Homogeneous solutions and Pattern formation in Ring cavities with composite and negative refractive materials

MARTIN DANIEL ALEJANDRO; HOYUELOS MIGUEL

Villa Carlos Paz

Workshop; XIII Latin American Workshop on Nonlinear Phenomena, LAWNP; 2013

We study a ring cavity filled with a Kerr-like medium. We allow the nonlinear medium to be a composite material with positive (PRM) or negative (NRM) refractive index [1]. The cavity is subject to a linearly or elliptically polarized incoming field. If the nonlinear material is a composite material, either NRM or PRM, intense nonlinear efects may be expected. Also, the coupling parameter between dierent polarizations, B, may take a wide range of possible values (at odds with "classical" materials) [2]. Also, nonlinear magnetization may arise [3], so that equations for magnetic eld propagation (which should also couple to the electric eld) are expected [4]. In the mean eld limit, we found that Electric and Magnetic eld should remain proportional [5], so that cavity can be described by means of two coupled Lugiato Lefever [6] equations. For linearly polarized fields, We find that considering a NRM does not bring cualitatively diferent behaviour, althoug stability of some solutions may vary [7]. Results for NRM can be related to results for PRM. We have found that, depending on B and the cavity detuning, steady homogeneous solutions may present multiple bifurcations: up to two pitchfork birurcations and more that two saddle node biurcations for some xed parameters as we vary the imput intensity [8]. Pattern formation is also studied. Examples of marginal instability diagrams are shown. It is shown that, within the model, instabilities cannot be of codimension higher than 3. A method for nding parameters for which codimension 2 or 3 takes place is given. The method allows us to choose parameters for which unstable wavenumbers fulll diferent relations. Numerical integration results where diferent instabilities coexist and compete are shown [9]. References [4] [5] [6] [7] [1] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000); R.A. Shelby, D.R. Smith and S. Schultz, Science 292, 77 (2001). [2] J.E. Sipe and R.W. Boyd, Phys. Rev. A 46, 1614 (1992); V. Yannopapas, Opt. Commun. 283, 1647 (2010); M. Lapine and M. Gorkunov, Phys. Rev. E 70, 066601 (2004); [3]A.A. Zharov, I.V. Shadrivov, and Y.S. Kivshar, Phys. Rev. Lett. 91, 037401, (2003); S. O´Brien, D. McPeake, S.A. Ramakrishna, and I.V. Shadrivov, A.B. Kozyrev, D.W. van der Weide, and Y.S. Kivshar, Opt. Express 16, 20266 (2008); [4] I. Kourakis and P.K. Shukla, Phys. Rev. E 72, 016626 (2005); N. Lazarides and G.P. Tsironis, Phys. Rev. E 71, 036614 (2005). [5] Daniel A Mártin and Miguel Hoyuelos, Physical Review E 80, (056601) 2009. [6] L.A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987). [7] P. Kockaert, P. Tassin, G. Van der Sande, I. Veretennicof and M. Tlidi, Phys. Rev. A 74, 033822 (2006); P. Tassin, L. Gelens, J. Danckaert, I. Veretennicof, P. Kockaert and M. Tlidi, Chaos 17, 037116 (2007). [8] Daniel A. Mártin and Miguel Hoyuelos, Physical Review A 82 (033841) 2010. [9] Daniel A. Mártin and Miguel Hoyuelos, Accepted for publication in PHYSICA D, http://dx.doi.org/10.1016/j.physd.2013.05.008.