Theory of polarization attraction in parametric amplifiers based on telecommunication fibers

Transcription

1 27 J. Ot. Soc. Am. B / Vol. 29, No. / October 22 Guasoni et al. Theory of olarization attraction in arametric amlifiers based on telecommunication fibers Massimiliano Guasoni,, * Victor V. Kozlov,,2 and Stefan Wabnitz Deartment of Information Engineering, Università di Brescia, Via Branze 38, 2523 Brescia, Italy 2 Deartment of Physics, St.-Petersburg State University, Petrodvoretz, St.-Petersburg 9854, Russia *Corresonding author: massimiliano.guasoni@ing.unibs.it Received May 25, 22; revised July 3, 22; acceted August 6, 22; osted August 6, 22 (Doc. ID 69394); ublished Setember, 22 We develo from first rinciles the couled wave equations that describe olarization-sensitive arametric amlification based on four-wave mixing (FWM) in standard (randomly birefringent) otical fibers. We show that in the small-signal case these equations can be solved analytically, and ermit us to redict the gain exerienced by the signal beam as well as its state of olarization (SOP) at the fiber outut. We find that, indeendently of its initial value, the outut SOP of a signal within the arametric gain bandwidth is solely determined by the um SOP. We call this effect of ulling the olarization of the signal towards a reference SOP the olarization attraction, and we call the arametric amlifier the FWM olarizer (which can equivalently be called the fiber-otic arametric amlifier olarizer). Our theory is valid beyond the zero olarization mode disersion (PMD) limit, and it takes into account moderate deviations of the PMD from zero. In articular, our theory is caable of analytically redicting the rate of degradation of the efficiency of the arametric amlifier, which is caused by the detrimental PMD effect. 22 Otical Society of America OCIS codes: , 6.437, 23.5, INTRODUCTION Recent years have witnessed a substantial growth of interest in develoing nonlinear-otical techniques for the control of the state of olarization (SOP) of light beams. The motivation behind such research activities is twofold. First of all, nonlinear-otical techniques may ermit relacing the inefficient and lossy method of olarizing a light beam with conventional assive linear olarizers with the lossless olarization attraction of an arbitrary initial SOP towards the desired SOP at the outut of a nonlinear medium. A key advantage of using lossless olarization attraction is that, in contrast with assive linear olarizers, inut signal SOP changes do not lead to outut signal intensity fluctuations or relative intensity noise (RIN). The second goal is to find efficient ways to exercise all-otical control over the SOP of a signal beam by exloiting its nonlinear interaction with a um beam with a well-determined SOP. Here we analyze a novel method for achieving the allotical control of the SOP of a signal beam, namely exloiting the four-wave-mixing (FWM)-mediated rocess of arametric amlification in a standard telecom otical fiber. In short, nonlinear-otical methods allow for designing novel tyes of olarizers with much greater functionality than conventional assive linear olarizers. So far, two distinctly different tyes of nonlinear-otical olarizers were roosed. The first class comrises the so-called nonlinear lossless olarizers (NLPs), which are based on the cross-olarization modulation (XPolM) of two intense beams in a Kerr medium. To the second class belong the so-called Raman olarizers, which are based on the olarization-sensitive Raman amlification of a signal beam in a Raman-active medium. These two tyes of olarizers exloit the two comlementary manifestations of the cubic nonlinearity of fibers conservative for inducing XPolM effect, and dissiative, which is resonsible for the Raman effect. Here, we exloit the same cubic nonlinearity, more recisely its conservative art, for initiating the rocess of olarization-sensitive FWM between three beams. The first NLP was roosed and exerimentally demonstrated by Heebner et al. in []. It was based not on the Kerr nonlinearity, but on a hotorefractive effect. This olarizer was caable of transforming, in a lossless manner, a light beam with an arbitrary initial SOP into a beam with one and the same SOP towards its outut. The rincile of oeration of this device was the conversion of energy from one olarization comonent of the beam into its orthogonal olarization comonent. Photorefractive materials are characterized by a nonlinear resonse that is far too slow to be useful in contemorary ultrafast otics. In contrast, the Kerr nonlinearity of silica is virtually instantaneous, which makes otical fibers a romising medium for imlementing lossless olarizers within high-bit-rate telecom networks. The rogress in develoing fiber-based NLPs started from imractical isotroic fibers [2 4] and evolved towards chea and reliable telecom fibers [5 8] or secialty fibers such as highly birefringent and sun fibers [9]. The mathematical asects of the roblem were studied in [ 4], and allowed us to get further insight into the hysics of fiberbased NLPs, whose rincile of oeration is different from that of hotorefractive lossless olarizers. Instead of the selfinteraction of a single beam in a hotorefractive material, a two-beam cross-interaction (namely, XPolM) is used in the Kerr medium. Namely, an auxiliary um beam with a welldefined SOP is emloyed, serving as a olarization reference for the signal beam with arbitrary initial SOP. As reviously outlined, when using lossless olarizers, inut signal SOP fluctuations do not lead to outut RIN [5] /2/27-$5./ 22 Otical Society of America

2 Guasoni et al. Vol. 29, No. / October 22 / J. Ot. Soc. Am. B 27 Another tye of nonlinear-otical olarizer is the Raman olarizer. It is different from conventional Raman amlifiers by its sensitivity to the SOP of the um beam. The signal that exeriences Raman amlification acquires an SOP that is dictated by the SOP of the um. In this way we may exercise an all-otical control over the olarization of the signal beam. Note that conventional fiber-otic Raman amlifiers oerate in the regime where the outut SOP of the signal is indeendent on the um SOP. Moreover, most conventional fiberotic Raman amlifiers are driven by unolarized ums. The first Raman olarizer was demonstrated by Martinelli et al. in [5], followed by a number of theoretical aers [6 23]. Similar olarization-sensitive amlification was redicted theoretically and confirmed exerimentally in [24] for the Brillouin amlification of a signal beam in standard otical fibers. These devices can be similarly called Brillouin olarizers. Since they are based on a gain mechanism, which is maximum whenever the signal and um SOPs are aligned and zero when they are orthogonal, in general both Raman and Brillouin-based olarizers suffer from severe outut RIN in the resence of inut signal SOP fluctuations. A common feature uniting all of these nonlinear fiber-otic olarizers is that they can oerate efficiently only in the limit of vanishing olarization mode disersion (PMD). PMD is the effect that is caused by random variations of the magnitude and/or orientation of the birefringence along the fiber length, and it is acquired as a result of inevitable technical imerfectnesses in the rocess of drawing a fiber from a reform. Recent rogress in fiber manufacturing brought to the market fibers with much lower values of PMD than was reviously available. It is this technological breakthrough that made ossible the observation of the reviously discussed olarizationsensitive effects in otical fibers. Theoretical estimates show that the smaller the PMD coefficient, the shorter the total fiber length, and the smaller the frequency searation of the signal and the um beams, the better the erformance of all of the above described olarizers. It is one of the main goals of a theory to be able to redict the degradation rate of useful olarization attraction effects, which is caused by PMD. Such degradation rates for NLPs and Raman olarizers have been calculated analytically in [22,25]. It is imortant to note that the concet of all of these smart olarizers is not limited to fiber-otics alications only. Indeed, nonlinear olarizers can be imlemented with any otical waveguide exhibiting Kerr and/or Raman nonlinearity. Using integrated otics waveguides may lift the roblems that are associated with fiber PMD, and even make nonlinear olarizers less bulky and more comact, roviding that the waveguide material exhibits nonlinear coefficients that are much larger than silica. For examle, the silicon-based Raman olarizer roosed in [26] is free of the PMD-induced degradation and has a centimeter-long size as comared to the kilometer-long fibers, thanks to 3 4 orders of magnitude Raman gain enhancement in silicon with resect to silica. The resent theoretical study extends the concet of nonlinear olarizers to the FWM rocess in telecom fibers. The goal here is to find the conditions uon which the rocess of arametric amlification is sensitive to the SOP of the um beam. In this way we arrive to the notion of an FWM olarizer, meaning that the SOP of the amlified signal beam is determined by the SOP of the fully olarized um beam. We derive here the couled wave equations for the um, idler, and signal beams. In the limit of zero PMD, these equations reduce to the equations that were reviously derived by McKinstrie et al. in [27] for describing degenerate FWM in standard fibers. The major advantage of our theory is its alicability (slightly) beyond the zero-pmd limit, in the sense that it is caable of redicting the degradation rate of the efficiency of the FWM olarizer for low-pmd fibers as well. Knowing this degradation rate allows one to roerly design ractical fiber-based nonlinear olarizers. The resent work substantially extends to the case of random birefringence telecom fibers a revious study of olarization attraction in deterministic, highbirefringence otical fibers [28]. Note that the olarizationsensitive arametric amlification in otical fibers was studied theoretically by Lin and Agrawal [29,3], and also theoretically and exerimentally by Freitas et al. in [3], and resulted in a roosal of a fiber-based olarization switch. As discussed in [28], FWM-based olarizers are based on the olarization sensitivity of arametric gain. Such gain is maximum for a signal SOP that is aligned with that of the um, and zero for a signal SOP orthogonal to the um. Thus FWM olarizers are not immune from outut RIN resulting from inut signal olarization fluctuations. Nevertheless, since arametric gain is generally larger than Raman gain in silica fibers, FWM olarizers may emloy shorter fibers or lower um owers than Raman olarizers. In addition, since the reolarization caability of FWM olarizers is based on arametric gain, these devices rovide a more flexible control over the gain and reolarization bandwidth. Indeed, such bandwidth may be extended u to 7 nm and even include the normal disersion regime by roerly engineering the wavelength deendence of the fiber disersion and by adjusting the um ower [32]. 2. EQUATIONS OF THE MODEL We shall consider the rocess of degenerate FWM. This rocess involves three continuous waves with frequencies that satisfy the matching condition 2ω ω s ω i. Pum, signal, and idler waves are labeled corresondingly as, s, i. All three waves are coroagating along the z direction in a telecommunication (i.e., randomly birefringent) fiber. The vectorial theory of arametric amlification in fibers was develoed in [27,29,3], basing on the tensorial roerties of silica in the telecom band. The starting equation is derived under standard for nonlinear otics aroximations, from Maxwell s equation with a olarization that takes into account the nonlinear cubic resonse of silica and the birefringence of the fiber. Utilizing the Jones reresentation, the equations for the Jones vectors of the um and the signal read as i U z ΔBω ;zu 2 h 3 γ U U U i 2 U U U 2 3 γu s U s U U s U U s U U s U s U i U i U U i U U i U U i U i 2 3 γ exiδkzu i U s U U i U U s U s U U i ; ()

3 272 J. Ot. Soc. Am. B / Vol. 29, No. / October 22 Guasoni et al. i U s z ΔBω s;zu s 2 h 3 γ U s U s U s i 2 U s U s U s 2 3 γu U U s U U s U U s U U U i U i U s U i U s U i U s U i U i 2 3 γ ex iδkz 2 U U U i U U i U ; (2) while the idler equation is obtained from Eq. (2) by exchanging labels s and i. The Jones vectors U f u xf ;u yf T (with f f; s; ig) are two-comonent vectors with u xf z and u yf z being the amlitudes of the olarization comonents in a fixed laboratory reference frame x; y. Note that the last terms in the left-hand sides of Eqs. () and (2) reresent the so-called energyexchange terms. They are resonsible for the transfer of energy between different waves, and as such are most imortant for our analysis of arametric amlification. The wave vector mismatch Δk β ei β es 2β e β oi β os 2β o, where β ef and β of are the roagation constants of the modes aligned with extraordinary (e), or slow, and ordinary (o), or fast, axes at frequency ω f, resectively. The 2 2 matrix ΔBω f reference frame, which is defined by the olarization modes e x and e y. σ 3 diag; and σ adiag; are known as Pauli matrices, where diag and adiag stand for diagonal and antidiagonal matrices, resectively. The orientation angle θ is randomly varying in fibers used for telecommunication alications, which exlains the term randomly birefringent fibers that is alied to them. In rincile, the magnitude of the birefringence bω f also varies stochastically along z. However, as noticed in [33], the two aroaches, one in which θ is the only stochastic variable, and the second, where both θ and b are stochastic variables, roduce nearly identical results. Thus, here we shall develo our theory by assuming the single stochastic variable θ. The angle θ is driven by a white noise rocess z θ g θ z, where hg θ zi and hg θ zg θ z i 2L c δz z. Here L c is the correlation length that characterizes the tyical distance over which θ changes randomly. The theory develoed below is the natural extension of the one-beam theory of Wai and Menyuk in [33] and the two-beam theory of Kozlov et al. [8] to the case of three interacting beams. All details of the derivations of the final equations of motion with deterministic coefficients starting from Eqs. () and (2) with stochastic coefficients, as well as the aroximations that aeared on the way, can be found in Aendices A and B. Here we write down the final result: i ϕ z C az ϕ i ϕ ϕ i 8 9 ϕ2 ϕ 2 2C 3 a z 2 3 C bz ϕ i ϕ s ϕ eiδkz C az ϕ ϕ s ϕ s C az ϕ ϕ 2i ϕ 2i 8 9 C azϕ i ϕ 2 ϕ 2i 2 C 3 a z 3 C bz ϕ s ϕ 2i ϕ 2 eiδkz 8 9 ϕ ϕ 2 ϕ 2 C a z 3 C bz ϕ i ϕ 2s ϕ 2 eiδkz 8 9 C azϕ s ϕ 2 ϕ 2s C az ϕ ϕ 2s ϕ 2s ; (3) i ϕ s z C a z 3 C bz ϕ 2 ϕ i e iδkz C az ϕ i ϕ s ϕ i C az ϕ ϕ s ϕ 8 9 ϕ2 s ϕ s C az ϕ s ϕ 2i ϕ 2i 2 C 3 a z 3 C bz ϕ ϕ 2 ϕ 2i e iδkz 8 9 C azϕ i ϕ 2s ϕ 2i C az ϕ s ϕ 2 ϕ C azϕ ϕ 2s ϕ ϕ sϕ 2s ϕ 2s ; (4) i ϕ 2s z C a z 3 C bz ϕ 2 2 ϕ 2i e iδkz C az ϕ 2i ϕ 2s ϕ 2i C az ϕ 2 ϕ 2s ϕ ϕ2 2s ϕ 2s C az ϕ 2s ϕ i ϕ i 2 C 3 a z 3 C bz ϕ 2 ϕ ϕ i e iδkz 8 9 C azϕ 2i ϕ s ϕ i C az ϕ 2s ϕ ϕ 8 9 C azϕ 2 ϕ s ϕ 8 9 ϕ 2sϕ s ϕ s. (5) bω f cos θσ 3 sin θσ reresents the birefringence tensor, where bω f 2 β ef β of is half of the value of the fiber birefringence at frequency ω f. Moreover, θ is the angle of orientation of the axis of the birefringence with resect to the fixed Here C a z ex 8 3Δ 2 L c z and C b z ex 3 Δ 2 L c z, where Δ bω bω i. The equation for ϕ i is obtained from Eq. (4) by exchanging the labels i and s; the equations for ϕ 2 and ϕ 2i are obtained from the equations for

4 Guasoni et al. Vol. 29, No. / October 22 / J. Ot. Soc. Am. B 273 ϕ and ϕ i, resectively, by exchanging the labels and 2. The olarization comonents ϕ f and ϕ 2f are obtained from the original comonents u xf and u yf by means of a unitary transformation of the reference frame; see Aendices A and B for details. When the value of Δ is zero, the z-deendent coefficients C a z and C b z are both equal to unity, and we restore the model equations derived in [27] starting from the Manakov equation. This limit corresonds to vanishing PMD, and it is quite natural to call it the Manakov limit. In this limit the conversion of the um energy into the signal is maximally efficient, and therefore arametric amlifiers should be designed in such a way that the PMD diffusion length L d 8 3Δ 2 L c is much longer than the fiber length L. To the best of our knowledge, our theory for the first time analytically redicts the length scale of degradation of the rocess of arameteric amlification in telecom fibers. Strictly seaking, our theory is valid in two limits: L L d and L L d. In the oosite limit (the limit of large PMD, which we call the diffusion limit) where L L d, the FWM rocess is totally suressed. Therefore this regime is not interesting from the viewoint of frequency conversion. Most likely, the intermediate case of L L d can be adequately treated only numerically; however, we believe that the exonential decay of the nonlinear coefficients rovides a qualitatively correct descrition of the rate of degradation. Note that the PMD diffusion length L d also enters the theory of two-beam nonlinear interactions: it was introduced by Lin and Agrawal in [25] in the context of fiber-otic Raman amlifiers, and it was identified as the tyical length at which the mutual orientation of the states of olarization of the um and signal beams is scrambled as a result of PMD. It is quite remarkable that the same length scale not only characterizes the olarization memory of Raman interactions, but also the degradation of XPolM mediated Kerr interactions of the two beams, as shown in [22]. In order to overcome the PMDinduced degradation of FWM efficiency, an exerimentalist needs to select a low-pmd highly nonlinear fiber. As the degradation rate deends quadratically on the frequency difference between signal and um, the effect of PMD can be also viewed as setting an uer limit to the bandwidth of the simultaneous arametric amlification and reolarization rocess. In order to bring the definition of the PMD diffusion length to a more standard form, we introduce the PMD coefficient as roosed by Wai and Menyuk in [33]: D 2 2 π L c : (6) L B ω Here L B ω 2π β e β o is the beat length at the um frequency. Then, we can write L d 3 D Δω 2, where Δω ω ω i. Note that Eqs. (3) (5) are alicable in the undeleted as well as um-deleted regimes. In this aer we are aiming at the roof-of-rincile demonstration of the FWM olarizer, and therefore limit ourselves to the study of the undeletedum regime only. 3. POLARIZATION ATTRACTION: ANALYTICAL RESULTS In this section, we shall aly Eqs. (3) (5) to describe the effect of olarization attraction of a signal or idler wave towards the SOP of a coroagating um beam by means of FWM in a randomly birefringent telecom fiber. We will limit our analysis here to the small-signal case; i.e., we make the undeletedum aroximation. As we shall see, this aroximation ermits us to obtain relatively simle analytical results for the effective bandwidth and gain of the olarization attraction rocess. From Eq. (3) one obtains the two olarization comonents of the um amlitude as ϕ P exiθ iγ8 9P tot z; ϕ 2 Q exiθ2 iγ8 9P tot z; (7) where P jϕ j 2, Q jϕ 2 j 2, P exθ, and Q ex θ 2 are the inut um amlitudes in the fiber and P tot P Q is the conserved total um ower. For vanishing PMD (Manakov limit) Δ, C a z C b z ; eak sideband gain is obtained at a frequency detuning Δω Δω m such that the disersive mismatch is comensated by the uminduced nonlinear hase shift, i.e., β 2 Δω 2 m γ6 9P tot. In the absence of higher-order disersion, this condition can only be reached in the anomalous disersion regime (i.e., with β 2 ). In the oosite case of large PMD (diffusion limit), one has C a z C b z, so that the FWM terms are effectively suressed. Yet, aroaching the diffusion limit is equivalent to reducing the effective um ower to zero, which corresondingly leads to eak gain for sideband detunings Δω. Let us consider now the intermediate case of L L d, where eak gain is observed for sideband detunings Δω such that Δω Δω m. In order to quantify the sideband gain and evaluate their SOP relative to the um, we need to solve Eqs. (3) (5). Let us aly the change of variables ϕ ;2i;s ~ϕ ;2i;s ex ivz 2 iθ ;2, where v β 2 Δω 2 6 9γP tot. By linearizing Eqs. (4) and 5) for the sidebands, one obtains where ~ϕ ~ϕ i ; ~ϕ s; ~ϕ 2i ; ~ϕ 2s T, and ~ϕ z i 8 9 Mz ~ϕ ; (8) 2 F A zp F B zq v 2 F C zp F D z PQ F Mz 6 B zp F A zp F B zq v 2 F C z PQ 4 F D z PQ F C z PQ F B zp F A zq v 2 F C z PQ F D z PQ F C zq 3 F C z PQ F D z PQ 7 F C zq 5 F B zp F A zq v 2

5 274 J. Ot. Soc. Am. B / Vol. 29, No. / October 22 Guasoni et al. with F A z 4γ 3 C a z 3, F B z 4γ 3 C a z 3, F C z 2γ 3C a zc b z 3; F D z 8γC a z 9. The solution of Eqs. (8) may be written as ~ϕ z L exωl ~ϕ z ; where Ωz is constructed from Mz as a Magnus series exansion [34]. Whenever the z-deendent coefficients F A;B;C;D z are slowly varying over L (i.e., L d L), we may truncate the exansion after the first term Z L ΩL Ω L Mzdz M (9) z so that we simly relace F A;B;C;D z with their average values F A;B;C;D L Z L z F A;B;C;D zdz; () which can be analytically calculated since C a k a ex k a L and Cb k b ex k bl, where k a 8 3Δ 2 L c and k b 3Δ 2 L c. In the anomalous disersion regime and for sideband frequency detunings Δω below a certain cutoff value Δω c, M has an eigenvalue with ositive imaginary art, leading to the effective (or average) sideband gain coefficient g e g 2 e 4 9 γ2 P 2 tot 4 5 C 2 a C 2 b 8 C a 6 C a Cb 4 β2 2 Δω4 4 9 β 2 C a γp tot Δω 2. () From Eq. (), we obtain the cutoff frequency Δω c of the gain band, the eak frequency detuning Δω, and the effective gain g e;eak as Δω 2 c 4c 27jβ 2 jc 2 L c Δω Δω c 2 ; g 2 e;eak 8 3 γ 2 P 2 tot3jβ 2 jc 2 L c 27β 2 c 2 L c 6γL c P tot 4D 2 γp tot c 2 L ; c L D 2 γp tot Lc 2 L c 4D 2 γp tot c 2 L ; (2) c L where c is the seed of light in vacuum. Let us briefly discuss the role of the different hysical arameters in determining the sideband gain and its bandwidth. First of all, increasing the PMD coefficient D or the fiber length L reduces the eak gain coefficient as well as the otimal sideband detuning. In order to study the olarization roerties of the sidebands, we need to consider the eigenvectors of M. For any frequency detuning Δω within the gain band, let us denote by the eigenvector of M, which grows as exg e z. After a relatively short distance into the fiber, we may well aroximate the sideband fields as ~ϕ C exg e z, where C is the rojection or scalar roduct (which we suose nonzero for simlicity) of the inut sidebands olarization vector ~ϕ z on. The comonents of are such that P Q. Idler amlitudes ϕ~ i and ϕ~ 2i corresond to the first and third comonents of ~ϕ, resectively. Thus their ratio can be exressed as ϕ~ i ϕ~ 2i 3 P Q. Since ϕ i ϕ 2i ϕ~ i ϕ~ 2i e iθ P iθ Q, we obtain that ϕ i P e iθ ϕ iθ 2 z ϕ 2i Q ϕ 2 z. (3) A similar treatment can be develoed for the signal amlitudes too, which roves the olarization attraction of both the signal and the idler to the inut olarization of the um. In ractice, since for a given sideband frequency detuning the effective gain coefficient g e decreases as the fiber length L grows larger, the corresonding strength of olarization attraction will be reduced whenever the fiber length aroaches L d. As a matter of fact, in the diffusion limit g e and FWM-induced olarization attraction is no longer observed. In the next section, we will rovide a quantitative descrition of the fiber length deendence of the olarization attraction efficiency. 4. POLARIZATION ATTRACTION: EXAMPLES Let us study the efficiency of olarization attraction as a function of fiber length L, hence of PMD. Consider a fiber with the nonlinear coefficient γ.9 W km and disersion β 2.5 s 2 km at the wavelength λ 55 nm; the PMD correlation length is set to L c m. The chosen arameters are tyical for highly nonlinear-otical fibers. As is well known, the SOP of each interacting wave may be reresented by means of its corresonding unitary dimensionless Stokes vector as S j S j S j ϕ j ϕ 2j ϕ j ϕ 2j, S 2j S j iϕ j ϕ 2j iϕ j ϕ 2j, S 3j S j jϕ jj 2 jϕ 2j j 2, j i; ; s, where S j jϕ j j 2 jϕ 2j j 2 2. In this notation the rincial SOPs [,, ], [,, ], and [,, ] reresent a linear olarization at 45, a right-handed circular olarization, and a linear olarization at from the local birefringence axes, resectively; the SOPs ; ;, ; ;, and ; ; reresent a linear olarization at 45, a lefthanded circular olarization, and a linear olarization at 9, resectively. The inut CW um beam ower is set to P tot S W, and its SOP is defined by the Stokes vector S.5 ;.4 ;.. We set the inut signal ower to Ps;in mw, whereas the idler is zero at the fiber inut, as in tyical FWM exeriments. We comared the numerical solution of Eq. (8) with the analytical solution of Eq. (9). As the initial condition we emloyed a set of inut signal SOPs, whose corresonding Stokes vectors are uniformly distributed over the Poincaré shere. Figures and 2 illustrate the deendence on sideband detuning of the signal gain g s and its outut degree of olarization (DOP), resectively, for four different values of D (namely, D ;.5 s km 2,.75 s km 2, and D 5 s km 2 ), and the fiber length L 3 m. The signal gain was comuted as g s 2L logp s;out P s;in, where P s;out is the outut signal ower. The outut DOP was calculated as discussed in [35]. In Figs. and 2, the curves refer to numerical solutions, and the dots to analytical solutions: as can be seen, the first-order term of the Magnus exansion rovides an excellent aroximation of the exact solution. Figure shows that, as the PMD grows larger, the signal gain g s is rogressively degraded; at the same time, both the eak gain frequency detuning Δω and the cutoff frequency Δω c shrink towards zero. In addition, Fig. 2 shows that the signal DOP is maximum for sideband frequencies close to eak gain values; however, the eak DOP raidly dros from

6 Guasoni et al. Vol. 29, No. / October 22 / J. Ot. Soc. Am. B g s [Km ] 4 3 DOP ν [THz] Fig.. Deendence of signal gain g s on its frequency detuning from the um, with L 3 m. Curves and circles were obtained with z- varying or average M coefficients, resectively. Moreover D (solid curve), D.5 s km 2 (dashed curve), D.75 s km 2 (dotted curve), and D 5 s km 2 (dash-dotted curve). unity as the PMD strength is increased (i.e., for D.5 s km 2 ). It is interesting to oint out that, in contrast with the case of the signal, the outut DOP of the idler (not shown here) remains close to unity throughout the entire gain bandwidth. The increased attraction of the idler towards the um is due to the fact that the idler grows from zero at the fiber inut; hence its rojection on the growing eigenvector is much larger than for the signal. In the second examle of Fig. 3, we show the signal DOP as a function of the fiber length L, for four different values of the sideband frequency detuning Δν Δω 2π (i.e., Δν.255 THz,.35 THz,.365 THz,.38 THz); here the PMD value is ket fixed to D.75 s km 2. As can be seen, for Δν.255 THz (which corresonds to the eak gain value), the DOP is monotonically increasing with distance, and it aroaches the unit value for L 5 m; in this case, L d 2. km. On the other hand, Fig. 3 shows that for other values of the sideband detuning, the outut DOP exhibits a damed oscillating behavior and it converges to relatively low values after fiber lengths of the order of km. Corresondingly, L d decreases from.2 km down to km. Indeed, the evolution of the signal olarization as described by Eqs. (3) (5) in the small-signal limit is determined by two distinct hysical effects: namely, arametric gain, which ulls the signal olarization towards the um, and um-induced nonlinear Kerr birefringence, which turns the signal SOP around the reresentative oint of the um SOP on the Poincaré shere L [m] Fig. 3. Signal DOP versus fiber length L with D.75 s km 2, and different values of the sideband detuning frequency: Δν.255 THz (solid curve), Δν.35 THz (dashed curve), Δν.365 THz (dotted curve), and Δν.38 THz (dash-dotted curve). (see also the discussion in [3]). Thus, arametric gain and nonlinear birefringence lead to a motion of the signal SOP along two orthogonal directions on the Poincaré shere. Hence, unless the arametric gain is so strong that the signal is immediately attracted towards the um, siral trajectories for the signal SOP may result on the shere, which exlains the DOP oscillations that are observed in Fig DOP ν [THz] Fig. 2. Same as Fig., but for the signal DOP. Fig. 4. (Color online) Tis of inut (a) and outut (b) signal Stokes vectors on the Poincaré shere for a fiber length L 5 m, D.75 s km 2, and Δ ν.255 THz. For the sake of clarity, only 225 vectors are reresented instead of the used in the simulations. Inut vectors are distributed uniformly over the shere. The black triangle reresents the inut um Stokes vector.

7 276 J. Ot. Soc. Am. B / Vol. 29, No. / October 22 Guasoni et al. arametric otical amlification in a standard telecom fiber with randomly varying birefringence. In the FWM olarizer, the SOP of the amlified signal (or idler) beam is attracted to the SOP of the coroagating, fully olarized um wave. We have derived the couled wave equations that describe the roagation of the um, the idler, and the signal in the resence of weak PMD. Our model substantially extends revious theory of FWM in otical fibers, since it may analytically describe the rate of degradation of FWM efficiency and olarization attraction for low-pmd fibers. Knowing the satial rate of PMD-induced degradation ermits the roer design of ractical nonlinear olarizers based on otical arametric amlification in kilometer-long nonlinear-otical fibers. Polarization attraction and control by arametric amlification in fibers is otentially alicable to frequency-conversion and hase-sensitive amlification devices when combined with olarization-sensitive otical rocessing devices (e.g., a heterodyne receiver). In addition, codirectional arametric reolarizers based on low-pmd telecom fibers may be used for comensating ultrafast inut signal SOP fluctuations. Although FWM-based olarizers suffer from outut RIN, RIN suression could be obtained when oerating the amlifier in the deleted um regime, as it occurs with Raman olarizers [36]. Fig. 5. (Color online) Distribution of the outut signal Stokes vectors with L 5 m and Δν.35 THz. Panels (a) and (b) dislay oosite views of the Poincaré shere. It is useful to visualize the effectiveness of olarization attraction by means of arametric gain or FWM by lotting on the Poincaré shere the end oints of the Stokes vectors corresonding to either the inut or the outut distributions of signal SOPs, corresonding to the results of Figs. 3. In Fig. 4, we comare the distribution of inut signal SOPs, which uniformly covers the shere [Fig. 4(a)], to the outut signal SOP distribution [Fig. 4(b)] from a fiber of length L 5 m with D.75 s km 2 ; here the signal detuning is Δν.255 THz. These arameters corresond to the sideband detuning for eak signal gain (see the dotted curve in Fig. 2). As shown by the solid curve in Fig. 3, the outut DOP is as high as.97, which means a nearly full attraction towards the inut um Stokes vector S. On the other hand, in Fig. 5, we show the outut distribution of signal SOPs when the sideband detuning is increased u to Δν.35 THz. Figure 3 shows that the outut DOP is only.73 in this case, which results in a relatively oor olarization attraction. It is imortant to oint out that the olarization attraction (to the um SOP) behavior that is illustrated in Figs. 4 and 5 does not deend uon the secific inut um SOP that is selected; indeed, the strength of olarization attraction only deends on the um ower level. 5. CONCLUSIONS In our study we roosed and analyzed a novel tye of nonlinear olarizer, exloiting the degenerate FWM rocess or APPENDIX A: STOCHASTIC THEORY OF PARAMETRIC AMPLIFICATION Our goal is to convert the initial equations for the field () and (2) with stochastic coefficients into corresonding equations with deterministic coefficients. In other words, we need to find a way to average the initial equations over the ensemble of fibers, which reresents all ossible realizations of the random fiber birefringence with a given statistics. Since both initial and final equations are nonlinear, our rocedure cannot be done exactly and it will require a number of aroximations. Thus, the final equations will have a limited range of alicability. We use the aroach first introduced into the fiber-otics theory by Wai and Menyuk in [33]. This aroach was formulated for a single beam (or ulse), and lead to the derivation of the celebrated Manakov equation and its generalization in the form of the Manakov-PMD equation. An extension of this theory for the two-beam configuration was undertaken in [8,7] and led to the formulation of the theoretical basis of XPolMinduced olarization attraction effect in telecom fibers and of Raman olarizers. Here, we need to extend this theory even further by fully taking into account the three interacting beams. Given that all these theories have very much in common, we shall omit many reetitions and where aroriate we simly refer to rior literature for more details. We start with the transformation of field vectors from the laboratory x; y frame into the local reference frame (, 2), which is defined by the z-deendent orientation of the axis of birefringence: Ψ f MzU f, where Mz is the 2 2 rotation matrix defined in Eq. (4) of [8]. Here Ψ f ψ f ; ψ 2f. All terms excet one in the field equations stay immune to this transformation. The only change is the form of the birefringence matrix, which now becomes bωf ΔBω f i 2 g θ i 2 g θ : (A) bω f

8 Guasoni et al. Vol. 29, No. / October 22 / J. Ot. Soc. Am. B 277 The next transformation: Φ f T f zψ f, is aimed at the decouling of the linear ortions of the field equations. This goal is reached if the transformation matrix a z T z a 2 z obeys the following equation: a 2 z a z i T z ΔB T : (A2) (A3) Matrices T s and T i are defined in a similar way, with b ;2 and c ;2 elements used instead of a ;2. The unitarity of this transformation is reserved by requiring that ja zj 2 ja 2 zj 2 jb zj 2 jb 2 zj 2 jc zj 2 jc 2 zj 2. Initial conditions for the elements of the T f z matrices are to be determined from the requirement that Φ f Ψ f at z. Thus, a b c and a 2 b 2 c 2. The transformation associated with the T f z matrix brings the equations for three fields in the form i Φ f z γn sm N xm N ex f : (A4) As exected, in this reference frame, the fields are couled by nonlinearity only through three tyes of cubic terms: SPolM terms N sm, XPolM terms N xm, and energy-exchange terms N ex. The number of these nonlinear terms is very large, and we do not rovide here their detailed structure. Instead, we refer to Eqs. (9) (2) in [8], where the SPolM and XPolM nonlinear terms are written down exlicitly. In our resent theory, we have all these terms as well, and in addition get energy-exchange terms in the form of cubic roducts involving three different fields. Coefficients rior to these terms are some self- and crossfourth-order olynomials comosed of a ;2 z, b ;2 z, c ;2 z, and their comlex conjugates. It is convenient to work with quadratic coefficients u m and u m (m 3). Coefficients u m with m 4 are identical to those introduced immediately below Eq. (2) in [8]. They are divided into self-terms u ja j 2 ja 2 j 2, u 2 a a 2 a a 2, u 3 ia a 2 a a 2, u 4 2a a 2, u 5 a 2 a2 2, u 6 ia 2 a2 2 and crossterms u 7 a b a 2 b 2, u 8 b a 2 b 2 a, u 9 ib a 2 a b 2, u ia b a 2 b 2, u a b 2 b a 2, u 2 a b a 2 b 2, u 3 ia b a 2 b 2, and u 4 ia b 2 a 2 b. The three-beam theory additionally brings 6 new coefficients. Coefficients u m with m 5 22 are the same as u m with m 7 4 but with b ;2 relaced with c ;2. Coefficients u m with m 23 3 are the same as u m with m 7 4,wherea ;2 is relaced with b ;2, and simultaneously b ;2 is relaced with c ;2. Nonlinear coefficients in Eq. (A3) are roducts of the tye u m u n or u m u n. They are z-deendent random coefficients, because they deend on the stochastic variable g θ z. We need to find average values of all nonlinear terms, which are of the form, for instance, u 2 9 ϕ sϕ 2 ϕ 2s. This is the lace where the most imortant aroximation comes into lay. We assume that the following factorization is valid: hu 2 9 ϕ sϕ 2 ϕ 2s i hu 2 9 ihϕ sϕ 2 ϕ 2si. This factorization is justified whenever the satial evolution of the fields is much slower than the satial evolution of the nonlinear coefficients, or vice versa, whenever the satial evolution of the fields is much faster than the satial evolution of the nonlinear coefficients. In the context of arametric amlification and in the absence of grouvelocity disersion, the nonlinear evolution of the fields scales with the nonlinear length L NL γp, where P is the um ower. In its turn, the z-deendence of hu m u n i or hu m u ni is governed by two different length scales. On the one hand, we have the relatively short satial scales that are associated with the correlation length L c and the beat length L B, both of which are tyically less than m. On the other hand, we have the relatively long satial scale, which is associated with the PMD diffusion length L d. For ractically interesting situations, we need to rovide the following hierarchy of scales: L c, L B L, L NL L d. In this range, the factorization aroximation is well justified; with this limitation in mind, we may roceed further. In order to find averages of the tye u 2 m and ju m j 2,itis convenient to grou coefficients as G fu ;u 2 ;u 3 g, G 2 fu 4 ;u 5 ;u 6 g, G 3 fu 7 ;u 8 ;u 9 ;u g, G 4 fu ;u 2 ;u 3 ;u 4 g, G 5 fu 5 ;u 6 ;u 7 ;u 8 g, G 6 fu 9 ;u 2 ;u 2 ;u 22 g, G 7 fu 23 ;u 24 ;u 25 ; u 26 g, and G 8 fu 27 ;u 28 ;u 29 ;u 3 g. For each grou we were able to formulate a closed system of linear first-order differential equations by using Eq. (A3). For an examle of such a system, we may refer to Eq. (3) in [8]. Next we need to know the average values of quadratic forms comosed by these coefficients. They can be found from the solutions to the equations of motion for the average of the generic function F. For instance, for Fu ;u 2 ;u 3 ; θ,we need to solve the equation z hfi hgfi. The generator G is to be constructed by a rocedure described in the aendix of [33]. For a secific examle of G, we may refer to Eqs. (A) and (A7) in [8]. Note also that the average over different realizations of the fiber birefringence can be relaced by a satial average as hf ilim z z Z z dz f z ; (A5) by assuming that the ergodicity hyothesis is valid. With this rocedure at hand, we are able to find the mean values of u 2 m, with m 9,, 3, 4, 7, 8, 2, 22, 25, 26, 29, 3 by solving the equation z V A M A V A for the vector V A hs 2 i; hs2 2 i; hs2 3 i; hs2 4 i; hs 2S 3 i; hs S 4 i T, where fs ;S 2 ;S 3 ;S 4 g is any of the grous G i with i 4 8, and with the matrix M A given by 2L c 2L c 2Δ 2L c 2Δ 2L c 2Δ 2Δ. (A6) Δ Δ L C c A Δ Δ L c Here Δ bω bω i and Δ bω bω i.itisa straightforward calculation to get an estimate Δ Δ Δω ω, where Δω ω ω i ω s ω. For tyical fiber arameters, the evolution associated with Δ is very fast, while Δ defines a much slower satial scale. Setting Δ to zero defines the Manakov limit, and brings us back to the formulation of couled wave equations with constant in z nonlinear coefficients. The difference of Δ from zero

9 278 J. Ot. Soc. Am. B / Vol. 29, No. / October 22 Guasoni et al. means the inclusion of effects caused by the PMD. In this case we are dealing with z-deendent nonlinear coefficients. Next we calculate the averages of the tye ju m j 2, with m 9,, 3, 4, 7, 8, 2, 22, 25, 26, 29, 3. Thereto we formulate the equation of motion z V B M B V B for the vector V B hjs j 2 i; hjs 2 j 2 i; hjs 3 j 2 i; hjs 4 j 2 i; hs 2 S 3 i; hs 2 S 3i; hs S 4 i; hs S 4i T, where fs ;S 2 ;S 3 ;S 4 g is any of the grous G i, with i 4 8, and where the matrix M B reads as 2L c 2L c Δ Δ 2L c 2L c Δ Δ Δ Δ Δ Δ Δ Δ L c. Δ Δ L c Δ Δ L c C A Δ Δ L c (A7) Note that initial conditions for the averages of the tye hu m zu n zi and hu m zu nzi can be found from the initial conditions for the coefficients a ;2, b ;2, and c ;2, and by observing that hu m u n i u m u n. Thus we find u ;5;7;2;5;2;25;28 and u 6;;3;7;2;26;29 i, while the remaining coefficients are all zero. Next, we turn to cross-terms like hu m u n i with m n. Many of these terms are zero, mainly because of the imosed zero initial conditions. Nonzero coefficients are hu 4 u 22 i, hu 4 u 22 i, hu u 8 i, hu u 8 i, hu 6u 29 i, hu 6 u 29 i, hu 3u 25 i, and hu 3 u 25i. The first four of these coefficients can be found by solving the equation z V B M B V B with the matrix M B defined as in Eq. (A7), and where the vector V B is identified with any of the following vectors: hu u 9 i; hu 2 u 2 i; hu 3 u 2 i; hu 4 u 22 i; hu 3 u 2 i; hu 2 u 2 i; hu 4 u 9 i; hu u 22 i T ; hu u 9 i; hu 2u 2 i; hu 3u 2 i; hu 4u 22 i; hu 3u 2 i; hu 2u 2 i; hu 4 u 9 i; hu u 22 it ; hu 7 u 5 i; hu 8 u 6 i; hu 9 u 7 i; hu u 8 i; hu 9 u 6 i; hu 8 u 7 i; hu u 5 i; hu 7 u 8 i T ; hu 7 u 5 i; hu 8u 6 i; hu 9u 7 i; hu u 8 i; hu 9u 6 i; hu 8u 7 i; hu u 5 i; hu 7u 8 it. In turn, the coefficients hu 6 u 29 i, hu 6 u 29 i, hu 3u 25 i, and hu 3 u 25 i can be found from the equation zv C M C V C with the matrix M C defined as Δ Δ Δ L c Δ B Δ L c Δ C Δ Δ 2L c 2L c 2L c 2L c Δ L c C A A8 when we associate the vector V C with any of the following vectors: hu 6 u 29 i; hu 5 u 29 i; hu 6 u 28 i; hu 5 u 28 i; hu 4 u 27 i; hu 4 u 3 i T ; hu 6 u 29 i; hu 5u 29 i; hu 6u 28 i; hu 5u 28 i; hu 4u 27 i; hu 4u 3 it ; hu 3 u 25 i; hu 3 u 24 i; hu 2 u 25 i; hu 2 u 24 i; hu u 23 i; hu u 26 i T ; hu 3 u 25 i; hu 3u 24 i; hu 2u 25 i; hu 2u 24 i; hu u 23 i; hu u 26 it : APPENDIX B: ANALYTIC ESTIMATION OF THE NONLINEAR COEFFICIENTS In this aendix, we look for aroximate analytical solutions to the linear systems of equations for the vectors V A, V B, and V C. This task is equivalent to finding the eigenvalues and eigenvectors of the matrices M A;B;C. Additionally, we need to find the decomosition of the initial vectors V A;B;C in the basis of the corresonding eigenvectors. In this way, we may determine the z-deendence of the nonlinear coefficients. We shall give a detailed analysis for the M A matrix, and sketch only briefly the results for the other matrices. We develo a erturbative aroach, by assuming that Δ is much smaller than Δ and L c. First, setting Δ to zero, we get a much simler matrix M ~ A. The difference ΔM A M A M ~ A is therefore a small correction. The matrix M ~ A has a doubly degenerate eigenvalue ~λ A and two corresonding eigenvectors ~e A ; ; ; ; ; T and ~e A2 ; ; ; ; ; T. The other eigenvalues of M ~ A all have relatively large negative real arts, in the sense that the corresonding eigenvectors vanish with distance very quickly. The satial scale of this decay is determined by the correlation length L c and the beat length L B, both of which are tyically less than m. So, the characteristic decay rate is estimated as L transient m. After the transient decay is over, we can write the solution of z V ~ A M ~ A V ~ A as V ~ A z C ~ ~e A C ~ 2 ~e A2 ex~λ A z C ~ ~e A C ~ 2 ~e A2, where C~ V A ~e A ~e A ~e A and C ~ 2 V A ~e A2 ~e A2 ~e A2, thanks to the orthogonality of the set of eigenvectors of M ~ A. When Δ is different from zero, the degeneracy is lifted and the doubly degenerate eigenvalue ~λ A slit into two different eigenvalues λ A and λ A2. Let us find λ A2 by way of develoing the erturbative analysis. First we find the eigenvalue equation for the exact M A matrix. It is detm A λi, where I is the unity matrix and detm A λi 32Δ 2 Δ 2 L c λ 6Δ 2 Δ 2 2Δ 2 L 2 c 2Δ 2 L 2 c λ 2 6Δ 2 L c 6Δ 2 L c 4L 3 c λ 3 4Δ 2 4Δ 2 9L 2 c λ 4 6L c λ 5 λ 6. Since Δ is small, we exect that the correction to the unerturbed zero eigenvalue ~λ A is also small. By keeing in the eigenvalue equation terms no higher than second order in λ, we get after some simlifications the aroximated solution λ A2 8 3Δ 2 L c L d. The eigenvector corresonding to eigenvalue λ A (λ A2 )ise A ~e A ~e A2 (e A2 ). The erturbed solution is V A z C e A exλ A zc 2 e A2 exλ A2 z C e A C 2 e A2 ex z L d. Here C V A e A e A e A. The exact exression for C 2 e A2 is cumbersome; however, under the condition Δ Δ ;L c, we can use the equality

10 Guasoni et al. Vol. 29, No. / October 22 / J. Ot. Soc. Am. B 279 of V ~ A and V A in the limit of Δ, and write C 2 e A2 ~C ~e A C ~ 2 ~e A2 C e A. Now we can turn to evaluation of the nonlinear coefficients. Let us start with the averages hu 2 9 i and hu2 i. Coefficients u 9 and u belong to the grou of coefficients denoted earlier as G 3. For this grou, the vector V A contains hu 2 9 i and hu2 i as the third and the fourth element, resectively. The initial condition reads as V A ; ; ; ; ;i T. Thus, for L L transient,we find hu 2 9 i 3ex z L d and hu 2 i ex z L d. Similarly, we find hu 2 7 ihu2 25 ihu2 9 i and hu2 8 ihu2 26 i hu 2 i. With initial conditions V A ; ; ; ; i; T, we get hu 2 3 ihu2 4 ihu2 2 ihu2 2 ihu2 29 ihu2 3 i. The matrix M B can be considered similarly. Again, in the limit Δ, this matrix ossesses a doubly degenerate zero eigenvalue ~λ B, while the other eigenvalues have large negative real arts, so that the corresonding eigenvectors vanish after a certain roagating distance, say L transient. Whenever Δ is different from zero, the degeneracy is lifted and the doubly degenerate eigenvalue ~λ B is slit into λ B and λ B2 L d. Thus, for L L transient, we find hju 9 j 2 ihju 7 j 2 ihju 2 25ji 2 6ex z L d, hju j 2 ihju 2 8 ji hju 26j 2 i 2 2ex z L d, hju 3 j 2 ihju 2 j 2 ihju 2 29ji 2 6 ex z L d,andhju 4 j 2 ihju 2 22 ji hju 3j 2 i 2 2 ex z L d. By using the same eigenvectors and eigenvalues of matrix M B, we find also that hu 4 u 22 i, hu 4 u 22 i 2 2 ex z L d, hu u 8 i ex z L d, and hu u 8 i 2 2ex z L d. Finally, matrix M C ossesses a nondegenerate eigenvalue ~λ C in the limit Δ, with the other eigenvalues vanishing for z L transient. The erturbative aroach yields the correction to the zero eigenvalue: λ C 8L d. 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