Scalar Diffraction Theory -- from Eric Weisstein's World of Physics

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Scalar Diffraction Theory

Given two scalar functions u and v, Green's second identity Eric Weisstein's World of Math asserts that

If u and v are both solutions of the Helmholtz equation, then

where k is the wavenumber, and

so

Now define

where is a wavefunction and r is the distance from a given point, and combine (1) with (5) to obtain

where is a small sphere surrounding the central singular point.

Diffraction, Fraunhofer Diffraction, Fresnel Diffraction, Vector Diffraction Theory