Page 266 - Phase Space Optics Fundamentals and Applications

Rays and Waves 247

Eq. (8.29) has an intuitive interpretation: the total flux entering the
bundle segment at one end equals the flux exiting at the other end.
This means that the rays behave as infinitesimal conduits of power.
When the bundle expands, causing its transverse area at the exit to be
bigger than that at the entrance, the flux density (and therefore the in-
tensity) becomes smaller in the same proportion, since the conserved
power spreads over a larger area.
Now, notice that the area elements are given by

(X) ∂ X ∂Y ∂ X ∂Y
a j = 1 2 = − 1 2 (8.30)
( ) ∂ 1 ∂ 2 ∂ 2 ∂ 1
z j z j
where the Jacobian (X)/ ( ) is the determinant of the stability matrix
∂X/∂ . By replacing z 1 with z, we can solve Eq. (8.29) for A 0 (X,z)
which, after using Eq. (8.30), gives
9
−1
H(z 0 , ) [X(z 0 , )] [X(z, )]
A 0 [X(z, ),z] = A 0 [X(z 0 , ),z 0 ]
H(z, ) ( ) ( )
(8.31)
This expression is the final piece that is needed to estimate the field.

8.3.3 The Field Estimate and Its Problems
at Caustics
All the pieces are now put together for the construction of the field
estimate. The estimate of the solution to the Helmholtz equation is
found in terms of the parameteric equation
U[X(z, ),z] ≈ A 0 [X(z, ),z] exp[ikL(z, )]
9
−1
H(z 0 , ) [X(z 0 , )] [X(z, )]
=
H(z, ) ( ) ( )
× A 0 [X(z 0 , ),z 0 ] exp[ikL(z, )] (8.32)

This approximation results from neglecting all terms but the first in
the Debye series for A, which is justified by the fact that k is large.
To use the formula in Eq. (8.32), one must determine, from the ini-
tial conditions of the field, the initial conditions for the rays. Once
this is done, the rays can be traced by using the ray equations, and
from there the amplitude and phase of the field can be computed.
A distinctive aspect of this formula is that the field information is
transmitted through the rays: the value of the field at any point is
determined only by the infinitesimal bundle(s) of rays surrounding