Kirchhoff's diffraction formula

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.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 surfaceas the boundary of some volume including P. The solution provided by the integral theorem for a monochromatic source isis the spatial part of the solution of the homogeneous scalar wave equation (i.e.,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, anddenotes differentiation along the integral surface element normal unit vectorConsider 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.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.in the aperture area A1 are the same as when the screen is not present, so at the position Q,is the angle between a straightly extended version of P0Q and the (inward) normal to the aperture.is the angle between a straightly extended version of PQ and the (inward) normal to the aperture.The contribution from the hemisphere A3 to the integral is expected to be zero, and it can be justified by one of the following reasons.As a result, finally, the integral above, which represents the complex amplitude at P, becomesThe Huygens–Fresnel principle can be derived by integrating over a different closed surface (the boundary of some volume having an observation point P).The area A1 above is replaced by a part of a wavefront (emitted from a P0) at r0, which is the closest to the aperture, and a portion of a cone with a vertex at P0, which is labeled A4 in the right diagram.On this new A1, the inward (toward the volume enclosed by the closed integral surface, so toward the right side in the diagram) normalwhere the integral is done over the part of the wavefront at r0 which is the closest to the aperture in the diagram.This integral leads to the Huygens–Fresnel principle (with the obliquity factorAssume that the aperture is illuminated by an extended source wave.in A2 are zero (Kirchhoff's boundary conditions) and that the contribution from A3 to the integral are also zero.This is the most general form of the Kirchhoff diffraction formula.To solve this equation for an extended source, an additional integration would be required to sum the contributions made by the individual points in the source.If, however, we assume that the light from the source at each point in the aperture has a well-defined direction, which is the case if the distance between the source and the aperture is significantly greater than the wavelength, then we can writeIn spite of the various approximations that were made in arriving at the formula, it is adequate to describe the majority of problems in instrumental optics.This is mainly because the wavelength of light is much smaller than the dimensions of any obstacles encountered.Analytical solutions are not possible for most configurations, but the Fresnel diffraction equation and Fraunhofer diffraction equation, which are approximations of Kirchhoff's formula for the near field and far field, can be applied to a very wide range of optical systems.One of the important assumptions made in arriving at the Kirchhoff diffraction formula is that r and s are significantly greater than λ.Another approximation can be made, which significantly simplifies the equation further: this is that the distances P0Q and QP are much greater than the dimensions of the aperture.