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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 24 — Nov. 21, 2011
  • pp: 24268–24279
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Activated polarization pulling and de-correlation of signal and pump states of polarization in a fiber Raman amplifier

Sergey V. Sergeyev  »View Author Affiliations


Optics Express, Vol. 19, Issue 24, pp. 24268-24279 (2011)
http://dx.doi.org/10.1364/OE.19.024268


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Abstract

We report on a theoretical study of activated polarization pulling and de-correlation of signal and pump states of polarization based on an advanced vector model of a fiber Raman amplifier accounting for random birefringence and two-scale fiber spinning. As a result, we have found that it is possible to provide de-correlation and simultaneously suppress PDG and PMD to 1.2 dB and 0.035 ps/km1/2 respectively.

© 2011 OSA

1. Introduction

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 s^=(s^1,s^2,s^3)andp^=(p^1,p^2,p^3), respectively, pointing to positions on the Poincaré sphere (Fig. 1
Fig. 1 Evolution of the pump p^ and the signal s^ states of polarization on the Poincaré sphere, as well as the fluctuations of the local birefringence vector Wi = (2bicosθ, 2bisinθ,0)T . Vectors s^ and p^ rotate around the local axis (W) at rates bp and bs, while vector (W) rotates randomly in the equatorial plane at the rate σ = Lc-1/2 (Lc is the correlation length), Φ is an angle between pump and signal SOPs.
). Due to birefringence, these vectors rotate on the Poincaré sphere around the birefringence vector Wi = (2bicosθ, 2bisinθ,0)T in the same direction, but at different rates bs and bp which are birefringence strengths (bi = π/Lbi where Lbi is the beat length) at signal λs and pump λp wavelengths. We assume that the birefringence strength 2bi is fixed and the orientation angle θ is driven by a white-noise process (fixed-modulus model [2

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

])
dθdz=β(z),β(z)=0,β(z)β(z')=σ2δ(zz'),
(1)
where <…> means averaging over the birefringence fluctuations along the fiber, δ(z) is a Dirac delta-function, and σ2 = 2/Lc (Lc is the birefringence correlation length).

We neglect herein the fiber twist and, therefore, the birefringence vector for the spun fiber takes the form of Wi = R3[2A(z)]Wi,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

9. 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]

]

R3(γ)=[cosγsinγ0sinγcosγ0001].
(2)

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:

S=s0Gaves^,P=P0(z)p^.
(3)

PDG10log(s0,max(L)/s0,min(L)),
(5)

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

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

]:

R=5log(s0,max(min)2(L)s0,max(min)(L)2/s0,max(min)(L)2).
(6)

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

4. 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]

]:

dsdz=g2P0(z)s0p^+(Ws+Ws(NL))×s,dp^dz=(Wp+Wp(NL))×p^.
(7)

Heres=s0s^, and Wp(NL), Ws(NL) describes the nonlinear SOP evolution caused by self- and cross-phase modulation (SPM and XPM): Wp(NL)=2γp/3(2S1,2S2,p^3P0(z)), Ws(NL)=2γs/3(2p^1P0(z),2p^2P0(z),S3). Kerr coupling constants are γi = 2πn2/(λiAeff) (i = s,p), where n2 is the nonlinear Kerr coefficient and Aeff 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 W˜i,un=(2bi,0,0) and p˜^(0)=(1,0,0), i.e. oriented along the X-axis on the Poincaré sphere. This can be accomplished by a suitable transformation W˜i=R31(θ+2A(z))Wi, and s˜=R31(θ+2A(z))s and p˜^=R31(θ+2A(z))p^, provided the variable s0 and scalar product x=p^s are invariant under the rotation. In addition, we consider cases which correspond to the maximum and minimum of Raman gain, viz. s˜max(0)=(1,0,0)and s˜min(0)=(1,0,0). Applying an averaging procedure considered in [5

5. S. Sergeyev, S. Popov, and A. T. Friberg, “Spun fiber Raman amplifiers with reduced polarization impairments,” Opt. Express 16(19), 14380–14389 (2008). [CrossRef] [PubMed]

,6

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]

] to Eqs. (7), we find the system of equations:

ds0dz=ε1exp(ε2z)x,dxdz=ε1exp(ε2z)s0ε3y(δpδs)exp(ε2z)p˜^3(p˜^2s˜1p˜^1s˜2),dydz=ε3[xp˜^1s˜1]2α(z')uyL2Lc+ε4p˜^2s˜1ε5p˜^1s˜2(δpδs)exp(ε2z)p˜^3p˜^1s˜3δsexp(ε2z)p˜^2(p˜^2s˜1p˜^1s˜2),dudz=2α(z')yu2+(δpδs)exp(ε2z)p˜^3p˜^2s˜3δsexp(ε2z)p˜^1(p˜^2s˜1p˜^1s˜2).
(8)

Here δp = 2γpLPin/3 δs = 4γsLPin/3, x=p˜^1s˜1+p˜^2s˜2+p˜^3s˜3, y=p˜^3s˜2p˜^2s˜3, u=p˜^3s˜1p˜^1s˜3, α(z')=A(z')/z' is the spin rate, z′ = z/L, ε1=gPinL/2,ε2=αsL, ε3=2πL/Lbp(λs/λp1), ε4=(2πL)/Lbp, ε5=(2πL)/Lbs.

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 W˜i), we can average over the fast rotations. It can be done by the transformations h=Rs(z)s˜ and f=Rp(z)p˜^, where

Ri(z)=(1000cos(2biz)sin(2biz)0sin(2biz)cos(2biz)),(i=s,p).
(9)

Using this transformation, Eq. (7) and averaging over the fast oscillations, we find the following equation for h1f1

dh1f1dz'=ε1exp(ε2z)s0f12LLch1f1.
(10)

If we choose parameters for Raman amplifier as, g = 2.3 dB W−1km−1, Pin = 5W, L = 10 km, Lc = 110 m, we find that ε1Lc/L=0.15<<1. Thus, we can neglect the first term on the right-hand side of Eq. (10). As a result,

p˜^1s˜1=h1f1=p˜^1(0)s˜1(0)exp(z'L/Lc).
(11)

In addition, this procedure leads to p˜^2s˜10, p˜^1s˜20, p˜^2p˜^3s˜20, p˜^1p˜^3s˜20,

p˜^3(p˜^2s˜1p˜^1s˜2)0 and so self- and cross-phase modulation (SPM and XPM) can be neglected [4

4. 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]

7

7. S. Sergeyev and S. Popov, “Two-section fiber optic Raman polarizer for high-speed transmission systems,” in the 13th International Conference on Transparent Optical Networks (June 26–30, 2011) Stockholm, Sweden, Th.A6.7.

]. A similar result has been obtained in [10

10. 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]

,12

12. 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]

] by direct modeling of stochastic Eqs. (7), viz. it was obtained that for the pump power Pin < 10 W and PMD parameter Dp>0.01 ps/km1/2 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):

ds02dz=2ε1exp(ε2z)s0x,ds0xdz=ε1exp(ε2z)(s02+x2)ε3ys0,ds0ydz=ε1exp(ε2z)xy+ε3[s0xy2s0p˜^1s˜1]2α(z')s0us0yL2Lc,dx2dz=2ε1exp(ε2z)s0x2ε3xy,dxydz=ε1exp(ε2z)s0y+ε3[x2xp˜^1s˜1]2α(z')xuxyL2Lc,dus0dz=ε1exp(ε2z)xu+2α(z')ys0us0L2Lc,dxudz=ε1exp(ε2z)s0uε3yu+2α(z')xyxuL2Lc,dyudz=ε3(xuup˜^1s˜1)+2α(z')(y2u2)+Lyu2Lc,du2dz=2α(z')yu+LLc(y2u2),dy2dz=2ε3[yxyp˜^1s˜1]2α(z')yuLLc(y2u2).
(12)

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

9. 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]

]
dΩdz=Wsω+Ws×Ω,SIRF=|Ω(L)|sp2/|Ω(L)|un2,
(13)
where |Ω(L)|sp2 and |Ω(L)|un2are 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 [9

9. 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]

]. As follows from [5

5. S. Sergeyev, S. Popov, and A. T. Friberg, “Spun fiber Raman amplifiers with reduced polarization impairments,” Opt. Express 16(19), 14380–14389 (2008). [CrossRef] [PubMed]

,6

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]

,9

9. 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]

], after averaging over fluctuations caused by random birefringence, equations for SIRF take the following form:

dSIRF2dz=Ω^1,dΩ^1dz=Ω^1L/Lc+2α(z)Ω^2+L/Lc,dΩ^2dz=2α(z)Ω^1Ω^2L/Lcε5Ω^3,dΩ^3dz=ε5Ω^2.
(14)

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

9. 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]

]

Dps=λs2LcLbscSIRF.
(15)

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

αam(z')=A0khfLcos(klfLz')cos(khfLz'),αpm(z')=A0khfLcos(khf(Lz'+a0cos(klfLz'))).
(16)

Here A0 is amplitude of the fiber spinning in rad, klf and khf 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

8. M. J. Li and D. A. Nolan, “Fiber spin-profile designs for producing fibers with low polarization mode dispersion,” Opt. Lett. 23(21), 1659–1661 (1998). [CrossRef] [PubMed]

].

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

17. 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]

19

19. 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]

], viz. Kramers length <Lk> and intrawell relaxation length <LR>. In the absence of low-frequency modulation (klf = 0), Eqs. (8) can be simplified for the case of khf>>Lc, Lb with the help of averaging over fast periodic fiber spinning [9

9. 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]

]. As a result, we have

dx^dz=ε1exp(ε2z)(1x^2)ε3y^,dy^dz=(J0(2A0)2ε3ε1exp(ε2z)y^)x^y^L2Lc.
(17)

Here y^=y/s0, x^cosΦ=x/s0 is the variable which indicates polarization pulling if x^1 or de-correlation pump and signal SOPs if x^0; and J0(2A0) is the zero-order Bessel function of the first kind. For exp(ε2z)<<1, solutions of Eqs. (17) x^0,y^0 independent on z, can be found from equations:

Δx^023+Δx^02+(Δ12Δ2)x^0Δ=0,y^0=J0(2A0)Δ1x^0Δx^0+1.
(18)

Here Δ=2Lcε1exp(ε2z)/L,Δ1=2LcJ0(2A0)ε3/L. Linear stability analysis of Eqs. (17) near the states x^0,y^0 results in the following eigenvalues:

Λ1,2=14Lc(3Δ<x^0>+1)±14Lc1+Δ2<x^0>22Δ<x^0>4Δ12+ΔΔ1y^0.
(19)

Thus, relaxation length can be defined as LR=1/Re(Λ1,2). If Im(Λ1,2)0then the system oscillates around the states x^0,y^0and so the averaged length at which the system is close to the state can be defined as Lk=2πL/Im(Λ1,2) (Kramers length).

3. Results and discussion

We have found x^0,y^0, Re{Λ1,2}, Im{Λ1,2} numerically from Eqs. (18), (19). We find also PDG, R and PMD parameter Dp 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, αs = 0.2 dB/km, λp = 1460 nm, λs = 1550 nm, g = 2.3 dBW−1km−1, Pin = 5 W, Lb = 8.3 m, A0 = 3 (in units of rad), khf = 6π/Lbp, klf = [0…300/L], Lc = [5m…205m], a0 = 100 m. Results for x^0,y^0 and Re{Λ1}, Im{Λ1} are shown in Fig. 2 (a, b)
Fig. 2 a):x^0 (solid line) and y^0(dotted line) and b): Im(Λ1),Im(Λ2), (dotted line), Re(Λ1) (solid line) and Re(Λ2) (dashed line) as a function of correlation length Lc. Parameters: L = 10 km, αs = 0.2 dB/km, λp = 1460 nm, λs = 1550 nm, g = 2.3 dBW−1km−1, Pin = 5 W, Lb = 8.3 m, A0 = 3 (in units of rad), khf = 6π/Lbp, klf = 0, Lc = [5m…205m].
. The results for PDG, R and PMD parameters with p^(0)=(1,0,0), s^min(0)=(1,0,0),s^max(0)=(1,0,0) are shown in Fig. 3 (a-f)
Fig. 3 Polarization dependent gain PDG (a, b), parameter R for the p˜^(0)=s˜max(0)=(1,0,0) (c, d) and PMD parameter Dp (e, f) as a function of fiber spinning frequency klf and correlation length Lc . 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): Lc = 105 m and klf = 6 km−1 (Figs. 3 (a, c)), Lc = 110 m and klf = 3 km−1 (Figs. 3 (b, d)); polarization pulling (maximum in PDG with an arrow): Lc = 35 m and klf = 30 km−1 (Figs. 3 (a, c)), Lc = 110 m and klf = 22 km−1 (Figs. 3 (b, d)).
.

For the parameters listed, Eqs. (18) have only one real solution with |x^0|1. As follows from Fig. 2 (a) for Lc→0 (high birefringence fluctuations) x^01 and so we have polarization pulling and for Lc>>200 m de-correlation of SOPs with x^00. It is clear from Fig. 2 (b) that oscillations of angle between SOPs start when Lc≥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 Lc and spinning frequency klf. For slow amplitude modulation of the fiber spinning (Fig. 3 (a)), the maximum PDG corresponds to the case when Lc, klf →0. As follows from the results of [8

8. M. J. Li and D. A. Nolan, “Fiber spin-profile designs for producing fibers with low polarization mode dispersion,” Opt. Lett. 23(21), 1659–1661 (1998). [CrossRef] [PubMed]

,9

9. 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]

], fast periodic fiber spinning results in PMD mitigation and so leads to polarization pulling [7

7. S. Sergeyev and S. Popov, “Two-section fiber optic Raman polarizer for high-speed transmission systems,” in the 13th International Conference on Transparent Optical Networks (June 26–30, 2011) Stockholm, Sweden, Th.A6.7.

13

13. 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]

]. However, with increased frequency of amplitude modulation PDG reaches a minimum of 1.5 dB for Lc = 105m and klf = 6 km−1 and for Lc = 55 m and klf = 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 Lc = 110 m and klf = 3 km−1 and a maximum of 14.5 dB for Lc = 110 m and klf = 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
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: s˜max(0)=(1,0,0), solid lines˜min(0)=(1,0,0). Thick lines and thin lines correspond to the maximum and minimum in PDG shown by arrows.
. As follows from Figs. 3 (a, b) and Figs. 4, the maximum of PDG corresponds to the polarization pulling withcosΦ1, and the minimum of PDG corresponds to the case of SOPs de-correlation for which pump and signal SOPs are freely rotating and so cosΦ0. 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).

To characterize polarization pulling and SOPs de-correlation in terms of SR phenomena, we present in Fig. 5 (a-d)
Fig. 5 PDG (a,b), parameter R (c, d) and PMD parameter Dp (e, f) as a function of modulation frequency kl,f (a, c) and correlation length Lc (b, d) for amplitude (thick lines) and phase modulated (thin lines) of fiber spinning. Parameters in Figs. 5 (a, c, d): Lc = 35 m (thick solid line), Lc = 105 m (thick dotted line), Lc = 55 m (thin solid line), Lc = 115 m (thin dotted line), Parameters in Figs. 5 (b, d, f): klf = 6 km−1 (thick solid line), klf = 30 km−1 (thick dotted line), klf = 3 km−1 (thin solid line), klf = 22 km−1 (thin dotted line).
the results for PDG, parameter R and PMD parameter Dp as a function of modulation frequency kl,f and correlation length Lc. For low-frequency amplitude modulation, PDG has a minimum and parameter R has a maximum at frequency of klf,max = 6 km−1 which coincides with the resonance frequency for excitable systems calculated from Eq. (19).

A maximum and minimum at this frequency is evidence of Stochastic Resonance where random birefringence fluctuations are synchronized with an external modulation [18

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

]. 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

19. 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]

]. 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/Lc) 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/Lc 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

17. 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]

,18

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

]. As follows from Fig. 3(a), PDG is almost independent of modulation frequency klf 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

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

]. 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/Lc and modulation frequency klf 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 kl,f and Lc−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/km1/2 respectively.

4. Conclusion

Acknowledgment

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

References and links

1.

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers,” Electron. Lett. 22(19), 1029–1030 (1986). [CrossRef]

2.

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

3.

R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron. 15(10), 1157–1160 (1979). [CrossRef]

4.

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]

5.

S. Sergeyev, S. Popov, and A. T. Friberg, “Spun fiber Raman amplifiers with reduced polarization impairments,” Opt. Express 16(19), 14380–14389 (2008). [CrossRef] [PubMed]

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.

S. Sergeyev and S. Popov, “Two-section fiber optic Raman polarizer for high-speed transmission systems,” in the 13th International Conference on Transparent Optical Networks (June 26–30, 2011) Stockholm, Sweden, Th.A6.7.

8.

M. J. Li and D. A. Nolan, “Fiber spin-profile designs for producing fibers with low polarization mode dispersion,” Opt. Lett. 23(21), 1659–1661 (1998). [CrossRef] [PubMed]

9.

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]

10.

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]

11.

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]

12.

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]

13.

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]

14.

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]

15.

J. Fatome, S. Pitois, P. Morin, and G. Millot, “Observation of light-by-light polarization control and stabilization in optical fibre for telecommunication applications,” Opt. Express 18(15), 15311–15317 (2010). [CrossRef] [PubMed]

16.

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]

17.

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]

18.

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

19.

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]

20.

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]

21.

S. A. Babin, V. Karalekas, E. V. Podivilov, V. K. Mezentsev, P. Harper, J. D. Ania-Castanon, and S. K. Turitsyn, “Turbulent broadening of optical spectra in ultralong Raman fiber lasers,” Phys. Rev. A 77(3), 033803 (2008). [CrossRef]

OCIS Codes
(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(190.5650) Nonlinear optics : Raman effect

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: August 31, 2011
Revised Manuscript: September 21, 2011
Manuscript Accepted: September 23, 2011
Published: November 14, 2011

Citation
Sergey V. Sergeyev, "Activated polarization pulling and de-correlation of signal and pump states of polarization in a fiber Raman amplifier," Opt. Express 19, 24268-24279 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-24-24268


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References

  1. C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers,” Electron. Lett.22(19), 1029–1030 (1986). [CrossRef]
  2. P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol.14(2), 148–157 (1996). [CrossRef]
  3. R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron.15(10), 1157–1160 (1979). [CrossRef]
  4. Q. Lin and G. P. Agrawal, “Vector theory of stimulated Raman scattering and its application to fiber-based Raman amplifiers,” J. Opt. Soc. Am. B20(8), 1616–1631 (2003). [CrossRef]
  5. S. Sergeyev, S. Popov, and A. T. Friberg, “Spun fiber Raman amplifiers with reduced polarization impairments,” Opt. Express16(19), 14380–14389 (2008). [CrossRef] [PubMed]
  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. S. Sergeyev and S. Popov, “Two-section fiber optic Raman polarizer for high-speed transmission systems,” in the 13th International Conference on Transparent Optical Networks (June 26–30, 2011) Stockholm, Sweden, Th.A6.7.
  8. M. J. Li and D. A. Nolan, “Fiber spin-profile designs for producing fibers with low polarization mode dispersion,” Opt. Lett.23(21), 1659–1661 (1998). [CrossRef] [PubMed]
  9. 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]
  10. M. Martinelli, M. Cirigliano, M. Ferrario, L. Marazzi, and P. Martelli, “Evidence of Raman-induced polarization pulling,” Opt. Express17(2), 947–955 (2009). [CrossRef] [PubMed]
  11. 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]
  12. 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]
  13. N. J. Muga, M. F. S. Ferreira, and A. N. Pinto, “Broadband polarization pulling using Raman amplification,” Opt. Express19(19), 18707–18712 (2011). [CrossRef] [PubMed]
  14. 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]
  15. J. Fatome, S. Pitois, P. Morin, and G. Millot, “Observation of light-by-light polarization control and stabilization in optical fibre for telecommunication applications,” Opt. Express18(15), 15311–15317 (2010). [CrossRef] [PubMed]
  16. A. Zadok, E. Zilka, A. Eyal, L. Thévenaz, and M. Tur, “Vector analysis of stimulated Brillouin scattering amplification in standard single-mode fibers,” Opt. Express16(26), 21692–21707 (2008). [CrossRef] [PubMed]
  17. 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]
  18. B. Lindner, J. García-Ojalvo, A. Neimand, and L. Schimansky-Geier, “Effects of noise in excitable systems,” Phys. Rep.392(6), 321–424 (2004). [CrossRef]
  19. 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,” Chaos11(3), 587–594 (2001). [CrossRef] [PubMed]
  20. 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]
  21. S. A. Babin, V. Karalekas, E. V. Podivilov, V. K. Mezentsev, P. Harper, J. D. Ania-Castanon, and S. K. Turitsyn, “Turbulent broadening of optical spectra in ultralong Raman fiber lasers,” Phys. Rev. A77(3), 033803 (2008). [CrossRef]

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