Analytical formula is derived for the M(2)-factor of coherent and partially coherent dark hollow beams (DHB) in turbulent atmosphere based on the extended Huygens-Fresnel integral and the second-order moments of the Wigner distribution function. Our numerical results show that the M(2)- factor of a DHB in turbulent atmosphere increases on propagation, which is much different from its invariant properties in free-space, and is mainly determined by the parameters of the beam and the atmosphere. The relative M(2)-factor of a DHB increases slower than that of Gaussian and flat-topped beams on propagation, which means a DHB is less affected by the atmospheric turbulence than Gaussian and flat-topped beams. Furthermore, the relative M(2)-factor of a DHB with lower coherence, longer wavelength and larger dark size is less affected by the atmospheric turbulence. Our results will be useful in long-distance free-space optical communications.

The propagation of phase-locked and non-phase-locked laser array beams of radial and rectangular symmetries in a turbulent atmosphere are investigated based on the extended Huygens-Fresnel integral. The beamlet used in our paper for constructing the laser array beams is of elliptical Gaussian mode. Analytical formulae for the average irradiance of phase-locked and non-phase-locked radial and rectangular laser array beams are derived through vector integration and tensor operation. The irradiance properties of these laser array beams in a turbulent atmosphere are studied numerically. It is found that both phase-locked and non-phase-locked radial and rectangular laser array beams eventually become circular Gaussian beams in a turbulent atmosphere, which is much different from their propagation properties in free space. The propagation properties are closely related to the parameters of laser array beams and the structure constant of the turbulent atmosphere

The scintillation indices of optical plane and spherical waves propagating in underwater turbulent media are evaluated by using the Rytov method, and the variations in the scintillation indices are investigated when the rate of dissipation of mean squared temperature, the temperature and salinity fluctuations, the propagation distance, the wavelength, the Kolmogorov microscale length, and the rate of dissipation of the turbulent kinetic energy are varied. Results show that as in the atmosphere, also in underwater media the plane wave is more affected by turbulence as compared to the spherical wave. The underwater turbulence effect becomes significant at 5-10 m for a plane wave and at 20-25 m for a spherical wave. The turbulence effect is relatively small in deep water and is large at the surface of the water. Salinity-induced turbulence strongly dominates the scintillations compared to temperature-induced turbulence.

In a turbulent atmosphere, starting with a cos-Gaussian excitation at the source plane, the average intensity profile at the receiver plane is formulated. This average intensity profile is evaluated against the variations of link lengths, turbulence levels, two frequently used free-space optics wavelengths, and beam displacement parameters. We show that a cos-Gaussian beam, following a natural diffraction, is eventually transformed into a cosh-Gaussian beam. Combining our earlier results with the current findings, we conclude that cos-Gaussian and cosh-Gaussian beams act in a reciprocal manner after propagation in turbulence. The rates (paces) of conversion in the two directions are not the same. Although the conversion of cos-Gaussian beams to cosh-Gaussian beams can happen over a wide range of turbulence levels (low to moderate to high), the conversion of cosh-Gaussian beams to cos-Gaussian beams is pronounced under relatively stronger turbulence conditions. Source and propagation parameters that affect this reciprocity have been analyzed.

Analytical formulas are derived for the average irradiance and the degree of polarization of a radially or azimuthally polarized doughnut beam (PDB) propagating in a turbulent atmosphere by adopting a beam coherence-polarization matrix. It is found that the radial or azimuthal polarization structure of a radially or azimuthally PDB will be destroyed (i.e., a radially or azimuthally PDB is depolarized and becomes a partially polarized beam) and the doughnut beam spot becomes a circularly Gaussian beam spot during propagation in a turbulent atmosphere. The propagation properties are closely related to the parameters of the beam and the structure constant of the atmospheric turbulence.

The analytical formulae for the wave structure functions (WSF) and the spatial coherence radius of plane and spherical waves propagating through oceanic turbulence are derived. It is found that the Kolmogorov five-thirds power law of WSF is also valid for oceanic turbulence in the inertial range. The changes of the WSF and the spatial coherence radius versus different parameters of oceanic turbulence are examined.

The scintillation index is formulated for a flat-topped Gaussian beam source in atmospheric turbulence. The variations of the on-axis scintillations at the receiver plane are evaluated versus the link length, the size of the flat-topped Gaussian source, and the wavelength at selected flatness scales. The existing source model that represents the flat-topped Gaussian source as the superposition of Gaussian beams is employed. In the limiting case our solution correctly matches with the known Gaussian beam scintillation index. Our results show that for flat-topped Gaussian beams scintillation is larger than that of the single Gaussian beam scintillation when the source sizes are much smaller than the Fresnel zone. However, this trend is reversed and scintillations become smaller than the Gaussian beam scintillations for flat-topped sources with sizes much larger than the Fresnel zone.

For an incidence composed of partially coherent multiple Gaussian beams, Huygens-Fresnel principle-based on-axis scintillation index is formulated in a weakly turbulent homogeneous horizontal atmospheric path. Our general formulation is applied to two examples of partially coherent annular and partially coherent flat-topped Gaussian beams. Compared to partially coherent single Gaussian beam scintillations, annular beam scintillations seem to possess higher values for all partial coherence levels, whereas flat-topped Gaussian beam intensity fluctuations are slightly larger, especially at lower coherence levels and at larger source sizes. At the same source partial coherence, annular beams exhibit smaller scintillations for larger ring sizes. For flat-topped Gaussian beams, except for very small and very large source sizes, as the number of Gaussian beams forming the flatness increases, intensity fluctuations also increase, a trend applicable for different degrees of coherence. A trend valid for both single and multiple Gaussian incidence, except for certain annular beams of large primary beam sizes, is that the scintillations decrease as the source becomes less coherent. Being applicable for all degrees of source coherences, for both beams examined, scintillations increase steadily as the Rytov plane wave scintillation index increases.