Page 32 - Phase Space Optics Fundamentals and Applications

Wigner Distribution in Optics 13

We mentioned already that the integral over the frequency variable

W(r, q) dq = (r, r) represents the intensity of the signal, whereas

the integral over the space variable W(r, q) dr = ¯ (q, q) yields the
intensity of the signal’s Fourier transform; the latter is, apart from the
2
usual factor cos (where is the angle of observation with respect to
the z axis), proportional to the radiant intensity. 42,43 The total energy E
of the signal follows from the integral over the entire space-frequency
domain:

E = W(r, q) dr dq (1.24)

The real symmetric 4 × 4 matrix M of normalized second-order
moments, defined by

1 r t t
M = [r , q ]W(r, q) dr dq
E q
t t
1 rr rq
= t t W(r, q) dr dq
E qr qq
⎡ ⎤
m xx m xy m xu m xv

Mrr Mrq ⎢ m yy m yu m yv⎥
⎥
⎢ m xy
= t = ⎢ ⎥ (1.25)
M rq Mqq ⎣m xu m yu m uu m uv ⎦
m xv m yv m uv m vv
√
yields such quantities as the effective width d x = m xx of the intensity
(r, r) in the x direction

1 2 1 2 2
m xx = x W(r, q) dr dq = x (r, r) dr = d x (1.26)
E E
√
and similarly the effective width d u = m uu of the intensity ¯ (q, q)in
the u direction, but it also yields all kinds of mixed moments. It will be
clear that the main-diagonal entries of the moment matrix M, being
interpretable as squares of effective widths, are positive. As a matter
of fact, it can be shown that the matrix M is positive definite; see, for
instance, Refs. 44 to 46
The radiant emittance 42,43 is equal to the integral

2 2 t
k − (2 ) q q
j z (r) = W(r, q) dq (1.27)
k
where k = 2 / o represents the usual wave number. When we com-
bine the radiant emittance j z with the two-dimensional vector

2 q
j (r) = W(r, q) dq (1.28)
r
k