The work in [

10] proposed a new type of multilevel-polarization modulation (“bands of polarization” modulation) that goes beyond the classical concept of symbols belonging to a rigid constellation in the Euclidean space. This modulation model takes advantage of an intrinsic characteristic of optical fibers such as the birefringence. As a matter of fact, with a suitable twisting process, the induced circular birefringence

β_{3} becomes predominant with respect to the linear birefringence components

β_{1} and

β_{2}. In this case, the evolution of the SOP during its spatial propagation along the fiber is confined latitudinally within specific physical tracks (called “bands of polarization”).

Figure 1 shows the spatial evolution in the Poincaré sphere of five different SOPs in their own “bands of polarization”; the simulated twisted fiber has a twist rate of 6 rad/m (~1 turn/m).

A fundamental benefit of this system consists of the reduced complexity of the receiver. Its simplicity derives from the simple need to detect only the S_{3} component of the received SOP. Therefore, there is no need to implement a specific circuit to track the birefringence’s variations.

#### 3.2. Statistical Analysis

The five transmitted symbols were chosen in such a way that the relative “bands of polarization” were symmetrical around the starting value of the latitude. To achieve this objective, we analyzed the behavior of different SOPs that belonged to the same band of polarization; the chosen test band was the equatorial band that included the linear polarizations.

Figure 2 shows the spatial evolution of different linear polarizations. It can be seen that the cycloidal spatial trajectory of the SOP is not symmetrical with respect to the equatorial plane if the starting value of

S_{2} is null, facing downwards if

S_{1} is positive (

Figure 2a) and upwards in the opposite case (

Figure 2b); conversely, the trajectory is symmetrical with respect to the equatorial plane if the starting value of

S_{2} is equal to one (

Figure 2c). Moreover, the trajectories in

Figure 2a,b are prolate cycloids while that in

Figure 2c is a curtate cycloid.

To enhance the visualization of the cycloidal patterns, we chose a weak twist rate of 1.5 rad/m. In fact, with a low twist rate, the spatial trajectory is a prolate cycloid, while when increasing the twist rate, it becomes first an ordinary cycloid and then a curtate cycloid. The same behavior holds true for the elliptical polarizations. On the contrary, circular polarization is flattened towards the pole because of the presence of strong spatial constraints (

Figure 3).

The first objective is to study the dependency of the transmitted SOP spatial evolution from the propagation distance of the optical field along the fiber for different values of the twisting process. For each simulation cycle, we calculated the probability density function (hereinafter referred to as the pdf) of the third Stokes vector component

S_{3} relative to the transmitted symbols. In fact, as demonstrated by Equation (8),

S_{3} depends directly on the latitude angle and its variance is closely related to the width of its associated “band” (

Figure 1).

Afterwards, in order to consider all the cycles’ contributions, we derived the average pdf as the mean curve of all the executed simulations. In

Figure 4, the behavior of the above-described mean pdf for different propagation distances is shown.

These plotted functions were obtained with different fiber distances but with the same value of the twist rate (6 rad/m). The transmitted symbol had a 45° linear polarization SOP with a Stokes vector equal to [0,1,0]. The

S_{3} variance, and consequently the width of the bands of polarization, widens with increasing the distance of propagation. This variance growth has a linear dependence on the propagation distance as shown in

Figure 5, which shows a comparison between the simulated values and a linearly fitted curve.

Figure 6, instead, shows the dependency of the

S_{3} pdf on different values of the twisting process. These plotted functions were obtained by considering a fixed value of the propagation distance equal to 500 m. In this case, the transmitted symbol also had a 45° linear polarization SOP with a Stokes vector equal to [0,1,0]. The

S_{3} variance, and consequently the width of the bands of polarization, narrows with increasing the twisting value. Therefore, an increase of the twisting process generates a potential throughput rise, with the same available bandwidth, for this type of multilevel polarization modulation, because it allows for the presence of a greater number of bands (and consequently, symbols) on the Poincaré sphere.

The variance decrease has an exponential dependence on the twist rate, as shown in

Figure 7, which shows a comparison between the simulated values and an exponentially fitted curve. Therefore, the statistical results show how the width of the polarization bands has a dependence on the twisting process that is much stronger (exponential) than that of the propagation distance (linear).

The data originating from these simulations show that the twisting process gives rise to a physical track even tighter for circular polarization than for those equatorial and elliptical polarizations. Considering the same values of distance (500 m) and twist rate (6 rad/m),

Figure 8a shows the comparison between a linear and an elliptical SOP, while in

Figure 8b a circular SOP is added. It is clear from

Figure 8b, how large the difference is between the pdf curves of circular polarization on one side and those of the equatorial and elliptical polarizations on the other side.

Another important result that can be deduced by

Figure 8a,b is that the width of the “bands” decreases, starting from the equator to the pole. This behavior proves how a transmitted circular SOP is physically advantaged with respect to the other SOPs, in terms of a less probable deviation from its initial position, during the spatial propagation in a twisted optical fiber. It is reasonable to assume that this behavior is determined by the greater strength of the circular polarization with respect to the symmetry, also circular, of the fiber core.