1. Introduction

The propagation properties of various laser beams through the turbulent atmosphere have been extensively investigated. It is well known that partially coherent beams are less influenced by turbulent atmosphere. Many studies have been done on the characterization of partially coherent beams propagating through turbulent atmosphere [1–6]. Quite recently a class of stochastic electromagnetic beams became of great interest. With the help of the unified theory of coherence and polarization [7], the propagation properties of stochastic electromagnetic beams through turbulent atmosphere have been investigated [8–13].

On the other hand, a variety of linear, rectangular and radial laser arrays have been developed to achieve high system powers, high-energy weapons, etc. The propagation of various laser arrays in atmosphere has been investigated [14–16]. In this paper, we investigate a two-dimensional (2D) rectangular array beam, composed by electromagnetic Gaussian Schell-model sources, propagating through turbulent atmosphere. Two kinds of superposition, i.e., the correlated superposition and the uncorrelated superposition are considered. The analytical propagation formulas for the array beam are obtained with the help of vector integration. The changes in the spectral density and in the spectral degree of polarization of this class of beams on propagation through turbulent atmosphere are investigated in details.

2. Theory

Suppose that the beam array consists of M×N stochastic electromagnetic Gaussian Schell-model sources, positioned at the plane of z=0 (see Fig. 1). We assume that M and N are odd numbers, the extension to the even numbers is straightforward.

 

Fig. 1. 2D model of the M×N stochastic electromagnetic Gaussian Schell-model sources array.

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2.1. Uncorrelated superposition

Suppose that the beamlets of the array are uncorrelated with each other, then the intensity distribution in the source plane can be considered as the simple superposition of intensity of each beamlet. The elements of the cross-spectral density matrix of the beamlet with subscription m and n centered at the point ρ _0mn can be expressed as

where ${\mathbf{\rho }}_1^{\text{'}}=\left(\begin{array}{c}{x}_1^{\text{'}}\\ y_1^{\text{'}}\end{array}\right),{\mathbf{\rho }}_2^{\text{'}}=\left(\begin{array}{c}{x}_2^{\text{'}}\\ y_2^{\text{'}}\end{array}\right),{\mathbf{\rho }}_{0\mathrm{mn}}=\left(\begin{array}{c}m{x}_0\\ n{y}_0\end{array}\right)$, x ₀ and y ₀ are the separation distances of the beamlets. The coefficients A_i, A_j, B_ij and the variances σ_i, σ_j, σ_ij are independent of position but may depend on frequency [9]. Written in the tensor form, Eq. (1) can be expressed as:

where ${\overline{\mathbf{\rho }}}_{12}^{\text{'}}=\left(\begin{array}{c}{\mathbf{\rho }}_1^{\text{'}}\\ {\mathbf{\rho }}_2^{\text{'}}\end{array}\right),{\overline{\mathbf{\rho }}}_{0\mathrm{mn}}=\left(\begin{array}{c}{\mathbf{\rho }}_{0\mathrm{mn}}\\ {\mathbf{\rho }}_{0\mathrm{mn}}\end{array}\right)$, and M′^-1 _ij is a 4×4 matrix of the following form:

where I is a 2×2 unitary matrix. Then the elements of the cross-spectral density matrix of the laser array source in the plane z=0 can be expressed as:

The propagation formula for the elements of the cross-spectral density matrix in the turbulent atmosphere has the following form [3, 13]:

Consider the Kolmogorov power spectrum, the last term in Eq. (5) can be written as [1, 3, 13]:

with ρ′_d=ρ′₁-ρ′₂, ρ _d=ρ ₁ ρ ₂°ρ ₀(0.545C ² _nk ² z)^-3/5 is the coherence length of a spherical wave propagating through the turbulent medium and C ² _n is the structure parameter of the refractive index. We have employed a quadratic approximation [1] for the Rytov’s phase structure function in order to obtain simple and viewable analytical result.

On substituting from Eq. (6) into Eq. (5) and using the tensor method, we obtain the integral formula for the elements of cross-spectral density matrix in the half-space z>0 as follows [13]:

where ρ̄′^T ₁₂=(ρ′^T ₁,ρ′^T ₂)=(x′₁,y′₁,x′₂,y′₂) and ρ̄′^T ₁₂=(ρ′^T ₁,ρ′^T ₂)=(x′₁,y′₁,x′₂,y′₂) are fourdimensional vectors, T stands for the matrix transposition, Det is the determinant, and

On substituting from Eq. (4) into Eq. (7), and after performing vector integration, the elements of the cross-spectral density matrix of the laser array beam in the half-space z>0 turn out to be

where

2.2 Correlated superposition

Suppose that the beamlets of the laser array are correlated with each other. Similar to the expression for 1D array in [17], the elements of the cross-spectral density matrix of the laser array source can be expressed as:

where ρ _0mn and ρ _0pq are the center positions of two different beamlets, E_i(ρ′₁-ρ _0mn) and E_j(ρ′₂-ρ _0pq) are the field components of two different beamlets. If 〈E ^* _i(ρ′₁-ρ _0mn)E_j(ρ′₂-ρ _0pq)〉=0 for ρ _0mn≠ρ _0pq, Eq. (11) reduces to the expression for uncorrelated superposition. Equation (11) can be written in the form

where ${\overline{\mathbf{\rho }}}_{0\mathrm{mnpq}}=\left(\begin{array}{c}{\mathbf{\rho }}_{0\mathrm{mn}}\\ {\mathbf{\rho }}_{0\mathrm{pq}}\end{array}\right),{{\mathbf{M}}_{\mathrm{ij}}^{\text{'}}}^{-1}$ is given by Eq. (3).

On substituting from Eq. (12) into Eq. (7), one can obtain the elements of the cross-spectral density matrix of the laser array beam in the half-space z>0:

where M ^-1 _2ij has the same form with Eq. (10). With the choice of ρ ₁=ρ ₂=ρ, one can obtain the elements of the cross-spectral density matrix W_ij(ρ,z,ω) at any two coincident points in the half-space z>0. The spectral density and the spectral degree of polarization of the array beam at the point (ρ,z) with ρ̄^T ₁₂=(ρ ^T,ρ ^T) are given by [7, 13, 18]

3. Numerical calculations and analyses

Figure 2 shows the normalized spectral density of the stochastic electromagnetic Gaussian Schell-model array beam composed by uncorrelated superposition (solid curve) and correlated superposition (dashed curve) propagating in free space and through the turbulent atmosphere. Figure 2(a) shows the free space propagation, it can be seen that the normalized spectral density always has fringe pattern on propagation for the correlated superposition, this phenomenon is similar to the fringe generating by multiple-beams interference. However, the normalized spectral density always has a Gaussianlike profile for the uncorrelated superposition. Figure 2(b) shows the propagation through the turbulent atmosphere. It can be seen that the fringe disappears and becomes a Gaussianlike profile when the array beam propagates into the sufficiently far field.

 

Fig. 2. Normalized spectral density for the array beam composed by uncorrelated superposition (solid curve) and correlated superposition (dashed curve) propagating in (a) free space and (b) turbulent atmosphere. The source parameters for each beamlet are: λ=632.8 nm, A_x=2, A_y=1, B_xy=0.2exp(iπ/3), σ_x=1 cm, σ_y=1 cm, σ_xx=σ_yy=3 mm, σ_xy=6 mm. The other parameters are: x ₀=y ₀=σ_y, M=N=3, C ² _n=10^-15m^-2/3.

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The physical reason for the above fringe generating in the case of correlated superposition can be explained as follows: the optical path difference from each beamlet in the source plane to the observation point is stationary, which induces the stationary phase deviation. When the array beam propagates through atmosphere, the random turbulence destroys the wavefront configuration, i.e., disturbs the stationary phase deviation, and then the fringe disappears when the array beam propagates into the sufficiently far field.

Figure 3 shows the changes in the spectral degree of polarization along the z-axis (ρ̄₁₂=0) of an electromagnetic Gaussian Schell-model array beam passing through the turbulent atmosphere. One can find that the spectral degree of polarization has the same value in the source plane or in the sufficiently far field for the two types of superposition, while it has different values on propagation. In Fig. 3(a), σ_i and σ_j are assumed to be equal, the spectral degree of polarization returns to its value in the source plane after the beam propagates a sufficiently long distance. This phenomenon can be considered as a consequence of the fact that the polarization components of the source field have different spatial coherence properties, i.e., the coherence-induced polarization change. In Fig. 3(b), we show the spectral degree of polarization of an array beam generated by the beamlets of anisotropic source, i.e., the source with σ_i≠σ_j. It can be seen that the spectral degree of polarization always changes on propagation even though each beamlet satisfies δ_xx=δ_yy=δ_xy, which can be considered as the anisotropic-induced polarization change. In Fig. 3(c), the source of each beamlet is anisotropic and has different coherent properties of the different components of the electromagnetic field. It can be seen that the spectral degree of polarization in the sufficiently far field approaches a fixed value that is different from the source plane. This phenomenon can be considered as the combinations of the coherence-induced polarization change and the anisotropic-induced polarization change.

 

Fig. 3. Changes in the spectral degree of polarization P along the z-axis of electromagnetic Gaussian Schell-model array beam propagating through the turbulent atmosphere. Solid curves: the uncorrelated superposition. Dashed curves: the correlated superposition. The source parameters are the same as in Fig. 2, but (a) σ_x=σ_y=1 cm, σ_xx=σ_yy=3 mm, σ_xy=6 mm; (b)σ_x=2cm, σ_y=1 cm, σ_xx=σ_yy=σ_xy=6 mm; (c) σ_x=2 cm, σ_y=1 cm, σ_xx=σ_yy=3 mm, σ_xy=6 mm;. The other parameters are: x ₀=y ₀=σ_y, M=N=3, C ² _n=10^-15m^-2/3.

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4. Conclusions

In summary, generalized analytical expressions for the elements of the cross-spectral density matrix of stochastic electromagnetic Gaussian Schell-model array beam composed by uncorrelated superposition and correlated superposition propagating in the homogeneous and isotropic turbulent atmosphere have been derived with the help of vector integration. By using numerical calculations we have studied the changes in the spectral density and the spectral degree of polarization of the stochastic electromagnetic array beams. We found that, for the correlated superposition, the spectral density of the array beam has fringe pattern on propagation, and the atmosphere can destroy the fringe pattern when the array beam propagates into the sufficiently far field. For the uncorrelated superposition, the spectral density of the array beam has Gaussianlike profile in the far field. For the array beam composed by isotropic Gaussian-Schell model sources that satisfy σ_x=σ_y (uniform polarized beamlet sources) and have different correlation coefficients of the x and y components of the field, the spectral degree of polarization in the sufficiently far field returns to the value of the array source; for the array beam composed by anisotropic Gaussian-Schell model sources that satisfy σ_x≠σ_y (non-uniform polarized beamlet sources) and have different or the same correlation coefficients of the x and y components of the field (see Ref. [18], Appendix III), the spectral degree of polarization in the sufficiently far field approaches a fixed value that is different from the source.