Vector analysis of stimulated Brillouin scattering amplification in standard single-mode fibers

Transcription

1 Vector analysis of stimulated rillouin scattering amplification in standard single-mode fibers Avi Zadok,,, Elad Zilka, Avishay Eyal, uc hévenaz 3, and Moshe ur School of Electrical Engineering, Faculty of Engineering, el-aviv University, el-aviv 69978, Israel Currently with the Department of Applied Physics, MC 8-95, California Institute of echnology, Pasadena, CA 95, USA 3 Ecole Polytechnique Fédérale de ausanne, Institute of Electrical Engineering, SI-GR-SCI Station, 5 ausanne, Switzerland Corresponding author: avizadok@caltech.edu Abstract: he polarization properties of stimulated rillouin scattering (SS) amplification or attenuation in standard single-mode fibers are examined through vectorial analysis, simulation and experiment. Vector propagation equations for the nal wave, incorporating SS and birefringence, are derived and analyzed in both the Jones and Stokes spaces. he analysis shows that in the undepleted 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 nal, which, respectively, provide the nal with maximum and minimum SS amplification (or attenuation). Under high rillouin gain conditions and excluding zero-probability cases, the output SOP of arbitrarily polarized input nals, would tend to converge towards that of maximum SS gain. In the case of high SS attenuation the output SOP of an arbitrarily polarized nal would approach the output SOP corresponding to minimum attenuation. It is found that for a wide range of practical powers ( mw ) and for sufficiently long fibers with typical SS and birefringence parameters, the nal aligned for maximum SS interaction will enter/emerge from the fiber with its electric field closely tracing the same ellipse in space as that of the at the corresponding side of the fiber, albeit with the opposite sense of rotation. he analytic predictions are experimentally demonstrated for both Stokes (amplification) and anti-stokes (attenuation) nals. 8 Optical Society of America OCIS codes: (9.59) Scattering, Stimulated rillouin; (.54) Polarimetry. References and links.. Horiguchi,. Kurashima, and M. ateda, "A technique to measure distributed strain in optical fibers," IEEE Photon. echnol. ett., , (99).. M. Nikles,. hévenaz, and P. Robert, "rillouin gain spectrum characterization in single-mode optical fibers," J. ightwave echnol. 5, 84-85, (997). 3. X. ao, D. J. Webb, and D. A. Jackson, 3-km distributed temperature sensor using rillouin loss in optical fiber, Opt. ett. 8, , (993). 4. J. C. Yong,. hévenaz, and. Y. Kim, rillouin fiber laser ed by a DF laser diode, J. ightwave echnol., , (3). 5. A. oayssa, D. enito, and M. J. Grade, Optical carrier-suppression technique with a rillouin-erbium fiber laser, Opt. ett. 5, 97-99, (). 6. Y. Shen, X. Zhang, and K. Chen, Optical single side-band modulation of GHz RoF system using stimulated rillouin scattering, IEEE Photon. echnol. ett. 7, 77-79, (5). 7. A. Zadok, A. Eyal, and M. ur, GHz-wide optically reconfigurable filters using stimulated rillouin scattering, J. ightwave echnol. 5, 68-74, (7). 8. A. oayssa, and F. J. ahoz, roadband RF photonic phase shifter based on stimulated rillouin scattering and single side-band modulation, IEEE Photon. echnol. ett. 8, 8-, (6). 9. A. oayssa, J. Capmany, M. Sagues, and J. Mora, Demonstration of incoherent microwave photonic filters with all-optical complex coefficients, IEEE Photon. echnol. ett. 8, , (6). #564 - $5. USD Received 9 Oct 8; revised 8 Nov 8; accepted Dec 8; published 6 Dec 8 (C) 8 OSA December 8 / Vol. 6, No. 6 / OPICS EXPRESS 69

2 . Z. Zhu, D. J. Gauthier, and R. W. oyd, Stored light in an optical fiber via Stimulated rillouin Scattering, Science 38, , (7).. M. González-Herráez, K.-Y. Song, and. hévenaz, Optically controlled slow and fast light in optical fibers using stimulated rillouin scattering, Appl. Phys. ett. 87, 83, (5).. M. D. Stenner, M. A. Neifeld, Z. Zhu, A. M. C. Dawes, and D. J. Gauthier, Distortion management in slowlight pulse delay, Opt. Express 3, 9995-, (5). 3. Z. Zhu, A. M. C. Dawes, D. J. Gauthier,. Zhang, and A. E. Willner, "roadband SS slow light in an optical fiber," J. ightwave echnol. 5, -6, (7). 4. M. González-Herráez, K.-Y. Song, and. hévenaz, Arbitrary-bandwidth rillouin slow light in optical fibers, Opt. Express 4, 395-4, (6). 5. K. Y. Song, M. Gonzalez Herraez, and. hévenaz, Observation of pulse delay and advancement in optical fibers using stimulated rillouin scattering, Opt. Express 3, 8-88, (5). 6. A. Zadok, A. Eyal, and M. ur, Extended delay of broadband nals in stimulated rillouin scattering slow light using synthesized chirp, Opt. Express 4, , (6). 7. R. W. oyd, Nonlinear optics, (San Diego, CA: Academic Press, 3) chapter 9, A. Yariv, Optoelectronics, (Orlando F: Saunders College Publishing, 4th Edition, 99), chapter 9, Z. Zhu, D. J. Gauthier, Y. Okawachi, J. E. Sharping, A.. Gaeta, R. W. oyd, and A. E. Willner, Numerical study of all-optical slow-light delays via stimulated rillouin scattering in an optical fiber, J. Opt. Soc. Am., , (5)... Horiguchi, M. ateda, M. Shibata, and Y. Azuma, rillouin gain variation due to a polarization-state change of the or Stokes field in standard single mode fibers, Opt. ett. 4, 39-33, (989).. M. O. van Deventer, and A. J. oot, Polarization properties of stimulated rillouin scattering in single mode fibers, J. ightwave echnol., , (994).. In [], the and probe SOPs are defined in two different reference frames, corresponding to opposite directions of propagation. In this work, as well as in most of the literature on polarization [3,4], a single reference frame is used. herefore, we defer the mathematical description of the conditions for maximum/minimum SS gain to Section. 3. R. C. Jones, A new calculus for the treatment of optical system, J. Opt. Soc. Am. 37, 7-, (947). 4. E. Collett, Ed., Polarized light fundamentals and applications. (New York: Marcel Dekker, 993). 5.. hévenaz, A. Zadok, A. Eyal, and M. ur, All-optical polarization control through rillouin amplification, paper OM7 in OFC/NFOEC 8, San Diego, Ca, (8). 6. J. P. Gordon and H. Kogelnik, PMD fundamentals: polarization mode dispersion in optical fibers, P. Natl. Acad. Sci. USA 97, , (). 7. R. H. Stolen, Polarization effects in fiber Raman and rillouin lasers, IEEE J. of Quantum Electron. 5, 57-6, (979). 8. F. Corsi, A. Galtarossa, and. 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Palmieri, M. Santagiustina,. Schenato, and. Ursini, Polarized rillouin amplification in randomly birefringent and unidrectionally spun fibers, IEEE Photon echnol. ett, 4-4, (8). #564 - $5. USD Received 9 Oct 8; revised 8 Nov 8; accepted Dec 8; published 6 Dec 8 (C) 8 OSA December 8 / Vol. 6, No. 6 / OPICS EXPRESS 693

3 . Introduction Stimulated rillouin Scattering (SS) requires the lowest activation power of all non-linear effects in silica optical fibers. SS has found numerous applications, including distributed sensing of temperature and strain [-3], fiber lasers [4], optical processing of high frequency microwave nals [5-9], and even optical memories []. Over the last three years, SS has been highlighted as the underlying mechanism in many demonstrations of variable group delay setups [-6], often referred to as slow and fast light. In all of the above applications, SS has been a favorable mechanism for its robustness, simplicity of implementation and low power in standard fibers at room temperature. In SS, a strong wave and a typically weak, counter-propagating nal wave optically interfere to generate, through electrostriction, a traveling longitudinal acoustic wave. he acoustic wave, in turn, couples these optical waves to each other [7,8]. he SS interaction is efficient only when the difference between the optical frequencies of the and nal waves is very close (within a few tens of MHz) to a fiber-dependent parameter, the rillouin shift ν, which is of the order of - GHz in silica fibers at room temperature and at telecommunication wavelengths [7,8]. An input nal whose frequency is ν lower than that of the (Stokes wave) experiences SS amplification. If the input nal frequency is ν above that of the (anti-stokes wave), SS-induced nal attenuation is obtained instead. he strength of the interaction is often quantified in terms of an exponential gain coefficient, which is defined as the logarithm of the nal linear power gain (or loss), normalized to a unit power and unit fiber length [W m] -. (he coefficient equals the rillouin gain factor g [9], divided by the fiber effective area). Since SS originates from optical interference between the and nal waves, the SS interaction, at a given point along the fiber, is most efficient when the electric fields of the and nal 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 SS interaction at that point averages to zero over an optical period. Consequently, in the presence of birefringence, the overall nal gain (or loss) depends on the birefringent properties of the fiber, as well as on the input states of polarization (SOPs) of both and nal. Following initial work by Horiguchi et al. [], van Deventer and oot [] have studied in detail the nal SOPs leading to maximum and minimum gain. ased on the statistical properties of the evolution of the and nal SOPs in optical fibers much longer than the polarization beat length, but implicitly ignoring any influence of the rillouin 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 /3 of the maximum gain coefficient in a birefringence-free fiber. Furthermore, maximum gain is achieved when the and nal have identical polarizations (in their respective directions of propagation), while minimum gain is obtained for the corresponding orthogonal case [-4]. heir analysis was nicely corroborated by an experiment, which showed that for a given, there were indeed two input SOPs, chosen by the experimenters to be identical or orthogonal to that of the, with one providing an exponential gain twice that of the other one. However, the SS amplification of an arbitrarily polarized input nal SOP was not discussed, nor was the role played by the rillouin effect itself in the evolution of the nal SOP considered. In this paper, the pioneering work of van Deventer and oot is analytically substantiated and extended, using a vector formulation of the SS amplification process in the presence of birefringence. A vector differential equation, combining both effects, is studied in the Jones and Stokes spaces. ased on the Jones space representation, it is shown that in the undepleted regime, the input nal SOPs which lead to maximum/minimum SS gain are always orthogonal, regardless of power and the statistics of the and nal SOPs along the fiber. hese maximum and minimum gain SOPs, therefore, provide a convenient vector #564 - $5. USD Received 9 Oct 8; revised 8 Nov 8; accepted Dec 8; published 6 Dec 8 (C) 8 OSA December 8 / Vol. 6, No. 6 / OPICS EXPRESS 694

4 base for the examination of an arbitrarily polarized input nal wave. In the Stokes space, the evolution of the magnitude and the SOP of the nal wave along the fiber are described by a pair of coupled, rather simple differential equations. Using this representation, it is analytically shown that the SS overall gain coefficient is determined by an average over a local mixing coefficient, similar (but not identical) to that of [], for any input SOP, polarization statistics, or power. In addition, we show that in the presence of a, the evolution of the nal SOP is controlled not only by the fiber birefringence but also by the local SS interaction, which drags the nal SOP towards that of the. he equations provide inht into the relation between the nal SOPs, which lead to maximum and minimum gain, and the SOP of the. he magnitude and SOP of the nal are then studied numerically. he maximum and minimum SS gain coefficient for a general birefringent fiber are found to always be ( ± μ), μ. Even for a weakly birefringent fiber, μ does not necessarily need to be /3, although in the limit of a fiber embodying the fully developed statistics of [], it tends to this value. he SOPs of input nals, which experience maximum/minimum gains, are then studied as a function of the power. he nal SOP is also examined experimentally, for both Stokes and anti-stokes nal waves. As predicted by the analysis, the output nal SOP is seen to converge towards a specific, preferred SOP, which is practically independent of both the input nal SOP and polarization transformations along the fiber [5]. hat preferred output SOP could be arbitrarily varied, however, by changing the input SOP. he remainder of this paper is organized as follows: Section presents vector theoretical analysis of the nal wave, subject to both SS and birefringence, in the undepleted regime. Section 3 is dedicated to numerical simulations. Section 4 provides experimental results, and brief concluding remarks are given in section 5.. heory et us denote the column Jones vector of a monochromatic nal wave as E () z, z indicating position along the fiber, with the launch and exit points at z = and z =, respectively ( is the fiber length). With no, the propagation of () z can be described by: E () z () z () E E = () with () z a unitary Jones matrix representing the effect of fiber birefringence. he wave, whose Jones vector is denoted by () z, is launched into the fiber at z =. E hroughout this paper, we work in the same right-handed coordinate system { x y, z},, where the nal propagates in the positive z direction, while the propagates in the negative z direction. hus, if both E () z and E equal the X vector [ j] ( stand for transpose), they represent a right-handed circularly polarized nal and a left-handed circularly polarized wave, respectively [3,4]. We also neglect linear polarizationdependent power losses in the fiber, although such losses can be easily included in the analysis. Further, since the rillouin shift ν is merely ~GHz, and only a few kilometers of modern fibers are concerned, polarization mode dispersion can be ignored and, therefore, shifting the optical frequency by ν has a negligible effect on the Jones matrix of the fiber. Hence, the propagation of the wave (in the absence of a probe) can also be expressed using () z : E () = () z E () z E () z = () z E (); () where inv () z = z. [ ] ( ) #564 - $5. USD Received 9 Oct 8; revised 8 Nov 8; accepted Dec 8; published 6 Dec 8 (C) 8 OSA December 8 / Vol. 6, No. 6 / OPICS EXPRESS 695

5 (7) (6) When both the probe and waves are present, the local evolution of () z and E is driven by both the fiber birefringence and the SS effect to give (see [6] for the birefringence term, and [7] for the SS term): de d (3a) = + [ E E ] E de () z d = + E E E [W m] - is the SS gain per unit length per a unit of 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 optical spectrum and the frequency offset between the and nal waves. We dedicate most of the analysis to the Stokes wave scenario, so that is positive, but the analysis and results, properly interpreted, are equally valid for the anti-stokes case, where the optical frequency of the nal is ν above that of the. he anti-stokes nal surrenders its power to the, thereby becoming attenuated with an SS attenuation per unit length of. Note that ( / ) [ E () z E () z ] is a matrix, representing the outer product of a column vector ( E () z ) with a row one (the transpose conjugate of E () z ). From now on it will be assumed that SS-induced nal amplification or attenuation negligbly affect the (i.e., the so-called undepleted approximation []). hus, the SS term in Eq. (3b) can be ignored and Eq. (3a) becomes linear in () z. herefore, E ( ) ( ) E E E (3b) = H, (4) where H is a matrix, which depends on the fiber birefringence, the fiber length, the power, and its SOP at z =. he matrix H is generally non-unitary. Nevertheless, it can be processed using the singular value decomposition (SVD) technique: G (5) H = U S V = U V, G where U and V are unitary matrices, G, G are real and positive and satisfy G > G > in the case of SS amplification and > G > G in the case of SS attenuation. Using this decomposition two orthogonal input nal Jones vectors can be identified, which provide the maximum and minimum nal output powers, namely: E = [ V ] = V ; E = V he corresponding output Jones vectors are given by: E = U S V V = U S = G U E out _ min = U S V V = U S = G U #564 - $5. USD Received 9 Oct 8; revised 8 Nov 8; accepted Dec 8; published 6 Dec 8 (C) 8 OSA December 8 / Vol. 6, No. 6 / OPICS EXPRESS 696

6 (9) and are, therefore, also orthogonal. It is thus convenient to represent an arbitrarily polarized input nal using the orthogonal base of E, E : in E = α E + β E (8) Using Eqs. (7) and (8), the output nal Jones vector and the nal power are: out E = α G U G + β U P out = α G + β G When G >> G, Eq. (9) suggests that unless α is negligible, an arbitrarily polarized input nal will be drawn towards the SOP of E. hese predictions are supported by experiments, to be described in section 4. Next, we try to relate E, E to the SOP of the wave. o that end, we have transformed Eq. (3) to the Stokes space (see Appendix): ds P (a) _ = ( + ) S _ Here d = β( z) S _ = β( z) + + P P ( ) [ ( ) ] is the nal power, [ ],, 3 normalized Stokes vectors ( s s + s ) (b) s ˆ = s s s and similarly ŝ, are 3X + =, describing the evolution of the,,,, 3,, polarizations of the counter-propagating nal and waves, respectively, and finally, P denotes the power, which for the undepleted, lossless case is z-independent. he three-dimensional vector β(z) where σ describes the fiber birefringence in Stokes space [6]: d β σ j, is a row vector of Pauli spin matrices [6] (see also in the Appendix). he vector β(z) is aligned with the Stokes space representation of the local slow axis of birefringence [6]. Note that we express both Stokes vectors in the same right handed coordinate system, in which the nal wave propagates in the positive z direction. herefore, the Stokes vector = [ ] represents a right-handed circular polarization for the nal wave, but a lefthanded circular polarization for the. Eq. (a) is easily cast into a form: d ln( S ) P () z () _ = ( + ). In the undepleted regime, the solution is readily obtained: () #564 - $5. USD Received 9 Oct 8; revised 8 Nov 8; accepted Dec 8; published 6 Dec 8 (C) 8 OSA December 8 / Vol. 6, No. 6 / OPICS EXPRESS 697

7 S out _ P in = S exp _ = S in _ exp P ( + ) ' ( + ) (3) is the scalar product of the and nal Stokes vectors, averaged over the fiber length. hus, for any input SOP one can define an effective SS gain, given by: (4) = ( + ) Obviously, depends on the nal input SOP, as well as on the SOP. Equation (b) specifies two driving forces that control the evolution of the SOP along the fiber. he first, β ŝ, describes the birefringence-induced evolution of the nal SOP [6]. he same term also governs the evolution of the SOP, albeit in the opposite direction. he second term, ( / ) P [ ( ) ], represents the effect of SS amplification on the nal SOP. his second term has a very interesting physical interpretation on the Poincare sphere: it is a vector, orthogonal to ŝ, and tangentially (on the sphere surface) pointing towards ŝ. his term nifies a force pulling ŝ towards ŝ. he magnitude of this pulling force scales with the power and depends on the local projection of on, vanishing when either ŝ is parallel to ŝ ( and nal SOPs aligned) or anti-parallel to it (in the Stokes space, namely: orthogonal in the Jones space). Several special cases are of particular interest. et us consider a fiber with no birefringence, so that the evolution of the nal SOP is governed by SS alone. If the input in nal SOP is aligned with that of the, ( s ˆ = ), then it follows from Eq. (b) that the d =. he nal SOP, therefore, remains aligned with that of the throughout the fiber. Since in this case s ˆ =, the SS gain coefficient of Eq. (4) equals. Alternatively, when the nal input SOP is orthogonal to that of the in s ˆ ˆ =, we still obtain =, and the and nal remain orthogonal for ( ) all s d z. Now s ˆ ˆ = and the SS gain coefficient is zero. hus, s E out _ min ( E ) in a birefringence-free fiber is parallel (perpendicular) to E ( z = ) (using our conventions, E for a right-handed circularly polarized is left-handed polarized). When the input nal is arbitrarily polarized, the SS polarization pulling term of Eq. (b) is nonzero, so that ŝ is gradually drawn towards ŝ. he slope of the gain coefficient versus power curve, determined by, will increase with power (or fiber length), eventually approaching its maximum value of, and the SOP of the emerging nal will draw nearer and nearer that of the wave. hese trends are, of course, fully consistent with the Jones space description of Eq. (9). We now turn to the more prevalent scenario of standard single-mode fibers, where the birefringence term Eq. (b) is larger than the SS term (for a an average beat length of 4 m, β z ~.6 m whereas P for =. [ m W], P.W / ~.m = ). While being relatively small, the SS term cannot be ignored. High differential gains ( G / G > ) are easily observed, and according to Eq. (9), any nal, whose input SOP even slightly #564 - $5. USD Received 9 Oct 8; revised 8 Nov 8; accepted Dec 8; published 6 Dec 8 (C) 8 OSA December 8 / Vol. 6, No. 6 / OPICS EXPRESS 698

8 deviates from that of E, will emerge with its SOP being pulled towards that of E. While the polarization pulling is due to the SS term, the final nal SOP is not that of E ( z = ). he relation between the SOP of and that of E ( z = ) are studied below, first analytically, in the low power limit, and then, numerically for the general case. et us assume first a very weak so that the rillouin term in Eq. (b) can be ignored ( G / G P ). In this limit, the forward evolution of ŝ and the backward evolution of ŝ are solely governed by the birefringence term. We denote the maximum value of ˆ over all possible SOPs of the input nal ˆ ( z = ), but for a given s ( z = ), as max { } ( z= ) SOP ( ) Ensemble Average = = E. ut: () M s () M M ˆ s () (5) M Ensemble Average Ensemble Average () = () (). Here (z M ) and M are the Mueller matrices representing (z) and, respectively ( 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 ( ) was taken from [8]. M z One can easily conclude from Eq. (5) that max { } ( z= ) is /3, resulting in a maximum achievable gain coefficient of ( / 3) (Eq. 4, see also a discussion in []). his maximum is attained when ( z = ) is the image of ( z = ) on the Poincare sphere, max max with the equatorial plane acting as a mirror, namely: s ˆ ( ) = ( ), s ˆ ( ) = ( ),,,, max and s ( ) = ( ). his s ˆmax ( z = ) is the normalized Stokes representation of the ˆ 3, 3, complex conjugate of the Jones vector at z =, namely, E ( z = ), rather than that of E ( z = ) (as in the birefringence-free case). Conversely, { ˆ ˆ ˆ min ( ) s } s = 3, s z= corresponding to a minimum gain coefficient of ( / 3). his minimum value is attained for min max ( z = ) = ( z = ), which is the Stokes representation of a polarization orthogonal to that of ( z = ) z =. It is easily proven from Eqs. (-) that for E, to be denoted by ( ) E a unitary (z) (and ignoring the rillouin term), if E they will continue to be parallel for all z polarization as that of E ( z = ) and E, so that E ( z ) are a parallel pair at z =, = has the same. hese analytically obtained results are no different than the seemingly intuitively-drived conclusions of [], when carefully noting the difference in the reference frame convention, but both approaches are strictly valid only in the limit of very max weak power. he relation between E and level of power is investigated in the next section. E in the presence of non-negligible #564 - $5. USD Received 9 Oct 8; revised 8 Nov 8; accepted Dec 8; published 6 Dec 8 (C) 8 OSA December 8 / Vol. 6, No. 6 / OPICS EXPRESS 699

9 3. Simulations for Signal amplification (Stokes) and attenuation (anti-stokes) he SS amplification/attenuation and the output SOP of the amplified/attenuated nal for different power levels were numerically examined, using Eqs. (), (3) and (a-b). Simulations were based on the commonly used concatenated random wave-plate model [9], with the three components of β of Eq. () normally distributed with zero mean and the same standard deviation, chosen so that the average beat length π β equals 4 m. Although a broad set of parameters was numerically investigated, results below were obtained for a fiber length of 5 m comprising, plates and =.[ W ]. m z Pump power [mw] - - Pump power [mw], of the input nal (normalized) Stokes vector Fig.. (a). he projection, ˆ s ( z = ) for maximum SS gain, onto the (normalized) Stokes vector corresponding to ( z = ) E, as a function of power, for different fiber realizations. (b) he power dependence max for the same realizations. he beat length in all realizations was 4 m. of { } ( z= ) _ max _ max Figure (a) shows the projection of s ˆin on ( z = ), where s ˆin and s ˆ () are the normalized Stokes-space counterparts of E and E ( z = ), respectively. Figure (b) shows the calculated for an input nal SOP aligned with E, as a function of power. he figures include several different fiber realizations, each with different random drawings of β values for the concatenated wave-plates, though all with an average beat length of 4 m. At low powers ( z = ) does not depend on, as discussed above, and the relatively small misalignments between and P E (), as well as the deviations of s E ˆ from the predicted value of /3, are due to the finiteness and discreteness of the model. It is clearly seen that aligned with ( z = ) E remains closely E, even for a power as high as mw. While not shown, this close alignment also holds at the fiber output, z =, where the SOP of E lies in a E z =. similar close vicinity that of ( ) Clearly, max { } ( z= ), while not exactly /3 (Fig. (b)), depends very weakly on P, resulting in a practically linear relationship between the achievable max/min gain coefficient and power. Figure shows the nal power gain as a function of #564 - $5. USD Received 9 Oct 8; revised 8 Nov 8; accepted Dec 8; published 6 Dec 8 (C) 8 OSA December 8 / Vol. 6, No. 6 / OPICS EXPRESS 7

10 power, for a nal SOP aligned with either E (maximum gain), or E (minimum gain). According to Eq. (4) 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 P, while not exactly, is always + max { } max { } ( z= ) ( z= ) 8. Signal gain [d] Pump power [mw] Fig.. Signal gain as a function of power for different SOPs of the input nal. he linear curves are calculated for an input SOP, aligned with either the -dependent E (blue-top line), or orthogonal to it, i.e., parallel to E (green-bottom line). he red-dashed line is for the case where the input SOP deviates from E ( P = 5mW) by a π rad rotation about the ŝ axis on the Poincare sphere Yet, SS still has a dramatic effect on the output SOP of an input nal, whose SOP only slightly deviates from that of E, (see Fig. ). As the difference between the minimum and maximum gains increases, the red-dashed gain curve in Fig. changes its slope, approaching that of the maximum gain case. Incidentally [], the same argument applies to amplified spontaneous SS, which, therefore, under high differential gain conditions, emerges from the fiber with the SOP of E ( z = ). hus, under high differential gain conditions the SOP of amplified spontaneous SS at the fiber output ( z = ) coincides with that of E! For arbitrarily polarized input nals and for powers above 5 mw, the output nal SOPs are clearly seen in Fig. 3(a) and Fig. 3(b) to converge towards the SOP of E, which is almost unaffected by power. Figure 3(c) shows the evolution of as a function of the position coordinate z along the fiber, for different powers, when the nal input SOP is E ( z = ). hat input SOP is close to, but not quite equal to E, especially at high powers. Note the gradual pulling of nal SOP towards that of E () z, requiring many beat lengths before the effect becomes quite distinct at high powers. #564 - $5. USD Received 9 Oct 8; revised 8 Nov 8; accepted Dec 8; published 6 Dec 8 (C) 8 OSA December 8 / Vol. 6, No. 6 / OPICS EXPRESS 7

11 a: b: c: Position [m] Fig. 3. (a) and (b): Scatter plots of output amplified nal SOP on the Poincare sphere, corresponding to random input nal SOPs, for a specific fiber realization. he input Stokes vector ŝ was chosen as [ ]. he horizontal and vertical axes in all figures correspond to the Stokes s and s axes, respectively. Red closed circles indicate SOPs 3 for which s is positive, whereas open blue squares indicate a negative s. X denotes the location E in Stokes space. he power was 5 mw (a) and 5 mw (b). (c): Stokes space projection of the nal SOP on the conjugate of the SOP, ( ) ˆ z s, as a function of position for an input nal SOP exactly orthogonal to that of ( z = ) power was 5 mw (red dashed), 4 mw (black dotted) and 5 mw (blue solid). E. Pump he 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 SS interaction for an anti- Stokes nal is still maximum for an input SOP aligned with E (). However, this interaction results in efficient nal attenuation, rather than nal gain. We can, therefore, expect stronger attenuation for that input nal field component aligned with E (), and weaker attenuation ( maximum gain ) for the orthogonal component. Correspondingly, the SOP of the emerging nal is dominated by that of the maximum power SOP, and is expected to be closely aligned with that of E ( z = ). Figure 4 shows the scatter plots of the attenuated anti-stokes nal, for s ˆ = [ ], indicating convergence towards [ ], which is the (normalized) Stokes-spaced representation of E ( z = ). oth scenarios are visited in the experiment, to be described next. Fig. 4. Scatter plots of attenuated output nal SOP on the Poincare sphere, corresponding to random input nal SOPs, for a specific fiber realization. he input Stokes vector ŝ was chosen as [ ]. he horizontal and vertical axes in all figures correspond to Stokes s and s axes, respectively. Red closed circles indicate SOPs for which s 3 is positive, whereas open blue squares indicate a negative s. X denotes the location space. he power was 5 mw (left), 5 mw (center) and 5 mw (right). E in Stokes #564 - $5. USD Received 9 Oct 8; revised 8 Nov 8; accepted Dec 8; published 6 Dec 8 (C) 8 OSA December 8 / Vol. 6, No. 6 / OPICS EXPRESS 7

12 4. Experiment he experimental setup for characterizing polarization related properties of SS is shown in Fig. 5. ight emitted from a tunable laser source was split by a 5% coupler. In the lower () branch, the light was amplified by a high-power Erbium-doped fiber amplifier (EDFA), and directed into the fiber under test via a circulator. he length of the fiber under test was 5 m, and its rillouin frequency shift was ν =.57 GHz. he power was controlled by a variable optical attenuator (VOA). In the upper (nal) branch, the laser light was modulated by an electro-optic intensity modulator (EOM). he modulation frequency was tuned to ν, and the EOM bias voltage was adjusted to suppress the optical carrier []. Following the EOM, the nal was filtered by a narrow-band Fiber ragg grating (FG). For SS nal amplification measurements, the frequency of the tunable laser was adjusted so that the lower modulation sideband matched the FG reflection frequency [3]. his way, the frequency of the nal propagating in the fiber under test was ν below that of the. For SS attenuation measurements, the tunable laser frequency was modified so that the upper modulation sideband was retained by the FG [3]. Following the SS interaction, the nal was routed to a power meter, followed by a lock-in amplifier to filter out spontaneous SS, or to a polarization analyzer for the measurement of the nal output power and SOP. A second FG in the detection path was used to filter out the backscattered, as well as the spontaneous rillouin scattering amplified by the Stokes process in the SS loss scenario. For each power, the input nal SOPs which corresponded to minimum and maximum nal output power were found using the following procedure: First, a programmable polarization controller (Prog. PC) in the nal path was set to four nondegenerate SOPs, and the output nal power was recorded for each. ased on these four measurements, the top row of the 4X4 Mueller matrix describing the ed fiber under test was extracted [3], and nal SOPs for minimum and maximum output power could be calculated. Next, the programmable PC was set to these two input SOPs and the output nal power was recorded. Fig. 5. Experimental setup for characterizing the polarization dependence of SS. A: Optical attenuator. VOA: Variable optical attenuator. FG: Fiber ragg grating. DS: Double side band modulation. SS: single side band modulation. PC: Polarization controller. EDFA: Erbium-doped fiber amplifier. EOM: electro-optic modulator. ν p denotes the optical frequency of the Figure 6(a) shows the logarithm of the nal power gain (Stokes nal) as a function of power, for three different SOPs of the input nal wave. In the upper and lower curves, the nal SOP is adjusted for each power level to achieve maximum and minimum gain, respectively. In these curves, the logarithmic SS gain appears to be linearly proportional to the power over the entire measurement range, indicating a power- #564 - $5. USD Received 9 Oct 8; revised 8 Nov 8; accepted Dec 8; published 6 Dec 8 (C) 8 OSA December 8 / Vol. 6, No. 6 / OPICS EXPRESS 73

13 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 []. hese results indicate that our.km fiber comprises many correlation lengths of the random birefringence [, 8]. he third curve of Fig. 6(a) shows the logarithm of the SS gain for a in _ near _ min nal, whose input SOP, E, is azimuthally 4 away from E ( Stokes). Initially, for ( Stokes) relatively low power, the gain slope is that of the minimum gain curve. However, for higher powers, the measured gain increases rapidly and its slope approaches that of the maximum gain curve, as discussed in Sec. 3. a: b: c: Gain [d] Gain [d] Pump Power [mw] d: Pump Power [mw] Fig. 6. (a). SS gain (Stokes nal) in d as a function of power, for a 5 m long fiber. ower curve (Green) optimized for minimum gain, Upper curve (lue) optimized for maximum gain, Dashed curve (Red) for an input SOP in the vicinity of E, rotated ( Stokes) from it by 4 around the s 3 (R) axis (the black squares are explained in the text). (b) he SOPs of the emerging amplified nals for the three cases of (a): maximum (blue solid circles), minimum (green open diamonds), and red squares for the intermediate case. Open symbols denote SOPs in the back of the sphere. he size of the square is a measure of the nal power, increasing with power for Stokes nals. he black + is the SOP of the spontaneous SS. he straight line through the center of the sphere connects this SOP to its orthogonal counterpart. (c) SS attenuation (anti-stokes nal) in d as a function of power. ower curve (Green) optimized for minimum output power (maximum attenuation), Upper curve (lue) optimized for maximum output power (minimum attenuation), Dashed curve (Red) for an input SOP in the vicinity of E, rotated from it by 4 around the ( A Stokes) s 3 (R) axis. (d) he SOPs of the emerging attenuated nals for the cases of (c): maximum (blue open circles), minimum (green solid diamonds), and red squares for the intermediate case. he straight line through the center of the sphere is that of (b), shown here for reference. #564 - $5. USD Received 9 Oct 8; revised 8 Nov 8; accepted Dec 8; published 6 Dec 8 (C) 8 OSA December 8 / Vol. 6, No. 6 / OPICS EXPRESS 74

14 As a consistency check, we used Eqs. (8)-(9) first to project in _ near _ min E ( Stokes) on the measured E, E ( Stokes)), and then used the measured values for G ( Stokes) (maximum gain) and G in _ near _ min (minimum gain) to analytically predict the gain experienced by E. he results are ( Stokes)) 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 out _ min out _ near _ min E ( Stokes), E and E ( Stokes) for all powers. Also shown on the sphere is the SOP ( Stokes) of spontaneously amplified rillouin scattering, which was obtained by turning off the nal input and measuring the SOP of the rillouin-scattered light at ν = ν ν. Note that as out _ min P spans the 5-35mW range, { } and { } E ( Stokes ) E ( Stokes ) s p 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 rillouin scattering, leading to a E E ( Stokes ) coincides, as expected, with the SOP ( Stokes ) of the spontaneously amplified rillouin scattering. Also shown is the evolution of the nal in _ near _ min SOP for. Figure out _ min larger spread near { }). Furthermore, { } E ( Stokes), clearly indicating the pulling of its SOP towards that of { } E ( Stokes ) 6(c) shows the logarithm of the maximum and minimum attenuation of an anti-stokes nal. 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 nal, albeit with a minus n. he figure also shows the measured in _ near _ min and calculated logarithmic loss of an anti-stokes nal with an input SOP E. Finally, ( A stokes) out _ min out _ near _ min Fig. 6(d) shows the output SOPs corresponding to E, E and E for all ( A Stokes) ( A Stokes) powers. Polarization pulling towards the SOP of { E } ( A Stokes) out _ min { E } ( A Stokes) { } E ( Stokes ) ( A Stokes) is observed. It is seen that (solid diamonds in Fig. 6(d)), which suffers maximum attenuation are parallel to (solid circles in Fig. 6(b)), which enjoys the maximum possible gain. Figure 7 shows the nal output SOP for twenty different input SOPs, which were evenly distributed on the Poincare sphere. As the power is increased, the output nal SOPs converge to a particular, preferred state. hus, the converging effect is effective for both SS nal gain and nal loss, in the undepleted regime. a: b: c: Fig. 7. Measured output nal SOP for SS nal gain and SS nal loss for twenty evenly distributed input nal SOPs. (a) Stokes SOP, power is 5 mw. (b) Stokes SOP, power is 45 mw. (c) Anti-Stokes SOP, power is mw (SOP measurements in the nal attenuation scenario were difficult due to the presence of spontaneous SS, which competed with the attenuated nal. hus, reliable readings could not be obtained for powers above 5 mw.) #564 - $5. USD Received 9 Oct 8; revised 8 Nov 8; accepted Dec 8; published 6 Dec 8 (C) 8 OSA December 8 / Vol. 6, No. 6 / OPICS EXPRESS 75

15 5. Conclusions In this work, the analysis of SS in birefringent fibers was extended to include arbitrarily polarized nals. A vector propagation equation for the nal wave in the undepleted regime was provided, both in Jones and in Stokes spaces. he equations and their subsequent analysis provide expressions for the output nal vector, regardless of the polarization statistics of the and nal waves along the fiber. he analysis showed that SS in the undepleted 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 nal will be closely aligned with that of the maximum gain axis at the fiber output. his prediction was experimentally confirmed, for both Stokes and anti-stokes nal waves. Somewhat similar polarization attraction between counter propagating waves, based on the Kerr effect in short, highly non-linear fibers, was recently reported [3], but the effect was restricted to circular SOPs. he vector properties of SS can give rise to an arbitrary polarization synthesis. he analysis also shows that the maximum and minimum input nal SOPs for the Stokes wave in long, standard single-mode fibers correspond to the conjugate of the outgoing, and the orthogonal of that conjugate, respectively. his correspondence is practically valid for powers up to tens of milliwatts over fibers a few km long. he roles of the two SOPs are reversed for the anti-stokes wave. he polarization and birefringence dependence of SS has already been used in rillouin 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 SS based slow light setups. In addition, birefringence was observed to cause a nonlinear response in the delay- power transfer function [37]. In one recent example, the polarization sensitivity of SS-induced delay was overcome using a Faraday rotator mirror [38]. Clearly, the polarization related properties of SS in long, standard single-mode fibers continue to be of large interest. he tools developed in this work provide a broad and comprehensive framework for the study of SS and polarization and open new horizons for applications. A possible application, in which the spread of the nal 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 nal propagation equation in the presence of SS and birefringence, Eq. (a-b), is derived (see [39] for a different, though equivalent, formulation, of the equations governing the evolution of the non-normalized nal Stokes vector). he starting point for the derivation is Eq. (3) repeated here for convenience: de () z d (A) = () z + E E E de () z d = + E E E (A) o obtain the evolution of the nal power and its SOP in terms of the Stokes parameters we use the common definitions [6, Eqs. (.6), (3.5)] ( σ is a vector of the Pauli spin matrices): S E E ; S E σ E j (A3) ; σ =, σ = and σ = 3 j and S and ŝ denote, respectively, the power and normalized Stokes vector which correspond to E. Using Eq. () which reads: #564 - $5. USD Received 9 Oct 8; revised 8 Nov 8; accepted Dec 8; published 6 Dec 8 (C) 8 OSA December 8 / Vol. 6, No. 6 / OPICS EXPRESS 76

16 d β σ j, and using the expansion of the projection operator in terms of Pauli matrices [6]: S EE ( I + σ), with I denoting the X identity matrix, we obtain: de j S = β σ + ( I + σ) E. 4 E Eq. (A6) can be used to obtain an equation for the evolution of the nal power: ds ( ) d E E j S = = E β σ + ( I + σ) E + 4 j S β σ + 4 ( I + σ) E = ( + ) S S he equation governing the propagation of the normalized nal Stokes vector can be derived using Eqs. (A.3)-(A7) of [6]: d ( ) ( ) ( ) ( ) d E E (A8) σ E E d E σe E σe d E E = = EE ( E E ) S = d( E ) ( ) σe S + S d d S = E E E E ( ) S σ + σ + S S = β + [ + ] ( + ) S S S = β + ( ) = β + ( ) Following similar steps the corresponding equations for the can be derived. he resulting set of coupled equations is (with β ~ ~ ~ ~ defined by β = β ; β = β β = +β ): 3 3 ds ds (A9) = ( + ) S S = ( + ) S S ; d S d S ~ = β + ( ) = β + ( ) Acknowledgments his work has been carried out within the framework of the European COS Action 99 FIDES. M. ur, 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. (A4) (A5) (A6) (A7) #564 - $5. USD Received 9 Oct 8; revised 8 Nov 8; accepted Dec 8; published 6 Dec 8 (C) 8 OSA December 8 / Vol. 6, No. 6 / OPICS EXPRESS 77