## 1 Introduction  
Free-space optical (FSO) communication is a widely used method of sending high data rate signals over long distances. It allows for wide bandwidths, licence free spectra, and high bit rates [1; 2; 3; 4; 5]. Atmospheric turbulence, however, poses a significant threat to FSO communication [1]. Signal attenuation resulting from turbulent fluctuations [6; 7; 8; 9] can result in increased bit-error rates and data security concerns. Work is constantly being done to improve FSO communications such that it can become resistant to turbulence [10], for which many techniques require cumbersome active optics, advanced algorithms, and/or machine learning [11; 12; 13; 14; 15]. It has been shown that polarization can be used for atmospheric FSO [16]. The use of optical polarization in communication opens up the realm of quantum communication and quantum internet [17]. However, turbulence in the atmosphere can potentially produce impurities in quantum states. Understanding how turbulence affects polarization lays the groundwork for better understanding how turbulence may affect quantum states that are realized via optical polarization states. Turbulence in the atmosphere due to the temperature differences between packets of air results in random fluctuations of the index of refraction \(n\). These variations can be modeled as [18]. \[n(\vec{r})=n_{0}+n_{1}(\vec{r}), \tag{1}\]  
where \(\vec{r}\) denotes the position, \(n_{0}\approx 1\) is the mean value of the refractive index, and \(n_{1}(\vec{r})\) are fluctuations. For atmospheric turbulence, \(n_{1}(\vec{r})\) is typically several orders of magnitude smaller than \(n_{0}\)[19]. The effects that these fluctuations have on light passing through the atmosphere can be studied by considering the wave equation for the electric field \(\vec{E}(\vec{r},t)\). For convenience, we consider monochromatic light with frequency \(\omega\) and time dependence \(e^{i\omega t}\) propagating through a source-free region of atmosphere. Then, [PERSON]'s equation for the electric fields are reduced to [18]. \[\
abla^{2}\vec{E}(\vec{r})+\frac{\omega^{2}n(\vec{r})^{2}}{c^{2}}E(\vec{r})+2 \
abla(\vec{E}(\vec{r})\cdot\
abla\ln[n(\vec{r})])=0, \tag{2}\]  
where \(c\) is the speed of light in vacuum. The third term in Equation (2) couples the different components of the electric field, which weakens the polarization of the field. Because \(n_{1}(\vec{r})\ll n_{0}\), however, this depolarizing term tends to be very small and can often be ignored for visible light passing through the atmosphere [20]. We also confirm this via simulations in Appendix A. This would suggest that the polarization of light could serve as a useful degree of freedom in FSO communications, particularly given that polarization transformers have been shown to be able to manipulate the state and degree of polarization (SOP and DOP, respectively) [21] in ways that may be suitable for FSO communications [22]. It has also been shown that polarization can be used to encode and transmit information securely [23], and that states with a DOP of 1 are preserved in the presence of turbulence [16; 24]. Utilizing these properties with methods like coherent binary polarization shift keying [25] may lead to improvements in FSO communications [26]. However, it can be difficult to develop these methods, as the experiments generally require the implementation of long distance atmospheric testing. An alternative approach could be the use of air bubbles in water, which are known to create intensity fluctuations that are often well-described by the log-normal distribution commonly used in studies of atmospheric turbulence [27; 28; 29; 30]. Directly applying the atmospheric turbulence models to bubbles in water can be problematic as it overlooks key differences between the environments [31]. In particular, the refractive index model will be very different for a region of water with bubbles.