Page 286 - Phase Space Optics Fundamentals and Applications

Rays and Waves 267

where the functional coefficients C mn are those in Eqs. (8.82a) through
(8.82f), with all derivatives of set to zero.
The main feature of SAFE is that the Gaussians are not independent
beams but interrelated contributions. This is due to the presence of the
integral in Eq. (8.88), which plays a fundamental role in the derivation.
The key for this is a trick based on the fact that the derivative of g
with respect to is given by

g = k[ X + i(L − PX + P )]g = k( X + iP ) g (8.89)

where, in the last step, we chose L to be the area under the PSC, so
that Eq. (8.21) is satisfied. Equation (8.89) can be written as
g
g = (8.90)
kY
where the shorthand Y = X + iP is used in what follows. By using
this expression, a factor of multiplying g can be removed at the

cost of turning g into g (and including an extra factor of 1/kY ).

The derivative in g can then be removed through integration by
parts. This process can be repeated to remove higher powers of in
the form

m n m−1 C mn n−1 m−1 C mn n−1
k C mn g d = k g d =− k

Y Y
× g d

X
= k m−1 C mn n−2 − C mn n−1 g d

Y Y
(8.91)
where the integrated terms resulting from the integration by parts
are dropped by assuming that the magnitude of the integrands goes
to zero at the limits. By using this trick repeatedly, we can rewrite
Eq. (8.88) as

$ %

X C 21 0
2
k C 20 + k C 22 + C 10 − + O(k ) g d = 0 (8.92)
Y Y
2
As for the Gaussian beams, setting the leading term, k C 20 , to zero
leads to the laws of ray optics, which also make C 21 vanish. Then forc-
ing the remaining part of the next term to vanish amounts to choosing
u so that X C 22 + Y C 10 = O(k −1 ). The derivation that follows re-

quires a few steps. Let us start by substituting Eqs. (8.82c) and (8.82d)