Page 42 - Phase Space Optics Fundamentals and Applications

Wigner Distribution in Optics 23

1.6.5 Transport Equations
With the tools of this section, we could study the propagation of the
Wigner distribution through free space by considering a section of
free space as an optical system with an input plane and an output
plane. It is possible, however, to find the propagation of the Wigner
distribution through free space directly from the differential equation
that the signal must satisfy. We therefore let the longitudinal variable
z enter into the formulas and remark that the propagation of coherent
light in free space (at least in the Fresnel approximation) is governed
by the differential equation (see, for instance, Ref. 15, p. 358)
2
∂ f 1 ∂
−i = k + f (1.55)
∂z 2k ∂r 2
2
2
2
2
2
2
with ∂ /∂r representing the scalar operator ∂ /∂x +∂ /∂y and with
k the wave number. The propagation of the Wigner distribution is now
described by a so-called transport equation 7,8,67–70 which in this case
takes the form
t
2 q ∂W ∂W
+ = 0 (1.56)
k ∂r ∂z
with ∂/∂r =∇. The transport equation (1.56) has the solution

2 q
W(r, q; z) = W r − z, q;0 (1.57)
k
which is equivalent to the result Eq. (1.44) in Sec. 1.6.1, with the special
choice A = D = I.
In a weakly inhomogeneous medium, the optical signal must satisfy
the Helmholtz equation

∂ f ∂ 2
2
−i = k (r,z) + f (1.58)
∂z ∂r 2
with k = k(r,z). In this case, we can again derive a transport equation
for the Wigner distribution; the exact transport equation is rather com-
plicated, but in the geometric-optical approximation, i.e., restricting
ourselves to first-order derivatives, it takes the simple form

t
t
2
2 t
2 q ∂W k − (2 ) q q ∂W ∂k ∂W
+ + = 0 (1.59)
k ∂r k ∂z 2 ∂r ∂q
which, in general, cannot be solved explicitly. With the method of
characteristics, however, we conclude that along a path defined by

2 t
2
dr 2 q dz k − (2 ) q q dq ∂k
= = = (1.60)
ds k ds k ds 2 ∂r