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Abstract
The Fried parameter is a fundamental coherence length measure that characterizes the spatial resolution effects of atmospheric turbulence on imaging and beam propagation. Its expression was originally derived with the Rytov approximation and a near-field assumption, which are only strictly valid for weak turbulent scattering. However, the actual range of validity of the Fried parameter formulation remains controversial. Here, wave optic simulations are conducted for modeling the long-exposure point spread function through atmospheric turbulence. The results indicate that analytic expressions for the Fried parameter are accurate throughout the weak to strong turbulent scattering regimes.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Atmospheric turbulence, with its associated random refractive index variations, disturbs the propagation of light through the atmosphere. The disturbance generally takes the form of distortion of the wavefront shape and variations of the intensity across the wavefront. In the mid-1960s, D. L. Fried published a series of papers that presented an analysis of the effect of atmospheric turbulence on long- and short-exposure imaging [1–4]. His work developed the wave structure function (WSF) for relating the statistics of wave distortion to optical resolution. In addition, a fundamental spatial coherence length measure, which is now commonly known as the Fried parameter, was introduced to provide a characterization of the spatial resolution effects of atmospheric turbulence on imaging and beam propagation. For Kolmogorov turbulence (i.e., zero inner scale and infinite outer scale) with horizontal propagation over a distance z, analytic expressions of the long-exposure Fried parameter r0 are [1,5]
where k = 2π/λ is the optical wave number and λ is the wavelength. Cn2 is the refractive index structure constant that characterizes the homogeneous turbulence strength along the horizontal path of propagation.The r0 expressions in Eq. (1) were originally derived with the Rytov approximation to the wave equation and a near-field assumption applied to the WSF. Specifically, the effects of the refractive index inhomogeneities on wave propagation were described as a multiplicatively perturbed version of the free-space solution, with the assumption that the first-order field perturbation term is small (i.e., weak scattering theory). In addition, the most important eddies are assumed to be within the near-field of the imaging system so that each ray traversing the atmosphere is only delayed by the refractive index variations; i.e., there is no ray bending, no amplitude fluctuations (scintillation) and negligible diffraction effects [1–2,5]. Therefore, the wave structure function (WSF) is simply equal to the phase structure function and the r0 expressions are strictly valid for weak scattering. But some studies argue that the phase fluctuation statistics have a wider range of validity than the log-amplitude statistics [6], and the WSF for long paths, which includes the combination of log-amplitude and phase fluctuation effects, leads to the same formulations as the phase structure function in the near-field regime [5,7]. Consequently, it is presumed that the analytic expression of the Fried parameter r0 in Eq. (1) can be extended to more general conditions than the very restrictive weak scattering regime. However, the range of turbulence scattering conditions where the Rytov perturbation analysis is valid remains unclear and therefore the range of applicability of the Fried parameter formulation is also uncertain.
A preferred approach to test the range of applicability of the r0 theory would be to make independent experimental measurements of r0 and Cn2. However, commercial turbulence instruments that report both values, such as scintillometers, wave front sensors, or differential tilt devices, typically measure one quantity (e.g., Cn2) and rely on the analytic theory to find the other (r0) [8–11]. Furthermore, simultaneous Cn2 measurements reported by different instruments can by inconsistent by factors of more than 100% in strong turbulent scattering situations [12,13].
Given the difficulties with experimental results, in this paper we apply a numerical simulation approach to address the question of whether the classical formulation of the Fried parameter is accurate in the strong scattering regime. The optical propagation simulation is applied to a range of turbulence scattering strengths and long-exposure Fried parameter results are compared with the theoretical expressions. To our knowledge, this is the first confirmation through simulation of the validity of the Fried parameter theory in the strong scattering regime.
2. Simulation approach
The split-step wave optics simulation (WOS) approach has been used extensively to study scintillation, beam size, anisoplanatism, angle-of-arrival, wavefront discontinuities and other properties of optical beams propagating through atmospheric turbulence [14–20]. In this study, this method is employed to simulate the propagation of the field from a point source (spherical wave) or a plane wave through homogenous Kolmogorov turbulence, focusing of the field by a lens and recording of the intensity point spread function (PSF) at the imaging plane. The basic components of the simulation are illustrated in Fig. 1. The field is propagated a distance z to the pupil via Fresnel diffraction through a sequence of phase screens that model atmospheric turbulence. The focusing element at the pupil is a thin positive lens that is described by a complex pupil function incorporating both a circular aperture and quadratic phase focus. A final Fresnel propagation is applied to the field exiting the lens to reach the image plane where the intensity PSF is calculated, with the assumption of no distortion along this back focal path zxp (i.e., vacuum). A long-exposure result is found by repeating the process 104 times with independent turbulence screens and averaging the PSFs. The full width at half maximum (FWHM) value of the average PSF is measured and used to calculate r0 using [20–21]
It is important to recognize that our simulations assume homogeneous Kolmogorov turbulence along the propagation path, a Gaussian probability distribution of the refractive index fluctuations, statistically independent turbulence phase screens (Markov approximation) and paraxial field propagation (Fresnel propagation). Furthermore, the construction of an individual turbulence phase screen is based on the theory introduced by Fried but accordingly, the propagation interval distance (Δz) associated with each screen is chosen to be short enough that the weak scattering requirement is satisfied. On the other hand, the numerical propagation of the optical field between the screens includes the effects of diffraction, the interaction of the field with each screen effectively involves “ray bending” and the field/intensity that arrives at the pupil exhibits scintillation. Thus, the processes that are ignored in the classical derivation of the Fried parameter are included in the simulation result. This type of numerical simulation has been used extensively to investigate the strong turbulent scattering regime and study effects such as the saturation of scintillation [14,17].
The simulation parameters of interest are defined in Table 1 and the values used for the results presented here are listed. The choices of the path length z, the wavelength λ and the refractive index structure parameter Cn2 ranging from 10−16 m−2/3 to 10−13 m−2/3, correspond to Rytov variance $\sigma _R^2\; $values that span from 0.0267 (weak scattering) to 26.7 (strong scattering), where $\sigma _R^2$ = 1.23Cn2k7/6z11/6. Three key configuration points to consider for the simulation are the source modelling, the application of phase screens and the numerical sampling. These are discussed in the following paragraphs.
Table 1. Simulation parameter values
The point source is modeled by assigning a non-zero value to a single pixel in the center of the source grid. We experimented with other point source models (e.g., windowed Gaussian [22]), which are designed to reduce interference (ringing) artifacts at the edges of the grid as the field is propagated, but we found there were no significant differences in the results. The plane wave source is modeled by assigning a unit value to all the pixels in the source grid. The numerical fields for both the point source and plane wave interact with the grid boundaries to produce ringing artifacts. However, any effects of the artifacts are largely avoided in our simulation by only utilizing a center portion of the grid at the pupil plane. We also investigated the method of applying absorbing boundaries at each intermediate plane to dampen the ringing [22] but again found no significant differences with the results shown here.
To generate the phase screens that model atmospheric turbulence, we use the technique of Fourier filtering white Gaussian noise to conform to the Kolmogorov spectrum [22–23]. This spectrum exhibits zero inner scale and an infinite outer scale. To model the infinite outer scale, a random tilt component is added to each screen that compensates for turbulent wavefront tilts with length scales greater than the grid width [14,18]. As shown in Fig. 1, the total propagation path z is evenly divided into equal length segments Δz and a turbulence phase screen is positioned at the center of each segment. At the i-th intermediate plane, the phase screen ${\psi _i}$ is applied in the usual way as$\; u_i^{\prime} = {u_i}{e^{j{\psi _i}}}$, where ${u_i}\; $is the incident field and $\; u_i^{\prime}\; $is the field that exits the plane. The number of the screens N required along the path is an integer that should be larger than (10$\sigma _R^2$)6/11, which ensures the Rytov variance for each interval distance Δz is less than 0.1, and therefore phase-only screens are adequate [14,18,22]. The minimum values of N for the weakest and strongest turbulence cases we studied are 2 and 22, respectively. Fewer screens can be used in the weaker turbulence regimes (Cn2 = 10−16∼10−15 m−2/3) to reduce computation time. However, we implemented more than the required minimum number of screens for the weak turbulence cases to provide a better representation of the continuous volume turbulence along the path.
Parameter choices related to the sampling of the spatial grid are important in WOS to avoid numerical artifacts. Appropriate sampling criterion are discussed in various publications [22-25, and references therein]. Here, uniform grid spacing is used where Δx = Δy = λz/L = (λz/M)1/2 for all the planes (source, screens, pupil and image) where L is the grid side length and M is the linear sample number. This grid spacing definition, referred to as the ideal, or critical sampling condition, typically provides the most accurate results where any related artifacts are sufficiently small to be negligible [24–25]. Note that even though the split-step approach for the turbulence modeling involves a sequence of short propagation segment, the critical sampling is associated with the total distance z rather than the interval distance Δz. It is convenient and without loss of generality to utilize a 1:1 (unit) magnification imaging system (i.e., zxp = z) to also maintain the critical sampling criterion for the pupil-to-image path. Thus, the lens focal length f is equal to z/2 for the point source and z for the plane wave. We also show results for a relatively larger pupil diameter (D = L/2 = 0.3536 m) and a smaller diameter (D = L/8 = 0.0884 m) to illustrate the effect of diffraction on the PSF.
3. Numerical results
Figure 2 displays examples of simulated average PSF results using the point source and large aperture under four turbulence conditions (Cn2 = 10−16, 10−15, 10−14 and 10−13 m−2/3). Normalized intensity profiles through the center of each PSF are shown to the right of each image panel. The PSFs appear Gaussian in shape and the width increases with turbulence strength, which is consistent with expectations [1,5]. The full width at half maximum (FWHM) value of the PSFs ranges from 4.258 mm to 0.166 m. Note that for the strongest scattering [Fig. 2(d)], the wings of the intensity profile elevate slightly toward the edges of the array. This artifact, related to the periodic extension property of the fast Fourier transform used in the numerical propagation, does not affect the PSF FWHM measurements in this study. For our simulation grid and sampling arrangement, this artifact can become problematic for Cn2 values larger than the maximum value in our study (Cn2 = 10−13 m−2/3). However, the maximum value already corresponds to scattering conditions that are relatively strong ($\sigma _R^2 = 26.7$) for real scenarios [11,26–27]. Therefore, we did not pursue increasing the grid size and sample number to model stronger scattering.
Fig. 2. Simulated average PSFs for the point source and larger aperture with (a) Cn2 = 10−16, (b) Cn2 = 10−15, (c) Cn2 = 10−14, and (d) Cn2 = 10−13 m−2/3.
The Fried parameter values are calculated by applying Eq. (2) to the FWHM measurements of the PSF simulation results. Figure 3 shows comparisons of r0 values obtained from the simulation data and from the analytic expressions. The results are presented in a log scale as a function of the Rytov standard deviation,$\; {\sigma _R} = {(\sigma _R^2)^{1/2}}$, which is commonly used as a measure of scattering strength. Specifically, Figs. 3(a) and 3(b) show results for the spherical wave and the two pupil aperture diameters and Figs. 3(c) and 3(d) show the results for the plane wave. The simulation and theory results agree extremely well in all four cases except for a “droop” in the simulation curves in the weak scattering regime. This effect, most notable for the small aperture [Figs. 3 (b) and 3(d)], is expected and is due to the diffractive effect of the pupil aperture where the aperture becomes smaller than r0.
Fig. 3. Simulation and theory results of the Fried parameter r0 as a function of the Rytov standard deviation: (a) spherical wave, D = L/2 (0.3536 m), (b) spherical wave, D = L/8 (0.0884 m), (c) plane wave, D = L/2 (0.3536 m), and (d) plane wave, D = L/8 (0.0884 m).
The WOS imaging arrangement detailed in this paper (Fig. 1) actually models a simple experimental approach for recording PSFs and making measurements of r0 that are independent from other turbulence strength measurements. An alternative approach to obtain the Fried parameter from the simulation data is to calculate the average mutual coherence function of the field at the pupil plane (in the absence of an aperture) and extract the proper correlation width. To corroborate the image-based results presented here, we also implemented this approach and found that the r0 expressions again show excellent agreement with the simulation results throughout the scattering regimes we studied. To avoid repetition, the details and results for the mutual coherence function approach are not presented here.
4. Conclusion
This paper presents numerical WOS results of the long-exposure point spread function and Fried parameter through atmospheric turbulence. The numerical results show that the classical expressions for r0, for both spherical and plane waves in Kolmogorov turbulence, are valid throughout the weak to strong scattering regimes. This conclusion fully supports the argument by previous researchers that the analytic expressions of the Fried parameter r0 are applicable to more general conditions than the weak scattering regime. This work advances our understanding of the combined effects of phase and amplitude variations caused by turbulence and supports the use of r0 as an optical system design parameter even in strong turbulent scattering conditions. Our results also provide confidence in the theoretical and simulation foundation for further exploration of real-world turbulence characteristics such as inhomogeneity, intermittency, and anisotropy, under strong scattering conditions.
Funding
Air Force Office of Scientific Research (FA9550-17-C-0021); Office of Naval Research (N00014-17-1-2535).
Disclosures
The authors declare no conflicts of interest.
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