Kirchhoff integral theorem

Summary

Kirchhoff's integral theorem (sometimes referred to as the Fresnel–Kirchhoff integral theorem)[1] is a surface integral to obtain the value of the solution of the homogeneous scalar wave equation at an arbitrary point P in terms of the values of the solution and the solution's first-order derivative at all points on an arbitrary closed surface (on which the integration is performed) that encloses P.[2] It is derived by using the Green's second identity and the homogeneous scalar wave equation that makes the volume integration in the Green's second identity zero.[2][3]

Integral edit

Monochromatic wave edit

The integral has the following form for a monochromatic wave:[2][3][4]

 

where the integration is performed over an arbitrary closed surface S enclosing the observation point  ,   in   is the wavenumber,   in   is the distance from an (infinitesimally small) integral surface element to the point  ,   is the spatial part of the solution of the homogeneous scalar wave equation (i.e.,   as the homogeneous scalar wave equation solution),   is the unit vector inward from and normal to the integral surface element, i.e., the inward surface normal unit vector, and   denotes differentiation along the surface normal (i.e., a normal derivative) i.e.,   for a scalar field  . Note that the surface normal is inward, i.e., it is toward the inside of the enclosed volume, in this integral; if the more usual outer-pointing normal is used, the integral will have the opposite sign.

This integral can be written in a more familiar form

 

where  .[3]

Non-monochromatic wave edit

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  , and the following expression is obtained:[2]

 

where the square brackets on V terms denote retarded values, i.e. the values at time ts/c.

Kirchhoff showed that the above equation can be approximated to a simpler form in many cases, known as the Kirchhoff, or Fresnel–Kirchhoff diffraction formula, which is equivalent to the Huygens–Fresnel equation, except that it provides the inclination factor, which is not defined in the Huygens–Fresnel equation. The diffraction integral can be applied to a wide range of problems in optics.

Integral derivation edit

Here, the derivation of the Kirchhoff's integral theorem is introduced. First, the Green's second identity as the following is used.

 
where the integral surface normal unit vector   here is toward the volume   closed by an integral surface  . Scalar field functions   and   are set as solutions of the Helmholtz equation,   where   is the wavenumber (  is the wavelength), that gives the spatial part of a complex-valued monochromatic (single frequency in time) wave expression. (The product between the spatial part and the temporal part of the wave expression is a solution of the scalar wave equation.) Then, the volume part of the Green's second identity is zero, so only the surface integral is remained as
 
Now   is set as the solution of the Helmholtz equation to find and   is set as the spatial part of a complex-valued monochromatic spherical wave   where   is the distance from an observation point   in the closed volume  . Since there is a singularity for   at   where   (the value of   not defined at  ), the integral surface must not include  . (Otherwise, the zero volume integral above is not justified.) A suggested integral surface is an inner sphere   centered at   with the radius of   and an outer arbitrary closed surface  .

Then the surface integral becomes

 
For the integral on the inner sphere  ,
 
and by introducing the solid angle   in  ,
 
due to  . (The spherical coordinate system which origin is at   can be used to derive this equality.)

By shrinking the sphere   toward the zero radius (but never touching   to avoid the singularity),   and the first and last terms in the   surface integral becomes zero, so the integral becomes  . As a result, denoting  , the location of  , and   by  , the position vector  , and   respectively,

 

See also edit

References edit

  1. ^ G. Kirchhoff, Ann. d. Physik. 1883, 2, 18, p. 663.
  2. ^ a b c d Max Born and Emil Wolf, Principles of Optics, 7th edition, 1999, Cambridge University Press, Cambridge, pp. 418–421.
  3. ^ a b c Hecht, Eugene (2017). "Appendix 2: The Kirchhoff Diffraction Theory". Optics (5th and Global ed.). Pearson Education. p. 680. ISBN 978-1292096933.
  4. ^ Introduction to Fourier Optics J. Goodman sec. 3.3.3

Further reading edit

  • 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