Kirchhoff's diffraction formula
Kirchhoff's diffraction formula (also Fresnel–Kirchhoff diffraction formula) can be used to model the propagation of light in a wide range of configurations, either analytically or using numerical modelling. It gives an expression for the wave disturbance when a monochromatic spherical wave passes through an opening in an opaque screen. The equation is derived by making several approximations to the Kirchhoff integral theorem which uses Green's theorem to derive the solution to the homogeneous wave equation.
Kirchhoff's integral theorem, sometimes referred to as the Fresnel–Kirchhoff integral theorem, uses Green's identities to derive the solution to the homogeneous wave equation at an arbitrary point P in terms of the values of the solution of the wave equation and its first order derivative at all points on an arbitrary surface which encloses P.
The solution provided by the integral theorem for a monochromatic source is:
where U is the complex amplitude of the disturbance at the surface, k is the wavenumber, and s is the distance from P to the surface.
The assumptions made are:
Consider a monochromatic point source at P0, which illuminates an aperture in a screen. The energy of the wave emitted by a point source falls off as the inverse square of the distance traveled, so the amplitude falls off as the inverse of the distance. The complex amplitude of the disturbance at a distance r is given by
where a represents the magnitude of the disturbance at the point source.
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