# Kirchhoff integral theorem

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.

## Equation

### Monochromatic waves

The integral has the following form for a monochromatic wave: 

$U(r)={\frac {1}{4\pi }}\int _{S}\left[U{\frac {\partial }{\partial n}}\left({\frac {e^{iks}}{s}}\right)-{\frac {e^{iks}}{s}}{\frac {\partial U}{\partial n}}\right]dS$ where the integration is performed over the whole of the arbitrary surface S, s is the distance between the point r and the surface S, and ∂/∂n denotes differentiation along the normal on the surface with direction into the surface.

Please note: It may be confusing because most used to normal direction pointing outwards of the surface. In that case the eq. shall be multiplied by minus.

### Non-monochromatic waves

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:

$V(r,t)={\frac {1}{\sqrt {2\pi }}}\int U_{\omega }(r)e^{-i\omega t}d\omega$ where, by Fourier inversion, we have:

$U_{\omega }(r)={\frac {1}{\sqrt {2\pi }}}\int V(r,t)e^{i\omega t}dt$ The integral theorem (above) is applied to each Fourier component Uω, and the following expression is obtained

$V(r,t)={\frac {1}{4\pi }}\int _{S}\left\{[V]{\frac {\partial }{\partial n}}\left({\frac {1}{s}}\right)-{\frac {1}{cs}}{\frac {\partial s}{\partial n}}\left[{\frac {\partial V}{\partial t}}\right]-{\frac {1}{s}}\left[{\frac {\partial V}{\partial n}}\right]\right\}dS$ where the square brackets on V terms denote retarded values, i.e. the values at time t − s/c.

Kirchhoff showed the above equation can be approximated in many cases to a simpler form, known as the Kirchhoff, or Fresnel-Kirchhoff diffraction formula, which is equivalent to the Huygens-Fresnel equation, but provides a formula for the inclination factor which is not defined in the latter. The diffraction integral can be applied to a wide range of problems in optics.