Page 217 - Phase Space Optics Fundamentals and Applications

198 Chapter Six

Random mask Aperture Random Image plane
Object F mask
plane F

F F F F

FIGURE 6.2 Schematic sketch of the imaging setup.

mask is positioned near the detector at which the output image is
obtained. The detector performs time, wavelength, or other types of
averaging to decode the spatial information that was encrypted by the
encoding input mask. The encoding mask can be a random spatial dis-
tribution, time-varying spatial distribution, or wavelength-dependent
filter, etc. The dependency of this mask is related to the domain into
which we aim to convert the degrees of freedom.
As mentioned before, we assume that the spatial encoding mask
which has some polarization, wavelength, or temporal dependency
is positioned near the input object u(x). We denote this mask as
g(x, p(t, ), ,t). The meaning of this notation is that at the different
spatial positions along the mask we may have wavelength dependen-
cies (denoted by ), time dependencies (denoted by t), or polarization
dependency (denoted by p) while the polarization dependency can
have time and wavelength dependency as well: p(t, ).
We denote the spatial spectrum of this mask as

G( x ,p( ,t), ,t) = g (x, p( ,t), ,t) exp (−2 ix x x) dx (6.10)

and the spatial spectrum of the object by

U( x ) = u(x) exp (−2 ix x x) dx (6.11)
Since the encoding mask is attached to the object (i.e., mathemat-
ically there is a multiplication operation between the two distribu-
tions in the space domain), in the spectrum we obtain a convolution
operation between the spectrum of the input object and the Fourier
transform of the mask:

U( )G( x − ,p( ,t), ,t) d (6.12)

x x x