Kirchhoff integral theorem


Kirchhoff's integral theorem is a surface integral to obtain the value of the solution of the homogeneous scalar wave equation at an arbitrary point P in terms of the values of the solution and the solution's first-order derivative at all points on an arbitrary closed surface that encloses P. It is derived by using Green's second identity and the homogeneous scalar wave equation that makes the volume integration in Green's second identity zero.

Integral

Monochromatic wave

The integral has the following form for a monochromatic wave:
where the integration is performed over an arbitrary closed surface S enclosing the observation point ', in is the wavenumber, in is the distance from an integral surface element to the point ', is the spatial part of the solution of the homogeneous scalar wave equation, is the unit vector inward from and normal to the integral surface element, i.e., the inward surface normal unit vector, and denotes differentiation along the surface normal i.e., for a scalar field. Note that the surface normal is inward, i.e., it is toward the inside of the enclosed volume, in this integral; if the more usual outer-pointing normal is used, the integral will have the opposite sign.
This integral can be written in a more familiar form
where.

Non-monochromatic wave

A more general form can be derived for non-monochromatic waves. The complex amplitude of the wave can be represented by a Fourier integral of the form
where, by Fourier inversion, we have
The integral theorem is applied to each Fourier component, and the following expression is obtained:
where the square brackets on V terms denote retarded values, i.e. the values at time t − s/''c''.
Kirchhoff showed that the above equation can be approximated to a simpler form in many cases, known as the Kirchhoff, or Fresnel–Kirchhoff diffraction formula, which is equivalent to the Huygens–Fresnel equation, except that it provides the inclination factor, which is not defined in the Huygens–Fresnel equation. The diffraction integral can be applied to a wide range of problems in optics.

Integral derivation

Here, the derivation of the Kirchhoff's integral theorem is introduced. First, the Green's second identity as the following is used.
where the integral surface normal unit vector here is toward the volume closed by an integral surface. Scalar field functions and are set as solutions of the Helmholtz equation, where is the wavenumber, that gives the spatial part of a complex-valued monochromatic wave expression. Then, the volume part of the Green's second identity is zero, so only the surface integral is remained as
Now is set as the solution of the Helmholtz equation to find and is set as the spatial part of a complex-valued monochromatic spherical wave where is the distance from an observation point in the closed volume. Since there is a singularity for at where, the integral surface must not include. A suggested integral surface is an inner sphere centered at with the radius of and an outer arbitrary closed surface.
Then the surface integral becomes
For the integral on the inner sphere,
and by introducing the solid angle in,
due to.
By shrinking the sphere toward the zero radius, and the first and last terms in the surface integral becomes zero, so the integral becomes. As a result, denoting, the location of, and by, the position vector, and respectively,