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Polarization evolution in Brillouin scattering with a partially polarized pump

Junhe Zhou, Tengyuan Liu, and Yunheng Wang

Open Access Open Access

Abstract

Polarization sensitivity has been a major issue in Brillouin scattering-based optical fiber sensing systems. Randomization of the polarization state of the pump is one of the ways to circumvent the problem. However, there could exist a residual degree of polarization (DOP) for the pump after polarization randomization, and hence, a model to characterize the polarization evolution in Brillouin scattering with a partially polarized pump is essential for the performance evaluation. In this work, a comprehensive theoretical model to characterize the beam variation with the partially polarized pump wave and Stokes wave is proposed, which is based on a set of stochastic differential equations (SDEs). The polarized part of the pump wave and the Stokes wave, as well as the total powers of the waves, are incorporated in the coupled SDE simultaneously, which enables the comprehensive simulation of the polarization evolution in the fiber. It is revealed in the study that the DOPs of the pump wave and the Stokes wave affect the gain stability and should be reduced simultaneously by polarization scrambling to ensure a fixed Brillouin gain without fluctuations.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Brillouin scattering in the optical fiber is a phenomenon caused by the interaction between the optical wave and the acoustic wave, which transfers the power of the forward propagating pump wave to the backward propagating probe or the Stokes wave [1]. It has various applications in fiber optics such as fiber sensing [2] in infrastructure (like buildings, bridges, ports etc.) monitoring, microwave photonic subsystems [3], narrow linewidth opto-mechanical oscillator [4], and all optical signal processing [5].

Brillouin scattering is polarization dependent [1,6–11], which is the result of random polarization state rotation of the pump and the Stokes wave caused by the polarization mode dispersion (PMD) in the fiber [12–19]. This induces the power fluctuations in the optical systems which implement Brillouin scattering, and brings performance degradation for the applications.

Polarization randomization has been proposed to circumvent the difficulty [6,7]. By randomizing the polarization states of the reference signal or the pump wave, one may obtain a relatively stable probe or the Stokes wave. However, there could be residual degree of polarization (DOP) for the pump after polarization randomization and it brings the residual Brillouin gain fluctuations. Characterization of the pump wave and the Stokes wave polarization evolution under such a partially polarized pump is valuable and essential for the Brillouin scattering based system analysis.

Pioneering researches have been conducted on this topic. A. Galtarossa et. al. proposed a vector model, and an analytical solution was obtained by neglecting the pump depletion [8]. L. Ursini et. al. proposed a polarization dependent gain (PDG) vector model in the undepleted pump regime [9]. A. Zadok et. al. proposed a full vector model which incorporated the pump depletion, and the polarization pulling effect was revealed by the model [10]. C. Wang et al. investigated the polarization dependent fluctuation of Brillouin frequency shift [11]. In these models, both the pump wave and the Stokes wave were assumed to be fully polarized.

Despite these inspiring works on the polarization effect in Brillouin scattering, there has been no model to characterize the Brillouin scattering effect with a partially polarized pump, which is very likely in the practical situations. A simple treatment to separate the polarized part and the depolarized part of the pump wave in the simulation is not valid in the pump depletion regime, because a depolarized pump might be repolarized by a polarized Stokes wave, which is a result of polarization pulling. The complex coupling effect between the polarized parts and the depolarized parts of the pump wave and the Stokes wave suggests that a comprehensive model for Brillouin scattering with a partially polarized pump in the pump depletion regime is highly demanded.

In this work, such a theoretical model is proposed by extending our recent work on Raman amplifiers with partially polarized pumps [20], which is based on a set of stochastic differential equations (SDEs). The coupled equation is derived in the Stokes space with the power of the wave and the Stokes vector components of the wave as independent variables. The polarized part of the waves is indicated by the Stokes vector, while the depolarized part of the wave is contained in the total power. The coupled SDE takes the complex nonlinear coupling and the random PMD induced polarization rotation into account, and a thorough study is performed on a Brillouin amplifier with a partially polarized pump.

2. Theory

The pump wave field and the Stokes wave field can be described by either the Jones vectors $|p \rangle$ and $|s \rangle$ or the Stokes vectors $\vec{p}$ and $\vec{s}$, which are related by the following [14]

$$\begin{array}{l} \vec{p} = \left\langle p \right|\vec{\sigma }|p \rangle ,\\ \vec{s} = \left\langle s \right|\vec{\sigma }|s \rangle , \end{array}$$
where $\vec{\sigma }$ is a vector with its elements as three Pauli matrices. The Jones vector describes the full wave with its inner product as the total power of the polarized and the depolarized parts of the wave. The polarized part of the wave is described by the Stokes vector, as the depolarized part becomes zero in it. The Stokes vector magnitude represents the power of the polarized part of the wave, which is related to the total power by:
$$\begin{array}{l} P = \left\langle {p} \mathrel{|{\vphantom {p p}} } {p} \right\rangle ,\\ P = |{\vec{p}} |+ {P^d},\\ S = \left\langle {s} \mathrel{|{\vphantom {s s}} } {s} \right\rangle ,\\ S = |{\vec{s}} |+ {S^d}, \end{array}$$
where P and S stand for the total powers of the pump wave and the Stokes wave, Pd and Sd stand for the powers of the depolarized parts, and $|{\vec{p}} |$ and $|{\vec{s}} |$ stand for the powers of the polarized parts. The DOPs of the waves are defined as:
$$\begin{array}{l} DO{P_{pump}} = \frac{{|{\vec{p}} |}}{P},\\ DO{P_{Stokes}} = \frac{{|{\vec{s}} |}}{S}. \end{array}$$

The evolution of the polarized waves in Brillouin scattering inside the fiber can be described by the equation for the Jones vectors [10]:

$$\begin{array}{l} - \frac{{d|p \rangle }}{{dz}} ={-} \frac{\alpha }{2}|p \rangle - \frac{1}{{4{A_{eff}}}}{g_B}({\vec{s}\cdot \vec{\sigma } + S} )|p \rangle - \frac{j}{2}({({\mathbf{M}\vec{b}} )\cdot \vec{\sigma }} )|p \rangle \\ \mathbf{M} = \left( {\begin{array}{ccc} 1&{}&{}\\ {}&1&{}\\ {}&{}&{ - 1} \end{array}} \right)\\ \frac{{d|s \rangle }}{{dz}} ={-} \frac{\alpha }{2}|s \rangle + \frac{1}{{4{A_{eff}}}}{g_B}({\vec{p}\cdot \vec{\sigma } + P} )|s \rangle - \frac{j}{2}({\vec{b}\cdot \vec{\sigma }} )|s \rangle . \end{array}$$
where the – sign corresponds to the backward propagation of the pump wave, z is the propagation distance, α is the attenuation coefficient, $\vec{b}$ is the PMD vector, gB is the Brillouin gain coefficient and Aeff is the fiber effective mode area. The matrix M exists because the backward propagating pump wave experiences different birefringence with respect to the forward propagating Stokes wave [8–10].

The PMD vector $\vec{b}$ fulfills [16]:

$$\begin{array}{l} \left\langle {\vec{b}(z )} \right\rangle = 0,\\ \left\langle {\vec{b}({{z_1}} )\vec{b}({{z_2}} )} \right\rangle = \frac{{{l_c}}}{2}{\sigma ^2}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over I} \exp \left( { - \frac{{|{{z_1} - {z_2}} |}}{{{l_c}}}} \right), \end{array}$$
where ${\sigma ^2} = \frac{1}{3}{\omega ^2}D_p^2$, with Dp as the fiber PMD parameter, and ω as the angular frequency of the pump wave or the Stokes wave (since they are quite close to each other, the same angular frequency is used). $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over I}$ is the second order unit tensor, and lc is the PMD correlation length. When lc approaches 0, the correlation function approaches the Dirac function.

It should be noted that Eq. (4) is a static model, which does not include the temporal variation of the beam. Generally, it is adequate to study the Brillouin gain fluctuation. It is possible to be extended for the transient case by simply adding the velocity term and including the equation for the acoustic wave. The analysis will be far more complicated and the research is expected to be accomplished in the future.

Based on Eq. (4) and the property of the Pauli matrix, one may derive the following equations (the detailed derivation process is shown in the Appendix):

$$\begin{array}{l} - \frac{{dP}}{{dz}} ={-} \alpha P - \frac{1}{{2{A_{eff}}}}{g_B}({\vec{s}\cdot \vec{p} + SP} ),\\ - \frac{{d\vec{p}}}{{dz}} ={-} \alpha \vec{p} - \frac{1}{{2{A_{eff}}}}{g_B}({P\vec{s} + S\vec{p}} )+ ({\mathbf{M}\vec{b}} )\times \vec{p},\\ \frac{{dS}}{{dz}} ={-} \alpha S + \frac{1}{{2{A_{eff}}}}{g_B}({\vec{s}\cdot \vec{p} + SP} ),\\ \frac{{d\vec{s}}}{{dz}} ={-} \alpha \vec{s} + \frac{1}{{2{A_{eff}}}}{g_B}({P\vec{s} + S\vec{p}} )+ \vec{b} \times \vec{s}. \end{array}$$

In Eq. (6), the total powers, which include the polarized parts and the depolarized parts of the waves, couple with the Stokes vectors, which solely represent the polarized parts of the waves. The complex nonlinear coupling in Brillouin scattering makes the polarized parts and the depolarized parts of the waves inseparable. For example, we may conclude from the 2nd equation in Eq. (6) that a fully depolarized pump wave could be possibly repolarized by a polarized Stokes wave after the propagation, because the term $P\vec{s}$ is non-zero in this case.

3. Results and discussions

We have implemented the proposed model on a Brillouin fiber amplifier. The fiber Brillouin gain parameters of a non-zero dispersion-shifted fiber (NZDSF) is used for modeling: Aeff = 50 µm2, gB = 5 × 10−11 m/W. The reason for the choice of the NZDSF is that the fiber has a smaller core size and a higher Brilouin gain compared to the standard single mode fiber. Certainly, the fiber could be switched without affecting the conclusion of the work. The fiber length is assumed to be 100 m. The pump is located at 1550 nm and the Stokes wave propagates backwardly with the frequency detuning from the Brillouin gain peak Δω as 0. The fiber PMD is assumed to be 0.1 ps/sqrt(km). 1000 realizations are used to study the random Brillouin gain. The initial pump power is 100 mW.

To verify the correctness of the proposed formulas, comparison has been made between the proposed formulas and the undepleted pump model used in [8,9]. Both the pump DOP and the Stokes wave DOP are assumed as 100%. The input Stokes wave varies from -50 dBm to -10 dBm. The results are shown in Fig. 1. From the figure, it can be seen that the average gain and the gain standard deviation predicted by the proposed formulas and the undepleted pump model agree well, particularly for the case with a low input Stokes wave power. As the Stokes wave power grows, the undepleted pump model predicts an unaltered gain because the pump power evolution is not affected by the Stokes wave power (the slight change of the gain by the undepleted pump model in Fig. 1 under different Stokes wave power is caused by the slight fluctuation in the Monte Carlo simulations). However, such an impact can not be ignored as the Stokes wave power grows and the Brillouin gain enters the saturation regime as indicated by the results from Eq. (6), which indicates the necessity to propose the formulas in this work.

 figure: Fig. 1.

Fig. 1. The Brillouin gain under different Stokes wave input power with the pump DOP as 100% and the Stokes DOP as 100% (a) the average Brillouin gain (b) the gain standard deviation.

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It is reported that even a fully depolarized pump could still have Brillouin gain fluctuation. To study this phenomenon, we assume a depolarized pump and a polarized input Stokes wave. The Stokes wave power is assumed as 1 mW. Due to the nonlinear interactions in Brillouin amplification, the polarization of the Stokes wave could affect the DOP of the pump and make the pump to be partially polarized after the propagation and therefore it results in the gain fluctuations. The correlation length of the PMD is in the order of meters [16], and it is assumed to be 1 m and 10 m. The Brillouin gain distributions and the repolarized output pump wave DOP distributions under the two different correlation lengths are shown in Fig. 2. As is shown in the figure, the PMD in the fiber causes random rotation which induces the gain fluctuations, and a probability distribution is formed. When the fiber PMD correlation length is short, e.g. 1 m in Fig. 2(a), the gain fluctuates more significantly and the gain distribution has a much wider range. When the fiber PMD correlation length is relatively longer, e.g. 10 m in Fig. 2(c), the gain experiences less fluctuations. The gain fluctuation is caused by the repolarization of the fully depolarized input pump as is shown in Fig. 2(b, d), i.e., a pump with a zero initial DOP has evolve to a pump with a non-zero output DOP.

 figure: Fig. 2.

Fig. 2. The distribution of the Brillouin gain and the output pump DOP with the input pump DOP as 0% and the input Stokes wave DOP as 100%. (a,b) The fiber PMD correlation length as 1 m (c,d) the fiber PMD correlation length as 10 m.

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Without loss of generality, the PMD correlation length of 1 m is used in the rest of the study.

It is difficult to fully depolarize a pump, and the residual DOP usually occurs. In Fig. 3, the average Brillouin gain and the gain standard deviation under different pump DOP with a polarized Stokes wave are shown. It can be seen from the figure that the average Brillouin gain and the gain standard deviation increase as the pump DOP increases. When the pump and the Stokes wave are both fully polarized, the gain standard deviation is as high as 1.1 dB. The gain standard deviation decreases remarkably as the pump is depolarized. When the residual DOP is 10%, which is usually the case when we try to randomize the pump polarization, the gain standard deviation is about 0.2 dB.

 figure: Fig. 3.

Fig. 3. The Brillouin gain under different pump wave DOP with the Stokes DOP as 100% (a) the average Brillouin gain (b) the gain standard deviation.

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To study the case with a partially polarized input Stokes wave and find the impact of the Stokes DOP on the Brillouin gain fluctuation, the DOP of the Stokes wave is assumed to be 0% to 100% while the DOP of the pump maintains 0%. The results are shown in Fig. 4. The average Brillouin gain decreases and the gain fluctuation increases as the DOP of the Stokes wave increases as is demonstrated in Fig. 4(a, b). For a fully polarized Stokes wave, a depolarized pump could generate an average Brillouin gain of about 15.4 dB with a gain standard deviation of about 0.2 dB. Only with a fully depolarized Stokes wave and a fully depolarized pump wave, one could achieve an average Brillouin gain of about 15.6 dB and 0 dB gain standard deviation. Therefore, controlling the DOP of the Stokes wave in addition to the pump DOP could become a further Brillouin gain fluctuation reduction methodology.

 figure: Fig. 4.

Fig. 4. The Brillouin gain under different Stokes DOP with the pump wave DOP as 0% (a) the average Brillouin gain (b) the gain standard deviation.

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Afterwards, we considered the case when the Stokes wave is fully depolarized and its DOP is 0%. The pump DOP varies from 0% to 100% and the average Brillouin gain and the gain standard deviation are shown in Fig. 5. As can be seen from the figure, the gain standard deviation still increases as the pump DOP increases. The results further confirm the conclusion that the DOPs of the pump and the Stokes should be minimized simultaneously to reduce the Brillouin gain fluctuation, albeit the overall Brillouin gain will be reduced accordingly due to polarization scrambling.

 figure: Fig. 5.

Fig. 5. The Brillouin gain under different pump wave DOP with the Stokes DOP as 0% (a) the average Brillouin gain (b) the gain standard deviation.

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Finally, we consider the impact of pump power and fiber length on the Brillouin gain and its standard deviation. The pump DOP is 0% and the Stokes wave DOP is 100%. The results are shown in the Figs. 6,7. It can be seen from Fig. 6 that when the pump power grows higher, the pump will be depleted more significantly and the gain will be more saturated while it is growing with the increase of the power. The gain standard deviation will reach a peak value as the pump power grows and decrease afterwards. This may be attributed to the effect of polarization pulling while a Stoke wave with a higher power could be more likely to be aligned with the pump wave on the polarization state and thus it reduces the possible gain fluctuation caused by the random polarization rotation between the pump and the Stokes wave. The results in Fig. 7 also show the similar trend. When the fiber length grows, the gain will be more saturated because the Stokes wave experiences more pump gain in the propagation distance. The gain standard deviation also reaches a peak as the fiber length grows, which can also be attributed to the effect of polarization pulling.

 figure: Fig. 6.

Fig. 6. The Brillouin gain under different pump power with the pump DOP as 0% and the Stokes DOP as 100% (a) the average Brillouin gain (b) the gain standard deviation.

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 figure: Fig. 7.

Fig. 7. The Brillouin gain under different fiber length with the pump DOP as 0% and the Stokes DOP as 100%. The pump power is 100 mW. (a) the average Brillouin gain (b) the gain standard deviation.

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It is interesting to compare the two amplifiers, namely the Raman amplifiers [20] and the Brillouin amplifiers, on the PDG with a partially polarized pump. While the Raman amplifiers have broader amplification bandwidths and can have multiple pumps, the Brillouin amplifiers have narrower bandwidths and only one pump can be applied. Astonishingly, both the amplifiers have a similar polarization pulling effect, which will induce DOP transfer from the pump to the Stokes wave and enhance the PDG. While the Raman amplifiers require more than tens of kilometers of fiber to see such a trend, the Brillouin amplifiers require only tens of meters of fiber due to the higher gain coefficient comparing with the Raman amplifiers.

4. Conclusion

In summary, we have proposed a comprehensive model to describe the polarization behavior of the Brillouin scattering waves with a partially polarized pump wave. A SDE which incorporates the complex interactions between the polarized and the depolarized pump wave and the Stokes wave as well as the random PMD vector, is proposed. The proposed model could be valuable to study the performance of a fiber system with Brillouin scattering. A Brillouin amplifier is studied, and the study shows that the DOPs of the pump wave and Stokes wave affect the Brillouin gain stability and should be minimized simultaneously by polarization randomization, which may be accomplished by scrambling the pump wave directly at the input, and meanwhile, scrambling the probe wave in the middle of the fiber by inserting a polarization scrambler if the probe is initiated by the noise. As is mentioned previously, the model is a static one and therefore, it is not possible to perform the transient simulation in the time domain. One may try to find the characteristics of the gain bandwidth by making the Brillouin gain coefficient frequency dependent as the Stoke wave frequency moves away from the center of the Brillouin gain peak. The analysis for the gain bandwidth could be performed in the frequency domain accordingly.

Appendix

In the appendix, the derivation of Eq. (6) is provided in detail. We may rewrite the first equation in Eq. (4) as:

$$- \frac{{d|p \rangle }}{{dz}} ={-} \frac{\alpha }{2}|p \rangle - \frac{1}{{4{A_{eff}}}}{g_B}({\vec{s}\cdot \vec{\sigma } + S} )|p \rangle - \frac{j}{2}({({\mathbf{M}\vec{b}} )\cdot \vec{\sigma }} )|p \rangle .$$

Taking complex conjugate on both sides of the equation, we have:

$$- \frac{{d\left\langle p \right|}}{{dz}} ={-} \frac{\alpha }{2}\left\langle p \right|- \frac{1}{{4{A_{eff}}}}{g_B}\left\langle p \right|({\vec{s}\cdot \vec{\sigma } + S} )+ \left\langle p \right|\frac{j}{2}({({\mathbf{M}\vec{b}} )\cdot \vec{\sigma }} ).$$

We may obtain:

$$\begin{array}{l} - \frac{{dP}}{{dz}} ={-} \frac{{d\left\langle {p} \mathrel{|{\vphantom {p p}} } {p} \right\rangle }}{{dz}} ={-} \left\langle p \right|\frac{{d|p \rangle }}{{dz}} - \frac{{d\left\langle p \right|}}{{dz}}|p \rangle \\ ={-} \frac{\alpha }{2}\left\langle {p} \mathrel{|{\vphantom {p p}} } {p} \right\rangle - \frac{1}{{4{A_{eff}}}}{g_B}S\left\langle {p} \mathrel{|{\vphantom {p p}} } {p} \right\rangle - \frac{1}{{4{A_{eff}}}}{g_B}\left\langle p \right|\vec{s}\cdot \vec{\sigma }|p \rangle - \frac{j}{2}\left\langle p \right|({({\mathbf{M}\vec{b}} )\cdot \vec{\sigma }} )|p \rangle \\ - \frac{\alpha }{2}\left\langle {p} \mathrel{|{\vphantom {p p}} } {p} \right\rangle - \frac{1}{{4{A_{eff}}}}{g_B}S\left\langle {p} \mathrel{|{\vphantom {p p}} } {p} \right\rangle - \frac{1}{{4{A_{eff}}}}{g_B}\left\langle p \right|\vec{s}\cdot \vec{\sigma }|p \rangle + \frac{j}{2}\left\langle p \right|({({\mathbf{M}\vec{b}} )\cdot \vec{\sigma }} )|p \rangle \\ ={-} \alpha \left\langle {p} \mathrel{|{\vphantom {p p}} } {p} \right\rangle - \frac{1}{{2{A_{eff}}}}{g_B}S\left\langle {p} \mathrel{|{\vphantom {p p}} } {p} \right\rangle - \frac{1}{{2{A_{eff}}}}{g_B}\left\langle p \right|\vec{s}\cdot \vec{\sigma }|p \rangle . \end{array}$$

Using the property [14],

$$\left\langle p \right|({\vec{s}\cdot \vec{\sigma }} )|p \rangle = \vec{s}\cdot \vec{p},$$
we have:
$$- \frac{{dP}}{{dz}} ={-} \alpha P - \frac{1}{{2{A_{eff}}}}{g_B}SP - \frac{1}{{2{A_{eff}}}}{g_B}\vec{s}\cdot \vec{p},$$
which is the first equation of Eq. (6).

We may also obtain:

$$\begin{array}{l} \frac{{d\vec{p}}}{{dz}} ={-} \frac{{d\left\langle p \right|\vec{\sigma }|p \rangle }}{{dz}} ={-} \left\langle p \right|\vec{\sigma }\frac{{d|p \rangle }}{{dz}} - \frac{{d\left\langle p \right|}}{{dz}}\vec{\sigma }|p \rangle \\ ={-} \frac{\alpha }{2}\left\langle p \right|\vec{\sigma }|p \rangle - \frac{1}{{4{A_{eff}}}}{g_B}S\left\langle p \right|\vec{\sigma }|p \rangle - \frac{1}{{4{A_{eff}}}}{g_B}\left\langle p \right|\vec{\sigma }({\vec{s}\cdot \vec{\sigma }} )|p \rangle - \frac{j}{2}\left\langle p \right|\vec{\sigma }({({\mathbf{M}\vec{b}} )\cdot \vec{\sigma }} )|p \rangle \\ - \frac{\alpha }{2}\left\langle p \right|\vec{\sigma }|p \rangle - \frac{1}{{4{A_{eff}}}}{g_B}S\left\langle p \right|\vec{\sigma }|p \rangle - \frac{1}{{4{A_{eff}}}}{g_B}\left\langle p \right|({\vec{s}\cdot \vec{\sigma }} )\vec{\sigma }|p \rangle + \frac{j}{2}\left\langle p \right|({({\mathbf{M}\vec{b}} )\cdot \vec{\sigma }} )\vec{\sigma }|p \rangle . \end{array}$$

Since [14]:

$$\begin{array}{l} \vec{\sigma }({\vec{s}\cdot \vec{\sigma }} )= \vec{s}\mathbf{I} + j\vec{s} \times \vec{\sigma },\\ ({\vec{s}\cdot \vec{\sigma }} )\vec{\sigma } = \vec{s}\mathbf{I} - j\vec{s} \times \vec{\sigma },\\ \vec{\sigma }({({\mathbf{M}\vec{b}} )\cdot \vec{\sigma }} )= ({\mathbf{M}\vec{b}} )\mathbf{I} + j({\mathbf{M}\vec{b}} )\times \vec{\sigma },\\ ({({\mathbf{M}\vec{b}} )\cdot \vec{\sigma }} )\vec{\sigma } = ({\mathbf{M}\vec{b}} )\mathbf{I} - j({\mathbf{M}\vec{b}} )\times \vec{\sigma }, \end{array}$$
we have:
$$\begin{array}{l} \vec{\sigma }({\vec{s}\cdot \vec{\sigma }} )+ ({\vec{s}\cdot \vec{\sigma }} )\vec{\sigma } = 2\vec{s}\mathbf{I},\\ \vec{\sigma }({({\mathbf{M}\vec{b}} )\cdot \vec{\sigma }} )- ({({\mathbf{M}\vec{b}} )\cdot \vec{\sigma }} )\vec{\sigma } = 2j({\mathbf{M}\vec{b}} )\times \vec{\sigma }. \end{array}$$

Therefore, we have

$$\frac{{d\vec{p}}}{{dz}} ={-} \alpha \left\langle p \right|\vec{\sigma }|p \rangle - \frac{1}{{2{A_{eff}}}}{g_B}S\left\langle p \right|\vec{\sigma }|p \rangle - \frac{1}{{2{A_{eff}}}}{g_B}\left\langle {p} \mathrel{|{\vphantom {p p}} } {p} \right\rangle \vec{s} + \left\langle p \right|({\mathbf{M}\vec{b}} )\times \vec{\sigma }|p \rangle .$$

Since [14],

$$\left\langle p \right|({({\mathbf{M}\vec{b}} )\times \vec{\sigma }} )|p \rangle = ({\mathbf{M}\vec{b}} )\times \vec{p},$$
we have:
$$\frac{{d\vec{p}}}{{dz}} ={-} \alpha \vec{p} - \frac{1}{{2{A_{eff}}}}{g_B}S\vec{p} - \frac{1}{{2{A_{eff}}}}{g_B}P\vec{s} + \mathbf{M}\vec{b} \times \vec{p},$$
which is the second equation of Eq. (6). In a similar manner, the equation for the Stokes wave can be obtained, i.e., Eq. (6) is derived.

Funding

National Science and Technology Major Project (2022ZD0119302); National Natural Science Foundation of China (62375206).

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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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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Figures (7)
Fig. 1.
Fig. 1. The Brillouin gain under different Stokes wave input power with the pump DOP as 100% and the Stokes DOP as 100% (a) the average Brillouin gain (b) the gain standard deviation.

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Fig. 2.
Fig. 2. The distribution of the Brillouin gain and the output pump DOP with the input pump DOP as 0% and the input Stokes wave DOP as 100%. (a,b) The fiber PMD correlation length as 1 m (c,d) the fiber PMD correlation length as 10 m.

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Fig. 3.
Fig. 3. The Brillouin gain under different pump wave DOP with the Stokes DOP as 100% (a) the average Brillouin gain (b) the gain standard deviation.

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Fig. 4.
Fig. 4. The Brillouin gain under different Stokes DOP with the pump wave DOP as 0% (a) the average Brillouin gain (b) the gain standard deviation.

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Fig. 5.
Fig. 5. The Brillouin gain under different pump wave DOP with the Stokes DOP as 0% (a) the average Brillouin gain (b) the gain standard deviation.

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Fig. 6.
Fig. 6. The Brillouin gain under different pump power with the pump DOP as 0% and the Stokes DOP as 100% (a) the average Brillouin gain (b) the gain standard deviation.

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Fig. 7.
Fig. 7. The Brillouin gain under different fiber length with the pump DOP as 0% and the Stokes DOP as 100%. The pump power is 100 mW. (a) the average Brillouin gain (b) the gain standard deviation.

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Equations (17)

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p → = ⟨ p | σ → | p ⟩ , s → = ⟨ s | σ → | s ⟩ ,
P = ⟨ p | p p p ⟩ , P = | p → | + P d , S = ⟨ s | s s s ⟩ , S = | s → | + S d ,
D O P p u m p = | p → | P , D O P S t o k e s = | s → | S .
− d | p ⟩ d z = − α 2 | p ⟩ − 1 4 A e f f g B ( s → ⋅ σ → + S ) | p ⟩ − j 2 ( ( M b → ) ⋅ σ → ) | p ⟩ M = ( 1 1 − 1 ) d | s ⟩ d z = − α 2 | s ⟩ + 1 4 A e f f g B ( p → ⋅ σ → + P ) | s ⟩ − j 2 ( b → ⋅ σ → ) | s ⟩ .
⟨ b → ( z ) ⟩ = 0 , ⟨ b → ( z 1 ) b → ( z 2 ) ⟩ = l c 2 σ 2 I ↔ exp ⁡ ( − | z 1 − z 2 | l c ) ,
− d P d z = − α P − 1 2 A e f f g B ( s → ⋅ p → + S P ) , − d p → d z = − α p → − 1 2 A e f f g B ( P s → + S p → ) + ( M b → ) × p → , d S d z = − α S + 1 2 A e f f g B ( s → ⋅ p → + S P ) , d s → d z = − α s → + 1 2 A e f f g B ( P s → + S p → ) + b → × s → .
− d | p ⟩ d z = − α 2 | p ⟩ − 1 4 A e f f g B ( s → ⋅ σ → + S ) | p ⟩ − j 2 ( ( M b → ) ⋅ σ → ) | p ⟩ .
− d ⟨ p | d z = − α 2 ⟨ p | − 1 4 A e f f g B ⟨ p | ( s → ⋅ σ → + S ) + ⟨ p | j 2 ( ( M b → ) ⋅ σ → ) .
− d P d z = − d ⟨ p | p p p ⟩ d z = − ⟨ p | d | p ⟩ d z − d ⟨ p | d z | p ⟩ = − α 2 ⟨ p | p p p ⟩ − 1 4 A e f f g B S ⟨ p | p p p ⟩ − 1 4 A e f f g B ⟨ p | s → ⋅ σ → | p ⟩ − j 2 ⟨ p | ( ( M b → ) ⋅ σ → ) | p ⟩ − α 2 ⟨ p | p p p ⟩ − 1 4 A e f f g B S ⟨ p | p p p ⟩ − 1 4 A e f f g B ⟨ p | s → ⋅ σ → | p ⟩ + j 2 ⟨ p | ( ( M b → ) ⋅ σ → ) | p ⟩ = − α ⟨ p | p p p ⟩ − 1 2 A e f f g B S ⟨ p | p p p ⟩ − 1 2 A e f f g B ⟨ p | s → ⋅ σ → | p ⟩ .
⟨ p | ( s → ⋅ σ → ) | p ⟩ = s → ⋅ p → ,
− d P d z = − α P − 1 2 A e f f g B S P − 1 2 A e f f g B s → ⋅ p → ,
d p → d z = − d ⟨ p | σ → | p ⟩ d z = − ⟨ p | σ → d | p ⟩ d z − d ⟨ p | d z σ → | p ⟩ = − α 2 ⟨ p | σ → | p ⟩ − 1 4 A e f f g B S ⟨ p | σ → | p ⟩ − 1 4 A e f f g B ⟨ p | σ → ( s → ⋅ σ → ) | p ⟩ − j 2 ⟨ p | σ → ( ( M b → ) ⋅ σ → ) | p ⟩ − α 2 ⟨ p | σ → | p ⟩ − 1 4 A e f f g B S ⟨ p | σ → | p ⟩ − 1 4 A e f f g B ⟨ p | ( s → ⋅ σ → ) σ → | p ⟩ + j 2 ⟨ p | ( ( M b → ) ⋅ σ → ) σ → | p ⟩ .
σ → ( s → ⋅ σ → ) = s → I + j s → × σ → , ( s → ⋅ σ → ) σ → = s → I − j s → × σ → , σ → ( ( M b → ) ⋅ σ → ) = ( M b → ) I + j ( M b → ) × σ → , ( ( M b → ) ⋅ σ → ) σ → = ( M b → ) I − j ( M b → ) × σ → ,
σ → ( s → ⋅ σ → ) + ( s → ⋅ σ → ) σ → = 2 s → I , σ → ( ( M b → ) ⋅ σ → ) − ( ( M b → ) ⋅ σ → ) σ → = 2 j ( M b → ) × σ → .
d p → d z = − α ⟨ p | σ → | p ⟩ − 1 2 A e f f g B S ⟨ p | σ → | p ⟩ − 1 2 A e f f g B ⟨ p | p p p ⟩ s → + ⟨ p | ( M b → ) × σ → | p ⟩ .
⟨ p | ( ( M b → ) × σ → ) | p ⟩ = ( M b → ) × p → ,
d p → d z = − α p → − 1 2 A e f f g B S p → − 1 2 A e f f g B P s → + M b → × p → ,
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