## Abstract

In twisted single-mode optical fibers the polarization of light is affected by an elastooptically induced optical activity and by a modification of any linear birefringence present. These effects are discussed theoretically and demonstrated experimentally. The activity/twist ratio is *α*/*τ* ≃ 0.13 … 0.16 universally in weakly guiding silica fibers. Twisted fibers may be used as polarization rotators. A fiber with a ±68° double twist operates as a fast/slow mode interchanger, suitable for delay equalization.

© 1979 Optical Society of America

## I. Introduction

Single-mode optical fibers promise to find applications not only in very wide bandwidth optical communication systems, but also in current monitoring,[1] rotation sensing,[2] and other interferometric devices. In those applications the fibers operate with coherent polarized light, and a thorough understanding of their polarization properties is fundamental to the successful construction and operation of all devices mentioned. In this paper we discuss the influence of twist on the evolution of the polarization along single-mode fibers, because twist is one of the imperfections usually present on any real, installed fiber. Moreover, useful polarization optical devices can be constructed in a fiber by applying controlled twists to it.

The influence of twist is intimately linked with the influence of the linear birefringence that exists in any real fiber[3],[4] due to deviations from a circular shape of its core or due to internal stress. Three typical situations may be distinguished, depending on the relative magnitudes of the twist rate *τ* and the linear birefringence *β*. In the case of weak twist, |*τ*| ≪ |*β*|, the mentioned linear birefringence is the dominant effect. Its principal axes are rotated by the twist. Referring to these axes, the polarization evolves essentially identically as it would do in the nontwisted fiber. Thus, the polarization is simply twisted with the fiber. At medium twist, |*τ*| ≃ |*β*|, the shear strain in the twisted fiber gives rise to a circular birefringence *α* proportional to the twist, *α* = *gτ*. In combination with the linear birefringence, elliptical birefringence results and causes a rather complicated evolution of the polarization. At strong twist, |*τ*| ≫ |*β*|, the situation becomes simple again, because the evolution is then dominated by the induced circular birefringence *α*.

For a detailed description of these effects, we start in Sec. II by considering the electromagnetic fields in a general, imperfect fiber. The evolution of the fields along the fiber is described by a set of coupled-mode equations. In the single-mode fiber, only two guided modes contribute to this evolution. Fiber imperfections like twist-induced shear stress or a deformation of the core may lift the degeneracy of phase velocities of the two modes, resulting in a beating, phenomenon. Simultaneously, the imperfections may couple the modes. Both effects are characterized by coupling coefficients *κ** _{mn}*. Their calculation from the details of the imperfection is deferred to the Appendix.

In Sec. III the coupled wave equations are integrated by interpreting them geometrically as rotations of a Poincaré sphere. This Poincaré representation permits a direct, intuitively appealing description of the evolution of polarization along a real single-mode fiber. In particular, it provides a clear insight into the roles of linear and circular birefringence effects and their combinations.

In Sec. IV we give experimental results illustrating. and confirming the above concepts. The evolution of polarization has been measured along fibers subjected to various twists, using electrooptic and magnetooptic modulation methods. Finally, in Sec. V we utilize the twist-induced optical activity to devise a simple in-line fiber-optical element which interchanges the powers in the fast and slow modes of an imperfect single-mode fiber. Positioned midway on a short uniform fiber transmission line, such an element can compensate the influence of mode dispersion.

## II. Coupled Mode Theory

We describe a straight, real optical fiber by the spatial distribution of its relative dielectric permeability,

Here, *∊*_{0}(*r*) is a scalar function which depends only on the radial distance *r* = (*x*^{2} + *y*^{2})^{1/2} from the fiber axis and characterizes the ideal fiber [Fig. 1(a)]. All imperfections of the fiber are contained in the perturbation
$\tilde{\u220a}$(*x*,*y*,*z*). That term may be a scalar (describing a deformed core, for example) or a tensor in the case of stress- or twist-induced optical anisotropy. In the following we only postulate |
$\tilde{\u220a}$| ≪ 1 so that we may apply perturbation theory to determine the influence of
$\tilde{\u220a}$ on the local propagation properties of the fiber.

In the ideal fiber, the electric and magnetic fields of a mode *m* have the general form *A** _{m}*(

*z*)

**E**

*(*

_{m}*x*,

*y*) and

*A*

*(*

_{m}*z*)

**H**

*(*

_{m}*x*,

*y*), respectively, where

A monochromatic time dependence exp(−*ikct*) of all field quantities is tacitly assumed here, with *k* = 2*π*/λ_{vac} denoting the propagation constant of vacuum and *k** _{m}* (possibly complex), that of mode

*m*.

The fields in the imperfect fiber [Eq. (1)] are represented as superposition of the modes of the ideal fiber. The transverse field distributions
${\mathbf{E}}_{m}^{t}(x,y)$ of the latter form a complete orthogonal set of functions,[5] suitable to express the general transverse electric field in the imperfect fiber in any given plane *z* = const. as

The symbol *m* and the summation here include the discrete, guided modes as well as the continuum of radiation modes. The transverse component of a field vector is denoted by the superscript *t*, the longitudinal one by a superscript *z*.

In an ideal fiber, the evolution of the mode amplitudes *A** _{m}*(

*z*) is determined by Eq. (2) with constant

*a*

*. In the imperfect fiber, the evolution of the*

_{m}*A*

*(*

_{m}*z*) is more complicated because the perturbation $\tilde{\u220a}$ modifies and couples the ideal modes. We may maintain the general form [Eq. (2)] of the

*A*

*(*

_{m}*z*) if we permit the

*a*

*to vary along the fiber,*

_{m}*a*

*=*

_{m}*a*

*(*

_{m}*z*). As shown in the Appendix, then, the wave equation in the perturbed fiber [Eq. (1)] reduces to a set of coupled-mode equations for the

*a*

*(*

_{m}*z*),

Here, the prime indicates the derivative with respect to *z*. The coupling coefficients *κ** _{mn}* depend on the perturbation
$\tilde{\u220a}$.

To evaluate Eq. (4) for a single-mode fiber, we note that it can support two guided modes of orthogonal polarizations.[3]–[6] Their propagation constants are real and degenerate (*k*_{1} = *k*_{2}) in the ideal fiber if we neglect absorption. We also ignore here losses which may result from coupling to the radiative modes. Even if those modes should become excited, their power is lost continuously from the core region. Therefore, the general electric field in an imperfect single-mode fiber can be well represented by the superposition of the two guided modes only,

The form of the modal fields **E*** _{m}*(

*x*,

*y*) is given in Eqs. (A9)and (A10) of the Appendix. Likewise, for the reasons just mentioned, only two of the coupled-mode equations [Eq. (4)] are sufficient for the further discussion of polarization. They describe the evolution of

*a*

_{1}(

*z*) and

*a*

_{2}(

*z*).

In the practically most important limit of a weakly guiding[6] fiber, perturbation theory (Appendix) gives the following expressions for the coupling coefficients.

The asterisk * indicates complex conjugation; *J*(*r*) is the radial wave function of the fiber, discussed in the Appendix; and
${n}_{0}\simeq {\u220a}_{0}^{1/2}$ is a mean refractive index of the fiber.

From Eqs. (4)–(8) we recognize the way in which a given perturbation
$\tilde{\u220a}$ influences the evolution of the mode amplitudes. In the electric field **E*** _{n}* of a mode,
$\tilde{\u220a}$ produces an extra dielectric polarization (
$\tilde{\u220a}$

**E**

*) which then couples to the same or another mode*

_{n}*m*according to the overlap with

**E**

*, expressed by the integral (7). Similarly, the quantity ∇( $\tilde{\u220a}$*

_{m}**E**

*) in Eq. (8) may be interpreted as a ficticious space charge produced by $\tilde{\u220a}$, coupling only to the longitudinal component ${\mathbf{E}}_{m}^{z}$.*

_{n}In the coupled-wave equations [Eq. (4)], the diagonal coefficients *κ*_{11} and *κ*_{22} describe a detuning of the two previously degenerate modes. If *κ*_{11} ≠ *κ*_{22}, the degeneracy of their propagation constants is lifted and the modes show a spatial beating effect with their spatial difference frequency (*κ*_{11} − *κ*_{22}). The off-diagonal coefficients *κ*_{12} and *κ*_{21} describe the mutual coupling of the modes, resulting in an interchange of power. Because the fiber was assumed as lossless,
${\kappa}_{12}={\kappa}_{21}^{*}$ holds.

In a general situation, the beating and the coupling effects exist simultaneously, leading to the well-known phenomenon of coupled-mode propagation. It should be noted here that the relative importance of beating versus coupling depends only on the choice of the base functions, here the **E*** _{m}*(

*x*,

*y*). This had been discussed with respect to single-mode fibers in Ref. [7]. A given perturbation $\tilde{\u220a}$ may lead for one set of base functions to uncoupled, asynchronous propagation of the two modes so that they beat. For another choice of base functions, the same $\tilde{\u220a}$ may yield coupled, synchronous propagation with periodic interchange of power. Both viewpoints have been used, separately, in the literature. A clear discussion of these problems appears necessary here, because we are dealing with the general, intermediate case, and we are going to consider the twisted fiber both in a laboratory-fixed coordinate system (

*x*,

*y*,

*z*) and in a local system (

*x*

^{0},

*y*

^{0},

*z*

^{0}) twisted with the fiber [Fig. 1(a)].

#### A. Twist

To calculate the coupling coefficients *κ** _{mn}* resulting from twist, we assume that the elastic properties and the elastooptic[8] tensor

*p*

*are uniform throughout the fiber. The twist causes a rotation*

_{rs}*ϕ*

*=*

_{B}*τz*of the cross-sectional plane

*z*[see Fig. 1(a)]. A positive

*τ*denotes a right-handed twist. Evaluation of the elastic deformation field gives the strain tensor

*e*

*. Its only non-vanishing components are, in abbreviated[8] notation,*

_{rs}*e*

_{4}=

*τx*and

*e*

_{5}= −

*τy*. They cause changes ∑

*p*

_{rs}*e*

*of the impermeability tensor, which may be converted to changes in the dielectric permeability, i.e., to the perturbation tensor $\tilde{\u220a}$. In the amorphous material of the fiber we obtain*

_{s}With this
$\tilde{\u220a}$, the coupling integrals (7) vanish if *m* = *n*. For *m* ≠ *n*, they contain two equal contributions, resulting from the couplings
${\mathbf{E}}_{m}^{z}\to {\mathbf{E}}_{n}^{t}$ and
${\mathbf{E}}_{m}^{t}\to {\mathbf{E}}_{n}^{z}$. The radial integrals can be converted by integration by parts. In evaluating Eq. (6), then, they cancel against the normalization integrals *Q** _{m}*, regardless of the nature of the radial wave function

*J*(

*r*) and of the (weakly guiding) index profile

*∊*

_{0}(

*r*). The remaining coupling integrals (8) are negligibly small for practical fibers. Hence, we obtain, universally for weakly guiding fibers of arbitrary index profile, the influence of twist.

As we shall see, these coefficients cause circular birefringence (optical activity). The reason is the imaginary sign on *κ*_{12} and *κ*_{21}. It results from the coupling between a longitudinal (
${\mathbf{E}}_{m}^{z}$) and a transverse field component (
${\mathbf{E}}_{n}^{t}$), which are *π*/2 out of phase. In deriving Eqs. (11) we have ignored all scalar variations
$\tilde{\u220a}$ which may result from the twist. They detune both modes identically (*κ*_{11} = *κ*_{22}) but do not couple them. Therefore, they leave the polarization unchanged.

#### B. Core Deformation

In real fibers, the cross section of the core is often not perfectly round. The resulting linear birefringence combines with the circular one derived above and is therefore discussed here briefly. One of the simplest deviations is an elliptical deformation of the index profile. We consider a relative compression *η* of the profile *∊*_{0}(*r*) along the azimuthal direction *ϕ** _{B}* and a corresponding dilatation in the orthogonal direction [Fig. 2(a)]. The resulting perturbation is scalar,

and couples only the
${\mathbf{E}}_{m}^{z}$ components by Eqs. (7) and (8). Because these
${\mathbf{E}}_{m}^{z}$ are in phase, the *κ** _{mn}* turn out real, causing linear birefringence.

The quantity *β* introduced here is proportional to *η* and contains certain integrals over *∊*_{0}(*r*). An evaluation of Eqs. (6)–(9) leads to the result obtained already by Marcuse[5] in a different way. For the present discussion, Eqs. (12) show that the deformation causes maximum detuning and beating of the modes when its axes are parallel to the polarizations **E**_{1} and **E**_{2}. When the axes lie at 45° to these polarizations, there is no detuning, but maximum coupling. This illustrates our earlier remarks about the relative importance of beating and coupling.

#### C. External Fields

Birefringence in the fiber can also be caused by the application of electric or magnetic fields. A transverse electric field of azimuthal direction *ϕ** _{K}* [see Fig. 2(b)] produces a homogeneous field strength

*E*

*in the fiber. By the quadratic (Kerr) electrooptic effect, coupling coefficients like Eqs. (12) result, but with*

_{K}*ϕ*

*instead of*

_{K}*ϕ*

*and with*

_{B}Here, *B** _{K}* is the Kerr electrooptic constant of the fiber material, assumed to be homogeneous throughout the fiber.

A magnetic field with a component *H** _{F}* along the direction of propagation produces a gyroelectric perturbation

Here *V* is the Verdet constant of the material, like *H** _{F}* assumed to be homogeneous throughout the crosssection of the weakly guiding fiber. The integrals

*I*

^{(3)}are negligibly small, and from Eq. (7) the coupling coefficients are

They are imaginary and therefore cause circular birefringence (Faraday rotation). Contrary to the twist-induced circular birefringence [Eq. (11)] the coupling exists here only between the transverse components of the electrical fields, but the perturbation $\tilde{\u220a}$ itself is imaginary now.

## III. Evolution of Polarization

In discussing the polarization in a single-mode fiber, one should distinguish between a local and a global distribution of polarization. At each point (*x*,*y*,*z*) in the fiber, the electric field [Eq. (5)] has a specific local polarization. Generally, this field has a longitudinal component also and varies within a given cross section *z* = const., for example, as indicated in Fig. 1(a). The variation is completely determined by the modal wave functions **E*** _{m}*(

*x*,

*y*) if their amplitudes

*a*

_{1}and

*a*

_{2}are specified. Actually, because the total power (|

*a*

_{1}|

^{2}+ |

*a*

_{2}|

^{2}) and the common phase of the two modes are irrelevant for polarization, it is sufficient to specify only the ratio

*a*

_{1}/

*a*

_{2}in a given cross-sectional plane to characterize there the distribution of local polarization. The global distribution of polarization, on the other hand, is described then by the

*z*dependence of the ratio

*a*

_{1}(

*z*)/

*a*

_{2}(

*z*). In our perturbational approach, only this global evolution of polarization along the fiber is affected by the various birefringence effects, and it alone is discussed further.

An important aspect of the global evolution is that it is deterministic if the light is strictly monochromatic and the fiber is at rest. If, in that case, *a*_{1}(*z*)/*a*_{2}(*z*) has a certain, constant value at one point along a given fiber (e.g., at the input), it is fully determined at all other points of the fiber, too, and does not vary in time. The reason is simply that the solution of Maxwell’s equations is completely determined by specifying fixed initial and boundary conditions. Stated differently, this means that the imperfect single-mode fiber does not depolarize strictly monochromatic light. It can only alter the state of polarization along the fiber, not the degree of polarization. For polarized light of finite spectral bandwidth, however, the correlation between the two modes may decrease along the imperfect fiber because of the nondegenerate propagation constants, so that the light becomes depolarized.[12] Depolarization may also occur by thermal or rapid mechanical fluctuations of the fiber properties.

For each cross section *z* we represent the ratio *a*_{1}(*z*)/*a*_{2}(*z*) by a point *C*(*z*) or by the vector **C**(*z*) = *OC* on a generalized Poincaré sphere[7]
$\mathcal{S}$ of unit radius (Fig. 3). Referring first to Fig. 3(a), the angular coordinates of *C*(*z*) are

This representation has been chosen so that it becomes identical with the conventional Poincaré representation[9] of polarization in the limit of a large core diameter. In that limit, the *z* components of the wave functions **E*** _{m}*(

*x*,

*y*) vanish, and these functions approach plane waves of uniform local polarization, representable in the usual way.[9] Moreover, our representation [Eqs. (14)] of the ratio

*a*

_{1}(

*z*)/

*a*

_{2}(

*z*) coincides with a conventional Poincaré representation of the local polarization of the field

**E**(

*o*,

*o*,

*z*) on the fiber axis. There, the longitudinal component

**E**

*vanishes, and the modal fields*

^{z}**E**

_{1}(

*o*,

*o*) and

**E**

_{2}(

*o*,

*o*) can be identified with the horizontal and vertical field components of a plane wave. With this representation of

*a*

_{1}(

*z*)/

*a*

_{2}(

*z*) we can take over the entire formalism of the Poincaré representation of plane-wave polarization for discussing the global evolution of polarization in the fiber.

In that sense, then, any global state of polarization (SOP) with real *a*_{1}/*a*_{2} is called a linear state and corresponds to a point *C* on the equator of the sphere
$\mathcal{S}$. Points *H* and *V* correspond to horizontal and vertical polarizations, while *P* and *Q* correspond to linear SOPs of +45° and −45° azimuth, respectively. The poles *R* and *L* of the spheres
$\mathcal{S}$ represent circular polarizations, with the electric vector rotating clockwise or counterclockwise, respectively, when looking toward the source.

The evolution of any initial SOP along the fiber can be represented by a trajectory **C**(*z*) on
$\mathcal{S}$. The local velocity and the direction of evolution, **C**′ ≡ *d***C**/*dz*, are governed by the coupled wave equations [Eqs. (4)]. In order to find the evolution of polarization over a longer piece of fiber, these equations must be integrated. The representation on the Poincaré sphere is well suited for this integration, because in the two-mode case considered here, Eqs. (4) describe simply a rotation[7] of the sphere
$\mathcal{S}$ with a certain angular velocity *ω*(*z*). According to Ref. [7], we have

Thus, an infinitesmal length *dz* of fiber rotates all SOPs about the direction ** ω** through an angle

*ωdz*. Equations (18) express the spherical coordinates of the direction

**(**

*ω**z*) in terms of the coupling coefficients

*κ*

*(*

_{mn}*z*) of Sec. II. Therefore,

**(**

*ω**z*) is known, in principle, for all points along the fiber. It is possible then to construct for any given input SOP the trajectory

**C**(

*z*) on $\mathcal{S}$. This geometrical construction of

**C**(

*z*) as a succession of rotations permits a clear insight into the various polarization effects. This construction is particularly convenient if

**(**

*ω**z*) is constant or varies systematically along the fiber. We discuss this construction now for various polarization effects, considering them first singly and then in combinations.

#### A. Linear Birefringence

When *κ*_{12} is real, as, for example, in the case of a deformed core or of a transversely applied field [Eqs. (12)], the axis of rotation lies in the equatorial plane according to Eq. (18b). As in crystal optics, this rotation characterizes linear birefringence. We denote it by a vector ** β**. Its magnitude |

**| is identical with the**

*β**β*defined earlier in Eqs. (12) and (13). The position of

**in the equatorial plane follows from Eq. (18a) as 2**

*β**χ*

*=*

_{B}*π*/2 − 2

*ϕ*

*. For further discussion we switch to the more conventional[9] coordinate system of Fig. 3(b). In that system, the vector*

_{B}**has simply the longitude 2**

*β**ϕ*

*. When a fiber of length*

_{B}*L*has only linear birefringence

*β*, a general input state

**C**(

*o*) is rotated through the angle

*βL*about

**. Thus, all trajectories**

*β***C**(

*z*) are circular arcs about

**, as indicated in Fig. 4(a). We recognize here that in the Poincaré representation the coupling and beating effects discussed earlier merge most naturally.**

*β*#### B. Circular Birefringence (Optical Activity)

Circular birefringence is characterized by an imaginary *κ*_{12} in combination with *κ*_{11} = *κ*_{22}, as, for example, in a twisted fiber [Eqs. (11)] and in the Faraday rotation [Eqs. (14)]. Here, the rotation vector ** ω** coincides with the polar axis

*RL*of Fig. 3(b) and we shall call it

**. The trajectories are parallel to the equator [see Fig. 4(b)]. From Eqs. (17) and (18b), the angular velocity is**

*α**α*= −2

*iκ*

_{12}. For a linear SOP, the plane of polarization rotates at the rate

*ρ*=

*α*/2 per unit length of the fiber. We define the sign of

*α*as positive and

**as pointing north, if the plane of polarization appears to rotate counterclockwise when looking in the direction to the source (**

*α**l*-rotatory activity).

In the twisted fiber, the strain-induced optical activity is proportional to the twist. According to Eq. (11),

with
$g=-{n}_{0}^{2}{p}_{44}$. Using the literatures[10] value *p*_{44} = (*p*_{11} − *p*_{12})/2 = −0.075 and *n*_{0} ≃ 1.46 for fused silica, we find that a right-handed twist should induce *l*-rotatory optical activity in the fiber, and that *g* ≃ 0.16 should hold universally for weakly doped single-mode fibers of arbitrary index profile. Different values of *g* may be expected, however, if the doping modifies *p*_{44}.

The twist-induced optical activity predicted by Eq. (19) was confirmed quantitatively by our experiments (see Sec. IV below). The existence of this activity is surprising in view of the symmetry considerations of the piezorotatory effect in Ref. [11]. For an isotropic material twisted according to Eq. (10) and a plane wave propagating along the *z* direction, that theory predicts vanishing optical activity. Thus a contradiction arises to the result of Eq. (19) above, which remains unchanged if we go to the limit of a fiber with infinite core diameter, i.e., to a plane wave. It appears that the plane-wave approach in Ref. [11] may be overidealized to encompass the twist-induced optical activity.

#### C. Elliptical Birefringence

Elliptical birefringence results from the superposition of linear and circular birefringences. Because the corresponding rotations *β**dz* and *α**dz* are infinitesimal, they commute and can simply be added vectorially. In a fiber having both linear birefringence (e.g., due to deformation) and circular birefringence (due to twist), the resulting angular velocity is

For the case where ** ω** does not depend on

*z*, the trajectories are indicated in Fig. 4(c). We recognize the existence of two eigenstates of elliptical polarizations, ±

**/**

*ω**ω*, which propagate unchanged. The state +

**/**

*ω**ω*propagates at a greater phase velocity than the state −

**/**

*ω**ω*. The azimuthal orientations of these eigenstates are those of

**, i.e., they coincide with the symmetry axes (if present) of the linear birefringence. Under the influence of a constant elliptical birefringence**

*β***, any input SOP is reproduced at the output of the fiber if its length is**

*ω**L*= 2

*π*(

*α*

^{2}+

*β*

^{2})

^{−1/2}or a multiple thereof, so that the evolution goes through an integral number of revolutions.

#### D. Birefringence and Twist

When a general birefringent fiber is twisted, the evolution **C**(*z*) is complicated by the fact that the rotation vector ** ω**(

*z*) is not constant but is itself moving on the Poincaré sphere. We consider only the practically most important case in which a uniform twist

*τ*is applied to a fiber of initially uniform linear birefringence

**. When the fiber is twisted, the azimuth of this linear birefringence becomes**

*β**ϕ*

*=*

_{B}*τz*, and the twist-induced circular birefringence is given by Eq. (19). Consequently,

**(**

*ω**z*) moves along a parallel circle on the sphere, and the trajectories became cycloidical curves [Fig. 4(d)].

To understand these trajectories, we note that the motion of ** ω**(

*z*) can be expressed by the vector product

The twist vector ** τ** has the magnitude

*τ*and is directed parallel to the

*RL*axis. For a right-handed twist

**is pointing north. For each cross section of the fiber, we introduce an auxiliary, local coordinate system**

*τ**ℛ∘*whose axes

*x∘*and

*y∘*are parallel to the local fast and slow axes of birefringence, respectively, while

*z∘*=

*z*. Hence, this system

*ℛ∘*revolves about the fiber axis at the rate of the twist. Associated with

*ℛ∘*there is a Poincaré sphere $\mathcal{S}\xb0$, rotating about

*RL*at the rate 2

*τ*relative to the laboratory-fixed sphere $\mathcal{S}$ considered so far. We mark all quantities referring to

*ℛ∘*and $\mathcal{S}\xb0$ by a superscript degree sign. Thus, the vectors

**and**

*β∘***=**

*α∘***are constant, with**

*α***pointing along the**

*β∘**x∘*direction. Moreover,

**=**

*ω∘***+**

*β∘***is constant.**

*α∘*We want to determine now the trajectories **C∘**(*z*) on
$\mathcal{S}\xb0$. It may seem that the evolution on
$\mathcal{S}\xb0$ is a uniform rotation about the fixed vector ** ω∘**. This view is a fallacy, however, as becomes clear by considering the case

*τ*= 0,

*α*= 0,

*β*→ 0. In that limit, all polarization effects vanish in the original coordinate system

*ℛ*, so that they must not vanish in

*ℛ∘*.

Yet, the evolution on $\mathcal{S}\xb0$ is a uniform rotation indeed, but with the angular velocity vector

This vector **Ω∘** is fixed on
$\mathcal{S}\xb0$, and all trajectories **C∘**(*z*) are circles on
$\mathcal{S}\xb0$ about **Ω∘** as axis. We recognize this evolution by regarding the two states
$+{\mathbf{C}}_{\mathrm{\Omega}}^{\circ}$ which lie on
$\mathcal{S}\xb0$ in the fixed directions ±**Ω∘**.

These states propagate unchanged in *ℛ∘*, because these states read **C**_{Ω} = (** ω** − 2

**)/Ω when expressed in**

*τ**ℛ*. They are simultaneous solutions of Eqs. (16) and (21). Hence, $\pm {\mathbf{C}}_{\mathrm{\Omega}}^{\circ}$ are the eigenstates of polarization in the twisted system

*ℛ∘*. All other states revolve about the axis connecting $+{\mathbf{C}}_{\mathrm{\Omega}}^{\circ}$ and $-{\mathbf{C}}_{\mathrm{\Omega}}^{\circ}$, because the angular distance [ $\mathbf{C}\xb0,{\mathbf{C}}_{\mathrm{\Omega}}^{\circ}$] of any general state

**C∘**from ${\mathbf{C}}_{\mathrm{\Omega}}^{\circ}$ must be conserved[7] during the evolution. The mentioned rotation is the only motion conserving this distance generally for all points.

To find the trajectories **C**(*z*) we switch back from the rotating sphere
$\mathcal{S}\xb0$ to the fixed
$\mathcal{S}$. The discussion just preceding shows that a general vector **C**(*z*) moves along a cone of constant semiaperture
$[\mathbf{C}\xb0(z),\hspace{0.17em}{\mathbf{C}}_{\mathrm{\Omega}}^{\circ}]=[\mathbf{C}(z),\hspace{0.17em}{\mathbf{C}}_{\mathrm{\Omega}}(z)]$ with angular velocity **Ω**, while the axis of that cone is moving along a parallel circle [dashed line in Fig. 4(d)] of latitude

Therefore, **C**(*z*) is generally a cycloid on
$\mathcal{S}$. It is characterized by equal excursions to the north and south of the mean latitude 2*ψ*_{Ω}. The cycloid originating from an input state **C**(*o*) may be looping [**C**_{1} in Fig. 4(d)] or may be stretched [**C**_{2} in Fig. 4(d)], depending on whether the distance of **C**(*o*) from the local axis **C**_{Ω}(*o*) is larger or less than the distance of ** ω**(

*o*) from the axis

**C**

_{Ω}(

*o*), respectively.

Corresponding points on successive periods of the cycloid are spaced 2*π*/Ω apart on the fiber, equivalent to one full revolution of
$\mathcal{S}\xb0$. Relative to the local axes *H∘* and *V∘* of the fiber, such successive states are equal. In the laboratory coordinate system, however, the second state is rotated by 2*πτ*/Ω with respect to the first one, just like the fiber. When *τ*/Ω is a rational number, the **C**(*z*) are closed curves.

It is interesting to consider this general, cycloidal evolution in the limits of weak and strong twists. The twist is weak when |*τ*| ≪ *β*. Then also |*α*| ≪ *β*, and **Ω∘** ≃ ** β∘**, so that the evolution in
$\mathcal{S}\xb0$ remains practically unaffected by the twist. Therefore, the polarization is rotated as if it were rigidly attached to the fiber. This limit is realized in the recently reported polarization-maintaining fiber,[13] whose operation is based on its particularly large linear birefringence

*β*. The limit of weak twist corresponds to the situation in a birefringent crystal with weak optical activity. As is well known,[8] that activity is generally completely quenched by the linear birefringence.

A condition of strong twist exists if |*α*| ≫ *β* so that 2*ψ*_{Ω} → *π*/2. From Eq. (22) we obtain **Ω∘** ≃ ** α** − 2

**= (**

*τ**g*− 2)

**. In $\mathcal{S}\xb0$ the evolution therefore approaches a uniform rotation about the polar axis at the rate (**

*τ**α*− 2

*τ*). Correspondingly, on the laboratory fixed sphere $\mathcal{S}$, the SOPs rotate at the rate

*α*=

*gτ*. In real space, the plane of polarization rotates at half that rate. We recognize that in this limit the linear birefringence is quenched by the twist-induced circular birefringence. A circular input state will remain circular even in the presence of weak linear birefringence (

*β*≪

*α*). Hence, a strongly twisted fiber is polarization-maintaining for circular polarization in the same sense as the above-mentioned fiber[13] for linear polarization. The mechanical strength of modern fibers easily leads to the condition of strong twist. We have twisted silica fibers of 115-

*μ*m outer diameter up to 30 rev/m without breakage, reaching 2

*ψ*

_{Ω}≥ 80°.

These considerations provide guidelines for the mechanical precision required in the handling and installation of fibers in interferometric and other polarization-sensitive devices. To fully utilize the low birefringence of high-quality fibers (e.g., *β* ≤ 10^{−2} rad/m), it is necessary to keep their twist at a comparably low value, lest a disturbing ellipticity 2*ψ*_{Ω} be introduced.

## IV. Experimental Results

We have measured the evolution of polarization on straight, birefringent, silica single-mode fibers under various conditions of twist and have compared the results with the above theory. The fibers have a core diameter of ~5 *μ*m, an outer diameter of 115 *μ*m, a single-mode cutoff near 580 nm, and they were measured at 633-nm wavelength. To determine nondestructively the SOP at any point along the fiber, we used the technique of electrooptic (Kerr) and magnetooptic (Faraday) modulation of the unknown polarization by externally applied fields.[14] In brief, this technique is based on the conservation of distances between SOPs on the Poincaré sphere. The modulating field is applied over a short length *l* at the point of interest. It induces a weak extra birefringence *δ**ω***,** as discussed in Sec. II above. The unknown state **C**(*z*) is modulated by *δ***C**(*z*) ≃ *lδ*** ω** ×

**C**(

*z*). At all points beyond

*z*, the SOPs are modulated by the same amount |

*δ*

**C**|. In particular, at the output end of the fiber, |

*δ*

**C**(

*L*)| = |

*δ*

**C**(

*z*)|. This quantity can be detected even for the relatively weak Kerr effect. The observed modulation |

*δ*

**C**| is largest when

*δ*

**and**

*ω***C**(

*z*) are

*π*/2 apart on $\mathcal{S}$, and it vanishes when they are parallel.

In the electrooptic determination of an SOP, the azimuthal direction of the modulating field is rotated slowly about the fiber [Fig. 2(b), with *l* ≃ 1 cm]. From the relative heights of the observed minima and maxima of |*δ***C**| the ellipticity 2*ψ* of **C**(*z*) [see Fig. 3(b)] can be determined. The angular position of the minima marks the longitude 2*ϕ* of the SOP. Figure 5 shows two examples of an evolution measured in this way, point by point, along a twisted fiber. We recognize the general type of cycloids discussed above. By comparison with the theory given, these curves can be analyzed for three types of imperfections. First, the total birefringence is found [Fig. 5(b)] from the fiber length per loop (*L** _{p}* = 0.142 m) as Ω = 44.2 rad/m. Second, the progression of the cycloid from loop to loop yields the twist rate

*τ*. In Fig. 5(b), the progression is ~103° per loop or 12.6 rad/m. The latter value compares well with 12.2 rad/m, calculated from the initial twist rate in Fig. 5(a),

*τ*= 1.9 rad/m, and the externally applied twist rate,

*τ*= 4.2 rad/m. The observed irregularity of the measured evolution is believed to result from nonuniformities of the inherent twist and/or birefringence. Third, the mean latitude 2

*ψ*

_{Ω}= 15° of the cycloid Fig. 5(b) permits calculation of the linear and circular components of the total birefringence

**Ω∘**,

In the example of Fig. 5(b), we find *β* = 42.7 rad/m. The effective twist rate is found as (*α* − 2*τ*) ≃ 11.4 rad/m, compared to the value of 11.6 rad/m which would be expected from *τ* and *α* by Eq. (19). The agreement is good in view of the mentioned nonuniformities of this fiber.

In the magnetooptic measurements, a modulating axial magnetic field (~400 Hz) is applied to the fiber by a coil wound around the fiber, using a ferrite shell core with internal air gap (~2 mm) as indicated in Fig. 6. This arrangement is preferable to our earlier setup with a coil by the side of the fiber,[14] because it permits simple, straightforward evaluation. The modulating magnetic field *H** _{F}*(

*z*) is essentially confined to the short gap. Hence, in lowest order of approximation, the modulation is a periodic rotation of all SOPs about the polar axis through the angle

For the coil of Fig. 6, this integral is equal to the current enclosed by the fiber, i.e., to the number of ampere-turns of the coil. With an externally arranged coil, as in Fig. 2(a) of Ref. [14] (erasing head), a substantial magnetooptic modulation can be obtained, too, although the above-line integral vanishes. In that case, the modulation results mainly from the stray magnetic field and is not the simple rotation about the polar axis.

By its symmetry, the magnetic modulation with the coil of Fig. 6 yields information only on the evolution of the latitude of an SOP. Yet, when the longitude is not required, this magnetic measurement is more convenient for an analysis of fiber birefringence because it permits direct recording of |2*ψ*(*z*)|. Figure 6 is an example. Each maximum of this curve corresponds to an SOP on the equator. The minima mark, alternatingly, the extreme latitudes on the northern and southern hemispheres. From their difference the mean latitude 2*ψ*_{Ω} of the cycloid can be determined. The periodicity of the curve gives Ω directly.

It may appear possible to modulate the SOP also by a transverse magnetic field (Cotton-Mouton effect) in analogy to the electric field. In principle, such measurements should permit the determination of the azimuth of an SOP also. They have been attempted, but without success, because the much stronger longitudinal effect by stray fields could not be sufficiently suppressed.

By taking magnetooptic measurements like Fig. 6 on a fiber at different twists *τ* it is possible to accurately determine the ratio *α*/*τ*. In Fig. 7 we have plotted Ω(*τ*) in a form which permits direct comparison with Eq. (19). The measurements at two fibers of different birefringence agree within the experimental uncertainties, yielding for *g* the value

The positive sign of *α*/*τ* was verified independently by static polarization measurements like those of Fig. 8 below. The measured *g* ratio [Eq. (27)] is noticeably smaller than the value *g* = 0.16 calculated by Eq. (25) from the published elastooptic coefficients of silica.[12] This discrepancy may be caused by a wavelength dependence of the *p** _{rs}* coefficients and/or by the presence of doping elements in the fiber core. In an earlier investigation on the polarization properties of a twisted fiber[4] the induced optical activity had not been observed because the twists employed there were too small compared to the birefringence (

*τ*/

*β*≲ 0.03) to produce a noticeable ellipticity or extra rotation.

## V. Applications

#### A. Polarization Rotator

A question of practical importance in the construction of polarizing fiber-optical devices is the variation of the output SOP of a fiber of given length *L* for a given input SOP when the fiber is gradually twisted by rotating its output end. Such a twisting can be used advantageously to adjust the output polarization **C**(*L*) to a desired azimuth. However, the associated variations of the total birefringence Ω and of the ellipticity 2*ψ*_{Ω} of the axes have to be taken into account.

We describe these variations, which go beyond the simple, rigid rotation of the output polarization, by a trajectory
${\mathbf{C}}_{L}^{\xb0}(\tau )$ on the sphere
$\mathcal{S}\xb0$(*L*) which is twisted with the fiber end. The polarizing effect of the fiber is a rotation through Ω*L* about the axis
${\mathbf{C}}_{\mathrm{\Omega}}^{\xb0}$. At small twists, *τ* ≪ *β*, the latitude 2*ψ*_{Ω} of that axis increases according to Eqs. (25) and (19) approximately linearly with *τ*, while the variation of the angle Ω*L* is small in second order according to Eq. (23). Hence, in the laboratory system
$\mathcal{S}$ such a weakly twisted fiber simply rotates the azimuths *ϕ* of all SOPs through the twist angle *τL*. As an illustration, we consider the aforementioned twist experiment of Ref. [4]. There, a 16-λ fiber (length *L* ≃ 32*π*/*β*) was twisted through half a revolution (−*π*/2 < *τL* < *π*/2). It was found that linearly polarized input into states *H∘* or *V∘* produced an *H∘* or *V∘* output, i.e., the output polarization was rotated through *τL* within the accuracy of the experiment. This observation is in agreement with Eqs. (22)–(25). They predict a maximum ellipticity |2*ψ*_{Ω}| < 0.03, which means in the terms of Ref. [3] a degree of polarization of better than 0.998 and an azimuthal deviation of less than 1° for input into *H* or *V*.

More generally we calculate for such a rotation of the output azimuth of a twisted Nλ fiber that the maximum deviation of any output SOP on $\mathcal{S}\xb0$ is (in radians)

provided |*τL*| < *π*/2. The factor 1.63 in Eq. (28) stands for (1 − *g*/2)(1 + *π*/4 − *πg*/8). The operation of such a rotator is particularly simple if *N* is the chosen integer.

At large twists, |*τ*| ≫ *β*, the axis
${\mathbf{C}}_{\mathrm{\Omega}}^{\circ}$ moves close to a pole on
$\mathcal{S}\xb0$ and the rotation angle Ω*L* increases nearly in proportion to *τ*. In that limit, the trajectory
${\mathbf{C}}_{L}^{\xb0}(\tau )$ runs nearly parallel to the equator. This effect is recognized from Fig. 8. The
${\mathbf{C}}_{L}^{\xb0}(\tau )$ curve of Fig. 8(a) has been measured with an ~60-cm-long fiber of birefringence *β* = 45.5 rad/m. He–Ne laser light was coupled approximately into the *H∘* state so that the output is also *H∘* without twist. .A section of *L* = 25 cm at the output end of this fiber was then twisted through up to two turns. The output SOP was analyzed and is shown in Fig. 8(a). For *τ* ≲ *β* the output SOP is seen to vary only weakly on
$\mathcal{S}\xb0$, i.e., it is rotated with the fiber end. For large twists, however, the term (*α* − 2*τ*) in Eqs. (22) and (23) becomes dominant. For comparison, we show in Fig. 8(b) the trajectory as calculated from Eqs. (22)–(27). To get the best agreement with Fig. 8(a), it was necessary to assume the input SOP for the twisted section to have the coordinates 2*ϕ* = 0.25; 2*ψ* = −0.15. This slight deviation from *H∘* may have been caused by the section lying before the twist or by the fixtures (epoxy) of the fiber required for twisting. In Fig. 8(b), the output SOP passes repeatedly through the input SOP whenever Ω*L* is a multiple of 2*π*, i.e., when the fiber becomes an *N*λ fiber.

#### B. A Mode Interchanger

The twist-induced optical activity is used specifically in another application to interchange the powers propagating in the fast and slow modes (states *H∘* and *V∘*) of a birefringent fiber, [see Fig. 9(a)]. The fiber is fixed at two points *z*_{1} and *z*_{3} and its center at *z*_{2} is rotated through an angle *θ*. Therefore, the twist rates *τ* of the two sections *z*_{1}*z*_{2} and *z*_{2}*z*_{3} are equal but of opposite signs. The angle *θ* = *τL* should be chosen so that 2*ψ*_{Ω} = ±*π*/4, and the lengths of the two sections should make Ω*L* = *π*. From Eqs. (23), (25), and (27) we find that these conditions are satisfied for a fiber of given birefringence *β* if *L* = *π*/(2)^{1/2}*ϕ* and *θ* = *π*/(2)^{1/2}(2 − *g*) ≃ 68°. The polarization characteristics of the double-twist arrangement are explained on the Poincaré sphere
$\mathcal{S}\xb0$ in Fig. 9(b). The first section causes half a revolution about the axis **Ω**_{1}, interchanging the states *H∘* ↔ *L∘; V∘* ↔ *R∘*; and *P∘* ↔ *Q∘*. In the second section, the rotation about **Ω**_{2} interchanges *L∘* ↔ *V∘; R∘* ↔ *H∘*; and *Q∘* ↔ *P∘*. Consequently, the combination of both rotations interchanges *H∘* ↔ *V∘* and *L∘* ↔ *R∘*, whereas *P∘* and *Q∘* remain fixed. We recognize that the effect of the two sections (*z*_{1} − *z*_{3}) is to rotate
$\mathcal{S}\xb0$ through one-half revolution about the axis *P∘Q∘*. In particular, the light arriving at *z*_{1} in the fast mode (state *H∘*) leaves at *z*_{3} in the slow mode (state *V∘*).

An equivalent effect could be achieved without twist by breaking the fiber, rotating one of the two ends by *π*/2 about its axis, and rejoining the ends. The advantage of the ±68° double twist is that it can be applied to the fiber without breaking it. When this device is installed midway on a transmission line of uniformly birefringent single-mode fiber, it equalizes the transit times of phase and group through the line for *H∘* and *V∘* polarizations and therefore for all polarizations. Such an equalization is essential for interferometric applications of single-mode fibers when a moderate or wide spectral bandwidth is used, because otherwise the mode dispersion associated with birefringence could cause severe depolarization in these cases.[12]

## VI. Conclusions

We have discussed the evolution of polarization along twisted single-mode fibers. Starting from first principles, we have described a perturbational method to calculate how the two degenerate modes of an ideal fiber become detuned and coupled by imperfections. Specifically, we have considered twist-induced shear strain and external electric and magnetic fields. Other kinds of imperfection may be treated by the same formalism. The global evolution of the modal amplitude ratio *a*_{1}(*z*)/*a*_{2}(*z*) is conveniently represented as a trajectory *C*(*z*) on a generalized Poincaré sphere. In perfect analogy to the polarization optics of plane waves, the fiber imperfections appear as birefringence effects of various kinds (linear, circular, elliptical). A fiber subject to right-handed twist *τ* exhibits a strain-induced, *l*-rotatory optical activity *α* = *gτ*. The proportionality factor
$g=-{n}_{0}^{2}{p}_{44}$ should hold universally for weakly doped silica fibers.

The evolution of polarization along a twisted fiber is determined by the relative magnitudes of linear birefringence and twist. In the general case, the evolution follows cycloidical trajectories. Under conditions of weak or strong twists, the trajectories reduce approximately to circles about an equatorial or polar axis, respectively, characterizing the dominance of the linear birefringence or of the twist-induced circular birefringence.

Our experimental results have confirmed this theoretical description and roughly also the calculated *g* factor. The validity of the theory is further demonstrated in the discussion of polarization rotation by a twisted fiber end and of the ±68° double twist interchanging the fast and slow modes of a fiber.

## Appendix: Derivation of Coupled-Mode Equations

For the sake of generality we did not specify explicitly the ideal refractive index profile. We postulate, however, that *∊*_{0}(*r*) is a smooth function of radius *r* [Fig. 1(b)], and that
$\tilde{\u220a}$ → 0 as *r* → ∞. We thus avoid the need to discuss boundary conditions. Step-index fibers may be treated by taking suitable limits.

The field in the imperfect fiber must satisfy the wave equation for the inhomogeneous and possibly anisotropic medium [Eq. (1)],

Here, ∇ is the nabla operator. We consider only the transverse part of this vector equation. The last term of it is transformed using the identity

Inserting here Eqs. (3) and (2) with variable amplitudes *a** _{m}*(

*z*) we obtain

In Eq. (A2), terms containing the product $\tilde{\u220a}{a}_{m}^{\prime}$ have been omitted as they are small as second order. Similarly, inserting Eq. (A2) now into the wave equation (A1), we may neglect the ${a}_{m}^{\u2033}$ terms. Two groups of terms remain. The first group exists even with $\tilde{\u220a}$ = 0. It defines the mode functions ${\mathbf{E}}_{m}^{t}\hspace{0.17em}(x,y)$ of the ideal fiber and therefore vanishes. The other group is

Applying here the orthogonality relations,
$\int \int {[{\mathbf{E}}_{m}^{t}\times {\mathbf{H}}_{n}^{t*}]}^{z}\hspace{0.17em}dxdy=0$ for *m* ≠ *n*, we arrive directly at the system [Eqs. (4)] of coupled-mode equations.

The coupling coefficients have the form [Eq. (6)] with

All integrations extend over the full cross section of the fiber. The integrals (A7) and (A8) given here have been converted from their original form, containing ${\mathbf{H}}_{m}^{t*}$, by an integration by parts and by substitution from Maxwell’s equations.

This perturbation formalism holds for a general, imperfect fiber. Specializing it to the two modes of a weakly guiding single-mode fiber, their polarizations are chosen arbitrarily to lie symmetrically about the *x* and *y* axes, as shown in Fig. 1(c). All field components of both modes (*m* = 1,2) can be expressed by one real, radial wave function *J*(*r*) which is determined by *∊*_{0}(*r*). Their Cartesian components [*x*,*y*,*z* in Fig. 1(a)] are

Here, the dot indicates ∂/∂*r*. In the limit of a step-index fiber, *J* equals the cylinder function *J*_{0} in the core and *K*_{0} in the cladding. With these fields, the normalization integral (A5) can be partially evaluated, yielding Eq. (9), and the integrals (A6) and (A7) can be combined to give Eq. (7).

We thank W. Eickhoff and O. Krumpholz of AEG-Telefunken, Ulm, for providing us some of the fibers, and M. Johnson for his independent, accurate determination of *α*/*τ*.

## Figures

## References

**1. **A. M. Smith, Appl. Opt. **17**, 53 (1978).

**2. **V. Vali, R. W. Shorthill, and M. F. Berg, Appl. Opt. **16**, 2605 (1977) [CrossRef] [PubMed] .

**3. **F. P. Kapron, N. B. Borelli, and D. B. Keck, IEEE J. Quantum Electron. **QE-8**, 222 (1972); [CrossRef] W. A. Gambling, D. N. Payne, and H. Matsumara, in *Digest of Topical Meeting on Optical Fiber Transmission II*, (Optical Society of America, Washington, D.C., 1977), p. TuD5.

**4. **H. Papp and H. Harms, Appl. Opt. **14**, 2406 (1975) [CrossRef] [PubMed] .

**5. **D. Marcuse, *Theory of Dielectric Optical Waveguides* (Academic Press, New York, 1974).

**6. **D. Gloge, Appl. Opt. **10**, 2252 (1971) [CrossRef] [PubMed] .

**7. **R. Ulrich, Opt. Lett. **1**, 109 (1977) [CrossRef] [PubMed] .

**8. **J. F. Nye, *Physical Properties of Crystals* (Clarendon, Oxford, 1960).

**9. **G. N. Ramachandra and S. Ramaseshan, in *Handbuch der Physik*, S. Flügge, Ed. (Springer, Berlin, 1962), Vol. 25/1.

**10. **D. Gray, Ed. *American Institute of Physics Handbook* (McGraw Hill, New York, 1976), p. 6–236.

**11. **G. S. Ranganath and S. Ramaseshan, J. Opt. Soc. Am. **59**, 1229 (1969) [CrossRef] .

**12. **S. C. Rashleigh and R. Ulrich, Opt. Lett. **3**, 60 (1978) [CrossRef] [PubMed] .

**13. **R. H. Stolen, V. Ramaswamy, P. Kaiser, and W. Pleibel, Appl. Phys. Lett. **33**, 699 (1978) [CrossRef] .

**14. **A. Simon and R. Ulrich, Appl. Phys. Lett. **31**, 517 (1977) [CrossRef] .