Page 289 - Introduction to Information Optics

274 5. Transformation with Optics

This is the original Wigner distribution function sheared in the x direction.
When an image is magnified by the optical system with a factor of s, fix )
becomes f(x/s), and the Wigner distribution function becomes

f
W f(x, (o) = I /((x + x'/2)/s)/((x - x'/2)/s) GXp(-wjx')dx'

= sWj-(x/s, soj).

The original Wigner distribution function is then dilated along the x-axis in
the space and is shrunk along the co-axis in the frequency.
In geometrical optics, the relation between the input and the output of an
optical system is commonly described by

where (c, v) and (x, co) are the position and the propagation direction of the
rays in the output and input planes, respectively, and M is the ray propagation
matrix. According to geometrical optics, optical systems, such as those involv-
ing the Fourier transform, lens, free space, and magnifier, have ray propagation
matrices M as
1 1
M,=
2n/// 1
~s 0 "
M. =
0 Ms

respectively. These optical systems implement affine transforms in the (x, LO)
space.
According to the preceding calculation for the Wigner distribution func-
tions, when an optical field passes through an optical system, its Wigner
distribution function does not change the values, but they are modified by an
affine coordinate transform, which corresponds exactly to that given by the ray
propagation matrices in geometrical optics. One of the links between physical
optics and geometrical optics is then established. Both show that an optical
system performs affine transforms in the space-frequency joint space, in which
each location corresponds to the position and orientation of rays.
According to ray optics, when optical systems are cascaded, the ray
propagation matrices are simply multiplied. The cascaded optical system is
described by an overall ray propagation matrix. Similarly, in terms of the
Wigner distribution function, subsystems perform the affine transforms of the