Des

in which F(x,y) is the pupil function defined in (3.488). Although formally the range of integration is infinite, the pupil function is, in fact, unity (normalized) within the area of the pupil and zero outside it. The autocorrelation function of (3.494) corresponds to a shear of the pupil of the amounts -ARs, -ARt in the directions x and y respectively, as shown in Fig. 3.108. Although the pupil is represented as a complete circle, the shear principle is equally valid for a pupil with vignetting or central obstruction. Clearly, if the vector shear in Fig. 3.108 reaches the diameter of the pupil, there is no common area and no signal at higher frequencies can be transmitted through the system, i.e. the contrast according to (3.483) is zero and the optical system is the equivalent of a low bandpass filter. In Fig. 3.109, the shear situation is represented in one dimension. The shear is ARs. Clearly, the limit frequency is given by

ARsmax D ;

where D is the diameter of the pupil, so that

where 'umax is the semi-angular aperture of the exit pupil and 2umax = 1/N, the reciprocal of the f/no. Now smax = 1/Amin, where Amin is the minimum wavelength of transmission of a sinusoidal spatial intensity function. Then

This result can be compared directly with the linear resolution based on the Rayleigh criterion of the radius of the first diffraction minimum given

Fig. 3.109. One-dimensional pupil shear to demonstrate the low bandpass filter function of an optical system

in (3.450). The results are identical except for the factor 1.220 in the latter case, which referred to the PSF for a circular aperture. Here we are dealing with the Line Spread Function, LSF. Referring to the case of the PSF at a rectangular aperture, Eq. (3.436) shows that the linear resolution corresponding to the radius of the first minimum is identical to (3.497).

Referring again to Fig. 3.109, we have cos 0

ARs D

The common area A of the sheared pupils, normalized to the area of the pupil, is

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