Obliquity factor in Huygens principle

Question

What is the derivation of the obliquity factor (intensity proportional to 1+cosø,where ø is the angle between direction of wave propagation and wavefront tangent at some point on the envelop) used in the field of wave optics under Huygens principle?

Answer 1

Here's a derivation from the most elementary principles, including an abstract and a conclusion: "Consistent derivation of Kirchhoff's integral theorem and diffraction formula and the Maggi-Rubinowicz transformation using high-school math".

A theory of diffraction can be constructed on three principles: causality, superposition, and the assumption that the wave function had a beginning. Given a region R containing no sources and separated from the remaining region R' by a surface S, these principles suffice to show that if the ("primary") sources in R' are replaced by a distribution of ("secondary") sources on S, such that the step-changes in the wave function and its normal derivative, in crossing from R' to R, are respectively equal to the original wave function and its normal derivative on S, then we get the original wave function in R and a null wave function in R' (no backward secondary waves). By further assuming the form of the wave function due to a monopole source, we obtain the distribution of monopole and dipole sources over S that causes the desired step-changes ("saltus conditions"). Adding the contributions from these secondary sources yields an integral expressing the wave function in R in terms of the boundary conditions at the R side of S. The Kirchhoff integral theorem follows by elementary rules of differentiation. In the case of a single monopole primary source, the spatial derivatives in the Kirchhoff integrand can be expressed in terms of angles, yielding the Kirchhoff diffraction formula and its far-field obliquity factor, which reduces to the familiar ${(1+\cos\chi)\big/2}$  if S is a spherical primary wavefront.