This research began as a continuation of work on the propagation of planar electromagnetic (EM) waves through a turbulent atmosphere, specifically a form of refractive index based phase turbulence modeled by the Modified von Karman Spectrum (MVKS). In the previous work within our group, EM propagation through a turbulent atmosphere under the MVKS model was investigated for essentially isoplanatic propagation, whereby the propagation from the source to the receiver progressed along a horizontal path, such that the effective structure parameter associated with the turbulence remained unchanged along the propagation. The problem was numerically set up by using the split-step propagation model, whereby the EM wave from the source (sometimes interpreted as a planar aperture) propagates alternately through non-turbulent regions (governed by standard Fresnel-Kirchhoff diffraction), and thereafter through MVKS regions where the phase turbulence occurs. A narrowly turbulent layer is described by a random 2D phase screen in the transverse plane; extended turbulence is modeled by a series of planar phase screens placed along the propagation path.
In the above analyses, propagation of both uniform as well as profiled plane waves was considered. The present research commenced with investigating uniform, Gaussian and Bessel beam propagation along a turbulent path, and detailed numerical simulations were carried out relative to infinite as well as finite apertures in the source plane (including single and double slits, and single and double circular apertures), considering both non-turbulent and turbulent paths for comparison. Results were obtained in the near, far and deep far fields.
The problem was further developed to include the case of anisoplanatic plane EM wave propagation over a slanted path. The turbulence structure function (Cn2) in this environment was considered to be altitude dependent, and for this purpose the Hufnagel-Valley (HV) model for the structure function. A standard prototype tested for this system consisted of propagation along a slanted path with a fixed horizontal distance, and made up along the propagation path of a diffractive (LD) and a turbulent (LT) section. The effect of turbulence was examined for test 2D images/transparencies under two environments: (a) the 2D image, under digital encoding converted to time signals, being used to modulate a carrier (typically optical) wave, which is thereafter propagated across the LD+LT path, and recovered in the “image” plane using heterodyne-type communications strategies. Of special note here is the fact that since MVKS and most other turbulence models are intrinsically spatial in nature, a method has been developed within the group whereby the time-statistics of the turbulence is derived from received intensities (typically on-axis) as the phase screen(s) is/are varied at a specific rate corresponding to the average turbulence frequency (in the range 20 Hz-200 Hz). Using this statistical information, the modulated wave propagation across the turbulence is examined; and (b) the source image/transparency is treated as a spatial amplitude distribution through which an unmodulated carrier wave (in the phasor domain) is propagated, and later the object transparency is recovered via a positive thin lens in its back focal plane (assuming thereby that the object transparency is essentially located at infinity relative to the lens). Of the two strategies, it was found that the carrier modulation method yielded better image cross-correlation (CC) products than the method using the thin lens, in the presence of turbulence
Overall, it is seen that recovered EM signals (2D object transparencies, modulated plane waves, and also dynamic/video scenes) are adversely affected by MVKS turbulence (which incidentally is limited in its applicability to only cases where in the Rytov criterion is satisfied, and therefore in many cases works for only weak to moderate levels of turbulence; some cases involving strong turbulence have been investigated nevertheless), and the degree of drop in the CC product goes up as the strength of the turbulence increases. In view of this, a strategy was adopted later whereby the goal was to ascertain if by “packaging” the incident signal/digitized image inside a chaotic carrier, and thereafter propagating the encrypted chaos wave across the turbulent path might help mitigate the loss of CC product (leading to image distortion) during propagation through (MVKS) turbulence. This concept has thereafter been tested for several 2D image scenarios, using an acousto-optically generated chaotic carrier for the encryption prior to turbulent propagation. The corresponding recovered signals (obtained via two levels of demodulation) consistently indicate improvements in the CC products of the recovered images relative to the source. Additionally, the MVKS turbulent system used along the slanted path is also examined under an interchanging of the source and receiver positions.
Following extended examinations of the altitude-dependent propagation along an MVKS turbulent path, this work next focused on an alternative turbulence model, viz., the gamma-gamma turbulence (also refractive index) model, which it turns out actually is valid for all atmospheric turbulence (weak through strong) conditions. The use of the HV model for Cn2 assumes, however, that much of the turbulence considered is within a relatively low-altitude limit. For the gamma-gamma problem as well, applications similar to those used for the MVKS cases (i.e., propagation of modulated EM carriers with message signals transmitted along a turbulent (LT) path) using the gamma-gamma time statistics. This problem was analyzed via numerical simulation for both non-chaotic and chaotic carriers. Once again, use of a chaotic carrier is consistently found to improve the bit error rates (BERs) of the recovered image relative to the source image.