Kirchhoff's diffraction formula^[1][2] (also called Fresnel–Kirchhoff diffraction formula) approximates light intensity and phase in optical diffraction: light fields in the boundary regions of shadows. The approximation can be used to model light propagation in a wide range of configurations, either analytically or using numerical modelling. It gives an expression for the wave disturbance when a monochromatic spherical wave is the incoming wave of a situation under consideration. This formula is derived by applying the Kirchhoff integral theorem, which uses the Green's second identity to derive the solution to the homogeneous scalar wave equation, to a spherical wave with some approximations.

The Huygens–Fresnel principle is derived by the Fresnel–Kirchhoff diffraction formula.

Derivation of Kirchhoff's diffraction formula

Kirchhoff's integral theorem, sometimes referred to as the Fresnel–Kirchhoff integral theorem,^[3] uses Green's second identity to derive the solution of the homogeneous scalar wave equation at an arbitrary spatial position P in terms of the solution of the wave equation and its first order derivative at all points on an arbitrary closed surface ${\displaystyle S}$ as the boundary of some volume including P.

The solution provided by the integral theorem for a monochromatic source is

where ${\displaystyle U}$ is the spatial part of the solution of the homogeneous scalar wave equation (i.e., ${\displaystyle V(\mathbf {r} ,t)=U(\mathbf {r} )e^{-i\omega t}}$ as the homogeneous scalar wave equation solution), k is the wavenumber, and s is the distance from P to an (infinitesimally small) integral surface element, and ${\displaystyle {\frac {\partial }{\partial n}}}$ denotes differentiation along the integral surface element normal unit vector ${\displaystyle n}$ (i.e., a normal derivative), i.e., ${\displaystyle {\frac {\partial f}{\partial n}}=\nabla f\cdot n}$. Note that the surface normal or the direction of ${\displaystyle n}$ 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. And also note that, in the integral theorem shown here, ${\displaystyle n}$ and P are vector quantities while other terms are scalar quantities.

For the below cases, the following basic assumptions are made.

• The distance between a point source of waves and an integral area, the distance between the integral area and an observation point P, and the dimension of opening S are much greater than the wave wavelength ${\displaystyle \lambda }$.
• ${\displaystyle U}$ and ${\displaystyle {\frac {\partial U}{\partial n}}=\nabla U\cdot n}$ are discontinuous at the boundaries of the aperture, called Kirchhoff's boundary conditions. This may be related with another assumption that waves on an aperture (or an open area) is same to the waves that would be present if there was no obstacle for the waves.

Point source

Consider a monochromatic point source at P₀, which illuminates an aperture in a screen. The intensity of the wave emitted by a point source falls off as the inverse square of the distance traveled, so the amplitude falls off as the inverse of the distance. The complex amplitude of the disturbance at a distance ${\displaystyle r}$ is given by

where ${\displaystyle a}$ represents the magnitude of the disturbance at the point source.

The disturbance at a spatial position P can be found by applying the Kirchhoff's integral theorem to the closed surface formed by the intersection of a sphere of radius R with the screen. The integration is performed over the areas A₁, A₂ and A₃, giving

To solve the equation, it is assumed that the values of ${\displaystyle U}$ and ${\displaystyle {\frac {\partial U}{\partial n}}}$ in the aperture area A₁ are the same as when the screen is not present, so at the position Q,

where ${\displaystyle r}$ is the length of the straight line P₀Q, and ${\displaystyle (n,r)}$ is the angle between a straightly extended version of P₀Q and the (inward) normal to the aperture. Note that ${\displaystyle 0<(n,r)<{\frac {\pi }{2}}}$ so ${\displaystyle \cos(n,r)}$ is a positive real number on A₁.

At Q, we also have

where ${\displaystyle s}$ is the length of the straight line PQ, and ${\displaystyle (n,s)}$ is the angle between a straightly extended version of PQ and the (inward) normal to the aperture. Note that ${\displaystyle {\frac {\pi }{2}}<(n,s)<{\frac {3\pi }{2}}}$ so ${\displaystyle \cos <!--\; \; -->(n,s)}$Zdroj:https://en.wikipedia.org?pojem=Kirchhoff's_diffraction_formula
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