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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 6 — Mar. 12, 2012
  • pp: 6385–6399
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Polarization dependence of Brillouin linewidth and peak frequency due to fiber inhomogeneity in single mode fiber and its impact on distributed fiber Brillouin sensing

Shangran Xie, Meng Pang, Xiaoyi Bao, and Liang Chen  »View Author Affiliations


Optics Express, Vol. 20, Issue 6, pp. 6385-6399 (2012)
http://dx.doi.org/10.1364/OE.20.006385


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Abstract

The dependence of Brillouin linewidth and peak frequency on lightwave state of polarization (SOP) due to fiber inhomogeneity in single mode fiber (SMF) is investigated by using Brillouin optical time domain analysis (BOTDA) system. Theoretical analysis shows fiber inhomogeneity leads to fiber birefringence and sound velocity variation, both of which can cause the broadening and asymmetry of the Brillouin gain spectrum (BGS) and thus contribute to the variation of Brillouin linewidth and peak frequency with lightwave SOP. Due to fiber inhomogeneity both in lateral profile and longitudinal direction, the measured BGS is the superposition of several spectrum components with different peak frequencies within the interaction length. When pump or probe SOP changes, both the peak Brillouin gain and the overlapping area of the optical and acoustic mode profile that determine the peak efficiency of each spectrum component vary within the interaction length, which further changes the linewidth and peak frequency of the superimposed BGS. The SOP dependence of Brillouin linewidth and peak frequency was experimentally demonstrated and quantified by measuring the spectrum asymmetric factor and fitting obtained effective peak frequency respectively via BOTDA system on standard step-index SMF-28 fiber. Experimental results show that on this fiber the Brillouin spectrum asymmetric factor and effective peak frequency vary in the range of 2% and 0.06MHz respectively over distance with orthogonal probe input SOPs. Experimental results also show that in distributed fiber Brillouin sensing, polarization scrambler (PS) can be used to reduce the SOP dependence of Brillouin linewidth and peak frequency caused by fiber inhomogeneity in lateral profile, however it maintains the effects caused by fiber inhomogeneity in longitudinal direction. In the case of non-ideal polarization scrambling using practical PS, the fluctuation of effective Brillouin peak frequency caused by fiber inhomogeneity provides another limit of sensing frequency resolution of distributed fiber Brillouin sensor.

© 2012 OSA

1. Introduction

Brillouin scattering in optical fiber has been extensively studied over decades. Originated from the interaction between lightwave and moving acoustic wave, the backscattered light has a Brillouin frequency shift around 11GHz from input light at wavelength 1550nm in standard step-index SMF. The exponential damping of the acoustic wave provides the Lorentzian shape of the Brillouin gain spectrum (BGS). The damping time, i.e. the phonon lifetime determines the spectrum linewidth [1

1. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press 2007), Chap. 9.

]. Several applications of Brillouin scattering require the measurement of BGS in optical fibers, e.g. distributed fiber Brillouin sensing [2

2. X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, and D. A. Jackson, “Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering,” J. Lightwave Technol. 13(7), 1340–1348 (1995). [CrossRef]

5

5. M. N. Alahbabi, Y. T. Cho, and T. P. Newson, “150-km-range distributed temperature sensor based on coherent detection of spontaneous Brillouin backscatter and in-line Raman amplification,” J. Opt. Soc. Am. B 22(6), 1321–1324 (2005). [CrossRef]

], therefore it has been an important subject of studying the features of BGS in optical fibers over years. The inhomogeneous broadening of the BGS in optical fiber has been reported [6

6. B. Y. Zel’dovich and A. N. Pilipetskii, “Influence of sound diffraction on stimulated Brillouin scattering in a single-mode waveguide,” Sov. J. Quantum Electron. 16(4), 546–548 (1986). [CrossRef]

8

8. N. Shibata, R. G. Waarts, and R. P. Braun, “Brillouin-gain spectra for single-mode fibers having pure-silica, GeO2-doped, and P2 05-doped cores,” Opt. Lett. 12(4), 269–271 (1987). [CrossRef] [PubMed]

]. It shows that both fiber waveguide structure and inhomogeneity lead to the broadening of Brillouin linewidth from ~16MHz@1550nm in bulk material to ~30MHz@1550nm in optical fibers. Recently, it is reported that the BGS can be varied by changing fiber refractive index profiles [9

9. A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photon. 2(1), 1–59 (2010). [CrossRef]

15

15. B. Ward and J. Spring, “Finite element analysis of Brillouin gain in SBS-suppressing optical fibers with non-uniform acoustic velocity profiles,” Opt. Express 17(18), 15685–15699 (2009). [CrossRef] [PubMed]

]. Several new types of fiber with specially designed BGS were manufactured in order to increase their stimulated Brillouin scattering (SBS) threshold [9

9. A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photon. 2(1), 1–59 (2010). [CrossRef]

, 10

10. M. Li and D. A. Nolan, “Optical transmission fiber design evolution,” J. Lightwave Technol. 26(9), 1079–1092 (2008). [CrossRef]

].

SBS is associated with electrostriction which is a polarization sensitive process. For the three parameters of BGS (gain, linewidth and peak frequency), the polarization dependence of Brillouin gain has been studied over decades [16

16. M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single mode fibers,” J. Lightwave Technol. 12(4), 585–590 (1994). [CrossRef]

18

18. T. Gogolla and K. Krebber, “Distributed beat length measurement in single-mode optical fibers using Stimulated Brillouin-Scattering and Frequency-Domain Analysis,” J. Lightwave Technol. 18(3), 320–328 (2000). [CrossRef]

]. It is shown that besides the energy conservation and phase matching condition, the maximum Brillouin gain occurs when pump and probe waves are polarization matched, i.e. their Stokes vectors trace identical ellipses and in the same sense of rotation on Poincaré Sphere [2

2. X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, and D. A. Jackson, “Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering,” J. Lightwave Technol. 13(7), 1340–1348 (1995). [CrossRef]

, 16

16. M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single mode fibers,” J. Lightwave Technol. 12(4), 585–590 (1994). [CrossRef]

, 17

17. 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]

]. However when pump or probe SOP is varied, besides the variation of Brillouin gain, the phase matching condition that determines the Brillouin resonance also varies due to the existence of fiber birefringence, which will further leads to the variation of Brillouin linewidth and peak frequency. To the best of our knowledge, this effect has not been reported in SMF. The detailed theoretical analysis in Section 2 and 3 show due to the finite interaction length of pump and probe wave, not only fiber birefringence but also the sound velocity variation caused by fiber inhomogeneity contribute to this effect. Recently the effect of polarization pulling is reported based on the vector analysis of Brillouin gain [17

17. 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]

]. It shows the output SOP of the scrambled input probe wave will be converged to the SOP of the strong pump wave due to Brillouin interaction. This phenomenon is caused by the Brillouin gain difference of the two polarization components of probe wave. From the aspect of Brillouin linewidth and peak frequency, in this work, it is shown that the measured BGS is the superposition of several Brillouin spectrum components with different peak frequencies which is caused by both fiber birefringence and sound velocity variation. The variation of Brillouin peak efficiencies (which is given by the ratio of Brillouin gain and the overlapping area of optical and acoustic mode profile) of all spectrum components with pump or probe SOP causes the polarization dependent Brillouin linewidth and peak frequency. In distributed fiber Brillouin sensing, local Brillouin peak frequency is recovered by the Lorentzian fitting of the measured BGS to achieve distributed temperature and strain monitoring [2

2. X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, and D. A. Jackson, “Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering,” J. Lightwave Technol. 13(7), 1340–1348 (1995). [CrossRef]

5

5. M. N. Alahbabi, Y. T. Cho, and T. P. Newson, “150-km-range distributed temperature sensor based on coherent detection of spontaneous Brillouin backscatter and in-line Raman amplification,” J. Opt. Soc. Am. B 22(6), 1321–1324 (2005). [CrossRef]

]. The influence of lightwave SOP on BGS measurement can affect the accuracy of Brillouin peak frequency measurement and thus the sensing resolution of the measurands.

This paper is organized as follows: in Section 2, by considering both the contribution of fiber birefringence and sound velocity variation, we demonstrate fiber inhomogeneity leads to the existence of several Brillouin spectrum components superimposed to form the measured BGS within the interaction length (half of spatial resolution), which causes the broadening and asymmetry of the spectrum; in Section 3, the SOP dependence of the spectrum asymmetric factor and the effective Brillouin peak frequency obtained by single peak Lorentzian fitting is studied. It is shown that these two parameters are determined by the Brillouin peak efficiencies (i.e. peak height) of all spectrum components. For different pump or probe SOPs, Brillouin peak efficiencies vary among the spectrum components within the interaction length; in Section 4, a BOTDA system was constructed to experimentally demonstrate and quantify the SOP dependence of the spectrum asymmetric factor and effective Brillouin peak frequency on standard step-index SMF-28 fiber. 40 times’ repeat measurement and average was used to further reduce the measurement uncertainty in order to observe the effect of fiber inhomogeneity. Also in this section the impact of this effect on distributed fiber Brillouin sensing is discussed; in Section 5, we draw the conclusion.

2. Existence of Brillouin spectrum components with different peak frequencies

Based on energy conservation and phase matching condition, under the first order approximation, the Brillouin peak frequency νBrelated to the mth order longitudinal acoustic mode is given by [1

1. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press 2007), Chap. 9.

, 9

9. A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photon. 2(1), 1–59 (2010). [CrossRef]

]:
νBm=2noeffVameffλp
(1)
where the subscript ‘o’ stands for optical mode, ‘am’ stands for the mth order acoustic mode. noeff is the effective refractive index of the fundamental optical mode traveling in the fiber, Vameff is the effective sound velocity of the mth order acoustic mode that scatters lightwave, λp is pump wavelength. Therefore for a fixedλp, νBm is dependent on both noeffand Vameff:

δνBmνBm=δnoeffnoeff+δVameffVameff
(2)

In real fiber, there exists inhomogeneity which includes geometrical asymmetry and density non-uniformity in both lateral profile (r, θ) and longitudinal (z) direction. Fiber inhomogeneity modifies the properties of fiber material, thus makes material optical refractive index (nomat(r,θ,z)), Young’s modulus (G(r,θ,z)) and density (ρ(r,θ,z)) be the function of (r, θ, z). Due to the interaction of optical and acoustic waves in Brillouin scattering, the influence of fiber inhomogeneity under Brillouin scattering is more obvious than that in lightwave transmission case where only the contribution of noeffis involved.

2.1 Contribution of noeff

Originated from Maxwell equations, the lateral profile (f) and effective optical index (noeff) of optical mode are simultaneously obtained by solving the eigenvalue equation with boundary condition given by the material optical refractive index nomat(r,θ) [9

9. A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photon. 2(1), 1–59 (2010). [CrossRef]

, 12

12. M. J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007). [CrossRef] [PubMed]

]:
2f(r,θ)+k2((nomat(r,θ))2(noeff)2)f(r,θ)=0
(3)
where 2 is the transverse Laplacian operator, kis the vacuum optical wave vector. The asymmetric nomat(r,θ) in Eq. (3) caused by fiber inhomogeneity in lateral profile lifts the degeneracy of the two orthogonal polarization optical modes and makes them having different optical profiles (fx(r,θ),fy(r,θ)) and different effective refractive indices (no(x)eff,no(y)eff).The contribution of this effect is commonly referred as fiber birefringence [1

1. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press 2007), Chap. 9.

,19

19. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000). [CrossRef] [PubMed]

]. Meanwhile due to fiber inhomogeneity in z direction (nomat(r,θ,z)), both optical mode profile and effective refractive index also vary along the fiber and can be written as fx(r,θ,z), fy(r,θ,z), no(x)eff(z), no(y)eff(z). Considering pump and probe waves are launched from opposite ends of the fiber to create SBS via electrostriction. The corresponding electric fields can be expressed as:
|Ep(r,θ,z,t)=Apxfpx(r,θ,z)exp[i(0zkpx(u)duωpt)]|x^p(z)+Apyfpy(r,θ,z)exp[i(0zkpy(u)duωpt)]|y^p(z)
(4)
|Er(r,θ,z,t)=Arxfrx(r,θ,z)exp[i(Lzkrx(u)duωrt)]|x^r(z)+Aryfry(r,θ,z)exp[i(Lzkry(u)duωrt)]|y^r(z)
(5)
where the subscript ‘p’ stands for pump wave, ‘r’ stands for probe wave.x^p(r)(z)and y^p(r)(z)are the local principal axes of pump (probe) wave; Ap(r)x and Ap(r)y are the electric field partition of the pump (probe) wave on two axes; ωp(r) is the frequency of pump (probe) wave; L is fiber length. The beating via electrostriction in the fiber due to above two lightwaves can be written as:
Er(r,θ,z,t)|Ep(r,θ,z,t)=ApxArx*fpx(r,θ,z)frx*(r,θ,z)exp[i(0z[kpx(u)krx(u)]du(ωpωr)t+0Lkrx(u)du)]x^r(z)|x^p(z)+ApyAry*fpy(r,θ,z)fry*(r,θ,z)exp[i(0z[kpy(u)kry(u)]du(ωpωr)t+0Lkry(u)du)]y^r(z)|y^p(z)+ApxAry*fpx(r,θ,z)fry*(r,θ,z)exp[i(0z[kpx(u)kry(u)]du(ωpωr)t+0Lkry(u)du)]y^r(z)|x^p(z)+ApyArx*fpy(r,θ,z)frx*(r,θ,z)exp[i(0z[kpy(u)krx(u)]du(ωpωr)t+0Lkrx(u)du)]x^r(z)|y^p(z)
(6)
where ‘*’ stands for complex conjugate. If the fiber is purely linear birefringent or purely circular birefringent, the 3rd and 4th terms in the right-hand side of Eq. (6) are to be zero (assuming zero dispersion for both principal axes), i.e. y^r(z)|x^p(z)=x^r(z)|y^p(z)=0. In this case each principal axis component beats with its corresponding counter propagating beam to excite two moving acoustic waves with the same angular frequency Ωd=ωpωr. In the most general case of optical fibers with elliptical birefringence, the Jones Matrix of local birefringent section has different eigenvalues for the forward and backward directions [19

19. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000). [CrossRef] [PubMed]

], the x (y) component of pump wave is not orthogonal with the y (x) component of probe wave, i.e. y^r(z)|x^p(z) and x^r(z)|y^p(z) are not zero and are different with each other, therefore there are four moving acoustic waves with different Brillouin peak frequencies shown in Eq. (7.1)-(7.4):
νBm(1)(z)=Vameff2πc(no(px)eff(z)ωp+no(rx)eff(z)ωr)2Vameffλpno(px)eff(z),no(rx)eff(z)=2Vameffλpno(1)eff(z)
(7.1)
νBm(2)(z)=Vameff2πc(no(py)eff(z)ωp+no(ry)eff(z)ωr)2Vameffλpno(py)eff(z),no(ry)eff(z)=2Vameffλpno(2)eff(z)
(7.2)
νBm(3)(z)=Vameff2πc(no(px)eff(z)ωp+no(ry)eff(z)ωr)2Vameffλpno(px)eff(z),no(ry)eff(z)=2Vameffλpno(3)eff(z)
(7.3)
νBm(4)(z)=Vameff2πc(no(py)eff(z)ωp+no(rx)eff(z)ωr)2Vameffλpno(py)eff(z),no(rx)eff(z)=2Vameffλpno(4)eff(z)
(7.4)
where the “” notation means the average of the two effective refractive indices. In Eq. (7.1)-(7.4), we neglected the tiny frequency difference of pump and probe waves for simplicity. We also neglected the sound velocity difference of the four acoustic waves caused by fiber acoustic dispersion and thus considered Vameffas a constant. Equation (7.1)-(7.4) show that due to fiber inhomogeneity in lateral profile, the four Brillouin spectrum components with different peak frequencies coexist in the fiber, meanwhile the four spectrum components vary over distance due to fiber inhomogeneity along z direction.

2.2 Contribution of Vameff

Analogous to the optical case, the mode profile (ξm(r,θ)) and effective velocity (Vameff) of the mth order acoustic mode can be obtained by solving the eigenvalue equation with boundary condition determined by G(r,θ)and ρ(r,θ) [9

9. A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photon. 2(1), 1–59 (2010). [CrossRef]

]:
2ξm(r,θ)+q2(ρ(r,θ)(Vameff)2G(r,θ)1)ξm(r,θ)=0
(8)
where q is the acoustic wave vector given by the phase matching condition. Due to the z dependence of G(r,θ,z) and ρ(r,θ,z),Vameff becomes z dependent and can be written as Vameff(z), which leads to the z dependence of νBmvia Eq. (1). Therefore fiber inhomogeneity modifies sound velocity over distance, which also causes the existence of Brillouin spectrum components with different peak frequencies along the fiber.

3. Dependences of Brillouin spectrum asymmetric factor (AF) and effective peak frequency (νBeff) on lightwave SOP

From Eq. (9) we know that within the interaction length W, the SOP dependence of SA(ν,SOP)comes from the SOP dependence of γB1(νB,SOP). According to Eq. (10), there are two mechanisms that make γB1 varying with lightwave SOP:
  • 1) The optical powers of the two orthogonal polarization modes vary with lightwave SOP, then the peak Brillouin gain gBm(z,SOP) of the four spectrum components with νB given by Eq. (7.1)-(7.4) vary with lightwave SOP;
  • 2) The overall optical mode profile f varies with lightwave SOP, as
    f(r,θ,z,SOP)=κx(SOP)fx(r,θ,z)+κy(SOP)fy(r,θ,z)
    (12)
where κx(SOP)andκy(SOP)are the normalized weight factors of the two polarization mode profiles respectively. Because of Eq. (12), according to Eq. (11), the overlap area (A1ao(z,SOP)) of optical and acoustic mode profile varies with lightwave SOP.

It is worth noting that the existence of z direction fiber inhomogeneity provides the third mechanism that makesSA(ν,SOP) varying with lightwave SOP: 3) when the lightwave SOP varies at the beginning of the interaction length, the lightwave SOP at each position in the interaction length changes accordingly. Thus the values of γB1(νB,SOP) vary over distance within the interaction length, which changes the weight of each spectrum component and then the superimposed BGS.

According to Eq. (9), in the most general case, SA(ν,SOP) will be broader than that of each spectrum component and deviate from the symmetric spectrum. We define a factor AFto quantify the asymmetric property of SA(ν,SOP):
AF=ΔνBrΔνBl
(13)
where ΔνBrand ΔνBl are the right and left part of the Brillouin linewidth (full width half maxima (FWHM)) as shown in Fig. 1. AF=1 corresponds to the case of symmetric spectrum as shown in Fig. 1(a). For different SOPs of pump or probe wave, the variation of γB1(νB,SOP) of each spectrum component within the interaction length W leads to the changing of AFas shown in Fig. 1(b) and 1(c). Therefore the above three mechanisms together make spectrum asymmetric factor AFvarying with lightwave SOP.

In distributed fiber Brillouin sensing, for the case where the range of νBvariation ([νBmin,νBmax]) is much smaller than the value of FWHM (this is the real case according to experiment), the effective Brillouin peak frequency νBeffis recovered by the single peak Lorentzian fitting of the BGS. In this way, νBeffcan be considered as the weighted average of the peak frequencies of all spectrum components. By considering both the contribution of noeffand Va1eff, the general expression of the measured νBeff within interaction length W is given by:
νBeff(z,SOP)|W=2λpzz+W[j=14(αj(τ,SOP)no(j)eff(τ))(η(τ,SOP)Va1eff(τ))]dτ
(14)
where αj(τ,SOP)andη(τ,SOP)are the weight factors proportional to the local Brillouin peak efficiencyγB1(νB,SOP)of the corresponding spectrum component. Because of the above three mechanisms, when pump or probe SOP is varied, both the weight factors α and η vary, then νBeff varies.

When polarization scrambler (PS) is applied on pump and probe waves, both of their optical mode profiles are averaged over SOP, then νBeffcan be written as:
νB(PS)eff(z)|W=2λpzz+W[j=14(αj(τ,SOP)SOPno(j)eff(τ))(η(τ,SOP)SOPVa1eff(τ))]dτ
(15)
=2λpn¯oeff(z)V¯a1eff(z)
(16)
where “SOP” stands for the average over SOP. n¯oeff(z) is the averaged effective refractive index over SOP of all local non-degenerate polarization modes within the distance range [z, z+W]; V¯a1eff(z) is the averaged effective sound velocity over SOP of all acoustic modes within the distance range [z, z+W]. Equation (15) and (16) mean that ideal PS can eliminate the SOP dependence of νBeffwhile maintaining its z dependence. This indicates even if ideal PS is applied, νB(PS)eff still varies over distance which remains several spectrum components coexisting within the interaction length W, therefore PS cannot eliminate the asymmetry of the measured BGS due to the existence of z direction fiber inhomogeneity.

4. Experimental results and discussion

A single laser BOTDA system [2

2. X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, and D. A. Jackson, “Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering,” J. Lightwave Technol. 13(7), 1340–1348 (1995). [CrossRef]

,4

4. M. Niklès, L. Thévenaz, and P. A. Robert, “Simple distributed fiber sensor based on Brillouin gain spectrum analysis,” Opt. Lett. 21(10), 758–760 (1996). [CrossRef] [PubMed]

] shown in Fig. 2
Fig. 2 Experimental setup
was used to measure the SOP dependence of AF and νBeff. A 1550nm distributed feedback (DFB) laser with 2MHz linewidth was split into two paths. The first path with 95% light intensity was modulated by an electro-optic modulator (EOM) which was controlled by a tunable radio frequency (RF) source as well as a pulse generator. The output of EOM was a pulsed light with two sidebandsν0±νRF. A narrow bandwidth fiber Bragg grating filter (FBG1, 3GHz bandwidth) was used to retain the anti-Stokes sidebandν0+νRF. After optical amplification by EDFA and ASE noise attenuation by FBG2, the pulse was sent into a section of fiber under test (FUT) through a circulator as pump wave. The second path from laser with 5% intensity was launched into fiber from the other side as probe wave which was amplified by SBS process. The amplified probe wave was detected to recover Brillouin spectrum. The sampling rate of digitizer card was set 100MHz. The pulse duration was 50ns, corresponding to 5m spatial resolution (2.5m interaction length). The input power of the CW probe wave was −2dBm, the peak power of the pulsed pump wave was 21dBm. The FUT was 86m standard step-index SMF-28 fiber set in loose state (strain free) and was put into an oven to keep temperature at 25.0⁰C with accuracy of ± 0.1⁰C.

Two polarization scramblers (PS1 and PS2, General Photonics® PSM-001) were applied in series on pulsed pump wave to minimize the impact of polarization induced Brillouin gain fluctuation. The scrambling frequencies of PS1 and PS2 were 12kHz. In this way, the effective optical mode profile of pump wave was averaged over SOPs. A polarization controller (PC) was applied on CW probe wave to adjust its input SOP. A polarimeter was used to monitor the output SOP of probe wave. We chose two SOPs (SOP1 and SOP2) of input probe wave which were orthogonal with each other. This was realized by launching probe input SOP on fiber principal states of polarization (PSPs) that provide the maximum and minimum overall Brillouin gain by using CW pump and probe waves (pump wave was launched on one of the PSPs) [2

2. X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, and D. A. Jackson, “Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering,” J. Lightwave Technol. 13(7), 1340–1348 (1995). [CrossRef]

, 20

20. M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol. 12(4), 585–590 (1994). [CrossRef]

]. The time needed for fiber PSP launching was about 10 minutes, which was the time interval between the measurement of SOP1 and SOP2 cases. 2000 times’ trace average was taken to reduce random noise. The value of AF was estimated according to Eq. (13) by searching the maximum and FWHM value of the measured spectrum after 20 points’ interpolation and 5 points’ smoothing. The measurement uncertainty of AF was about ±0.02 which was estimated by the ratio of sampling frequency interval and half of FWHM. We applied single peak Lorentzian fitting to obtain νBeffof SOP1 and SOP2 cases. The fitting uncertainty of νBeff was about ± 0.03MHz which was estimated by the standard deviation of the fitted curve from the measured data points. As the fitting uncertainty of νBeffwas smaller than the corresponding νBeff measurement uncertainty given by the temperature accuracy of the oven (about ± 0.1MHz), the final uncertainty of single νBeffmeasurement resulted ± 0.1MHz. The measurement uncertainties of AF and νBeffare summarized in Table 1

Table 1. Measurement Uncertainties of AF and νBeff

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.

At different probe input SOPs (SOP1 and SOP2), we measured AFas the function of distance. In order to further reduce the measurement uncertainty, for the same input SOP of the probe wave we repeated the measurement of AFfor 40 times then made an average. The results are shown in Fig. 3(a)
Fig. 3 40 times’ repeat measurements of (a)AFand (b)νBeff, and the averaged curves for the same probe input SOP
. The blue background curves are the results of 40 times’ measurement, and it can be seen that the value of AF at each position varies within the range of ± 0.02, which validates our uncertainty estimation of single AF measurement. The red solid curve is their average. In this way, the uncorrelated noises of the measurement, e.g. the random fluctuation of oven temperature, the random drift of FBG filter center frequency, the random fluctuation caused by the non-ideal PS, the random fitting error, etc., are further reduced. The time needed for single measurement of AF was about 5 minutes, therefore for 40 times’ repeat measurement, the time needed was about 3.5 hours. Within the time of 3.5 hours, according to the SOP monitoring by polarimeter, the input SOP of probe wave can be considered as unchanged. The measurement uncertainty of AFof the averaged curve can be estimated as ±0.02/40±0.003. Fig. 4(a)
Fig. 4 (a) AF curves for two orthogonal probe input SOPs measured in three days (b) |δAF|varies over distance (c) AFPS curves for the scrambled probe input SOP measured in three days (d) |δAFPS|varies over distance
shows the curves of AF(after 40 times’ average) for two orthogonal probe SOPs measured on consecutive three days with the same condition. The same probe input SOP was guaranteed by monitoring the SOP of the output probe light with a polarimeter (adjusted by PC if needed). It can be seen that the AFvalues for both SOP cases are well repeatable. In some regions, e.g. from 20m to 30m (from 40m to 50m), the AFvalues of SOP1 case are larger (smaller) than that of SOP2 case in all three days. Figure 4(b) shows the distributed AFdifference (|δAF|) obtained by the absolute difference of local AFbetween SOP1 and SOP2 cases (after averaging the three days’ data). The variation of |δAF|is within the range of 0.02 (2% relative to unity), which is larger than the measurement uncertainty. Therefore the dependence of AFon lightwave SOP is demonstrated. At the same time the small difference (<2%) between AFand unity indicates the suitability of using single peak Lorentzian fitting to obtainνBeffin sensing application. From Fig. 4(a), it is worth noting that at some positions (e.g. at 6m), for the same probe input SOP (e.g. SOP1), the AF difference over three days is larger than the measure uncertainty. This is mainly caused by the variation of the polarization characteristic of FUT over three days, as fiber polarization characteristic may vary over time and with oven switching on and off in a random fashion. While by comparing Fig. 4(a) and Fig. 4(b) it can be seen that the value of this AF difference is smaller than the variation of AF caused by probe input SOP change at most positions over distance, therefore the polarization dependence of AF can also be demonstrated.

Then we measured νBeff over distance at different probe input SOPs. Again curves of 40 times’ measurement were averaged to reduce random noise as shown in Fig. 3(b). The fluctuation range of νBeffat each position also validates the uncertainty estimation of ±0.1MHz for single νBeffmeasurement. The measurement uncertainty of the averaged curve can be estimated as ±0.1/40±0.016MHz. Figure 5(a)
Fig. 5 (a) νBeff curves for two orthogonal probe input SOPs measured in three days (b) |δνBeff|varies over distance (c) νB(PS)eff curves for the scrambled probe input SOP measured in three days (d) |δνB(PS)eff|varies over distance
shows the curves of νBeff (after 40 times’ average) for the two orthogonal probe SOPs measured on three days with the same condition. It is shown that for each SOP the values of νBeffare well repeatable. The difference of νBeffbetween two SOP cases can be demonstrated from two aspects:

One is the distributed νBeffdifference (|δνBeff|) between two SOP cases which is shown in Fig. 5(b). The curve was obtained by the absolute difference of local νBeff between SOP1 and SOP2 cases after averaging the three days’ data. It shows that |δνBeff|varies within the range around 0.06MHz, which is larger than the measurement uncertainty. The maximum δνBeff occurs at 80m with the value of about 0.06MHz.

The other is the statistical property of νBeff. Figures 6(a)
Fig. 6 PDF of νBeff(a) SOP1 (b) SOP2 (c) scrambled
and 6(b) show the probability density function (PDF) of νBeff at the cases of SOP1 and SOP2 respectively over entire fiber length and over three days. Because the sampling rate of the digitizer card was set 100MHz (i.e. 1m per date point), for 86m fiber the number of data points was 86, therefore for all three days’ data the total number of data points in the PDF was 86 × 3 = 258. It can be seen that for the same FUT, the shapes of the obtained PDFs are different for the two SOP cases. The statistical parameters of the PDFs are listed in Table 2

Table 2. Statistical Parameters of νBeffPDF of SOP1, SOP2 and Scrambled Case

table-icon
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. It can be seen that the PDF is left-skewed (with negative skewness) in SOP1 case and it is right-skewed (with positive skewness) in SOP2 case, and the most frequently occurred νBeff for the two SOP cases have a difference of 0.05MHz. The PDF difference reflects the different fiber inhomogeneities (i.e. νBefffluctuation) along z direction measured at SOP1 and SOP2. From these two aspects, the dependence of νBeffon lightwave SOP is clearly demonstrated. It is worth noting that the 4σbounds (95%) of the PDFs of the two SOP cases are almost the same, which indicates the similar variation range of νBeffover distance attributed by the z direction fiber inhomogeneity.

In distributed Brillouin sensing, PS is commonly used to average the SOP effect, therefore we further replaced the PC on the probe arm by a PS (PS3 in Fig. 1) to average the SOP of the probe wave. The measuredAFand νBeff curves in three days (after 40 times’ average) are shown in Fig. 4(c) and Fig. 5(c) respectively. It can be seen that the repeatable trend difference of AFand νBeffcurves between the two SOP cases (i.e. the SOP dependence of AF and νBeff) is averaged by using PS3. The PDF of the measured νBeff in PS case is shown in Fig. 6(c) and the statistical parameters are also listed in Table.2.We notice that: 1) the skewness of PS case is much closer to zero compared to that of SOP1 and SOP2 cases, which means the PDF of PS case is more symmetric and can be considered as the average of the PDFs of the two SOP cases, 2) the observed effective Brillouin peak frequency in PS case is in between SOP1 and SOP2 cases, and 3) the PDF of PS case has the largest kurtosis value which means the samples are more centralized to the mass center. These results indicate PS3 effectively averages the probe light over SOP and compromises the influence of probe SOP on νBeff. It is worth noting that the 4σ bound of the PS case is almost the same as that of the specific SOP cases, which indicates PS3 maintains the νBeffvariation over distance caused by the fiber inhomogeneity along z direction. Those results agree with our theoretical predictions discussed in Section 3.

It can be seen in Fig. 4(c) that although PS3 was used, at some regions (e.g. from 5m to 10m on DAY3) the value of AF is biased from unity, i.e. the measured BGS is still asymmetric. This is because the existence of z direction inhomogeneity (which causes the variation of noeffand Va1effover distance) within the interaction length that cannot be averaged by PS3, which is predicted in Section 3. Figure 4(d) shows the local AF fluctuation over three days in PS3 case (|δAFPS|) which is obtained by the maximum difference of AF over three days at each position. It shows even with PS3, the value of |δAFPS|is still larger than the measurement uncertainty at some regions (e.g. from 40m to 50m). This is because PS3 has finite scramble frequency and thus the incomplete Poincaré Sphere coverage which leads to the non-ideal SOP average. For the same reason, local νBeff with PS3 also fluctuates over three days (|δνB(PS)eff|) as shown in Fig. 5(d). It shows the similar trend as that of AF in Fig. 4(d). At some regions (e.g. near position 10m, 20m and from 40 m to 50m) the value of νBefffluctuation (|δνB(PS)eff|) is clearly larger than the measurement uncertainty. This means in distributed fiber Brillouin sensor using practical PS with non-ideal polarization scrambling, even if the measurement uncertainty of νBeffis reduced to a very low level (here it is ± 0.016MHz through 40 times’ repeat measurement and average), the accuracy of νBeff measurement is still limited by the fluctuation caused by fiber inhomogeneity. The usage of practical PS can reduce the range of νBefffluctuation, while by comparing Fig. 5(b) with Fig. 5(d), it can be seen that the reduction effect of PS3 is position dependent, e.g. the reduction effect in the range of 70m~80m is obviously larger than that in 10m~20m. One possible explanation is the input SOP dependence of the Poincaré Sphere coverage of the practical PS with finite scrambling frequency, which has been discussed in [21

21. F. Heismann, “Compact electro-optic polarization scramblers for optically amplified lightwave systems,” J. Lightwave Technol. 14(8), 1801–1814 (1996). [CrossRef]

]. In fact, it has been revealed that the interaction of polarization scrambling with fiber itself (e.g. with PDL and PMD) will offset the performance of polarization scrambling [21

21. F. Heismann, “Compact electro-optic polarization scramblers for optically amplified lightwave systems,” J. Lightwave Technol. 14(8), 1801–1814 (1996). [CrossRef]

25

25. N. S. Bergano and C. R. Davidson, “Polarizdtion-scrambling-induced timing jitter in optical-amplifier systems,” Conf. Opt. Fiber Commun. 8, 122–123 (1995).

]. In BOTDA configuration, at each position over distance, the SBS process will induce a nonlinear refractive index by pump and probe waves [1

1. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press 2007), Chap. 9.

]. Because this modulated refractive index is not purely from fiber birefringence as described by Eq. (7.1)-(7.4), PS will not be able to completely remove its polarization dependence, i.e. in BOTDA configuration the nonlinear interaction of the light fields with fiber would essentially make the PS not 100% effective, therefore the usage of practical PS can reduce but not eliminate the νBeff fluctuation caused by fiber inhomogeneity. As a comparison, under Brillouin optical time domain reflectometry (BOTDR) configuration, the refractive index change from SBS can be neglected, i.e. PS is more effective to reduce the fluctuation of νBeffcaused by fiber inhomogeneity, while in this case the weak spontaneous Brillouin scattering leads to low signal to noise ratio (SNR) of the system. Detailed analysis of the influence of SBS interaction on the performance of polarization scrambling as well as the comparison of the cases of BOTDA and BOTDR are currently under study.

Figure 7
Fig. 7 |δνB(PS)eff|maxvaries with scrambling frequency of PS3
shows the experimental results of the maximum νBefffluctuation over 3 days (|δνB(PS)eff|max) at three different scrambling frequencies of PS3 (100Hz, 1kHz, 12kHz). For the three cases, 2000 times’ trace average was taken, and the measurement uncertainties were the same given by the temperature accuracy of the oven. It can be seen that the fluctuation range of νBeffdecreases with the increase of scrambling frequency, which means the limit of sensing frequency resolution caused by fiber inhomogeneity varies with the performance of PS. However according to Fig. 7, it can be seen that this limit tends to be decreased asymptotically at high scrambling frequency, which verifies the fact that practical PS cannot totally eliminate the fluctuation of νBeffdue to the non-ideal polarization scrambling. Because the inhomogeneity of the sensing fiber exists regardless of the changing of temperature (T) and strain (ε) as well as the improvement of system SNR, in the case of non-ideal polarization scrambling using practical PS, the fluctuation of νBeffprovides another limit of the measurement resolution of the sensor. The measured results in Fig. 7 show the value of this limit at different scrambling frequencies. For the case of 12kHz scrambling frequency of PS3, by using the maximum νBefffluctuation over three days (|δνB(PS)eff|max=0.051MHz at position 10m), according to the temperature and strain coefficients CBT=1.12MHz/°Cand CBε=0.05MHz/με [26

26. Y. Dong, L. Chen, and X. Bao, “High-spatial-resolution time-domain simultaneous strain and temperature sensor using Brillouin scattering and birefringence in a polarization-maintaining fiber,” IEEE Photon. Technol. Lett. 22(18), 1364–1366 (2010). [CrossRef]

], the temperature and strain measurement resolution of the constructed BOTDA system on the tested SMF-28 fiber can be estimated as 0.06°C and 1.02μεrespectively.

5. Conclusion

In this paper we demonstrate the SOP dependence of Brillouin linewidth and peak frequency which are experimentally quantified by measuring the spectrum asymmetric factor and effective peak frequency respectively via BOTDA system on standard step-index SMF-28 fiber. The SOP dependence of the two factors originates from fiber inhomogeneity which introduces the existence of Brillouin spectrum components with different peak frequencies within the interaction length. In distributed fiber Brillouin sensor using practical polarization scramblers with non-ideal scrambling efficiency, the effective Brillouin peak frequency fluctuation caused by fiber inhomogeneity provides another limit of the temperature and strain measurement resolution. It is worth noting that the standard step-index SMF-28 fiber has the simplest waveguide structure with the smallest uncertainty caused by fiber inhomogeneity. If different types of the fiber with complicated waveguide structures are used, the fluctuation of effective peak frequency caused by fiber inhomogeneity will be larger. As a result, the strain or temperature resolution of the sensor should be varied with the type of sensing fiber regardless of system SNR. The comparison among different types of fiber is our future work.

Acknowledgment

Shangran Xie is supported by China Scholarship Council (2010621132). This research is supported by Natural Sciences and Engineering Research Council (NSERC) Discovery Grants and Canada Research Chair Program.

References and links

1.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press 2007), Chap. 9.

2.

X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, and D. A. Jackson, “Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering,” J. Lightwave Technol. 13(7), 1340–1348 (1995). [CrossRef]

3.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995). [CrossRef]

4.

M. Niklès, L. Thévenaz, and P. A. Robert, “Simple distributed fiber sensor based on Brillouin gain spectrum analysis,” Opt. Lett. 21(10), 758–760 (1996). [CrossRef] [PubMed]

5.

M. N. Alahbabi, Y. T. Cho, and T. P. Newson, “150-km-range distributed temperature sensor based on coherent detection of spontaneous Brillouin backscatter and in-line Raman amplification,” J. Opt. Soc. Am. B 22(6), 1321–1324 (2005). [CrossRef]

6.

B. Y. Zel’dovich and A. N. Pilipetskii, “Influence of sound diffraction on stimulated Brillouin scattering in a single-mode waveguide,” Sov. J. Quantum Electron. 16(4), 546–548 (1986). [CrossRef]

7.

B. Ya. Zel’dovich and A. N. Pilipetskii, “Role of a “soundguide” and “antisoundguide” in stimulated Brillouin scattering in a single-mode waveguide,” Sov. J. Quantum Electron. 18(6), 818–822 (1988). [CrossRef]

8.

N. Shibata, R. G. Waarts, and R. P. Braun, “Brillouin-gain spectra for single-mode fibers having pure-silica, GeO2-doped, and P2 05-doped cores,” Opt. Lett. 12(4), 269–271 (1987). [CrossRef] [PubMed]

9.

A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photon. 2(1), 1–59 (2010). [CrossRef]

10.

M. Li and D. A. Nolan, “Optical transmission fiber design evolution,” J. Lightwave Technol. 26(9), 1079–1092 (2008). [CrossRef]

11.

A. Kobyakov, S. Kumar, D. Q. Chowdhury, A. B. Ruffin, M. Sauer, S. R. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Opt. Express 13(14), 5338–5346 (2005). [CrossRef] [PubMed]

12.

M. J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007). [CrossRef] [PubMed]

13.

A. H. McCurdy, “Modeling of stimulated Brillouin scattering in optical fibers with arbitrary radial index profile,” J. Lightwave Technol. 23(11), 3509–3516 (2005). [CrossRef]

14.

Y. Koyamada, S. Sato, S. Nakamura, H. Sotobayashi, and W. Chujo, “Simulating and designing Brillouin gain spectrum in single-mode fibers,” J. Lightwave Technol. 22(2), 631–639 (2004). [CrossRef]

15.

B. Ward and J. Spring, “Finite element analysis of Brillouin gain in SBS-suppressing optical fibers with non-uniform acoustic velocity profiles,” Opt. Express 17(18), 15685–15699 (2009). [CrossRef] [PubMed]

16.

M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single mode fibers,” J. Lightwave Technol. 12(4), 585–590 (1994). [CrossRef]

17.

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]

18.

T. Gogolla and K. Krebber, “Distributed beat length measurement in single-mode optical fibers using Stimulated Brillouin-Scattering and Frequency-Domain Analysis,” J. Lightwave Technol. 18(3), 320–328 (2000). [CrossRef]

19.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000). [CrossRef] [PubMed]

20.

M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol. 12(4), 585–590 (1994). [CrossRef]

21.

F. Heismann, “Compact electro-optic polarization scramblers for optically amplified lightwave systems,” J. Lightwave Technol. 14(8), 1801–1814 (1996). [CrossRef]

22.

D. Waddy, L. Chen, and X. Bao, “State of polarization bias in aerial fibers,” Electron. Lett. 38(19), 1086–1087 (2002). [CrossRef]

23.

E. Lichtman, “Limitations imposed by polarization-dependent gain and loss on all-optical ultralong communication systems,” J. Lightwave Technol. 13(5), 906–913 (1995). [CrossRef]

24.

F. Bruyere and O. Audouin, “Penalties in long-haul optical amplifier systems due to polarization dependent loss and gain,” IEEE Photon. Technol. Lett. 6(5), 654–656 (1994). [CrossRef]

25.

N. S. Bergano and C. R. Davidson, “Polarizdtion-scrambling-induced timing jitter in optical-amplifier systems,” Conf. Opt. Fiber Commun. 8, 122–123 (1995).

26.

Y. Dong, L. Chen, and X. Bao, “High-spatial-resolution time-domain simultaneous strain and temperature sensor using Brillouin scattering and birefringence in a polarization-maintaining fiber,” IEEE Photon. Technol. Lett. 22(18), 1364–1366 (2010). [CrossRef]

OCIS Codes
(060.2370) Fiber optics and optical communications : Fiber optics sensors
(190.5890) Nonlinear optics : Scattering, stimulated

ToC Category:
Sensors

History
Original Manuscript: December 20, 2011
Revised Manuscript: February 26, 2012
Manuscript Accepted: February 29, 2012
Published: March 5, 2012

Citation
Shangran Xie, Meng Pang, Xiaoyi Bao, and Liang Chen, "Polarization dependence of Brillouin linewidth and peak frequency due to fiber inhomogeneity in single mode fiber and its impact on distributed fiber Brillouin sensing," Opt. Express 20, 6385-6399 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-6-6385


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References

  1. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press 2007), Chap. 9.
  2. X. Bao, J. Dhliwayo, N. Heron, D. J. Webb, and D. A. Jackson, “Experimental and theoretical studies on a distributed temperature sensor based on Brillouin scattering,” J. Lightwave Technol.13(7), 1340–1348 (1995). [CrossRef]
  3. T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol.13(7), 1296–1302 (1995). [CrossRef]
  4. M. Niklès, L. Thévenaz, and P. A. Robert, “Simple distributed fiber sensor based on Brillouin gain spectrum analysis,” Opt. Lett.21(10), 758–760 (1996). [CrossRef] [PubMed]
  5. M. N. Alahbabi, Y. T. Cho, and T. P. Newson, “150-km-range distributed temperature sensor based on coherent detection of spontaneous Brillouin backscatter and in-line Raman amplification,” J. Opt. Soc. Am. B22(6), 1321–1324 (2005). [CrossRef]
  6. B. Y. Zel’dovich and A. N. Pilipetskii, “Influence of sound diffraction on stimulated Brillouin scattering in a single-mode waveguide,” Sov. J. Quantum Electron.16(4), 546–548 (1986). [CrossRef]
  7. B. Ya. Zel’dovich and A. N. Pilipetskii, “Role of a “soundguide” and “antisoundguide” in stimulated Brillouin scattering in a single-mode waveguide,” Sov. J. Quantum Electron.18(6), 818–822 (1988). [CrossRef]
  8. N. Shibata, R. G. Waarts, and R. P. Braun, “Brillouin-gain spectra for single-mode fibers having pure-silica, GeO2-doped, and P2 05-doped cores,” Opt. Lett.12(4), 269–271 (1987). [CrossRef] [PubMed]
  9. A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photon.2(1), 1–59 (2010). [CrossRef]
  10. M. Li and D. A. Nolan, “Optical transmission fiber design evolution,” J. Lightwave Technol.26(9), 1079–1092 (2008). [CrossRef]
  11. A. Kobyakov, S. Kumar, D. Q. Chowdhury, A. B. Ruffin, M. Sauer, S. R. Bickham, and R. Mishra, “Design concept for optical fibers with enhanced SBS threshold,” Opt. Express13(14), 5338–5346 (2005). [CrossRef] [PubMed]
  12. M. J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express15(13), 8290–8299 (2007). [CrossRef] [PubMed]
  13. A. H. McCurdy, “Modeling of stimulated Brillouin scattering in optical fibers with arbitrary radial index profile,” J. Lightwave Technol.23(11), 3509–3516 (2005). [CrossRef]
  14. Y. Koyamada, S. Sato, S. Nakamura, H. Sotobayashi, and W. Chujo, “Simulating and designing Brillouin gain spectrum in single-mode fibers,” J. Lightwave Technol.22(2), 631–639 (2004). [CrossRef]
  15. B. Ward and J. Spring, “Finite element analysis of Brillouin gain in SBS-suppressing optical fibers with non-uniform acoustic velocity profiles,” Opt. Express17(18), 15685–15699 (2009). [CrossRef] [PubMed]
  16. M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single mode fibers,” J. Lightwave Technol.12(4), 585–590 (1994). [CrossRef]
  17. 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]
  18. T. Gogolla and K. Krebber, “Distributed beat length measurement in single-mode optical fibers using Stimulated Brillouin-Scattering and Frequency-Domain Analysis,” J. Lightwave Technol.18(3), 320–328 (2000). [CrossRef]
  19. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A.97(9), 4541–4550 (2000). [CrossRef] [PubMed]
  20. M. O. van Deventer and A. J. Boot, “Polarization properties of stimulated Brillouin scattering in single-mode fibers,” J. Lightwave Technol.12(4), 585–590 (1994). [CrossRef]
  21. F. Heismann, “Compact electro-optic polarization scramblers for optically amplified lightwave systems,” J. Lightwave Technol.14(8), 1801–1814 (1996). [CrossRef]
  22. D. Waddy, L. Chen, and X. Bao, “State of polarization bias in aerial fibers,” Electron. Lett.38(19), 1086–1087 (2002). [CrossRef]
  23. E. Lichtman, “Limitations imposed by polarization-dependent gain and loss on all-optical ultralong communication systems,” J. Lightwave Technol.13(5), 906–913 (1995). [CrossRef]
  24. F. Bruyere and O. Audouin, “Penalties in long-haul optical amplifier systems due to polarization dependent loss and gain,” IEEE Photon. Technol. Lett.6(5), 654–656 (1994). [CrossRef]
  25. N. S. Bergano and C. R. Davidson, “Polarizdtion-scrambling-induced timing jitter in optical-amplifier systems,” Conf. Opt. Fiber Commun. 8, 122–123 (1995).
  26. Y. Dong, L. Chen, and X. Bao, “High-spatial-resolution time-domain simultaneous strain and temperature sensor using Brillouin scattering and birefringence in a polarization-maintaining fiber,” IEEE Photon. Technol. Lett.22(18), 1364–1366 (2010). [CrossRef]

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