Kirchhoff integral theorem

Kirchhoff's integral theorem (sometimes referred to as the Fresnel–Kirchhoff integral theorem) 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. 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.

Integral

Monochromatic wave

The integral has the following form for a monochromatic wave:

$U(\mathbf {r} )={\frac {1}{4\pi }}\int _{S}\left[U{\frac {\partial }{\partial {\hat {\mathbf {n} }}}}\left({\frac {e^{iks}}{s}}\right)-{\frac {e^{iks}}{s}}{\frac {\partial U}{\partial {\hat {\mathbf {n} }}}}\right]dS,$

where the integration is performed over an arbitrary closed surface S enclosing the observation point $\mathbf {r}$, $k$ in $e^{iks}$ is the wavenumber, $s$ in ${\frac {e^{iks}}{s}}$ is the distance from an (infinitesimally small) integral surface element to the point $\mathbf {r}$, $U$ is the spatial part of the solution of the homogeneous scalar wave equation (i.e., $V(\mathbf {r} ,t)=U(\mathbf {r} )e^{-i\omega t}$ as the homogeneous scalar wave equation solution), ${\hat {\mathbf {n} }}$ is the unit vector inward from and normal to the integral surface element, i.e., the inward surface normal unit vector, and ${\frac {\partial }{\partial {\hat {\mathbf {n} }}}}$ denotes differentiation along the surface normal (i.e., a normal derivative) i.e., ${\frac {\partial f}{\partial {\hat {\mathbf {n} }}}}=\nabla f\cdot {\hat {\mathbf {n} }}$ 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

$U(\mathbf {r} )={\frac {1}{4\pi }}\int _{S}\left(U\nabla \left({\frac {e^{iks}}{s}}\right)-{\frac {e^{iks}}{s}}\nabla U\right)\cdot d{\vec {S}},$

where $d{\vec {S}}=dS{\hat {\mathbf {n} }}$.

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

$V(r,t)={\frac {1}{\sqrt {2\pi }}}\int U_{\omega }(r)e^{-i\omega t}\,d\omega ,$

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:

$V(r,t)={\frac {1}{4\pi }}\int _{S}\left\{[V]{\frac {\partial }{\partial n}}\left({\frac {1}{s}}\right)-{\frac {1}{cs}}{\frac {\partial s}{\partial n}}\left[{\frac {\partial V}{\partial t}}\right]-{\frac {1}{s}}\left[{\frac {\partial V}{\partial n}}\right]\right\}dS,$

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 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.

$$\int _{V}\left(U_{1}\nabla ^{2}U_{2}-U_{2}\nabla ^{2}U_{1}\right)dV=\oint _{\partial V}\left(U_{2}{\partial U_{1} \over \partial {\hat {\mathbf {n} }}}-U_{1}{\partial U_{2} \over \partial {\hat {\mathbf {n} }}}\right)dS,$$ where the integral surface normal unit vector ${\hat {\mathbf {n} }}$ here is toward the volume $V$ closed by an integral surface $\partial V$. Scalar field functions $U_{1}$ and $U_{2}$ are set as solutions of the Helmholtz equation, $\nabla ^{2}U+k^{2}U=0$ where $k={\frac {2\pi }{\lambda }}$ is the wavenumber ($\lambda$ 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 $$\oint _{\partial V}\left(U_{2}{\partial U_{1} \over \partial {\hat {\mathbf {n} }}}-U_{1}{\partial U_{2} \over \partial {\hat {\mathbf {n} }}}\right)dS=0.$$ Now $U_{2}$ is set as the solution of the Helmholtz equation to find and $U_{1}$ is set as the spatial part of a complex-valued monochromatic spherical wave $U_{1}={\frac {e^{iks}}{s}}$ where $s$ is the distance from an observation point $P$ in the closed volume $V$. Since there is a singularity for $U_{1}={\frac {e^{iks}}{s}}$ at $P$ where $s=0$ (the value of ${\frac {e^{iks}}{s}}$ 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 $S_{1}$ centered at $P$ with the radius of $s_{1}$ and an outer arbitrary closed surface $S_{2}$.

Then the surface integral becomes $$\oint _{S_{1}}\left(U_{2}{\partial \over \partial {\hat {\mathbf {n} }}}{\frac {e^{iks}}{s}}-{\frac {e^{iks}}{s}}{\partial \over \partial {\hat {\mathbf {n} }}}U_{2}\right)dS+\oint _{S_{2}}\left(U_{2}{\partial \over \partial {\hat {\mathbf {n} }}}{\frac {e^{iks}}{s}}-{\frac {e^{iks}}{s}}{\partial \over \partial {\hat {\mathbf {n} }}}U_{2}\right)dS=0.$$ For the integral on the inner sphere $S_{1}$, $${\frac {\partial }{\partial {\hat {\mathbf {n} }}}}{\frac {e^{iks}}{s}}=\nabla {\frac {e^{iks}}{s}}\cdot {\hat {\mathbf {n} }}=\left({\frac {ik}{s}}-{\frac {1}{s^{2}}}\right)e^{iks},$$ and by introducing the solid angle $d\Omega$ in $dS=s^{2}d\Omega$, $$\oint _{S_{1}}\left(U_{2}{\partial \over \partial {\hat {\mathbf {n} }}}{\frac {e^{iks}}{s}}-{\frac {e^{iks}}{s}}{\partial \over \partial {\hat {\mathbf {n} }}}U_{2}\right)dS=\oint _{S_{1}}\left(U_{2}\left({\frac {ik}{s}}-{\frac {1}{s^{2}}}\right)e^{iks}-{\frac {e^{iks}}{s}}{\partial \over \partial {\hat {\mathbf {n} }}}U_{2}\right)s^{2}d\Omega =\oint _{S_{1}}\left(iksU_{2}-U_{2}-s{\frac {\partial }{\partial {\hat {\mathbf {n} }}}}U_{2}\right)e^{iks}d\Omega$$ due to ${\frac {\partial }{\partial {\hat {\mathbf {n} }}}}U_{2}=\nabla U_{2}\cdot {\hat {\mathbf {n} }}={\frac {\partial }{\partial s}}U_{2}$. (The spherical coordinate system which origin is at $P$ can be used to derive this equality.)

By shrinking the sphere $S_{1}$ toward the zero radius (but never touching $P$ to avoid the singularity), $e^{iks}\to 1$ and the first and last terms in the $S_{1}$ surface integral becomes zero, so the integral becomes $-4\pi U_{2}$. As a result, denoting $U_{2}$, the location of $P$, and $S_{2}$ by $U$, the position vector $\mathbf {r}$, and $S$ respectively, $$U(\mathbf {r} )={\frac {1}{4\pi }}\oint _{S}\left(U{\partial \over \partial {\hat {\mathbf {n} }}}{\frac {e^{iks}}{s}}-{\frac {e^{iks}}{s}}{\partial \over \partial {\hat {\mathbf {n} }}}U\right)dS.$$

See also

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

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