Scintillation index of a multiwavelength beam in turbulent atmosphere

We characterize the scintillation index of a multiwavelength plane-wave optical beam that is subjected to a turbulent optical channel. It is assumed that the level of turbulence in the atmosphere ensures a weak-turbulence scenario and that the turbulence is due to the fluctuations in the index of refraction of the medium. It is assumed that the propagation path is nearly horizontal and that the heights of the transmitter and receiver justify a near-ground propagation assumption.     

Bit error rate of a Gaussian beam propagating through biological tissue

ABSTRACT

The scintillation index and bit error rate (BER) of a Gaussian beam propagating in a weakly turbulent soft tissue are formulated and analysed numerically. The scintillation indices are plotted against half of the measured slope in the range of power-law scaling at different tissue parameters, such as the random variations in the refractive index of the tissue, outer scale of the tissue turbulence and the tissue length between the optical source and the detector. Moreover, BERs of Gaussian beams against the signal to noise ratio (SNR) are examined for different tissue parameters. Our graphical results show that the scintillation index and BER increase with larger outer scales, longer tissue lengths and larger random variations in the refractive index of the tissue. In comparison with the spherical wave propagation, it was found that Gaussian beam yields larger scintillation index and BER values.     

Scintillation index and bit error rate of hollow Gaussian beams in atmospheric turbulence

Based on the Huygens–Fresnel principle and Rytov method, the on-axis scintillation index is derived for hollow Gaussian beams (HGBs) in weak turbulence. The relationship between bit error rate (BER) and scintillation index is found by only considering the effect of atmosphere turbulence based on the probability distribution of intensity fluctuation, and the expression of the BER is obtained. Furthermore, the scintillation and the BER properties of HGBs in turbulence are discussed in detail. The results show that the scintillation index and BER of HGBs depend on the propagation length, the structure constant of the refractive index fluctuations of turbulence, the wavelength, the beam order and the waist width of the fundamental Gaussian beam. The scintillation index, increasing with the propagation length in turbulence, for the HGB with higher beam order increases more slowly. The BER of the HGBs increases rapidly against the propagation length in turbulence. For propagating the same distance, the BER of the fundamental Gaussian beam is the greatest, and that of the HGB with higher order is smaller.     

Scintillations of cos-Gaussian and annular beams

Based on the generalized beam formulation, we derive the scintillation index and selectively evaluate it for cos-Gaussian and annular beams propagating in weak atmospheric turbulence. Dependence of the scintillation index on propagation length, focusing and displacement parameters, wavelength of operation, and source size are individually investigated. From our graphical outputs, it is observed that a cos-Gaussian beam exhibits lower scintillations and thus has a tendency to be advantageous over a pure Gaussian beam particularly at lower propagation lengths. It is also found that at longer propagation lengths, this advantage switches to the side of the annular beam. Furthermore, the scintillation index of a focused annular beam will be below those of both Gaussian and cos-Gaussian beams starting at earlier propagation distances. When analyzed against source sizes, it is seen that cos-Gaussian beams will offer advantages at relatively large source sizes, while the reverse will be applicable for annular beams.     

Scintillation index of optical spherical wave propagating through biological tissue

Effects of the tissue turbulence on the propagation of an optical spherical wave are analysed. For this purpose, scintillation index of an optical spherical wave which is propagating in a soft tissue is formulated and evaluated in weakly turbulent soft tissue. Scintillation index of the optical spherical wave is examined against the changes in the tissue parameters which are the tissue length between the optical spherical wave source and the detector, random variations in the refractive index of the tissue and the outer scale of the tissue turbulence. According to our graphical outputs, it is observed that increase in the random variations of the refractive index of the tissue results in an increase in the scintillation index at a certain realization of the turbulence spectrum. On the other hand, larger outer scales and longer tissue lengths yield larger scintillations. The variation of the scintillation index of the optical spherical wave versus the wavelength is also investigated. It is found that at small tissue lengths, wavelength has almost no effect on the scintillations; however, when the tissue length reaches a certain value, shorter wavelengths give rise to larger intensity fluctuations.     

Light propagation through anisotropic turbulence

A wealth of experimental data has shown that atmospheric turbulence can be anisotropic; in this case, a Kolmogorov spectrum does not describe well the atmospheric turbulence statistics. In this paper, we show a quantitative analysis of anisotropic turbulence by using a non-Kolmogorov power spectrum with an anisotropic coefficient. The spectrum we use does not include the inner and outer scales, it is valid only inside the inertial subrange, and it has a power-law slope that can be different from a Kolmogorov one. Using this power spectrum, in the weak turbulence condition, we analyze the impact of the power-law variations α on the long-term beam spread and scintillation index for several anisotropic coefficient values ?. We consider only horizontal propagation across the turbulence cells, assuming circular symmetry is maintained on the orthogonal plane to the propagation direction. We conclude that the anisotropic coefficient influences both the long-term beam spread and the scintillation index by the factor ?(2-α).     

Scintillation of astigmatic dark hollow beams in weak atmospheric turbulence

The scintillation properties of astigmatic dark hollow beams (DHBs) in weak atmospheric turbulence were investigated in detail. An explicit expression for the on-axis scintillation index of an astigmatic DHB propagating in weak atmospheric turbulence was derived. It was found that the scintillation index value of an astigmatic DHB with suitable astigmatism (i.e., ratio of the beam waist size in the x direction to that in the y direction), dark size, beam waist size, and wavelength can be smaller than that of a stigmatic DHB and that of stigmatic and astigmatic flat-topped, annular, and Gaussian beams in weak atmospheric turbulence particularly at long propagation ranges. Our results will be useful in long-distance free-space optical communications.     

A transport model for broadening of a linearly polarized,coherent beam due to inhomogeneities in a turbulent atmosphere

Traditional models for beam broadening include both a diffractive term, owing to the source aperture, and a beam ‘wandering’ term that stems from refractive index variations in the atmosphere. Here we derive a novel beam broadening term that depends on the properties of atmospheric turbulence. The derivation rests on a transport formulation of the propagation problem whereby the magnitude of the electric field is viewed as the density of a fluid, moving in a flow that is driven by the refractive index perturbations. Properties of the transport solutions are obtained using Lagrangian coordinates and are demonstrated to be entirely consistent with existing theory on the subject. The new factor predicts appreciable (25% in our example) increases in beam broadening for applications requiring propagation over very long optical paths and heavy turbulence.     

Scintillation and beam-wander analysis in an optical ground station-satellite uplink

In an optical communication link between an optical ground station and a geostationary satellite the main problems appear in the uplink and are due to beam wander and to scintillation. Reliable methods for modeling both effects simultaneously are needed to provide an accurate tool with which the robustness of the communication channel can be tested. Numerical tools, especially the split-step method (also referred to as the fast-Fourier-transform beam propagation method), have demonstrated their ability to deal with problems of optical propagation during atmospheric turbulence. However, obtaining statistically significant results with this technique is computationally intensive. We present an analytical-numerical hybrid technique that provides good information on the variance in optical irradiance with an important saving of time and computational resources.     

Common omissions and misconceptions of wave propagation in turbulence: discussion

This review paper addresses typical mistakes and omissions that involve theoretical research and modeling of optical propagation through atmospheric turbulence. We discuss the disregard of some general properties of narrow-angle propagation in refractive random media, the careless use of simplified models of turbulence, and omissions in the calculations of the second moment of the propagating wave. We also review some misconceptions regarding short-exposure imaging, propagation of polarized waves, and calculations of the scintillation index of the beam waves.     

Scintillation characteristics of cosh-Gaussian beams

By using the generalized beam formulation, the scintillation index is derived and evaluated for cosh-Gaussian beams in a turbulent atmosphere. Comparisons are made to cos-Gaussian and Gaussian beam scintillations. The variations of scintillations against propagation length at different values of displacement and focusing parameters are examined. The dependence of scintillations on source size at different propagation lengths is also investigated. Two-dimensional scintillation index distributions covering the entire transverse receiver planes are given. From the graphic illustrations, it is found that in comparison to pure Gaussian beams cosh-Gaussian beams have lower on-axis scintillations at smaller source sizes and longer propagation distances. The focusing effect appears to impose more reduction on the cosh-Gaussian beam scintillations than those of the Gaussian beam. The distribution of the off-axis scintillation index values of the Gaussian beams appears to be uniform over the transverse receiver plane, whereas that of the cosh-Gaussian beam is arranged according to the position of the slanted axis.     

Scintillation index for two Gaussian laser beams with different wavelengths in weak atmospheric turbulence

We study the propagation of the two lowest-order Gaussian laser beams with different wavelengths in weak atmospheric turbulence. Using the Rytov approximation and assuming a slow detector, we calculate the longitudinal and radial components of the scintillation index for a typical free-space laser communication setup. We find the optimal configuration of the two laser beams with respect to the longitudinal scintillation index. We show that the value of the longitudinal scintillation for the optimal two-beam configuration is smaller by more than 50% compared with the value for a single lowest-order Gaussian beam with the same total power. Furthermore, the radial scintillation for the optimal two-beam system is smaller by 35%-40% compared with the radial scintillation in the single-beam case. Further insight into the reduction of intensity fluctuations is gained by analyzing the self- and cross-intensity contributions to the scintillation index.     

Laser beam propagation in atmospheric turbulence

The optical effects of atmospheric turbulence on the propagation of low power laser beams are reviewed in this paper. The optical effects are produced by the temperature fluctuations which result in fluctuations of the refractive index of air. The commonly-used models of index-of-refraction fluctuations are presented. Laser beams experience fluctuations of beam size, beam position, and intensity distribution within the beam due to refractive turbulence. Some of the observed effects are qualitatively explained by treating the turbulent atmosphere as a collection of moving gaseous lenses of various sizes. Analytical results and experimental verifications of the variance, covariance and probability distribution of intensity fluctuations in weak turbulence are presented. For stronger turbulence, a saturation of the optical scintillations is observed. The saturation of scintillations involves a progressive break-up of the beam into multiple patches; the beam loses some of its lateral coherence. Heterodyne systems operating in a turbulent atmosphere experience a loss of heterodyne signal due to the destruction of coherence.     

Scintillation index of flat-topped Gaussian beams

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.     

Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence

Results obtained on the intensity fluctuations of flat-topped Gaussian beams in weakly turbulent non-Kolmogorov horizontal atmospheric optics links are represented. Effects on the scintillation index of the power law α that describes the non-Kolmogorov spectrum are examined. Our results correctly reduce to the existing intensity fluctuations of flat-topped beams in Kolmogorov turbulence. Variation of the scintillation index against non-Kolmogorov power law α exhibits a peak at the worst power law α(w), which happens to be smaller than the Kolmogorov power law of 11/3. If the power law is smaller (larger) than α(w), increase in α will increase (decrease) the intensity fluctuations. Evaluation of the scintillation index at the worst power law results in smaller fluctuations for a Gaussian beam at short propagation distances; however, at long propagation distances flatter beams happen to possess smaller fluctuations. The scintillation change versus the source size follows a similar trend regardless whether the flat-topped beam propagates in a Kolmogorov or non-Kolmogorov medium.     

Effects of time averaging on optical scintillation in a ground-to-satellite atmospheric propagation

Temporal natures for a variance of turbulence-induced log-intensity fluctuations are obtained. The variance of the optical fluctuation is reduced when the optical signals are integrated in a photodetector, and we express the index of reduction (called the time-averaging factor) by using an autocovariance function of the optical fluctuation. The optical fluctuations for a ground-to-satellite path are caused by both atmospheric turbulence and the beam-pointing jitter error of the optical transmitter. The turbulence-induced optical scintillation can be discriminated from the fluctuation that is due to the beam-pointing jitter error. The compared result from the probability density function of the optical signal reveals good agreement. The temporal autocovariance functions of optical scintillation are obtained and used to calculate the time-averaging factor. The analytically expected effects of time averaging are verified by the experimental results. The estimations contribute to the link budget design for the optical tracking channel through atmospheric turbulence.     

Inferring path average Cn2 values in the marine environment

Current mathematical scintillation theory describing laser propagation through the atmosphere has been developed for terrestrial environments. Scintillation expressions valid in all regimes of optical turbulence for propagation in the maritime environment, based on what we believe to be a newly developed marine atmospheric spectrum, have been developed for spherical waves. Path average values of the structure parameter, C(n)(2), were inferred from optical scintillation measurements of a diverged laser beam propagating in a marine environment, using scintillation expressions based on both terrestrial and marine refractive index spectra. In the moderate-to-strong fluctuation regime, the inferred marine C(n)(2) values were about 20% smaller than inferred terrestrial C(n)(2) values, but a minimal difference was observed in the weak fluctuation regime. Measurements of angle-of-arrival fluctuations were used to infer C(n)(2) values in the moderate-to-strong fluctuation regime, resulting in values of the structure parameter that were at least an order of magnitude larger than the two scintillation-inferred C(n)(2) values.     

Effects of finite inner and outer scales on the scintillation index of turbulent slant path

In this paper, the effects of nonzero inner scales and finite outer scales are investigated, in the context of Gaussian beam propagation along a slant path under general turbulence conditions. Theoretical expressions for the cut-off spatial frequencies are derived with an approach method, and thereby a modified scintillation model is developed to incorporate inner scale and outer scale parameters in the analysis. Then, inner and outer scale effects on the downlink are analysed with respect to the zenith angle, the altitude of the transmitter, the initial beam radius, as well as the turbulence strength. Numerical results indicate that the effects of a finite outer scale mainly influence transmission that occurs at large zenith angles or high altitudes, while the inner scale effects are more prevalent. This study may be helpful to improve the accuracy of calculation of slant-path scintillation index, and thus benefit the characterization and optimization of space/air-ground laser communication systems.     

Beam scintillations for ground-to-space propagation. Part 2: Gaussian beam scintillation

On the basis of the analytic techniques presented in the first of these two companion papers [J. Opt. Soc. Am. A27, 2169 (2010)] we present the complete asymptotic analysis of the axial beam scintillation index for coherent Gaussian beams on the ground-to-space propagation paths. The ratio of turbulence layer thickness to overall propagation path length contributes an additional small parameter to the analysis. We show that it is possible to use three dimensionless parameters to describe the problem and that the general arrangement of the asymptotic regions established in our earlier work [Waves Random Media 4, 243 (1994)]) is preserved. We find that on a slant propagation path, collimated beams can experience the unusual double-scattering-dominated scintillation found originally for focused beams.     

Polarization changes in a partially coherent radially polarized doughnut beam through turbulent ocean

On the basis of the extended Huygens–Fresnel integral principle and unified theory of coherence and polarization of light, we studied the effects of oceanic turbulence on polarization properties of a partially coherent radially polarized doughnut (PCRPD) beam. The ocean-induced fluctuations in the refractive index are assumed be driven by temperature and salinity fluctuations. Numerical examples of changes in polarization properties, such as the degree of polarization, the degree of ellipticity, and the orientation angle in the oceanic turbulence for the PCRPD beam, are given. Our analysis demonstrates how polarization of the PCRPD beam is affected by statistical properties of the source and by several parameters of oceanic turbulence. We find that the propagation of the PCRPD beam is different from that of stochastic beams in oceanic turbulence. The degree of polarization for the PCRPD beam approaches a certain steady value, and the elliptical polarized state of the fully polarized portion of the beam will become fully linear in the far field.