Turbulence-induced Phase Distortions Research Articles

Generalized approach to compensate for low- and high-frequency errors in fast Fourier transform-based phase screen simulations

Fast Fourier transform-based phase screen simulations give accurate results only when the screen size (G) is much larger than the outer scale parameter (L0). Otherwise, they fall short in correctly predicting both the low and high frequency behaviors of turbulence-induced phase distortions. Subharmonic compensation is a commonly used technique that aids in low-frequency correction but does not solve the problem for all values of screen size to outer scale parameter ratios (G / L0). A subharmonics-based approach will lead to unequal sampling or weights calculation for subharmonics addition at the low-frequency range and patch normalization factor. We have modified the subharmonics-based approach by introducing a Gaussian phase autocorrelation matrix that compensates for these shortfalls. We show that the maximum relative error in structure function with respect to theoretical value is as small as 0.5% to 3% for (G / L0) ratio of 1/1000 even for screen sizes up to 100 m diameter.

Entanglement protection of high-dimensional states by adaptive optics

We theoretically study the potential of adaptive optics (AO) to protect entanglement of high-dimensional photonic orbital-angular-momentum (OAM) states against turbulence-induced phase distortions. We demonstrate that AO is able to reduce crosstalk among the OAM modes and, consequently, the entanglement decay as well as photon losses. A test of the AO-stabilized output state against high-dimensional Bell inequalities shows that the transmitted entanglement allows for secure communication, even in the strong scintillation regime.

Karhunen–Loeve functions for non-Kolmogorov turbulence

ABSTRACTKarhunen–Loeve functions for the case of non-Kolmogorov turbulence are calculated. The calculations are performed using the covariance matrix of Zernike coefficients which is derived in analytical form. It is shown that the Karhunen–Loeve expansion is more efficient than the Zernike one for simulations of high-order, turbulence-induced phase distortions.

Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence

Numerical modeling of optical wave propagation in atmospheric turbulence is traditionally performed by using the so-called ‘split’-operator method, where the influence of the propagation medium's refractive index inhomogeneities is accounted for only within a set of infinitely narrow phase distorting layers (phase screens). These phase screens are generated on a numerical grid of finite size, which corresponds to a rather narrow slice (spatial area) of atmospheric turbulence. In several important applications including laser target tracking, remote sensing, adaptive optics, and atmospheric imaging, optical system performance depends on atmospheric turbulence within an extended area that significantly exceeds the area associated with the numerical grid. In this paper we discuss methods that allow the generation of a family of long (including infinitely long) phase screens representing an extended (in one direction) area of atmospheric turbulence-induced phase distortions. This technique also allows the generation of long phase screens with spatially inhomogeneous statistical characteristics.

Methods for correcting tilt anisoplanatism in laser-guide-star-based multiconjugate adaptive optics.

Multiconjugate adaptive optics (MCAO) is a technique for correcting turbulence-induced phase distortions in three dimensions instead of two, thereby greatly expanding the corrected field of view of an adaptive optics system. This is accomplished with use of multiple deformable mirrors conjugate to distinct ranges in the atmosphere, with actuator commands computed from wave-front sensor (WFS) measurements from multiple guide stars. Laser guide stars (LGSs) must be used (at least for the forseeable future) to achieve a useful degree of sky coverage in an astronomical MCAO system. Much as a single LGS cannot be used to measure overall wave-front tilt, a constellation of multiple LGSs at a common range cannot detect tilt anisoplanatism. This error alone will significantly degrade the performance of a MCAO system based on a single tilt-only natural guide star (NGS) and multiple tilt-removed LGSs at a common altitude. We present a heuristic, low-order model for the principal source of tilt anisoplanatism that suggests four possible approaches to eliminating this defect in LGS MCAO: (i) tip/tilt measurements from multiple NGS, (ii) a solution to the LGS tilt uncertainty problem, (iii) additional higher-order WFS measurements from a single NGS, or (iv) higher-order WFS measurements from both sodium and Rayleigh LGSs at different ranges. Sample numerical results for one particular MCAO system configuration indicate that approach (ii), if feasible, would provide the highest degree of tilt anisoplanatism compensation. Approaches (i) and (iv) also provide very useful levels of performance and do not require unrealistically low levels of WFS measurement noise. For a representative set of parameters for an 8-m telescope, the additional laser power required for approach (iv) is on the order of 2 W per Rayleigh LGS.

Comparison study of the performance of piston, thin plate and membrane mirrors for correction of turbulence-induced phase distortions

In the framework of the Kolmogorov turbulence model and analytical and numerical models of deformable mirrors, we investigated and intercompared the correction performances of piston, membrane and flexible faceplate (with either free edge, supported edge or clamped edge) adaptive mirrors featuring orthogonal, hexagonal and segmented ring actuator structures. It was shown that all modal correctors provide similar correction performance, which is close to the performance of the ideal corrector with Karhunen–Loève-type influence functions.

Efficiency of the Hartmann test with different subpupil forms for the measurement of turbulence-induced phase distortions

The reconstruction quality of turbulence-induced phase distortions from Hartmann data is calculated for masks with different subpupil forms by means of computer simulations. Four subpupil forms are considered: the circle, the square, the hexagon, and the polar segment. We show that, for the case of a circular aperture, the mask with polar segment subpupils provides the best reconstruction quality.

Power series evaluation of covariances for turbulence-induced phase distortions including outer scale and servo lag effects

Power series formulas are developed to efficiently compute the covariance of the integrated turbulence-induced phase distortions along a pair of ray paths through the atmosphere from two points in a telescope aperture to a pair of sources at finite or infinite range. These covariances may be used to evaluate and optimize the predicted performance of adaptive optical (AO) systems. The power series formulas apply when one or both of the phase distortions is temporally filtered by the closed-loop impulse response function of the AO control loop, thereby allowing the effects of a finite servo bandwidth to be included in AO system performance modeling without the introduction of additional numerical integrations. Results are presented for the Kolomogorov turbulence spectrum with an infinite outer scale, as well as for the case of a finite outer scale with the von Karman turbulence spectrum. Amplitude scintillation effects are neglected. The Taylor, or frozen flow, hypothesis is used to model the temporal behavior of the turbulence, by using a fixed windspeed profile w(z) and a random wind direction profile θw(z) for which the mean values of the quantities cos[kθw(z)] and sin[kθw(z)] can be computed. The resulting formulas for the covariances are weighted integrals of the refractive-index structure constant Cn2(z), where the weighting functions are power series in from one to three indices depending on the choices made regarding the atmospheric turbulence spectrum and the direction of the wind. The integral with respect to z may also be evaluated analytically, provided that (i) the Cn2(z) profile is a sum of terms of the form zp exp(-cz) and (ii) the phase distortion profiles are not temporally filtered.

Including outer scale effects in zonal adaptive optics calculations

Mellin transform techniques are applied to evaluate the covariance of the integrated turbulence-induced phase distortions along a pair of ray paths through the atmosphere from two points in a telescope aperture to a pair of sources at finite or infinite range. The derivation is for the case of a finite outer scale and a von Karman turbulence spectrum. The Taylor hypothesis is assumed if the two phase distortions are evaluated at two different times and amplitude scintillation effects are neglected. The resulting formula for the covariance is a power series in one variable for the case of a fixed atmospheric wind velocity profile and a power series in two variables for a fixed wind-speed profile with a random and uniformly distributed wind direction. These formulas are computationally efficient and can be easily integrated into computer codes for the numerical evaluation of adaptive optics system performance. Sample numerical results are presented to illustrate the effect of a finite outer scale on the performance of natural and laser guide star adaptive optics systems for an 8-m astronomical telescope. A hypothetical outer scale of 10 m significantly reduces the magnitude of tilt anisoplanatism, thereby improving the performance of a laser guide star adaptive optics system if the auxiliary natural star used for full-aperture tip/tilt sensing is offset from the science field. The reduction in higher-order anisoplanatism that is due to a 10-m outer scale is smaller, and the off-axis performance of a natural guide star adaptive optics system is not significantly improved.

Dual-wavelength adaptive optical systems

A method for decomposition of phase difference error between measurements of atmospheric turbulence-induced phase distortion at two different wavelengths is described. Calculations are made of the phase difference errors in the first five Zernike radial modes for both ground-to-ground and ground-to-space transmission of laser radiation. It is found that the phase difference error compared with the uncorrected wavefront phase error is relatively insignificant in the first (tilt) Zernike mode but increases in significance with the order of the Zernike mode. Relative phase difference error is also found to depend on transmitted and received wavelengths, aperture diameter, propagation path, and strength of turbulence.

Optical resolution through a turbulent medium with adaptive phase compensations*

The theory of short-term average modulation transfer function developed by Fried is extended by considering the correlation between the turbulence-induced phase distortion and the residual phase aberrations after adaptive phase compensations. Calculation of the optical resolution shows that the original approximation is valid for the normalized lens diameter D/r0 less than 4.0; conversely, for D/r0 greater than 4.0 the optical resolution becomes arbitrarily large because the MTF is unrealistically overcorrected at high spatial frequencies. The extended formulation is then applied to the cases with higher-order phase compensations such as focus and astigmatism. The peak optical resolution occurs at larger values of D/r0 when the higher-order corrections are included.