Does the degree of polarization of vector beams remain unchanged on atmospheric propagation?

Zhiwei Tao; Zhiwei Tao; Zhiwei Tao; Azezigul Abdukirim; Azezigul Abdukirim; Azezigul Abdukirim; Congming Dai; Congming Dai; Congming Dai; Pengfei Wu; Pengfei Wu; Haiping Mei; Haiping Mei; Chuankai Luo; Chuankai Luo; Yunsong Feng; Ruizhong Rao; Ruizhong Rao; Heli Wei; Heli Wei; Yichong Ren

doi:10.1364/OE.502352

1. Introduction

One of the essential properties of light, such as polarization, was proven to be a paramount resource in many cutting-edge experiments [1–7], which offers a proper way to understand the real physical world and remains to be the theme of much fundamental research today. The historical survey of understanding polarization can be dated back to the underlying theory, bearing Stokes’s name, in 1852 [8]. He claims that polarization can be characterized in terms of four intensity parameters or represented by a point on a unit sphere, now known as the Poincare sphere, around the origin [9,10], which leads to a straightforward description of degree of polarization (DOP) [11–14] and provides an elegant geometrical picture to analyze the impacts of polarization transformations [15].

Over past decades, the question of whether beams whose DOP changes on propagation has generated considerable interest because the polarization degree of freedom is easily accessible for manipulating, measuring, and delivering information both theoretically and experimentally [16–22]. In 1967, Clifford and Strohbeln are first concerned with the polarization change of a plane wave propagating through a turbulent medium [23]. They derived an analytical estimate of the mean square polarization fluctuation and pointed out that the condition where the polarization keeps unchanged is that the wavelength of a plane wave is much smaller than the inner scale of turbulence (Schmidt cited their conclusions in his own book [24], who considers the third term of propagation equation in a random medium is the primary factor that causes a change in polarization, see Eq. (9.20) of his book), which paves the way toward the more systematic investigation of vector beams propagation through atmospheric turbulence.

In 2013, Ma et al. extensively investigated the polarization change of a laser beam with linear and circular properties propagating through a horizontal atmospheric layer by using the so-called Monter-Carlo ray-tracing method [25,26]. They view atmospheric turbulence as a series of turbulent eddies with different sizes and refractive indices and specific distribution [27,28]. Based on this assumption, they consider each photon in a beam experiences a couple of reflections and refractions instead of scattering when encountering one turbulent eddy because the wavelength of the vector beam is much smaller than the smallest turbulent eddy (In atmospheric optics, the size of the smallest turbulent eddy is equal to the inner scale of turbulence [29]).

It is worth highlighting that these arguments cannot describe the random nature of the light emission process. For this reason, Wolf presents a statistical framework which bridges the connection between coherence and polarization of random electromagnetic beams [30–33]. He and his students generalized the investigation of DOP for a general partially coherent vector beam on the basis of this statistical method formulated in terms of the cross-spectral density matrix [34–38] and found that DOP can keep unchanged on propagation, including a vacuum channel [39–48] or a turbulent channel [49–53], but it commonly needs to satisfy some specific conditions. In other words, the coherence and polarization theory provide a broader approach to examine the variations in the DOP of both partially coherent and fully coherent light during propagation, beyond the limitations of the plane wave configuration. This allows for the possibility of DOP changes that differ from the assumption that a plane wave maintains a constant DOP throughout propagation. Especially, to investigate the effects of atmospheric turbulence on DOP, they have focused their efforts primarily on how to introduce a random phase into the cross-spectral density matrix by introducing the extended Huygens Fresnel principle [49–53]. In summary, all these conclusions are obtained through the coherence theory of light.

In this article, we aim to find another road that leads to Rome. We propose a novel theoretical framework and demonstrate vector beams whose DOP does not change on atmospheric propagation from another perspective, which is based on the continuity of electromagnetic field [54,55] instead of starting from the propagation equation of a random medium or the coherence theory of light. The main difference between our method and others is that we suggest that vector beams propagation through atmospheric turbulence can be seen as several executions of a couple of procedures consisting of vacuum diffraction, phase modulation, and the reflection and refraction exerted by a single turbulent cell, as depicted in Fig. 1. Concretely, we propose that in the case of a vector beam propagating through the same medium, the thickness of the phase screen can be disregarded, and the phase fluctuations can be considered as part of the wave vector contribution. With this assumption, even when the two homogeneous media are separated by a single turbulent cell and have the same optical properties, it is possible for reflection and refraction to occur without following Snell’s law. This can result in a change in the DOP of the incident vector beam.

Fig. 1. The source generates a single vector beam and sends it through a turbulent atmosphere toward a detector. A turbulent channel is divided into multiple turbulent cells, each cell introduces a random contribution $\varphi$ to the phase, and the effects of reflection and refraction to the vector beam. The process of vector beams propagating between two cells is built by the vectorial diffraction theory. $\Delta z$ represents the diffractive distance. In free-space links, we set $\Delta z$ constant and in satellite-mediated links, we set $\Delta z$ (we employ another notation $\Delta h$ in satellite-mediated calculation instead of $\Delta z$) in terms of the rule of equivalent Rytov-index interval phase screen, see more details in Appendix A. The zoomed-in diagram illustrates the physical image of a vector beam undergoing refraction and reflection after propagating through a phase screen, where the direction of electric field $\mathbf {E}^{\left ( i\right ) }$ is always perpendicular to that of the wave vector $\mathbf {k}^{\left ( i\right ) }$. The process of refraction and reflection is investigated under the $xy$ coordinate and is generated from the change of wave vector because of the introduction of the gradient of the random contribution $\partial _{m}\varphi$ ($m=x,y$). The phase screen array is ladder-shaped distributed because the divergence angle becomes larger as propagation distance increases.

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We become aware of those aforementioned groundbreaking works that advance our knowledge to investigate whether vector beams whose polarization changes when propagating through atmospheric turbulence and make a significantly solid contribution to the community of atmospheric propagation. This leads to crucial inquiries such as the importance of our work and how it differs from prior research efforts. In this regard, we explicate the research significance and divergences between our framework and earlier theoretical explorations by outlining the following points:

• When comparing Ma’s Monte Carlo ray-tracing method [25,26] with our proposed framework, we identify two aspects of their method that needs to be improved: Firstly, the ray-tracing method leads to a highly time consuming and computational complexity because of the calculation demands of each photon and turbulent eddy. Secondly, the ray-tracing method cannot obtain the optical field distribution at the receiver due to the limitation of geometrical optics. Herein, we resort to the idea of the split-step beam propagation method for obtaining the optical field distribution more than the total optical power, phase, and polarization distribution at the receiver (Actually, our framework is the combination of geometrical optics and the physical ones). Moreover, the computational complexity is significantly degraded because we can calculate the polarization properties of vector beams by averaging the results of several realizations of atmospheric propagation (see another perspective in Refs. [56,57]).
• When comparing the work done by Clifford and Strohbeln [23] with our proposed method, there are three key differences to note: Firstly, they started by exploring the vector propagation equation of a random medium (see Eq. (9.20) of Schmidt’s book), whereas our method leverages the idea of the numerical stochastic parabolic equation, namely, the so-called split-step beam propagation method; Secondly, they employed the vector propagation equation together with spectral expansion to derive an analytical form of polarization fluctuation and give a necessary condition where the polarization remains unchanged during atmospheric propagation. In contrast, our method generalizes the continuity of electromagnetic field and consider the polarization change exerted by the refraction and reflection of a phase screen because of the non-vectorization of stochastic parabolic equation. Actually, we introduce the process of refraction and reflection for a phase screen and the vectorial diffraction between two phase screens to make up for the deficiency that the stochastic parabolic equation cannot deal with the problem of vector beam propagation, because one should meet the condition that the wavelength of a vector beam is much smaller than the inner scale of turbulence if one wants to employ the idea of the Fresnel equation to describe the refraction of a phase screen. As such, our theoretical framework is essentially equivalent to that of Clifford and Strohbeln. Lastly, they focused their efforts primarily on the investigation of the polarization change of a plane wave propagating through a turbulent medium, whereas our method can obtain the time-variation of polarization at the receiver for any type of vector beams in the different turbulence strengths regimes (The reason our method is applicable to vector beams of any shape is because we make two key assumptions: we consider only the positive $\mathit {z}$-direction as the propagation direction, and we require that the electromagnetic field satisfies the paraxial approximation. In this context, the wavefront of the beam can be defined or customized to take any desired shape).
• When comparing the coherence and polarization theory [49–53] with our method, it is important to acknowledge the importance and indispensability of coherence theory in the polarization research. This theory unified the investigation of coherence and polarization of random electromagnetic beams, providing a general method to study how vector beams whose degree of freedom of polarization change on atmospheric propagation. Furthermore, we comprehend the intricacy of solving general vectorial polarization problems, whereas the coherence theory can convert the investigation of polarization properties to that of scalar coherence ones by the use of cross-spectral density functions. Lastly, in contrast to previous investigations into the polarization of plane waves, coherence theory enables the exploration of DOP variation for a partially coherent vector beam or random beam during atmospheric propagation. This approach is particularly applicable due to the partially spatial coherence exhibited by a light source. In our proposed method, we take the process of refraction and reflection exerted by a phase screen into consideration and replace the scalar diffraction theory with the vectorial one to compensate for the inability of the conventional split-step method when dealing with a general vectorial polarization problem.

The structure of the main text can be delineated as follows. In Section 2, we provide an introduction to the preliminary knowledge on turbulence description and the split-step beam propagation method for a scalar beam. In Section 3, we initially discuss the theoretical framework pertaining to a generally polarized vector beam propagating through atmospheric turbulence, which is the generalization of the scalar beam propagation method as it incorporates the vectorial diffraction between two turbulent cells and the refractive characteristics of a single one. Additionally, we carry out an analytical derivation of the reflection and refraction processes that occur on a random phase screen on the basis of the continuity of the electromagnetic field, which is the key step of our propagation model. Frequently asked skepticisms regarding our theoretical framework and a more detailed explanation of how our model handles the process of refraction and its analogy to the Fresnel equations are provided in Section 4. In Section 5, we apply our theoretical framework to explore the propagation of vector beams across atmospheric turbulence, both in free-space links and satellite-mediated links and quantify the change of DOP for different parameters and parameter settings. Moreover, we also provide a reliability certification of our theoretical approach by the comparison of historical experimental data obtained in an urban-to-campus and a satellite-to-ground laser communication environment. Finally, in Section 6, we summarize our findings.

2. Preliminary

2.1 Turbulence description

The inhomogeneity and anisotropy of temperature and pressure in the atmosphere give rise to stochastic fluctuations in the refractive index along the light beam’s propagation path. These variations in the refractive index degrade the beam profile and steer the beam off axis, both of which dramatically affects the quality of vector beams at the receiver. In order to introduce our propagation model for a generally polarized vector beam, an in-depth understanding of the characteristics of the turbulent media is of utmost importance, either its connection to the models of atmosphere.

A standard theory of turbulence has been established by the Russian mathematician Kolmogorov on the basis of probability statistics, which assumes that the statistical properties of the atmospheric turbulence are homogeneous and isotropic within a specific scale [58–60]. Under this assumption, the stochastic statistical characteristics of turbulence can be characterized by the refractive index power density spectrum, which illustrates the distribution of kinetic energy of atmospheric turbulence across different frequencies. Moreover, this spectrum is directly associated with the phase fluctuations of a beam during atmospheric propagation path [61] and is subsequently employed to generate random phase screens by using the well-known subharmonic-compensation-based fast-Fourier-transform algorithm [24,62–64]. In this article, we realize the corresponding random phase screens by employing the von-Karman spectrum of refractive-index fluctuation, which is implemented on the python library named AOtools [65].

2.2 Split-step beam propagation method

A scalar beam propagation through a turbulent channel is depicted by the paraxial approximation to the wave equation, which is under the assumption that the depolarization effects are negligible on atmospheric propagation [66]:

(1)$$\nabla _{{\perp} }^{2}E+2i\left\vert \mathbf{k}\right\vert \partial_{z}E+2\left\vert \mathbf{k}\right\vert ^{2}\delta nE=0$$

where $E\left ( \mathbf {r}\right )$ represents a scalar electromagnetic field, $\mathbf {r}\equiv \left ( x,y,z\right )$, $\delta n=n-\left \langle n\right \rangle$, $\left \vert \mathbf {k}\right \vert =2\pi /\lambda$ is the wave number. $\lambda$ is the wavelength, $\nabla _{\perp }^{2}$ represents the Laplacian for transverse field, $n$ is the refractive index of the turbulent channel and $\left \langle n\right \rangle$ donates to the spatial average value of $n$. Equation (1) can be solved numerically using the Fourier split-step methods for its accuracy and speed [67]. The split-step solution of $E$ at $z+\Delta z$ can be expressed as:

(2)$$E\left( z+\Delta z\right) =e^{i\varphi }\mathcal{F}^{{-}1}\left\{ E\left( z,\left\vert \mathbf{k}\right\vert _{{\perp} }\right) e^{{-}i\left\vert \mathbf{k}\right\vert _{{\perp} }^{2}\Delta z/2\left\vert \mathbf{k}\right\vert}\right\}$$

where $\varphi =\left \vert \mathbf {k}\right \vert \int _{z}^{z+\Delta z}\delta n\left ( \mathbf {r}\right ) dz$ represents the phase difference between $z$ and $z+\Delta z$, $\left \vert \mathbf {k}\right \vert _{\perp }=\sqrt { k_{x}^{2}+k_{y}^{2}}$ is the wave number in the transverse plane. From Eq. (2), it can be seen that a turbulent channel can be regarded as a series of regularly spaced turbulent cells, the effects of turbulence on propagation can be split into several iterations of the same modulation. In other words, each cell introduces a random contribution $\varphi$ to the phase, but essentially no change in the amplitude; besides, intensity fluctuations build up by diffraction over many cells. It is for this reason that this model is also known as the multiple phase screen method and each random contribution of phase fluctuations is generated by the use of power spectrum of turbulence mentioned above.

3. Theory

It is worth noting that the conventional theoretical framework of atmospheric propagation is an incomplete characterization for vector beams. In other words, the split-step method will lose its usefulness when studying the influence of turbulence on a vector beam. Hence, if one wants to investigate whether the DOP of beams changes by employing the idea of this method, a significant change is needed for compensating the inability of the split-step beam propagation method to deal with a general vectorial polarization problem. Here, we adopt the wave refraction and together with the vectorial diffraction theory [68,69] to construct the atmospheric propagation model for a general polarized vector beam. The conceptual diagram of a vector beam propagating through turbulence that is depicted by our theoretical model is given in Fig. 1. Without loss of generality, we summarize and divide the procedure of this model into two steps as follows: Firstly, we decompose the electric field of a general vector beam into horizontal and vertical components. Secondly, for each component of the electric field, the modulation by the random phase and refraction function of the phase screen and the subsequent diffraction together are repeated several times during the simulation. The crucial procedure in the above model that leading to a change in the DOP of electric field is the several refractions of multiple phase screen.

To quantify the adverse impact of phase screen on DOP, we report the refraction of a single random phase screen for the incident electric field $\mathbf {E}\left ( \mathbf {r}\right )$ of wave vector $\mathbf {k}$. Generally, the relationship between $\mathbf {k}$ and $\phi \left ( \mathbf {r}\right )$ obeys

(3)$$\mathbf{k}=\nabla \phi \left( \mathbf{r}\right)$$

with $\phi \left ( \mathbf {r}\right )$ denoting the phase of one components of $\mathbf {E}\left ( \mathbf {r}\right )$, where $\nabla$ stands for the Laplacian for the whole field. More explicitly, if we assume that $E_{x}\left ( \mathbf {r}\right )$ and $E_{y}\left ( \mathbf {r}\right )$ represent the horizontal and vertical components of $\mathbf {E}\left ( \mathbf {r}\right )$ and indentify $\phi \left ( \mathbf {r}\right )$ as one of arguments of $\mathbf {E}\left ( \mathbf {r}\right )$ (i.e., $\phi \left ( \mathbf {r}\right ) =\arg \left ( E_{x}\left ( \mathbf {r}\right ) \right )$ or $\arg \left ( E_{y}\left ( \mathbf {r}\right ) \right )$), then $\mathbf {k}=\left ( k_{x},k_{y},k_{z}\right )$ with $k_{m}=\partial _{m}\phi$ ($m=x,y$), $k_{z}^{2}=\left \vert \mathbf {k}\right \vert ^{2}-k_{x}^{2}-k_{y}^{2}$. As shown in Fig. 1, supposing the positive direction of $z$ as the propagation direction, Eq. (3) can be further re-evaluated in terms of incident, reflected and refracted components

(4)$$\begin{aligned} \mathbf{k}^{\left( i\right) } &=\left( k_{x}^{\left( i\right) },k_{y}^{\left( i\right) },\sqrt{\left\vert \mathbf{k}\right\vert ^{2}-k_{x}^{\left( i\right) 2}-k_{y}^{\left( i\right) 2}}\right)\\ \mathbf{k}^{\left( r\right) } &=\left( k_{x}^{\left( r\right) },k_{y}^{\left( r\right) },-\sqrt{\left\vert \mathbf{k}\right\vert ^{2}-k_{x}^{\left( r\right) 2}-k_{y}^{\left( r\right) 2}}\right) \\ \mathbf{k}^{\left( t\right) } &=\left( k_{x}^{\left( t\right) },k_{y}^{\left( t\right) },\sqrt{\left\vert \mathbf{k}\right\vert ^{2}-k_{x}^{\left( t\right) 2}-k_{y}^{\left( t\right) 2}}\right) \end{aligned}$$

with the subscripts $i$, $r$, $t$ referring to incident, reflected and refracted components, respectively, where $k_{m}^{\left ( l\right ) }$ ($m=x,y$, $l=i,r,t$) represent the partial derivative of one phase components of $\mathbf {E}^{\left ( l\right ) }\left ( \mathbf {r}\right )$ (i.e., $\arg \left ( E_{x}^{\left ( l\right ) }\left ( \mathbf {r}\right ) \right )$ or $\arg \left ( E_{y}^{\left ( l\right ) }\left ( \mathbf {r}\right ) \right )$) in $x$ and $y$ directions, namely, $k_{m}^{\left ( l\right ) }=\partial _{m}\phi ^{\left ( l\right ) }$, $k_{z}^{\left ( l\right ) }=\sqrt {\left \vert \mathbf {k} \right \vert ^{2}-k_{x}^{\left ( l\right ) 2}-k_{y}^{\left ( l\right ) 2}}$. Notably, we can see in Eq. (4) that we consider the two homogeneous media split by a single phase screen has the same optical properties (i.e., $n_{i}$, $n_{t}$ have same portion with average value of unity, where $n_{l}$ ($l=i,r,t$) denotes the refractive index of two homogeneous media) and the incident and refracted field have the same wavelength (see more detailed explanations for using this assumption in the following text). However, it is the well-known result in the Fresnel equations that the two homogeneous media has the different optical properties, which means that reflection and refraction are functions of the angle of incidence of the incoming wave with respect to some surface of discontinuous change of the index of refraction [70]. Hence, if we follow the idea of the Fresnel equations, a vector beam propagating through a phase screen may undergo two processes of refraction, namely, if we assume that a thin phase screen has a certain thickness, then the beam will undergo two refractions when propagating through the boundary of phase screen, see Fig. 2(a). In this circumstance, we should calculate two processes that a vector beam propagates from vacuum to the media of phase screen because of the fluctuation of refraction index in a turbulent media and from the turbulent media to vacuum. The refractive index of the turbulent media can be specified with $1+ {\lambda } {\varphi }/2\pi d$, where $d$ is the thickness of a phase screen. Unlike the Fresnel equations, we make an assumption, see Fig. 2(b), that the thickness of a single phase screen is negligible and incorporate the phase fluctuation of a turbulent media into the contribution of a wave vector instead of adopting $k_{m}^{\left ( t\right ) }=k_{m}^{\left ( i\right ) }$ ($m=x,y$) in the Fresnel equations, then the change of the wave vector direction and the magnitudes of its individual components caused by a phase screen is the main reason for the process of reflection and refraction (note that the wave number stays the same before and after the refraction process). Considering the phase modulation effect exerted by a single phase screen, we suggest $k_{m}^{\left ( t\right ) }$ and $k_{m}^{\left ( i\right ) }$ must obey the following relationship:

(5)$$k_{m}^{\left( t\right) }=k_{m}^{\left( i\right) }+\partial _{m}\varphi$$

where the above equation only holds on the transverse plane instead of $z$ direction, which is mainly because of the same optical properties of two homogeneous media. Here, Eq. (5) represents the wave vector components in the $x$ and $y$ directions, respectively, that undergo changes. If, at this point, the wave vector component in the $z$ direction remains unchanged, it will inevitably result in a disparity in the wave numbers before and after refraction. This would lead to a shift in the frequency of the vector beam, creating an inherent contradiction. Therefore, to maintain the frequency of the vector beam unaltered before and after refraction, it becomes imperative for the $z$-direction component of the wave vector to play a reconciliatory role. In reality, owing to the continuity of the electromagnetic field in the time direction, it is assured that the wave number remains unaltered both before and after refraction (because the two homogeneous media share the same refractive index, and the continuity of the electromagnetic field in the time direction guarantees that the beam frequency stays the same). In summary, the wave vectors $\mathbf {k}^{\left ( t\right ) }$ and $\mathbf {k}^{\left ( i\right ) }$ exhibit different directions due to the consistent wave number before and after refraction, the mediating role of the $z$-direction component of the wave vector, and the alteration of individual components of the wave vector. This is the fundamental reason why our theoretical framework, which assumes that two homogeneous media with the same refractive index, can still lead to the process of refraction. By means of the orthogonal formula $\mathbf {E} ^{\left ( l\right ) }\left ( \mathbf {r}\right ) \cdot \mathbf {k}^{\left ( l\right ) }=0$ ($l=i,r,t$, note that the orthogonal formula strictly holds because the direction of electric field is always perpendicular to that of the wave vector), we can express the incident, reflected and refracted components of $\mathbf {E}\left ( \mathbf {r}\right )$ as

(6)$$\begin{aligned} \mathbf{E}^{\left( i\right) } &=\left( E_{x}^{\left( i\right) },E_{y}^{\left( i\right) },-\left( E_{x}^{\left( i\right) }k_{x}^{\left( i\right) }+E_{y}^{\left( i\right) }k_{y}^{\left( i\right) }\right) /k_{z}^{\left( i\right) }\right)\\ \mathbf{E}^{\left( r\right) } &=\left( E_{x}^{\left( r\right) },E_{y}^{\left( r\right) },\left( E_{x}^{\left( r\right) }k_{x}^{\left( r\right) }+E_{y}^{\left( r\right) }k_{y}^{\left( r\right) }\right) /k_{z}^{\left( r\right) }\right) \\ \mathbf{E}^{\left( t\right) } &=\left( E_{x}^{\left( t\right) },E_{y}^{\left( t\right) },\left( E_{x}^{\left( t\right) }k_{x}^{\left( t\right) }+E_{y}^{\left( t\right) }k_{y}^{\left( t\right) }\right) /k_{z}^{\left( t\right) }\right) \end{aligned}$$

where $k_{z}^{\left ( l\right ) }=\sqrt {\left \vert \mathbf {k}\right \vert ^{2}-k_{x}^{\left ( l\right ) 2}-k_{y}^{\left ( l\right ) 2}}$ ($l=i,r,t$). It is worth noting that because the vector beam, after refraction by the phase screen, needs to undergo subsequent vacuum diffraction before being used again as the incident field interacting with the next phase screen for refraction, it remains crucial to determine whether the wave vector of the refracted field satisfies the paraxial approximation. For more in-depth information on this matter, please refer to Ref. [71]. Other than that, the components of magnetic field are obtained by combining the relation $\mathbf {H}=n\left \vert \mathbf {k}\right \vert ^{-1}\mathbf {k}\times \mathbf {E}$ ($n$ represents the refractive index) and Eq. (6), we give

(7)$$\begin{aligned} \mathbf{H}^{\left( i\right) } &=\frac{1}{\left\vert \mathbf{k}\right\vert k_{z}^{\left( i\right) }}\left( \begin{array}{c} -\left[ E_{x}^{\left( i\right) }k_{x}^{\left( i\right) }k_{y}^{\left( i\right) }+E_{y}^{\left( i\right) }\left( \left\vert \mathbf{k}\right\vert ^{2}-k_{x}^{\left( i\right) 2}\right) \right] \\ E_{y}^{\left( i\right) }k_{x}^{\left( i\right) }k_{y}^{\left( i\right) }+E_{x}^{\left( i\right) }\left( \left\vert \mathbf{k}\right\vert ^{2}-k_{y}^{\left( i\right) 2}\right) \\ \left( E_{y}^{\left( i\right) }k_{x}^{\left( i\right) }-E_{x}^{\left( i\right) }k_{y}^{\left( i\right) }\right) k_{z}^{\left( i\right) } \end{array} \right)\\ \mathbf{H}^{\left( r\right) } &=\frac{1}{\left\vert \mathbf{k}\right\vert k_{z}^{\left( r\right) }}\left( \begin{array}{c} E_{x}^{\left( r\right) }k_{x}^{\left( r\right) }k_{y}^{\left( r\right) }+E_{y}^{\left( r\right) }\left( \left\vert \mathbf{k}\right\vert ^{2}-k_{x}^{\left( r\right) 2}\right) \\ -\left[ E_{y}^{\left( r\right) }k_{x}^{\left( r\right) }k_{y}^{\left( r\right) }+E_{x}^{\left( r\right) }\left( \left\vert \mathbf{k}\right\vert ^{2}-k_{y}^{\left( r\right) 2}\right) \right] \\ \left( E_{y}^{\left( r\right) }k_{x}^{\left( r\right) }-E_{x}^{\left( r\right) }k_{y}^{\left( r\right) }\right) k_{z}^{\left( r\right) } \end{array} \right) \\ \mathbf{H}^{\left( t\right) } &=\frac{1}{\left\vert \mathbf{k}\right\vert k_{z}^{\left( t\right) }}\left( \begin{array}{c} -\left[ E_{x}^{\left( t\right) }k_{x}^{\left( t\right) }k_{y}^{\left( t\right) }+E_{y}^{\left( t\right) }\left( \left\vert \mathbf{k}\right\vert ^{2}-k_{x}^{\left( t\right) 2}\right) \right] \\ E_{y}^{\left( t\right) }k_{x}^{\left( t\right) }k_{y}^{\left( t\right) }+E_{x}^{\left( t\right) }\left( \left\vert \mathbf{k}\right\vert ^{2}-k_{y}^{\left( t\right) 2}\right) \\ \left( E_{y}^{\left( t\right) }k_{x}^{\left( t\right) }-E_{x}^{\left( t\right) }k_{y}^{\left( t\right) }\right) k_{z}^{\left( t\right) } \end{array} \right) \end{aligned}$$

It is well-known that boundary conditions of electromagnetic field demand that across the boundary the tangential components of $\mathbf {E}$ and $\mathbf {H}$ should be continuous, namely [38,39]

(8)$$\begin{aligned} E_{x}^{\left( i\right) }+E_{x}^{\left( r\right) } &=E_{x}^{\left( t\right) },E_{y}^{\left( i\right) }+E_{y}^{\left( r\right) }=E_{y}^{\left( t\right) }\\ H_{x}^{\left( i\right) }+H_{x}^{\left( r\right) } &=H_{x}^{\left( t\right) },H_{y}^{\left( i\right) }+H_{y}^{\left( r\right) }=H_{y}^{\left( t\right) } \end{aligned}$$

Hence, substituting Eq. (3) and Eq. (4) into Eq. (5) and undergoing a series of algebraic operations, we can immediately calculate the relationship between $E_{m}^{\left ( i\right ) }$ and $E_{m}^{\left ( t\right ) }$ ($m=x,y$)

(9)$$\begin{aligned} E_{x}^{\left( t\right) } &=\frac{M_{12}M_{23}-M_{13}M_{22}}{ M_{11}M_{22}-M_{12}M_{21}}\\ E_{y}^{\left( t\right) } &=\frac{M_{13}M_{21}-M_{11}M_{23}}{ M_{11}M_{22}-M_{12}M_{21}} \end{aligned}$$

where

(10)$$M_{11}=M_{22}=\frac{1}{\left\vert \mathbf{k}\right\vert k_{z}^{\left( i\right) }}k_{x}^{\left( i\right) }k_{y}^{\left( i\right) }+\frac{1}{ \left\vert \mathbf{k}\right\vert k_{z}^{\left( t\right) }}k_{x}^{\left( t\right) }k_{y}^{\left( t\right) }$$

(11)$$M_{12}=F\left( k_{x}^{\left( i\right) },k_{x}^{\left( t\right) }\right) ,M_{21}=F\left( k_{y}^{\left( i\right) },k_{y}^{\left( t\right) }\right)$$

(12)$$M_{13}={-}\frac{2}{\left\vert \mathbf{k}\right\vert k_{z}^{\left( i\right) }} \left[ E_{x}^{\left( i\right) }k_{x}^{\left( i\right) }k_{y}^{\left( i\right) }+E_{y}^{\left( i\right) }\left( \left\vert \mathbf{k}\right\vert ^{2}-k_{x}^{\left( i\right) 2}\right) \right]$$

(13)$$M_{23}={-}\frac{2}{\left\vert \mathbf{k}\right\vert k_{z}^{\left( i\right) }} \left[ E_{y}^{\left( i\right) }k_{x}^{\left( i\right) }k_{y}^{\left( i\right) }+E_{x}^{\left( i\right) }\left( \left\vert \mathbf{k}\right\vert ^{2}-k_{y}^{\left( i\right) 2}\right) \right]$$

with

(14)$$F\left( k_{1},k_{2}\right) =\frac{1}{\left\vert \mathbf{k}\right\vert k_{z}^{\left( i\right) }}\left( \left\vert \mathbf{k}\right\vert ^{2}-k_{1}^{2}\right) +\frac{1}{\left\vert \mathbf{k}\right\vert k_{z}^{\left( t\right) }}\left( \left\vert \mathbf{k}\right\vert ^{2}-k_{2}^{2}\right)$$

Fig. 2. A schematic diagram illustrating the two methods for handling refraction and reflection of a phase screen. (a) The refraction and reflection of a phase screen follow the rules of Fresnel equations, assuming a certain thickness of the phase screen. In this regime, a vector beam propagates through the phase screen undergoes two refraction processes: from vacuum to the phase screen medium, and from the phase screen medium back to vacuum. The refracted field and incident field are related through the equations $k_{m}^{\left ( t\right ) }=k_{m}^{\left ( i\right ) }$ ($m=x,y$) and Snell’s law. If we assume the thickness of the phase screen is $d$, the refractive index of the phase screen medium is given by $1+ {\lambda } {\varphi } /2\pi d$. (b) The refraction and reflection processes of a phase screen arise from variations in the wave vector direction and the magnitudes of its individual components due to the phase fluctuation $\varphi$ (note that the wave number remains unchanged before and after the refraction process). The thickness of the phase screen can be neglected, and the refracted field is related to the incident field through the equation $k_{m}^{\left ( t\right ) }=k_{m}^{\left ( i\right )}+\partial _{m}\varphi$ ($m=x,y$).

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So far, we have quantified the change in the electric field of $\mathbf {E} ^{\left ( i\right ) }$ after refraction through the phase screen, which is the key step to determine whether DOP changes after the electric field is propagated through turbulence. It is important to stress that during the above calculation, either $E_{x}^{\left ( i\right ) }$ or $E_{y}^{\left ( i\right ) }$ of the electric field needs to be set to zero due to the condition $\phi \left ( \mathbf {r}\right ) \equiv \arg \left ( E_{x}\right )$ or $\arg \left ( E_{y}\right )$. In other words, if we identify $\phi ^{\left ( i\right ) }\equiv \arg \left ( E_{x}^{\left ( i\right ) }\right )$, we should set $E_{y}^{\left ( i\right ) }$ equals to zero (i.e., $E_{y}^{\left ( i\right ) }=0$ in Eq. (6)) when calculating the refracted field of $E_{x}^{\left ( i\right ) }$. Consequently, the horizontal component of refracted field is, in fact, a superposition of $x$ component generated by $E_{x}^{\left ( i\right ) }$ and $E_{y}^{\left ( i\right ) }$ (i.e., $E_{x}^{\left ( t\right ) }=E_{x\rightarrow x}^{\left ( t\right ) }+E_{y\rightarrow x}^{\left ( t\right ) }$, where $E_{x\rightarrow x}^{\left ( t\right ) }$ and $E_{y\rightarrow x}^{\left ( t\right ) }$ represent $x$ component of refracted field calculating in the $E_{y}^{\left ( i\right ) }=0$ and $E_{x}^{\left ( i\right ) }=0$ regime, respectively). This can be easily demonstrated from Eq. (9) that even if the incident light is polarized solely in the direction of the $x$ or $y$ coordinate axis, the resulting propagated light, refracted by a phase screen, still exhibits two polarization components along the $x$ and $y$ directions. After obtaining the refracted and reflected field in the transverse plane, we add the random contribution $\varphi$ to the phase of the refracted field and establish intensity modulation to the refracted field between two turbulent cells through vectorial diffraction theory.

4. Discussions

4.1 Frequently asked questions

We would like to address some technical details regarding our theoretical framework that may raise doubts. Firstly, our framework proposes that the polarization change of vector beams is predominantly caused by the reflection and refraction processes exerted by multiple phase screens. This assertion may be met with skepticism by some readers. It is theoretically relevant to ask whether the reflection of radiance by each phase screen implies a loss of radiance as the beam propagates through the screens, which would be in contradiction with the assumption of atmospheric transparency. To assuage any doubts held by our readers, we offer the following two explanations:

• It is widely recognized that a single realization of beam propagating through atmospheric turbulence is a unitary process, which means that if we only consider the process of interaction between a beam and atmospheric turbulence, there is no energy loss during the propagation of the beam [73,74] (see the proof in Appendix B for the unitarity of turbulence operator). Hence, the primary factor that leads to atmospheric transmittance being less than unity is the process of reflection and refraction exerted by multiple phase screens. As an example, we present a simulation that a general vector beam propagating through the atmospheric turbulence which is divided into 5 equally spaced turbulent cells and calculate the intensity reflectance of this vector beam as a function of the number of turbulence realization, as depicted in Fig. 3. We found that the average intensity reflectance is almost close to zero under different realizations of turbulence, which means that the reflection of multiple phase screens cannot lead to the loss of radiance as it propagates through these phase screens.
• It is worth highlighting that our proposed method for a general vector beam propagating through atmospheric turbulence leverages the idea of the numerical stochastic parabolic equation together with the continuity of electromagnetic field. Additionally, our proposed method suggest that the random fluctuation caused by atmospheric turbulence can be attributed to the introduction of a random phase screen, which may lead to a significant change of wave vector. Hence, if the atmosphere is always transparent, namely, there is no reflection between two homogeneous media, then the law of conservation of energy and the continuity conditions of electromagnetic field would not hold, which is obviously contradictory to our understanding of the physical world.

Fig. 3. An example of a general vector beam propagating through the atmospheric turbulence which is divided into 5 equally spaced turbulent cells, where the intensity reflectance $R$ of the vector beam is calculated under different realizations of turbulence and $N$ is the number of turbulence realization.

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As such, from these two perspectives, we suggest that although the effect of atmospheric turbulence on beam propagation can be considered a unitary transformation, it does not imply that the atmosphere is always transparent. Rather, it signifies that the average atmospheric transmittance is significantly close to unity. Secondly, our theoretical framework calculates the process of refraction and reflection under the condition that the two homogeneous media split by a single phase screen has the same optical properties, namely, $n_{i}=n_{t}=1$. Nevertheless, it is well-known that reflection and refraction are functions of the angle of incidence of the incoming wave with respect to some surface of discontinuous change of the index of refraction. Then, one may subconsciously realize the serious problem and pose another theoretical important question: how does the phase screen reflect the portion of radiance between the same medium? We hereby provide an explanation as to why the same medium can indeed lead to noteworthy reflections:

• It is the well-known results that the Fresnel equations is derived on the basis of the continuity conditions of electromagnetic field and the fundamental Snell’s refraction law. It means that if we follow the idea of the Fresnel equations, a vector beam propagating through the same medium split by a single phase screen will not lead to the change of beam propagation direction and the reflected portion of radiance unless two homogeneous media have the different optical properties. Otherwise, the beam will undergo two refractions when propagating through the thin phase screen if we assume that a thin phase screen has a certain thickness, see Fig. 2(a). In this regime, the calculation becomes more complicated. The reader can calculate an example of a vector beam propagating from vacuum with refraction index of $1$ to the media of phase screen with refraction index of $1+{\lambda } {\varphi } /2\pi d$. In our proposed model, we, without loss of generality, calculate the refractive process under the assumption that the two homogeneous media have the same optical properties. we make an assumption that a single phase screen is thin and incorporate the phase fluctuation of a turbulent media into the contribution of a wave vector instead of adopting $k_{m}^{\left ( t\right ) }=k_{m}^{\left ( i\right ) }$ ($m=x,y$) and Snell’s law in the Fresnel equations, which is the main reason that leads to the phenomenon of reflection and refraction even a vector beam propagates through the same medium (a more straightforward illustration is presented in Fig. 2(b)). In the other words, the foundation of the Fresnel equations and our theory are based on the continuity of the electromagnetic field. Our theory posits that the process of refraction arises from the continuity of the electromagnetic field in both spatial and temporal directions, coupled with an effective modification to the wave vector (the temporal continuity of the electromagnetic field ensures that the wave number remains constant before and after refraction, but the magnitudes of the individual components of the wave vector change, inevitably leading to alterations in the wave vector’s direction). In contrast, the Fresnel equations utilize the continuity of the electromagnetic field in conjunction with Snell’s law of refraction to change the direction of the refracted field.

4.2 Analogue of the Fresnel equations

Lastly, it is inevitable that one may question the differences between our theory and the Fresnel equations. In order to address this and provide a detailed comparison, in this subsection we give a better description of the analogue of the Fresnel equations and highlight the specific points of distinction as follows:

• Selection of coordinate system: The Fresnel equations consider the variation of the two directional components of the electric field, namely the $s$-component and $p$-component, within the plane perpendicular to the wave vector, in relation to the changes before and after refraction. In contrast, our theory examines the changes in the two directional components of the electric field within the $xy$ plane before and after refraction.
• Incorporation of phase fluctuations: We incorporate the phase fluctuations of the turbulent medium into the contribution of the wave vector (see Eq. (5)). This is a departure from the Fresnel equations, which typically assume a deterministic behavior of wave vector (the equation $k_{m}^{\left ( t\right ) }=k_{m}^{\left ( i\right ) }$ ( $m=x,y$) is set in the derivation of the Fresnel equations) without accounting for random fluctuations.
• Generation of refraction and reflection: The derivation of the Fresnel equations is based on the different optical properties of two isotropic media. By incorporating Snell’s law, the relationship between the refracted field and the incident field is obtained. In contrast, our theory does not impose any requirements on the isotropic optical properties. The phenomenon of refraction in the incident field is ensured through variations in the wave vector.

5. Results

5.1 Numerical details and notes

In this section, we present the simulation details of our theoretical framework, which serve as a crucial foundation for the subsequent presentation of our results. Due to the incorporation of the split-step beam propagation concept in our method, we can precisely capture the variations in turbulence along the propagation path by strategically placing and controlling the number of phase screens. We employ the two condition of atmospheric scintillation described in Ref. [24,61] to accurately break down the light propagation into discrete steps through numerical researches. By carefully positioning the phase screens, we are able to accurately simulate the changes induced by turbulence throughout the propagation process. Other than that, it is crucial to appropriately allocate the turbulence strength for each phase screen corresponding to the turbulence strength along the entire propagation path. A measure of the strength of atmospheric turbulence along the propagation path is the refractive index structure parameter $C_{n}^{2}\left ( z\right )$ (note that in the satellite-mediated atmospheric channel, turbulence strength can be described by $C_{n}^{2}$ as a function of altitude $h$) or the integrated strength parameter of turbulence $r_{0}$ [75]$.$ The relationship between these two turbulence parameters is given by [76,77]

(15)$$r_{0}=\left[ \alpha \left\vert \mathbf{k}\right\vert ^{2}\int_{0}^{z}C_{n}^{2}\left( z\right) dz\right] ^{{-}3/5}$$

where $\alpha =0.423$ is a constant number which derived in the case of the phase variance is approximately one. The integral is taken over the overall propagation path from the transmitter plane to the receiver plane. For free-space links, we employ $r_{0}$ to describe the turbulence strength and keep $C_{n}^{2}\left ( z\right )$ constant during the propagation. We achieve this by setting the proportion of turbulence strength for each segment of the path within the entire propagation link, which determines the magnitude of turbulence strength required for each phase screen calculation. Specifically, we can accomplish the calculation of each segment by using the formula: $r_{0j}=\left ( \beta _{j}r_{0}^{-5/3}/\sum _{j}\beta _{j}\right ) ^{-3/5}$, where $\beta _{j}$ represents the weight of turbulence strength for each segment. For satellite-mediated links, we employ $C_{n}^{2}\left ( h\right )$ to describe the turbulence strength by using the widely used Hufnagel-Valley (HV) model [78] and adjust the distribution of turbulent cells by following the so-called rule of equivalent Rytov-index interval phase screen (ERPS). More details about this rule and the turbulent link modeling used to perform the satellite-mediated calculation is further discussed in the Appendix A. Once we know the height distribution of phase screen, we can obtain the corresponding $r_{0j}$ for each segment based on Eq. (15).

Additionally, it is, in the numerical simulation process, necessary to specify the type and size of the phase screens, which is typically done based on the specific propagation scene. In our simulation, we utilize the von-Karman model [24] of atmospheric turbulence to generate Kolmogorov turbulence phase screens, accurately simulating turbulence under different conditions. Based on the predetermined size, intensity, distribution, number, and type of the phase screens, we sample the incident vector beams to complete one iteration simulation of atmospheric propagation. The result of numerical simulation in one iteration, representing a single realization of turbulence, can be repeated several times. The ensemble average of different iterations of these discrete steps represents the result of vector beams propagation through atmospheric turbulence under one group of parameter settings and can be employed to calculate the DOP of the electric field by using the following relationship [79]:

(16)$$\begin{aligned} \mathbf{I} &=\left( I,Q,U,V\right) ^{T}=\left( \begin{array}{c} \left\langle E_{x}E_{x}^{{\ast} }+E_{y}E_{y}^{{\ast} }\right\rangle \\ \left\langle E_{x}E_{x}^{{\ast} }-E_{y}E_{y}^{{\ast} }\right\rangle \\ \left\langle E_{x}E_{y}^{{\ast} }+E_{y}E_{x}^{{\ast} }\right\rangle \\ i\left\langle E_{x}E_{y}^{{\ast} }-E_{y}E_{x}^{{\ast} }\right\rangle \end{array} \right) \end{aligned}$$

(17)$$\begin{aligned} P &=\frac{\sqrt{Q^{2}+U^{2}+V^{2}}}{I} \end{aligned}$$

where $I$ is the total intensity of the beam, $Q$ and $U$ describe the linear polarization status, $V$ denotes the status of the circular polarization of vector beams. The notation $\mathbf {I}$, $P$ and $\left \langle \cdot \right \rangle$ represent the Stokes vector, the DOP of the electric field and ensemble average over different realizations of turbulence, respectively.

At last, it is important to highlight the implementation of random but differentiable phase screens in our numerical simulation. This crucial step enables the efficient modification of the wave vector, considering the gradient of the random contribution $\partial _{m}\varphi$, and facilitates potential changes in the DOP of vector beams. Unlike previous researches that use the three-point method [72] and Fourier-series method [80] to calculate the gradient of a phase screen, our approach is based on a simple idea: performing differential operations on the phase screen in the $x$ and $y$ directions. However, there is an issue where directly applying differential operations to the phase screen would result in a gradient matrix with a different pixel-number compared to the original phase screen. To address this problem, we first generate a phase screen $\Phi$ with a larger pixel-number (denoted as $\Xi$), and set an edge-number for the substantial calculations (denoted as $\xi$, in our calculations, we set $\xi =1$). Next, we extract rows $\xi$ through $\Xi -\xi$ from $\Phi$ (denoted as $\Phi _{p}$) to calculate the column random gradient contribution. The specific method involves performing differential operations on two new phase screens: one formed from columns $\xi -1$ through $\Xi -\xi -1$ of $\Phi$ (denoted as $\Phi _{p_{1}}$), and the other formed from columns $\xi +1$ through $\Xi -\xi +1$ of $\Phi$ (denoted as $\Phi _{p_{2}}$). This yields a matrix with a size of $\left ( \Xi -2\xi \right ) \times \left ( \Xi -2\xi \right )$, which we call the gradient matrix of the phase screen $\partial _{y}\varphi$. The calculation of the gradient matrix $\partial _{x}\varphi$ is similar to $\partial _{y}\varphi$, but it involves extracting columns $\xi$ through $\Xi -\xi$ of $\Phi$ and performing differential operations in the row direction. Finally, we extract rows $\xi$ through $\Xi -\xi$ and columns $\xi$ through $\Xi -\xi$ from $\Phi$ for the propagation calculation. This process allows us to obtain a phase screen (denoted as $\varphi$) with the same pixel-number as $\partial _{m}\varphi$ ($m=x,y$).

5.2 Typical propagation scenes

In the preceding sections, we have introduced the basic framework of our theory and highlighted its applicability to vector beams with arbitrary wavefronts. This versatility arises from the fact that our method only requires the electromagnetic field to satisfy the paraxial approximation. In this subsection, we will conduct simulation studies for vector beams with a specific shape of wavefront in two distinct propagation scenes: free-space links and satellite-mediated links. Concretely, we will employ Gaussian vector beams for these calculations (for specific reasons, we refer the reader to Ref. [81]).

5.2.1 Free-space links

Figure 4 illustrates how the DOP of circularly polarized light changes with respect to $r_{0}$ under different off-axis magnitudes [82]. Each plot from upper to bottom represents the results of different values of wavelength (In the calculation of wavelength, we keep the propagation distance $z$ and the spacing $\Delta z$ between two adjacent phase screens constant and set $z=1000m$, $\Delta z=100m$. Moreover, we adopt the side length at transmitter plane and receiver plane of $30cm$ with a spatial resolution of $1.2mm$ and calculate the DOP averaged over $1000$ realizations of turbulence). It can be seen that in the strong turbulence regime, the DOP of circularly polarized light rapidly decreases with the increase of turbulence strength, whereas in the weak-to-moderate turbulence regime, DOP almost does not change as $r_{0}$ gradually decreases. We found that, overall, the change of DOP affected by atmospheric turbulence mainly fluctuates around the order of $10^{-13}$, which can be almost ignored. Moreover, we observe that the polarization properties of the optical field offset from the center of the optical axis are strongly affected by turbulence. In other words, atmospheric turbulence has smaller effect on the DOP of on-axis optical field compared to that of off-axis one, which leads to a reduction of DOP as the off-axis magnitude becomes larger. Finally, from the comparison between Fig. 4(a) to 4(e), we see that the DOP at the center of the optical axis hardly varies with the wavelength, yet the polarization properties of the off-axis optical field may become smaller as the wavelength increases, which is mainly caused by the fact that a short wavelength polarized light undergoes a larger effect of atmospheric turbulence.

Fig. 4. DOP of a circularly polarized light as a function of $r_{0}$ for different values of wavelength: (a) $\lambda =532nm$; (b) $\lambda =633nm$; (c) $\lambda =671nm$; (d) $\lambda =845nm$; (e) $\lambda =1064nm$. All curves in each plot correspond to the DOP calculated under different off-axis magnitudes. The standard errors are plotted by the shaded area around each curve.

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The DOP of vector beams as a function of $r_{0}$ under different parameter settings, including beam parameters and turbulence parameters, are shown in Fig. 5, where each row from upper to bottom represents the results with different propagation distances $z$ (Fig. 5(a) to (c)), beam waists $w_{0}$ (Fig. 5(d) to (f)), polarization types $\delta$ (Fig. 5(g) to (i)), outer scales $L_{0}$ (Fig. 5(j) to (l)) and inner scale $l_{0}$ (Fig. 5(m) to (o)), respectively, where $\delta$ stands for the phase difference between $E_{x}^{\left ( i\right ) }$ and $E_{y}^{\left ( i\right ) }$. All curves in each plot correspond to the DOP under different off-axis magnitudes. As depicted in Fig. 5(a) to 5(c), we observe that for vector beams that are perturbed while propagating over longer and more turbulent links the compound effect on the DOP is more damaging than the effect of simply accumulating distance or turbulence over the link. In other words, we found that in the short propagation distance or weak turbulence regime, the polarization properties remain almost constant and is hardly affected by the off-axis magnitude. Other than that, from the results presented in Fig. 5(d) to (f), we found that a smaller beam waist of vector beams may lead to a strong effect of atmospheric turbulence. The primary reason because a vector beam with a smaller beam waist possesses a larger divergence angle so that it is more susceptible to atmospheric turbulence [83]. The other conclusions about the effects of turbulence strength and off-axis magnitude are the same as those of Fig. 4. The effects of different polarization types at the transmitter on DOP with respect to $r_{0}$ are illustrated in Fig. 5(g) to (i). We notice that in the moderate-to-strong turbulence regime, the DOP of circularly polarized light may be more vulnerable to atmospheric turbulence than that of linearly polarized light; besides, when $\delta =45^{\circ }$, elliptically polarized light has the worst performance when propagating in atmospheric turbulence (see more explanations in Ref. [84]). Finally, we reveal the effects of outer scale and inner scale of turbulence on the DOP of vector beams in Fig. 5(j) to 4(o). From the comparison between different outer scales and inner scales, we observe that a large value of $L_{0}$ and a small value of $l_{0}$ may lead to a significant reduction of DOP, which is partly because inner scale and outer scale of turbulence form the lower limit and upper limit of the inertial range, a smaller value of $l_{0}$ and a larger value of $L_{0}$ are obtained with the increase of turbulence strength [61,85,86]. In the other words, the decreasing of $l_{0}$ or increasing of $L_{0}$ is equivalent to increase the number of turbulent cells along the turbulent channel, which causes a vector beam meets more turbulence during the atmospheric propagation.

Fig. 5. DOP of vector beams as a function of $r_{0}$ for different beam parameters and turbulence parameters, where each row from upper to bottom represents the results calculated under different propagation distances ((a) $z=100m$; (b) $z=500m$; (c) $z=1000m$), beam waists ((d) $w_{0}=3cm$; (e) $w_{0}=5cm$; (f) $w_{0}=8cm$), polarization types ((g) $\delta =0^{\circ }$; (h) $\delta =45^{\circ }$; (i) $\delta =90^{\circ }$), outer scales ((j) $L_{0}=10m$; (k) $L_{0}=100m$; (l) $L_{0}=1000m$) and inner scales ((m) $l_{0}=5mm$; (n) $l_{0}=8mm$; (o) $l_{0}=10mm$), respectively. All curves in each plot correspond to the DOP calculated under different off-axis magnitudes. The standard errors are plotted by the shaded area around each curve.

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5.2.2 Satellite-mediated links

After introducing the DOP of vector beams changes under different parameter settings in the free-space links, we now turn our attention to the satellited-mediated links. We investigate whether the DOP of vector beams changes in the vertical atmospheric links, which serves as exemplary scenarios for atmospheric propagation in long-distance and nonuniform turbulent links. Notably, we adjust the side length at transmitter plane and receiver plane with a specific value according to the propagation distance and beam waist during the satellite-mediated simulations because of the divergence properties of the vector beams propagating through atmospheric turbulence (e.g., when the propagation distance increases from $10km$ to $200km$, we set the side length at receiver plane from $1m$ to $3m$).

In Fig. 6, we illustrate the variation curves of the on-axis DOP of vector beams propagating through satellite-mediated links as a function of zenith angle $\theta _{Z}$ under different parameters (averaged over $1500$ realizations of turbulence). These parameters are the same as in Fig. 5. All curves in each plot correspond to the on-axis DOP under different parameter settings. We see in Fig. 6 that, overall, the DOP of vector beams remains almost unchanged with respect to $\theta _{Z}$ except when $\theta _{Z}$ becomes larger, however, the change of DOP in satellite-mediated links affected by atmospheric turbulence mainly fluctuates around the order of $10^{-5}$ to $10^{-7}$, which is negligible, but is more affected compared to the results achieved from free-space atmospheric propagation. As depicted in Fig. 6(a), we compare the on-axis DOP under different propagation distances (expressed as $h$). We found that in the large $\theta _{Z}$ regime, the DOP of long-distance propagation shows a significant reduction compare to that of short-distance one, which indicates that the DOP of on-axis optical field is gradually affected by atmospheric turbulence as the increase of propagation distance. In Fig. 6(b), we move our concern to the circumstance of different beam waists. We clearly observe the same conclusion obtained in the free-space links where the DOP of vector beams possessing a larger beam waist outperforms that of vector beams with a smaller one and once again verify the primary reason that a smaller beam waist has a larger divergence angle so that it is more vulnerable in atmospheric turbulence whatever in free-space or satellite-mediated links. In Fig. 6(c), we calculate the DOP of a circularly-polarized vector beam propagating through vertical atmosphere under different values of wavelength. We clearly observe that the polarization properties are deeply affected by atmospheric turbulence with the decreasing wavelength (we use the word "deeply" to describe because the effect of wavelength on DOP varies in several orders of magnitude). In addition, we notice that the DOP is more affected by the wavelength for a larger $\theta _{Z}$, which may be caused by the combined effect of propagation distance and wavelength. We investigate the effects of different polarization types at the transmitter on DOP with respect to $\theta _{Z}$ in Fig. 6(d). We can easily achieve the same conclusion obtained in Fig. 5(g) to (i), namely, the elliptically polarized vector beam is more fragile compared to linearly and circularly polarized light. In a word, it should be emphasized that vector beams with different polarization types remain almost unaffected by atmospheric turbulence even in the satellite-mediated links. Figure 6(e) and (f) plot the DOP of vector beams propagating in the Kolmogorov turbulence as a function of $\theta _{Z}$ for different values of $l_{0}$ and $L_{0}$, where the calculation is shown for a distance of $150km.$ It can be seen that the smaller $l_{0}$ or the larger $L_{0}$ will always lead to a vector beam meet more turbulence either in the free-space or satellite-mediated propagation regime, for the reason we have already explained an intuitive way in the previous section.

Fig. 6. DOP of vector beams as a function of $\theta _{Z}$ for different beam parameters and turbulence parameters: (a) propagation distance; (b) beam waist; (c) wavelength; (d) polarization types; (e) inner scale; (f) outer scale. All curves in each plot correspond to the DOP calculated under different parameter settings. The error bars represent the standard error.

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5.3 Validation of theoretical framework

In the previous two subsections, we present the simulation details of our theoretical framework and showcased the computational results for various typical scenes. To validate the reliability of our theoretical approach, this subsection focuses on comparing our simulation results with historical experimental data. For the free-space propagation scene, our simulation data can be qualitatively compared with the experimental data that have been taken in Ref. [26]. Figure 7(a) presents a schematic diagram of the experimental setup for this system, where the transmitter end is located in a residential building on the northern bank of the Songhua River in Harbin City, while the receiver end is situated on the rooftop of Building 2A in the Harbin Institute of Technology Science Park on the southern bank of the Songhua River. The straight-line distance between the two points is $11.16km$. The incident light has a wavelength of $1550nm$ and a divergence angle of $230 \mu rad$. A polarization beam splitter (PBS) and a quarter-wave plate (QWP) are used in the transmitter to generate circularly polarized light. The incident light, after propagating through a free-space link, enters the Cassegrain telescope with an aperture size of $12.7cm$ and is split into two beams using a beam splitter. One beam is directed towards the CMOS for further analysis, while the other beam is sent to the polarization analyzer to measure its polarization characteristics. Measurements of turbulence strength along the propagation link are conducted with the record of beam centroid drift by using a CMOS camera and the calculation of arrival-angle fluctuation at the receiver. Figure 7(b) presents a comparative graph showing the temporal variation of the numerical simulation results and experimental measurements. It can be observed that the calculated results closely match the experimental data (with an average measured DOP of $1.0357$), thereby validating the reliability of our theoretical framework. However, it should be noted that the variation of the calculated DOP does not exhibit the same level of fluctuations as the measured data. This discrepancy arises from the fact that the numerical calculations do not account for various sources of experimental errors, such as the non-ideal angle between the HWP and the PBS at the transmitter, platform vibrations during the experiment, and measurement errors in the polarization analyzer. Concretely, the reason why the average measured value of DOP exceeding 1 may be because of measurement errors in the polarization analyzer. The deviation from a precise $45^{\circ }$ angle between the QWP and the PBS could lead to a vector beam generated at the transmitter not being strictly circularly polarized light during the experiment. Moreover, a smaller platform vibration between the transmitter and receiver could introduce the misalignment error during the experiment, causing the received light at the receiver to be partially off-axis [26].

Fig. 7. (a) A schematic diagram of the experimental setup for the free-space link. The PBS and QWP employed in the transmitter are used to generate circularly polarized light, The received beam, after propagating through a free-space link, enters the Cassegrain telescope and undergoes beam splitting using a beam splitter. This division generates two beams: one is directed towards the CMOS for the measurement of turbulence strength, while the other is routed to the polarization analyzer for the measurement of its polarization properties. (b) The temporal variation of DOP at the receiver, after the propagation of the vector beam through the $11.16km$ free-space link from the transmitter end. A series of circular scatter points represent the experimental result provided in Ref. [26] , while the square scatter points denote the results calculated on the basis of our theoretical framework.

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After discussing the free-space propagation scenario, we will now delve into the comparative results for satellite-mediated links. Figure 8(a) presents a geometric representation of the satellite-mediated communication link, while the experimental data are obtained from the polarization measurement experiments conducted by the National Institute of Information and Communications Technology (NICT) in Japan from October 2008 to February 2009 [87,88]. The NICT, formerly known as the Communications Research Laboratory (CRL), conducted polarization measurement experiments using a laser source deployed in space. These experiments utilized the Optical Inter-Orbit Communications Engineering Test Satellite (OICETS) Kirari, a Low Earth Orbit (LEO) satellite. The polarization measurement experimental data between the NICT ground station (located in Koganei of downtown Tokyo) and OICETS were achieved in the night from 16:16:08 to 16:21:58 in the Universal Time on December 23, 2008. The minimum distance between the ground station and the satellite was $959.8km$, with a minimum zenith angle of $54.7^{\circ }$. The experiment duration was $350$ seconds, with an elevation angle above $15^{\circ }$ for the satellite. The atmospheric scintillation indices varied from $0.05$ to $0.4$, depending on the elevation angles, and there were no clouds present during the measurements. The downlink laser beam from the satellite had a beam divergence of approximately $6 \mu rad$ and a wavelength of $847nm$. The DOP data was directly measured using a polarimeter and recorded at a rate of $10Hz$. As illustrated in Fig. 8(b), we compare the numerical simulation results with the measured data from the satellite-mediated experiment, which shows a close agreement between the calculated outcomes and the experimental observations (with an average measured DOP of 0.994). This further validates the reliability of our theoretical framework. It is worth noting that, similar to the previous scenario, the calculated polarization variations do not exhibit the same level of fluctuations as the measured data, which can be explained by the fact that most factors, such as the instrumental error, the backscattered light from the uplink beacon light, and atmospheric polarization effects, do not accounted for in the numerical calculations [87] (Although the sky was cloudless on the day of the experiment, the presence of aerosol distribution at different vertical atmospheric layers resulted in the uplink beacon laser exhibiting polarization characteristics due to backscattering from atmospheric particles. Furthermore, as the atmospheric particles are distributed differently at various altitude layers and possess distinct scattering properties, the atmospheric background radiation after particle scattering leads to specific polarization distribution characteristics at different azimuth and elevation angles in the sky).

Fig. 8. (a) A geometric representation of satellite-mediated polarization measurement experiments in the night from 16:16:08 to 16:21:58 in the Universal Time on December 23, 2008. The NICT ground station performed polarization measurements in Koganei of downtown Tokyo using a laser source installed on the OICETS/Kirari satellite. On that day, the minimum and maximum elevation angles of the measurement were $15^{\circ }$ and $35.3^{\circ }$, respectively. The minimum distance of the measurement between the NICT and the Kirari satellite was $959.8km$. (b) The temporal variation of DOP observed at the ground station during the experiment. A series of circular scatter points represent the experimental result provided in Ref. [87], while the square scatter points denote the theoretical result calculated under the same recording rate with the experiment.

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6. Conclusions

In this paper, we have studied the propagation of vector optical beams inside atmospheric turbulence, taking into account the change of polarization properties both in free-space links and satellite-mediated links. Unlike previous researches derived from Maxwell’s equations and evaluated by the root-mean-square variation of polarization angle [23,89,90], we propose a novel propagation model for a generally polarized vector beam in analogy with the well-known split-step beam propagation method. The main idea behind our method is that we consider the process of reflection and refraction exerted by a phase screen, which is derived on the generalization of the continuity of electromagnetic field. Additionally, we employ the vectorial diffraction formula to describe the vacuum diffraction between two phase screens. After making such revisions to the conventional propagation model, we investigate the change of polarization on atmospheric propagation under different parameters and parameter settings. It is found that the changes of DOP in free-space links and satellite-mediated links are mainly surrounded by the order of $10^{-13}$ and $10^{-6}$ and can nearly be ignored. Our results further confirm the fact that vector beams whose DOP does not change on atmospheric propagation and will be useful for free-space optical communications and quantum communications.

Appendix A: satellite-mediated turbulent link modeling

We divide the satellite-mediated link into $N_{S}$ turbulent cells bounded by specific altitudes $h_{j}$ with $j$ ranging from $1$ to $N_{S}$ (note that turbulent cells are arranged from lower altitudes to higher altitudes). The altitude $h_{j}$ of each turbulent cell is calculated by the rule of ERPS, as depicted in Fig. 9. For convenience of presentation, we summarize and divide the procedure of ERPS’s execution into four steps as follows (we employ the Rytov index $\sigma _{R}^{2}\left ( \Delta h_{j}\right )$ to characterize the scintillation between two turbulent cells, see more detailed reasons in Ref. [91]):

1) We set the constant $c$ such that the Rytov index of two adjacent phase screens is equal to $c$ (i.e., $\sigma _{R}^{2}\left ( \Delta h_{j}\right ) \equiv c$, where $\Delta h_{j}=h_{j}-h_{j-1}$), where $c$ is commonly set by a small value (In our calculation, we set $c=5\times 10^{-4}$. For more detailed explanations, we refer the reader to Ref. [92]).
2) We calculate the altitude of the first phase screen based on $C_{n}^{2}\left ( h_{0}\right )$ and the Rytov equation: $\sigma _{R}^{2}\left ( \Delta h_{1}\right ) =1.23C_{n}^{2}\left ( h_{0}\right ) k^{7/6}\left ( \Delta h_{1}\right ) ^{11/6}\equiv c$ for setting the initial value, where $C_{n}^{2}\left ( h_{0}\right )$ represent the near-surface refractive index structure parameter, $h_{0}$ denote the ground station altitude (In our calculation, we assume $h_{0}=0$).
3) We calculate the spacing $\Delta h_{j}$ by using $C_{n}^{2}\left ( h_{j-1}\right )$ and solving the identity: $1.23C_{n}^{2}\left ( h_{j-1}\right ) k^{7/6}\times \left ( \Delta h_{j}\right ) ^{11/6}\equiv c$.
4) We repeat step 3) several times and terminate the iteration until the sum of the spacing of phase screen is greater than the total propagation distance (i.e., we should decide whether $\sum _{j=1}^{N_{S}}\Delta h_{j}$ is not less than to $h$. If not, we repeat step 3; if yes, we terminate the loop).

Fig. 9. The modeling of a satellite-mediated turbulent link. A vector beam is generated within a satellite, and propagated over a simulated turbulent link to a ground station. The turbulent link is segmented into $N_{S}$ turbulent cells bounded by specific altitudes $h_{j}$ with $j$ ranging from $1$ to $N_{S}$. The altitude $h_{j}$ of each turbulent cell, depicted in a dashed line, is determined using the rule of ERPS. It is important to note that, in general, when the zenith angle $\theta _{Z}$ is non-zero, the altitudes of each turbulent phase screen should be divided by the cosine of $\theta _{Z}$.

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So far, we have determined the exact number of $N_{S}$ and obtained the specific altitudes of $h_{j}$. However, it is worth emphasizing that the above calculation is performed assuming the condition that $\theta _{Z}=0$. If $\theta _{Z}\neq 0$, the specific altitudes of turbulent cells should be adjusted to $h_{1}\sec \theta _{Z}$, $h_{2}\sec \theta _{Z}$, $\cdots$, $h_{N_{S}}\sec \theta _{Z}$.

Appendix B: unitarity of turbulence operator

We revisit and rearrange the stochastic parabolic equation (see Eq. (1)) which describe a scalar beam propagating through atmospheric turbulence into the following way [74]:

(18)$$\partial _{z}E=\left( \frac{i}{2\left\vert \mathbf{k}\right\vert }\nabla _{{\perp} }^{2}+i\left\vert \mathbf{k}\right\vert ^{2}\delta n\right) E=i\left( \widehat{D}+\widehat{S}\right) E$$

where the two linear operators $\widehat {D}$ and $\widehat {S}$ on the right-hand side control the diffraction and refraction of $E$ over increments $\delta n$, respectively. By expressing the stochastic parabolic equation in the form of Eq. (1), its solution can be written as follows:

(19)$$E\left( z+\Delta z\right) =e^{i\int_{z}^{z+\Delta z}\left( \widehat{D}+ \widehat{S}\right) dz}E\left( z\right) =\widehat{T}\left( \Delta z\right) E\left( z\right)$$

where $\widehat {T}\left ( \Delta z\right )$ is the turbulence operator and characterizes the propagation of an electric field through atmospheric turbulence over a distance $\Delta z$. Due to the hermiticity of the linear operators $\widehat {D}$ and $\widehat {S}$, we can see that $\widehat {T} \left ( \Delta z\right )$ is unitary (i.e., $\widehat {T}^{\dagger }=\widehat {T} ^{-1}$), further, the power transfer efficiency is proven to be unity.

Funding

National Natural Science Foundation of China (11904369, 62301530); Open Fund of Infrared and Low Temperature Plasma Key Laboratory of Anhui Province, NUDT (IRKL2023KF05); the HFIPS Director’s Foundation (YZJJ2022QN07, YZJJ2023QN05); National Key Research and Development Program of China (2019YFA0706004).

Acknowledgments

We would like to express our sincere gratitude to Prof. Jing Ma and his Master’s student Chuangli Yu for their polarization measurement data in the free-space link and also like to extend our appreciation to Prof. Morio Toyoshima and his research group for sharing the polarization measurement data obtained between the NICT Ground Station and the OICETS/Kirari satellite. Their valuable contribution has significantly enhanced the reliability and credibility of our research.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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92. The constant c and the height profile of $C_{n}^{2}\left ( h\right )$, jointly determine the phase screen distribution in satellite-mediated links. Here, c determines the number of phase screens, while $C_{n}^{2}\left ( h\right )$ governs both their heights and strengths. In practice, when computing the distribution of phase screens in satellite-mediated links, it is sufficient to adhere to the two conditions introduced in Section 5.1 regarding atmospheric scintillation. Typically, a smaller value of c implies more phase screens along the propagation path, leading to more precise simulation results. However, this also significantly increases the computational time for a single realization of turbulence. Therefore, it is usually necessary to select a compromise value for the constant c to balance these two factors.

Does the degree of polarization of vector beams remain unchanged on atmospheric propagation?

Abstract

1. Introduction

2. Preliminary

2.1 Turbulence description

2.2 Split-step beam propagation method

3. Theory

4. Discussions

4.1 Frequently asked questions

4.2 Analogue of the Fresnel equations

5. Results

5.1 Numerical details and notes

5.2 Typical propagation scenes

5.2.1 Free-space links

5.2.2 Satellite-mediated links

5.3 Validation of theoretical framework

6. Conclusions

Appendix A: satellite-mediated turbulent link modeling

Appendix B: unitarity of turbulence operator

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (19)

Optics Express