Page 189 - Phase Space Optics Fundamentals and Applications

170 Chapter Five

road map for visualizing the basic integral transformations that define
phase-space representations. See Chaps. 1 and 2 of this book for the
conceptual developments sketched in Fig. 5.4.

5.3 Optical Imaging Systems
Optical imaging devices are commonly analyzed using linear system
theory. Under coherent illumination, a space-invariant optical system
maps linearly the complex amplitude distribution at the input u 0 (x)
into the complex amplitude distribution at the output u(x). If the input
isapinhole-sizesource,thenthecomplexamplitudedistributionatthe
output is the coherent point-spread function, or the coherent impulse
response, q(x). Hence, if one has a space-invariant optical system, then
the imaging process is represented by the convolution integral

∞

u(x) = q(x − x 0 )u 0 (x 0 ) dx 0 (5.8)
−∞

In what follows we explore the use of the WDF for linking ray optics
with wave optics. To that end, we rephrase the linear mapping, in Eq.
(5.8), in terms of the WDF of the input, W 0 (x, ), and the WDF of the
impulse response, W q (x, ). Then we rewrite Eq. (5.8) as

∞

W(x, ) = W 0 (x 0 , )W q (x − x 0 , ) dx 0 (5.9)
−∞
Now, if the input is a pinhole-size source u 0 (x) = (x), then in the
paraxial regime the WDF of the input represents a bundle of rays with
the same amplitude, for any possible angle = , that is, W 0 (x, ) =
(x). Then, according to Eq. (5.9), the output WDF is, W q (x, ).By
adding the amplitudes of any possible ray, associated to the output
WDF, we obtain the output irradiance distribution

∞

I (x) = W q (x, ) d (5.10)
−∞
We illustrate the use of Eq. (5.10) by considering the WDF of a clear
pupil aperture with cutoff spatial frequency

W q (x, ) = 4( −| |) sinc[4( −| |)x] rect (5.11)
2