Complex degree of coherence for partially coherent general beams in atmospheric turbulence

With the use of the general beam formulation, the modulus of the complex degree of coherence for partially coherent cosh-Gaussian, cos-Gaussian, Gaussian, annular and higher-order Gaussian optical beams is evaluated in atmospheric turbulence. For different propagation lengths in horizontal atmospheric links, the moduli of the complex degree of coherence at the source and receiver planes are examined when reference points are taken on the receiver axis and off-axis. In the on-axis case, it is observed that in propagation, the moduli of the complex degree of coherence are symmetrical and look like the intensity profile of the related coherent beam propagating in a turbulent atmosphere. For all the beams considered, the moduli of the complex degree of coherence profiles turn into Gaussian shapes beyond certain propagation lengths. In the off-axis case, the moduli of complex degree of coherence patterns become drifted at the earlier propagation lengths. Among the beams investigated, the cos-Gaussian beam is found to be almost independent of the changes in the source partial coherence parameter, and the annular beam seems to be affected the most against the variations of the source partial coherence parameter.     

Spatial correlation properties and correlation vortices of partially coherent vortex beams diffracted by an aperture

Taking the Gaussian–Schell model vortex beam as a typical example of partially coherent vortex beams, the spatial correlation properties and correlation vortices of partially coherent vortex beams diffracted by an aperture are studied. It is shown that the off-axis displacement and spatial coherence affects the spectral degree of coherence. The number and position of correlation vortices depend on the off-axis displacement, spatial coherence, aperture truncation and propagation distance, where the effect of aperture diffraction on the correlation vortices is stressed. The number of correlation vortices decrease as the truncation parameter increases. The correlation vortices in the diffracted field result from the vortex embedded in partially coherent beams at the source plane rather than from the aperture diffraction. The correlation vortices in the diffracted field appear even when the vortex core is stopped by the aperture.     

Scintillations of incoherent flat-topped Gaussian source field in turbulence

The intensity fluctuations of incoherent flat-topped Gaussian beams are evaluated when such sources are used in weakly turbulent horizontal atmospheric links. The formulation is developed for a detector having a response time much longer than the source coherence time. The flat-topped Gaussian profile is obtained by superposing many Gaussian beams, then the incoherence is introduced through delta correlation in space. The scintillation index of the incoherent flat-topped Gaussian beams is found to be smaller than the scintillation index of the corresponding incoherent Gaussian beams at the same link length, source size, and wavelength. When compared with the coherent counterparts, the intensity fluctuations of the incoherent flat-topped Gaussian beams are much smaller, yielding the same value only at the spherical wave limit, as expected. Transmitter aperture averaging is a special case of our solution.     

Propagation of a partially coherent higher-order cosh-Gaussian beam through a paraxial ABCD optical system

Based on the generalized Huygens–Fresnel integral, analytical expressions for the mutual coherence function, the spatial complex degree of coherence, and the effective size of a partially coherent higher-order cosh-Gaussian beam through a paraxial ABCD optical system have been derived. As a numerical example, the propagation of a partially coherent higher-order cosh-Gaussian beam through an optical Fourier-transforming system with a limiting aperture is illustrated. The normalized intensity distribution, the spatial complex degree of coherence, and the effective beam size for the partially coherent higher-order cosh-Gaussian beam are numerically demonstrated in the observation plane. The influences of the spatial coherence length and the limiting aperture on the normalized intensity distribution, the spatial complex degree of coherence, and the effective beam size are also examined in detail.     

Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam

The truncated fractional Fourier transform (FRT) is applied to a partially coherent Gaussian Schell-model (GSM) beam. The analytical propagation formula for a partially coherent GSM beam propagating through a truncated FRT optical system is derived by using a tensor method. Furthermore, we report the experimental observation of the truncated FRT for a partially coherent GSM beam. The experimental results are consistent with the theoretical results. Our results show that initial source coherence, fractional order, and aperture width (i.e., truncation parameter) have strong influences on the intensity and coherence properties of the partially coherent beam in the FRT plane. When the aperture width is large, both the intensity and the spectral degree of coherence in the FRT plane are of Gaussian distribution. As the aperture width decreases, the diffraction pattern gradually appears in the FRT plane, and the spectral degree of coherence becomes of non-Gaussian distribution. As the coherence of the initial GSM beam decreases, the diffraction pattern for the case of small aperture widths gradually disappears.     

Performance analysis of a free-space laser communication system with a Gaussian Schell model

In practice, due to reasons related to the characteristics of the laser device and the inevitable error of the processing technique, a laser beam emitted from a communication terminal is represented by the Gaussian–Schell model. The incident optical intensity at the receiver aperture is affected by the source coherence parameter through atmospheric turbulence. With full consideration of both the average optical intensity and the scintillation, the statistical distribution of the optical intensity and the average bit error rate (BER) of an on-off keying (OOK) receiver is obtained. The results indicate that the effect on the degradation of the average optical intensity is reduced with a smaller beamwidth. The performance of the OOK receiver degrades drastically with the increasing source coherence parameter. Moreover, when the beamwidth becomes larger, the BER₀ with a consistent source coherence parameter shifts to the side of lower BER. The goal of this work is to improve the redundancy design of the laser communication receiver system.     

Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere

Based on the Huygens–Fresnel principle and the unified theory of coherence and polarization of partially coherent beams, we investigate the propagation characteristics of a partially coherent radially polarized doughnut (PCRPD) beam in a turbulent atmosphere. It is found that, after propagating through a turbulent atmosphere, the doughnut beam spot is changed into a circular Gaussian beam. Moreover, the degree of coherence, the degree of polarization and the degree of cross-polarization of the beam will change on propagation, and this change is dependent upon the degree of coherence of the source and atmospheric turbulence.     

Reliability analysis of the flat-topped array beam FSO communication link

Free-space optical communication systems are affected by turbulent atmosphere. The atmospheric transmission is affected by absorption, scattering and turbulence. In this paper, the effects of absorption and scattering are taken into account using Beer’s law and the effects of turbulence are considered in calculating the average intensity distribution. An analytical expression for the average intensity distribution of a partially coherent flat-topped array (PCFTA) beam in turbulent atmosphere is derived based on the extended Huygens–Fresnel principle. The average intensity, power in bucket, signal to noise ratio, and bit error rate of this kind of beam are investigated in details. It is shown by numerical results and analytical methods that the average intensity and link parameters of PCFTA beams change during propagation and these changes are dependent upon both source parameters and weather conditions.     

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.     

Propagation of partially coherent flat-topped beams through a turbulent atmosphere

Based on the modified beam model for flat-topped beams and the Schell model for partially coherent light, an expression for partially coherent flat-topped (PCFT) beams has been proposed. The propagation characteristics of PCFT beams with circular symmetry through a turbulent atmosphere have been studied. By using the generalized Huygens-Fresnel integral and Fourier transform method, the expressions for the cross-spectral density function and the average intensity have been given and the analytical expression for the root-mean-square width has been derived. The effects of the beam order, the spatial coherence, and the turbulent parameter on the intensity distributions and beam spreading have been discussed in detail. Our results show that the on-axis intensity of the beams decreases with increasing turbulence and decreasing coherence of the source, whereas the on-axis intensity of the beams in the far field decreases slightly with increasing beam order. The relative spreading of PCFT beams is smaller for beams with a higher order, a lower degree of global coherence of the source, a larger inner scale, and a smaller outer scale of the turbulence.     

Focal shift and focal switch of partially coherent flat-topped beams passing through an alignment and misalignment lens system with aperture

The focal shift and focal switch of partially coherent flat-topped (PCFT) beams passing through a lens system with an aperture are studied in detail. We have shown that a focal shift is also present for beams propagating through an aligned optical system and the amount of the focal shift depends not only on the radius aperture of the focusing system, but also on the spatial coherence and order of the flat-topped beam of the incident partially coherent light. A new phenomenon called ‘focal switch’ occurs for misaligned optical systems, i.e. the focal shift experiences a sudden transition as the aligned optical system becomes misaligned, and the influence of the spatial coherence, order of flat-topped beam and aperture size reduction, on the focal switch is investigated in detail. Finally, the necessary conditions for the focal switch to take place are investigated.     

Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence

The closed-form expression for the mean-squared beam width of partially coherent Hermite-Gaussian (H-G) beams propagating through atmospheric turbulence is derived. The influence of turbulence on the spreading of partially coherent H-G beams is studied quantitatively by examining the mean-squared beam width. It is found that the smaller the coherence length sigma(0) of the source is, and the larger the beam order m and the wavelength lambda are, the less partially coherent H-G beams are affected by the turbulence, although the beams with smaller sigma(0), larger m, and larger lambda have greater spreading in free space. In addition, it is shown that two partially coherent H-G beams may generate the same angular spread and that there exist equivalent partially coherent H-G beams that may have the same directionality as a fully coherent Gaussian beam in free space and also in turbulence. The results are illustrated by examples, and a comparison with previous work is also made.     

Evolution properties of partially coherent four-petal Gaussian beams in oceanic turbulence

Based on the extended Huygens–Fresnel integral formula, the analytical expressions for partially coherent four-petal Gaussian beam propagating in oceanic turbulence are derived, and the influences of coherence length, beam order N and the parameters of oceanic turbulence (the rate of dissipation of turbulent kinetic energy per unit mass of fluid, the rate of dissipation of mean square temperature and the relative strength of temperature and salinity fluctuations) on average intensity properties are investigated using numerical examples in detail. The results show that the beam with the higher beam order N or coherence length will lose its initial four-petal profiles slower. It is also indicated that the beam will evolve into a Gauss-like beam more rapidly with increasing oceanic turbulence strength. The results have the potential application in underwater laser communication using a partially coherent four-petal Gaussian beam.     

Scattering of a partially coherent Gaussian-Schell beam from a diffuse target in slant atmospheric turbulence

On the basis of the extended Huygens-Fresnel principle, the scattering of partially coherent Gaussian-Schell-model (GSM) beams from a diffuse target in slant double-passage atmospheric turbulence is studied and compared with that of fully coherent Gaussian beams. Using the cross-spectral density function of the GSM beams, we derive the expressions of the mutual coherence function, angle-of-arrival fluctuation, and covariance and variance of the intensity of the scattered field, taking into account the fluctuations of both the log-amplitude and phase. The numerical results are presented, and the influences of the wavelength, propagation distance, and waist radius on scattering properties are discussed. The perturbation region of the normalized intensity variance of the partially coherent GSM beam is smaller than that of the fully coherent Gaussian beam at the middle turbulence level. The normalized intensity variance of long-distance beam propagation is smaller than that of beam propagation along a short distance.     

Propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere

Based on the generalized Huygens–Fresnel integral, propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere was investigated. Analytical propagation formulae were derived for the cross-spectral densities of partially coherent Lorentz and Lorentz–Gauss beams. As an application example, the focusing properties of partially coherent Gaussian, Lorentz and Lorentz–Gauss beams in a turbulent atmosphere and in free space were studied numerically and comparatively. It is found that the focusing properties of such beams are closely related to the initial coherence length and the structure constant of turbulence. By choosing a suitable initial coherence length, a partially coherent Lorentz beam can be focused more tightly than a Gaussian or Lorentz–Gauss beam in free space or in a turbulent atmosphere with small structure constant at the geometrical focal plane.     

Effect of turbulence on the beam quality of apertured partially coherent beams

The effects of turbulence on the beam quality of apertured partially coherent beams have been studied both analytically and numerically. Taking the Gaussian Schell-model (GSM) beam as a typical example of partially coherent beams, closed-form expressions for the average intensity, mean-squared beam width, power in the bucket, beta parameter, and Strehl ratio of apertured partially coherent beams propagating through atmospheric turbulence are derived. It is shown that the smaller the beam truncation parameter is, the less affected by turbulence the apertured partially coherent beams are. Furthermore, the apertured partially coherent beams are less sensitive to the effects of turbulence than unapertured ones. The main results are interpreted physically.     

Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication

A partially coherent quasi-monochromatic Gaussian laser beam propagating in atmospheric turbulence is examined by using a derived analytic expression for the cross-spectral density function. Expressions for average intensity, beam size, phase front radius of curvature, and wave-front coherence length are obtained from the cross-spectral density function. These results provide a model for a free-space laser transmitter with a phase diffuser used to reduce pointing errors.     

Statistical distribution of the optical intensity obtained using a Gaussian Schell model for space-to-ground link laser communications

Abstract

Based on the characteristics of the laser device and the inevitable error of the processing technique, a laser beam emitted from a communication terminal can be represented by the Gaussian Schell model (GSM). In space-to-ground link laser communications, the optical intensity is affected by the source coherence parameter and the zenith angle. With full consideration of these two parameters, the statistical distribution model of the optical intensity with a GSM laser in both downlink and uplink is derived. The simulation results indicate that increasing the source coherence parameter has an effect on the statistical distribution of the optical intensity; this effect is highly similar to the effect of a larger zenith angle. The optical intensity invariably degrades with increasing source coherence parameter or zenith angle. The results of this work can promote the improvement of the redundancy design of a laser communication receiver system.     

Heterodyne lidar returns in the turbulent atmosphere: performance evaluation of simulated systems

Simulations of beam propagation in three-dimensional random media were used to study the effects of atmospheric refractive turbulence on coherent lidar performance. By use of the two-beam model, the lidar return is expressed in terms of the overlap integral of the transmitter and the virtual (backpropagated) local oscillator beams at the target, reducing the problem to one of computing irradiance along the two propagation paths. This approach provides the tools for analyzing laser radar with general refractive turbulence conditions, beam truncation at the antenna aperture, beam-angle misalignment, and arbitrary transmitter and receiver configurations. Simplifying assumptions used in analytical studies, were tested and treated as benchmarks for determining the accuracy of the simulations. The simulation permitted characterization of the effect on lidar performance of the analytically intractable return variance that results from turbulent fluctuations as well as of the heterodyne optical power and system-antenna efficiency.     

M2-factor of truncated partially coherent beams propagating through atmospheric turbulence

A method of studying the M2-factor of truncated partially coherent beams both in free space and in turbulence is proposed, i.e., the method of the window function being expanded into a finite sum of complex-valued Gaussian functions. Taking the Gaussian Schell-model (GSM) beam as a typical example of partially coherent beams, the analytical formula of the M2-factor of truncated GSM beams propagating through atmospheric turbulence is derived. It is shown that the M2-factor decreases as the truncation parameter δ and the coherence parameter α increase. However, the M2-factor in turbulence is more sensitive to δ than that in free space. On the other hand, the M2-factor of truncated partially coherent beams with smaller δ is more affected by turbulence. In addition, the effect of turbulence on the M2-factor of truncated GSM beams is less sensitive to the coherence parameter α than that of nontruncated GSM beams.