Variance in image fluctuations in propagation through a turbulent medium

As a result of turbulence in the atmosphere, the image of an object in the focal plane of a lens fluctuates with time. In this paper we discuss the variance of the image intensity, a measure of this fluctuation. In order to calculate the variance, we require a knowledge of the fourth-order coherence function of the light impinging upon the lens. We shall assume, in our analysis, that the object is located far above the earth, emits incoherent light with Gaussian statistics and is small enough to insure isoplanicity. Under these conditions, the fourth-order coherence function is the product of two factors. The first factor is the fourth-order coherence function resulting from a point source passing through a turbulent medium. The second factor is the fourth-order coherence function of the light from the incoherent object after it propagates through free space. The fourth-order coherence function of the light emitted by a Gaussian incoherent source is easily determined. On the other hand, it is very difficult to determine the fourth-order coherence function of the light from a point source passing through a turbulent medium, unless it is possible to neglect intensity fluctuations at the lens or to use perturbation theory. Therefore, in this paper we shall discuss what is known about the general case and when approximations may be made. In conclusion, we shall calculate the variance of the image of a point source when intensity fluctuations at the lens are neglected.