Page 287 - Introduction to Information Optics

272 5. Transformation with Optics

interpreted when one considers the local frequency as the ray propagation
direction. According to the physical diffraction when the light is localized by a
pinbole of a size Ax in the space, it would be diffracted within a range of
diffraction angles, Aco, which is inversely proportional to the pinhole size A.x.

5.6.4, WTGNER DISTRIBUTION OPTICS

The Wigner distribution function can be used to describe light propagation
in space in a similar way as geometrical ray optics does.
In Wigner distribution function interpretation, a parallel beam having only
a single spatial frequency /(x) = exp(za> 0x) is a horizontal line normal to the
o)-axis in the space-frequency joint space, because

W f(x, co) = exp(/to 0(x + x72)) exp( — m> 0(x — x'/2)) exp(icox') dx' — d(co — •(%).
«/
A spatial pulse passing through a location x 0 is described as /(x) = S(x — x 0),
its Wigner distribution function is a vertical line normal to the x-axis in the
space- frequency joint space, because

x 0 + x'/2)6(x — x 0 — x'/2) exp(icox') dx' — d(x — x 0).

In both cases we keep the localization of the signal in frequency and space,
respectively.
The Wigner distribution function definition in space, Eq. (5.22), and in
frequency, Eq. (5.23), which are completely symmetrical. The former Wigner
distribution function of /(x) is computed in space, resulting in W f(x, co), and
the latter Wigner distribution function of the Fourier transform F(co) is
computed in frequency, resulting in \V f(co, x). In the two equations, the roles
of x and co are interchanged. The Wigner distribution function is then rotated
by 90° in the space-frequency joint representation by the Fourier transform of
the signal.
When the optical field /(x) passes through a thin lens, it is multiplied by a
2
quadratic phase factor, exp( — inx /tf), where / is the focal length of the lens.
The Wigner distribution function of the optical field behind the lens becomes

2
W f(x,co) = /(x + x72)exp(-m(x + x72) )/*(x-x72)
*/
2
x exp( — m(x —- x'/2) ) exp( — icox') dx'
= W f(x, to + (2n/Af)x).