Kirchhoff integral theorem
Kirchhoff's integral theorem (sometimes referred to as the Fresnel-Kirchhoff integral theorem) 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.
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.
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 Uω, and the following expression is obtained
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.
- Kirchhoff's diffraction formula
- Vector calculus
- Huygens–Fresnel principle
- G. Kirchhoff, Ann. d. Physik. 1883, 2, 18, p663
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- Introduction to Fourier Optics J. Goodman sec. 3.3.3
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