Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence

Zhangrong Mei; Elena Shchepakina; Olga Korotkova

doi:10.1364/OE.21.017512

1. Introduction

Several models for scalar random sources have been recently introduced in addition to well-explored Gaussian-Schell model source [1] differing by the form of their spectral degree of coherence. Among such classes are the Bessel-correlated sources [2, 3], the non-uniformly correlated sources [4] and the Multi-Gaussian Schell-model sources [5, 6]. Apart from these classes a family of Schell-model sources stands out in which the classic Gaussian degree of coherence is modulated by a well-behaved function, leading to Bessel-Gaussian-correlated, Laguerre-Gaussian-correlated [7] and Cosine-Gaussian-correlated Schell-model sources [8]. The sources in [7] and [8] all generate random beams which can have arbitrary intensity distribution in the source plane (say, Gaussian) but acquire propagation-invariant ring-shaped intensity profile in the far-field (on propagation in free space). Such behavior distinguishes these random beams from the well-known deterministic and random dark-hollow beams (DHB) which possess ring-shaped intensity profiles already at the source plane but loose them on free-space propagation (see [9] and references wherein). It is shown that the DHB fills in the center and takes on the Gaussian form at shorter distances if the atmosphere is present [10]. One should not confuse the Cos-Gaussian-correlated Schell-model beams with the beams of Ref. [11, 12] in which the degree of coherence is just Gaussian but the intensity is Gaussian function modulated by cosine function.

The purpose of this paper is to tackle the clear-air atmospheric propagation of Cos-Gaussian-correlated Schell-model beams which we will briefly refer to as CGC beams and to show whether and how the optical turbulence modifies the rings. Compared to other sources producing the ring-shaped beams [7] the CGC beam [8] is the most attractive because its modulation by cosine function leads to analytic results for beams propagating in free-space and in the atmosphere with quite arbitrary power spectra. Further, the ability of the CGC sources will be demonstrated to produce beams which build up and sustain robust intensity rings at intermediate distances in the turbulent medium. Moreover, we will show that the filling-in-the-center effect occurring at large distances imply that a 3D bottle-like beam [13] can be obtained which circumvents objects on propagation in random medium. This feature makes the CGC beams attractive in situations when an optical signal must not be intercepted by a detector located on the optical axis of the beam. The other goal of this work is to explore the evolution in coherence properties of the CGC beam on its atmospheric propagation. In order to cover a wide scope of atmospheric conditions we employ the atmospheric spectrum in which in addition to the structure constant, the inner and the outer scales [14] the slope of the spectrum in the inertial range can be tuned [15].

2. Analytic solutions for the CGSM propagating in turbulent atmosphere

The cross-spectral density function of a wide-sense statistically stationary field generated by the CGC source located in the plane z = 0, at points specified by two-dimensional position vectors ρ′₁ = (x′₁, y′₁) and ρ′₂ = (x′₂, y′₂) and oscillating at angular frequency ω has form [8]

W^{(0)} (ρ_{1}^{'}, ρ_{2}^{'}) = exp (- \frac{{| ρ_{1}^{'} |}^{2} + {| ρ_{2}^{'} |}^{2}}{4 σ^{2}}) cos [\frac{n \sqrt{2 π} (ρ_{2}^{'} - ρ_{1}^{'})}{δ}] exp [- \frac{{| ρ_{2}^{'} - ρ_{1}^{'} |}^{2}}{2 δ^{2}}],

where σ is the r.m.s. width, δ is the r.m.s. correlation width and index n controls the number of minima and maxima of the source degree of coherence and, hence, the width of the dark center of the beam.

The paraxial form of the extended Huygens-Fresnel principle which describes the interaction of waves with isotropic and homogeneous random medium implies that the elements of the cross-spectral density matrix at two positions r₁ = (ρ₁, z) and r₂ = (ρ₂, z) in the same transverse plane of the half-space z > 0 are related to those in the source plane as [6]

\begin{array}{l} W (ρ_{1}, ρ_{2}, z) = {(k / 2 π z)}^{2} \iint d^{2} ρ_{1}^{'} \iint d^{2} ρ_{2}^{'} W^{(0)} (ρ_{1}^{'}, ρ_{2}^{'}) \\ \times exp {- (i k z / 2 z) [{(ρ_{1} - ρ_{1}^{'})}^{2} - {(ρ_{2} - ρ_{2}^{'})}^{2}]} {〈 exp [ϕ^{*} (ρ_{1}, ρ_{1}^{'}, z) + ϕ (ρ_{2}, ρ_{2}^{'}, z)] 〉}_{M}, \end{array}

here ϕ denotes the complex phase perturbation induced by the random medium and 〈...〉_M denotes averaging over the ensemble of its realizations. Under the strong fluctuation conditions of turbulence, the last term in the integrand of the right-hand side of Eq. (2) can be given by the expression [6]

\begin{array}{l} {〈 exp [ϕ^{*} (ρ_{1}, ρ_{1}^{'}, z) + ϕ (ρ_{2}, ρ_{2}^{'}, z)] 〉}_{M} = exp {- (π^{2} k^{2} z / 3) \\ \times [{(ρ_{1} - ρ_{2})}^{2} + (ρ_{1} - ρ_{2}) (ρ_{1}^{'} - ρ_{2}^{'}) + {(ρ_{1}^{'} - ρ_{2}^{'})}^{2}] \int_{0}^{\infty} κ^{3} Φ_{n} (κ) d κ}, \end{array}

where Φ_n(κ) is the spatial power spectrum of the refractive-index fluctuations of the turbulent medium.

Substituting Eqs. (2) and (3) into Eq. (1) and calculating the resulting integral we arrive at the formulas

W (ρ_{1}, ρ_{2}, z) = \frac{k^{2} σ^{2}}{4 z^{2} Δ (z)} exp [- \frac{{(ρ_{1} - ρ_{2})}^{2}}{R (z)} - \frac{i k}{2 z} (ρ_{1}^{2} - ρ_{2}^{2})] [exp (\frac{γ_{+}^{2}}{Δ (z)}) + exp (\frac{γ_{-}^{2}}{Δ (z)})],

where

\begin{matrix} \frac{1}{R (z)} = \frac{k^{2} σ^{2}}{2 z^{2}} + \frac{k^{2} π^{2} z}{3} I, I = \int_{0}^{\infty} κ^{3} Φ_{n} (κ) d κ, Δ (z) = \frac{1}{R (z)} + \frac{1}{2 ζ^{2}}, \\ \frac{1}{ζ^{2}} = \frac{1}{4 σ^{2}} + \frac{1}{δ^{2}}, γ_{\pm} = (\frac{3 k^{2} σ^{2}}{4 z^{2}} - \frac{1}{2 R (z)}) (ρ_{1} - ρ_{2}) + \frac{i k}{4 z} (ρ_{1} + ρ_{2}) \pm \frac{in \sqrt{2 π}}{2 δ} . \end{matrix}

We will assume that the power spectrum Φ_n(κ) entering Eq. (5) is represented by model in [15, 16], in which the slope 11/3 of the conventional van Karman spectrum is generalized to an arbitrary parameter α, i.e.

Φ_{n} (κ) = A (α) {\tilde{C}}_{n}^{2} exp [- κ^{2} / κ_{m}^{2}] / {(κ^{2} + κ_{0}^{2})}^{α / 2}, 0 \leq κ < \infty, 3 < α < 4,

where κ₀ = 2π/L₀ and κ_m = c(α)/l₀, L₀ and l₀ being the outer and the inner scales of turbulence, and

c (α) = {Γ [(5 - α) / 2] A (α) (2 π / 3)}^{1 / (α - 5)}, A (α) = Γ (α - 1) cos (α π / 2) / (4 π^{2}),

with Γ(·) being the Gamma function. The term

{\tilde{C}}_{n}^{2}

in Eq. (7) is a generalized refractive-index structure parameter with units m³⁻^α. With the power spectrum in Eq. (6) the integral in Eq. (5) becomes [16]

I = \int_{0}^{\infty} κ^{3} Φ_{n} (κ) d κ = \frac{A (α) {\tilde{C}}_{n}^{2}}{2 (α - 2)} [κ_{m}^{2 - α} β exp (\frac{κ_{0}^{2}}{κ_{m}^{2}}) Γ (2 - \frac{α}{2}, \frac{κ_{0}^{2}}{κ_{m}^{2}}) - 2 κ_{0}^{4 - α}],

where

β = 2 κ_{0}^{2} - 2 κ_{m}^{2} + α κ_{m}^{2}

and Γ(·,·) denotes the incomplete Gamma function.

From the cross-spectral function (4), the spectral density S and the spectral degree of coherence μ, of the CGSM beams on passing in the turbulent atmosphere can be calculated by the expressions [1]

S (ρ, z) = W (ρ, ρ, z),

μ (ρ_{1}, ρ_{2}, z) = W (ρ_{1}, ρ_{2}, z) / \sqrt{W (ρ_{1}, ρ_{1}, z)} \sqrt{W (ρ_{2}, ρ_{2}, z)} .

3. Numerical examples

Now we will analyze the behavior of the second-order statistical properties of the CGC beams on propagation in free space and in turbulent atmosphere by a set of numerical examples based on the analytical formulas derived in the previous section. Without loss of generality the parameters of the source and of atmosphere are chosen to be σ = 1 cm, δ = 1 mm, λ = 632.8 nm, ${\tilde{C}}_{n}^{2} = 10^{- 13} m^{3 - α}$ , L₀ = 1 m, l₀ = 1 mm, α = 3.67 unless other values are specified in figure captions.

In Fig. 1 the evolution of the spectral density S calculated from Eq. (9) of several CGC beams with different values of index n, along the optical axis (ρ = 0) is shown. Free-space Fig. 1(a) and atmospheric Fig. 1(b) clearly show the qualitative difference in the beam behavior. While on propagation in free space the on-axis intensity decreases monotonically for all values of n, on propagation in turbulence at the intermediate distances from the source the dark region is formed, but it is gradually filled in at sufficiently large distances, on the order of tens of kilometers. The higher the value of n the more efficiently the dark center is formed: while for n = 0 (classic Gaussian Schell-model beam) and n = 1 the on-axis intensity drop does not occur, for n = 4 it becomes significant. The distances at which local minima and maxima are reached for each order n can be found analytically. Indeed on finding the derivative of S(ρ = 0, z) with respect to z, setting it to zero and making some simplifications one obtains the fourth-order equation of the form

z^{4} + z^{3} (\frac{5}{ζ^{2}} - \frac{π n^{2}}{δ^{2}}) \frac{1}{2 π^{2} k^{2} I} + z^{2} \frac{3}{2 π^{4} k^{4} I^{2} ζ^{4}} + z \frac{2 σ^{2}}{π^{2} I} + \frac{3 σ^{2}}{2 π^{4} k^{2} I^{2}} (\frac{1}{ζ^{2}} - \frac{π n^{2}}{δ^{2}}) = 0.

Using Ferrari’s formulas or numerical methods one finds that for n > 1 apart from the two complex-valued (unphysical) roots Eq. (11) has two positive real roots, the smaller/larger root corresponding to the local intensity minimum/maximum, respectively. For example, for the parameters used for the curves in Fig. 1 one finds that the local minima/maxima of on-axis spectral density S occur at distances z_min(n = 2) = 3891.37 m and z_max(n = 2) = 28210.98 m; z_min(n = 3) = 1304.26 m and z_max(n = 3) = 97622.85 m; z_min(n = 4) = 912.36 m and z_max(n = 4) = 191510.70 m. There are no maxima and minima for orders n < 2.

Fig. 1 The evolution of spectral density S [Eq. (9)] of the CGSM beam on propagation (a) in free space; (b) in the atmosphere: n = 0 (thick solid brown curve), n = 1 (dotted black curve), n = 2 (dashed green curve), n = 3 (dash-dotted red curve) and n = 4 (thin solid blue curve).

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In Fig. 2 the evolution of the whole transverse cross-section of the CGC beams with different values of parameter n is shown at several selected distances z from the source plane. Unlike for mode n = 0, which remains Gaussian for all distances, the beams with n > 0 first form the dark centers, but at larger distances regain the Gaussian shape. The higher the order n the wider and the sooner the ring-shaped profile is formed and remains more robust to optical turbulence. As is seen from Fig. 2(d) plotted for distance z = 1000 km the CGC beams’ intensity still remain a ring-like for n > 1, while the beam with n = 1 reconstructs its original Gaussian profile. We note that the horizontal and the vertical scales of parts (a), (b), (c) and (d) are different addressing the substantial expansion of the beams on propagation.

Fig. 2 The evolution of transverse cross-sections of the spectral density S for the same five CGSM beams as in Fig. 1, for selected distances from the source: (a) z = 0 m; (b) z = 30 m; (c) z = 10³ m and (d) z = 10⁵ m.

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Figure 3 illustrates the behavior of the transverse cross-sections of the beam with n = 2 as a function of α (a) and ${\tilde{C}}_{n}^{2}$ (b) and implies that the influence of both parameters on the intensity distribution is crucial. In Fig. 4 the contour plots of the spectral density as a function of z and ρ are shown in two regions: close to the source (a) and sufficiently far from the source (b). The formation and disappearance of the ring structure is evident.

Fig. 3 The behavior of the transverse cross-sections of the spectral density S for n = 2, z = 10000 m for free space (black curve) and (a) ${\tilde{C}}_{n}^{2} = 10^{- 13}$ and α = 3.1 (blue dash-dotted curve), α = 3.67 (red dashed curve) ; (b) α = 3.1 and ${\tilde{C}}_{n}^{2} = 10^{- 13}$ (blue dash-dotted curve), ${\tilde{C}}_{n}^{2} = 10^{- 14}$ (red dashed curve).

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Fig. 4 Contour plots of the spectral density S as a function of propagation distance z [m] from the source (horizontal axis) and transverse position ρ [m] from the beam axis (vertical axis) for the CGSM beam with n = 4 and α = 3.1.

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We now turn our attention to the changes in the spectral degree of coherence calculated with the help of Eq. (10). Figure 5 illustrates the evolution of the transverse degree of coherence evaluated at separation distance ρ_d between two points ρ₁ = ρ/2 and ρ₂ = −ρ/2 located symmetrically with respect to the optical axis: ρ_d = |ρ₁ − ρ₂|. The degree of coherence is shown for the same five CGC modes as the spectral density in Fig. 1, at several selected distances. In the source plane [Fig. 5(a)] the degree of coherence of the beam differs with parameter n, having Gaussian form for n = 0 and starting to resemble a sinc-function as n increases. For relatively short ranges from the source [Fig. 5(b)] the degree of coherence changes its form to Gaussian for all n. At a distance of approximately 600 m (for the given source and atmospheric parameters) they converge to a single Gaussian distribution [Fig. 5(c)]. This modification can be attributed to the source correlations. Further, the degree of coherence is being qualitatively modified for the second time by the turbulence: as is seen from Figs. 5(d)– 5(f) it goes through another change depending on the mode n, constituting a novel effect in atmospheric propagation. At distances of tens of kilometers the sinc-like profile is formed [Fig. 5(e)], which later converts to Gaussian profile [Fig. 5(f)]. In contrast, it can be readily seen from Fig. 5 that for the GSM beams (n = 0, brown solid thick curve) propagating in the atmosphere the degree of coherence goes through a single cycle: while the Gaussian profile is always preserved its transverse r.m.s. correlation width first grows due to source correlations and then reduces due to turbulence.

Fig. 5 The transverse spectral degree of coherence μ [Eq. (10)] for the same beams as in Fig. 1 as a function of separation between two points at distances: (a) z = 0 m; (b) z = 300 m; (c) z = 600 m; (d) z = 1500 m; (e) z = 10⁴ m and (f) z = 10⁶ m.

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4. Concluding remarks

In this article we have investigated the effect of atmospheric turbulence on propagation characteristics of a recently introduced class of Schell-model beams with Cos-Gaussian degree of coherence (CGC beams). The analytical formula for the cross-spectral density function of such a beam for propagation in the atmosphere with general, non-classic power spectrum of the refractive index fluctuations is derived and used to explore the evolution of the spectral density and the spectral degree of coherence in various atmospheric conditions.

We have found that the CGSM source can produce an annular beam in weak turbulence, robust at least for several kilometers in typical Kolmogorov atmosphere. As the propagation distance grows and/or atmospheric fluctuations become more severe (α reduces and/or ${\tilde{C}}_{n}^{2}$ grows) the hollow beam profile gradually disappears being again converted into a wide Gaussian distribution. Thus on propagation in optical turbulence the CGC beam has the 3D bottle-beam-like structure, convenient for circumventing objects at intermediate distances. The dimensions and the shape of such a 3D optical bottle can be adjusted by the choice of source parameters σ, δ and n.

Another discovered effect that is not pertinent to other deterministic and random beams is the double-cycle qualitative change of the transverse spectral degree of coherence. Namely, for all modes with n > 0, after changing the profile from cos-Gaussian to Gaussian at some intermediate distance, the degree of coherence suffers another round of transformation: it losses the Gaussian profile at distances on the order of several kilometers (to a function resembling a sinc-Gaussian function with low orders) and regains it only at very large distances. The appearance of the double cycle in the evolution of the degree of coherence can be explained by two mechanisms, source correlations and atmospheric turbulence, subsequently affecting the beam. At small propagation distances the beam is mostly modified by the source correlations, leading to the ring-shaped intensity and the Gaussian degree of coherence. In the absence of atmospheric fluctuations this would be the single cycle everywhere in the far field. However, as the beam passes to larger distances through turbulence, the latter modifies the beam intensity distribution from ring back to Gaussian while narrowing the degree of coherence. The combination of these two atmospheric effects produces the second cycle. Note that for classic Gaussian Schell-model beams, when neither intensity nor degree of coherence are modified in shape there is still double modification in the width of the transverse degree of coherence: it first widens due to source correlations and then gradually becomes narrower on propagation at sufficiently long distances in the atmosphere [17]. Thus, our new effect can be regarded as the generalization of the one discussed in [17], relating not to scale change but to shape transformation.

Thus, the analytic and numerical results that we obtained relating to the atmospheric propagation of recently introduced CGC beams [8] revealed that the turbulence can suppress transformation in the intensity profile (e.g. Gaussian to annular), initially formed by source correlations, converting it back (from annular to Gaussian) at large distances. Such complex intensity transformation was found to be accompanied by the double cycle change in the shape of the degree of coherence. Our finding about the reconstruction mechanism of the random beam intensity on propagation in atmospheric turbulence supports the similar results reported for other statistics, such as spectral composition [18] and polarization [19].

Acknowledgments

Z. Mei’s research is supported by the National Natural Science Foundation of China (NSFC) ( 11247004) and Zhejiang Provincial Natural Science Foundation of China ( Y6100605). E. Shchepakina is supported in part by the Russian Foundation for Basic Research (grants 12-08-00069a, 13-01-97002-p) and AFOSR ( FA9550-12-1-0449). O. Korotkova’s research is supported by US AFOSR (FA9550-12-1-0449).

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Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence

Abstract

1. Introduction

2. Analytic solutions for the CGSM propagating in turbulent atmosphere

3. Numerical examples

4. Concluding remarks

Acknowledgments

References and links

Cited By

Figures (5)

Equations (11)

Optics Express