Page 309 - Phase Space Optics Fundamentals and Applications

290 Chapter Nine

n n

Δn
x x

Md x p x s
(a) (b)

FIGURE 9.5 Phase-space interpretation of the walk-off effect.

Fresnel diffraction corresponds to a horizontal shear of the phase-
space distribution, and only over a region x s , in Fig. 9.5b, where all
plane waves can interfere, we expect the self-image to resemble the
input signal. The region x p marks a transition region that can be associ-
ated with edge diffraction from the grating aperture as the dominating
effect.
To derive a limiting-conditions analog to Abbe’s theory of the mi-
croscope, we consider a cosine pattern as input signal with = 2/d,
giving rise to three propagating plane waves only (the analysis does
not change if we assume a periodic pattern with higher-order nonzero
Fourier coefficients; in this case defines the frequency band for
which truncated plane waves have not yet completely moved out of
the signal window).
With a grating aperture of size Md we can now estimate the maxi-
mum distance over which self-imaging can be observed as the point
where x s = 0. With x p = z max = Md/2wefind

Md 2
z max = (9.19)

which corresponds to the estimate given in Ref. 30.

9.6 The Fractional Talbot Effect
9
While forming the basis for various applications, the Talbot effect
is also interesting from a mathematical perspective. For the major-
ity of functions that are commonly used to model optical systems, the
Fresnel diffraction integral has no simple analytic solution. The Talbot