## 3 Results  
We obtain experimental results for five input states that are linearly polarized with different DOPs. In Figure 2, we plot the values of the input and output DOP. The diagonal black line in Figure 2 represents equal input and output DOP. The experimental results are compared with the results from the simulation with stationary bubbles. The individual experimental Stokes parameters are shown in Table 2. We note that there is significant intensity attenuation of the beam as it propagates through the bubbles and water, which is to be expected. We see a reduction in average intensity by a factor of about 20, while the DOP is well-maintained. In Figure 3, we show the location of the input and output states in the Poincare sphere representation of normalized in three-dimensional Stokes space. We see the corresponding input and output states lie very close to each other on the Poincare sphere.  
While we measure all Stokes parameters, a full tomography of the polarization matrix is not necessarily required for communication purposes, given appropriate encoding and decoding protocols are chosen. If we use the input states shown in Figure 3, only measurement along the \(s_{1}\) Stokes parameter is necessary to determine the encoded value. Thus, the measurement can be done in a more robust manner, with only two intensity measurements, \(I(0^{\circ},\,0^{\circ})\) and \(I(0^{\circ},\,90^{\circ})\). Due to our choice of basis, where \(s_{2}\) and \(s_{3}\) are approximately 0, we can recover the DOP with only \(s_{0}\) and \(s_{1}\). This measurement can be generalized to all directions on the Poincare sphere by exploiting its symmetry, as there is no preferred direction on the Poincare sphere.  
Figure 3: Input and output states in the Poincare sphere representation. Experimental input states are orange triangles, experimental output states are green diamonds, stationary bubbles simulation input states are red squares, stationary bubbles simulation output states are blue circles. On the Poincare sphere: D = diagonal, A = anti-diagonal, L = left, R = right, V = vertical, H = horizontal.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**State**} & \multicolumn{2}{c}{\(s_{1}\)/\(s_{0}\)} & \multicolumn{2}{c}{\(s_{2}\)/\(s_{0}\)} & \multicolumn{2}{c}{\(s_{3}\)/\(s_{0}\)} & \multicolumn{2}{c}{**DOP**} \\ \cline{2-9}  & **Input** & **Output** & **Input** & **Output** & **Input** & **Output** & **Input** & **Output** \\ \hline
0 & 0.09 & 0.14 & \(-\)0.04 & 0.01 & \(-\)0.04 & \(-\)0.05 & 0.11 & 0.15 \\
1 & 0.36 & 0.39 & \(-\)0.05 & \(-\)0.07 & \(-\)0.03 & \(-\)0.07 & 0.36 & 0.40 \\
2 & 0.53 & 0.55 & \(-\)0.04 & \(-\)0.08 & \(-\)0.02 & \(-\)0.10 & 0.53 & 0.57 \\
3 & 0.74 & 0.71 & \(-\)0.05 & \(-\)0.03 & \(-\)0.02 & \(-\)0.06 & 0.74 & 0.72 \\
4 & 0.99 & 0.92 & \(-\)0.06 & \(-\)0.10 & \(-\)0.02 & \(-\)0.08 & 0.99 & 0.93 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Measured mean normalized Stokes parameters of the input and output partially polarized states on the horizontal/vertical axis through experimental underwater turbulence.  
Figure 2: Measured DOP of the output states through underwater bubbles compared to the measured DOP of the input states for different partially polarized states. Input and output DOP are equal at the black like (\(y=x\)). Experimentally obtained data is shown in red and simulation results are shown in blue. The error bars represent 99% confidence interval.  
We can also choose some arbitrary basis to generate arbitrary states. The individual Stokes parameters, when measuring states along an arbitrary basis are shown in Table 3. Thus, our results suggest that SOP is also preserved. The input and output DOPs, along with the locations of each state on the Poincare sphere are shown in Figure 4.