词条 | Kirchhoff integral theorem |
释义 |
EquationMonochromatic wavesThe 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 (a normal derivative). Note that in this equation the normal points to the inner of the enclosed volume; if the more usual outer-pointing normal is used, the integral will have the opposite sign. Non-monochromatic wavesA 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 , 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 that 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
References1. ^G. Kirchhoff, Ann. d. Physik. 1883, 2, 18, p. 663. 2. ^1 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
3 : Diffraction|Optics|Gustav Kirchhoff |
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