Propagation of a finite optical beam in an inhomogeneous medium

The first part of this paper is devoted to extending the Huygens-Fresnel principle to a medium that exhibits a spatial (but not temporal) variation in index of refraction. Utilizing a reciprocity theorem for a monochromatic disturbance in a weakly inhomogeneous medium, it is shown that the secondary wave-front will be determined by the envelope of spherical wavelets from the primary wavefront, as in the vacuum problem, but that each wavelet is now determined by the propagation of a spherical wave in the refractive medium. In the second part, the above development is applied to the case in which the index of refraction is a random variable; a further application of the reciprocity theorem results in a formula for the mean intensity distribution from a finite aperture in terms of the complex disturbance in the aperture and the modulation transfer function (MTF) for a spherical wave in the medium. The results are applicable for an arbitrary complex disturbance in the transmitting aperture in both the Fresnel and Fraunhofer regions of the aperture. Using a Kolmogorov spectrum for the index of refraction fluctuations and a second-order expression for the MTF, the formula is used to calculate the mean intensity distribution for a plane wave diffracting from a circular aperture and to give approximate expressions for the beam spreading at various ranges.