Università degli studi dell’Insubria Como, 22 settembre, 2005

The hunt for The hunt for 3D global or localized structures in a 3D global or localized structures in a

semiconductor resonatorsemiconductor resonator

Ph.D student: Ph.D student: Lorenzo ColumboLorenzo Columbo

Supervisor: Prof. Luigi Lugiato Supervisor: Prof. Luigi Lugiato External supervisor: Prof. Massimo Brambilla External supervisor: Prof. Massimo Brambilla

((Politecnico di BariPolitecnico di Bari))

OutlineOutlineOutlineOutline

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.

Future agenda and Conclusions

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.

3D Pattern formation in a semiconductor resonator driven by a coherent injected field.

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.

PPattern formation in a semiconductor resonatorattern formation in a semiconductor resonator PPattern formation in a semiconductor resonatorattern formation in a semiconductor resonator

EE RR

(reflected field)

Active layer (MQWs

GaAs-GaAlAs)

n-contact

Bragg reflector

Bragg reflector

EE InIn

(injected field plane wave))

p-contact

n-contact

VCSEL device (Bottom Emitter)

GaAs Substrate

~150 m

~

m Rolls

Numerical simulations: Global and localized structures

in |ER|2 transverse profile

Transverse intensity field profile on the exit window

Honeycombs Cavity Solitons

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 resonator’s feedback\dissipation action.

AApplications of Cavity Solitons to thepplications of Cavity Solitons to the optical information technology optical information technology

AApplications of Cavity Solitons to thepplications of Cavity Solitons to the optical information technology optical information technology

Nonlinear effects: self-focusing, saturable absorption..

Linear effects: diffraction

Resonator’s action: feedback, dissipation

- -

|ER|2

xy

We don’t have itin the Spatial Solitons case

Intensity field profile of a single CS

x

y

Ideal scheme of a binary optical memory.

Es: CSs based parallel optical memory

From a mere applicative point of view, since CSs can be externally excited, erased and drifted by means of suitable addressing beams (as it has been predicted and very recently observed (Nature 419 699, 2002)), these micro pixels are candidate to realize all optical devices for parallel information storage and processing.

Nonlinear mediumGaussian pulse

Plane wave EIPlane wave EI

ER

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.)

BBeyondeyond Single Longitudinal Mode Approximation!!Single Longitudinal Mode Approximation!! BBeyondeyond Single Longitudinal Mode Approximation!!Single Longitudinal Mode Approximation!!

x

y

z

Plane wave

Plane wave

z ?

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?

??

CS: Transverse localization.

Starting from 2002 we tried to answer to the previous questions by considering first a optical prototype:

1. B1. Beyond SLMA: two level systemeyond SLMA: two level system1. B1. Beyond SLMA: two level systemeyond SLMA: two level system

EI

ER

ET

4

3

1 T=0

T=0 0a nm

c

a = atomic transition frequency0 = input field frequencyn = generic empty cavity mode

Unidirectional ring resonator filled with a vapour of two level atoms and driven by a coherent injected beam

2

Nonlinear medium

EI = injected field (Plane wave)ET= transmitted fieldER= reflected field

Nonlinear mediumE E

1.1. 1.1. MMaxwell-Bloch equationaxwell-Bloch equation 1.1. 1.1. MMaxwell-Bloch equationaxwell-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:

Boundary condition:

Ei|E|1

)i1(E

T

L

z

E

T

1

t

E 22

Aa

)t,1z(EReTY)t,0z(E 0ai

E = normalized envelope of the intracavity field Y = normalized envelope of the injected field LA = resonator length=nonlinear medium lengtha = normalized absorption coefficient at resonance T = transmission coefficient (R=1-T) =(a-0)/ a0= (c-0) LA /c z = normalized propagation coordinate x, y = normalized transverse coordinate t = normalized time coordinate

1.2. 3D global and 3D localized structures1.2. 3D global and 3D localized structures 1.2. 3D global and 3D localized structures1.2. 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 Ysome filaments contract into stable fully

localized structures. We then answered to question A

a) 3D filaments b) Cavity Light Bullets

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.

x

z

CLB external control CLB external control (2+1) dim(2+1) dimCLB external control CLB external control (2+1) dim(2+1) dim

Switching on of one or more CLBs

We then answered to question B

a) Switching on of a single CLB b) Switching on of two parallel CLBs c) Switching on of a CLB train

OutlineOutlineOutlineOutline

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.

Future agenda and Conclusions

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.

3D Pattern formation in a semiconductor resonator driven by a coherent injected field.

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.

??

2. B2. Beyond SLMA: semiconductor resonatorseyond SLMA: semiconductor resonators2. B2. Beyond SLMA: semiconductor resonatorseyond SLMA: semiconductor resonators

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:

)NN(cn

Ai 00

)i1( e

)i1(

with , N= carrier density, N0 = transparency carrier density, A = absorption\gain coefficient, n = background refractive index, e = half width of the excitonic absorption line, e = central frequency of the excitonic absorption line, = linewidth enhancement factor

e0ee /)(

Fast carrier dynamics → we cannot adiabatically eliminate carrier dynamics

ET

4

3

1 T=0

T=0

EI

2

Nonlinear mediumNonlinear medium

E E

)1/(AA 2e

AA

Nonlinear mediumE E

ER

2.1. 2.1. MMaxwell-Bloch equationsaxwell-Bloch equations 2.1. 2.1. MMaxwell-Bloch equationsaxwell-Bloch equations

D = normalized difference between N and N0 = normalized cavity detuning d = diffusion coefficient = nonradiative decay constant photon life time = pump parameter (<0→absorber; 0<<1→amplifier; >1→laser)

)Dd)|E|1(D(t

D

EiDEz

E

T

1

t

E

22

2

)t,1z(EReTY)t,0z(E 0i

(1a)

(1b)

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:

Intensity field profile for a fixed (x,y) value

In the general case, the nonlinear character of eq. (1a)-(1b) prevents us to solve them analyticallyEquating 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.

Linear Stability AnalysisLinear Stability AnalysisLinear Stability AnalysisLinear Stability Analysis

0,0 0,2 0,4 0,6 0,8 1,00

50

100

150

200

250In

tens

ity (

norm

aliz

ed u

nits

)

z

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:

with X<<Xs we cannot derive for each modalamplitude an equation for describing its the temporal evolution. Then, extending the results obtained in the two level system, we adopt an alternative approach:

yxzt)ykxkzk(i

k,k,k0ss dkdkdkeeX)z(XXXX yxz

yxz

we expand X 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.

yxt)ykxk(i

k,k0 dkdkee)z(XX yx

yx

yx k,k0 )z(E )z(E0

Step1

Fourier expansion

The easiest way to proceed at this point is to introduce the polar representationofEsandE0

where s, s, , are real quantities. After some simple algebra, we then get:

where k=(kx2 + ky

2)1/2, (z)2s(z) and r and u are auxiliary variables linked to

and trough the linear transformation:

)i(eE

eE

s

s

i0

iss

Step 2

2

1u

)dk1(2

1kr

d

du

2

1ku

)dk1(2

1r

d

dr

22

22

T

er

tz

T

eu

tz

(2a)

(2b)

Combining eq. (2a) and (2b) we derive the following 2nd

order linear differential equation for r

where the coefficients A, B, Hi, i=1..5 depend on the physical parameters, Xs, k and also on .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 as superposition of the two linearly independent series solutions r1 and r2

where c1, c2are arbitrary complex constants. We also get for u from (2a) and (2b):

222

55

44

33

2210

2

2

A)1(B

HHHHHHr

)1(A

A

d

dr

d

rd

Step 3

)(rc)(rc)(r 2211

)(uc)(uc)(u 2211

Keeping fixed the other quantities, it represents a nonlinear implicit equation for the 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” of the function C with Re0.

Step 4

0)),1z(u),,1z(r),,0z(u),,0z(r,,,T(C iiii0

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.

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:

Observation:

From a computational point of view the implicit character of the equation C()=0 represents and additional CPU time consuming factor. This forced us to implement a parallel numerical code for LSA.

2.2. Numerical simulations2.2. Numerical simulations2.2. Numerical simulations2.2. 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 (d=0) and after adiabatic elimination of the carrier density variable in the limit >>1, 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 d=0 and >>1 in parametric regimes linked to those in which we observed CLBs through relations:

Ei|E|1

)i1(E

z

E

T

1

t

E 22e

T

LAae 0a0

Ei|E|1

)i1(E

T

L

z

E

T

1

t

E 22

Aa

Two level system

Do you remember?

Self-defocusing passive parametric regimesSelf-defocusing passive parametric regimesee=2, =2, 00=-0.3, =-0.3, T=0.1,d=0 and T=0.1,d=0 and →3→300.0)00.0)

Longitudinal filaments……and fully localized structuresLongitudinal filaments……and fully localized structures

Self-defocusing passive parametric regimesSelf-defocusing passive parametric regimesee=2, =2, 00=-0.3, =-0.3, T=0.1,d=0 and T=0.1,d=0 and →3→300.0)00.0)

Longitudinal filaments……and fully localized structuresLongitudinal filaments……and fully localized structures

When, as happens in this case, d=0 and Im<<1<<the instability domains are independent from . In spite of this, still plays a role in influencing system’s dynamical evolution. We observe in the general case highly correlated longitudinal filaments at regime.

Stationary transversely homogeneousstates curves (independent from )

x

z

Two fully localized structures for =50.0

Two stable fully localized structures obtained by cutting two longitudinal filaments and letting the system evolve.They are not independent from each other.

Y=22.975

Self-focusing passive parametric regimesSelf-focusing passive parametric regimesee=-2, =-2, 00=-0.4, =-0.4, T=0.1,d=0 and T=0.1,d=0 and →→500.0)500.0)

Longitudinal filaments ……Longitudinal filaments ……((2+1) dim)((2+1) dim)

Self-focusing passive parametric regimesSelf-focusing passive parametric regimesee=-2, =-2, 00=-0.4, =-0.4, T=0.1,d=0 and T=0.1,d=0 and →→500.0)500.0)

Longitudinal filaments ……Longitudinal filaments ……((2+1) dim)((2+1) dim)

When, as happens in this case, d=0 and Im<<1<<the instability domains are independent from . In spite of this, still plays a role in influencing system dynamical evolution.

Longitudinal filaments ~300Stationary transversely homogeneousstates curves (independent from )

17.0 11.0YY

x

z

8 10 12 14 16 18 20 22 24

0

5

10

15

20

25

30

35

40

45

unstable states

I=|E

(z=1)

|2

Y

((2+1) dim)((2+1) dim)

0,0 0,2 0,4 0,6 0,8 1,0

0

20

40

60

80

100

120

t.u.

I_M

ax

I_M

in

0.1 t.u. cavity round trip time

…… ……....and fully confined structuresand fully confined structures …… ……....and fully confined structuresand 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 (~300)

x

z

Intensity field profile on the exit window

! The localized structure disappearsThe localized structure disappearswhen we decrease when we decrease under a under a

certain thresholdcertain threshold

ObservationObservationObservationObservation

Since we have: = nr/kp where nr 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 by increasing LA.

a) We could consider for example Edge Emitter configurations

l~250

b) Moreover, we can get the same result by considering the case: medium length ≠ cavity length and increasing the latter

Semiconductor sample

Input mirror Output mirror

lLA

Ei ET

LA<<l

a) b)

We put >>1

OutlineOutlineOutlineOutline

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.

Future agenda and Conclusions

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.

3D Pattern formation in a semiconductor resonator driven by a coherent injected field.

Future AgendaFuture AgendaFuture AgendaFuture Agenda

9.65 t.u.1.3 t.u.0.35 t.u.

x

z

Switching on process of a single localized structure by using an external addressing beam

Passive case

Looking for fully localized structures in less critical parametric domains and\or configurations.

Systematic study of the proprieties of these localized structures in analogy to what we did for CLBs.

Future AgendaFuture AgendaFuture AgendaFuture Agenda

Active case

We can also consider the active configurations below or above lasing threshold.

In the laser configuration should we remove the rate equation approximation? (We already did some calculations about this! )?[ ]

+

=The Vertical External Cavity Surface Emitting Laser (VECSEL) configuration is for example already used to produce mode locking laser operation.→ (longitudinal localization)

The VCSEL configuration is already used to observe CSs even above lasing threshold.→ (transverse localization)

?

ConclusionsConclusionsConclusionsConclusions

FunfacsFunfacs European Project European ProjectFunfacsFunfacs European Project European Project

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.

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.

We looked for 3D pattern formation and 3D self-confinement in a semiconductor resonator driven by a coherent injected field.

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.

L. A. Lugiato, Prog. in Opt. 21 71, 1984.L. A. Lugiato, L. M. Narducci and M. F. Squicciarini, Phys. Rev. A, 34 3101, 1986.M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli and W. J. Firth, Phys. Rev. Lett., 79 2042, 1997.L. A. Lugiato, M. Brambilla and A. Gatti, Optical pattern formation, in Advances in Atomic, Molecular and Optical Physics 40 229, 1998.S. Barland, et al., Nature 419 699, 2002.L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, Phys. Rev. A, 58 2542, 1998; L. Spinelli and M. Brambilla, The European Physical Journal D, 6 523, 1999.M. Brambilla, L. Columbo and T. Maggipinto, J. Opt. B: Quantum Semiclass. Opt., 6 S197, 2004.M. Brambilla, T. Maggipinto, G. Patera and L. Columbo, Phys. Rev. Lett. 93 2042, 2004.M. Brambilla, T. Maggipinto, G. Patera and L. Columbo, Proceeding of SPIE (Photonic West) 2005), 2005.

MPI site (on of the most popular): http://www-unix.mcs.anl.gov.

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