Optical bright and dark states in side-coupled resonator structures

Abstract

We analyze side-coupled standing-wave cavity structures consisting of Fabry-Perot and photonic crystal resonators coupled to two waveguides. We show that optical bright and dark states, analogous to those observed in coherent light-matter interactions, can exist in these systems. These structures may be useful for variable, switchable delay lines. © 2007 Optical Society of America OCIS codes: 230.5750, 230.7370. In this Letter, we show the unique dispersion properties of single-mode, side-coupled (SC) cavity structures can be used to generate large variable, switchable delays. Using a transfer-matrix method, we first calculate the dispersion and transmission properties of SC structures based on coupled Fabry-Perot (FP) resonators. We show these structures support "bright" and "dark" states, accessible by tuning the relative phase of the input fields. In analogy with electromagnetically induced transparency (EIT) [1], here a dark state is one in which the resonators are not excited by the optical signal even on resonance. We then show that these bright and dark states also exist in photonic crystal waveguide-cavity structures. Finally, we illustrate how these dark and bright states can be used to construct variable, switchable delay lines. We first consider structures consisting of two waveguides side coupled to a periodic chain of FP resonators. They can take the form of either (i) a FP coupled-resonator optical waveguide (CROW) side coupled to two waveguides (FP-SC-CROW) as in Fig. 1(a) or (ii) a SC integrated spaced sequence of resonators (SCISSOR) with FP cavities (FP-SCISSOR) as shown in Fig. 1(b) [2,3]. The primary difference between the devices in Figs. 1(a) and 1(b) is the degree of coupling between adjacent cavities: the FP-SC-CROW reduces to the FP-SCISSOR in the limit of zero interresonator coupling. In general, the FP resonators in the middle waveguide can be replaced with other single-mode cavities, as long as the resonators support only one mode on resonance. In contrast, microrings, for example, support two degenerate modes on resonance.To analyze the coupled waveguide structures in Fig. 1, we employ a transfer-matrix method in which we propagate the incoming and outgoing fields along z. In general, the transmission properties of each unit cell in Fig. 1 can be described by a 6 ϫ 6 matrix; the field at a particular point in z can be described by ͓e 1 e 2¯e6 ͔ T . The vector includes both the forward-and backward-propagating fields in the resonator and the two waveguides [4]. To relate the fields at z = 0 and z = ⌳ for a structure that has inversion symmetry with respect to the center of the unit cell, we use the scattering matrix ͑1͒where e + = ͓e 1 e 3 e 5 ͔ T and e − = ͓e 2 e 4 e 6 ͔ T . T and R are 3 ϫ 3 matrices describing the transmission and reflection properties of the unit cell, and their explicit forms are presented in [4]. The scattering matrices can be made more tractable by accounting for nearest waveguide coupling only and neglecting the coupling between the outer waveguides and t...

Dark modes, slow modes, and coupling in multimode systems

“…The significance to related topics (light-matter interaction, imaginary parts, physics of eigenmodes in DBR) has also been given and possible paradoxes or semantic caveats addressed. We believe that this discussion helps in distinguishing and classifying among emerging proposals, be it simply by terming the interaction scheme an "N − N F " scheme; e.g., N − N F ϵ 5 − 3 in [34]; N − N F ϵ n − 1 in [23], etc. Grating-induced back reflection in a broad multimode waveguide also falls within the scope of the present description on the optical side.…”

“…For example, in our system that has N = N p + N q modesstates, indices p (modes in family 1) and q (modes in family 2) may refer to forward and backward modes [41], or to backward and quasi-localized modes such as coupled resonator optical waveguide (CROW) modes [34][35][36][37]. Families p and q may also be photon modes and atom-excitonpolaron states (among many others, see [19]), often with negligible dispersion, interacting together.…”

“…In the general case, we can infer that there are F bright states and N d = N − N F dark states. In [34], N F = 3 and N = 5 states are present (two waveguides with degenerate forward and backward modes, plus a nondegenerate cavity or CROW mode), hence N d =5−3 = 2 dark states are formed. We propose to term this "an N − N F classification," in order to compare the numerous emerging proposals with a basic synthetic tool.…”

“…Let us now turn to the reasons for our contribution, which is the increased use of the dark state or dark mode terminology in the area of waveguides, nanophotonics, photonic crystals, and plasmonics [33][34][35][36][37][38][39][40]. For man-made physical objects such as waveguides instead of atoms, we have to consider modes localized in a specific dielectric (or metallodieletric) structure.…”

Design and engineering of metasurfaces for high-directivity antenna and sensing applications

“…A dark state refers to a state of an atom or molecule that cannot absorb or emit photons. The concept has been intensively exploited during the last decade in the fields of photonics such as integrated optics [10][11][12], photonic crystals [13,14], metamaterials [15] or plasmonics [9,16]. The interests to this domain were principally driven so as to obtain sharp spectral transmission features within an absorption band, which are highly desirable in sensing applications [15].…”