Page 314 - Phase Space Optics Fundamentals and Applications

Self-Imaging in Phase Space 295

will allow us to extend the context of our discussion and further high-
light the importance of the fractional Talbot.

9.7 Matrix Formulation of the Fractional
Talbot Effect
In Sec. 9.6 we made no assumption regarding the groove shape of the
grating structure to illustrate that the fractional Talbot effect can be
expressed as a superposition of shifted and modulated copies of the
grating’s transmission function.
We will now consider the case in which each period of the input
function can be written as

Q−1
, qd
u p (x) = u int (x) ∗ a q x − (9.28)
Q
q=0
with Q being an integer number. We can interpret Eq. (9.28) as
a sampling expansion with u int (x) defining the interpolation func-
tion. In particular, with u int (x) = sinc(xQ/d), Eq. (9.28) turns into
the well-known Shannon-Whitaker sampling theorem. For u int (x) =
rect(xQ/d) we obtain a model most suitable to describe binary diffrac-
tive optical elements or images with rectangular pixels.
We are again at liberty to explore Fresnel diffraction without speci-
fying the interpolation function explicitly. It is sufficient to investigate
Fresnel propagation of the discrete sampled version of the input func-
tion. Given our discussion in Sec. 9.6, we expect to find fractional
Talbot planes where the diffraction amplitude is described as a su-
perposition of Q copies laterally shifted by a multiple of d/Q. This,
in turn, implies that for an input function consisting of modulated
functions at intervals d/Q, the output function also has to be a mod-
ulated comb function with the same spacing between pulses. With
the interpolation functions remaining unaffected, each period of the
Fresnel diffracted wave can be expressed as

Q−1
, qd
u p (x, z M,N ) = u int (x) ∗ b q x − (9.29)
Q
q=0
In other words, we can solve the propagation problem by express-
ing coefficients b q as a linear superposition of the coefficients a q .
This, in turn, means we can interpret the a q and b q as vectors in a
Q-dimensional vector space and determine a linear discrete trans-
formation to relate both sets of coefficients. 14,15 The transformation
matrix has to be a function of the respective Talbot coefficients which