1. Introduction

Polarization impairments, viz. polarization mode dispersion (PMD) and polarization dependent gain (PDG), are among major factors limiting the progress in further increase of transmission rates and overall capacity of the next generation of optical networks based on distributed fiber Raman amplification. PMD leads to the pulse broadening caused by varying group velocities for the pulses with different states of polarization (SOPs) [1,2] whereas PDG leads to an uncontrollable output gain variation caused by its dependence on the input signal SOP [3–7]. By the traditional approach, spinning the fiber periodically, it is possible to reduce PMD to below 0.04 ps/km^1/2 [8,9], but this is accompanied with a simultaneous increase in Raman PDG [3–7]. Sergeyev et al. have shown that it is possible to mitigate both PDG and PMD simultaneously by using a two-section fiber (‘two-section approach’) in which the first section has no spin and the second one is periodically spun [5–7]. As follows from [7], this approach demonstrated polarization pulling of the signal SOP to the pump SOP and so can be considered as a two-section Raman polarizer which is the one of the techniques of all-optical polarization control [10–16]. Unlike the one-section fiber Raman polarizer demonstrated by Martinelli et al. [10], Kozlov et al. [11], L. Ursini et al. [12] and N. J. Muga et al. [13], a two-section fiber provides almost the same polarization pulling for all input signal states of polarization and so results in very small polarization dependent gain [7]. In view of increasing interest in polarization multiplexing of signals along with Raman amplification in fiber optic communication, polarization pulling approaches cannot be used straightforwardly and so have to be modified. In this paper, we have suggested a new approach based on application of fiber spinning with two different scales, viz. fast spinning with a period of 2.7 m and slow spinning with a period of more than 200 m for 10 km of single mode fiber. In view of the similarity of a two-scale periodically spun fiber Raman amplifier (SFRA) to a periodically driven excitable system (ES), we have found resonance-like de-correlation of pump and signal SOPs and polarization pulling equivalent to the Stochastic Resonance (SR) and Dynamic Localization (DL) in ES [17–19]. As a result, we have demonstrated that it is possible to provide de-correlation of pump and signal SOPs and simultaneously suppress PDG and PMD to the 1.2 dB and 0.035 ps/km^1/2 respectively. The practical implementation of the obtained results can be in the design of a fiber Raman amplifier-based quasi-isotropic spatially and spectrally transparent media for high-speed fiber optic communication [20,21].

2. Model of a fiber Raman amplifier with random birefringence and arbitrary spin profile

The evolution of signal and pump states of polarization in a single mode fiber is typically described in terms of the unit vectors $\widehat{s}=({\widehat{s}}_1,{\widehat{s}}_2,{\widehat{s}}_3)$and$\widehat{p}=({\widehat{p}}_1,{\widehat{p}}_2,{\widehat{p}}_3)$, respectively, pointing to positions on the Poincaré sphere (Fig. 1 ). Due to birefringence, these vectors rotate on the Poincaré sphere around the birefringence vector W_i = (2b_icosθ, 2b_isinθ,0)^T in the same direction, but at different rates b_s and b_p which are birefringence strengths (b_i = π/L_bi where L_bi is the beat length) at signal λ_s and pump λ_p wavelengths. We assume that the birefringence strength 2b_i is fixed and the orientation angle θ is driven by a white-noise process (fixed-modulus model [2])

 

Fig. 1 Evolution of the pump $\widehat{p}$ and the signal $\widehat{s}$ states of polarization on the Poincaré sphere, as well as the fluctuations of the local birefringence vector W_i = (2b_icosθ, 2b_isinθ,0)^T . Vectors $\widehat{s}$ and $\widehat{p}$ rotate around the local axis (W) at rates b_p and b_s, while vector (W) rotates randomly in the equatorial plane at the rate σ = L_c^-1/2 (L_c is the correlation length), Φ is an angle between pump and signal SOPs.

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We neglect herein the fiber twist and, therefore, the birefringence vector for the spun fiber takes the form of W_i = R₃[2A(z)]W_i,un, where A(z) is the spin profile, and R₃(γ) represents rotation in the equatorial plane by angle γ around the z-axis on the Poincaré sphere [9]

Stimulated Raman scattering leads to amplification of the signal wave and changes its direction and so evolution of the signal S and pump P SOPs can be found as follows:

Here s₀ is the part of the signal amplitude related to the pump-signal SOPs interaction, $G_{ave}=\exp \left({\displaystyle \underset{0}{\overset{z}{\int }}g{P}_0\left(z\text{'}\right)/2dz\text{'}-{\alpha }_{s}z}\right)$is the averaged Raman gain, g is the Raman gain coefficient, the forward pump power$P_0\left(z\right)=P_{in}\exp \left(-{\alpha }_{p}z\right)$; ${\alpha }_s$ and${\alpha }_p$are the signal and pump losses, respectively; L is the fiber length. The part of the Raman gain ΔG related to polarization evolution due to random birefringence and fiber spinning depends on s₀ as follows [4–7]:

To characterize de-correlation of pump and signal SOPs and polarization pulling in terms of the Stochastic Resonance phenomenon we introduce parameter R which is similar to the signal-to-noise ratio in the models of SR [18]:

To calculate s₀(L), we use a vector model of a fiber Raman amplifier for forward pump with allowance for fiber spin profile and random birefringence and neglecting pump depletion [4]:

Here$s=s_0\widehat{s}$, and W_p^(NL), W_s^(NL) describes the nonlinear SOP evolution caused by self- and cross-phase modulation (SPM and XPM): $W_p^{(NL)}=2{\gamma }_p/3\left(-2S_1,-2S_2,{\widehat{p}}_3{P}_0\left(z\right)\right),$ $W_s^{(NL)}=2{\gamma }_s/3\left(-2{\widehat{p}}_1{P}_0\left(z\right),-2{\widehat{p}}_2{P}_0\left(z\right),S_3\right)$. Kerr coupling constants are γ_i = 2πn₂/(λ_iA_eff) (i = s,p), where n₂ is the nonlinear Kerr coefficient and A_eff is the effective core area of the fiber.

We choose the reference frame in the Stokes space in such a way that the local birefringence vector is ${\tilde{W}}_{i,un}=\left(2b_i,0,0\right)$ and $\widehat{\tilde{p}}\left(0\right)=(1,0,0)$, i.e. oriented along the X-axis on the Poincaré sphere. This can be accomplished by a suitable transformation ${\tilde{W}}_i=R_3^{-1}\left(\theta +2A(z)\right)W_i$, and $\tilde{s}=R_3^{-1}\left(\theta +2A(z)\right)s$ and $\widehat{\tilde{p}}=R_3^{-1}\left(\theta +2A(z)\right)\widehat{p}$, provided the variable $s_0$ and scalar product $x=\widehat{p}\cdot s$ are invariant under the rotation. In addition, we consider cases which correspond to the maximum and minimum of Raman gain, viz. ${\tilde{s}}_{\max }\left(0\right)=(1,0,0)$and ${\tilde{s}}_{\min }\left(0\right)=(-1,0,0)$. Applying an averaging procedure considered in [5,6] to Eqs. (7), we find the system of equations:

Here δ_p = 2γ_pLP_in/3 δ_s = 4γ_sLP_in/3, $〈x〉=〈{\widehat{\tilde{p}}}_1{\tilde{s}}_1+{\widehat{\tilde{p}}}_2{\tilde{s}}_2+{\widehat{\tilde{p}}}_3{\tilde{s}}_3〉$, $〈y〉=〈{\widehat{\tilde{p}}}_3{\tilde{s}}_2-{\widehat{\tilde{p}}}_2{\tilde{s}}_3〉$, $〈u〉=〈{\widehat{\tilde{p}}}_3{\tilde{s}}_1-{\widehat{\tilde{p}}}_1{\tilde{s}}_3〉$, $\alpha (z\text{'})=\partial A(z\text{'})/\partial z\text{'}$ is the spin rate, z′ = z/L, ${\epsilon }_1=g{P}_{in}L/2,{\epsilon }_2={\alpha }_{s}L$, ${\epsilon }_3=2\pi L/L_{bp}\left({\lambda }_s/{\lambda }_p-1\right)$, ${\epsilon }_4=(2\pi L)/L_{bp}$, ${\epsilon }_5=(2\pi L)/L_{bs}$.

In view of ε₃<< ε₄, ε₅ (rotation of the signal SOP with respect to the pump SOP is much slower than the SOPs rotation with respect the local birefringence vector ${\tilde{W}}_i$), we can average over the fast rotations. It can be done by the transformations $h=R_s(z)\tilde{s}$ and $f=R_p(z)\widehat{\tilde{p}}$, where

Using this transformation, Eq. (7) and averaging over the fast oscillations, we find the following equation for $〈h_1{f}_1〉$

If we choose parameters for Raman amplifier as, g = 2.3 dB W⁻¹km⁻¹, P_in = 5W, L = 10 km, L_c = 110 m, we find that ${\epsilon }_1{L}_c/L=0.15<<1$. Thus, we can neglect the first term on the right-hand side of Eq. (10). As a result,

In addition, this procedure leads to $〈{\widehat{\tilde{p}}}_2{\tilde{s}}_1〉\to 0$, $〈{\widehat{\tilde{p}}}_1{\tilde{s}}_2〉\to 0$, $〈{\widehat{\tilde{p}}}_2{\widehat{\tilde{p}}}_3{\tilde{s}}_2〉\to 0$, $〈{\widehat{\tilde{p}}}_1{\widehat{\tilde{p}}}_3{\tilde{s}}_2〉\to 0$,

$〈{\widehat{\tilde{p}}}_3\left({\widehat{\tilde{p}}}_2{\tilde{s}}_1-{\widehat{\tilde{p}}}_1{\tilde{s}}_2\right)〉\to 0$ and so self- and cross-phase modulation (SPM and XPM) can be neglected [4–7]. A similar result has been obtained in [10,12] by direct modeling of stochastic Eqs. (7), viz. it was obtained that for the pump power P_in < 10 W and PMD parameter D_p>0.01 ps/km^1/2 SPM and XPM has no contribution to polarization pulling. Using transformation (9) and averaging over the birefringence fluctuations and fast oscillations we find the following equations which we use to calculate parameter R from Eq. (6):

To calculate the spin induced reduction factor (SIRF) for the case of spun fiber we use the standard model of PMD [9]

Finally, the PMD parameter for the spun fiber D_ps can be found as follows [9]

In addition, we use fast periodic spinning along with slow- amplitude and phase modulation:

Here A₀ is amplitude of the fiber spinning in rad, k_lf and k_hf are low and high frequencies of the fiber spinning. Practical realization of amplitude- and phase-modulated fiber spinning can be achieved based on the modern technology of direct fiber spinning instead of spinning the preform [8].

To quantify polarization pulling and SOPs de-correlation in terms of SR phenomena, we calculate parameters equivalent to the parameters used in the models of excitable systems [17–19], viz. Kramers length <L_k> and intrawell relaxation length <L_R>. In the absence of low-frequency modulation (k_lf = 0), Eqs. (8) can be simplified for the case of k_hf>>L_c, L_b with the help of averaging over fast periodic fiber spinning [9]. As a result, we have

Here $〈\widehat{y}〉=〈y〉/〈s_0〉$, $〈\widehat{x}〉\equiv \cos \Phi =〈x〉/〈s_0〉$ is the variable which indicates polarization pulling if $〈\widehat{x}〉\to 1$ or de-correlation pump and signal SOPs if $〈\widehat{x}〉\to 0$; and J₀(2A₀) is the zero-order Bessel function of the first kind. For $\exp \left(-{\epsilon }_2{z}^{\prime }\right)<<1$, solutions of Eqs. (17) $〈{\widehat{x}}_0〉,〈{\widehat{y}}_0〉$ independent on $z^{\prime }$, can be found from equations:

Here $\Delta =2L_c{\epsilon }_1\exp \left(-{\epsilon }_2{z}^{\prime }\right)/L,{\Delta }_1=2L_c{J}_0\left(2A_0\right){\epsilon }_3/L$. Linear stability analysis of Eqs. (17) near the states $〈{\widehat{x}}_0〉,〈{\widehat{y}}_0〉$ results in the following eigenvalues:

Thus, relaxation length can be defined as $〈L_R〉=1/\mathrm{Re}\left({\Lambda }_{1,2}\right)$. If $\mathrm{Im}\left({\Lambda }_{1,2}\right)\ne 0$then the system oscillates around the states $〈{\widehat{x}}_0〉,〈{\widehat{y}}_0〉$and so the averaged length at which the system is close to the state can be defined as $〈L_k〉=2\pi L/\mathrm{Im}\left({\Lambda }_{1,2}\right)$ (Kramers length).

3. Results and discussion

We have found $〈{\widehat{x}}_0〉,〈{\widehat{y}}_0〉$, Re{Λ_1,2}, Im{Λ_1,2} numerically from Eqs. (18), (19). We find also PDG, R and PMD parameter D_p numerically from Eqs. (5), (6), (8) and (12)–(15) for two-scale fiber spinning (16). We use parameters typical for a distributed fiber Raman amplifier: L = 10 km, α_s = 0.2 dB/km, λ_p = 1460 nm, λ_s = 1550 nm, g = 2.3 dBW⁻¹km⁻¹, P_in = 5 W, L_b = 8.3 m, A₀ = 3 (in units of rad), k_hf = 6π/L_bp, k_lf = [0…300/L], L_c = [5m…205m], a₀ = 100 m. Results for $〈{\widehat{x}}_0〉,〈{\widehat{y}}_0〉$ and Re{Λ₁}, Im{Λ₁} are shown in Fig. 2 (a, b) . The results for PDG, R and PMD parameters with $\widehat{p}\left(0\right)=(1,0,0)$, ${\widehat{s}}_{\min }\left(0\right)=(-1,0,0),$${\widehat{s}}_{\max }\left(0\right)=(1,0,0)$ are shown in Fig. 3 (a-f) .

 

Fig. 2 a):$〈{\widehat{x}}_0〉$ (solid line) and $〈{\widehat{y}}_0〉$(dotted line) and b): $\mathrm{Im}\left({\Lambda }_1\right),\text{}\mathrm{Im}\left({\Lambda }_2\right),$ (dotted line), $\mathrm{Re}\left({\Lambda }_1\right)$ (solid line) and $\mathrm{Re}\left({\Lambda }_2\right)$ (dashed line) as a function of correlation length L_c. Parameters: L = 10 km, α_s = 0.2 dB/km, λ_p = 1460 nm, λ_s = 1550 nm, g = 2.3 dBW⁻¹km⁻¹, P_in = 5 W, L_b = 8.3 m, A₀ = 3 (in units of rad), k_hf = 6π/L_bp, k_lf = 0, L_c = [5m…205m].

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Fig. 3 Polarization dependent gain PDG (a, b), parameter R for the $\widehat{\tilde{p}}\left(0\right)={\tilde{s}}_{\max }\left(0\right)=(1,0,0)$ (c, d) and PMD parameter D_p (e, f) as a function of fiber spinning frequency k_lf and correlation length L_c . Low-frequency amplitude (a, c, e) and phase modulated (b, d, f) fiber spinning. Parameters are the same as for Fig. 2. De-correlation of pump and signal SOPs (minimum in PDG with an arrow): L_c = 105 m and k_lf = 6 km⁻¹ (Figs. 3 (a, c)), L_c = 110 m and k_lf = 3 km⁻¹ (Figs. 3 (b, d)); polarization pulling (maximum in PDG with an arrow): L_c = 35 m and k_lf = 30 km⁻¹ (Figs. 3 (a, c)), L_c = 110 m and k_lf = 22 km⁻¹ (Figs. 3 (b, d)).

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For the parameters listed, Eqs. (18) have only one real solution with $\left|〈{\widehat{x}}_0〉\right|\le 1$. As follows from Fig. 2 (a) for L_c→0 (high birefringence fluctuations) $〈{\widehat{x}}_0〉\to 1$ and so we have polarization pulling and for L_c>>200 m de-correlation of SOPs with $〈{\widehat{x}}_0〉\to 0$. It is clear from Fig. 2 (b) that oscillations of angle between SOPs start when L_c≥35 m.

As a result of nonlinearity in the form of Raman amplification and presence of random birefringence and periodic fiber spinning, PDG has maxima and minima as a function of correlation length L_c and spinning frequency k_lf. For slow amplitude modulation of the fiber spinning (Fig. 3 (a)), the maximum PDG corresponds to the case when L_c, k_lf →0. As follows from the results of [8,9], fast periodic fiber spinning results in PMD mitigation and so leads to polarization pulling [7–13]. However, with increased frequency of amplitude modulation PDG reaches a minimum of 1.5 dB for L_c = 105m and k_lf = 6 km⁻¹ and for L_c = 55 m and k_lf = 30 km⁻¹ PDG reaches the maximum of 6.9 dB (Fig. 3 (a)). In the case of slow phase modulation, PDG reaches a minimum of 1.2 dB for L_c = 110 m and k_lf = 3 km⁻¹ and a maximum of 14.5 dB for L_c = 110 m and k_lf = 22 km⁻¹ (Fig. 3 (b)).

The evolution of angle Φ between pump and signal SOPs for the cases of minima and maxima is shown in Figs. 4 . As follows from Figs. 3 (a, b) and Figs. 4, the maximum of PDG corresponds to the polarization pulling with$\cos \Phi \to 1$, and the minimum of PDG corresponds to the case of SOPs de-correlation for which pump and signal SOPs are freely rotating and so $\cos \Phi \to 0$. In addition, fiber spinning with low-frequency phase modulation provides better polarization pulling as compared to the case of amplitude modulation (thick lines in Figs. 4).

 

Fig. 4 Polarization pulling metrics cosΦ as a function of distance along the fiber z for low-frequency amplitude (a) and phase modulated (b) fiber spinning. Dotted line: ${\tilde{s}}_{\max }\left(0\right)=(1,0,0)$, solid line${\tilde{s}}_{\min }\left(0\right)=(-1,0,0)$. Thick lines and thin lines correspond to the maximum and minimum in PDG shown by arrows.

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To characterize polarization pulling and SOPs de-correlation in terms of SR phenomena, we present in Fig. 5 (a-d) the results for PDG, parameter R and PMD parameter D_p as a function of modulation frequency k_l,f and correlation length L_c. For low-frequency amplitude modulation, PDG has a minimum and parameter R has a maximum at frequency of k_lf,max = 6 km⁻¹ which coincides with the resonance frequency for excitable systems calculated from Eq. (19).

 

Fig. 5 PDG (a,b), parameter R (c, d) and PMD parameter D_p (e, f) as a function of modulation frequency k_l,f (a, c) and correlation length L_c (b, d) for amplitude (thick lines) and phase modulated (thin lines) of fiber spinning. Parameters in Figs. 5 (a, c, d): L_c = 35 m (thick solid line), L_c = 105 m (thick dotted line), L_c = 55 m (thin solid line), L_c = 115 m (thin dotted line), Parameters in Figs. 5 (b, d, f): k_lf = 6 km⁻¹ (thick solid line), k_lf = 30 km⁻¹ (thick dotted line), k_lf = 3 km⁻¹ (thin solid line), k_lf = 22 km⁻¹ (thin dotted line).

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A maximum and minimum at this frequency is evidence of Stochastic Resonance where random birefringence fluctuations are synchronized with an external modulation [18]. In view of the maximum in R coinciding with the minimum in PDG, SR for this case corresponds to de-correlation of the pump and input SOPs. For the off-resonance conditions (low and high frequencies of spinning), the random birefringence fluctuations have no synchronization with the external modulation and so there is no activation of de-correlation of pump and signal SOPs. For the case of phase modulated fiber spinning, PDG and parameter R have maxima and minima at frequencies different from SR frequencies (thin lines in Figs. 5 (a, c)). It is quite similar to the case of Dynamic Localization (DL) considered by Dykman et al., where the right and the left escape rates become different to each other depending on the phase of non-adiabatic aperiodic modulation [19]. Thus, DL phenomena take the form of either polarization pulling or de-correlated SOPs depending on the phase of phase modulated fiber spinning. It is likely that increased noise power σ² (σ² = 1/L_c) leads to increased asymmetry of probabilities and so to the further increased PDG and parameter R (solid lines in Figs. 5 (a, c)).

PDG has a minimum as a function of noise power σ² = 1/L_c for amplitude and phase modulation (Fig. 5 (b)). When the birefringence fluctuations are large (small correlation lengths) and low (large correlation lengths) polarization pulling is dominating and PDG is high (Fig. 5 (b)). If fluctuations are large, escape events happen at very small intervals and so at larger intervals the fiber looks isotropic with small changes in pump and signal SOPs along the fiber. The case of small birefringence fluctuations corresponds to the polarization maintaining (PM) fiber where pump and signal SOPs initially oriented along the fast or slow axes preserve their orientations along the fiber. For an optimal level of noise above the activation threshold, random hops are synchronized with modulation frequency and so SR takes the form of de-correlation of SOPs (minimum in PDG) [17,18]. As follows from Fig. 3(a), PDG is almost independent of modulation frequency k_lf for amplitude modulation in the limit of high frequency modulation. It is likely that if the fiber spinning oscillates so fast that the system feels mainly the averaged spinning which can be small and not enough to activate de-correlation of SOPs [18]. As a result, with increasing modulation frequency polarization pulling occurs instead of SOPs de-correlation (thick dotted line in Fig. 5 (b)). As mentioned before, low-frequency phase-modulation of fiber spinning can result in a DL phenomenon in the form of either polarization pulling or de-correlated SOPs depending on the phase. This results herein in a minimum in PDG as a function of correlation length (thin lines in Figs. 5 (b)). Parameter R can have maxima and minima as a function of a noise power σ² = 1/L_c and modulation frequency k_lf and so it is likely that our case corresponds to more complex cases of SR and DL. PMD parameter has maxima and minima as function of k_l,f and L_c⁻¹ which can be an indication of SR and DL presence for polarization properties of fiber described by Eqs. (14).

Thus, application of fast (2.7 m period) and slow (200-600 m period) amplitude or phase-modulated fiber spinning can result in activated polarization pulling or de-correlation of pump and signal SOPs. De-correlation is accompanied with simultaneous suppression of PDG and PMD to the 1.2 dB and 0.035 ps/km^1/2 respectively.

4. Conclusion

In conclusion, we have developed an advanced vector model of a fiber Raman amplifier accounting for fiber spinning and random birefringence. Based on this model and the PMD model previously developed in [9], we characterize a Raman amplifier in terms of PDG, gain fluctuations and PMD. Unlike previous approaches, we demonstrated that it is possible simultaneously to de-correlate pump and signal SOPs and to suppress PDG and PMD. This new technique is based on application of fiber spinning with two different scales: fast spinning with period of 2.7 m and slow spinning with period of 200 – 6000 m for 10 km of single mode fiber. In view of the similarity of a two-scale periodically spun fiber Raman amplifier to a periodically driven excitable system, we have found resonance-like de-correlation of pump and signal SOPs and polarization pulling equivalent to the Stochastic Resonance and Dynamic Localization in excitable systems [17–19]. As a result, we demonstrate that it is possible to provide de-correlation of pump and signal SOPs and simultaneously suppress PDG and PMD to the 1.2 dB and 0.035 ps/km^1/2 respectively. The practical implementation of the obtained results can be in the design of fiber Raman amplifier-based quasi-isotropic spatially and spectrally transparent media for high-speed fiber optic communication [20,21].