Cavity Soliton Dynamics William J Firth Department of

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Cavity Soliton Dynamics William J Firth Department of Physics, University of Strathclyde, Glasgow, Scotland Cavity Soliton Dynamics William J Firth Department of Physics, University of Strathclyde, Glasgow, Scotland Acknowledgements: Thorsten Ackemann, Damia Gomila, Graeme Harkness, John Mc. Sloy, Gian-Luca Oppo, Andrew Scroggie, Alison Yao (Strathclyde) Fun. FACS and PIANOS partners Cavity Soliton Dynamics, Cargese, May 2006

MENU (Lugiato) - Science behind Cavity Solitons: Pattern Formation - Cavity Solitons and their 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 Cavity Soliton Dynamics, Cargese, May 2006

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

Cavity Soliton Dynamics Peak Height WJF + Andrew Scroggie, PRL 76, 1623 (1996) Background Cavity Soliton Dynamics Peak Height WJF + Andrew Scroggie, PRL 76, 1623 (1996) Background Intensity Cavity Soliton Simulation: Saturable Absorber with phase pattern on drive field Bifurcation diagram for such cavity solitons. Note unstable branch, bifurcating from MI point. Cavity Soliton Dynamics, Cargese, May 2006

1 D Kerr Cavity Sech-Roll Solitons Computed (full) and analytic (dashed) (unstable) branches of 1 D Kerr Cavity Sech-Roll Solitons Computed (full) and analytic (dashed) (unstable) branches of subcritical rolls and cavity solitons emerging from MI point of the 1 D Kerr Cavity (i. e. Lugiato-Lefever Equation). Quantitative analytics runs out here: need to rely on numerics: simulation – or solution-finding methods Cavity Soliton Dynamics, Cargese, May 2006

Our Approach – Newton Method model equation stationary states discretise algebraic system Newton method Our Approach – Newton Method model equation stationary states discretise algebraic system Newton method solutions A/ t = -[1+i( - I)] A + ia 2 A + i. I( A+A*+A 2+2|A|2+|A|2 A ) F=0 = -[1+i( - I)] A + ia 2 A + i. I( A+A*+A 2+2|A|2+|A|2 A ) Instead of A(x, y) we keep Aj on some grid points j. Compute spatial derivatives in Fourier space: Aj FFT Ak x -|k|2 Bk (FFT)-1 Involves the Jacobian matrix, Jij= Fi/ Aj Cavity Soliton Dynamics, Cargese, May 2006 ( 2 A)j

Example: Semiconductor Cavity Solitons T Maggipinto, M Brambilla, G K Harkness, WJF; PRE 62, 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 Cavity Soliton Dynamics, Cargese, May 2006

Experimental confirmation that CS exist as stableunstable pairs LCLV feedback system: A Schreiber et Experimental confirmation that CS exist as stableunstable pairs LCLV feedback system: A Schreiber et al, Opt. Comm. 136 415 (1997) Unstable branch identified with marginal switch-pulse Cavity Soliton Dynamics, Cargese, May 2006

Newton Method 2 Newton method solutions stability linear response The Jacobian matrix, used in Newton Method 2 Newton method solutions stability linear response The Jacobian matrix, used in the Newton method, gives solution’s linearisation. Its eigenvectors are solution’s eigenmodes, and its eigenvalues give the solution’s stability with respect to perturbations, , supported on the grid. Generalise to stability with respect to spatial modulations: (x, y) ei. K. r (R) eim cartesian coordinates cylindrical coordinates Thus we can find solution’s response to perturbations: translation, deformation, etc. due to noise, interactions, gradients etc. Cavity Soliton Dynamics, Cargese, May 2006

Example: Semiconductor Cavity Solitons T Maggipinto, M Brambilla, G K Harkness, WJF; PRE 62, 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 Cavity Soliton Dynamics, Cargese, May 2006

Neutral Mode T Maggipinto, M Brambilla, G K Harkness, WJF; PRE 62, 8726 (2000) 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. . Cavity Soliton Dynamics, Cargese, May 2006

Azimuthal Eigenmodes: m=0 and 1 T Maggipinto, M Brambilla, G K Harkness, WJF; PRE Azimuthal Eigenmodes: m=0 and 1 T Maggipinto, M Brambilla, G K Harkness, WJF; PRE 62, 8726 (2000) m=0 m=1 Cylindrically-symmetric (m=0) mode determines low-intensity limit (saddle-node). Neutral mode is m=1 in cylindrical coords. Cavity Soliton Dynamics, Cargese, May 2006

Azimuthal Eigenmode: m=2 T Maggipinto, M Brambilla, G K Harkness, WJF; PRE 62, 8726 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. Cavity Soliton Dynamics, Cargese, May 2006

Kerr Cavity Solitons WJF, G. K. Harkness, A. Lord, J. Mc. Sloy, D. Gomila, Kerr Cavity Solitons WJF, G. K. Harkness, A. Lord, J. Mc. Sloy, D. Gomila, P. Colet, JOSA B 19 747 -751 (2002) Lugiato-Lefever eqn in Kerr cavity: perturbed NLS: 1 st and 3 rd 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 Cavity Soliton Dynamics, Cargese, May 2006

2 D Kerr Cavity Soliton WJF, G. K. Harkness, A. Lord, J. Mc. Sloy, 2 D Kerr Cavity Soliton WJF, G. K. Harkness, A. Lord, J. Mc. Sloy, D. Gomila, P. Colet, JOSA B 19 747 -751 (2002) Cavity Soliton Dynamics, Cargese, May 2006

Stability of 2 D Kerr Cavity Solitons WJF, G. K. Harkness, A. Lord, J. Stability of 2 D Kerr Cavity Solitons WJF, G. K. Harkness, A. Lord, J. Mc. Sloy, D. Gomila, P. Colet, JOSA B 19 747 -751 (2002) hex sol 2 D 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 … Cavity Soliton Dynamics, Cargese, May 2006

Hopf-unstable Kerr Cavity Soliton WJF, G. K. Harkness, A. Lord, J. Mc. Sloy, D. Hopf-unstable Kerr Cavity Soliton WJF, G. K. Harkness, A. Lord, J. Mc. Sloy, D. Gomila, P. Colet, JOSA B 19 747 -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 • =1. 3; I= 0. 9. Cavity Soliton Dynamics, Cargese, May 2006

Dynamics of 2 D Kerr Cavity Solitons W. J. Firth et al JOSA B Dynamics of 2 D Kerr Cavity Solitons W. J. Firth et al JOSA B 19 747 -752 (2002). • 2 D 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 Cavity Soliton Dynamics, Cargese, May 2006

Stability of 2 D Kerr Cavity Solitons W. J. Firth et al JOSA B Stability of 2 D Kerr Cavity Solitons W. J. Firth et al JOSA B 19 747 -752 (2002). 2 D KCS exist above lowest curve. STABLE in WHITE region Hopf unstable in RED, Pattern-unstable in YELLOW/GREEN. Cavity Soliton Dynamics, Cargese, May 2006

Dynamics of 2 D Kerr Cavity Solitons W. J. Firth et al JOSA B Dynamics of 2 D Kerr Cavity Solitons W. J. Firth et al JOSA B 19 747 -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) • Cavity Soliton Dynamics, Cargese, May 2006

Dynamics of 2 D Kerr Cavity Solitons W. J. Firth et al JOSA B Dynamics of 2 D Kerr Cavity Solitons W. J. Firth et al JOSA B 19 747 -752 (2002). 6 -fold instability on “ring” • produces hexagonal pattern • again oscillates. Cavity Soliton Dynamics, Cargese, May 2006

Hopf Unstable CS: 1 D D Michaelis et al OL 23 1814 (1998) Oscillating Hopf Unstable CS: 1 D 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. Cavity Soliton Dynamics, Cargese, May 2006

Multi-Solitons in a Saturable Absorber Cavity G. K. Harkness, WJF, G. -L. Oppo and Multi-Solitons in a Saturable Absorber Cavity G. K. Harkness, WJF, G. -L. Oppo and J. M. Mc. Sloy, Phys. Rev. E 66, 046605/1 -6 (2002). Lossy, mistuned, driven, diffractive, single longitudinal-mode cavity, containing saturable absorber of “density” 2 C Analysis predicts instability to pattern with for strong enough driving and C big enough. Will look at “localised patterns” or multi-solitons, states intermediate between soliton and pattern. Cavity Soliton Dynamics, Cargese, May 2006

Newton Method - Numerics J Mc. Sloy, G K Harkness, WJF, G-L Oppo; PRE Newton Method - Numerics J Mc. Sloy, 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 128 x 128 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 R 14000 processors, with additional speedup obtainable via Open. MP 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 1 D, minutes in 2 D 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. Cavity Soliton Dynamics, Cargese, May 2006

Multi-Solitons in a Saturable Absorber Cavity G. K. Harkness, WJF, G. -L. Oppo and Multi-Solitons in a Saturable Absorber Cavity G. K. Harkness, WJF, G. -L. Oppo and J. M. Mc. Sloy, Phys. Rev. E 66, 046605/1 -6 (2002). Branches of multi-solitons with even/odd numbers of peaks. Cavity Soliton Dynamics, Cargese, May 2006

Multi-Solitons in a Saturable Absorber Cavity G. K. Harkness, WJF, G. -L. Oppo and Multi-Solitons in a Saturable Absorber Cavity G. K. Harkness, WJF, G. -L. Oppo and J. M. Mc. Sloy, Phys. Rev. E 66, 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 Cavity Soliton Dynamics, Cargese, May 2006

Cavity Solitons linked to Patterns Coullet et al (PRL 84, 3069 2000) argued that 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 1 D and 2 D). Harkness et al, Phys. Rev. E 66, 046605/1 -6 (2002) Gomila et. al. , Physica D (submitted). Cavity Soliton Dynamics, Cargese, May 2006

Multi-Solitons in a Saturable Absorber Cavity G. K. Harkness, WJF, G. -L. Oppo and Multi-Solitons in a Saturable Absorber Cavity G. K. Harkness, WJF, G. -L. Oppo and J. M. Mc. Sloy, Phys. Rev. E 66, 046605/1 -6 (2002). In two dimensions, qualitatively similar to 1 D (in some ways). Cavity Soliton Dynamics, Cargese, May 2006

Multi-Solitons in a Saturable Absorber Cavity G. K. Harkness, WJF, G. -L. Oppo and Multi-Solitons in a Saturable Absorber Cavity G. K. Harkness, WJF, G. -L. Oppo and J. M. Mc. Sloy, Phys. Rev. E 66, 046605/1 -6 (2002). Eigenmodes of fundamental 2 D 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) Cavity Soliton Dynamics, Cargese, May 2006

Cavity Soliton modes and dynamics Most cavity+medium systems to date described by Field • 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”. Cavity Soliton Dynamics, Cargese, May 2006

Cavity Soliton modes and dynamics • To find the response of an eigenmode to 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. Cavity Soliton Dynamics, Cargese, May 2006

Clusters of solitons A. G. Vladimirov et al Phys. Rev. E 65, 046606/1 -11 Clusters of solitons A. G. Vladimirov et al Phys. Rev. E 65, 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. Cavity Soliton Dynamics, Cargese, May 2006

Clusters of solitons A. G. Vladimirov et al Phys. Rev. E 65, 046606/1 -11 Clusters of solitons A. G. Vladimirov et al Phys. Rev. E 65, 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. Cavity Soliton Dynamics, Cargese, May 2006

Four-Clusters of solitons A. G. Vladimirov et al Phys. Rev. E 65, 046606/1 -11 Four-Clusters of solitons A. G. Vladimirov et al Phys. Rev. E 65, 046606/1 -11 ( 2002) Each eigenmode has the potential to distort the structure to a neighboring square ( ) rectangular () rhomboid ( ) or trapezoid ( ) configuration. Cavity Soliton Dynamics, Cargese, May 2006

Soliton Clusters in Feedback Mirror System Schäpers et al PRL 85 748 (2000) • Soliton Clusters in Feedback Mirror System Schäpers et al PRL 85 748 (2000) • Clusters show preferred distances, as in theory Cavity Soliton Dynamics, Cargese, May 2006

Spontaneous Complexes of Cavity Solitons S. Barbay et al (2005) Cavity Soliton Dynamics, Cargese, Spontaneous Complexes of Cavity Solitons S. Barbay et al (2005) Cavity Soliton Dynamics, Cargese, May 2006

Cavity Soliton Pixel Arrays John Mc. Sloy, private commun. Stable square cluster of cavity Cavity Soliton Pixel Arrays John Mc. Sloy, private commun. Stable square cluster of cavity solitons which remains stable with several solitons missing – pixel function. Theory? Cavity Soliton Dynamics, Cargese, May 2006

Arbitrary Cavity Soliton Complexes? Do arbitrary sequences of solitons and holes exist, as needed 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. ) MAYBE – In Kerr cavity model (1 D) we find high complexity, some evidence for spatial chaos. Gomila et. al. , NLGW 2004: Physica D (submitted) Cavity Soliton Dynamics, Cargese, May 2006

Cavity Soliton modes and dynamics • To find the response of an eigenmode to 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. Cavity Soliton Dynamics, Cargese, May 2006

CS Drift Dynamics: All-optical delay line inject train of pulses here read out at CS Drift Dynamics: All-optical delay line inject train of pulses here read out at other side parameter gradient Ø time delayed version of input train all-optical delay line/ buffer register Ø a radically different approach to „slow light“ Ø thrown in: serial to parallel conversion and beam fanning Ø won‘t work for non-solitons – beams diffract Cavity Soliton Dynamics, Cargese, May 2006

Experimental realisation Schäpers et. al. , PRL 85, 748 (2000), Proc. SPIE 4271, 130 Experimental realisation Schäpers et. al. , PRL 85, 748 (2000), Proc. SPIE 4271, 130 (2001); AG Lange, WWU Münster sodium vapor driven in vicinity of D 1 -line with single feedback mirror t = 0 ms adressing beam AOM Na tilt of mirror soliton drifts holding beam B t = 16 ms t = 32 ms t = 48 ms t = 64 ms t = 80 ms ignition of soliton by addressing beam proof of principle, quite slow, will be much faster in a semiconductor microresonator Cavity Soliton Dynamics, Cargese, May 2006

Drift velocity Maggipinto et al. , Phys. Rev. E 62, 8726, 2000 predicted velocity Drift velocity Maggipinto et al. , Phys. Rev. E 62, 8726, 2000 predicted velocity of CS: 5 µm/ns = 5000 m/s no evidence of saturation Experimental speed: 18 µm in 38 ns 470 m/s Hachair et al. , PRA 69 (2004) 043817 assume diameter of CS of 10 µm strength of gradient transit time 2 ns some 100 Mbit/s Cavity Soliton Dynamics, Cargese, May 2006

Non-instantaneous Kerr cavity log (velocity / gradiant) A. Scroggie (Strathclyde) unpublished 0. 01 semiconductor Non-instantaneous Kerr cavity log (velocity / gradiant) A. Scroggie (Strathclyde) unpublished 0. 01 semiconductor slope 1 • 1 D, perturbation analysis • velocity affected by response time of medium log ( ) • limits to ideal response for fast medium >1 Cavity Soliton Dynamics, Cargese, May 2006

Pinning of Cavity Solitons Hachair et al. , PRA 69 (2004) 043817 Experiment (left) Pinning of Cavity Solitons Hachair et al. , PRA 69 (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. ) Cavity Soliton Dynamics, Cargese, May 2006

Cavity Solitons in Reverse Gear A. Scroggie et al. PRE (2005) CS in OPO: 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(im. Kx) for fixed m v(K) REVERSE GEAR r K a ge e s r ve Re Along this curve CS are stationary EVERYWHERE despite background phase modulation K Cavity Soliton Dynamics, Cargese, May 2006 E

Kerr Media and Saturable Absorbers Phase Modulator E 0 E Medium v(K) General but Kerr Media and Saturable Absorbers Phase Modulator E 0 E Medium v(K) General but not Universal Kerr Saturable REVERSE GEAR Cavity Soliton Dynamics, Cargese, May 2006 K

“Sweeping” Cavity Solitons INLN 2005, using 200µm diameter Ulm Photonics VCSEL Fun. FACS experiment “Sweeping” Cavity Solitons INLN 2005, using 200µm diameter Ulm Photonics VCSEL Fun. FACS experiment in new VCSEL amplifier. Hold beam is progressively blocked by shutter, moves soliton several diameters. Cavity Soliton Dynamics, Cargese, May 2006

Digression: snowballs G D'Alessandro and WJF, Phys Rev A 46, 537 -548 (1992) Challenge: Digression: snowballs G D'Alessandro and WJF, Phys Rev A 46, 537 -548 (1992) Challenge: find/explain these “solitons”! Cavity Soliton Dynamics, Cargese, May 2006 Non-local – diffusion.

Inertia of Cavity Solitons CS can acquire inertia if a second mode becomes degenerate 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 become identical to translational mode leading to spontaneous motion Cavity Soliton Dynamics, Cargese, May 2006

Self-propelled cavity solitons A J Scroggie et al, PRE 66 036607 (2002), following work 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 a dynamic gradient force. Cavity solitons can self-drive. a=3 Stationary cavity solitons (right) are unstable to a stable moving cavity soliton (left) with similar amplitude. Cavity Soliton Dynamics, Cargese, May 2006

Self-propelled cavity solitons A J Scroggie et al, PRE 66 036607 (2002), following work Self-propelled cavity solitons A J Scroggie et al, PRE 66 036607 (2002), following work by Spinelli et al (PRA 2001) Initialised with stationary cavity soliton, noise induces transition to the stable moving soliton. Shown on left is temperature (which has a slow recovery time). Cavity Soliton Dynamics, Cargese, May 2006

What causes spontaneous motion? A J Scroggie et al, PRE 66 036607 (2002 When What causes spontaneous motion? A J Scroggie et al, PRE 66 036607 (2002 When peak of E is displaced from Tmin it tends to move on the detuning gradient (controlled by T) in which it finds itself, and also lowers T at its new location. If not, the intensity peak keeps moving, cooling the material it meets while the temperature behind relaxes to ambient level. =0. 04 If the movement is slow enough for the temperature to respond, the soliton simply establishes itself in a new location. =0. 03 =0. 05 Spontaneous motion bifurcation point Cavity Soliton Dynamics, Cargese, May 2006

Equation of motion A J Scroggie et al, PRE 66 036607 (2002 Spontaneously moving Equation of motion A J Scroggie et al, PRE 66 036607 (2002 Spontaneously moving solitons emerge from stationary soliton in a supercritical bifurcation. Destabilising mode is identical to translational mode Passive. Pump Input Device + Gaussian (cf. Michaelis et al 2001, Skryabin et al 2001) The soliton’s velocity v obeys: y 0 is the adjoint null eigenmode, f. C the unstable eigenmode, n 3 and w 2 nonlinear terms Cavity Soliton Dynamics, Cargese, May 2006

Self-propelled gas-discharge solitons A. W. Liehr et al, New Journal of Physics, 5, 89. Self-propelled gas-discharge solitons A. W. Liehr et al, New Journal of Physics, 5, 89. 1 (2003). Gas discharges can form solitonlike filaments, which show a bifurcation to spontaneous motion. (Purwins group, Muenster) Cavity Soliton Dynamics, Cargese, May 2006

Self-propelled soliton “laser” J Mc. Sloy, thesis. Field (white) and temperature (red) of self-propelled Self-propelled soliton “laser” J Mc. Sloy, thesis. Field (white) and temperature (red) of self-propelled soliton confined by dip in pump field - note inertia. In conditions where moving pattern forms in dip, a “bump” added to dip induces emission of soliton train. Cavity Soliton Dynamics, Cargese, May 2006

Absorbing medium: dark cavity solitons A J Scroggie et al, PRE 66 036607 (2002) Absorbing medium: dark cavity solitons A J Scroggie et al, PRE 66 036607 (2002) move – 2 D collisions Field Temperature Cavity Soliton Dynamics, Cargese, May 2006

Inertia of Self-Propelled Cavity Solitons A. Scroggie (Strathclyde) unpublished CS can acquire inertia if Inertia of Self-Propelled Cavity Solitons A. Scroggie (Strathclyde) unpublished CS can acquire inertia if a second mode becomes degenerate with the translational mode. But dynamics is not Newtonian. Temperature field through a collision, showing CS inertia. “Volley” of address pulses creates CS unstable to motion: merge on collision Neither number nor “mass” of CS is conserved (but speed is) Cavity Soliton Dynamics, Cargese, May 2006

Beyond mean-field cavity models … In the linear limit, long-time evolution of the field Beyond mean-field cavity models … In the linear limit, long-time evolution of the field at a given plane in a cavity can be exactly described by: +N(E) Elements of “ABCD matrix” (complex) obey AD-BC=1; 2 cos D. T: “slow” evolution time: TR: round-trip time; k: optical wavevector; : round-trip linear gain/loss and/or phase shift; Ein: input field • Can maybe capture nonlinearity by simply adding N(E) • “C” term is lens-like, forces confinement (gaussian mode) • Cavity soliton must be much more tightly self-confined than C-confined • “B” term describes diffraction (+ diffusion if complex) Cavity Soliton Dynamics, Cargese, May 2006

Beyond mean-field cavity models 2 In the linear limit, long-time evolution of the field Beyond mean-field cavity models 2 In the linear limit, long-time evolution of the field at a given plane in a cavity can be exactly described by: +N(E) Elements of “ABCD matrix” (complex) obey AD-BC=1; 2 cos D. T: “slow” evolution time: TR: round-trip time; k: optical wavevector; : round-trip linear gain/loss and/or phase shift; Ein: input field • This “master equation” can describe arbitrary numbers of modes (longitudinal as well as transverse) • Hence pulse envelopes (may need to add k-dispersion) • Also asymmetry (prisms, gratings, misalignments) with additional terms in x, ∂/∂x. (WJF and A Yao, J Mod Opt, in press) Cavity Soliton Dynamics, Cargese, May 2006

3 D Cavity Light Bullets in a Nonlinear Optical Resonator M. Brambilla, L. Columbo, 3 D Cavity Light Bullets in a Nonlinear Optical Resonator M. Brambilla, L. Columbo, T. Maggipinto, G. Patera, Phys. Rev. Lett. 93, 203901 (2004) Focusing regime: D=-2, C=50, 0=-0. 4, T=0. 1 Adding time dimension to two-level cavity soliton model, longitudinal filaments spontaneously contract to 3 D localized structures called cavity light bullets (CLBs). x z They endlessly travel the cavity strafing pulses from the output mirror. CLB stability tested versus different choices of the integration grid and use of an additive white noise in space and time. Cavity Soliton Dynamics, Cargese, May 2006

Conclusion: Time for something edible! Syrup Solitons on German TV! Cavity Soliton Dynamics, Cargese, Conclusion: Time for something edible! Syrup Solitons on German TV! Cavity Soliton Dynamics, Cargese, May 2006




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