Parametric resonance of a defect mode in a 2D photonic crystal Vladimir V. Konotop * Departamento de Fı ´sica and Centro de Fı ´sica da Mate ´ria Condensada, Faculdade de Cie ˆncias, Universidade de Lisboa, Complexo Interdisciplinar, Av. Professor Gama Pinto2, P-1649-003, Lisboa, Portugal Vladimir Kuzmiak ² Institute of Radio Engineering and Electronics, Czech Academy of Sciences, Chaberska 57, 182 51 Prague 8, Czech Republic Received 1 December 2000; revised manuscript received 20 April 2001; published 11 September 2001 The possibility of enhancement of defect modes in photonic band gap structures through the parametric resonance is predicted when periodic modulation of the dielectric constant of the impurity is assumed. The effect is illustrated on an example of a defect mode in a two-dimensional triangular photonic crystal fabricated of GaAs rods embedded in vacuum with InSb defect. The amplitude of the electric field associated with the defect mode obtained from the numerical simulation based on finite-difference time-domain method reveals exponentially growing behavior which is in qualitative agreement with the theoretical prediction. DOI: 10.1103/PhysRevB.64.125120 PACS numbers: 42.79.Dj I. INTRODUCTION It is well established now that the photonic crystals are of great importance for numerous practical applications. De- vices based on such structures can operate in wide range of wavelengths, spanning the from microwaves to infrared and optical regime. 1 One of practically useful system based on the photonic crystal is high-Q photonic band gap PBG resonant cavity that can be typically realized by introducing a point or line defect into otherwise regular photonic lattice. In analogy to defect states within the energy band gap in semiconductors the defects in photonic crystals may induce the existence of the exponentially decaying states that appear within the stop band. The nature of the localized modes has been under intensive theoretical and experimental investiga- tion for their potential applications, e.g., in semiconductor lasers, resonators and single-mode emitting diodes as well as for many interesting physical phenomena that occur as a re- sult of their unique properties. The great majority of existing both theoretical and experi- mental works explored stationary properties of photonic crystals as potentially useful technology basis in design of passive devices. Interest in construction of active elements has led to investigation of properties of periodic structures possessing nonlinearity and suggestions on possibility of ex- ploiting effects associated with the generation of higher har- monics and induced interband transitions have been investigated. 2–7 Qualitatively new approach which in con- trast to static photonic crystals employs rigid vibrations of a crystal fabricated from linear material has been recently suggested. 8 It has been reported that by tuning the driving frequency of vibrations to the frequency of the interband transition leads to coupling of the modes. 9 In this paper we propose yet another principle potentially useful in design of active elements that is based on the pos- sibility of driving characteristics of a defect mode. In particu- lar, we are interested in amplification of the amplitude asso- ciated with a defect mode by relatively weak parametric signal, which eventually can be of different physical nature guided electromagnetic field, acoustic wave, etc. We dem- onstrate that such effect is a manifestation of parametric resonance resulting in exponential growth of the amplitude of the defect mode. The organization of the paper is as follows. In Sec. II we discuss a realization of a PBG structure containing time- dependent defect. In Sec. III we argue that the resonance observed can be identified as a parametric resonance in the conventional sense which subject to definite assumptions corresponds to the parametric resonance of a linear oscillator. In Sec. IV we describe the finite-difference time domain FDTD technique based on the excitation of the defect mode and in Sec. V we present the results of the numerical simu- lation obtained by using this method. In Sec. VI we discuss the conditions under which the parametric resonance occurs and summarize both theoretical and numerical results. II. A PHYSICAL MODEL In general, for the phenomenon of parametric amplifica- tion of the defect mode to occur, one has to employ a physi- cal mechanism that leads to the modulation of dielectric per- mittivity of the medium taking place locally, as it is in the case considered below, or globally, as it was suggested in Refs. 8,9. This can be done in different ways, depending on the value of the frequency of the defect mode. To illustrate this phenomenon for a specific structure we focus on the optical frequency range of frequencies and therefore in order to scale the normalized frequency we consider the photonic lattice with the lattice constant of the order of 1  m. To achieve the most effective interaction of the defect mode with the PBG crystal one has to impose the resonant condi- tions, i.e., modulate crystal parameters with the frequency comparable with the frequency of the defect mode. Then it is natural to arrange the resonant interaction in an optical re- gime by varying system parameters by using the electromag- netic wave of optical wavelength. As an example we consider a structure depicted in Fig. 1 in which one linear rod is substituted by a nonlinear one possessing Kerr type nonlinear permittivity and having the refractive index larger than that of the surrounding medium. PHYSICAL REVIEW B, VOLUME 64, 125120 0163-1829/2001/6412/1251207/$20.00 ©2001 The American Physical Society 64 125120-1