Page 332 - Phase Space Optics Fundamentals and Applications

Sampling and Phase Space 313

where k represents the spatial frequency, denotes complex conjuga-
∗
tion, and {u(x)}(x, k) denotes the WDF operator. The WDF can also
be equivalently defined in terms of the Fourier transform (FT) of u(x),
which is denoted U(k).
∞

{u(x)}(x, k) = {U(k)}(x, k) = U k − U ∗ k +
2 2
∞
× exp(+ j2 k ) d (10.2)
∞

U(k) = u(x) exp(− j2 kx) dx (10.3)
−∞
The real-valued WDF of a function has double the number of di-
2
mensions of the function. To find the intensity I (x) =|u(x)| ,wein-
tegrate {u(x)}(x, k) over k; similarly, to find the spatial frequency
2
distribution ˜ I(k) =|U(k)| , we integrate over x. The WDF is real-
valued and it is reversible with the exception of a constant phase
factor. The WDF of a shifted signal is given by a simple shift
{u(x − )}(x, k) = {u(x)}(x − ,k). Similarly, if we multiply u(x)
by a harmonic function exp( j2 x), the resultant WDF is shifted in
k, {u(x) exp(+ j2 x)}(x, k) = {u(x)}(x, k − ). If two signals u and
h are convolved along the x axis, the WDF of the resultant signal is
given by the convolution of the individual WDFs along the same x
axis. Conversely, if two signals u and v are multiplied in the x domain,
the WDF of the resultant signal is given by the convolution of the in-
dividual WDF along the same k axis. The WDF of a convolution and
a product are given by Eqs. (10.4) and (10.5) respectively,

u( )˙v(x − ) d (x, k) = {u(x)}(x − x ,k) {v(x)}(x ,k) dx

x
= {u(x)}(x, k) ∗ {v(x)}(x, k) (10.4)

{u(x)v(x)}(x, k) = {u(x)}(x, k − k ) {v(x)}(x, k ) dk
k
= {u(x)}(x, k) ∗ {v(x)}(x, k) (10.5)
x
In the above equations we have introduced the notation ∗ and
k
∗ to denote convolution along the x and k axes, respectively. The
similarities between these properties and the convolution property of
the FT are obvious. Another property of the WDF is that it is bilinear;

{u(x) +v(x)}(x, k) = W uu (x, k) + W vv (x, k) + W uv (x, k) + W vu (x, k)
∗
∗
∗
∗
(10.6)