Spiral attractor created by vector solitons

Mode-locked lasers emitting a train of femtosecond pulses called dissipative solitons are an enabling technology for metrology, high-resolution spectroscopy, fibre optic communications, nano-optics and many other fields of science and applications. Recently, the vector nature of dissipative solitons has been exploited to demonstrate mode locked lasing with both locked and rapidly evolving states of polarisation. Here, for an erbium-doped fibre laser mode locked with carbon nanotubes, we demonstrate the first experimental and theoretical evidence of a new class of slowly evolving vector solitons characterized by a double-scroll chaotic polarisation attractor substantially different from Lorenz, Rössler and Ikeda strange attractors. The underlying physics comprises a long time scale coherent coupling of two polarisation modes. The observed phenomena, apart from the fundamental interest, provide a base for advances in secure communications, trapping and manipulation of atoms and nanoparticles, control of magnetisation in data storage devices and many other areas. Vector solitons exhibiting unique polarization states with complex trajectories have been observed in a mode-locked fibre laser. The polarization dynamics of these solitons evolve slowly to follow a double-scroll chaotic attractor on the Poincare sphere (a visual model for representing polarization states). Sergey Sergeyev and co-workers from Aston University in the UK generated the solitons using an erbium-doped fibre laser that was mode-locked using carbon nanotubes as a fast saturable absorber. The mechanism for the observed polarization behaviour is postulated to be due to coherent coupling between two polarization modes. In addition to providing valuable insight into laser operation, these findings could also be useful for creating controllable dynamic states of polarization, with applications in secure communications, spintronics and the optical trapping of atoms and nanoparticles.


INTRODUCTION
Vector solitons (VSs) in mode-locked lasers comprise a train of stabilized short pulses (dissipative solitons 1-13 ) with the specific shape defined by a complex interplay and balance between the effects of gain/ loss, dispersion and nonlinearity. The state of polarisation (SOP) of the solitons either rotates with a period of a few round trips or is locked. [4][5][6][7][8][9][10][11][12][13] The stability of VSs at the different time scales from femtosecond to microseconds is an important issue to be addressed for increased resolution in metrology, 14 spectroscopy 15 and suppressed phase noise in high speed fibre optic communication. 16 In addition, there is considerable interest in achieving high flexibility in the generation and control of dynamic SOPs in the context of trapping and manipulation of atoms and nanoparticles, [17][18][19] control of magnetisation 20 and secure communications. 21 The stability and evolution of VSs at a time interval from a few to thousands of cavity round trips is defined by asymptotic states (attractors) which the laser SOP approaches at a long time scale, viz. fixed point, periodic, quasiperiodic and chaotic dynamics. Soto-Crespo and Akhmediev 1,2 predicted theoretically based on the Ginzburg-Landau scalar model that dissipative solitons can generate strange attractors at the time scale of thousands of cavity round trips. Quantitative characterisation of strange attractors is usually based on the determination of geometric properties such as the fractal dimensionality, entropy and Lyapunov exponents. [22][23][24][25][26] For experimental data obtained in the form of a one-dimensional waveform (output power vs. time), such analysis requires the accumulation of a large amount of low noise data that is very difficult to achieve in systems where the output power is corrupted by noise. In addition, a reconstruction procedure has to be applied to obtain the attractor in the embedded three-dimensional space from the measured one-dimensional data. 22,26 High signal-to-noise ratio (.40 dB) measurement in the case of mode-locked lasers and application of a polarimeter, gives an opportunity of direct observation of attractors embedded in three-dimensional space in terms of the Stokes parameters S 1 , S 2 and S 3 . 12,13 Though simple forms of polarisation attractors generated by VSs, fixed points and limit cycles, have already been observed 3-13 more complex types of polarisation attractors such as chaotic ones have not yet been demonstrated.
In this article, we demonstrate the first experimental evidence of a new chaotic polarisation attractor, which results from the microsecond scale evolution of vector solitons in an erbium-doped fibre (EDF) laser mode locked with carbon nanotubes. This attractor takes the form of a double-scroll on the Poincaré sphere and has a correlation dimensionality of about 1.6. To characterize the attractor theoretically, we develop a new vector model that goes beyond the limitations of the previously used models based on either coupled nonlinear Schrödinger or Ginzburg-Landau equations. 1-11 By accounting for the vector nature of the interaction between an optical field and an active medium, the new model demonstrates that coherent coupling of two polarisation modes at a long time scale results in a double-scroll attractor with the similar correlation dimension of about 1.6 in agreement with the experiment.

Experiment details
We use mode-locked ring cavity fibre laser the same as was studied in our previous publications 12,13 (Figure 1). As follows from our previous publications, 12 at pump current I p 5310 mA laser operates in the mode-locking regime with pulses having a fundamental soliton shape of sech 2 (t/T p ) where T p is a pulse width (T p 5583 fs). 12 By tuning intracavity and pump laser diode polarisation controllers, we observed for the first time VSs slowly evolving on the surface of the Poincaré sphere on a double spiral trajectory ( Figure 2a). We use polarimeter with 1 ms resolution and measurement interval of 1 ms (25-25 000 round trips) to detect normalized Stokes parameters s 1 , s 2 and s 3 and the degree of polarisation (DOP). Normalized Stokes parameters and DOP are functions of the output powers of a two linearly cross-polarized SOPs u j j 2 and v j j 2 and the phase difference between them Dw: The slow, as compared to the round trip, dynamics on the double spiral trajectory includes residence near the orthogonal states of polarisation for approximately 200 ms and relaxation oscillations with a period of about 8 ms (Figure 2a, 2d and 2g). In view of the residence time and oscillation period being much longer than the pulse width of 583 fs and round trip time of 38.9 ns, we have a new type of vector soliton, viz. polarisation precessing VS with a spiral structure quite similar to the double scroll polarisation attractors demonstrated theoretically for dye laser, vertical-cavity semiconductor laser and degenerate two-level optical medium. [27][28][29] The double-scroll attractor is sensitive to the external perturbations caused by small fluctuations of the lasing field due to spontaneous emission and fluctuating fibre birefringence. In the same way as for spiral chaotic attractors, [27][28][29][30] this sensitivity can lead to completely unpredictable behaviour with large variations of output dynamics (red and blue lines in Figure 2a).
By tuning the intracavity and pump laser diode polarisation controllers, we have observed a polarisation attractor in the form of a fixed point (Figure 2b). It corresponds to the conventional polarisationlocked VS with a very high degree of polarisation of 92% 12 (Figure 2e). Increasing the pump current to 330 mA and tuning the polarisation controllers, we have also found a new polarisation attractor in the form of a double semicircle (Figure 2c). In view of photodetector resolution of 1 ms, it integrates an optical signal over 25 round trips. The pulse width is of 600 fs, round trip time is of 40 ns and so amplified spontaneous emission contributes to the integrated optical signal when lasing pulse is absent. DOP for amplified spontaneous emission is close to zero and, therefore, integrated by photodetector signal will have maximum DOP ,100%.

Vector model of an EDF laser model locked with carbon nanotubes
The polarisation attractors presented in the previous sections cannot be characterized in terms of coupled Schrödinger or Ginzburg-Landau equations 1-11 which are widely used to describe mode-locked lasers. The attractors have been observed in the time frame of 25-25 000 round trips. Therefore the model of the active medium has to be considered more accurately than in the typical simplified general laser models. [1][2][3][4][5][6][7][8][9][10][11] To address this issue, we have developed a new vector model of EDF lasers mode locked with carbon nanotubes acting as a fast saturable absorber (see Supplementary Information). The new model accounts for the dipole mechanism of the light absorption and emission in erbium and carbon nanotube (CNT), slow relaxation dynamics of erbium ions and absorption of erbium at the lasing wavelength. 27,[31][32][33] It also includes a low-pass filter effect caused by the photodetector with a bandwidth of 1 MHz.
To simplify consideration of the dipole mechanism of the light absorption and emission in erbium and CNT, we suggest that dipole moments of the transition with absorption and emission for erbiumdoped silica (m a ,m e ) and the dipole moment of the transition with absorption for CNT m a are located in the plane defined by the orthogonal components of the lasing field E5E x e x 1E y e y transverse to the direction of propagation as shown in Figure 3 The absorption cross-section of an elliptically polarized pump is s a,p~s p ð Þ a m a e p À Á 2 and as a result, the pump light creates an orientational distribution of the population inversion n(h) 27,32,33 ( Figure 3). The lasing field E5E x e x 1E y e y burns a hole in this distribution with depth 27,32,33 As follows from Equation (3), the depth of a polarisation hole depends on the phase difference Dw that results in coherent coupling of polarisation modes through gain sharing. 32,33 Note that in the scalar model the lasing field E5E x e x 1E y e y burns a hole in the field distribution with depth h~E x j j 2 z E y 2 and so there is no coherent coupling of the polarisation modes caused by polarisation hole burning. 1-11,27 As a result, the polarisation mode coupling is incoherent in this case, and the scalar model, which works well for the soliton description, cannot govern the observed complex dynamics. Excitation migration and upconversion in high-concentration EDFs [34][35][36] can result in orientation relaxation, 27 leading to transformation of the anisotropic distribution of the population inversion to isotropic. However, these processes are much slower than the time of the pulse duration and so can be neglected in the limits considered. Numerical simulations are based on equations averaged over the short-scale dynamics for the amplitudes of the soliton pulses for two linearly polarized states of polarisation, u and v. The equations Output power (a.u.) Output power (a.u.) Output power (a.u.) Coefficients D ij can be found as follows: for a vector model of EDF and CNT: for a vector model of EDF and a scalar model of CNT: for a scalar model of EDF and a vector model of CNT: for a scalar model of EDF and CNT.
Here t s~z V g t R À Á is a slow-time variable, e5t R c d , c d is the rate of relaxation for population inversion in EDF, t R 5L/V g is the photon round-trip time, V g is the group velocity, L is the cavity length, u j j 2 and v j j 2 are normalized to the saturation power I ss , the pump power I p is normalized to the saturation power I ps ( , C L and C p are the confinement factors of the EDF at the lasing and pump wavelengths, s (L) a(e) and s (p) a are absorption and emission cross-sections at the lasing wavelength l L and at the pump wavelength l p , A is the fibre core cross-section area, d is the ellipticity of the pump wave, D is the detuning of the lasing wavelength with respect to the maximum of the gain spectrum, a 1~sa C L r and a 2 are EDF and CNT absorption coefficients at the lasing wavelength, r is the concentration of erbium ions, x~s (L) a zs (L) e À Á s (L) a , a 3 is the ratio of saturation powers for CNT and EDF, a 4 is the linear loss coefficient, the Kerr coupling constant is c52pn 2 /(l L A), where n 2 is the nonlinear Kerr coefficient and A eff is the effective core area of the fibre.
We consider the case when the intracavity polarisation controller compensates the phase and amplitude anisotropy caused by any fibre birefringence and polarisation-dependent losses. The angular distribution of erbium ions at the first excited level n(h) can be expanded into a Fourier series as follows: 32,33 n h ð Þ~n In the experiment, a fibre patchcord of 5 m has been placed between the output of the laser and the polarimeter. We account for SOP distortions caused by the patchcord as the composition of POC and a birefringent element placed at the output of the laser: 37 A~{ cos y 1 cos y 2 , B~{ sin y 3 sin y 1 , C~{ cos y 1 sin y 2 , D~{ sin y 1 cos y 3 , Here u L(pol) and v L(pol) are the electric field components at the output of the laser and at the input of the polarimeter; a/2, c/2 and (a1j)/ 2 are the orientation of the first quarter wave plate, half wave plate and the second quarter wave plate with respect to vertical axis, L b is a birefringence beat length. We have used parameters which are quite close to the experimental ones, viz. L510 m, a 1 L5200/ln 10, a 2 L50.136, a 3 510 24 , a 4 L550/ln 10, x55/3, D50.1, I p 530, cLI ss 5 2310 26 , e510 24 , L/L c 51.05, A50.05, B50.2 and C50.05.
We have applied two-level model instead of three-and four-level models [34][35][36]38,39 that is justified for EDFs in view of populations of the first n, second n 2 and third n 3 excited levels are related as follows n..n 2 ..n 3 . 38, 39 We also neglected excited state absorption (ESA) in Equation (4) that is justified if inequality I p (s ESA 1s a (p) )/ I ps s a (p) ,,c d t 2 holds. 35 Here s ESA is excited state absorption crosssection and t 2 is the second excited level lifetime. Specifications of high-concentration EDF Er8-8/125 are the following: s ESA ,s a (p) , 39 t 2 510 ms, c d 5100 s 21 , r56.3310 25 ions m 23 , and mode filed diameter d59.5 mm. As follows from these data and notations to Equations (4)-(9), I ps ,4.4 mW and c d t 2 510 3 and so for I p 5170 mW used in experiments aforementioned inequality holds. We have also neglected the dynamics of the absorption in CNT that holds when saturable absorber relaxation t a time is smaller than the pulse width T p . In our experiments, t a ,300 fs and T p ,600 fs and so approximation of fast saturable absorber will be still valid if we make change of variables a 2 Ra 2 (12exp (2T p / t a )) for the case of a 2 ,,1. Though ansatz (7) in Supplementary Information can be used only if the pulse width of the fundamental solion is fixed that is confirmed by our experimental study, ansatz which separates fast and slow time variables can be used for different pulse shapes with fixed pulse widths (for example, Gaussian in the case of normal dispersion). If states of polarisation for adjacent pulses in multipulse or in bound state soliton regimes are the same and phase shift and pulse separation for bound state are fixed, Equations (4)-(9) can be still valid to describe SOP evolution for such multipulse and bound state regimes. To describe SOP evolution for the soliton regimes with evolving pulse parameters, the more general equation (6) in Supplementary Information can be used after minor revisions to account for gain spectral bandwidth and cavity anisotropy caused by in-cavity polarisation controller. Sergeyev et al. [34][35][36] have shown that migration assisted upconversion in high-concentration EDF results in decreasing first excited lifetime more than 10 times and in change of pump power distribution and so gain distribution along the fibre. 34 Model based on Equations (4)-(9) is discrete rather than distributed one and so we are able to mimic effect of upconversion accelerated by migration by decreasing the lifetime only.

RESULTS AND DISCUSSION
Numerical simulations using the developed vector model are shown in Figure 4. All the results in Figure 4 correspond to the case when the polarisation controller compensates all anisotropy in the cavity and only an external fibre patchcord transforms the output lasing SOP. Anisotropy in the cavity is caused by the elliptically polarized pump. Figure 4 shows that a double-scroll attractor appears if laser cavity is symmetrical, i.e., when the pump is circularly polarized, with ellipticity d51 (Figure 4a). When the pump SOP becomes more elliptical with d50.8 and the system becomes more anisotropic, the system approaches the steady state solution (Figure 4b). Increased pump and weak deviation of the pump SOP from the circular to the elliptical (d50.99) distorts the symmetry and the trajectories fill more densely the surface of the Poincaré sphere (Figure 4c). The orientation distribution of the inversion can be written as Equation (10) where, according to Equation (3), n 22 t ð Þ~cos Dw t ð Þ. As follows from Figure 4d, n 22 t ð Þ=0 and so polarisation hole burning with a depth proportional to n 22 (t) results in coherent coupling of the polarisation modes and, therefore, in the complex behaviour shown in Figure 4a.
We have found that the low-pass filter effect of a photodetector with cutoff frequency of 1 MHz can affect the appearance of the attractors in Figure 4. By using a low-pass filter with a Hanning window (trans- have processed the time domain waveforms shown in Figure 4. As a result of the low-pass filter effect, the double-scroll attractor in Figure 4a is slightly modified showing remarkable similarity to the attractor observed experimentally (Figures 2a, 5a, 5d and 5g). Note that such filtering does not affect a polarization-locked attractor (Figure 5b, 5e and 5h). However, the low-pass filter transforms the fast trajectory rotation on the Poincaré sphere shown in Figure 4c to the slow evolution on the double semicircle trajectory similar to the experimentally observed one (Figures 2c, 2f , 2i, 5c, 5f and 5i). Presented in Figures 2a, 2d, 2g, 5a, 5d and 5g, spiral attractor comprises dynamics related to the relaxation oscillations. Period of the oscillations can be estimated in terms of round trip time and notations to Equation (4) as T osc ,e 1/2 . 40 With 40 ns round trip time and e510 24 , we have T osc ,4 ms. If gain dynamics is neglected, then eR' in Equation (4) and so T osc R' that means the absence of auto-oscillations and, therefore, spiral attractor cannot be found. Thus, Equation (4) cannot be further simplified and slow gain dynamics has to be accounted for.
The results can be interpreted in terms of coupled oscillators' theory where weak coupling leads to a complex behaviour, while increase in the coupling strength leads to the oscillation quenching and appearance of the globally stable steady state (the Bar-Eli effect 41 ). Thus, the coupling in our system is determined by the pump SOP and power. In addition to this, orthogonally polarized laser modes are coupled through gain sharing, detuning of the lasing wavelength with respect to the maximum of the gain spectrum and the self-phase modulation caused by the Kerr nonlinearity.
To find the contribution of each of these factors to the origin of the spiral attractor, we have investigated three distinct cases: CNT at D50 (D is detuning of the lasing wavelength with respect to the maximum of the gain spectrum). First, we linearized the equations by substituting u j j~u 0 j jzx 1 , v j j~v 0 j jzx 1 ,Dw~Dw 0 zx 3 , f 1~f10 zx 4 , f 2~f20 zx 5 , f 3~f30 zx 5 and account for the different coefficients for the cases i-iii. Here u 0 j j, v 0 j j, Dw 0 , f 10 , f 20 and f 30 are steady-state solutions for the cases i-iii: where u~u j j exp w u ð Þ, v~v j j exp w v ð Þ and Dw~w u {w v is the phase difference between two linearly polarized states of polarisation. The equations are valid for a 3 (juj 2 1jvj 2 )/p,,1. Substituting x i~x x i exp lt s ð Þ into the linearized equations, we find the following eigenvalues: ,a 2~2 cLI ss u 0 j j 2 3 ,a 3~a 1 L 1zD 2 , b 1~e xf 10 2 u 0 j j 2 1zD 2 À Á p ,b 2~e 1z Next, we have introduced a saddle index as follows 30 n~r=c j j, If l 1~c w0, l 2,3~{ r+iv, v=0,rw0 ð Þ , or l 1~{ cv0, l 2,3~r +iv ðv=0,rw0Þ ð15Þ Equation (15) describes saddle focus where limit cycles can emerge in the homoclinic bifurcation. According to the Shilnikov theorem, the stability and number of the limit cycles depends on the saddle index n. 30 If n.1, homoclinic bifurcations results in one stable limit cycle. Unlike this, for n,1, an infinite number of unstable cycles can emerge and form a chaotic attractor. 30 The saddle index as a function of pump power I p is shown in Figure 6a. As follows from Figure 6a, the doublescroll attractor cannot exist for the first and the second case. For the third case, we have obtained eigenvalues l i~{ r i +iv i v i =0, r i w0 ð Þ and so the double-scroll attractor cannot exist as well. Thus, the detuning of the lasing wavelength with respect to the maximum of the gain spectrum along with the coherent mode coupling through the gain sharing must have lead to the complex dynamics.
The attractors shown in Figures 2 and 5 are different from the classical double-scroll strange Lorenz attractor obtained for single mode lasers. 40,42 The Lorenz attractor is embedded in three-dimensional space, while the attractors in Figures 2 and 4 are located on the surface of the Poincaré sphere, i.e., are embedded in two-dimensional space. Strange attractors are typically characterized by fractal dimensionality d, which is smaller than the dimensionality of the space in which the attractors are embedded. 24,25 Herein we applied the Grassberger-Procaccia algorithm for the calculation of the correlation dimension n, which is related to the dimension d as nfd. 24,25 This measure is obtained based on a consideration of correlation between points of the time series on the attractor, i.e., for our case, {x kj }, k50, 1, …, N; j50, 1, …, N (N is the number of points), where x kj~a rccos s 1,k s 1,j zs 2,k s 2,j zs 3,k s 3,j À Á , As a result, the correlation dimension n can then be found as follows: 24,25 n~lim The results are shown in Figure 6b. We found that n,1.6 for the double-scroll attractors in Figures 2a and 4b. Thus, the dimension obtained indicates that the observed double-scroll attractor could be a new strange attractor.

CONCLUSIONS
In summary, we demonstrate experimentally and theoretically a novel type of VS in EDF lasers mode locked with carbon nanotubes. Slow polarisation dynamics of this soliton measured in the time frame of 25-25 000 round trips takes the shape of a double-scroll chaotic attractor on the Poincaré sphere. Applying the Grassberger-Procaccia algorithm, we calculated a correlation dimension of 1.6 that is evidence of a new strange attractor in addition to only a few already experimentally observed in laser physics, viz. Lorenz, Rössler and Ikeda strange attractors. [22][23][24][25][26]40,42 The descriptions of new structures are found to be beyond the standard models of modelocked lasers based on coupled Schrödinger or Ginzburg-Landau equations. Therefore, we have developed a new model accounting for the dipole mechanism of the light absorption and emission in erbium and CNT, slow relaxation dynamics of erbium ions and absorption of erbium at the lasing wavelength. As a result, a new chaotic attractor has been reproduced theoretically for parameters close to the experimental ones. By unveiling the origin of new types of unique states of polarisation evolving on very complex trajectories, our theoretical and experimental studies pave the way to new techniques in metrology, 14 high-resolution femtosecond spectroscopy, 15 high-speed and secure fibre optic communications, 16,21 nanooptics (trapping and manipulation of nanoparticle and atoms [17][18][19] ) and spintronics (vector control of magnetisation 20 ). With further advances in the development of fast polarimetry in the context of speed (.1 GHz) and the number of stored samples (.10 M samples), the mode-locked laser with polarisation controller will become a demonstrator of the complex dynamics including dynamic chaos like a generic Chua circuit. 23 It will open the possibility for creating fundamentally new types of lasers with controlled dynamical states of polarisation.