Experimental study of stochastic phenomena

in vertical cavity lasers

Istituto Nazionale Ottica Applicata Largo Enrico Fermi 6 50125 Firenze (Italy) www.inoa.it

Giovanni Giacomelligiacomelli@inoa.itwww.inoa.it/~gianni

Coworkers

Sylvain Barbay, LPN-CNRS, Marcoussis (France)

Vyacheslav Chizhevsky, B.I. Stepanov Institute of Physics, Minsk (Belarus)

Stefano Lepri, INFM, UdR Firenze (Italy)

Francesco Marin, Physics Dept., Univ. of Firenze (Italy)

Ivan Rabbiosi, ENS, Lyon (France)

Alessandro Zavatta, Systems and Informatics Dept., Univ. of Firenze (Italy)

Summary

1. The vertical cavity laser (VCSEL)

2. Noise-driven polarization dynamics

3. Basics: a Langevin description

4. Periodic modulation I: stochastic resonance

5. Periodic modulation II: phase synchronization

6. Random modulation: aperiodic stochastic resonance

7. Vibrational resonance

8. Back to basics: a microscopic Langevin model

9. Conclusions

The edge emitting laser (EEL)

Wavelength: near IR (~800 nm), but also red, 1330 and 1550 nm ...

Output power: ~mW.... (and much more!)

Single transverse mode, multiple longitudinal modes (long and narrow cavity)

Single polarization emission

BAD optical quality of the beam

High frequency pump modulation

1. The vertical cavity laser (VCSEL)

Wavelength: near IR (~800 nm),.....

Output power: ~mW....

Single longitudinal mode, multiple transverse modes (short and wide cavity – High Fresnel number)

TWO LINEAR polarizations emission (symmetrical cavity)

GOOD beam quality

High frequency modulation

Polarization bistability in VCSELs

2 polarization directions selected by the crystal axis

AND

symmetrical cavity (circular, rectangular)

===>

laser action on both polarizations

BUT

impurities, inhomogeneities: ONE polarization lasing at threshold

.....for particular values of the pump current, the symmetry can be restored:

POLARIZATION BISTABILITY!

Experimental setup

Remark: very stable system (~hour), fast timescales (~µs)==> GOOD STATISTICS

Polarization bistability I

Scanning the pump current across a bistable point...

Polarization bistability II

(a): total intensity(b), (c): intensity in the two polarizations

2. Noise-driven polarization dynamics

Reduction to two level: residence time distributions

(Kramers statistics)

Kramers rates, changing theadded noise power

3. Basics: a Langevin description

Given the Langevin equation dx/dt = - U'(x) + ξ

where (white, gaussian) < ξ (t) ξ (t') > = 2 D δ (t-t')

the prob. distribution for x is P(x) = P0 exp( -U(x)/D )

A direct evaluation of the probability distribution allows for the reconstruction of the potential involved in the Langevin process:

QUASIPOTENTIAL

REMARK: A consistent description requires that the quasipotential is not affected by the amount of noise added to the system

Quasipotentials I

Experimental quasipotentials, for different added noise powers

Quasipotentials II

Probability distributions evaluated atdifferent working points (bias currents)

A dynamical remark: is it stochastic?

Polinomial fit of quasipotentials + Langevin simulation:Statistics of residence times (simulated) = Statistics of residence times (exp)

Return maps: Xn+1

MAX vs Xn

MAX

Structure? No!

Evaluation of correlation dimension (Grassberger-Procaccia)Increses linearly with the embedding dimension....

4. Periodic modulation I: stochastic resonance

Time series (noise increasesfrom bottom to top)

Response at the modulation frequency.Curve: linear response theory withexperimentally measured parameters

Stochastic resonance: statistical characterization

Two-level reduction: Residence Time Probability Distributions

Resonances: first, second, ... peak

Bona-fide SR

Linear response theory: single pole response (dots: exp)

BUT: something happens at a well-defined frequency.....

Bona fide SR: statistical analysis

Area of the peaks in the RTPDs, with background subtraction:left: upper level; right: lower level: resonances!

5. Periodic modulation II: phase synchronization

Definition of phase φ of a (real) signal x(t) with the Hilbert transform H:

X(t) = x(t) + i (Hx)(t) = A(t) exp( iφ (t) )

(in the case of sinusoidal signal, φ = Ωt)

Study of the phase difference: φ = φ OUT

- φ IN

Feasible with experimental (or numerical...) time series

Spectral power amplification

SIN input: modulation amplitude increases from top to bottom

Time series: SIN input

Noise increases from bottom to top.Inset: at the resonant noise value, for a small modulation amplitude.

Time series: SQR input

Noise increases from bottom to top.Inset: at the resonant noise value, for a small modulation amplitude.

Phase synchronization:statistical indicators

Average frequency (first moment):

<ωOUT

> = <d/dt (φOUT

)>

Diffusion coefficient (second moment):

DEFF

= ½ d/dt ( <φ2OUT

> - <φOUT

>2)

Average output frequency (SIN and SQR)

SIN: filled symbols; SQR: empty symbols. A

SQR = A

SIN/sqrt(2)

Effective diffusion coefficient(SIN and SQR)

SIN: filled symbols; SQR: empty symbols.The SQR amplitude A

SQR = A

SIN/sqrt(2)

Duration of locking episodes

Scaling of the locking times: TLOCK

~ exp(A/D) ~ (Deff

)-1

(evaluated at the noise corresponding to the match of the average output frequency with the input frequency => no drift)

Phenomenological model – I

dφ/dt = ∆ – ∆s sin(φ

) + ξ

Adler equation: describes the motion of a particle in a (tilted) potential

U(φ) = −∆φ – ∆s cos(φ

);

Phase locking <=> fluctuations in the bottom of the potential wells (when ∆ < ∆

s)

Phase slip <=> jump from a well to the next one

Phenomenological model – II

Langevin model with bistable potential and modulation:

dx/dt = - U(x) + µ(t) + ξ

where µ(t) is SIN, SQR or SAW with the same RMS amplitude:A

SQR = A

SIN/sqrt(2); A

SAW = A

SIN/sqrt(3)

Average output frequency(SIN, SQR and SAW)

SIN: filled symbols; SQR: empty symbols; SAW: grey symbols.A

SQR = A

SIN/sqrt(2); A

SAW = A

SIN/sqrt(3)

Effective diffusion coefficient(SIN, SQR and SAW)

SIN: filled symbols; SQR: empty symbols; SAW: grey symbols.A

SQR = A

SIN/sqrt(2) = A

SAW /sqrt(3) (same RMS value)

Phenomenological model – III

Time-dependent Kramers' rates: a1,2

= a0(t) exp(- ∆V

1,2(t)/D)

Small modulation amplitudes: ∆V1,2

(t) ~ ∆V ± µ(t); a0(t) ~ 2sqrt(2)/π

Approximate rates: a1,2

~ 2sqrt(2)/π exp(-(∆V ± µ(t))/D)

= rK exp(± µ(t))/D)

SQR rates: a1,2

SQR= rK exp(± ΑSQR/D)

Effective rates (e.g. SIN mod.): a

1,2SIN ~ < r

K exp(± Α sin(Ωt)/D)>

T/2

Adiabatic rates (averaged)

Averaged Kramers' rates for SIN and SQR modulations,( A

SQR = A

SIN/sqrt(2) )

6. Random modulation: aperiodic stochastic resonance

A different input signal? Bit-stream

Random sequence of ¨low¨ (0) and ¨high¨ (1) level amplitudes, with a fixed rate (clock)

Advantages:

Typical digital communication signal (applications...)

Analytical evaluation of statistical indicators

Time series: ASR

Definition of the rates

Kramers times (symmetrical case): W1

- = W0

+ = 1/T

s ; W1

+ = W0

- = 1/T

l

ASR: in-out correlation

Dots: experimental correlationsBoxes: analytic theory with experimentally measured (4) rates

Bit Error Rate (BER)

BER = (#wrong trasmitted bits)/(#total input bits)

7. Vibrational resonance

IDEA: (partially....) replacing the white, gaussian noise with

high frequency modulation

Expected features:

SR-like resonance changing the HF modulation amplitude;

More efficient device than a SR-based system for signal regeneration

Unavoidable internal noise: SNR?

Vibrational vs stochastic resonance

Left: laser response for increasing HF modulation amplitude;right: increasing the added noise power (SR)

Gain and SNR

Left: gain; right: SNRBlue: VR; red: SR

Low signal detection I

Left: output signal without modulation or added noise;Center: with optimal added noise (SR)Right: with optimal HF modulation (VR)

Low signal detection II

Left: gain; right: SNRBlue: VR; red: SR

Phenomenological approach to VR

The HF modulation parametrically modifies the potential ruling the process: the effective potential function is obtained through averaging over the HF period.

The barrier height can therefore be reduced by a suitable choice of the HF modulation

==> jumping induced by the internal noise.

The output signal is enhanced using a deterministic, non-resonant modulation

==> improvement of the gain and of the SNR.

8. Back to basics: a microscopic Langevin model

Reduction of the San Miguel – Feng – Moloney (SFM) model with multiple scale methods ==> rate equations for the polarizations; further reduction to a single Langevin equation (I

X + I

Y ~ const) [1];

Derivation of a potential and comparison with the experiment [2];

Comparison of stochastic resonance features in numerical and analytical model [3].

[1] Phys. Rev. A 67, 013809 (2003) [VUB group, Brussels (Belgium)];

[2] Phys. Rev. A 68, 013813 (2003) [INOA, INFM, FI-PhysDept, VUB];

[3] Phys. Rev. E 67, 056112 (2003) [VUB group, Brussels (Belgium)].

Comparison

Dwell times: experimental data and fit with analytic expression for proton implanted (top) and air-post (top) samples

Experimental and numerical quasipotentials, for different values of the pump current

Conclusions

A flexible and powerful experimental system for the study of stochastic phenomena

Evidence and characterization of standard and aperiodic stochastic resonance

Study of noise-induced phase synchronization

Evidence and characterization of vibrational resonance

Langevin description: beyond phenomenology

Applications in communications?

Key references:http://www.inoa.it/~gianni

Polarization bistability: Quantum Semiclass. Opt. 10, 469 (1998)

! Stochastic resonance: - PRL 82, 675 (1999)

- PRE 61, 157 (2000)

" Aperiodic SR: - PRL 85, 4652 (2000)

- PRE 63, 051110-1 (2001)

# Phase synchronization: PRE 68, 020101(R) (2003)

$ Vibrational resonance: PRL 91, 220602 (2003)

% µLangevin model: PRA 68, 013813 (2003)