Statistics of the random phase screens used for the modeling of beam propagation and imaging through the turbulent atmosphere is currently based on the Markov Approximation (MA) for wave propagation. This includes the phase structure functions of individual screens and the use of the statistically-independent screens for the multi-screen splitstep simulation of wave propagation. As the propagation modeling progresses to address the deep turbulence conditions, the increased number of phase screens is required to accurately describe the multiple scattering. This makes the MA a critical limitation, both because phase statistic of the thin turbulent layer does not follow MA, and because the closely space screens cannot be considered as statistically and functionally independent. A recently introduced Sparse-Spectrum (SS) model of statistically homogeneous random fields makes it possible to generate 3-D samples of refractive-index fluctuations with prescribed spectral density at a very reasonable computational cost. This leads to generation of samples of the phase screen sets that are free from the limitations of the MA. We investigated statistics of the individual phase screens and cross-correlations between the pairs of phase screens and found that the thickness Δz of the turbulent layer replaced by the phase screen is a new parameter defining the phase statistics in the non-Markov case. SS-based numerical algorithms for generation of the 3-D samples of the turbulent refractive index, and for the phase screen sets are presented. We also compare the split-step simulation results for the traditional MA and non-Markov screens.
We evaluate the accuracy of recently published, tractable yet approximate, analytic theory for describing the centroid-centered mean irradiance of beams propagated through atmospheric turbulence. Such theory, if accurate, is a highly desirable candidate for use in optical system performance calculations that perform active tracking.
Beam spread and beam wandering are the most perceptible effects of atmospheric turbulence on propagating laser beams. The width of the mean irradiance profile is typically used to characterize the beam spread. This so-called Long- Term (LT) statistic allows for a relatively simple theoretical description. The LT beam size is not a very practical measure of the beam spread because its measurements are sensitive to the movements of the source and detector, and to the large-scale variations of the refractive index that are not associated with turbulence. The Short-Term (ST) beam spread is measured relative to the instantaneous position of the beam center and is free of these drawbacks, but has not been studied as thorough as the LT spread. We use a Markov approximation-based theoretical model for the ST beam irradiance that is valid for the wide range of turbulent conditions. Additional approximations are invoked to allow introduction of the isoplanatic ST Point Spread Function (PSF). Unlike the LT PSF, the ST PSF depends on the overall beam geometry. Adjustments of the initial beam width and focal distance make it possible to increase the contribution of the LT beam spread that is attributed to the beam wander and minimize the ST beam size at the observation plane for any given turbulence level. Analytical calculations of the optimal beam geometry are presented for the simple case of the coherent Gaussian beam, and Kolmogorov turbulence. We present the results of direct numerical simulation of beam wave propagation that confirm the existence of the optimal beam geometry.
We evaluate the accuracy of 3 analytic approaches for computing the mean irradiance of beams propagated through atmospheric turbulence. The 3 approaches, representing major classes available in the literature, are the Markov approach, a modified Markov approach, and the Rytov Method. Accuracy is ultimately determined by comparison to results from a numerical solution of turbulent beam propagation. Of the 3 analytic offerings, the Markov approach yielded the most accurate and most generally applicable result.
Reciprocity principle for the optical wave propagation in turbulence suggests that scintillations in the focal point of a
coherent optical beam and in the center of the point spread function (PSF) of the imaging system are identical, provided
that the imaging aperture and initial beam irradiance are matched. Rigorous weak and strong scintillation asymptotes of
the scintillation index (SI) in the beam focus indicate that the relatively simple extended Huygens-Fresnel (HF)
approximation is accurate in both asymptotic cases. This motivated us to use the HF approximation for calculation of the
SI in the moderate turbulence case when SI reaches its maximum. The 8 - fold integral representing the SI was
calculated using Mont-Carlo technique. We compare the HF results to the direct numeric wave optics simulation results
and find some discrepancies that can be attributed to the finite grid sampling used in simulation.
In practical situation the exact position of the beam focal point at the end of the long propagation path is rarely available,
but instantaneous, short-term (ST) beam center can be estimated by the beam centroid position. For imaging problems,
the short-exposure (SE) PSF and its scintillation are of great interest. We used the combination of the HF approximation
and available SE imaging model to calculate the short-term SI for the focused beams under weak strong and intermediate
turbulence conditions using the same numeric integration technique as for the Long-Term (LT) case. Calculations show
up to 500% increase in the average irradiance and substantial reduction of scintillation for the SE case.
Asymptotic theory of the finite beam scintillations (Charnotskii, WRM, 1994, JOSA A, 2010) provides an exhaustive
description of the dependence of the beam scintillation index on the propagation conditions, beam size and focusing.
However the complexity of the asymptotic configuration makes it difficult to apply these results for the practical
calculations of the scintillation index (SI). We propose an estimation technique and demonstrate some examples of the
calculations of the scintillation index dependence on the propagation path length, initial beam size, wavelength and
turbulence strength for the beam geometries and propagation scenarios that are typical for applications. We suggest
simple analytic bridging approximations that connect the specific asymptotes with the accuracy sufficient for the
engineering estimates. Proposed technique covers propagation of the wide, narrow, collimated and focused beams under
the weak and strong scintillation conditions.
Direct numeric simulation of the beam waves propagation through turbulence expediently complements the
asymptotic theory being most efficient when the governing scales difference is not very large. We performed numerical
simulations of the beam wave propagation through turbulence for conditions that partially overlap with the major
parameter space domains of the asymptotic theory. The results of the numeric simulation are used to confirm the
asymptotic theory and estimate the accuracy of the bridging approximations.
We present a survey of apodization functions intended to suppress diffraction effects over a finite propagation distance and within a
finite "score" region on the final plane. Such apodization functions would be suitable for use in numerical simulations as part of a
scheme to reduce numerical computation cost without sacrificing numerical accuracy.
We present a simulation of a 200 km air-to-air link in the presence of aero-optical boundary layers. The boundary
layer is shown to be the dominant phase aberrator. The random tilt content in the boundary layer is minimal,
which reduces the performance gain of a fast steering mirror. Higher order adaptive optics are shown to provide
a significant performance improvement provided it can run at high enough bandwidths.
The two approximate solutions to the stochastic wave equation governing propagation through atmospheric turbulence applicable in
weak scintillation conditions are reviewed. Then, an extensive set of numerical solutions are shown to test the ability of the 2
approximate solutions in predicting scintillation and the irradiance probability density function for a wide variety of beam
propagation examples. The non-log normal irradiance behavior associated with one of the approximate solutions is noted and
verified by the numerical data.
A new analytic approach for the description of deformable mirror facesheet deformation is presented. This new approach contains a high degree of physics fidelity, yet is relatively simple to implement and quick to use when compared with the equally-accurate finite element approach. Modeled physics include thin plate treatment for the deformable mirror facesheet and full mechanical coupling between the facesheet and the underlying actuators. Example influence functions for a circular DM are presented.
The effects upon imaging due to varying the spatial coherence of the illumination in an optical system are studied. A rotating diffuser is located directly behind the object in an optical system and is trans-illuminated with spatially coherent monochromatic light. The statistical properties of the diffuser surface determine the scattering cone angle and the partial coherence effects in the image. A model is presented that can be used to determine the diffuser properties required to yield incoherent imaging. Two metrics are used to determine if an image is incoherent: the apparent transfer function and image contrast.
Low-order turbulence effects dominate the random irradiance fluctuations in a weakly-scintillated Gaussian beam subject to a certain
set of initial conditions, leading to a natural departure from log normal irradiance behavior. This departure conflicts with earlier
theoretical studies of weakly scintillated beams which have traditionally assumed log normal behavior. The dominance of low order
effects leads to an increase in the theoretical scintillation and probability of fade relative to predictions based on the assumption of
log normal behavior. This paper recounts a detailed derivation of a low order turbulence model that successfully captures the non-log
normal behavior, and reviews theoretical scintillation and probability of fade predictions that follow from the model.
Recent papers comparing weak-scintillation data from wave-optics simulations with predictions derived from the Rytov perturbation method for ground-to-space Gaussian beams have revealed a region of inaccuracy in the Rytov-based predictions. The discrepancy region is defined by beam diameter and focus settings that place the target in the beam far field and the turbulence in the transmitter near field. Under such conditions turbulence-induced beam wander dominates the scintillation at the target. We develop a solution to the turbulent propagation physics that is applicable in the discrepancy region, and demonstrate agreement in scintillation behavior with our own wave-optics simulation data and with predictions from a rigorous extended Huygens-Fresnel analysis. A combination of our solution and Rytov-based scintillation theory can be used to yield accurate scintillation predictions throughout the weak-scintillation regime for ground-to-space beams. Separately, we show that Rytov-based scintillation theory best describes the physics of a wander-tracking transmitter (where beam wander has been removed from the propagation physics) at the mean transmitter aim point on the target plane, as opposed to the physics of a stationary transmitter.
The five distinct regions of on-axis weak scintillation behavior exhibited by Gaussian beam propagation in the turbulent atmosphere are reviewed. One of the regions (D1) is unique, in that the random irradiance across much of the target plane is non log normal, scintillation predictions derived from the Rytov approximation are inaccurate, and no generally applicable and computationally efficient scintillation theory is currently available. The D1 region cannot be easily dismissed, since it includes horizontal path focused beams and ground to space collimated beams. A survey of the D1 region is provided, including physics insight from past publications, the role of the phase screen approximation, and the impact of distributed low order turbulence. An overview of a new approach that yields accurate, computationally efficient scintillation predictions within D1 is provided.
Predictions of scintillation for ground to space collimated Gaussian beams generated from a numerical wave optics simulation are compared with recent weak scintillation theory developed from the Rytov perturbation approach (L.C. Andrews, R.L. Phillips, P.T. Yu, Ap Opt 34, p 7742-7751, 1995; J.D. Shelton, JOSA A12, p 2172-2181, 1995). Significant discrepancies are revealed for intermediate-sized beams, defined as beams whose initial diameters place the near ground turbulence in the transmitter near field and the remote space target in the transmitter far field. By adding wander tracking to the wave optics simulation, and by developing a separate analytic model of the beam wander scintillation mechanism, we show that the scintillation for intermediate-sized beams is dominated by turbulence-induced beam wander at the target, and that the results from the wave optics simulation are accurate. We conclude that the analytic theory’s treatment of beam wander is incomplete, leading to the output of incorrect predictions for the second moment of irradiance. The error is most severe at the target point on the transmitter’s optical axis.
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