Page 214 - Phase Space Optics Fundamentals and Applications

Super Resolved Imaging in Wigner-Based Phase Space 195

phase space. We discuss the use of Wigner for code, time, polarization,
wavelength, and gray-level multiplexing SR. Section 6.4 concludes the
chapter.

6.2 General Definitions
The Wigner phase space is a bilinear distribution defined as

x x

W u (x, x ) = u x + u ∗ x − exp (−2 i x x ) dx (6.1)
2 2
where u(x) is the signal that is being transformed while x and x are
the space and spatial-frequency coordinates, respectively.
The projection of the Wigner distribution contains the spatial and
the spectral distribution of the signal u:

2
|u(x)| = W u (x, x ) d x
(6.2)

2
|U( x )| = W u (x, x ) dx
where U( x ) is the Fourier transform of u(x). Another important prop-
erty is that the area of the Wigner equals the total energy of the signal:

2 2
|u(x)| dx = |U( x )| d x = W u (x, x ) d x dx (6.3)
Because of those important and fundamental properties, as well as
the fact that basic optical modules can be represented as simple and
well-defined geometric operations over the Wigner chart, 41 this rep-
resentation became a very useful tool for analyzing optical systems
and especially when dealing with optical SR.
One important parameter is the number of degrees of freedom, also
called the space-bandwidth product (SW). This number equals
N = x · (6.4)

where x is the spatial size of the signal and is its spectral width. In
the general case also when the Wigner distribution is not a rectangular
function, it can still be shown that the number of degrees of freedom
is related to the area of the Wigner. 36

N = W u (x, x ) dx d x
The proof is done by dividing the Wigner into infinitesimal rectangles
with each representing a single degree of freedom and showing that
their overall contribution equals the overall area of the distribution.