Finite Outer Scale Research Articles

Integral field spectroscopy supports atmospheric optics to reveal the finite outer scale of the turbulence

The spatial coherence wavefront outer scale ( characterizes the size of the largest turbulence eddies in Earth's atmosphere, determining low spatial frequency perturbations in the wavefront of the light captured by ground-based telescopes. Advances in adaptive optics (AO) techniques designed to compensate for atmospheric turbulence emphasize the crucial role of this parameter for the next generation of large telescopes. The motivation of this work is to introduce a novel technique for estimating \ from seeing-limited integral field spectroscopic (IFS) data. This approach is based on the impact of a finite \ on the light collected by the pupil entrance of a ground-based telescope. We take advantage of the homogeneity of IFS observations to generate band filter images spanning a wide wavelength range, enabling the assessment of image quality (IQ) at the telescope's focal plane. Comparing the measured wavelength-dependent IQ variation with predictions derived from a first-order analytical approach based on turbulence statistics simplifications using the von Kármán model provides valuable insights into the prevailing \ parameter during the observations. We applied the proposed technique to observations from the Multi-Unit Spectroscopic Explorer (MUSE) in the wide-field mode obtained at the Paranal Observatory. Our analysis successfully validates the first-order analytical expression, which combines the seeing ($ $) and the \ parameters, to predict the IQ variations with the wavelength in ground-based astronomical data. However, we observed some discrepancies between the measured and predictions of the IQ that are analyzed in terms of uncertainties in the estimated $ $ and dome-induced turbulence contributions. This work constitutes the empirical validation of the analytical expression for estimating IQ at the focal plane of ground-based telescopes under specific $ $ and finite \ conditions. Additionally, we provide a simple methodology to characterize the \ and dome-seeing ($ dome $) as by-products of IFS observations routinely conducted at major ground-based astronomical observatories.

Equalization equal gain combining for a single-input to multiple-output underwater wireless optical communication system under aGaussian beam.

In this study, we examined the performance of an underwater wireless optical communication (UWOC) system employing a single-input to multiple-output (SIMO) scheme and proposed an equalization equal gain combining (EEGC) algorithm for it under Gaussian beam conditions. Furthermore, based on a Yue spectrum with the instability of oceanic water stratification and a finite outer scale, we derived the closed analytical formulas for the scintillation index and spatial coherence radius in weak oceanic turbulence for a Gaussian beam, from which we could obtain the threshold of the detector spacing and the strength of oceanic turbulence. We then derived the closed-form formula for the upper bound average bit error rate of the EEGC SIMO system with ON-OFF keying modulation by using the hyperbolic tangent distribution function. Our simulations demonstrate two issues if oceanic water stratification is treated as a steady state: the performance of the diversity receiver system will be significantly underestimated in salinity-dominated weak oceanic turbulence channels and will be significantly overestimated in temperature-dominated weak oceanic turbulence channels. Additionally, the SIMO system performance improvement using the proposed EEGC algorithm was more evident with increasing detector spacing, and the EEGC algorithm reduced the impact of the layout of the avalanche photodiode arrays on the UWOC system performance, in contrast to the equal gain combining algorithm.

Low-wavenumber compensation with Zernike tilt for non-Kolmogorov turbulence phase screens.

The fast-Fourier-transform-based filtering method for phase screen generation remains popular for numerical simulation of optical propagation through turbulence; however, these screens inherently underrepresent the spectral density at low wavenumbers. Here, the "Z-tilt" approach is explored to augment the spectral density at low wavenumbers by adding a random phase tilt, which is derived from the wavefront phase statistics of a Zernike polynomial basis. This approach is computationally efficient and can be applied to any statistically homogeneous and isotropic refractive index field. An analytic result is provided for the von Kármán spectrum with finite outer scale. In a quantitative comparison with phase screens compensated for using a common subharmonic approach, the Z-tilt method shows the best agreement with the analytical structure function when the outer scale is greater than about three times the screen dimension. For outer scales of the order of the screen dimension, the subharmonic and a modified Z-tilt method give the most accurate results. A propagation simulation demonstrates that the aperture-averaged angle-of-arrival variance is accurately predicted using the Z-tilt method.

Equivalent refractive-index structure constant of non-Kolmogorov turbulence: comment.

The paper by Y. Li, W. Zhu, X. Wu, and R. Rao entitled "Equivalent refractive-index structure constant of non-Kolmogorov turbulence," Opt. Express23(18), 23004 (2015).10.1364/OE.23.023004 relates the non-Kolmogorov turbulence structure constant to the classical structure constant for Kolmogorov turbulence by imposing equality of their respective structure functions at large separation distances, higher than the outer scale. As opposed to previous attempts to relate the two structure constants, the approach of Li et al. is anchored on a measurable meteorological parameter, the outer scale. The error lies in the fact that the authors have used a default Kolmogorov structure function with an infinite outer scale. A subsequent assumption of a finite outer scale is not compatible with the initial assumption. In this paper we show the correct procedure to obtain the relationship between the non-Kolmogorov and Kolmogorov structure constants which is based on an explicitly finite outer scale used throughout all calculations.

Analysis of Scintillation Effects on Free Space Optical Communication Links in South Africa

The performance of free space optical communication (FSOC) systems is severely degraded by certain atmospheric conditions prevalent in places where they are deployed, in spite of their numerous advantages. In clear weather conditions, the random fluctuation in the atmosphere’s refractive index causes substantial scintillation losses to transmitted optical signals. It is therefore imperative to estimate the potential losses due to atmospheric turbulence in locations where FSOC links are to be deployed. This will provide the necessary fade margin for FSOC systems so that designed links withstand such atmospheric disturbances. In this paper, statistical analysis of wind speed data collected for various cities of South Africa is used for calculating the corresponding refractive index structure parameter (Cn2). These Cn2 values, as well as the zero inner scale and infinite outer scale model and finite inner and finite outer scale model, are used in computing the scintillation indices not exceeding 50%, 99%, 99.9%, and 99.99% of the time for the investigated locations. The Lognormal and Gamma–gamma distribution models are then employed for the computational analysis of the irradiance fluctuations and channel characteristics while considering the effect of pointing errors for weak and moderate to strong turbulence regimes. Finally, derived mathematical expressions for outage probabilities and bit error rate (BER) performances for FSOC links, employing various intensity modulation and direct detection (IM/DD) schemes, are presented.

A series approximation to the Kirchhoff integral for Gaussian and exponential roughness covariance functions.

The Kirchhoff integral is a fundamental integral in scattering theory, appearing in both the Kirchhoff approximation and the small slope approximation. In this work, a functional Taylor series approximation to the Kirchhoff integral is presented, under the condition that the roughness covariance function follows either an exponential or Gaussian form-in both the one-dimensional and two-dimensional cases. Previous approximations to the Kirchhoff integral [Gragg, Wurmser, and Gauss (2001) J. Acoust. Soc. Am. 110(6), 2878-2901; Drumheller and Gragg (2001) J. Acoust. Soc. Am. 110(5), 2270-2275] assumed that the outer scale of the roughness was very large compared to the wavelength, whereas the proposed method can treat arbitrary outer scales. Assuming an infinite outer scale implies that the root mean square (rms) roughness is infinite. The proposed method can efficiently treat surfaces with finite outer scale and therefore finite rms height. This series is shown to converge independently of roughness or acoustic parameters and converges to within roundoff error with a reasonable number of terms for a wide variety of dimensionless roughness parameters. The series converges quickly when the dimensionless rms height is small and slowly when it is large.

Angular anisoplanatism in non-Kolmogorov turbulence over horizontal paths.

We use Mellin-transform techniques to derive generalized expressions for the piston-removed and piston-and-tilt-removed anisoplanatic error in non-Kolmogorov turbulence with a finite outer scale. We use these expressions to investigate the behavior of the anisoplanatic error when imaging over long horizontal paths where the angular extent of the scene is often many times the isoplanatic angle. By evaluating these expressions, we first find that in many cases, the anisoplanatic error saturates to a value less than 1rad2. Next, as power law increases, the contributions due to piston and tilt dominate the anisoplanatic error expression. Last, the size of the outer scale contributes primarily to the piston and tilt terms. Together, these behaviors imply that when piston and tilt are removed, anisoplanatism is reduced by as much as 60%.

Turbulence-induced changes in the state of polarization of partially polarized, partially coherent pulsed electromagnetic beams propagating through anisotropic turbulence

The changes in the state of polarization of partially polarized, partially coherent pulsed electromagnetic beams propagating along a horizontal path through anisotropic turbulence are investigated. Unpolarized and partially polarized, partially coherent pulsed electromagnetic Gaussian Schell-model beams are compared in terms of the turbulence-induced changes in the spectral degree of polarization. Our analysis is focused on models pertaining to electromagnetic Gaussian Schell-model pulsed beams but can either be readily reduced to scalar or be readily generalized to other beam classes. Within the framework of the anisotropic generalized exponential spectrum, considering simultaneously the finite inner and outer scales of turbulence and the asymmetric property of turbulence eddies in the orthogonal x y -plane throughout the propagation path, we derive analytical expressions for the cross-spectral density matrix, spectral degree of polarization, orientation angle, and degree of ellipticity. Finally, we study the effects of the anisotropic turbulence parameter on the spectral degree of polarization, orientation angle, and degree of ellipticity of the beams. Our results can be useful for applications involving partially polarized, partially coherent pulsed beams propagating through atmospheric turbulence.

Randomized spectral sampling for efficient simulation of laser propagation through optical turbulence

We present a new method for the generation of atmospheric turbulence phase screens based on the frequency shift property of the Fourier transform. This method produces low spatial frequency distortions without additional computation time penalties associated with methods using subharmonic subgrids. It is demonstrated that for simulations of atmospheric turbulence with finite outer scales, the performance of our method with respect to the statistical phase structure function of the screen meets or exceeds other methods with respect to agreement with theory. We outline small-scale accuracy issues associated with modelling non-Kolmogorov spectral power laws using existing techniques, and propose a solution. For simulations of long-range propagation through atmospheric optical turbulence, our method provides various advantages over standard methods.

Influence of anisotropy of marine turbulence on the variance of optical wave AOA fluctuations.

The optical wave angle of arrival (AOA) fluctuations in marine turbulence severely degrade long-range imaging quality by introducing image distortion. Theoretical investigations of optical wave AOA fluctuations are critical to improve the performances of long-range imaging systems in marine turbulence conditions. In this work, analytic expressions for the variance of optical plane and spherical wave AOA fluctuations in anisotropic marine turbulence will be derived. Compared with the previous models that focus on the isotropic and Kolmogorov properties of marine turbulence, both the anisotropy and non-Kolmogorov features of marine turbulence are considered. The anisotropic factor, general spectral power law instead of constant value of 11/3, finite turbulence inner and outer scales, and variable wavelength are included in the models developed in this work. Also, comparative analyses are performed for the models derived separately in marine and terrestrial anisotropic turbulence conditions. It turns out the anisotropy of turbulence cells degrades the effects of marine turbulence on the optical wave AOA fluctuations. Also, the marine atmosphere humidity fluctuations increase the impacts of anisotropic marine turbulence on the final derived models.

Beam-wander analysis in turbulent ocean with the effect of the eddy diffusivity ratio and the outer scale.

In this paper, we theoretically analyze the beam wander variance for the Gaussian beam propagation through weak oceanic turbulence. Particularly, we consider the effect of the eddy diffusivity ratio of salinity to temperature as well as the outer scale on the beam wander. To get to the tractable theoretical results, we first develop an approximate oceanic refractive-index spectrum with the outer scale and variable eddy diffusivity ratio. Based on this spectrum in weak turbulence, we then derive the closed-form expression for the beam wander variance. We present numerical results to show that the beam wander is either overestimated or underestimated as compared to the previous analyses which set the eddy diffusivity ratio to one. In addition, the finite outer scale of turbulence can significantly reduce the beam wander for Gaussian beams.

Effects of anisotropic turbulence on the long term beam spread and beam wander of Gaussian beam

Experiments and theoretical investigations have shown that the isotropic turbulence is not the only possible turbulence in the atmosphere. Researches about the anisotropic non-Kolmogorov have attracted more and more attentions recently. In this work, the spreading and wandering of Gaussian beam under weak anisotropic turbulence are investigated. New theoretical expressions for these two kinds of turbulence effects are developed. They are observed against the variations in the effective anisotropic factor, finite turbulence inner and outer scales, and the general spectral power law. It is found that the increased effective anisotropic factor decreases the influence of anisotropic turbulence on the long term beam spread and beam wander. The increased turbulence inner scale is advantageous for decreasing the long term beam spread. While the increased turbulence outer scale is disadvantageous for alleviating the beam wander. The theoretical investigations performed in this work are helpful to better understand the optical waves’ propagation under anisotropic turbulence.

Further analysis of scintillation index for a laser beam propagating through moderate-to-strong non-Kolmogorov turbulence based on generalized effective atmospheric spectral model

A new expression of the scintillation index (SI) for a Gaussian-beam wave propagating through moderate-to-strong non-Kolmogorov turbulence is derived, using a generalized effective atmospheric spectrum and the extended Rytov approximation theory. Finite inner and outer scale parameters and high wave number “bump” are considered in the spectrum with a generalized spectral power law in the range of 3–4, instead of the fixed classical Kolmogorov power law of 11/3. The obtained SI expression is then used to analyze the effects of the spectral power law and the inner scale and outer scale on SI under various non-Kolmogorov fluctuation conditions. These results will be useful in future investigations of optical wave propagation through atmospheric turbulence.

Temporal spectrum of beam wander for Gaussian Schell-model beams propagating in atmospheric turbulence with finite outer scale: publisher's note.

This publisher's note corrects a typo in the title in Opt. Lett.38, 1887 (2013)OPLEDP0146-959210.1364/OL.38.001887.

Irradiance scintillation models of plane and spherical waves considering the general distribution of anisotropic turbulence cells

Theoretical investigations and experiments have shown the atmosphere turbulence exhibit anisotropic properties. In this work, the irradiance scintillation of optical plane and spherical waves in weak anisotropic turbulence will be investigated. Previously derived irradiance scintillation index of plane and spherical waves in weak anisotropic turbulence assumed the circular symmetric distribution of turbulence cells in the plane orthogonal to the direction of propagation. This assumption is very special for the real anisotropic turbulence. To consider the general distribution of turbulence cells in anisotropic turbulence, new expressions for the irradiance scintillation index of plane and spherical waves in weak anisotropic non-Kolmogorov turbulence have been derived with the Rytov approximation theory. In the investigations, two anisotropic factors are introduced to parameterize the asymmetric distribution of turbulence cells in anisotropic turbulence. In addition, the general spectral power law α in the range 3–4 and the finite turbulence inner and outer scales are considered in the modeling. Calculations are performed to analyze these parameters’ influences on the derived irradiance scintillation index models of optical waves.

Effects of inner and outer scale on the modulation transfer function for a Gaussian wave propagating through anisotropic non-Kolmogorov turbulence

Both experimental results and empirical research have shown that the atmospheric turbulence can present the anisotropic property not only at a few meters above the ground but also at high altitudes of up to several kilometers. This paper investigates the modulation transfer function of a Gaussian beam propagating along a horizontal path in weak anisotropic non-Kolmogorov turbulence. Mathematical expressions are obtained based on the generalized exponential spectrum for anisotropic turbulence, which includes the spectral power law value, the finite inner and outer scales of turbulence, the anisotropic factor, and other essential optical parameters of the Gaussian beam. The numerical results indicate that the atmospheric turbulence would produce less negative effects on the wireless optical communication system with an increase in the anisotropic factor.

Effects of finite inner and outer scales on the scintillation index of turbulent slant path

In this paper, the effects of nonzero inner scales and finite outer scales are investigated, in the context of Gaussian beam propagation along a slant path under general turbulence conditions. Theoretical expressions for the cut-off spatial frequencies are derived with an approach method, and thereby a modified scintillation model is developed to incorporate inner scale and outer scale parameters in the analysis. Then, inner and outer scale effects on the downlink are analysed with respect to the zenith angle, the altitude of the transmitter, the initial beam radius, as well as the turbulence strength. Numerical results indicate that the effects of a finite outer scale mainly influence transmission that occurs at large zenith angles or high altitudes, while the inner scale effects are more prevalent. This study may be helpful to improve the accuracy of calculation of slant-path scintillation index, and thus benefit the characterization and optimization of space/air-ground laser communication systems.

Irradiance scintillation of Gaussian beam considering the concept of anisotropy at different scales in weak anisotropic turbulence

In this work, taking the concept of anisotropy at different scales of turbulence cells or eddies into considerations, new analytic expression of the irradiance scintillation index of a Gaussian beam is derived for Gaussian beam propagating through weak anisotropic non-Kolmogorov turbulence. The effective anisotropic factor is introduced to parameterize the asymmetry of turbulence cells or eddies in the horizontal and vertical directions. Also, the finite turbulence inner and outer scales, and the general spectral power law deviated from the classic 11/3 spectral power law behavior for Kolmogorov turbulence are considered in the investigations. The derived expression of irradiance scintillation index for a Gaussian beam can reduce correctly to the previously published results in the special cases of effective anisotropic factor equaling one. Calculations are conducted to analyze the influences of turbulence parameters on the irradiance scintillation index of Gaussian beam under weak anisotropic non-Kolmogorov turbulence.

Irradiance scintillation index for Gaussian beam based on the generalized anisotropic von Karman spectrum

Experiments and theoretical investigations have shown that the atmosphere turbulence exhibits both anisotropic and non-Kolmogorov properties. In this paper, based on the anisotropic generalized von Karman spectrum and the Rytov approximation theory, new expression for the irradiance scintillation index of optical waves is derived for Gaussian beam propagating through weak anisotropic non-Kolmogorov turbulence. Compared with previously published results, it considers simultaneously the asymmetry property of turbulence cells or eddies in the orthogonal xy-plane, the general spectral power law in the range 3–4 instead of constant value of 11/3 for the Kolmogorov turbulence, and the finite turbulence inner and outer scales. Two anisotropic factors are introduced to parameterize the anisotropy of turbulence cells or eddies in horizontal and vertical directions. In the special cases of these two anisotropic factors equaling one and the finite turbulence inner and outer scales equaling separately zero and infinite, the derived expression can reduce correctly to the previously published results. Calculations are performed to analyze the derived results.

Temporal power spectra of plane wave considering finite turbulence inner and outer scales in anisotropic turbulence

Theoretical investigations and experiments have shown that the atmosphere turbulence exhibits both anisotropic and non-Kolmogorov properties compared with traditional atmospheric turbulence which takes the hypotheses of isotropy and homogeneity. There is an increasing interest for modeling atmosphere turbulence with a power spectrum that incorporates anisotropy and general spectral power law deviated from the standard value of 11/3 for the Kolmogorov turbulence. In this study, new analytic expressions for the temporal power spectra of irradiance fluctuations and angle of arrival (AOA) fluctuations have been derived based on the Rytov approximation theory for optical plane wave propagating through weak anisotropic non-Kolmogorov atmosphere turbulence. Compared with the previously published results, the effective anisotropic factor is introduced to the final results to include the anisotropy for different turbulence cells or eddies. The finite turbulence inner and outer scales and the general spectral power law in the range of 3–4 are also considered in the final derived models. The results in this work will help better understand the optical waves’ propagation through anisotropic non-Kolmogorov turbulence.