Page 54 - Phase Space Optics Fundamentals and Applications

Wigner Distribution in Optics 35

and extremum values of arise for the angle o , defined by
xx
/4 1 0 /2 0 /2
m xu − m x m u m xx − 2 m xx + m xx − m m x
x
tan 2 o = 2 = 2 2
/2
xx − uu m 0 xx − m xx − m 0 x + m x /2 2
(1.85)

Note that o corresponds to the minimum value of ,if o is cho-
xx
/2
0
1
sen such that cos 2 o has the same sign as xx − ; then o +
xx
2

1
corresponds to the maximum value of . The angles o and o +
xx
2
determine the principal axes of the moment ellipse in phase space.
Kernels can be optimized by rotating them and aligning them to these
principal axes. 89
1.8.4 Rotated Version of the Smoothed
Interferogram
Wewillapplythealigningofthekerneltothesmoothedinterferogram,
which can be best derived from the pseudo-Wigner distribution. With

∗
S f (x, u; w) = f (x + x o )w (x o ) exp(−i2 ux o ) dx o (1.86)
denoting the windowed Fourier transform, the pseudo-Wigner dis-
tribution P f (x, u; w), i.e., the Wigner distribution with the additional
window w( x )w (− x ) in its defining integral [see Eq. (1.17)] can
1
1
∗
2 2
also be represented as

1 1
P f (x, u; w) = S f x, u + t; w S ∗ f x, u − t; w dt (1.87)
2 2
The smoothed interferogram, also known as the S method, is now
defined as 90

1 1
P f (x, u; w, z) = S f x, u + t; w z(t)S ∗ x, u − t; w dt (1.88)
2 f 2
It is based on the pseudo-Wigner distribution written in the form
(1.87), but with an additional smoothing window z(t) in the u direc-
tion. The resulting distribution is of the Wigner distribution form,
with significantly reduced cross-terms of multicomponent signals,
while the auto-terms are close to those in the pseudo-Wigner dis-
tribution. For z(t) = (t), the bilinear representation P f (x, u; w, z) =
2
|S f (x, u; w)| is known as the spectrogram: the squared modulus of the
windowed Fourier transform. For z(t) = 1, P f (x, u; w, z) reduces to
the pseudo-Wigner distribution (1.87).
Since the window z(t) controls the behavior of P f (x, u; w, z)—more
Wigner-type or more spectrogram-type—we spend one paragraph on