RAPID COMMUNICATIONS PHYSICAL REVIEW A 84, 021808(R) (2011) Ab initio dissipative solitons in an all-photonic crystal resonator Christoph Etrich, 1 Rumen Iliew, 1 Kestutis Staliunas, 2 Falk Lederer, 1 and Oleg A. Egorov 1 1 Institute of Condensed Matter Theory and Solid State Optics, Friedrich-Schiller-Universit¨ at Jena, Max-Wien-Platz 1, D-07743 Jena, Germany 2 Instituci´ o Catalana de Recerca i Estudis Avanc ¸ats (ICREA), Departament de F´ ısica i Enginyeria Nuclear, Universitat Polit` ecnica de Catalunya, Colom 11, E-08222 Terrassa, Barcelona, Spain (Received 11 April 2011; published 25 August 2011) We identify dissipative solitons in a Kerr-nonlinear all-photonic crystal resonator by solving Maxwell’s equations directly. The photonic crystal allows for diffraction management, leading to solitons with unique properties. These results are compared to a mean-ﬁeld model based on Bloch waves, ﬁnding excellent agreement even for a high-contrast photonic crystal. By adjusting the quality factor and resonance frequencies of the resonator, optimal Bloch cavity solitons in terms of width and pump energy are identiﬁed. In particular, the width is independent of the quality factor, in contrast to the usual homogeneous cavity. DOI: 10.1103/PhysRevA.84.021808 PACS number(s): 42.65.Tg, 42.60.Da, 42.65.Hw, 42.70.Qs The existence of localized solutions or cavity solitons (CSs) in passive planar or ring resonators ﬁlled with a nonlinear material is an intriguing fact. Typically they are found in regions of bistability of the homogeneous solutions. Here they may be associated with a modulational instability of the upper branch. In the pioneering work [1] CSs are regarded as locked switching waves of stable homogeneous solutions. Bistability is not necessary to have CSs [2–4]. CSs belong to the class of dissipative solitons, because of the permanent energy exchange between the cavity and the environment. Cavity solitons were found in resonators ﬁlled with different nonlinear materials. In Refs. [1] and [3] the Maxwell-Bloch equations for a two-level system were reduced to an equation for the ﬁeld with a saturable Kerr nonlinearity. One- and two-dimensional (2D) stable CSs were found. In the simpler case of a Kerr medium, stable 2D CSs were identiﬁed in a narrow region in parameter space in Ref. [5]. In the case of the full Maxwell-Bloch equations for two levels, CSs were found in the limit of nascent bistability, where the dynamics can be described by the Swift-Hohenberg equation [6]. A more complicated case is a resonator ﬁlled with a material with a second-order nonlinearity. A variety of CSs was found. The system can be pumped either at the fundamental [4,7] or the second harmonic frequency (for an overview, see Ref. [8]). An important example is a resonator ﬁlled with a semicon- ductor since it is more or less easily accessible to experiments. 2D CSs were predicted in Refs. [9] and [10] and found experimentally in Refs. [11] and [12]. Earlier experimental evidence of CSs was found in degenerate four-wave mixing [13]. Recently the idea of combining a nonlinear cavity with a photonic crystal (PhC) to tailor the diffraction properties of light was proposed [14,15]. For a weak modulation of the refractive index only in the transverse direction, one- dimensional CSs were found for third- or second-order nonlinearity [16,17]. More recently, for a cavity weakly modulated in the transverse and the longitudinal direction, one-dimensional so-called Bloch cavity solitons (BCSs) were identiﬁed [14,15]. The periodicity of the weak modulation has to be relatively large compared to the wavelength. This is detrimental to the advantage of a diffraction-managed soliton, allowing, for instance, for the reduction of the soliton width. In order to exploit the diffraction management at most, also the modulation should be on the smallest possible spatial scale. This leads to the idea of spatial solitons in an intracavity PhC with a strong index modulation on the wavelength scale. Up to now CSs were treated as envelope solitons of either paraxial roundtrip models for forward and backward propagating waves, together with appropriate cavity boundary conditions, or a mean-ﬁeld approach for the transmitted ﬁeld derived from this. In this Rapid Communication we aim at a direct simulation of Maxwell’s equations without an approximation. This was performed by means of a nonlinear version of the ﬁnite-difference time-domain (FDTD) method [18], allowing also for the treatment of structures with a strong dielectric modulation and for a direct proof of the above idea of diffraction-managed solitons. We identify different types of Bloch cavity solitons as a direct solution of Maxwell’s equations. They are supported by a focusing or defocusing nonlinearity and may have an extremely small width. We then introduce a mean-ﬁeld model based on Bloch waves to see how this compares to the direct approach. Maxwell’s equations with a Kerr nonlinearity in the time domain are ∇ × H(r,t ) = ∂ D(r,t ) ∂t , ∇ × E(r,t ) =−μ 0 ∂ H(r,t ) ∂t , D(r,t ) = ε 0 ε(r)E(r,t ) + ε 0 χ (3) (r)|E(r,t )| 2 E(r,t ), (1) where E and H are the real electric and magnetic ﬁelds, ε 0 and μ 0 are the permittivity and permeability constants, ε is the relative permittivity, and χ (3) is the nonlinear coefﬁcient. Note that within the FDTD method the divergence equations are automatically fulﬁlled for all times. Both the linear and nonlinear response are assumed to be instantaneous, i.e., material dispersion effects are neglected. The conﬁguration to start with is displayed in Fig. 1(a) (for details of the geometry parameters, see the caption). This can be considered as an effective index distribution of a membrane with ﬁnite thickness, for instance, in air. We restrict ourselves here to a 2D system in order to keep the computation time within a reasonable amount. Apart from possible out-of-plane losses which reduce the quality factor of the cavity, the results 021808-1 1050-2947/2011/84(2)/021808(4) ©2011 American Physical Society