# Kirchhoff integral theorem

**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]}

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:

where, by Fourier inversion, we have:

The integral theorem (above) is applied to each Fourier component U_{ω}, and the following expression is obtained:^{[2]}

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.

## See also

- Kirchhoff's diffraction formula
- Vector calculus
- Integral
- Huygens–Fresnel principle
- Wavefront
- Surface

## References

## Further reading

*The Cambridge Handbook of Physics Formulas*, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.*Introduction to Electrodynamics (3rd Edition)*, D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3*Light and Matter: Electromagnetism, Optics, Spectroscopy and Lasers*, Y.B. Band, John Wiley & Sons, 2010, ISBN 978-0-471-89931-0*The Light Fantastic – Introduction to Classic and Quantum Optics*, I.R. Kenyon, Oxford University Press, 2008, ISBN 978-0-19-856646-5*Encyclopaedia of Physics (2nd Edition)*, R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3*McGraw Hill Encyclopaedia of Physics (2nd Edition)*, C.B. Parker, 1994, ISBN 0-07-051400-3