## Abstract

Multiple-scale analysis is used to study linear wave propagation in a rapidly-spun fiber and its predictions are shown to be consistent with results obtained by other methods. Subsequently, multiple-scale analysis is used to derive a generalized Schroedinger equation for nonlinear wave propagation in a rapidly-spun fiber. The consequences of this equation for pulse propagation and four-wave mixing are discussed briefly.

©2006 Optical Society of America

## 1. Introduction

It has been known for many years that one can reduce the birefringence of an optical fiber by spinning it during the drawing process [1]. Spun fibers are used in interferometers and current (magnetic-field) sensors based on Faraday rotation, and as links in optical communication systems. One can also use spun fibers in parametric devices based on four-wave mixing (FWM).

There are three-types of FWM [2]: In the modulation instability (MI), which is driven by one pump wave, the signal wave is amplified and an idler wave is generated, which is a frequency-shifted and conjugated copy of the signal. In the phase-conjugation (PC) process, which is driven by two pumps, the signal is amplified and a frequency-shifted and conjugated idler is generated. In the frequency-conversion (FC) process, which is also driven by two pumps, the signal is not amplified. Instead, its power is transferred to an idler, which is a frequency-shifted, but non-conjugated copy of the signal. These processes have many uses in optical communication systems [3,4].

Because transmission fibers (links) are not polarization-maintaining, parametric devices in communication systems must operate in the same way for all input-signal polarizations. PC driven by orthogonal (perpendicular) pumps in a highly-nonlinear fiber (HNF) provides amplification and conjugation that does not depend on the signal polarization [5, 6, 7]. In contrast, FC in a HNF depends sensitively on the signal polarization [7]. It was predicted recently that FC driven by co-rotating, circularly-polarized pumps in a rapidly-spun fiber (RSF) does not depend on the signal polarization [8, 9]. Although this prediction was validated approximately by an experiment [8], it was based on the idealization that a RSF behaves like a perfectly-isotropic fiber. The goal of this paper is to determine the conditions under which the theoretical predictions are accurate (and are likely to agree with other experimental results).

This paper is organized as follows: In Section 2 the transfer matrix associated with the propagation of a monochromatic wave through a uniformly-spun birefringent fiber is derived by solving the linear wave equation exactly. This transfer matrix is shown to be equivalent to the transfer matrix associated with a retarder followed by a rotator. The spatial evolution of the wave polarization is also determined. In Section 3 multiple-scale analysis (MSA) is used to reproduce the results of Section 2 approximately, for a RSF. This reproduction validates the perturbation method. Subsequently, in Section 4 MSA is used to derive a generalized Schroedinger equation (GSE) for nonlinear pulse propagation in a RSF. The implications of this equation for pulse propagation and FC are discussed briefly in Section 5. Finally, in Section 6 the main results of this paper are summarized.

## 2. Linear wave propagation

Let *A* = [*A*_{x}
,*A*_{y}
]^{t} be the amplitude vector of a light wave. Then, in the absence of nonlinearity, the effects of propagation through a fiber are described by the input-output relation

At any distance *z*, the wave power *P* = *A*
^{†}
*A*, where † denotes a hermitian conjugate. If the fiber is lossless, power is conserved. This condition requires that *T*
^{†}
*T* = *I*: The transformation matrix *T* is unitary. The associated condition |det(*T*)| = 1 implies that det(*T*) = *e*_{iψ}
. If one defines *U* = *Te*
^{-iψ/2}, then *U*
^{†}
*U* = 1 and det(*U*) = 1: The modified transformation matrix *U* is unimodular. Physically, *ψ*/2 is the average phase shift of the polarization components, which we neglect, and *U* characterizes the changes in polarization, on which we focus. Unimodular transformations in Jones space, and the associated rotations in Stokes space, are reviewed in [10].

The following discussion pertains to the reduced transformation

A unimodular matrix has 4 complex (8 real) coefficients. However, the conditions *U*
^{†}
*U* = *I* and det(*U*) = 1 impose 5 real conditions on these coefficients, so only 3 real coefficients are independent. Hence, a unimodular matrix can be written in the (Cayley–Klein) form

where the elements (transfer functions) *μ* and *ν* satisfy the auxiliary condition |*μ*|^{2} +|*ν*|^{2} = 1.

It is useful to review the transfer functions associated with some common optical devices. For a linear phase-shifter (retarder)

where *ϕ*
_{1} is the angle between the *ξ* (slow) axis of the retarder and the *x*-axis (in the laboratory frame), and *ϕ*
_{2} = (*ϕ*
_{ξ} -*ϕ*
_{η})/2 is half the phase difference between amplitude components that are defined relative to the linearly-polarized eigenvectors *E*_{ξ}
= [1,0]^{t} and *E*_{η}
= [0,1]^{t} (in the retarder frame). For the special case in which *ϕ*
_{1} = 0 (aligned retarder), Eqs. (4) and (5) reduce to *μ* = exp(*iϕ*
_{2}) and *ν* = 0, respectively.

For a circular phase-shifter (rotator)

where *ϕ*
_{3} = (*ϕ*
_{+} - *ϕ*
_{−}) is half the phase difference between amplitude components that are defined relative to the circularly-polarized eigenvectors *E*_{+}
= [1,*i*]^{t}/2^{1/2} and *E*_{−}
= [1,-*i*]^{t}/2^{1/2}.Equations (6) and (7) describe the (active) rotation of a linearly-polarized input vector through the angle -*ϕ*
_{3}.

For a retarder followed by a rotator, the composite transfer functions

This composite transformation is characterized by 3 real parameters (*ϕ*
_{1}, *ϕ*
_{2} and *ϕ*
_{3}). Hence, a unimodular transformation (transmission through a lossless fiber) is mathematically equivalent to transmission through a retarder followed by a rotator. This representation of fiber transmission is common in classical optics [11]. The composite transfer functions associated with a rotator followed by a retarder are also defined by Eqs. (8) and (9), with 2*ϕ*
_{1} - *ϕ*
_{3} replaced by 2*ϕ*
_{1} + *ϕ*
_{3}. This alternative representation is also common in classical optics. Other concatenated transformations are described briefly in Appendix A.

For every unimodular transformation in Jones space there is an associated rotation in Stokes space. Despite their physical merits, Eqs. (8) and (9) do not provide a convenient mathematical link between Jones transformations and Stokes rotations. Fortunately, there is another approach. Let *λ*
_{±} be the eigenvalues of the Jones matrix *U* and let *E*
_{±} be the associated eigenvectors. Then the Jones matrix can be written in the canonical form

Two complex eigenvectors have 4 complex (8 real) components. However, the orthonormality relations ${E}_{\pm}^{\u2020}$
*E*
_{±} = 1 and ${E}_{\pm}^{\u2020}$
*E*
_{∓} = 0 impose 4 real conditions on these components, and the phases of the eigenvectors are arbitrary, so only two of these components are independent: The eigenvectors can be written in the form

where *θ*
_{1} and *θ*
_{2} are real parameters. Because the eigenvalues have unit modulus, they can be written in the form *λ*
_{±} = exp(∓*iθ*
_{3}/2), where *θ*
_{3} is another real parameter. By using these definitions, one can write the canonical transfer matrix (10) in the form of Eq. (3), in which the transfer functions

Equations (13) and (14) are different from, but equivalent to, Eqs. (8) and (9).

For every Jones vector *A* there is an associated Stokes vector *A*⃗ = (*A*
_{1},*A*
_{2},*A*
_{3}), where

It follows from Eqs. (11) and (12), and Eqs. (15)–(17), that

Equation (18) implies that *θ*
_{1} is the angle between *E*⃗_{+} and 1-axis, and *θ*
_{2} is the angle between the projection of *E*⃗_{+}on the 23-plane and the 2-axis: *θ*
_{1} and *θ*
_{2} are spherical polar coordinates in Stokes space. In [10] it is shown that the canonical transfer matrix (10) is associated with rotation about the axis *E*⃗_{+} through the angle *θ*
_{3}: Each parameter in representation (10) has a simple interpretation in Stokes space.

For example, by equating Eqs. (4) and (5) to Eqs. (13) and (14), one finds that *θ*
_{1} = 2*ϕ*
_{1}, *θ*
_{2} = 0 and *θ*
_{3} = -2*ϕ*
_{2}: The retarder transformation corresponds to rotation about an axis, which lies in the 12-plane and is inclined at an angle 2*ϕ*
_{1} to the 1-axis, through the angle -2*ϕ*
_{2} [rotation about the anti-parallel axis (*θ*
_{1} = *π* - 2*ϕ*
_{1} and *θ*
_{2} = *π*) through the angle 2*ϕ*
_{2}]. By equating Eqs. (6) and (7) to Eqs. (13) and (14), one finds that *θ*
_{1} = *π*/2, *θ*
_{2} = *π*/2 and *θ*
_{3} = -2*ϕ*
_{3}: The rotator transformation corresponds to rotation about the positive 3-axis through the angle -2*ϕ*
_{3} [rotation about the negative 3-axis (*θ*
_{1} = *π*/2 and *θ*
_{2} = -*π*/2) through the angle 2*ϕ*
_{3}]. In general, the relations between the retarder-rotator parameters ϕ_{1}-*ϕ*
_{3} and the canonical parameters *θ*
_{1}-*θ*
_{3} are complicated.

Now consider propagation through a spun birefringent fiber. If the fiber is spun during the drawing process, the finished fiber is stress free. It is customary to model such a fiber as a stack of infinitesimal birefringent plates, whose birefringence axes (*ξ* and *η*) are rotated by infinitesimal angles relative to those of the preceding plates [12]. In this model, the birefringence axes of the plate located at the distance *z* are rotated by the angle *ρz* relative to the coordinate axes of the laboratory (*x* and *y*). (As *z* increases, the tips of unit vectors parallel to the birefringence axes trace out right-handed screws.)

The rotating-frame amplitude vector is related to the laboratory-frame amplitude vector by the equation

where the (passive) rotation matrix

In the rotating frame, the amplitude vector is governed by the equation

where *d*_{z}
= *d*/*dz*, the coefficient matrix

*δ* = *β*_{ξ}
- *β*_{η}
)/2 is the birefringence strength (half the wavenumber difference) and *ρ* is the rotation rate. (The birefringence beat-length is *π*/*δ* and the rotation length is 2*π*/*ρ*.) The transfer matrix is the solution of the equation

which is an extension of Eq. (21), subject to the boundary (initial) condition *U*(0) = *I*. By solving Eq. (23), one finds that the transfer functions

where *k* = (*δ*
^{2} + *ρ*
^{2})^{1/2}. If *ρ* = 0, *μ*_{r}
(*z*) = exp(*iδz*) and *ν*_{r}
(*z*) = 0, so the principal axes of the transformation are the birefringence axes of the fiber, and the phase difference between the principal amplitude components is 2*δz*. Conversely, if *δ* = 0,*μ*_{r}
(*z*) = cos(*ρz*) and *ν*_{r}
(*z*) = sin(*ρz*), so the Jones vector is rotated through an angle of -*ρz* relative to the birefringence axes of the fiber.

It follows from Eqs. (2) and (19) that the laboratory-frame transfer matrix is related to the rotating-frame transfer matrix by the equation

By combining Eqs. (20) and (24)–(26), one finds that the transfer functions

If *ρ* = 0, *μ*_{l}
(*z*)= exp(*iδz*) and *ν*_{l}
(*z*) = 0: In the absence of spinning, the rotating frame coincides with the laboratory frame. Conversely, if *δ* = 0, *μ*_{l}
(*z*)= 1 and *ν*_{r}
(*z*) = 0: In the absence of birefringence, the wave does not feel the effects of spinning, so the Jones vector is constant in the laboratory frame.

One can determine the parameters of the equivalent retarder-rotator by comparing Eqs. (24) and (25), or Eqs. (27) and (28), to Eqs. (8) and (9). In the rotating frame, one finds that

Equations (29) and (31) reflect the fact that 2*ϕ*
_{1} - *ϕ*
_{3} = *nπ*. One can make *ϕ*
_{1} and *ϕ*
_{3} continuous by adding appropriate multiples of *π* at the appropriate distances.

The retarder parameters *ϕ*
_{1} and *ϕ*
_{2} have the same values in the laboratory frame as they do in the rotating frame. In contrast, the rotation parameter *ϕ*
_{3} is smaller by *ρz*, because two successive rotations through the angles -*ϕ*
_{3} and *ρz* are equivalent to one rotation through the angle -(*ϕ*
_{3} -*ρz*). It follows from Eq. (31) and the preceding statement that

Equations (29), (30) and (32) are equivalent to Eqs. (3), (1) and (2) of [1], respectively, which were based on the results of [12]. For a RSF (|*ρ*|≫|*δ*|), these equations imply that

where *k* ≈ *ρ* and |*δ*|≪*k*. Equations (33) and (34) are equivalent to Eqs. (11) and (10) of [1], respectively, and Eq. (35) is an improvement of Eq. (10) of [1]. Equation (33) implies that the slow axis of the equivalent retarder increases monotonically and rapidly with distance. However, Eq. (34) implies that |*ϕ*
_{2}|≪1, so the effects of the equivalent retarder are weak, and can be neglected. (Rapid spinning also reduces the polarization-mode dispersion of a randomly-birefringent fiber [13].) The rotation parameter of the equivalent rotator has a contribution that increases steadily and a contribution that oscillates. For long distances the latter contribution can be neglected: The effects of a RSF are equivalent to those of a rotator with *ϕ*
_{3}(*z*) ≈ *δ*
^{2}
*z*/2*ρ*. One can obtain the same result from Eqs. (27) and (28). In the aforementioned limit, the transfer functions *μ*_{l}
(*z*)≈cos[(*k* - *ρ*)*z*] and *ν*_{l}
(*z*)≈sin[(*k* - *ρ*)*z*], which represent a rotation with the parameter (*k*-*ρ*)*z*~≈*δ*
^{2}
*z*/2*ρ*.

The laboratory-frame rotation angle -*ϕ*
_{3} is plotted as a function of the distance parameter *kz* in Fig. 1, for the case in which the birefringence-to-spin ratio *δ*/*ρ* = 0.15. The predictions of the approximate formula (35) are accurate.

One can visualize the wave evolution in Stokes space by combining Eqs. (24) and (25), or Eqs. (27) and (28), with definitions (15)-(17). The rotating-frame Stokes components are plotted as functions of the distance parameter *kz* in Fig. 2, for the case in which *δ*/*ρ* = 0.15, and the trajectory of the rotating-frame Stokes vector is plotted in Fig. 3. By comparing Eqs. (24) and (25) to Eqs. (13) and (14), one finds that *θ*
_{1} = tan^{-1} (-*ρ*/*δ*), *θ*
_{2} = -π/2 and *θ*
_{3} = 2*kz*. The tip of the vector rotates about the axis (-0.15,0.00, -0.99), which is approximately equal to the negative 3-axis. In the rotating frame, the trajectory is simple.

The laboratory-frame Stokes components are plotted as functions of the distance parameter *kz* in Fig. 4, for the case in which *δ*/*ρ*= 0.15, and the trajectory of the laboratory-frame Stokes vector is plotted in Fig. 5. The 3-component undergoes small, but fast, oscillations. In contrast, the 1- and 2-components undergo large, but slow, oscillations (on which are superposed the very small, but fast, oscillations that are required to conserve the length of the vector). Although the global transformation is complicated (as are the relations between the transfer functions and the canonical parameters), the associated local transformation (rate of change) is simple. Equations similar to (21) and (23) apply to the laboratory-frame amplitude vector *A*_{l}
and transfer matrix *U*_{l}
. It follows from (the laboratory-frame) Eq. (23) that

where *U*_{l}
is known. By combining Eqs. (26) and (36) with the identity *R*†*R* = *I*, one obtains the alternative equation

where *R* and *L*_{r}
are known. It follows from either of the preceding equations that

In [10] it is shown that the coefficient matrix (38) is associated with rotation about the axis -(cos(2*ρz*),sin(2*ρz*),0) at the rate 2*δ* [rotation about the axis (cos(2*ρz*),sin(2*ρz*),0) at the rate -2*δ*]. In the laboratory frame, the trajectory is complicated because the rotation axis is not fixed: It rotates quickly about the 3-axis. The fast oscillations of the 3-component, which are evident in Figs. 4 and 5, are caused by the *δ* terms in Eqs. (27) and (28). If these contributions to the transfer functions are neglected, the tip of the vector rotates slowly about the negative 3-axis, as implied by the discussion that precedes Fig. 1.

In summary, the effects of propagation through a RSF are described approximately by the laboratory-frame transfer matrix

This approximate transfer matrix is the solution of the equation

where the approximate coefficient matrix

and the initial condition *U*_{l}
(0) = *I*. Equations ((39)–(41)) describe the rotation of the Jones vector, with rate -*δ*
^{2}/2*ρ*. Their analogs in Stokes space describe rotation about the 3-axis, with rate *δ*
^{2}/*ρ*.

## 3. Multiple-scale analysis of linear wave propagation

Typical single-mode fibers (SMFs) have beat lengths of order 10 m. In a recent FWM experiment with a twisted SMF [8], the twist length was of order 0.1 m. It is reasonable to consider spun fibers with comparable spin (rotation) lengths. Let *ε* denote the birefringence-to-spin ratio *δ*/*ρ* and *ζ* denote the distance variable *ρz*. Then, for the aforementioned RSFs, *ε* ≪ 1 and one can rewrite the transfer functions (24) and (25) approximately as

It is convenient to work in the rotating frame, because the wave equation has constant coefficients in that frame. (The subscript r is omitted for simplicity of notation.) Formulas (42) and (43) are correct to order *ε*, and conserve power to order *ε*, for distances of order 1/*ε*. One can also derive these formulas systematically, by using MSA [14].

Equations (21) and (22) imply that the propagation equation has the dimensionless form

where the differential operator *D* = *d*/*dζ*, the amplitude vector *A* = [*A*_{ξ}
,*A*_{η}
]^{t} and the matrix operator

One can solve Eq. (44) perturbatively by writing *A*=*A*
_{0} + *εA*
_{1} + *ε*
^{2}
*A*
_{2} and *L* = *L*
_{0} + *εL*
_{1}, and introducing the independent distance scales *ζ*
_{0} and *ζ*
_{2} so that *D* = *D*
_{0} + *ε*
^{2}
*D*
_{2}, where *D*_{j}
= *∂*/*∂ζ*_{j}
. By making these substitutions in Eq. (44), one finds that

Equation (46) can be solved order-by-order in powers of *ε*, subject to the initial conditions *A*
_{0}(0) = *A*(0) and *A*_{n}
(0) = 0 for *n* ≤ 1.

The zeroth-order equation is

where

The eigenvalues of *L*
_{0} are ±*i*, and the associated eigenvectors *E*
_{±} are [1,*i*]^{t}/2^{1/2} and [1,-*i*]^{t}/2^{1/2}, respectively, which correspond to (left and right) circularly-polarized eigenstates. Because the eigenvectors of *L*
_{0} are complete, one can write the zeroth-order amplitude in the form

where the amplitude components

At this stage, the dependences of *A*¯_{±} on *ζ*
_{2} are unknown. The first-order equation is

where

By making the substitution

in Eq. (51), and using the facts that *L*_{1}*E*
_{±} = *iE*
_{±}, one finds that

Equations (54) do not contain terms [proportional to exp(±*iζ*
_{0})] that drive the first-order components *B*
_{±} resonantly, so their solutions

are bounded (and no *ζ*
_{1} is necessary).

The second-order equation is

By making the substitution

in Eq. (56), and using Eqs. (49), (50), (53) and (55), one finds that

The resonant contributions to the right sides of Eqs. (58) will be absent if

It follows from these conditions that

To obtain a solution that is accurate to first order in *ε*, it is not necessary to calculate the bounded contribution to the second-order amplitude.

By combining the preceding formulas for *A*
_{0} and *A*
_{1}, one finds that

$$\phantom{\rule{2.5em}{0ex}}\left\{\mathit{i\epsilon}{\overline{A}}_{+}\left(0\right)\mathrm{sin}\left(z\right)+{\overline{A}}_{-}\left(0\right)\mathrm{exp}\left[-i\left(1+\frac{{\epsilon}^{2}}{2}\right)z\right]\right\}{E}_{-}.$$

Equation (61) describes the approximate solution in terms of circularly-polarized basis vectors. By using the facts that *E*
_{±} = (*E*
_{ξ} ± *iE*
_{η})/2^{1/2}, where *E*
_{ξ} = [1,0]^{t} and *E*
_{η} = [0,1]^{t}, and the related facts that *A*¯_{±}(0) = [*A*_{ξ}
(0) ∓ *iA*_{η}
(0)]/2^{1/2}, one can rewrite Eq. (61) in the form

$$\phantom{\rule{2.7em}{0ex}}\left\{-{A}_{\xi}\left(0\right)\mathrm{sin}\left(\mathit{kz}\right)+{A}_{\eta}\left(0\right)\left[\mathrm{cos}\mathit{\left(}\mathit{kz}\right)-\mathit{i\epsilon}\phantom{\rule{.2em}{0ex}}\mathrm{sin}\left(\mathit{kz}\right)\right]\right\}{E}_{\eta},$$

where *k* ≈ 1 + *ε*
^{2}/2. Solution (62) is consistent with Eqs. (42) and (43). Hence, MSA reproduces the known results for linear wave propagation in a RSF. By continuing MSA to third order in *ε*, one can prove that the first-order amplitude has the same wavenumber correction as the zeroth-order amplitude. This extended analysis is described briefly in Appendix B.

## 4. Multiple-scale analysis of nonlinear pulse propagation

Now consider nonlinear pulse propagation in a RSF. In the rotating frame, the amplitude component *A*
_{ξ} is governed by the generalized Schroedinger equation (GSE)

In Eq. (63) *∂z *= *∂*/*∂z*, *∂*_{τ}
= *∂*/*∂τ*, *τ* is the retarded time *t* - *sz*, where *s* is the polarization-averaged group slowness of the carrier wave, *δ* and *ρ* are the birefringence and spin coefficients, respectively, and *β*
_{1} and *β*
_{2} are the differential-convection and second-order dispersion coefficients *dβ*/*dω* - *s* and *d*
^{2}
*β*/*dω*
^{2}, respectively, evaluated at the carrier frequency. The Kerr nonlinearity

where *γ* is the Kerr coefficient, is based on the approximations that the nonlinear response is instantaneous and isotropic [15]. One obtains the equation for *A*_{η}
from Eq. (63) by interchanging the subscripts *ξ* and *η*, and changing the signs of the birefringence, spin and convection terms (*β*
_{1η} = -*β*
_{1ξ}). To include the effects of higher-order dispersion, one replaces the second-order term -*β*
_{2}
*∂*
_{ττ}
*A*/2 by the Taylor expansion ∑_{n≤2}
*β*_{n}
(*i∂*_{τ}
)^{n}
*A*/*n*!.

In typical FWM experiments, the dephasing lengths (associated with dispersion) and mixing lengths (associated with nonlinearity) are of order 1 Km. To model FWM in RSFs, it is reasonable to assume that the spin, birefringence and mixing lengths have the ratios 1:10^{2}:10^{4}. Similar ratios apply to the propagation of short intense pulses in RSFs. For such ratios, one can solve Eq. (63), and its analog for *A*_{η}
, perturbatively. The spin terms in these GSEs are of order 1, the birefringence terms are of order *ε*, and the differential-convection, dispersion and nonlinearity terms are of order *ε*
^{2}. (In this section the variables and parameters are dimensional, and *ε* is a book-keeping parameter.)

By proceeding as described in Section 3, one can write the GSEs in the matrix form

By defining the unitary matrices

one can write the zeroth- and first-order linear operators in the compact forms *L*
_{0} = *ρK* and *L*_{1}
= *iδJ*, respectively. It is convenient to write the second-order linear operator in the form *L*
_{2} = *L*
_{2a} + *L*
_{2d}, where the average and half-difference operators

respectively, *δ*_{a}
= (*δ*_{ξ}
+ *δ*_{η}
)/2 and *δ*_{d}
= (*δ*_{ξ}
- *δ*_{η}
)/2. The nonlinear vector

Equation (65) can be solved order-by-order in powers of *ε*. Because the new (convection, dispersion and nonlinearity) terms are of second order, the zeroth- and first-order equations are equivalent to the corresponding equations in Section 3. The zeroth-order equations (and the analysis of Section 3) imply that

and the first-order equations imply that

The second-order equation

contains the basic terms -*D*
_{2}
*A*
_{0} and *L*
_{1}
*A*
_{1}, like Eq. (56), and the extra terms *L*
_{2}
*A*
_{0} and *N*
_{2}(*A*
_{0}), which were defined in Eqs. (67)–(69). It follows from Eqs. (72) and (73) that the second term

Hence, the + component has the resonant part *i*(*δ*
^{2}/2*ρ*)*A*¯_{+} exp(*iρz*
_{0}) and the nonresonant part -*i*(*δ*
^{2}/2*ρ*)*A*¯_{+} exp(-*iρz*
_{0}), which can be neglected. It follows from Eq. (67) that

Because *A*
_{+} is proportional to exp(*iρz*0) [Eq. (71)], the whole + component is resonant. It follows from Eq. (68) that

Because *A*
_{-} is proportional to exp(-*iρz*0) [Eq. (71)], no part of the + component is resonant, so the whole component can be neglected. It follows from Eq. (69) that

$$\phantom{\rule{1.5em}{0ex}}+i\left(\frac{2\gamma}{3}\right)\left({2\mid {A}_{+}\mid}^{2}+{\mid {A}_{-}\mid}^{2}\right){A}_{-}{E}_{-}.$$

Because *A*
_{±} is proportional to exp(±*iρz*0),|*A*
_{+}|^{2}+2|*A*
_{-}|^{2} is constant on the *z*0-scale and the whole + component is resonant. Similar statements can be made about the - components in Eqs. (75)–(78). It follows from these observations that the resonant contributions to the right side of Eq. (74) will be absent if

$$\phantom{\rule{3em}{0ex}}+i\left(\frac{2\gamma}{3}\right)\left({\mid {\overline{A}}_{\pm}\mid}^{2}+2{\mid {\overline{A}}_{\mp}\mid}^{2}\right){\overline{A}}_{\pm}.$$

In the absence of dispersion and nonlinearity, Eqs. (79) reduce to Eqs. (59). It follows from Eqs. (71) and (79) that

$$\phantom{\rule{3em}{0ex}}+i\left(\frac{2\gamma}{3}\right)\left({\mid {A}_{\pm}\mid}^{2}+2{\mid {A}_{\mp}\mid}^{2}\right){A}_{\pm}.$$

Equations (80) apply to components that are defined relative to circularly-polarized basis vectors. By using the relations stated after Eq. (61), one finds that

$$\phantom{\rule{3em}{0ex}}+\mathit{i\gamma}\left({\mid {A}_{\xi}\mid}^{2}{A}_{\xi}+2{\mid {A}_{\eta}\mid}^{2}\frac{{A}_{\xi}}{3}+\frac{{A}_{\xi}^{*}{A}_{\eta}^{2}}{3}\right),$$

where the components are defined relative to linearly-polarized basis vectors. By comparing Eq. (81) to Eqs. (63) and (64), one finds that rapid spinning eliminates differential convection and dispersion, for distances of order 1/*ε*
^{2}, but has no effect on the Kerr nonlinearity, which is intrinsically isotropic. One obtains the equation for *A*_{η}
, from Eq. (81) by interchanging the subscripts *ξ* and *η*, and changing the sign of the (rotation) coefficient *ρ* + *δ*
^{2}/2*ρ*.

Equations (80) and (81) are valid in the rotating frame. One obtains the laboratory-frame equations by removing the terms proportional to *ρ*, as explained in Section 2. By doing so, one obtains the vector GSE

where *L* is the rotation matrix defined by Eq. (41) and *β*_{a}
(*i∂*_{τ}
) = ∑_{n≤2}
*β*_{na}
(*i∂*_{τ}
)^{n}/*n*!. The dispersion and nonlinearity terms have the same forms in the rotating and laboratory frames.

## 5. Discussion

By comparing Eq. (81) to Eq. (63), one finds that spinning a fiber rapidly suppresses the beating of the polarization components of the pulse, and their differential convection and dispersion, without affecting their nonlinear interaction. The only trace of birefringence and spinning that remains is polarization rotation at the (laboratory-frame) rate -*δ*
^{2}/2*ρ*. Because this rotation rate is the same for every frequency component of the pulse, the residual rotation has no significant effect on pulse propagation or FWM.

It is instructive to consider this result in detail. In a linearly-polarized basis, the residual effects of birefringence and spinning manifest themselves as slow rotation. Let *L* be the rotation matrix in Eq. (82) and let *U* be the solution of the equation *d*_{z}*U* = *LU*, subject to the boundary condition *U*(0) = *I*. Then, by making the substitution *A* = *UB* in Eq. (82), and using the fact that *U*
^{*} = *U*, one obtains the transformed vector GSE

in which the effects of rotation are absent. Alternatively, in a circularly-polarized basis, the residual effects of birefringence and spinning manifest themselves as weak circular birefringence (wavenumber shifts ±*δ*
^{2}/2*ρ*). By making the substitutions *A*
_{±} = *B*
_{±}exp(±${\delta}_{z}^{2}$/2*ρ*) in (the laboratory-frame) Eqs. (80), and using the fact that these scalar GSEs are incoherently-coupled (|*A*
_{±}|^{2} = |*B*
_{±}|^{2}), one obtains transformed GSEs for *B*
_{±}, in which the birefringence terms are absent.

The predictions of [8, 9], that FC driven by co-rotating pumps in a RSF does not depend on the signal polarization, were based on the assumptions that the linear response of the fiber is perfectly isotropic, and the nonlinear response is the full Kerr response (not modified by constant [16] or random [17, 18] birefringence). These physical assumptions led to a vector GSE that is mathematically equivalent to Eq. (83): Although the *a priori* omission of polarization rotation was not justified physically, the predictions made by the aforementioned GSE are correct.

If a fiber is twisted after the drawing process, the residual stress produces linear optical activity. One can model the effects of optical activity on pulse propagation by replacing the rotating-frame spin parameter *ρ* with the twist parameter *ρ* - *α*, where *α* ≈ 0.1*ρ* [1]. In the laboratory frame, slow polarization rotation at the rate -*δ*
^{2}/2*ρ* is replaced by moderately-fast rotation at the rate *α* - *δ*
^{2}/2(*ρ* - *α*). However, the rotation rate is still the same for every frequency component of the pulse. If one were to assume that the residual stress does not change the Kerr nonlinearity significantly, one would conclude that nonlinear pulse propagation and FWM in a rapidly-twisted fiber are similar to propagation and FWM in a RSF. This conclusion is consistent with the results of a recent experiment [8].

## 6. Summary

In this paper, multiple-scale analysis was used to derive a generalized Schroedinger equation that models optical pulse propagation in a rapidly-spun fiber [Eq. (82)]. This analysis was based on the assumption that (apart from the carrier wavelength of the pulse) the shortest length scale is the spin length, the intermediate scale is the birefringence beat-length, and the longest scale is the (common) dispersion and nonlinearity length. Because spinning rotates the birefringence axes of the fiber rapidly, the effects of birefringence (beating, differential convection and differential dispersion of the polarization components) are reduced significantly. In contrast, spinning does not change the nonlinear (Kerr) response of the fiber. For distances of interest, the only trace of birefringence and spinning that remains is the slow rotation of the pulse polarization. Because the rotation rate is the same for every frequency component of the pulse, this residual rotation has no significant effect on pulse propagation or four-wave mixing: Previous predictions [8, 9], that frequency-conversion driven by co-rotating pumps in a rapidly-spun fiber does not depend on the signal polarization, are correct.

## Appendix A: Other concatenated transformations

In Section 2, wave propagation through a lossless fiber was shown to be equivalent to wave propagation through a nonaligned retarder and a rotator. This equivalence exists because the transfer matrix of the fiber can be written as the product of the transfer matrices associated with the aforementioned devices. The transfer matrix of the fiber can be decomposed in other ways. For example, the composite transfer functions associated with a rotator, located between two aligned retarders, are

where *ϕ*
_{2} is the rotation parameter (half phase-difference) associated with the rotator (-*ϕ*
_{2} is the rotation angle), and *ϕ*
_{1} and *ϕ*
_{3} are the phase shifts (half phase-differences) associated with the initial and final retarders, respectively. Because the composite transformation is characterized by three real parameters, it is mathematically equivalent to transmission through a fiber. This representation of fiber transmission is common in quantal optics [19], because the rotator transfer functions (6) and (7) are the same as those for a beam-splitter, whose properties are well known.

By comparing Eqs. (84) and (85) with Eqs. (24) and (25), one finds that

and *ϕ*
_{3} = *ϕ*
_{1}. These results are valid in the rotating frame. One can convert them to laboratory-frame results by multiplying the transfer matrix associated with Eqs. (84) and (85) by the transfer matrix associated with a rotation through the angle *ρz*. However, because the last transformation associated with Eqs. (84) and (85) is a retardation, rather than a rotation, the laboratory-frame results are complicated. In the present context, the nonaligned retarder–rotator representation of fiber transmission is more convenient.

## Appendix B: Third-order multiple-scale analysis

In Section 3 it was shown that the zeroth- and first-order amplitudes can be written in the forms of Eqs. (49) and (53), respectively, where the amplitude components *A*
_{±} and *B*
_{±} satisfy Eqs. (50) and (54). The first-order equations have the solutions

where *B*¯_{±}(0) = *A*¯_{±}(0). At this stage, the dependences of *B*¯_{±} on *ζ*2 are unknown. If one makes the *a priori* assumptions that *B*¯_{±}(*ζ*2) = *A*¯_{±}(*ζ*2), which are consistent with the initial conditions, then Eqs. (88) reduce to Eqs. (55). However, these assumptions are unnecessarily restrictive.

The second-order amplitude can be written in the form of Eq. (57), where the components *C*
_{±} satisfy the equations

The resonant contributions to the right sides of Eqs. (89) will be absent if

It follows from these conditions that

Equations (90) and (91) are identical to Eqs. (59) and (60), respectively. It follows from Eqs. (89) that the bounded contributions to the second-order components are

where *C*¯_{±}(0)=*B*¯_{∓}(0).

The third-order equation is

By making the substitution

in Eq. (93), one finds that

$$\phantom{\rule{5em}{0ex}}-\frac{i\left[{\overline{C}}_{\mp}\mathrm{exp}\left(\mp i{\zeta}_{0}\right)-{\overline{B}}_{\pm}\mathrm{exp}\left(\pm i{\zeta}_{0}\right)\right]}{4}.$$

The resonant contributions to the right sides of Eqs. (95) will be absent if

It follows from these conditions, and the initial conditions stated after Eqs. (88), that

By combining Eqs. (88) and (97), one finds that

as stated in Section 3.

## Acknowledgment

We thank L. Schenato for his constructive comments on the manuscript.

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