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Section DYNAMICS IN LIFE SCIENCES, NEUROSCIENCE APPLICATIONS WORKSHOP
Distributed Delay Differential Equation Models in Laser Dynamics
A. G. Vladimirov12, G. Huyet3,4,5 and A. Pimenov1
1 Weierstrass Institute, Mohrenstr 39, Berlin, Germany;
2 Lobachevsky State University of Nizhny Novgorod, 23 Gagarina av., 603950 Russia;
3 Tyndall National Institute, University College Cork, Lee Maltings, Dyke Parade, Cork,Ireland;
4 Centre for Advanced Photonics and Process Analysis, and Department of Applied Physics and Instrumentation, Cork Institute of Technology, Cork, Ireland;
5 National Research University of Information Technologies, Mechanics and Optics, 199034 St.Petersburg, Russia. * Presenting e-mail: vladimir@wias-berlin.de
OM&P
Introduction
Mode-locking is a powerful technique to generate ultrashort optical pulses, which are used in numerous applications [1]. The most common way to model semiconductor mode-locked lasers is based on the use of the so-called traveling wave equations [2], a system of PDEs governing the space-time evolution of the amplitudes of two counter-propagating waves in the laser cavity coupled to the carrier density in the semiconductor medium. An alternative and simpler approach to the analysis of mode-locking based on a system of delay differential equations (DDEs) was proposed in [3]. Later a modification of DDE model was applied to describe the dynamics of Fourier domain mode-locked (FDML) [4] and sliding frequency mode-locked [5] lasers used in optical coherence tomography. However, despite of a remarkable success of the DDE model in describing the dynamics of mode-locked and FDML lasers, this model does not take into account such important phenomenon as chromatic dispersion of the intracavity media. In order to fill this gap, here we develop a new model of an FDML laser that takes into account chromatic dispersion of the fiber delay line. This is a system of DDEs, which in addition to a fixed delay contains a distributed delay term and can be reduced to an infinite chain of delay differential equations with a single fixed delay.
Model equations and CW stability analysis
We consider a multimode ring-cavity laser with the round trip time T consisting of a short semiconductor optical amplifier (SOA) gain section, linear frequency selective spectral filter, and a long dispersive fiber delay line [4]. We assume that this delay line is described by linear equations that account for the dispersion produced by a single detuned Lorenteian absorption line. Using the lumped element method together with the approach described in [3] we derive the following set of two differential equations for the amplitude of the electric field A and saturable gain G:
Here the „polarization" P(t) is given by p(t) = -aLY
J —CO
yj aLY it - s)
(3)
Using the DDE model (1)-(3) we have performed analytical stability analysis of the continuous wave (CW) laser operation regimes in the limit of large delay time T, when the eigenvalues belonging to the so-called pseudo-continuous spectrum can be represented in the form: A=i|a+A/T+O(1/T2) [6]. We have shown that in the case of sufficiently strong anomalous dispersion (Q<0) all the CW solutions are modulationally unstable. Furthermore, both normal and anomalous dispersion can cause instability close to the lasing thresghold. Finally, we have derived a necessary condition for the stability of CW solutions with respect to modulational instability, which resembles similar condition for the complex Ginzburg-Landau equation.
Conclusion
We have developed a DDE model of a multimode laser taking into account the chromatic dispersion of the fiber delay line and studied analytically the stability of CW regimes in this model in the limit of large delay time. We have demonstrated that in the anomalous dispersion regime a CW operation can be destabilized. Our DDE FDML laser model (1)-(3) satisfies automatically the causality principle and contains a distributed delay term similar to the one introduced in [7].
Section DYNAMICS IN LIFE SCIENCES, NEUROSCIENCE APPLICATIONS WORKSHOP
However, unlike the model discussed in [7], our model provides an explicit analytical description of the linear response function in (3) and, therefore, allows performing analytical stability analysis of the CW solutions. Apart from FDML lasers the approach discussed here can be applied to study the dynamics of mode-locked photonic crystal [7] and other types of multimode lasers.
Acknowledgements
We gratefully acknowledge useful discussions with Julien Javaloyes, Svetlana Gurevich, Svetlana Slepneva, and Shal-va Amiranashvili. A.G.V. and A.P. acknowledge the support of SFB 787 of the DFG. A.G.V. acknowledges the support of the grant 14-41-00044 of Russian Scientific Foundation.
References
1. H. A. Haus, IEEE J. Sel. Top. Quantum Electron., 2000, 6(6), 1173-1185.
2. U. Bandelow, M. Radziunas, J. Sieber, and M. Wolfrum, IEEE J. Quantum Electron., 2001, 37, 183-188.
3. A. G. Vladimirov and D. Turaev, Phys. Rev. A, 2005, 72, 033808.
4. S. Slepneva, B. Kelleher, B. O'Shaughnessy, S. Hegarty, A. G. Vladimirov, and G. Huyet, Opt. Express, 2013, 21(16), 19240-19251.
5. S. Slepneva, B. O'Shaughnessy, B. Kelleher, S. Hegarty, A. G. Vladimirov, H. Lyu, K. Karnowski, M. Wojtkowski, and G. Huyet, Opt. Express, 2014, 22(15), 18177-18185.
6. S. Yanchuk and M. Wolfrum, SIAM J. Appl. Dyn. Syst., 2010, 9, 519-535.
7. M. Heuck, S. Blaaberg, and J. Mark, Opt. Express, 2010, 18(17), 18003-18014.
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Optimal Extraction of Collective Rhythmicity from Unreliable EEG Channels
Justus Schwabedal*
Max-Planck-Institute for the Physics of Complex Systems,Germany. * Presenting e-mail: jschwabedal@gmail.com
I present a novel data-processing method that facilitates the detection and analysis of the irregular-oscillatory dynamics. The method is particularly useful to EEG analysis, as I will demonstrate in polysomnographic EEG recordings. By design, the method copes well with unreliable recordings showing fluctuating signal amplitude, phase offsets, and substantial amounts of measurement noise. Under such relatively general conditions, I will show that the method optimally enhances a rhythm of interest, and demonstrate its use by the detection and analysis of EEG sleep spindles.
Active Wireless Networks for Experimental Study in Neuroscience
AS. Dmitriev*, R.Y. Emelyanov, M.Yu. Gerasimov
Institute of Radio Engineering and Electronics. VA Kotelnikov RAS, Moscow Institute of Physics and Technology * Presenting e-mail: chaos@cplire.ru
The report examines the active wireless network, which can serve as an experimental tool in the study of various objects in neurodynamics. The network combines the nodes on which digital or analog neuron model (if necessary this may be living neurons), and programmable connections between them, which are implemented through wireless channels can be implemented. The latter circumstance allows the implementation of any type of connection (Linear. Non-linear, with delay, etc.) with any desired topology As an example, the modeling of the phenomenon of chimeras in the system of coupled oscillators is presented.
Chimeras - a popular and interesting phenomenon in the oscillator system [1], which are mainly studying by computer simulation. Experimental study of chimeras, in particular, in small ensembles requires special experimental setups. The active wireless network [2] is used as such experimental equipment in the report. Experiments were carried out with small ensemble of coupled oscillators. Ensemble of six phase oscillators [3] was using as the study system:
