Page 285 - Phase Space Optics Fundamentals and Applications

266 Chapter Eight

dimensions, the corresponding field estimate then takes the form of
an integral over the PSC

U(x, z) ≈ u(z, )g (z, ) (x, z, ) d (8.85)

where

k 2
g (x, z, ) = exp − [x−X(z, )] +ik{L(z, )+P(z, )[x−X(z, )]}
2
(8.86)
and u(0, ) and (0, ) must be chosen so that the superposition
matches the initial field at z = 0. Again, this method is valid as long
as the parabasal approximation holds for each independent Gaussian
beam.

8.9 Stable Aggregates of Flexible Elements
A different approach for building wave field estimates based on rays
throughthesuperpositionofGaussiancontributionsisnowdiscussed.
Like Eq. (8.85), this framework, referred to as stable aggregates of flexi-
ble elements (SAFE), 39–43 takes the form of a continuous superposition
of Gaussian components around the rays in the Lagrange manifold.
However, in SAFE, the Gaussian contributions are not independently
propagating parabasal Gaussian beams; rather, they are interrelated
contributions, where is not constrained to have a specific depen-
dance on z or :

U(x, z) = u(z, )g (x, z, ) d (8.87)

In what follows, it is assumed for simplicity that is a real and positive
constant, although the results remain valid if this parameter has an
imaginary part and/or is allowed to vary slowly with z, , or even x.

8.9.1 Derivation of the Estimate
To find the specific form of this estimate, we substitute Eq. (8.87) into
Eq. (8.1). By following the same steps as in Sec. 8.8, we obtain a result
that is identical to that in Eq. (8.81), except for the presence of an
integral in :

2
2
[k (C 20 +C 21 +C 22 +· · ·)+k(C 10 +C 11 )+C 00 ] g d = 0 (8.88)