January 15, 2015 / Vol. 40, No. 2 / OPTICS LETTERS

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Two-dimensional simulation of optical wave propagation through atmospheric turbulence Milo W. Hyde IV,1,* Santasri Basu,1 and Jason D. Schmidt2 1 2

Air Force Institute of Technology, 2950 Hobson Way, Dayton, Ohio 45433, USA

MZA Associates Corporation, 1360 Technology Court Suite 200, Dayton, Ohio 45430, USA *Corresponding author: [email protected] Received October 8, 2014; revised December 4, 2014; accepted December 4, 2014; posted December 8, 2014 (Doc. ID 224545); published January 12, 2015

A methodology for the two-dimensional simulation of optical wave propagation through atmospheric turbulence is presented. The derivations of common statistical field moments in two dimensions, required for performing and validating simulations, are presented and compared with their traditional three-dimensional counterparts. Wave optics simulations are performed to validate the two-dimensional moments and to demonstrate the utility of performing two-dimensional wave optics simulations so that the results may be scaled to those of computationally prohibitive 3D scenarios. Discussions of the benefits and limitations of two-dimensional atmospheric turbulence simulations are provided throughout. © 2015 Optical Society of America OCIS codes: (010.1300) Atmospheric propagation; (010.1330) Atmospheric turbulence; (010.7060) Turbulence; (030.7060) Turbulence; (070.7345) Wave propagation; (350.5500) Propagation. http://dx.doi.org/10.1364/OL.40.000233

Two-dimensional (2D) approximations to threedimensional (3D) problems have been utilized in physics and electromagnetics for several decades [1,2]. In the field of electromagnetics, many problems of interest are invariant (or approximately so) in a direction and therefore can be more easily solved in 2D, e.g., determining cutoff frequencies of waveguiding structures and scattering from cylinders, wedges, and rough surfaces [3]. Because of the reduction in dimensionality, 2D analysis is extremely useful in the field of computational electromagnetics, where optically large problems can be solved with relative ease. In optics, 2D theory is typically used in geometrical optics and in the burgeoning fields of optical metamaterials and computational photonics, where full-wave electromagnetic techniques, e.g., the method of moments [2], finite-difference timedomain, are typically used [4]. Although common in a number of related fields, surprisingly, 2D analysis has not been applied to simulations of optical wave propagation through atmospheric turbulence. While any one instance of the field, after propagating through turbulence, is certainly not invariant in a direction, typically, the source field (e.g., plane wave, point source, Gaussian beam) and the underlying statistical behavior of the atmosphere (because of the common assumption of isotropy) are invariant. This leads to the statistical moments of the field being invariant in a direction as well (in 3D analysis, the moments are typically azimuthally invariant). Therefore, a 2D simulation, in these common cases, should capture the same physical behavior of the field as the full 3D simulation with a significant reduction in computational burden [3]. Propagation through turbulence scenarios, where the computational requirements are especially demanding and thus good candidates for 2D analysis, include studies of anisoplanatism [5], strong turbulence, partially coherent sources [6], and scattering from objects embedded in turbulence [7,8]. Research investigating the scattering of Gaussian beams from rough surfaces 0146-9592/15/020233-04$15.00/0

embedded in turbulence was the impetus for this work [8]. Considering that the full 3D scattering problem is very computationally demanding, 2D simulations were performed to gain physical insight into the problem. Where possible, 2D results were compared to 3D simulation results, and good agreement was noted between the 2D and 3D average scattered irradiances and the scattered-field mutual coherence functions (MCFs). In this Letter, a methodology for the 2D simulation of optical wave propagation through atmospheric turbulence is presented. Common statistical field moments, the MCF Γ, spatial coherence radius ρ0 , atmospheric coherence diameter r 0 , and the log-amplitude variance (Rytov number) σ 2χ , in 2D are derived and discussed. These moments are compared to their 3D counterparts, such that the results of 2D turbulence simulations can be converted to 3D values and vice versa. The results of wave optics simulations are presented and discussed. The presented results compare 2D and 3D ρ0 and σ 2χ versus turbulence strength and serve to validate the 2D theory as well as to demonstrate the utility of 2D simulations. In addition to the simulation results, the generation of 2D phase profiles, akin to 3D phase screens, is discussed. Last, this Letter concludes with a summary of the work presented. The germane propagation geometry is shown in Fig. 1. The figure depicts light emitted from the source plane at

Fig. 1. Two-dimensional (y-invariant) optical propagation geometry. The turbulence strength is characterized by C 2n , which, in general, is a function of z. © 2015 Optical Society of America

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z  0 propagating through atmospheric turbulence to the observation plane at z  L. The source field and the turbulence are assumed to be invariant in the y direction, making this a 2D problem. The strength of the turbulence is given by the index of refraction structure constant C 2n , which, in general, is a function of z. The atmospheric turbulence is assumed to be statistically homogeneous and isotropic. As in the 3D case, the field at the observation plane can be found by solving the scalar stochastic Helmholtz wave equation (here in 2D): ∇2t Uρ  k2 1  2n1 ρUρ  0;

(1)

ˆ  zˆ z, ∇2t  ∂2 ∕∂x2  ∂2 ∕∂z2 , k  2π∕λ, and where ρ  xx n1 is the zero-mean, random component of the total index of refraction n ≈ 1  n1 . An approximate solution to Eq. (1) can be found using the Rytov perturbation method [9–11]. The forms of the 2D first- and second-order Rytov perturbations are nearly identical to the 3D expressions and are therefore omitted for brevity. With the first- and second-order Rytov perturbation solutions, moments of the observed field can be computed. The moments of interest in this work are the MCF Γ and the Rytov number σ 2χ . From Γ, one can obtain expressions for the modulus of the complex degree of coherence (MDoC) μ, the spatial coherence radius ρ0 , and the atmospheric coherence diameter r 0 . Assuming that the source field is a 2D point source (more accurately termed a line source), Γ and σ 2χ take the form    Z LZ ∞ exp j Lk pr exp −8πk2 Γ2D L; x1 ; x2   jAj F n κ; z 8πL 0 0    p κz dκdz sin2 2L   Z LZ ∞ L−z 2 κ z dκdz; F n κ; zsin2 σ 2χ;2D ≈ 4πk2 2kL 0 0 2

(2)

Z F n κ x ; κ y ; z 



−∞

Φn κ x ; κy ; κz  cosκz zdκz :

(3)

Tatarskii evaluated this integral using the 3D Kolmogorov PSD, Φn κ  0.033C 2n κ−11∕3 , and obtained F n κ; z ≈ 0.0494κz4∕3 K 4∕3 κzC 2n κ −8∕3 ;

(4)

where κ  κ 2x  κ 2y 1∕2 (κy  0 for the 2D analysis presented here) and K ν is a modified Bessel function of the second kind [11]. Like in 3D turbulence theory, the index of refraction in 2D is assumed to be delta correlated in the direction of propagation (the 2D equivalent of the Markov approximation) [9,10]; thus, Eq. (4) is simplified by substituting in the small argument expression for K ν [12], namely, F n κ; z ≈ 0.0555C 2n zκ−8∕3 :

(5)

It should be noted that Eq. (5) can also be obtained by using 1 F n κ  4πκ 2

Z

∞ 0

  d d r D r J 0 κrdr; dr dr n

(6)

where J 0 is a zeroth-order Bessel function of the first kind and Dn r  C 2n r 2∕3 is the inertial subrange Kolmogorov index of refraction structure function [9–11]. Equation (6) is derived under the assumption that F n is not a function of z. Since Dn is assumed to have the same r 2∕3 behavior in 2D as it does in 3D, 2D C 2n is equivalent to 3D C 2n . With a form for F n , Γ2D and σ 2χ;2D can be computed. Substituting Eq. (5) into Eq. (2) and simplifying produces   exp jLk pr Γ2D L;x1 ;x2 ≈jAj 8πL  5∕3   ZL z dz ×exp −1.4572p5∕3 C 2n z L 0   5∕6  ZL z z 5∕6 1− dz: σ 2χ;2D ≈0.3016k7∕6 L5∕6 C 2n z L L 0 2

(7) where A is the amplitude of the source field, p  x1 − x2 , r  x1  x2 ∕2, and F n κ; z is the 2D index of refraction power spectral density (PSD). Note that F n only depends on z via C 2n . The relations assuming a plane wave source field can be obtained from the 2D spherical wave (more accurately termed a cylindrical wave) statistics above by removing z∕L from both Γ2D and σ 2χ;2D and eliminating the complex exponential term and the 1∕8πL from Γ2D . Note that Γ2D and σ 2χ;2D are similar to their traditional 3D counterparts; however, here they are inverse Fourier cosine transforms instead of inverse Hankel transforms. This result is intuitive considering the reduction in dimensionality. To further simplify Eq. (2), a form for the 2D index of refraction PSD F n is required. F n is related to the 3D index of refraction PSD Φn by the Fourier transform relation [9–11]

Note that Γ2D is identical to its 3D counterpart with the exception of the amplitude factor 1∕8πL. Therefore, the MDoC μ and ρ0 in 2D and 3D are equal. The expression for σ 2χ;2D is also identical to its 3D counterpart with the exception of the 0.3016 (versus 0.563) scaling factor. It is hypothesized that ρ0;2D  ρ0;3D because the 3D problem (from which ρ0;3D is derived) is statistically azimuthally invariant (arising from the ϕ-invariant source field and the common assumption of isotropic turbulence). This simplifies the 3D problem (originally, ρ, ϕ, and z) to an equivalent 2D problem (ρ and z). It is believed that this ρ0 relationship holds for 3D problems in which the above source field and turbulence conditions are met—both conditions are commonly met in practice. This statement has yet to be rigorously proved; however, simulation results do support this hypothesis.

January 15, 2015 / Vol. 40, No. 2 / OPTICS LETTERS

The expressions for Γ2D and σ 2χ;2D given in Eq. (7) are the main theoretical results of this Letter. Since it is common to characterize 3D turbulent paths by their r 0 ≈ 2.1ρ0 and Rytov number values, Eq. (7) provides a means of converting 3D ρ0 and σ 2χ to their equivalent 2D quantities. This is extremely useful when one desires to simulate a scenario in which the full 3D simulation is too computationally intensive. In these instances, performing the simulation in 2D may provide the required information. Considering its very common use in characterizing 3D turbulent paths, it is worth presenting the 2D equivalent of r 0 . Following the analysis of Fried [5,13], the 2D long-exposure optical transfer function (OTF) is     νλf 5∕3 ; ℋLE ν  exp − 2D ρ0;2D

(8)

where ν is the spatial frequency and f is the focal length of the imaging lens. The 2D diffraction-limited OTF is ℋDL 2D ν

 νλf  Λ D jνj ≤ λfD ;  0 else

(9)

where Λx is the triangle function and D is the aperture dimension. From these, the 2D equivalent of Fried’s resolution metric can be obtained, i.e., 2D RD  λf  1 ≈ λf

Z 0

1

    D 5∕3 5∕3 dt 1 − t exp − t ρ0;2D D

2ρ0;2D 35 Γ

3 5

D ≪ ρ0;2D D ≫ ρ0;2D

;

(10)

where Γ in this context is the gamma function [12]. As physically expected, resolution in 2D increases linearly with D (for D ≪ ρ0;2D ) vice D2 in 3D. It should be noted that the integral for R can be evaluated analytically yielding an expression in terms of gamma and incomplete gamma functions. This relation is not shown here because insight into the behavior of R versus D is lost. Like in Fried’s 3D analysis, r 0;2D is defined as the x coordinate of the intersection point of the two asymptotes of R shown in Eq. (10). Performing this analysis, one obtains r 0;2D ≈ 1.7870ρ0;2D ≈ 0.8510r 0;3D . Considering that r 0;2D ≠ r 0;3D , it is more convenient here to characterize turbulent paths using ρ0 since ρ0;2D  ρ0;3D . Note that r 0;2D ≠ r 0;3D because R ∼ D in 2D (for D ≪ ρ0;2D ) versus D2 in 3D. Having completed the relevant 2D turbulence theory, attention can now be turned to the wave optics simulations. A 2D line source and a 3D point source were propagated, L  10 km through constant C 2n turbulence. The spatial coherence radii, ρ0;2D and ρ0;3D , as well as σ 2χ;2D and σ 2χ;3D were computed from 1000 Monte Carlo trials. To show how these parameters vary versus turbulence strength, the 1000 propagations described above were repeated 13 times using different C 2n values chosen such that σ 2χ;3D varied from 0.05 to 1.25 (i.e., weak to strong amplitude fluctuations). Note that while these

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simulations were easily performed in 3D, the 2D simulations executed 1200 times faster. The line and point sources were created using the spatially filtered point source technique presented in Ref. [14]. The simulated wavelength of the sources was λ  532 nm. Utilizing the optical propagation sampling analysis also presented in Ref. [14], 29 points per side and 210 points per side were used to discretize the source and observation planes for the first eight and last five C 2n values, respectively. The source plane and observation plane sample spacings were chosen so that, at a minimum, 2.5 samples existed across the central lobes of the spatially filtered line and point sources and across ρ0  ρ0;2D  ρ0;3D , respectively. For all 13 C 2n simulations, the diameter of the receiving aperture was 50 cm. Eleven and six 2D phase profiles and 3D phase screens, spaced equally along the 10 km path, were used to simulate the turbulence for the first eight and last five C 2n values, respectively. The coherence radii of the individual phase profiles and screens, necessary to discretely model the continuous turbulence, were determined using the method discussed in Ref. [14]. The Fourier transform (FT) method, in combination with the subharmonic method, was used to generate the 2D phase profiles and 3D phase screens [14,15]. The 2D phase PSD is required to generate phase profiles using the FT method. To the authors’ knowledge, the 2D phase PSD does not exist in previous literature. Following the 3D analysis of Tatarskii [11], the 2D phase PSD takes the form F ϕ κ  2πk2 LF n κ  0.2395ρ0−5∕3 κ −8∕3 ;

(11)

where ρ0 in this context is the desired ρ0 of the individual phase profile (not the full-path ρ0 ). Figures 2(a) and 2(b) show the 2D and 3D simulation results as well as the 2D and 3D theoretical expressions plotted versus C 2n for ρ0 and σ 2χ , respectively. Note that in Fig. 2(a), ρ0−5∕3 is plotted because ρ0−5∕3 ∝ C 2n .

Fig. 2.

(a) ρ0−5∕3 and (b) σ 2χ versus C 2n .

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The agreement between simulation and theory is quite excellent for ρ0 . Minor discrepancies are noted in the strong turbulence regime, i.e., σ 2χ > 0.5. Under weak turbulence conditions (σ 2χ < 0.2), the agreement between σ 2χ simulation and theory is very good. The results begin to deviate as the turbulence transitions from weak to moderate (0.2 < σ 2χ < 0.5—the focusing regime), and then from moderate to strong where σ 2χ;2D and σ 2χ;3D approach their asymptotic values (the saturation regime) [9]. Note that the σ 2χ theoretical expressions discussed in this Letter are applicable in weak turbulence; thus, the σ 2χ strongturbulence discrepancy evident in Fig. 2(b) is expected. The spatial coherence radius ρ0 does not experience a similar discrepancy because the 2D and 3D strong turbulence MCFs are equal to their weak turbulence counterparts discussed above [9]. In conclusion, a methodology for the 2D simulation of optical wave propagation through atmospheric turbulence was presented. The derivations of the 2D MCF, ρ0 , r 0 , and σ 2χ , necessary for performing and verifying atmospheric turbulence simulations, were presented and compared with their 3D counterparts. To validate the 2D theory and demonstrate the utility of 2D simulations, wave optics simulations were performed comparing 2D and 3D ρ0 and σ 2χ versus turbulence strength. The results showed good agreement with the predictions of the 2D theoretical analysis, thus validating the proposed approach. For turbulence scenarios with heavy sampling requirements or where many Monte Carlo trials are required for convergence, e.g., situations involving anisoplanatism, strong turbulence, partially coherent sources, scattering from objects embedded in turbulence, a 2D simulation should be considered as a possible alternative to the traditional 3D wave optics simulation. The views expressed in this Letter are those of the authors and do not reflect the official policy or position

of the U.S. Air Force, the Department of Defense, or the U.S. Government. This research was supported in part by an appointment to the Postgraduate Research Participation Program at the Air Force Institute of Technology administered by the Oak Ridge Institute for Science and Education through an interagency agreement between the U.S. Department of Energy and AFIT. References 1. R. Harrington, Time-Harmonic Electromagnetic Fields (IEEE, 2001). 2. R. Harrington, Field Computation by Moment Methods (IEEE, 1993). 3. A. A. Maradudin, ed., Light Scattering and Nanoscale Surface Roughness (Springer, 2007). 4. M. S. Wartak, Computational Photonics: An Introduction with MATLAB (Cambridge, 2013). 5. J. W. Goodman, Statistical Optics (Wiley, 1985). 6. O. Korotkova, Random Light Beams: Theory and Applications (CRC, 2014). 7. M. Tateiba and Z. Q. Meng, Prog. Electromagn. Res. 14, 317 (1996). 8. S. Basu, M. W. Hyde IV, J. E. McCrae, Jr., M. F. Spencer, and S. T. Fiorino, Proc. SPIE 9224, 92240L (2014). 9. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005) 10. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1999). 11. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961). 12. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972). 13. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966). 14. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010). 15. R. G. Lane, A. Glindemann, and J. C. Dainty, Wave Random Media 2, 209 (1992).

Two-dimensional simulation of optical wave propagation through atmospheric turbulence.

A methodology for the two-dimensional simulation of optical wave propagation through atmospheric turbulence is presented. The derivations of common st...
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