Given the recurrence propagation expression, we deduce the approximate analytical expressions of the beam propagation factor M² in terms of the generalized truncated second-order moments. Under an optical system with multiple hard-edged apertures in a cylindrical coordinate system, the recurrence propagation expression is derived for the controllable dark-hollow beams (CDHBs) by expanding the hard-aperture function into a finite sum of complex Gaussian functions. We also briefly discuss some of the currently developing underwater technologies such as imagers, wireless communications, and LIDARs whose operation might be affected by the oceanic turbulence. We overview in detail the recent theoretical studies on interaction of various light waves with the turbulent oceans and outline the corresponding computer simulations and experiments. In addition, for some beam-like waves, with specially prescribed source correlations the spectral composition and the polarization properties can also be shown to change in the oceanic turbulence in a manner different from that in vacuum. If light waves pass through such water volumes their phase statistics and, hence, intensity statistics may become severely affected, causing scintillations, wandering, diffraction additional to that in vacuum, and changes in the state of coherence. Such situations include turbulence in the ocean boundary layer during rains, around thermoclines, in the vicinity of melting ice and underground rivers flowing into open ocean waters, etc. The oceanic optical turbulence is primarily caused by fluctuations in temperature and salinity concentration and may become substantial in regions where the mechanical mixing of cold and warm (and/or fresh and salty) waters occurs. In addition to molecular absorption and scattering from organic and non-organic matter, light waves propagating through oceanic waters may be affected by optical turbulence, i.e., relatively mild and fast variations in the refractive index. It is expected that the results obtained in this paper may be useful for the application of partially coherent beams in tissue imaging and biomedical diagnosis. Moreover, under the same condition, the HGCSM beam is less affected by turbulence than of Gaussian Schell-model (GSM) beam. The larger the beam orders, the fractal dimension, and the small length-scale factor are, or the smaller the transverse coherence width and the characteristic length of heterogeneity are, the smaller the normalized propagation factor is, and the better the beam quality of HGCSM beams in turbulence of biological tissue is. It is shown that the HGCSM beam does not exhibit self-splitting properties on propagation in tissues due to the strong turbulence in the refractive index of biological tissue. ![]() The average intensity, spectral degree of coherence, and the dependence of the propagation factors on the beam orders, transverse coherence width, fractal dimension, characteristic length of heterogeneity, and small length-scale factor are numerically investigated. The propagation characteristics of a Hermite-Gaussian correlated Schell-model (HGCSM) beam in the turbulence of biological tissue are analyzed.
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