## Abstract

The polarization properties of stimulated Brillouin scattering (SBS) amplification or attenuation in standard single-mode fibers are examined through vectorial analysis, simulation and experiment. Vector propagation equations for the signal wave, incorporating SBS and birefringence, are derived and analyzed in both the Jones and Stokes spaces. The analysis shows that in the undepleted pump regime, the fiber may be regarded as a polarization-dependent gain (or loss) medium, having two orthogonal input SOPs, and corresponding two orthogonal output SOPs, for the signal, which, respectively, provide the signal with maximum and minimum SBS amplification (or attenuation). Under high Brillouin gain conditions and excluding zero-probability cases, the output SOP of arbitrarily polarized input signals, would tend to converge towards that of maximum SBS gain. In the case of high SBS attenuation the output SOP of an arbitrarily polarized signal would approach the output SOP corresponding to minimum attenuation. It is found that for a wide range of practical pump powers (≤ 100mW) and for sufficiently long fibers with typical SBS and birefringence parameters, the signal aligned for maximum SBS interaction will enter/emerge from the fiber with its electric field closely tracing the same ellipse in space as that of the pump at the corresponding side of the fiber, albeit with the opposite sense of rotation. The analytic predictions are experimentally demonstrated for both Stokes (amplification) and anti-Stokes (attenuation) signals.

©2008 Optical Society of America

## 1. Introduction

Stimulated Brillouin Scattering (SBS) requires the lowest activation power of all non-linear effects in silica optical fibers. SBS has found numerous applications, including distributed sensing of temperature and strain [1–3], fiber lasers [4], optical processing of high frequency microwave signals [5–9], and even optical memories [10]. Over the last three years, SBS has been highlighted as the underlying mechanism in many demonstrations of variable group delay setups [11–16], often referred to as slow and fast light. In all of the above applications, SBS has been a favorable mechanism for its robustness, simplicity of implementation and low pump power in standard fibers at room temperature.

In SBS, a strong pump wave and a typically weak, counter-propagating signal wave optically *interfere* to generate, through electrostriction, a traveling longitudinal acoustic wave. The acoustic wave, in turn, couples these optical waves to each other [17,18]. The SBS interaction is efficient only when the difference between the optical frequencies of the pump and signal waves is very close (within a few tens of MHz) to a fiber-dependent parameter, the Brillouin shift *ν _{B}*, which is of the order of 10–11 GHz in silica fibers at room temperature and at telecommunication wavelengths [17,18]. An input signal whose frequency is

*ν*lower than that of the pump (Stokes wave) experiences SBS amplification. If the input signal frequency is

_{B}*ν*above that of the pump (anti-Stokes wave), SBS-induced signal attenuation is obtained instead. The strength of the interaction is often quantified in terms of an exponential gain coefficient

_{B}*γ*, which is defined as the logarithm of the signal linear power gain (or loss), normalized to a unit pump power and unit fiber length [W·m]

^{-1}. (The coefficient

*γ*equals the Brillouin gain factor

*g*[19], divided by the fiber effective area).

Since SBS originates from optical interference between the pump and signal waves, the SBS interaction, at a given point along the fiber, is most efficient when the electric fields of the pump and signal are aligned, i.e., their vectors trace parallel ellipses and in the same sense of rotation. Conversely, if the two ellipses are again similar, but traced in opposite senses of rotation, with their long axes being orthogonal to each other, then the SBS interaction at that point averages to zero over an optical period. Consequently, in the presence of birefringence, the overall signal gain (or loss) depends on the birefringent properties of the fiber, as well as on the input states of polarization (SOPs) of both pump and signal. Following initial work by Horiguchi *et al*. [20], van Deventer and Boot [21] have studied in detail the signal SOPs leading to maximum and minimum gain. Based on the statistical properties of the evolution of the pump and signal SOPs in optical fibers much longer than the polarization beat length, but implicitly ignoring any influence of the Brillouin interaction on SOP evolution, they argued that for *standard*, low birefringence single-mode fibers, the maximum gain coefficient is twice that of the minimum one, and equals 2/3 of the maximum gain coefficient in a birefringence-free fiber *γ*
_{0}. Furthermore, maximum gain is achieved when the pump and signal have identical polarizations (in their respective directions of propagation), while minimum gain is obtained for the corresponding ‘orthogonal’ case [21–24]. Their analysis was nicely corroborated by an experiment, which showed that for a given pump, there were indeed two input SOPs, chosen by the experimenters to be identical or orthogonal to that of the pump, with one providing an exponential gain twice that of the other one. However, the SBS amplification of an arbitrarily polarized input signal SOP was not discussed, nor was the role played by the Brillouin effect itself in the evolution of the signal SOP considered.

In this paper, the pioneering work of van Deventer and Boot is analytically substantiated and extended, using a vector formulation of the SBS amplification process in the presence of birefringence. A vector differential equation, combining both effects, is studied in the Jones and Stokes spaces. Based on the Jones space representation, it is shown that in the undepleted pump regime, the input signal SOPs which lead to maximum/minimum SBS gain are *always* orthogonal, regardless of pump power and the statistics of the pump and signal SOPs along the fiber. These maximum and minimum gain SOPs, therefore, provide a convenient vector base for the examination of an arbitrarily polarized input signal wave. In the Stokes space, the evolution of the magnitude and the SOP of the signal wave along the fiber are described by a pair of coupled, rather simple differential equations. Using this representation, it is analytically shown that the SBS overall gain coefficient is determined by an average over a local mixing coefficient, similar (but not identical) to that of [21], for *any* input SOP, polarization statistics, or pump power. In addition, we show that in the presence of a pump, the evolution of the signal SOP is controlled not only by the fiber birefringence but also by the *local* SBS interaction, which drags the signal SOP towards that of the pump. The equations provide insight into the relation between the signal SOPs, which lead to maximum and minimum gain, and the SOP of the pump. The magnitude and SOP of the signal are then studied numerically. The maximum and minimum SBS gain coefficient for a general birefringent fiber are found to always be $\frac{1}{2}{\gamma}_{0}(1\pm \mu )$, 0≤µ≤1. Even for a weakly birefringent fiber, µ does not necessarily need to be 1/3, although in the limit of a fiber embodying the fully developed statistics of [21], it tends to this value. The SOPs of input signals, which experience maximum/minimum gains, are then studied as a function of the pump power.

The signal SOP is also examined experimentally, for both Stokes and anti-Stokes signal waves. As predicted by the analysis, the output signal SOP is seen to converge towards a specific, preferred SOP, which is practically independent of both the input signal SOP and polarization transformations along the fiber [25]. That preferred output SOP could be arbitrarily varied, however, by changing the input pump SOP.

The remainder of this paper is organized as follows: Section 2 presents vector theoretical analysis of the signal wave, subject to both SBS and birefringence, in the undepleted pump regime. Section 3 is dedicated to numerical simulations. Section 4 provides experimental results, and brief concluding remarks are given in section 5.

## 2. Theory

Let us denote the column Jones vector of a monochromatic signal wave as *E⃗ _{sig}*(

*z*),

*z*indicating position along the fiber, with the launch and exit points at

*z*=0 and

*z*=

*L*, respectively (

*L*is the fiber length). With no pump, the propagation of

*E⃗*can be described by:

_{sig}(z)with **T**(*z*) a unitary Jones matrix representing the effect of fiber birefringence. The pump wave, whose Jones vector is denoted by *E⃗ _{pump}(z)*, is launched into the fiber at

*z=L*. Throughout this paper, we work in the same right-handed coordinate system {

*x, y, z*}, where the signal propagates in the positive

*z*direction, while the pump propagates in the negative

*z*direction. Thus, if both

*E⃗*and

_{sig}(z)*E⃗*equal the 2X1 vector [1

_{pump}(z)*j*]

*(*

^{T}*T*stand for transpose), they represent a right-handed circularly polarized signal and a left-handed circularly polarized pump wave, respectively [23,24]. We also neglect linear polarization-dependent power losses in the fiber, although such losses can be easily included in the analysis. Further, since the Brillouin shift

*ν*is merely ~10GHz, and only a few kilometers of modern fibers are concerned, polarization mode dispersion can be ignored and, therefore, shifting the optical frequency by

_{B}*ν*has a negligible effect on the Jones matrix of the fiber. Hence, the propagation of the pump wave (in the absence of a probe) can also be expressed using

_{B}**T**(

*z*):

where inv[TT(z)]=T*(z).

When both the probe and pump waves are present, the *local* evolution on of *E⃗sig(z)* and *E⃗ _{pump}(z)* is driven by both the fiber birefringence and the SBS effect to give (see [26] for the birefringence term, and [27] for the SBS term):

*γ _{0}* [W·m]

^{-1}is the SBS gain per unit length per a unit of pump power for a scalar interaction (i.e., for a fiber with no birefringence), and depends on the fiber material properties, the mode field diameter, the pump optical spectrum and the frequency offset between the pump and signal waves. We dedicate most of the analysis to the Stokes wave scenario, so that

*γ*is positive, but the analysis and results, properly interpreted, are equally valid for the anti-Stokes case, where the optical frequency of the signal is

_{0}*ν*above that of the pump. The anti-Stokes signal surrenders its power to the pump, thereby becoming attenuated with an SBS attenuation per unit length of -

_{B}*γ*

_{0}. Note that (

*γ*

_{0}/2)[

*E⃗*is a 2×2 matrix, representing the outer product of a column vector (

_{pump}(z)E⃗†_{pump}(z)]*E⃗*) with a row one (the transpose conjugate of

_{pump}(z)*E⃗*).

_{pump}(z)From now on it will be assumed that SBS-induced signal amplification or attenuation negligbly affect the pump (i.e., the so-called undepleted pump approximation [21]). Thus, the SBS term in Eq. (3b) can be ignored and Eq. (3a) becomes linear in *E⃗ _{sig}(z)*. Therefore,

where **H** is a 2×2 matrix, which depends on the fiber birefringence, the fiber length *L*, the pump power, and its SOP at *z=L*. The matrix **H** is generally non-unitary. Nevertheless, it can be processed using the singular value decomposition (SVD) technique:

where **U** and **V** are unitary matrices, *G _{1}, G_{2}* are real and positive and satisfy

*G*in the case of SBS amplification and 1

_{1}>G_{1}>1*>G*in the case of SBS attenuation.

_{1}>G_{2}Using this decomposition two orthogonal input signal Jones vectors can be identified, which provide the maximum and minimum signal output powers, namely:

The corresponding *output* Jones vectors are given by:

$${\overrightarrow{E}}_{\mathrm{sig}}^{\mathrm{out}\_min}=\mathbf{U}\xb7\mathbf{S}\xb7{\mathbf{V}}^{\u2020}\xb7\mathbf{V}\left[\begin{array}{c}0\\ 1\end{array}\right]=\mathbf{U}\xb7\mathbf{S}\left[\begin{array}{c}0\\ 1\end{array}\right]={G}_{2}\mathbf{U}\left[\begin{array}{c}0\\ 1\end{array}\right]$$

and are, therefore, also orthogonal. It is thus convenient to represent an arbitrarily polarized input signal using the orthogonal base of *E⃗ ^{in}_max_{sig}, E⃗^{in}_^{min}_{sig}*:

Using Eqs. (7) and (8), the output signal Jones vector and the signal power are:

$${P}_{\mathrm{sig}}^{\mathrm{out}}={\mid {\alpha}_{0}\mid}^{2}{G}_{1}^{2}+{\mid {\beta}_{0}\mid}^{2}{G}_{2}^{2}$$

When *G*
_{1}≫*G*
_{2}, Eq. (9) suggests that unless *α _{0}* is negligible, an arbitrarily polarized input signal will be drawn towards the SOP of

*E⃗out_*. These predictions are supported by experiments, to be described in section 4.

^{max}_{sig}Next, we try to relate *E⃗in_ ^{max}_{sig}, E⃗in_^{min}_{sig}* to the SOP of the pump wave. To that end, we have transformed Eq. (3) to the Stokes space (see Appendix):

$$=\overrightarrow{\beta}\left(z\right)\xb7{\hat{s}}_{\mathrm{sig}}\left(z\right)+\frac{{\gamma}_{0}{P}_{\mathrm{pump}}\left(z\right)}{2}\left[{\hat{s}}_{\mathrm{pump}}\left(z\right)-\left({\hat{s}}_{\mathrm{pump}}\left(z\right)\xb7{\hat{s}}_{\mathrm{sig}}\left(z\right)\right){\hat{s}}_{\mathrm{sig}}\left(z\right)\right]$$

Here *S _{0_sig}* is the signal power,

*ŝ*=[

_{sig}*s*]

_{1,sig}s_{2,sig}s3,_{sig}*and similarly*

^{T}*ŝ*are 3X1 normalized Stokes vectors (

_{pump}*s2*), describing the evolution of the polarizations of the counter-propagating signal and pump waves, respectively, and finally,

_{1,sig,pump}+s2s_{2,sig,pump}+s^{2}3,_{sig,pump}*P*denotes the pump power, which for the undepleted, lossless case is

_{pump}*z*-independent. The three-dimensional vector β⃗(

*z*) describes the fiber birefringence in Stokes space [26]:

where *σ*⃗ is a row vector of Pauli spin matrices [26] (see also in the Appendix). The vector β⃗(*z*) is aligned with the Stokes space representation of the local slow axis of birefringence [26]. Note that we express both Stokes vectors in the same right handed coordinate system, in which the signal wave propagates in the positive *z* direction. Therefore, the Stokes vector *s*̂=[0 0 1]* ^{T}* represents a right-handed circular polarization for the signal wave, but a left-handed circular polarization for the pump.

Eq. (10a) is easily cast into a form:

In the undepleted pump regime, the solution is readily obtained:

$$={S}_{0\_\mathrm{sig}}^{\mathrm{in}}\mathrm{exp}\left[\frac{{\gamma}_{0}{P}_{\mathrm{pump}}}{2}L\left(1+{\u3008{\hat{s}}_{\mathrm{pump}}\xb7{\hat{s}}_{\mathrm{sig}}\u3009}_{L}\right)\right]$$

〈*ŝ _{pump};·ŝ_{sig}*〉L is the scalar product of the pump and signal Stokes vectors, averaged over the fiber length. Thus, for any input SOP one can define an effective SBS gain, given by:

Obviously, *γ* depends on the signal input SOP, as well as on the pump SOP.

Equation (10b) specifies two driving forces that control the evolution of the SOP along the fiber. The first, * _{sig}* β⃗×s⃗sig;, describes the birefringence-induced evolution of the signal SOP [26]. The same term also governs the evolution of the pump SOP, albeit in the opposite direction. The second term, (

*γ*

_{0}/2)

*P*⌊-(

_{pump}*ŝ*·

_{pump}*ŝ*)

_{sig}*ŝ*⌋, represents the effect of SBS amplification on the signal SOP. This second term has a very interesting physical interpretation on the Poincare sphere: it is a vector, orthogonal to

_{sig}*ŝ*, and tangentially (on the sphere surface) pointing towards

_{sig}*ŝ*. This term signifies a force pulling

_{pump}*ŝ*towards

_{sig}*ŝ*. The magnitude of this pulling force scales with the pump power and depends on the local projection of

_{pump}*ŝ*on

_{pump}(z)*ŝ*(

_{sig}*z*), vanishing when either

*ŝ*is parallel to

_{sig}*ŝ*(pump and signal SOPs aligned) or anti-parallel to it (in the Stokes space, namely: orthogonal in the Jones space).

_{pump}Several special cases are of particular interest. Let us consider a fiber with no birefringence, so that the evolution of the signal SOP is governed by SBS alone. If the input signal SOP is aligned with that of the pump, (*ŝ _{pump}·ŝin_{sig}=1*), then it follows from Eq. (10b) that the

*dŝ*=0. The signal SOP, therefore, remains aligned with that of the pump throughout the fiber. Since in this case 〈

_{sig}/dz*ŝ*=1, the SBS gain coefficient of Eq. (14) equals

_{pump};·ŝ_{sig}〉L*γ*

_{0}. Alternatively, when the signal input SOP is orthogonal to that of the pump (

*ŝ*=-1), we still obtain

_{pump}·ŝin_{sig}*dŝ*=0, and the pump and signal remain orthogonal for all 0≤

_{sig}/dz*z*≤L. Now 〈

*ŝ*=-1 and the SBS gain coefficient is zero. Thus,

_{pump};·ŝ_{sig}〉L*E⃗*in a birefringence-free fiber is parallel (perpendicular) to

^{out}_max_{sig}(E⃗^{out}_min_{sig})*E⃗*(using our conventions,

_{pump}(z=L)*E⃗*for a right-handed circularly polarized pump is left-handed polarized). When the input signal is arbitrarily polarized, the SBS polarization pulling term of Eq. (10b) is nonzero, so that

^{out}_^{max}_{sig}*ŝ*is gradually drawn towards

_{sig}*ŝ*;. The slope of the gain coefficient versus pump power curve, determined by 〈

_{pump}*ŝ*, will increase with pump power (or fiber length), eventually approaching its maximum value of

_{pump};·ŝ_{sig}〉L*γ*, and the SOP of the emerging signal will draw nearer and nearer that of the pump wave. These trends are, of course, fully consistent with the Jones space description of Eq. (9).

_{0}LWe now turn to the more prevalent scenario of standard single-mode fibers, where the birefringence term Eq. (10b) is larger than the SBS term (for a an average beat length of 40 m, 〈|β⃗|〉z~0.16 m^{-1} whereas *γ _{0}/P_{pump}*/2~0.01m-1 for γ0=0.2 [m·W]

^{-1},

*P*=0.1W). While being relatively small, the SBS term cannot be ignored. High differential gains (

_{pump}*G*

_{1}/

*G*

_{2}>10) are easily observed, and according to Eq. (9), any signal, whose input SOP even slightly deviates from that of

*E⃗*, will emerge with its SOP being pulled towards that of

^{in_min}_{sig}*E⃗*. While the polarization pulling is due to the SBS term, the final signal SOP is

^{out_max}_{sig}*not*that of

*E⃗*. The relation between the SOP of

_{pump}(z=L)*E⃗*and that of

^{out_max}_{sig}*E⃗*are studied below, first analytically, in the low pump power limit, and then, numerically for the general case.

_{pump}(z=L)Let us assume first a very weak pump so that the Brillouin term in Eq. (10b) can be ignored ${G}_{1}\u2044{G}_{2}\underset{{P}_{\mathrm{pump}}\to 0}{\to}1$. In this limit, the forward evolution of *ŝ _{sig}* and the backward evolution of

*ŝ*are

_{pump}*solely*governed by the birefringence term. We denote the maximum value of 〈

*ŝ*over all possible SOPs of the input signal

_{pump}·ŝ_{sig}〉_{L}*ŝ*, but for a given pump SOP (

_{sig}(z=0)*ŝ*(

_{pump}*z=L*)), as $\underset{{\hat{s}}_{\mathrm{sig}}\left(z=0\right)}{max}\left\{{\u3008{\hat{s}}_{\mathrm{pump}}\xb7{\hat{s}}_{\mathrm{sig}}\u3009}_{L}\right\}$. But:

$$={\hat{s}}_{\mathrm{pump}}^{T}\left(0\right){\u3008{\mathbf{M}}_{{\mathbf{T}}^{*}}^{T}\xb7{\mathbf{M}}_{\mathbf{T}}\left(z\right)\u3009}_{\begin{array}{c}\mathrm{Ensemble}\\ \mathrm{Average}\end{array}}{\hat{s}}_{\mathrm{sig}}\left(0\right)$$

$$={\hat{s}}_{\mathrm{pump}}^{T}\left(0\right)\left[\begin{array}{ccc}\frac{1}{3}& \phantom{\rule{.2em}{0ex}}& \phantom{\rule{.2em}{0ex}}\\ \phantom{\rule{.2em}{0ex}}& \frac{1}{3}& \phantom{\rule{.2em}{0ex}}\\ \phantom{\rule{.2em}{0ex}}& \phantom{\rule{.2em}{0ex}}& -\frac{1}{3}\end{array}\right]{\hat{s}}_{\mathrm{sig}}\left(0\right).$$

Here **M**T(*z*) and **M**
_{T}*(*z*) are the Mueller matrices representing **T**(*z*) and **T***(*z*), respectively (*T* stands for transpose), and the fiber is assumed to be long enough so that most *z* values are much larger than many correlation lengths of the random birefringence. Finally, the ensemble averaged value of 〈**M**
_{T}T*(*z*)·**M**
*T*(*z*)〉 was taken from [28].

One can easily conclude from Eq. (15) that $\underset{{\hat{s}}_{\mathrm{sig}}\left(z=0\right)}{max}\left\{{\u3008{\hat{s}}_{\mathrm{pump}}\xb7{\hat{s}}_{\mathrm{sig}}\u3009}_{L}\right\}$ is 1/3, resulting in a maximum achievable gain coefficient of (2/3)*γ*
_{0} (Eq. 14, see also a discussion in [21]). This maximum is attained when *ŝ _{sig}*(

*z*=0) is the image of

*ŝ*(

_{pump}*z*=0) on the Poincare sphere, with the equatorial plane acting as a mirror, namely:

*ŝmax1,*, and ŝ

_{sig}(0)=ŝ1,_{pump}(0), ŝmax2,_{sig}(0)=ŝ2,_{pump}(0)^{max}

_{3,sig}(0)=ŝ

_{3,pump}(0). This

*ŝ*is the normalized Stokes representation of the complex conjugate of the pump Jones vector at

^{max}_{sig}(z=0)*z*=0, namely,

*E⃗**, rather than that of

_{pump}(z=0)*E⃗*(

_{pump}*z*=0) (as in the birefringence-free case). Conversely, $\underset{{\hat{s}}_{\mathrm{sig}}\left(z=0\right)}{max}\left\{{\u3008{\hat{s}}_{\mathrm{pump}}\xb7{\hat{s}}_{\mathrm{sig}}\u3009}_{L}\right\}=-1\u20443$, corresponding to a minimum gain coefficient of (1/3)

*γ*

_{0}. This minimum value is attained for

*ŝmin*=-

_{sig}(z=0)*ŝ*, which is the Stokes representation of a polarization orthogonal to that of

^{max}_{sig}(z=0)*E⃗**, to be denoted by

_{pump}(z=0)*E⃗*⊥*. It is easily proven from Eqs. (1–2) that for a unitary

_{pump}(z=0)**T**(

*z*) (and ignoring the Brillouin term), if sig

*E⃗*are a parallel pair at

_{sig}and E⃗*_{pump}*z*=0, they will continue to be parallel for all 0≤

*z*≤L, so that

*E⃗out_*has the same polarization as that of

^{ma}x_{sig}*E⃗**. These analytically obtained results are no different than the seemingly intuitively-drived conclusions of [21], when carefully noting the difference in the reference frame convention, but both approaches are strictly valid only in the limit of very weak pump power. The relation between

_{pump}(z=L)*E⃗*and

^{max}_{sig}*E⃗**in the presence of non-negligible level of pump power is investigated in the next section.

_{pump}## 3. Simulations for Signal amplification (Stokes) and attenuation (anti-Stokes)

The SBS amplification/attenuation and the output SOP of the amplified/attenuated signal for different pump power levels were numerically examined, using Eqs. (2), (3) and (10a–10b). Simulations were based on the commonly used concatenated random wave-plate model [29], with the three components of β⃗ of Eq. (11) normally distributed with zero mean and the same standard deviation, chosen so that the average beat length *L _{B}*≡2

*π*/〈|β⃗|〉

*equals 40 m. Although a broad set of parameters was numerically investigated, results below were obtained for a fiber length*

_{z}*L*of 2250 m comprising 10,000 plates and

*γ*

_{0}=0.2[W·m]

^{-1}.

Figure 1(a) shows the projection of *ŝ ^{in_max}_{sig}* on

*ŝ*, where

_{pump}*(z=0)*ŝ*and

^{in_max}_{sig}*ŝ**(0) are the normalized Stokes-space counterparts of

_{pump}*E⃗*_minsig and

_{in}*E⃗**, respectively. Figure 1(b) shows the calculated 〈

_{pump}(z=0)*ŝ*for an input signal SOP aligned with

_{pump}·ŝ_{sig}〉L*E⃗*, as a function of pump power. The figures include several different fiber realizations, each with different random drawings of β⃗ values for the concatenated wave-plates, though all with an average beat length of 40 m. At low pump powers

^{in_max}_{sig}*ŝ*does not depend on

^{in_max}_{sig}·ŝ_{pump}*(z=0)*P*, as discussed above, and the relatively small misalignments between

_{pump}*E⃗*, as well as the deviations of 〈

^{in_max}_{sig}and E⃗*_{pump}(0)*ŝ*from the predicted value of 1/3, are due to the finiteness and discreteness of the model. It is clearly seen that

_{pump}·ŝ_{sig}〉_{L}*E⃗*remains closely aligned with

^{in_max}_{sig}*E⃗**, even for a pump power as high as 100 mW. While not shown, this close alignment also holds at the fiber output,

_{pump}(z=0)*z=L*, where the SOP of

*E⃗*lies in a similar close vicinity that of

^{out_max}_{sig}*E⃗**.

_{pump}(z=L)Clearly, $\underset{{\hat{s}}_{\mathrm{sig}}\left(z=0\right)}{max}\left\{{\u3008{\hat{s}}_{\mathrm{pump}}\xb7{\hat{s}}_{\mathrm{sig}}\u3009}_{L}\right\}$, while not exactly 1/3 (Fig. 1(b)), depends very weakly on *P _{pump}*, resulting in a practically linear relationship between the achievable max/min gain coefficient and pump power. Figure 2 shows the signal power gain as a function of pump power, for a signal SOP aligned with either

*E⃗*(maximum gain), or

^{in_max}_{sig}*E⃗*(minimum gain). According to Eq. (14) and regardless of the particular fiber realization, the ratio of the slope of the maximum gain curve to that of the minimum gain at a particular

^{in_min}_{sig}*P*, while not exactly 2, is always $\left(1+\underset{{\hat{s}}_{\mathrm{sig}}\left(z=0\right)}{max}\left\{{\u3008{\hat{s}}_{\mathrm{pump}}\xb7{\hat{s}}_{\mathrm{sig}}\u3009}_{L}\right\}\right)\u2044\left(1-\underset{{\hat{s}}_{\mathrm{sig}}\left(z=0\right)}{max}\left\{{\u3008{\hat{s}}_{\mathrm{pump}}\xb7{\hat{s}}_{\mathrm{sig}}\u3009}_{L}\right\}\right)$.

_{pump}Yet, SBS still has a dramatic effect on the output SOP of an input signal, whose SOP only slightly deviates from that of *E⃗ ^{in_min}_{sig}*, (see Fig. 2). As the difference between the minimum and maximum gains increases, the red-dashed gain curve in Fig. 2 changes its slope, approaching that of the maximum gain case. Incidentally [21], the same argument applies to amplified spontaneous SBS, which, therefore, under high differential gain conditions, emerges from the fiber with the SOP of

*E⃗**. Thus, under high differential gain conditions the SOP of amplified spontaneous SBS at the fiber output (

_{pump}(z=L)*z=L*) coincides with that of

*E⃗*!

^{out_max}_{sig}For arbitrarily polarized input signals and for pump powers above 25 mW, the output signal SOPs are clearly seen in Fig. 3(a) and Fig. 3(b) to converge towards the SOP of *E⃗ ^{out_max}_{sig}*, which is almost unaffected by pump power. Figure 3(c) shows the evolution of

*ŝ*as a function of the position coordinate z along the fiber, for different pump powers, when the signal input SOP is

_{pump}(z)·ŝ_{sig}(z)*E⃗*⊥*(

_{pump}*z*=0). That input SOP is close to, but not quite equal to

*E⃗*, especially at high pump powers. Note the gradual pulling of signal SOP towards that of

^{in_min}_{sig}*E⃗**, requiring many beat lengths before the effect becomes quite distinct at high pump powers.

_{pump}(z)The above analysis was provided for the Stokes wave. However, the anti-Stokes case can be treated very similarly. As in the Stokes wave scenario, the SBS interaction for an anti-Stokes signal is still maximum for an input SOP aligned with *E⃗* _{pump}*(0). However, this interaction results in efficient signal attenuation, rather than signal gain. We can, therefore, expect stronger attenuation for that input signal field component aligned with

*E⃗**(0), and weaker attenuation (‘maximum gain’) for the orthogonal component. Correspondingly, the SOP of the emerging signal is dominated by that of the maximum power SOP, and is expected to be closely aligned with that of

_{pump}*E⃗*⊥*. Figure 4 shows the scatter plots of the attenuated anti-Stokes signal, for

_{pump}(z=L)*ŝ*, indicating convergence towards [0-1 0]

_{pump}=[0 1 0]^{T}*, which is the (normalized) Stokes-spaced representation of*

^{T}*E⃗*⊥*. Both scenarios are visited in the experiment, to be described next.

_{pump}(z=L)## 4. Experiment

The experimental setup for characterizing polarization related properties of SBS is shown in Fig. 5. Light emitted from a tunable laser source was split by a 50% coupler. In the lower (pump) branch, the light was amplified by a high-power Erbium-doped fiber amplifier (EDFA), and directed into the fiber under test via a circulator. The length of the fiber under test was 2250 m, and its Brillouin frequency shift was *ν _{B}*=10.57GHz. The pump power was controlled by a variable optical attenuator (VOA). In the upper (signal) branch, the laser light was modulated by an electro-optic intensity modulator (EOM). The modulation frequency was tuned to

*ν*, and the EOM bias voltage was adjusted to suppress the optical carrier [2]. Following the EOM, the signal was filtered by a narrow-band Fiber Bragg grating (FBG). For SBS signal amplification measurements, the frequency of the tunable laser was adjusted so that the lower modulation sideband matched the FBG reflection frequency [30]. This way, the frequency of the signal propagating in the fiber under test was

_{B}*ν*below that of the pump. For SBS attenuation measurements, the tunable laser frequency was modified so that the upper modulation sideband was retained by the FBG [30]. Following the SBS interaction, the signal was routed to a power meter, followed by a lock-in amplifier to filter out spontaneous SBS, or to a polarization analyzer for the measurement of the signal output power and SOP. A second FBG in the detection path was used to filter out the backscattered pump, as well as the spontaneous Brillouin scattering amplified by the Stokes process in the SBS loss scenario.

_{B}For each pump power, the input signal SOPs which corresponded to minimum and maximum signal output power were found using the following procedure: First, a programmable polarization controller (Prog. PC) in the signal path was set to four non-degenerate SOPs, and the output signal power was recorded for each. Based on these four measurements, the top row of the 4X4 Mueller matrix describing the pumped fiber under test was extracted [31], and signal SOPs for minimum and maximum output power could be calculated. Next, the programmable PC was set to these two input SOPs and the output signal power was recorded.

Figure 6(a) shows the logarithm of the signal power gain (Stokes signal) as a function of pump power, for three different SOPs of the input signal wave. In the upper and lower curves, the signal SOP is adjusted for each pump power level to achieve maximum and minimum gain, respectively. In these curves, the logarithmic SBS gain appears to be linearly proportional to the pump power over the entire measurement range, indicating a power-independent gain coefficient, as obtained in simulations. Furthermore, the slope of the maximum gain curve is extremely close to twice that of the minimum gain curve [21]. These results indicate that our 2.2km fiber comprises many correlation lengths of the random birefringence [21, 28]. The third curve of Fig. 6(a) shows the logarithm of the SBS gain for a signal, whose input SOP, *E⃗ ^{in_near_min}_{sig(Stokes)}*, is azimuthally 400 away from

*E⃗*. Initially, for relatively low pump power, the gain slope is that of the minimum gain curve. However, for higher pump powers, the measured gain increases rapidly and its slope approaches that of the maximum gain curve, as discussed in Sec. 3.

^{in_min}_{sig(Stokes)}As a consistency check, we used Eqs. (8)–(9) first to project *E⃗ ^{in_near_min}_{sig(Stokes)}* on the measured

*E⃗*),

^{in_max}_{sig(Stokes)}*E⃗*, and then used the measured values for

^{in_min}_{sig(Stokes)}*G*

_{1}(maximum gain) and

*G*

_{2}(minimum gain) to analytically predict the gain experienced by

*E⃗*. The results are shown as open squares on the dashed (red) curve in Fig. 6(a), demonstrating excellent agreement with the measured gain. Figure 6(b) shows the output SOPs corresponding to

^{in_near_min}_{sig(Stokes)}*E⃗*for all pump powers. Also shown on the sphere is the SOP of spontaneously amplified Brillouin scattering, which was obtained by turning off the signal input and measuring the SOP of the Brillouin-scattered light at

^{out_max}_{sig(Stokes)}, E⃗^{out_min}_{sig(Stokes)}and E⃗^{out_near_min}_{sig(Stokes)}*ν*. Note that as

_{s}=ν_{P}-ν_{B}*P*spans the 5–35mW range, {

_{pump}*E⃗*} hardly change and they are fairly orthogonal to one another (the SOP readings of the polarization analyzer in the minimum gain case were contaminated by the spontaneously amplified Brillouin scattering, leading to a larger spread near {

^{out_max}_{sig(Stokes)}} and {E⃗^{out_min}_{sig(Stokes)}*E⃗*}. Furthermore, {

^{out_min}_{sig(Stokes)}*E⃗*} coincides, as expected, with the SOP of the spontaneously amplified Brillouin scattering. Also shown is the evolution of the signal SOP for

^{out_max}_{sig(Stokes)}*E⃗*, clearly indicating the pulling of its SOP towards that of {

^{in_near_min}_{sig(Stokes)}*E⃗*}. Figure 6(c) shows the logarithm of the maximum and minimum attenuation of an anti-Stokes signal. As obtained for the Stokes wave, the curves for maximum and minimum are linear, and the ratio of their slopes is close to two. Note that the obtained curves replicate those of the corresponding Stokes signal, albeit with a minus sign. The figure also shows the measured and calculated logarithmic loss of an anti-Stokes signal with an input SOP

^{out_max}_{sig(Stokes)}*E⃗*. Finally, Fig. 6(d) shows the output SOPs corresponding to

^{in_near_min}_{sig(A-Stokes)}*E⃗*,

^{out_max}_{sig(A-Stokes)}*E⃗*and

^{out_min}_{sig(A-Stokes)}*E⃗*for all pump powers. Polarization pulling towards the SOP of {

^{out_near_min}_{sig(A-Stokes)}*E⃗*} is observed. It is seen that {

^{out_max}_{sig(A-Stokes)}*E⃗*} (solid diamonds in Fig. 6(d)), which suffers maximum attenuation are parallel to {

^{out_min}_{sig(A-Stokes)}*E⃗*} (solid circles in Fig. 6(b)), which enjoys the maximum possible gain.

^{out_max}_{sig(Stokes)}Figure 7 shows the signal output SOP for twenty different input SOPs, which were evenly distributed on the Poincare sphere. As the pump power is increased, the output signal SOPs converge to a particular, preferred state. Thus, the converging effect is effective for both SBS signal gain and signal loss, in the undepleted pump regime.

## 5. Conclusions

In this work, the analysis of SBS in birefringent fibers was extended to include arbitrarily polarized signals. A vector propagation equation for the signal wave in the undepleted pump regime was provided, both in Jones and in Stokes spaces. The equations and their subsequent analysis provide expressions for the output signal vector, regardless of the polarization statistics of the pump and signal waves along the fiber. The analysis showed that SBS in the undepleted pump regime may be modeled as a pseudo-linear partial polarizer, whose input states for maximum and minimum gain are orthogonal. Due to the large difference in gain between these maximum and minimum states, it is expected that the SOP of an arbitrarily polarized input signal will be closely aligned with that of the maximum gain axis at the fiber output. This prediction was experimentally confirmed, for both Stokes and anti-Stokes signal waves. Somewhat similar polarization attraction between counter propagating waves, based on the Kerr effect in short, highly non-linear fibers, was recently reported [32], but the effect was restricted to circular SOPs. The vector properties of SBS can give rise to an arbitrary polarization synthesis. The analysis also shows that the maximum and minimum input signal SOPs for the Stokes wave in long, standard single-mode fibers correspond to the conjugate of the outgoing pump, and the orthogonal of that conjugate, respectively. This correspondence is practically valid for pump powers up to tens of milliwatts over fibers a few km long. The roles of the two SOPs are reversed for the anti-Stokes wave.

The polarization and birefringence dependence of SBS has already been used in Brillouin fiber lasers [33,34] and distributed birefringence measurements [35]. On the other hand, the same dependence can hinder the performance of distributed strain and temperature sensors [36], and SBS based slow light setups. In addition, birefringence was observed to cause a nonlinear response in the delay-pump power transfer function [37]. In one recent example, the polarization sensitivity of SBS-induced delay was overcome using a Faraday rotator mirror [38]. Clearly, the polarization related properties of SBS in long, standard single-mode fibers continue to be of large interest. The tools developed in this work provide a broad and comprehensive framework for the study of SBS and polarization and open new horizons for applications. A possible application, in which the spread of the signal output SOP for random input polarization serves as a measure of the fiber beat length, is currently under study.

## Appendix

In this appendix, the Stokes space representation of the signal propagation equation in the presence of SBS and birefringence, Eq. (10a-10b), is derived (see [39] for a different, though equivalent, formulation, of the equations governing the evolution of the non-normalized signal Stokes vector). The starting point for the derivation is Eq. (3) repeated here for convenience:

To obtain the evolution of the signal power and its SOP in terms of the Stokes parameters we use the common definitions [26, Eqs. (2.6), (3.5)] (σ⃗ is a vector of the Pauli spin matrices):

and *S ^{0}* and

*s*̂ denote, respectively, the power and normalized Stokes vector which correspond to

*E*⃗. Using Eq. (11) which reads:

and using the expansion of the projection operator in terms of Pauli matrices [26]:

with **I** denoting the 2X2 identity matrix, we obtain:

Eq. (A6) can be used to obtain an equation for the evolution of the signal power:

$${\overrightarrow{E}}_{\mathrm{sig}}^{\u2020}\left[\frac{j}{2}\overrightarrow{\beta}\xb7\overrightarrow{\sigma}+\frac{{\gamma}_{0}{S}_{\mathrm{pump}}^{0}}{4}\left(I+{\hat{s}}_{\mathrm{pump}}\xb7\overrightarrow{\sigma}\right)\right]{\overrightarrow{E}}_{\mathrm{sig}}=\frac{{\gamma}_{0}}{2}\left(1+{\hat{s}}_{\mathrm{pump}}\xb7{\hat{s}}_{\mathrm{sig}}\right){S}_{\mathrm{pump}}^{0}{S}_{\mathrm{sig}}^{0}$$

The equation governing the propagation of the normalized signal Stokes vector can be derived using Eqs. (A.3)–(A7) of [26]:

$$=\left[d\left({\overrightarrow{E}}_{\mathrm{sig}}^{\u2020}\overrightarrow{\sigma}{\overrightarrow{E}}_{\mathrm{sig}}\right)\u2044\mathrm{dz}-{S}_{\mathrm{sig}}^{0}{\overrightarrow{s}}_{\mathrm{sig}}\frac{{\gamma}_{0}{S}_{\mathrm{pump}}^{0}}{2}\left(1+{\hat{s}}_{\mathrm{pump}}\times {\hat{s}}_{\mathrm{sig}}\right)\right]\u2044{S}_{\mathrm{sig}}^{0}$$

$$=\frac{1}{{S}_{\mathrm{sig}}^{0}}[\left(\frac{d}{\mathrm{dz}}{\overrightarrow{E}}_{\mathrm{sig}}^{\u2020}\right)\overrightarrow{\sigma}{\overrightarrow{E}}_{\mathrm{sig}}+{\overrightarrow{E}}_{\mathrm{sig}}^{\u2020}\overrightarrow{\sigma}\left(\frac{d}{\mathrm{dz}}{\overrightarrow{E}}_{\mathrm{sig}}\right)]-\frac{{\gamma}_{0}{S}_{\mathrm{pump}}^{0}}{2}\left(1+{\hat{s}}_{\mathrm{pump}}\times {\hat{s}}_{\mathrm{sig}}\right){\hat{s}}_{\mathrm{sig}}$$

$$=\overrightarrow{\beta}\times {\hat{s}}_{\mathrm{sig}}+\frac{{\gamma}_{0}{S}_{\mathrm{pump}}^{0}}{2}[{\hat{s}}_{\mathrm{sig}}+{\hat{s}}_{\mathrm{pump}}]-\frac{{\gamma}_{0}{S}_{\mathrm{pump}}^{0}}{2}\left(1+{\hat{s}}_{\mathrm{pump}}\times {\hat{s}}_{\mathrm{sig}}\right){\hat{s}}_{\mathrm{sig}}$$

$$=\overrightarrow{\beta}\times {\hat{s}}_{\mathrm{sig}}+\frac{{\gamma}_{0}{S}_{\mathrm{pump}}^{0}}{2}{\hat{s}}_{\mathrm{pump}}-\frac{{\gamma}_{0}{S}_{\mathrm{pump}}^{0}}{2}\left({\hat{s}}_{\mathrm{pump}}\times {\hat{s}}_{\mathrm{sig}}\right){\hat{s}}_{\mathrm{sig}}=\overrightarrow{\beta}\times {\hat{s}}_{\mathrm{sig}}+\frac{{\gamma}_{0}{S}_{\mathrm{pump}}^{0}}{2}{\hat{s}}_{\mathrm{sig}}\times \left({\hat{s}}_{\mathrm{pump}}\times {\hat{s}}_{\mathrm{sig}}\right)$$

Following similar steps the corresponding equations for the pump can be derived. The resulting set of coupled equations is (with $\tilde{\beta}$
defined by $\tilde{\beta}$
_{1}=-β_{1};$\tilde{\beta}$
=-β_{2}
$\tilde{\beta}$
_{3}=+β_{3}):

## Acknowledgments

This work has been carried out within the framework of the European COST Action 299 — FIDES. M. Tur, E. Zilka and A. Eyal also wish to acknowledge the support of the Israeli Science Foundation (ISF). A. Zadok acknowledges the support of a doctoral research fellowship from the Clore Foundation, Israel, a post-doctoral research fellowship from the Center of Physics in Information (CPI), Caltech, and the Rothschild post-doctoral fellowship from Yad-Hanadiv Foundation, Israel.

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