Transcript Document 7583782

Cavity Soliton Dynamics
William J Firth
Department of Physics,
University of Strathclyde, Glasgow, Scotland
Acknowledgements: Thorsten Ackemann, Damia Gomila, Graeme Harkness,
John McSloy, Gian-Luca Oppo, Andrew Scroggie, Alison Yao (Strathclyde)
FunFACS and PIANOS partners
Cavity Soliton Dynamics, Cargese, May 2006
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 + 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
Cavity Soliton Dynamics, Cargese, May 2006
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
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
Cavity Soliton Dynamics, Cargese, May 2006
Our Approach – Newton Method
model
equation
stationary
states
discretise
algebraic
system
Newton
method
solutions
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
FFT
Ak
x -|k|2
Bk
(FFT)-1
Involves the Jacobian matrix, Jij= Fi/Aj
Cavity Soliton Dynamics, Cargese, May 2006
(2A)j
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 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 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)  eiK.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, 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)
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 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 (2000)
m=2
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. McSloy, D. Gomila, P. Colet, JOSA B19 747-751 (2002)
Lugiato-Lefever eqn in Kerr cavity: perturbed NLS:
E 1  2 E
2
i

 E E  i(E  iE  E in )
2
t 2 x
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
Cavity Soliton Dynamics, Cargese, May 2006
2D Kerr Cavity Soliton
WJF, G. K. Harkness, A. Lord, J. McSloy, D. Gomila, P. Colet, JOSA B19 747-751 (2002)
Cavity Soliton Dynamics, Cargese, May 2006
Stability of 2D Kerr Cavity Solitons
WJF, G. K. Harkness, A. Lord, J. McSloy, D. Gomila, P. Colet, JOSA B19 747-751 (2002)
hex
sol
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 …
Cavity Soliton Dynamics, Cargese, May 2006
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
• =1.3; I= 0.9.
Cavity Soliton Dynamics, Cargese, May 2006
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
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
• “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 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.
Cavity Soliton Dynamics, Cargese, May 2006
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)
•
Cavity Soliton Dynamics, Cargese, May 2006
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.
Cavity Soliton Dynamics, Cargese, May 2006
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.
Cavity Soliton Dynamics, Cargese, May 2006
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).
E
2C
2
 E  iE  E in  i E 
E
2
t
1 E
Lossy, mistuned, driven, diffractive, single
longitudinal-mode cavity, containing saturable

absorber of “density” 2C
2
Analysis predicts instability to pattern with k c  
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 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.
Cavity Soliton Dynamics, Cargese, May 2006
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.
Cavity Soliton Dynamics, Cargese, May 2006
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
Cavity Soliton Dynamics, Cargese, May 2006
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).
Cavity Soliton Dynamics, Cargese, May 2006
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).
Cavity Soliton Dynamics, Cargese, May 2006
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)
Cavity Soliton Dynamics, Cargese, May 2006
Cavity Soliton modes and dynamics
Most cavity+medium systems to date described by
E
F
2
 Ein  G(E,F, );
 H(E,F, 2 )
t
t
Field
•
•
•
•
•
Medium
 e.g. Newton method to find time-independent CS solutions
Use,
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 a perturbation, project
the perturbation on to the mode
• BUT the modes are not orthogonal – bi-orthogonal to adjoint set
aÝn  n an  u†n ,P
 an  
1
n
†
n
u ,P
u†n ,un
†
n
u ,un
• 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

aÝ0  u†0,P
u†0,un  vCS
• 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. 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.
G0
GN 
2
N

j l
e
 Re( k ) R jl
cos Imk R jl   
R jl
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. 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:
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
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. E65, 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)
• 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, May 2006
Cavity Soliton Pixel Arrays
John McSloy, 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
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
(1D) 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 a perturbation, project
the perturbation on to the mode
• BUT the modes are not orthogonal – bi-orthogonal to adjoint set
aÝn  n an  u†n ,P
 an  
1
n
†
n
u ,P
u†n ,un
†
n
u ,un
• 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

aÝ0  u†0,P
u†0,un  vCS
• 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
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
Cavity Soliton Dynamics, Cargese, May 2006
QuickTime™ and a
Animation decompressor
are needed to see this picture.
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
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 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
g  0.01  semiconductor
slope 1
• 1D,
perturbation analysis
• velocity affected by response time of medium
log (g)
• limits to ideal response for fast medium g>1
Cavity Soliton Dynamics, Cargese, May 2006
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: predicted and “measured” CS-velocity v(K) induced by driving
field phase modulation exp(imKx) for fixed m
v(K)
REVERSE GEAR
K
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
E0
E
Medium
v(K)
E
 (1 i) E  E 0e im cos(Kx )  G(| E |2 )E  ixx E
t
G(| E |2 )  i | E |2
Kerr Medium

G(| E | )  
1 | E |2
2
Saturable Medium
REVERSE GEAR
Cavity Soliton Dynamics, Cargese, May 2006
General but
not Universal
Kerr
Saturable
K
“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.
Cavity Soliton Dynamics, Cargese, May 2006
Digression: snowballs
G D'Alessandro and WJF, Phys Rev A46, 537-548 (1992)
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
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 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 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.
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 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 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 solitons
emerge from stationary soliton in
a supercritical bifurcation.
Passive
Device
Input Pump
+ Gaussian
   i  N  1
 
2  x 2  
 2  
E I  E I 1 1
exp
Destabilising mode is identical to
translational mode
(cf. Michaelis et al 2001,
Skryabin et al 2001)
0


J 0
 

  1 .6


The soliton’s velocity v obeys:
 ,Lˆ  Lˆ     ,n  

v
v

0
t
0
 0 ,  C 
C
0
3
 ,  
 0 C

, w2  3
v

 0, C  
0
0 is the adjoint null eigenmode, C the unstable eigenmode, n3 and w2 nonlinear terms
Cavity Soliton Dynamics, Cargese, May 2006
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 McSloy, thesis.
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
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)
move – 2D collisions
Field
Temperature
Cavity Soliton Dynamics, Cargese, May 2006
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 at a given
plane in a cavity can be exactly described by:
E
i
B 2
 1
TR

{ 2  i(A  D)(x  )  kCx 2}E  E  Ein +N(E)
T 2sin  k x
x 2
Elements of “ABCD matrix” (complex) obey AD-BC=1; 2cos
T: “slow” evolution time: TR: round-trip time; k: optical wavevector;
 : round-trip linear gain/loss and/or phase shift; Ein: input field
D.
• 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 at a given
plane in a cavity can be exactly described by:
E
i
B 2
 1
TR

{ 2  i(A  D)(x  )  kCx 2}E  E  Ein +N(E)
T 2sin  k x
x 2
Elements of “ABCD matrix” (complex) obey AD-BC=1; 2cos
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
D.
3D Cavity Light Bullets in a Nonlinear Optical Resonator
M.Brambilla, L.Columbo, T.Maggipinto, G.Patera, Phys.Rev.Lett. 93, 203901 (2004)
Focusing regime: =-2, C=50, 0=-0.4, T=0.1
filaments
localized structures
140
120
I=|F ST (L)|
2
100
80
40
20
0
6
8
10
12
14
yinj
x
16
18
20
22
Adding time dimension to two-level cavity
soliton model, longitudinal filaments
spontaneously contract to 3D localized
structures called cavity light bullets (CLBs).
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, May 2006