Hopf bifurcation in parallel polarized Nd:YAG laser M.R. Parvathi a, * , Bindu M. Krishna b , S. Rajesh a , M.P. John a , V.M. Nandakumaran c,1 a International School of Photonics, Cochin University of Science and Technology, Cochin, India b Sophisticated Test and Instrumentation Centre, Cochin University of Science and Technology, Cochin, India c Indian Institute of Science Education and Research, Thiruvananthapuram, India article info Article history: Accepted 19 January 2009 Communicated by Prof. Ji-Huan He abstract Dynamics of Nd:YAG laser with intracavity KTP crystal operating in two parallel polarized modes is investigated analytically and numerically. System equilibrium points were found out and the stability of each of them was checked using Routh–Hurwitz criteria and also by calculating the eigen values of the Jacobian. It is found that the system possesses three equilibrium points for (I j , G j ), where j = 1, 2. One of these equilibrium points undergoes Hopf bifurcation in output dynamics as the control parameter is increased. The other two remain unstable throughout the entire region of the parameter space. Our numerical analysis of the Hopf bifurcation phenomena is found to be in good agreement with the analytical results. Nature of energy transfer between the two modes is also studied numerically. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Laser systems have been the all time favorite of researchers because of the rich variety of dynamics they exhibit. A lot of research has been carried out to study the chaotic fluctuations in the output dynamics of laser systems. Among them, insta- bilities in multimode solid state lasers were of special interest. Intracavity doubled continuous wave infrared lasers are effi- cient sources of coherent visible light. Nd:YAG lasers can be developed as cw visible sources using high power laser diode arrays for pumping and also doubling crystals with large nonlinear gain coefficient. It was Baer [1] who first reported large amplitude fluctuations in this laser system. He observed that large amplitude fluctuations and longitudinal mode instabil- ities arise in the output of diode pumped Nd:YAG laser in the presence of an intracavity doubling crystal. Coupling of various longitudinal modes of the laser by sum frequency generation was found to be the origin of these instabilities. He developed a deterministic rate equation model to explain these fluctuations. Using this model, he was able to predict the dependence of these fluctuations on the pump level, the nonlinear coupling constant and the number of oscillating modes. A detailed analysis of the periodic and the chaotic fluctuations in the output intensity of multimode solid state laser was carried out by Bracikowski and Roy [2]. They studied the Nd:YAG laser with intracavity KTP crystal and obtained several interesting results. It is possible to eliminate the chaotic fluctuations in this laser and also obtain complex periodic wave- forms such as antiphase states by varying the relative orientation of the YAG and KTP crystals. They have also made a sta- tistical study of these chaotic fluctuations. Studies by Oka and Kubota have shown that the laser dynamics is strongly influenced by the polarizations of the cavity modes [3]. This dependence is due to the fact that the amount of green light produced by sum frequency generation depends on whether the contributing fundamental modes are polarized parallel or orthogonal to each other. It has been shown that when the laser is operating with two orthogonally polarized modes the chaotic fluctuations in the output intensity can be stabilized through a reverse period doubling bifurcation by varying a particular control parameter (relative orientation of 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.01.013 * Corresponding author. E-mail address: parvathiabhin@gmail.com (M.R. Parvathi). 1 On leave from International School of Photonics, CUSAT. Chaos, Solitons and Fractals 42 (2009) 515–521 Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos