Kirchhoff integral theorem

The integral has the following form for a monochromatic wave:{{Cite book|last=Hecht|first=Eugene|title=Optics|publisher=Pearson Education|year=2017| isbn=978-1292096933 | edition=5th and Global|pages=680|chapter=Appendix 2: The Kirchhoff Diffraction Theory}}Introduction to Fourier Optics J. Goodman sec. 3.3.3

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

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

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

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

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

:$V(r,t)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; \backslash int\; U\_\backslash omega(r)\; e^\{-i\backslash omega\; t\}\; \backslash ,d\backslash omega,$

where, by Fourier inversion, we have

:$U\_\backslash omega(r)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; \backslash int\; V(r,t)\; e^\{i\backslash omega\; t\}\; \backslash ,dt.$

The integral theorem (above) is applied to each Fourier component $U\_\backslash omega$, and the following expression is obtained:

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

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.

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

$$\backslash int\_V\; \backslash left(\; U\_1\; \backslash nabla^2\; U\_2\; -\; U\_2\; \backslash nabla^2\; U\_1\backslash right)\; dV\; =\; \backslash oint\_\{\backslash partial\; V\}\; \backslash left(\; U\_2\; \{\backslash partial\; U\_1\; \backslash over\; \backslash partial\; \backslash hat\{\backslash mathbf\{n\}\}\}\; -\; U\_1\; \{\backslash partial\; U\_2\; \backslash over\; \backslash partial\; \backslash hat\{\backslash mathbf\{n\}\}\}\; \backslash right)\; dS,$$

where the integral surface normal unit vector $\backslash hat\{\backslash mathbf\{n\}\}$ here is toward the volume $V$ closed by an integral surface $\backslash partial\; V$. Scalar field functions $U\_1$ and $U\_2$ are set as solutions of the Helmholtz equation, $\backslash nabla^2\; U\; +\; k^2U\; =\; 0$ where $k\; =\; \backslash frac\{2\backslash pi\}\{\backslash lambda\}$ is the wavenumber ($\backslash lambda$ 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

$$\backslash oint\; \_\{\backslash partial\; V\}\; \backslash left(U\_2\; \{\backslash partial\; U\_1\; \backslash over\; \backslash partial\; \{\backslash hat\; \{\backslash mathbf\; \{n\}\; \}\}\}\; -\; U\_1\; \{\backslash partial\; U\_2\; \backslash over\; \backslash partial\; \{\backslash hat\; \{\backslash mathbf\; \{n\}\; \}\}\}\backslash right)dS\; =\; 0.$$

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 $U\_1\; =\; \backslash frac\{e^\{iks\}\}\{s\}$ where $s$ is the distance from an observation point $P$ in the closed volume $V$. Since there is a singularity for $U\_1\; =\; \backslash frac\{e^\{iks\}\}\{s\}$ at $P$ where $s=0$ (the value of $\backslash frac\{e^\{iks\}\}\{s\}$ 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

$$\backslash oint\; \_\{S\_1\}\; \backslash left(U\_2\; \{\backslash partial\; \backslash over\; \backslash partial\; \backslash hat\backslash mathbf\{n\}\; \}\backslash frac\{e^\{iks\}\}\{s\}\; -\; \backslash frac\{e^\{iks\}\}\{s\}\{\backslash partial\; \backslash over\; \backslash partial\; \{\backslash hat\; \{\backslash mathbf\; \{n\}\; \}\}\}\; U\_2\; \backslash right)\; dS\; +\; \backslash oint\; \_\{S\_2\}\; \backslash left(U\_2\; \{\backslash partial\; \backslash over\; \backslash partial\; \backslash hat\backslash mathbf\; n\; \}\; \backslash frac\{e^\{iks\}\}\{s\}\; -\; \backslash frac\{e^\{iks\}\}\{s\}\{\backslash partial\; \backslash over\; \backslash partial\; \backslash hat\backslash mathbf\{n\}\; \}\; U\_2\; \backslash right)dS\; =\; 0.$$

For the integral on the inner sphere $S\_1$,

$$\backslash frac\{\backslash partial\}\{\backslash partial\backslash hat\{\backslash mathbf\{n\}\}\}\; \backslash frac\{e^\{iks\}\}\{s\}\; =\; \backslash nabla\; \backslash frac\{e^\{iks\}\}\{s\}\; \backslash cdot\; \backslash hat\{\backslash mathbf\{n\}\}\; =\; \backslash left(\backslash frac\{ik\}\{s\}\; -\; \backslash frac\{1\}\{s^2\}\backslash right)\; e^\{iks\},$$

and by introducing the solid angle $d\backslash Omega$ in $dS\; =\; s^2\; d\backslash Omega$,

$$\backslash oint\; \_\{S\_1\}\; \backslash left(U\_\{2\}\{\backslash partial\; \backslash over\; \backslash partial\; \{\backslash hat\; \{\backslash mathbf\; \{n\}\; \}\}\}\; \backslash frac\{e^\{iks\}\}\{s\}\; -\; \backslash frac\{e^\{iks\}\}\{s\}\; \{\backslash partial\; \backslash over\; \backslash partial\; \{\backslash hat\backslash mathbf\; \{n\}\; \}\}\; U\_2\; \backslash right)dS\; =\; \backslash oint\; \_\{S\_1\}\; \backslash left(U\_2\; \backslash left(\backslash frac\{ik\}\{s\}\; -\; \backslash frac\{1\}\{s^2\}\backslash right)e^\{iks\}\; -\; \backslash frac\{e^\{iks\}\}\{s\}\; \{\backslash partial\; \backslash over\; \backslash partial\; \{\backslash hat\; \{\backslash mathbf\; \{n\}\; \}\}\}\; U\_2\; \backslash right)s^2d\backslash Omega\; =\; \backslash oint\; \_\{S\_1\}\; \backslash left(iksU\_2\; -\; U\_2\; -\; s\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash hat\{\backslash mathbf\{n\}\}\}\; U\_2\; \backslash right)\; e^\{iks\}\; d\backslash Omega$$

due to $\backslash frac\{\backslash partial\}\{\backslash partial\backslash hat\{\backslash mathbf\{n\}\}\}\; U\_2\; =\; \backslash nabla\; U\_2\; \backslash cdot\; \backslash hat\{\backslash mathbf\{n\}\}\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; s\}U\_2$. (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), $e^\{iks\}\; \backslash to\; 1$ and the first and last terms in the $S\_1$ surface integral becomes zero, so the integral becomes $-4\; \backslash pi\; U\_2$. As a result, denoting $U\_2$, the location of $P$, and $S\_2$ by $U$, the position vector $\backslash mathbf\{r\}$, and $S$ respectively,

$$U(\backslash mathbf\{r\})=\; \backslash frac\{1\}\{4\backslash pi\}\backslash oint\; \_\{S\}\; \backslash left(U\{\backslash partial\; \backslash over\; \backslash partial\; \{\backslash hat\; \{\backslash mathbf\; \{n\}\; \}\}\}\backslash frac\{e^\{iks\}\}\{s\}\; -\; \backslash frac\{e^\{iks\}\}\{s\}\{\backslash partial\; \backslash over\; \backslash partial\; \{\backslash hat\; \{\backslash mathbf\; \{n\}\; \}\}\}\; U\; \backslash right)dS.$$

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

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Category:Physical optics

Category:Gustav Kirchhoff