# Kirchhoff integral theorem

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Kirchhoff's integral theorem (sometimes referred to as the Fresnel-Kirchhoff integral theorem)[1] 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 that encloses P.[2]

## Equation

### Monochromatic waves

The integral has the following form for a monochromatic wave:[2][3]

${\displaystyle U(\mathbf {r} )={\frac {1}{4\pi }}\int _{S}\left[U{\frac {\partial }{\partial {\hat {\mathbf {n} }}}}\left({\frac {e^{iks}}{s}}\right)-{\frac {e^{iks}}{s}}{\frac {\partial U}{\partial {\hat {\mathbf {n} }}}}\right]dS,}$

where the integration is performed over an arbitrary closed surface S (enclosing r), s is the distance from the surface element to the point r, and ∂/∂n denotes differentiation along the surface normal. Note that in this equation the normal points inside the enclosed volume; if the more usual outer-pointing normal is used, the integral has the opposite sign.

### 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:

${\displaystyle V(r,t)={\frac {1}{\sqrt {2\pi }}}\int U_{\omega }(r)e^{-i\omega t}\,d\omega ,}$

where, by Fourier inversion, we have:

${\displaystyle U_{\omega }(r)={\frac {1}{\sqrt {2\pi }}}\int V(r,t)e^{i\omega t}\,dt.}$

The integral theorem (above) is applied to each Fourier component Uω, and the following expression is obtained:[2]

${\displaystyle V(r,t)={\frac {1}{4\pi }}\int _{S}\left\{[V]{\frac {\partial }{\partial n}}\left({\frac {1}{s}}\right)-{\frac {1}{cs}}{\frac {\partial s}{\partial n}}\left[{\frac {\partial V}{\partial t}}\right]-{\frac {1}{s}}\left[{\frac {\partial V}{\partial n}}\right]\right\}dS,}$

where the square brackets on V terms denote retarded values, i.e. the values at time ts/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.

## References

1. G. Kirchhoff, Ann. d. Physik. 1883, 2, 18, p663
2. Max Born and Emil Wolf, Principles of Optics, 1999, Cambridge University Press, Cambridge, pp. 417-420
3. Introduction to Fourier Optics J. Goodman sec. 3.3.3