Page 203 - Phase Space Optics Fundamentals and Applications

184 Chapter Five

f f f f

FIGURE 5.10 Optical setup for using a phase conjugate pair, at the Fourier
plane.

Now, let us suppose that the pupil mask is formed with a pair of con-
jugate phase elements, and that these elements are laterally displaced
in opposite directions, say, by the spatial frequency /2 along the hor-
izontal axis. Then the complex amplitude transmittance of the pupil
mask can be expressed in terms of the product spectrum representa-
tion P T ( , )as

S( , ; ) = T + T ∗ − rect
2 2 2

= P T ( , ) rect (5.51)
2
Equation (5.51) tells us how a generalized pupil function can be syn-
thesized from a pair of conjugate phase elements. To find the cor-
responding synthesized PSF, we take the two-dimensional inverse
Fourier transform of S( , ; ). Except for the normalization factor
1/(2 ), this gives

⎡ ⎤
∞

s(x, y; ) = sinc (2 y) ⎣ T + T ∗ − exp (i2 x ) d ⎦
2 2
−∞
(5.52)
The integral in Eq. (5.52) can be readily recognized as the AF, A T ( ,x),
of the mask T( ). Hence, the modulus of s(x, y; ) can be written as
|s(x, y; )|=|sinc (2 y)||A T ( ,x)| (5.53)