Beam wander of J 0- and I 0-Bessel Gaussian beams propagating in turbulent atmosphere

Root mean square (rms) beam wander of J ₀-Bessel Gaussian and I ₀-Bessel Gaussian beams, normalized by the rms beam wander of the fundamental Gaussian beam, is evaluated in atmospheric turbulence. Our formulation is based on the first and the second statistical moments obtained from the Rytov series. It is found that after propagating in atmospheric turbulence, the collimated J ₀-Bessel Gaussian and the I ₀-Bessel Gaussian beams have smaller rms beam wander than that of the Gaussian beam, regardless of the choice of Bessel width parameter. However, the extent of such an advantage depends on the chosen width parameter, Gaussian source size, propagation distance and the wavelength. Focusing at finite distances of the considered beams causes the rms beam wander to decrease sharply at the propagation distances equal to the focusing parameter.     

Effective Rayleigh range of Gaussian array beams propagating through atmospheric turbulence

The analytical expressions for the effective Rayleigh range z_R of Gaussian array beams in turbulence for both coherent and incoherent combinations are derived. It is shown that z_R of Gaussian array beams propagating through atmospheric turbulence depends on the strength of turbulence, the array beam parameters and the type of beam combination. For the coherent combination z_R decreases due to turbulence. However, for the incoherent combination there exists a maximum of z_R as the strength of turbulence varies. The z_R of coherently combined Gaussian array beams is larger than that of incoherently combined Gaussian array beams, but for the coherent combination case z_R is more sensitive to turbulence than that for the incoherent combination case. The larger the beam number is, the longer z_R is, and the more z_R is affected by turbulence. For the coherent combination z_R is not monotonic versus the relative beam separation distance, and the effect of turbulence on z_R is appreciable within a certain range of the relative beam separation distance.     

Polarization fluctuations of Gaussian Schell-model type photon beams in a slant channel with non-Kolmogorov turbulence

Polarization properties of Gaussian Schell-mode type photon beams propagating through the non-Kolmogorov turbulence in a slant channel are studied which are based on the model of quantum field and the effective photon annihilation/creation operator. Our numerical results show that the degree of polarization increases with the increasing of the transverse coherent width of source ρ_s0, the source's transverse size ω₀ and the power law of the turbulent spectrum, but the degree of polarization P decreases as the zenith angle θ increases. The effect of the wavelength of light beam on the degree of polarization of Gaussian Schell-model beams is small.     

Average capacity for non-Kolmogorov turbulent slant optical links with beam wander corrected and pointing errors

The effects of beam wander corrected and pointing errors on the average capacity of a non-Kolmogorov turbulent atmosphere communication channel are studied. The gamma–gamma irradiance probability density function model has been considered to include the effects of non-Kolmogorov turbulence. An expression for the average capacity under beam wander corrected, pointing errors and the spectral index of refractive-index fluctuation. Simulation results for effects of circular detection aperture of radius, atmosphere turbulence strength and wavelength on average capacity are presented. Our results show that the average capacity has an oscillation in the case of short propagation distance, but in the case of long propagation distance, it has gradually reduced curves. And the turbulence strength has a strong influence on average capacity. In general, turbulence reduces the average capacity; it can reduce average capacity by increasing of the power law α. We can ignore the little effects of wavelength increases on the average capacity.     

Beam wander of dark hollow, flat-topped and annular beams

Benefiting from the earlier derivations for the Gaussian beam, we formulate beam wander for dark hollow (DH) and flat-topped (FT) beams, also covering the annular Gaussian (AG) beam as a special case. Via graphical illustrations, beam wander variations of these beams are analyzed and compared among themselves and to the fundamental Gaussian beam against changes in propagation length, amplitude factor, source size, wavelength of operation, inner and outer scales of turbulence. These comparisons show that in relation to the fundamental Gaussian beam, DH and FT beams will exhibit less beam wander, particularly at small primary beam source sizes, lower amplitude factors of the secondary beam and higher beam orders. Furthermore, DH and FT beams will continue to preserve this advantageous position all throughout the considered range of wavelengths, inner and outer scales of turbulence. FT beams, in particular, are observed to have the smallest beam wander values among all, up to certain source sizes.     

Beam quality of truncated laser beams with amplitude modulations and phase fluctuations in turbulence

The closed-form expressions for the Rayleigh range z_R and the M²-factor of truncated laser beams with amplitude modulations (AMs) and phase fluctuations (PFs) in turbulence are derived, and the beam quality is studied by taking the z_R and the M²-factor as the characteristic parameters of beam quality. The M²-factor of truncated laser beams with AMs and PFs is always larger than that of truncated Gaussian beams both in free space and in turbulence. However, in turbulence the beam quality of truncated laser beams with AMs and PFs may be better than that of truncated Gaussian beams if the z_R is taken as the characteristic parameter of beam quality. For laser beams with AMs and PFs in turbulence, the beam quality expressed in terms of z_R is consistent with that in terms of the M²-factor versus the phase fluctuation parameter α, but not versus the intensity modulation parameter σ_A. The beam quality of truncated laser beams with AMs and PFs is less sensitive to turbulence than that of truncated Gaussian beams. The beam quality of laser beams with smaller α and larger σ_A is less affected by turbulence than those with larger α and smaller σ_A.     

Influence of turbulence on the beam propagation factor of Gaussian Schell-model array beams

The analytical expression for the beam propagation factor (M²-factor) of Gaussian Schell-model (GSM) array beams propagating through atmospheric turbulence is derived. It is shown that the M²-factor of GSM array beams depends on the beam number, the relative beam separation distance, the beam coherence parameter, the type of beam superposition, and the strength of turbulence. The turbulence results in an increase of the M²-factor. However, for the superposition of the intensity the M²-factor is less sensitive to turbulence than that for the superposition of the cross-spectral density function. The M²-factor of GSM array beams is larger than that of the corresponding Gaussian array beams. However, the M²-factor of GSM array beams is less affected by turbulence than that of the corresponding Gaussian array beams. For the superposition of the cross-spectral density function a minimum of the M²-factor of GSM array beams may appear in turbulence, which is even smaller than that of the corresponding single GSM beams.     

大气湍流对径向分布高斯列阵光束扩展和方向性的影响

采用积分变换的技巧,推导出了径向分布高斯列阵光束通过湍流大气传输的二阶矩束宽和远场发散角的解析公式,并详细研究了大气湍流对光束扩展和方向性的影响.研究表明,相干合成情况下,子光束数N越小、径向分布半径r₀越大,列阵光束扩展受湍流影响越小.相干较非相干合成时列阵光束的扩展小,但非相干合成时列阵光束扩展受湍流的影响比相干合成时的小.特别地,N足够小或r₀足够大时,相干与非相干合成列阵光束的远场束宽相等.另一方面,还给出了相干和非相干合成径向分布高斯列阵 关键词： 径向分布高斯列阵光束 大气湍流 相干和非相干合成 二阶矩束宽     

Turbulence distance of radial Gaussian Schell-model array beams

The effect of turbulence on the spreading of radial Gaussian Schell-model (GSM) array beams is studied quantitatively by examining the mean-squared beam width. The analytical expression for the turbulence distance z _T of radial GSM array beams is derived by using the integral transform technique, which indicates within what ranges radial GSM array beams will be less affected by turbulence. It is shown that the effect of turbulence on the spreading of radial GSM array beams can be reduced by choosing the suitable array beam parameters and the type of the beam superposition. In addition, a comparison with the previous work is also made.     

Block error rate of FSO links over non-Kolmogorov turbulence with exponentiated Weibull distribution

In this paper, we model block error rate (BLER) performance of optical wireless communication systems in strong turbulence slant channels with very slow fading and additive Gaussian noise, using the Exponentiated Weibull distribution models. In a communication system, which transmits data in blocks of N bits, the BLER probability P(M, N) of more than M bit errors in a block is particularly useful in evaluating the wireless optical channel performance. The joint effects of the beam wander and spread, pointing errors and the spectral index of non-Kolmogorov turbulence on system's performance are included. The obtained results can be essential for the designing of such links under real circumstances.     

Influence of atmospheric turbulence on the propagation of superimposed partially coherent Hermite-Gaussian beams

The influence of atmospheric turbulence on the propagation of superimposed partially coherent Hermite-Gaussian (H-G) beams is studied in detail. The closed-form propagation equation of superimposed partially coherent H-G beams through atmospheric turbulence is derived. It is shown that the turbulence accelerates the evolution of three stages which superimposed partially coherent H-G beams undergo. The turbulence results in a beam spreading and a decrease of the maximum intensity. However, the larger the beam number M, the beam order m, the separate distance x_d, and the smaller the beam correlation length σ₀ are, the less the power focusability of superimposed partially coherent H-G beams is affected by the turbulence. Specially, superimposed partially coherent H-G beams are less sensitive to turbulence than superimposed fully ones, and than partially coherent H-G beams if the beam power focusability and the maximum intensity are taken as beam criterions. However, the maximum intensity of superimposed partially coherent H-G beams is less sensitive or more sensitive to turbulence than that of superimposed Gaussian Schell-model (GSM) beams depending on σ₀.     

大气湍流中光束束宽扩展和角扩展的比较研究

以厄米-双曲余弦-高斯（H-ChG）光束为例,对H-ChG光束通过大气湍流传输时的束宽扩展和角扩展做了详细研究.用相对束宽和相对角扩展代替束宽和角扩展来研究湍流对光束影响的灵敏程度.研究表明,折射率结构常数C²_n越小,光束束宽扩展和角扩展越小.有较大阶数m,n,较小参数Ω₀和束腰宽度w₀ H-ChG光束的角扩展受湍流影响较小.当传输距离足够远时,这一结论对H-ChG光束的束宽扩展也成立.当传输距离不长时,对H-ChG光束相对束宽随Ω₀和w₀的变化规律做了分析.用数值计算例做了说明,并对结果的正确性做了物理解释.厄米-高斯,双曲余弦高斯和高斯光束在大气湍流中的扩展可作为H-ChG光束的特例来处理. 关键词： 束宽扩展和角扩展 大气湍流 厄米-双曲余弦-高斯光束     

Polarization fluctuations of partially coherent Hermite–Gaussian beams in a slant turbulent channel

The propagation of a partially coherent Hermite–Gaussian beam in a slant turbulent channel is studied. The analytical formula for the quantum degree of polarization of a partially coherent Hermite–Gaussian beam is derived based on the quantum Stokes operators and the generalized Huygens–Fresnel principle. It is shown that the zenith angle slightly affects the polarization degree of partially coherent Hermite–Gaussian beams, and the changes of polarization degree are affected by the coherence length, the detection photon-number and the ground refractive index structure parameter. Furthermore, the numerical simulations show that a partially coherent Hermite–Gaussian beam with bigger coherence length, higher photon-number level, lower beam order, shorter wavelength is less affected by the turbulence. These results indicate that one can choose the partially coherent Hermite–Gaussian beam with bigger coherence length, lower beam order, shorter wavelength, higher detection photon-number and set the size of transmitting aperture w₀$w_0$ as about 0.065 m to improve the performance of a polarization-encoded free-space quantum communication system.     

Spreading of spatially partially coherent polychromatic beams in atmospheric turbulence

Taking the polychromatic Gaussian Schell-model (GSM) beam as a typical example of spatially partially coherent polychromatic beams, the spreading of polychromatic GSM beams in atmospheric turbulence is studied. The mean-squared width of polychromatic GSM beams in turbulence is derived by using the effective source and the strong fluctuation models. It is shown that the same result is obtained using both the models. The diffraction, atmospheric turbulence and beam polychroism result in a spreading of polychromatic GSM beams. If the scaling law fails, the spreading of polychromatic GSM beams increases with increasing bandwidth Γ, but the influence of Γ on the spreading of polychromatic GSM beams becomes small as the structure constant C_n² of the refractive index and spatial correlation parameter α increase. The spreading of polychromatic GSM beams increases as C_n² increases and α decreases. Spatially partially coherent polychromatic beams are less sensitive to the effects of atmospheric turbulence than spatially fully coherent polychromatic beams.     

Characteristics of a partially coherent Gaussian Schell-model beam propagating in slanted atmospheric turbulence

On the basis of the extended Huygens-Fresnel principle and the model of the refractive-index structure constant in the atmospheric turbulence proposed by the International Telecommunication Union-Radio Communication Sector,the characteristics of the partially coherent Gaussian Schell-model(GSM) beams propagating in slanted atmospheric turbulence are studied.Using the cross-spectral density function(CSDF),we derive the expressions for the effective beam radius,the spreading angle,and the average intensity.The variance of the angle-of-arrival fluctuation and the wander effect of the GSM beam in the turbulence are calculated numerically.The influences of the coherence degree,the propagation distance,the propagation height,and the waist radius on the propagation characteristics of the partially coherent beams are discussed and compared with those of the fully coherent Gaussian beams.     

Propagation analysis of flattened circular Gaussian beams with a circular aperture in turbulent atmosphere

The propagation properties of flattened Gaussian beam with aperture in turbulent atmosphere have been studied by using the extended Huygens-Fresnel principle. From the study and numerical calculation, the effects of aperture on the propagation of flattened Gaussian beams in turbulent atmosphere have been illuminated. It shows that when the value of the truncation parameter δ is bigger, for example δ?2, the effects of aperture on the propagation properties are too small to be neglected. But when the truncation parameter δ is smaller, for example δ<2, the effects of aperture are complex. The peak value of the average intensity descends more rapidly and the beam spot spreads quicker with aperture than that without aperture when the propagation distance increases. Meanwhile, with the propagation distance increasing, the average intensity profiles of flattened Gaussian beams gradually convert into Gaussian average intensity profiles. In addition, some limiting cases are also discussed. It agrees with the existing results.     

The effects of tilt corrected residual aberration on the detection probability of single-photon propagation in a slant path turbulent atmosphere

Based on the assumption of a pulse laser beam with an initial Gaussian temporal shape and a collimated fundamental-model Gaussian beam, the Rytov approximation and Kolmogorov spectrum model for the index-of-refraction fluctuation of atmosphere, the effect of turbulence on the probability density, acquisition transmittance probability, transmittance probability density, acquisition probability of single-photon propagation in atmospheric communication channel with z-tilt and centroid-tilt aberration corrected are studied theoretically. The probability density, acquisition transmittance probability, transmittance probability density and acquisition probability models for single-photon propagation in uplink path and downlink path are derived. Our results shown that the detection probability and the acquisition transmittance probability of the single-photon are obvious increase, when the beams are propagation in the z-tilt corrected communication channel.     

部分相干艾里光在大气湍流中的光束扩展与漂移

基于Andrews和Philips的经典漂移方差模型,结合部分相干艾里光的交叉谱密度函数和Tatarskii功率谱,推导出部分相干艾里光在大气中传输时的光束扩展和漂移解析式,对比分析了不同的衰减因子、湍流强度和相干度等参数对部分相干艾里光的光束扩展和漂移的影响,并与高斯光束所得结果进行了比较.研究表明:衰减因子、湍流强度和相干度等对光束漂移影响较大,当它们的取值增大时,漂移越严重;内尺度对光束漂移几乎没有影响;在相同的条件下,艾里光束的光束漂移要比高斯光束小两个数量级,且其本身具有自恢复特性和较强的抑制湍流特性.     

Influence of oceanic turbulence on propagation characteristics of Gaussian array beams

Based on the power spectrum of oceanic turbulence proposed by Nikishov, the influence of oceanic optical turbulence on propagation of Gaussian array beams is studied in detail by using five propagation characteristic parameters (i.e., mean-squared beam width, Rayleigh range, turbulence distance, power in the bucket and Strehl ratio). It is shown that the influence of oceanic optical turbulence on propagation of laser beams becomes stronger as τ (ratio of temperature to salinity contribution to the refractive index spectrum) and χ_T (rate of dissipation of mean-squared temperature) increase and ɛ (rate of dissipation of kinetic energy per unit mass of fluid) decreases, which is in agreement with the further analysis of oceanic turbulence shown in this paper. Furthermore, it is concluded that propagation of laser beams is more affected by oceanic optical turbulence in abyssal region than that in active region or oceanic surface.     

Bit error rate analysis of MISO FSO systems

Abstract

Multiple-input single-output (MISO) systems are employed in free space optical (FSO) links to mitigate the degrading effects of atmospheric turbulence. In this paper, we consider a MISO FSO system with practical transmitter and receiver configuration that consists of radial laser array with Gaussian beams and a Gaussian receiver aperture function. We have employed our previously derived formulation of the power scintillation in which Huygens–Fresnel principle was employed. Therefore, we choose system parameters within the range of validity of the wave structure functions. Using the on-off keying modulation and the log-normal probability distribution function, we quantify the average bit error rate (〈BER〉) of laser array beams in weak turbulence. It is observed that the radial array beams at the transmitter are more advantageous than the single Gaussian beam. However, increasing the number of array beamlets to more than three seems to have negligible effects on 〈BER〉 . It is further observed that 〈BER〉 decreases when the source size, the ring radius and the receiver aperture radius increase.