1. Coupled nonlinear Schrödinger equations with dissipative terms and control of soliton propagation in broadband optical waveguide systems Avner Peleg Racah Institute of Physics Workshop on Quantum (and Classical) Physics with Non-Hermitian Operators The Israel Institute for Advanced Studies The Hebrew University of Jerusalem, July 2015 In collaboration with: Q.M. Nguyen, Y. Chung, D. Chakraborty, J.-H. Jung, T.P. Tran, T.T. Huynh, M. Chertkov, and I. Gabitov

2. Outline Introduction to broadband (multichannel) optical waveguide transmission. The nonlinear Schrödinger (NLS) equation and soliton pulses. Perturbative description of single-soliton propagation and of a single two-pulse collision. Effects of dissipative perturbations and crosstalk. Coupled-NLS models for soliton propagation. N-dimensional Lotka-Volterra (LV) models for amplitude dynamics. Stability and bifurcation analysis for the LV models for stabilization and switching. Comparison between the coupled-NLS models and the LV models. Further stabilization by frequency dependent linear gain-loss Conclusions.

3. Broadband (multichannel) optical waveguide systems and crosstalk Broadband (multichannel) transmission is used for enhancing transmission rates in optical waveguide links. In these broadband systems, many pulse sequences propagate through the same waveguide. Pulses from different sequences (frequency channels) propagate with different group velocities. => Collisions between pulses from different channels are very frequent, and can severely limit transmission quality. Interchannel crosstalk – energy exchange in collisions between pulses from different frequency channels. Two main mechanisms: (a) delayed Raman response, (b) nonlinear loss or gain (cubic or higher order). Example: Raman crosstalk in an on-off-keyed optical fiber transmission system with 101 frequency channels. [AP, Phys. Lett. A 360, 533 (2007)]

4. Two broadband (multichannel) optical waveguide experiments A 109-channel fiber optics system operating at 10 Gbits/s per channel. Dispersion-managed solitons. Experiment: Mollenauer et al. [Opt. Lett. (2003)]. Standard requirement: BER<10 -9. Crosstalk in silicon nanowaveguides A 2-channel silicon nanowaveguide transmission system at 10 Gbits/s per channel. BER increases with increasing input power. Experiment by Okawachi et al. [IEEE Photon. Technol. Lett., (2012)].

5. Nonlinear Schrödinger equation and solitons Hasegawa and Tappert (1973) - pulse propagation in optical waveguides can be described by the nonlinear Schrödinger (NLS) equation:  (t,z) – proportional to the envelope of the electric field z – distance along the waveguide, t – time d 2 – second-order dispersion coefficient κ – Kerr nonlinearity coefficient In dimensionless form: The single soliton solution of the NLS equation in a frequency channel β: η β, y β, α β - the soliton amplitude, position, and phase. Solitons are stable and shape-preserving => Soliton-based transmission is advantageous compared with other transmission formats.

6. Soliton collisions In an ideal fiber soliton collisions are elastic: the amplitude, frequency, and shape do not change as a result of the collision. In real optical fibers this elastic nature of the collisions breaks down due to the presence of perturbations (corrections to the ideal NLSE). In this case soliton collisions might lead to: emission of radiation, change in the soliton amplitude and group velocity, corruption of the shape, etc.

7. Effects of delayed Raman response on single-soliton propagation Pulse propagation is described by the perturbed NLSE: Use adiabatic perturbation theory for the NLSE soliton [D.J. Kaup, Phys. Rev. A (1990) and (1991)]. Look for a solution:, where is the soliton part with slowly varying parameters, and v rad (t,z) is the radiation part. Substitute into the perturbed NLSE to obtain Project both sides on the eigenmodes of the linear operator describing small perturbations about the NLSE soliton. The only effect of delayed Raman response on soliton parameters in O(ε R ) is a frequency downshift: Raman self-frequency shift (Gordon 1986)

8. Effects of delayed Raman response on a single collision Consider a single collision between a soliton in the reference channel (β=0) and a soliton in the β channel (Chi & Wen 1989, Malomed 1991, Kumar 1998, Lakoba and Kaup 1999, Chung and AP 2005, Nguyen and AP 2010). Assumptions: 1/| β | « 1, ε R « 1 An O(ε R ) change in the soliton amplitude (Raman-induced crosstalk) An O(ε R / β) frequency change (Raman induced cross frequency shift) Assuming ε R « 1/ | β | « 1 we can neglect effects of O(ε R 2 ) or higher. independent of the magnitude of β

9. Analysis of a single two-soliton collision (1) Example – perturbed NLS equation with delayed Raman response: Consider a single collision between a soliton in the reference channel (β=0) and a soliton in the β channel. Assume: 1/|β| « 1, ε R « 1 (typical for broadband transmission). Look for a two-soliton solution of the perturbed equation in the form ψ 0, ψ β – single-pulse solutions of the perturbed NLS equation in channels 0 and β. φ 0, φ β – collision effects in channels 0 and β. Solve, for example, for the pulse in the 0 channel. Substitute Use resonant approximation (|β|»1), and neglect terms rapidly oscillating in z. [Y. Chung and AP, Nonlinearity (2005); Q.M. Nguyen and AP, JOSA B (2010)]

10. Analysis of a single two-soliton collision (2) Expand Φ 0 in a perturbation series: The equation in O(ε R ): Integrate over z from -∞ to ∞ So On the other hand f j (x 0 ), j=0, …, 3 – the four localized eigenmodes of the linear operator describing small perturbations about the NLS soliton. Project (ΔΦ(x 0 ),ΔΦ*(x 0 )) T onto the localized eigenmode f 0 (x 0 ) to obtain the collision- induced amplitude shift (Raman crosstalk): [Q.M. Nguyen and AP, JOSA B (2010)] from Raman term

11. Crosstalk in broadband (multichannel) waveguide systems In amplitude-keyed transmission, crosstalk leads to severe transmission degradation due to the interplay between collision-induced amplitude shifts and amplitude-pattern randomness. A method for overcoming crosstalk – encode information in the phase => phase-shift-keyed (PSK) transmission. In PSK transmission all time slots are occupied => Crosstalk-induced amplitude dynamics is deterministic. Is it possible to achieve stable stationary transmission with nonzero amplitudes in all channels? Answer this question by obtaining a reduced ODE model for pulse amplitudes. (a) Analyze perturbation effects on a single collision. (b) Use (a) and collision-rate calculations to obtain the reduced ODE model. (c) Analyze stability of equilibrium points of reduced ODE model. (d) Compare predictions of ODE model with numerical solution of corresponding coupled-NLS model. (e) Study role of high-order effects and find ways to control them.

12. A Lotka-Volterra model for Raman-induced amplitude dynamics (1) A broadband fiber optics system with 2N+1 channels and frequency difference Δβ between adjacent channels. The amplitude shift of a jth-channel soliton due to a collision with a kth-channel soliton: ε R – Raman coefficient [ε R =0.006/τ 0, τ 0 - pulse width in picoseconds] η j, η k – initial amplitudes; β j, β k – initial frequencies. f(|j-k|) – a constant describing the strength of the Raman interaction. Assumptions: (1) ε R « 1/ | β | « 1; (2) Deterministic pulse sequences; (3) Sequences are infinitely long (long-haul transmission) or are subject to periodic temporal boundary conditions (closed fiber loop experiments). g j – net linear gain-loss in jth channel. Δz c – inter-collision distance for collisions between solitons from adjacent channels. Take into account amplitude shift due to: (a) single-pulse propagation, (b) collisions. [Q.M. Nguyen and AP, Optics Communications (2010)]

13. A Lotka-Volterra model for Raman-induced amplitude dynamics (2) The change in the jth-channel soliton’s amplitude in the interval (z,z+Δz c ): Taking the continuum limit A predator-prey model with 2N+1 species! Determine g j values by looking for an equilibrium state with equal nonzero amplitudes in all channels: The gain required for maintaining an equilibrium state with equal amplitudes is not “flat” (constant) with respect to frequency. Model takes the form: [Q.M. Nguyen and AP, Optics Communications (2010)]

14. Equilibrium states of the Lotka-Volterra model and their stability Equilibrium states with non-zero amplitudes are determined by “Trivial” equilibrium state: η j (eq) =η for -N≤j≤N. For an odd # of channels there are infinitely many steady states => infinitely many possibilities for stationary transmission (with unequal amplitudes). Show stability by constructing Lyapunov functions for the model: V L satisfies: (a) dV L /dz=0 (along trajectories of the model); (b) ; (c) for any with positive amplitudes; => Equilibrium state is stable for any initial condition. Stability is robust – it is independent of the details of the approximation for the Raman interaction [the exact values of the f(|j-k|) coefficients].

15. Example: numerical solution of the LV model with 3 channels Channels: j=0 (middle), j=50 (highest), and j=-50 (lowest). Equations for amplitude dynamics Equilibrium states: With equal amplitudes: η 50 =η 0 =η -50 =1. With unequal amplitudes: η 50 =η -50 = (3-η 0 )/2 => a line segment of equilibrium states. For input amplitude values that are off the equilibrium states dynamics is oscillatory => Stable transmission. Phase portrait for η 50, η 0 and η -50 Pulse amplitudes vs propagation distance

16. Comparison with full-scale coupled-NLS simulations The LV model neglects high-order effects, such as radiation emission, which can be important at large distances, and can lead to the breakdown of the LV model’s description. Need to compare the LV model’s predictions with simulations with the full NLS model. The dynamics involves a large number of fast collisions. => Amplitude measurements are difficult to perform with a single NLS model. We therefore work with the following equivalent coupled-NLS model: ψ j – envelope of the electric field of the jth sequence g j – linear gain-loss coefficient for the jth sequence We numerically solve the coupled-NLS model with periodic boundary conditions and an initial condition consisting of 2N+1 periodic soliton sequences: [AP, Q.M. Nguyen and T.P. Tran, arXiv:1501.06300]

17. Comparison with coupled-NLS simulations: 2 channels A 2-channel system with ε R =0.0024, T=18, Δβ=40, and 5 solitons in each sequence. Good agreement between the LV model predictions and the coupled-NLS simulations up to a distance z=2500. Frequency difference also oscillates due to coupling to amplitude dynamics. The oscillations are captured by the following perturbed predator-prey model: The shapes of the solitons are retained up to z=2500, but at larger distances, soliton shapes degrade, due to radiation emission. [AP, Q.M. Nguyen, T.P. Tran, arXiv:1501.06300]

18. Comparison with coupled-NLS simulations: 4 channels A 4-channel system with ε R =0.0018, T=20, Δβ=15, and 2 solitons in each sequence. Good agreement between the LV model and the coupled-NLS simulations up to a z=800. The shapes of the solitons are retained up to z=800. At larger distances, soliton shapes degrade, due to radiation emission. In a four-channel system with Δβ=15, radiative sidebands for the jth sequence develop at frequencies β k (z) of the other soliton sequences. These sidebands can be suppressed by increasing Δβ or by using a fiber coupler with frequency dependent linear gain-loss. [AP, Q.M. Nguyen, T.P. Tran, arXiv:1501.06300]

19. Further stabilization by frequency dependent linear gain-loss The Fourier transform of the soliton part of ψ j (t,z): Since |β k -β j |»1, the are well-separated and this can be used to suppress radiation emission. Use frequency-dependent linear gain-loss. For example, in a nonlinear N-waveguide coupler, we can choose: where g L <0. The coupled-NLS model Stable propagation extended to z=5000. No generation of radiation sidebands. [Q.M. Nguyen, AP, and T.P. Tran, arXiv:1501.06300]

20. Transmission switching in the presence of nonlinear gain-loss Example: A Ginzburg-Landau gain-loss profile in waveguide lasers Consider a nonlinear waveguide with weak linear loss, cubic gain, and quintic loss, i.e., with a Ginzburg-Landau gain-loss profile. Perturbed NLS equation: Amplitude shift in a single two-soliton collision in the presence of quintic loss: The LV model for crosstalk-induced amplitude dynamics in a two-channel system: The corresponding coupled-NLS model: Amplitude shift is quartic in soliton amplitudes [AP and Y. Chung, Phys. Rev. A (2012)]

21. Amplitude dynamics with a Ginzburg-Landau gain-loss profile Stable propagation for a wide range of ε 5 values, including ε 5 =0.5, i.e., outside of the perturbative regime. On-off (off-on) transmission switching: turning off (on) transmission of one of the soliton sequences, using bifurcations of the equilibrium state with equal amplitudes in both channels. Example: use the saddle-node bifurcation of (1,1) at κ c =(8T-15)/(5T-15) to turn off (on) transmission of sequence 2. In on-off switching, κ is increased from κ i κ c, such that (1,1) becomes unstable, while (η s,0) is stable. As a result, η 2 and η 1 tend to 0 and η s after the switching. In off-on switching, κ is decreased from κ i >κ c to κ f <κ c, such that (1,1) becomes stable. As a result, both η 2 and η 1 tend to 1 after the switching. on-off switching off-on switching [D. Chakraborty, AP, and J.-H. Jung, Phys. Rev. A (2013)]

22. Conclusions We developed a general framework for transmission control in broadband (multichannel) soliton-based optical waveguide systems. Using single-collision analysis and collision-rate calculations, we showed that amplitude dynamics in an N-channel waveguide system can be described by N-dimensional Lotka-Volterra (LV) models, where the form of the LV model depends on the physical perturbation. Stability and bifurcation analysis of the steady states of the LV models is used to develop methods for achieving robust transmission stabilization and switching for the main nonlinear dissipative processes, including delayed Raman response and nonlinear loss and gain. The method can find applications in a variety of waveguide systems, including fiber optics communication systems, data transfer on computer processors, and multiwavelength waveguide lasers.

23. Main Publications Crosstalk-induced dynamics in broadband waveguide systems Q.M. Nguyen and AP, Opt. Commun. 283, 3500 (2010). AP, Q.M. Nguyen, and Y. Chung, Phys. Rev. A 82, 053830 (2010). AP and Y. Chung, Phys. Rev. A 85, 063828 (2012). D. Chakraborty, AP, and J.-H. Jung, Phys. Rev. A 88, 023845 (2013). Q.M. Nguyen, AP, and T.P. Tran, Phys. Rev. A 91, 013839 (2015). AP, Q.M. Nguyen, and T.P. Tran, submitted, arXiv:1501.06300. AP, Q.M. Nguyen, and T.T. Huynh, submitted, arXiv:1506.01124. Single-collision analysis AP, M. Chertkov, and I. Gabitov, Phys. Rev. E 68, 026605 (2003). J. Soneson and AP, Physica D 195, 123 (2004). Y. Chung and AP, Nonlinearity 18, 1555 (2005). Q.M. Nguyen and AP, J. Opt. Soc. Am. B 27, 1985 (2010). AP, Q.M. Nguyen, and P. Glenn, Phys. Rev. E 89, 043201 (2014).

24. Back of the envelope derivation of the NLSE E(t,z) – the envelope of the electric field Taylor expansion of the wave number slow varying envelope approximation: 1/(ω 0 τ 0 ) «1