## Abstract

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

## 1. Introduction

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. [19]).

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 [25] 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 [27]. 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 [31]. 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 [20]. 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 [34] defined at two positions **r**_{1}^{0} = (*x*_{1}^{0}, *y*_{1}^{0}, 0) and **r**^{0}_{2}=(*x*_{2}^{0}, *y*_{2}^{0}, 0) and angular frequency *ω* as

where *E _{i}* and

*E*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 [35] and square brackets are used to denote the 2×2 matrix components. The elements of the cross-spectral density matrix propagating to points

_{j}**r**

_{1}= (

*x*

_{1},

*y*

_{1},

*z*

_{1}) and

**r**

_{2}= (

*x*

_{2},

*y*

_{2},

*z*

_{2}) of the half-space

*z*> 0, filled with turbulent atmosphere can then be given by the formula [36]

where *k* = *c*/*ω* = 2*π*/*λ* is the wave number of the optical wave, and *K* (**r**_{1}^{0}, **r**^{0}_{2}, **r**_{1}, **r**_{2}; *ω*) is the propagator, depending on the Green’s function of the random medium, of the form

$$\times \mathrm{exp}\left\{-\genfrac{}{}{0.1ex}{}{{\pi}^{2}{k}^{2}z}{3}\left[{\left({\mathbf{r}}_{1}-{\mathbf{r}}_{2}\right)}^{2}+\left({\mathbf{r}}_{1}-{\mathbf{r}}_{2}\right)\left({\mathbf{r}}_{1}^{0}-{\mathbf{r}}_{2}^{0}\right)+{\left({\mathbf{r}}_{1}^{0}-{\mathbf{r}}_{2}^{0}\right)}^{2}\right]\underset{0}{\overset{\infty}{\int}}{\kappa}^{3}{\mathrm{\Phi}}_{n}\left(\kappa \right)d\kappa \right\},$$

where Φ_{n}(*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 Φ_{n}(*κ*) has the van Karman form, in which the slope 11/3 is generalized to arbitrary parameter *α*, i.e. [20,21]:

where *κ*_{0} = 2*π*/*L _{o}*,

*L*

_{0}being the outer scale of turbulence,

*κ*=

_{m}*c*(

*α*)/

*l*

_{0},

*l*

_{0}being the inner scale of turbulence, and

The term *C*̃_{2}_{n} in Eq. (3) is a generalized refractive-index structure parameter with units *m*_{3-α}, and

with Γ(*x*) being the Gamma function. For the power spectrum (3) the integral in expression Eq. (2) becomes

where *β* = 2*κ*_{0}^{2} − 2*κ _{m}*

^{2}+

*α*

*κ*

_{m}^{2}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

**W**

^{0}(

**r**

_{1}

^{0},

**r**

^{0}

_{2};

*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 [34]

the spectral degree of coherence

and the spectral degree of polarization

In Eqs. (5)–(7) Tr and Det stand for trace and determinant of the cross-spectral density matrix with components defined by Eq. (1).

## 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 *I _{x}*,

*I*are the intensities along

_{y}*x*- and

*y*- axes,

*B*= ∣

_{ij}*B*∣

_{ij}*e*is the single-point correlation coefficient between

^{iφij}*i*and

*j*field components,

*φ*being its phase,

_{ij}*σ*is the r.m.s. width of the beam,

*δ*and

_{xx}*δ*are the r.m.s. widths of auto-correlations of

_{yy}*E*and

_{x}*E*field components, and

_{y}*δ*,

_{xy}*δ*are the r.m.s. widths of the cross-correlations of

_{yx}*E*and

_{x}*E*. The fact that

_{y}*σ*does not depend on indexes

*i*and

*j*implies that the single-point polarization properties are uniform across the source [37]. In addition, the following set of conditions should be satisfied by some parameters entering the model [28,38]

$$max\{{\delta}_{\mathit{xx}};{\delta}_{\mathit{yy}}\}\le {\delta}_{\mathit{xy}}\le min\{\genfrac{}{}{0.1ex}{}{{\delta}_{\mathit{xx}}}{\sqrt{{B}_{\mathit{xy}}}};\genfrac{}{}{0.1ex}{}{{\delta}_{\mathit{yy}}}{\sqrt{{B}_{\mathit{xy}}}}\}.$$

These relations all follow from the non-negative definiteness and quasi-hermiticity of the cross-spectral density matrix [34]. It was shown in [39] that after propagation at distance *z* from the source the elements of the cross-spectral density matrix of an EMGSM beam take the form

$$\times \mathrm{exp}\left\{-\left[\genfrac{}{}{0.1ex}{}{1}{2{\mathrm{\Delta}}_{\mathit{ij}}^{2}\left(z\right)}\left(\genfrac{}{}{0.1ex}{}{1}{4{\sigma}^{2}}+\genfrac{}{}{0.1ex}{}{1}{{\delta}_{\mathit{ij}}^{2}}\right)+\genfrac{}{}{0.1ex}{}{1}{3}{\pi}^{2}{k}^{2}\mathit{zI}\left(1+{\sigma}^{2}\right)-\genfrac{}{}{0.1ex}{}{{\pi}^{4}{k}^{2}{z}^{4}{I}^{2}}{18{\sigma}^{2}{\mathrm{\Delta}}_{\mathit{ij}}^{2}\left(z\right)}\right]{\left({\mathbf{r}}_{1}-{\mathbf{r}}_{2}\right)}^{2}\right\},$$

where the spreading coefficients and curvature terms are given respectively by the expressions

$${R}_{\mathit{ij}}\left(z\right)=\genfrac{}{}{0.1ex}{}{{\sigma}^{2}{\mathrm{\Delta}}_{\mathit{ij}}^{2}\left(z\right)z}{{\sigma}^{2}{\mathrm{\Delta}}_{\mathit{ij}}^{2}\left(z\right)+\genfrac{}{}{0.1ex}{}{1}{3}{\pi}^{2}{z}^{3}I-{\sigma}^{2}},$$

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: *C*̃^{2}_{n} = 10^{−13}m^{3−α}, *A _{y}* = 1,

*l*

_{0}= 10

^{−3}m,

*L*

_{0}= 1 m,

*λ*= 0.6328 × 10

^{−6}m,

*σ*= 0.025 m,

*δ*= 5 × 10

_{xx}^{−3}m,

*δ*= 5 × 10

_{yy}^{−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. *S _{N}*(

**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 [27] 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 [40]. 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**∣ = ∣**r**_{1}-**r**_{2}∣ between two fixed points symmetrically situated about the optical axis, **r**_{1} = **r**/2 and **r**_{2} = -**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.

## Acknowledgments

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).

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