Kirchhoff's diffraction formula (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 can be 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, 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 S {\displaystyle S} as the boundary of some volume including P. The solution provided by the integral theorem for a monochromatic source is
U ( P ) = 1 4 π ∫ S [ U ∂ ∂ n ( e i k s s ) − e i k s s ∂ U ∂ n ] d S , {\displaystyle U({\textbf {P}})={\frac {1}{4\pi }}\int _{S}\left[U{\frac {\partial }{\partial {\textbf {n}}}}\left({\frac {e^{iks}}{s}}\right)-{\frac {e^{iks}}{s}}{\frac {\partial U}{\partial {\textbf {n}}}}\right]dS,}
where U {\displaystyle U} is the spatial part of the solution of the homogeneous scalar wave equation (i.e., V ( r , t ) = U ( r ) e − i ω t {\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 ∂ f ∂ n {\displaystyle {\frac {\partial f}{\partial {\textbf {n}}}}} denotes a derivative of a function f {\displaystyle f} along the normal unit vector n {\displaystyle {\textbf {n}}} of an integral surface element d S {\displaystyle dS} (i.e., a normal derivative), i.e., ∂ f ∂ n = ∇ f ⋅ n {\displaystyle {\frac {\partial f}{\partial {\textbf {n}}}}=\nabla f\cdot {\textbf {n}}} . Note that the surface normal unit vector n {\displaystyle {\textbf {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, n {\displaystyle {\textbf {n}}} and P are vector quantities while other terms are scalar quantities. (In below sections, these vector quantities are often written n and P as italic for writing simplicity.) For the below sections, following 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 (aperture) S are much greater than the wave wavelength λ {\displaystyle \lambda } . Waves on the aperture is same to waves that would be present if there was no obstacle (e.g., a screen having the aperture) for the waves.
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