Page 229 - Phase Space Optics Fundamentals and Applications

210 Chapter Six

occupying the field of view of x and having spectral bandwidth of
3 . If we divide the high spatial resolution over a 3 times larger field
of view (i.e., reduces the spatial resolution by a factor of 3 by spreading
it along the field of view), we obtain 3 x for the spatial region and 3
times smaller spectral bandwidth of only . In this figure every one
out of the three spatial regions is reduced by 3 times in its resolution
(thus, the spectral bandwidth of each one of the three spatial regions
is reduced by a factor of 3 while their spatial region is increased by a
factor of 3 and the entire area of each one of them is preserved). An
effect similar to that is obtained in the case of optical magnification or
zooming.
Another type of field of view multiplexing approach is the tech-
nique in which every one out of the three spectral slots (each one
of the three slots has the bandwidth of ) is multiplexed by be-
ing shifted to different spatial positions, transmitted through the
resolution-limiting imager, and later on demultiplexed back to com-
pose the high-resolution image. This multiplexing/demultiplexing is
done using proper gratings. The grating can redirect or reposition
the different spectral slots (modulation and demodulation operation).
This operation is described in Fig. 6.9c and 6.9d. In this case the vari-
ous spectral slots are not changed in their shape as before (reduced in
the spectral domain and expanded in the space domain), but rather
only repositioned along the spatial axis. The optical realization of the
setup that is using the grating to perform the relocation of the spectral
slots while sacrificing the field of view can be achieved by positioning
2 or 3 gratings in predetermined locations along the imaging system,
as described in Ref. 28.
In many cases such as those presented in Refs. 31 and 32 where the
object is a one-dimensional object, one may use the second spatial axis
to improve the imaging resolution. In this case the schematic sketch
of the phase-space distribution is very similar to the one presented in
Fig. 6.8, while in this case the axis that is denoted as or as the dynamic
range axis in Fig. 6.8 (e.g., in Fig. 6.8a) will be now the spectral axis
corresponding to the second spatial dimension (y instead of x).
In Fig. 6.10 we perform a true numerical simulation for the Wigner
distribution in the case of a time multiplexing super resolution ap-
proach. In Fig. 6.10a we see the Wigner transform of a Gaussian signal.
In Fig. 6.10b we plot the Wigner distribution of the lowpass Gaussian
signal being low passed with a rectangular spectral window that is
approximately 3 times narrower than the original width occupied by
the input Gaussian.
In Fig. 6.10c we show the Wigner transform of the lowpass signal
after it is multiplied by the time-varying random decoding mask. The
chart in Fig. 6.10d is the computed Wigner distribution of the recon-
struction, i.e., the signal after being averaged in the time domain.