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Polarization dependent gain in Raman fiber amplifiers with multiple pumps

Junhe Zhou, Wang Yuheng, and Tengyuan Liu

Open Access Open Access

Abstract

In this paper, the polarization dependent gain (PDG) in Raman fiber amplifiers (RFAs) with multiple pumps is studied thoroughly. A comprehensive model, which takes the random polarization mode dispersion, the nonlinear coupling between the pumps, and the degree of polarization (DOP) of the pumps into account, is proposed. The complex nonlinear and random coupling inside the optical fiber is described by a set of nonlinear stochastic differential equations (SDEs), which enable co-simulation of the polarized part and the depolarized part of the multiple pumps. It is revealed that the average PDG and the PDG standard deviation are linearly proportional to the DOP of the pumps, which agrees with the single mode case. More importantly, when the pump wavelength is far away from the signal amplification range (pump-signal wavelength difference larger than 100 nm), its DOP still affects the PDG of the signal. Such a phenomenon is caused by the fact that the pumps interact with each other and the pump DOP could transfer among the pumps, which could enhance the PDG of the RFA. The findings in the work will have important implications for the design of the optical transmission systems with the multi-pump RFAs.

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

1. Introduction

Raman fiber amplifiers (RFAs) have been widely applied in the long-haul transmission systems due to their wide amplification bandwidth and ultra-low equivalent noise figure (NF) [1–4]. It is known that the Raman gain is polarization dependent if the pump laser is polarized. The signal and the pump waves undergo different polarization rotations in the fiber due to the polarization mode dispersion (PMD) effect, and therefore, the relative polarization orientation between the pump and the signal changes, which results in randomly varying polarization dependent gain (PDG) along the fiber. A depolarized pump will significantly reduce the PDG of the RFA [5–6]. However, usually the depolarizer cannot fully depolarize the pump and the residual degree of the polarization (DOP) could be close to 5% or 10% [6–7].

Characterization such a behavior is essential for the RFA analysis and it has caught the attention of the research community. Several pioneering researches have been carried out. Q. Lin et. al. [8–10] proposed a comprehensive theory on this topic by assuming an un-depleted polarized pump based on the stochastic differential equation (SDE) theory. The variable gain and the gain statistics were obtained analytically. The studies suggested that the Raman gain was polarization dependent and the PDG varied randomly. Also, the amount of average PDG scaled linearly with respect to the pump DOP [10]. J. Cheng et. al [11] proposed a general model to characterize the polarization dependent Raman gain in the sense of Jones matrix. It was also found that the varying PDG in RFAs would enhance the relative intensity noise (RIN) transfer [12–13].

Despite the outstanding researches in the field, most of the existing studies on the topic focused on the single pump RFAs [8–13]. However, the RFAs with multiple pumps have a wider and flatter gain spectrum and are more widely used in the practical systems [1–4]. Although the single pump RFAs has an analytical formulation under the un-depleted pump assumption, the pump power evolution and gain variation along the fiber becomes quite complicated if the depletion is considered [14]. It was found in [14] that even a depolarized pump could be repolarized through depletion by a polarized signal, which could be the result of polarization pulling [15–18], i.e., the Stokes wave polarization state will evolve towards the pump polarization state. Such phenomenon was also referred to as the “Raman polarizer” effect [19–21]. This suggests that the pumps in a multi-pump RFA undergo complex nonlinear interactions and the power transferring from the shorter wavelength pump to the longer wavelength pump might bring DOP transferring. Such transferring suggests the multi-pump RFAs could have a higher PDG in comparison to the single pump RFAs. Upon such a complicated yet important problem, few studies have been performed. D. Wang et. al. investigated a RFA with multiple forward pumps [22], however, neither the DOP of the pumps nor the random PMD were considered in the model. In [14], only the pump depletion by the signal was considered as the cause of the PDG, the intrinsic DOP of the pumps were not included.

As suggested above, a model to co-simulate the polarized part and the depolarized part of the multiple pumps in the RFAs is essential, which is currently not available. In this paper, such a model is derived rigorously from the Jones matrix formulation, which considers the power and the Stokes vector components of the pumps to be independent variables so that they form a set of coupled random SDEs to characterize the random PMD effect and the complex interactions between the pump powers and the pump Stokes vector components. A four-pump RFA is studied by the model with the SDEs solved by the Monte Carlo simulations. It is revealed that even the pumps far away from signal amplification range could have a significant impact on the signal PDG in the RFA.

2. Theory

In this work, the mth pump field is described by either the Jones vector $|{{p_m}} \rangle$ or the Stokes vector ${\vec{p}_m}$, which are related by the following [23]

$${\vec{p}_m} = \left\langle {{p_m}} \right|\vec{\sigma }|{{p_m}} \rangle ,$$
where $\vec{\sigma }$ is a vector with its elements as three Pauli matrices. While the Jones vectors contains the polarized and depolarized part of the pump, the Stokes vector represents the polarized part and its magnitude does not equal the total power Pm if the light is partially polarized:
$$\begin{array}{l} {P_m} = \left\langle {{{p_m}}} \mathrel{|{\vphantom {{{p_m}} {{p_m}}}}} {{{p_m}}} \right\rangle ,\\ {P_m} = |{{{\vec{p}}_m}} |+ {P_m}^d, \end{array}$$
where Pmd stands for the depolarized part of the pump power. Due to the nature that the Jones vector contains the full power of the pump, we start with the vector theory of the RFAs in the form of Jones matrix form [10–11]. Without loss of generality, we have assumed that the pumps co-propagate with the signal. One may easily convert the equation for the backward pumping scheme by adding the negative sign. Since the signal power is relatively small comparing with the pumps, its contribution to the pump evolution can be ignored [10–11]. Under these conditions, we have the equations for the pump and signal Jones vectors as:
$$\begin{array}{l} \frac{{d|{{p_m}} \rangle }}{{dz}} ={-} \frac{{{\alpha _{pm}}}}{2}|{{p_m}} \rangle + \frac{1}{4}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} )({{{\vec{p}}_n}\cdot \vec{\sigma } + {P_n}} )|{{p_m}} \rangle } - \frac{j}{2}{\omega _m}({\vec{b}\cdot \vec{\sigma }} )|{{p_m}} \rangle \\ m = 1 \cdots M,\\ \frac{{d|s \rangle }}{{dz}} ={-} \frac{{{\alpha _s}}}{2}|s \rangle + \frac{1}{4}\sum\limits_{n = 1}^M {{g_R}({{\omega_n},{\omega_s}} )({{{\vec{p}}_n}\cdot \vec{\sigma } + {P_n}} )|s \rangle } - \frac{j}{2}{\omega _s}({\vec{b}\cdot \vec{\sigma }} )|s \rangle , \end{array}$$
where $|s \rangle$ is the signal Jones vector, z is the propagation distance, αpm and αs are the attenuation coefficients on the corresponding pump frequency ωm and the signal frequency ωs, $\vec{b}$ is the PMD vector, and ${g_R}({{\omega_n},{\omega_m}} )$ is the Raman gain coefficient between frequency ωmand frequency ωn [1]:
$${g_R}({{\omega_n},{\omega_m}} )= \left\{ {\begin{array}{{c}} {{\gamma_R}({{\omega_n},{\omega_m}} ),\quad \quad \quad {\omega_n} > {\omega_m},}\\ { - \frac{{{\omega_n}}}{{{\omega_m}}}{\gamma_R}({{\omega_m},{\omega_n}} ),\quad {\omega_n} < {\omega_m},} \end{array}} \right.$$
where γR indicates the Raman gain between frequency ωmand frequency ωn. The PMD vector $\vec{b}$ fulfills [10]:
$$\begin{array}{l} \left\langle {\vec{b}(z )} \right\rangle = 0,\\ \left\langle {\vec{b}({{z_1}} )\vec{b}({{z_2}} )} \right\rangle = \frac{1}{3}D_p^2\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over I} \delta ({{z_1} - {z_2}} ), \end{array}$$
where Dp is the fiber PMD parameter, and $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over I}$ is the second order unit tensor. Based on Eq. (3) and following the detailed derivation process in the Appendix, one may rigorously derive two important coupled equations:
$$\frac{{d{P_m}}}{{dz}} ={-} {\alpha _{pm}}{P_m} + \frac{1}{2}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} ){P_n}{P_m}} + \frac{1}{2}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} ){{\vec{p}}_n}\cdot {{\vec{p}}_m}} .$$
$$\frac{{d{{\vec{p}}_m}}}{{dz}} ={-} {\alpha _{pm}}{\vec{p}_m} + \frac{1}{2}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} ){P_n}{{\vec{p}}_m}} + \frac{1}{2}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} ){{\vec{p}}_n}{P_m}} + {\omega _m}\vec{b} \times {\vec{p}_m}.$$

Eq. (7) resembles the equation derived in [10]. However, there is a remarkable difference between the two, i.e., the pump power ${P_m} = |{{{\vec{p}}_m}} |$ in [10] while it is not the case in Eq. (7) as demonstrated in Eq. (2).

If the pump has been assumed to be fully polarized as in [10], one has ${\vec{p}_m} = {P_m}{\hat{p}_m}$, where ${\hat{p}_m}$ is the unit Stokes vector of the mth pump. In this case, Eq. (6) becomes a subsidiary equation, i.e., it can be derived from Eq. (7). If the pump is not fully polarized, and one has ${\vec{p}_m} \ne {P_m}{\hat{p}_m}$. Eq. (6) becomes an independent equation and forms a coupled equation together with Eq. (7). Combining Eqs. ((6)–(7)), one may comprehensively characterize the complex and random pump power evolution in the RFAs.

Compared with the Jones vector model in [11], the model proposed in this work enables the direct simulation of a Raman amplifier with multiple partially depolarized pumps, which has not been discussed before.

It is attractive to find an analytical solution to the equations. However, it seems difficult with the complex nonlinear interaction between the different pumps. The existing analytical solution in Refs. [20–21] had assumed an un-depleted pump, which can not be valid in the present case.

The two coupled equations have significant indications, i.e., the depolarized light could be repolarized again through Raman amplification. For example, if the mth pump is fully depolarized at the input while one of the other pumps, e.g., pump n is not fully depolarized, the term ${g_R}({{\omega_n},{\omega_m}} ){\vec{p}_n}{P_m}$ will be non-zero and it makes ${\vec{p}_m}$ non-zero after the evolution in the optical fiber, i.e., pump m is repolarized after the propagation in the RFA.

To characterize the PDG, the equation for the polarization transfer matrix, i.e., the Jones matrix of the signal is used [11]:

$$\frac{{d{{\mathbf T}_{s}}}}{{dz}} ={-} \frac{{{\alpha _s}}}{2}{{\mathbf T}_s} + \frac{1}{4}\sum\limits_{n = 1}^M {{g_R}({{\omega_n},{\omega_s}} )({{{\vec{p}}_n}\cdot \vec{\sigma } + {P_n}} ){{\mathbf T}_s}} - \frac{j}{2}{\omega _s}({\vec{b}\cdot \vec{\sigma }} ){{\mathbf T}_s},$$
where Ts is the Jones matrix of the signal, and the PDG in dB can characterized as:
$$\begin{array}{l} PDG = 10{\log _{10}}\left( {\frac{{{\lambda_{\max }}}}{{{\lambda_{\min }}}}} \right),\\ {[{{\lambda_{\max }},{\lambda_{\min }}} ]}= eig({{{\mathbf T}_s}{{\mathbf T}_s}^H} ), \end{array}$$
where eig stands for the eigen value of the matrix and H stands for the Hermitian transpose.

3. Results and discussions

We have performed a thorough study on a four-pump RFA with an 80 km-long Corning SMF28e fiber. The Raman gain spectrum and the attention spectrum of the fiber can be found in [24–25]. The four pump wavelengths are 1420 nm, 1430 nm, 1450 nm, and 1460 nm, which are selected to ensure the C band amplification. The corresponding pump powers are 270 mW, 160 mW, 260 mW and 180 mW, which have been optimized for gain flatness. The backward pumping scheme is assumed. Firstly, the pumps are assumed to be fully depolarized with the gain spectrum of the RFA being optimized to be flat in the C band, i.e., 1528 nm to 1564 nm. The optimized on-off gain spectrum is shown in Fig. 1 which suggests the attenuation of the 80-km fiber has been almost fully compensated after the distributed amplification. As shown in Fig. 1, the four pumps create several peaks in the C band. The first peak appears around 1530 nm, which is mostly created by the 2nd pump at the wavelength of 1430 nm. The last peak appears around 1560 nm, which is mostly created by the 4th pump at the wavelength of 1460 nm. It is shown in Fig. 1 that the Raman gain profile decreases rapidly if the pump and the signal wavelength difference is larger than 100 nm, as can be seen from the gain spectrum within the wavelength range of 1560 nm to 1565 nm. This indicates that the 1st pump and the 2nd pump at the wavelength of 1420 nm and 1430 nm could hardly provide amplification alone for the signal at the wavelength of 1550 nm.

 figure: Fig. 1.

Fig. 1. The on-off gain spectrum of the Raman fiber amplifier with four pumps in an 80 km SMF28e fiber span.

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The impact of the pump DOP on the PDG of the RFA is studied afterwards. The signal wavelength is set as 1550 nm. The fiber PMD is assumed to be 0.1 ps/sqrt(km). Firstly, the pumps are assumed with the same amount of DOP ranging from 0% to 20%. The average PDG and the standard deviation of the PDG are computed from the two coupled Eqs. ((6)-(7)) through Monte Carlo simulations with 3000 different realizations. As can be seen from Fig. 2, the average PDG and the standard deviation of the PDG grow linearly as the DOP increases from 0% to 20%. Such a linear scaling relationship agrees with the results for the single pump RFAs [10]. It can be seen from Fig. 2 that when the pump DOP is 10%, the average PDG is around 0.15 dB and its standard deviation is around 0.06 dB. This indicates the PDG could possibly exceed 0.2 dB for the multi-pump RFA, which is higher than the one pump RFA case [7]. This is because the complex interactions between the multiple pumps could enhance the DOP of the pumps.

 figure: Fig. 2.

Fig. 2. The average PDG and PDG standard deviation under different DOP of the four pumps.

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Fig. 3 shows the distribution of the 3000 realizations of the FRA PDG. Fig. 3(a-d) have the DOP of the four pumps ranging from 5% to 20%. From Fig. 3(a) and Fig. 3(b), it can be seen more clearly that one requires the pump DOP to be below 5% to ensure the PDG to be smaller than 0.2 dB. When the DOP reaches 20%, the PDG could reach as high as 0.5 dB with some possibility as is in Fig. 3(d).

 figure: Fig. 3.

Fig. 3. The PDG distribution with the four pump DOP as (a) 5% (b) 10% (c) 15% (d) 20%.

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A more thorough study on the impact of the individual pump DOP on the PDG is shown in Figs. 4–6. The four pumps are assumed with the DOP from 0% to 20% individually while the rest of the three pumps are fully depolarized. The signal wavelength is still 1550 nm and the fiber PMD is still 0.1 ps/sqrt(km). 3000 realizations were computed in the Monte Carlo simulations.

 figure: Fig. 4.

Fig. 4. The average PDG under different DOP of an individual pump.

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 figure: Fig. 5.

Fig. 5. The PDG standard deviation under different DOP of an individual pump.

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 figure: Fig. 6.

Fig. 6. The PDG distribution with one pump DOP as 5% (a) the 1420 nm pump (b) the 1430 nm pump (c) the 1450 nm pump (d) the 1460 nm pump.

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The average PDG is shown in Fig. 4 and the standard deviation of the PDG is shown in Fig. 5. It is interesting to see while the DOPs of the 1450 nm pump and the 1460 nm pump affect the PDG of the signal at 1550 nm, the DOPs of the 1420 nm pump and the 1430 nm pump also affect the PDG as well. As is discussed in the previous paragraph, the Raman gain range is within 100 nm in the C band and the 1420 nm pump and the 1430 nm pump can not provide sufficient gain for the signal at 1550 nm. However, their DOPs still have significant impact on the signal PDG. This is because the amplification from the shorter wavelength pumps to the longer wavelength pumps could transfer the DOP of the former to the latter. Thus, although the 1450 nm pump and the 1460 nm pump are fully depolarized at the input, the DOPs of the 1420 nm pump and the 1430 nm pump could still cause remarkable PDG. It can be seen from the figures that the 1450 nm pump has the most significant impact on the signal PDG at 1550 nm while the 1430 nm pump has the least impact.

A second important phenomenon to notice is that if the average PDG or the PDG standard deviation caused by each pump adds up, one has the overall average PDG or the PDG standard deviation caused by the four pumps with the same DOP as shown in Fig. 2. Therefore, gathering the information of the DOP of each pump is quite essential to characterize the PDG of a RFA with multiple pumps whose DOPs are not identical, which is very likely to be true in the practical applications.

Fig. 6 shows the distribution of the PDG under the 5% DOP of an individual pump from the 1420 nm pump to the 1460 nm pump. It can be seen from the figure that a partially polarized 1450 nm pump has the highest probability to reach a high PDG, which is in accordance with the results in Figs. 4–5. Furthermore, as shown in Figs. 6 (a-b), the 1420 nm pump and the 1430 nm pump do have contributions to the signal PDG although the wavelength difference between the pump and the signal is more than 100 nm.

In case the four pumps have different DOPs, which is likely the case in the practical applications, the signals at different wavelengths will experience different PDGs, albeit they have similar average gain, which is a unique feature in the multi-pump FRAs. Fig. 7 demonstrates the PDG with respect to different wavelengths. The four pumps are assumed to be with the DOP of 5% at 1420 nm, 10% at 1430 nm, 15% at 1450 nm, and 20% at 1460 nm. It can be seen from the figure that the signals experience quite different PDG levels. The PDG peak for the average PDG and the PDG standard deviation occurs at the wavelength of 1560 nm, where all the four pumps contribute to the PDG at this peak wavelength.

 figure: Fig. 7.

Fig. 7. The average PDG and PDG standard deviation under different PMD values. The four pumps are assumed to be with the DOP of 5% at 1420 nm, 10% at 1420 nm, 15% at 1420 nm, and 20% at 1420 nm.

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The reason for the DOP of the pump outside the gain range could affect the signal PDG is the DOP transferring phenomenon, which is clearly demonstrated in Fig. 8. The 1420 nm pump is assumed to be with a 100% initial DOP while the rest three pumps have the intial DOPs of 0%. The average DOPs along the optical fiber are shown in Fig. 8. As is shown in the figure, the DOPs of the three pumps grows from 0% to 10% or 25% as the pump waves evolve in the fiber. The phenomenon is unique in the multi-pump FRA. The DOP evolution suggests that the DOP transferring concentrates in the first 20-km length of the fiber, because the pump power attenuates significantly after the 20-km propagation and the nonlinear interaction becomes less significant.

 figure: Fig. 8.

Fig. 8. The average DOP evolution of the four pumps when the 1420 nm pump DOP is 100% and the rest three pumps’ DOPs are 0%.

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The impact of the fiber PMD on the PDG is studied afterwards, the four pumps have the fixed DOP of 10% and the fiber parameters remain the same except that the PMD value varies from 0.01 ps/sqrt(km) to 10 ps/sqrt(km). As one may see from Fig. 9, the PMD value does have significant impact on the PDG. When the PMD is low, the PDG is high and so is its standard deviation. As the PMD grows, the PDG decreases and reaches a relatively stable value when the PMD exceeds 0.1 ps/sqrt(km). The results are in accordance with the published results for the one pump Raman amplifiers [9]. The results in Fig. 9 suggest that the typical PMD value of 0.1 ps/sqrt(km) used in the simulations of this work is reasonable and representative.

 figure: Fig. 9.

Fig. 9. The average PDG and PDG standard deviation under different PMD values.

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Finally, it is worth mentioning that the work studies the weak signal case, i.e., the depletion of the pump by the signal is neglected. In the case of a relatively large signal, one should consider the signal as a part of the pump wave in the equation and the nonlinear coupling between the pumps and the signal will affect results.

4. Conclusion

In summary, we have derived a set of coupled SDEs to characterize the PDG in a RFA with multiple pumps. Monte Carlo simulations are performed to study a four-pump RFA in a span with the fiber length of 80 km. It is shown in the study that the PDG scales linearly with respect to the DOP of each pump. The scaling factor differs from one pump to another, and even the partially polarized pump far away from the signal amplification range has a significant impact on the PDG. The proposed model can be useful in the design and analysis of the optical transmission systems with the multi-pump RFAs.

Appendix

In the appendix, the derivation of Eqs. ((6)-(7)) is provided in detail. We may rewrite Eq. (3) as:

$$\begin{array}{l} \frac{{d|{{p_m}} \rangle }}{{dz}} ={-} \frac{{{\alpha _{pm}}}}{2}|{{p_m}} \rangle + \frac{1}{4}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} )({{{\vec{p}}_n}\cdot \vec{\sigma } + {P_n}} )|{{p_m}} \rangle } - \frac{j}{2}{\omega _m}({\vec{b}\cdot \vec{\sigma }} )|{{p_m}} \rangle ,\\ m = 1 \cdots M. \end{array}$$

Taking Hermitian transpose on both sides of the equation, we have:

$$\begin{array}{l} \frac{{d\left\langle {{p_m}} \right|}}{{dz}} ={-} \frac{{{\alpha _{pm}}}}{2}\left\langle {{p_m}} \right|+ \frac{1}{4}\left\langle {{p_m}} \right|\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} )({{{\vec{p}}_n}\cdot \vec{\sigma } + {P_n}} )} + \frac{j}{2}\left\langle {{p_m}} \right|{\omega _m}({\vec{b}\cdot \vec{\sigma }} ).\\ m = 1 \cdots M. \end{array}$$

We may obtain:

$$\begin{array}{l} \frac{{d{P_m}}}{{dz}} = \frac{{d\left\langle {{{p_m}}} \mathrel{|{\vphantom {{{p_m}} {{p_m}}}}} {{{p_m}}} \right\rangle }}{{dz}} = \frac{{d\left\langle {{p_m}} \right|}}{{dz}}|{{p_m}} \rangle + \left\langle {{p_m}} \right|\frac{{d|{{p_m}} \rangle }}{{dz}}\\ ={-} \frac{{{\alpha _{pm}}}}{2}\left\langle {{{p_m}}} \mathrel{|{\vphantom {{{p_m}} {{p_m}}}}} {{{p_m}}} \right\rangle + \frac{1}{4}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} ){P_n}\left\langle {{{p_m}}} \mathrel{|{\vphantom {{{p_m}} {{p_m}}}}} {{{p_m}}} \right\rangle } \\ + \frac{1}{4}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} )} \left\langle {{p_m}} \right|({{{\vec{p}}_n}\cdot \vec{\sigma }} )|{{p_m}} \rangle + \frac{j}{2}{\omega _m}\left\langle {{p_m}} \right|({\vec{b}\cdot \vec{\sigma }} )|{{p_m}} \rangle \\ - \frac{{{\alpha _{pm}}}}{2}\left\langle {{{p_m}}} \mathrel{|{\vphantom {{{p_m}} {{p_m}}}}} {{{p_m}}} \right\rangle + \frac{1}{4}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} ){P_n}\left\langle {{{p_m}}} \mathrel{|{\vphantom {{{p_m}} {{p_m}}}}} {{{p_m}}} \right\rangle } \\ + \frac{1}{4}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} )} \left\langle {{p_m}} \right|({{{\vec{p}}_n}\cdot \vec{\sigma }} )|{{p_m}} \rangle - \frac{j}{2}{\omega _m}\left\langle {{p_m}} \right|({\vec{b}\cdot \vec{\sigma }} )|{{p_m}} \rangle . \end{array}$$

Using the property of Pauli matrices [16]:

$$\left\langle {{p_m}} \right|({{{\vec{p}}_n}\cdot \vec{\sigma }} )|{{p_m}} \rangle = {\vec{p}_n}\cdot {\vec{p}_m},$$
we have:
$$\frac{{d{P_m}}}{{dz}} ={-} {\alpha _{pm}}{P_m} + \frac{1}{2}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} ){P_n}{P_m}} + \frac{1}{2}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} ){{\vec{p}}_n}\cdot {{\vec{p}}_m}} ,$$
which is Eq. (6). We may also obtain:
$$\begin{array}{l} \frac{{d{{\vec{p}}_m}}}{{dz}} = \frac{{d\left\langle {{p_m}} \right|\vec{\sigma }|{{p_m}} \rangle }}{{dz}} = \frac{{d\left\langle {{p_m}} \right|}}{{dz}}\vec{\sigma }|{{p_m}} \rangle + \left\langle {{p_m}} \right|\vec{\sigma }\frac{{d|{{p_m}} \rangle }}{{dz}}\\ ={-} \frac{{{\alpha _{pm}}}}{2}\left\langle {{p_m}} \right|\vec{\sigma }|{{p_m}} \rangle + \frac{1}{4}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} ){P_n}\left\langle {{p_m}} \right|\vec{\sigma }|{{p_m}} \rangle } \\ + \frac{1}{4}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} )\left\langle {{p_m}} \right|({{{\vec{p}}_n}\cdot \vec{\sigma }} )\vec{\sigma }|{{p_m}} \rangle } + \frac{j}{2}{\omega _m}\left\langle {{p_m}} \right|({\vec{b}\cdot \vec{\sigma }} )\vec{\sigma }|{{p_m}} \rangle \\ - \frac{{{\alpha _{pm}}}}{2}\left\langle {{p_m}} \right|\vec{\sigma }|{{p_m}} \rangle + \frac{1}{4}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} ){P_n}\left\langle {{p_m}} \right|\vec{\sigma }|{{p_m}} \rangle } \\ + \frac{1}{4}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} )\left\langle {{p_m}} \right|\vec{\sigma }({{{\vec{p}}_n}\cdot \vec{\sigma }} )|{{p_m}} \rangle } - \frac{j}{2}{\omega _m}\left\langle {{p_m}} \right|\vec{\sigma }({\vec{b}\cdot \vec{\sigma }} )|{{p_m}} \rangle . \end{array}$$

Using the following properties of the Pauli matrices [16]:

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

Therefore, we may derive the following equation based on Eq. (15):

$$\begin{array}{l} \frac{{d{{\vec{p}}_m}}}{{dz}} ={-} {\alpha _{pm}}\left\langle {{p_m}} \right|\vec{\sigma }|{{p_m}} \rangle + \frac{1}{2}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} ){P_n}\left\langle {{p_m}} \right|\vec{\sigma }|{{p_m}} \rangle } \\ + \frac{1}{2}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} ){{\vec{p}}_n}\left\langle {{{p_m}}} \mathrel{|{\vphantom {{{p_m}} {{p_m}}}}} {{{p_m}}} \right\rangle } + \frac{1}{2}{\omega _m}\left\langle {{p_m}} \right|({\vec{b} \times \vec{\sigma }} )|{{p_m}} \rangle . \end{array}$$

Furthermore, we have the following property for the Pauli matrices [16]:

$$\left\langle {{p_m}} \right|({\vec{b} \times \vec{\sigma }} )|{{p_m}} \rangle = \vec{b} \times {\vec{p}_m},$$
and hence, we have
$$\frac{{d{{\vec{p}}_m}}}{{dz}} ={-} {\alpha _{pm}}{\vec{p}_m} + \frac{1}{2}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} ){P_n}{{\vec{p}}_m}} + \frac{1}{2}\sum\limits_{n = 1,n \ne m}^M {{g_R}({{\omega_n},{\omega_m}} ){{\vec{p}}_n}{P_m}} + {\omega _m}\vec{b} \times {\vec{p}_m},$$
which is Eq. (7).

Funding

National Key Research and Development Program of China (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.

References

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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 (9)
Fig. 1.
Fig. 1. The on-off gain spectrum of the Raman fiber amplifier with four pumps in an 80 km SMF28e fiber span.

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Fig. 2.
Fig. 2. The average PDG and PDG standard deviation under different DOP of the four pumps.

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Fig. 3.
Fig. 3. The PDG distribution with the four pump DOP as (a) 5% (b) 10% (c) 15% (d) 20%.

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Fig. 4.
Fig. 4. The average PDG under different DOP of an individual pump.

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Fig. 5.
Fig. 5. The PDG standard deviation under different DOP of an individual pump.

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Fig. 6.
Fig. 6. The PDG distribution with one pump DOP as 5% (a) the 1420 nm pump (b) the 1430 nm pump (c) the 1450 nm pump (d) the 1460 nm pump.

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Fig. 7.
Fig. 7. The average PDG and PDG standard deviation under different PMD values. The four pumps are assumed to be with the DOP of 5% at 1420 nm, 10% at 1420 nm, 15% at 1420 nm, and 20% at 1420 nm.

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Fig. 8.
Fig. 8. The average DOP evolution of the four pumps when the 1420 nm pump DOP is 100% and the rest three pumps’ DOPs are 0%.

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Fig. 9.
Fig. 9. The average PDG and PDG standard deviation under different PMD values.

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Equations (20)

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p → m = ⟨ p m | σ → | p m ⟩ ,
P m = ⟨ p m | p m p m p m ⟩ , P m = | p → m | + P m d ,
d | p m ⟩ d z = − α p m 2 | p m ⟩ + 1 4 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) ( p → n ⋅ σ → + P n ) | p m ⟩ − j 2 ω m ( b → ⋅ σ → ) | p m ⟩ m = 1 ⋯ M , d | s ⟩ d z = − α s 2 | s ⟩ + 1 4 ∑ n = 1 M g R ( ω n , ω s ) ( p → n ⋅ σ → + P n ) | s ⟩ − j 2 ω s ( b → ⋅ σ → ) | s ⟩ ,
g R ( ω n , ω m ) = { γ R ( ω n , ω m ) , ω n > ω m , − ω n ω m γ R ( ω m , ω n ) , ω n < ω m ,
⟨ b → ( z ) ⟩ = 0 , ⟨ b → ( z 1 ) b → ( z 2 ) ⟩ = 1 3 D p 2 I ↔ δ ( z 1 − z 2 ) ,
d P m d z = − α p m P m + 1 2 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) P n P m + 1 2 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) p → n ⋅ p → m .
d p → m d z = − α p m p → m + 1 2 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) P n p → m + 1 2 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) p → n P m + ω m b → × p → m .
d T s d z = − α s 2 T s + 1 4 ∑ n = 1 M g R ( ω n , ω s ) ( p → n ⋅ σ → + P n ) T s − j 2 ω s ( b → ⋅ σ → ) T s ,
P D G = 10 log 10 ( λ max λ min ) , [ λ max , λ min ] = e i g ( T s T s H ) ,
d | p m ⟩ d z = − α p m 2 | p m ⟩ + 1 4 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) ( p → n ⋅ σ → + P n ) | p m ⟩ − j 2 ω m ( b → ⋅ σ → ) | p m ⟩ , m = 1 ⋯ M .
d ⟨ p m | d z = − α p m 2 ⟨ p m | + 1 4 ⟨ p m | ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) ( p → n ⋅ σ → + P n ) + j 2 ⟨ p m | ω m ( b → ⋅ σ → ) . m = 1 ⋯ M .
d P m d z = d ⟨ p m | p m p m p m ⟩ d z = d ⟨ p m | d z | p m ⟩ + ⟨ p m | d | p m ⟩ d z = − α p m 2 ⟨ p m | p m p m p m ⟩ + 1 4 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) P n ⟨ p m | p m p m p m ⟩ + 1 4 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) ⟨ p m | ( p → n ⋅ σ → ) | p m ⟩ + j 2 ω m ⟨ p m | ( b → ⋅ σ → ) | p m ⟩ − α p m 2 ⟨ p m | p m p m p m ⟩ + 1 4 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) P n ⟨ p m | p m p m p m ⟩ + 1 4 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) ⟨ p m | ( p → n ⋅ σ → ) | p m ⟩ − j 2 ω m ⟨ p m | ( b → ⋅ σ → ) | p m ⟩ .
⟨ p m | ( p → n ⋅ σ → ) | p m ⟩ = p → n ⋅ p → m ,
d P m d z = − α p m P m + 1 2 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) P n P m + 1 2 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) p → n ⋅ p → m ,
d p → m d z = d ⟨ p m | σ → | p m ⟩ d z = d ⟨ p m | d z σ → | p m ⟩ + ⟨ p m | σ → d | p m ⟩ d z = − α p m 2 ⟨ p m | σ → | p m ⟩ + 1 4 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) P n ⟨ p m | σ → | p m ⟩ + 1 4 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) ⟨ p m | ( p → n ⋅ σ → ) σ → | p m ⟩ + j 2 ω m ⟨ p m | ( b → ⋅ σ → ) σ → | p m ⟩ − α p m 2 ⟨ p m | σ → | p m ⟩ + 1 4 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) P n ⟨ p m | σ → | p m ⟩ + 1 4 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) ⟨ p m | σ → ( p → n ⋅ σ → ) | p m ⟩ − j 2 ω m ⟨ p m | σ → ( b → ⋅ σ → ) | p m ⟩ .
σ → ( b → ⋅ σ → ) = b → I + j b → × σ → , ( b → ⋅ σ → ) σ → = b → I − j b → × σ → , σ → ( p → n ⋅ σ → ) = p → n I + j p → n × σ → , ( p → n ⋅ σ → ) σ → = p → n I − j p → n × σ → ,
σ → ( p → n ⋅ σ → ) + ( p → n ⋅ σ → ) σ → = 2 p → n I , ( b → ⋅ σ → ) σ → − σ → ( b → ⋅ σ → ) = − 2 j b → × σ → .
d p → m d z = − α p m ⟨ p m | σ → | p m ⟩ + 1 2 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) P n ⟨ p m | σ → | p m ⟩ + 1 2 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) p → n ⟨ p m | p m p m p m ⟩ + 1 2 ω m ⟨ p m | ( b → × σ → ) | p m ⟩ .
⟨ p m | ( b → × σ → ) | p m ⟩ = b → × p → m ,
d p → m d z = − α p m p → m + 1 2 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) P n p → m + 1 2 ∑ n = 1 , n ≠ m M g R ( ω n , ω m ) p → n P m + ω m b → × p → m ,
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