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Spring School on Solitons in Optical Cavities
Cargèse, May 8-13, 2006
Introduction to Cavity Solitons and Experiments
in Semiconductor Microcavities
Luigi A. Lugiato
Dipartimento di Fisica e Matematica, Università dell’Insubria, Como (Italy)
Collaborators:
- F. Prati, G. Tissoni, L. Columbo (Como)
- M. Brambilla, T. Maggipinto, I.M. Perrini (Bari)
- X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J.R. Tredicce, INLN (Nice)
- R. Jaeger (Ulm)
- R. Kheradmand (Tabriz)
- M. Bache (Lingby)
- I Protsenko (Moscow)
Program
- Science behind Cavity Solitons: Pattern Formation (Maestoso)
- Cavity Solitons and their properties (Andante con moto)
- Experiments on Cavity Solitons in VCSELs (Allegro)
Future: the Cavity Soliton Laser (Allegro vivace)
- My lecture will be “continued” by that of Willie Firth
-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
Optical Pattern Formation
y
x
z
Optical pattern formation: old history
- J. V. Moloney
- A huge and relevant Russian literature
(A.F. Sukhov, N.N. Rosanov, I. Rabinovich, S.A. Akhmanov,
M.A. Vorontsov etc.)
In particular, N.N. Rosanov introduced and studied “Diffractive Autosolitons”,
precursors of Cavity Solitons
A recent review:
LL, Brambilla, Gatti, Optical Pattern Formation
in Advances in Atomic, molecular and optical physics, Vol. 40,
p 229, Academic Press, 1999
Nonlinear Optical Patterns 1
 The mechanism for spontaneous optical pattern formation from a homogeneous
state is a modulational instability, exactly as e.g. in hydrodynamics,
nonlinear chemical reactions etc
 Modulational instability: a random initial spatial modulation, on top of
a homogeneous background, grows and gives rise to the formation of a
pattern
0
23
 In optical systems the modulational instability is produced by the
combination of nonlinearity and diffraction.
In the paraxial approximation diffraction is described by the transverse Laplacian:

2

2
2
 2 2
x y
Nonlinear Optical Patterns 2
 Optical patterns may arise
 in propagation
 in systems with feedback, as e.g.
optical resonators
or single feedback mirrors
 Optical patterns arise for many kinds of nonlinearities ((2), (3), semiconductors,
photorefractives..)
 There are stationary patterns and time-dependent patterns of all kinds
Optical Pattern Formation
Nonlinear media in cavities
Nonlinear
Medium
Nonlinear Medium
nlnl
Input
Cavity
(Plane Wave )
Hexagons
Honeycomb
Output
(Pattern )
Rolls
MEAN FIELD MODELS
Mean field limit  thin sample, high cavity finesse
The purely dispersive case (L.L., Lefever PRL 58, 2209 (1987))
E
 
t
 E  E  i  E  i E E  i  a  E
2
I
2

normalized slowly varying envelope of the electric field
cavity damping rate (inverse of lifetime of photons in the cavity)
input field of frequency 0
cavity detuning parameter
 
c  0

, c = longitudinal cavity frequency nearest to 0
cubic, purely dispersive, Kerr nonlinearity
diffraction parameter
The purely absorptive case (LL, Oldano PRA 37, 96 (1988) ;
Firth, Scroggie PRL 76, 1623 (1996))
E
 
t


2C
2
E

E

i

E

E

i

a



I
E
2
1 E


saturable absorption, C = bistability parameter
MEAN FIELD MODELS as “simple” as pattern formation models in nonlinear
chemical reactions, hydrodynamics, etc.
The “ideal” configuration for mean field models (mean field limit, plane mirrors)
has been met in broad area VCSELs (Vertical Cavity Surface Emitting Lasers).
Kerr slice with feedback mirror (Firth, J.Mod.Opt.37, 151 ( 1990))
B|F
F
thin Kerr slice
Plane Mirror
- Crossing the Kerr slice, the radiation undergoes phase modulation.
- In the propagation from the slice to the mirror and back, phase modulation
is converted into an amplitude modulation
- Beautiful separation between the effect of the nonlinearity and that of
diffraction, only one forward-backward propagation  Simplicity
- Strong impact on experiments
Encoding a binary number in a 2D pattern??
1
0
1
0
0
1
1
1
0
Problem: different peaks of the pattern are strongly correlated
The solution to this problem lies in the concept of
Localised Structure
The concept of Localised Structure is general in the field of pattern formation:
- it has been described in Ginzburg-Landau models (Fauve Thual 1988)
and Swift-Hohenberg models (Glebsky Lerman 1995),
- it has been observed in fluids (Gashkov et al., 1994), nonlinear chemical
reactions (Dewel et al., 1995), in vibrated granular layers (Tsimring
Aranson 1997; Swinney et al, Science)
Solution: Localised Structures
1D case
Spatial structures concentrated in a relatively small region
of an extended system, created by stable fronts connecting
two spatial structures coexisting in the system
Theory 1D: P. Coullet, C. Riera, and C. Tresser, Phys. Rev. Lett. 84, 3069-3072
(2000)
Solution: Localised Structures
1D case
Spatial structures concentrated in a relatively small region
of an extended system, created by stable fronts connecting
two spatial structures coexisting in the system
Theory 1D: P. Coullet, C. Riera, and C. Tresser, Phys. Rev. Lett. 84, 3069-3072
(2000)
Localised Structures
Tlidi, Mandel, Lefever
- Localised structure = a piece of a pattern
- The scenario of localised structures corresponds to a pattern
“broken in pieces”
E.g. a Cavity Soliton corresponds to a single peak of a hexagonal pattern
(Firth, Scroggie PRL 76, 1623 (1996))
-WARNING: there is a smooth continuous transition from a pattern
(in the rigid sense of complete pattern or nothing at all) to a scenario
of independent localised structures (see e.g. Firth’s lecture)
Program
- Science behind Cavity Solitons: Pattern Formation (Maestoso)
- Cavity Solitons and their properties (Andante con moto)
- Experiments on Cavity Solitons in VCSELs (Allegro)
Future: the Cavity Soliton Laser (Allegro vivace)
- My lecture will be “continued” by that of Willie Firth
-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 SOLITONS
Holding beam
Output field
Nonlinear medium
nl
The cavity soliton persists after the passage of the pulse.
Each cavity soliton can be erased by re-injecting the
writing pulse.
Intensity profile
Intensity
Writing
pulses
x
y
- Cavity solitons are independent of one another (provided they are not too
close to one another) and of the boundary.
- Cavity solitons can be switched on and off independently of one another.
- What is the connection with standard solitons?
Solitons in propagation problems
Solitons are localized waves that propagate (in nonlinear media) without change of form
Temporal Solitons: no dispersion broadening
u
 2u
2
i  u u  2 0
z
t
“Temporal” NLSE:
z
propagation
dispersion
Spatial Solitons: no diffraction broadening
u
 2u
2
i  u u 2 0
z
x
x
z
“Spatial” NLSE:
diffraction
u
 u  u
2
 u u  2  2 0
z
x  y
2
i
y
1D
2
2D
Cavity Solitons are dissipative !
E.g. they arise in the LL model, which is equivalent to a “dissipative NLSE”
u
2
2
i  u u iu   u  i uinj  0
t
dissipation
diffraction
Dissipative solitons are “rigid”, in the sense that, once the values
of the parameters have been fixed, they have fixed characteristics
(height, radius, etc)
Typical scenario: spatial patterns and Cavity Solitons
Cavity Solitons
Roll pattern
Honeycomb pattern
2,5
|E|
2,0
1,5
1,0
Stable hom. branch
Unstable hom. branch
0,5
-
0,0
0,0
0,5
1,0
| EI |
1,5
2,0
On/off switching of Cavity Solitons
- Coherent switching: the switch-on is obtained by injecting a writing beam
in phase with the holding beam; the switch-off by injecting a writing beam
in opposition of phase with respect to the writing beam
- Incoherent switching: the switch-on and the switch-off are obtained
independently of the phase of the holding beam.
E.g. in semiconductors, the injection of an address beam with a frequency
strongly different from that of the holding beam has the effect
of creating carriers, and this can write and erase CSs.
(See Kuszelewicz’s lecture)
~2ns
CS off
CS on
CS on
~5 ns
CS off
The incoherent switching is more convenient, because it does not require
control of the phase of the writing beam
Motion of Cavity Solitons
KEY PROPERTY: Cavity Solitons move in presence of external gradients, e.g.
1) Phase Gradient in the holding beam,
2) Intensity gradient in the holding beam,
3) temperature gradient in the sample,
In the case of 1) and 2) usually the motion is counter-gradient, e.g. in the case
of a modulated phase profile in the holding beam, each cavity soliton tends to
move to the nearest local maximum of the phase
Possible applications:
realisation of reconfigurable
soliton matrices, serial/parallel
converters, etc
Phase profile
A complete description of CS motion, interaction, clustering etc. will be given
in Firth’s lecture.
Review articles on Cavity Solitons
- L.A.L., IEEE J. Quant. Electron. 39, 193 (2003).
- W.J. Firth and Th. Ackemann, in Dissipative solitons, Springer Verlag
(2005), p. 55-101.
Experiments on Cavity Solitons
- in macroscopic cavities containing e.g. liquid crystals,
photorefractives, saturable absorbers
- in single feedback mirror configuration (Lange et al.)
- in semiconductors
The semiconductor case is most interesting because of:
- miniaturization of the device
- fast response of the system
Program
- Science behind Cavity Solitons: Pattern Formation (Maestoso)
- Cavity Solitons and their properties (Andante con moto)
- Experiments on Cavity Solitons in VCSELs (Allegro)
Future: the Cavity Soliton Laser (Allegro vivace)
- My lecture will be “continued” by that of Willie Firth
-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
The experiment at INLN (Nice)
and its theoretical interpretation
was published in
Nature 419, 699
(2002)
Experimental Set-up
S. Barland, M. Giudici and J. Tredicce, Institut Non-lineaire de Nice (INLN)
L
aom
L
Holding beam aom
M
M
Tunable Laser
Writing beam
BS
L
VCSEL
L
BS
C
CCD
C
BS
BS
Detector linear array
BS: beam splitter, C: collimator, L: lens, aom: acousto-optic modulator
The VCSEL
Th. Knoedl, M. Miller and R. Jaeger, University of Ulm
p-contact
Bottom Emitter (150m)
Bragg reflector
Active layer (MQW)
Bragg reflector
GaAs Substrate
n-contact
ER
E In
Features
1) Current crowding at borders (not critical for CS)
2) Cavity resonance detuning (x,y)
3) Cavity resonance roughness (layer jumps) See R.Kuszelewicz et al.
"Optical self-organisation in bulk and MQW GaAlAs Microresonators", Phys.Rev.Lett. 84,
6006 (2000)
Experimental results
Below threshold,
injected field
Above threshold,
no injection (FRL)
x
x
Intensity (a.u.)
Interaction disappears on the right side
of the device due to cavity resonance
gradient (400 GHz/150 m, imposed
by construction)
Observation of different
structures (symmetry and
spatial wavelength)
in different spatial regions
Frequency (GHz)
Frequency (GHz)
Intensity (a.u.)
x (m)
x (m)
In the homogeneous region:
formation of a single spot of about
10 m diameter
Experimental demonstration of independent writing and erasing
of 2 Cavity Solitons in VCSELS below threshold,
obtained at INLN Nice
S. Barland et al, Nature 419, 699 (2002)
The Model
M. Brambilla, L. A. L., F. Prati, L. Spinelli, and W. J. Firth, Phys. Rev. Lett. 79, 2042 (1997).
L.Spinelli, G.Tissoni, M. Brambilla, F. Prati and L. A. L., Phys.Rev.A 58 , 2542 (1998)


E
2
  1  i  ( x, y )  E  EI ( x, y )  i 2 C  N  E  i a   E ,
t

N
2
2
  N  Im  N  E  I ( x, y )  d   N
t
Where
 N     i N  1

E = normalized S.V.E. of the intracavity field
EI = normalized S.V.E. of the input field
N = carrier density scaled to transp. value
 = cavity detuning parameter
 = linewidth enhancement factor
2C = bistability parameter
(x,y) = (C - 0) /  + (x,y)
EI ( x, y)
Broad Gaussian (twice the VCSEL)
Choice of a simple model: it describes the basic physics and more refined models
showed no qualitatively different behaviours.
Theoretical interpretation
Patterns (rolls, filaments)
Cavity Solitons
37.5
0
75
3
x (m)
112.5
150
-1.50
-1.25
x (m)
0 37.5 75 112.5 150
|ES|
2
1
0
-2.25
-2.00
-1.75

-2.25 -2.00 -1.75 -1.50 -1.25

The vertical line corresponds to the MI boundary
CS form close to the MI boundary, on the red side
Pinning by inhomogeneities
Experiment
Numerics
 (x,y)
Broad beam only
Add local perturbation
Cavity Solitons
appear close to the MI boundary,
Final Position is imposed by roughness
of the cavity resonance frequency
Broad beam only
7 Solitons: a more recent achievement
X. Hachair, et al., Phys. Rev. A 69, 043817 (2004).
CS can also appear spontaneously ...........
Experiment
Numerics
In this animation we reduce the injection level of the holding beam starting from values where
patterns are stable and ending to homogeneous solutions which is the only stable solution for
low holding beam levels. During this excursion we cross the region where CSs exist. It is
interesting to see how pattern evolves into CS decreasing the parameters. Qualitatively this
animation confirms the interpretation of CS as “elements or remains of bifurcating patterns”.
VCSEL above threshold
Depending on current injection level two different scenarios are possible
(Hachair et al. IEEE Journ. Sel. Topics Quant. Electron., in press)
 = - 2,  = 3, J = 1.01, d = 0.052,  / p = 0.01, ||/p = 0.0001
1,6
|ES|
5% above threshold
1,2
CS
g
Turin
0,8
unsta
ble
 = - 2,  = 3, J = 1.2, d = 0.052,  / p = 0.01, ||/p = 0.0001
2,5
0,4
|ES|
CS on unstable background
2,0
0,0
0,0
Hopf unstable
0,2
0,4
0,6
0,8
1,0
EI
1,2
1,5
le
nstab
gu
Turin
1,0
20% above threshold
0,5
Hopf unstable
0,0
0,0
0,5
1,0
1,5
EI
2,0
Despite the background oscillations, it is perfectly possible to create
and erase solitons by means of the usual techniques of WB injection
Program
- Science behind Cavity Solitons: Pattern Formation (Maestoso)
- Cavity Solitons and their properties (Andante con moto)
- Experiments on Cavity Solitons in VCSELs (Allegro)
Future: the Cavity Soliton Laser (Allegro vivace)
- My lecture will be “continued” by that of Willie Firth
-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 Laser
- A cavity soliton laser is a laser which may support cavity solitons (CS)
even without a holding beam : simpler and more compact device!
- A cavity soliton emits a set of narrow be18ams (CSs), the number and
position of which can be controlled
CSL
CS are embedded
in a dark background:
maximum visibility.
- In a cavity soliton laser the on/off switching must be incoherent
The realization of Cavity Soliton Lasers is the main goal of the
FET Open project FunFACS.
LPN Marcoussis
INLN Nice
INFM Como, Bari
USTRAT Glasgow
ULM Photonics
LAAS Toulouse
- CW Cavity Soliton Laser
- Pulsed Cavity Soliton Laser (Cavity Light Bullets)
Approaches:
- Laser with saturable absorber
- Laser with external cavity or external grating
Conclusion
Cavity Solitons are
interesting !
Control of two independent spots
50 W writing beam
(WB) in b,d. WB-phase
changed by  in h,k
All the circled states
coexist when only the broad
beam is present
Spots can be
interpreted
as CS