experimental study of stochastic phenomena in vertical cavity lasers - cnr · 2006-06-28 ·...
TRANSCRIPT
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Experimental study of stochastic phenomena
in vertical cavity lasers
Istituto Nazionale Ottica Applicata Largo Enrico Fermi 6 50125 Firenze (Italy) www.inoa.it
Giovanni [email protected]/~gianni
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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)
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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
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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
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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
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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!
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Experimental setup
Remark: very stable system (~hour), fast timescales (~µs)==> GOOD STATISTICS
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Polarization bistability I
Scanning the pump current across a bistable point...
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Polarization bistability II
(a): total intensity(b), (c): intensity in the two polarizations
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2. Noise-driven polarization dynamics
Reduction to two level: residence time distributions
(Kramers statistics)
Kramers rates, changing theadded noise power
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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
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Quasipotentials I
Experimental quasipotentials, for different added noise powers
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Quasipotentials II
Probability distributions evaluated atdifferent working points (bias currents)
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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....
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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
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Stochastic resonance: statistical characterization
Two-level reduction: Residence Time Probability Distributions
Resonances: first, second, ... peak
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Bona-fide SR
Linear response theory: single pole response (dots: exp)
BUT: something happens at a well-defined frequency.....
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Bona fide SR: statistical analysis
Area of the peaks in the RTPDs, with background subtraction:left: upper level; right: lower level: resonances!
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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
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Spectral power amplification
SIN input: modulation amplitude increases from top to bottom
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Time series: SIN input
Noise increases from bottom to top.Inset: at the resonant noise value, for a small modulation amplitude.
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Time series: SQR input
Noise increases from bottom to top.Inset: at the resonant noise value, for a small modulation amplitude.
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Phase synchronization:statistical indicators
Average frequency (first moment):
<ωOUT
> = <d/dt (φOUT
)>
Diffusion coefficient (second moment):
DEFF
= ½ d/dt ( <φ2OUT
> - <φOUT
>2)
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Average output frequency (SIN and SQR)
SIN: filled symbols; SQR: empty symbols. A
SQR = A
SIN/sqrt(2)
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Effective diffusion coefficient(SIN and SQR)
SIN: filled symbols; SQR: empty symbols.The SQR amplitude A
SQR = A
SIN/sqrt(2)
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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)
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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
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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)
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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)
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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)
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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
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Adiabatic rates (averaged)
Averaged Kramers' rates for SIN and SQR modulations,( A
SQR = A
SIN/sqrt(2) )
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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
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Time series: ASR
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Definition of the rates
Kramers times (symmetrical case): W1
- = W0
+ = 1/T
s ; W1
+ = W0
- = 1/T
l
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ASR: in-out correlation
Dots: experimental correlationsBoxes: analytic theory with experimentally measured (4) rates
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Bit Error Rate (BER)
BER = (#wrong trasmitted bits)/(#total input bits)
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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?
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Vibrational vs stochastic resonance
Left: laser response for increasing HF modulation amplitude;right: increasing the added noise power (SR)
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Gain and SNR
Left: gain; right: SNRBlue: VR; red: SR
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Low signal detection I
Left: output signal without modulation or added noise;Center: with optimal added noise (SR)Right: with optimal HF modulation (VR)
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Low signal detection II
Left: gain; right: SNRBlue: VR; red: SR
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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.
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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)].
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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
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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?
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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)