1. Introduction

It has been known that polarization properties of stochastic electromagnetic beams, in particular, their spectral degree of polarization, can experience changes on propagation in free space [1–3] and through turbulent atmosphere [4–6]. We have also studied the changes in polarization of stochastic electromagnetic beams propagating through axially nonsymmetrical optical systems [7] and misaligned optical systems [8].

Wolf has shown in his latest work that under sufficiency conditions the degree of polarization of a beam may be the same throughout the far zone and in the source plane [9], and a light beam can be considered as the sum of a completely polarized and a completely unpolarized beam [10]. We are interested in these conclusions and put question to the propagation of completely unpolarized beams and completely polarized beams: will they keep their spectral degree of polarization or become partially polarized?

Recently the polarization invariance of the stochastic electromagnetic Gaussian Schell-model beam has been studied [11], Salem and Wolf have also demonstrated that the coherence of the field in the source plane affects the polarization of the beam on propagation [12]. But both of these investigations are performed under free space propagation condition and the stochastic electromagnetic beams are assumed to be generated by partially polarized sources.

In this paper we analyze the cross-spectral density matrixes of electromagnetic Gaussian Schell-model sources, then the completely unpolarized source and the completely polarized source are given. Our purpose is to find out the key that leads the completely unpolarized or completely polarized stochastic electromagnetic Gaussian Schell-model beams to be partially polarized ones during their propagation not only in free space but also through turbulent atmosphere. Some typical examples are illustrated and necessary explanations are also given for the physical phenomena. All the work will be done within the framework of the paraxial approximation.

 

Fig. 1. Illustrating the notation.

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2. Theoretical analysis

Let us consider a stochastic, statistically stationary, electromagnetic beam generated by a source located in the plane z=0 and propagating into the positive half-space z>0 (see Fig. 1). The second-order correlation properties of such a beam at a pair of points r ₁, r ₂ can be characterized by the 2×2 electric cross-spectral density matrix [13]

where the asterisk denotes the complex conjugate and the angular brackets mean the ensemble average. x and y are two mutually orthogonal directions perpendicular to the beam axis. E=(E_x, E_y) is a statistical ensemble of the fluctuating component of the transverse electric field.

In terms of the cross-spectral density matrix W ^⃡(0) (ρ′₁,ρ′₂ ω) of the source, the cross-spectral density matrix of the beam at r ₁≡(ρ ₁, z) and r ₂≡(ρ ₂,z) in any plane z=const. can be obtained by the formula [14]

where

and G(ρ-ρ′, z,ω) is the Green’s function of the Helmholtz operator for paraxial propagation in free space. If the positive half-space is a medium, which is linear and random but static such as the turbulent atmosphere, Eq. (3) should be replaced by

where 〈…_rm〉 implies that the average is taken over the ensemble of the random medium.

To judge the completely unpolarized or completely polarized beams propagating in free space or through turbulent atmosphere, we should use the formula of the spectral degree of polarization at the point (ρ, z) [13, 14]:

where Det and Tr denote the determinant and the trace of the matrix, respectively.

For the completely unpolarized source, the cross-spectral density matrix can be expressed by [10]

For the completely polarized source, the cross-spectral density matrix has the form [10]

where e_x and e_y are of the general forms of e ₁ and e ₂ which are “one point” constants shown in Ref. [15], Eq. (6.3–12). The sources whose cross-spectral density matrixes are given by Eqs. (6) or (7) can be proved as completely unpolarized P(ρ′,ρ′, z=0,ω)=0 or completely polarized P(ρ′,ρ′, z=0,ω)=1 with the help of Eq. (5).

We now consider stochastic electromagnetic beams generated by electromagnetic Gaussian Schell-model sources. The elements of the cross-spectral density matrix for such sources are given by [3–5]

where the coefficients A_i, A_j, B_ij and the variances σ_i, σ_j, δ_ij are independent of position but may depend on frequency (Ref. [15], Sec. 5.3.2), and B_ii≡1. In order to calculate the spectral degree of polarization in the source plane, Eq. (8) can be simplified into one point (ρ′, z=0) as

For the completely unpolarized source, the parameters should be chosen as A_x=A_y=A, B_xy=0, σ_x=σ_y=σ, and the matrix is obtained as

The spectral correlation width δ_xx and δ_yy can be selected freely. For the completely polarized source, we should choose |B_xy|=1 to make B_xyB_yx=B_xxB_yy. The spectral correlation width δ_ij must correspond the realizable conditions as [16]

so δ_xx, δ_yy and δ_xy should be equal. Other parameters can be selected freely. The matrix is then obtained as

The beam sources characterized by Eqs. (10) or (12) can also be proved as uniformly completely unpolarized or uniformly completely polarized with the help of Eq. (5).

The cross-spectral density matrixes of electromagnetic Gaussian Schell-model sources that are completely unpolarized or completely polarized have been derived, but what will happen on propagation? By using the tensor method, we have investigated the propagation of general stochastic electromagnetic Gaussian Schell-model beams through turbulent atmosphere in our recent work [6] and obtained the following analytical expression for the elements of the cross-spectral density matrix as

where k=2π/λ is the wave number, ρ ₁₂ ^T=(ρ ₁ ^T,ρ ₂ ^T)=(x ₁,y ₁,x ₂,y ₂) is a four-dimensional vector in the output plane, and T means the matrix transpose operation. M′_ij ^-1 is a 4×4 complex matrix characterized by [6–8]

Ī is a 4×4 unitary matrix, and

where I is a 2×2 unitary matrix. ρ ₀(0.545C ² _n k ² _z)^-3/5 is the coherence length of a spherical wave propagating in the turbulent medium based on Kolmogorov spectrum model, and ² _n C is the structure parameter of the refractive index. If C ² _n=0, we can obtain that P̄=0 and Eq. (13) can be simplified as

which represents the condition of propagating in free space.

With the help of Eqs. (16), (13) and (5), we can investigate the changes in the spectral degree of polarization P of stochastic electromagnetic Gaussian Schell-model beams generated by completely unpolarized or completely polarized sources on propagation in free space or through turbulent atmosphere.

3. Numerical calculations and discussions

In this section the criterion for keeping completely unpolarized or completely polarized stochastic electromagnetic Gaussian Schell-model beams on propagation will be derived from both numerical calculation and theoretical analysis.

In Fig. 2, the source is assumed to be uniformly completely unpolarized electromagnetic Gaussian Schell-model source and the beam is propagating in free space. The profile of P across the plane y=0 is chosen, so it is a function of x-coordinate. We can find in Fig. 2(a) that the spectral degree of polarization at the initial plane has the same extreme value, zero. If δ_xx=δ_yy, P will not change when the beam propagates in free space as shown by the solid lines, so we can keep the completely unpolarized beam. This case that P dose not change is in accord with the condition derived in Ref. [11]. If δ_xx≠δ_yy, P will have different distributions at different propagation distances but become changeless in the far-field. Under the condition of δ_xx≠δ_yy, the beam generated by a completely unpolarized source becomes partially polarized on propagation, and it can be considered as coherence-induced polarization changes [12].

 

Fig. 2. Changes in the spectral degree of polarization P as a function of the scaled variable x of the stochastic electromagnetic beam propagating in free space at different propagation distances. The source is assumed to be completely unpolarized electromagnetic Gaussian Schell-model source with the parameters: λ=632.8 nm, A_x=A_y=1, B_xy=0, σ_x σ_y,=1cm, δ_xx=2mm, and δ_yy is shown in the figure.

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In Fig. 3, the beam generated by completely unpolarized electromagnetic Gaussian Schell-model source is passing through turbulent atmosphere. The distributions of P at the initial plane and the near-field shown in Fig. 3(a) and (b) are the same as those shown in Fig. 2(a) and (b), but in the far-field there will be obvious differences. It depends on the effect of the atmospheric turbulence, which occurs with the increase of the propagation distance. We find that, in the propagation environment of turbulent atmosphere, the criterion on δ_xx=δ_yy for keeping the completely unpolarized beam is the same as the condition of free space.

 

Fig.3. . s Fig. 2, but passing through the turbulent atmosphere with C ² _n=10^-12 m^-2/3.

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We will give the necessary theoretical explanation to the physical phenomena shown in Figs. 2 and 3, especially under the condition of the beam propagating through turbulent atmosphere. For the completely unpolarized stochastic electromagnetic beam, if the parameters are chosen as A_x=A_y=A, B_xy=0, σ_x=σ_y, and δ_xx=δ_yy, we can find from Eq. (14) that M′_xx ^-1=M′_yy ^-1, and if the function F_ij(ρ ₁₂, z, ω) is defined as

it is obvious that F_xx(ρ ₁₂,z,ω)=F_yy(ρ ₁₂,z,ω)=F(ρ ₁₂,z,ω), then Eq. (13) can be expressed briefly as

On substituting from Eq. (18) into the formula (5), we obtain

where ρ ₁₂ ^T=(ρ ^T, ρ ^T). The spectral degree of polarization is the same at every point, and the criterion δ_xx=δ_yy for keeping completely unpolarized stochastic electromagnetic beam is derived theoretically. The instance is the same when the beam propagates in free space described by Eq. (16).

 

Fig. 4. Changes in the spectral degree of polarization P as a function of the scaled variable x of the stochastic electromagnetic beam propagating in free space at different propagation distances. The source is assumed to be completely polarized electromagnetic Gaussian Schell-model source with the parameters: λ=632.8 nm, A_x=2, A_y=1, B_xy=exp(iπ/3), σ_x=1cm, δ_xx=δ_yy=δ_xy=2mm, and σy is shown in the figure.

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In Fig. 4, the source is assumed to be uniformly completely polarized electromagnetic Gaussian Schell-model source and Fig. 4(a) shows that the spectral degree of polarization at the initial plane has the other extreme value, one. If σ_x=σ_y, P will not change when the beam propagates in free space, so we can keep the completely polarized beam. If σ_x≠σ_y, the beam will become partially polarized on propagation as shown in Fig. 4(b)–(d). It is known that the parameter σ_i represents the effective width of the spectral density, i.e. that [3–5]

The numerical examples indicate that spectral density properties of the electromagnetic field at the source plane affect the polarization properties of the beam on propagation. We may describe this phenomenon as spectral density-induced polarization changes.

In Fig. 5, we illustrate the beam generated by completely polarized electromagnetic Gaussian Schell-model source passing through turbulent atmosphere. The spectral density-induced polarization changes are also existent.

The theoretical explanation for the physical phenomena shown in Figs. 4 and 5 will also be given. For the completely polarized stochastic electromagnetic beam, if the parameters are chosen en as |B_xy|=1, δ_xx=δ_yy=δ_xy, and σ_x=σ_y, we can find from Eq. (14) that M′_xx ^-1=M′_xy ^-1=M′_yx ^-1=M′_yx ^-1=M′_yy ^-1, then F_xx(ρ ₁₂,z,ω=F_yx(ρ ₁₂,z,ω=F_yy(ρ ₁₂,z,ω)=F(ρ ₁₂,z,ω), Eq. (13) can be expressed briefly as

On substituting from Eq. (21) into the formula (5), we find that

so the spectral degree of polarization

P is the same at every point, and the criterion σ_x=σ_y for keeping completely polarized stochastic electromagnetic beam is derived theoretically. The instance is also the same when the beam propagates in free space described by Eq. (16).

 

Fig. 5. As Fig. 4, but passing through turbulent atmosphere with C ² _n=10^-12 m^-2/3.

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4. Conclusions

In conclusion, to judge the completely unpolarized or completely polarized stochastic electromagnetic Gaussian Schell-model beams propagating in free space or through turbulent atmosphere, we start with the analysis of cross-spectral density matrixes of electromagnetic Gaussian Schell-model sources. We find that the completely unpolarized beams will keep their spectral degree of polarization on propagation if δ_xx=δ_yy in the source plane, and the completely polarized beams will keep their spectral degree of polarization on propagation if σ_x=σ_y in the source plane, otherwise both of them will become partially polarized. The phenomena are explained as coherence-induced polarization changes and spectral density-induced polarization changes. Finally, it is to be noted that these results may be important for many applications, such as tracking, remote sensing, and free-space optical communications.