# 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 which encloses P.[2]

## Equation

### Monochromatic waves

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

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

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:

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

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 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.

## 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

• 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