Scintillation properties of a rectangular dark hollow beam

The scintillation properties of a rectangular dark hollow beam (DHB) in a weak turbulent atmosphere are investigated. Explicit expression for the on-axis scintillation index of a rectangular DHB is derived. It is found that the scintillation index value of a rectangular DHB can be smaller than that of Gaussian, elliptical Gaussian and rectangular flat-topped beams in a weak turbulent atmosphere under certain conditions. Our results will be useful in long-distance free-space optical communications. The scintillation properties of a rectangular DHB are closely controlled by its initial beam parameters.     

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 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.     

Average intensity and spreading of an astigmatic sinh-Gaussian beam with small beam width propagating in atmospheric turbulence

Propagation properties of astigmatic sinh-Gaussian beams (ShGBs) with small beam width in turbulent atmosphere are investigated. Based on the extended Huygens–Fresnel integral, analytical formulae for the average intensity and the effective beam size of an astigmatic ShGB are derived in turbulent atmosphere. The average intensity distribution and the spreading properties of an astigmatic ShGB propagating in turbulent atmosphere are numerically demonstrated. The influences of the beam parameters and the structure constant of atmospheric turbulence on the propagation properties of astigmatic ShGBs are also discussed in detail. In particular, for sufficiently small beam width and sinh-part parameter as well as suitable astigmatism, we show that the average intensity pattern converts into a perfect dark-hollow profile from initial two-petal pattern when ShGBs with astigmatic aberration propagate through atmospheric turbulence.     

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 index of flat-topped Gaussian laser beam in strongly turbulent medium

In a strongly turbulent medium, the scintillation index of flat-topped Gaussian beams is derived and evaluated. In the formulation, unified solution of Rytov method is utilized. Our results correctly reduce to the existing strong turbulence scintillation index of the Gaussian beam, and naturally to spherical and plane wave scintillations. Another checkpoint of our result is the scintillation index of flat-topped Gaussian beams in weak turbulence. Regardless of the order of flatness, scintillations of flat-topped Gaussian beams in strong turbulence are found to be determined mainly by the small-scale effects. For large-sized beams in moderate and strongly turbulent medium, flatter beams exhibit smaller scintillations.     

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.     

Scintillation optimization of linear Gaussian beam array propagating through weak turbulence

The analytical expression for the scintillation index of the linear Gaussian beam array propagating through weak turbulence in the horizontal path is derived, where the coherent combination is considered. By simulating this expression, the on-axis scintillation index of the linear Gaussian beam array is investigated. The results indicate that the beam number could be optimized to reduce the scintillation index of the uniformly-spaced linear beam array, and the optimum beam number decreases with the separation distance between the adjacent beams increasing. With a given beam array length, the scintillation index of the non-uniformly-spaced linear beam array in the far field can be reduced by symmetrically adjusting the locations of the inner beams.     

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.     

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.     

Scintillation of pseudo-Bessel correlated beams in atmospheric turbulence

The concept of pseudo-Bessel correlated beams is introduced, and their scintillation properties on propagation through turbulence are investigated. By using the Rytov approximation, the scintillation index of pseudo-Bessel correlated beams is formulated in weak turbulence. The study of scintillation is extended into strong turbulence by numeric simulations. It is shown that by choosing an appropriate coherence parameter, pseudo-Bessel correlated beams have lower scintillation than comparable fully coherent beams in both weak and strong turbulence. In addition, the configuration of pseudo-Bessel correlated beams is modified by adding a horizontal beamlet; the scintillation properties of these modified beams are also discussed.     

Effect of atmospheric spectrum models on scintillation in moderate turbulence

Experimental studies have shown that a ‘bump’ occurs in the atmospheric spectrum just prior to turbulence cell dissipation. In weak optical turbulence, this bump affects calculated scintillation. The purpose of this study was to determine if a simpler non-bump atmospheric power spectrum can be used to model scintillation for plane waves and spherical waves in moderate to strong optical turbulence regimes. Scintillation expressions were developed from an ‘effective’ von Karman spectrum using an approach similar to that used by Andrews et al. in developing expressions from an effective modified (bump) spectrum. The effective spectrum extends the Rytov approximation into all optical turbulence regimes using filter functions to eliminate mid-range turbulent cell size effects to the scintillation index. Filter cutoffs were established by matching to known weak and saturated scintillation results. The resulting new expressions track those derived from the effective bump spectrum fairly closely. In extremely strong turbulence, differences are minimal.     

Effective beam parameters and the turbulent beam waist for convergent Gaussian beams

Expressions are developed for the location and the size of the beam waist for a convergent Gaussian beam in statistically homogeneous and isotropic atmospheric turbulence. Subsidiary expressions are presented that lead to the maximum distance from the transmitter at which the beam waist can be located under given optical turbulence conditions and the optimal initial radius of curvature required for placing the beam waist at a desired location. The free-space beam radius W of a Gaussian beam satisfies the relationship ?W/?z = - W/R, where z represents the path length and R is the phase-front radius of curvature at z. By enforcing this relation on the effective beam spot size in turbulence W(e), we can define an effective radius of curvature R(e). In addition to specifying the beam waist, R(e) leads to a pair of effective beam parameters θ(e) and Λ(e) that provide a natural extension to the complex amplitude plane. Within this context, general propagation characteristics may be described, including the coherence properties of a Gaussian beam in both weak and strong optical turbulence.     

Observation of the Gouy phase anomaly in astigmatic beams

The Gouy phase anomaly, well established for stigmatic beams, is validated here for astigmatic beams. We simulate the predicted Gouy phase anomaly near astigmatic foci using a beam propagation algorithm integrated within lens design software. We then compare computational results with experimental data acquired using a modified Mertz-Sagnac interferometer. Both in simulation and in experiment, results show that a π/2-phase change occurs as the beam passes through each of the astigmatic foci, experimentally validating results derived in a recent paper by Visser and Wolf [Opt. Commun. 283, 3371-3375 (2010)].     

Comparative study of the beam-width spreading of partially coherent Hermite-sinh-Gaussian beams in atmospheric turbulence

Taking the partially coherent Hermite-sinh-Gaussian (H-ShG) beam as a more general type of partially coherent beams, a comparative study of the beam-width spreading of partially coherent H-ShG beams in atmospheric turbulence is performed by using the relative width, normalized beam width, and turbulence length. It is shown that the relative width versus the beam parameters, such as the spatial correlation length sigma(0), beam orders m, n, Sh-part parameter Omega(0), and waist width w(0), provides a simple and intuitive insight into the beam-width spreading of partially coherent H-ShG beams in turbulence, and the results are consistent with those using the turbulence length. The validity of our results is interpreted physically.     

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.     

Coupling characteristics and kurtosis parameter of partially coherent beams in turbulent atmosphere

Coupling properties and kurtosis parameter (K parameter) of arbitrary beams propagating through atmospheric turbulence are investigated. A correlation factor (C⁴-factor) is introduced to describe the influence of turbulence on coupling characteristics. The general analytical expression for C⁴-factor of arbitrary beams in atmospheric turbulence is derived. It is shown that C⁴-factor of arbitrary beams in the turbulent atmosphere depends on the initial second-order moments and fourth-order moments and turbulence quantities. Taking the partially coherent anomalous elliptical hollow Gaussian (PCAEHG) beam as an example, we can obtain that C⁴-factor decreases as structure constant of the refractive index fluctuations and inner scale increase, and waist width and transverse coherence length decrease when z?>?5?km. Moreover, K parameter of PCAEHG beam in turbulent atmosphere converges to 2 when propagation distance is large enough. It indicates that the profile of PCAEHG beams in turbulent atmosphere finally evolves into fundamental Gaussian distribution.     

Scintillation of initially convergent Gaussian beams in the vicinity of the geometric focus

As an initially convergent Gaussian beam enters the vicinity of the geometric focus, weak fluctuation theory predicts a drop in the longitudinal component of the log-irradiance variance and an increase in the radial component off the beam center. The phenomenon intensifies as the beam nears the geometric focus, also with decreasing magnitude of the focusing parameter. Precisely at the geometric focus, first-order weak fluctuation theory further predicts that as the initial beam size continues to increase, the longitudinal component of the log-irradiance variance decreases toward zero, while the radial component increases without bound. This eventually entails a rapid change in scintillation across the beam surface that has yet to be verified experimentally, to our knowledge. We demonstrate that when diffractionlike effects produced by optical turbulence are introduced, predicted log-irradiance variance exhibits such extremes in behavior only in the case of weak turbulence. Also, at the exact geometric focus, scintillation does not vanish with increasing initial beam size but achieves a value determined by and growing with turbulence strength and nearly independent of initial beam size. The radial component of log-irradiance quickly loses significance as turbulence strength increases. In fact, general extremal behavior of the log-irradiance variance in the vicinity of the geometric focus is drastically curtailed. Differences across the diffractive beam surface become small and exhibit only a modest dependence on the initial beam size.     

On the scintillation index of a multiwavelength Gaussian beam in a turbulent free-space optical communications channel

The scintillation statistics of a multiwavelength Gaussian optical beam are characterized when the beam 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 fluctuations in the signal intensity are due to variations in the refractive index of the medium, which in turn are caused by regional temperature variations due to atmospheric turbulence. Furthermore, 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. The Rytov approximation is used to arrive at the desired results. Furthermore, it is assumed that the first- as well as second-order perturbation terms are present in modeling the impact of atmosphere-induced scintillation. Numerical results are presented to shed light on the performance of multiwavelength optical radiation in weak turbulence and to underscore the benefits of the proposed approach as compared with its single-wavelength counterpart in combating the effect of turbulence. Furthermore, it is shown that if the separation of wavelengths used is sufficiently large, wavelength separation affects the scintillation index in a measurable way.