David L. Fried: bringing vision to atmospheric optics [Invited].

This paper recognizes the life work of  David L. Fried. The goal is to put the relevance of his contributions to atmospheric optics in context from three perspectives: solutions to real world problems as seen by an experimentalist, insights of his analytical process as perceived by co-workers, and his role as a mentor to bright, young minds passionate about a deeper understanding of the governing physics. During his professional career of sixty years from 1962 to 2022, Fried published 106 refereed papers, cited over 6000 times. This paper reviews the impact of Fried's work, especially addressing his contributions to ameliorating the effects of turbulence on optical system performance and the insight this provides into his analytical approach and interactions with colleagues.

Linear analysis of thermal blooming compensation instabilities in laser propagation.

Thermal blooming compensation instabilities are examined. The linearized system of thermal blooming compensation (TBC) equations is studied to develop parameters that characterize the stability of phase-only and full-wave (amplitude and phase) compensation for the effects of thermal blooming. The stabilizing effects of microscale wind shear are included in the analysis to provide a mechanism to stabilize the TBC equations. Stability is equated to existence of bounded solutions of the linear TBC equations, and appropriate dimensionless parameters are developed that ensure existence and uniqueness of bounded solutions to the TBC equations. Parameters characterizing stability are expressed in forms analogous to conventional scaling laws.

Practical configuration for wave-front sensing with two-beam phase retrieval.

A wave-front sensing approach based on two-beam phase retrieval is described. Light from an aberrated beam is split into two paths. A random phase and amplitude perturbation is applied to the beam in one path, and the interference patterns of the resultant two beams are measured in two planes along the axis of propagation. By modulation of one of the two beams, the intensity of each beam and the phase difference between the two beams are recovered in each plane. A rapidly convergent phase-retrieval algorithm is formulated by the method of sequential projections onto constraint sets. Examples are given illustrating the convergence properties of the approach.

Multigrid approach to predictive wave-front reconstruction in adaptive optical systems.

A computationally efficient approach, based on the principles of multigrid methods, to predictive wave-front reconstruction in adaptive optical systems is described. Local predictive estimators are computed by use of recursive least squares on multiple grids. Each grid is increasingly coarse, allowing for temporal prediction of the behavior of both high- and low-spatial-frequency aberrations. Example numerical simulation results are given, showing that implementing the recursive least-squares algorithm for predictive estimation in a multigrid fashion greatly accelerates convergence to the steady-state optimal estimator condition. By implementation of the multigrid predictive reconstructor in parallel, the computational cost of implementing a predictive wave-front reconstruction scheme that uses recursive least squares for each processor at each cycle can be reduced from [symbol: see text](m2) to [symbol: see text](2m), where m is the number of actuators.

Rapidly convergent phase-retrieval strategy for use with reflected laser light.

An approach for wave-front sensing using reflected laserlight from a rough object is proposed. Light from a single laser beam is split into two beams, and the beams are launched from spatially separated apertures to illuminate an object. The reflected laser light is measured in the pupil plane of a receive telescope and in a plane conjugate to the object. By modulation of one of the two illuminator beams, the intensity pattern associated with each beam, as well as the field cross product of the two beams, is measured in each plane. A phase-retrieval algorithm is formulated by using projections onto constraint sets to recover the complex field associated with each illuminator. The algorithm is found to converge rapidly to the correct solution, particularly when compared with the convergence rates of more conventional phase-retrieval approaches. The new algorithm exhibits excellent performance in strong scintillation and is very tolerant to noise, exhibiting only a very small noise gain.

Optimal control of laser beams for propagation through a turbulent medium.

Concerning the problem of transmitting a laser beam from one telescope to another telescope through a turbulent medium, it is established that using an adaptive optical system on both telescopes to precompensate an outgoing laser beam based on the aberrations measured on the received laser beam leads to an iteration that maximizes the transmission (neglecting attenuation losses) of laser power between the telescopes. Simulation results are presented demonstrating the effectiveness of this technique when the telescopes are equipped with either phase-only or full-wave compensation systems. Simulation results are shown that indicate that for a uniform distribution of the strength of turbulence, 95% transmission of laser power is attained when both telescopes can achieve full-wave compensation provided that the aperture diameter D of the two telescopes is greater than twice the Fresnel length square root of lambdaL, where A is the wavelength of propagation and L is the distance between the two telescopes.

Evaluation of the performance of a shearing interferometer in strong scintillation in the absence of additive measurement noise.

A simulation study is presented that evaluates the ability of a unit-shear, shearing interferometer to estimate a complex field resulting from propagation through extended turbulence. Performance is defined in terms of the Strehl ratio achieved when the estimate of the complex field obtained from reconstruction is used to correct the distorted wave front presented to the wave-front sensor. A series of evaluations is performed to identify the strengths and weaknesses of the shearing interferometer in the two-dimensional space of the Fried parameter r0 and the Rytov number. The performance of the shearing interferometer is compared with that of a Hartmann sensor in the Fried and Hutchin geometries. Although the effects of additive measurement noise (such as read noise, shot noise, amplifier noise) are neglected, the fundamental characteristics of the measurement process are shown to distinguish the performance of the various wave-front sensors. It is found that the performance of a shearing interferometer is superior to that of a Hartmann sensor when the Rytov number exceeds 0.2.

Convergence rates for iterative vector space projection methods for control of two deformable mirrors for compensation of both amplitude and phase fluctuations.

The control of two deformable mirrors for compensation of time-varying fluctuations in the complex field that results from wave propagation through a turbulent medium is considered. Iterative vector space projection methods are utilized to determine the control commands to be applied to the two deformable mirrors. Convergence of the iterative algorithm is accelerated when the algorithm is initialized, at each measurement period, with the values for the phase commands obtained from the previous measurement period. Furthermore, it is found that, if the sample frequency is sufficiently greater than the Greenwood frequency, then only a single iterative step at each measurement period is required to obtain good compensation of both amplitude and phase fluctuations.

Closed-loop stable control of two deformable mirrors for compensation of amplitude and phase fluctuations.

A method for closed-loop stable control of two deformable mirrors for compensation of both amplitude and phase fluctuations is described. A generic implementation is described as well as an implementation that integrates the concept behind a point diffraction interferometer with a two-deformable-mirror system. The relationship of the closed-loop control algorithm to previously developed open-loop iterative algorithms is described. Simulation results are presented that indicate that the system is stable and provides superior performance over that of a single-deformable-mirror system. The impact of finite servo bandwidth on control of two deformable mirrors is evaluated by means of wave optical simulation, and it is found that to achieve a performance improvement attributable to compensation of amplitude fluctuations, the bandwidth of the two-deformable-mirror system must be at least twice the Greenwood frequency.

Evaluation of the performance of Hartmann sensors in strong scintillation.

A simulation study is presented that evaluates the performance of Hartmann wave-front sensors with measurements obtained with the Fried geometry and the Hutchin geometry. Performance is defined in terms of the Strehl ratio achieved when the estimate of the complex field obtained from reconstruction is used to correct the distorted wave front presented to the wave-front sensor. A series of evaluations is performed to identify the strengths and the weaknesses of Hartmann sensors used in each of the two geometries in the two-dimensional space of the Fried parameter r0 and the Rytov parameter. We found that the performance of Hartmann sensors degrades severely when the Rytov number exceeds 0.2 and the ratio l/r0 exceeds 1/4 (where l is the subaperture side length) because of the presence of branch points in the phase function and the effect of amplitude scintillation on the measurement values produced by the Hartmann sensor.

Application of the parallel generalized projection algorithm to the control of two finite-resolution deformable mirrors for scintillation compensation.

A modification of the parallel generalized projection algorithm is presented that allows for the use of projections in a weighted norm. Convergence properties of the modified algorithm, denoted the weighted parallel generalized projection algorithm, are developed. The weighted parallel generalized projection algorithm is applied to the control of two finite-resolution deformable mirrors to compensate for both the amplitude and the phase fluctuations that result from propagation through a turbulent medium. Numerical results are shown that indicate that a two-deformable-mirror system can provide improved performance over that of a single-deformable-mirror system.

Evaluation of phase-shifting approaches for a point-diffraction interferometer with the mutual coherence function.

The estimation accuracy of a point-diffraction interferometer is examined with two phase-shifting schemes: spatial and temporal. Under the assumption of plane- or spherical-wave propagation through isotropic turbulence that can be accurately represented as a series of thin phase screens, results that are valid for any scintillation regime are obtained by use of the invariance with a propagation of the mutual coherence function. It is established that the estimation accuracy of the spatial phase-shifting point-diffraction interferometer is invariant with scintillation. Upper and lower bounds on the perfor mance of the temporal phase-shifting point-diffraction interferometer are developed. Wave optical simulation results are presented that validate the analytic predictions for the two phase-shifting schemes. The results and techniques presented can be used to assess the appropriate phase-shifting scheme given finite resources, such as a limited number of pixels in a detector array or a restricted detector frame rate.