Page 286 - Introduction to Information Optics

5.6. Wigner Distribution Function 2 / 1

5.6.2, INVERSE TRANSFORM

The projection of the Wigner distribution function W f(x, co) in the space
frequency joint space along the frequency co-axis gives the square modulus of
the signal f(t), because according to Eq. (5.22) the projection of W f(x, co) along
the co-axis is

x
W f (x, co) dco = / x + /* - - ex P( -J (OX> dx da) =
)
'
The projection of W f(x, co) in the space -frequency joint space along the
space axis x gives the square modulus of the Fourier transform F(co) of the
signal, because according to Eq. (5.23) the projection along the x-axis is
i r c / '\ / r \
W f(x, co) dx = — F f to 4- y ) F* ( co - ~- ) exp(jto'x) da)' dx = \F(co)
*/ */ \ / \ *"* /
In addition, we have the energy conservation of the Wigner distribution
function in the space-frequency joint representation:

I f i f f
2
2
— W f(x, co)dxdco = — F(co)\ dco = f(x)\ dx.
J
In J 2nj J '
5.6.3. GEOMETRICAL OPTICS INTERPRETATION

Let f(x) denote the complex amplitude of the optical field. Its Wigner
distribution function W f(x, o>) describes the propagation of the field in the
space-frequency joint representation, where the frequency co is interpreted as
the direction of a ray at point x.
The concept of spatial frequency is introduced in the Fourier transform. In
the optical Fourier transform described in Eq. (5.7), the spatial frequency is
defined as co — 2nu/Az, where u is the spatial coordinate in the Fourier plane.
Therefore, u/z is the angle of ray propagation. In this case, the spatial frequency
co can be considered the local frequency. When an optical field is represented
in the space-frequency joint space, its location x and local frequency must
obey the uncertainty principle. One can never represent a signal with an infinite
resolution in the space-frequency joint space, but can only determine its
location and frequency within a rectangle of size

AxAeo ^ 1/2.

This Heisenberg inequality familiar in quantum mechanics can be easily