Bak, A E

Brown, C S

Ishiekwene, G C

Mensah, S Y

CERN Document Server

Available at: http://www.ictp.trieste.it/~pub off IC/2000/86United Nations Educational Scientic and Cultural OrganizationandInternational Atomic Energy AgencyTHE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSTHE USE OF LORENTZ GROUP FORMALISM IN SOLVINGPOLARIZATION EFFECTS OF A BIREFRINGENT SINGLEMODE OPTICAL FIBERG.C. Ishiekwene1Laser and Fiber Optics Center, University of Cape Coast, Cape Coast, GhanaandThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy,C.S. BrownLucent Technologies, Transmission Media Laboratory,2000 Northeast Expressway, Norcross, GA 30071, USA,S.Y. MensahLaser and Fiber Optics Center, University of Cape Coast, Cape Coast, GhanaandA.E. Bak2Department of Physics, Morehouse College,830 Westview Drive SW, Atlanta, GA 30314, USA.MIRAMARE { TRIESTEJuly 20001Regular Associate of the Abdus Salam ICTP.Permanent address: University of Liberia, Monrovia, Liberia. E-mail: ishiekwene@usa.net2Formerly Fuad Muhammad.ext-2000-22201/11/2000AbstractA theoretical analysis on the polarization eects of a light beam propagating in a birefrigentsingle-mode ber is presented. We derive a system of dierential equations representing theevolution of Stokes parameters and illustrate their application to polarization eects in a straightbirefringent single mode optical ber. The solutions to the set of equations are obtained usingspecically the methods of the unied formalism for polarization optics which adopt the use ofthe Stokes-Mueller equation and the Lorentz group to model polarization phenomena in mediasuch as optical bers. The analytical results presented using this approach are identical toresults obtained from other conventional methods. We observe the characteristic exponentialdecrease in the total intensity of the input light due to atteuation by the ber.11 INTRODUCTIONThe evolution of the state of polarization of a light beam propagating along a birefringentsingle mode optical ber can be described using several analytical and graphical methods. Oneapproach involves using a straightforward method of calculating how the electric eld variesas the wave propagates along the ber. In such an approach, an understanding of how thepolarization evolves as the beam propagates is not immediately obvious from the complex slowlyvarying electric eld amplitude [1]. The Jones matrix formalism is another approach that hasbeen used to study the rotational eects of polarization in optical bers [2]. Svirko and Zheludev[3] used tensor analysis to obtain expressions that describe the response of a birefringent berto the propagating beam. Several other authors use an alternative method of the dynamicalequation for Stokes parameters to describe the polarization changes of a wave propagating inan optical medium [4] and [5]. The use of Stokes parameters has an advantage of providing arather straightforward analytical denition of dierent polarization states of light. Daino et al[6] analysed the evolution of the state of polarization along a nonlinear single-mode birefringentber using Stokes parameters and gave exact solutions illustrated by means of a Poincare sphererepresentation.In this work, the polarization state of a lightwave is expressed in terms of Stokes parameters.Using the method of ordered exponential operator [7], a formal solution to the system of dier-ential equations is obtained in terms of the Stokes-Mueller equation which has a Mueller matrixcontaining the information necessary to characterize the ber so that its eect on the state ofpolarization can be deduced and the Stokes parameters of the outgoing and incident light beamsbecome known. The elements of the Mueller matrix for dierent sources of perturbations arethen obtained using the Lorentz group formalism.2 THEORYTo analyze the polarization dynamics of a light beam propagating along a birefringent singlemode optical ber, we begin by deriving a system of four coupled dierential equations repre-senting the evolution of polarization along the optical ber. We use the methods of Svirko andZhedulev [3] to derive the system of coupled dierential equations which represent the evolutionequations for Stokes parameters for the case in which both linear birefringence and dichroismare present in the ber and are parallel to one another in Stokes space2dS0dz=  S0  200cos 2S1  200sin 2S2dS1dz=  S1  200cos 2S0  20sin 2S3dS2dz=  S2  200sin 2S0+ 20cos 2S3(1)dS3dz=  S3  20cos 2S2+ 20sin 2S1Eq.(1) can be expressed in matrix form asddz0BB@S0S1S2S31CCA=0BB@   200cos 2 200cos 2   200sin 2 00 20sin 2 200sin 2 00  20sin 2  20cos 2 20cos 2  1CCA0BB@S0S1S2S31CCA(2) is the isotropic loss of the ber, 0represents anisotropy in the ber, and 00relates to itsdiattenuation eects. fS0; S1; S2; S3; g are the Stokes parameters and z is an axis that coincideswith the axis of the ber along which the elds propagate.  is an angle between the laboratoryand natural coordinate frames of the ber.Solutions to the set of equations can be expressed compactly in terms of the Stokes-Muellermatrix equation in the form)S=cM)S(0) (3)wherecM is the Mueller matrix of the perturbed single mode ber.)Sand)S(0) represent theStokes 4-vector notation for the input and output Stokes parameters, respectively.After expanding in terms of the ordered exponential operators [7], the 4 4 Mueller matrixcM can be explicitly written for uniform birefringence and dichroism as [8]cM (;d;; z) = exp ( z) expd^Dz +* ^Bz(4)where d = fd1;d2;d3g is the dichroism 3-vector related to the polarization dependent losseects of the ber and*=*1;*2;*3is the birefringence 3-vector related to the beranisotropy. We note that d =n200cos 2; 200sin 2; 0oand*=n20cos 2; 20sin 2; 0o.Also,^D=^D1;^D2;^D3and^B=^B1;^B2;^B3are six matrices which form an appropriate setof Lorentz generators for dichroism and birefringence, respectively.The Mueller approach involves the Stokes parameters collectively behaving similar to aLorentz 4-vector. In this case, the Mueller matrices are essentially four-dimensional Lorentz3transformations acting on the Stokes 4-vector [9]. A Lorentz transformation is any matrix thatcan be expressed as [10]L (v;u) = exp [(v K) + (u  J)] (5)where v is a real 3-vector (with cartesian components v1;v2;v3) and u is a real 3-vector (withcartesian components u1;u2;u3). A transformation for which u = 0 is a rotation matrix and atransformation for which v = 0 is a boost matrix. The notations K and J are each shorthand forthree matrices: fKm;Jmm = 1; 2; 3g. These matrices are called generators, and they satisfythe following commutation relations[Jl;Jm] =  3Xn=1"lmnJn[Kl;Km] =3Xn=1"lmnJn(6)[Jl;Km] =  3Xn=1"lmnKnwhere "lmnis component of the three-dimensional totally antisymmetric tensor with ("123= +1).We note that the matries^Dand^Bin Eq.(4) are the generators of the irrep 12;12of theLorentz group such that^D= K(12;12);^B= J(12;12)(7)where the irrep 12;12is the vector representation of the Lorentz group. It is a four-dimensionalreal representation with generatorsJ(12;12)1=0BB@0 0 0 00 0 0 00 0 0 10 0  1 01CCA=^B1; J(12;12)2=0BB@0 0 0 00 0 0 10 0 0 00  1 0 01CCA=^B2J(12;12)3=0BB@0 0 0 00 0 1 00  1 0 00 0 0 01CCA=^B3; K(12;12)1=0BB@0  1 0 0 1 0 0 00 0 0 00 0 0 01CCA=^D1(8)K(12;12)2=0BB@0 0 00 0 0 0 1 0 0 00 0 0 01CCA=^D2; K(12;12)3=0BB@0 0 0  10 0 0 00 0 0 1 1 0 0 01CCA=^D34Thus the Mueller matrix of the perturbed single mode ber is essentially a Lorentz vectortransformation. Then, the Mueller matrix for a purely birefringent single mode ber is a rotationmatrix. That is,cM0; 0;* ; z="^I+e^B2# e^B2cos z +e^Bsinz (9)where  = 20is the magnitude of*, ande=*= 1e1+ 2e2+ 3e3(10)is a unit vector in the direction of the total birefringence.We now writecM0; 0;* ; zmore explicitly ascM0; 0;*; z=0BB@1 0 0 00 cos 2+ sin22 cos z sin 2 cos 2 (1  cos z)   sin 2 sinz0 sin 2 cos 2 (1  cos z) sin22+ cos22 cos z cos 2 sinz0 sin2 sinz   cos 2 sinz cos z1CCA(11)Also, the Mueller matrix for a purely dichroic single mode ber is the product of a scalarexponential factor and a boost matrix. Thus,cM (;d; 0; z) = exp ( z)"^I ed^D2#+ed^D2cosh dz +ed^Dsinhdz (12)where d = 200is the magnitude of d, anded=dd= d1e1+ d2e2+ d3e3(13)is a unit vector in the direction of the total dichroism.^Iis the 4 4 identity matrix. Similarly,Eq.(12) can be written more explicitly ascM (;d; 0; z) =exp ( z)0BB@Ch  cos 2 sinh dz   sin 2 sinhdz 0  cos 2 sinhdz sin22+ cos22 cosh dz sin 2 cos 2 (cosh dz   1) 0  sin 2 sinhdz sin 2 cos 2 (cosh dz   1) cos22+ sin22 cosh dz 00 0 0 11CCA(14)The Mueller matrices for pure birefringence and dichroism are Lorentz transformations andtherefore have specic geometrical interpretations. In the next section, we discuss specic casesof the Stokes-Mueller matrix equation for dierent input polarization of the propagating lightbeam.53 RESULTSThe Mueller matrix which characterizes the perturbations due to the presence of linear bire-fringence and dichroism in the ber can therefore be obtained and expressed as the compositeof the individual matrices due to only birfringence and only dichroism because the individualMueller matrices commute. In other words, the product of equations (11) and (14) yields theMueller matrix for the ber. Hence, the Stokes Mueller equation can be written as0BB@S0S1S2S31CCA= e z0BB@Ch C2Sh S2Sh0 C2ShC22Ch+ S22CC2S2(Ch  C)  S2S S2ShC2S2(Ch  C) S22Ch+ C22CC2S0 S2S C2SC1CCA0BB@S00S10S20S301CCA(15)where Ch= cosh dz, Sh= sinhdz, S2= sin2, C2= cos 2, C= cosz and S= sinz.The elements of the matrix in Eq.(15) can be expressed in functional forms to obtain theoutput Stokes parameters for dierent input polarization of a light beam propagating alonga birefringent single mode ber. Using these functional forms, graphical results showing thevariation in the output Stokes parameters for dierent input polarization of a beam whichpropagates along a birefringent ber having small losses can be obtained. Figure (1) showsthe change in output Stokes parameters for right circularly polarized input light as a functionof the ber length for a specic case in which circular birefringence and dichroism are bothassumed absent in the ber [11]. In this gure, the total intensity of the input light is seen tobe dissipated by the ber. This is due to attenuation by the ber. Also, the output beam isobserved to emerge elliptically polarized.We now discuss specic cases of the Stokes-Mueller relation Eq.(15) and analyze some ofthe resulting polarization eects which emerge from dierent input polarization states of lightpropagating along a birefringent ber. If the ber is assumed to have negligible loss with smallanisotropy along its length, then there is no dichroism and the linear birefringence relates to theanisotropy. In addition, we assume that there are external perturbations due to twists along thelength of the ber so that circular birefringence is present. Again, the corresponding Muellermatrix can be expanded in terms of the exponential of the birefringence vector and Lorentzgenerators to yield the following Stokes-Mueller relation0BB@S0S1S2S31CCA=0BBBBBBBB@1 0 0 00C+C22C22(1  C)S2S+C22S2C2'(1  C) S2C2S+C2C2S2(1  C)0 S2S+C2C22S2(1  C)C+C22S22(1  C)C2C2S+S2C2S2(1  C)0S2C2S+C2C2S2(1  C) C2C2S+S2C2S2(1  C)C+S22(1  C)1CCCCCCCCA0BB@S00S10S20S301CCA(16)where C2= cos 2; C2= cos 2; S2= sin 2; S= sinz and C= cos z.Using Eq.(16), numerical results for the output Stokes parameters for dierent cases of inputlight (initially linearly polarized light, linear +45polarized light, right circularly polarized lightor elliptically polarized light) as a function of ber length can be obtained. Figure (2) illustrates6the variation in the output Stokes parameters as a function of ber length for elliptically polarized(45azimuth and 22.5ellipticity) input light [11]. It is observed that for this input polarizationand with circular birefringence present, the Stokes parameters vary sinusoidally with length.However, it is further observed that when loss eects are neglected and circular birefringence isassumed absent, the Stokes parameters remain constant for any input polarization. Thus, thepresence of circular birefringence induces a variation in the polarization state of the propagatinglight in a lossless ber. It is also observed that the beam emerging from the ber has the sameform as the input polarization for all cases of input polarization regardless of the presence orabsence of circular birefringence. That is, for linearly horizontally polarized input light, theoutput light is observed to be linearly polarized on-axis and similarly for o-axis, circular andelliptical input polarizations.4 CONCLUSIONSIn this paper, we have used the Lorentz group formalism to obtain analytical solutions to theevolution equations for the Stokes parameters. These solutions can be used to numerically obtainthe polarization dynamics of a light beam along a birefringent single mode optical ber subjectedto a bend or twist. To test the validity of the solutions, we applied them to the test case of astraight ber. The basic polarization eects of linear and circular birefringence and dichroismhave been observed and presented. We have noted that when loss eects in a birefringent singlemode optical ber are neglected and circular birefringence is further assumed absent, the Stokesparameters remain constant for any input polarization. In addition, the presence of circularbirefringence induces a variation in the polarization state of the propagating light in a losslessber. As expected, when losses are considered, the total intensity of the input light decaysexponentially. The total intensity is dissipated due to attenuation by the ber.5 ACKNOWLEDGEMENTSThis 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. G. C. Ishiekwene acknowledges support of theInternational Atomic Energy Agency.76 REFERENCES1. M. V. Tratnik and J. E. Sipe, \Nonlinear polarization dynamics I. The single-pulse equations,"Phys. Rev. A, 35 (7), 2965-2975, (1987).2. R. Dandliker, \Rotational eects of polarization in optical bers," in Anisotropic andNonlinear Waveguides, C. G. Somenda and G. Stegeman, (Editors), Elsevier Science Pub-lishers B. V., 39-76, (1992).3. Y. P. Svirko and N. I. Zheludev, Polarization of Light in Nonlinear Optics, JohnWiley and Sons, (1998).4. K. L. Sala, \Nonlinear refractive-index phenomena in isotropic media subjected to a dcelectric eld: Exact solutions," Phys. Rev. A, 29 (4), 1944-1956, (1984).5. G. Gregori and S. Wabnitz, \New eaxct solutions and bifurcations in the spatial distri-bution of polarization in third-order nonlinear optical interactions," Phys. Rev. Lett. 56 (6),600-603, (1986).6. B. Daino, G. Gregori, and S. Wabnitz, \New all-optical devices based on third-ordernonlinearity of birefringent bers, " Opt. Lett. 11, 42-44, (1986).7. R. F. Fox, \Gaussian stochastic processes in physics," Phys. Rep. 48, 179-283, (1978).8. C. S. Brown and A. E. Bak, \Unied formalism for polarization optics with applicationto polarimetry on a twisted optical ber," Opt. Eng. 34 (6), 1625-1635, (1995).9. A. E. Bak, C. S. Brown, G. C. Ishiekwene, M. W. Shute Sr, \A review of the uniedpolarization calculus based on the lorentz group," unpublished.10. A. E. Bak, F. K. A. Allotey, C. S. Brown, G. C. Ishiekwene, \The unied polarizationcalculus applied to analogous dynamical systems" unpublished.11. G. C. Ishiekwene, \Nonlinear polarization eects in a birefringent single modeoptical ber," Ph.D. Thesis, (2000).8

The use of Lorentz group formalism in solving polarization effects of a birefringent single mode optical fiber

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