Page 289 - Phase Space Optics Fundamentals and Applications

270 Chapter Eight

where in the second step the integration-by-parts trick in Eq. (8.91)
was used to remove the ’s. This means that the variation of SAFE’s
estimate due to changes of the contributions’ widths is asymptotically
small, provided that a 0 is chosen to be independent of . This is the
reason for the name of the method: while each of the contributions (or
“elements”) is flexible in width, their superposition (or “aggregate”)
is stable.

8.9.3 Phase-Space Interpretation
Insight into SAFE and the insensitivity of the estimate on (as well
as the limitations of this insensitivity) can be gained through a phase-
space picture. The windowed Fourier transform (WFT) like the one de-
fined in Eq. (1.86) is a linear phase-space representation of a function.
Here, let us consider a WFT where the window is chosen as a Gaussian
of width (k ) −1/2 (with > 0):

k 1/2 (x − x) 2

S f (x, p; ) = f (x ) exp −k
2 2
x
! "

× exp −ikp x − dx (8.101)
2
The squared modulus of this transform is known as the spectrogram or
Husimi function. 46 The application of this transformation to the field
estimate in Eq. (8.98) gives
(x, p; z, )
S U
9
k 1/2 X + iP (x − X) 2 ( p − P) 2

= a 0 exp −k −1 −1 − k
2 ( + ) H 2( + ) 2( + )
! "
ik p x
× exp − x P − − p X − − XP d
+ 2 2
(8.102)
That is, the contribution from each ray is a Gaussian (times a linear
phase factor) localized around the corresponding phase-space point,

√
−1
with rms widths ( −1 + )/2k in the x direction and ( + )/2k
in the p direction, as shown in Fig. 8.6a. This leads to the following
intuitive picture illustrated in Fig. 8.6b: the evaluation of Eq. (8.102) is
like spray-painting the wave field’s phase-space distribution over the
rays’ traced line (the PSC). The characteristic footprint of the spray
can has the widths mentioned earlier. However, the appearance of
the final thicker fuzzy line painted over the PSC is roughly indepen-
dent of the widths of the footprint, 47 as long as this footprint is fine
enough to resolve the sections of the PSC where the curvature is tight.