Diffraction integrals

We note that the term quasioptics implies that it is not sufficient to borrow familiar optical concepts, such as point focus, the lensmaker s equation, etc. without modification. In fact, diffraction plays a crucial role in characterizing system behavior. Fortunately, the quasioptics formalism allows us to avoid the time-consuming computation of diffraction integrals that would otherwise be necessary for a complete system analysis. We will concentrate instead on those aspects of quasioptics that are readily amenable to calculation in the paraxial approximation (see subsequent text). In particular, we will study the propagation of Gaussian beams. 

As we have said, the diffracted integrated intensity is proportional to the quantity per volume of the phase. However, in a polyciystalline sample irradiated with a... 

In a later passage of the chapter we shall consider the short-wave limit for the Helmholtz equation, describing optical phenomena, and for the Schrodinger equation, describing quantum phenomena. The relation between the obtained oscillation integrals and the elementary catastrophe functions will be revealed and straightforward examples of application of the canonical diffraction integrals (3.25) discussed. 

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... 

Fig. 49. Diffraction elliptic umbilic catastrophe (D4 ) diffraction integral D4 in the b, c, a = 0 plane.

During the reconstruction process, we illuminate with a zero order Bessel beam, the optical field can be obtained from the diffraction integral representation, where a set of optical beams can be identified. The amplitude function for the diffraction optical field is... 

When the field point approaches a caustic, then two or more rays having almost equal directions pass through P (see Fig. 6). In this case the diffraction integral I for a two-dimensional field takes the form... 

A. Calculation of Diffraction Integrals by Expansion of the Field in Gaussian Beams... 

On the exit side (see Fig. 11.11) at a distance Z from the film, the intensity distribution on the observation plane P is given by the Kirchhoff diffraction integral ... 

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