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 U_ω, 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

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

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

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