Page 218 - Phase Space Optics Fundamentals and Applications

Super Resolved Imaging in Wigner-Based Phase Space 199

After the coded spatial information is passed through the finite
aperture of the imaging lens, we have a multiplication of this aperture
by the overall spectral distribution:

x

rect U( )G( x − ,p( ,t), ,t) d x (6.13)
x
x
x
We denoted the spatial width of the aperture by x . The decoding
mask is attached to the output plane, and thus its Fourier transform
performs an additional convolution operation with the overall expres-
sion of Eq. (6.13).

x

G d ( x − ,p( ,t), ,t) rect
x
x
(6.14)

× U( )G( − ,p( ,t), ,t) d d
x x x x x
where G d is the Fourier transform of the decoding mask g d :

G d ( x ,p( ,t), ,t) = g d (x, p( ,t), ,t) exp (−2 ix x x) dx (6.15)
Assuming that the decoding mask is the complex conjugate of the
encoding mask, we have
g d (x) = g (x) (6.16)
∗
which means that

∗
G d ( x ) = G (− x ) (6.17)
and thus the expression of Eq. (6.14) becomes

x
∗

U R ( x ) = G (− x + ,p( ,t), ,t) rect
x
x
(6.18)

× U( )G( − ,p( ,t), ,t) d d
x x x x x
where U R is the spectrum of the reconstructed image u R :

U R ( x ) = u R (x) exp (−2 ix x x) dx (6.19)