# Difference between revisions of "Kirchhoff integral theorem"

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− | '''[[Gustav Kirchhoff|Kirchhoff]]'s integral theorem''' (sometimes referred to as the Fresnel-Kirchhoff integral theorem)<ref>G. Kirchhoff, Ann. d. Physik. 1883, 2, 18, p663</ref> 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 | + | '''[[Gustav Kirchhoff|Kirchhoff]]'s integral theorem''' (sometimes referred to as the Fresnel-Kirchhoff integral theorem)<ref>G. Kirchhoff, Ann. d. Physik. 1883, 2, 18, p663</ref> 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'''.<ref name = "Born and Wolf">Max Born and Emil Wolf, Principles of Optics, 1999, Cambridge University Press, Cambridge, pp. 417-420</ref> |

==Equation== | ==Equation== | ||

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===Monochromatic waves=== | ===Monochromatic waves=== | ||

− | The integral has the following form for a [[monochromatic]] wave:<ref name = "Born and Wolf"/> | + | The integral has the following form for a [[monochromatic]] wave:<ref name = "Born and Wolf"/><ref>Introduction to Fourier Optics J. Goodman sec. 3.3.3</ref> |

− | <ref>Introduction to Fourier Optics J. Goodman sec. 3.3.3</ref | ||

− | |||

− | + | :<math>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,</math> | |

− | + | 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 [[Normal (geometry)#Uniqueness of the normal|outer-pointing normal]] is used, the integral has the opposite sign. | |

===Non-monochromatic waves=== | ===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: | + | 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: |

− | :<math> V(r,t)= \frac {1}{\sqrt{2 \pi}} \int U_ \omega(r) e^{-i \omega t}d \omega</math> | + | :<math>V(r,t) = \frac{1}{\sqrt{2\pi}} \int U_\omega(r) e^{-i\omega t} \,d\omega,</math> |

where, by [[Fourier inversion theorem|Fourier inversion]], we have: | where, by [[Fourier inversion theorem|Fourier inversion]], we have: | ||

− | :<math> U_\omega (r)= \frac {1}{\sqrt{2 \pi}} \int V(r,t) e^{i \omega t}dt</math> | + | :<math>U_\omega(r) = \frac{1}{\sqrt{2\pi}} \int V(r,t) e^{i\omega t} \,dt.</math> |

− | The integral theorem (above) is applied to each Fourier component {{math|U<sub>ω</sub>}}, and the following expression is obtained<ref name = "Born and Wolf"/> | + | The integral theorem (above) is applied to each Fourier component {{math|U<sub>ω</sub>}}, and the following expression is obtained:<ref name="Born and Wolf"/> |

− | :<math>V(r,t) | + | :<math>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,</math> |

− | where the square brackets on ''V'' terms denote retarded values, i.e. the values at time ''t − s/c''. | + | 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's diffraction formula|Kirchhoff, or | + | Kirchhoff showed the above equation can be approximated in many cases to a simpler form, known as the [[Kirchhoff's diffraction formula|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== | ==See also== |

## Latest revision as of 01:14, 13 October 2014

**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