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  1. 1. J. Phys. D: Appl. Phys. 32 (1999) 1618–1625. Printed in the UK PII: S0022-3727(99)01288-7 Measurement of optical fibre parameters using an optical polarimeter and Stokes–Mueller formalismP Olivard, P Y Gerligand, B Le Jeune, J Cariou and J LotrianLaboratoire de Spectrom´ trie et Optique Laser, Universit´ de Bretagne Occidentale, 6, e eAvenue Le Gorgeu BP 809, 29285 Brest Cedex, FranceE-mail: Pascal.Olivard@univ-brest.frReceived 26 January 1999Abstract. Although the method based on the Mueller matrix for the experimentaldetermination of optical-device polarization behaviour is a powerful tool, it has rarely beenapplied to optical fibre. This paper introduces an experimental methodology for measuringthe Mueller matrices of monomode fibre under uniform strains. Using a theoretical modelderived from coupled-mode equations, we were able to first estimate the physical parametersof the fibre, then to use them to both test the model validity and assess their influence on thepolarimetric behaviour of the fibre.1. Introduction 2. Theoretical foundationsThe polarization characteristics of optical fibre are of In the ideal case of an isotropic monomode optical fibre twoparticular interest in systems like optical sensors or degenerate modes can propagate. This degeneracy indicatesinterferometers using fibre. Although the Mueller matrix- that their propagation constants are identical, and so all thebased technique has rarely been used to describe polarization states of polarization launched at the input face of the guidephenomena in monomode optical fibres, Mueller matrix will not be transformed during the propagation process. Inpolarimetry is powerful for the experimental determination the case of external perturbations or of imperfections in theof the polarization properties of optical devices like fibre core the degeneracy is lifted, the value of the propagationlinear or circular birefringence, dichroism, losses, rotary constant is modified and then the two modes are coupled. Ifpower or also depolarization. Thus, an experimental ax and ay are the amplitudes of these modes which are nowMueller matrix enables one to predict the transformation z-dependent, the total electric field in the weakly guidingof every state of polarization by propagation through approximation can be described by the following expressionthe medium characterized by the matrix. Furthermore, where the time dependence exp(−jωt) is assumed and notthe coupled-mode theory well known in the Jones expressed:space (amplitude terms), transformed into Stokes space(energetic terms), has rarely been investigated experimentally E(x, y, z) = ax (z)Ex (x, y) + ay (z)Ey (x, y). (1)[1, 2]. The present paper is organized as follows: the theoretical The coupling process can be described [3] by a set of twodevelopment permitting the description of polarization first-order differential equations on the hypothesis that theevolution in a uniformly perturbed fibre is recalled in variations be small versus z, the distance travelled along thesection 2 where the theoretical Mueller matrix used is guide. This set is written as followspresented. Section 3 reports on some simulation results in thecase of elliptical birefringence, whereas section 4 describes d ax (z) k k12 ax (z) = −j 11 (2)the experimental method applied to measure the Mueller dz ay (z) k21 k22 ay (z)matrix. Some experimental results and the methodologyto extract physical parameters, i.e. coupling coefficients, are where kij are the coupling coefficients and d/dz indicates thepresented in sections 5 and 6. Conclusions are presented in derivative with regard to z. This system can be directly solvedsection 7. [4, 5], and its solutions are then expressed as a function of0022-3727/99/141618+08$30.00 © 1999 IOP Publishing Ltd
  2. 2. Measurement of optical fibre parametersboth the initial conditions, i.e. the input state of polarization,and the coupling coefficients: y y ax = f (ax (0), ay (0), z, kij ). (3) τ x ayUnfortunately, this formulation has some drawbacks for z xcarrying out experimental investigations: first, the quantitiesare complex and thus difficult to measure and second,depolarization is not considered. These considerations led us Figure 1. A representation of the laboratory (x, y) and rotaryto integrate this system in Stokes space where all quantities (x , y ) frame.have an energy dimension. In consequence, the four Stokes Cparameters can be expressed as a function of amplitude terms S D 3by the following relations: P1 ∗ ∗ S0 (z) = ax (z)ax (z) + ay (z)ay (z) -45 ∗ ∗ S1 (z) = ax (z)ax (z) − ay (z)ay (z) H O V ∗ ∗ S 2θ S2 (z) = ax (z)ay (z) + ay (z)ax (z) 1 ∗ ∗ P0 S3 (z) = j[ax (z)ay (z) − ax (z)ay (z)]. (4) +45Introducing the system (4) into (2) and assuming insignificant S 2losses, which is consistent with the conservation of energyalong the fibre, i.e. dS0 /dz = 0 ⇒ S0 (z) = S0 (0) = C Gconstant, the differential system becomes  dS1  Figure 2. Evolution of the eigenpolarization in the rotary frame dz 0 2k2 2k1 S1 (z) versus the external twist rate.  dS2  = −2k2 0 S2 (z) (5) dz dS3 −2k1 − 0 S3 (z) along the fibre or at its output versus perturbation variation. dz ∗ At this level of development, thanks to an experimentalwhere K = k12 = k21 = k1 + jk2 and = k22 − k11 in the measurement of the Stokes vectors, one can determine thelossless case. physical parameters of the fibre, i.e. the coupling coefficients, Without depolarization, i.e. S0 = S1 + S2 + S3 the 2 2 2 2procedure developed by Franceschetti and Smith [6] enables and then find whether the model used is valid or not. It shouldone to solve this system and obtain the evolution of the be underlined that this measurement will be valid only in theStokes parameters versus the distance z. When the coupling case of a given incident state of polarization. For example,coefficients are independent of z, and only in that case, the predicting from one measurement how another polarizationsolutions take the following form state will behave will be quite impossible. The Mueller matrix has allowed us to get round this disadvantage.S0 (z) = S0 (0) = 1 Let us note that the output Stokes vector is linearly bound ˆ ˆS1 (z) = S1 N + (S1 (0) − S1 N) cos(δβz) to the input one, leading thus directly to the Mueller matrix ±(Sˆ3 S2 (0) − S2 S3 (0)) sin(δβz) ˆ in the following form: ˆ ˆ S2 (z) = S2 N + (S2 (0) − S2 N) cos(δβz) 1 0 0 ˆ1 S3 (0) − S3 S1 (0)) sin(δβz) ˆ 0 ˆ2 ˆ2 S1 + (1 − S1 )C ˆ ˆ ˆ S1 S2 (1 − C) ± S3 S ±(S [M(z)] =  ˆ ˆ  0 S1 S2 (1 − C) S3 S ˆ ˆ ˆ S2 + (1 − S2 )C 2 2 ˆ ˆS3 (z) = S3 N + (S3 (0) − S3 N) cos(δβz) ˆ ˆ ˆ ˆ ˆ 0 S1 S3 (1 − C) ± S2 S S2 S3 (1 − C) S1 S ˆ ˆ ˆ ±(S2 S1 (0) − S1 S2 (0)) sin(δβz) (6)  0 ˆ ˆ ˆwhere N = S1 S1 (0) + S2 S2 (0) + S3 S3 (0) and S = ˆ ˆ ˆ ˆ S1 S3 (1 − C) S2 S  ˆ ˆ ˆ[1, S1 , S2 , S3 ] is the Stokes vector of the eigenpolarization ˆ2 S3 (1 − C) ± S1 S  (9) S ˆ ˆmode, i.e. the states of polarization that propagate without ˆ2 ˆ2 S3 + (1 − S3 )C ˆtransformation between the fibre ends. The Si componentsare expressed as follows: with C = cos(δβz) and S = sin(δβz). This matrix describes the most general case of elliptical ˆ ˆ 2k1 ˆ 2k2 birefringence without losses and depolarization. As the S1 = ± S2 = S3 = ± (7) δβ δβ δβ measurement of the Mueller matrix allows one to predict the behaviour of each incident state of polarization, the δβ = 2 + 4|K|2 . (8) Poincar´ sphere representation can also be used to evaluate, eIn equation (8), δβ is the propagation constant difference for example, the influence of various parameters such as theof the two eigenpolarization modes. The set of equations (6) external strain on polarization evolution.permits us to describe the evolution of the state of polarization Two particular cases will be discussed in the following: 1619
  3. 3. P Olivard et al 1 m11 0.8 m12 0.6 0.4 m13 0.2 m21 0 m22 -0.2 0 m23 -0.4 m31 -0.6 m32 -0.8 m33 -1 0.25 0.5 0.75 1 1.25 1.5 Twist rate (turns per meter) Figure 3. Evolution of the Mueller matrix elements in the rotary frame versus the external twist rate.2.1. Linearly-birefringent fibre C D S 3 P1Let us suppose a monomode optical fibre in which thereis a linear birefringence β expressed in radians per length -45unit, so that its fast axis makes an angle θ with respect tothe horizontal x-axis, the coupling coefficients are then [7] O2k11 = −2k22 = β cos 2θ and 2k12 = −2k21 = β sin 2θ . H V S 2θ 1One can easily compute the parameters contained in the P0theoretical Mueller matrix defined in (9) so that +45 S 2k1 = β sin 2θ k2 = 0 2 C = −β cos 2θ δβ = β. GThe eigenpolarization Stokes vectors are then [1, cos 2θ, Figure 4. Evolution of the eigenpolarization in the laboratory frame versus the external twist rate. sin 2θ, 0] and the Mueller matrix takes the following wellknown form with C = cos 2θ and S = sin 2θ : takes the following form:  1 0 0    0 C 2 + S 2 cos βz CS(1 − cos βz) 1 0 0 0[M(z)] =   0 cos(gτ z) sin(gτ z) 0  0 CS(1 − cos βz) S 2 + C 2 cos βz [M(z)] =  . (11) 0 −S sin βz C sin βz 0 − sin(gτ z) cos(gτ z) 0  0 0 0 1 0 S sin βz  This is an ideal case. However, if the fibre exhibits a small . (10) −C sin βz intrinsic birefringence because of core imperfection, it will cos βz have significant effects on the polarization properties of the guide. Elliptical birefringence will occur which must beThis matrix corresponds to a linear birefringent network. carefully considered.This form is valid only in an invariant frame and with auniform birefringence, i.e. with independence of z. Thismatrix can be used to study the effect of bending with regard 3. Elliptical birefringenceto the internal intrinsic birefringence of the fibre. However, Elliptical birefringence results from the superposition ofin the calculation of the birefringence, one should be cautious linear and circular birefringence. This becomes the caseof the beat length (transition at 2π ) and take into account the when a linearly-birefringent monomode fibre is uniformlypossible rotation of the fibre [8] local axis (see section 3). twisted. It then requires one to take into account the rotation of the fibre axes which, in the laboratory frame x, y, z, makes2.2. Circular-birefringent fibre the coupling coefficients dependent on z. As previously mentioned, the integration method giving the Mueller matrixAn isotropic monomode fibre under uniform twist τ (in is not valid in this case. To circumvent this problem, oneradians per length unit) holds a circular birefringence gτ , must refer to the local frame of the fibre (x , y , z) depictedwhere g is the elasto-optic coefficient of the material. In in figure 1. The coupling coefficients are then independentsuch a case, the coupling coefficients can be expressed [9] of z and the Mueller matrix can be computed by the method ∗as k11 = k22 = 0 and k12 = k21 = jgτ/2, so that k1 = 0, described above.2k2 = gτ , = 0 and δβ = gτ . The eigenpolarization Let us consider a monomode fibre with a linear uniformStokes vectors are then [1, 0, 0, ±1] and the Mueller matrix retardance β expressed in radians per length unit so that its1620
  4. 4. Measurement of optical fibre parameters 1 m11 0.8 m12 0.6 0.4 m13 0.2 m21 0 m22 -0.2 0 m23 -0.4 m31 -0.6 m32 -0.8 m33 -1 1 2 3 4 5 6 7 8 Twist rate (turns per meter) Figure 5. Evolution of the Mueller matrix elements in the laboratory frame versus the external twist rate. (a) (b)Figure 6. Evolution of the state of polarization for two different incident states (in the laboratory frame). (a) Horizontal incident state:[1, 1, 0, 0]. (b) Elliptical incident state: [1, 0.578, −0.21, 0.788] (θ = −10◦ , ε = 26◦ ). vertical Quarter-wave Quarter-wave Horizontal polarizer plate plate polarizer ν ν Laser Sample Detector P1 L1 L2 P2 Figure 7. The schematic layout of the axis makes an angle θ with respect to the horizontal A similar procedure was also followed by Sakai and Kimurax -axis. If this fibre is submitted to an external twist τ (in [4] in Jones space.radians per length unit), the coupling coefficients can be Let [M] be the Mueller matrix in the laboratory frameexpressed in the rotary frame by and [M ] that in the rotary frame. The following relation can then be written: 2k11 = −2k22 = β cos 2θ ∗ [M(z)] = [R(−2τ z)][M (z)] 2k12 = k21 = β sin 2θ − jτ (2 − g) (12)and the theoretical Mueller matrix takes the general form [M (z)] = [R(2τ z)][M(z)]. (13)of equation (9). In the most general case, experimental In the same way, let [S] be the Stokes vector in the laboratoryMueller matrices are measured in the laboratory frame, so frame and [S ] that in the rotary frame. One can thus writethey are not directly comparable to the theoretical modeland this does not allow an easy estimation of the physical [S(z)] = [R(−2τ z)] (14a)parameters. There are two possible ways of solving thisproblem: either the theoretical matrices are converted from and consequentlythe rotary to the laboratory frame or, conversely, experimentalresults are converted from the laboratory to the rotary frame. [S(z)] = [R(−2τ z)][M (z)][S (0)]. (14b) 1621
  5. 5. P Olivard et al P L P1 L1 O1 F O2 L2 P 2 M3 CAMERA M1 M2 LASER P, P1, P2 : Linear polarizers M 1 , M 2 , M 3 : Dielectric mirrors L1 , L2 : Quarter - wave plates O1 , O2 : Injection optics L : Half - wave plate F : Optical fiber Figure 8. Experimental set-up. 1.20 0.80 m11 m12 0.40 m13 m21 0.00 m22 m23 -0.40 m31 m32 -0.80 m33 -1.20 0.00 3.14 6.28 α z (rad) 9.42 Figure 9. Evolution of experimental Mueller matrice coefficients (in the laboratory frame).When considering that the rotary frame coincides with the monomode optical fibre of 1 m in length was considered andlaboratory frame at τ = 0 where there is no twist, the previous the following physical parameters were used: g = 0.16,relation becomes β = 0.52 rad m−1 , θ = 0.52 rad. The Mueller matrix coefficients were computed by introducing equation (12) into [S(z)] = [R(−2τ z)][M (z)][S(0)]. (15) (9) and applying relation (13). Equations (14) and (15)Introducing equations (12) into (9) with the help of equations were used for the Stokes vectors. The simulation results(13)–(15) enables one to simulate the evolution of every are presented by a couple of figures. Figures 2 and 3 werecoefficient of the Mueller matrix or of the Stokes vectors obtained in the rotary frame and figures 4 and 5 in thein the local or laboratory frame. These simulations can be laboratory frame.made versus z, the distance propagated in the guide, versus The arrows in figures 3 and 5 indicate some characteristicthe physical parameters such as β or versus the external twist points where the fibre can be considered as a pure polarizationstrain τ . Thus, the Poincar´ sphere can be used to draw e rotator. One should note that all the coefficients of the last rowthe trajectory of the polarization state using the normalized and last column of the matrix (m33 being excepted) becomeStokes vector coefficients. rapidly negligible, and the rotation characteristics are then preponderant. This fact was corroborated by the evolution4. Simulations of the eigenstate of polarization which converged toward the north of the sphere. The two trajectories simulated on theTo illustrate the developments described in the previous Poincar´ sphere (figure 6) indicate a reduction of oscillation esection we will now present some simulation results. A with twist and the rotation also appears to be preponderant.1622
  6. 6. Measurement of optical fibre parameters 500 Therefore, the 64 measured intensities were expressed by the (δβ)² Experimental points following matrix 400 Quadratic interpolation k = [0.15] [Ik ] = [B][m1 ] with (18) l = [4i + j ] 300 where the coefficients of matrix B depend on the orientation angles of the quarter-wave plates L1 and L2 . The mij (i, j = 200 0.3) were represented by [m1 ] = ([B]T [B])−1 [B]T [Ik ] (19) 100 where []T indicates the transposed matrix. 0 This equation can then be computed for each 0.00 3.14 6.28 αz (rad) 9.42 measurement, and the expression ([B]T [B])−1 [B]T remains valid from one experience to each other. Figure 10. Evolution of (δβ)2 . Figure 8 displays the experimental set-up. The light source is an ionized argon laser emitting at 514 nm. The These simulation results have shown that the intrinsic detection is made by a charged coupled device (CCD) camera.linear birefringence of the fibre can exert a strong influence The polarimeter was calibrated without the fibre and theon the guided state of polarization even when this fibre is injection devices (O1 , O2 ). The axes of the various opticalsubmitted to a small twist strain. elements were aligned with an accuracy of 0.01◦ using a dichotomous method at null intensity. A ‘χ 2 test’ [10] was carried out to quantify the influence of noise on the Mueller5. Experimental methodology and measurement matrix coefficients, which was thus minimized with respectset-up to these coefficients. The estimated standard deviation on each mij was then less than 0.5%.The Mueller matrix was measured with an optical polarimeteras schematized in figure 7. The input polarization encoding 6. Results and analysissystem is composed of a vertical (y-axis) linear polarizer fol-lowed by a quarter-wave plate whose fast axis makes an angle The experimental results presented were obtained with aν with the y-axis. The output polarization decoding system 0.865 m long monomode optical fibre submitted to a uniformis composed of a quarter-wave plate whose fast axis makes small twist, i.e. less than 1.5 turn. Figure 9 illustrates thean angle ν with respect to the reference y-axis, followed by evolution of the experimental Mueller matrix coefficientsa linear horizontal polarizer. The two polarizers P1 and P2 (birefringent elements only). Some characteristic points likewere initially crossed in order to obtain a null intensity. those defined in section 3 are viewed, so the fibre is then The Stokes vector S emerging from the last polarizer a pure rotator of polarization. The depolarization index ofcan be expressed by the following matrix product each matrix was computed. As the depolarization index was always greater than 0.95, the depolarization was considered S = [P2 ][L2 ][M][L1 ][P1 ]S(0) (16) as negligible. Moreover, it indicated that the theoretical model defined above could be used both to describe evolutionwhere [P2 ], [L2 ], [L1 ] and [P1 ] are the Mueller matrices of and to estimate the physical parameters.each of the devices constituting the polarimeter, [M] is the The experimental Mueller matrices represented byunknown Mueller matrix and S(0) is the Stokes vector of the equation (9) were transferred from the laboratory frame (seeincident light. The measurable intensity is enclosed in the figure 9) to the rotary frame using equation (13). The physicalfirst term of the output Stokes vector and, in the case of perfect parameters g and β are enclosed in δβ, which was easilyoptical devices, i.e. quarter-wave plate linear retardance equal computed to give the following expression:to 90◦ , can be expressed as a function of the orientations of 3 3both the two quarter-wave plates (ν, ν ) and the 16 unknown δβ = tan−1 (mij − mj i )1/2 miiMueller matrix coefficients: i=j =1 i=1S0 (ν, ν ) = I (ν, ν ) = m00 + m01 C 2 + m02 CS + m03 S = (β 2 + τ 2 (2 − g)2 )1/2 . (20) 2 2 +(m10 + m11 C + m12 CS + m13 S)(−C ) These two parameters, g and β, were estimated with a +(m20 + m21 C 2 + m22 CS + m23 S)(−C S ) quadratic method from the curve presented in figure 10. In +(m30 + m31 C 2 + m32 CS + m33 S)(S ) (17) this particular case we obtained g = 0.147 and β = 0.46 rad. The estimated elasto-optic coefficient was in good agreementwith C = cos 2ν, S = sin 2ν, C = cos 2ν and S = sin 2ν . with published results [5, 7, 9]. One needs only 16 equations to obtain the Mueller matrix The direction of the linear birefringent axis, θ , can becoefficients. To minimize errors we used an overdetermined computed in the rotary frame and takes the following form:system of 64 equations corresponding to 64 combinations ofthe angles (ν, ν ). These angles were multiples of 22.5◦ so m31 − m13 m23 + m32 2θ = tan−1 = tan−1 . (21)that they were capable of describing specific test positions. m23 − m32 m13 + m31 1623
  7. 7. P Olivard et al 1.00 1.00 1.00 0.80 0.80 0.80 0.60 0.60 0.60 0.40 0.40 0.40 0.20 0.20 0.20 m11 m12 m13 0.00 0.00 0.00 -0 .2 00.00 3.14 6.28 9.42 -0 .2 00.00 3.14 6.28 9.42 -0 .2 00.00 3.14 6.28 9.42 -0 .4 0 -0 .4 0 -0 .4 0 -0 .6 0 -0 .6 0 -0 .6 0 -0 .8 0 -0 .8 0 -0 .8 0 -1 .0 0 -1 .0 0 -1 .0 0 1.00 1.00 1.00 0.80 0.80 0.80 0.60 0.60 0.60 0.40 0.40 0.40 0.20 0.20 0.20 m21 m22 m23 0.00 0.00 0.00 -0 .2 00.00 3.14 6.28 9.42 -0 .2 00.00 3.14 6.28 9.42 -0 .2 00.00 3.14 6.28 9.42 -0 .4 0 -0 .4 0 -0 .4 0 -0 .6 0 -0 .6 0 -0 .6 0 -0 .8 0 -0 .8 0 -0 .8 0 -1 .0 0 -1 .0 0 -1 .0 0 1.00 1.00 1.00 0.80 0.80 0.80 0.60 0.60 0.60 0.40 0.40 0.40 0.20 0.20 0.20 m31 m32 m33 0.00 0.00 0.00 -0 .2 00.00 3.14 6.28 9.42 -0 .2 00.00 3.14 6.28 9.42 -0 .2 00.00 3.14 6.28 9.42 -0 .4 0 -0 .4 0 -0 .4 0 -0 .6 0 -0 .6 0 -0 .6 0 -0 .8 0 -0 .8 0 -0 .8 0 -1 .0 0 -1 .0 0 -1 .0 0Figure 11. Measured Mueller matrices versus the external twist αz (in rad). The curves represent the theoretical constructions obtainedwith equation (13). Experimental theoretical (a) (b)Figure 12. Evolution of the state of polarization for two different incident states. Theoretical and experimental trajectories. (a) Horizontalincident state: [1, 1, 0, 0]. (b) Elliptical incident state: [1, 0.578, −0.21, 0.788] (θ = −10◦ , ε = 26◦ ).However, the value obtained from this expression is not 7. Conclusionconstant, and this implies that the assumption of uniformlinear birefringence is not valid. Consequently, to compare An experimental method to determine the physicalour measurements with the theoretical model, we computed parameters of a monomode optical fibre has been presented.the mean value of θ and obtained θ = −0.52 rad. The theoretical model derived from the coupled-mode With the estimated physical parameters we adjusted equations applied to the Stokes–Mueller formalism gavethe theoretical model and compared it to the experimental good results. It showed that the non-uniformity of the linearresults (figure 11). The agreement between theory and birefringence could influence polarization behaviour.measurement was quite good. Nevertheless, there were The Stokes–Mueller formalism is a powerful tool fordiscrepancies in the coefficients m13 , m23 , m31 and m32 ; they carrying out experimental studies of polarization phenomenawere likely to be due to the non-uniformity of the intrinsic in optical devices. The Poincar´ sphere, directly connected elinear birefringence of the fibre. The comparison between to the Stokes–Mueller formalism, also constitutes a usefultheoretical and measured trajectories on the Poincar´ sphere e representation to describe the evolution of the polarization(figure 12) showed identical rotary behaviour and a decrease state. The results reported here have shown that theseof the ellipticity with twist. This result corroborates the non- formalisms are very relevant in the characterization of opticaluniformity of the linear intrinsic birefringence. fibres.1624
  8. 8. Measurement of optical fibre parametersReferences [6] Franceschetti G and Smith C P 1981 Representation of the polarization of single-mode fibers using Stokes[1] Eftimov T A and Bock W J 1992 Experimental investigation parameters J. Opt. Soc. Am. 71 1487–91 of single-mode single-polarization optical fiber J. Opt. [7] Rashleigh S C 1983 Origins and control of polarization Soc. Am. 17 1061–3 effects in single-mode fibres IEEE J. Light. Technol. 1[2] Brown C S, Shute M W, Williams D D and Muhammed F 312–31 1994 The development and calibration of an optical fibre [8] Olivard P, Cariou J, Le Jeune B and Lotrian J 1995 polarimeter Proc. SPIE 2265 62–9 e ` D´ termination, a partir de matrices de Mueller[3] Yariv A 1973 Coupled-mode theory for guided-wave optics exp´ rimentales, de la birefringence induite par courbure e IEEE J. Quantum Electron. 9 919–33 ` dans une fibre optique monomode 15 eme Journ´ es. Nat. e[4] Sakai J and Kimura T 1981 Birefringence and Opt. Guid´ e (Palaiseau, France, November 1995) p 27 e polarization characteristics of single-mode optical fibre [9] Ulrich R and Simon A 1979 Polarization optics of twisted under elastic deformations IEEE J. Quantum Electron. 17 single-mode fibres Appl. Opt. 18 2241–51 1041–51 [10] Le Jeune B, Marie J P, Gerligand P Y, Cariou J and Lotrian J[5] Monerie M and Jeunhomme L 1980 Polarization mode 1994 Mueller matrix formalism in imagery. An coupling in long single mode fibres Opt. Quantum experimental arrangement for noise reduction Proc SPIE Electron. 12 449–61 2265 443–51 1625