Page 281 - Phase Space Optics Fundamentals and Applications

262 Chapter Eight

representations must be equal to or greater than 1/(2k). Since the min-
imum product of widths is achieved by Gaussians, field contributions
with transverse Gaussian amplitude profile are considered here. Let
X(z) and P(z) be the centroids in x and p of such a field contribution.
This contribution, called a Gaussian beam, can then be written as

(x − X) 2
U G (x, z) = u(z) exp − exp{ik[L(z) + P(x − X)]} (8.76)
2w 2
where u(z) is a complex amplitude and L(z) is a phase accumulated
under propagation. In what follows, it is shown that the beam cen-
troids X and P evolve according to geometrical optics.
The transverse rms width of the Gaussian beam in Eq. (8.76) is
√
w/ 2. It is easy to show that the Fourier transform in x of this beam
is indeed a Gaussian in p centered at P, with rms width equal to
√
1/( 2kw). Since the case of large k is considered, we choose w =
√
1/ k , where has units of inverse length. This way, the width of
√
the beam is proportional to 1/ k in both the position and momentum
representations, leading to comparable levels of localization in phase
space in the x and p directions. The Gaussian beam in Eq. (8.76) can
then be written as
U G (x, z) = u(z)g (x, z) (8.77)

where

k 2
g (x, z) = exp − [x − X(z)] +ik{L(z) + P(z)[x − X(z)]} (8.78)
2

Of course, the real part of must be positive.
The next step is to substitute U G into the Helmholtz equation. First,
the second partial derivatives of U G can be found to be

2
∂ U G 2 2
= [−k + k (iP − ) ]ug (8.79a)
∂x 2
$
2
∂ U G 2
= ¨ u +˙uk[2iH + 2( ˙ X + i ˙ P) −˙ ]
∂z 2
2

+uk i ˙ H − ( ˙ X + i ˙ P) ˙ X + ( ¨ X + i ¨ P + 2˙ ˙ X) − ¨
2
2
2 %
+uk 2 iH + ( ˙ X + i ˙ P) −˙ g (8.79b)
2