Vector Theory of Optical Nonlinearities in Birefringent Fibers

Understanding the effects of optical nonlinearities on wave propagation is crucial for applications such as fiber optic communication, optical signal processing, frequency generation, signal amplification, optical sensing, and quantum optics. While optical nonlinearities such as self-phase modulation, cross-phase modulation, four-wave mixing, Brillouin scattering, and Raman scattering are extensively studied in optical fibers, a comprehensive model considering all of these nonlinearities for the most general case of a birefringent fiber has not been reported. In this paper, we present a vectorial model that considers the effects of these nonlinearities on co-propagating and counter-propagating fields for the most general case of birefringent fibers, i.e., elliptically birefringent fibers. Additionally, unlike previous studies, we represent the vector interaction using the frequency-dependent polarization eigenmodes of the fiber. We then use this model and show how elliptically birefringent fibers enable frequency-dependent control over the nonlinear interaction between counter-propagating fields.

In practical optical fibers, these nonlinearities co-exist and influence each other.Therefore, there is a need for a comprehensive model that can describe their combined effects for the most general case of fibers, i.e., elliptically birefringent fibers.Furthermore, elliptically birefringent fibers have two distinct characteristics in contrast with linearly and circularly birefringent fibers.First, its polarization eigenmodes are different for forward and backward propagation [39], [40], and second, these polarization eigenmodes are frequency-dependent [15], [38].Notably, no reported model currently addresses the combined effects of these nonlinearities on co-propagating and counterpropagating fields within these fibers.To highlight this gap in the existing literature, we provide a brief overview of relevant studies.
In [14], Crosignani and Yariv discussed the effects of SPM, XPM, and FWM on co-propagating and counter-propagating in linearly and circularly birefringent fiber.However, they did not consider the general case of elliptically birefringent fiber.Furthermore, they excluded the effects of SBS and SRS in their model.Lin and Agrawal formulated the vector theories of SPM and XPM [41], and FWM [26] for isotropic birefringent fibers, but they did not consider the general case of elliptically birefringent fibers, and analyzed these nonlinearities separately.In [34] Zadok et al. examined SBS in linearly birefringent fibers, but did not include SPM, XPM, and FWM.Moreover, they did not consider the general case of elliptical birefringence.In [35] Williams et al. discussed the vector theory of SBS in elliptically birefringent fibers, but they did not account for the frequency-dependence of the polarization eigenmodes.Additionally, they ignore SPM, XPM, and FWM in their model.In [13], [15], Menyuk discussed the effects of SPM, XPM, and FWM on co-propagating fields in elliptically birefringent fibers, but did not include an analysis of counter-propagating fields.In [16], [27], [28], [30], [31] authors have considered the effects of different nonlinearities on unidirectional propagation in a randomly birefringent fiber, but they did not consider its effects on bidirectional propagation.Additionally, they did not account for the frequency dependence of the polarization modes of these fibers.In [23], the authors discussed the effects of SPM, XPM, and FWM on bidirectional propagation in randomly birefringent fibers.They showed that polarization changes in these fibers can be averaged, which effectively reduces the fiber's nonlinear parameter.However, these models do not consider the frequency-dependence of the polarization modes and are unable to explain the selective suppression of SBS observed in [38].
To bridge this gap in the literature, we present a model explaining the vector interaction of co-propagating and counterpropagating fields for the most general case of elliptically birefringent fibers.An elliptically birefringent fiber accounts for 0733-8724 © 2024 IEEE.Personal use is permitted, but republication/redistribution requires IEEE permission.
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the linear birefringence arising due to axial asymmetry in the refractive index and the circular birefringence arising due to a twist in the fiber.Linearly and circularly birefringent fibers can be considered special cases of elliptically birefringent fibers, depending on whether they have no twist or are highly twisted (We will quantify "highly" in Section II).We represent the optical fields using the polarization eigenmodes of an elliptically birefringent fiber as the basis vectors.Additionally, unlike previous studies, we consider the frequency-dependence of these elliptically polarized eigenmodes [15], [38] in our model.Such a model can be potentially used as a one-stop solution to study the effects of different nonlinearities encountered by counterpropagating fields in general birefringent fibers.
In our model, we consider continuous-wave propagation in single-mode fibers and exclude the time dynamics.We focus on third-order nonlinearities (the lowest order of nonlinearity in fibers) such as SPM, XPM, FWM, SBS, and SRS, and ignore higher-order nonlinearities and spontaneous scattering effects.Additionally, we assume that nonlinear effects are small enough and they do not alter the polarization eigenmodes of the fiber.
In the following sections, first, we discuss the polarization eigenmodes of elliptically birefringent fiber.Then we present a vector theory that describes the interaction of co-propagating and counter-propagating fields in these fibers.Next, we use these theories to formulate a framework that encompasses various optical nonlinearities in general birefringent media.

II. POLARIZATION EIGENMODES OF BIREFRINGENT FIBERS
A typical birefringent fiber exhibits elliptical birefringence as it possesses both twist-induced birefringence and linear birefringence due to its axial asymmetry in the core.Wave propagation in an elliptically birefringent media can therefore be characterized by a propagation constant k bt , which is influenced by both the inherent linear birefringence (characterized by k b = 2π l b , where l b is the beat length of the untwisted fiber) and the twist rate (characterized by k t = 2π l t , where l t is the twist period of the fiber) The form of k bt can then be written as [39], [40].In some literature [15], [39], [40], an ellipticity parameter e is used to characterize these fibers, and the relationship between e, k b , and A linearly birefringent fiber has k b >> k t , and e approaches 0, whereas a circularly birefringent fiber is highly twisted, i.e., k t >> k b , and e approaches 1.Note that unlike linearly and circularly birefringent fibers, for elliptically birefringent fibers, e is dependent on the frequency of the optical fields [15], [38], [42] because it is a function of the frequency-dependent beat length of the fiber.
We use a rotating right-handed coordinate system (ξ, η, z) to model the twist in the fiber (see Fig. 1(a)).Here, the slow (ξ) and fast axes (η) of this system rotate along the length of the fiber, according to the twist rate of the fiber.Without loss of generality, we assume that at z = 0, the (ξ, η, z) coordinate system is aligned with the (x, y, z) coordinate system.The Jones vector of an optical field in the rotating coordinate system (ξ, η, z) can be obtained from the Jones vector in the right-handed Cartesian Fig. 1.(a) Illustration of the rotating coordinate system (ξ, η, z) used for our analysis.The slow and fast axes of (ξ, η, z) system rotates along the fiber, according to the twist rate (k t ).At z = 0, we assume that (ξ, η, z) is aligned with the Cartesian coordinate system (x, y, z).Polarization eigenmodes of (b) linearly (c) circularly, and (d) elliptically birefringent fibers for forward and backward propagation.Here, H, V, LC, and RC denote the horizontal, vertical, left-circular and right-circular polarizations.The eigenmodes of linearly and circularly birefringent fibers are identical for both forward and backward propagation but differ for elliptically birefringent fibers.Furthermore, the eigenmodes of elliptically birefringent fibers are frequency-dependent, and changing the frequency changes the polarization eigenmode (e).
coordinate system (x, y, z) using the coordinate rotation matrix R(θ), where θ = k t z [39], [40].The transfer matrix (T f ) of an elliptically birefringent fiber in (ξ, η, z) coordinate system for propagation in +z direction is given by [39], [40], [43], [44]: The transfer matrix of the fiber for propagation in the backward direction (−z) in the same right-handed (ξ, η, z) coordinate system is the transpose of the transfer matrix for the propagation in the forward direction (T b = T T f )1 [39], [40] (where T denotes the transpose of the matrix).We denote the eigenmodes of T f by Jones vectors (J 1 , J 2 ), and their eigenvalues by λ 1 and λ 2 , respectively.The eigenmodes of T b are conjugates of those of T f .We denote them by Jones vectors (J 1 , J 2 ), and their eigenvalues by λ 1 and λ 2 , respectively.The values of these eigenmodes and their eigenvalues are given as follows: , and ( 2) , and ( 4) where ω denotes the angular frequency of an optical field.We denote the propagation constants of the eigenmodes J 1 (ω) and J 2 (ω) by k 1 (ω) (k ; and for the backward-propagating modes J 1 (ω) and J 2 (ω) by k 1 (ω) (k The propagation vectors for forward and backward directions are pointing in different directions (+z and −z), and k 0 is the corresponding propagation constant.
The polarization eigenmodes of linearly birefringent fiber (e = 0) are linearly polarized (see Fig. 1(b).Note in the example provided in Fig. 1(b), without the loss of generality, we use H and V polarizations as the eigenmodes of the linearly birefringent fiber), and the polarization eigenmodes of circularly birefringent fiber (e = 1) are right and left-circularly polarized (see Fig. 1(c)).Since the ellipticity parameter of these fibers is frequency-independent, their polarization modes are also frequency-independent (see ( 2) and ( 4) for e = 0, 1).Additionally, the eigenmodes of these fibers are identical for forward and backward propagation (compare (2) and ( 4) for e = 0, 1).In contrast, elliptically birefringent fibers have two distinct characteristics: 1) their polarization eigenmodes are different for forward and backward propagation (see ( 2) and (4) for 0 < e < 1, and Fig. 1(d)), and 2) these eigenmodes are frequency-dependent (see (2) and (4) for 0 < e < 1, and Fig. 1(e)).
In the following sections, we discuss the case of the nonlinear interaction of co-propagating and counter-propagating fields in an elliptically birefringent fiber.We represent the forward (+z) and backward (−z) propagating fields using the basis vectors (J 1 , J 2 ) and (J 1 , J 2 ), respectively throughout our analysis.Furthermore, we assume that each field has two frequency components, but the analysis can be extended to include multiple frequency components.Additionally, to simplify our model, we consider uniform linear birefringence and twist along the length of the fiber, but the model can be extended to account for random birefringence by breaking the fiber into segments with uniform twist and birefringence.

III. VECTOR THEORY OF CO-PROPAGATING FIELDS IN ELLIPTICALLY BIREFRINGENT FIBERS
The case of co-propagating fields in elliptically birefringent fibers has already been discussed in [13], [15], and we include it here briefly as a review, as well as serving a comparison with the case of counter-propagating fields.
We consider an electric field E propagating in +z direction of the (ξ, η, z) coordinate system, oscillating at two distinct angular frequencies2 (see Fig. 2(a)), and write it as a Fig. 2. Schematic illustrating the modelling condition for (a) co-propagating, and (b) counter-propagating fields in an elliptically birefringent fiber.For an elliptically birefringent fiber, the polarization eigenmodes are different in forward (J 1 , J 2 ) and backward directions (J 1 , J 2 ), and they are frequencydependent.In (a) we launch two fields in +z direction with their polarization aligned with the forward-propagating modes J 1 and J 2 .In (b) we launch two fields in the +z direction and two fields in the −z direction.The polarization of forward-propagating fields is aligned with the forward-propagating modes J 1 and J 2 , and the polarization of backward-propagating fields is aligned with the backward-propagating modes J 1 and J 2 .For both (a) and (b), we consider two frequency components for each field, oscillating at angular frequencies ω m and ω n .
superposition of fields aligned with the forward-propagating eigenmodes (J 1 , J 2 ), i.e., E = E 1 + E 2 + c.c.Here, Here i = 1, 2, j = m, n, and F ij and A ij denotes transverse mode profile and the amplitude of the ith polarization mode oscillating at angular frequency ω j .Next, we are focusing on the C-band in this paper, and our calculations show that the effective mode area changes by 1.9% only over the C-band.To simplify our calculations, we assume that the effective mode area is nearly the same for the polarization modes oscillating at different frequencies, and this leads to assuming that F ij is nearly the same for these modes (i.e., Next, the electric field satisfies the following Helmholtz equation: [1], [3]. ) where i = 1, 2, j = m, n, c is the speed of light in free space, 0 is the permittivity of free space, P L i is the linear polarization driving the field, and P NL i is the nonlinear polarization driving the field E i .The nonlinear polarization P NL i is a sum of Raman (P R i ), Brillouin (P B i ), and nonlinear optical contributions (P (3) i , P (4) i , . . .), and it can written as follows: For more information regarding the nonlinear polarization contributions from Raman and Brillouin interactions (third-order contributions), readers can refer to [1], [3], [31].Note that the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

contribution of P
(2) i is zero in silica fibers as they exhibit inversion symmetry.Furthermore, we only consider the contribution of third-order nonlinearities to the nonlinear polarization and ignore higher-order terms (i.e., P NL i We list all the phase-matched nonlinear polarization driving the co-propagating fields in Table I of Appendix A. Here, we neglect the nonlinearities with a phase mismatch because their contributions average out to zero provided their period of oscillation is much smaller than the length of the fiber [1], [3].Next, we can simplify the nonlinear polarization by applying different symmetries of the fiber such as the Kleinman symmetry and intrinsic permutation symmetry [1], [3], [4], [8], [9], [45] by considering the relationships between the forward and backward-propagating modes of the fiber (see Appendix C where we consider different birefringent fibers and apply these symmetries of fiber to simplify the nonlinear susceptibilities).
Next, we solve (7) using the method of variable separation, apply the slowly varying envelope approximation, and obtain the following equation for the amplitude of the propagation optical fields [1].
Here γ is the nonlinear parameter, and it is defined in Appendix C. Note that in (9), is independent of the transverse mode profile F (ξ, η), and it can be obtained by rewriting P i NL in terms of the field amplitude.For example, if P NL has a term (E 1 (ω j ).E * 2 (ω j ))E 2 (ω j ), then the corresponding contribution to P NL is ((A 1j J 1 ) T (A 2j J 2 ) * )(A 2j J 2 ).Furthermore, note that there are four differential equations for the co-propagating fields which can be obtained by changing the mode index i and frequency index j in (9).

IV. VECTOR THEORY OF COUNTER-PROPAGATING FIELDS IN ELLIPTICALLY BIREFRINGENT FIBERS
To analyze the interaction of counter-propagating fields, we need to consider the interaction of the forward and backwardpropagating modes (see Fig. 2(b)).Note that as mentioned in Section II, the forward and the backward polarization eigenmodes are conjugates of each other (see ( 2)-( 5)).For a linearly (e = 0) or circularly birefringent (e = 1) fiber, the polarization eigenmodes in the forward and backward directions are identical (see Fig. 1(a) and (b)), and thus their analysis can be done using a similar approach as for the case of co-propagating fields [1], [3].For an elliptically birefringent fiber, the forward and backwardpropagating eigenmodes are different (see (2)-( 5)), and this makes the analysis of counter-propagating fields in elliptically birefringent fibers more complicated.Until now, such a theory explaining the effects of nonlinearities on counter-propagating fields for an elliptically birefringent fiber has not been reported in the literature, and it is the main contribution of this paper.
We consider a forward (+z) propagating field E and a backward (−z) propagating field E in the (ξ, η, z) coordinate system of an elliptically birefringent fiber, oscillating at angular frequencies ω m and ω n (see Fig. 2(b)).and E (10) Here, fields E i (E i ) are aligned to forward (backward) propagating eigenmodes and can be expressed similarly as (6).For the counter-propagating fields as well, one can write the Helmholtz equation and the nonlinear polarization in the same way as (7) and (8).Here also, we only consider third-order nonlinearities and ignore the higher-order terms.We list the nearly and perfectly phase-matched third-order nonlinear polarization driving the counter-propagating fields in Tables II and III of Appendix B. We neglect the phase-mismatched contributions for the same reason as stated in Section III.Next, we write down the differential equations for the amplitude of the counter-propagating fields using the Helmholtz equation and the slowly varying envelope approximation [1], [3].
where i = 1, 2, j = m, n, and γ is the nonlinear parameter (see Appendix C).In ( 11) and ( 12), ) is independent of the transverse mode profile F (ξ, η), and it can be obtained from P i NL (P NL i ) following a similar approach as stated in Section III.Furthermore, note that there are a total of eight differential equations for the counter-propagating fields and they can be obtained by changing the mode indices i and i , and frequency index j in (11) and (12).
In the following subsection, we present an application of this model to analyze the effects of these nonlinearities on counterpropagating fields in birefringent fibers with different e values.We also show how a deviation in the frequency influences these nonlinear interactions.To keep our analysis simple, we focus on SPM, XPM, FWM, and SBS as an example to highlight the impact of frequency deviation.

A. Effects of SPM, XPM, FWM, and SBS on Counter-Propagating Fields in Elliptically Birefringent Fibers
We consider a linearly birefringent fiber (e = 0, no twist), an elliptically birefringent fiber with equal values of l b and l t (e = 0.41, k t = k b ), an elliptically birefringent fiber with different values of l b and l t (e = 0.82, k t > k b ), and a circularly birefringent fiber (e = 1) for our analysis.The Stokes vectors of the forward and backward polarization eigenmodes of these fibers are plotted on the Poincaré sphere in Fig. 3.
For linearly and circularly birefringent fibers, the eigenmodes are identical in both directions and don't vary with frequency (see the orange and green vectors in Fig. 3).However, for elliptically birefringent fibers (e = 0.41, 0.82), the polarization modes are frequency-dependent, elliptically polarized, and different for each direction (see the maroon and blue vectors in Fig. 3).Note that for the elliptically birefringent fiber with e = 0.82, the forward and backward propagation are nearly orthogonal (i.e., . For elliptically birefringent fiber with equal values of l b and l t (e = 0.41), Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.the polarization overlap of forward and backward propagation modes are equal (i.e., J † . For fibers with e = 0.41 and 0.82, the input polarizations of E and E are aligned to one of their eigenmodes J 1 and J 1 , respectively (i.e., amplitude of E 2 and E 2 are zero).For fibers with e = 0, their polarizations are aligned to horizontal (H) and vertical (V ) polarizations, respectively; whereas for fibers with e = 1, their polarizations are aligned to left (LC) and rightcircular (RC) polarizations, respectively.We set the frequency difference between fields E and E (i.e., ω n − ω m ) equal to the Brillouin resonance frequency of these fibers (9.87 GHz) since the phase and amplitude changes are maximum at the resonance frequency.However, note that the effects presented in the following text can be observed at any frequency difference within the Brillouin gain bandwidth of the medium.
In Appendix C, we relate the nonlinear susceptibilities listed in Tables I-III, by using the symmetries of silica fiber and the ellipticity parameter e of the fiber.Then, we write the corresponding third-order nonlinear polarization in (18) and (19) of Appendix B and derive the following differential equations that govern the amplitude of the counter-propagating fields.
Here i, j = 1, 2 such that i = j and i , j = 1 , 2 such that i = j .In ( 13) and ( 14), the first term on the right-hand side denotes the contribution of SPM, XPM, and FWM, and the second term denotes the contribution of SBS.Here g B (W −1 m −1 ) is the Brillouin gain coefficient, and it has a Lorentzian lineshape [1], [3].For all our simulations in this section, we consider the following input parameters: the input power of the forward propagating field = −10 dBm, the input power of the backward-propagating field = 15 dBm, g B = 0.9 W −1 m −1 , and γ = 5 W −1 km −1 .Next, we solve these differential equations for specified input polarizations and compute the nonlinear gain and nonlinear phase difference 2 ) of two forward-propagating modes.We exclude the birefringence-induced phase difference from our calculations because it is fast varying, and its value can be easily computed by using the eigenvalues (see ( 3) and ( 5)) of these modes.Additionally, since the power of backwardpropagating modes is much higher than the forward ones, they acquire negligible gain and phase, and thus we have not plotted them.Next, we will examine how adjusting the frequency of one of the bidirectional fields impacts the phase and amplitude of those fields in three simulation scenarios: 1) SPM, XPM, and FWM are on, but SBS is off, 2) SBS is on, but SPM, XPM, and FWM are turned off, and 3) All effects are on.
1) Case 1. SPM, XPM, and FWM on, SBS Off: When the optical frequencies of the fields E and E are kept constant, and their input polarizations are aligned to the eigenmodes of the fiber, their polarizations are maintained along the length of all the fibers under consideration.Consequently, there is no FWM since only two modes are interacting.Additionally, the calculation of nonlinear phase difference is not applicable for E since only one mode is present.
Next, we change the angular frequency of E by Δω while keeping the polarizations of both the fields, and the frequency of E constant.For linearly (e = 0) and circularly birefringent fibers (e = 1), the polarization remains aligned with the eigenmode despite frequency deviation (see Section II), and thus there is no change in the nonlinear interaction with frequency deviation.However, for elliptically birefringent fibers, the polarization eigenmodes are frequency-dependent.Thus the polarization of E is no longer an eigenmode for this new angular frequency (ω m + Δω) and is not maintained along the fiber.We can write its polarization as a superposition of eigenmodes at frequency ω m + Δω, i.e., J 1 (ω m ) = αJ 1 (ω m + Δω) + βJ 2 (ω m + Δω) (where α and β are constants).The counter-propagating fields E 1 (ω m + Δω), E 2 (ω m + Δω), and E 1 (ω n ) influence the phase and amplitude with each other due to SPM, XPM and FWM.Furthermore, the FWM interaction between these fields gives rise to field E 2 (ω n ).Thus, with a frequency deviation, the forward-propagating field in elliptically birefringent fibers acquires a nonlinear phase and gain due to SPM, XPM, and FWM3 (see Fig. 4).Note that for fiber with e = 0.41, since both the forward-propagating modes have an equal overlap with the backward-propagating mode, 2 ) is calculated as the phase difference of the two forward-propagating modes.When the optical frequency of these fields is fixed and their input polarizations are aligned to the eigenmodes of the fiber, their polarizations are maintained along the fiber.Thus the phase difference calculation is inapplicable and there is no FWM because only one mode is present in the forward-propagating field.When we change the frequency of E by Δν (Δν = Δω/2π), the polarizations do not remain aligned to the eigenmode and are not maintained along the fiber.they acquire nearly the same phase and the phase difference is nearly zero (phase difference is not exactly zero since the amplitudes of the modes are different, see the dashed blue curve in Fig. 4(b)).For a fiber with e = 0.82, mode J 2 acquires a larger phase compared to mode J 1 because it is nearly parallel to the backward-propagation mode J 1 .Hence the phase difference is negative (see the dashed red curve in Fig. 4(b)).
2) Case 2. SBS on, SPM, XPM, and FWM Off: When there is no deviation in the frequency or polarizations of fields E and E , there is no SBS interaction in fibers with e = 0 and 1, since the fields are aligned to the orthogonal polarization eigenmodes of the fiber and they maintain their orthogonality throughout the fiber (see the dashed green curve in Fig. 5(a)).For a fiber with e = 0.41, when the frequency of E is kept constant, E maintains its polarization.But, it acquires a significant nonlinear gain due to the SBS interaction as the polarization overlap of E and E is large (J † 1 J 1 = 0.7) (see the dashed sky blue curve in Fig. 5(a)).For a fiber with e = 0.82, the polarizations are maintained when the frequencies of the bidirectional fields are kept constant.The polarizations of forward and backward-propagating fields are nearly orthogonal (J † 1 J 1 ≈ 0), and as a result, it acquires a ) of E is calculated as the phase difference of two forward-propagating modes.When the optical frequency of these fields is fixed and their input polarizations are aligned to the eigenmodes of the fiber, their polarizations are maintained along the fiber.Thus, their phase difference calculation is inapplicable since only one mode is present in E. When we change the frequency of E by Δν (Δν = Δω/2π), for fibers with e = 0.41 and 0.82, the polarizations do not remain aligned to the eigenmode and are not maintained along the fiber.small nonlinear gain (see the dashed pink curve in Fig. 5(a)).However, the nonlinear phase difference is not applicable for all the fibers under consideration because only one mode is present.
With a frequency deviation of the field E by Δω, the polarization of E is maintained in fibers with e = 0 and 1 since they are aligned to the eigenmodes of the fiber.As a result, there is no SBS interaction since the forward and backward-propagating fields are orthogonal (see the dashed magenta curve in Fig. 5(a)).For fibers with e = 0.41 and 0.82, the polarization of the field E deviates from the eigenmode when we deviate the E field's frequency by Δω, and they can be written as a superposition of the polarization eigenmodes at the new frequency 4 similar to Case 1.As a result, the amplitude of modes J 2 becomes non-zero.For a fiber with e = 0.41, both the forward modes J 1 and J 2 have an equal polarization overlap with the backward mode J 1 field.As a result, both the modes acquire equal nonlinear gain, but the overall gain of E remains the same as in the case of no frequency deviation because the amplitude of J 2 is much smaller compared to J 1 (see the dashed blue curve in Fig. 5(a)).Furthermore, both the modes acquire nearly the same nonlinear phase, and this makes their phase difference nearly zero (see the dashed blue curve in Fig. 5(b)).For a fiber with e = 0.82, J 1 is nearly orthogonal to J 1 , and J 2 is nearly parallel to it.As a result, mode J 2 acquires a high nonlinear gain and phase, whereas mode J 1 does not (see the dashed red curve in Fig. 5(a) and (b)).Consequently, the overall nonlinear gain and phase difference of E is higher than the case without frequency deviation.
3) Case 3.All Effects on: For fibers with e = 0 and 1, the forward-propagating field remains orthogonal to the backwardpropagating field with and without frequency deviation of the bidirectional fields, and hence it does not acquire any nonlinear gain due to FWM and SBS (see the dashed green and magenta curves in Fig. 6(a)).Additionally, since the polarization of the forward-propagating field remains aligned to one of the eigenmodes (H) of the fiber, the calculation of nonlinear phase difference is not applicable.
For fibers with e = 0.41 and 0.82, when we do not deviate the frequency of the field E, its polarization remains aligned to the input eigenmode, and there is no FWM interaction.The nonlinear gain remains the same as in Case 2. Additionally, the nonlinear phase difference calculation for E is again inapplicable since only one mode is present.
For fibers with e = 0.41 and 0.82, when the frequency of the field E is varied by Δω, while keeping its polarization fixed, its polarization no longer remains an eigenmode of the fiber at this new frequency.We can again write it as a superposition of the eigenmodes at this new frequency similar to Case 1.The nonlinear gain in E for all the fibers is the sum of nonlinear gain due to FWM in Case 1 and SBS gain in Case 2 (see Fig. 6(a)).For fiber with e = 0.41, since both the forward-propagating modes have an equal polarization overlap with the backwardpropagating mode, they acquire nearly equal phases, making the phase difference nearly zero (see the dashed blue curve in Fig. 6(b)).For a fiber with e = 0.82, forward-propagation mode J 1 is nearly orthogonal to J 1 , and J 2 is nearly parallel to it.As a result, mode J 2 acquires a high nonlinear phase compared to mode J 1 , making the phase difference negative (see the dashed red curve in Fig. 6(b)).
In summary, the strength of nonlinear interaction does not change for linearly and circularly birefringent fibers with a frequency deviation since their eigenmodes are frequencyindependent.For an elliptically birefringent fiber with equal values of l b and l t , the forward and backward propagation modes have an equal polarization overlap.Thus, the strength of nonlinear interaction remains nearly the same with frequency deviation, even though the eigenmodes are frequency-dependent.
In elliptically birefringent fibers with e = 0.82, the forward and backward modes are nearly orthogonal.Thus when we launch the input polarizations into these nearly orthogonal eigenmodes, they remain nearly orthogonal along the fiber and experience a weak nonlinear interaction.With a deviation in 2 ) of E is calculated as the phase difference of two forward-propagating modes.When the optical frequency of these fields is fixed and their input polarizations are aligned to the eigenmodes of the fiber, their polarizations are maintained along the fiber.Thus, the phase difference calculation is inapplicable since only one mode is present in the forward-propagating field.When we change the frequency of E by Δν (Δν = Δω/2π), for fibers with e = 0.41 and 0.82, the polarizations do not remain aligned to the eigenmode and are not maintained along the fiber.frequency, the polarizations no longer remain aligned to the eigenmode and experience a strong nonlinear interaction.Such a type of control over nonlinear interaction has potential applications in fiber optic communication, optical signal processing, and frequency synthesis.In fact, in [38], authors have experimentally demonstrated such control over SBS using a similar elliptically birefringent fiber.

V. CONCLUSION
In summary, we presented a vectorial model explaining the effects of all the third-order nonlinearities -SPM, XPM, FWM, SBS, and SRS on the co-propagating and counter-propagating fields for the most general case of birefringent fibers i.e., elliptically birefringent fibers.This model can be used to analyze linearly and circularly birefringent fibers as well by varying the beat length and the twist period of the fiber.Additionally, one can also use this model to study random birefringent fibers, by considering a cascade of segments of elliptically birefringent fibers with a uniform twist and linear birefringence.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Furthermore, in our model, we represented the fields using the basis of polarization eigenmodes of the fiber and considered the frequency-dependence of these eigenmodes as well.The polarization eigenmodes of a linearly and circularly birefringent fiber are identical in the forward and backward directions and do not change with frequency.As a result, when we launch counter-propagating fields aligned to the orthogonal eigenmodes of these fibers, then the strength of nonlinear interaction does not change with a frequency deviation.
However, the polarization eigenmodes of an elliptically birefringent fiber are different in forward and backward directions and are frequency-dependent.Thus when we launch counterpropagating fields aligned with one of the forward and backward eigenmodes of the fiber, then the strength of nonlinear interaction varies with frequency deviation.In fact, in an elliptically birefringent fiber with e = 0.82, for such an input condition, the fields experience a weak nonlinear interaction when the frequency and polarizations are kept fixed, and the strength of their nonlinear interaction increases with a change in frequency.We can leverage this behaviour to control different nonlinear interactions between the counter-propagating fields of similar elliptically birefringent fibers.Such a type of control over nonlinear interactions can have significant implications in fiber optic communication, optical signal processing, and frequency synthesis.It is important to note that this type of control is exclusive to elliptically birefringent fiber with different twist periods and beat lengths and cannot be observed linearly or circularly birefringent fibers cannot provide similar control.
In this paper, we analyzed birefringent fibers with a single transverse mode.However, our approach can be expanded to study nonlinear effects in multi-mode birefringent fibers, drawing from the existing literature on multi-mode fibers [46], [47].Additionally, in this paper, we only considered continuous-wave operations, and therefore nonlinear effects such as SPM, XPM, and FWM are not significant compared to SBS and SRS.In future, it would be interesting to study the bidirectional pulse propagation in elliptically birefringent fibers where optical nonlinearities such as SPM, XPM, and FWM can play a significant role.

APPENDIX A THIRD-ORDER NONLINEAR SUSCEPTIBILITIES FOR CO-PROPAGATING FIELDS IN ELLIPTICALLY BIREFRINGENT FIBERS
In this appendix, we list the nonlinear polarizations driving the fields E 1 and E 2 , co-propagating in an elliptically birefringent fiber (see Section III in the main text for their definitions).Note that both E 1 and E 2 have two frequency components ω m and ω n , and their polarizations are aligned to the polarization eigenmodes J 1 and J 2 , respectively.Since an elliptically birefringent fiber has frequency-dependent polarization modes, there are a total of four fields E 1 (ω m ), E 1 (ω n ), E 2 (ω m ), E 2 (ω n ), co-propagating in this fiber.
In Table I, we list the nearly and perfectly phase-matched third-order nonlinear polarizations that drive field E i (ω m ), where i = 1, 2 and indices 1, 2 denote the forward propagating fields E 1 and E 2 .We neglect the terms with a phase-mismatch because their contribution averages out to zero if their period of oscillation is much smaller than the length of the fiber [1], [3]. Here , and Δk 2 = −Δk 1 .Using the propagation constants listed in Section II of the main text, we find that the phases Δk 1 and Δk 2 are nearly zero when ω m ≈ ω n , and thus they have a non-zero contribution to the field.Now, following a similar approach, one can write the nonlinear polarization driving the field E i (ω n ).

APPENDIX B THIRD-ORDER NONLINEAR SUSCEPTIBILITIES FOR COUNTER-PROPAGATING FIELDS
In this appendix, we list the third-order nonlinear polarizations driving the fields E and E , propagating in the forward (+z) and backward (−z) direction of an elliptically birefringent fiber (see Section IV of the main text for their definitions).Forward-propagating field E is written as the superposition of fields E 1 and E 2 , aligned with the forward-propagating modes J 1 and J 2 , respectively; and the backward-propagating field E can be written as a superposition of fields E 1 and E 2 , aligned with the backward-propagating modes J 1 and J 2 .Furthermore, all of these fields have two frequency components ω m and ω n .Now, since an elliptically birefringent fiber has frequencydependent polarization eigenmodes, we need to consider the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE II PHASE-MATCHED NONLINEAR POLARIZATION DRIVING E i
interaction of eight fields: four propagating in +z direction (E 1 (ω m ), E 1 (ω n ), E 2 (ω m ), and E 2 (ω n )), and four propagating in −z direction (E 1 (ω m ), E 1 (ω n ), E 2 (ω m ), and E 2 (ω n )).At first, we consider nonlinear contributions to field E 1 (ω m ).Note that this field is influenced by four copropagating fields and four counter-propagating fields.The phase-matched nonlinear polarization contributions from the copropagating fields are listed in Table I, and in the Table II, we list the nearly and perfectly phase-matched nonlinear polarization contributions from the counter-propagating fields.Note that we have again neglected the terms with a phase-mismatch for the same reasoning as stated in Appendix A.Here i, j = 1, 2 and i , j = 1 , 2 such that i = j and i = j.Furthermore, indices 1, 2 denote the forward propagating fields E 1 and E 2 , and indices 1 , 2 denote the backward-propagating fields E 1 and E 2 .
In these tables, phase Δk This phase is nearly zero when ω m ≈ ω n , and thus the corresponding nonlinear polarization is nearly phase-matched.Now, one can follow a similar approach as taken for the tables above, and write the nonlinear susceptibilities for the remaining fields oscillating at angular frequency ω n .

APPENDIX C RELATIONSHIPS BETWEEN NONLINEAR SUSCEPTIBILITIES FOR BIREFRINGENT FIBER
We can relate the nonlinear susceptibilities listed in Tables I-III using different symmetries of the silica fiber such as Kleinman symmetry and intrinsic permutation symmetry [1], [3], [4], [8], [9], [45].Using these symmetries, we relate the following nonlinear susceptibilities.
χ i i i i = 3χ i j i j = 3χ i i j j = 3χ i j i j (17) Now, it is not straightforward to apply these symmetries to relate the remaining nonlinear susceptibilities in Tables I-III since for elliptically birefringent fibers the forward and backward propagation modes are different.However, we can use the ellipticity parameter e of the fiber (or equivalently the relationships between forward and backward propagation modes) and approximate these susceptibilities.In Section IV-A of the main text, we have considered three values of e for simulation purposes, and for these cases, we can follow the following approach to relate the nonlinear susceptibilities. 1) For e = 0, the fiber has no twist and is linearly birefringent.Thus, the forward and backward modes are identical and are horizontally and vertically polarized.To relate the remaining susceptibilities we can replace the mode index i with i and j with j in Tables I-III 2) For e = 0.41, the fiber is elliptically birefringent, and the twist rate k t is equal to linear birefringence k b .The forward and backward modes are different.For this value Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.  of e, it is difficult to estimate the strength of nonlinear susceptibilities theoretically, and we assume that the strength of all the remaining susceptibilities in Tables I-III is 3 times lower than χ iiii .
3) For e = 0.82, the fiber is elliptically birefringent, and its twist rate k t is five times the linear birefringence k b .Using ( 2) and ( 4) of the main text, we find that eigenmodes J 1 and J 1 (J 2 and J 2 ) of this fiber are nearly orthogonal.(i.e., J † 1 J 1 → 0 and J † 2 J 2 → 0).Thus, we can approximate the remaining susceptibilities in Tables I-III by replacing the mode index i (j) with j (i ) and vice-a-versa.Next, we consider the simulation scenario presented in Section IV-A.Then we determine the relationships between different susceptibilities for this scenario using the above-mentioned approach and ( 15)- (17), and write the P (3) for different values of e.

Fig. 3 .
Fig. 3. Poincaré sphere representation of the (a) forward and (b) backward propagation eigenmodes of a linearly birefringent fiber (e = 0, no twist), an elliptically birefringent fiber with equal values of l b and l t (e = 0.41, k t = k b ), an elliptically birefringent fiber with different values of l b and l t (e = 0.82, k t > k b), and a circularly birefringent fiber (e = 1).The polarization eigenmodes of linearly birefringent fibers are horizontal (H) and vertical (V ) polarizations, and for circularly birefringent fiber they are left (LC) and right circular (RC) polarizations.The eigenmodes of a linearly (e = 0) and circularly (e = 1) birefringent fiber are identical for forward and backward directions, and are frequency-independent, whereas, the eigenmodes of an elliptically birefringent fiber (e = 0.41, 0.82) are frequency-dependent, elliptically polarized, and different for each direction.We denote the forward and backward-propagation modes of elliptically birefringent fibers by (J 1 , J 2 ) and (J 1 , J 2 ), respectively.

Fig. 4 .
Fig. 4. (a) Nonlinear gain and (b) nonlinear phase of the forward-propagating field acquired via SPM, XPM, and FWM interaction in birefringent fibers with different e parameters.Input polarization of E(ω m ) and E (ω n ) fields are aligned to one of their polarization eigenmodes J 1 (ω m ) and J 1 (ω n ), respectively.The nonlinear phase difference (φ NL 1 − φ NL2 ) is calculated as the phase difference of the two forward-propagating modes.When the optical frequency of these fields is fixed and their input polarizations are aligned to the eigenmodes of the fiber, their polarizations are maintained along the fiber.Thus the phase difference calculation is inapplicable and there is no FWM because only one mode is present in the forward-propagating field.When we change the frequency of E by Δν (Δν = Δω/2π), the polarizations do not remain aligned to the eigenmode and are not maintained along the fiber.

Fig. 5 .
Fig. 5. (a) Nonlinear gain and (b) nonlinear phase of the forward-propagating field E acquired via SBS interaction in birefringent fibers with different e parameters.Input polarization of E(ω m ) and E (ω n ) fields are aligned to one of their polarization eigenmodes J 1 (ω m ) and J 1 (ω n ), respectively for e = 0.41 and 0.82.For e = 0 the input polarizations of E(ω m ) and E (ω n ) are aligned to horizontal (H) and vertical (V ) polarizations, whereas for e = 1, they are aligned to left (LC) and right-circular (RC) polarizations.The nonlinear phase difference (φ NL 1 − φ NL2 ) of E is calculated as the phase difference of two forward-propagating modes.When the optical frequency of these fields is fixed and their input polarizations are aligned to the eigenmodes of the fiber, their polarizations are maintained along the fiber.Thus, their phase difference calculation is inapplicable since only one mode is present in E. When we change the frequency of E by Δν (Δν = Δω/2π), for fibers with e = 0.41 and 0.82, the polarizations do not remain aligned to the eigenmode and are not maintained along the fiber.

Fig. 6 .
Fig. 6.(a) Nonlinear gain and (b) nonlinear phase of the forward-propagating field acquired via SPM, XPM, FWM, and SBS interaction in birefringent fibers with different e parameters.Input polarization of E(ω m ) and E (ω n ) fields are aligned to one of their polarization eigenmodes J 1 (ω m ) and J 1 (ω n ), respectively for e = 0.41 and 0.82.For e = 0 the input polarizations of E(ω m ) and E (ω n ) are aligned to horizontal (H) and vertical (V ) polarizations, whereas for e = 1, they are aligned to left (LC) and right-circular (RC) polarizations.The nonlinear phase difference (φ NL 1 − φ NL2 ) of E is calculated as the phase difference of two forward-propagating modes.When the optical frequency of these fields is fixed and their input polarizations are aligned to the eigenmodes of the fiber, their polarizations are maintained along the fiber.Thus, the phase difference calculation is inapplicable since only one mode is present in the forward-propagating field.When we change the frequency of E by Δν (Δν = Δω/2π), for fibers with e = 0.41 and 0.82, the polarizations do not remain aligned to the eigenmode and are not maintained along the fiber.

1 =
Δk 1 = −Δk 2 = −Δk 2 , with Δk 1 = (k 2 − k 1 ) + (k 2 − k 1 ).Additionally, c 1 , c 2 , c 3 , and c 4 are real constants and they denote the relative strengths of different susceptibilities.Their values are listed in TableIV.Next, it is quite common practice to normalize the amplitudes such that |A i | 2 denotes the optical power in the ith mode.To do this, we redefine|A i | 2 as |A i | 2 = 1 2 nc 0 A m |A i | 2, where A m denotes the effective mode area.Then, we define nonlinear Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE I PHASE
-MATCHED NONLINEAR POLARIZATIONS DRIVING E i (ω m ) TABLE III PHASE-MATCHED NONLINEAR POLARIZATION DRIVING E i

TABLE IV RELATIVE
STRENGTHS OF SUSCEPTIBILITIES FOR FIBERS WITH DIFFERENT e ω j ) cA m, andγ i = ω n n 2 (ω n ) cA m, where, (