Transcript Document

The hunt for
3D global or localized structures in a
semiconductor resonator
Ph.D student: Lorenzo Columbo
Supervisor:
External supervisor:
Prof. Luigi Lugiato
Prof. Massimo Brambilla
(Politecnico di Bari)
Università degli studi dell’Insubria
Como, 22 settembre, 2005
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.
3D Pattern formation in a semiconductor resonator driven by a
coherent injected field.
Beyond the Single Longitudinal Mode Approximation:
The dynamical model, the Linear Stability Analysis and the first
numerical results.
Fully localized structure in a self-focusing passive regime.
Future agenda and Conclusions
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)
p-contact
~ mm
Bragg reflector
Active layer
(MQWs
GaAs-GaAlAs)
Bragg reflector
Numerical simulations:
Global and localized structures
in |ER|2 transverse profile
Rolls
Honeycombs
Cavity Solitons
~150 mm
GaAs Substrate
n-contact
n-contact
ER
E In
(reflected field) (injected field plane wave)
Transverse intensity field profile on the exit window
Applications of Cavity Solitons to the
optical information technology
From
applicative
point
of view, since
CSs
can befrom
externally
excited,
Even aif mere
a complete
physical
interpretation
is still
missing,
a fundamental
erased
andview
drifted
means of suitable
addressing
beams (as it has
been
point of
thesebyphenomena
result from
the competition\balance
between
predicted
and
very recently
(Nature 419 699, interaction
2002)), these
micro pixels
linear and
nonlinear
effectsobserved
in the radiation-matter
combined
with are
the
candidate
realize all optical action.
devices for parallel information storage and
resonator’stofeedback\dissipation
processing.
Linear effects: diffraction
Es: CSs based parallel optical memory
Nonlinear effects: self-focusing, saturable absorption..  Resonator’s action: feedback, dissipation

Plane
waveEEI
Plane wave
I
Nonlinear
medium
Gaussian pulse
|ER|2
ER
We don’t have it
in the Spatial
Ideal scheme of a binary optical
memory.
Solitons
case
y
x
y
Intensity field profile of a single CS
x
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 system’s 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
z
x
CS: Transverse localization.
?
z
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
1
E
Nonlinear medium
Nonlinear medium
E
4
wc
T=0

EI
g
ET
2
3
EI = injected field (Plane wave)
ET= transmitted field
ER= reflected field
T=0
wm
wa
w0
wn
wa = atomic transition frequency
w0 = input field frequency
wn = generic empty cavity mode
1.1.
Maxwell-Bloch equation
Maxwell-Bloch equation describing system dynamics in the slowly varying
envelope approximation (SVEA), paraxial approximation and after
adiabatic elimination of the atomic variables, but without introducing any
hypothesis on the longitudinal field profile:
 aL A E(1  i )
E 1 E
2



i

E
2
t T z
T 1 | E |
Boundary condition:
E(z  0, t )  TY  Re ia0 E(z  1, t )
E = normalized envelope of the intracavity field
Y = normalized envelope of the injected field
LA = resonator length=nonlinear medium length
a = normalized absorption coefficient at resonance
T = transmission coefficient (R=1-T)
=(wa-w0)/g a0= (wc-w0) LA /c
z = normalized propagation coordinate
x, y = normalized transverse coordinate
t = normalized time coordinate
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
For particular value of the injected field Y
some filaments contract into stable fully
localized structures.
We then answered to question A
a) 3D filaments
b) Cavity Light Bullets
CLB external control (2+1) 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
z
x
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
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.
3D Pattern formation in a semiconductor resonator driven by a
coherent injected field.
Beyond the Single Longitudinal Mode Approximation:
The dynamical model, the Linear Stability Analysis and the first
numerical results.
Fully localized structure in a self-focusing passive regime.
Future agenda and Conclusions
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.
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:
cn
  iA
(N  N0 )
w0
ER
1
E Nonlinear mediumE
Nonlinear medium
Nonlinear medium
4
T=0
EI
where in the passive configuration:
  (1  i e )
A  A /(1  2e )
while in the active configuration:
  (1  i)
AA
ET
2
3
e
T=0
with  e  (we  w0 ) / g e , 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,  = linewidth enhancement factor
Fast carrier dynamics → we cannot adiabatically eliminate carrier dynamics
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:
E 1 E

 DE  i 2E
t T z
D
  g(D(1 | E |2 )  m  d 2D)
t
Boundary condition:
E(z  0, t )  TY  Re i0 E(z  1, t )
D = normalized difference between N and N0
0 = normalized cavity detuning
d = diffusion coefficient
g = nonradiative decay constant  photon life time
m = pump parameter (m<0→absorber; 0<m<1→amplifier; m>1→laser)
(1a)
(1b)
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
(normalized units)
250
200
150
100
50
0
0,0
0,2
0,4
0,6
0,8
1,0
z
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).
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:
  
Fourier expansion
X  X s  X  X s ( z ) 

X 0k z ,k x ,k y e
i(k z z k x x k y y ) lt
e dk z dk x dk y
  
with X<<Xs 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:
Step1
we expand X on the transverse Fourier basis keeping implicit its
z-dependence :
 
X 

X 0 ( z )k x ,k y e
i(k x x k y y ) lt
e dk x dk y
 
Thus we get for each (kx, ky) a system of two linear ordinary differential
equations for E 0 ( z)k ,k , that we rename E0 ( z) , and its c.c.
x
y
Step 2
The easiest way to proceed at this point is to introduce the polar
representation of Es and E0
E s  r s eiqs
E0  eiqs (r  iq )
where rs, qs, r, q are real quantities. After some simple algebra, we then get:
 1
  2  1   
dr
g
  u k  
 
 r


2



d
 2 l  g(1    dk  )    2m  
 2  1  
  1 
du
g

  u
 
 r  k  

2


d
 2m  l  g(1    dk  )   2 

(2a)
(2b)
where k=(kx2 + ky2)1/2, (z)r2s(z) and r and u are auxiliary variables linked to
r and q trough the linear transformation:
e zlt
r  r
T
e zlt
u  q
T
Step 3
Combining eq. (2a) and (2b) we derive the following 2nd
order linear differential equation for r:
  H0  H1  H2  2  H3  3  H4  4  H5  5 
dr 
Ag

  r

2
2



d

A

g

(


1
)
d
B2  2 (1   )A  g 

 

d2r
where the coefficients A, B, Hi, i=1..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:
r()  c1r1()  c 2r2 ()
where c1, c2 are arbitrary complex constants.
We also get for u from (2a) and (2b):
u()  c1u1( )  c 2u2 ( )
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
C(T, 0 , l,ri (z  0, l), ui (z  0, l), ri (z  1, l), ui (z  1, l))  0
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>0.
NOTE:
Observation:
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.
From a computational point of view the implicit character of the
equation C(l)=0 represents and additional CPU time consuming
factor. This forced us to implement a parallel numerical code
for LSA.


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.
Do you remember?
First stage of investigation: close to the atomic system
 aL A E(1  i )
E 1 E

 we take advantage2 ofthe
i 2results
E obtained in the
In this first stage of investigation
z eq. (1a)-(1b)
T 1neglecting
|E|
atomic system; 
int factTfrom
diffusion (d=0) and after
Twolimit
levelg>>1,
system we get in the
adiabatic elimination of the carrier density variable in the
passive case:
E(1  i e )
E 1 E
2

m

i

E
2
t T z
1 | E |
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 d=0 and g>>1 in parametric regimes linked to those in which we
observed CLBs through relations:
 e   0  a0 m  
 aL A
T
Self-defocusing passive parametric regimes
e=2, 0=-0.3, m30, T=0.1,d=0 and g>>1 →300.0)
Longitudinal filaments……and fully localized structures
When, as happens in this case, d=0 and Iml<<1<<g the instability domains are
independent from g. In spite of this, g still plays a role in influencing system’s
dynamical evolution.
We observe in the general case highly correlated longitudinal filaments at regime.
Stationary transversely homogeneous
states curves (independent from g)
Two fully localized structures for g=50.0
Y=22.975
120
110
Two stable fully localized
structures obtained by cutting
two longitudinal filaments and
letting the system evolve.
They are not independent from
each other.
unstable states
100
I=|E(z=1)|
2
90
80
70
60
50
40
30
20
10
z
0
20
22
24
26
28
Y
30
32
34
36
x
Self-focusing passive parametric regimes
e=-2, 0=-0.4, m20, T=0.1,d=0 and g>>1 →500.0)
Longitudinal filaments ……((2+1) dim)
When, as happens in this case, d=0 and Iml<<1<<g the instability domains are
independent from g. In spite of this, g still plays a role in influencing system
dynamical evolution.
Longitudinal filaments g~300
Stationary transversely homogeneous
states curves (independent from g)
Y
17.0
11.0
45
unstable states
40
35
I=|E(z=1)|
2
30
25
20
15
10
5
z
0
8
10
12
14
16
Y
18
20
22
24
x
((2+1) dim)
……..and
fully confined structures
Although we still don’t observe phenomena of spontaneous structures localization
in the propagation direction, we proved that a longitudinal confined portion of a
longer filament represents a stable system’s solution for a sizable interval of
Y values.
Fully localized structure (g~300)
!
The localized structure disappears
when we decrease g under a
certain threshold
120
0.1 t.u. cavity round trip time
I_Max I_Min
100
80
60
40
20
z
0
0,0
x
0,2
0,4
0,6
0,8
1,0
t.u.
Intensity field profile on the exit window
Observation
We put g>>1
Since we have: g = gnr/kp where gnr is the nonradiative carrier density decay
constant, while kp=cT/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)
Input mirror
Ei
Output mirror
Semiconductor
sample
ET
l
LA
LA<<l
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.
3D Pattern formation in a semiconductor resonator driven by a
coherent injected field.
Beyond the Single Longitudinal Mode Approximation:
The dynamical model, the Linear Stability Analysis and the first
numerical results.
Fully localized structure in a self-focusing passive regime.
Future agenda and Conclusions
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.
0.35 t.u.
1.3 t.u.
9.65 t.u.
z
x
Switching on process of a single localized structure by using an external addressing beam
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)
=
?
[
?
In the laser configuration should we remove the rate equation approximation?
(We already did some calculations about this! )
]
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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1997.
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