# 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]

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

where the integration is performed over 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[2]

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

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. ^ a b c 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