Page 284 - Phase Space Optics Fundamentals and Applications

Rays and Waves 265

Gaussian beams. These schemes are known collectively as Gaussian
beam summation methods. They were proposed independently in
many areas starting with geophysics, for the description of seismic
waves. 24–28 Similar methods followed for quantum mechanics 29–31
(where the beams are referred to as Gaussian wave packets) and elec-
tromagnetic waves 32–37 (particularly in optics and radio science). One
of the main advantages of this approach is that it is free of problems
at caustics.
Gaussian beam summation methods rely on the expression of an
initial field as a Gabor representation, 38 i.e., as a weighted superposi-
tion of Gaussian functions of a given width, with different locations
and linear phase factors. These superpositions can be discrete or con-
tinuous. A discrete Gabor representation, for example, allows us to
write a function f (x) in the form

2
, −k 0 (x − m X )
f (x) = a m,n exp + ikn P (x − m X ) (8.84)
2
m,n
Here, the sampling spacings X and P must satisfy the relation
X P ≤ 2 /k, where the basis is complete if the equality holds and is
overcomplete otherwise. (The standard complete Gabor basis results
√ √
from choosing X = 2 /k 0 and P = 2 0 /k.) This basis of
Gaussians is not orthogonal, so the expansion coefficients a m,n must
be found through the use of a biorthogonal basis. 32,33 Each Gaussian
√ √
roughly occupies a phase-space area of size 2 /k 0 by 2 0 /k
centered at (x, p) = (m X ,n P ), so the Gabor representation can be
thought of as a subdivision of phase space into a Cartesian grid of cells
whosesizeissmallerthanorequaltotheminimum-uncertaintyphase-
space area. The procedure for the propagation of a field U(x, z) in two
dimensions is as follows: First the coefficients a m,n for the specified
initial field U(x, 0) are found. Then each Gaussian is propagated as
a Gaussian beam with u(0) = a m,n , (0) = 0 ,X(0) = m X ,P(0) =
n P , and L(0) = 0. Finally, the field away from the initial plane is
approximated as the sum of the propagated Gaussian beams. This
method relies on the validity of the parabasal approximation for each
beam.
Continuous Gaussian beam summation methods result from ex-
pressing the field as a continuous superposition of Gaussian beams.
These can involve all initial positions and directions, although then
the superposition is not unique, since the continuous set of Gaus-
sians constitutes an overcomplete basis. Another form of continuous
superposition can be used for fields associated with a known initial
Lagrange manifold. In this case, the initial field is not composed of
all possible Gaussians but only of those whose central initial position
and direction fall within this Lagrange manifold. For fields in two