Page 288 - Phase Space Optics Fundamentals and Applications

Rays and Waves 269

The field can be estimated by approximating a ≈ a 0 , giving
U( ) ≈ U ( r)
r

k X + iP

= a 0 ( )
2 H

k 2
× exp − (x − X) + ik[L + (x − X)P] d (8.98)
2
This result is SAFE’s basic field estimate. Like the Gaussian beam
summation methods, this estimate does not fail at caustics of any
kind. Its only divergence occurs when rays turn around in z, that is,
if H vanishes. Unlike the Gaussian beam summation methods, this
estimate does not depend on the parabasal approximation. In fact, its
results have been shown 43 to remain valid beyond the point at which
the parabasal approximation fails, and the corresponding continuous
Gaussian beam superposition [as given in Eq. (8.85)] breaks down. The
generalization of this result to three dimensions is straightforward:

k 1 ( · X + iP)
r
U( ) ≈ U ( r) = a 0 ( )
2 H ( )

k
× exp − (x − X) · · (x − X) + ik[L + (x − X) · P] d 1 d 2
2
(8.99)
In the most general form of this result, isa2 × 2 matrix (whose
eigenvalues must have positive real parts).
Related propagation methods have been proposed in the quantum-
mechanical context. Heller 44 proposed estimating the temporal evo-
lution of a wave function as a superposition of “frozen Gaussians.”
Later, Herman and Kluk 45 found that a prefactor was missing from
Heller’s formulation. Their corrected result, known as the HK-IVR
(initial value representation) method, has become a standard tool in
quantum chemistry. Unlike SAFE, these methods involve integration
over all phase space, and not only over a Lagrange manifold.
8.9.2 Insensitivity to
It would seem that SAFE’s estimate depends strongly on the choice of
the width parameter . However, it is easy to show that this is not the
case. Consider the derivative of Eq. (8.98) with respect to , that is,

2

∂U Y X
= a 0 − k gd
∂ H 2Y 2

Y X X −1 −1
= a 0 − + O(k ) gd = O(k )U (8.100)

H 2Y 2Y