## Abstract

We numerically investigate polarization instability of soliton fission and the polarization dynamics of Raman solitons ejected during supercontinuum generation in a photonics crystal fiber using the coupled vector generalized nonlinear Schrödinger equations for both linear and circular birefringent fibers. The evolution of the state of polarizations of the ejected Raman soliton as representated on the Poincaré sphere is affected by both nonlinear and linear polarization rotations on the Poincaré sphere. The polarization dynamics reveal the presence of a polarization separatrix and the emergence of stable slow and unstable fast eigen-polarizations for the Raman solitons ejected in the supercontinuum generation process. Circularly birefringent fiber is investigated and found to simplify the nonlinear polarization dynamics.

© 2015 Optical Society of America

## 1. Introduction

Polarization instability (PI) of solitons in birefringent fiber has been the subject of numerous investigations because of its fascinating underlying physics and potential for practical applications in nonlinear optics. In a birefringent nonlinear fiber, the intensity-dependent refractive index leads to PI for intense continuous-wave (CW) laser field polarized along the fast-axis [1, 2]. The state of polarization (SOP) of the light during propagation along the birefringent fiber can be represented on a phase plane graph which reveals a polarization separatrix, at which the nonlinear index is commensurate with the birefringence. Similarly, interesting optical bifurcation phenomena associated with this fast-axis instability has been reported [3] and the SOP trajectories were represented on the Poincaré sphere [4].

Unlike the case of CW light, optical solitons require lower average pulsed laser power to observe PI [5, 6]. Unlike quasi-CW light, optical solitons evolve as a unit [7, 8] that can be characterized by a single SOP qualitatively following the CW theory except the soliton stabilizes due to the interplay of dispersion and nonlinearity [9]. The propagation of solitons in birefringent fiber was investigated [9, 10] considering the interactions of its two orthogonal polarization components. Subsequently, vector solitons in birefringent nonlinear fiber were theoretically demonstrated using coupled nonlinear Schrödinger equations [11]. Almost a decade later, the experimental observation of PI of vector solitons in a weakly birefringent fiber was reported [6].

Since the discovery of supercontinuum generation (SC), soliton fission based SC in birefringent photonic crystal fiber (PCF) was found to have polarization-dependent properties [12–20], indicating a nonlinear coupling between the two orthogonally polarized components [19]. PI and the vectorial nature of soliton fission and Raman solitons produced during SC have been demonstrated in a nearly isotropic tapered PCF for a slightly elliptical input polarization [21]. Localized mode solutions continuously down-shift due to the Raman terms in the generalized nonlinear Schrödinger equation, so they should be referred to as “Raman solitary waves” since they are not true soliton. However, following the convention in the optics literature, in this paper we refer to them as “Raman solitons” [19]. Complex PI of Raman solitons can significantly impact not only the general spectrum but also the output SOP across the SC spectrum.

While the previous investigation laid the ground work for the study of PI of Raman solitons generated by SC in a birefringent fiber, we extend these results to different linear and elliptical input SOPs for both linear and circular fiber birefringence. In this work, we investigate numerically the polarization dynamics of SC soliton fission and the ejected Raman soliton by systematically varying the input SOP and fiber birefringence. We numerically solve the coupled generalized nonlinear Schrödinger equations (GNLSE) using a vector beam propagation algorithm, in which the dispersion evolution in Fourier space is performed in the fiber eigen-mode basis and the nonlinear evolution is performed in a circular basis. We analyze the SOP evolution of the first emitted Raman soliton after the soliton fission as it propagates down the fiber as a function of input SOP and fiber birefringence. To comprehensively understand the complex nature of PI in SC, the polarization evolution of the soliton fission process and Raman soliton are represented on the Poincaré sphere. Much of the previous work was focused on nonlinear polarization dynamics in linearly birefringent fibers, in this work, we also demonstrate Raman soliton polarization evolution in circularly birefringent PCF using the Poincaré sphere representation.

## 2. Numerical model

We begin with a study of linearly birefringent PCF, but we use a representation in the circular basis where the phase dependent four wave mixing term (FWM) is absent, which simplifies both the analytic representation and the numerical scheme. To reveal the underlying polarization effects of soliton fission and Raman soliton in SC, a model which takes into account the fiber birefringence effect requires the coupled GNLSE [17, 19, 22]. This model also incorporates all orders of dispersion, optical shock, Raman and Kerr nonlinearities:

*A*

_{+}and

*A*are the electric field amplitudes of the two circularly polarized components. The circular polarization electric fields are related to the linear polarization electric fields as,

_{−}*T* is the time reference in the group velocity frame, *β _{n}* is the

*n*-th order dispersion parameter centered at frequency

*ω*

_{0},

*α*is the loss coefficient which we neglect for a short length of fiber. We define linear birefringence

*B*=

*n*, where

_{x}− n_{y}*n*and

_{x}*n*are refractive indexes for the

_{y}*x*and

*y*polarization eigen-modes, respectively. Δ

*β*

_{0}=

*β*−

_{x}*β*=

_{y}*Bω*

_{0}

*/c*represents the phase mismatch due to fiber birefringence. $\mathrm{\Delta}{\beta}_{1}=\frac{1}{{v}_{gx}}-\frac{1}{{v}_{gy}}=\epsilon B/c={\beta}_{1x}-{\beta}_{1y}$ represents the group velocity mismatch between the slow and fast axes, where

*ε*is a constant which typically varies between 0.38 to 1.3 [22]. In our simulation we assume

*ε*= 1 for the case of equal dispersion slopes between

*β*(

_{x}*ω*) and

*β*(

_{y}*ω*) since separate dispersion curves for x and y polarization components are not available.

*R*(

*T*) is the Raman response function taking into account both the instantaneous electronic and delayed molecular response of fused silica. They are well approximated by [23],

The characteristic time constants and weighting factors we used are *t _{b}* = 96 fs,

*t*

_{1}= 12.2 fs,

*t*

_{2}= 32 fs,

*f*= 0.75,

_{a}*f*= 0.21,

_{b}*f*= 0.04 and

_{c}*f*= 0.245 [23,24].

_{R}Following [12], dispersion parameters of the PCF are given by *β*_{1} = 4.9579 ×10^{3} ps/m, *β*_{2} = *−*1.3504 *×* 10^{−}^{2} ps^{2}/m, *β*_{3} = 8.2385 × 10^{−}^{5} ps^{3}/m, *β*_{4} = −9.1713 × 10^{−8} ps^{4}/m, *β*_{5} = 1.7589*×*10^{−}^{10} ps^{5}/m, *β*_{6} = *−*3.8095*×*10^{−}^{13} ps^{6}/m and *β*_{7} 9.4138 × 10^{−}^{16} ps^{7}/m, *λ*_{0} = 800 nm. These parameters correspond to a fiber with a zero dispersion wavelength at 743 nm and anomalous regime at the laser pulse central wavelength *λ*_{0} = 800 nm. The nonlinear coefficient is *γ* = 80 kW^{−}^{1} m^{−}^{1}, the peak input power *P _{peak}* = 10 kW is sufficient to produce 13th order solitons of the GNLSE for a Sech input pulse width

*T*= 100 fs. The loss term

_{FWHM}*α*= 0 for the simulated fiber length of

*L*= 8 cm.

_{fiber}We model the coupled GNLSE using a vectorized symmetric split-step Fourier method. Notice the coupled GNLSE is expressed in a circular basis where the nonlinear terms decouple, although this then gives coupling due to the linear birefringence. We solve the dispersion propagation step in the linear polarization basis to diagonalize the Fourier evolution operator, and it is propagated in Fourier space. Then we transform into a circular polarization basis for the nonlinear propagation step. The nonlinear step is propagated in real space, while the causal Raman response convolution integral is performed with an additional temporal Fourier transform of the intensities of the circular basis field components. The temporal derivative of the nonlinear source (optical shock) is evaluated using a centered finite difference in time. To avoid numerical instabilities, we check the convergence of the simulation by decreasing the step size of the propagation distance and the temporal and frequency sampling interval until no further changes occurred. At 2^{15} temporal and samples over a *T* = 32 ps simulation window separated by Δ*t* = 1 fs and Δ*z* = 40 *µ*m spatial step size, we found the simulation reliably reaches a stable solution.

## 3. Simulation results and discussions

A linearly polarized 100 fs chirp-free Sech pulse with 10 kW peak power is launched into the anomalous-dispersion region of the PCF. This high power input pulse corresponds to a 13th order soliton. The log-scale temporal and spectral evolutions of the SC process for fiber birefringence *B* = 1 *×* 10^{−5} and input polarization angle *θ* = 45° (to the slow axis) are plotted in Figs. 1(a) and 1(f), showing the *x* and *y* eigen-polarization components separately as well as the total power. Initially, the input field experiences pulse compression and spectral broadening due to self-phase modulation (SPM). The compressed pulse reaches a maximum peak power at around 1.7 cm, which agrees with the fission length calculation *L _{fiss}* =

*L*/

_{D}*N*= 1.7 cm [25], where the dispersion length

*L*=

_{D}*T*

_{FWHM}^{2}/3.11|

*β*

_{2}| for Sech input pulse. Beyond this distance, due to nonlinear effects such as Raman induced-frequency shift (RISF), self-steepening (SS), and linear effects such as third order dispersion (TOD), the higher order soliton experiences soliton fission and breaks up into dispersive waves and multiple fundamental soliton-like pulses with varying peak powers and temporal widths (see Fig. 1(g), which shows a zoomed-in of Fig. 1(c)). From Figs. 1(a) and 1(b) and Figs. 1(d) and 1(e), we notice the

*x*and

*y*polarization components periodically couple and form a bound pair, i.e. exhibits soliton trapping, as they propagate in the PCF. Soliton trapping owes its existence to cross phase modulation (XPM), represented by the last term on the right side of Eq. (1). The XPM induced coupling between the

*x*and

*y*polarized solitons propagate at a common group velocity [19].

Soliton fission plays a critical role in higher order soliton SC [18,21,26]. When soliton fission occurs (Fig. 1(g)), multiple pulses are ejected and they form a bundle of interacting and interfering soliton components, except the shortest ejected pulse(s) which collides with the bundle of pulses, escapes with nearly half the total energy, and travels away from the main pulse as it slowly shifts towards longer wavelength. This complex fission mechanism related to TOD was recently analogized to Newton’s Cradle [27]. After the fission finishes (indicated by the dotted line in Fig. 1(g)), the ejected soliton pulses shed excess energies to evolve into fundamental Raman solitons as they travel more slowly than the input pulse in the group velocity frame and simultaneously shift to longer wavelengths as a result of RISF. The direction of pulse energy transport is determined by the sign of TOD. In this simulation since the TOD is positive, it results in a long wavelength shift of energy. The soliton ejected earliest has the shortest temporal width, highest peak power and and the largest group velocity difference relative to the pump pulse and the computational group velocity frame. This first Raman soliton is responsible for the most spectral shift (indicated by black arrows in Fig. 2) and energy transfer to the long wavelength regime. Its SOP evolves as a unit upon propagation in the fiber [6]. Successive solitons can escape the bundle of interacting pulses depending on conditions such as fiber length and input power. As shown in Fig. 2, as the input polarization angle changes, the center of the soliton spectrum (indicated by arrows) at the end of the 8 cm fiber shifts too.

#### 3.1. Soliton fission output polarization dynamics in linearly birefringent PCF

We investigate the SOP after soliton fission at a distance of *z* = 2.4 cm shown as dotted black line in Fig. 1(g)) and plot them on the Poincaré sphere as a function for various input polarizations and fiber birefringence. The results are shown in Fig. 3 for *B* = 1 × 10^{−}^{8}, 1×10^{−}^{5}, 2.5 *×* 10^{−}^{5} and 5 *×* 10^{−}^{4}, respectively. *B* = 1 *×* 10^{−}^{8} is the isotropic case, *B* = 1 × 10^{−}^{5} is a typical value for a weakly birefringent PCF used in the literature, and *B* = 5 × 10^{−}^{4} is a typical value for a highly birefringent commercial PCF. *B* = 2.5 × 10^{−}^{5} corresponds to the intermediate case where the ejected Raman soliton is near the polarization instability critical power and the polarization dynamics are most clearly revealed on the Poincaré sphere. In the linear regime, the SOP of a monochromatic beam evolves as a rotation about the *x*-*y* axis on the Poincaré sphere periodically with a period equal to the beat length for *λ* = 800 nm, *L _{B}* =

*λ/B*= 80 m,8 cm, 3.17 mm and 1.6 mm, respectively. The input polarizations used in these simulations are linear (Fig. 3(a)) and elliptical (Fig. 3(f)), spanning from 0° to

*±*90° in

*±*5° steps. A PI critical power

*P*= 3

_{cr}*πB/λγ*was predicted for CW light in nonlinear birefringent fiber using the Stokes parameters and the nonlinear dynamics on the Poincaré sphere [4]. Averaging the nonlinearity across the Sech profile gives an additional numerical factor of 4.5 for nonlinear Schrödinger equations [5] which can be used to estimate the critical power for Raman solitons in PCF. We simulated Raman soliton polarization dynamics for several fiber birefringence between

*B*= 1

*×*10

^{−}^{8}and

*B*= 5

*×*10

^{−}^{4}and presented them on the Poincaré sphere. In particular, we used

*B*= 1

*×*10

^{−}^{5}and

*B*= 2.5

*×*10

^{−}^{5}, where the ejected Raman soliton exceeds the critical power, as will be discussed in Sections (3.1) and (3.2). The SOPs of the ejected Raman soliton vary as the birefringence B increases as seen from Figs. 3 (b) to 3(e) for linear input SOP (Fig. 3(a)). Note that the different colors simply encode the different input SOP.

At low B (Fig. 3(b)), the Raman soliton SOP remains the same as the input SOP, since for these linear inputs, nonlinear polarization rotation, also known as self-induced ellipse rotation [4,19], leaves the SOP fixed on the equator.

As the fiber birefringence B is increased (Figs. 3(c) and 3(d)), the soliton fission SOP is governed by both linear and nonlinear polarization rotations. A cluster of soliton fission SOPs appear around the slow-axis (+*s*_{1}-axis) close to the equator, so this represents an attraction of the polarization dynamics. In contrast, the soliton fission SOP are repelled away from the fast-axis (*−s*_{1}-axis). For low birefringence, some soliton fission SOPs appear at the +*s*_{3} and *−s*_{3}-axis, depending on the sign of input SOP angle *θ*.

For B = 2.5 *×* 10^{−}^{5} (Fig. 3(d)), the fast-axis instability at the back of the sphere is clearer, and we see an attraction of the ejected soliton SOP around the slow-axis at the front of the sphere. For high B = 5 *×* 10^{−}^{4} (Fig. 3(e)), the critical power is well in excess of the ejected Raman solitons and linear polarization rotation dominates, although polarizations are clearly clustered into a spiral around +*s*_{1}-axis they also cluster at −*s*_{1}-axis.

For elliptical input SOPs (Fig. 3(f)), the output SOPs (Figs. 3(g) to 3(j)) are different from the linear case especially for low B (Fig. 3(g)), because for B = 1 *×* 10^{−}^{8}, the majority of the soliton fission output SOPs are attracted to the poles (i.e. RHC and LHC states) instead of being fixed on the equator (Fig. 3(b)). In Fig. 3(i), B = 2.5 *×* 10^{−}^{5}, the SOPs disperse around the back of the sphere. Comparing with the linear case (Fig. 3(d)), the clustering around the slow-axis is less pronounced. However, at high B (Fig. 3(j)), the clustering around the slow and fast axis appear again, indicating linear polarization rotation dominance similar to the linear case of Fig. 3(d).

#### 3.2. Raman soliton polarization evolution linearly birefringent PCF

In the previous section, we discussed the SOP of the ejected Raman soliton at *z* = 2.4 cm after soliton fission in SC from low to high linear birefringence, for linear and elliptical input SOPs. When soliton fission occurs, as seen in Fig. 1(g), the first Raman soliton collides with the remaining bundle of solitons, escapes and transforms itself into a fundamental soliton by shedding excess energy as it propagates away from the main pulse and shifts towards longer wavelength. We now investigate how the SOP of this Raman soliton develops as it propagates down the fiber and how the different fiber birefringence and input SOP influence the evolution of the SOP of Raman soliton.

To investigate this, we locate the Raman soliton trajectory and fit it to an analytic polynomial function. Then we calculate the Stokes parameter [*S*_{0}, *S*_{1}, *S*_{2}, *S*_{3}] at each time sample across the width of the Raman soliton and average these values across the temporal width of the soliton. If the polarization would vary across the temporal profile of the Raman soliton, then this would result in a depolarization since
$\sqrt{{S}_{1}{}^{2}+{S}_{2}{}^{2}+{S}_{3}{}^{2}}$ would be less than *S*_{0}. However, we found that this was not the case and the entire Raman soliton would have a consistent polarization that would evolve as it propagate and would remain on the surface of the Poincaré sphere, with a degree of polarization DOP *≈* 1. The polarization dynamics are then represented as trajectories on the surface of the Poincaré sphere parameterized by the propagation distance. In Fig. 4, we plot the SOP of the Raman soliton for both linear and elliptical SOP for *θ* spanning from *±*5° to *±* 85° in *±*10° steps coded with the same colors used in Figs. 3 (a) and 3(f).

First, for the case of very low B and linear input SOP (Fig. 4(a)), the evolution of Raman soliton SOP remains the same as input SOP due to vanishing nonlinear angular rotation on the equator. Therefore, for an almost isotropic fiber with high input intensity, the SOP of the Raman soliton is predictable for the entire length of propagation. Although, such a simple polarization evolution would be perturbed by random birefringence variations along the fiber length or the inclusion of quantum noise in the evolution equations.

However, with elliptical input SOP ((Fig. 4(e)), since both *x* and *y* polarization components are excited, the trajectories of the SOP are seen to rotate around *s*_{3}-axis nonlinearly with speeds determined by the ellipticities that result from soliton fission ((Fig. 3(g)). The rotation on the Poincaré sphere due to ellipse rotation is proportional to *S*_{3} = *sin*(2*χ*) and the speed is maximized when *χ* = 22.5°, half way between the *s*_{3}-axis and the *s*_{1}–*s*_{2} plane, then decreases as the ellipticity of SOP moves toward more circular or more linear. The directions of rotation on the northern and southern hemispheres are the opposite, agreeing with an earlier study on polarization dynamics in low birefringence tapered fiber with slightly elliptical input polarization [21]. Two fixed points of nonlinear rotation in this case are RHC and LHC, i.e. the north and south poles of the *s*_{3}-axis.

For B = 1 *×* 10^{−}^{5} (Figs. 4(b) and 4(f)), polarizations in the vicinity of the slow-axis rotate around the slow-axis (+*s*_{1}) in elliptical trajectories, which is a signature of a mixed state of linear and nonlinear polarization rotations [4]. As light intensity increases, nonlinear birefringence is induced because of the intensity-dependent refractive index. Asymmetry between the slow-and fast-axis arises when the induced nonlinear birefringence becomes comparable with the fiber intrinsic birefringence B. Furthermore, because for the slow-axis, the induced nonlinear birefringence adds to the intrinsic birefringence B, the slow-axis remains stable for input SOP close to it. On the other hand, along the fast-axis, because the induced birefringence reduces the intrinsic birefringence, as a result, the fast-axis becomes less stable. Interestingly, from the top views, the center of rotation for nonlinear ellipse rotation around the poles of the sphere are shifted toward the fast-axis, indicating formation of new nonlinear elliptical eigen-polarizations.

For *B* = 2.5*×*10^{−}^{5}, the induced nonlinear birefringence is comparable to the linear intrinsic birefringence, the effective beat length can become infinite for some states of polarization [19], and the polarization instability is most clearly revealed. The shift of the center of the nonlinear eigen-axis of rotation is evident and a new elliptical axis of nonlinear rotation is formed. The trajectories for the Raman soliton on the sphere is separated into three regions, one rotates around the stable slow-axis, two around the stable elliptical axis near the top and bottom of the sphere and a figure 8 separatrix crosses through the unstable fast axis. The number of fixed centers of rotation points is 3 in this case. For elliptical input SOPs (Fig. 4(g)), the separatrix is revealed by the nearby trajectories.

For high B (Figs. 4(d) and 4(h)), the ejected Raman soliton is below the critical power for polarization instability, the polarization trajectories rotate around the stable slow and fast axis because linear polarization rotation around +*s*_{1}-axis dominates over the nonlinear ellipse rotation around +*s*_{3}-axis. The SOP trajectories rotate around the fast and slow axis in upright circles due to the dominance of linear birefringence over nonlinear rotation with only a slight nonlinear curvature seen from the side view. The number of fixed points (e.g. centers of rotation) change from 3 to 2.

High linear birefringence can be used to suppress the unintentional birefringence or nonlinear polarization effects, and perhaps residual depolarization [22], when the incident polarization is aligned to the slow or fast-axis of the fiber. On the other hand, however, as seen in Fig. 4(a), in the low birefringence case, the output SOPs are fixed on the equator remaining exactly the same as input SOPs. This tells us that with an isotropic fiber, with high input power, linear SOP of soliton fission and Raman soliton is predictable and controllable over the entire length of nonlinear propagation. But residual birefringence variations along the fiber would destroy this simple picture. Random birefringence could be included in the model to simulate the polarization dynamics for propagation in a long fiber.

#### 3.3. Raman soliton polarization evolution in circularly birefringent PCF

The complex polarization evolution of the Raman solitons ejected during the SC process in birefringent fiber and the resulting polarization instability can be interpreted on the Poincaré sphere as the interplay between linear rotations around the +*s*_{1}-axis and nonlinear rotations proportional to *S*_{3} around the +*s*_{3}-axis. The interplay of these orthogonal rotations on the Poincaré sphere depend on the power and polarization of the Raman soliton ejected from the SC process. This dynamic process could simplified if instead the linear rotations also occur around the *s*_{3}-axis. This would be the case for circularly birefringent fiber, which can be produced by twisting the fiber as it is drawn [28]. We are thus motivated to simulate the SC process and the evolution of the polarization of the ejected Raman soliton for circularly birefringent fiber.

We simulated the SC process and the polarization dynamics of Raman solitons in a circularly birefringent PCF using coupled-GNLSE for both the linear and elliptical input SOP used in the linear birefringent fiber simulation. We define circular birefringence *B _{c}* =

*n*−

_{s}*n*, where

_{f}*n*and

_{s}*n*are the refractive indices for the slow and fast eigen-polarization components, respectively. For the circularly birefringent PCF case, instead of using Δ

_{f}*β*

_{1}=

*β*

_{1}

*−*

_{x}*β*

_{1}

*in Eq. (1), we use Δ*

_{y}*β*

_{1}=

*β*

_{1}

_{s}−β_{1}

*to represent the group velocity mismatch between slow and fast circular eigen-polarization components, where*

_{f}*β*

_{1}

*=*

_{s}*εn*/

_{s}*c*,

*β*

_{1}

*=*

_{f}*εn*/

_{f}*c*and

*ε*= 1 assuming equal slope between

*β*(

_{s}*ω*) and

*β*(

_{f}*ω*) as discussed in Section 2. The decoupling of the linear propagation of the circular eigen-polarization components and the phase independent nonlinear terms imply that Δ

*β*

_{0}does not affect the propagation, so it is set to zero. Hence, the fourth term in Eq. (1) becomes $i\left(\frac{\mathrm{\Delta}{\beta}_{1}}{2}\frac{\partial}{\partial T}\right){A}_{\pm}$, while all the other terms remain the same. We then numerically solved the circularly birefringent version of Eq. (1) using a vector symmetric split-step Fourier method. In this case, both the dispersion and the nonlinear steps are performed in the circular basis. In Fig. 5, the temporal evolution of both the slow and fast eigen-polarization component intensities are shown. The Raman soliton travels away from the main pulse and is delayed even more than in the case of linear birefringence shown in Figs. 1(a) and 1(b). The spectral evolution plots in Fig. 6 show that the Raman soliton shifts toward an even longer wavelength compared to the linear birefringence PCF case in Figs. 1(d) and 1(e). The appearance of a weak artifact at 7.5 cm due to slow circular wrapping around and becoming fast circular may be due to aliasing of the complex field.

The polarization dynamics of the ejected Raman solitons for various circular birefringence of *B _{c}* = 10

^{−}^{8}, 10

^{−}^{6}and 10

^{−}^{5}are represented on the Poincaré sphere in Fig. 7. The essentially isotropic case,

*B*= 10

_{c}

^{−}^{8}, is nearly identical to the isotropic linearly birefringent case,

*B*= 10

^{−}^{8}, in Fig. 4(e). Circularly birefringent PCF is not commercially available yet and twisting (either during fiber drawing or after) is likely to yield a small birefringence in the range of

*B*= 10

_{c}

^{−}^{6}[28]. However, the interesting case of

*B*= 10

_{c}

^{−}^{5}is also explored in our simulation since this leads to a beat length of

*L*= 8 cm, which is equal to our simulated fiber length. For linear input SOPs (Figs. 7 (a) to 7(c)), in the essentially isotropic case (

_{beat}*B*= 10

_{c}

^{−}^{8}), the output SOPs of the Raman soliton are fixed on the equator similar to the linear birefringent PCF case (Fig. 4(a)). However, as the circular birefringence is increased to 10

^{−}^{6}, we see the SOP of the Raman soliton rotates about the

*s*

_{3}-axis on the Poincaré sphere, in this case at a constant rate due to the circular birefringence. For

*B*= 10

^{−}^{5}, the SOP of the Raman soliton rotates about

*s*

_{3}-axis and spirals toward the south pole, indicating a stable slow eigen-mode on the Poincaré sphere. Note that all the trajectories for the evolution of these linear polarizations are identical downward spirals on the Poincaré sphere converging towards the slow eigen-mode. Unlike the linear birefringent PCF case, the soliton fission process for the circularly birefringent fiber has not scrambled the SOP of the ejected Raman soliton.

Figures 7 (d) to 7(f) show the polarization evolution of the ejected Raman solitons for the various elliptically polarized Sech pulses injected into the fiber. We see purely nonlinear polarization rotation around the *s*_{3}-axis which is proportional to the *S*_{3} stokes component of the ejected Raman soliton, which vanishes on the equator for the linear input SOP. However as the circular birefringence is increased (*B _{c}* = 10

^{−}^{6},10

^{−}^{5}), the output SOPs rotate around the

*s*

_{3}-axis and spiral downwards toward the slow eigen-mode. Notice that the rate that the spiral drifts downwards increase with increasing birefringence. The rotation rate around the +

*s*

_{3}-axis is different on the northern hemisphere where linear and nonlinear effects counteract, while on the southern hemisphere these two effects cooperate. The fast eigen-mode at the north pole of the Poincaré sphere is an unstable state since the trajectories spiral away from it.

## 4. Conclusion

In this paper, we have explored the complex polarization dynamics of the Raman solitons ejected during supercontinuum generation (SC) in photonic crystal fibers (PCF) through representations of polarization trajectories on the Poincaré sphere. We use numerical vector beam propagation simulations of the coupled generalized nonlinear Schrödinger equations (C-GNLSE) which include all orders of dispersion, Kerr and delayed Raman nonlinearities as well as optical shock. The simulation utilized a dispersive linear step in the frequency domain in the fiber eigen-basis combined with a nonlinear step in the time domain using the circular basis, where the nonlinear terms simplify since the phase dependent four wave mixing term disappears. This allowed the calculation of the SC process for both linearly and circularly birefringent (e.g. twisted) PCF with varying birefringence.

Different states of linearly and elliptically polarized 100 fs Sech pulses are applied to the simulated PCF input to investigate the polarization of the ejected Raman soliton. We also studied the dynamics of its polarization evolution on the Poincaré sphere for various fiber birefringence. The Raman solitons were seen to have a nearly constant state of polarization across their temporal profile, with an averaged Stokes vector that remained on the surface of the Poincaré sphere and exhibited no depolarization during propagation.

For essentially isotropic PCF, the nonlinear ellipse rotation vanishes for linearly polarized inputs. In this case, the ejected Raman soliton remains linearly polarized, although this would be susceptible to perturbations caused by random birefringence variations along the fiber length. While for elliptical input polarizations, the ejected Raman soliton reveal nonlinear ellipse rotation about the *s*_{3} axis of the Poincaré sphere in proportion to the ellipticity *S*_{3}. Highly linear birefringent fiber led to Raman solitons with various polarization evolutions dominated by linear polarization rotation about the *s*_{1} axis on the Poincaré sphere. Intermediate linear birefringence leads to complicated interplay of these orthogonal rotations at which the nonlinear refractive index due to the ejected soliton is close to the fiber birefringence. On the Poincaré sphere, this polarization instability was manifested with the emergence of 3 stable polarization states (along +*s*_{1} corresponding to the slow-axis and tilted slightly away from *±s*_{3}), a fast axis instability (along *−s*_{1}), and a polarization separatrix.

Circularly birefringent fiber, made by twisting during fiber drawing, will exhibit both linear rotations of the Poincaré sphere about *s*_{3} and nonlinear rotation about *s*_{3} proportional to *S*_{3}. In this case, it leads to much simpler polarization evolution as pure rotations around *s*_{3}, however with a downward spiral on the Poincaré sphere away from the unstable fast circular eigenstate towards the stable slow circular eigenstate. This suggests that if twisted PCF can be fabricated with sufficient circular birefringence to dominate over unwanted linear birefringence induced by bendings or stress, it may lead to the most stable and repeatable single polarization output SC by launching the stable slow circular polarization into the twisted PCF. For applications requiring a broadband single polarization, this could effectively double the available output power of the SC process as compared with unpolarized outputs often produced by SC in PCF.

## Acknowledgments

This work was supported by National Science Foundation (award number NSF-CBET-1134561).

## References and links

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