Fiber Raman amplification in a two-scale spun fiber

Sergey V. Sergeyev

doi:10.1364/OME.2.001683

1. Introduction

All-optical polarization stabilization based on nonlinear effects (Raman and Brillouin-based polarization pulling (RBPP and BPPP) [1–10], photo-refractivity [11], and four-wave mixing [12,13]) have been recently paid much attention in the context of applications in fiber optic communications. Raman-based polarization pulling is enhanced in low PMD fibers and so stabilization of the signal state of polarization (SOP) is accompanied by an increased Raman gain [1–9]. However, RBPP suffers from an increased polarization dependent gain (PDG), viz. an uncontrollable output gain variation caused by its dependence on the input signal SOP, which makes difficult an application of the advanced modulation formats using polarization multiplexing (POLMUX) of the input signals [1–9]. To make POLMUX along with a high Raman gain and suppressed PDG possible, we suggested in our previous paper a technique for de-correlation of pump and signal SOPs based on a two-scale periodically spun fiber Raman amplifier (SFRA) [4].

In view of similarity of two-scale SFRA to the periodically driven excitable system (ES) [8–10], we demonstrated resonance-like de-correlation of pump and signal SOPs equivalent to the Stochastic Resonance (SR) in ES [4]. Potential of the pump and the signal SOPs interaction is determined by a short spinning period of 1 m, noise is defined by the random birefringence in a fiber, and a long period (>100 m) fiber spinning plays a role of an adiabatic forcing [4]. As a result, it was possible to provide de-correlation of pump and signal SOPs with simultaneous suppression of PDG and PMD to the 1.2 dB and 0.035 ps/km^1/2 correspondently. However, the spin profile considered in [4] didn’t allow to have simultaneously increased Raman gain along with decreased PDG and PMD. Herein we present a new spinning profile for which it is possible to have pump-signal SOPs de-correlation along with suppression of PDG to 0.11 dB, PMD to 0.037 ps/km^1/2 and gain increase up to 15 dB.

The practical implementation of the obtained results can be in an extension of the transmission distance and reducing complexity of the fiber Raman amplifier-based un-repeatered high-capacity transmission systems [14].

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

The difference in gains (polarization dependent gain, PDG), averaged part of gain related to the pump-signal SOPs interaction G and mean-square gain fluctuations (MSGF) σ can be found as follows [4–6]

\begin{array}{l} P D G \equiv 10 \log (〈 s_{0, \max} (L) 〉 / 〈 s_{0, \min} (L) 〉), \\ G = 10 \log (〈 s_{0} (L) 〉 / 〈 s_{0} (0) 〉), \\ σ_{\max (\min)}^{} = \sqrt{〈 s_{0, \max (\min)}^{2} (L) 〉 - {〈 s_{0, \max (\min)} (L) 〉}^{2}} / 〈 s_{0, \max (\min)} (L) 〉, \end{array}

where <…> means averaging over the birefringence fluctuations along the fiber. To calculate s₀(L), we use the vector model of a fiber Raman amplifier for forward pump whilst accounting for fiber spin profile and random birefringence and neglecting the pump depletion [15]:

\begin{array}{l} \frac{d s}{d z} = \frac{g}{2} P_{0} (z) s_{0} \hat{p} + (W_{s} + W_{s}^{(N L)}) \times s, \\ \frac{d \hat{p}}{d z} = (W_{p} + W_{p}^{(N L)}) \times \hat{p}, \end{array}

Here

s = s_{0} \hat{s}

. Due to the linear birefringence, unit vectors

\hat{s}

and

\hat{p}

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_b_i where L_b_i is the beat length) at signal λ_s and pump λ_p wavelengths. The random birefringence in a single mode fiber can be represented in terms of a fixed modulus model (FMM) where the birefringence strength 2b_i is fixed and the orientation angle θ is driven by a white-noise process [16].

\begin{array}{l} \frac{d θ}{d z} = β (z), 〈 β (z) 〉 = 0, \\ 〈 β (z) β (z') 〉 = σ^{2} δ (z - z'), \end{array}

where <…> means averaging over the birefringence fluctuations along the fiber, δ(z) is a Dirac delta-function, and σ² = 2/L_c (L_c is the birefringence correlation length). 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 [17]:

R_{3} (γ) = [\begin{matrix} \cos γ & - \sin γ & 0 \\ \sin γ & \cos γ & 0 \\ 0 & 0 & 1 \end{matrix}] .

W_p^(NL), W_s^(NL) describes the nonlinear SOP evolution caused by self- and cross-phase modulation (SPM and XPM):

W_{p}^{(N L)} = 2 γ_{p} / 3 (- 2 S_{1}, - 2 S_{2}, {\hat{p}}_{3} P_{0} (z)),

W_{s}^{(N L)} = 2 γ_{s} / 3 (- 2 {\hat{p}}_{1} P_{0} (z), - 2 {\hat{p}}_{2} P_{0} (z), S_{3})

. Kerr coupling constant is 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. To calculate the spin induced reduction factor (SIRF) and PMD parameter D_p for the case of spun fiber we use the standard model of PMD [16,17]

\begin{array}{l} \frac{d Ω}{d z} = \frac{\partial W_{s}}{\partial ω} + W_{s} \times Ω, \\ S I R F = \sqrt{〈 {| Ω (L) |}_{s p}^{2} 〉 / 〈 {| Ω (L) |}_{u n}^{2} 〉}, \\ D_{p s} = \frac{λ_{s} \sqrt{2 L_{c}}}{L_{b s} c} S I R F \end{array}

where

〈 {| Ω (L) |}_{s p}^{2} 〉

and

〈 {| Ω (L) |}_{u n}^{2} 〉

are the mean-square differential group delays (DGD) for two orthogonal SOPs in the case of long-length spun fiber and the same fiber without spin, respectively. In addition, we use fast periodic spinning along with slow- amplitude modulation:

α_{a m} (z') = A_{0} k_{h f} L (\cos (k_{l f} L z') + a) \cos (k_{h f} L z') .

Here

α (z') = \partial A (z') / \partial z'

is the spin rate, z’ = z/L, A₀ is amplitude of the periodic fiber spinning in rad, k_lf and k_hf are low and high frequencies of the fiber spinning, a is an offset.

We have also introduced a new variable _{$\cos Φ = 〈 \hat{p} s 〉 / 〈 s_{0} 〉$} which indicates polarization pulling if $〈 \hat{x} 〉 \to 1$ or de-correlation of pump and signal SOPs if _{$〈 \hat{x} 〉 \to 0$}.

Applying an averaging procedure considered in [4–6] to Eqs. (2), we find the system of equations:

\begin{array}{l} \frac{d 〈 s_{0} 〉}{d z^{'}} = ε_{1} \exp (- ε_{2} z^{'}) 〈 x 〉, \\ \frac{d 〈 x 〉}{d z^{'}} = ε_{1} \exp (- ε_{2} z^{'}) 〈 s_{0} 〉 - ε_{3} 〈 y 〉, \\ \frac{d 〈 y 〉}{d z^{'}} = ε_{3} [〈 x 〉 - 〈 {\hat{\tilde{p}}}_{1} {\tilde{s}}_{1} 〉] - 2 α (z') 〈 u 〉 - \frac{〈 y 〉 L}{2 L_{c}}, \\ \frac{d 〈 u 〉}{d z^{'}} = 2 α (z') 〈 y 〉 - \frac{〈 u 〉 L}{2 L_{c}}, \\ \frac{d 〈 s_{0}^{2} 〉}{d z^{'}} = 2 ε_{1} \exp (- ε_{2} z^{'}) 〈 s_{0} x 〉, \frac{d 〈 s_{0} x 〉}{d z^{'}} = ε_{1} \exp (- ε_{2} z^{'}) (s_{0}^{2} + x^{2}) - ε_{3} 〈 y s_{0} 〉, \\ \frac{d 〈 s_{0} y 〉}{d z^{'}} = ε_{1} \exp (- ε_{2} z^{'}) 〈 x y 〉 + ε_{3} [〈 s_{0} x 〉 - 〈 y^{2} 〉 - 〈 s_{0} 〉 〈 {\hat{\tilde{p}}}_{1} {\tilde{s}}_{1} 〉] - 2 α (z') 〈 s_{0} u 〉 - \frac{〈 s_{0} y 〉 L}{2 L_{c}}, \\ \frac{d 〈 x^{2} 〉}{d z^{'}} = 2 ε_{1} \exp (- ε_{2} z^{'}) 〈 s_{0} x 〉 - 2 ε_{3} 〈 x y 〉, \\ \frac{d 〈 x y 〉}{d z^{'}} = ε_{1} \exp (- ε_{2} z^{'}) 〈 s_{0} y 〉 + ε_{3} [〈 x^{2} 〉 - 〈 x 〉 〈 {\hat{\tilde{p}}}_{1} {\tilde{s}}_{1} 〉] - 2 α (z') 〈 x u 〉 - \frac{〈 x y 〉 L}{2 L_{c}}, \\ \frac{d 〈 u s_{0} 〉}{d z^{'}} = ε_{1} \exp (- ε_{2} z^{'}) 〈 x u 〉 + 2 α (z') 〈 y s_{0} 〉 - \frac{〈 u s_{0} 〉 L}{2 L_{c}}, \\ \frac{d 〈 x u 〉}{d z^{'}} = ε_{1} \exp (- ε_{2} z^{'}) 〈 s_{0} u 〉 - ε_{3} 〈 y u 〉 + 2 α (z') 〈 x y 〉 - \frac{〈 x u 〉 L}{2 L_{c}}, \\ \frac{d 〈 y u 〉}{d z^{'}} = ε_{3} (〈 x u 〉 - 〈 u 〉 〈 {\hat{\tilde{p}}}_{1} {\tilde{s}}_{1} 〉) + 2 α (z') (〈 y^{2} 〉 - 〈 u^{2} 〉) + \frac{L 〈 y u 〉}{2 L_{c}}, \\ \frac{d 〈 u^{2} 〉}{d z^{'}} = 2 α (z') 〈 y u 〉 + \frac{L}{L_{c}} (〈 y^{2} 〉 - 〈 u^{2} 〉), \\ \frac{d 〈 y^{2} 〉}{d z^{'}} = 2 ε_{3} [〈 y x 〉 - 〈 y 〉 〈 {\hat{\tilde{p}}}_{1} {\tilde{s}}_{1} 〉] - 2 α (z') 〈 y u 〉 - \frac{L}{L_{c}} (〈 y^{2} 〉 - 〈 u^{2} 〉) . \end{array}

Here

〈 x 〉 = 〈 {\hat{\tilde{p}}}_{1} {\tilde{s}}_{1} + {\hat{\tilde{p}}}_{2} {\tilde{s}}_{2} + {\hat{\tilde{p}}}_{3} {\tilde{s}}_{3} 〉

,

〈 y 〉 = 〈 {\hat{\tilde{p}}}_{3} {\tilde{s}}_{2} - {\hat{\tilde{p}}}_{2} {\tilde{s}}_{3} 〉

,

〈 u 〉 = 〈 {\hat{\tilde{p}}}_{3} {\tilde{s}}_{1} - {\hat{\tilde{p}}}_{1} {\tilde{s}}_{3} 〉

, z′ = z/L,

ε_{1} = g P_{i n} L / 2, ε_{2} = α_{s} L

,

ε_{3} = 2 π L / L_{b p} (λ_{s} / λ_{p} - 1)

,

ε_{4} = (2 π L) / L_{b p}

,

ε_{5} = (2 π L) / L_{b s}

.

Herein we have neglected self- and cross-phase modulation (SPM and XPM). Justification of such approximation is found in [4–6]. The similar result have been obtained in [2] by direct modeling of stochastic equations (Eqs. (2)), 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.

As follows from [16,17], after averaging over fluctuations caused by random birefringence, equations for SIRF take the following form:

\begin{array}{l} \frac{d S I R F^{2}}{d z^{'}} = 〈 {\hat{Ω}}_{1} 〉, \\ \frac{d 〈 {\hat{Ω}}_{1} 〉}{d z^{'}} = - 〈 {\hat{Ω}}_{1} 〉 L / L_{c} + 2 α (z) 〈 {\hat{Ω}}_{2} 〉 + L / L_{c}, \\ \frac{d 〈 {\hat{Ω}}_{2} 〉}{d z^{'}} = - 2 α (z) 〈 {\hat{Ω}}_{1} 〉 - 〈 {\hat{Ω}}_{2} 〉 L / L_{c} - ε_{5} 〈 {\hat{Ω}}_{3} 〉, \\ \frac{d 〈 {\hat{Ω}}_{3} 〉}{d z^{'}} = ε_{5} 〈 {\hat{Ω}}_{2} 〉 . \end{array}

Finally, the PMD parameter for the spun fiber D_ps can be found as follows [4–6,17]

D_{p s} = \frac{λ_{s} \sqrt{2 L_{c}}}{L_{b s} c} S I R F .

3. Results and discussion

We find PDG, an averaged gain related to the pump-signal SOPs interaction G, mean-square gain fluctuations σ and PMD parameter D_p based on Eqs. (7) and (8). As shown by Sergeyev [4], potential of the interaction between the pump and the signal SOPs, noise and an adiabatic forcing are determined by high frequency of the fiber spinning k_hf, the correlation length L_c and low frequency of the fiber spinning k_lf, correspondently. In view of escape probability depends on noise and an adiabatic forcing [18–20], we study de-correlation of the pump and the signal SOPs (escape from polarization pulling) in terms of PDG, gain, mean-square gain fluctuations and PMD as a function of the frequency of an adiabatic forcing k_lf and the noise level σ² = 1/L_c [4]. We use the following parameters: λ_p = 1460 nm, λ_s = 1550 nm, P_in = 5 W, L_b = 8.3m, g = 2.3 dBW⁻¹km⁻¹, A₀ = 3 (in units of rad), k_hf = 6π/L_bp, k_lf = [0…300/L], L_c = [5m…205m], a = −2, L = 10 km, α_s = 0.2 dB/km. The results are shown in Figs. 1(a) -1(d) and Figs. 2(a) -2(d).

Fig. 1 a): Polarization dependent gain PDG, b): averaged part of the gain related to the pump-signal SOPs interaction, c): mean-square gain fluctuations σ, d): PMD parameter D_p as a function of fiber spinning frequency k_lf and correlation length L_c. Parameters: 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], a = −2, L = 10 km, α_s = 0.2 dB/km.

Download Full Size | PDF

Fig. 2 a), b): Averaged part of the gain related to the pump-signal SOPs interaction, polarization dependent gain PDG, mean-square gain fluctuations σ; c, d): PMD parameter D_p as a function of fiber spinning frequency k_lf and correlation length L_c. Parameters are the same as for Fig. 1. Gain (solid line), PDG (dotted line), σ (dashed line): a) L_c = 30 m (thin line), L_c = 65 m (thick line); b) k_lf = 3 km⁻¹ (thin line), k_lf = 5 km⁻¹ (thick line). PMD: c) L_c = 20 m (solid line), L_c = 30 m (dashed line); L_c = 55 m (dotted line); d) k_lf = 3 km⁻¹ (solid line), k_lf = 5 km⁻¹ (dashed line), k_lf = 15 km⁻¹ (dotted line).

Download Full Size | PDF

As follows from Figs. 1 and 2, PDG, averaged gain and gain fluctuations have maxima and minima as a function of correlation length L_c and spinning frequency k_lf. As is seen in Fig. 1(a) and Figs. 2(a) and 2(b), the minimum of PDG of 0.11 dB is found for L_c = 55m and k_lf = 3 km⁻¹. Unlike our previous result [4], the spin rate shown in Eq. (6) leads to growing averaged part of the gain related to the pump-signal SOPs interaction up to 15 dB (Fig. 1(c) and Figs. 2(a) and 2(b)).As follows from Fig. 2(b), PDG has a minimum as a function of noise power σ² = 1/L_c . When the birefringence fluctuations are large (small correlation lengths) and low (large correlation lengths) polarization pulling is dominating and PDG is high. If fluctuations are large, escape events happen at very small lengths and so at larger lengths 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 de-correlation of SOPs results in PDG suppression (Fig. 2(b)). Thus, the minimum of PDG as a function of correlation length L_c and spinning frequency k_lf leading to the activation of de-correlation of pump and signal SOPs which looks like Stochastic Resonance in the excitable systems [18,19].

As shown in Fig. 2(a), the local maximum for the mean-square gain fluctuations σ as a function of k_lf coincides with local minimum for PDG. In addition, gain fluctuations have a maximum as a function of correlation length that is in contrast to our previous results [4]. The main difference of the suggested fiber spinning in the form of Eq. (6) from the fiber spinning considered in [4] is the presence of the offset a = −2. In view of gain fluctuations have maximum when the difference in the birefringence strengths for pump and signal |b_b – b_s| is of order of the de-correlation rate 1/L_c [15,21], we suggest that by adding an offset into the fiber spinning profile can lead to the condition mentioned. Though the mean-square gain fluctuations are still high of 100%, the spin profile in the form of Eq. (6) gives opportunity to de-correlate pump and signal SOPs more effectively in terms of suppressed PDG of 0.11 dB, PMD of 0.037 ps/km^1/2 along with to increase gain up to 15 dB (Figs. 1(a)-1(d) and Figs. 2(a)-2(d)).

The evolution of the angle Φ between the pump and the signal SOPs for the fiber parameters corresponding to the case of minimum in Fig. 1(a) and Figs. 2(a) and 2(b), viz. L_c = 55m, k_lf = 3 km⁻¹, is shown in Fig. 3 . Unlike our previous results [4], by adding an offset it is possible to provide much better control of polarization pulling and de-correlation. As is seen in Fig. 3 for the minimum of PDG value, polarization pulling with _{$\cos Φ \to 1$} and SOPs de-correlation with $\cos Φ \to 0$ are swapping along the fiber length. In addition, an averaged value is shifted from _{$\cos Φ = 0$}to $\cos Φ \approx 0.3$ .

Fig. 3 Evolution of angle Φ between pump and signal SOPs and averaged part of gain related to the pump-signal SOPs interaction (L_c = 55m and k_lf = 3 km⁻¹), initial signal SOP (linearly polarized) coincide with the pump SOP (solid line) and is orthogonal to the pump SOP (dotted line).

Download Full Size | PDF

4. Conclusions

Using the vector model of a fiber Raman amplifier accounting for fiber spinning and random birefringence [4], we demonstrate for the first time effective suppression of PDG and PMD to 0.11 dB, and 0.037 ps/km^1/2 along with gain enhancement in 15 dB by tailoring properties of two-scale periodic fiber spinning. The practical implementation of the obtained results can be in extension of the transmission distance and reducing complexity of the fiber Raman amplifier-based un-repeatered high-capacity transmission systems [14].

Acknowledgments

S. Sergeyev acknowledges financial support from the European Union program FP7 PEOPLE-2009-IEF (grant 253297).

References and links

1. V. V. Kozlov, J. Nuño, J. D. Ania-Castañón, and S. Wabnitz, “Theory of fiber optic Raman polarizers,” Opt. Lett. 35(23), 3970–3972 (2010). [CrossRef] [PubMed]

2. M. Martinelli, M. Cirigliano, M. Ferrario, L. Marazzi, and P. Martelli, “Evidence of Raman-induced polarization pulling,” Opt. Express 17(2), 947–955 (2009). [CrossRef] [PubMed]

3. L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23(4), 254–256 (2011). [CrossRef]

4. S. V. Sergeyev, “Activated polarization pulling and de-correlation of signal and pump states of polarization in a fiber Raman amplifier,” Opt. Express 19(24), 24268–24279 (2011). [CrossRef] [PubMed]

5. S. Sergeyev and S. Popov, “Two-section fiber optic Raman polarizer,” IEEE J. Quantum Electron. 48(1), 56–60 (2012). [CrossRef]

6. S. Sergeyev, S. Popov, and A. T. Friberg, “Virtually isotropic transmission media with fiber Raman amplifier,” IEEE J. Quantum Electron. 46(10), 1492–1497 (2010). [CrossRef]

7. N. J. Muga, M. F. S. Ferreira, and A. N. Pinto, “Broadband polarization pulling using Raman amplification,” Opt. Express 19(19), 18707–18712 (2011). [CrossRef] [PubMed]

8. P. Morin, S. Pitois, and J. Fatome, “Simultaneous polarization attraction and Raman amplification of a light beam in optical fibers,” J. Opt. Soc. Am. B 29(8), 2046–2052 (2012). [CrossRef]

9. V. V. Kozlov and S. Wabnitz, “Suppression of relative intensity noise in fiber-optic Raman polarizers,” IEEE Photon. Technol. Lett. 23(15), 1088–1090 (2011). [CrossRef]

10. A. Zadok, E. Zilka, A. Eyal, L. Thévenaz, and M. Tur, “Vector analysis of stimulated Brillouin scattering amplification in standard single-mode fibers,” Opt. Express 16(26), 21692–21707 (2008). [CrossRef] [PubMed]

11. J. E. Heebner, R. S. Bennink, R. W. Boyd, and R. A. Fisher, “Conversion of unpolarized light to polarized light with greater than 50% efficiency by photorefractive two-beam coupling,” Opt. Lett. 25(4), 257–259 (2000). [CrossRef] [PubMed]

12. P. Morin, J. Fatome, C. Finot, S. Pitois, R. Claveau, and G. Millot, “All-optical nonlinear processing of both polarization state and intensity profile for 40 Gbit/s regeneration applications,” Opt. Express 19(18), 17158–17166 (2011). [CrossRef] [PubMed]

13. M. Guasoni and S. Wabnitz, “Nonlinear polarizers based on four-wave mixing in high-birefringence optical fibers,” J. Opt. Soc. Am. B 29(6), 1511–1520 (2012). [CrossRef]

14. J. D. Ania-Castañón, V. Karalekas, P. Harper, and S. K. Turitsyn, “Simultaneous spatial and spectral transparency in ultralong fiber lasers,” Phys. Rev. Lett. 101(12), 123903 (2008). [CrossRef] [PubMed]

15. Q. Lin and G. P. Agrawal, “Vector theory of stimulated Raman scattering and its application to fiber-based Raman amplifiers,” J. Opt. Soc. Am. B 20(8), 1616–1631 (2003). [CrossRef]

16. P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996). [CrossRef]

17. A. Galtarossa, L. Palmieri, A. Pizzinat, B. S. Marks, and C. R. Menyuk, “An analytical formula for the mean differential group delay of randomly-birefringent spun fibers,” J. Lightwave Technol. 21(7), 1635–1643 (2003). [CrossRef]

18. F. Marino, M. Giudici, S. Barland, and S. Balle, “Experimental evidence of stochastic resonance in an excitable optical system,” Phys. Rev. Lett. 88(4), 040601 (2002). [CrossRef] [PubMed]

19. B. Lindner, J. García-Ojalvo, A. Neiman, and L. Schimansky-Geier, “Effects of noise in excitable systems,” Phys. Rep. 392(6), 321–424 (2004). [CrossRef]

20. M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11(3), 587–594 (2001). [CrossRef] [PubMed]

21. S. Sergeyev, S. Popov, and A. T. Friberg, “Polarization dependent gain and gain fluctuations in a fiber Raman amplifier,” J. Opt. A, Pure Appl. Opt. 9(12), 1119–1122 (2007). [CrossRef]

Fiber Raman amplification in a two-scale spun fiber

Abstract

1. Introduction

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

3. Results and discussion

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (3)

Equations (9)

Optical Materials Express