Page 65 - Phase Space Optics Fundamentals and Applications

46 Chapter Two

The intensity distribution is nevertheless insufficient to describe
the optical field; the phase correlation between any pair of points
must be included in the description. For this purpose, it seems natural
∗
to generalize I (x) by the mutual intensity (x, x) which contains 2
times more variables and is reduced to I (x) in the particular case x =
x . The ambiguity function AF is defined, in the frame of phase-space

optics (PSO), as the Fourier transform with respect to x of the mutual
intensity written as (x + a/2, x − a/2):

a a

A(f, a) = dx exp (−i2 x · f) x + , x − (2.2)
2 2
The equivalent representation

f f
A(f, a) = dm exp (i2 m · a)˜ m + , m − (2.3)
2 2
is obtained by replacing the mutual intensity by its double Fourier
expansion

a a a
!
x + , x − = dg dh exp i2 g · x +
2 2 2
a
"#
− h · x − ˜ (g, h) (2.4)
2
Formula (2.2) shows that A(f, 0) is equal to the intensity spectrum
˜ I(f) which, as well as I (x), represents the experimental data recorded
by a digital detector. Formula (2.3) shows that A(0, a) is the inverse
Fourier transform of the intensity distribution ˜ (m, m) in Fourier
space.
The Wigner distribution function (WDF) is defined as the Fourier
†
transform of (x + a/2, x − a/2) with respect to a, instead of x; the
WDF formulas similar to Eqs. (2.2) and (2.3) are

a a

W(x, g) = da exp (−i2 a · g) x + , x −
2 2

m m
= dm exp (i2 m · x)˜ f + , f − (2.5)
2 2
The mutual intensity can be obtained from the AF or from the WDF as

a a

x + , x − = df exp (i2 f · x)A(f, a)
2 2

= dg exp (i2 g · a)W(x, g) (2.6)
∗ See for instance Refs. [1] and [14] for a detailed definition of the mutual intensity
function.
† The WDF is discussed in detail in Chapter 1 by Martin Bastiaans.