Kirchhoff integral theorem

Short description: Method to solve scalar wave equation

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 Green's second identity and the homogeneous scalar wave equation that makes the volume integration in Green's second identity zero.[2][3]

## Contents

* 1 Integral
* 1.1 Monochromatic wave
* 1.2 Non-monochromatic wave
* 1.3 Integral derivation
* 2 See also
* 3 References
* 4 Further reading

## Integral

### Monochromatic wave

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

U(𝐫)=14π∫S[U∂∂𝐧^(eikss)−eikss∂U∂𝐧^]dS,

where the integration is performed over an arbitrary closed surface S enclosing the observation point 𝐫, k in eiks is the wavenumber, s in eikss is the distance from an (infinitesimally small) integral surface element to the point 𝐫, U is the spatial part of the solution of the homogeneous scalar wave equation (i.e., V(𝐫,t)=U(𝐫)e−iωt as the homogeneous scalar wave equation solution), 𝐧^ is the unit vector inward from and normal to the integral surface element, i.e., the inward surface normal unit vector, and ∂∂𝐧^ denotes differentiation along the surface normal (i.e., a normal derivative) i.e., ∂f∂𝐧^=∇f⋅𝐧^ for a scalar field f. 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

U(𝐫)=14π∫S(U∇(eikss)−eikss∇U)⋅dS→,

where dS→=dS𝐧^.[3]

### Non-monochromatic wave

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)=12π∫Uω(r)e−iωtdω,

where, by Fourier inversion, we have

Uω(r)=12π∫V(r,t)eiωtdt.

The integral theorem (above) is applied to each Fourier component Uω, and the following expression is obtained:[2]

V(r,t)=14π∫S{[V]∂∂n(1s)−1cs∂s∂n[∂V∂t]−1s[∂V∂n]}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 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

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

∫V(U1∇2U2−U2∇2U1)dV=∮∂V(U2∂U1∂𝐧^−U1∂U2∂𝐧^)dS, where the integral surface normal unit vector 𝐧^ here is toward the volume V closed by an integral surface ∂V. Scalar field functions U1 and U2 are set as solutions of the Helmholtz equation, ∇2U+k2U=0 where k=2πλ is the wavenumber (λ 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 ∮∂V(U2∂U1∂𝐧^−U1∂U2∂𝐧^)dS=0. Now U2 is set as the solution of the Helmholtz equation to find and U1 is set as the spatial part of a complex-valued monochromatic spherical wave U1=eikss where s is the distance from an observation point P in the closed volume V. Since there is a singularity for U1=eikss at P where s=0 (the value of eikss not defined at s=0), the integral surface must not include P. (Otherwise, the zero volume integral above is not justified.) A suggested integral surface is an inner sphere S1 centered at P with the radius of s1 and an outer arbitrary closed surface S2.

Then the surface integral becomes ∮S1(U2∂∂𝐧^eikss−eikss∂∂𝐧^U2)dS+∮S2(U2∂∂𝐧^eikss−eikss∂∂𝐧^U2)dS=0. For the integral on the inner sphere S1, ∂∂𝐧^eikss=∇eikss⋅𝐧^=(iks−1s2)eiks, and by introducing the solid angle dΩ in dS=s2dΩ, ∮S1(U2∂∂𝐧^eikss−eikss∂∂𝐧^U2)dS=∮S1(U2(iks−1s2)eiks−eikss∂∂𝐧^U2)s2dΩ=∮S1(iksU2−U2−s∂∂𝐧^U2)eiksdΩ due to ∂∂𝐧^U2=∇U2⋅𝐧^=∂∂sU2. (The spherical coordinate system which origin is at P can be used to derive this equality.)

By shrinking the sphere S1 toward the zero radius (but never touching P to avoid the singularity), eiks->1 and the first and last terms in the S1 surface integral becomes zero, so the integral becomes −4πU2. As a result, denoting U2, the location of P, and S2 by U, the position vector 𝐫, and S respectively, U(𝐫)=14π∮S(U∂∂𝐧^eikss−eikss∂∂𝐧^U)dS.

## See also

* Kirchhoff's diffraction formula
* Vector calculus
* Integral
* Huygens–Fresnel principle
* Wavefront
* Surface integral

## References

1. ↑ G. Kirchhoff, Ann. d. Physik. 1883, 2, 18, p. 663.
2. ↑ 2.0 2.1 2.2 2.3 Max Born and Emil Wolf, Principles of Optics, 7th edition, 1999, Cambridge University Press, Cambridge, pp. 418–421.
3. ↑ 3.0 3.1 3.2 Hecht, Eugene (2017). "Appendix 2: The Kirchhoff Diffraction Theory". Optics (5th and Global ed.). Pearson Education. pp. 680. ISBN 978-1292096933.
4. ↑ Introduction to Fourier Optics J. Goodman sec. 3.3.3

## 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

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