Examples of higher-order diffraction catastrophes

Diffraction integrals are an essential tool of the description of a variety of diffraction phenomena in opitcs and quantum mechanics. The intensity of scattered light or the probability of finding a particle may be represented by integrals of the form (3.25). Let us recall that d 2 can be interpreted as the light intensity or the density of probability of finding a particle. 

For example, the diffraction integral (3.25) containing the potential function of a cusp catastrophe (zf3), F(x c) = x4 + ax2 + bx, describes ligth scattering on a two-dimensional diffraction grating, see Fig. 48. A function defined by such an integral is called the Pearcey function. 

There exist many other phenomena described by diffraction integrals. For example, in optics the description of light scattering on water droplets 

On the other hand, in connection with theoretical analysis of the problems related to scattering of atoms on crystals and in the model of scattering of atoms on rigid rotators appears the canonical integral (3.25) containing the potential function F(x c) of the elementary catastrophe of hyperbolic umbilic (D4+) F(x c) = x3 + y3 + axy + bx + cy. 

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