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; Liang Chen

doi:10.1364/OE.20.006385

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]. Several applications of Brillouin scattering require the measurement of BGS in optical fibers, e.g. distributed fiber Brillouin sensing [2–5], 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 [6–8]. 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 [9–15]. Several new types of fiber with specially designed BGS were manufactured in order to increase their stimulated Brillouin scattering (SBS) threshold [9, 10].

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 [16–18]. 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, 16, 17]. 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]. 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 [2–5]. 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 $ν_{B}$ related to the m^th order longitudinal acoustic mode is given by [1, 9]:

ν_{B m} = \frac{2 n_{o}^{e f f} V_{a m}^{e f f}}{λ_{p}}

where the subscript ‘o’ stands for optical mode, ‘am’ stands for the m^th order acoustic mode.

n_{o}^{e f f}

is the effective refractive index of the fundamental optical mode traveling in the fiber,

V_{a m}^{e f f}

is the effective sound velocity of the m^th order acoustic mode that scatters lightwave,

λ_{p}

is pump wavelength. Therefore for a fixed

λ_{p},

ν_{B m}

is dependent on both

n_{o}^{e f f}

and

V_{a m}^{e f f} :

\frac{δ ν_{B m}}{ν_{B m}} = \frac{δ n_{o}^{e f f}}{n_{o}^{e f f}} + \frac{δ V_{a m}^{e f f}}{V_{a m}^{e f f}}

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 ( $n_{o}^{m a t} (r, θ, z)$ ), 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 $n_{o}^{e f f}$ is involved.

2.1 Contribution of $n_{o}^{e f f}$

Originated from Maxwell equations, the lateral profile ( $f$ ) and effective optical index ( $n_{o}^{e f f}$ ) of optical mode are simultaneously obtained by solving the eigenvalue equation with boundary condition given by the material optical refractive index $n_{o}^{m a t} (r, θ)$ [9, 12]:

\nabla_{⊥}^{2} f (r, θ) + k^{2} ({(n_{o}^{m a t} (r, θ))}^{2} - {(n_{o}^{e f f})}^{2}) f (r, θ) = 0

where

\nabla_{⊥}^{2}

is the transverse Laplacian operator,

k

is the vacuum optical wave vector. The asymmetric

n_{o}^{m a t} (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 (

f_{x} (r, θ), f_{y} (r, θ)

) and different effective refractive indices (

n_{o (x)}^{e f f}, n_{o (y)}^{e f f}

).The contribution of this effect is commonly referred as fiber birefringence [1,19]. Meanwhile due to fiber inhomogeneity in z direction (

n_{o}^{m a t} (r, θ, z)

), both optical mode profile and effective refractive index also vary along the fiber and can be written as

f_{x} (r, θ, z),

f_{y} (r, θ, z),

n_{o (x)}^{e f f} (z),

n_{o (y)}^{e f f} (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:

\begin{array}{l} | E_{p} (r, θ, z, t) 〉 = A_{p x} f_{p x} (r, θ, z) \exp [i (\int_{0}^{z} k_{p x} (u) d u - ω_{p} t)] | {\hat{x}}_{p} (z) 〉 \\ + A_{p y} f_{p y} (r, θ, z) \exp [i (\int_{0}^{z} k_{p y} (u) d u - ω_{p} t)] | {\hat{y}}_{p} (z) 〉 \end{array}

\begin{array}{l} | E_{r} (r, θ, z, t) 〉 = A_{r x} f_{r x} (r, θ, z) \exp [i (\int_{L}^{z} k_{r x} (u) d u - ω_{r} t)] | {\hat{x}}_{r} (z) 〉 \\ + A_{r y} f_{r y} (r, θ, z) \exp [i (\int_{L}^{z} k_{r y} (u) d u - ω_{r} t)] | {\hat{y}}_{r} (z) 〉 \end{array}

where the subscript ‘p’ stands for pump wave, ‘r’ stands for probe wave.

{\hat{x}}_{p (r)} (z)

and

{\hat{y}}_{p (r)} (z)

are the local principal axes of pump (probe) wave;

A_{p (r) x}

and

A_{p (r) y}

are the electric field partition of the pump (probe) wave on two axes;

ω_{p (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:

\begin{array}{l} 〈 E_{r} (r, θ, z, t) | E_{p} (r, θ, z, t) 〉 \\ = A_{p x} A_{r x}^{*} f_{p x} (r, θ, z) f_{r x}^{*} (r, θ, z) \exp [i (\int_{0}^{z} [k_{p x} (u) - k_{r x} (u)] d u - (ω_{p} - ω_{r}) t + \int_{0}^{L} k_{r x} (u) d u)] 〈 {\hat{x}}_{r} (z) | {\hat{x}}_{p} (z) 〉 \\ + A_{p y} A_{r y}^{*} f_{p y} (r, θ, z) f_{r y}^{*} (r, θ, z) \exp [i (\int_{0}^{z} [k_{p y} (u) - k_{r y} (u)] d u - (ω_{p} - ω_{r}) t + \int_{0}^{L} k_{r y} (u) d u)] 〈 {\hat{y}}_{r} (z) | {\hat{y}}_{p} (z) 〉 \\ + A_{p x} A_{r y}^{*} f_{p x} (r, θ, z) f_{r y}^{*} (r, θ, z) \exp [i (\int_{0}^{z} [k_{p x} (u) - k_{r y} (u)] d u - (ω_{p} - ω_{r}) t + \int_{0}^{L} k_{r y} (u) d u)] 〈 {\hat{y}}_{r} (z) | {\hat{x}}_{p} (z) 〉 \\ + A_{p y} A_{r x}^{*} f_{p y} (r, θ, z) f_{r x}^{*} (r, θ, z) \exp [i (\int_{0}^{z} [k_{p y} (u) - k_{r x} (u)] d u - (ω_{p} - ω_{r}) t + \int_{0}^{L} k_{r x} (u) d u)] 〈 {\hat{x}}_{r} (z) | {\hat{y}}_{p} (z) 〉 \end{array}

where ‘*’ stands for complex conjugate. If the fiber is purely linear birefringent or purely circular birefringent, the 3^rd and 4^th terms in the right-hand side of Eq. (6) are to be zero (assuming zero dispersion for both principal axes), i.e.

〈 {\hat{y}}_{r} (z) | {\hat{x}}_{p} (z) 〉 = 〈 {\hat{x}}_{r} (z) | {\hat{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

Ω_{d} = ω_{p} - ω_{r} .

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], the x (y) component of pump wave is not orthogonal with the y (x) component of probe wave, i.e.

〈 {\hat{y}}_{r} (z) | {\hat{x}}_{p} (z) 〉

and

〈 {\hat{x}}_{r} (z) | {\hat{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):

ν_{B m (1)} (z) = \frac{V_{a m}^{e f f}}{2 π c} (n_{o (p x)}^{e f f} (z) ω_{p} + n_{o (r x)}^{e f f} (z) ω_{r}) \approx \frac{2 V_{a m}^{e f f}}{λ_{p}} 〈 n_{o (p x)}^{e f f} (z), n_{o (r x)}^{e f f} (z) 〉 = \frac{2 V_{a m}^{e f f}}{λ_{p}} n_{o (1)}^{e f f} (z)

ν_{B m (2)} (z) = \frac{V_{a m}^{e f f}}{2 π c} (n_{o (p y)}^{e f f} (z) ω_{p} + n_{o (r y)}^{e f f} (z) ω_{r}) \approx \frac{2 V_{a m}^{e f f}}{λ_{p}} 〈 n_{o (p y)}^{e f f} (z), n_{o (r y)}^{e f f} (z) 〉 = \frac{2 V_{a m}^{e f f}}{λ_{p}} n_{o (2)}^{e f f} (z)

ν_{B m (3)} (z) = \frac{V_{a m}^{e f f}}{2 π c} (n_{o (p x)}^{e f f} (z) ω_{p} + n_{o (r y)}^{e f f} (z) ω_{r}) \approx \frac{2 V_{a m}^{e f f}}{λ_{p}} 〈 n_{o (p x)}^{e f f} (z), n_{o (r y)}^{e f f} (z) 〉 = \frac{2 V_{a m}^{e f f}}{λ_{p}} n_{o (3)}^{e f f} (z)

ν_{B m (4)} (z) = \frac{V_{a m}^{e f f}}{2 π c} (n_{o (p y)}^{e f f} (z) ω_{p} + n_{o (r x)}^{e f f} (z) ω_{r}) \approx \frac{2 V_{a m}^{e f f}}{λ_{p}} 〈 n_{o (p y)}^{e f f} (z), n_{o (r x)}^{e f f} (z) 〉 = \frac{2 V_{a m}^{e f f}}{λ_{p}} n_{o (4)}^{e f f} (z)

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

V_{a m}^{e f f}

as 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 $V_{a m}^{e f f}$

Analogous to the optical case, the mode profile ( $ξ_{m} (r, θ)$ ) and effective velocity ( $V_{a m}^{e f f}$ ) of the m^th order acoustic mode can be obtained by solving the eigenvalue equation with boundary condition determined by $G (r, θ)$ and $ρ (r, θ)$ [9]:

\nabla_{⊥}^{2} ξ_{m} (r, θ) + q^{2} (\frac{ρ (r, θ) {(V_{a m}^{e f f})}^{2}}{G (r, θ)} - 1) ξ_{m} (r, θ) = 0

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),

V_{a m}^{e f f}

becomes z dependent and can be written as

V_{a m}^{e f f} (z),

which leads to the z dependence of

ν_{B m}

via 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 ( $A F$ ) and effective peak frequency ( $ν_{B}^{e f f}$ ) on lightwave SOP

In distributed fiber Brillouin sensing (e.g. BOTDA), pulse is applied with duration of several or tens of nanoseconds to achieve centimeter or meter spatial resolution [3]. Within the interaction length W (half of pulse width), as illustrated in Fig. 1 , the measured BGS (the red dashed profile) is the superposition of all spectrum components (the blue solid curves) with different Brillouin peak frequencies caused by fiber inhomogeneity mentioned in Section 2. The superimposed profile of the measured Brillouin spectrum $S_{A} (ν, S O P)$ can be expressed as:

S_{A} (ν, S O P) = \int_{ν_{B \min}}^{ν_{B \max}} γ_{B 1} (ν_{B}, S O P) S (ν, ν_{B}) d ν_{B}

where

ν_{B}

varies within the range of

[ν_{B \min}, ν_{B \max}],

S (ν, ν_{B})

is the symmetric shape of each individual normalized spectrum component attributed by acoustic damping and waveguide structure [6, 7],

γ_{B 1} (ν_{B}, S O P)

is the Brillouin peak efficiency of each spectrum component related to the first order acoustic mode. The Brillouin peak efficiency related to the m^th order acoustic mode can be expressed as [9]:

γ_{B}_{m} = \frac{g_{B}_{m}}{A_{m}^{a o}}

where

g_{B}_{m}

is peak Brillouin gain,

A_{m}^{a o}

is the effective acousto-optic area given by [9]:

A_{m}^{a o} = {[\frac{{〈 f^{2} 〉}_{⊥}}{{〈 ξ_{m} \cdot f^{2} 〉}_{⊥}}]}^{2} {〈 ξ_{m}^{2} 〉}_{⊥}

where “

{〈 〉}_{⊥}

” stands for averaging over cross-section.

A_{m}^{a o}

determines the overlap area of optical and the m^th order acoustic mode profile. Here in Eq. (9) (and in the following) only the first order acoustic mode that has the largest overlap area (

A_{1}^{a o}

) with optical mode profile was considered, because in standard step-index SMF-28 fiber the value of

A_{m}^{a o}

(m>1) is at least 40 times larger than

A_{1}^{a o}

[9].

Fig. 1 Illustration of the superimposed Brillouin spectrums with different $A F$ values: (a) $A F = 1$ (b) $A F < 1$ (c) $A F > 1$

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From Eq. (9) we know that within the interaction length W, the SOP dependence of $S_{A} (ν, S O P)$ comes from the SOP dependence of $γ_{B 1} (ν_{B}, S O P) .$ According to Eq. (10), there are two mechanisms that make $γ_{B 1}$ varying with lightwave SOP:

1) The optical powers of the two orthogonal polarization modes vary with lightwave SOP, then the peak Brillouin gain $g_{B m} (z, S O P)$ of the four spectrum components with $ν_{B}$ given by Eq. (7.1)-(7.4) vary with lightwave SOP;
2) The overall optical mode profile $f$ varies with lightwave SOP, as $f (r, θ, z, S O P) = κ_{x} (S O P) f_{x} (r, θ, z) + κ_{y} (S O P) f_{y} (r, θ, z)$

where

κ_{x} (S O P)

and

κ_{y} (S O P)

are the normalized weight factors of the two polarization mode profiles respectively. Because of Eq. (12), according to Eq. (11), the overlap area (

A_{1}^{a o} (z, S O P)

) 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 makes $S_{A} (ν, S O P)$ 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 $γ_{B 1} (ν_{B}, S O P)$ 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, $S_{A} (ν, S O P)$ will be broader than that of each spectrum component and deviate from the symmetric spectrum. We define a factor $A F$ to quantify the asymmetric property of $S_{A} (ν, S O P) :$

A F = \frac{Δ ν_{B r}}{Δ ν_{B l}}

where

Δ ν_{B r}

and

Δ ν_{B l}

are the right and left part of the Brillouin linewidth (full width half maxima (FWHM)) as shown in Fig. 1.

A F = 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

γ_{B 1} (ν_{B}, S O P)

of each spectrum component within the interaction length W leads to the changing of

A F

as shown in Fig. 1(b) and 1(c). Therefore the above three mechanisms together make spectrum asymmetric factor

A F

varying with lightwave SOP.

In distributed fiber Brillouin sensing, for the case where the range of $ν_{B}$ variation ( $[ν_{B \min}, ν_{B \max}]$ ) is much smaller than the value of FWHM (this is the real case according to experiment), the effective Brillouin peak frequency $ν_{B}^{e f f}$ is recovered by the single peak Lorentzian fitting of the BGS. In this way, $ν_{B}^{e f f}$ can be considered as the weighted average of the peak frequencies of all spectrum components. By considering both the contribution of $n_{o}^{e f f}$ and $V_{a 1}^{e f f}$ , the general expression of the measured $ν_{B}^{e f f}$ within interaction length W is given by:

ν_{B}^{e f f} (z, S O P) |_{W} = \frac{2}{λ_{p}} \int_{z}^{z + W} [\sum_{j = 1}^{4} (α_{j} (τ, S O P) n_{o (j)}^{e f f} (τ)) (η (τ, S O P) V_{a 1}^{e f f} (τ))] d τ

where

α_{j} (τ, S O P)

and

η (τ, S O P)

are the weight factors proportional to the local Brillouin peak efficiency

γ_{B 1} (ν_{B}, S O P)

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

ν_{B}^{e f f}

varies.

When polarization scrambler (PS) is applied on pump and probe waves, both of their optical mode profiles are averaged over SOP, then $ν_{B}^{e f f}$ can be written as:

ν_{B (P S)}^{e f f} (z) |_{W} = \frac{2}{λ_{p}} \int_{z}^{z + W} [\sum_{j = 1}^{4} ({〈 α_{j} (τ, S O P) 〉}_{S O P} n_{o (j)}^{e f f} (τ)) ({〈 η (τ, S O P) 〉}_{S O P} V_{a 1}^{e f f} (τ))] d τ

= \frac{2}{λ_{p}} {\bar{n}}_{o}^{e f f} (z) {\bar{V}}_{a 1}^{e f f} (z)

where “

{〈 〉}_{S O P}

” stands for the average over SOP.

{\bar{n}}_{o}^{e f f} (z)

is the averaged effective refractive index over SOP of all local non-degenerate polarization modes within the distance range [z, z+W];

{\bar{V}}_{a 1}^{e f f} (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

ν_{B}^{e f f}

while maintaining its z dependence. This indicates even if ideal PS is applied,

ν_{B (P S)}^{e f f}

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 [2,4] shown in Fig. 2 was used to measure the SOP dependence of AF and $ν_{B}^{e f f} .$ 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 sidebands $ν_{0} \pm ν_{R F} .$ A narrow bandwidth fiber Bragg grating filter (FBG1, 3GHz bandwidth) was used to retain the anti-Stokes sideband $ν_{0} + ν_{R F} .$ 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.

Fig. 2 Experimental setup

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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, 20]. 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 $A F$ 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 $A F$ 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 $ν_{B}^{e f f}$ of SOP₁ and SOP₂ cases. The fitting uncertainty of $ν_{B}^{e f f}$ 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 $ν_{B}^{e f f}$ was smaller than the corresponding $ν_{B}^{e f f}$ measurement uncertainty given by the temperature accuracy of the oven (about ± 0.1MHz), the final uncertainty of single $ν_{B}^{e f f}$ measurement resulted ± 0.1MHz. The measurement uncertainties of AF and $ν_{B}^{e f f}$ are summarized in Table 1 .

Table 1. Measurement Uncertainties of AF and $ν_{B}^{e f f}$

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At different probe input SOPs (SOP₁ and SOP₂), we measured $A F$ as 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 $A F$ for 40 times then made an average. The results are shown in Fig. 3(a) . 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 $A F$ of the averaged curve can be estimated as $\pm 0.02 / \sqrt{40} \approx \pm 0.003.$ Fig. 4(a) shows the curves of $A F$ (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 $A F$ values for both SOP cases are well repeatable. In some regions, e.g. from 20m to 30m (from 40m to 50m), the $A F$ values of SOP₁ case are larger (smaller) than that of SOP₂ case in all three days. Figure 4(b) shows the distributed $A F$ difference ( $| δ A F |$ ) obtained by the absolute difference of local $A F$ between SOP₁ and SOP₂ cases (after averaging the three days’ data). The variation of $| δ A F |$ is within the range of 0.02 (2% relative to unity), which is larger than the measurement uncertainty. Therefore the dependence of $A F$ on lightwave SOP is demonstrated. At the same time the small difference (<2%) between $A F$ and unity indicates the suitability of using single peak Lorentzian fitting to obtain $ν_{B}^{e f f}$ in 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.

Fig. 3 40 times’ repeat measurements of (a) $A F$ and (b) $ν_{B}^{e f f},$ and the averaged curves for the same probe input SOP

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Fig. 4 (a) AF curves for two orthogonal probe input SOPs measured in three days (b) $| δ A F |$ varies over distance (c) $A F_{P S}$ curves for the scrambled probe input SOP measured in three days (d) $| δ A F_{P S} |$ varies over distance

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Then we measured $ν_{B}^{e f f}$ 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 $ν_{B}^{e f f}$ at each position also validates the uncertainty estimation of ±0.1MHz for single $ν_{B}^{e f f}$ measurement. The measurement uncertainty of the averaged curve can be estimated as $\pm 0.1 / \sqrt{40} \approx \pm 0.016$ MHz. Figure 5(a) shows the curves of $ν_{B}^{e f f}$ (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 $ν_{B}^{e f f}$ are well repeatable. The difference of $ν_{B}^{e f f}$ between two SOP cases can be demonstrated from two aspects:

Fig. 5 (a) $ν_{B}^{e f f}$ curves for two orthogonal probe input SOPs measured in three days (b) $| δ ν_{B}^{e f f} |$ varies over distance (c) $ν_{B (P S)}^{e f f}$ curves for the scrambled probe input SOP measured in three days (d) $| δ ν_{B (P S)}^{e f f} |$ varies over distance

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

The other is the statistical property of $ν_{B}^{e f f} .$ Figures 6(a) and 6(b) show the probability density function (PDF) of $ν_{B}^{e f f}$ 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 . 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 $ν_{B}^{e f f}$ for the two SOP cases have a difference of 0.05MHz. The PDF difference reflects the different fiber inhomogeneities (i.e. $ν_{B}^{e f f}$ fluctuation) along z direction measured at SOP₁ and SOP₂. From these two aspects, the dependence of $ν_{B}^{e f f}$ on lightwave SOP is clearly demonstrated. It is worth noting that the $4 σ$ bounds (95%) of the PDFs of the two SOP cases are almost the same, which indicates the similar variation range of $ν_{B}^{e f f}$ over distance attributed by the z direction fiber inhomogeneity.

Fig. 6 PDF of $ν_{B}^{e f f}$ (a) SOP₁ (b) SOP₂ (c) scrambled

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Table 2. Statistical Parameters of $ν_{B}^{e f f}$ PDF of SOP₁, SOP₂ and Scrambled Case

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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 measured $A F$ and $ν_{B}^{e f f}$ 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 $A F$ and $ν_{B}^{e f f}$ curves between the two SOP cases (i.e. the SOP dependence of AF and $ν_{B}^{e f f}$ ) is averaged by using PS3. The PDF of the measured $ν_{B}^{e f f}$ 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 $ν_{B}^{e f f} .$ It is worth noting that the $4 σ$ bound of the PS case is almost the same as that of the specific SOP cases, which indicates PS3 maintains the $ν_{B}^{e f f}$ variation 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 $n_{o}^{e f f}$ and $V_{a 1}^{e f f}$ over 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 ( $| δ A F_{P S} |$ ) which is obtained by the maximum difference of AF over three days at each position. It shows even with PS3, the value of $| δ A F_{P S} |$ 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 $ν_{B}^{e f f}$ with PS3 also fluctuates over three days ( $| δ ν_{B (P S)}^{e f f} |$ ) 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 $ν_{B}^{e f f}$ fluctuation ( $| δ ν_{B (P S)}^{e f f} |$ ) 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 $ν_{B}^{e f f}$ is reduced to a very low level (here it is ± 0.016MHz through 40 times’ repeat measurement and average), the accuracy of $ν_{B}^{e f f}$ measurement is still limited by the fluctuation caused by fiber inhomogeneity. The usage of practical PS can reduce the range of $ν_{B}^{e f f}$ fluctuation, 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]. 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 [21–25]. In BOTDA configuration, at each position over distance, the SBS process will induce a nonlinear refractive index by pump and probe waves [1]. 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 $ν_{B}^{e f f}$ 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 $ν_{B}^{e f f}$ caused 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 shows the experimental results of the maximum $ν_{B}^{e f f}$ fluctuation over 3 days ( ${| δ ν_{B (P S)}^{e f f} |}_{\max}$ ) 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 $ν_{B}^{e f f}$ decreases 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 $ν_{B}^{e f f}$ due 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 $ν_{B}^{e f f}$ provides 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 $ν_{B}^{e f f}$ fluctuation over three days ( ${| δ ν_{B (P S)}^{e f f} |}_{\max} = 0.051$ MHz at position 10m), according to the temperature and strain coefficients $C_{B}^{T} = 1.12 M H z / ° C$ and $C_{B}^{ε} = 0.05 M H z / μ ε$ [26], 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 1.02 $μ ε$ respectively.

Fig. 7 ${| δ ν_{B (P S)}^{e f f} |}_{\max}$ varies with scrambling frequency of PS3

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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

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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]

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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]

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Probe input SOP	Mean value $μ$  (MHz)	Standard derivation $σ$  (MHz)	Skewness (normalized 3rd order moment)	Kurtosis (normalized 4th order moment)	Mode value (MHz)	$4 σ$ bound (MHz)
SOP₁	10861.01	0.045	−0.18	2.88	10861.03	0.18
SOP₂	10861.00	0.043	0.27	2.70	10860.98	0.17
Scrambled	10861.00	0.040	0.043	3.21	10861.00	0.16

Probe input SOP	Mean value $μ$  (MHz)	Standard derivation $σ$  (MHz)	Skewness (normalized 3rd order moment)	Kurtosis (normalized 4th order moment)	Mode value (MHz)	$4 σ$ bound (MHz)
SOP₁	10861.01	0.045	−0.18	2.88	10861.03	0.18
SOP₂	10861.00	0.043	0.27	2.70	10860.98	0.17
Scrambled	10861.00	0.040	0.043	3.21	10861.00	0.16

Polarization dependence of Brillouin linewidth and peak frequency due to fiber inhomogeneity in single mode fiber and its impact on distributed fiber Brillouin sensing

Abstract

1. Introduction

2. Existence of Brillouin spectrum components with different peak frequencies

2.1 Contribution of $n_{o}^{e f f}$

2.2 Contribution of $V_{a m}^{e f f}$

3. Dependences of Brillouin spectrum asymmetric factor ( $A F$ ) and effective peak frequency ( $ν_{B}^{e f f}$ ) on lightwave SOP

4. Experimental results and discussion

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (7)

Tables (2)

Equations (19)

Optics Express

Abstract

1. Introduction

2. Existence of Brillouin spectrum components with different peak frequencies

2.1 Contribution of noeff

2.2 Contribution of Vameff

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

4. Experimental results and discussion

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (7)

Tables (2)

Equations (19)

Optics Express

2.1 Contribution of $n_{o}^{e f f}$

2.2 Contribution of $V_{a m}^{e f f}$

3. Dependences of Brillouin spectrum asymmetric factor ( $A F$ ) and effective peak frequency ( $ν_{B}^{e f f}$ ) on lightwave SOP