1. Introduction

The Kerr nonlinear interaction of the two polarizations of the propagating modes of a waveguide leads to a host of physical effects that are significant from both fundamental and application points of view. Here, we develop a model of nonlinear interactions of the two polarizations using full vectorial nonlinear pulse propagation equations, with which we analyze the nonlinear interactions in the emerging class of subwavelength and high index optical waveguides. Based on this model we predict an anisotropy that originates solely from the waveguide structure, and which leads to switching states that can in principal be used to construct optical devices such as switches or logical gates. We derive the underlying nonlinear Schrödinger equations of the vectorial model with explicit integral expressions for the nonlinear coefficients. We analyze solutions of these nonlinear pulse propagation equations and the associated switching states by means of a Lagrangian formulation, which enables us to determine stability properties of the steady states; this formulation provides a global view of all solutions and their properties by means of the potential function and leads, for example, to the emergence of kink solitons as solutions to the model equations. We also use a Hamiltonian formalism in order to identify periodic and solitonic trajectories, including solutions that allow polarization flipping, and find conditions under which the unstable states and associated switching solutions are experimentally accessible. In order to provide examples of parameter values for which the predicted behavior occurs, we perform numerical calculations for waveguides with elliptical cross sections, although the underlying model is applicable to arbitrary fiber geometries.

The nonlinear interactions of the two polarizations of the propagating modes of a waveguide have been studied extensively over the last 30 years [1–13]. Different aspects of the interactions have been investigated, for example Stolen et al. [1] used the induced nonlinear phase difference between the two polarizations to discriminate between high and low power pulses. In the context of counterpropagating waves, the nonlinear interactions have been shown to lead to polarization domain wall solitons, [8–10] which are described as kink solitons representing a polarization switching between different domains with orthogonal polarization states. The nonlinear interactions can also lead to polarization attraction [9, 11–13, 15, 16] where the state of the polarization of a signal is attracted towards that of a pump beam. For twisted birefringent optical fibers, polarization instability [2, 5] and polarization domain wall solitons [17] have been reported. The nonlinear interactions also induce modulation instability which results in dark-soliton-like pulse-train generation [6,7]. Large-signal enhanced frequency conversion [18], cross-polarization modulation for WDM signals [10], and polarization instability [3] have also been reported and attributed to nonlinear polarization interactions. Stability behavior has been studied in anisotropic crystals [19].

The nonlinear interactions of the two polarizations can also be studied in the context of either nonlinear coherent coupling or nonlinear directional coupling in which the amplitudes of two or more electric fields, either the two polarizations of a propagating mode of a waveguide or different modes of different waveguides, couple to each other through linear and nonlinear effects [20–22]. Nonlinear directional coupling is relevant to ultrafast all-optical switching, such as soliton switching [23–27] and all-optical logic gates [28–30]. The interaction of ultrafast beams, with different frequencies and polarizations, in anisotropic media has also been studied and the conditions for polarization stability have been identified [24, 31].

In previous work ([32], Chapter 6), the nonlinear interactions of the two polarizations are described by two coupled Schrödinger equations. These equations employ the weak guidance approximation, which assumes that the propagating modes of the two polarizations of the waveguide are purely transverse and orthogonal to each other within the transverse x, y plane, perpendicular to the direction of propagation z. Based on this, the electric fields are written as

The weak guidance approximation works well only for waveguides with low index contrast materials, and large dimension structure compared to the operating wavelength. This approximation is, however, no longer appropriate for high index contrast subwavelength scale waveguides (HIS-WGs) [33–35]. These waveguides have recently attracted significant interest mainly due to their extreme nonlinearity and possible applications for all optical photonic-chip devices. Examples include silicon, chalcogenide, or soft glass optical waveguides, which have formed the base for three active field of studies: silicon photonics [36–40], chalcogenide photonics [41–43], and soft glass microstructured photonic devices [44–48].

In order to address the limitations of the scalar models in describing nonlinear processes in HIS-WGs, we have developed in [33] a full vectorial nonlinear pulse propagation model. Important features of this model are: (1) the propagating modes of the waveguide are not, in general, transverse and have large z components and, (2) the orthogonality condition of different polarizations over the cross section of the waveguide is given by $\int {\mathbf{e}}_1\left(x,y\right)\times {\mathbf{h}}_2^{*}\left(x,y\right)\blacksquare \widehat{\mathbf{z}}\hspace{0.17em}\text{d}A=0$, rather than simply e₁(x, y) ▪ e₂(x, y) = 0 as in the scalar models. These aspects lead to an improved understanding of many nonlinear effects in HIS-WGs; it was predicted in [33], for example, that within the vectorial model the Kerr effective nonlinear coefficients of HIS-WGs have higher values than those predicted by the scalar models due to the contribution of the z-component of the electric field, as later confirmed experimentally [46]. Similarly, it was also predicted that modal Raman gain of HIS-WGs should be higher than expected from the scalar model [49].

Here, we extend the vectorial model to investigate the nonlinear interaction of the two polarizations of a guided mode. The full vectorial model leads to an induced anisotropy on the dynamics of the nonlinear interaction of the two polarizations [50], which we refer to as structurally induced anisotropy, in order to differentiate this anisotropy from others, such as those for which the anisotropy originates from isotropic materials. The origin of the anisotropy is the structure of the waveguide rather than the waveguide material.

The origin of this anisotropy in subwavelength and high index contrast waveguides has also been reported by Daniel and Agrawal [35], who considered nonlinear interactions of the two polarizations in a silicon rectangular nanowire including the effect of free carriers. In their analysis, however, they ignore the coherent coupling of the two polarizations, considering the dynamics of the Stokes parameters only for a specific waveguide and ignore the linear phase.

This anisotropy in turn leads to a new parameter space in which the interaction of the two polarizations shows switching behavior, which is a feature of the vectorial model not accessible through the scalar model with the underlying weak guidance approximation. We also show that the resulting system of nonlinear equations, for the static case, can be solved analytically. Due to the underlying similarity of the nonlinear interaction of the two polarizations and the nonlinear directional coupling of two waveguides, the anisotropy discussed here can be also applied to the case of nonlinear directional coupling, in which the two waveguides have different effective nonlinear coefficients for the propagating modes.

This work develops and expands on results reported in [50,51], but in addition we derive (in Section 2) the equations that describe the nonlinear interactions of the two polarizations within the framework of the vectorial model, including all relevant nonlinear terms, with explicit integral expressions for all the nonlinear coefficients. In Section 3 we determine properties of the static solutions, classify the steady state solutions, and determine their stability using a Lagrangian formalism. We also discuss a Hamiltonian approach and how the phase space portrait provides a complete picture of the trajectories of the system, including the periodic and solitonic solutions (Section 3.5), and the relevance of the separatrix to the switching solutions. We derive analytical periodic solutions by direct integration of the system of equations in Section 4, and then discuss switching solutions and their properties. We relegate to the Appendix a mathematical analysis of the exact soliton solutions, which are relevant to the switching solutions, with concluding remarks in Section 5.

2. Nonlinear differential equations of the model

In the vectorial model the nonlinear pulse propagation of different modes of a waveguide is described by the equations:

The coupled Eqs. (2) describe the full vectorial nonlinear interaction of the two polarizations. There are two fundamental differences between these equations and the typical scalar coupled Schrödinger equations (see for example Chapter 6 in [32]). Firstly, the additional terms $A_{\mu }^{*}{A}_{\nu }^2$, A_μ |A_ν|², A_μ|A_μ|² on the right hand side of Eq. (2) represent interactions between the two polarizations. These do not appear in the scalar model since the effective nonlinear coefficients associated with these terms, ${\gamma }_{\mu \nu }^{(1)}$, ${\gamma }_{\mu \nu }^{(2)}$, ${\gamma }_{\mu \nu }^{(3)}$ as given in Eqs. (6)–(8), contain factors such as e_μ ▪ e_ν which are zero in the scalar model, since the modes are assumed to be purely transverse. All possible third power combinations of the two polarization fields, namely |A_ν|² A_ν, |A_μ|² A_ν, $A_{\mu }^2{A}_{\nu }^{*}$, $A_{\mu }^{*}{A}_{\nu }^2$, A_μ |A_ν|² and A_μ |A_μ|² occur on the right hand side of Eq. (2), due to the z-component of the modal fields. Secondly, in all effective nonlinear coefficients given by Eqs. (3)–(8), the modal fields e and h have both transverse and longitudinal components, unlike the scalar model in which modal fields have only transverse components. The terms containing nonzero e_μ ▪ e_ν provide a mechanism for the interaction of the two polarizations since they allow for exchange of power between the two modes through the z-components of their fields. The last term on the right hand side of Eq. (2), for example, indicates a coupling of power into a polarization, even if initially no power is coupled into that polarization.

Although the terms on the right hand side of Eq. (2) that contain e_μ ▪ e_ν are nonzero, they are generally significantly smaller than the remaining terms and are therefore neglected in the following; further investigation of the effects of these terms, and a discussion of their physical significance, will be presented elsewhere. The focus of this paper is to investigate the effect of the z-components of the fields e and h, which influence the values of the effective coefficients, and therefore also the nonlinear interactions of the two polarizations. Hence, from Eq. (2), we obtain the equations:

 

Fig. 1 Contour plots as functions of the elliptical waveguide dimensions x, y of (i) log₁₀ b; (ii) a as defined in Eq. (21) for P₀ = 1W; (iii) C = (γ₂ − γ_c)/γ′_c where C < 0 to the left of the white line; (iv) the birefringence Δβ.

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3. Static equations

We find now all solutions of Eq. (10) for the static case, in which the fields A₁, A₂ are functions of z only. We have therefore the two equations

In terms of these parameters we obtain the two equations:

At the initial time we have P₁, P₂ > 0 and so we always choose v₀ such that 0 < v₀ < 1. It may be shown from Eqs. (22) and (23) that 0 < v(τ) < 1 is then maintained for all τ > 0, i.e. the powers P₁, P₂ remain strictly positive at all later times. The constraint 0 < v₀ < 1 implies that the initial speed θ̇₀ is restricted, since it follows from Eq. (23) that |θ̇| ⩽ |a| + 2|b| + 1 at all times τ.

3.1. Properties of a, b

Of the two dimensionless parameters a, b, evidently b depends only on the optical fiber parameters, whereas a depends also on the total power P₀, unless Δβ = 0. For the scalar model, when Eqs. (11) are satisfied, we have b = 1 but generally b ≠ 1. In this case a set of steady state solutions appears (the states Eq. (24) discussed in Section 3.2 below) which for certain values of a, b are unstable. For fibers with elliptical cross sections we find that b > 1 and the unstable steady states exist provided 1 < a < 2b − 1. We have not, however, been able to eliminate the possibility that b < 1 for other geometries, and so in the following we also analyze the case b < 1. The parameter a can be positive or negative depending on the sign of Δβ and on the value P₀; when Eqs. (11) are satisfied we have a = −3Δβ/(P₀γ₁)+ 1 and hence a can take large positive or negative values for small P₀.

As an example, we have evaluated b using the definitions Eqs. (3)–(5) for step-index, air-clad glass waveguides with elliptical cross sections where the major/minor axes are denoted x, y. The host glass is taken to be chalcogenide with linear and nonlinear refractive indices of n = 2.8 and n₂ = 1.1 × 10⁻¹⁷m²/W at λ = 1.55μm (as in [52]). Figure 1(i) shows a contour plot of log₁₀ b as a function of x, y. We see, as expected, that b approaches 1 as the waveguide dimensions x, y increase towards the operating wavelength. For small core waveguides, however, we find b > 1 with values as large as b ≈ 200. The parameter a, on the other hand, depends on both the structure and the total input power P₀. For low input powers, specifically for P₀γ′_c ≪ |Δβ|, a can take large negative values (for Δβ > 0) or positive values (for Δβ < 0) as shown in Fig. 1(ii). For large values of P₀, however, a approaches the constant C = (γ₂ − γ_c)/γ′_c, whose contours for elliptical core waveguides are shown in Fig. 1(iii); most such waveguides have positive C values ranging up to 400, but some, those in the region on the left side of the white curve in Fig. 1(iii), have negative or small values of C. The contour plot for Δβ in Fig. 1(iv) shows that Δβ takes a wide range of positive and negative values as x, y vary.

3.2. Steady state solutions

There are four classes of steady state solutions of Eqs. (22) and (23), each of which exist only for values of a, b within certain limits, as follows:

Of these four classes, the values Eqs. (26) and (27) lie on the boundary of the physical region 0 < v < 1, but nevertheless influence properties of nearby nontrivial trajectories, and also play a role in soliton solutions. The states Eq. (24) lie within the physical region only if the parameters (a, b) belong to either the red or green region of the a, b plane shown in Fig. 2(i). Similarly the solutions Eq. (25) satisfy 0 < v < 1 only in the disjoint regions of the a, b plane defined by either 2b + 1 < a < −1 or −1 < a < 2b + 1. If a, b lie outside these regions, and also outside the strips given by |a| ⩽ 1 and |a − 2b| ⩽ 1, there are no steady state solutions.

 

Fig. 2 The a, b plane showing: (i) the regions of existence for the solutions Eq. (24), either 1 < a < 2b−1 (red), or 2b−1 < a < 1 (green); (ii) the regions of existence for the unstable solutions consisting of Eq. (24) (red), and Eq. (25) for which 2b + 1 < a < −1 (orange), together with Eqs. (26) and (27) for which |a| < 1 or |a − 2b| < 1 (light blue).

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For special values of a, b these steady states can coincide, for example if a = 1 the solution Eq. (26) coincides with the boundary value of Eq. (24). Steady states for values of a, b on the boundary of the regions shown in Fig. 2 may need to be considered separately; for example if a = b = 1 then all steady states are given either by Eq. (25), or else by cosθ = 1 and any constant v.

In practice, the values of a, b are determined by the waveguide structure, the propagating mode and, in the case of a, the input power P₀, and hence only restricted regions of the a, b plane are generally accessible. For example, Fig. 1(i) shows that for the fundamental mode of elliptical core fibers we have log₁₀ b ⩾ 0, and so the attainable values of b are limited to b ⩾ 1. We nevertheless include the case b < 1 in our analysis, since this possibility cannot be excluded for other fiber geometries. We discuss the accessible regions for the case of unstable steady states in Section 3.4.

3.3. Lagrangian formulation

We wish to determine the stability properties of each of the four classes of steady state solutions, in particular we look for unstable steady states. These are of interest because polarization states which lie close to these unstable states are very sensitive to small changes in parameters such as the total power P₀, and so can flip abruptly as a function of the optical fiber length z. Although we may determine stability properties by investigating perturbations about the constant solutions, we find it convenient to reformulate the defining Eqs. (22) and (23) as the Euler-Lagrange equations of a Lagrangian L which is a function of θ, θ̇, and depends otherwise only on the parameters a, b. This also provides insight into the properties and solutions of these equations, and we may then investigate stability by examining the corresponding potential function. From Eq. (23) we have

We consider Lagrangians L of the form

We may therefore investigate all possible solutions θ(τ) by analyzing properties of the periodic potential V(θ); every solution of the system of Eqs. (22) and (23) corresponds to the trajectory θ(τ) of a particle of variable mass M in the potential V. Steady state solutions are zeroes of V′(θ), and stability is determined by whether these zeroes are local maxima or minima of V, subject to the constraint that the associated function v should always satisfy 0 < v < 1. Trajectories which begin near a local minimum, with a small initial speed θ̇(0), oscillate periodically with a small amplitude. On the other hand, trajectories which begin near an unstable point, i.e. near a local maximum of V, can display periodic oscillations of large amplitude with abrupt transitions between adjacent local maxima; we refer to these as switching solutions (previously bistable solutions [50]) since cosΔϕ = cos(θ/2) switches periodically between two distinct values. Soliton trajectories also occur in which the particle moves between adjacent local maxima of V, see for example the discussion in [53], Section 2 and [54] for properties of solitons in optical fibers. As mentioned in Section 3.5, soliton trajectories also appear as the separatrix in phase plane plots.

We plot V as a function of θ and either a or b in Fig. 3, showing that V defines a complex surface with valleys and peaks which change suddenly as a or b are varied. Periodic solutions occur for trajectories restricted to a local valley, but there are also unbounded trajectories, in which θ increases or decreases indefinitely, depending on a, b and on whether θ̇(0) is sufficiently large. The potential, as a function of θ and a, has saddle points which indicate that a stable solution can become unstable as a is varied; according to the definition Eq. (21) we may vary a within certain limits by varying the total power P₀.

 

Fig. 3 The potential V plotted as a function of (i) θ, a for b = 0.8; (ii) θ, b for a = 0.

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For a = b the potential is essentially that of the nonlinear pendulum under the influence of gravity, namely a simple cosine potential, but with a mass that depends on θ. Provided b > 1 this mass varies between two positive, finite limits. The unstable steady states correspond to a pendulum balanced upright, while the switching states (discussed in Section 4) correspond to trajectories which begin with the pendulum positioned near the top, possibly with a small initial speed, then swinging rapidly through θ = 2π to reach the adjacent unstable steady state. During this motion $\text{cos}\mathrm{\Delta }\varphi =\frac{\text{cos}\theta }{2}$ flips rapidly between the values ±1. The soliton discussed in the Appendix is the trajectory in which the pendulum begins at the unstable upright position and, over an infinite time, moves through the stable minimum to the adjoining unstable steady state.

Although both M and V are singular when cosθ = b, which occurs only if |b| ⩽ 1, this singularity is an artifact of the Lagrangian formulation, as is evident from Eqs. (22) and (23), which have smooth bounded right hand sides for any b. In particular v, which is obtained from Eq. (28) given θ, is a smooth function of τ even if cosθ = b for some τ.

The energy $T+V=\frac{1}{2}M(\theta ){\dot{\theta }}^2+V(\theta )$ is a constant of the motion. Hence we may integrate Eq. (29) to obtain

A limitation of the Lagrangian formulation is that the constraint 0 < v < 1 is not easily implemented. Whereas every solution of the system Eqs. (22) and (23) defines a trajectory θ(τ) in the Lagrangian system Eq. (29), the converse is not true, i.e not all trajectories in this system satisfy 0 < v < 1. The initial speed θ̇(0) must be restricted to only those values allowed by Eq. (23), in which 0 < v(0) < 1, and similarly the constant solutions of Eq. (29) are valid steady states for the original Eqs. (22) and (23) only in certain regions of the a, b plane. Trajectories which violate 0 < v < 1, while not physical in the context of optical fiber configurations, can nevertheless be viewed as acceptable motions of the mechanical system defined by the Lagrangian Eq. (30). We investigate an alternative Hamiltonian formulation in terms of v in Section 3.5.

3.4. Stability of steady state solutions

The stability of each of the four classes of steady state solutions in Section 3.2 is determined by the sign of V″ for that solution; a positive sign implies that the solution lies at a local minimum of V and is therefore stable, whereas a negative sign implies that the solution is unstable.

For the steady states Eq. (24) we have V″ = (a − 1)(a − 2b + 1)(b − 1)/|b − 1|³ and so these states are stable for points a, b such that (a−1)(a−2b+1)(b−1 > 0, shown as the green region in the a, b plane in Fig. 2(i), and are unstable in the red region, where 1 < a < 2b − 1. For the steady states Eq. (25) we have V″ = (a+1)(−a+2b+1)(b+1)/|b+1|³ and so these solutions are stable for −1 < a < 2b+ 1 and are unstable in in the orange region 2b+ 1 < a < −1 shown in Fig. 2(ii).

For the remaining steady states Eqs. (26) and (27), for which v = 0 or v = 1, we have V″ = −2sin²θ/|a − b| which in all cases is negative, and so these states are unstable whenever they exist. This is consistent with the observation that v(τ) cannot attain the values 0, 1 at any time τ, provided 0 < v₀ < 1. The regions in the a, b plane where the unstable states exist are shown in Fig. 2(ii).

Next, we determine conditions under which the unstable steady state solutions Eq. (24) are accessible. For elliptical core step index fibers, for which b > 1 as shown in Fig. 1(i), the region of instability is indeed accessible and leads to properties such as nonlinear self-polarization flipping, discussed in Section 4. The region of unstable solutions is given by 1 < a < 2b − 1, equivalently

 

Fig. 4 (i) a as a function of P₀ for Δβ < 0 (blue solid line) and Δβ > 0 (red solid line). The green lines mark the boundaries of the (red) region of instability in the a, b plane shown in Fig. 2(i); (ii) contour plot of log₁₀( $P_0^{\text{min}}$) as a function of x, y, showing the minimum total power $P_0^{\text{min}}$ (in units W) required to access unstable steady states, where they exist.

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First we consider fibers for which 1 < C < 2b − 1, where C = (γ₂ − γ_c)/γ′_c takes the value shown by the dashed line in Fig. 4(i). Then a has two branches associated with either Δβ < 0 or Δβ > 0; for the branch corresponding to Δβ < 0 (the solid blue line), a is large and positive for small P₀ and asymptotically approaches C for large P₀. The intersection of this branch with the boundary a = 2b − 1 determines the minimum power P_min1 required in order to access the unstable region. In this case, only part of the unstable region corresponding to C < a < 2b−1 is accessible, as shown by the blue region. For the Δβ > 0 branch (red solid curve) a is large and negative for small P₀ and asymptotically approaches C for large P₀. For this branch, P₀ needs to be larger than a value P_min2. The unstable region is accessible provided 1 < a < C and is a subset (red shaded) of the whole unstable solution region. Figure 4(i) allows one to determine the minimum and maximum values of a and the minimum power to access the unstable solution region, once Δβ and C are known. For elliptical core fibers these two values are completely determined by the dimensions x, y, see Fig. 1(iii,iv) for plots of C and Δβ.

Besides fibers for which 1 < C < 2b − 1, there are the possibilities C > 2b − 1 or C < 1. From Fig. 1(iii,iv) one can show that these combinations (with Δβ positive or negative) either do not exist, or do not lead to unstable solutions, since the possible values of a do not lie in the unstable region 1 < a < 2b − 1. In summary, the only elliptical core fibers that allow unstable solutions are those with 1 < C < 2b − 1 with either positive or negative Δβ. The case in which Δβ = 0 is discussed separately in [55, 56].

Based on the above discussion, one can find the minimum power $P_0^{\text{min}}$ required to generate unstable solutions for elliptical core fibers. Figure 4(ii) plots log₁₀( $P_0^{\text{min}}$) (where $P_0^{\text{min}}$ is measured in watts) as a function of x, y, where the white region corresponds to fibers for which there are no unstable solutions, and the regions below and above the diagonal line correspond to P_min1 and P_min2, respectively, which have been obtained for the two branches of the function a(P₀) shown in Fig. 4(i). In these examples of elliptical fibers very high powers, in the range 1 − 10kW, are required in order to observe switching solutions, as we discuss further in our concluding remarks in Section 5. The underlying theory shows, however, that this behavior can occur at much lower powers for waveguides with small birefringence Δβ; specifically, it is necessary only that the inequalities Eq. (34) be satisfied, and so the total power P₀ required is small provided that Δβ is sufficiently small [55].

3.5. Hamiltonian function

Although the Lagrangian formulation in terms of θ is convenient for an analysis of the steady states and their stability, and also for a qualitative understanding of all solutions including solitons, the constraint 0 < v < 1 is more easily implemented by means of a direct formulation in terms of v. This automatically eliminates unphysical trajectories for which one of the input powers P₁, P₂ is negative. Such a formulation follows by construction of a Hamiltonian function which, being conserved, allows us to firstly integrate the nonlinear equations and obtain analytical solutions and, secondly, to interpret physically the possible states of polarizations within an optical waveguide from the phase plane contours. Corresponding to the conserved energy T +V which follows from Eq. (30) there is a Hamiltonian function H defined by

We can gain insight into possible solutions by plotting contours of constant H(v, θ) in the v, θ plane, which supplies essentially a phase portrait of the system. Solutions for which both v, θ are periodic in τ form closed loops, and lie close to a stable steady state, whereas nonperiodic trajectories lie outside the separatrix which defines soliton solutions, as we discuss in the Appendix. Figure 5 shows two examples in which stable steady states are marked in green, and unstable steady states are shown in red or orange. Periodic solutions are evident as closed loops surrounding stable steady states, whereas the separatrix marks soliton trajectories which connect unstable steady states. Apart from these solitons, all other solutions v, cosθ (but not necessarily θ) are periodic in τ. The switching solutions of particular interest, in which the state of polarization inside the waveguide flips between two well-defined states, are those close to the separatrix.

 

Fig. 5 Contours in the θ, v plane of constant H for (i) a = 1, b = 4; (ii) a = b = 2, with steady states marked by green dots (stable) and red or orange dots (unstable). The separatrix, which identifies the soliton trajectories, is shown in red.

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4. Periodic solutions

Periodic solutions v of Eq. (37) attain both minimum and maximum values, denoted v_min, v_max respectively, with 0 < v_min ⩽ v_max < 1. Since v̇ = 0 at a maximum or minimum of v, both v_min, v_max are roots of Q. We can factorize Q as a product of quadratic polynomials,

Having found v, cosθ is obtained from Eq. (36) and is also periodic in τ, as is θ̇ which is obtained from Eq. (23), however θ itself need not be periodic. Although it is straightforward to find v,θ numerically as functions of τ, for specified numerical values of a, b and initial values v₀, θ₀, the exact solutions are useful because they display the exact dependence of the solution on all parameters, such as the total power P₀; it is not necessary therefore to solve the equations numerically for every choice of P₀, rather the exact solution gives the explicit periodic solution and the period as known functions of P₀.

For switching solutions, the phase difference between the two polarization vectors experiences abrupt phase shifts through π as the light propagates within the waveguide. As a result, the state of polarization flips between two well-defined polarization states, where the flipping angle depends on a, b and on θ₀, v₀. The following are two examples of switching solutions.

As the first example we choose a = 1, b = 4 with the initial values v₀ = ε, θ₀ = 0, where ε = 10⁻⁴, in which case the input laser beam is linearly polarized and the polarization state is close to one of the principle axes of the waveguide. Hence, the trajectory starts near the unstable steady states Eq. (24) or Eq. (26), which lie on the boundary of the red region shown in Fig. 2(i). We plot v and $\text{cos}\frac{\theta }{2}=\text{cos}\mathrm{\Delta }\varphi$ as a function of τ in Fig. 6(i), showing switching behavior for $\text{cos}\frac{\theta }{2}$, which is periodic and flips abruptly between the values ±1; θ, however, is an increasing function of τ, with jumps through 2π at periodic intervals. The polarization vector experiences an angular flipping associated with the abrupt flipping of cosΔϕ, however, since v₀ = ε and θ₀ = 0, the flipping angle is very small, as depicted in the inset of Fig. 6(i). Regarded as the trajectory of a particle of mass M in the potential V in Eq. (32) this motion corresponds to a particle moving slowly over the peaks of the potential, which are the unstable steady states, then sliding quickly down the valleys through the minimum values of V and back to the peaks. For a = 1 the potential is flat at its maximum values, since in this case V′ = 0 = V″ = V′″, hence v, θ̇ are each close to zero except when θ moves to an adjoining maximum of V. In terms of the contour plots shown in Fig. 5(i) this trajectory corresponds to the contour which begins just above the unstable steady state (orange dot) and closely follows the separatrix shown in red (which is the soliton solution discussed in the Appendix) with a maximum value ∼ 0.4 for v.

 

Fig. 6 Switching solutions v and $\text{cos}\frac{\theta }{2}=\text{cos}\mathrm{\Delta }\varphi$ as functions of τ for: (i) a = 1, b = 4 and v₀ = ε, θ₀ = 0; (ii) a = b = 2 and $v_0=\frac{1}{2}$, θ₀ = ε where ε = 10⁻⁴. The insets show the polarization vectors associated with the values cosΔϕ = ±1.

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As a second example of switching behavior we choose a = b = 2 with v₀ = 1/2, θ₀ = ε, where ε = 10⁻⁴, which corresponds to a linearly polarized input laser beam in which the polarization vector makes an angle of 45° to either of the principle axes of the waveguide. Again, the initial value lies close to an unstable steady state Eq. (24) and a, b lie within the red region of instability in Fig. 2. We plot v and cos(θ/2) as functions of τ in Fig. 6(ii), showing the periodicity of these functions and the switching behavior of cos(θ/2). Since v₀ = 1/2, the angular flipping of the polarization vector is π/2, because cos(θ/2) flips between values ±1, as shown in the inset of Fig. 6(ii). Unlike the previous example, θ is also periodic in τ with a trajectory that corresponds to the motion of a particle in the potential V, starting slowly near the unstable steady state Eq. (24) but sliding rapidly through the potential minimum to approach an adjoining unstable steady state. This motion is similar to the periodic oscillations of a nonlinear pendulum (since a = b, see the definition of V in Eq. (32)) with a large amplitude of almost 2π, and v attains nearly all values between 0, 1. In terms of the phase space contours shown in Fig. 5(ii), the motion corresponds to a periodic trajectory which begins near the red dot (unstable steady state) and again closely follows the separatrix which marks the soliton trajectory.

5. Discussion and conclusion

Switching states, as defined and demonstrated here through simulation by means of a full vectorial model, are attractive for practical applications, since they allow nonlinear self-flipping of the polarization states of light propagating in an optical waveguide. This flipping is due to the nonlinear interactions of the two polarizations, and has properties that depend on the total optical power and on the specific fiber parameters. These properties can in principle be employed to construct devices such as optical logic gates [58], fast optical switches and optical limiters [55, 56], in which small controlled changes in the input parameters lead to sudden changes in the polarization states.

The minimum power necessary to generate such switching states is determined for any waveguide by the inequalities Eq. (34) and, for chalcogenide optical nanowires with elliptical core cross sections, is summarized in Fig. 4(ii). The minimum power required in such nanowires is in the range 1 – 10kW which, although not practicable for CW lasers, can be achieved in pulsed lasers. Although we have limited our analysis to the static case, ignoring the temporal variation of laser light, it is still applicable to slow pulses with pulse widths in the order of nanoseconds depending on the dispersion of the waveguide. A more practical minimum power requirement that achieves switching behavior is by means of asymmetric waveguides, such as rib waveguides, for which Δβ can be reduced to very small values while still having different field distributions for the two polarizations, as discussed in [55, 56].

The nonlinear interactions of the two polarizations can be impacted by two factors that have not yet been investigated: (1) interactions with higher order modes in few-mode waveguides and, (2) contributions from nonlinear terms containing different forms of e₁ ▪ e₂, i.e., nonzero values for the coefficients ${\gamma }_{\mu \nu }^{(1)}$, ${\gamma }_{\mu \nu }^{(2)}$, ${\gamma }_{\mu \nu }^{(3)}$ in Eqs. (6)–(8). (This applies only when e₁ ▪ e₂ is no longer very small, as assumed in this paper). In few-mode waveguides, higher order modes contribute to the nonlinear phase of each polarization of the fundamental mode through cross phase and coherent mixing terms. Inspection of Eq. (2) reveals that nonzero ${\gamma }_{\mu \nu }^{(1)}$, ${\gamma }_{\mu \nu }^{(2)}$, ${\gamma }_{\mu \nu }^{(3)}$ coefficients significantly change the dynamics of nonlinear interactions of the two polarizations and most likely lead to different parameter regimes for the existence of periodic and solitonic solutions. These factors will be the subject of further studies.

In summary, we have developed the theory of nonlinear interactions of the two polarizations using a full vectorial model of pulse propagation in high index subwavelength waveguides. This theory indicates that there is an anisotropy in the nonlinear interactions of the two polarizations that originates solely from the waveguide structure. We have found all static solutions of the nonlinear system of equations by finding exact constants of integration, which leads to expressions for the general solution in terms of elliptic functions. We have analyzed the stability of the steady state solutions by means of a Lagrangian formalism, and have shown that there exist periodic switching solutions, related to a class of unstable steady states, for which there is an abrupt flipping of the polarization states through an angle determined by the structural parameters of the waveguide and the parameters of the input laser. By means of a Hamiltonian formalism we have analyzed all solutions, including solitons which we have shown are close to the switching solutions of interest.