We report a novel vector model of Raman amplification in a fiber with randomly varying birefringence and unidirectional spin profile. Applying the model, we demonstrate for the first time simultaneous mitigation of polarization mode dispersion and polarization dependence of the Raman gain.
© 2008 Optical Society of America
Interest in optical fibers with the polarization mode dispersion (PMD) suppressed by a fiber spinning arose more than twenty years ago and was driven by the need to preserve the signal state of polarization (SOP) for the development of sensing devices, such as magneto-optic current sensors utilizing the Faraday effect [1-3], fiber Bragg grating sensors insensitive to transverse force , polarization transformers , and chiral fiber Bragg gratings for optical filters and in-line polarizers .
Strong demand in increasing bandwidth, distance, and bit rate of signal transmission for broadcasting media and medical applications has renewed the attention to such optical fibers with the PMD reduced by fiber spinning [7-20]. Modern technology of direct fiber spinning instead of spinning the preform  provides high-speed fiber drawing and flexible control of different spin profiles for better PMD suppression [7, 8] (Fig. 1). Quantitative analyses based on fixed modulus and random modulus models (FMM and RMM, respectively [9-17]) together with the spin profile measurements [18-20] lead to the conclusion that a unidirectional periodic spin profile is optimal, if strong PMD suppression is required [9-20].
Application of broadband fiber Raman amplifiers (FRAs) in optical telecom links appears as the next natural step to reduce the transmission system cost, increase the amplification bandwidth over 100 nm, and extend the link span over 1500 km . With the help of Raman amplification, the transmission fiber becomes a distributed amplifier due to stimulated Raman scattering (SRS), a process by which energy is transferred from a short-wavelength pump source to a longer-wavelength signal (100 nm shift) through nonlinear inelastic photon scattering on optical phonons. However, in Raman amplifiers that make use of fibers with suppressed PMD, the amplified optical pulse is severely distorted due to the polarization dependent gain (PDG) [22-26], i.e., dependence of the Raman gain on pump and signal SOPs, as illustrated in Fig. 2.
For simplicity, we show in Fig. 2 the Raman amplification in polarization maintaining (PM) fiber. For PM fiber, if the pump electric-field vector is oriented along a birefringence axis, the difference in the Raman gain values of the orthogonal signal components is maximal and the PDG takes its maximum value (Fig. 2 (a)). If the pump field is equally shared initially between the two orthogonal states of polarization, the pump SOP evolves through all the possible polarization states and returns to its original state after a beat length L b. In average, the pump power is the same along both orthogonal axes. Hence, there is no difference in the Raman gain values and the PDG becomes a zero.
For low PMD fibers, birefringence fluctuations can be simulated with the help of a set of concatenated pieces of fixed-birefringence fiber with a normal length distribution and a uniform distribution of fast/slow axis orientations. In the limit of strong fluctuations, the overall birefringence approaches zero and the fiber becomes isotropic. The other way to reach the small PMD value is spinning the fiber, for example periodically. Consequently, in both cases there is no absolute anisotropy axis with reference to the whole fiber. Thus the Raman gain depends only on the relative orientation between the input signal and pump polarizations. In this situation, co-polarized pump and signal waves give the maximum Raman gain and cross-polarized waves result in the minimum gain. Because the pump and signal SOPs are not changing along the length of a low PMD fiber, maximum and minimum PDGs converge at the limit value of PDG which is more than 20 dB [22-26]. To enable high-speed and long-distance transmission of broadband optical signals, one needs to find a solution that employs a Raman amplifier with minimum dependence on the pump and signal polarizations, while simultaneously keeping the suppressed PMD. At a first glance, these seem as mutually excluding requirements if the realization is attempted in the same fiber.
Most of the existing schemes to mitigate the PDG are rather expensive (polarization multiplexing of pump laser diodes , application of depolarizers [21, 27]) or not effective (backward pumping , linear-circular converter of the pump polarization [24-26, 28]) in the case of low PMD fibers. Finding an efficient method of simultaneous suppression of the PMD and PDG requires a rigorous vector model of the Raman fiber amplifier that accounts for the random birefringence of the fiber and for the fiber spin profile. Recently, theoretical characterization of the PDG in the case of backward pumped periodically spun fiber has been done by Bettini et al. . The approach was based on implementing Monte-Carlo simulation and the vector model of FRA suggested by Lin and Agrawal .
Apart from this specific situation of backward pump, a new model has to be applicable for characterization of PDG in more practical case of forward pumping. In addition, the model should reproduce the PDG dependence on pump polarization that was demonstrated experimentally for fibers without spinning . Furthermore, unlike time consuming Monte-Carlo simulation of stochastic equations, new model has to be based on numerical solution of the ordinary differential equations derived from stochastic equations by means of Stochastic Generator Approach [23-25]. In an advanced vector model recently formulated by Sergeyev et al., all features mentioned has been included except of the fiber spinning profile [24, 25]. The model was confirmed experimentally as a tool for the characterization of fiber birefringence (evaluating the beat length, correlation length, and PMD) using measured data on the maximum and minimum PDG values [25,26]. Updating this model by accounting for a certain particular spin profile, we report in this paper a novel method for cost-effective mitigation of both PMD and PDG simultaneously.
2. Simplified approach to the simultaneous mitigation of the PDG and PMD in spun fiber Raman amplifier
The vector model describing the polarization dependence of the Raman gain in fibers with random or regular birefringence can instructively be demonstrated using the Poincaré sphere. Here, the states of the signal and pump polarization are considered in terms of vectors s and p pointing to positions on the Poincaré sphere (Fig. 3). Fiber birefringence can be shown as a rotation of the s and p vectors on the Poincaré sphere around the birefringence vector w, which, in turn, can itself experience regular or/and random fluctuations depending on the specific origin of the birefringence. Regular (for example, induced by fiber spinning) or random oscillations of the birefringence vector lead to a slowing down of the SOP rotation [7-20], i.e., to suppression of the PMD. As is illustrated in Fig. 3, the signal and pump states of polarization revolve on the Poincaré sphere in the same direction but at different rates b s and b p around the birefringence vector w. If the difference b p-b s is much higher than de-correlation rate L c -1, then the s and p vectors periodically reach mutually parallel and orthogonal orientations and oscillatory behavior with a spatial period T occurs for pump to signal SOP projection <x>=<s p> .
As shown in Ref. 24, projections <x max> and <x min> associated with the maximum and minimum Raman gain values, i.e. s0max and s0min, respectively, oscillate in anti-phase along the fiber and coincide at the distances of z n=(n T)/2, where n is an integer number. It is known that the Raman PDG (in units of dB) is defined in terms of the averaged projections <x max> and <x min> as follows 
where L is the fiber length, g is the Raman gain coefficient, P in is the input pump power, and α p describes losses at the pump wavelength.
Our novel approach to simultaneously mitigate the PMD and PDG is based on combining a standard short-length fiber of length L 1 with a long section L 2 of periodically spun fiber. If choose the length of the short-length fiber as L 1=n T/2 (n=1,2…), then the projections <x max> and <x min> are the same at the output of section L 1 (or the input of periodically spun fiber L 2), and according to Eq. (1), do not contribute to the polarization dependent gain along this fiber. This result holds for a wide range of spinning amplitudes. Thus, the maximum and minimum PDG values for the case of forward pumping in the two-fiber combination can be calculated based on the results obtained for the FRA without spin 
where ε 1=g P in L c/2, ε 2=αs L c, , α s is the losses at the signal wavelength, c is the speed of light, D p is the PMD parameter for the fiber without spin, λ p and λ s are the pump and signal wavelengths. The calculated values are shown in Fig. 4(a), and give 0.27 dB and 0.13 dB, respectively. It will be shown in the next sections that much higher PDG values would be obtained for the periodically spun fiber used alone.
In the periodically spun fiber (section L 2), the spinning profile (in units of rad) is given by
where f 0 is the spin frequency, A 0 is the spin amplitude, and z is a distance along the fiber (Fig. 1). To describe random birefringence fluctuations, two models are commonly applied: fixed modulus model (FMM), where the length of the birefringence vector is fixed and its orientation is driven by a white-noise process, and more complex random modulus model (RMM) [12-17], in which both the orientation and the length of the birefringence vector are changing randomly. In our approach, we use the FMM which is simpler, but nonetheless demonstrates results for spin induced reduction factor (SIRF) similar to those obtained with the RMM [12-17] for the case when the spin frequency f 0 is higher than birefringence beat frequency 1/L b, where L b is the beat length . The SIRF is used for estimating the PMD suppression according to the well-known expression [12-15, 17]
where and 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 the spin, respectively . The latter is given by 
where L c is the correlation length. It is known that the SIRF for the periodic spin rate given by Eq. (3) can reach the minimum value of less than 0.01 for the phase-matching condition A 0≈1.2 [9-20]. For this condition, leading to 〈Δτ 2 un (L 1)〉>>〈Δτ 2(L 2)〉 in two-section fiber, the DGD for a long-length periodically spun fiber is much less than the DGD for a short section of the fiber without spin. The SIRF for the combination of these two sections of fiber is
The results of the SIRF calculation based on Eq. (6) are shown in Fig. 4(b). As one can see from the plot, it is possible to maintain a low value of SIRF around 0.16 in the two-section fiber, i.e., to reduce the PMD value down to 0.03 ps·km-1/2.
3. An advanced model of spun fiber Raman amplifier
To justify and to optimize the method of simultaneous mitigation of PMD and PDG, as discussed above, we update the vector model of FRA given in Refs. 22-25 by including the fiber spin profile (Eq. (3)). To calculate the SIRF for the case of spun fiber we use the standard model of PMD presented in Refs. 9-20. As a result, the evolution of the SOPs for signal and pump waves, and PMD vector (s, p and Ω vectors on Poincaré sphere) can be expressed by the following equations [9-20, 22-25]:
Here P 0(z) is the pump power (P 0(z)=P in exp(-α p z) for forward pump and P 0(z)=P in exp(-α p(L-z)) for backward pump), s=s 0 ŝ, ŝ and p̂ are unit Stokes vectors for the signal and pump waves (is the length of s), η=1 for forward pumping and η=-1 for backward pumping, ω is the angular frequency. The length of the PMD vector is equal to the DGD value, i.e. [12-15, 17]. Pump depletion in Eq. (7) is neglected as usual, since the pump power is much larger than the signal power.
We neglect herein the fiber twist and, therefore, the birefringence vector for the spun fiber takes the form of W i=R 3[2A(z)]W i,un, where W i,un=(2b i cos ϕ, 2b i sinϕ,0)T is the birefringence vector for the same fiber without spin, index i=p,s denotes the pump and signal components respectively, 2b i=2π/L bi is the birefringence strength, ϕ is the orientation angle of the birefringence axis with respect to the x-axis on the Poincaré sphere, L bi is the beat length, A(z) is the spin profile, and R 3(γ) represents the rotation in equatorial plane by angle γ around the z-axis on the Poincaré sphere 
To represent the random birefringence, we use the fixed modulus model (FMM) in which the birefringence strength 2b i is fixed and the orientation angle ϕ is driven by a white-noise process [12-15, 17]
We consider the more instructive and practical example of forward pumping, i.e. η=1. Using this notation and the reference frame where the local birefringence vector is W̃i,un=(2b i, 0, 0), i.e., it is oriented along the x-axis on the Poincaré sphere, and applying an averaging procedure (explained in Refs. 24) to Eq. (7), we find the system of equations for 〈s 0〉:
Here 〈x〉=〈p̃1 s̃1+p̃2 s̃2+p̃3 s̃〉, 〈y〉=〈p̃3 s̃2-p̃2 s̃3〉, 〈u〉=〈p̃3 s̃1-p̃1 s̃3〉 α(z)=∂A(z)/∂z is the spin rate, z′=z/L c. The system of equations for SIRF can be derived based on the averaging of PMD vector Ω and results of Ref. 14, 17
Here 〈Ω̂i〉 (i=1,2,3) are normalized components of the PMD vector, ε 4=L c/L, ε 5=(2πL c)/L bs.
4. Results and discussion
The results for the PDG and SIRF have been obtained for the following parameters: the pump SOP for minimum PDG is (0,0,1), the pump SOP for maximum PDG is (1,0,0), f o L b,s=3, , L 2=10 km, L c=30 m, L b,s=8.17 m, α p=0.27 dB/km, D p=0.2 ps·km-1/2, λ p=1460 nm, λ s=1550 nm, g=1.8 W-1km-1, and P=1 W. The resulting values for max/min PDG and SIRF as a function of the spin amplitude for two-section fiber are shown in Figs. 4(a), 4(b). The results of the calculations of the signal-to-pump SOP projections are shown in Fig. 4(c). As follows from Figs. 4(a)-4(c), the simplified approach considered in previous section (Eqs. (2), (6)) and advanced model (Eqs. (10), (11)) lead to quite close values for PDG in the wide range of spin amplitudes A0 and for SIRF at A0≈1.2. Thus, both approaches demonstrate opportunity to reduce the PDG and SIRF to the values acceptable for high-speed telecommunication, namely PDG=0.13 dB and SIRF=0.16, i.e., D p=0.032ps·km-1/2. As follows from Fig. 4(c), results of the advanced model confirm also the conclusion that the pump to signal SOP projections associated with the maximum and minimum Raman gain keep the same values along the periodically spun fiber and, therefore, make almost no contribution to the polarization dependent gain with phase-matching condition A 0≈1.2.
For comparison, the results for the SIRF, pump to signal SOP projection and PDG have been calculated for the case of a single section of periodically spun fiber (Figs. 4(a)-4(c)). The results shown in Fig. 4(a) demonstrate the possibility to reach the minimum value of 0.013 for the SIRF. However, as follows from Fig. 4(c), the pump to signal SOP projections diverge along the fiber and, therefore, the PDG increases up to the value of 22.4 dB (Fig. 4(b)).
To suppress the SIRF and PDG further we have applied a fiber spin tailoring approach based on application of the frequency modulated (FM) and exponentially varying (EXV) spin rates in Eqs. (10) and (11):
Finally, SIRF, pump to signal SOP projection <x> and PDG have been calculated for the following parameters: the pump SOP for minimum PDG is (0,0,1), the pump SOP for maximum PDG is (1,0,0), f o L b,s=3, L b,s=8.17 m, , L 2=10 km, L c=30 m, Ω=1/L b,s, α p=0.27 dB/km, D p=0.2 ps·km-1/2, λp=1460 nm, λs=1550 nm, g=1.8 W-1km-1, and P=1 W. The results of calculations are shown in Figs. 4(b), 4(d). As it is seen from Fig. 4(b) (dotted line) for frequency-modulated spin rate, spectrum of the spin profile includes the set of spatial frequencies and, therefore, the PMD reduction can be reached for the wide range of spinning amplitudes A 0. Because the minimum SIRF=0.16 can be reached for low fiber spin amplitude A 0≈0.4, it results in low maximum and minimum PDG values of 0.27 dB and 0.14 dB correspondently (Fig. 4(d)). As follows from Figs. 4(b), 4(d) for the exponentially varying spin rate, the SIRF and maximum and minimum PDGs are slightly less than the SIRF obtained for two-section fiber in the vicinity of phase-matching condition A 0≈1.2. To explain the suppression of PDG and SIRF for this case, it is necessary to perform a more complex analysis that is the subject of separate investigation.
We report a novel vector model of a spun fiber Raman amplifier. Based on this model, we demonstrate for the first time simultaneous suppression of polarization mode dispersion and polarization dependent gain to the values acceptable for high-speed telecommunication, namely PDG=0.13 dB and D p=0.032ps·km-1/2. The polarization dependence of the Raman gain is defined by the ratio of the averaged pump to signal SOP projections that are associated with the maximum and minimum Raman gain. These projections oscillate in anti-phase and converge at some characteristic distance which defines the length of the shorter section of the fiber. Thus, a combination of the short-length fiber with the long periodically spun fiber allows mitigating the PDG and PMD simultaneously. As a result, the projections are the same at the input of periodically spun fiber, and therefore they do not contribute to the polarization dependent gain along this fiber over a wide range of spinning amplitudes. Since the first section is much shorter than the periodically spun fiber, the composition of the two fibers will lead to suppressed PMD value for phase matching condition.
We have also considered additional ways of simultaneous suppression of PMD and PDG based on application of frequency modulated and exponentially varying spin rates.
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