Rytov Approximation Research Articles

Temporal averaging of atmospheric turbulence-induced optical scintillation.

Based on the Rytov approximation we have developed for weak scintillation conditions a general expression for the temporal averaged variance of irradiance. The present analysis provides, for what we believe is the first time, a firm theoretical basis for the often-observed reduction of irradiance fluctuations of an optical beam due to atmospheric turbulence. Accurate elementary analytic approximations are presented here for plane, spherical and beam waves for predicting the averaging times required to obtain an arbitrary value of the ratio of the standard deviation to the mean of an optical beam propagating through an arbitrary path in the atmosphere. In particular, a novel application of differential absorption measurement for the purpose of measuring column-integrated concentrations of various so-called greenhouse gas (GHG) atmospheric components is considered where the results of our analysis indicates that relatively short averaging times, on the order of a few seconds, are required to reduce the irradiance fluctuations to a value precise enough for GHG measurements of value to climate related studies.

Influence of anisotropic turbulence on the long-range imaging system by the MTF model

Theoretical and experimental investigations have shown that the atmospheric turbulence exhibits both anisotropic and non-Kolmogorov properties. In this paper, new analytic expressions for the anisotropic non-Kolmogorov turbulence modulation transfer function (MTF) based on Rytov approximation theory have been derived for optical plane and spherical waves propagating through weak anisotropic non-Kolmogorov atmospheric turbulence. Compared with the previously published results where the turbulence inner and outer scales were set separately to zero and infinite for calculation convenience, the concept of anisotropy at different turbulence cell scales and finite turbulence inner and outer scales are introduced to study the MTF models. Also, deviations from the classic 11/3 spectral power law behavior for Kolmogorov turbulence are allowed by assuming spectral power law value variations between 3 and 4. To reduce the complexity and calculation time of the analytic results, the asymptotic-fit expressions are also derived and they fit well with the closed-form ones. Calculations are performed to analyze the anisotropic non-Kolmogorov turbulence’s influence on the long-range imaging system.

Quantification and normalization of noise variance with sparsity regularization to enhance diffuse optical tomography.

Conventional reconstruction of diffuse optical tomography (DOT) is based on the Tikhonov regularization and the white Gaussian noise assumption. Consequently, the reconstructed DOT images usually have a low spatial resolution. In this work, we have derived a novel quantification method for noise variance based on the linear Rytov approximation of the photon diffusion equation. Specifically, we have implemented this quantification of noise variance to normalize the measurement signals from all source-detector channels along with sparsity regularization to provide high-quality DOT images. Multiple experiments from computer simulations and laboratory phantoms were performed to validate and support the newly developed algorithm. The reconstructed images demonstrate that quantification and normalization of noise variance with sparsity regularization (QNNVSR) is an effective reconstruction approach to greatly enhance the spatial resolution and the shape fidelity for DOT images. Since noise variance can be estimated by our derived expression with relatively limited resources available, this approach is practically useful for many DOT applications.

Fluctuations of refractivity as a systematic error source in radio occultations

AbstractThe fact that fluctuations of refractivity may result in a systematic negative shift of the phase of waves propagating in a random medium is known for a long time. Tatarskii was the first to reveal it, and von Eshleman put this into the context of the radio occultation sounding of planetary atmospheres. In this paper, we show that this effect may also be one of the causes of the negative bias of refractivity retrieved for radio occultation observations of the Earth's atmosphere. We perform theoretical estimates of this effect based on the Rytov approximation. These estimates, however, do not consider the regular refraction, which may significantly change the magnitude of this effect. We perform numerical simulations of radio occultations, based on the Kolmogorov‐von Kármán isotropic spectrum of refractivity fluctuations, with the internal and external scales and magnitude tuned so as to reproduce the realistic level of the variance of retrieved refractivity and the amplitude fluctuations of the modeled signals. The model of the regular atmosphere is based on analyses of the European Centre for Medium‐Range Weather Forecasts. We show that it is possible to set up a vertical profile of the structural constant of the fluctuation spectrum such that it will result in a systematic shift and variances of the retrieved refractivity consistent with those observed for COSMIC measurements.

Analysis of angle of arrival fluctuations for optical waves' propagation through weak anisotropic non-Kolmogorov turbulence.

Analytical expressions for the variance of angle of arrival (AOA) fluctuations based on the Rytov approximation theory are derived for plane and spherical waves' propagation through weak anisotropic non-Kolmogorov turbulence atmosphere. The anisotropic spectrum model based on the assumption of circular symmetry in the orthogonal plane throughout the path is adopted and it includes the same degree of anisotropy along the direction of propagation for all the turbulence cells size in the inertial sub-range. The derived expressions consider a single anisotropic coefficient describing the turbulence anisotropic property and a general spectral power law value in the range 3 to 4. They reduce correctly to the previously published analytic expressions for the cases of plane and spherical waves' propagation through weak isotropic non-Kolmogorov turbulence for the special case of anisotropic factor equaling one. To reduce the complexity of the analytical results, the asymptotic-fit expressions are also derived and they fit well with the close-form ones. These results are useful for understanding the potential impact of deviations from the standard isotropic non-Kolmogorov turbulence atmosphere.

Computational and experimental study on accuracy of off-axis reconstructions in optical diffraction tomography

We present a study on spatial changes in the accuracy of tomographic reconstructions obtained with two of the most popular tomographic reconstruction algorithms for diffraction tomography—filtered backprojection (FBPJ) and Rytov-based filtered backpropagation (FBPP). We find out that not only FBPJ but also FBPP suffers from a significant loss of accuracy in the off-axis regions of a tomographic reconstruction and this effect is stronger for objects with a high refractive index contrast. Moreover, we propose some modifications to FBPP which allow for significant improvement of the off-axis performance of the algorithm. In the modified algorithm, called the extended depth of focus filtered backpropagation (EDOF-FBPP), scattered waves are backpropagated using a rigorous propagation algorithm, and then the Rytov approximation is applied on extended EDOF images. This modification (1) prevents violation of the Rytov validity condition due to the defocus of scattered waves and (2) suppresses unwrapping errors. The tomographic reconstruction algorithms FBPJ, FBPP, and EDOF-FBPP are extensively tested with numerical simulations supported with rigorous wave scattering methods. The experimental evaluation of the performance of the tomographic algorithms is provided with a tomographic measurement of an optical microtip located 21 μ m from the central axis of the reconstruction.

Correlation between the phase and the log-amplitude of a wave through vertical atmospheric propagation

Expressions of the correlation between the log-amplitude and the phase of a wavefront propagating through atmospheric turbulence are presented. These expressions are useful to evaluate the feasibility of proposed methods to increase the confidence level of the detection of faint transient astronomical objects. The properties of the derived angular correlation functions are discussed using usual synthetic turbulence profiles. The close formulation between the phase and the log-amplitude allows an analytic formulation in the Rytov approximation. Equations contain the product of an arbitrary number of hypergeometric functions that are evaluated using the Mellin transforms integration method.

Validity of the Rytov Approximation in the Form of Finite-Frequency Sensitivity Kernels

The first-order (or linear) Rytov or Born approximation is the foundation for formulation of wave-equation tomography and waveform inversion, so the validity of the Rytov/Born approximation can substantially affect the applicability of these theories. However, discussions and research reported in literature on this topic are insufficient or limited. In this paper we introduce five variables in scattering theory to help us discuss conditions under which the Rytov approximation, in the form of the finite frequency sensitivity kernels (RFFSK), the basis of waveform inversion and tomography, is valid. The five variables are propagation length L, heterogeneity scale a, wavenumber k, anisotropy ratio ξ, and perturbation strength ɛ. Combined with theoretical analysis and numerical experiments, we conclude that varying the conditions used to establish the Rytov approximation can lead to uninterpretable or undesired results. This conclusion has two consequences. First, one cannot rigorously apply the linear Rytov approximation to all theoretical or practical cases without discussing its validity. Second, the nonlinear Rytov approximation is essential if the linear Rytov approximation is not valid. Different from previous literature, only phase (or travel time) terms for the whole wavefield are discussed. The time shifts of two specific events between the background and observed wavefields measured by cross-correlation will serve as a reference for evaluation of whether the time shifts predicted by the FFSKs are reasonably acceptable. Significantly, the reference “cross-correlation” should be regarded as reliable only if the condition “two specific similar signals” is satisfied. We cannot expect it to provide a reasonable result if this condition is not met. This paper reports its reliability and experimental limitations. Using cross-correlation (CC) samples as the X axis and sensitivity kernel (SK) or ray tracing (RT) samples as the Y axis, a chart of cross validation among the three is established. According to experimental analysis for a random and deterministic medium, the RFFSK works well for diffraction and geometric optics enclosed by the boundaries ka > 1 and (Λ 1, Φ < 2π), where Λ is a diffraction variable and Φ is a variable used to measure scattering strength. Both Λ and Φ depend on the five variables mentioned above. However, a large perturbation will limit the applicability of the Rytov approximation and narrow the range in which it is valid, because larger ɛ results in higher-amplitude fluctuations and larger back-scattering. Therefore, large ɛ is important in making SK/RT predictions discrepant from correct CC measurements. We also confirm that the RFFSK theory has wider range of application than ray theory. All the experiments point out the importance, both practical and theoretical, of respecting the conditions used to establish the Rytov approximation.

Effects of multi-Schell beams on orbital angular momentum of entangled signal photons in low-order turbulence aberration channels

We model the detection and crosstalk probability of orbital angular momentum (OAM) states of the entangled signal photon in the Kolmogorov channels of the low-order turbulence aberrations and by the Rytov approximation. The results show that lower OAM mode number of signal photons and larger sub-beam number of multi-Gaussian Schell-model pump beam, the less susceptible of the detection probability of the signal photon to spatial coherence of source and turbulence aberrations is achieved. The maximum crosstalk probability is decrease as the decreasing of the sub-beam number of multi-Gaussian Schell-model. Enlarging OAM difference value or decreasing sub-beam number of multi-Gaussian Schell-model pump beam results in a lower crosstalk probability of the OAM of entangled signal photons.

The effects of near-source heterogeneity on shear-wave evolution

The Born and Rytov approximation, radiative transfer theory, and other related techniques are commonly used to model features of wave propagation through heterogeneous geologic media such as scattering, attenuation, and pulse-broadening. However, due to the underlying assumptions about the scattering direction and the reference Green’s function, these methods overlook important features of the wavefield such as mode conversion and near-field term coupling. These effects are particularly important within the predicted S-wave nodes of a seismic source, so we analyzed the problem of wave propagation beneath a vertical-point force on the surface of a heterogeneous, elastic half space. To do this, we generated a suite of 3D synthetic heterogeneous geologic models using fractal statistics and simulated the wave propagation using the finite-difference method. We derived an estimate for the effective source radiation patterns, and we used these to compare the results of the models. Our numerical results showed that, due to a combination of mode conversion and near-source coupling effects, S-wave energy on the order of 10% of the P-wave energy is generated within the shear-radiation node. In some cases, this S-wave energy may occur as a coherent pulse and may be used to enhance seismic imaging.

Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence

We develop a novel model of the probability density of the orbital angular momentum (OAM) modes for Hankel-Bessel beams in paraxial turbulence channel based on the Rytov approximation. The results show that there are multi-peaks of the mode probability density along the radial direction. The peak position of the mode probability density moves to beam center with the increasing of non-Kolmogorov turbulence-parameters and the generalized refractive-index structure parameters and with the decreasing of OAM quantum number, propagation distance and wavelength of the beams. Additionally, larger OAM quantum number and smaller non-Kolmogorov turbulence-parameter can be selected in order to obtain larger mode probability density. The probability density of the OAM mode crosstalk is increasing with the decreasing of the quantum number deviation and the wavelength. Because of the focusing properties of Hankel-Bessel beams in turbulence channel, compared with the Laguerre-Gaussian beams, Hankel-Bessel beams are a good light source for weakening turbulence spreading of the beams and mitigating the effects of turbulence on the probability density of the OAM mode.

Design and evaluation of a reflectance diffuse optical tomography system

We have designed a continuous wave, back-reflection diffuse optical tomography system; and developed a new practical calibration method including both optode efficiency and positional error corrections. System design, data acquisition and calibration protocols are described in detail. Monte Carlo (MC) simulations of photon distribution for tissue phantoms have been used to obtain the weight matrix to be used in the Rytov approximation to the photon diffusion equation. The system has been evaluated by acquiring data from a tissue phantom with a background scattering coefficient (\(\upmu _{\mathrm{s}}^{\prime }\)) of \(10\,\hbox {cm}^{-1}\) and absorption coefficient (\(\upmu _{\mathrm{a}})\) of \(0.04\,\hbox {cm}^{-1}\). An inclusion made of 1 % Intralipid and indocyanine green with \(\upmu _{s}^{\prime }=10\,\hbox {cm}^{-1}\) and \(\upmu _{\mathrm{a}}= 0.16\,\hbox {cm}^{-1}\) was placed at a 2 cm depth from the tissue phantom surface. After calibration, the average value of the measurements over source–detector pairs at the same distance for each neighborhood was calculated. Perturbation data were obtained by subtracting the average data from the measurements with the same source–detector separation. In the reconstruction, weight matrixes obtained from MC simulation for \(\upmu _{\mathrm{s}}^{\prime }= 7\) to \(12\,\hbox {cm}^{-1}\) were used. The Depth Compensation Algorithm was used in the Tikhonov regularization to identify the location of the inclusion correctly in the reconstruction.

Effect of background turbulence in lidar investigations of clear air turbulence

The effect of atmospheric turbulence on power fluctuations of backscattered laser radiation at the lidar receiving aperture is estimated for divergent laser beams with the Gaussian intensity distribution in the Rytov approximation. Simple analytic formulas are suggested as estimators of the variance of received power fluctuations. The estimates showed that, for paths about 10 km long and sounding region radii of about 1 m, root mean square values of relative power fluctuations are comparable with the relative variation in the power received when changing from sounding of atmospheric regions with background turbulence to highly turbulent regions (clear air turbulence).

Free Space Optical Wireless Communications under Turbulence Channel Effect

This article is focused on the turbulence attenuation and their effect on optical beam in free space. Analysis has been done for comparison the atmospheric turbulence effect between two classical methods, Rytov approximation and Andrews's method. The SNR was calculated for not weak turbulence employing Tailor series at different strength of scintillation. These calculations run for communications of wavelengths 850nm and 1550nm. The distance between the transmitter and receiver o horizontal link set to value range (0-2000) m. Keywords: Rytov variance, Atmospheric turbulence, Rytov approximation, Andrews's method. Signal to noise ratio.

Analytical expressions for the log-amplitude correlation function for spherical wave propagation through anisotropic non-Kolmogorov atmosphere

An analytical expression for the log-amplitude correlation function based on the Rytov approximation is derived for spherical wave propagation through an anisotropic non-Kolmogorov refractive turbulent atmosphere. The expression reduces correctly to the previously published analytic expressions for the case of spherical wave propagation through isotropic Kolmogorov turbulence. These results agree well with a wave-optics simulation based on the more general Fresnel approximation, as well as with numerical evaluations, for low-to-moderate strengths of turbulence. These results are useful for understanding the potential impact of deviations from the standard isotropic Kolmogorov spectrum.

The influence of non-Kolmogorov turbulence on the entanglement of spatial two-qubit states in a slant channel

The effects of atmospheric turbulence on the entanglement of spatial two-qubit states that are prepared using the signal and idler photons produced by parametric down-conversion are studied. Utilizing the non-Kolmogorov model for atmospheric turbulence and Rytov approximation method, we quantify the effects of atmospheric turbulence on the entanglement of the two-qubit state in terms of Wootters's concurrence. Our results show that the effects of the zenith angle of communication channel and the outer scale of turbulence on the concurrence of a spatial two-qubit state can be ignored and the smaller inner scale of turbulence, the smaller refractive-index power α, the shorter wavelength of beams and the longer propagation distance will lead to the larger fluctuations of the concurrence of a spatial two-qubit state.

Effects of atmospheric turbulence on the mode weight of the Laguerre-Gaussian Schell beams

Based on the Rytov approximation, we analyze the effects of the turbulence on the crosstalk and spatial spread of orbital angular momentum modes of the Laguerre-Gaussian Schell (LGS) beams. Our results show that the spatial incoherence of light source increases the crosstalk effect of atmospheric turbulence among the orbital angular momentum states and the spatial spread of modes. The mode weight decays with the mode quality factor increasing and the refractive index structure parameter increasing. The quality factor of LGS modes increases with the increasing of refractive index structure parameter in approximate linear form.

Wave equation tomographic velocity inversion method based on the Born/Rytov approximation

This paper discusses Born/Rytov approximation tomographic velocity inversion methods constrained by the Fresnel zone. Calculations of the sensitivity kernel function and traveltime residuals are critical in tomographic velocity inversion. Based on the Born/Rytov approximation of the frequency-domain wave equation, we derive the traveltime sensitivity kernels of the wave equation on the band-limited wave field and simultaneously obtain the traveltime residuals based on the Rytov approximation. In contrast to single-ray tomography, the modified velocity inversion method improves the inversion stability. Tests of the near-surface velocity model and field data prove that the proposed method has higher accuracy and Computational efficiency than ray theory tomography and full waveform inversion methods.

Full-wave approach for x-ray phase imaging

We present a rigorous forward model for phase imaging of a 3-D object illuminated by a cone-shaped x-ray beam. Our model is based on a full-wave approach valid under the first Rytov approximation, and thus can be used with large and thick objects, e.g., luggage and human patients. We unify light-matter interaction and free-space propagation into an integrated wave optics framework. Therefore, our model can accurately calculate x-ray phase images formed with sources of arbitrary shape, and it can be effectively incorporated into x-ray phase tomography as a forward model. Within the best of our knowledge, this is the first non-paraxial, full-wave model for X-ray phase imaging.

Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels

Based on the Rytov approximation, we analyze the effect of the pump beam's space-coherence of parametric down-conversion on entangled orbital angular momentum (OAM) states propagation in slant low-order turbulence aberration channels. The detection probability of signal photon of entangled OAM states is modeled. Our numerical evaluation shows that the signal photon detection probability and the crosstalk probability decay nonlinearly with increasing number of space coherent speckle and the OAM quantum number of signal photon in the channels of Z-tilt aberration, astigmatism aberration, defocus aberration and coma aberration declines. The OAM entanglement states of low spatial coherence are improper to be used for the carrier wave of the encoding of OAM. The signal photon detection probability decreases when the power-law exponent of non-Kolmogorov spectrum increases from 3 to 4, in the turbulence Z-tilt, astigmatism and coma aberrations channels.