Laser cavity with a Van der Pol dynamics

M. Lozano, A. Kir’yanov, A. Pisarchik, V. Aboites


In this article, a beam within a ring phase conjugated laser is described by means of a Van der Pol bidimensional dynamic map using an ABCD matrix approach. Explicit expressions for the intracavity chaos-generating matrix elements were obtained; furthermore, computer calculations for different values of Van der Pol map’s parameters were made. The rich dynamic behavior displays periodicity when the parameter ¹ (which determines the non-inearity term) takes values around zero. These results were observed in phase diagrams and in diagrams of the optical thickness of the intracavity element.


Van der Pol map, phase conjugate, Laser Dynamics

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B. Van der Pol and J. Van der Mark, Frequency demultiplication, Nature, 120 (1927), 363–

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membranes.

BiophysicsJ, 1 (1961), 445–466.

J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating

nerve axon. Proc. IRE, 50 (1962), 2061–2070.

J. Cartwright, V. Eguiluz, E. Hernandez-Garcia, and O. Piro, Dynamics of elastic excitable

media. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 2197–2202.

A.D. Case, P.J. Soan, M.J. Damzen and M.H.R. Hutchinson, Coaxial flash-lamp-pumped

dye laser with a stimulated Brillouin scattering reflecto. J.Opt. Soc. Am. B 9 (1992), 374.

H.-J. Eichler, R. Menzel, and D. Schumann. Appl. Opt.,31 No. 24 (1992), 5038-5043.

M. O’Connor, V. Devrelis, and J. Munch,in Proc. Int. Conf. on Lasers’95 (1995), pp.500-

M.J. Damzen, V.I. Vlad, V. Babin, and A. Mocofanescu, Stimulated Brillouin Scattering:

Fundamentals and Applications, Institute of Physics, Bristol (2003)

V. Aboites, Dynamic of a Laser Resonator, IJPAM 36(4) (2007), 352-354.

M. Wilson, V. Aboites, Optical resonators and dynamic maps, Proceedings SPIE-The

International Society for Optical Engineering. 8011 (2011), 215.

M. Wilson, V. Aboites, Y. Barmenkov, A. Kir’yanov, Optical Devices Chapter XX, Optical

Resonators and DynamicMaps. Ed. Intech, (2012). 17-36.

V. Aboites, Y. Barmenkov, A.V. Kiryanov, M. Wilson, Bidimensional dynamic maps in

Optical Resonators, Rev.Mex.Fis. 60 (2014), 13-23.

V. Aboites, M. Huicochea, Hénon Beams. International Journal of Pure and Applied

Mathematics. 65 (2010), 129-136.

V. Aboites, D. Liceaga, R. Jaimes-Reategui, J. Garcia, Bogdanov Map for Modelling a

Phase-Conjugated Ring Resonator. Entropy. 21 (2019), 384.

V. Aboites, D. Liceaga, A. Kir’yanov, M. Wilson, Ikeda Map and Phase Conjugated Ring

Resonator Chaotic Dynamics. Applied Mathematics Information Sciences. 10 (2016), 1-

D. Dignowity, M. Wilson, P. Rangel-Fonseca, V. Aboites, Duffing spatial dynamics

induced in a double phase-conjugated resonator. Laser Physics. 23 (2013), 5002.

V. Aboites, A. Pisarchik, A. Kir’yanov, X. Gomez-Mont, Dynamic Maps in

PhaseConjugated Optical Resonators. Optics Communications. 283 (2010), 3328-3333.

V. Aboites, M. Wilson, F. Lomeli, Standard Map Spatial Dynamics in a RingPhaseConjugated Resonator. Applied Mathematics Information Sciences. 1 (2015), 1-5.

V. Aboites, Y. Barmenkov, A. Kir’yanov, M. Wilson, Tinkerbell beams in a non-linear

ringresonator. Results in Physics. 2 (2012), 216-220.

V. Aboites, M. Wilson, Tinkerbell chaos in a ring phase-conjugated resonator.Int. J. Pure

Appl. Math. 54 (2009),429–435.

Azarkhalili, Behrooz Moghadas, Peyman Rasouli, Mohammad, Approximation Behavior

of Van der Pol Equation: Large and Small Nonlinearity Parameter. Proceeding of the

International MultiConference of Engineers and Computer Scientists 2011, Vol. II.

Nguyen-Van, Triet and N. Hori, A new discrete-time model for a van del Pol Oscillator.

Proceedings of the SICE Annual Conference.10 (2010), 2699 – 2704.

Y Hafeez, Hafeez. Analytical Study of the Van der Pol Equation in the Autonomous

Regime. Progressin Physics. 11 (2015), 252-255.

Hallbach, K. Matrix. Representation of Gaussian Optics. Am. J. Phys. (1964), 32, 90.

WikipediaList of chaotic maps.

W. Albarakati, N. Lloyd, J. Pearson, Transformation to Lienard form. Electronic Journal of

Differential Equations. 2000 (2000).

T. Hará, T. Yoneyama, J. Sugie, A necessary and sufficient condition for oscillation of the

generalized Lienard equation. Annali di Matematica pura ed applicata 154 (1989), 223–

A.O. Ignat’ev, Estimate for the amplitude of the limit cycle of the Lienard equation.

Diff Equat 53 (2017), 302–310.

K. Nipp, D. Stoffer, Invariant curves for the discretised van der Pol equation .Bit

Numer Math 57 (2017), 463–497



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