Self-localised structures in semiconductor microcavitiesM.Brambilla

www.funfacs.org

I.Perrini, T.Maggipinto, CNR-INFM/CNISM Bari, ItalyL.Columbo, G.Tissoni, L.A.Lugiato, CNR-INFM Como, ItalyS.Barbay, R.Kuszelewicz, CNRS-UPR20 Marcoussis, France

Les Houches 11-16 Jan 2007

Outline of the talk

Patterns and CS in MQD microcavities

3D pattern formation in s.c. extended resonators

6,5 7,0 7,50

5

10

15

Injected Intensity

Patterns & homogeneous solution coexist for Iinj fixed

unstable homog state

Fiel

d In

tens

ity• Diffractive patterns are “global” structures, with high internal spatial correlation.• Self-localization: portions of patterns and (e.g) a hom. emission can coexist in the spatial profile of the emitted field: Localised structures

• High intensity spots : Cavity Solitons

Loc. Strs. in Kerr [Tlidi, Mandel,1994]

Spatial patterns

Cavity Solitons in a VCSEL(INLN, 2002)

Microscopic model for a broad-area MQD-μRes

Objective: describe pattern formation and CS in amplified systems (currrent pump in the WL)- well isolated resonance - moderate but significant e,h densities in WL- high QD density

50 nm 200 nm

50 nm 200 nm

50 nm 200 nm

50 nm 200 nm

50 nm 200 nm

50 nm 200 nm

8 25.10A cmρ −=

10 25.10B cmρ −=

11 22.10C cmρ −=

InAlAs/GaAlAs samples grown at LPN-CNRS, J-M. Benoit, A. Lemaître, G. Patriarche, K. Meunier, S. Barbay and R. Kuszelewicz, in preparation

MQD vertical resonators

MQD devices have risen a broad interest

-All carriers at quasi 2-lev state-High carrier densities, -Self-focusing regimes readily accessible-Lasers: low I thr, high pwrs, selectable λ

Main MQD mechanisms relevant for the s-t dynamics of the coherent field

- Inhomogeneous broadening - Capture and escapes (WL QD)-Auger many-body processes A.V. Uskov, J. McInerney, F. Adler, H. Schweizer, M.H. Pilkuhn, Appl. Phys. Lett. 72, 58 (1998)

- Statistical distribution of WL/QD cross sections

Basics of the QD model

( ) ( ) ( )[ ]{ }( ) ( )( )[ ]2

2

21

EGnLntn

GnLdCEiEEitE

i

i

eree

eI

Δ−Δ+−=∂

∂

Δ−ΔΔ+∇+++−=∂∂

Δ

Δ⊥ ∫

γ

θ

E = normalized field envelopene(ωa) = excited state spectral prob.θ = cavity detuningC = bistability parameterγ = rescaled carrier decay rate

Inhomogeneous broadening of a populationof 2-level systems

Γ−

=Δ 0ωωCa

i γωω 0−

=Δ aBarbay, J. Koehler, R. Kuszelewicz and M. Brambilla, T. Maggipinto, I. M. Perrini, “Optical patterns and cavity solitons in quantum dot microresonators”, IEEE Journ.Q.El., 39, 245 (2003)

Energy

γWL

γTEd σ

γrad

QD γNR

QD

Energy

γWL

γTEd σ

γrad

QD γNR

QD

Basic WL and QD dynamics

WL/QD interaction (Auger processes)After A.V. Uskov, et al. Appl.Phys. Lett. 72, 58 (1998)

Second order in N(WL), relevant far above lasing threshold

First order in N(WL) – Included in our model

Inclusion of WL – QD interaction

MQD model equations: field, carrier spectral probabilities in QDs, Carrier density in the wetting layer

Details in I.M. Perrini, et al., “Model for optical pattern and cavity soliton formation in a microresonator with self-assembled semiconductor quantum dots”, Appl. Phys. B 81 (7), 905 (2005).

Study of the lasing and transparency thresholds both for focusing and defocusing regimes

15 20 25 30 35 40 452,34

2,36

2,38

2,40

2,42

2,44

2,46

2,48

2,50

2,52

2,54

2,56

2,58

2,60

Lasing Threshold Trasparency Threshold

C

Defocusing case Δ i = 0.5

-2,0 -1,5 -1,0 -0,5 0,0

2,4

2,6

2,8

3,0

3,2

3,4 Lasing Threshold Trasparency Threshold

Δ i

10 15 20 25 30 35 402,3

2,4

2,5

2,6

2,7

2,8

2,9

3,0

Lasing Threshold Trasparency Threshold

Lasi

ng T

hres

hold

C

Self Focusing Case Δ i = -1

The transparency depends on σcap and γesc.

0 20 40 60 80

0

10

20

30

40

|E|

EIN

Pump Variation

0 0.5 1.0 1.5 1.8 2.0

The SS curves lose their bistablecharacter close to transparency

Symmetric gain leads to a vanishing or null Henry factorT.C. Newell, D.J. Bossert, A. Stintz, B. Fuchs, K.J.Malloy, L.F. Lester, IEEE Photon. Tech. Lett. 11 (1999) 1527.

Linewidth enhancement factor

( )∫ ΔΔΔΠ−+Δ++Δ

−= dGnnii

he ),(1 2χ

( ) ( )∫ ΔΔΔΠ−+Δ+

Δ−= dGnn i

he ),(1

Re 2χ

( ) ( )∫ ΔΔΔΠ−+Δ+

−= dGnn ihe ),(

11Im 2χ

( )

( )χ

χα

Im

Re

N

N

∂∂

∂∂

∝

An important asymmetry cause of QD susceptivity is the dispersion of the QD size - Distribution of exciton energies (Inhomogeneous broadening)- Analogous dependence of the WL/QD rates γesc and σcap The higher QD energy favors escape of the carriers towards the WL, and v.v. for capture from the WL.

By considering an equilibrium distribution we modeled a variation proportional to

⎟⎟⎠

⎞⎜⎜⎝

⎛ −∝

TkEE

B

WLESesc expγ

( )Δ⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

Γ−≈⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+Δ∝ ββ

γγωωγγ expexpexpexpexp 00

iBB

WL

B

WLesc TkTk

ETk

E

( )Δ−⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

Γ∝ ββ

γσ expexp icap

β is the homogeneous breadth to thermal energy ratio

See e.g. M.Giovannini et al. Opt.Quant.Elec. 38 p.381, 2006

The numerical simulations show a marked dependence on β

( )( ) ( ){ }∫ Δ−+ΔΔ−≈ GnnLd he 2χ

It is instructive to analyse how the frequencydistributed contributions to the susceptivity

change with beta (less carriers at highenergies, i.e. absorption, et v.v.)

Δ

absorption

contribs to gain

increasing QD energies →

J.Oksanen et al. J.Appl.Phys.94, 2003

Preliminary results show qualitative agreement with reported behaviourNegative, -2 ~10

Courtesy of G.Huyet,Tyndall Labs, and Univ.Cork

H.C. Schneider, W.W. Chow, and S.W. Koch, Phys. Rev. B, 66, 041310(R), 2002

N.C.Gerhardt et al. J.Phys.Cond.Mat. 16,2004

Steady state curves and spatial structures

The bistable regime, possibly withextended MI regions is not trivial tomeet at low pump

An accurate analysis shows suitableregimes where the lasing thresholdoccurs for moderate pumpsGuidelilne: in MQW devices, CS are favoured around 10% below L.Th. at α~ 5-6

Modulational Instabilities readily appearand lead to global pattern formationand stable Cavity Solitons

Pattern competition : evidences of rectangular symmetriesEffect of the S-defoc families ? In progress

For C = 20 Nqd ~ 4.7 1012 cm-2

• Reduce C

• Control Inhomogeneous width

QD density must be resonable

• Study of the optical susceptivity• Prediction of favourable regimes for pattern formation and CS • Guidelines for sample growth and experiment• The knowledge can be exploited towards extended cavities

contemplating a combined active and passive medium (LSA)

Conclusions and Perspectives

CS are 2D localised structures. They are unconfined in the propagation direction

y

x

z

In the transverse plane Inside the resonator

Next step ahead: access 3D self-confinementto achieve endlessly propagating pulse serieswith the same CS properties in the (x,y) planeCavity Light Bullets

LeBerre,Tallet, Patrascu, Tlidiet al.: Ring cavity + 2-lev abs. Medium dynamics. In the strongly dispersive limit. (CSF 10, 627, 1999). Negligible absorption [Opt. Comm. 91, 433, 1992]Chaotically oscillating 2D LS, Multiconical emission M. Tlidi, P. Mandel et al.: Kerr resonators, DOPO,Type-II SHG (no boundary conditions).Based on a 2nd order dispersion NLSE model Periodic 3D crystals, Localised dark light drops or cylinders.J.Opt.B 2, 438 (2000); Eur.Phys.Lett.55, 26 (2001)

Y.Silberberg, Opt. Lett. 15, 1282 (90)

N.N.Rosanov et al.2nd order dispersion in a dissipative medium,Laserw/ Sat.Abs., no resonator b.c.;Wavefront locking at Maxwell point3D Bullets,drift,interaction

Opt. Spectr. 76, 621 (94), 81,276 (96; , Opt. Spectr.89, 623 (2001)

Pattern Formation in 3D : previous researches and results

FiF

iFzF

tF

c⊥∇+

+Δ+Δ−

−=∂∂

+∂∂ 2

22 ||1)1(1 α

0),,,(),0,,( δietLyxRFTytyxF inj−+=

3D Pattern Formation in a saturable absorber ring cavity : beyond the mean field limit

1) Formation of longitudinally modulated “filaments” which underproper input field intensity undergo a contraction phase2) The 3D localised structures stabilise at regime

Homogeneous input field 1 or 2 transverse dimensions

3D pattern formation and self-localisation

M.Brambilla, L.Columbo, T.Maggipinto, G.Patera, "3D Cavity Light Bullets in a Nonlinear Optical Resonator", Phys.Rev.Lett. 93, 203901 (2005)

M. Brambilla, L. Columbo and T. Maggipinto, J. Opt. B: Quantum Semiclass. Opt., 6 S197, 2004.

The complex susceptivity is described by the same 2-level excitonic model as in (Phys. Rev. A, 58 2542, 1998):

where in the passive configuration:

while in the active configuration:

Extension of the 3D model to a Multi Quantum Well (MQW) semiconductor medium

)( 00

NNcnAi −Θ−=ω

χ

)1( eiΔ−=Θ

)i1( α−=Θ

with , N= carrier density, N0 = transparency carrier density, A = absorption\gain coefficient, n = background refractive index, ωe= central frequency of the excitonic absorption line, γe = FWHM of the excitonic absorption line, α = linewidth enhancement factor.

eee γωω /)( 0−=Δ

Carrier dynamics N(t) cannot be adiabatically eliminated

ε

ET

4

3

1 T=0

T=0

EI

2

Nonlinear mediumNonlinear medium

E E

)1/(AA 2eΔ+=

AA =

Nonlinear mediumE E

ER

Maxwell-Bloch equations (rate equations) describing system dynamics within the SVEA and paraxial approximations but without introducing any hypothesis on the longitudinal field profile:

Boundary condition:

D = normalized difference between N and N0δ0 = normalized cavity detuning d = diffusion coefficient γ = nonradiative decay constant photon life timeμ = pump parameter (μ<0→absorber; 0<μ<1→amplifier; μ>1→laser)

))||1((

1

22

2

DdEDtD

EiDEzE

TtE

⊥

⊥

∇−−+−=∂∂

∇+Θ=∂∂

+∂∂

μγ

(1a)

(1b)

),1(Re),0( 0 tzETYtzE i =+== − δ

Semiconductor Maxwell-Bloch Equations

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

50

100

150

200

250

Inte

nsity

(nor

mal

ized

uni

ts)

zIntensity field profile at a fixed (x,y) transverse location

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 solutionsXs, 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 Analysis

Expand δX on the transverse Fourier basis keeping implicit itsz-dependence :

Thus we have for each (kx, ky) a system of two linear ordinary differential equations for , and its c.c.

At difference from what happens in the Single Longitudinal Mode Approx. 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 δX<<Xs we cannot derive for each perturbation modal amplitude an equation for λ describing its the temporal evolution. Then, extending the results obtained in the two level system, we adopt analternative approach:

yxzt)ykxkzk(i

k,k,k0ss dkdkdkeeX)z(XXXX yxzyxz

λ+++∞

∞−

+∞

∞−

+∞

∞−∫ ∫ ∫ δ+=δ+=

yxt)ykxk(i

k,k0 dkdkee)z(XX yxyx

λ++∞

∞−

+∞

∞−∫ ∫ δ=δ

yx kkzE ,0 )(δ )z(E0δ

Fourier expansion

The easiest way to proceed at this point is to introduce the polar representation of Es and δE0

where ρs, θs, δρ, δθ are real quantities. After some simple algebra, we then get:

where k⊥=(kx2 + ky

2)1/2, Ψ(z)=ρ2s(z) and r and u are auxiliary variables linked to δρ

and δθ trough the linear transformation:

)i(eE

eEs

s

i0

iss

δθ+δρ=δ

ρ=θ

θ

⎟⎠

⎞⎜⎝

⎛Ψ

+⎟⎟⎠

⎞⎜⎜⎝

⎛

+Ψ+γ+λ

αγ+⎟⎟

⎠

⎞⎜⎜⎝

⎛ΨμΨ+

−=Ψ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ΨμΨ+

+⎟⎟⎠

⎞⎜⎜⎝

⎛

+Ψ+γ+λ

γ−

Ψ=

Ψ

⊥⊥

⊥⊥

21u

)dk1(21kr

ddu

21ku

)dk1(21r

ddr

22

22

Ter

tzλ

δρ=T

eutzλ

δθ=

(2a)

(2b)

Once the eigenvalue λ is known, one can ask the question:How can an actual 3D Modulational Instability (MI) can be discriminated from a 2D instability only involving transversemodes (k⊥≠0, kz=0)?

0

0

=

=

dzddz

d

δθ

δρ( ) ( )

( ) ( ) 0~12

0~12

22

22

222

2

=+−+⎟⎟⎠

⎞⎜⎜⎝

⎛

++++−

=+⎟⎟⎠

⎞⎜⎜⎝

⎛

+++−+−

⊥⊥

⊥⊥

Sst

Sst

st

SstS

Dkd

Dk

kkd

DD

λδθργλγαρδρ

δθργλγρλδρ

The answer appears to be negative: the criterion is always satisfied for small values of the carrier-to-photon life rate (carrier lifetimes 0.1-1 ns, photon 10-100 ps)

We thoroughly investigated the passive case and results will be presented here for the active case (pumped device)

The general indication is that for high γ (approx. 300) self confinement is working and CLBs can be addressed, while for lower values closer to realistic conditions (0.01-1.0) only transversally confined patterns are found

L.Columbo, I.M.Perrini, T.Maggipinto and M.Brambilla, “3D self-organized patterns in the field profile of a semiconductor resonator”, New. Journ. Phys. 8, 312 (2006)

Regime: T = 0.2 , α = 5, µ = 0.9, δo = -0.2, γ =320

1.10 0.951.40

The pattern/filament landscape showslongitudinal modulationHigher K_z visible in the zig-zagSelf-localization does stabilize to 3d self-loc. structures

CLBs can turned on and off by pulses. Parallel and serial encoding

The crucial limitation is γ>50

γ=320 γ=250 γ=200 γ=150 γ=50

γ=320-50

Developments and perspectives in s.c. CLBs

The actual 3D character of the M.I. and the CLB stability are hindered by the “carrier sleuth”.

Can broader MI ranges be found in credible s.c. parameter domains ? Possibly a higher nonlinear modal competition might be beneficial: inspect regimes of higher mirror transmission T

The passive/active monolithic non-MFL semiconductor could be not suited for 3D confinement

Move to different devices: Laser with fast SatAbs (MQD) and to Vertical Resonators with Extended Cavities

Experiment: Crucial to prove the preliminary effect of 1) filamentation, 2) filament decorrelation and possibly 3) zig-zag instability