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Cavity Soliton Dynamics

Acknowledgements: Thorsten Ackemann, Damia Gomila, Graeme Harkness, John McSloy, Gian-Luca Oppo, Andrew Scroggie, Alison Yao (Strathclyde) FunFACS and PIANOS partners. Cavity Soliton Dynamics. William J Firth Department of Physics, University of Strathclyde, Glasgow, Scotland.

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Cavity Soliton Dynamics

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  1. Acknowledgements: Thorsten Ackemann, Damia Gomila, Graeme Harkness, John McSloy, Gian-Luca Oppo, Andrew Scroggie, Alison Yao (Strathclyde)FunFACS and PIANOS partners Cavity Soliton Dynamics William J Firth Department of Physics, University of Strathclyde, Glasgow, Scotland

  2. MENU (Lugiato) - Science behind Cavity Solitons: Pattern Formation - Cavity Solitons and their properties • Experiments on Cavity Solitons in VCSELs • Future: the Cavity Soliton Laser - My lecture will be “continued” by that of Willie Firth A complete description of CS motion, interaction, clustering etc. will be given in Firth’s lecture. • The lectures of Paul Mandel and Pierre Coullet will elaborate • the basics and the connections with the general field of • nonlinear dynamical systems - The other lectures will develop several closely related topics

  3. Cavity Soliton Dynamics W. J. Firth + FunFACS partners, 8 May 2006 • Introduction: basics of Cavity Solitons (CS) • Existence of CS (Newton method) • Modes and Stability – Semicon; 2D Kerr CS • Complexes and clusters of CS – sat absorber • Dynamics of CS – response to “forces” • Spontaneous motion of Cavity Solitons • Conclusions

  4. Cavity Soliton Dynamics WJF + Andrew Scroggie, PRL 76, 1623 (1996) Peak Height Background Intensity Bifurcation diagram for such cavity solitons.Note unstable branch, bifurcating from MI point. Cavity Soliton Simulation: Saturable Absorber with phase pattern on drive field

  5. 1D Kerr Cavity Sech-Roll Solitons Computed (full) and analytic (dashed) (unstable) branches of subcritical rolls and cavity solitons emerging from MI point of the 1D Kerr Cavity (i.e. Lugiato-Lefever Equation). Quantitative analytics runs out here: need to rely on numerics: simulation – or solution-finding methods

  6. A/t = -[1+i( - I)] A + ia2 A + iI( A+A*+A2+2|A|2+|A|2A ) F=0 = -[1+i( - I)] A + ia2 A + iI( A+A*+A2+2|A|2+|A|2A ) Instead of A(x,y) we keep Aj on some grid points j. Compute spatial derivatives in Fourier space: Aj Ak Bk (2A)j x -|k|2 FFT (FFT)-1 Involves the Jacobian matrix, Jij= Fi/Aj Our Approach – Newton Method model equation stationary states algebraic system Newton method discretise solutions

  7. Example: Semiconductor Cavity Solitons T Maggipinto, M Brambilla, G K Harkness, WJF; PRE 62, 8726 (2000) Model couples (diffractive) intra-cavity field to (diffusive) photocarrier density Stationary solutions confirm simulations and give extra information

  8. Experimental confirmation that CS exist as stable- unstable pairs LCLV feedback system: A Schreiber et al, Opt.Comm. 136 415 (1997) Unstable branch identified with marginal switch-pulse

  9. Newton Method 2 Newton method linear response solutions stability The Jacobian matrix, used in the Newton method, gives solution’s linearisation. Its eigenvectors are solution’s eigenmodes, andits eigenvalues give the solution’s stability with respect to perturbations, , supported on the grid. Generalise to stability with respect to spatial modulations: (x,y)  eiK.r cartesian coordinates (R)  eimcylindrical coordinates Thus we can find solution’s response to perturbations: translation, deformation, etc. due to noise, interactions, gradients etc.

  10. Example: Semiconductor Cavity Solitons T Maggipinto, M Brambilla, G K Harkness, WJF; PRE 62, 8726 (2000) Eigenvalues of upper- and lower branch cavity solitons • upper branch (left) is well-damped (note neutral mode) • lower (right) – just one unstable mode

  11. Neutral Mode T Maggipinto, M Brambilla, G K Harkness, WJF; PRE 62, 8726 (2000) Assuming translational symmetry, the gradient of a cavity soliton is an eigenmode of its Jacobian, with eigenvalue zero. In this semiconductor model the CS is actually a composite of field E and photocarrier density N. Graphs verify that the neutral mode is indeed the gradient of (E, N)cs..

  12. m=0 m=1 Azimuthal Eigenmodes: m=0 and 1 T Maggipinto, M Brambilla, G K Harkness, WJF; PRE 62, 8726 (2000) Cylindrically-symmetric (m=0) mode determines low-intensity limit (saddle-node). Neutral mode is m=1 in cylindrical coords.

  13. m=2 Azimuthal Eigenmode: m=2 T Maggipinto, M Brambilla, G K Harkness, WJF; PRE 62, 8726 (2000) m=2 mode becomes unstable while m=0 modes all damped.This mode breaks symmetry, generates roll-dominated pattern.

  14. Kerr Cavity Solitons WJF, G. K. Harkness, A. Lord, J. McSloy, D. Gomila, P. Colet, JOSA B19 747-751 (2002) • Lugiato-Lefever eqn in Kerr cavity: perturbed NLS: • 1st and 3rd non-NLS terms on rhs: loss and driving. •  describes the cavity mistuning • Plane-wave intra-cavity intensity I is the other parameter (single-valued if  <√3) • Plane-wave solution unstable for I>1 • Solitons possible when I<1, with a coexisting pattern

  15. 2D Kerr Cavity Soliton WJF, G. K. Harkness, A. Lord, J. McSloy, D. Gomila, P. Colet, JOSA B19 747-751 (2002)

  16. Stability of 2D Kerr Cavity Solitons hex sol WJF, G. K. Harkness, A. Lord, J. McSloy, D. Gomila, P. Colet, JOSA B19 747-751 (2002) 2D KCS (left) and their (radial) perturbation eigenvalues (right). Lower branch (dotted trace) always has one unstable mode. Upper branch (solid trace) has all eigenvalues negative for low enough intensity, and is thus stable there. Hopf instability …

  17. Hopf-unstable Kerr Cavity Soliton WJF, G. K. Harkness, A. Lord, J. McSloy, D. Gomila, P. Colet, JOSA B19 747-751 (2002) • Initialise close to upper-branch • Inset shows the growth of amplitude of unstable eigenmode • which agrees very well with calculated eigenvalue • Fully-developed dynamics “dwells” at bottom of its oscillation • In fact comes close to middle-branch soliton • A is the deviation from background plane wave • q=1.3; I= 0.9.

  18. Dynamics of 2D Kerr Cavity Solitons W. J. Firth et al JOSA B19 747-752 (2002). • 2D Kerr cavity soliton does not collapse • but becomes Hopf-unstable • “dwells” close to related unstable soliton state. • but cannot cross manifold and decay without “kick” • Makes even unstable cavity solitons robust

  19. Stability of 2D Kerr Cavity Solitons W. J. Firth et al JOSA B19 747-752 (2002). 2D KCS exist above lowest curve. STABLE in WHITE region Hopf unstable in RED, Pattern-unstable in YELLOW/GREEN.

  20. Dynamics of 2D Kerr Cavity Solitons W. J. Firth et al JOSA B19 747-752 (2002). • Instability on “ring”, 5-fold case • view as MI of innermost ring, with above unstable mode • spawns growing pattern • hexagonal coordination, but 5-fold symmetry preserved • pattern oscillates (Hopf)

  21. Dynamics of 2D Kerr Cavity Solitons W. J. Firth et al JOSA B19 747-752 (2002). • 6-fold instability on “ring” • produces hexagonal pattern • again oscillates.

  22. Hopf Unstable CS: 1D D Michaelis et al OL 23 1814 (1998) Oscillating Dark Cavity Solitons Dark CS occur against “bright”, i.e. high intensity background. They have no phase singularity. Model: defocusing Kerr-like medium.

  23. Analysis predicts instability to pattern with for strong enough driving and C big enough. Multi-Solitons in a Saturable Absorber Cavity G.K. Harkness, WJF, G.-L. Oppo and J.M. McSloy, Phys. Rev. E66, 046605/1-6 (2002). Lossy, mistuned, driven, diffractive, single longitudinal-mode cavity, containing saturable absorber of “density” 2C Will look at “localised patterns” or multi-solitons,states intermediate between soliton and pattern.

  24. Newton Method - Numerics J McSloy, G K Harkness, WJF, G-L Oppo; PRE 66, 046606 (2002) Our numerical analysis of this system consists of three algorithms which we solve on a computational mesh of 128x128 grid points. The first directly integrates the spatiotemporal dynamics using a split-step operator integrator, in which nonlinear terms are computed via a Runge-Kutta method and the Laplacian by a fast Fourier transform. Our second algorithm is an enhanced Newton-Raphson method that can find all stable and unstable stationary solutions. A Newton-FFT method has been used, for evaluation of the Laplacian, but solution of the resultant dense matrix is computationally intensive, especially in two spatial dimensions.To overcome this problem, here we evaluate this spatial operator using finite differences, hence obtaining a sparse Jacobian matrix that can be inverted easily using library routines. As an extension to this algorithm we used an automated variable step Powell enhancement to the Newton-Raphson method, allowing it to be quasiglobally convergent, thus giving our algorithm very low sensitivity to initial conditions. All stationary, periodic or nonperiodic solutions in one and two spatial dimensions can hence be solved on millisecond and second time scales. Simulations were run on SGI, Origin 300 servers with 500 MHz R14000 processors, with additional speedup obtainable via OpenMP parallelization. The third algorithm is used to determine the stability of stationary structures from our Newton algorithm. It is a sparse finite-difference algorithm based on the ‘‘Implicitly Restarted Arnoldi Iteration’’ method. We use this algorithm to find the eigenvalues and corresponding eigenmodes of the Jacobian of derivatives of the solution in question. This allows us to calculate the eigenspectrum in a matter of seconds in 1D, minutes in 2D with approximately linear speed-up achievable across multiple processors via MPI parallelization. Although in this work these methods are applied to the solution of Jˆ with rank 32 768, we have used them efficiently when Jˆ has rank >262 144, and they could easily be modified to calculate stationary solutions and stability of fully three-dimensional problems.

  25. Multi-Solitons in a Saturable Absorber Cavity G.K. Harkness, WJF, G.-L. Oppo and J.M. McSloy, Phys. Rev. E66, 046605/1-6 (2002). Branches of multi-solitons with even/odd numbers of peaks.

  26. Multi-Solitons in a Saturable Absorber Cavity G.K. Harkness, WJF, G.-L. Oppo and J.M. McSloy, Phys. Rev. E66, 046605/1-6 (2002). Existence limits vs tuning and background intensity of multi-solitons with different numbers of peaks. Many-peaked structures seem to asymptote to definite limits. Coullet et al identified limits with “locking range” of interface between homogeneous solution and pattern. … Coullet lectures

  27. Cavity Solitons linked to Patterns Coullet et al (PRL 84, 3069 2000) argued that n-peak cavity solitons generically appear and disappear in sequence in the neighbourhood of the “locking range” (Pomeau 1984) within which a roll pattern and a homogeneous state can stably co-exist. We have verified this for both Kerr and saturable absorber models in general terms (in both 1D and 2D). Harkness et al, Phys. Rev. E66, 046605/1-6 (2002) Gomila et. al., Physica D (submitted).

  28. Multi-Solitons in a Saturable Absorber Cavity G.K. Harkness, WJF, G.-L. Oppo and J.M. McSloy, Phys. Rev. E66, 046605/1-6 (2002). In two dimensions, qualitatively similar to 1D (in some ways).

  29. Multi-Solitons in a Saturable Absorber Cavity G.K. Harkness, WJF, G.-L. Oppo and J.M. McSloy, Phys. Rev. E66, 046605/1-6 (2002). Eigenmodes of fundamental 2D cavity soliton. Corresponding eigenvalues plotted vs background intensity. At 1.53 they are 0, 0, 0.037, 0.035, 0.017, 0.015 for modes (b-g)

  30. Cavity Soliton modes and dynamics Most cavity+medium systems to date described by Field Medium • Use, e.g. Newton method to find time-independent CS solutions • Then eigenvalues of linearisation around solution give stability • Corresponding eigenvectors are the perturbation modes of the CS • Determine dynamics of response to other solitons and external forces • Localised patterns and other clusters of solitons as “bound states”.

  31. Cavity Soliton modes and dynamics • To find the response of an eigenmode to a perturbation, project the perturbation on to the mode • BUT the modes are not orthogonal – bi-orthogonal to adjoint set • Thus well-damped modes respond weakly - CS particle-like • BUT translational mode has zero eigenvalue: its amplitude is the displacement of the CS, and hence • This non-Newtonian dynamics of stable CS usually dominates.

  32. Clusters of solitons A. G. Vladimirov et al Phys. Rev. E65, 046606/1-11 ( 2002) Through modified Bessel functions the tails of N cavity solitons can create an effective potential GN. This system of scattered solitons will evolve towards a state where the soliton positions correspond to a minimum of the potential GN. The net force on a given soliton is simply the vector sum of its interaction forces with every other soliton, thus obeying the same superposition principle as Coulomb or gravitational forces.

  33. Clusters of solitons A. G. Vladimirov et al Phys. Rev. E65, 046606/1-11 ( 2002) Dynamics depend on overlaps, which happen in the soliton tails Use asymptotic expressions for the tails to get analytic positions Compare with simulations: Calculate exact eigenmodes of the cavity soliton cluster: including the translational mode, and unstable modes like the ones in these movies.

  34. Four-Clusters of solitons A. G. Vladimirov et al Phys. Rev. E65, 046606/1-11 ( 2002) Each eigenmode has the potential to distort the structure to a neighboring square() rectangular ()rhomboid () or trapezoid () configuration.

  35. Soliton Clusters in Feedback Mirror System Schäpers et al PRL 85 748 (2000) • Clusters show preferred distances, as in theory

  36. Spontaneous Complexes of Cavity Solitons S.Barbay et al (2005)

  37. Cavity Soliton Pixel Arrays John McSloy, private commun. Stable square cluster of cavity solitons which remains stable with several solitons missing – pixel function. Theory?

  38. MAYBE – In Kerr cavity model (1D) we find high complexity, some evidence for spatial chaos. Gomila et. al., NLGW 2004: Physica D (submitted) Arbitrary Cavity Soliton Complexes? Do arbitrary sequences of solitons and holes exist, as needed for information storage and processing? YES – P. Coullet et al, CHAOS 14, 193 (2004) NO – Only reversible sequences robust (e.g. Champneys et al.)

  39. Cavity Soliton modes and dynamics • To find the response of an eigenmode to a perturbation, project the perturbation on to the mode • BUT the modes are not orthogonal – bi-orthogonal to adjoint set • Thus well-damped modes respond weakly - CS particle-like • BUT translational mode has zero eigenvalue: its amplitude is the displacement of the CS, and hence • This non-Newtonian dynamics of stable CS usually dominates.

  40. read out at other side CS Drift Dynamics: All-optical delay line inject train of pulses here parameter gradient all-optical delay line/ buffer register • time delayed version of input train • a radically different approach to „slow light“ • thrown in: serial to parallel conversion and beam fanning • won‘t work for non-solitons – beams diffract

  41. adressing beam Na AOM tilt of mirror  soliton drifts holding beam B t = 0 ms t = 16 ms t = 32 ms t = 48 ms t = 64 ms t = 80 ms Experimental realisation Schäpers et. al., PRL 85, 748 (2000),Proc. SPIE 4271, 130 (2001); AG Lange, WWU Münster sodium vapor driven in vicinity of D1-line with single feedback mirror ignition of soliton by addressing beam proof of principle, quite slow, will be much faster in a semiconductor microresonator

  42. Drift velocity Maggipinto et al., Phys. Rev. E 62, 8726, 2000 predicted velocity of CS:5 µm/ns = 5000 m/sno evidence of saturation Experimental speed:18 µm in 38 ns  470 m/s Hachair et al., PRA69(2004)043817 • assume diameter of CS of 10 µm • transit time 2 ns • some 100 Mbit/s strength of gradient

  43. Non-instantaneous Kerr cavity A. Scroggie (Strathclyde) unpublished g 0.01  semiconductor slope 1 log (velocity / gradiant) • 1D, perturbation analysis • velocity affected by response time of medium • limits to ideal response for fast medium >1 log (g)

  44. Pinning of Cavity Solitons Hachair et al., PRA69(2004)043817 Experiment (left) and simulation (right) of solitons and patterns in a VCSEL amplifier agree provided there is a cavity thickness gradient and thickness fluctuations. (The latter stop the solitons drifting on the gradient.)

  45. Cavity Solitons in Reverse Gear A. Scroggie et al. PRE (2005) CS in OPO: predicted and “measured” CS-velocity v(K) induced by driving field phase modulation exp(imKx) for fixed m v(K) REVERSE GEAR K Reverse gear Along this curve CS are stationary EVERYWHERE despite background phase modulation K E

  46. Kerr Media and Saturable Absorbers Phase Modulator E E0 Medium v(K) General but not Universal Kerr Saturable K REVERSE GEAR

  47. “Sweeping” Cavity Solitons INLN 2005, using 200µm diameter Ulm Photonics VCSEL FunFACS experiment in new VCSEL amplifier. Hold beam is progressively blocked by shutter, moves soliton several diameters.

  48. Digression: snowballs • G D'Alessandro and WJF, Phys Rev A46, 537-548 (1992) Challenge: find/explain these “solitons”! Non-local – diffusion.

  49. Inertia of Cavity Solitons CS can acquire inertia if a second mode becomes degenerate with the translational mode.Even so, dynamics may not be Newtonian. • Galilean (boost) invariance leads to inertia proportional to energy (Rosanov) • Destabilising mode may becomeidentical to translational mode leading to spontaneous motion

  50. a=3 Self-propelled cavity solitons A J Scroggie et al, PRE 66 036607 (2002), following work by Spinelli et al (PRA 2001) Due to thermal cavity tuning, T is coupled to E, so there is adynamicgradient force. Cavity solitons can self-drive. Stationary cavity solitons (right) are unstable to a stable moving cavity soliton (left) with similar amplitude.

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