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

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, $$\nabla^2 U + k^2U = 0$$ 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 $$dS = s^2 d\Omega$$, $$\oint _{S_1} \left(U_{2}{\partial \over \partial {\hat {\mathbf {n} }}} \frac{e^{iks}}{s} - \frac{e^{iks}}{s} {\partial \over \partial {\hat\mathbf {n} }} U_2 \right)dS = \oint _{S_1} \left(U_2 \left(\frac{ik}{s} - \frac{1}{s^2}\right)e^{iks} - \frac{e^{iks}}{s} {\partial \over \partial {\hat {\mathbf {n} }}} U_2 \right)s^2d\Omega = \oint _{S_1} \left(iksU_2 - U_2 - s \frac{\partial}{\partial \hat{\mathbf{n}}} U_2 \right) e^{iks} d\Omega$$ 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, $$U(\mathbf{r})= \frac{1}{4\pi}\oint _{S} \left(U{\partial \over \partial {\hat {\mathbf {n} }}}\frac{e^{iks}}{s} - \frac{e^{iks}}{s}{\partial \over \partial {\hat {\mathbf {n} }}} U \right)dS.$$