Page 302 - Phase Space Optics Fundamentals and Applications

Self-Imaging in Phase Space 283

length f we find

x
W L (x, ) = W 0 x, + (9.10)
f
In Fig. 9.2 both operations are applied to a generic phase-space vol-
ume of rectangular shape. In addition, Fig. 9.2d shows the effect of a
Fourier transformation that corresponds to exchanging both phase-
space coordinates with a clockwise rotation of the WDF by 90 .
◦
Important operations are modulation and convolution of two sig-
nals. For the product of two functions u(x) = g(x) h(x), the corre-
sponding WDFs are convolved with respect to the frequency variable

∞

W u (x, ) = W g (x, )W h (x, − ) d = W g (x, ) ∗ W h (x, )
−∞
(9.11)
The symmetry between x and implies that a convolution between the
two signals is translated to a convolution between the corresponding
WDFs with respect to x.
Finally, we will also need the phase-space representation of a linear
chirp function
2
u ch (x) = exp[i2 ( x + x + )] (9.12)
n n
W (x, v) W Fr (x, v)
0

x x

(a) (b)

n
n W FT (x, v)
W (x, v)
L
x x

(c) (d)

FIGURE 9.2 Paraxial optics in phase space: (a) Generic phase-space
distribution of an optical signal, (b) signal after Fresnel diffraction, (c) after
modulation with a quadratic phase function, and (d) after Fourier
transformation.