Page 301 - Phase Space Optics Fundamentals and Applications

282 Chapter Nine

angle , where = sin , with being the wavelength of the coherent
wavefront.
The WDF can be regarded as a generalized ray distribution. If we
interpret each of the two plane waves as a bundle of rays, where each
bundle has a different propagation angle, we can map each plane
wave into the PSD. The two horizontal -lines, marked as 1 and 2
in Fig. 9.1b, correspond to the ray coordinates which we would in-
tuitively expect as the phase-space distribution of two plane waves.
The additional cosine modulated line at the intermediate frequency
1,2 is the so-called interference term or cross-term related to the bilin-
earity of the WDF. The fundamental period of the interference term
is represented as the period of the dashed line in the PSD. This ad-
ditional term ensures proper encoding of interference effects is not
considered by geometrical optics. It is also implicit that the inter-
ference term carries the information about the mutual coherence of
4
the two plane waves. For mutually incoherent waves the interfer-
ence term of the WDF vanishes, and for partially coherent signals
it is a weighted contribution related to the degree of mutual coher-
ence of the two plane waves. While not corresponding to rays in a
geometrical optics sense, the phase-space points associated with in-
terference terms behave exactly as points associated with ordinary
rays.
This means that the WDF allows us to study the phase space of rays
and how it changes as the light signal propagates through a paraxial
optical system. Then the same rules are applied to the generalized
phase-space distribution of the WDF to propagate wavefronts through
the optical system. Paraxial ray tracing is conveniently described with
matrix optics. In fact, matrix optics, which appears in many textbooks
(see, e.g., Ref. 5) is phase-space optics in disguise. Each optical element
or system can be represented by a 2 × 2 matrix, and the coordinates
transform according to

x A B x
= (9.7)
C D
out in
Thus any paraxial optical system that can be described by an ABC D
matrix amounts to a geometrical transformation of the WDF
W out (x, ) = W in (Ax + B ,Cx + D ) (9.8)

modifying the location of each point of the WDF, but not its value.
For instance, paraxial free-space propagation or Fresnel diffraction
corresponds to a shear of the WDF parallel to the x axis

W Fr (x, ) = W 0 (x − z , ) (9.9)
The Fourier dual operation is the phase modulation with a linear chirp
function, i.e., the function of a parabolic lens. For a convex lens of focal