Analytic solution for the two‐frequency mutual coherence function for spherical wave propagation

The quadratic approximation for the phase structure function is used to obtain the two‐frequency mutual coherence function Γ(Δ ρ , Δ ω ) for spherical wave propagation through a finite slab with transmitter and receiver located in free space on opposite sides of the slab. General analytic solutions are derived for two cases. In the first case the random slab is represented by a one‐dimensional power spectrum of electron density fluctuations corresponding to propagation through elongated irregularities as would occur for an equatorial satellite link to a ground station. In the second case the random slab consists of isotropic ionization irregularities. Both cases taken together represent the extremes of the range of results to be expected for propagation through ionospheric fluctuations, solar wind irregularities, or ionization irregularities caused by barium cloud instabilities. For both cases the complex general analytic results are simplified by use of the thin phase screen approximation to obtain useful analytic expressions for Γ as well as the resulting impulse response function. It is shown that the impulse response to a transmitted power delta function reduces to an exponential form in the limit of strong diffraction and to a Gaussian form in the geometrical optics limit. The Gaussian form corresponds to pulse wander while the exponential form corresponds to diffractive spreading produced by multipath effects. The relationship between the generalized power spectrum and the impulse response function is given, and results are presented for the mean time delay and time delay jitter. The accuracy of the thin phase screen calculation of these quantities is investigated in detail.