Vol. 71, No. 12/December 1981/J. Opt. Soc. Am. 1487

Representation of the polarization of single-mode fibersusing Stokes parameters

Giorgio Franceschetti

Istituto Elettrotecnico, Universit6 di Napoli, Via Claudio 21, Napoli, Italy, and Department of ElectricalSciences, University of California at Los Angeles, Los Angeles, California 90024

Cynthia Porter Smith*

Hughes Research Laboratories, 3011 Malibu Canyon Road, Malibu, California 90265

Received December 26, 1980; revised manuscript received May 27, 1981

A simple, analytic method for describing the evolution of polarization along a single-mode optical fiber is presented.A system of linear differential equations for the change of the Stokes parameters, and hence for the change in polar-ization, along a fiber is derived from coupled-mode theory and the definition of the Stokes parameters. The polar-ization eigenstates-the polarization states not affected by the perturbations-are determined from these equa-tions. The general solution of the system of equations is found. This solution is applied to several specific casesand is found to agree with results obtained from the geometric method of Ulrich and Simon [Appl. Opt. 18, 2241(1979)].

1. INTRODUCTION AND DISCUSSION OFPRINCIPAL RESULTS

It is well known that single-mode optical fibers are useful incommunication systems because of their large bandwidthcapabilities. Recently there have emerged other applicationsfor single-mode fibers, many of which depend on the fiber'spreserving a particular state of polarization over long dis-tances.' For example, the effective operation of single-modefiber interferometric devices, such as the fiber-optic gyroscopeand the fiber-optic hyrdrophone, depends on the interferenceof two coherent beams with identical polarizations. In ad-dition, many integrated-optic circuits, e.g., Afl reversal switch,are polarization sensitive, so that fibers used with these de-vices must preserve polarization. Also, some nonlinear effectsin fibers are polarization sensitive.

Although an ideal single-mode fiber will preserve polar-ization over long lengths, along a real fiber the state of polar-ization changes after a few meters.2-5 The polarization statein an optical fiber depends on modal birefringence, the dif-ference in effective refractive indexes for orthogonally pola-rized modes. 1 An ideal fiber propagates two degeneratemodes with orthogonal linear polarizations.6 Because themodes are degenerate, they have identical refractive indices.Therefore the polarization is not affected by the fiber. On theother hand, in a real fiber the degeneracy is lifted by pertur-bations of the fiber, such as geometric or dielectric imperfec-tions; strains that are due to bends, twists, or handling; andchanges in ambient temperature and pressure. Thus the twomodes have different phase velocities. They may also coupleand exchange power. These effects change the polarizationstate. Accordingly, along a real fiber the state of polarizationis modified. In this paper we present a simple, analyticmethod for determining how the state of polarization changesalong the fiber. From perturbation theory, coupled-mode

theory, and the definition of the Stokes parameters, we derivea system of equations for the change of the Stokes parametersand hence for the change in the polarization along the fiber.Ulrich7 and Ulrich and Simon7 ' 8 have represented the evolu-tion of polarization as trajectories on the Poincar6 sphere.Although their representation is geometrical, ours is analyt-ical.

The advantage of the use of the proposed equations com-pared with other methods can be summarized as follows. Themethod is analytical, yet more accurate than a graphical one.Field equations need not be solved since the proposed equa-tions directly relate Stokes parameters (i.e., the polarizationstate) to the propagating-mode parameters and guide per-turbation (i.e., phase mismatch and coupling constant).These, in turn, can be taken to be z dependent, thus allowinga space-varying perturbation (e.g., a twist) of the fiber; or astatistical perturbation can be considered, thus making theStokes parameters stochastic variables. In all cases the po-larization properties of the fiber are determined without needfor any intermediate step. For a space-invariant perturba-tion, the equations can be solved analytically. In more generalcases the numerical solution is easy to obtain. Furthermore,this solution permits an immediate determination of the po-larization eigenstates, i.e., the polarization states that prop-agate unperturbed down the fiber irrespective of the (as-sumed) perturbation (see Section 3). Even an experimentaltechnique for determining phase mismatch and couplingconstant can be devised (see Section 3).

The paper is organized as follows.In Section 2 we use coupled-mode theory together with the

definition of Stokes parameters to derive a system of equa-tions that describes the evolution of polarization along a sin-gle-mode fiber. We solve the equations for the lossless casein Section 3 and for the lossy case in Section 5. Also, we showin Section 4 that, in the suitable limiting cases, the equations

0030-3941/81/121487-05$00.50 © 1981 Optical Society of America

G. Franceschetti and C. P. Smith

1488 J. Opt. Soc. Am.IVol. 71, No. 12/December 1981

give results that are consistent with those obtained by thePoincar6 sphere method of Ulrich and Simon.8

2. ANALYSIS

Let us consider a fiber of circular cross section (see Fig. 1).For the moment, the dielectric constant is assumed to be zindependent and possibly a function of the radial coordinateonly. When no perturbation is present (see Section 1), thefiber can support two degenerate modes, with coincidentpropagation constant along the z axis. The polarization ofthe two modes can be chosen to be linear along the x and yaxes, respectively.

When perturbations do exist, the degeneracy of the twomodes is lifted, and they couple with each other. By lettingax and ay be the (normalized) amplitudes of these modes, thecoupling process can be described by a system of equations9' 2

that, for the case of a weakly guiding fiber,' 3 can be writtenas

a,' = ikxax + iXay,

G. Franceschetti and C. P. Smith

ax

Fig. 1. The two orthogonal modes of an ideal single-mode fiber.

AS,

,-4>20

R

ay' = ikyay + ix*ax, (1)Fig. 2. The Poincar6 sphere.

in which

k. = kl + ik. 2, ky = ky, + iky2 (2)

characterize their detuning;

X = X1 + iX2 (3)

is the coupling constant; a prime implies differentiation withrespect to the z coordinate, i.e., a' = da/dz; and a time de-pendence exp(-iwt) is assumed and suppressed. Note thatthe parameters kx and ky can be computed, in principle, whenthe type of perturbation of the fiber is known.'4 [The as-sumption that the coupling constants of the two modes arecomplex conjugates of each other is strictly valid for thelossless case. Although it is not necessary, it is convenient toretain this assumption in the low-loss case, the only one ofpractical interest. Physically, this implies that losses are notsignificant in the coupling process; see Eq. (33)].

Polarization properties of the fiber are completely deter-mined when the orientation angle ̂ t and the ellipticity angle0 of the polarization ellipsis (see Fig. 2) are given at any pointalong the fiber. These angles are related to the Stokes pa-rameters si, where i = 0, 1, 2, 3, by the system of equations

s, =socos 20 cos 24',

S2 = so cos 20 sin 2i,

S3 = so sin 20 (4)

SO = aaxa* + ayay*,

s, = axa,* - ayay*,

S2 = aw*ay + axay*,

is 3 = ax*ay - axay*. (5)

The Stokes parameters si, s2, and S3 form the Cartesiancoordinates of the Poincar6 sphere, whose radius is so. Anypoint on the sphere represents a definite polarization state.

The equator represents all the linearly polarized states, withpolarization directions given by the angle A (see Fig. 2). Thenorth pole represents left-hand circularly polarized light, andthe south pole corresponds to right-hand circularly polarizedlight. Any other point on the sphere corresponds to ellipti-cally polarized light that is either left-hand or right-handpolarized, depending on whether the point is north or southof the equator, respectively. The direction of polarization ofthe elliptically polarized light (i.e., the direction of the majoraxis) is A, and the degree of ellipticity is tan k.

It is now evident that, by taking the z derivative of system(5) and using Eq. (1), it is possible to obtain the equations thatdescribe the propagation of the Stokes parameters down thefiber. These in turn completely describe the polarizationproperties of the fiber since, from Eq. (4), we easily get

tan 24 = , sin 20 = .S1 So

It is convenient to introduce the new parameters

K = h.,-kyl,

+= kX2 + ky 2,

(6)

(7)

(8)a- = k2- ky2-

Then, after lengthy but straightforward computations, weget

SO' = -a+so - -sl,

s1 = -aso - a+s, - 2 X2S2-2X1S3,

SY = 2 X2S1 - a+S2 + KS3,

S3' = 2xisi - KS2 - at+s3- (9)

These are the fundamental equations that describe thechange of Stokes parameters along the fiber. Note that sys-tem (9) is linear and that the fiber can be lossy. A spatiallyslowly varying perturbation (along the z axis) can be accom-modated by taking the parameters of Eqs. (3), (7), and (8) tobe slowly varying functions of z (see also Section 6).

Vol. 71, No. 12/December 1981/J. Opt. Soc. Am. 1489

3. LOSSLESS CASE

As an important simpler case, let us assume that a+ ,- a- =0, i.e., a lossless fiber. Accordingly, system (9) decouples inthe two systems17 :

so' = 0

s1(O) = Ai + Bi = s°,

si'(0) = Ci,

s11'(0) = -3 2 B1 , (21)

(10) and the derivatives can be eliminated by using system (11).By letting

Si' = - 2X2S2 - 2X1S3,

S2 = 2 X2S1 + KS3,

S3' = 2 Xisi - KS2-

The solution to Eq. (10) is

sO(z) = so(0) = s00 = const,

KSIO ± 2Xis2O - 2 X2S3OX2- =

we get

(11) S1(Z) = Kt + (Sl° - K4)Cos AZ - 2 X1S3O + 2 X2820 isi /z

(12)

which is consistent with conservation of energy along thefiber.

Before solving system (11), let us remark that the polar-ization eigenstates, i.e., the polarization inputs that do notchange as we move down the fiber, are defined by

SI' =s 2' =s 3'=0- (13)

Forcing the condition in Eq. (13) into system (11), we cansolve for the eigenstates .9j, where i = 1, 2, 3, namely,

(22)

S2(Z) = 2xi + (s20 - 2Xt)cos O + KS3

0 + 2 X2 sin z,

S3(Z) = - 2 X24 + (S30 + 2X20)cos Oz

+ K sinm z. (23)

System (23) together with Eq. (12) provides the Stokesparameters, i.e., the polarization state, at any abscissa z of thefiber when the input state siO has been given. They can berecast in a compact, elegant form by using the eigenstate ex-pressions [system (16)]. Normalizing all Stokes parametersto sOO, i.e., by taking a unit power input and letting

2X, I ^ 2X2 A

2 = S1, S3 =- S.K K

And, since

S12 + S22 + S32 = So2,

(14)

(15)

by substituting Eq. (14) into Eq. (15) and solving for s1, wehave

S1 = i:LSOO,

£ 2 = =2X SOO,

S3 = w 2X2 SOO, (16)

in which the parameter

/ = (4 1XI2 + K2)1/2 (17)

is going to play an important role in the future developmentof the theory.

The solution to system (11) can readily be accomplished byassuming a functional z dependence of the type exp(iXz).Substituting in system (11) and equating the determinant tozero results in the following equation for X:

X(X2 - 41X12 - K2 ) = 0, (18)

i.e.,

X=0, X=±/3. (19)

Accordingly, the solutions are of the type

si(z) = Ai + Bi cos Oz + C, sin /z, i = 1,2,3, (20)

in which the constants are determined by the conditions

S2 = §1SIO + S2S20 + S3S30, (24)

we get

SO(Z) = 1,

S1(Z) = S2§s + (S10 - S2%)cos Oz I (s2

0s 3 -s 3092)sin fz,

S2 (Z) = S2 S2 + (S20 - S2 92 )cos OZ I (S3091 -s1 0 93 )sin Oz,

S3 (Z) = S2S3 + (S30

- S 2 33 )COS #Z

+ (s10. 2 - S20sl)sin fz. (25)

It can be again checked from system (25) that the .§i are thepolarization eigenstates of the fiber. In passing, let us remarkthat the experimental determination of these eigenstatespermits the determination of the parameters K, Xi, and X2- Asa matter of fact, we have, from Eq. (20),

sin B = Cj[sj(l) - A i] - Bi {B1 2 + C12 -[si(l) - Ai]2}'I2

I i Bi .2 + Ci2

in which the constants are known in terms of si° and 9i andsi (1) is the value of any one Stokes parameter measured at anyconvenient abscissa z = 1. Equation (26) permits the deter-mination of /. Then, use of system (16) gives K, Xi, and X2.

4. PARTICULAR CASES

Some particular cases are of interest.Let us first assume K to be zero or negligible, i.e., the phase

velocities of the two modes to be (at least approximately)coincident. We have s1 = 0, i.e., for the orientation eigenangle1 and eigenangle ¢,

4 -45O, tan20 = F X2 (27)Xi

G. Franceschetti and C. P. Smith

1490 J. Opt. Soc. Am./Vol. 71, No. 12/December 1981

S3 S3

A BFig. 3. Possible polarization trajectories on the Poincar6 sphere. A,zero mismatch and purely imaginary coupling coefficient; B, zeromismatch and purely real coupling coefficient.

Since the choice of reference axes is arbitrary, the orientationangle VI is irrelevant. And relations (27) state that the po-larization eigenstate is in general elliptical.

When, further, X2 = 0, the polarization eigenstate is linear,and, from system (23),

Si(z) = S10 cos 2xlz - S30 sin 2Xiz,

S2 (Z) = S2

S3 (Z) = s30 cos 2X1z + s, 0 sin 2 XIz. (28)

Then, using system (4),

cot 24 = (s10 /s20 )cos 2Xlz - (s30/s 2

0)sin 2X1z,

sin 20 = (S30 /o1) cos 2 Xiz + (s10/so)sin 2Xiz. (29)

Accordingly, both ellipticity and orientation angles changealong the fiber. An example of this case is a fiber with a de-formed core or one that is subject to a transversely applied dcelectric field. 8.' 4

On the contrary, when XI = 0, the polarization eigenstateis circular, and, from system (23),

sI(z) = s10 cos 2X2Z - S20 sin 2X2Z,

S 2 (Z) = S20 cos 2X2Z + sI0 sin 2X2Z,

S 3 (Z) = S30 . (30)

Then, by using system (4),

4 = '(0) + X2Z,

sin 20 = S30/So. (31)

Accordingly, the ellipticity angle remains constant, whereasthe orientation angle is linearly changing along the fiber. Anexample of this case is a fiber in an axial dc magneticfield. 8. 1 4

These same results can be presented using the Poincar6sphere8 ; our presentation is, however, much simpler and moregeneral. For example, for the case described by systems (30)and (31), S12 + S22 = constant along the fiber, and possibletrajectories corresponding to different values of S 3

0 are givenin Fig. 3B, in accordance with the results of Ref. (8).

For the case described by systems (28) and (29), s12 + s3

2

= constant along the fiber, and possible trajectories corre-sponding to different values of s2

0 are given in Fig. 3A, againin accordance with results of Ref. (8). If the polarization isoriginally linear, it will become elliptical one-handedness, thenlinear perpendicular to the original one, then elliptical with

opposite-handedness, and finally back to linear in the initialdirection.

5. LOSSY CASE

Let us now relax the assumption that a+ = a- = 0, i.e., let usconsider a lossy fiber.

By assuming, as in Section 3, a functional dependenceexp(i Xz) for the Stokes parameters, substituting into system(9), and equating the determinant to zero, we get

(X - ia+)2[(X - ia+)2 - (32]

Accordingly,

a 2[(X - ia+)2 - K2] = 0. (32)

(X- 2+)2 = [3 -| 2 )+ K2a-2]1

In the small-loss case, which is the only one of practicalinterest, we get the four solutions for X:

A J4 + ia+,

i [a+ 4- (KIO)a-]= i[KX2[l : (k/)] + Ky2[1 F (k/#l)]} = iX4. (33)

Accordingly, the general solution is of the type

si(z) = Ai+ exp(-X+z) + A,- exp(-X-z)+ Bi exp(-a+z)cos fiz + Ci exp(-a+z)sin Oz. (34)

When K is negligible, or the attenuation constants of the twomodes are comparable so that ar- 0, Eq. (34) simplifiesto

si(z) = exp(-a+z)(Ai + B. cos /z + Ci sin /3z). (35)

If a+ is so small that the exponential term in Eq. (35) canbe regarded as constant in computing the values of Ai, Bi, andC,, then the solution coincides with system (23), provided thatthe right-hand side is multiplied by exp(-a+z). The polar-ization eigenstates are also given by system (16).

Different results are obtained when the two attenuationconstants are different, say k, 2 we 0, k, 2 0. Under thosecircumstances XA < X+, a+; the term containing the expo-nential factor 2 becomes the dominant one at large observationpositions if KI/ is close to unity. Under those circumstances,the Stokes parameters stabilize, and polarization remainsconstant.

6. ALTERNATIVE REPRESENTATIONS

The fundamental system (9) is a convenient representationof the polarization properties of the single-mode fiber sincea closed-form solution does exist for a spatially invariant fiber,i.e., when K, XI, X2, a+, and a- are constant. When theproperties of the fiber are slowly varying along z, system (9)can still be used. For very smooth variations, we can take

si(z) = Ai + Bi cos[f /3(z)dzl

+ Ci sin[J' /3(z)dzj (36)

as a solution, where the lossless case has been considered forthe sake of simplicity.

G. Franceschetti and C. P. Smith

G. Franceschetti and C. P. Smith

No analytical solution can be expected in general. It istherefore convenient to derive alternative descriptions for thepolarization properties of the fiber, which could be useful forparticular space variations of the parameters. The mostnatural choice is to get the equations for the propagation ofthe ellipticity and orientation angles. By differentiating Eq.(6) with respect to z and using system (11), we get

X = X1 cos 2q -K/2 sin 2l,

't = X2 + tan 2(0(X1 sin 24 + K/2 cos 24'). (37)

System (37) is highly nonlinear, so that the linear system(11) in the Stokes parameters can be appreciated in compar-ison. As an example, let us assume that K = Xl = 0. Then,from system (37),

0(z = OM0,

(z) = (M) + JO X2(Z)dz, (38)

which is, in this case, an exact solution.

ACKNOWLEDGMENTS

C. P. Smith wishes to express her thanks to Lee Casperson ofthe University of California at Los Angeles (UCLA) for hisguidance. She did this work at UCLA under the HughesAircraft Company Fellowship Program in partial fulfillmentof the requirements of her master's degree.

* Present address, Edward L. Ginzton Laboratory, W. W.Hansen Laboratories of Physics, Stanford University, Stan-ford, California 94305.

Vol. 71, No. 12/December 1981/J. Opt. Soc. Am. 1491

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16. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon,Oxford, 1975).

17. G. Franceschetti and C. P. Smith, "Polarization properties ofsingle-mode fibers," presented at IEEE/AP-S Symposium,Quebec, Canada, 1980 [Note: Delete j's in Eq. (5) for S3].