Diapositiva 1

1
The hunt for 3D global or localized structures
in a semiconductor resonator Ph.D student
Lorenzo Columbo Supervisor Prof.
Luigi Lugiato External supervisor Prof.
Massimo Brambilla
(Politecnico di Bari)
Università degli studi dellInsubria
Como, 22 settembre, 2005
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Outline
Short introduction. 2D and 3D structures
localization in a dissipative optical system
Cavity Solitons in a VCSEL below lasing threshold
and Cavity Light Bullets in a nonlinear resonator
filled with a two level system.
Beyond the Single Longitudinal Mode
Approximation The dynamical model, the Linear
Stability Analysis and the first numerical
results.
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Pattern formation in a semiconductor resonator
The spontaneous formation in the transverse
profile of the field emitted by a Vertical Cavity
Surface Emitting Laser (VCSEL) driven by a
coherent injected field and slightly below lasing
threshold of highly spatial correlated structures
(global structures) or that of independent,
isolated intensity peaks (localized structures or
Cavity Solitons (CSs)) represents a valid example
of Pattern formation in Optics.
VCSEL device (Bottom Emitter)
Numerical simulations Global and localized
structures in ER2 transverse profile
p-contact
Bragg reflector
Cavity Solitons
Active layer (MQWs GaAs-GaAlAs)
Rolls
Honeycombs
mm
Bragg reflector
150 mm
GaAs Substrate
n-contact
n-contact
Transverse intensity field profile on the exit
window
E In (injected field plane wave)
E R (reflected field)
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Applications of Cavity Solitons to the optical
information technology
Even if a complete physical interpretation is
still missing, from a fundamental point of view
these phenomena result from the
competition\balance between linear and nonlinear
effects in the radiation-matter interaction
combined with the resonators feedback\dissipation
action.
? -
? -
?
We dont have it in the Spatial Solitons case
Intensity field profile of a single CS
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Beyond Single Longitudinal Mode Approximation!!
The previous results are valid in the Single
Longitudinal Mode Approximation (SLMA) according
to which the field profile is uniform in the
propagation direction in all the systems
configurations. Although this condition is well
verified in a VCSEL for example, we could ask
what would happen in the longitudinal field
profile when it is not fulfilled (long cavities,
high values of mirrors transmissivity etc.)
A. Is it possible to observe spontaneous 3D
confinement? B. In this case could we externally
control these new fully localized structures like
what happens with CSs?
?
y
?
Plane wave
x
z
CS Transverse localization.
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1. Beyond SLMA two level system
Starting from 2002 we tried to answer to the
previous questions by considering first a optical
prototype
Unidirectional ring resonator filled with a
vapour of two level atoms and driven by a
coherent injected beam
ER
Nonlinear medium
4
1
T0
EI
ET
3
2
T0
wa atomic transition frequency w0 input field
frequency wn generic empty cavity mode
EI injected field (Plane wave) ET transmitted
field ER reflected field
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1.1. Maxwell-Bloch equation
E normalized envelope of the intracavity
field Y normalized envelope of the injected
field LA resonator lengthnonlinear medium
length aa normalized absorption coefficient
at resonance T transmission
coefficient (R1-T) D(wa-w0)/g? da0
(wc-w0) LA /c z normalized propagation
coordinate x, y normalized transverse
coordinate t normalized time coordinate
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1.2. 3D global and 3D localized structures
We predicted in this case in more than one
parametric regime the formation of 3D global
structures and 3D self-confinement phenomena (M.
Brambilla et al., PRL 93 2042, 2004). We named
Cavity Light Bullets (CLBs) the fully localized
structures travelling along the resonator with a
constant spatial dimensions and a constant
period.
Isosurface plot of the intracavity intensity
field profile
a) 3D filaments
b) Cavity Light Bullets
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CLB external control (21) dim
We also demonstrated the possibility to excite or
erase one or more independent CLBs by means of
suitable addressing beams in both parallel or
serial configurations. We also managed to drift
transversely a single CLB.
Switching on of one or more CLBs
a) Switching on of a single CLB
b) Switching on of two parallel CLBs
c) Switching on of a CLB train
We then answered to question B
Even more exciting applications to optical
information storage and processing than CSs
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Outline
Short introduction. 2D and 3D structures
localization in a dissipative optical system
Cavity Solitons in a VCSEL below lasing threshold
and Cavity Light Bullets in a nonlinear resonator
filled with a two level system.
Beyond the Single Longitudinal Mode
Approximation The dynamical model, the Linear
Stability Analysis and the first numerical
results.
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2. Beyond SLMA semiconductor resonators
?
Why are semiconductors devices relevant?
They have a very fast dynamics Their growth and
hence their energy spectrum can be controlled
with high precision degree They can be
miniaturized They already have broad
applications in telecommunications and
optoelectronics etc.
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Unidirectional ring resonator filled with a Bulk
or a Multi Quantum Wells (MQWs) semiconductor
sample
Phenomenological model used to describe radiation
matter-interaction by means of a complex
susceptibility where in the passive
configuration while in the active
configuration
ER
4
1
T0
EI
ET
3
2
T0
e
with , N carrier
density, N0 transparency carrier density, A
absorption\gain coefficient, n background
refractive index, we half width of the
excitonic absorption line, ge central frequency
of the excitonic absorption line, a linewidth
enhancement factor
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2.1. Maxwell-Bloch equations
Maxwell-Bloch equations describing system
dynamics within the rate equation, SVEA and
paraxial approximations but without introducing
any hypothesis on the longitudinal field profile
Boundary condition
(1a)
(1b)
D normalized difference between N and N0
d0 normalized cavity detuning
d diffusion coefficient g nonradiative
decay constant ? photon life time m pump
parameter (mlt0?absorber 0ltmlt1?amplifier
mgt1?laser)
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Linear Stability Analysis
In the general case, the nonlinear character of
eq. (1a)-(1b) prevents us to solve them
analytically. Equating to zero the time
derivatives and the terms with the laplacian
operators we can get numerically their stationary
and transversely homogeneous solutions Xs, where
X stands for the generic variable it turns out
these solutions are associated to a non uniform
field profile in the propagation
direction.
Intensity field profile for a fixed (x,y) value
We study the stability of Xs against spatially
modulated perturbations by applying a well known
approximate method the Linear Stability Analysis
(LSA).
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LSA
Contrary to what happens in the Single
Longitudinal Mode Approximation, the a priori
unknown z-dependence of Xs introduces an high
degree of complexity in LSA. In particular,
looking for solution of Maxwell-Bloch equations
in the form with dXltltXs we cannot derive
for each modal amplitude an equation for l
describing its the temporal evolution. Then,
extending the results obtained in the two level
system, we adopt an alternative approach
Fourier expansion
we expand dX on the transverse
Fourier basis keeping implicit its
z-dependence Thus we get for each (kx, ky)
a system of two linear ordinary differential
equations for , that we
rename , and its c.c.
Step1
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LSA
The easiest way to proceed at this
point is to introduce the polar
representation of Es and dE0 where rs, qs,
dr, dq are real quantities. After some simple
algebra, we then get where k?(kx2
ky2)1/2, ?(z)r2s(z) and r and u are auxiliary
variables linked to dr and dq trough the linear
transformation
Step 2
(2a)
(2b)
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LSA
Combining eq. (2a) and (2b) we
derive the following 2nd order
linear differential equation for r where
the coefficients A, B, Hi, i1..5 depend on the
physical parameters, Xs, k? and also on l. We
then reduce the initial problem to that of
solving the previous equation. Since the
complicated expressions of the polynomial
coefficients it is not easy (possible?) to find
an analytical general solution of this equation
on the other hand we can approximate it around
the regular singular point ?0 as superposition
of the two linearly independent series solutions
r1 and r2 where c1, c2 are arbitrary complex
constants. We also get for u from (2a) and
(2b)
Step 3
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LSA
Finally, taking into account the boundary
conditions for r and u, we get an algebraic
homogeneous system for c1 and c2
which admits non trivial solution if and only if
the following condition is fulfilled
Step 4
Keeping fixed the other
quantities, it represents a nonlinear implicit
equation for the l which, solving our LSA
problem, tells us how evolves in time the generic
transverse mode amplitude of the perturbation
given a stationary transversely homogeneous
state it is unstable if exists at least one
zero l of the function C with Re(l)gt0.
NOTE We checked the validity of this LSA by
reproducing the results obtained in the SLMA
framework for a parametric regime which fulfils
the SLMA conditions.
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2.2. Numerical simulations
Using the indications of LSA we study system
dynamics by numerical integration of eq.
(1a)-(1b) with the relative boundary condition.
For this highly demanding computational task we
developed a parallel code. First stage of
investigation close to the atomic system In
this first stage of investigation we take
advantage of the results obtained in the atomic
system in fact from eq. (1a)-(1b) neglecting
diffusion (d0) and after adiabatic elimination
of the carrier density variable in the limit
ggtgt1, we get in the passive case which is
formally equivalent to the equation describing
system dynamics in the atomic case. Following
this analogy, the idea is to look for fully
confined structures still using eq. (1a)-(1b)
with d0 and ggtgt1 in parametric regimes linked to
those in which we observed CLBs through
relations
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Self-defocusing passive parametric regimesDe2,
d0-0.3, m-30, T0.1,d0 and ggtgt1 (?300.0)
Longitudinal filamentsand fully localized
structures
When, as happens in this case, d0 and
Im(l)ltlt1ltltg the instability domains are
independent from g. In spite of this, g still
plays a role in influencing systems dynamical
evolution. We observe in the general case highly
correlated longitudinal filaments at regime.
Two fully localized structures for g50.0
Stationary transversely homogeneous states curves
(independent from g)
Y22.975
Two stable fully localized structures obtained by
cutting two longitudinal filaments and letting
the system evolve. They are not independent from
each other.
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Self-focusing passive parametric regimesDe-2,
d0-0.4, m-20, T0.1,d0 and ggtgt1 (?500.0)
Longitudinal filaments ((21) dim)
When, as happens in this case, d0 and
Im(l)ltlt1ltltg the instability domains are
independent from g. In spite of this, g still
plays a role in influencing system dynamical
evolution.
Longitudinal filaments (g300)
Stationary transversely homogeneous states curves
(independent from g)
Y
17.0
11.0
((21) dim)
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..and fully confined structures
Although we still dont observe phenomena of
spontaneous structures localization in the
propagation direction, we proved that a
longitudinal confined portion of a longer
filament represents a stable systems solution
for a sizable interval of Y values.
The localized structure disappears when we
decrease g under a certain threshold
Fully localized structure (g300)
Intensity field profile on the exit window
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Observation
We put ggtgt1
Since we have g gnr/kp where gnr is the
nonradiative carrier density decay constant,
while kpcT/nLA is the inverse of the photon life
time, we can think to get large value of g by
increasing LA. a) We could consider for example
Edge Emitter configurations
b) Moreover, we can get the same result by
considering the case medium length ? cavity
length and increasing the latter
a)
b)
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Outline
Short introduction. 2D and 3D structures
localization in a dissipative optical system
Cavity Solitons in a VCSEL below lasing threshold
and Cavity Light Bullets in a nonlinear resonator
filled with a two level system.
Beyond the Single Longitudinal Mode
Approximation The dynamical model, the Linear
Stability Analysis and the first numerical
results.
25
Future Agenda
Passive case
Systematic study of the proprieties of these
localized structures in analogy to what we did
for CLBs.
Looking for fully localized structures in less
critical parametric domains and\or
configurations.
Few days ago...
Switching on process of a single localized
structure by using an external addressing beam
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Future Agenda
Active case We can also consider the active
configurations below or above lasing threshold.

The VCSEL configuration is already used to
observe CSs even above lasing threshold.?
(transverse localization)
The Vertical External Cavity Surface Emitting
Laser (VECSEL) configuration is for example
already used to produce mode locking laser
operation.? (longitudinal localization)

?
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Conclusions
We looked for 3D pattern formation and 3D
self-confinement in a semiconductor resonator
driven by a coherent injected field.
We extended the model describing system dynamics
in SLMA to include a generic intracavity field
longitudinal profile we applied to the LSA of
the stationary and transversely homogeneous field
configurations a semianalytical approach
developed in a prototype.
Even in this case, the first numerical
investigations show the existence of both global
(longitudinal filaments) and fully localized
structures the latter are candidate to be the
semiconductor analogous of CLBs.
Funfacs European Project
This work is supported by the Funfacs
(Fundamentals, Functionalities and Applications
of Cavity Solitons)- F.E.T. VI P.Q. UE. In the
framework of this European collaboration with
many other theoretical and experimental research
units, I am going to join the Computational
Nonlinear and Quantum Optics group at the
University of Strathclyde (Scotland) for a
visiting period of six months.
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