We present a detailed investigation, qualitative and quantitative, on how the atmospheric turbulence with a non-Kolmogorov power spectrum affects the major statistics of stochastic electromagnetic beams, such as the spectral composition and the states of coherence and polarization. We suggest a detailed survey on how these properties evolve on propagation of beams generated by electromagnetic Gaussian Schell-model sources, depending on the fractal constant α of the atmospheric power spectrum.
©2010 Optical Society of America
It has been experimentally shown in the last several decades that generally atmospheric turbulence might possess structure different from the classic Kolmogorov’s one [1, 2], i.e. it can have other energy distribution among differently-sized turbulent eddies, and exhibit non-homogeneity and/or anisotropy [3–18]. Such deviations are usually pertinent to higher atmospheric layers, being caused by gravity waves and the jet-stream, and they certainly affect the statistics of electromagnetic waves, especially at optical frequencies, as can be shown by direct data application in propagation equations (cf. ).
While it is generally impossible to characterize all the features of non-Kolmogorov’s atmosphere several analytical models have been recently suggested [20–22] for taking into account the slope variation of the atmospheric power spectrum. In particular, it was assumed that instead of classic power law 11/3 the power spectrum has a generalized law, defined by parameter α, in the range 3 < α < 5, as the one-dimensional fractal distribution stipulates. It was also shown how parameter α influences various statistics of monochromatic [23, 24] and partially coherent  optical waves. Since the atmosphere was shown to be layered in terms of the power spectra at different altitudes several studies were carried out on modeling of the non-Kolmogorov spectrum specifically for up/down/slant path propagation [6, 7, 26]. However, all the studies relating to wave propagation in such turbulence conditions were based on the scalar theory of propagation.
In this study we extend the previous analysis from scalar to electromagnetic stochastic beam-like fields and consider the main set of their properties, including spectral, coherence and polarization states. It was recently discovered that polarization properties of beams can change on propagation, even in free space . This effect is caused solely by correlation properties of the source. In the conditions of the Kolmogorov’s atmosphere such changes were demonstrated to depend on both source and medium fluctuations and occur, in contrast with the free-space effects, in a non-monotonic fashion [28–30]. In addition, depending on whether the source is uniformly polarized or not, in the former case all the single-point polarization properties self-reconstruct after traveling, in the Kolmogorov’s atmosphere, for sufficiently large distance . It will be of interest to test whether the same predictions still hold for polarization of beams propagating in the non-Kolmogorov’s turbulence.
We will illustrate our analytical results by calculating numerically the major statistics of the stochastic electromagnetic beams for the famous class of electromagnetic Gaussian Schell-model (EMGSM) beams, propagating in the non-Kolmogorov atmospheric turbulence with different values of parameter α and will point out to the differences in the results from those relating to the classic Kolmogorov’s theory.
On passing to the main part of the paper we would like to mention that our atmosphere-related study can also be of interest for optical beam interaction with other natural media, such as turbulent ocean and biological tissues, just to name a few. It is well known that human and animal tissues, for instance, can also be characterized by their spatial power spectra. As was recently shown in [32,33] polarization properties of beams trespassing human epidermis can be determined in a way similar to one used in atmospheric studies but have qualitatively different behavior. For instance, the beam only depolarizes with traveling distance.
2. Propagation of cross-spectral density matrix in non-Kolmogorov turbulence
We will now develop equations for the second-order properties of a stochastic electromagnetic beam-like field passing through a non-Kolmogorov turbulence . Suppose that the beam is generated in the source plane z = 0 and propagates into the half-space z > 0, nearly parallel to the positive z direction. The second-order statistical properties of such a beam may be characterized by a cross-spectral density matrix  defined at two positions r10 = (x10, y10, 0) and r02=(x20, y20, 0) and angular frequency ω as
where Ei and Ej are the mutually orthogonal components of the electric field, * stands for complex conjugate, angular brackets denote ensemble average in the sense of coherence theory in space-frequency domain  and square brackets are used to denote the 2×2 matrix components. The elements of the cross-spectral density matrix propagating to points r1 = (x1,y1,z1) and r2 = (x2,y2,z2) of the half-space z > 0, filled with turbulent atmosphere can then be given by the formula 
where k = c/ω = 2π/λ is the wave number of the optical wave, and K (r10, r02, r1, r2; ω) is the propagator, depending on the Green’s function of the random medium, of the form
where Φn(κ) is the one-dimensional power spectrum of fluctuations in the refractive index of the turbulent medium.
We will assume here that the turbulence is governed by non-Kolmogorov statistics, and that the power spectrum Φn(κ) has the van Karman form, in which the slope 11/3 is generalized to arbitrary parameter α, i.e. [20,21]:
where κ0 = 2π/Lo, L0 being the outer scale of turbulence, κm = c(α)/l0, l0 being the inner scale of turbulence, and
The term C̃2n in Eq. (3) is a generalized refractive-index structure parameter with units m3-α, and
with Γ(x) being the Gamma function. For the power spectrum (3) the integral in expression Eq. (2) becomes
where β = 2κ02 − 2κm2 + ακm2 and Γ denotes the incomplete Gamma function. Equations (1), (2) and (4) provide the theoretical basis for propagation of arbitrary stochastic electromagnetic beams in general, non-Kolmogorov turbulence. On substituting for W0(r10, r02; ω) into Eq. (1) one of the available models for stochastic electromagnetic beams one can determine their second-order statistical properties everywhere within random medium. Among the statistics of interest we will consider, in what follows, the spectral density 
the spectral degree of coherence
and the spectral degree of polarization
3. An example: electromagnetic Gaussian-Schell model beam
We will now apply the formulas developed in Sec. 2 to the important model beam, known in the literature as the electromagnetic Gaussian-Schell model [EMGSM] beam. Such a beam can be characterized in the source plane by a matrix with elements
where Ix, Iy are the intensities along x- and y- axes, Bij = ∣Bij∣eiφij is the single-point correlation coefficient between i and j field components, φij being its phase, σ is the r.m.s. width of the beam, δxx and δyy are the r.m.s. widths of auto-correlations of Ex and Ey field components, and δxy, δyx are the r.m.s. widths of the cross-correlations of Ex and Ey. The fact that σ does not depend on indexes i and j implies that the single-point polarization properties are uniform across the source . In addition, the following set of conditions should be satisfied by some parameters entering the model [28,38]
These relations all follow from the non-negative definiteness and quasi-hermiticity of the cross-spectral density matrix . It was shown in  that after propagation at distance z from the source the elements of the cross-spectral density matrix of an EMGSM beam take the form
where the spreading coefficients and curvature terms are given respectively by the expressions
and I having been defined in Eq. (4).
We will now numerically determine the behavior of statistics Eqs. (5)–(7) in the case of a typical EMGSM beam with diagonal matrix (without loss of generality) and will analyze their dependence on parameter α. We will assume below the following values of the parameters of the atmosphere and the beam: C̃2n = 10−13m3−α, Ay = 1, l0 = 10−3 m, L0 = 1 m, λ = 0.6328 × 10−6 m, σ = 0.025 m, δxx = 5 × 10−3 m, δyy = 5 × 10−4 m, unless other values are specified in the figure captions.
Figure 1 shows variation of the on-axis spectral density normalized by its value in the source plane, i.e. SN(r; ω) = S(0, 0, z; ω)/S(0, 0, 0; ω), and the spectral degree of polarization P(0, 0, z; ω) with parameter α at distance 1 km from the source. We note that in this figure the illumination beam is uniformly unpolarized across the source, but due to source correlations it becomes nearly polarized at 1 km the effect first noticed by James  for propagation in vacuum. In our case the atmospheric turbulence also modifies these statistics, the strength of the effect being dependent on α. The main trends of both beam properties are similar with maximum values at the ends of the interval 3 < α < 5 and one minimum inside. As also is seen from this figure the turbulence affects both statistics most when α is close to its lower limit.
In Fig. 2 we demonstrate the on-axis normalized spectral density as a function of distance z from the source for several values of parameter α, one of which corresponds to the Kolmogorov’s turbulence (α = 3.67) (dotted curves). We note that the spectral density remains almost the same up to about a kilometer and then decreases at rates depending on α.
Figure 3 shows the dependence of the on-axis values of the spectral degree of polarization as a function of propagation distance z from the source, for several fixed values of the parameter α. As is seen from the figure, the polarization of the initially unpolarized beam first grows to almost unity due to source correlations, independently of α, and then decreases depending on the value of α.
Figures 4–6 illustrate the variation of the modulus of the spectral degree of coherence η with propagation distance from the source, separation distance between two points in the transverse plane and the turbulence parameter α. The absolute value of the spectral degree of coherence is an important measurable quantity that can be related to the interference of fringes in the Young’s interference experiment, including the case when the beam is electromagnetic . In particular, Fig. 4 gives, on two scales (a) basic and (b) logarithmic, the dependence of the absolute value of the spectral degree of coherence with propagation distance from the source, calculated at distance r = ∣r∣ = ∣r1-r2∣ between two fixed points symmetrically situated about the optical axis, r1 = r/2 and r2 = -r/2. While this quantity grows to high values at distances close to the source, the effect being attributed to source correlations, it steadily starts to decrease, at about 1 km from the source, to lower values, the rate depending on α. The dependence on α is non-monotonic as before, for other statistics, the fastest drop corresponding to α = 3.1 among the four selected values. On the other hand for values of α close to 5 the turbulence does not practically affect the state of coherence and resembles the free-space scenario [29,30]. Figures 5 and 6 explore different perspectives of the same dependence: Fig. 5 shows the variation of the modulus of the spectral degree of coherence on parameter α for several fixed separation distances between points in the transverse plane z = 1 km. The most drastic variation of the coherence state can be noticed for the case corresponding to the dashed curve, r = 1 mm. Also, for α > 4 there is almost no effect of the atmosphere on the state of coherence of the beam. Finally, Fig. 6 presents the drop in the modulus of the degree of coherence with growing values of r at fixed distance 1 km from the source, for discrete values of α.
4. Concluding remarks
We have explored the variation of the three main statistics of a typical stochastic electromagnetic beam: the spectral density, and the states of coherence and polarization, on propagation in the turbulent atmosphere depending on the slope α (3 < α < 5) of the non-Kolmogorov power spectrum of refractive index fluctuations. We have found that a typical beam is affected the least for α close to 5 and the most in a region about 3.1, the dependence of all the statistics on α being, hence, non-monotonic. Our results may find uses in communication and sensing systems operating through atmospheric channels at high altitudes from the ground where the atmosphere does not possess the classic Kolmogorov’s structure. It was recently shown that the beams of the considered class may be efficiently used in both aforementioned applications (cf. [41,42]) in the classic case and can be now modified, based on our results, for the non-Kolmogorov’s statistics in a straightforward manner.
O. Korotkova’s research is funded by the US AFOSR (grant FA 95500810102) and US ONR (grant N0018909P1903). E. Shchepakina is supported by the Russian Foundation for Basic Research (grant 10-08-00154a).
References and links
1. A. N. Kolmogorov, “The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers,” C. R. Acad. Sci. URSS 30, 301–305 (1941).
2. A. N. Kolmogorov, “Dissipation of energy in the locally isotropic turbulence,” C. R. Acad. Sci. URSS 32, 16–18 (1941).
3. V. I. Tatarski, Wave Propagation in a Turbulent Medium (Nauka, Moscow, 1967).
4. J. R. Kerr, “Experiments on turbulence characteristics and multiwavelength scintillation phenomena,” J. Opt. Soc. Am. 62, 1040–1049 (1972). [CrossRef]
5. R. S. Lawrence, G. R. Ochs, and S. F. Clifford, “Measurements of atmospheric turbulence relevant to optical propagation,” J. Opt. Soc. Am. 60, 826–830 (1970). [CrossRef]
6. A. S. Gurvich and M. S. Belenkii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517–2522 (1995). [CrossRef]
7. M. S. Belenkii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12, 2517–2522 (1995). [CrossRef]
8. F. Dalaudier, M. Crochet, and C. Sidi, “Direct comparison between in situ and radar measurements of temperature fluctuation spectra: a puzzling results,” Radio Sci. 24, 311–324 (1989). [CrossRef]
9. F. Daludier and C. Sidi, “Direct evidence of sheets in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994). [CrossRef]
10. H. Luce, F. Daludier, M. Crochet, and C. Sidi, “Direct comparison between in situ and VHF oblique radar measurements of refractive index spectra: a new successful attempt,” Radio Sci. 31, 1487–1500 (1996). [CrossRef]
11. M. S. Belenkii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999). [CrossRef]
12. A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77 (2008). [CrossRef]
13. M. S. Belenkii, S. J. Karis, J. M. Brown II, and R. Q. Fugate, “Experimental study of the effect of non Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997). [CrossRef]
14. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995). [CrossRef]
15. D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994). [CrossRef]
16. B. Joseph, A. Mahalov, B. Nicolaenko, and K. L. Tse, “Variability of turbulence and its outer scales in a model tropopause jet,” J. Atm. Sci. 61, 621–643 (2004). [CrossRef]
17. A. Mahalov, B. Nicolaenko, K.L. Tse, and B. Joseph, “Eddy mixing in jet-stream turbulence under stronger stratification,” Geophys. Res. Lett. 31, L23111 (2004). [CrossRef]
18. C. Rao, W. Jiang, and N. Ling, “Adaptive-Optics Compensation by Distributed Beacons for Non-Kolmogorov Turbulence,” Appl. Opt. 40, 3441–3449 (2001). [CrossRef]
19. O. Korotkova, N. Farwell, and A. Mahalov, “The effect of the jet-stream on the intensity of laser beams propagating along slanted paths in the upper layers of the turbulent atmosphere,” Waves in Random Media , 19, 692–702 (2009). [CrossRef]
20. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Engineering 47, 026003 (February 2008). [CrossRef]
21. 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–1–12 (2007). [CrossRef]
22. C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47, 1111–1126 (2000). [CrossRef]
23. W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctiations for wave propagation trough non-Kolmogorov turbulence,” Opt. Commun. 282, 705–708 (2009). [CrossRef]
26. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47, 6385–6391 (2008). [CrossRef] [PubMed]
27. D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1649 (1994). [CrossRef]
28. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of partially coherent beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225–230 (2004). [CrossRef]
29. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent EM beams propagating through turbulent atmosphere,” Waves in Random Media 14, 513–523 (2004). [CrossRef]
30. O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere,” Waves in Random and Complex Media 15, 353–364 (2005). [CrossRef]
31. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007). [CrossRef] [PubMed]
32. W. Gao, “Changes of polarization of light beams on propagation trough tissue,” Opt. Commun. 260, 749–754 (2006). [CrossRef]
33. W. Gao and O. Korotkova, “Changes in the state of polarization of a random electro-magnetic beam propagating through tissue,” Opt. Commun. 270, 474–478 (2007). [CrossRef]
34. E. Wolf, Intoduction to the Theories of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).
35. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
36. J. Pu and O. Korotkova, “Propagation of the degree of cross-polarization in the turbulent atmosphere” Opt. Commun. 282, 1691–1698 (2009). [CrossRef]
37. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A: Pure Appl. Opt. 3, 1–9 (2001). [CrossRef]
38. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379–385 (2005). [CrossRef]
39. W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271, 1–8 (2007). [CrossRef]
40. E. Wolf, “The influence of Young’s interferences experiment on the development of statistical optics,” in Progress in Optics, E. Wolf, ed. (Elsevier B. V., 2007), 50 pp. 251–273.
41. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008).
42. O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94, 681–690 (2009). [CrossRef]