cavity soliton dynamics

Cavity Soliton Dynamics, Cargese, May 2006Cavity Soliton Dynamics, Cargese, May 2006

William J FirthDepartment of Physics,

University of Strathclyde, Glasgow, Scotland

Cavity Soliton Dynamics

Acknowledgements: Thorsten Ackemann, Damia Gomila, Graeme Harkness, John McSloy, Gian-Luca Oppo, Andrew Scroggie, Alison Yao (Strathclyde)

FunFACS and PIANOS partners


MENU (Lugiato)MENU (Lugiato)

- Science behind Cavity Solitons: Pattern Formation- Science behind Cavity Solitons: Pattern Formation

- Cavity Solitons and their properties- Cavity Solitons and their properties

- Experiments on Cavity Solitons in VCSELsExperiments on Cavity Solitons in VCSELs

- Future: the Cavity Soliton LaserFuture: the Cavity Soliton Laser

- My lecture will be “continued” by that of Willie Firth- My lecture will be “continued” by that of Willie Firth

-The lectures of Paul Mandel and Pierre Coullet will elaborateThe lectures of Paul Mandel and Pierre Coullet will elaborate the basics and the connections with the general field of the basics and the connections with the general field of nonlinear dynamical systemsnonlinear dynamical systems

- The other lectures will develop several closely related topics- The other lectures will develop several closely related topics

A A completecomplete description of CS motion, interaction, clustering description of CS motion, interaction, clustering etc. etc. will be givenwill be givenin Firth’s lecture.in Firth’s lecture.



• 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

W. J. Firth + FunFACS partners, 8 May 2006


Cavity Soliton Simulation: Saturable Absorber with phase

pattern on drive field

Bifurcation diagram for such cavity solitons.

Note unstable branch, bifurcating from MI point.


Background Intensity

Pea

k H

eig

ht

WJF + Andrew Scroggie, PRL 76, 1623 (1996)


1D Kerr Cavity Sech-Roll Solitons

Quantitative analytics runs out here: need to rely on numerics: simulation – or solution-finding methods

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).


Involves the Jacobian matrix, Jij= Fi/Aj

Instead of A(x,y) we keep Aj on some grid points j.Compute spatial derivatives in Fourier space:

Aj Ak Bk (A)j

FFT (FFT)-1x -|k|2

Our Approach – Newton Method

model equation

stationary states

discretisealgebraic system

Newton method

solutions

F=0 = -[1+i( - I)] A + iaA + iI( A+A*+A2+2|A|2+|A|2A )

A/t = -[1+i( - I)] A + iaA + iI( A+A*+A2+2|A|2+|A|2A )


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


Experimental confirmation that CS exist as stable- unstable pairs

Unstable branch identified with marginal switch-pulse

LCLV feedback system: A Schreiber et al, Opt.Comm. 136 415 (1997)


solutions stabilitylinear

response

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.

Newton Method 2Newton method


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


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..


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.

m=0 m=1


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.

m=2

Azimuthal Eigenmode: m=2


Kerr Cavity Solitons

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

€

i∂E

∂t+

1

2

∂ 2E

∂x 2+ E

2E = i(−E − iθE + E in )

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


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


Stability of 2D Kerr Cavity Solitons

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 …

hex

sol



• 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.


Hopf-unstable Kerr Cavity Soliton


Dynamics of 2D Kerr Cavity Solitons

• 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

W. J. Firth et al JOSA B19 747-752 (2002).

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.


2D KCS exist above lowest curve.

STABLE in WHITE region

Hopf unstable in RED,

Pattern-unstable in YELLOW/GREEN.

Stability of 2D Kerr Cavity SolitonsW. 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)

Dynamics of 2D Kerr Cavity SolitonsW. J. Firth et al JOSA B19 747-752 (2002).


6-fold instability on “ring”

• produces hexagonal pattern• again oscillates.

Dynamics of 2D Kerr Cavity SolitonsW. J. Firth et al JOSA B19 747-752 (2002).


Oscillating Dark Cavity Solitons

Dark CS occur against “bright”, i.e. high intensity background.

They have no phase singularity.

Model: defocusing Kerr-like medium.

Hopf Unstable CS: 1D D Michaelis et al OL 23 1814 (1998)


Multi-Solitons in a Saturable Absorber Cavity

Lossy, mistuned, driven, diffractive, single longitudinal-mode cavity, containing saturable absorber of “density” 2C€

∂E

∂t= −E − iθE + E in + i∇ 2E −

2C

1+ E2 E

Analysis predicts instability to pattern with for strong enough driving and C big enough.

€

kc2 = −θ

G.K. Harkness, WJF, G.-L. Oppo and J.M. McSloy, Phys. Rev. E66, 046605/1-6 (2002).

Will look at “localised patterns” or multi-solitons,states intermediate between soliton and pattern.


Newton Method - Numerics

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 additionalspeedup obtainable via OpenMP parallelization. The third algorithm is used to determine the stability ofstationary 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.

J McSloy, G K Harkness, WJF, G-L Oppo; PRE 66, 046606 (2002)


Multi-Solitons in a Saturable Absorber CavityG.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.



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 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).



In two dimensions, qualitatively similar to 1D (in some ways).



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)


Most cavity+medium systems to date described by

€

∂E

∂t= Ein + G(E,F,∇2);

∂F

∂t= H(E,F,∇2)

Cavity Soliton modes and dynamics

• 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”.

Field Medium


• 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.

€

˙ a n = λ nan + un† ,P un

†,un

→ an = −λ n−1 un

† ,P un†,un

€

˙ a 0 = u0†,P u0

†,un = vCS


Through modified Bessel functions the tails of N cavity solitons Through modified Bessel functions the tails of N cavity solitons can create an effective potential Gcan create an effective potential GNN. This system of scattered . This system of scattered

solitons will evolve towards a state where the soliton positions solitons will evolve towards a state where the soliton positions correspond to a minimum of the potential Gcorrespond to a minimum of the potential GNN..

( )( )∑≠

− +−=

N

lj jl

jlRk

NR

RkeGG

jl φImcos

2

)Re(

0

The net force on a given soliton is simply the vector sum of its The net force on a given soliton is simply the vector sum of its interaction forces with every other soliton, thus obeying the same interaction forces with every other soliton, thus obeying the same superposition principle as Coulomb or gravitational forces.superposition principle as Coulomb or gravitational forces.

Clusters of solitons A. G. Vladimirov et al Phys. Rev. E65, 046606/1-11 ( 2002)


Clusters of solitons

Dynamics depend on overlaps, which happen in the soliton tailsUse 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.



A. G. Vladimirov et al Phys. Rev. E65, 046606/1-11 ( 2002)




Each eigenmode has the potential to distort the structure to a neighboring Each eigenmode has the potential to distort the structure to a neighboring squaresquare (() rectangular ()) rectangular () rhomboid (rhomboid () or trapezoid () or trapezoid () configuration) configuration..

Four-Clusters of solitons A. G. Vladimirov et al Phys. Rev. E65, 046606/1-11 ( 2002)


Soliton Clusters in Feedback Mirror System Schäpers et al PRL 85 748 (2000)

• Clusters show preferred distances, as in theory


Spontaneous Complexes of Cavity SolitonsS.Barbay et al (2005)


Cavity Soliton Pixel Arrays

Stable square cluster of cavity solitons which remains stable with several solitons missing – pixel function. Theory?

John McSloy, private commun.


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.)

Gomila et. al., NLGW 2004: Physica D (submitted)

MAYBE – In Kerr cavity model (1D) we find high complexity, some evidence for spatial chaos.


• 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.

€

˙ a n = λ nan + un† ,P un

†,un

→ an = −λ n−1 un

† ,P un†,un

€

˙ a 0 = u0†,P u0

†,un = vCS


CS Drift Dynamics: All-optical delay line

inject train of pulses here

read out at other side

parameter gradient

time delayed version of input train all-optical delay line/ buffer register

a radically different approach to „slow light“

thrown in: serial to parallel conversion and beam fanning

won‘t work for non-solitons – beams diffract

QuickTime™ and aAnimation decompressor



Experimental realisation

B

Naadressing beam

holding beam

AOMsodium vapor driven

in vicinity of D1-line

with single feedback mirror

t = 0 s t = 80 st = 64 st = 48 st = 32 st = 16 s

ignition of soliton by addressing beam

proof of principle, quite slow, will be much faster in a semiconductor microresonator

Schäpers et. al., PRL 85, 748 (2000), Proc. SPIE 4271, 130 (2001); AG Lange, WWU Münster

tilt of mirror soliton drifts


Drift velocity

strength of gradient

Maggipinto et al., Phys. Rev. E 62, 8726, 2000

assume diameter of CS of 10 µm

transit time 2 ns

some 100 Mbit/s

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


Non-instantaneous Kerr cavityA. Scroggie (Strathclyde) unpublished

0.01 semiconductor

log

(ve

loci

ty /

gra

dia

nt)

log ()

slope 1

• 1D, perturbation analysis

• velocity affected by response time of medium

• limits to ideal response for fast medium >1


Pinning of Cavity Solitons

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.)

Hachair et al., PRA 69 (2004) 043817


Cavity Solitons in Reverse Gear

K

Rever

se gea

r

REVERSE GEARv(K)

K

E

CS in OPO: predicted and “measured” CS-velocity v(K) induced by driving field phase modulation exp(iKx) for fixed

Along this curve CS are stationary EVERYWHERE

despite background phase modulation

A. Scroggie et al. PRE (2005)


Kerr Media and Saturable Absorbers

€

∂E

∂t= −(1+ iϑ ) E + E 0e

iμ cos(Kx ) + G(|E |2)E + i∂xxE

G(| E |2) = i | E |2 Kerr Medium

G(| E |2) = −α

1+ | E |2Saturable Medium

MediumE0

E

PhaseModulator

General but not Universal

v(K)

KREVERSE GEAR

Kerr

Saturable


“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.




G D'Alessandro and WJF, Phys Rev A46, 537-548 (1992)

Digression: snowballs

Challenge: find/explain these “solitons”!Non-local – diffusion.


Inertia of Cavity SolitonsInertia of Cavity Solitons

CS can acquire inertia if a second mode becomes CS can acquire inertia if a second mode becomes degenerate with the translational mode.degenerate with the translational mode. Even so, dynamics may not be Newtonian. Even so, dynamics may not be Newtonian.

• Galilean (boost) invariance leads toGalilean (boost) invariance leads to inertia proportional to energy inertia proportional to energy (Rosanov) (Rosanov)

• Destabilising mode may becomeDestabilising mode may become identical identical to translational mode to translational mode leading to leading to spontaneous motionspontaneous motion


Self-propelled cavity solitons Self-propelled cavity solitons

Stationary cavity solitons (right) are unstable to a stable Stationary cavity solitons (right) are unstable to a stable movingmoving cavity soliton (left) with similar amplitude cavity soliton (left) with similar amplitude

..

Due to thermal cavity tuning, Due to thermal cavity tuning, T T is coupled tois coupled to E E, so there is a, so there is a dynamicdynamic gradient force. Cavity solitons can self-drive. gradient force. Cavity solitons can self-drive.

A J Scroggie et al, PRE 66 036607 (2002), following work by Spinelli et al (PRA 2001)


Initialised with stationary cavity soliton, noise induces Initialised with stationary cavity soliton, noise induces transition to the stable transition to the stable movingmoving soliton. soliton.

Shown on left is temperature Shown on left is temperature (which has a slow recovery time).(which has a slow recovery time).

Self-propelled cavity solitons Self-propelled cavity solitons A J Scroggie et al, PRE 66 036607 (2002), following work by Spinelli et al (PRA 2001)


When peak of When peak of EE is displaced from is displaced from TTminmin it tends to move on the it tends to move on the

detuning gradient (controlled by detuning gradient (controlled by TT) in which it finds itself, ) in which it finds itself, and also lowers T at its new locationand also lowers T at its new location..

Spontaneous motion bifurcation pointSpontaneous motion bifurcation point

If the movement is slow If the movement is slow enough for the temperature enough for the temperature

to respond, the soliton to respond, the soliton simply establishes itself in simply establishes itself in

a new location.a new location.

If not, the intensity peak If not, the intensity peak keeps moving, cooling the keeps moving, cooling the material it meets while the material it meets while the

temperature behind relaxes temperature behind relaxes to ambient level.to ambient level.

==

0.0

40

.04==0.030.03 ==0.050.05

What causes spontaneous motion?What causes spontaneous motion? A J Scroggie et al, PRE 66 036607 (2002


( )( )21

1

Δ+

−+Δ−=

Niχ

0=J 6.1=β

∂tv =ψ 0, ˆ L − ˆ L 0[ ]φC( )

ψ 0,φC( )v+

ψ 0,n3( )ψ 0,φC( )

+ψ 0,∂ξw2( )

ψ 0,φC( )

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ v3

Passive DevicePassive Device

00 is the adjoint null eigenmodeis the adjoint null eigenmode,, CC the unstable eigenmode the unstable eigenmode,, n n33 and and ww22 nonlinear terms nonlinear terms

√√↵

√√↵

−+=

2

2

exp10 ω

εx

EE II

Input PumpInput Pump+ Gaussian+ Gaussian

Spontaneously moving solitons Spontaneously moving solitons emerge from stationary soliton in emerge from stationary soliton in a supercritical bifurcation.a supercritical bifurcation.

Destabilising mode is Destabilising mode is identical identical to to translational modetranslational mode (cf. Michaelis (cf. Michaelis et alet al 2001, 2001, Skryabin Skryabin et al et al 2001)2001)

The soliton’s velocity The soliton’s velocity v v obeys:obeys:

Equation of motionEquation of motion A J Scroggie et al, PRE 66 036607 (2002


Self-propelled gas-discharge solitons Self-propelled gas-discharge solitons A. W. Liehr et al, New Journal of Physics, 5, 89.1 (2003).

Gas discharges can form Gas discharges can form soliton-like filaments, which soliton-like filaments, which show a bifurcation to show a bifurcation to spontaneous motion. spontaneous motion. (Purwins group, Muenster)(Purwins group, Muenster)


Field (white) and temperature (red) of self-propelled soliton Field (white) and temperature (red) of self-propelled soliton confined by dip in pump field - note inertia.confined by dip in pump field - note inertia.

Self-propelled soliton “laser” Self-propelled soliton “laser” J McSloy, thesis.



In conditions where moving pattern forms in dip, a “bump” In conditions where moving pattern forms in dip, a “bump” added to dip induces emission of soliton train.added to dip induces emission of soliton train.




FieldField TemperatureTemperature

Absorbing medium: dark cavity solitons Absorbing medium: dark cavity solitons move – 2D collisionsmove – 2D collisions A J Scroggie et al, PRE 66 036607 (2002)




Inertia of Self-Propelled Cavity SolitonsInertia of Self-Propelled Cavity Solitons

CS can acquire inertia if a second mode becomes degenerate CS can acquire inertia if a second mode becomes degenerate with the translational mode.with the translational mode. But dynamics is not Newtonian. But dynamics is not Newtonian.

Temperature field through a collision, Temperature field through a collision, showing CS inertia.showing CS inertia.

Neither number nor “mass” of CS is Neither number nor “mass” of CS is conserved (but conserved (but speedspeed is) is)

““Volley” of address pulses creates CSVolley” of address pulses creates CSunstable to motion: merge on collisionunstable to motion: merge on collision

A. Scroggie (Strathclyde) unpublished


In the linear limit, long-time evolution of the field at a given plane in a cavity can be exactly described by:

Elements of “ABCD matrix” (complex) obey AD-BC=1; 2cosD. T: “slow” evolution time: TR: round-trip time; k: optical wavevector; : round-trip linear gain/loss and/or phase shift; Ein: input field €

TR∂E

∂T=

iψ

2sinψ{B

k

∂2

∂x2+ i(A − D)(x

∂

∂x+

1

2) + kCx2}E + ΓE + Ein +N(E)

Beyond mean-field cavity models …

• 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)


In the linear limit, long-time evolution of the field at a given plane in a cavity can be exactly described by:

Elements of “ABCD matrix” (complex) obey AD-BC=1; 2cosD. T: “slow” evolution time: TR: round-trip time; k: optical wavevector; : round-trip linear gain/loss and/or phase shift; Ein: input field €

TR∂E

∂T=

iψ

2sinψ{B

k

∂2

∂x2+ i(A − D)(x

∂

∂x+

1

2) + kCx2}E + ΓE + Ein +N(E)

Beyond mean-field cavity models 2

• 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)


Focusing regime: =-2, C=50, 0=-0.4, T=0.1

z

x

Adding time dimension to two-level cavity soliton model, longitudinal filaments spontaneously contract to 3D localized

structures called cavity light bullets (CLBs).

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.

M.Brambilla, L.Columbo, T.Maggipinto, G.Patera, Phys.Rev.Lett. 93, 203901 (2004)

6 8 10 12 14 16 18 20 22

0

20

40

80

100

120

140 filamentslocalized structures

yinj

3D Cavity Light Bullets in a Nonlinear Optical Resonator


Conclusion: Time for something edible!

Syrup Solitons on German TV!

cavity soliton dynamics

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