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]

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 (a normal derivative). 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:

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.

See also

References

  1. ↑ G. Kirchhoff, Ann. d. Physik. 1883, 2, 18, p663
  2. ↑ 2.0 2.1 2.2 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

Further reading