Temporal radio frequency spectra of multifrequency waves in a turbulent atmosphere characterized by a complex refractive index

Ishimaru's formulation for the amplitude and phase spectra for radio waves propagating in a turbulent medium is extended to cover the case where the refractive index is complex and frequency dependent. New static limits for the amplitude and phase spectrum are derived. The temporal radio frequency spectra give an estimate of the short time variations (i.e., tenths to tens of seconds) that tend to occur when radio waves propagate in a turbulent medium. For example, the refractive index is complex in a dispersive medium with a Lorentzian line shape, such as the earth's atmosphere over the the range of 50 to 70 GHz, due to resonant frequencies of molecular oxygen. The results for the amplitude and phase spectrum corresponding to radio links on earth (altitudes less than 5 km) indicate that turbulence does not significantly alter the radio frequency spectrum even for radio frequencies separated by 3 GHz. By contrast, the impact of turbulence is to make the details of the spectrum due to an isolated oxygen line in the 60 GHz band time dependent and, therefore, not readily compensated for by an equalizer. An isolated line in the 60 GHz oxygen band does constitute the propagation medium for radio links at altitudes greater than 25 km.