Finite Outer Scale Research Articles

Features of optical image jitter in a random medium with a finite outer scale

Features of optical wave fluctuations while propagating through a randomly inhomogeneous turbulent medium with a finite outer scale are considered, including conditions when areas with dominating influence of one large scale coherent structure are observed in the atmosphere, for which the spectrum of atmospheric turbulence can differ significantly from the Kolmogorov model spectrum. Using an approximate model of the spectrum for coherent turbulence, described earlier in our works, the variance of jitter of an optical image is calculated (under the applicability condition for the smooth perturbation method). The comparison of these equations with known similar equations for Kolmogorov turbulence has shown that the variance of fluctuations is significantly weaker in coherent turbulence than in the Kolmogorov theory under similar conditions. This means that phase fluctuations of optical radiation decrease significantly in coherent turbulence. The importance of this conclusion is noted for interpretation of the results of optical sounding of atmospheric turbulence.

Atmospheric Phase Screen Simulation Using Zernike Polynomial

An algorithm is described which simulates atmospheric phase screen (PS) distorted by von Karman atmospheric turbulence by using Zernike expansion of randomly weighted Karhunen-Loeve function. The statistics of the PS generated by using power spectrum method which is most commonly used for generating PS poorly match up with the theoretical structure function, especially at low spatial frequency, while the algorithm described in the this paper can compensate for this shortcoming. And the accuracy of the PS is verified by comparing with the theoretical results. Furthermore, comparing with the existed methods which also use the Zernike expansion to simulate PS, algorithm presented in this paper is more accurate because the effects of the finite outer scale L0.is included

Expressions of the scintillation index for optical waves propagating through weak non-Kolmogorov turbulence based on the generalized atmospheric spectral model

The high frequency “bump” which occurs in the turbulence spectral model just prior to the turbulence cell dissipation is important for the analysis of the irradiance scintillation for optical wave propagating through atmospheric turbulence. In this study, expressions of the irradiance scintillation index are developed from the generalized modified atmospheric spectral model for optical waves propagating through weak non-Kolmogorov turbulence. Compared with the expressions of the irradiance scintillation index derived from the general non-Kolmogorov spectral model, the new expressions can consider the influences of finite turbulence inner and outer scales and the influence of finite diameter aperture receiver. As the irradiance scintillation is caused mainly by the small scale turbulence cells' diffractive effects for weak turbulence, the turbulence outer scale's influence can be ignored. Numerical simulations show that variable inner scale values produce obvious effects on the irradiance scintillation for non-Kolmogorov turbulence.

Inner- and outer-scale effects on the scintillation index of an optical wave propagating through moderate-to-strong non-Kolmogorov turbulence

By use of the generalized von Kármán spectrum model that features both inner scale and outer scale parameters for non-Kolmogorov turbulence and the extended Rytov method that incorporates a modified amplitude spatial-frequency filter function under strong-fluctuation conditions, theoretical expressions are developed for the scintillation index of a horizontally propagating plane wave and spherical wave that are valid under moderate-to-strong irradiance fluctuations. Numerical results show that the obtained expressions also compare well with previous results in weak-fluctuation regimes. Based on these general models, the impacts of finite inner and outer scales on the scintillation index of an optical wave are examined under various non-Kolmogorov fluctuation conditions.

Scintillation index and performance analysis of wireless optical links over non-Kolmogorov weak turbulence based on generalized atmospheric spectral model

Based on the generalized spectral model for non-Kolmogorov atmospheric turbulence, analytic expressions of the scintillation index (SI) are derived for plane, spherical optical waves and a partially coherent Gaussian beam propagating through non-Kolmogorov turbulence horizontally in the weak fluctuation regime. The new expressions relate the SI to the finite turbulence inner and outer scales, spatial coherence of the source and spectral power-law and then used to analyze the effects of atmospheric condition and link length on the performance of wireless optical communication links.

Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence

Kolmogorov turbulence theory based models cannot be directly applied in non-Kolmogorov turbulence case, which has been reported recently by increasing experimental evidence and theoretical investigation. In this study, based on the generalized von Karman spectral model, the theoretical expression of the irradiance scintillation index is derived for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence with horizontal path. In the derivation, the expression is divided into two parts for physical analysis purpose and mathematical analysis convenience. This expression considers the influences of finite turbulence inner and outer scales and has a general spectral power law value in the range 3 to 4 instead of standard power law value of 11/3 (for Kolmogorov turbulence). Numerical simulations are conducted to investigate the influences.

Wavefront propagation in turbulence: an unified approach to the derivation of angular correlation functions

A general expression of the spatial correlation functions of quantities related to the phase fluctuations of a wave that have propagated through the atmospheric turbulence are derived. A generalization of the method to integrand containing the product of an arbitrary number of hypergeometric functions is presented. The formalism is able to give the coefficients of phase-expansion functions orthogonal over an arbitrary circularly symmetric weighting function for an isotropic turbulence spectrum, as well as to describe the effect of the finite outer and inner scales of the turbulence and to describe the spherical propagation or to derive the effects of the analytical operators acting on the phase such as the derivatives of any order. The derivation of the generalized integrals with multiparameters is based on the Mellin transforms integration method.

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.

Theoretical expressions of the angle-of-arrival variance for optical waves propagating through non-Kolmogorov turbulence

Based on the generalized exponential spectrum for non-Kolmogorov atmospheric turbulence, theoretical expressions of the angle-of-arrival (AOA) variance are derived for plane and spherical optical waves propagating through weak turbulence. Without particular assumption, the new expressions relate the AOA variance to the receiver aperture, finite turbulence inner and outer scales, and the optical wavelength.

Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence

A generalized exponential spectrum model is derived, which considers finite turbulence inner and outer scales and has a general spectral power law value between the range 3 to 5 instead of standard power law value 11/3. Based on this generalized spectrum model, a new generalized long exposure turbulence modulation transfer function (MTF) is obtained for optical plane and spherical wave propagating through horizontal path in weak fluctuation turbulence. When the inner scale and outer scale are set to zero and infinite, respectively, the new generalized MTF is reduced to the classical generalized MTF derived from the non-Kolmogorov spectrum.

Atmospheric image blur with finite outer scale or partial adaptive correction

Seeing-limited resolution in large telescopes working over wide wavelength range depends substantially on the turbulence outer scale and cannot be adequately described by one "seeing" value. We attempt to clarify frequent confusions on this matter. We study the effects of finite turbulence outer scale and partial adaptive corrections by means of analytical calculations and numerical simulations. If a von Karman turbulence model is adopted, a simple approximate formula captures the dependence of atmospheric long-exposure resolution on the outer scale over the entire practically interesting range of telescope diameters and wavelengths. In the infrared (IR), the difference with the standard Kolmogorov seeing formula can exceed a factor of two. We find that low-order adaptive turbulence correction produces residual wave-fronts with effectively small outer scale, so even very low compensation order leads to a substantial improvement in resolution over seeing, compared to the standard theory. Seeing-limited resolution of large telescopes, especially in the IR, is currently under-estimated by not accounting for the outer scale. On the other hand, adaptive-optics systems designed for diffraction-limited imaging in the IR can improve the resolution in the visible by as much as two times.

Karhunen-Loève Basis Functions Of Kolmogorov Turbulence In The Sphere (Baltic Astronomy, 17, 383, 2008, Erratum)

AbstractThe previously published factor, which scales the matrix elements of the eigenvalue equation as a finite outer scale of the von-Kármán model is switched on, is corrected.

Theoretical wave structure function when the effect of the outer scale is significant

The wave structure function (WSF) for a plane wave, calculated from the basic Rytov theory, is usually expressed as 6.88(r/r(0))(5/3), but this does not include the effect of a finite outer scale (or of a nonzero inner scale) of turbulence. When separation distance r is only 5% of the outer scale, this expression overpredicts the WSF by a factor of approximately 2. Accurate evaluations of the Rytov formulas are given for the WSFs of plane and spherical waves in Kolmogorov and von Karman turbulence and for the structure function of the atmosphere's index of refraction. Simple formulas make the results easy to use.

Sky coverage estimates for adaptive optics systems from computations in Zernike space

A sky coverage model for laser guide star adaptive optics systems is proposed. The atmosphere is considered to consist of a finite number of phase screens, which are defined by Zernike basis polynomials, located at different altitudes. These phase screens are transformed to the aperture plane, where they are converted to laser and natural guide star wavefront sensing measurements. These transformations incorporate the cone effect due to guide stars at finite heights, anisoplanatism due to guide stars off axis with respect to the science object, and adaptive optics systems with multiple guide stars. The wavefront error is calculated tomographically with minimum variance estimators derived from the transformation matrices and the known statistical properties of the atmosphere. This sky coverage model provides fast Monte Carlo simulations over random natural guide star configurations, irrespective of telescope diameter. The Monte Carlo simulations outlined show that inclusion of a finite outer scale for the atmosphere significantly reduces the median wavefront error, that increasing the number of laser guide stars in the asterism reduces the median wavefront error, and that a larger natural guide star patrol field provides a smaller median wavefront error when there is a low star density in the field.

Simulating the phase fluctuations produced by many stars in the weak atmospheric turbulence limit

We propose a method allowing for a simulation of cross-correlated turbulence-induced phase fluctuations produced by many stars is proposed. It turns out that for weak-turbulence conditions which are important for many applications, it is possible to suggest a relatively simple and fast simulation procedure. Both the variation of C 2 n along a propagation path and an effect of finite outer scale of the turbulence are considered. The validity of the simulation is verified by comparing theoretical and simulated results, and it is shown that there is good agreement takes place even for big separations between the observation points. Adaptive optics, guide stars, speckle interferometry, long-base interferometry, imaging through the turbulence, as well as simulations of the image distortions of extended objects are among the possible applications of this method.

Non-Kolmogorov features of differential image motion restored from the Multichannel Astrometric Photometer data

In this paper we present a new restoration of atmospheric turbulent parameters using published data on image motion observations with the Multichannel Astrometric Photometer. These data have been previously analyzed by Han (1989) whose estimate of the power of the phase structure function α = 1.64 was close to the 5/3 value expected for the Kolmogorov atmospheric turbulence. Investigating the temporal image motion spectrum, we show, however, that the experimental data do not follow predictions of the Kolmogorov model. The best fitting of the observed spectrum is achieved with α = 1.9−2.0, which is typical of non-classic phase distortions. The outer scale of turbulence was found to be 1800m or longer. Expressions for the non-Kolmogorov temporal spectrum of differential image motion allowing for a finite outer scale length are given.

Effects of a finite outer scale on the measurement of atmospheric-turbulence statistics with a Hartmann wave-front sensor.

Phase-structure and aperture-averaged slope-correlated functions with a finite outer scale are derived based on the Taylor hypothesis and a generalized spectrum, such as the von Kármán modal. The effects of the finite outer scale on measuring and determining the character of atmospheric-turbulence statistics are shown especially for an approximately 4-m class telescope and subaperture. The phase structure function and atmospheric coherent length based on the Kolmogorov model are approximations of the formalism we have derived. The analysis shows that it cannot be determined whether the deviation from the power-law parameter of Kolmogorov turbulence is caused by real variations of the spectrum or by the effect of the finite outer scale.

Theory of optical scintillation: Gaussian-beam wave model

A scintillation model previously developed by the authors is extended in this paper to the case of a propagating Gaussian-beam wave. As in the previous model, we account for the loss of spatial coherence as the optical wave propagates through atmospheric turbulence by eliminating effects of certain turbulent scale sizes that exist between the scale size of the spatial coherence radius of the beam and that of the scattering disc. These mid-range scale-size effects are eliminated through the formal introduction of spatial frequency filters that continually adjust spatial cut-off frequencies as the optical wave propagates. Unlike the previous model, in this paper we include the effect of a finite outer scale in addition to the inner scale. With a finite outer scale, the scintillation index can be substantially lower in strong turbulence than that predicted by a model with an infinite outer scale. This particular behaviour of scintillation in strong turbulence, mostly associated with horizontal paths near the ground, cannot be explained on the basis of previous expressions deduced from the asymptotic theory. Comparisons of the scintillation models with published experimental and simulation data through weak and strong irradiance fluctuations show excellent fits.

Tilt angular anisoplanatism and a full-aperture tilt-measurement technique with a laser guide star

A method is presented for sensing atmospheric wave-front tilt from a laser guide star (LGS) by observing a laser beacon with auxiliary telescopes. The analysis is performed with a LGS scatter model and Zernike polynomial expansion of wave-front distortions. It is shown that integration of the LGS image over its angular extent and the position of the auxiliary telescope in an array reduce the tilt sensing error associated with the contribution from the downward path. This allows us to single out only the wave-front tilt of the transmitted beam on the uplink path that corresponds to the tilt for the scientific object. The tilt angular correlation is analyzed in the atmosphere with a finite turbulence outer scale. The tilt correlation angle depends on the angular size of the telescope and the outer scale of turbulence. The tilt sensing error increases with the auxiliary telescope diameter, suggesting that an auxiliary telescope must be small. The Strehl ratio associated with the contribution from the downward path is in the range from 0.1 to 0.9 when the relative telescope diameter D/r(0) varies from 4 to 93 and the turbulence outer scale is in the 10-150-m range. Tilt correction increases the Strehl ratio compared with the uncorrected image for all the system parameters and seeing conditions considered. The method discussed gives a higher performance than the conventional technique, which uses an off-axis natural guide star. A scheme for measuring tilt with a beam projected from a small aperture is described. This scheme allows us to avoid phosphorescence of the main optical train for a sodium LGS.

Isoplanatism in a multiconjugate adaptive optics system.

Turbulence correction in a large field of view by use of an adaptive optics imaging system with several deformable mirrors (DM's) conjugated to various heights is considered. The residual phase variance is computed for an optimized linear algorithm in which a correction of each turbulent layer is achieved by applying a combination of suitably smoothed and scaled input phase screens to all DM's. Finite turbulence outer scale and finite spatial resolution of the DM's are taken into account. A general expression for the isoplanatic angle thetaM of a system with M mirrors is derived in the limiting case of infinitely large apertures and Kolmogorov turbulence. Like Fried's isoplanatic angle theta0,thetaM is a function only of the turbulence vertical profile, is scalable with wavelength, and is independent of the telescope diameter. Use of angle thetaM permits the gain in the field of view due to the increased number of DM's to be quantified and their optimal conjugate heights to be found. Calculations with real turbulence profiles show that with three DM's a gain of 7-10x is possible, giving the typical and best isoplanatic field-of-view radii of 16 and 30 arcseconds, respectively, at lambda = 0.5 microm. It is shown that in the actual systems the isoplanatic field will be somewhat larger than thetaM owing to the combined effects of finite aperture diameter, finite outer scale, and optimized wave-front spatial filtering. However, this additional gain is not dramatic; it is less than 1.5x for large-aperture telescopes.