Page 311 - Phase Space Optics Fundamentals and Applications

292 Chapter Nine

n = -1 n = 1 n = 3
n = -2 n = 0 n = 2 n = 4
n
n¢ = 5
n¢ = 4
n¢ = 3
n¢ = 2
n¢ = 1
x n¢ = 0
n¢ = -1
n¢ = -2
1 n¢ = -3
d
n¢ = -4
n¢ = -5

d

FIGURE 9.6 The PSD of the comb function. The functions with a positive
(+) sign are interlaced with functions with a negative (−) sign.

comb into the definition of the WDF in Eq. (9.1), we find

∞ ∞
1 , , nn nd n
W comb (x, ) = (−1) x − − (9.21)
2d 2 2d

n=−∞ n =−∞
The corresponding PSD is shown in Fig. 9.6. The functions at loca-

tions (x m , m ) = (md, m /d), with m and m being integer numbers,

are interlaced with a grid of functions of alternating sign. The alter-
nating sign ensures that these interlaced terms do not contribute to
the marginals of WDF, i.e., the intensity and the power spectrum of
the comb function.
We can now use this PSD to study Fresnel diffraction by applying
a linear shear in x. As we increase the shear, we can identify cases
where points with a spatial frequency coordinate N/(2d) are laterally
shifted by a multiple of the period Md. Figure 9.7 shows the PSD cor-
responding to M = 1,N = 3. We observe registration of the horizontal
positions of functions forming columns of points with only posi-
tive sign, interlaced with columns of alternating sign. Without further
analysis we can deduce that the intensity distribution has to be again
a comb function. For N an odd integer, we find N delta functions
within an interval of size d. The corresponding diffraction plane is