LFFs in Vertical Cavity Lasers

LFFs in Vertical Cavity Lasers A. Torcini, G. Giacomelli, F. Marin torcini@inoa.it, S. Barland ISC - CNR - Firenze (I) Physics Dept - Firenze (I) INLN - Nice (F) FNOES - Nice, 14.9.5 p.1/21

Plan of the Talk Schematic description of the VCSEL and experimental setup FNOES - Nice, 14.9.5 p.2/21

Plan of the Talk Schematic description of the VCSEL and experimental setup The low frequency fluctuations (LFF) dynamical regime FNOES - Nice, 14.9.5 p.2/21

Plan of the Talk Schematic description of the VCSEL and experimental setup The low frequency fluctuations (LFF) dynamical regime Rate equations for single mode lasers with feedback: Lang-Kobayashi eqs. Deterministic versus stochastic dynamics of the LK eqs Experimental results for VCSEL with a moderate optical feedback FNOES - Nice, 14.9.5 p.2/21

VCSEL in brief Single laser Semiconductor laser Wavelength: near IR ( nm) (optimal coupling with optical fibers) Single longitudinal mode, multiple transverse modes. Two linearly polarized emissions (symmetrical cavity). Good beam quality. VCSEL with delayed optical feedback External cavity cm, round trip time ns Polarization selective optical feedback FNOES - Nice, 14.9.5 p.3/21

Dynamics of the VCSEL Power (W) Power (a.u.) 6e-5 5e-5 4e-5 3e-5 2e-5 1e-5 8 6 4 Solitary laser with feedback LFF 1 1.5 2 2.5 3 3.5 4 I (ma) I=2.64 ma I th = 2.76 ma Main Polarization Secondary Polarization for 6 ma the device emits linearly polarized light in the fundamental mode (the transverse modes are not active); for + feedback the VCSEL is a single mode on the main polarization and its dynamics exhibits LFFs ; the secondary polarization is absent. 2 2 4 6 8 Time (ns) FNOES - Nice, 14.9.5 p.4/21

VCSEL Phase Diagram We will examine the single mode LFF regime, where the dynamical behaviour is not particularly influenced by the phase delay induced by the feedback. FNOES - Nice, 14.9.5 p.5/21

LFF in a nutshell 1 I=2.7 ma Filtered at 1Ghz 1 5 1 P (a.u.) Filtered at 2Mhz P (a.u.) 5 5 T LFF 2 4 6 8 1 Time (ns) 3 4 5 6 7 8 9 1 11 12 Time (ns) LFFs are feedback induced instabilities; the light is emitted in short pulses ns ; the filtered intensity grows to a almost constant value over a cycle of duration ns then drops to a much lower value; the LFF cycles can coexist with stable emission on a high-gain mode. FNOES - Nice, 14.9.5 p.6/21

. %,% " % + + " - + 5 + A I A H A G F E B TO NO L O L L c K Z Y ` L L d Lang-Kobayashi eqs The dynamics of the VCSEL with delayed optical feedback is described for by the rate eqs.: " #,- " # % & " ' (*) & % "$#! % " = # :<; 2 1 4 32 / 1 & % & "$# 9-78 6 % & "$# %- % "$# % "$# 32 a?@ the carrier density, and?@ where is the complex field, Gaussian noise term (spontaneous emission).?@ A!CD?@ A?@ > ns, S K PRQ, NO K KML J. be a ` O A_^ X NO K The parameters have been experimentally measured: ps,, ns,? T []\ U KML P XML W N T PVU will be determined by comparison of numerical vs experimenal data. S. Barland, P. Spinicelli, G. Giacomelli, F. Marin, IEEE J. Quantum Electronics (25). FNOES - Nice, 14.9.5 p.7/21

" / - l k ji ) e ) Stationary solutions The deterministic LK eqs admit stationary solutions of the type: % "$# % h# % " # m g 5f The unstable solutions with a real positive eigenvalue are termed Antimodes, the other solutions can be stable or unstable depending on the parameters and are termed Modes. The Modes can be destabilized by different mechanisms: e.g. Modulational or Turing instabilities. 25 2 α=3.2 c =.93 θ=-.32 α=3.22 c =.93 θ=-.8 Im Λ (ns -1 ) 15 1 5 -.2 -.1.1.2.3.4 Re Λ (ns -1 ) FNOES - Nice, 14.9.5 p.8/21

d J Stationary solutions.2 c =.93 α=3.1 k=.25 Modes Maximum Gain Mode Antimodes.15 ρ 2.1.5..4.45.5.55.6.65 N Stable modes are more probable for small or -values. The maximum gain mode is stable and it coexists with the LFF dynamics. Two/three (maximum gain) modes can be stable at the same time. S. Yanchuk & M. Wolfrum, WIAS preprint n. 962, Berlin (24) FNOES - Nice, 14.9.5 p.9/21

Deterministic dynamics The LFF dynamics is due to the chaotic itinerancy from one quasi-attractor to another, where the quasi-attractors are the ruins of a local chaotic attractor emerged via a period doubling bifurcations from the corresponding mode. On each local quasi-attractor the dynamics is chaotic but transient, the trajectory jumps from one quasi-attractor to another climbing towards the maximum gain mode. When the trajectory collides with an anti-mode the intensity drop and the dynamics restart again from low gain modes. (Sisyphus Cycles ) This is the deterministic explanation of LFFs ( T. Sano, PRA 5 (1994) 2719 ).5 α=4. c =.95 k=.25.2.4 ρ S 2.3 ρ 2.2.1 α=3.2 c =.93.1 1.3e+5 1.35e+5 1.4e+5 1.45e+5 1.5e+5 Time (ns) -.6 -.4 -.2.2.4.6 θ FNOES - Nice, 14.9.5 p.1/21

+ q LFF as a transient phenomenon 1 9 < T s > (ns) 1 8 1 7 c =.93 c =.95 c =.97 c =.99 1 6 1 5 1 4 2 3 4 5 6 α Below the threshold of the solitary laser ( ) and for meaningful values of the LFFs are present only as a transient phenomenon. ( ) The average transient time npo diverges for increasing ( ) and. Preliminar indications have been reported in T. Heil et al., Optics Let. 24 (1999) 1275 FNOES - Nice, 14.9.5 p.11/21

J t J Lyapunov analysis Kaplan-Yorke Dimension Transient Lyapunov Exponent - averaged over 5 i.c. s 5.1 D KY <λ> c =.99 c =.97 c =.95 c =.93 4 3 α=5. α=4..1 2 1.1 2.5 3 3.5 4 4.5 5 α.9.95 1 1.5 c The transient dynamics is chaotic and the maximal Lyapunov increases with and. The number of active degrees of freedom (measured by dynamics increases with and. The system is not low dimensional. r3s ) involved in the FNOES - Nice, 14.9.5 p.12/21

u u Stochastic Dynamics I The presence of additive gaussian noise of variance destabilize the maximum gain mode leading in the LK eqs. can for small noise to an intermittent behaviour ; for larger values to a non transient LFF dynamics. α=3.3 -- c =.97 -- D=2 x 1-4 α=3.3 -- c =.97 -- D=3 x 1-3 Filter 8 MHz 1.5 Filter 8 MHz.6.6.5 1.2 D exp =(2.7 ±.6) x 1-3.4 ρ 2.4.3 T res ρ 2.9.6.2 185 19.2.3.1 3e+5 6e+5 9e+5 1.2e+6 Time (ns) 5e+5 1e+6 Time (ns) FNOES - Nice, 14.9.5 p.13/21

}~ {z J ƒ ` O Z L NO S L? T Stochastic Dynamics II 1e+7 Residence Times in the Maximum Gain Mode α=3.3 - c =.97 <T res > (ns) W=.187 (fitted value) 1e+6 1e+5 1 1 1 2 3 4 5 1/D The intermittent dynamics can be seen as a stochastic escape process from the MGM induced by noise fluctuations vxwy \ where the barrier height variance of the noise for K K L almost corresponds to the experimental value of the A r y U FNOES - Nice, 14.9.5 p.14/21

r J Lyapunov analysis.1.5 Transient - D= D=3x1-5 D=3x1-4 D=3x1-3 (a).25.2 3 4 5.15 λ.1.5 3 4 5 α For noise variance r y U Lyapunov exponent for any examined the asymptotic dynamics exhibits a positive maximal -values. FNOES - Nice, 14.9.5 p.15/21

L O X Jˆ r Š Jˆ Experiments vs numerics As a first comparison between experimental and numerical data the average duration of LFF is considered in the range for two sets of experimental data. v3 L 1 T LFF (ns) k=.25 -- D=3 x 1-3 Exp Data Exp Data Simulation α=3.3 + Noise Simulation α=4. + Noise Simulation α=5. 1 T LFF (ns) k=.35 -- D=3 x 1-3 Simulation α=4. + Noise Exp Data Simulation α=5. Simulation α=3.3 + Noise 1 1.95 1 1.5 1.1 1.15 c.9.92.94.96.98 1 1.2 1.4 1.6 c The agreemen is reasonably good for high values of without noise K KML with noise ( ) r y U FNOES - Nice, 14.9.5 p.16/21

PDFs of the Intensities Experimental data - Filtered at 2 MHz.9 < c < 1. 1 P(ρ 2 ).1.1.5 1 1.5 2 ρ 2 Numerical data - Filtered at 2 MHz α=5 - no noise -.9 < c <.99 Numerical data - Filtered at 2MHz α=4. + Noise -.9 < c <.99 1 P(ρ 2 ) 1 P(ρ 2 ).1.1.1.1.5 1 1.5 2 ρ 2.5 1 1.5 2 ρ 2 FNOES - Nice, 14.9.5 p.17/21

+ Œ % " u n PDFs of the LFF times n Ž % n Ž " % " Œ where is the average of n Ž and n Ž. 1 Experimental data I th =2.7614 ma 1 Experimental & simulation data α=4. + noise 1 1 1 2 1 3 P(x)=P(T LFF )*STD I=2.54 ma I=2.56 ma I=2.64 ma I=2.7 ma I=2.75 ma I=2.8 ma I=2.9 ma exp( x) P(x) 1 1 1 2 1 3 I=2.54 ma I=2.56 ma I=2.64 ma I=2.7 ma I=2.75 ma I=2.8 ma I=2.9 ma exp( x) sim c =.9 sim c =.94 sim c =.98 sim c =1.2 1 4 1 4 1 5 5. 5. 15. x=(t LFF <T LFF >)/STD 1 5 5.. 5. 1. 15. x FNOES - Nice, 14.9.5 p.18/21

I A? ž? Ÿ «} ª ` ` C Av? ƒv T ž Ÿ First passage times A Brownian motion with a drift can be written as A?@ I šœ A?@ with initial condition A?O, A?@ is a Gaussian noise with zero average. The average first passage time to reach a fixed threshold variance is. ƒ A is and its The PDF of the first passage times is the so-called inverse Gaussian : ª œ ž where. FNOES - Nice, 14.9.5 p.19/21

Comparison with the experiments PDF of the T LFF.1 PDF of T LFF Experimental data I=2.56 ma Inverse Gaussian.1 exp data I=2.64 ma Inverse gaussian.1.1.1.1 1e-5 5 1 T LFF (ns) 1 2 3 4 5 T LFF (ns) The inverse Gaussian describes reasonably well the experimental distributions FNOES - Nice, 14.9.5 p.2/21

Concluding Remarks The deterministic LK eqs exhibit LFF as a transient chaotic dynamics; the introduction of additive noise in the LK eqs leads via intermittency to sustained LFF; the experimentally observed Sisypho Cycles can be interpreted as a Brownian motion with drift plus a reset mechanism ; the role of noise appears to be essential in the modelization of the phenomenon. FNOES - Nice, 14.9.5 p.21/21