Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence.

Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence Zhangrong Mei1,* and Olga Korotkova2 1 2

Department of Physics, Huzhou Teachers College, Huzhou 313000, China Department of Physics, University of Miami, Coral Gables, FL 33146, USA * [email protected]

Abstract: A recently introduced class of scalar cosine-Gaussian SchellModel [CGSM] beams is generalized to electromagnetic theory. The realizability conditions and the beam conditions on the source parameters are derived. Analytical formulas for the cross-spectral density matrix elements of the electromagnetic cosine-Gaussian Schell-model [EM CGSM] beams propagating in isotropic random medium are derived. It is found that the EM CGSM beams possess single-ring or double-ring intensity profiles, depending of source parameters. As two examples, the statistical characteristics of the EM CGSM beams propagating in free space and non-Kolmogorov turbulent atmosphere are studied numerically. The effects of the fractal constant of the atmospheric spectrum and the refractive-index structure constant on such characteristics are analyzed in detail. ©2013 Optical Society of America OCIS codes: (030.1640) Coherence; (010.1300) Atmospheric propagation; (010.1330) Atmospheric turbulence; (260.5430) Polarization.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995). F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987). F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008). H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004). H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4–6), 379–385 (2005). F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009). 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), 1–9 (2001). O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1–3), 35–43 (2005). 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(25), 16909–16915 (2007). Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 025705 (2013). Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012). Z. Mei, Z. Tong, and O. Korotkova, “Electromagnetic non-uniformly correlated beams in turbulent atmosphere,” Opt. Express 20(24), 26458–26463 (2012). G. Gbur and T. D. Visser, “Can spatial coherence effects produce a local minimum of intensity at focus,” Opt. Lett. 28(18), 1627–1629 (2003).

#197263 - $15.00 USD Received 6 Sep 2013; revised 23 Oct 2013; accepted 25 Oct 2013; published 1 Nov 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.027246 | OPTICS EXPRESS 27246

18. Y. Cai and F. Wang, “Partially coherent anomalous hollow beam and its paraxial propagation,” Phys. Lett. A 372(25), 4654–4660 (2008). 19. M. Alavinejad, G. Taherabadi, N. Hadilou, and B. Ghafary, “Changes in the coherence properties of partially coherent dark hollow beam propagating through atmospheric turbulence,” Opt. Commun. 288, 1–6 (2013). 20. Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013). 21. Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013). 22. H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278(1), 17–22 (2007). 23. G. Zhou and X. Chu, “Propagation of a partially coherent cosine-Gaussian beam through an ABCD optical system in turbulent atmosphere,” Opt. Express 17(13), 10529–10534 (2009). 24. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Some limitations on optical communication reliability through Kolmogorov and non-Kolmogorov turbulence,” Opt. Commun. 283(7), 1229–1235 (2010). 25. E. Wolf, Introduction to the Theories of Coherence and Polarization of Light (Cambridge University, 2007). 26. X. Du and D. Zhao, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express 17(6), 4257–4262 (2009). 27. 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), 65510E (2007). 28. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010). 29. 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. Eng. 47(2), 026003 (2008). 30. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).

1. Introduction Recently the classic family of Gaussian Schell-Model (GSM) sources [1] has been augmented by other models, including Bessel correlated sources [2,3], non-uniformly correlated sources [4], Multi-Gaussian Schell-model (MGSM) sources [5,6], Bessel-Gaussian and LaguerreGaussian Schell-model sources [7], etc. The beams generated by these sources have been found to exhibit very interesting features on propagation. While the scalar sources and beams they radiate are to be specified by a single correlation function, the electromagnetic (EM) random beams are characterized by the 2 × 2 cross-spectral density (CSD) matrices which must satisfy additional restrictions, leading to restrictions for source parameters known as the realizability conditions [8,9]. As a rule, for specific model sources such conditions are not easily derived. The realizability conditions are direct consequence of the non-negative definiteness and the quasi-Hermiticity of the CSD matrix which must be directly employed, or an alternative integral representation [10] must exist. So far only three model random beams have enjoyed a comprehensive extension from scalar to electromagnetic versions: the EM GSM beams [11–13], the EM MGSM beams [14] and the EM non-uniformly correlated beams [15,16]. On the other hand, in recent years dark-hollow beams (DHB) have attracted a wealth of attention because of their wide applications in atomic optics. Also, partially coherent DHB have some advantages over the completely coherent DHB because of their low sensitivity to speckle. Therefore the partially coherent DHB may be more useful in atomic optics experiments involving atomic lenses, atom switches and optical tweezers [17]. Some theoretical models have been proposed to describe partially coherent DHB and analyze their propagation characteristics [18,19]. However, in these models the dark-hollow intensity profiles only remain invariant for short propagation distances. It has been shown that the transverse hollow cross-section of the beam disappears gradually in propagation and becomes Gaussian in the far field. Recently, we have introduced the scalar random sources to generate DHB [20]. Unlike the deterministic DHB models, the dark-hollow intensity profiles of new random sources are formed not at the source plane but in the far field, where they remain shape-invariant. This feature makes them particularly suitable for applications involving particle trapping in cases when the presence of propagation path between the source and the particle cannot be avoided. In this paper, we extend the results of Refs [20,21]. to electromagnetic domain, terming the novel class of beams the Electromagnetic cosine-Gaussian Schell-model (EM CGSM)


beams, in which all the correlations are prescribed with the help of the scalar CGSM distributions. We stress that unlike the existing partially coherent cosine Gaussian beams [22,23], the cosine function is now employed for modeling of the source correlations, rather than for the intensity distribution. That is why the CGSM source leads to the qualitatively distinct evolution in the beam’s spectral density on free-space propagation acquiring a robust dark-hollow intensity profile (see also [20,21]). As we illustrate by numerical examples, similar conclusion can be made regarding the EM CGSM beam’s intensity, coherence and polarization properties, with the only difference that for EM counterpart formation of two rings is also a possibility. The double ring intensity profile may be of use in applications dealing with particle manipulation. Studies of the propagation characteristics of stochastic electromagnetic beam-like fields in a turbulent atmosphere are of importance because of their direct applications in communication and sensing systems [24–29]. After introducing the EM CGSM sources and deriving their realizability and beam conditions our task in this study is to explore the behavior of the major second-order properties of the generated beams which propagate in free-space and in the non-Kolmogorov’s atmospheric turbulence with different fractal constant α of the atmospheric spectrum and refractive-index structure constant C n2 . 2. Electromagnetic cosine-Gaussian Schell-model source The 2 × 2 cross-spectral density (CSD) matrix of a statistically stationary electromagnetic field in the source plane, at points specified by two-dimensional position vectors ρ1′ = ( x1′, y1′) and ρ′2 = ( x2′ , y2′ ) and angular frequency ω , is defined by the expression [25] W (0) (ρ′ , ρ′ ; ω ) Wxy(0) (ρ1′ , ρ′2 ; ω )  Wˆ (0) (ρ1′ , ρ′2 ; ω ) =  xx(0) 1 2 . (0) Wyx (ρ1′ , ρ′2 ; ω ) Wyy (ρ1′ , ρ′2 ; ω ) 

(1)

The matrix elements are scalar correlation functions of the form Wαβ(0) (ρ1′ , ρ′2 ; ω ) =  Eα∗ (ρ1′ ) Eβ (ρ′2 );

(α = x, y; β = x, y ),

(2)

where Eα and Eβ denote the components of the electric field in two mutually orthogonal x and y directions perpendicular to the z -axis, and the angular brackets denote the ensemble average. In what follows the angular frequency dependence of all the quantities of interest will be omitted but implied. A genuine CSD matrix for any electromagnetic stochastic beam must be non-negative definite. This condition is fulfilled if the elements of the CSD matrix it can be written as an integral of the form [10, 15] Wαβ(0) (ρ1′ , ρ′2 ) =  pαβ (v) H α∗ (ρ1′ , v) H β (ρ′2 , v)dv,

(3)

where pαβ (v) is an arbitrary non-negative weight function, H α (ρ′, v) is an arbitrary kernel. A simple and significant class of the CSD matrices, leading to the vectorial Schell-model sources, can be obtained by assigning to functions H α (ρ′, v) a Fourier-like structure. More explicitly, we set: Hα (ρ1′ , v) = Aατ (ρ1′ ) exp(−2π ivρ1′ ),

(4)

H β (ρ′2 , v) = Aβ τ (ρ′2 ) exp(−2π ivρ′2 ),

(5)

where Aα is the amplitude of the field component, τ (ρ′) is a profile function.


The choice of pαβ (v) defines a family of sources with different correlation functions. We now consider a modulation to the conventional Gaussian Schell-model (GSM) coherence functions, by choosing (see also [20,21]) 2 2 pαβ (v) = 2π Bαβ δαβ cos h[n(2π )3/ 2 δαβ v]exp(−2π 2δαβ v − 2n 2π ),

(6)

where n and δ αβ are positive real constants, cosh( x) is the hyperbolic cosine function, Bαβ = Bαβ e

iϕαβ

is the single-point correlation coefficient and δαβ are the characteristic source

correlations. On substituting from Eqs. (4)–(6) into Eq. (5) and setting the Gaussian profile, exp[− | ρ′ |2 /(4σ 2 )] for function τ (ρ′) , one finds the explicit form of the CSD matrix elements:  ρ′2 + ρ′ 2 Wαβ(0) (ρ1′ , ρ′2 ) = Aα Aβ Bαβ exp  − 1 2 2 4σ 

 (ρ′2 − ρ1′ ) 2   n 2π (ρ′2 − ρ1′ )    exp  −  . (7)  cos  2 δαβ 2δαβ     

Equation (7) represents a new family of sources with cosine-Gaussian correlation function that may be named Electromagnetic cosine-Gaussian Schell-model (EM CGSM) sources. Our next step is to establish the restrictions for the source parameters guaranteeing that the mathematical model (7) describes a physically realizable field. From the condition that the (0) correlation matrix must be quasi-Hermitian [1], i.e. that Wαβ(0) (ρ1′ , ρ′2 ) = Wβα (ρ′2 , ρ1′ ) , it follows at once that Bxx = Byy = 1, | Bxy |=| Byx |, δ xy = δ yx .

(8)

Further, the function pαβ(v) must be non-negative definite [10], i.e., pαβ(v) ≥ 0 and pxx (v) p yy (v) − pxy (v) p yx (v) ≥ 0,

(9)

for any v . From Eq. (6), it follows that pαβ(v) is surely nonnegative, and substituting it into Eq. (9) implies that it is satisfied if

δ xx2 δ yy2 cos h[n(2π )3/ 2 δ xx v]cos h[n(2π )3/ 2 δ yy v]exp  −v 2 (δ xx2 + δ yy2 ) / 2  ≥| Bxy |2 δ xy4 {cos h[n(2π )3/ 2 δ xy v} exp ( −v 2δ xy2 ) . 2

(10)

Since function cosh( x) is monotonically increasing but function exp(− x 2 ) is monotonically decreasing functions of their argument the product of two functions is not monotonic. Therefore, it is difficult to obtain an analytical formula for the choice of parameters, but numerical solutions can be readily found. Applying inequality (10), we find several values of δ xy for different values of δ xx , δ yy , | Bxy | and n and summarize the results in Table 1. One can see that the minimum values of δ xy are almost independent of the values of | Bxy | and n , being approximately equal to

(δ xx2 + δ xy2 ) / 2 . This is due to the fact that the dependence of

the exponential function on its argument is quadratic, and, hence for v → ∞ , inequality (10) implies that δ xy ≥ (δ xx2 + δ yy2 ) / 2 . However, the maximum values of δ xy depend on the difference between δ xx and δ yy , as well as on the values of | Bxy | and n . That is, the dynamic range of δ xy values is larger for a smaller difference between δ xx and δ yy , and for smaller values of | Bxy | and n .


We will now derive the conditions that the parameters of the EM CGSM source should satisfy to generate a beam-like field. Recall that the spectral density at a point specified by a position vector r = rs ( s is a unit vector in its direction) in the far zone is given by the expression [25] S ∞ (r ) = (2π k / r ) 2 cos 2 θ W xx(0) (− ks ⊥ , ks ⊥ ) + W yy(0) (−ks ⊥ , ks ⊥ )  ,

(11)

where k is the wave number of the field, s ⊥ is the projection of s onto the source plane, θ is the angle that unit vector s makes with a positive z direction, and Wαα(0) (f1 , f 2 ) =

1

 Wαα (ρ , ρ (0)

(2π ) 4

1

2

) exp[−i (f1 ⋅ ρ1 + f 2 ⋅ ρ 2 )d 2 ρ1d 2 ρ 2 ,

(12)

is the four-dimensional Fourier transform of Wαα(0) . On substituting from Eq. (7) first into Eq. (12) and then into Eq. (11), one find that S ∞ (r ) =

k 2σ 2 cos 2 θ 2r 2

 Ax2  k 2 s⊥2 n 2π −  exp  − 2  4axx 2axxδ xx  axx

 k 2 s⊥2 n 2π exp  − + − 2  a yy  4a yy 2a yyδ yy Ay2

 n 2π ks⊥   cos h    2axxδ xx

  n 2π ks⊥  cos h    2a yy δ yy

  

 ,   

(13)

2 where aαα = 1/ (8σ 2 ) + 1/ (2δαα ) . In order for the matrix Wˆ (0) to generate a beam propagating close to a z axis, the spectral density in Eq. (13) must be negligible except when unit vector s lies in a narrow solid angle about the z axis. Since cosh( x) ≥ 1 for any values of x , Eq. (13) implies that this will be the case if

 k 2 s⊥2 exp  −  4axx

 k 2 s⊥2  0, exp ≈  −    4a yy

 2  ≈ 0, unless s⊥  1, 

(14)

leading to restrictions 4axx  k 2 ,

4a yy  k 2 ,

(15)

or in terms of source parameters 1 4σ 2

+

1

δ xx2



2π 2

λ2

,

1 4σ 2

+

1

δ yy2



2π 2

λ2

(16)

.

We note that the beam conditions expressed by Eq. (16) are the same as that for the classic EM GSM sources [8]. Table 1. Value interval of δxx δyy |Bxy| n min[δxy] max[δxy]

2.5 3.0 0.2 2 2.759 3.330

δ xy

for different values of

2.5 3.0 0.3 2 2.759 3.169

2.5 3.0 0.2 4 2.759 2.905

δ xx , δ yy , | Bxy |

2.5 3.0 0.3 4 2.759 2.862

3.0 4.0 0.2 2 3.535 4.214

and n. 3.0 4.0 0.2 4 3.535 3.669


3. EM CGSM beam propagating in free space and in linear random medium The paraxial form of the Huygens-Fresnel principle which describes the interaction of waves with free space or linear random medium implies that the elements of the cross-spectral density matrix at two positions r1 = (ρ1 , z ) and r2 = (ρ 2 , z ) in the same transverse plane of the half-space z > 0 are related to those in the source plane as [28] 2

 k  2 2 (0) Wαβ (ρ1 , ρ 2 , z ) =     d ρ1′   d ρ′2Wαβ (ρ1′ , ρ′2 ) z 2 π   (17)  ik 2 2  ∗ × exp − [(ρ1 − ρ1′ ) − (ρ 2 − ρ′2 ) ]  exp[φ (ρ1 , ρ1′ , z ) + φ (ρ 2 , ρ′2 , z )] M .  2z 

Here φ denotes the complex phase perturbation due to the random medium and  ⋅⋅⋅ M denotes averaging over the ensemble of its realizations. In the absence of random medium the term in angular brackets reduces to unity. In the presence of random medium, for points located sufficiently close to the optical axis, the last term in the integrand of the righthand side of Eq. (17) is shown to be approximated by the expression [28]  exp[φ ∗ (ρ1 , ρ1′ , z ) + φ (ρ 2 , ρ′2 , z )] M = ∞  π 2k 2 z  (18)  (ρ1 − ρ 2 ) 2 + (ρ1 − ρ 2 )(ρ1′ − ρ′2 ) + (ρ1′ − ρ′2 ) 2   κ 3 Φ n (κ )dκ  , exp −  0 3  

where Φ n (κ ) is the three-dimensional spatial power spectrum of the refractive-index fluctuations of the isotropic turbulent medium. We will investigate the major properties of the EM CGSM beams by discussing numerical examples involving their evolution in free space and in the isotropic, homogeneous turbulent atmosphere governed by statistics described by a model [29] for the power spectrum Ф(к), 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 )C n2 exp[−(κ 2 / κ m2 )] / (κ 2 + κ 02 )α / 2 ,

0 ≤ κ < ∞, 3 < α < 4, (19)

where κ 0 = 2π / L0 and κ m = c(α ) / l0 , L0 and l0 being the outer and the inner scale of turbulence, and c(α ) = [Γ(5 −

α 2

) A(α )

A(α ) = Γ(α − 1) ⋅

2π 1/ (α −5) ] , 3

cos(απ / 2) , 4π 2

(20) (21)

with Γ( x) being the Gamma function. The term C n2 in Eq. (19) is a generalized refractiveindex structure parameter with units m3−α . With the power spectrum in Eq. (19) the integral in Eq. (18) becomes



∞

0

κ 3 Φ n (κ )dκ =

 κ2 A(α )  2  2 −α α κ2 Cn κ m β exp( 02 )Γ(2 − , 02 ) − 2κ 04 −α  , 2(α − 2) 2 κm κm  

(22)

where β = 2κ 02 − 2κ m2 + ακ m2 and Γ denotes the incomplete Gamma function. After substituting Eqs. (7) and (18) into Eq. (17) and calculating the integral we obtain the formula:


Wαβ (ρ1 , ρ 2 , z ) =

k 2σ 2 Aα Aβ Bαβ

 (ρ1 − ρ 2 ) 2   ik 2 2  exp −  exp  − (ρ1 − ρ 2 )  R z z 4 z 2 Δαβ ( z ) ( ) 2    

  γ +2   γ −2   , × exp  + exp    Δ ( z)   Δ ( z )     αβ   αβ 

(23)

where 1 k 2σ 2 k 2π 2 z ∞ 3 = + κ Φ n (κ )dκ , R( z ) 2z2 3 0 Δαβ ( z ) =

1 1 1 + 2+ 2 , R( z ) 8σ 2δαβ

(24) (25)

 3k 2σ 2 1  ik in 2π − (26) .  (ρ1 − ρ 2 ) + (ρ1 + ρ 2 ) ± 2 2 R( z )  4z 2δαβ  4z From the components of the cross-spectral density matrix (23), the spectral density S, the spectral degree of coherence μ, and the spectral degree of polarization P in the turbulent atmosphere are calculated by the expressions [25] ˆ S (ρ, z ) = TrW (ρ, ρ, z ), (27)

γ± = 

μ (ρ1 , ρ 2 , z ) =

TrWˆ (ρ1 , ρ 2 , z ) , TrWˆ (ρ1 , ρ1 , z )TrWˆ (ρ 2 , ρ 2 , z )

P (ρ, z ) = 1 −

ˆ ρ, ρ, z ) 4DetW( , 2 ˆ ρ, ρ, z )   TrW(  

(28)

(29)

where Det and Tr stand for the determinant and the trace of the matrix. Further, the state of polarization of the polarized portion of the beam may be described in terms of the parameters of the polarization ellipse specified by the orientation angle φ and the degree of ellipticity [12]

φ (ρ, z ) = arc tan{2ℜ[Wxy (ρ, z )] / [Wxx (ρ, z ) − Wyy (ρ, z )]} / 2, ε ± (ρ, z ) =

A+ B , A− B

(30) (31)

where A = [Wxx (ρ, z ) − Wyy (ρ, z )]2 + 4 | Wxy (ρ, z ) |2 ,

(32)

B = [Wxx (ρ, z ) − Wyy (ρ, z )]2 + 4 | ℜ[Wxy (ρ, z )] |2 ,

(33)

and ℜ denotes the real part of a complex number.


Fig. 1. Three-dimensional transverse distributions and corresponding contour graphs of the spectral density of an EM CGSM beam with n = 2 at several different propagation distances in free space. (a) z = 0 m; (b) z = 100 m; (c) z = 250 m; (d) z = 400 m.

4. Examples: EM CGSM beams in free space

We will first consider the evolution of the spectral density of the EM CGSM beam in free space. Since the spectral density does not depend on the off-diagonal components of Wmatrix we will discuss the behavior of unpolarized source Ax = Ay = 1 . We also set values of the other source parameters as follows: σ = 1 cm , λ = 632.8nm , δ xx = 1 mm . Figure 1 illustrates typical evolution of the spectral density of the EM GSM beam normalized by its on-axis value in the source plane, in the transverse beam cross-sections, at several distances z from the source plane on propagation in free space with Bxy = 1 , δ yy = 1 cm and n = 2 . One clearly sees that in general the double-ring profile is gradually generated. Two rings correspond to the spectral densities’ distributions of x and y components of the electric field. In order to demonstrate the dependence of the spectral density behavior on index n and on the r.m.s. correlation widths we plot in Fig. 2 its evolution in the transverse beam crosssections, at several distances z from the source plane, for n = 0, 1, 2 and for δ yy = 1 mm, 1cm . Note that for the case δ yy = 1 mm the distribution is the same as for the scalar CGSM beam [20, 21] for all values of n. However, for δ yy = 1 cm the genuine electromagnetic beam is radiated, resulting in the partial of full appearance of the second ring. Also, while mode n = 0 does not lead to substantial deviation from Gaussian profile, starting from n = 1 single or double ring profiles are generated. For values of n > 2 (not shown) still only two rings are generated with maxima occurring at larger radial positions for larger values of n.


Fig. 2. Free-space evolution of the normalized spectral density for different n and

δ yy . Black

solid curves (source plane, z = 0), red dashed curves (z = 100 m), blue dotted curves (z = 500 m) and green dash-dotted curves (z = 1 km).

Figure 3 shows the behavior of the absolute value of the degree of coherence of the EM CGSM beam as a function of the separation half-distance ρ d =| ρ1 − ρ 2 | where the two points are chosen at locations symmetric with respect to the optical axis, i.e. ρ1 = −ρ 2 = ρ 2. The degree of coherence starts as a modulated by cosine function Gaussian distribution but gradually converts to wider Gaussian profile. As figure shows, for low modes n substantial quantitative changes in the degree of coherence start to occur at distances much shorter (< 1 m) than those for the spectral density (>10 m).


Fig. 3. Free-space evolution of degree of coherence for different n and

δ yy .

Black solid

curves (source plane, z = 0), red dashed curves (z = 0.1 m), blue dotted curves (z = 100 m) and green dash-dotted curves (z = 500 m).

In Fig. 4 typical distributions of the polarimetric properties are shown for the most general case of partially correlated electric field components, as beam propagates at 1 km in free space. The values of the source parameters correspond to the first column of Table 1, for δxy attaining its minimum value and while all r.m.s. widths being in mm. The spectral density distribution is provided for comparison. The characteristic feature of all polarization properties in the well-formed beam is the outer dark/bright ring and the dark/bright central spot. The values of the orientation angle Fig. 4(c) and the ellipticity parameter Fig. 4(d) rapidly change in the ring. 5. Examples: EM CGSM beams in atmospheric turbulence

We will now numerically analyze the characteristics of the EM CGSM beams in nonKolmogorov turbulence specified by Eqs. (19)–(21) and tackle their dependence on slope parameter α and structure constant C n2 . The inner and outer scales of the turbulent atmosphere are chosen to be l0 = 10−3 m and L0 = 1m, respectively, and other parameters are specified in figure captions.


Fig. 4. Transverse cross-sections of the spectral density (a) and three polarization properties: the degree of polarization (b), the orientation angle (c) and the degree of ellipticity (d) of the EM CGSM beams at the range 1 km from the source.

Figure 5 shows the transverse distribution of the spectral density of an EM CGSM beam with the same parameters as in Figs. 1 and 2 at propagation distance z = 10 km in the nonKolmogorov turbulence for different values of parameters α and C n2 . The first four parts Figs. 5(a)–5(d) indicate the changes in the Kolmogorov turbulence as the values of C 2 n

increase. For the case Fig. 5(a) of the weakest atmosphere the intensity in the inner ring dominates that in the outer ring, while for next two cases Figs. 5(b) and 5(c) the inner ring is being suppressed first. Finally, in Fig. 5(d) for substantially strong turbulence the inner and the outer rings are suppressed completely and the beams’ intensity resembles Gaussian profile. As is evident from parts Figs. 5(c), 5(e) and 5(f), provided the value of C n2 is kept fixed, the beam’s intensity distribution is the most robust for Kolmogorov case Fig. 5(c) and is destroyed the most when α is in the region of 3.1. This value was also shown to be critical for other beams [6,28].


Fig. 5. Transverse distribution of the spectral density of the same EM CGSM beam as in Fig. 1 propagating in the atmosphere at distance z = 10 km for different values of atmospheric parameters.

Fig. 6. Modulus of the spectral degree of coherence of the same EM CGSM beam as in Fig. 1 on propagation in the atmosphere as a function | ρ d | . Black thick solid curves (source plane, z = 0), red dotted curves (100 m); blue dashed curves (500 m), green dash-dotted curves (2 km); black thin solid curves (z = 10 km).

We will now turn to the analysis of the spectral degree of coherence of the EM CGSM beam (with the same parameters as in Figs. 1 and 2) as it travels in the atmosphere. In Fig. 6 the evolution of the modulus of the degree of coherence, as a function of modulus of


difference vector ρ d = (ρ 2 − ρ1 ) / 2 of two points symmetric with respect to the optical axis, at several propagation distances in the atmosphere for different parameters α and C 2 is shown. n

One sees that there are two effects occurring with the degree of coherence as either the propagation distance or the strength of turbulence increases. At first, the cosine modulation is being suppressed and the degree of coherence converts to Gaussian-like. Then the width of the Gaussian profile decreases under the influence of atmosphere.

Fig. 7. Changes in the polarization ellipse associated with the polarized part of the EM CGSM beam with n = 4 for different parameters α and C n at the plane z = 4 km in the nonKolmogorov turbulence and free space (z = 0 and 4km). The background is the transverse distribution of the spectral density of the beam. 2

Figure 7 shows the changes in the polarization ellipse associated with the polarized part of a typical EM CGSM beam for different parameters α and C n2 at the plane z = 4 km from the source plane. The source field and free space cases also are provided for comparison. The parameters of the beam for this figure have been set to be n = 4, Ax = Ay = 1 , σ = 2cm ,

λ = 632.8nm , δ xx = 2.5 , δ yy = 3 , δ xy = 2.8 , Bxy = 0.2 exp(iπ / 3) , unless other values are specified in figure captions. The background is the transverse distribution of the spectral density of the beam. One can see from this figure that both α and C n2 influence the


distribution of polarization state. More importantly, it is easy to verify that the polarization state on the ring (at positions of maximum intensity) is uniform. 6. Concluding remarks

In this article we have introduced a novel class of stochastic electromagnetic sources, in which the correlations are prescribed with the help of the cosine-Gaussian Schell-model coherence functions. The sufficient realizability conditions and the beam conditions for such sources are derived and analyzed. The analytical formula for the cross-spectral density matrix of the EM CGSM beam on propagation in free space and in linear random medium is derived and used to explore the evolution of its second-order characteristics. We have found that the novel source which can initially have any intensity distribution, say, Gaussian, as in our examples, can produce a robust double-ring intensity distribution in the far field in free place as well as at short distances in the turbulent atmosphere, depending on the values of the refractive-index structure parameter C n2 and the slope α of the turbulence power spectrum. For free-space propagation the double ring profile is preserved for any propagation distances but it is destroyed by the atmosphere. For sufficiently large C n2 and for α in the vicinity of value 3.1, the influence of the atmosphere to light field is the strongest: the inner and outer rings gradually disappear with the increase in the propagation distance and the Gaussian-like distribution is produced. The initial cosine modulation of the spectral degree of coherence is shown to be suppressed to Gaussian profile both in free space and in the atmosphere. The resulting Gaussian profile broadens in width in free space but shrinks in turbulence, just like for other beam classes. The EM CGSM beams can be produced with the help of the interferometric technique involving two spatial light modulators described in Ref [30]. For the EM CGSM the phase correlation functions of the modulators should take forms of Gaussian functions, modulated by cosine functions, i.e.,  ρ′2 − ρ1′ 2   n 2π (ρ′2 − ρ1′ )  exp − ,  2 δαβ 2δαβ    

(0) μαβ (ρ1′ , ρ′2 ) = Bαβ cos 

(34)

instead of being purely Gaussian, as suggested in [30] for generation of EM Gaussian Schellmodel beams. 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). O. Korotkova’s research is supported by US ONR (N00189-12-T-0136) and US AFOSR (FA9550-12-1-0449).


Propagation of Bessel and Airy beams through atmospheric turbulence.

Propagation of electromagnetic stochastic beams in anisotropic turbulence.

Free space self-similar beams.

Measuring the intensity fluctuation of partially coherent radially polarized beams in atmospheric turbulence.

Random wandering of laser beams with orbital angular momentum during propagation through atmospheric turbulence.

Turbulence distance for laser beams propagating through non-Kolmogorov turbulence.

Subluminal group velocity and dispersion of Laguerre Gauss beams in free space.

Effects of source spatial partial coherence on temporal fade statistics of irradiance flux in free-space optical links through atmospheric turbulence.

Random bits, true and unbiased, from atmospheric turbulence.

Laser differential image-motion monitor for characterization of turbulence during free-space optical communication tests.

Field correlations of laser arrays in atmospheric turbulence.

Electromagnetic sinc Schell-model beams and their statistical properties.

Propagation of rotational field correlation through atmospheric turbulence.

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

Note: A balloon-borne accelerometer technique for measuring atmospheric turbulence.

Hanbury Brown-Twiss effect with partially coherent electromagnetic beams.

Diffraction-free beams in fractional Schrödinger equation.

Accelerating diffraction-free beams in photonic lattices.

Upscale energy transfer and flow topology in free-surface turbulence.

Rotation of the polarization direction and reversal of helicity of ultrashort pulsed beams propagating in free space.

Propagation factors of cosine-Gaussian-correlated Schell-model beams in non-Kolmogorov turbulence.

Coherence evolution of partially coherent beams carrying optical vortices propagating in non-Kolmogorov turbulence.

Effects of astigmatism on spectra and polarization of aberrant electromagnetic nonuniformly correlated beams in turbulent ocean.

Effects of non-Kolmogorov turbulence on the spiral spectrum of Hypergeometric-Gaussian laser beams.