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 $U_{\omega }$ , 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 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.

References

↑ G. Kirchhoff, Ann. d. Physik. 1883, 2, 18, p. 663.
1 2 3 Max Born and Emil Wolf, Principles of Optics, 1999, Cambridge University Press, Cambridge, pp. 417–420.
↑ Introduction to Fourier Optics J. Goodman sec. 3.3.3

Kirchhoff integral theorem

Equation

Monochromatic waves

Non-monochromatic waves

See also

References

Further reading