Rytov Approximation Research Articles

Analysis of beam wander effects for a horizontal-path propagating Gaussian-beam wave: focused beam case

Failure of the first-order Rytov approximation to properly predict the scintillation index of a large-aperture focused beam, or an uplink collimated (or focused) beam, has been discussed in several recent publications, which cite beam wander effects as the main reason for this failure. We use computer simulations to examine several aspects of beam wander phenomena on a propagating convergent beam in the weak-fluctuation regime over a horizontal path at high altitude for which the refractive index structure parameter is on the order of C=1.39×10−16 m−2/3. Simulation results are presented at various ranges up to 10 km for (1) the beam wander centroid displacement, (2) the kurtosis excess of the irradiance profile, (3) the irradiance profile, (4) the mean-square hot spot displacement from the boresight and from the centroid, and (5) the scintillation index at the optical axis of the beam. In addition, simulation results are compared with theoretical models.

Solution for the Fourth Moment Equation of Waves in Random Continuum Under Strong Fluctuations: General Theory and Plane Wave Solution

The fourth moment plays an important role in wave propagation and scattering in random media. However, in the strong scattering regimes, it remains to be solved although a rich variety of methods have been proposed. The fourth moment equation is solved using the modified Gaussian solution method proposed in this paper. The fourth moment can be considered as a sum of the Gaussian solution and non-Gaussian correction term. The Gaussian solution is the fourth moment in the saturated regime, and can be obtained as a sum of products of the second-order moments. The derived equation of the non-Gaussian correction term is solved using the Rytov approximation and the Green's function. The fourth moment solution obtained is general for without restrictions on incident wave forms. Specifically, the solution for a plane wave is derived and applied to study the amplitude scintillation and intensity correlation function of trans ionospheric radio signals. The theoretical calculation on the scintillation index agrees well with the experimental results, which confirms the fourth moment solution to some extent.

Scintillation Reduction by Use of Multiple Gaussian Laser Beams With Different Wavelengths

We study the scintillation index of N partially overlapping collimated lowest order Gaussian laser beams with different wavelengths in weak atmospheric turbulence. Using the Rytov approximation, we calculate the initial beam separation that minimizes the longitudinal scintillation. Further reduction of the longitudinal scintillation is obtained by optimizing with respect to both beam separation and spot size. The longitudinal scintillation of the optimal N-beam configurations is inversely proportional to N, resulting in a 92% reduction for a nine-beam system compared with the single beam value. The radial scintillation values for the optimal N-beam configurations are significantly smaller than the corresponding single beam values

Analytical Solution to the $n$-$n$th Moment Equation of Wave Propagation in Continuous Random Media

Higher order symmetrical moments play an important role in wave propagation and scattering in random media, however it remains to be solved under strong fluctuations. In this paper, a modified Gaussian solution method is proposed for analytically solving the n-nth moment. After propagating through a random medium in the fully saturated regime, the higher order symmetrical moment of the received wave is the sum of products of the second moments, i.e., the Gaussian solution. In strong scattering regimes, the higher order symmetrical moment can be considered as a sum of the Gaussian solution and a non-Gaussian correction term, where the key issue is how to solve the derived equation of the correction term. Two methods are proposed, i.e., Green's function method and the Rytov approximation approach. Green's function method leads to a rigorous solution form, but it is complicated due to an integral equation. The approach using the Rytov approximation is found to be reasonable, as the correction is relatively small

Atmospheric Phase Error in Coherent Laser Radar

Coherent laser radar (ladar) deduces the shape, velocity and vibration of a remote target by measuring the round-trip optical phase or the Doppler shift of target-scattered light. Phase fluctuations from wind-driven atmospheric turbulence generate micro-Doppler or velocity error at mum/sec levels. The turbulence-induced noise spectra of all geometric phase components of the ladar, the average phase power spectral densities of arbitrary ladar receiver modes, and the resultant Doppler noise spectra, are derived in the Rytov approximation for any isotropic stationary turbulence, any wind velocity profile, any Fresnel number and any source illumination pattern. In Kolmogorov turbulence the piston phase noise spectrum, and the average phase shift over a time interval, are well-approximated by simple universal asymptotic forms dependent only on atmospheric coherence length, turbulence-weighted average wind speed Vmacr, and aperture size D. The spectrum slope changes at a critical frequency of order Vmacr/D. The approximate results do not depend on the usually-unknown outer scale of turbulence

Optimized multiemitter beams for free-space optical communications through turbulent atmosphere

Using laser beams with less than perfect spatial coherence is an effective way of reducing scintillations in free-space optical communication links. We report a proof-of-principle experiment that quantifies this concept for a particular type of a partially coherent beam. In our scaled model of a free-space optical communication link, the beam is composed of several partially overlapping fundamental Gaussian beams that are mutually incoherent. The turbulent atmosphere is simulated by a random phase screen imprinted with Kolmogorov turbulence. Our experiments show that for both weak-to-intermediate and strong turbulence an optimum separation between the constituent beams exists such that the scintillation index of the optical signal at the detector is minimized. At the minimum, the scintillation reduction factor compared with the case of a single Gaussian beam is substantial, and it is found to grow with the number of constituent beams. For weak-to-intermediate turbulence, our experimental results are in reasonable agreement with calculations based on the Rytov approximation.

Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence

The calculation of the aperture-averaged angle-of-arrival variance, observed with a telescope with a circular aperture, of a plane or spherical wave propagating through homogeneous and isotropic turbulence is one of the classical problems in the theory of wave propagation through random media. We present and discuss approximate closed-form solutions on the basis of the Rytov approximation. For both plane and spherical waves, the accuracy of the approximations is better than 0.25% for all ratios of aperture diameter and Fresnel length.

Inverting the power spectrum for a heterogeneous medium

SUMMARY The delta-correlated assumption of the random medium is of considerable significance in statistical wave propagation problems. Assuming the Rytov and parabolic approximations, this work completed an exact derivation of the transverse coherence functions without using the delta-correlated approximation. The error introduced by this approximation is shown to be negligible when the source‐receiver distance is large. Our new formulation shows that the TCF does have limited vertical spectral resolution, in contrast to previous results. We define a new observational variable whose Fourier transform directly yields the power spectral density function of the random medium. This enables a simultaneous determination of the scale and statistical distribution of heterogeneities without assuming the power spectral function a priori.

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.

Three-Dimensional Diffraction Tomography Using Filtered Backpropagation and Multiple Illumination Planes

In this paper, a three-dimensional (3-D) extension of the well-known filtered-backpropagation (FBP) algorithm is presented with the aim of taking into account scattered-field-data measurements obtained using incident directions not restricted in a single plane. The FBP algorithm has been extensively used to solve the two-dimensional inverse-scattering problem under the first-order Born and Rytov approximations for weak scatterers. The extension of this algorithm in three dimensions is not straightforward, because the task of collecting the data needed to obtain a low-pass filtered version of the scattering object, taking into account all spatial frequencies within a radius of radic2k0, and of incorporating these data to the FBP algorithm, needs to be addressed. A simple extension using incident field directions restricted to a single plane (illumination plane) leaves a region of spatial frequencies of the sphere of radius radic2k 0 undetermined. The locus of these spatial frequencies may be crucial for the accurate reconstruction of objects which do not vary slowly along the axis perpendicular to the illumination plane. The proposed 3-D FBP algorithm presented here is able to incorporate the data collected from more than one illumination plane and to ensure the reliability of the reconstruction results

A family of approximations spanning the Born and Rytov scattering series

A new hybrid scattering series is derived that incorporates as special cases both the Born and Rytov scattering series, and includes a parameter so that the behavior can be continuously varied between the two series. The parameter enables the error to be shifted between the Born and Rytov error terms to improve accuracy. The linearized hybrid approximation is derived as well as its condition of validity. Higher order terms of the hybrid series are also found. Also included is the integral equation that defines the exact solution to the forward scattering problem as well as its Fréchet derivative, which is used for the solution of inverse multiple scattering problems. Finally, the linearized hybrid approximation is demonstrated by simulations of inverse scattering off of uniform circular cylinders, where it is shown that the hybrid approximation achieves smaller error than either the Born or Rytov approximations alone.

An inverse scattering algorithm using the Rytov approximation for a transient three-dimensional problem

We present a novel algorithm to solve the inverse scattering problem in the time domain and to obtain the reconstruction of an unknown object in an area of interest. In its most general form, the inverse scattering problem is computationally very intensive due to the necessity of computing the internal field in parallel with the reconstruction of the characteristic object function. The algorithm presented here utilizes a Rytov-based approximation of the internal field, thus expanding to a considerable degree, the class of scatterers which can be reconstructed compared to the simple Born approximation.

Rytov approximation of tomographic receptions in weakly range-dependent ocean environments

An approximation method based on the Born and Rytov approximation is presented for the wave-theoretic prediction of acoustic arrival patterns associated with long-range pulse propagation in weakly range-dependent ocean environments. The environment is considered as a perturbation of a range-independent background state, and normal-mode theory is used, for the representation of the background Green’s function. Using the Born and Rytov approximations, the perturbed Green’s function corresponding to the range-dependent environment is expressed for each frequency within the source bandwidth in terms of the background Green’s function and the medium (sound-speed) perturbation. The actual arrival pattern in the time domain is then computed through the inverse Fourier transform. Using the normal-mode representation, closed-form expressions for the first and second Born and Rytov approximations are derived, generalizing previous range-independent results, and indicating that the effect of range dependence on the acoustic field in the case of adiabatic perturbations is of second order. To cope with the multimodal nature of ocean acoustic propagation, a variation of the standard Rytov method is applied, proposed by Keller, according to which each modal component must be treated independently. A number of numerical examples demonstrate an advantage of the Rytov approximation (over the Born approximation) for time-domain calculations.

Tangent linear and second-order models from the Born and Rytov approximations

Adjoint and higher-order methods for the solution of inverse problems in underwater acoustics rely on tangent linear and higher-order models of the forward propagation problem, with respect to the underlying parameters of interest. The use of Born and Rytov approximations is examined for the representation of perturbations of the Green’s function and perturbations of the phase, respectively, caused by range-dependent variations of the sound-speed distribution about a range-independent background environment. Using the normal-mode representation for the background Green’s function and applying the stationary-phase approach in combination with analytic evaluation at singular points, closed-form expressions are derived for the first- and second-order perturbations of the Green’s function, generalizing previous perturbation results derived under the assumption of range independence. The resulting expressions for the phase show the second-order character of the phase (and the associated travel times) in the case of range-dependent, zero-mean sound-speed perturbations.

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.

Multi-frequency diffuse optical tomography

The performance of diffuse optical tomography (DOT) is investigated when multi-modulation frequencies are used simultaneously in the inverse problem in order to cast the forward model. The forward model used was based on the first-order Rytov approximation of the diffusion equation. Numerical measurements were generated with a finite difference approach for configurations relevant to breast optical mammography. The impact of various frequency-set characteristics was evaluated. Furthermore, the effect of sensitivity matrix preconditioning was investigated. The criteria underlying this study were based on a ‘singular value analysis’ and parameter cross-talk between absorption and scattering reconstructions. It had been shown that preconditioning the matrix was necessary to provide accurate results and an even distribution of frequencies within the larger bound. Furthermore, correlating the different criterions used demonstrated that analysis based on optimization of condition number alone was providing misleading results.

Time domain surface impedance concept for low frequency electromagnetic problems—Part II: Application to transient skin and proximity effect problems in cylindrical conductors

A time domain boundary element formulation employing the surface impedance boundary conditions (SIBCs) is developed for the 3-dimensional transient eddy current problem of cylindrical conductors. SIBCs of different orders of approximation are implemented using the perturbation technique in the small parameter proportional to the ratio of the skin depth and characteristic size of the conductor cross-section. The formulation consists of a set of time domain surface integral equations that have identical left-hand sides and can be solved using the same program procedure. The number of equations is determined by the order of approximation of the SIBC, namely: solutions in the perfect electrical conductor (PEC) limit (lowest order) and in the so-called Rytov approximation (highest order) are given by one and four equations, respectively. It is demonstrated that each equation admits separation of variables into space and time components, a property that significantly reduces computational costs compared with traditional time domain formulations that require the integral equations to be solved at each time step. For the purpose of validation, a test problem is solved by the proposed formulation and by the ‘original’ BEM based on the time-dependent fundamental solution. Conditions of applicability are discussed and the effect of such factors as the shape of the incident current pulse and proximity effect are considered.

Microwave Imaging of Immersed Dielectric Bodies: An Experimental Survey

Abstract Diffraction tomography provides a good method for reconstructing cross-section images of weakly scattering dielectric objects. The Born and Rytov approximations are well-known approximations for the first-order diffraction tomography method. In this paper, the restrictions of the Born and Rytov approximations are experimentally investigated when the object is immersed in a medium. Microwave tomography images of several kinds of dielectrics of different shapes and dielectric constants are imaged both in the air and in the sand. It is shown experimentally from the resultant images that the Born approximation produces a better estimation for objects small in size and even for objects of large deviations in relative refractive index; on the other hand, Rytov approximation gives a more accurate estimation mainly for objects with small deviations in relative refractive index. The Rytov approximation is also valid for large-sized objects if the first condition is satisfied. The results also show that it...

Microwave Imaging of Immersed Dielectric Bodies: An Experimental Survey

Diffraction tomography provides a good method for reconstructing cross-section images of weakly scattering dielectric objects. The Born and Rytov approximations are well-known approximations for the first-order diffraction tomography method. In this paper, the restrictions of the Born and Rytov approximations are experimentally investigated when the object is immersed in a medium. Microwave tomography images of several kinds of dielectrics of different shapes and dielectric constants are imaged both in the air and in the sand. It is shown experimentally from the resultant images that the Born approximation produces a better estimation for objects small in size and even for objects of large deviations in relative refractive index; on the other hand, Rytov approximation gives a more accurate estimation mainly for objects with small deviations in relative refractive index. The Rytov approximation is also valid for large-sized objects if the first condition is satisfied. The results also show that it is possible to image objects located in medium by the diffraction tomography method and that the method can be applied in biomedical and NDT applications.

Sensitivity kernels for finite-frequency surface waves

Sensitivity kernels for fundamental mode surface waves at finite frequency for 2-D phase speed and 3-D shear wave speed are constructed based on the Born and Rytov approximations working with a potential representation for surface waves. The use of asymptotic Green's functions for scalar wave equations provides an efficient way to calculate the Born or Rytov kernels. The 2-D sensitivity kernels enable us to incorporate the finite-frequency effects of surface waves, as well as off-great-circle propagation, in tomographic inversions for phase-speed structures. We derive examples of the 2-D sensitivity kernels both for a homogeneous background model (or a spherically symmetric model), and for a laterally heterogeneous model. The resulting distortions of the shape of the sensitivity kernels for a heterogeneous background model indicate the importance of the use of proper kernels to account of the heterogeneity in the real Earth. By combining a set of 2-D sensitivity kernels with 1-D vertical sensitivity kernels for a particular frequency range and taking the inverse Fourier transform, we can derive 3-D sensitivity kernels for surface waves in the time domain. Such 3-D kernels are useful for efficient forward modelling of surface waveforms incorporating finite-frequency effects, and will also enable us to perform direct inversion of surface waveforms into 3-D structure taking account of finite-frequency effects.