Anisotropic non-Kolmogorov turbulence phase screens with variable orientation Jeremy P. Bos,1,* Michael C. Roggemann,2 and V. S. Rao Gudimetla3 1
NRC Research Associate, Air Force Research Laboratory, Directed Energy Directorate, Det. 15, 500 Lipoa Parkway, Kihei, Hawaii 96753, USA
2
Department of Electrical and Computer Engineering, Michigan Technological University, Houghton, Michigan 49931, USA 3
Air Force Research Laboratory, Directed Energy Directorate, Det. 15, 500 Lipoa Parkway, Kihei, Hawaii 96753, USA *Corresponding author: [email protected] Received 16 October 2014; revised 29 December 2014; accepted 20 January 2015; posted 2 February 2015 (Doc. ID 224882); published 6 March 2015
We describe a modification to fast Fourier transform (FFT)-based, subharmonic, phase screen generation techniques that accounts for non-Kolmogorov and anisotropic turbulence. Our model also allows for the angle of anisotropy to vary in the plane orthogonal to the direction of propagation. In addition, turbulence strength in our model is specified via a characteristic length equivalent to the Fried parameter in isotropic, Kolmogorov turbulence. Incorporating this feature enables comparison between propagating scenarios with differing anisotropies and power-law exponents to the standard Kolmogorov, isotropic model. We show that the accuracy of this technique is comparable to other FFT-based subharmonic methods up to three-dimensional spectral power-law exponents around 3.9. OCIS codes: (010.3310) Laser beam transmission; (010.1290) Atmospheric optics; (010.1330) Atmospheric turbulence; (010.1080) Active or adaptive optics; (010.1285) Atmospheric correction. http://dx.doi.org/10.1364/AO.54.002039
1. Introduction
Phase screen models are indispensable for modeling the effects of atmospheric turbulence on laser beam propagation and imaging. Though there have been significant advances in analytical models, computer simulations using the Fourier split-step method continue to provide the best match to experimental data [1]. Specifically, the split-step model accurately predicts the onset of saturation in intensity fluctuations. Apart from certain heuristic models [2], this saturation cannot be predicted analytically. Lately, there has been significant interest by researchers in understanding the effects of non-Kolmogorov and anisotropic [3–10] turbulence. These efforts focus on the development of analytical models for fluctuation statistics of the received field. While these models are useful, it follows that phase screen models are just as important in the practical modeling of non-Kolmogorov turbulence.
Our interest in non-Kolmogorov turbulence is well founded. There is strong evidence that nonKolmogorov turbulence is commonplace, especially in the upper atmosphere [11–14]. Researchers theorize that structures such as temperature sheets, von Kármán vortex streets, gravity, and littoral waves may all result in non-Kolmogorov turbulence. The geometry of these structures is complex and, from a modeling standpoint, may include a variety of power-law exponents and degrees and directions of anisotropy. Modeling these arbitrary and complex structures analytically is complicated and generally involves integrals without closed form solutions. On the other hand, modeling via simulation is usually straightforward. Though some work has been done in the area of simulating non-Kolmogorov [15] and anisotropic phase screens other limitations exists. Specifically, current phase screen models restrict the direction 10 March 2015 / Vol. 54, No. 8 / APPLIED OPTICS
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or orientation of anisotropy along one axis perpendicular to the direction of propagation. This limitation precludes modeling structures where the direction of anisotropy varies along the propagation path. In non-Kolmogorov turbulence research there is also no broad agreement on how to define turbulence strength [16]. In this paper, we describe a non-Kolmogorov phase screen model where turbulence strength is specified by a characteristic length equivalent to the Fried parameter in Kolmogorov turbulence. Our model also allows both degree and angle of anisotropy to vary arbitrarily in the plane perpendicular to the direction of propagation. Researchers have developed a number of ways of generating phase screens that simulate the effects of turbulence on optical wave propagation. Though other methods exist [17,18] the most popular involve filtering white Gaussian noise so that the simulated screen has the correct spatial statistics [19]. Our model follows the development of Sasiela [20] and Schmidt [21], and uses the model for anisotropy described by Wheelon [22]. Inner- and outer-scale effects are included via the generalized von Kármán spectrum and emphasized via subharmonic sampling [23]. Our method shows excellent agreement with theory departing by no more than 16% of the theoretical value out to separations of more than 12 times the Fried radius in the Kolmogorov case. In the next section, we describe our method of generating random optical turbulence phase screens starting with a structure function defined in terms of an arbitrary characteristic length. Once we have arrived at a generalized spectral model we then describe how anisotropy and rotation in anisotropy are included. A brief overview of the subharmonic approach is also provided in Section 2. In Section 3 we generate an ensemble of phase screens with power laws in the range of 3 to 4. For each ensemble, the structure function is evaluated and shown to compare favorably to theory. A similar approach is used to demonstrate variation in the degree and direction of anisotropy. In this case, we inspect lines of constant contour in the measured phase structure function observing the appropriate expansion, contraction, and rotation. Our results are summarized and conclusions are provided in Section 4. 2. Background
Under the Kolmogorov model, the mean squared difference in index of refraction, n⃗r, at two arbitrary points, r⃗ 1 and r⃗ 2 , depends only on the distance between the two points, ρ j⃗r1 − r⃗ 2 j, and is independent of direction [24]. That is, fluctuation statistics are isotropic, and can be described by a structure function according to a 2∕3 power law such that hn⃗r1 − n⃗r2 2 i Dρ C2n ρ2∕3 ;
(1)
where C2n is the refractive index structure parameter, an indicator of fluctuation strength, and has units m−2∕3. 2040
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So-called non-Kolmogorov turbulence describes any model for atmospheric turbulence that deviates from this description. Analytical descriptions of the fluctuations in amplitude or phase are further restricted under the Rytov and Markov approximations to the stochastic parabolic wave equation as described by Tatarskii [24]. These approximations are necessary if closed-form analytical descriptions of beam statistics are desired. Still, these approaches have limitations. For example, the Rytov model does not predict saturation in intensity fluctuations. Other solutions to the stochastic wave equation are fraught with similar difficulties either in their predictive ability, limitations on the solution space, or tractability. Among the alternative approaches, it is generally agreed that path-integral solutions [25] provide the most accurate analytical models for wave propagation in a random media. An alternative is simulation via the Fourier splitstep method [26]. The split-step method is essentially a discrete approximation of the path integral approach [1] and, if configured correctly, results in accurate representations of second- and fourth-order complex field statistics. The turbulence volume in this method is approximated using a series of thin phase screens. Each screen represents the summed index of refraction variations along a path through a random instance of the turbulence volume. Screens are statistically independent with spatial statistics that match a prescribed model of index of refraction fluctuations. Both in phase screen generation and analytical models it is common to use a spatial power spectral density (PSD) description of index of refraction fluctuations. The phase PSD, Φϕ κ, can be subsequently expressed in terms of its refractive index spectrum, Φn κ, wavenumber, k, and transverse dimension, L, as Φϕ κ 2π 2 k2 LΦn κ;
(2)
and in Kolmogorov turbulence Φn κ is Φn κ 0.033C2n κ −11∕3 :
(3)
In both Eqs. (2) and (3) κ is the PSD spatial frequency and has units of radians per meter. Our problem, then, is to develop a generalized model for the phase PSD in a non-Kolmogorov power-law medium. To start, consider a random media with index of refraction fluctuations described by a structure function, Dn ρ, in the form of a power law such that Dn ρ βργ :
(4)
In our case, index of refraction fluctuations are a function of separation, ρ, raised to a power γ with generalized strength parameter, β, with units of length raised to −γ. The corresponding index power spectrum is [24,27,28]
Φn κ
1 4π 2 κ 2
Z
∞ 0
sinκρ d 2 d ρ Dn ρ dρ: κρ dρ dρ
Z (5)
Using Eq. (4) in Eq. (5) under the constraint 0 < γ < 1 results in Φn κ
1 πγ −3−γ : βγγ 1κ Γγ sin 2 2 4π
(6)
Making the substitution α γ 3 and defining Aα
cosπα 2 Γα 2
− 1
4π
(8)
(9)
In Eq. (9) the constant 6.88 is the value of the plane wave, wave structure function (WSF) at a separation of r0. We can generalize this expression for arbitrary characteristic lengths, rˆ 0 , and power laws such that Dϕ ρ c1 α
α−2 ρ ; rˆ 0
(10)
where c1 α is a constant equal to the value of the WSF at a separation equal to the characteristic length. We aim to describe the 2D phase fluctuations by the power spectrum in terms of a characteristic length, rˆ 0 , such that Φϕ κ Bαc1 αˆr02−α κ−α ;
(11)
where Bα is a parameter than maintains consistency between the structure function and PSD descriptions. Expounding on the development by Noll [17] and others [20,30], we use the relation Z Dϕ ⃗r 2
∞
0
Z
∞ 0
Φϕ ⃗κ 1 − cos⃗κ · r⃗ d⃗κ:
0
(12)
Assuming the turbulence is isotropic we can rewrite Eq. (12) in polar coordinates ρ and θ and integrate over θ so that
Z
2π 0
κΦϕ κ × 1 − cosκρ cosθdκdθ: (13) R 2π
Using the identity, 2πJ 0 κρ 0 cosκρ cos θdθ [31, Eq. 9.1.18], the result of the integration is Z Dϕ ρ 4π
∞ 0
κΦϕ κ1 − J 0 κρdκ
(14)
after substituting Eq. (11) and using [17, Eq. 20] we find Dϕ ρ
with limits derived from Eq. (6) of 3 < α < 4. Observe that setting α 11∕3 results in Kolmogorov turbulence and a structure function with γ 2∕3. Assume instead that we start with a phase structure function described by a characteristic length, rˆ 0 . In Kolmogorov turbulence the most commonly used length scale is the Fried parameter, r0 [29]. In this specific case, the two-dimensional phase structure function is 5∕3 ρ Dϕ ρ 6.88 : r0
∞
(7)
provides the common result Φn κ Aαβκ −α ;
Dϕ ρ 2
22−α παΓ− α2 c1 αBαρα−2 rˆ 02−α : Γ2α
(15)
Comparing this result with Eq. (10) and solving for Bα provides Bα
2−α
2
Γα2 : παΓ− 2α
(16)
We note that this term is identical to the one provided by Charnotskii [18, Eq. 17] if the constant α is replaced by α 2. Similarly, if α 11∕3 and c1 α 6.88, the phase spectrum becomes Φϕ κ 0.49r0−5∕3 κ−11∕3 ;
(17)
which is identical to the result provided by Schmidt [21, p. 160, Eq. 9.49]. To be useful in producing phase screens, the motivation for this discussion, the frequency variable in Eq. (17) must be converted from radians per meter to cycles per meter. Also, from the presence of coherence length in Eq. (10), it is understood that the spatial dimension is scaled to the 2 − α power. Consequently, the generalized power spectrum can be written as Φϕ f 2π2−α Bαc1 αˆr02−α f −α :
(18)
A number of options exist here in terms of the characteristic length, rˆ 0 . In the traditional Kolmogorov structure function defined by the Fried parameter the constant, c1 α, is the value of the plane wave structure function at the coherence radius, r0 . Using Stribling’s definition for the Fried parameter in non-Kolmogorov turbulence [28] we can write the parameter c1 α as α−2 2 8 2 Γ : c1 α 2 α−2 α−2
(19)
Equation (19) evaluates to 6.88 when α 11∕3 and, theoretically, results in the same effective resolution as defined by Fried for a given coherence length rˆ 0 . Structure functions for a variety of power-law exponents as defined by Eq. (10) are plotted together in Fig. 1 for separations between ρ∕ˆr0 0 and 1. Each 10 March 2015 / Vol. 54, No. 8 / APPLIED OPTICS
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f 0x f x cos θ f y sin θ;
(22)
f 0y −f x sin θ f y cos θ:
(23)
Last of all, finite inner- and outer-scale effects are added by way of the modified von Kármán spectrum, which has generalized form Φκ c1 αBαˆr0
exp−κ2 ∕κ 2m : κ 2 κ 20 α∕2
(24)
In Eq. (24), the constant κm is the inner-scale wave number cutoff defined as κm cα∕l0 with cα 1 α−5 [4] and l 23 πΓ5−α 0 is the inner-scale size. 2 Aα We can now generate phase screens incorporating all of the above features using the Fourier transform method and the model for the power spectrum as
Fig. 1. Theoretical structure functions as a function of separation normalized by the coherence length rˆ 0 for α 3, 3.17, 3.33, 3.50, 3.67, 3.83, 3.98 the Kolmogorov case α 3.67, α 11∕3 is included for reference.
line intersects the left limit of the figure at the appropriate value of c1 α. For example, when α 11∕3 the line intersects the point, 6.88 which is the vertical limit in Fig. 1. Alternatively, we could arbitrarily fix c1 α at some meaningful value and allow α to vary. For example, if c1 α 2 then at ρ rˆ 0 the complex degree of coherence for an unbounded plane wave is expf−1∕2Dρ0 g expf−1g. This is the definition of the atmospheric coherence length, ρ0 [2]. Noting this alternative, in the remainder of this work we make use of Stribling’s c1 α parameter and the corresponding characteristic length rˆ 0 . A.
Including Anisotropy
The effect of anisotropy can be included through application of a simple affine transform in spatial frequency space. Following the approach used by Wheelon [22], the spatial spectrum in Eq. (3) can be rewritten as [3] Φn κ 0.033C2n ab11∕6 a2 κ 2x b2 κ2y −11∕6 :
(20)
An expression for the anisotropic phase spectrum results by a similar modification to Eq. (18), Φϕ f
c1 αBα2π2−α rˆ 02−α abα∕2 : a2 f 2x b2 f 2y −α∕2
(21)
For simplicity, anisotropy in our implementation is varied along only one axis such that a ϵ, the degree of anisotropy, and b 1. Rotation by angle θ is handled by a second affine transform in frequency space 2042
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Φϕ f c1 αBα2π2−α rˆ 02−α abα∕2 ×
expf−2π2 f 2 ∕κ2m g : 2 02 2 α∕2 a2 f 02 x b f y 1∕L0
(25)
Random phase screens are generated by first sampling Eq. (25) over a rectangular grid f m; n at interval of Δf 1∕L, where L is the screen physical side length using the desired number of samples N with m, n 1 to N. Next, a N × N matrix of zero mean, unit variance, complex circular Gaussian random numbers is generated, denoted here as W n m; n. The inverse Fourier transform of the element-wise product of W n and square root of the m; n sampled spectrum, ΦaVka ϕ m; n−1∕2 g ϕn x; y F −1 fW n m; nΦaVKa ϕ
(26)
results in two independent phase screens, one each from the real and imaginary parts. B. Subharmonic Compensation
The actual spectrum modeled by the FFT-based phase screen model described in Eq. (25) is discretely sampled in frequency space on the grid m n 1…N, where N is the number of samples in frequency space. Consequently, the realized spectrum is a sampled version of the complete spectrum such that ΦNK c1 αBα2π2−α rˆ 02−α m2 n2 −α∕2 : ϕ
(27)
The range of spatial frequencies covered by this spectrum is limited on the upper end by the grid spacing, Δf , such that f max 1∕2Δf , and at the lower end by the physical size of the phase screen, f min 1∕L. Assuming the sample spacing is smaller, or of the order of, the inner-scale size the upper end of the frequency range can be considered adequately sampled. To adequately sample low spatial frequencies, it is generally necessary that the screen size be much
larger than either the outer-scale size or the aperture size of interest. Screen sizes having both adequate size and adequate sampling can be prohibitively large, especially for larger apertures. An alternative proposed by Lane [23] is to break the lowest spatial frequency sample into subharmonic components. The spectrum is sampled at each of the subharmonic frequencies and contributions are appropriately weighted and added to the spectrum. In this work we match the implementation described by Frelich [32] and as outlined by Schmidt [21]. Extending this technique to non-Kolmogorov, anisotropic models requires only replacing the Kolmogorv PSD with the one described in Eq. (25). 3. Results
We now present examples of phase screens simulated using the technique developed in Section 2. Figure 2 shows the theoretical phase structure function as defined by Eq. (10) plotted together with the phase structure function evaluated over an ensemble of 500 phase screens for several values of α. In all cases the characteristic length was set to rˆ 0 0.1 m and is equivalent to the Fried parameter r0 when α 11∕3. Each phase screen was generated with a side length of 5 m on a 1024 × 1024 grid. Structure functions are evaluated over a 2.5 m diameter circular aperture at the center of each screen. Unless otherwise mentioned the outer scale as specified in Eq. (25) is set to infinity; the inner scale is set to zero. Seven subharmonics were included in each screen to ensure the best possible match to theory. The separation axis in Fig. 2 is normalized by the characteristic length rˆ 0 . In Fig. 2(a) the structure function is evaluated for power laws of α 3, 3.17, 3.33 and compared to the value predicted by Eq. (10) using Stribling’s constant, c1 α, in Eq. (19). The Kolmogorov case, α 11∕3, is also included for reference. The value of c1 α in this range varies from 5.65 to 6.09 and c1 11∕3 6.88. Similarly, the ensemble structure function for phase screens with power laws of α 3.5, 3.83, 3.98 is plotted in Fig. 2(b). Here c1 α varies from 6.44 to 7.94. In both cases, the match between phase screen values and theory agree within about 15% at separations out to 9r0 and 16% out to separations greater than 12r0 in the Kolmogorov case. We also observe that, with one exception, the slope of each curve varies according to the prescribed power law and approximately intercepts the prescribed value of c1 α at a normalized separation of ρ∕ˆr0 1. The exception in Fig. 2(b) is for the α 3.98 powerlaw case. We include it here to demonstrate the limitation of the subharmonic technique as the power law approaches the upper limit at α 4. Referring to the structure function definition in Eq. (4), we observe that as α approaches 4, γ α − 3 approaches 1. At this upper bound, phase fluctuations are a linear function of separation and physically manifested as tip and tilt fluctuations of the beam or image. Tip-tilt fluctuations originate from the very lowest spatial
Fig. 2. Comparison of the simulated (solid) and theoretical (dashed) structure functions as a function of separation and normalized by the coherence length rˆ 0 for (a) α 3, 3.17, 3.33 and (b) α 3.5, 3.83, 3.98. The Kolmogorov case α 3.67 is present in both (a) and (b) for reference.
frequencies and, even then, are limited by the size of the phase screen [27]. Thus, it is reasonable that as α approaches 4 we should observe deviations between our phase screens and the theoretical structure function. In exploring this matter, we find that if the number of subharmonics is increased to 9 or more, reasonable agreement with theory is possible up to around α ∼ 3.9. Next, we evaluate anisotropy in our phase screen model. In Fig. 3 the contour where the two-dimensional structure function evaluates to 10 March 2015 / Vol. 54, No. 8 / APPLIED OPTICS
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instances. Values of anisotropy in Fig. 3 are set as ϵ f2; 1.5; 1.33; 0.75; 0.66; 0.5g. The isotropic (ϵ 1) case is also included for reference. As we would expect there is a symmetry between inverse values of anisotropy. For example, in the ϵ 2 and ϵ 0.5 cases the coherence length is increased by a factor 2 in vertical and horizontal directions, respectively. Finally, we show that the angle of anisotropy in our model can be varied continuously. In Fig. 4 the angle of anisotropy, θ, is varied between 33 and 135 deg while anisotropy is fixed as ϵ 2. All other parameters are the same as Fig. 3. As expected, the contours are identical to those for the ϵ 2 case in Fig. 3 but with the minor axis of the constant value parabola rotated by the indicated angle. 4. Conclusion
Fig. 3. Contour lines of the 2D structure function at Dρ 6.88 for several values of the anisotropy factor ϵ. The isotropic, Kolmogorov case, ϵ 1 is emphasized for reference.
Dρ 6.88 is plotted for several values of anisotropy. The power-law exponent is fixed at α 11∕3 and r0 0.1 m. The screen size was increased to 10 m and sampled by a 2048 square grid. A finite outer-scale size of L0 10 m was used to observe the proper scaling. In each case, the structure function was evaluated over a 5 m circular aperture at the center of each screen and averaged over 500
Fig. 4. Contour plot of the structure function for phase screens with varied orientation of anisotropy. Each contour represents the separation, ρ, at which the structure function evaluates to 6.88. Both axes are normalized by the Fried parameter, r0 . For comparison purposes, the isotropic case appears in bold as a circle with unit radius. 2044
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We have described a useful modification to the traditional FFT-based method of generating turbulence phase screens. This modification allows for a variable direction of anisotropy, power-law exponent, turbulence strength, as well as inner and outer scale. Turbulence strength in this model is defined in terms of a single characteristic length parameter providing a way of readily comparing propagation paths with varied power laws and geometries to standard isotropic-Kolmogorov models. Subharmonic compensation is used to compensate for undersampling at low-spatial frequency fluctuations. The phase screen model described here is accurate in as much as it is equivalent to other popularly used FFT-based phase screen generation techniques. In the Kolmogorov isotropic case the structure function differs by less than 15% from the theoretical model up to separations of 9r0. However, achieving this level of accuracy requires using the center portion of a screen twice the size of interest and generated using seven subharmonic components. These steps may be omitted, but at the cost of accuracy relative to the theoretical model. The degree to which this accuracy is important depends on the application. A phase screen with a grid spacing of Δx 0.005 m and a square grid size of 16384 provides a physical screen size of around 80 m. Using only the center portion of this screen results in a usable side length of 40 m, which is of the order of the largest telescope apertures currently proposed. The results presented here are also for turbulence with an infinite outer-scale size. Considering most extreme estimates for the outer-scale size are no more than a couple hundred meters, it is likely that FFT-based approaches, like the one described here, will satisfy most needs. On the other hand generating time-correlated simulations in which frozen turbulence advects across an aperture requires long turbulence ribbons. Modeling these ribbons with the proper statistics, especially atmospheric piston, remains a challenge. This research was performed while the author held a National Research Council Research Associateship Award at the Air Force Research Lab. The views
expressed in this paper are those of the authors and do not necessarily represent the views of the Department of Defense or its components. Rao Gudimetla was partially supported during the progress of this work by the Air Force Office of Scientific Research. References 1. S. M. Flatté and J. S. Gerber, “Irradiance-variance behavior by numerical simulation for plane-wave and spherical-wave optical propagation through strong turbulence,” J. Opt. Soc. Am. A 17, 1092–1097 (2000). 2. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005). 3. V. S. R. Gudimetla, R. B. Holmes, C. Smith, and G. Needham, “Analytical expressions for the log-amplitude correlation function of a plane wave through anisotropic atmospheric refractive turbulence,” J. Opt. Soc. Am. A 29, 832–841 (2012). 4. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007). 5. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through nonKolmogorov turbulence,” Opt. Express 18, 10650–10658 (2010). 6. B. Xue, L. Cui, W. Xue, X. Bai, and F. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28, 912–916 (2011). 7. J. Cang and X. Liu, “Average capacity of free-space optical systems for a partially coherent beam propagating through non-Kolmogorov turbulence,” Opt. Lett. 36, 3335–3337 (2011). 8. C. Rao, W. Jiang, and N. Ling, “Adaptive-optics compensation by distributed beacons for non-Kolmogorov turbulence,” Appl. Opt. 40, 3441–3449 (2001). 9. L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through nonKolmogorov turbulence,” Opt. Express 18, 451–462 (2010). 10. L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013). 11. J. T. Beyer, M. C. Roggemann, L. J. Otten, T. J. Schulz, T. C. Havens, and W. W. Brown, “Experimental estimation of the spatial statistics of turbulence-induced index of refraction fluctuations in the upper atmosphere,” Appl. Opt. 42, 908– 921 (2003). 12. F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of sheets in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
13. M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown II, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999). 14. S. Gladysz, K. Stein, E. Sucher, and D. Sprung, “Measuring non-Kolmogorov turbulence,” Proc. SPIE 8890, 889013 (2013). 15. V. S. R. Gudimetla, R. B. Holmes, T. C. Farrell, and J. Lucas, “Phase screen simulations of laser propagation through nonKolmogorov atmospheric turbulence,” Proc. SPIE 8038, 803808 (2011). 16. M. Charnotskii, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence: comment,” J. Opt. Soc. Am. A 29, 1838–1840 (2012). 17. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976). 18. M. Charnotskii, “Sparse spectrum model for a turbulent phase,” J. Opt. Soc. Am. A 30, 479–488 (2013). 19. M. C. Roggemann, B. M. Welsh, and B. R. Hunt, Imaging through Turbulence (CRC Press, 1996). 20. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms (SPIE, 2007). 21. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010). 22. A. D. Wheelon, Electromagnetic Scintillation (Cambridge University, 2003). 23. R. Lane, A. Glindemann, and J. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209– 224 (2006). 24. V. I. Tatarskii, Wave Propagation in Turbulent Medium (McGraw-Hill, 1961), Vol. 1. 25. R. Dashen, G.-Y. Wang, S. M. Flatte, and C. Bracher, “Moments of intensity and log intensity: new asymptotic results for waves in power-law media,” J. Opt. Soc. Am. A 10, 1233–1242 (1993). 26. J. M. Martin and S. M. Flatté, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A 7, 838–847 (1990). 27. R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995). 28. B. E. Stribling, “Laser beam propagation in non-Kolmogorov atmospheric turbulence,” Master’s thesis (Air Force Institute of Technology, 1994). 29. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966). 30. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2. 31. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972), Vol. 1. 32. R. Frehlich, “Simulation of laser propagation in a turbulent atmosphere,” Appl. Opt. 39, 393–397 (2000).
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NRC Research Associate, Air Force Research Laboratory, Directed Energy Directorate, Det. 15, 500 Lipoa Parkway, Kihei, Hawaii 96753, USA
2
Department of Electrical and Computer Engineering, Michigan Technological University, Houghton, Michigan 49931, USA 3
Air Force Research Laboratory, Directed Energy Directorate, Det. 15, 500 Lipoa Parkway, Kihei, Hawaii 96753, USA *Corresponding author: [email protected] Received 16 October 2014; revised 29 December 2014; accepted 20 January 2015; posted 2 February 2015 (Doc. ID 224882); published 6 March 2015
We describe a modification to fast Fourier transform (FFT)-based, subharmonic, phase screen generation techniques that accounts for non-Kolmogorov and anisotropic turbulence. Our model also allows for the angle of anisotropy to vary in the plane orthogonal to the direction of propagation. In addition, turbulence strength in our model is specified via a characteristic length equivalent to the Fried parameter in isotropic, Kolmogorov turbulence. Incorporating this feature enables comparison between propagating scenarios with differing anisotropies and power-law exponents to the standard Kolmogorov, isotropic model. We show that the accuracy of this technique is comparable to other FFT-based subharmonic methods up to three-dimensional spectral power-law exponents around 3.9. OCIS codes: (010.3310) Laser beam transmission; (010.1290) Atmospheric optics; (010.1330) Atmospheric turbulence; (010.1080) Active or adaptive optics; (010.1285) Atmospheric correction. http://dx.doi.org/10.1364/AO.54.002039
1. Introduction
Phase screen models are indispensable for modeling the effects of atmospheric turbulence on laser beam propagation and imaging. Though there have been significant advances in analytical models, computer simulations using the Fourier split-step method continue to provide the best match to experimental data [1]. Specifically, the split-step model accurately predicts the onset of saturation in intensity fluctuations. Apart from certain heuristic models [2], this saturation cannot be predicted analytically. Lately, there has been significant interest by researchers in understanding the effects of non-Kolmogorov and anisotropic [3–10] turbulence. These efforts focus on the development of analytical models for fluctuation statistics of the received field. While these models are useful, it follows that phase screen models are just as important in the practical modeling of non-Kolmogorov turbulence.
Our interest in non-Kolmogorov turbulence is well founded. There is strong evidence that nonKolmogorov turbulence is commonplace, especially in the upper atmosphere [11–14]. Researchers theorize that structures such as temperature sheets, von Kármán vortex streets, gravity, and littoral waves may all result in non-Kolmogorov turbulence. The geometry of these structures is complex and, from a modeling standpoint, may include a variety of power-law exponents and degrees and directions of anisotropy. Modeling these arbitrary and complex structures analytically is complicated and generally involves integrals without closed form solutions. On the other hand, modeling via simulation is usually straightforward. Though some work has been done in the area of simulating non-Kolmogorov [15] and anisotropic phase screens other limitations exists. Specifically, current phase screen models restrict the direction 10 March 2015 / Vol. 54, No. 8 / APPLIED OPTICS
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or orientation of anisotropy along one axis perpendicular to the direction of propagation. This limitation precludes modeling structures where the direction of anisotropy varies along the propagation path. In non-Kolmogorov turbulence research there is also no broad agreement on how to define turbulence strength [16]. In this paper, we describe a non-Kolmogorov phase screen model where turbulence strength is specified by a characteristic length equivalent to the Fried parameter in Kolmogorov turbulence. Our model also allows both degree and angle of anisotropy to vary arbitrarily in the plane perpendicular to the direction of propagation. Researchers have developed a number of ways of generating phase screens that simulate the effects of turbulence on optical wave propagation. Though other methods exist [17,18] the most popular involve filtering white Gaussian noise so that the simulated screen has the correct spatial statistics [19]. Our model follows the development of Sasiela [20] and Schmidt [21], and uses the model for anisotropy described by Wheelon [22]. Inner- and outer-scale effects are included via the generalized von Kármán spectrum and emphasized via subharmonic sampling [23]. Our method shows excellent agreement with theory departing by no more than 16% of the theoretical value out to separations of more than 12 times the Fried radius in the Kolmogorov case. In the next section, we describe our method of generating random optical turbulence phase screens starting with a structure function defined in terms of an arbitrary characteristic length. Once we have arrived at a generalized spectral model we then describe how anisotropy and rotation in anisotropy are included. A brief overview of the subharmonic approach is also provided in Section 2. In Section 3 we generate an ensemble of phase screens with power laws in the range of 3 to 4. For each ensemble, the structure function is evaluated and shown to compare favorably to theory. A similar approach is used to demonstrate variation in the degree and direction of anisotropy. In this case, we inspect lines of constant contour in the measured phase structure function observing the appropriate expansion, contraction, and rotation. Our results are summarized and conclusions are provided in Section 4. 2. Background
Under the Kolmogorov model, the mean squared difference in index of refraction, n⃗r, at two arbitrary points, r⃗ 1 and r⃗ 2 , depends only on the distance between the two points, ρ j⃗r1 − r⃗ 2 j, and is independent of direction [24]. That is, fluctuation statistics are isotropic, and can be described by a structure function according to a 2∕3 power law such that hn⃗r1 − n⃗r2 2 i Dρ C2n ρ2∕3 ;
(1)
where C2n is the refractive index structure parameter, an indicator of fluctuation strength, and has units m−2∕3. 2040
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So-called non-Kolmogorov turbulence describes any model for atmospheric turbulence that deviates from this description. Analytical descriptions of the fluctuations in amplitude or phase are further restricted under the Rytov and Markov approximations to the stochastic parabolic wave equation as described by Tatarskii [24]. These approximations are necessary if closed-form analytical descriptions of beam statistics are desired. Still, these approaches have limitations. For example, the Rytov model does not predict saturation in intensity fluctuations. Other solutions to the stochastic wave equation are fraught with similar difficulties either in their predictive ability, limitations on the solution space, or tractability. Among the alternative approaches, it is generally agreed that path-integral solutions [25] provide the most accurate analytical models for wave propagation in a random media. An alternative is simulation via the Fourier splitstep method [26]. The split-step method is essentially a discrete approximation of the path integral approach [1] and, if configured correctly, results in accurate representations of second- and fourth-order complex field statistics. The turbulence volume in this method is approximated using a series of thin phase screens. Each screen represents the summed index of refraction variations along a path through a random instance of the turbulence volume. Screens are statistically independent with spatial statistics that match a prescribed model of index of refraction fluctuations. Both in phase screen generation and analytical models it is common to use a spatial power spectral density (PSD) description of index of refraction fluctuations. The phase PSD, Φϕ κ, can be subsequently expressed in terms of its refractive index spectrum, Φn κ, wavenumber, k, and transverse dimension, L, as Φϕ κ 2π 2 k2 LΦn κ;
(2)
and in Kolmogorov turbulence Φn κ is Φn κ 0.033C2n κ −11∕3 :
(3)
In both Eqs. (2) and (3) κ is the PSD spatial frequency and has units of radians per meter. Our problem, then, is to develop a generalized model for the phase PSD in a non-Kolmogorov power-law medium. To start, consider a random media with index of refraction fluctuations described by a structure function, Dn ρ, in the form of a power law such that Dn ρ βργ :
(4)
In our case, index of refraction fluctuations are a function of separation, ρ, raised to a power γ with generalized strength parameter, β, with units of length raised to −γ. The corresponding index power spectrum is [24,27,28]
Φn κ
1 4π 2 κ 2
Z
∞ 0
sinκρ d 2 d ρ Dn ρ dρ: κρ dρ dρ
Z (5)
Using Eq. (4) in Eq. (5) under the constraint 0 < γ < 1 results in Φn κ
1 πγ −3−γ : βγγ 1κ Γγ sin 2 2 4π
(6)
Making the substitution α γ 3 and defining Aα
cosπα 2 Γα 2
− 1
4π
(8)
(9)
In Eq. (9) the constant 6.88 is the value of the plane wave, wave structure function (WSF) at a separation of r0. We can generalize this expression for arbitrary characteristic lengths, rˆ 0 , and power laws such that Dϕ ρ c1 α
α−2 ρ ; rˆ 0
(10)
where c1 α is a constant equal to the value of the WSF at a separation equal to the characteristic length. We aim to describe the 2D phase fluctuations by the power spectrum in terms of a characteristic length, rˆ 0 , such that Φϕ κ Bαc1 αˆr02−α κ−α ;
(11)
where Bα is a parameter than maintains consistency between the structure function and PSD descriptions. Expounding on the development by Noll [17] and others [20,30], we use the relation Z Dϕ ⃗r 2
∞
0
Z
∞ 0
Φϕ ⃗κ 1 − cos⃗κ · r⃗ d⃗κ:
0
(12)
Assuming the turbulence is isotropic we can rewrite Eq. (12) in polar coordinates ρ and θ and integrate over θ so that
Z
2π 0
κΦϕ κ × 1 − cosκρ cosθdκdθ: (13) R 2π
Using the identity, 2πJ 0 κρ 0 cosκρ cos θdθ [31, Eq. 9.1.18], the result of the integration is Z Dϕ ρ 4π
∞ 0
κΦϕ κ1 − J 0 κρdκ
(14)
after substituting Eq. (11) and using [17, Eq. 20] we find Dϕ ρ
with limits derived from Eq. (6) of 3 < α < 4. Observe that setting α 11∕3 results in Kolmogorov turbulence and a structure function with γ 2∕3. Assume instead that we start with a phase structure function described by a characteristic length, rˆ 0 . In Kolmogorov turbulence the most commonly used length scale is the Fried parameter, r0 [29]. In this specific case, the two-dimensional phase structure function is 5∕3 ρ Dϕ ρ 6.88 : r0
∞
(7)
provides the common result Φn κ Aαβκ −α ;
Dϕ ρ 2
22−α παΓ− α2 c1 αBαρα−2 rˆ 02−α : Γ2α
(15)
Comparing this result with Eq. (10) and solving for Bα provides Bα
2−α
2
Γα2 : παΓ− 2α
(16)
We note that this term is identical to the one provided by Charnotskii [18, Eq. 17] if the constant α is replaced by α 2. Similarly, if α 11∕3 and c1 α 6.88, the phase spectrum becomes Φϕ κ 0.49r0−5∕3 κ−11∕3 ;
(17)
which is identical to the result provided by Schmidt [21, p. 160, Eq. 9.49]. To be useful in producing phase screens, the motivation for this discussion, the frequency variable in Eq. (17) must be converted from radians per meter to cycles per meter. Also, from the presence of coherence length in Eq. (10), it is understood that the spatial dimension is scaled to the 2 − α power. Consequently, the generalized power spectrum can be written as Φϕ f 2π2−α Bαc1 αˆr02−α f −α :
(18)
A number of options exist here in terms of the characteristic length, rˆ 0 . In the traditional Kolmogorov structure function defined by the Fried parameter the constant, c1 α, is the value of the plane wave structure function at the coherence radius, r0 . Using Stribling’s definition for the Fried parameter in non-Kolmogorov turbulence [28] we can write the parameter c1 α as α−2 2 8 2 Γ : c1 α 2 α−2 α−2
(19)
Equation (19) evaluates to 6.88 when α 11∕3 and, theoretically, results in the same effective resolution as defined by Fried for a given coherence length rˆ 0 . Structure functions for a variety of power-law exponents as defined by Eq. (10) are plotted together in Fig. 1 for separations between ρ∕ˆr0 0 and 1. Each 10 March 2015 / Vol. 54, No. 8 / APPLIED OPTICS
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f 0x f x cos θ f y sin θ;
(22)
f 0y −f x sin θ f y cos θ:
(23)
Last of all, finite inner- and outer-scale effects are added by way of the modified von Kármán spectrum, which has generalized form Φκ c1 αBαˆr0
exp−κ2 ∕κ 2m : κ 2 κ 20 α∕2
(24)
In Eq. (24), the constant κm is the inner-scale wave number cutoff defined as κm cα∕l0 with cα 1 α−5 [4] and l 23 πΓ5−α 0 is the inner-scale size. 2 Aα We can now generate phase screens incorporating all of the above features using the Fourier transform method and the model for the power spectrum as
Fig. 1. Theoretical structure functions as a function of separation normalized by the coherence length rˆ 0 for α 3, 3.17, 3.33, 3.50, 3.67, 3.83, 3.98 the Kolmogorov case α 3.67, α 11∕3 is included for reference.
line intersects the left limit of the figure at the appropriate value of c1 α. For example, when α 11∕3 the line intersects the point, 6.88 which is the vertical limit in Fig. 1. Alternatively, we could arbitrarily fix c1 α at some meaningful value and allow α to vary. For example, if c1 α 2 then at ρ rˆ 0 the complex degree of coherence for an unbounded plane wave is expf−1∕2Dρ0 g expf−1g. This is the definition of the atmospheric coherence length, ρ0 [2]. Noting this alternative, in the remainder of this work we make use of Stribling’s c1 α parameter and the corresponding characteristic length rˆ 0 . A.
Including Anisotropy
The effect of anisotropy can be included through application of a simple affine transform in spatial frequency space. Following the approach used by Wheelon [22], the spatial spectrum in Eq. (3) can be rewritten as [3] Φn κ 0.033C2n ab11∕6 a2 κ 2x b2 κ2y −11∕6 :
(20)
An expression for the anisotropic phase spectrum results by a similar modification to Eq. (18), Φϕ f
c1 αBα2π2−α rˆ 02−α abα∕2 : a2 f 2x b2 f 2y −α∕2
(21)
For simplicity, anisotropy in our implementation is varied along only one axis such that a ϵ, the degree of anisotropy, and b 1. Rotation by angle θ is handled by a second affine transform in frequency space 2042
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Φϕ f c1 αBα2π2−α rˆ 02−α abα∕2 ×
expf−2π2 f 2 ∕κ2m g : 2 02 2 α∕2 a2 f 02 x b f y 1∕L0
(25)
Random phase screens are generated by first sampling Eq. (25) over a rectangular grid f m; n at interval of Δf 1∕L, where L is the screen physical side length using the desired number of samples N with m, n 1 to N. Next, a N × N matrix of zero mean, unit variance, complex circular Gaussian random numbers is generated, denoted here as W n m; n. The inverse Fourier transform of the element-wise product of W n and square root of the m; n sampled spectrum, ΦaVka ϕ m; n−1∕2 g ϕn x; y F −1 fW n m; nΦaVKa ϕ
(26)
results in two independent phase screens, one each from the real and imaginary parts. B. Subharmonic Compensation
The actual spectrum modeled by the FFT-based phase screen model described in Eq. (25) is discretely sampled in frequency space on the grid m n 1…N, where N is the number of samples in frequency space. Consequently, the realized spectrum is a sampled version of the complete spectrum such that ΦNK c1 αBα2π2−α rˆ 02−α m2 n2 −α∕2 : ϕ
(27)
The range of spatial frequencies covered by this spectrum is limited on the upper end by the grid spacing, Δf , such that f max 1∕2Δf , and at the lower end by the physical size of the phase screen, f min 1∕L. Assuming the sample spacing is smaller, or of the order of, the inner-scale size the upper end of the frequency range can be considered adequately sampled. To adequately sample low spatial frequencies, it is generally necessary that the screen size be much
larger than either the outer-scale size or the aperture size of interest. Screen sizes having both adequate size and adequate sampling can be prohibitively large, especially for larger apertures. An alternative proposed by Lane [23] is to break the lowest spatial frequency sample into subharmonic components. The spectrum is sampled at each of the subharmonic frequencies and contributions are appropriately weighted and added to the spectrum. In this work we match the implementation described by Frelich [32] and as outlined by Schmidt [21]. Extending this technique to non-Kolmogorov, anisotropic models requires only replacing the Kolmogorv PSD with the one described in Eq. (25). 3. Results
We now present examples of phase screens simulated using the technique developed in Section 2. Figure 2 shows the theoretical phase structure function as defined by Eq. (10) plotted together with the phase structure function evaluated over an ensemble of 500 phase screens for several values of α. In all cases the characteristic length was set to rˆ 0 0.1 m and is equivalent to the Fried parameter r0 when α 11∕3. Each phase screen was generated with a side length of 5 m on a 1024 × 1024 grid. Structure functions are evaluated over a 2.5 m diameter circular aperture at the center of each screen. Unless otherwise mentioned the outer scale as specified in Eq. (25) is set to infinity; the inner scale is set to zero. Seven subharmonics were included in each screen to ensure the best possible match to theory. The separation axis in Fig. 2 is normalized by the characteristic length rˆ 0 . In Fig. 2(a) the structure function is evaluated for power laws of α 3, 3.17, 3.33 and compared to the value predicted by Eq. (10) using Stribling’s constant, c1 α, in Eq. (19). The Kolmogorov case, α 11∕3, is also included for reference. The value of c1 α in this range varies from 5.65 to 6.09 and c1 11∕3 6.88. Similarly, the ensemble structure function for phase screens with power laws of α 3.5, 3.83, 3.98 is plotted in Fig. 2(b). Here c1 α varies from 6.44 to 7.94. In both cases, the match between phase screen values and theory agree within about 15% at separations out to 9r0 and 16% out to separations greater than 12r0 in the Kolmogorov case. We also observe that, with one exception, the slope of each curve varies according to the prescribed power law and approximately intercepts the prescribed value of c1 α at a normalized separation of ρ∕ˆr0 1. The exception in Fig. 2(b) is for the α 3.98 powerlaw case. We include it here to demonstrate the limitation of the subharmonic technique as the power law approaches the upper limit at α 4. Referring to the structure function definition in Eq. (4), we observe that as α approaches 4, γ α − 3 approaches 1. At this upper bound, phase fluctuations are a linear function of separation and physically manifested as tip and tilt fluctuations of the beam or image. Tip-tilt fluctuations originate from the very lowest spatial
Fig. 2. Comparison of the simulated (solid) and theoretical (dashed) structure functions as a function of separation and normalized by the coherence length rˆ 0 for (a) α 3, 3.17, 3.33 and (b) α 3.5, 3.83, 3.98. The Kolmogorov case α 3.67 is present in both (a) and (b) for reference.
frequencies and, even then, are limited by the size of the phase screen [27]. Thus, it is reasonable that as α approaches 4 we should observe deviations between our phase screens and the theoretical structure function. In exploring this matter, we find that if the number of subharmonics is increased to 9 or more, reasonable agreement with theory is possible up to around α ∼ 3.9. Next, we evaluate anisotropy in our phase screen model. In Fig. 3 the contour where the two-dimensional structure function evaluates to 10 March 2015 / Vol. 54, No. 8 / APPLIED OPTICS
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instances. Values of anisotropy in Fig. 3 are set as ϵ f2; 1.5; 1.33; 0.75; 0.66; 0.5g. The isotropic (ϵ 1) case is also included for reference. As we would expect there is a symmetry between inverse values of anisotropy. For example, in the ϵ 2 and ϵ 0.5 cases the coherence length is increased by a factor 2 in vertical and horizontal directions, respectively. Finally, we show that the angle of anisotropy in our model can be varied continuously. In Fig. 4 the angle of anisotropy, θ, is varied between 33 and 135 deg while anisotropy is fixed as ϵ 2. All other parameters are the same as Fig. 3. As expected, the contours are identical to those for the ϵ 2 case in Fig. 3 but with the minor axis of the constant value parabola rotated by the indicated angle. 4. Conclusion
Fig. 3. Contour lines of the 2D structure function at Dρ 6.88 for several values of the anisotropy factor ϵ. The isotropic, Kolmogorov case, ϵ 1 is emphasized for reference.
Dρ 6.88 is plotted for several values of anisotropy. The power-law exponent is fixed at α 11∕3 and r0 0.1 m. The screen size was increased to 10 m and sampled by a 2048 square grid. A finite outer-scale size of L0 10 m was used to observe the proper scaling. In each case, the structure function was evaluated over a 5 m circular aperture at the center of each screen and averaged over 500
Fig. 4. Contour plot of the structure function for phase screens with varied orientation of anisotropy. Each contour represents the separation, ρ, at which the structure function evaluates to 6.88. Both axes are normalized by the Fried parameter, r0 . For comparison purposes, the isotropic case appears in bold as a circle with unit radius. 2044
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We have described a useful modification to the traditional FFT-based method of generating turbulence phase screens. This modification allows for a variable direction of anisotropy, power-law exponent, turbulence strength, as well as inner and outer scale. Turbulence strength in this model is defined in terms of a single characteristic length parameter providing a way of readily comparing propagation paths with varied power laws and geometries to standard isotropic-Kolmogorov models. Subharmonic compensation is used to compensate for undersampling at low-spatial frequency fluctuations. The phase screen model described here is accurate in as much as it is equivalent to other popularly used FFT-based phase screen generation techniques. In the Kolmogorov isotropic case the structure function differs by less than 15% from the theoretical model up to separations of 9r0. However, achieving this level of accuracy requires using the center portion of a screen twice the size of interest and generated using seven subharmonic components. These steps may be omitted, but at the cost of accuracy relative to the theoretical model. The degree to which this accuracy is important depends on the application. A phase screen with a grid spacing of Δx 0.005 m and a square grid size of 16384 provides a physical screen size of around 80 m. Using only the center portion of this screen results in a usable side length of 40 m, which is of the order of the largest telescope apertures currently proposed. The results presented here are also for turbulence with an infinite outer-scale size. Considering most extreme estimates for the outer-scale size are no more than a couple hundred meters, it is likely that FFT-based approaches, like the one described here, will satisfy most needs. On the other hand generating time-correlated simulations in which frozen turbulence advects across an aperture requires long turbulence ribbons. Modeling these ribbons with the proper statistics, especially atmospheric piston, remains a challenge. This research was performed while the author held a National Research Council Research Associateship Award at the Air Force Research Lab. The views
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