

10.8.2Incoherent Imaging System

Incoherent imaging systems are linear in intensity. The visual quality of an image is largely determined by the contrast of the relative intensity of the informationbearing details of the image to the ever-present background. The output image and the input ideal image can be normalized by the total image energy to reflect this property:

I0ðx0

; y0; d0

Þ ¼

1

Iðx0; y0; d0Þ

ðð Iðx0; y0; d0Þdx0dy0

1

ð10:8:2-1Þ

IG0 ðx; yÞ ¼

1

IGðx; yÞ

ðð IGðx; yÞdxdy

1

Let us denote the 2-D FT of I0ðx0; y0; d0Þ and IG0 ðx; yÞ by Jð fx; fyÞ and JGð fx; fyÞ, respectively. By convolution theorem, Eq. (10.7.2-2) can be written as

Jð fx; fyÞ ¼ HI ð fx; fyÞJGð fx; fyÞ

ð10:8:2-2Þ

where

ðð

jhðx; yÞj2e j2pð fxxþfyyÞ

1

HI ð fx; fyÞ ¼

1 1

dxdy

ð10:8:2-3Þ

ðð

jhðx; yÞj2dxdy

1

HI ð fx; fyÞ is called the optical transfer function (OTF). The modulation transfer function (MTF) is defined as jHI ð fx; fyÞj.

168

IMAGING WITH QUASI-MONOCHROMATIC WAVES

It is observed that HI ðfx; fyÞ is the normalized FT of jhðx; yÞj2, a nonnegative function. By Property 15 of the FT and Parseval’s theorem discussed in Section 2.5, HI ð fx; fyÞ is the normalized autocorrelation of Hð fx; fyÞ:

ðð

Hð fx0; fy0

ÞH ð fx þ fx0; fy þ fy0Þdfx0dfy0

1

HI ð fx; fyÞ ¼

1

1

ð10:8:2-4Þ

ðð

jHð fx; fyÞj2dfxdfy

1

The most important properties of the OTF are the following:

A.HI ð0; 0Þ ¼ 1

B.HI ð fx; fyÞ ¼ HI ð fx; fyÞ

C.jHI ð fx; fyÞj HI ð0; 0Þ

The last property is a consequence of Schwarz’ inequality, which states that for

any two complex-valued functions f and g,

ðð

fgdA 2 ðð j f j2dA ðð

jgj2dA

ð10:8:2-5Þ

with equality iff g

¼

Kf

where K is a complex constant.

Letting f and g be equal to Hð fx0; fy0Þ and H ð fx þ fx0; fy þ fy0Þ, respectively, and using Eq. (10.8.2-5) yields Property C above.

The coherent transfer function Hð fx; fyÞ is given by Eq. (10.8.1-2). Using this

result in Eq. (10.8.2-5) gives

ÞPðld0ð fx þ fx0ÞÞ; ld0ð fy þ fy0Þd fx0d fy0

ðð

Pðld0 fx0; ld0fy0

1

HI ð fx; fyÞ ¼

1

1

ð10:8:2-6Þ

ðð

Pðld0 fx; ld0 fyÞd fxd fy

1

where the fact P2 ¼ P is used in the denominator.

Incorporating a change of variables, Eq. (10.8.2-6) can be written as

ðð

P fx0

þ l

2 ; fy0

þ l 2

P fx0 l 2 ; fy0 l 2

d fx0d fy0

1

d0 fx

d0 fy

d0 fx

d0 fy

HI ð fx; fyÞ ¼

1

1

ðð Pðld0 fx0; ld0 fy0Þ

1

ð10:8:2-7Þ

FREQUENCY RESPONSE OF A DIFFRACTION-LIMITED IMAGING SYSTEM

169

The two pupil functions in the numerator above are displaced from each other by ðld0j fxj; ld0j fyjÞ. The integral equals the area of overlap between the two pupil functions. Hence, HI ðfx; fyÞ can be written as

HI ð fx; fyÞ ¼

area of overlap

ð10:8:2-8Þ

total area

where the areas are computed with respect to the scaled pupil function. The OTF is always real and nonnegative.

Note that the incoherent impulse response function jhðx; yÞj2 is similar to the power spectrum of a stationary 2-D random field. By the same token, HI ð fx; fyÞ is similar to the autocorrelation function of a 2-D stationary random field [Besag, 1974].

EXAMPLE 10.8 (a) Determine the OTF of a diffraction-limited optical system whose exit pupil is a square of width 2W, (b) Determine the cutoff frequency fc of the system.

Solution: The area of the pupil function equals 4W2. The area of overlap is illustrated in Figure 10.7.

The area of overlap is computed from Figure 10.7 as

A

fx; fy

8ð2W ld0j fxjÞð2W ld0j fyjÞ

fx

2W=ld0

ð

Þ ¼

j fyj

2W=ld0

>

j j

>

2

otherwise

<

0

When A

ð

f

Þ

is

normalized by 4W , the OTF is given by

x; fy

:

HI ðfx; fyÞ ¼ tri

fx

tri

fy

2fc0

2fc0

where tri( ) is the triangle function, and fc0 is the cutoff frequency for coherent illumination, equal to W=ld0.

(b) It is obvious that the cutoff frequency fc is given by

fc ¼ 2 fc0

Figure 10.7. The area of overlap for the computation of the OTF of a square aperture.

170

IMAGING WITH QUASI-MONOCHROMATIC WAVES

Figure 10.8. The aperture function for Example 10.9.

EXAMPLE 10.9 An exit aperture function consists of two open squares as shown in Figure 10.8.

Determine

(a)the coherent transfer function

(b)the coherent cutoff frequencies

(c)the amplitude impulse response

(d)the optical transfer function

Solution: (a) The coherent transfer function is the same as the scaled aperture function. Mathematically, the aperture function can be written as

P

x; y

rect

x 2s

;

y

rect

x þ 2s

;

y

Þ ¼

2s þ

2s

ð

2s

2s

Hð fx; fyÞ is given by

Hð fx; fyÞ ¼ Pðld0 fx; ld0 fyÞ

l

d0 fx

2s d0 fy

l

d0 fx

2s d0 fy

¼ rect

þ

;

l

þ rect

;

l

2s

2s

2s

2s

(b) The cutoff frequencies along the two directions are given by

fxc ¼

3s

;

fyc

¼

s

ld0

ld0

(c) The amplitude impulse response hðx; yÞ is the inverse Fourier transform of Hð fx; fyÞ. Let a be equal to ld0s. hðx; yÞ is computed as

hðx; yÞ ¼

ðð

Hð fx; fyÞ ¼ ej2pð fxxþfyyÞdfxdfy

1

1

¼ 2

3s

cosð2p fxxÞdfx 2

s

ð0

ð0

cosð2p fyyÞd fy

¼

1 sinð6psxÞ sinð2psyÞ

p2

x

y

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