Page 283 - Phase Space Optics Fundamentals and Applications

264 Chapter Eight

In analogy with those derivations, it might be expected that the am-
plitude u should be expanded into a Debye series and that this would
lead to a hierarchy of equations. However, given that the coefficients
of different powers of both k and must be made to vanish indepen-
dently, this would lead to a set of mutually inconsistent equations. For
this reason, the strategy employed here is to leave u as one function.
The equations that describe its evolution as well as that of follow
from considering only the next two most significant terms in Eq. (8.80),
i.e., from forcing C 22 and C 10 to vanish. The resulting equations are

2 2
i 1 ∂ n 2 2
˙ =− (X, z) + + ( ˙ X + i ˙ P)
H 2 ∂x 2
2 2 2 2
i 2 n (X, z) P ∂n 1 ∂ n
=− 2 + i 2 (X, z) + 2 (X, z)
H H H ∂x 2 ∂x
%
2
2
1 ∂n
− (X, z) (8.83a)
2H ∂x
u 2
˙ u = [ ˙ X ˙ P − ˙ H − i (1 + ˙ X )]
2H
2 2
u ∂n ∂n 2
= 3 P (X, z) − H (X, z) − 2i n (X, z) (8.83b)
4H ∂x ∂z
Equation (8.83a) is a nonlinear first-order differential equation of the
Riccati type, which can be expressed in terms of the solution to a
second-order linear differential equation. 23 Equation (8.83b) requires
the use of the solution of Eq. (8.83a). These equations do not lead to
closed-form solutions except for certain simple refractive index dis-
tributions such as free space, where the solutions are the standard
paraxial Gaussian beams. In three dimensions, these equations are
more complicated, since must be replaced by a 2 × 2 matrix.
Settingtheremainingcoefficientstozeroleadstoconstraintsthatare
inconsistent with the ones found earlier. Therefore, the fields U G that
result from the substitution of the solutions of Eqs. (8.83a) and (8.83b)
into Eq. (8.77) are only approximate solutions to the Helmholtz equa-
tion. These beams are sometimes called parabasal Gaussian beams, as
they are the result of an expansion around a “base” ray with phase-
space coordinates (X, P).

8.8.2 Sums of Gaussian Beams
The parabasal propagation of Gaussian beams can be calculated
through the methods outlined earlier. A large body of work has been
produced in the last few decades with the goal of modeling the prop-
agation of arbitrary fields by expressing them as superpositions of