Mutual coherence function for a double-passage retroreflected optical wave in atmospheric turbulence

The mutual coherence function in the source-receiver plane of a reflected Gaussian beam wave from a retroreflector is calculated and analyzed for two refractive-index spectral models and compared with similar results for the case of a plane mirror. Specific expressions are calculated for the mean irradiance and spatial coherence radius based on a Gaussian model for the finite reflector. Results that we obtained here using a modified spectrum with a high wave-number rise and inner scale generally show greater amplitude enhancements in the reflected wave than predicted by the pure power-law spectrum of Kolmogorov. In contrast, a finite outer scale in the spectral model leads to a reduction in the amount of beam spreading caused by turbulence and, in the case of a retroreflector, also leads to a reduction in the peak amplitude enhancement on the optical axis. This last result is in contrast with a plane mirror reflector, in which outer scale effects tend to increase the peak amplitude enhancement on the optical axis. The theory also predicts that, except for small reflectors, the coherence radius associated with a retroreflector can be as much as 1.4 times larger than that associated with a plane mirror, and 1.2 times that of a bistatic configuration for a plane mirror. All calculations are based on weak fluctuation theory and generalized spectral representations that use complex ABCD ray matrices.     

Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence

A new generalized modified atmospheric spectral model is derived theoretically for wave propagating through non-Kolmogorov turbulence, which has been reported recently by increasing experimental evidence and theoretical investigation. The generalized, modified atmospheric spectrum considers finite turbulence inner and outer scales and has a spectral power law value in the range of 3 to 5 instead of the standard power law value of 11/3. When the inner scale and outer scale are set to zero and infinity, respectively, this spectral model is reduced to the classical non-Kolmogorov spectrum.     

Statistics of the fractal structure and phase singularity of a plane light wave propagation in atmospheric turbulence

Numerical experiments are carried out for a plane wave propagating in the atmospheric turbulence for a weak to strong fluctuation condition, i.e., the Rytov index being in a large range of 2x10(-3) to 20. Mainly two categories of propagation events are explored for the same range of Rytov index. In one category the propagation distance and also the Fresnel length are kept fixed with the turbulence strength changing. In the other the turbulence strength is kept fixed with the distance changing. The statistical characteristics of the scintillation index, the maximum and minimum of the intensity, the fractal dimension of the intensity image, and the number density of the phase singularity are analyzed. The behaviors of the fractal dimension and the density of the phase singularity present obvious differences for the two categories of propagation. The fractal dimension depends both on the Rytov index and the Fresnel length. In both weak and strong fluctuation conditions the dimension generally increases with the Rytov index, but is at minimum at the onset region. The phase singularity density is coincident with the theoretical results under a weak fluctuation condition, and has a slowly increasing manner with the Rytov index in the strong fluctuation condition. The dependence on the Fresnel size is confident and there is no saturation for the phase singularity.     

Coupling plane wave received by an annular aperture into a single-mode fiber in the presence of atmospheric turbulence

The efficiency of coupling a plane wave into a single-mode fiber can be reduced by both the aperture obstruction of receivers and the turbulence-induced degradation of optical coherence. Using the Gaussian approximation to the mutual coherence function of the incident optical field, we derived an analytical solution for the fiber-coupling efficiency when a plane wave, propagating through atmospheric turbulence, is received by an annular-aperture receiver and coupled into a single-mode fiber. It is a function of the coupling geometry, the aperture-radius-to-coherence-radius ratio (ARCRR), and the aperture-obstruction parameter. It is found by the numerical optimization method that the optimal coupling geometry depends on both the ARCRR and the aperture-obstruction parameter. The results obtained are useful for analyzing and designing a fiber-coupling system influenced by atmospheric turbulence.     

Analytical solution for the covariance and for the decorrelation time of the angle of arrival of a wave front corrugated by atmospheric turbulence

The two-dimensional (2D) spatial covariance of the angle-of-arrival (AA) fluctuations is often used to investigate the properties of wave fronts corrugated by the atmosphere for high-angular-resolution techniques. Theoretical series expansions of this covariance are presented. The fast convergence of these series reduces the calculation time of the covariance done by numerical integration. The 2D covariance is a nonradial function. A physical interpretation of this anisotropy is proposed. The spatiotemporal correlation of the AA is deduced from the covariance assuming the "frozen-flow" hypothesis. The impact of the anisotropy on the evaluation of the number of predominant turbulent layers and on the corresponding winds is investigated, and an analysis of temporal correlations is performed. A simple theoretical approximation of the decorrelation time of the AA is given, which is found to be in agreement with experimental results.     

Analytical empirical expressions of the transverse coherence properties for monostatic and bistatic lidars in the presence of moderate atmospheric refractive-index turbulence

There have been many analyses of the reduction of lidar system efficiency in bistatic geometry caused by beam spreading and by fluctuations along the two paths generated by refractive-index turbulence. Although these studies have led to simple, approximate results that provide a reliable basis for preliminary assessment of lidar performance, they do not apply to monostatic lidars. For such systems, calculations and numerical simulations predict an enhanced coherence for the backscattered field. However, to the authors' knowledge, a simple analytical mathematical framework for diagnosing the effects of refractive-index turbulence on the performance of both bistatic and monostatic coherent lidars does not exist. Here analytical empirical expressions for the transverse coherence variables and the heterodyne intensity are derived for bistatic and monostatic lidars as a function of moderate atmospheric refractive-index turbulence within the framework of the Gaussian-beam approximation.     

Optimum truncation of a Gaussian beam for propagation through atmospheric turbulence

The mean on-axis far-field (or focal-plane) irradiance of a Gaussian beam that is truncated by a circular aperture in the presence of atmospheric turbulence is considered. In the absence of turbulence, an accurate analytic approximation for the irradiance distribution that is valid within the main central lobe of the beam is presented. Based on this approximation, the mean on-axis far-field irradiance and the corresponding turbulence Strehl ratio for the truncated Gaussian beam are then obtained. By maximization of the on-axis irradiance, the optimum ratio of the beam diameter to the aperture diameter in the presence of turbulence is obtained, and the results for the corresponding maximum on-axis irradiance as a function of the strength of turbulence are presented. In particular, for D/r(0) > 1, where D is the aperture diameter and r(0) is Fried's coherence length, optimum truncation of a Gaussian beam and uniform illumination of a circular aperture (where the same total power isuniformly distributed over the aperture) result in the same on-axis irradiance in the presence of uncompensated turbulence.     

Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence

High-speed free-space optical communication systems have recently used fiber-optic components. The received laser beam in such a system must be coupled into a single-mode fiber at the input of the receiver module. However, propagation through atmospheric turbulence degrades the spatial coherence of a laser beam and limits the fiber-coupling efficiency. We numerically evaluate the fiber-coupling efficiency for laser light distorted by atmospheric turbulence. We also investigate the use of a coherent fiber array as a receiver structure and find that a coherent fiber array that consists of seven subapertures would significantly increase the fiber-coupling efficiency.     

Scattering of a plane wave by an anisotropic ferrite-coated conducting sphere

On the basis of the three-dimensional spherical vector wave functions in ferrite anisotropic media, and the fact that the first and second spherical vector wave functions in ferrite anisotropic media satisfy the same differential equations, the electromagnetic fields in homogeneous ferrite anisotropic media can be expressed as the addition of the first and second spherical vector wave functions in ferrite anisotropic media. Applying the continue boundary condition of the tangential component of electromagnetic fields in the interface between the ferrite anisotropic medium and free space, and the tangential electric field vanishing in the interface of the conducting sphere, the expansion coefficients of electromagnetic fields in terms of spherical vector wave function in ferrite medium and the scattering fields in free space can be derived. The theoretical analysis and numerical result show that when the radius of a conducting sphere approaches zero, the present method can be reduced to that of the homogeneous ferrite anisotropic sphere. The present method can be applied to the analyses of related microwave devices, antennas and the character of radar targets.     

Atmospheric spectral model and theoretical expressions of irradiance scintillation index for optical wave propagating through moderate-to-strong non-Kolmogorov turbulence

A new atmospheric spectral model and expressions of irradiance scintillation index are derived theoretically for optical wave propagating through moderate-to-strong non-Kolmogorov turbulence. They are developed under Andrews' assumption that small-scale irradiance fluctuations are modulated by large-scale irradiance fluctuations of the wave, and the geometrical optics approximation is adopted for mathematical development. A wide range of turbulence strength is considered instead of a limited range for weak turbulence. The atmospheric spectral model has a spectral power law value in the range of 3 to 4 instead of the standard power law value of 11/3. Numerical calculations are conducted to analyze the influences of spectral power law and turbulence strength.     

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.     

On a linear spectral method for plane turbulence

A critical analysis and extension of the results of a linear spectral method for stationary, axially homogeneous, plane turbulent flow between two fixed boundaries, are presented. This method aims at the determination of the spatial variation of the spectral density function, with the knowledge of the mean flow and the spectral structure of turbulence at one point in the flow core. The spectra for axial fluctuating velocity components are satisfactorily predicted.     

Aperture averaging of the two-wavelength intensity covariance function in atmospheric turbulence

The influence of aperture averaging on the two-wavelength intensity covariance function was experimentally determined for visible (0.63 microm) and infrared (1.06 microm) collinear, approximately spherical beams which propagated through the earth's turbulent atmosphere. Range varied from 1300 to 3250 m, and due to the prevailing atmospheric conditions, most measurements were made in the strong turbulence regimes. Results show that (1) the covariance function monotonically decreases as the receiver aperture size increases; (2) the correlation coefficient attains high values > or = 0.7) even for a relatively small aperture size of 5 mm; (3) while the single wavelength probability distribution of the intensity is approximately lognormal, the experimental two-wavelength conditional probabilities are higher than those predicted by the lognormal model.     

Double passage enhancement of thin beam wandering through atmospheric turbulence

Abstract

Beam wandering (random lateral displacement) enhancement of thin beams by a double passage through turbulence is theoretically and experimentally investigated. A relationship is found between the random lateral displacements of double passage and single passage, in the case of homogeneous turbulence. Results of laboratory measurements are presented and suggestions for practical use given.     

Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams

The analytical expression for the spectral degree of coherence of partially coherent flat-topped beams propagating through the turbulent atmosphere is derived, and the spatial correlation properties are studied in detail. In particular, we find that the oscillatory behavior and phase singularities of the spectral degree of coherence may appear when partially coherent flat-topped beams propagate through the turbulent atmosphere; this behavior is very different from the behavior of Gaussian Schell-model beams. But the oscillatory behavior becomes weaker with increasing turbulence and even disappears when the turbulence is strong enough. The width of the spectral degree of coherence is always smaller than that of the spectral density in the far field when the turbulence is strong enough, whereas the width of the spectral degree of coherence in free space can be either larger or smaller than that of the spectral density in the far field.     

Set of modes for the description of wave propagation through slabs with a transverse variation of the refractive index

The scattering of electromagnetic waves by a slab whose refractive index is changing along its boundary planes is exactly calculated in a closed analytical form. The key feature of the calculation is the introduction of a new set of modes. As a specific example, we calculate the reflected and transmitted fields generated by the interaction of an incoming plane wave with an N-layered medium, the layers of which are perpendicular to the boundary planes of a slab.     

Spatial coherence function of partially coherent Gaussian beams in atmospheric turbulence

We introduce a new method of estimating the coherence function of a Gaussian-Schell model beam in the inertial subrange of atmospheric turbulence. It is compared with the previously published methods based on either the quadratic approximation of the parabolic equation or an assumed independence between the source's randomness and the atmosphere using effective beam parameters. This new method, which combines the results of the previous two methods to account for any random source/atmospheric coupling, was shown to more accurately estimate both the coherence radius and coherence functional shape across much of the relevant parameter space. The regions of the parameter space where one method or another is the most accurate in estimating the coherence radius are identified along with the maximum absolute estimation error in each region. By selecting the appropriate estimation method for a given set of conditions, the absolute estimation error can generally be kept to less than 5%, with a maximum error of 7%. We also show that the true coherence function is more Gaussian than expected, with the exponential power tending toward 9/5 rather than the theoretical value of 5/3 in very strong turbulence regardless of the nature of the source coherence.     

Analytical expressions for stress and displacement fields in viscoelastic axisymmetric plane problem involving time-dependent boundary regions

An analytical solution is developed in this paper for viscoelastic axisymmetric plane problems under stress or displacement boundary condition involving time-dependent boundary regions using the Laplace transform. The explicit expressions are given for the radial and circumferential stresses under stress boundary condition and the radial displacement under displacement boundary condition. The results indicate that the two in-plane stress components and the displacement under corresponding boundary conditions have no relation with material constants. The general form of solutions for the remaining displacement or stress field is expressed by the inverse Laplace transform concerning two relaxation moduli. As an application to deep excavation of a circular tunnel or finite void growth, explicit solutions for the analysis of a deforming circular hole in both infinite and finite planes are given taking into account the rheological characteristics of the rock mass characterized by a Boltzmann or Maxwell viscoelastic model. Numerical examples are given to illustrate the displacement and stress response. The method proposed in this paper can be used for analysis of earth excavation and finite void growth.     

Transmission of an inhomogeneous plane wave through an electrically small aperture in a perfectly conducting plane screen

Most solutions for electromagnetic diffraction by a circular aperture in a perfectly conducting plane screen are for an incident homogeneous (propagating) plane wave. When the aperture is electrically small (dimensions small compared to the wavelength), the well-known transmission coefficient behaves as the fourth power of the diameter/wavelength. We consider the case in which the incident field is an inhomogeneous (evanescent) plane wave. Numerical calculations for the electrically small circular aperture show that the transmission coefficient for an inhomogeneous plane wave can be substantially greater than for a homogeneous plane wave at the same frequency. This observation may be helpful in explaining the increased transmission recently reported for electrically small apertures in plane screens with modifications. The numerical calculations for the electrically small aperture are in agreement with results from approximate analytical expressions that are based on the equivalent electric and magnetic dipole moments for the electrically small complementary disk.