Page 269 - Phase Space Optics Fundamentals and Applications

250 Chapter Eight

Eq. (8.2) with real A and into the definition of the field’s flux:
∗
Im(U ∇U) 2
F = = A ∇ (8.36)
ik
That is, ∇ points now in the direction of the local flux. [Notice that
Eq. (8.35b) is simply the scalar version of Poynting’s theorem for a sta-
tionary field.] Therefore, the trajectories that result from this approach
are the flux lines of the field.
Equation (8.35a) strongly resembles the eikonal equation in Eq. (8.4),
2
2
except for the presence of the correction term −∇ A/k A. Since this
correction contains A, Eqs. (8.35a) and (8.35b) are coupled: one can
no longer first solve the eikonal equation to find the rays and then
use these rays to solve the transport equation. The correction term
is proportional to k −2 , so for large k this correction is usually very
small, meaning that the flux lines are very similar to the rays almost
everywhere. This correction has a similar effect to that of the refractive
index: its variation causes the flux lines to bend. This extra bending is
negligible except in regions where Avaries quickly, e.g., near a focus.
There, this term causes the flux lines to deflect away from one another
instead of crossing. (It also changes the spacing of the wavefronts,
giving rise to the Gouy phase shift.) Therefore, flux lines never cross,
and there is always only one at any point in space. It would therefore
appear that the estimation of the field in terms of flux lines is a better
alternative than the one in terms of rays, since there are no problems
with caustics. However, this approach has two disadvantages. First,
the fact that Eqs. (8.35a) and (8.35b) are coupled complicates the deter-
mination of the trajectories, which must be found numerically even in
very simple cases such as propagation in free space. (A result of this
coupling is that, unlike rays, flux lines do not propagate in a mutually
independent way.) Second, the method has problems at zeros of the
field, since the correction term in Eq. (8.35a) can diverge when A = 0.

8.5 Analogy with Quantum Mechanics
All the ideas presented so far can also be applied to the study of
quantum dynamics. 14 Let us concentrate on the case of nonrelativis-
tic quantum mechanics for a single particle of mass m moving in a
potential V( r, t). This problem is ruled by the Schr¨odinger equation

∂ ¯ h 2 2
i¯h ( r, t) =− ∇ ( r, t) + V( r, t) ( r, t) (8.37)
∂t 2m
where ¯h is the reduced Planck constant. The wavefunction plays
the role of the wave field U. As with the Helmholtz equation, let us