Brown, C S

Ishiekwene, G C

Mensah, S Y

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Available at: http://www.ictp.it/~pub−off IC/2004/83United Nations Educational Scientific and Cultural OrganizationandInternational Atomic Energy AgencyTHE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSKERR ELLIPTICITY EFFECT IN A BIREFRINGENTOPTICAL FIBERG.C. Ishiekwene1Institute of Research, University of Liberia, Monrovia, Liberia,Department of Physics, University of Cape Coast, Cape Coast, GhanaandThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy,S.Y. Mensah2Department of Physics, University of Cape Coast, Cape Coast, GhanaandC.S. BrownDepartment of Physics, Morehouse College, Atlanta, Georgia.AbstractAn intensity-dependent change in the ellipticity of an input light beam leads to a character-istic shift in polarization instability. Dichroism gives rise to a self-induced ellipticity effect in thepolarization state of an intense input light oriented along the fast axis of a birefringent opticalfiber. The critical power at which the fiber effective beat length becomes infinite is reducedconsiderably in the presence of dichroism.MIRAMARE – TRIESTESeptember 20041Regular Associate of ICTP.2Corresponding author. profsymensah@yahoo.co.uk1 IntroductionWhen two or more optical waves copropagate inside a birefringent single-mode fiber they candiffer not only in their wavelengths but also in their states of polarization. The optical pulsescan further couple with each other through fiber nonlinearity and the polarization of each fieldcan change during propagation as a result of optically induced nonlinear birefringence. Thecoupling of two waves with the same frequency but different polarizations gives rise to a numberof interesting effects in optical fibers. One such effect is the polarization instability whichdevelops when the nonlinear change in refractive index grows to be comparable with linearbirefringence. This instability manifests as large changes in the output state of polarizationwhen the input power or the input polarization state is changed slightly. Then, at a criticalvalue of input power, nonlinear birefringence can cancel intrinsic birefringence completely andthe effective beat length becomes infinite. It has been discovered and demonstrated that inducednonlinear birefringence is responsible for polarization instability [1-5].As early as 1964, it was observed that an intense elliptically polarized beam propagatingwithin an isotropic medium induces a nonlinear birefringence effect known as ellipse rotationwhich is a continuous precession of the orientation angle of the polarization ellipse, while leavingits shape and handedness unchanged assuming no dichroism [6]. Since then, self-induced polar-ization effects have been studied extensively, particularly in the context of optical kerr effect ornonlinear refractive index and remains a topic of much interest given its practical and importantconsequences for laser propagation characteristics as well as its use as the basis for a wide vari-ety of applications [7]. These nonlinear polarization effects in a birefringent single-mode opticalfiber have led to practical applications through intensity discriminators, fiber-optic gates andkerr shutters [8]. Also, a new class of devices, among which a linear coherent amplifier mixer andan optically activated polarization switch, have been envisaged due to the intensity-dependentchanges in the light polarization as it evolves along a lossless birefringent single mode opticalfiber [9]. The problem of the interaction between a nonlinear and a dc-induced birefringencehas been analytically solved by Sala for an isotropic medium [10]. The form of the refractiveindex change induced by the elliptically polarized pump beam corresponds to elliptical birefrin-gence with the important consequence that an arbitrarily polarized probe beam in addition toorientation experiences a change in the shape and handedness of its polarization ellipse. Win-ful presented exact solutions for the intensity-dependent polarization state of a light wave in abirefringent optical fiber taking into account both the intrinsic linear polarization evolution andthe nonlinear ellipse rotation and showed that self-induced polarization changes can occur withequal excitation of the fiber principal axes [11]. While these effects on optical properties andpropagation characteristics in birefringent fibers have long been known and examined in details,the problem of the resulting effects due to an intensity-dependent change in the ellipticity of thepolarization ellipse has received very little attention.2In their work, the authors assumed a lossless fiber thereby neglecting dichroism which isrelated to the imaginary part of the nonlinear susceptibility tensor and is also responsible forellipticity which is acquired by the different absorption rates of the right and left circularlypolarized modes due to the imginary part of the complex refractive index. Contrary to theabove assumption, dichroism is always present in a real fiber. The purpose of this paper is toevaluate the influence of dichroism on the polarization nonlinear evolution after including theeffects of both intrinsic linear birefringence and optically induced nonlinear birefringence. Wefind as was reported in [1] that if dichroism is neglected, the fiber’s effective beat length becomesinfinite when total input power P0 reaches a critical power level Pcr. However, we further observethat in the presence of dichroism, the effective beat length becomes infinite when P0 = 2Pcr.We note that this result is in agreement with the experiments conducted by [4]. Thus, there isa characteristic shift in the critical power value at which polarization instability occurs whendichroism is considered present in the fiber. We also want to state that the shift in polarizationinstability results from an intensity-dependent change in the ellipticity of the input beam. Wereport the presence of Kerr ellipticity whose origin is due mainly to the presence of dichroismin the fiber and manifested as a change in the shape of the polarization ellipse. Kerr ellipticitymodulates the critical input power value for polarization instability. Infact, the critical powerat which polarization instability occurs is decreased considerably in the presence of increaseddichroic effects. This in our opinion can be useful for efficient optical switching.The theoretical method used to calculate and show the polarization state dependence on thenonlinear susceptibility tensor is presented in section 2, while in section 3 we present and discussour main results. Our conclusions are summarized in section 4.2 Theoretical BackgroundThe monochromatic electric fields propagating along a birefringent single-mode optical fiber canbe expressed as a superposition of the two orthogonal polarization of the fundamental modeE =∑Aj (z, t)j=x,yEj (x, y) exp [i (kjz − ωt)] + c.c. (1)where Aj (z, t) are the field amplitudes along the principal axes of the fiber, Ej (x, y) are thetranseverse mode distributions, and ω, t, and kj have the usual interpretation of respectivelyfrequency, time, and propagation constant. The longitudinal coordinate z is chosen to coincidewith the fiber axis of symmetry along which the light wave propagates.The nonlinear wave equation which describes the transverse electric fields evolution along abirefringent optical fiber is expressed asd2Eidz2+(ωc)2ijEj = −4pi(ωc)2PNLi (ω, r) (2)3where P NLi (ω, r) is the spectral amplitude of the nonlinear induced polarization and ij is thedielectric tensor which is diagonal in the principal axis representation.Since the anisotropy of birefringent fibers is relatively weak(δn ' 10−4), the nonlinear po-larization will not differ much from that of an isotropic medium, at least for effects third orderin the electric field [11]. Thus, we can write the nonlinear induced polarization asPNLi (ω, r) = 3χ(3)ijkl (ω;ω, ω,−ω) Ej (ω, r) Ek (ω, r) E∗l (ω, r) (3)where the indices run over (x, y).The evolution equations describing the state of the light polarization can thus be expressedin compact form as [12, 13]dSdz= [ΩL (z) + ΩL (z,S)] × S (4)Here, S =(S1, S2, S3) is the Stokes vector which is associated with the polarization state of thefields and Ω with projections Ω1,Ω2, and Ω3 is a vector in stokes space related to the materialcharacteristics and depend on the Stokes parameters if the medium is nonlinear. In one of thestandard conventions [14], the Stokes parameters are defined asSi = A∗j (σi)jk Ak i = 1, 2, 3 (5)where σi are the Pauli spin matrices.If we assume small anisotropy along the direction of propagation and that the cubic nonlin-earity is isotropic, the nonlinear evolution equations will take the following formdS1dz= −(12piωncχ(3)xxyy)S2S3 (6)dS2dz=ω2nc(xx − zz + 24piχ(3)xxyyS1)S3 (7)dS3dz= −{ω2nc(xx − zz)}S2 (8)The exact analytical solutions to this system of equations have been obtained in terms of ellipticfunctions [10]S3 =2pkfqcn (R0fz + C ; k) (9)S2 =2pkf2q[sn (R0fz + C ; k)dn (R0fz + C ; k)] (10)S1 =f2q{1− 2m[sn2 (R0fz + C ; k)]}− 1 (11)where cn, sn, and dn are Jacobi elliptic functions and R0 = ω (xx − zz) /2nc with f =[(1 + qS10)2− q2S220] 14 . Also, q = 24piχ(3)xxyy/ (xx − zz) and p = sgn (S30). The Jacobimodulus k is given byk =[12+14f2(q2 − 1− f4)] 12(12)4and−Re [K (m)] ≤ C ≤ +Re [K (m)] (13)with [K (m)] denoting the Jacobian quarter-period. Here, the Jacobian parameter m = k2.Clearly, the output polarization state of the optical beam is dependent on the Jacobianmodulus k which in turn depends on the input polarization of the light as well as the nonlinearsusceptibility tensor and the anisotropic tensor.3 Results and DiscussionSince the period of the elliptic function determines the effective beat length of the fiber [1], thebeat length is dependent on both input intensity through the nonlinear susceptibility tensor andinput polarization. The variation of the effective beat length as a function of input power forbeams polarized along the fast and slow axes of the fiber has been obtained [1]. We obtain inFig.(1), as was deduced in [1] that, barring fiber losses, when the input beam is polarized alongthe fast axis of the fiber, the effective beat length becomes infinite at a critical power valuebecause of complete cancellation between the intrinsic and nonlinear birefringences. However,we further observe as is shown also in Fig.(1) that, in the presence of dichroism, the effective beatlength becomes infinite at a different critical power value. Infact, in terms of the normalizedinput power defined as p = P0/Pcr, when dichroism is absent p = 1 and in the presence ofdichroism p = 2. The latter result is in agreement with experiments conducted by [4]. Thus,there is a characteristic shift in the critical power value at which polarization instability occurswhen dichroism is considered present in the fiber. This shift in polarization instability resultsfrom an intensity-dependent change in the ellipticity of the polarization ellipse. The polarizationellipse of the propagating beam experiences an intensity-dependent change in its shape throughthe nonlinear susceptibility tensor due to dichroism. The change in ellipticity (Kerr ellipticity) ofpolarization is due mainly to the presence of dichroism in the fiber and manifested as a change inthe shape of the polarization ellipse. Kerr ellipticity will modulate the critical input power valuefor polarization instability. Infact, the critical power at which polarization instability occurs isreduced considerably in the presence of increased dichroic effects. In our opinion, this can beuseful for efficient optical switching.4 ConclusionsWe have studied the resulting effects of dichroism on polarization instability after includingboth intrinsic linear birefringence and optically induced nonlinear birefringence. It was foundthat the critical power at which polarization instability occured was reduced considerably in thepresence of dichroism. Our results are in agreement with the theory which neglects dichroism [1]and with experiments [4] when dichroic effects are intrinsically present and cannot be discarded.More importantly, we found that the shift in polarization instability is due to Kerr ellipticity5which results from an intensity-dependent change in the ellipticity of the propagating beam whendichroism is present. In our opinion, this can be useful for efficient optical switching. Therefore,eventhough polarization instability is a birefringence related nonlinear effect, we have been ableto stress the role played by the presence of dichroism.5 AcknowledgmentsThis work is supported by a grant from the Swedish International Cooperation DevelopmentAgency within the framework of the Associateship Scheme of the Abdus Salam InternationalCentre for Theoretical Physics, Trieste, Italy.6 References[1] H.G. Winful, Opt. Lett. 11, 33 (1986).[2] B. Daino, G. Grgori, and S. Wabnitz, Opt. Lett. 11, 42 (1986).[3] G. Gregori, and S. Wabnitz, Phys Rev. Lett. 56, 600 (1986).[4] S. Trillo, S. Wabnitz, R.H. Stolen, G. Assanto, C.T. Seaton, and G.I. Stegeman, Appl. Phys.Lett. 49, 1224 (1986).[5] R.E. Slusher, S. Spalter, B.J. Eggleton, S. Pereira, and J.E. Sipe, Opt. Lett. 25, 749 (2000).[6] P.D. Maker, RW. Terhune, and C.M. Savage, Phys. Rev. Lett. 12, 507 (1964).[7] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press (2001).[8] A. Vatarescu, Appl. Phys. Lett. 49, 61, (1986).[9] B. Daino, G. Gregori, and S. Wabnitz, Opt. Lett. 11, 42 (1986).[10] K.L. Sala, Phys. Rev. A 29, 1944 (1984).[11] H.G. Winful, Appl. Phys. Lett. 47, 61, (1985).[12] F. Matera and S. Wabnitz, Opt. Lett. 11, 467 (1986).[13] Y.P. Svirko and N.I. Zheludev, Polarization of Light in Nonlinear Optics, John Wiley andSons (1998).[14] M.V. Tratnik and J.E. Sipe, Phys. Rev. A 35, 2965 (1986).6y q( )x q( )q0 1 2 3 4 5012345INPUT POWER, pEFFECTIVE BEAT LENGTHFigure 1: Effective beat length as a function of input power when dichroism is neglected (solidline) and in the presence of dichroism (dashed line).7

Kerr ellipticity effect in a birefringent optical fiber

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