DOI: 10.1007/s00340-004-1504-8 Appl. Phys. B 79, 9–13 (2004) Lasers and Optics Applied Physics B a. di falco c. conti g. assanto ✉ Photonic crystal wires for optical parametric oscillators in isotropic media NOOEL-Nonlinear Optics and OptoElectronics Laboratory, National Institute for the Physics of the Matter (INFM), University “Roma Tre”, Via della Vasca Navale 84, 00186 Rome, Italy Received: 12 February 2004/ Revised version: 18 February 2004 Published online: 7 April 2004 • © Springer-Verlag 2004 ABSTRACT We investigate four wave mixing in photonic crystal wire microresonators realized in an isotropic medium. One- dimensional optical parametric oscillators are numerically ana- lyzed by solving Maxwell’s equations in all dimensions and including material dispersion as well as nonlinear polarization. PACS 42.65.Yj; 42.70.Qs; 42.82.Gw 1 Introduction The interest of photonic crystals (PC) relies mostly on the design versatility afforded by such structures in vari- ous materials for use in micro- and nano-optics and integrated applications. The engineering of the dispersion relation in PC-based devices can be achieved by periodic alternation of high and low refractive index materials in one, two or three dimensions, and has been extensively studied and demon- strated [1–4]. Several linear and nonlinear phenomena have been proposed for exploitation in PC structures, spanning from Bragg reflectors and bistable gates to filters, delay lines, waveguides, switches, lasers [5–11]. In the past few years, by introducing suitably located defects in the periodic pattern, researchers have been able to obtain high quality ( Q-) fac- tors for the modes resonating in PC structures, up to values Q > 10 4 [12]. More recently, the features of optical modes near the band-edges have been examined in two dimensional (2D-) PC exhibiting comparably high Q-factors [13]. One of the simplest PC geometries is the PC-wire, i.e., a strip of material with a limited number of holes able to open a band-gap in the transmission characteristic. Among them, air-bridge PC-wires are particularly interest- ing and potentially amenable to the highest degree of circuit integration [14]. In this paper we show for the first time how the micro- resonant structure of an air-bridge PC-wire can be exploited to efficiently confine light and obtain gain and optical para- metric oscillation of the cavity modes via four wave mixing (FWM), exploiting the cubic response of isotropic media. ✉ Fax: +39-065579078, E-mail: assanto@ele.uniroma3.it In the next section we describe the model and its numeri- cal implementation, while in Sect. 3 we perform the modal analysis of the structure. Finally, in Sect. 4 we explore the oscillatory behaviour of the PC-wire subject to high power pumping. We will show how the oscillation dynamics relates to the pumped mode, in terms of both symmetry constraints and quality factors. 2 Numerical Model Optical oscillations throughout energy exchange between PC modes via nonlinear four wave mixing can be studied by means of a finite-difference time-domain (FDTD) code. Maxwell’s equations were solved with no approxima- tions, except for discretization, in both the three spatial coor- dinates and time. An isotropic nonlinear medium was mod- eled by introducing a Lorentz oscillator yielding the cubic polarization P: ∇× E =−μ 0 ∂ H ∂t ∇× H = ε 0 ∂ E ∂t + ∂ P ∂t ∂ 2 P ∂t 2 + 2γ 0 ∂ P ∂t + ω 2 0 f ( P )P = ε 0 (ε s − 1) ˆ ω 2 0 E (1) where f ( P ) = 1 ( P 2 = P · P) represents a linear single-pole dispersive response. To describe an isotropic Kerr-like mate- rial we chose f ( P ) =[1 + ( P / P 0 ) 2 ] −3/2 as in [15]. As long as the ratio P / P 0 is small, a standard Kerr response is retrieved; as its size becomes appreciable, however, the model is able to account for higher order terms (χ (5) etc.). For the integration we employed a nonlinear generalization of the L-DIM1 ap- proach for dispersive media [16]. To enforce electromagnetic field continuity at the boundaries between different media and prevent spurious reflections, we adopted the Yee’s grid and uniaxial perfectly matched layers (UPML) [17]. The sourcing was arranged throughout a total-field/scattered-field (TF/SF) approach, allowing both single-cycle (SC) and continuous wave (cw) excitations, the latter mimicked by an “mnm” pulse. The “mnm” pulse allows for the avoiding of undesired spectral distortions by trailing and tailing with m cycles while holding the peak value for n cycles, respectively [18]. A linear waveguide provided the input to the nonlinear portion of the