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

Figure 10.5. Model of an optical imaging system.

systems as well as to quasi-monochromatic sources with spatially coherent and spatially incoherent illumination.

A general imaging system typically consists of a number of lenses. Such an imaging system can be characterized in terms of an entrance pupil and an exit pupil, which are actually both images of an effective system aperture [Goodman]. This is shown in Figure 10.5. Diffraction effects can be expressed in terms of the exit pupil and the distance d0 between the exit pupil and the image plane.

An optical system is called diffraction limited if a diverging spherical wave incident on the entrance pupil is mapped into a converging spherical wave at the exit pupil. For a real imaging system, this property is at best limited to finite areas of object and image planes. Aberrations are distortions modifying the spherical property. They are discussed in Section 10.9.

In a general imaging system with many lenses, Eqs. (9.4-17) and (9.4-18) remain valid provided that P(.,.) denotes the finite equivalent exit pupil of the system, the equivalent focal length of the system and the distance from the exit pupil to the image plane is used, and the system is diffraction limited [Goodman].

where Uðx; yÞ is the original image (wave field). Equation (9.4-16) is rewritten as

10.7IMAGING WITH QUASI-MONOCHROMATIC WAVES

In practical imaging systems, the conditions for quasi-monochromatic wave fields are usually satisfied. For example, in ordinary photography, films are sensitive to the visible range of the electromagnetic temporal spectrum. Then, it can be shown that the complex envelope of the analytic signal satisfies the imaging equations derived previously. Thus, Eq. (10.6-2) can be written as

UAðx0; y0; d0; tÞ ¼ hðx0; y0Þ UGðx0; y0; t tÞ

where hðx0; y0Þ is given by Eq. (10.6-3) except that l is replaced by lc, and UGðx0; y0; tÞ is the complex envelope of the analytic signal corresponding to the ideal image Uð xM0 ; yM0 ; tÞ predicted by geometrical optics; t is a time delay associated with propagation to the image plane.

The amplitude transfer function Hðfx; yyÞ is the FT of hðx; yÞ:

Detector systems are sensitive to intensity of the wave field, which can be written as

where h i indicates an infinite time average. Using Eq. (10.7.1), Iðx0; y0; d0Þ can be written as

¼hðx0 x; y0 yÞ h ðx0 x0; y0 y0ÞIGðx; y; x0; y0Þdx0dy0 dxdy

IG is known as the mutual intensity. In Eq. (10.7-5), t1 and t2 are approximately equal because the impulse response h is limited to a small region around the image point and hence can be neglected.

10.7.1Coherent Imaging

For perfectly coherent wave fields, we can write

where UGðx; yÞ is the phasor amplitude of UGðx; y; tÞ relative to the wave field at the origin.

We write

1

ð10:7:1-5Þ

Thus, the coherent imaging system is linear in complex amplitude of the analytic signal relative to the origin.

EXAMPLE 10.7 Determine the cutoff frequency of a diffraction-limited coherent imaging system with a circular effective pupil of radius R, assuming that the image forms at a distance d0 from the system.

Solution: The pupil function in this case is given by

p!

x2 þ y2

Pðx; yÞ ¼ circ

R

The amplitude transfer function is given by