Changes in the spectrum of radial array beams through turbulent atmosphere

According to the extended Huygens–Fresnel principle, the propagation formulas of the spectrum for correlated superposition radial array Gaussian Schell-model (RAGSM) beams in a turbulent atmosphere have been derived. The spectral properties of correlated superposition RAGSM beams in free space and in turbulence have also been compared and analyzed. The results show that there exist two identical spectral maxima for the normalized on-axis spectrum of RAGSM beams at one critical position against propagation distance. When the propagation distance is large enough, the on-axis relative spectral shifts in free space and those in turbulence tend to a steady nonzero and zero value, respectively. The off-axis relative spectral shift in free space exhibits a blue-shift alternating with red-shift against off-axis distance, and it depends on the array radius and the beamlet number; whereas the spectral shift has a rapid transition in turbulence. In addition, the on-axis and off-axis relative spectral shifts also depend on the characteristic parameters of turbulence, the beam parameters, and the array parameters.     

涡旋光束轨道角动量在大气湍流传输下的特性分析

从拉盖尔-高斯涡旋光束表达式出发,基于瑞利衍射理论,通过研究涡旋光束在大气湍流中传输时的旋转相干函数的变化规律,总结了涡旋光束在大气湍流中传输时各轨道角动量之间的串扰情况,使用了拓扑荷数探测概率描述串扰规律,并推导了拓扑荷数探测概率的解析表达式。研究了涡旋光束通过湍流后的拓扑荷数的分布情况,并将结果与涡旋光束通过大气随机相位屏的数值仿真结果进行了对比,给出了理论与仿真的拓扑荷数的探测概率随湍流强度以及初始涡旋光束拓扑荷数大小的关系图对比,验证了推导的拓扑荷数探测概率解析表达式的正确性。通过该表达式可进一步研究大气湍流与涡旋光束相互作用从而影响涡旋光束轨道角动量散射的本质,为涡旋光束的空间光通信中选择合适的拓扑荷数间隔,以及在不同湍流强度下选择合适束腰大小以减少串扰带来的误码率提供了理论依据。     

Spectral changes of polychromatic stochastic electromagnetic vortex beams propagating through turbulent atmosphere

The unified theory of coherence and polarization and the propagation law of 2 × 2 cross-spectral density are employed to investigate spectral changes of the polychromatic stochastic electromagnetic vortex beam propagating in turbulent atmosphere. It is shown that the spectral changes of a polychromatic stochastic electromagnetic vortex beam in turbulent atmosphere differ from those of the beam without vortex. Specially, the on-axis relative spectral shifts exhibit not only blue-shift, but also red-shift. It is also shown that the topological charge, the correlation length and the refractive index structure constant influence the spectral changes of polychromatic stochastic electromagnetic vortex beams in a turbulent atmosphere.     

Effects of atmosphere on spectrum of modified partially coherent flat-topped beams

The analytical formulas for the spectrum of modified partially coherent flat-topped (MPCFT) beams propagating in a turbulent atmosphere have been derived under strong fluctuation condition of turbulence. The spectral properties of MPCFT beams propagating in a strong turbulence have been investigated, and the changes of on-axis and off-axis spectral distributions have been analyzed quantitatively. The results show that the on-axis relative spectral shift of the modified partially coherent flat-topped beam exhibits spectral switch for the beam order M > 1, and with the increasing propagation distance the blue-shift gradually decreases in atmospheric turbulence. A rapid transition for the off-axis spectrum appears at one critical position in turbulence, and the position decreases with the increasing beam order for the lower beam order and the nearer propagation distance. The spectral properties and the spectral shifts of MPCFT beams also depend on the inner scale, the spatial coherence parameter.     

Beam spreading of vortex beams propagating in turbulent atmosphere

We present some results obtained by numerical modeling of the propagation of vortex beams LG(0l) through a randomly inhomogeneous medium. The vortex beams are the lower order Laguerre-Gaussian modes. Such beams, if propagated under conditions of weak turbulence, also experience distortions, like a Gaussian beam. However, the statistically averaged vortex beams (LG(0l)) conserve the central intensity dip with a nonzero intensity on the beam axis. The beam broadening of vortex beams is analyzed. The average vortex beams are found to be broadened less than the Gaussian beam while propagated through a randomly inhomogeneous medium. The higher the topological charge l is, the smaller the beam broadening is.     

Partially coherent dark hollow beams propagating through real ABCD optical systems in a turbulent atmosphere

A closed-form analytical expression is derived for a partially coherent dark hollow beam (DHB) propagating through an arbitrary real ABCD optical system in a turbulent atmosphere. The average intensity of the beam in the output plane is investigated in the presence of, respectively, a thin lens image system and a two-lens system along the optical path. For a special thin lens image system, the partially coherent DHB and the fully coherent DHB have the same evolution properties, and the comparative analysis is made between the propagation of the focused DHB and the collimated DHB for direct propagation in turbulence to show the effect of the thin lens on the average intensity. As for the two-lens system, the effects of the lens systems, the structure constant in the turbulent medium and the parameters of the incident beam on the average intensity are evaluated and illustrated. The result shows that different lens systems and propagation parameters can evidently affect the evolution properties of the beam.     

Propagation properties of flat-topped beams in a turbulent atmosphere

Based on the extended Huygens–Fresnel principle and the quadratic approximation for phase structure function, the analytical formulas for the average irradiance of the phase-locked and the non-phase-locked radial modified flat-topped array beams in turbulence have been derived. The propagation properties of radial modified flat-topped array beams have been investigated. It is found that the irradiance distributions of the radial flat-topped array beams propagating in turbulence are different from those in free space. The propagation properties depend on the refractive index structure constant, the beam order and the array radius. The power in the bucket (PIB) for the array beams increases with the decrease of the array radius for a given propagation distance. For a certain bucket radius, there exhibits a PIB maximum upon propagation. The optimal propagation distance becomes shorter along with the increasing refractive index structure constant, and the PIB maximum increases with bucket radius.     

Scintillation behaviour of vortex beams in strong turbulence region

Abstract

We study the scintillation behaviour of vectorial vortex beams in strong turbulence region. For this purpose, a list of vortex source beams is prepared. Then their scintillation performances are analysed one by one using the random phase screen approach. The results indicate that there will always be scintillation reductions with increasing values of topological charge, although its effect will diminish as we go towards higher values of the topological charge. The increases in the other specific beam parameters seem to have opposite effect. For the vectorial Hermite Gaussian beam, the use of higher orders will also aid scintillation reductions. It is foreseen that the outcome of this study will be useful for long haul optical communication links.     

Field correlation of flat-topped beams in anisotropic non-Kolmogorov turbulent atmosphere

Field correlation of flat-topped beams in anisotropic non-Kolmogorov turbulent atmosphere is formulated and evaluated. Larger anisotropic factor causes higher field correlations. Smaller field correlations are seen when the transverse distance at the receiver plane increases. Smaller field correlations are observed at large off-axis transverse receiver points, which are valid for any anisotropic factor and for any power-law exponent of non-Kolmogorov turbulence. When the flat-topped beam is composed of large number of Gaussian beams, the field correlation becomes smaller. In anisotropic non-Kolmogorov turbulence, longer propagation distances, larger structure constants, smaller inner scales and smaller source sizes decrease the field correlation. Larger power law exponent of non-Kolmogorov turbulence increases the field correlations at any anisotropic factor.     

Effect of quadratic radial variation of phase on single slit diffraction of Laguerre–Gauss vortex beams

Effect of quadratic radial phase variation in the plane of the aperture on Fraunhofer diffraction of Laguerre–Gauss vortex beams by a slit is studied experimentally. For slit positions near the incident beam waist, its effect is to shear the diffraction pattern relative to that at the waist. The magnitude and sense of shear depend on the topological charge and slit location relative to the incident beam waist. For slit positions far from the waist, the diffraction pattern evolves to be significantly different and is dominated by two strong peaks. A closed form analytical expression for the diffraction pattern is presented, which reproduces experimental results quite well for all slit positions.     

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.     

Evolution of decentred laser beams propagating through atmospheric turbulence

The changes of the average intensity, the centre of beam gravity and the position of intensity maximum of decentred laser beams propagating through atmospheric turbulence are examined in detail. It is shown that the decentred intensity distribution is amended gradually with increasing the propagation distance and the strength of turbulence, and it becomes an off-axis Gaussian-like beam when the propagation distance and the strength of turbulence become large enough. The centre of beam gravity is independent of both the propagation distance and the strength of turbulence. On the other hand, there are two intensity maxima, and their positions are symmetrical around the propagation z-axis when the propagation distance z is small. With increasing z, there is only one intensity maximum. As z further increases, position of the intensity maximum is further shifted towards the z-axis. When z is large enough, the position of the intensity maximum is unchanged. The unchanged position of the intensity maximum moves further away from the z-axis with an increase in the refraction index structure constant, the decentred parameter and the waist width.     

Propagation based on second-order moments for partially coherent Laguerre–Gaussian beams through atmospheric turbulence

Abstract

The Wigner distribution function (WDF) has been used to study the propagation properties of partially coherent Laguerre Gaussian (PCLG) beams through atmospheric turbulence. Based on the extended Huygens–Fresnel principle, an analytical formula of the propagation matrixes in terms of the second-order moments of the WDF for PCLG Beams in the receiving plane is derived. And then the analytical formulae for the curvature radii of PCLG Beams propagating in turbulence are given by the second-order moments of the WDF. The numerical results indicate that the curvature radius of PCLG Beams changes more rapidly in turbulence than that in the free space. The influence of the transverse coherence width and the beam waist width on the curvature radius of PCLG Beams is obvious, while the laser wavelength and the inner scale of turbulence have a slight effect. The study results may be useful for remote sensing and free space optical communications.     

Conversion of the high-mode solitons in strongly nonlocal nonlinear media

The conversion of high-mode solitons propagating in Strongly Nonlocal Nonlinear Media (SNNM) in three coordinate systems, namely, the elliptic coordinate system, the rectangular coordinate system and the cylindrical coordinate system, based on the Snyder–Mitchell Model that describes the paraxial beam propagating in SNNM, is discussed. Through constituting the trial solution with modulating the Gaussian beam by Ince polynomials, the closed-solution of Gaussian beams in elliptic coordinate is accessed. The Ince–Gaussian (IG) beams constitute the exact and continuous transition modes between Hermite–Gaussian beams and Laguerre–Gaussian (LG) beams, which is controlled by the elliptic parameter. The conditions of conversion in the three types of solitons are given in relation to the Gouy phase invariability in stable propagation. The profiles of the IG breather at a different propagating distance are numerically obtained, and the conversions of a few IG solitons are illustrated. The difference between the IG soliton and the corresponding LG soliton is remarkable from the Poynting vector and phase plots at their profiles along the propagating axis.     

Propagation characteristics of partially coherent anomalous elliptical hollow Gaussian beam propagating through atmospheric turbulence along a slant path

Based on the extended Huygens–Fresnel principal and the Wigner distribution function, the root mean square (rms) angular width and propagation factor (M²-factor) of partially coherent anomalous elliptical hollow Gaussian (PCAEHG) beam propagating through atmospheric turbulence along a slant path are studied in detail. Analytical formulae of the rms angular width and M²-factor of PCAEHG beam are derived. Our results show that the rms angular width increases with increasing of wavelength and zenith angle and with decreasing of transverse coherence length, beam waist sizes and inner scale. The M²-factor increases with increasing of zenith angle and with decreasing of wavelength, transverse coherence length, beam waist sizes and inner scale. The saturation propagation distances (SPDs) increase as zenith angle increases. The numerical calculations also indicate that the SPDs of rms angular width and M²-factor for uplink slant paths with zenith angle of π/12 are about 0.2 and 20 km, respectively.     

Analysing the behaviour of partially coherent divergent Gaussian beams propagating through oceanic turbulence

Theoretical study of propagation behaviour of partially coherent divergent Gaussian beams through oceanic turbulence has been performed. Based on the previously developed knowledge of propagation of a partially coherent beam in atmosphere, the spatial power spectrum of the refractive index of ocean water, extended Huygens–Fresnel principle and the unified theory of coherence and polarization, analytical formulas for cross-spectral density matrix elements are derived. The analytical formulas for intensity distribution, beam width and spectral degree of coherence are determined by using cross-spectral density matrix elements. Then, the effects of some source factors and turbulent ocean parameters on statistical properties of divergent Gaussian beam propagating through turbulent water are analysed. It is found that beam’s statistical propagation behaviour is affected by both environmental and source parameters variations.     

Coherence properties of a random electromagnetic vortex beam propagating in biological tissues

Based on the extended Huygens-Fresnel principle and the unified theory of polarization and coherence, the coherence properties of random electromagnetic Gaussian Schell model (GSM) vortex beams propagating through biological tissues were investigated. It was shown that with an increasing propagation distance, the change in the spectral degree of coherence |μ(ρ, ?ρ, z)| is more complex than that of μ(0, ρ, z). If the change in |μ(ρ, ?ρ, z)| is divided into the preceding and latter stages, then the change in μ(0, ρ, z) is similar to that of |μ(ρ, ?ρ, z)| at the latter stage. The wavelength of the far-infrared beam (λ?=?10.6?µm) is similar to that emitted by the biological tissues, and a resonance with water molecules occurs in the biological tissues. At the initial plane, the larger ρ_x corresponds to a smaller value of |μ(ρ, ?ρ, z)|, and during the propagation process, the spatial coherent interval is wider for |μ(ρ, ?ρ, z)| than for μ(0, ρ, z). The fluctuation change depth of |μ(ρ, ?ρ, z)| in the biological tissues is somewhat shallower than that of μ(0, ρ, z) under the same conditions, such as the same biological tissue, the same wavelength, or the same interval between two field points. In addition, the spatial self-correlation length σ_yy has an effect on |μ(ρ, ?ρ, z)| and on μ(0, ρ, z) to a different extent, while the changes in these two quantities are independent of the spatial mutual-correlation length σ_xy of the random electromagnetic GSM vortex beam.     

Simulation of a premixed turbulent flame with the discrete vortex method

The behaviour of a premixed turbulent flame is numerically studied in this paper. The numerical model is based on solving turbulent flow field by the discrete vortex method. The flame is considered to be of zero thickness boundary which separates burnt and unburnt regions with different constant density and propagates into the fresh mixture at a local curvature‐dependent flame speed. The flame front is located by means of level‐set algorithm. The flow turbulence is simulated through the unsteady vortex‐shedding mechanism. The computed velocity fields, turbulence scalar statistics as well as flame brush thickness for the turbulent V‐flame are well comparable to experimental results. The computed Reynolds stresses in the flame brush region based on unconditioned velocities are substantial, but the two conditioned Reynolds stresses are negligible. These results show that the intermittency effect is a major influence on turbulent statistics in premixed flame and should require careful consideration in numerical models. Copyright © 2000 John Wiley & Sons, Ltd.     

Focal shift of a focused partially coherent Laguerre-Gaussian beam of all orders

Abstract

We explore the focusing properties of a partially coherent Laguerre–Gaussian (LG) beam of all orders, particularly the focal shift (i.e. the shift of the actual focal plane away from the geometrical focal plane). We derive the analytical expressions for the average intensity and the effective beam width of a focused partially coherent LG beam, and we adopt the minimum effective beam width instead of the conventional maximum on-axis intensity to determine the actual focal plane. It is found that the focused beam shape, minimum effective beam width and the focal shift of a focused partially coherent LG beam are determined by its initial coherence width, radial mode order and azimuthal mode order (i.e. topological charge) together. Our results may be useful for optical trapping and micro-fabrication, where precise focal position and prescribed beam shape are required.