Rigorous simulations of bent and branched sections of coupled resonator optical waveguides (CROWs) composed of side-coupled whispering gallery (WG) mode microdisks are performed. Pre- and post-fabrication tuning capability of the designed structures is explored, and a novel concept of realization of tunable CROW-based routers and switches is introduced. The proposed tuning mechanism exploits the properties of CROW optical modes coupling with avoided crossing scenario rather than the previously used Vernier effect. Applications of spectrally-engineered branched CROW structures for controllable manipulation of coupling between spatially separated nano-emitters are also discussed.
©2007 Optical Society of America
Since a concept of the coupled-resonator optical waveguide (CROW) was introduced nearly a decade ago , such structures have been explored in a variety of material platforms and resonator types, including photonic-crystal defect cavities, microspheres, microdisks, and Fabry-Perot resonators. This research effort has been fueled by the interest in their various potential applications encompassing light slowing and storage [2–6], detection and measurement of rotation , group velocity compensation , etc. Optical characteristics of infinite CROW chains composed of identical microcavities have been extensively investigated and are well understood by now, and the focus of attention has recently shifted to studying more realistic finite-size CROWs with structural disorder [9–14]. Furthermore, potential capability of coupled-resonator waveguides to guide light around sharp corners  requires further detailed investigation [14, 15]. These new avenues of research not only provide estimates of the CROW fabrication tolerances [9, 12, 13] but also offer ways to optimize the properties of coupled-cavity structures at the pre-fabrication design stage by tuning their geometrical configurations [14, 16–18].
High-Q optical microcavities and coupled-cavity structures also hold high promise for the development of dynamically-tunable optical devices with extraordinarily small footprints. Variations in the microcavities refractive indices can be induced by using various physical mechanisms including free-carrier-plasma dispersion, thermo-optic and electro-optic effects. As has been shown both theoretically and experimentally, the effect of cavity refractive index change on the cavity spectral characteristics is significantly enhanced owing to the tight field confinement and long photon lifetimes in such cavities [19–22]. This makes possible realization of optical bistability in very compact integrated structures with relatively low power. Manipulation of optical bistability adds new functionalities to resonator-based optical devices such as signal modulation, switching, and memory functions, already demonstrated in the case of waveguide-coupled microresonator-based add/drop filters [22–27], double-cavity photonic molecule lasers [18, 28, 29], and multiple-resonator high-order filters [19, 30]. Here, I propose and theoretically explore CROW configurations with bends and branches, which can be pre-designed to enable easy modulation of the electromagnetic energy flow through them by using various external control impacts that cause the change of microcavities effective refractive index. Potential applications of coupled-cavity structures proposed and studied in this paper encompass CROW routers and splitters as well as dynamically-tunable coupled-microdisk photonic structures in which optical coupling between two spatially separated nano-emitters, e.g. quantum dots, can be manipulated in a controllable manner.
2. Optical modes in finite-size coupled-microdisk CROW sections
CROWs composed of side-coupled microdisk resonators studied in this paper offer certain advantages over other microcavity types, e.g., easy integration with other components on the optoelectronic chip. By using high-refractive-index-contrast material platforms such as silica-on-silicon (SOI), ultra-compact microring resonators of several micrometers in diameters supporting very high-Q whispering-gallery (WG) modes have been successfully realized [5, 6, 22–24]. However, it has been demonstrated that using microdisks rather than microrings enables further reduction of the resonators size without spoiling their high Q-factors [31, 32].
The WG modes supported by the coupled-microdisk CROW section are calculated by solving an integral form of the Maxwell’s equations in which the mode complex wavelength λ=λ ′+iλ ″ is the eigenvalue [14, 16–18]. The mode quality factor is calculated as Q=λ ′2λ ″. Once the modal eigenvalues are found, the corresponding eigenvectors can be constructed, which give the modal near-field portraits. The technique used in this paper takes into account all the electromagnetic interactions between the microdisks. This makes possible reliable modeling of strongly-coupled microdisks and also studying CROW sections with sharp bends and branches, where noticeable electromagnetic coupling can occur not only between neighboring resonators. For thin disks, the 3-D boundary-value problem of finding the modes eigenfrequencies can be replaced with a pair of equivalent 2-D formulations for the TE and TM polarized modes by using the effective-index method to account for the vertical field confinement. In this case, the disk refractive indices are replaced with their effective values computed as modal indices of the TM- or TE- fundamental guided modes of an equivalent slab waveguide of the same thickness as the microdisks . TM-polarized WG-modes (having electric field vector perpendicular to the plane of the disks) typically have significantly lower effective refractive indices (and thus higher losses) than the TE-polarized modes for a fixed disk size. Therefore, in the rest of the paper we will consider the TE-polarized WG-modes only. However, all the concepts and conclusions are equally applied to the TM modes , which can be launched in the CROW by using a polarized laser source . The effective refractive index of the disks at 1550 nm (n eff=2.9) used in this and the following sections corresponds to air-clad 250-nm thick Si disks located on a silica substrate.
First, we consider a finite-size linear CROW composed of 7 microdisks with effective refractive indices n eff=2.9 and diameters D=2.8 µm side-coupled via equal airgaps of width w as shown in the inset of Fig. 1. Clearly, moving the microcavities closer to each other shifts their resonant frequencies further apart (Fig. 1(a)). Optical coupling between neighboring resonators usually results in broadening of the CROW transmission resonance in comparison to the transmission resonance corresponding to a WG-mode of a single microcavity . Furthermore, experimental transmission spectra of finite-size CROWs feature periodic ripples, with ripple periodicity depending on the number of resonators in the chain . If the microcavities are strongly coupled via very narrow airgaps (or are touching each other) spectrally well-separated CROW modes can be observed experimentally, which constitutes the so-called “Bloch regime” . It can be seen in Fig. 1 that all spectrally-separated CROW modes group into nearly-degenerate doublets , which can be indistinguishable in the experimental CROW spectra. Such grouping of modes in doublets is a general feature of linear chains of coupled WG-mode resonators, observable even in the simplest case of a double-cavity photonic molecule . The modes in each doublet are not however truly degenerate, they have different field symmetry with respect to x and y axes, and their Q-factors (imaginary parts of complex modal eigenfrequencies) can differ drastically. The results presented in Fig. 1 reveal that the mode doublet with the shortest wavelength is most tolerant to the variations in the coupling airgap width. This doublet corresponds to the anti-bonding CROW modes  with even/even (EE) and odd/odd (OO) symmetry along the x and y axes, respectively (Fig. 1(c,d)). It can be seen that these two modes experience moderate frequency detuning when cavities are brought into contact with each other (Fig. 1(a)) and their Q-factors are an order of magnitude higher than those of other spectrally-separated CROW modes in this case (Fig. 1(b)). Furthermore, our previous research shows that the Q-factors of these modes remain high in bent CROWs (at least for certain values of the bend angle) .
3. Engineering low-loss CROW bends
To make possible integration of complex functionality on a small device footprint, unconventional complex CROW configurations featuring sharp turns and/or branching sections should be explored [1, 10, 14, 15]. A possibility of making low-loss or even loss-less bends in CROWs by choosing the bend angle value according to the angular symmetry of the WG-mode supported by the microdisks has been predicted in the pioneering paper . However, further studies revealed that this rule-of-thumb, which relates the CROW bend angle β and the azimuthal index of the WG-mode m as β=Mπ/m(M=0,1,2…), can only be reliably applied to CROWs composed of weakly-coupled microcavities supporting high-azimuthal-order tightly-confined WG modes . In the case of strong coupling between microresonators, designing low-loss bends in CROWs requires application of more accurate theoretical approaches. The red and blue lines in Figs. 2(a) and 2(b) represent the optical characteristics (namely, resonant frequencies and Q-factors) of the EE- and OO- anti-bonding modes supported by a bent CROW section with the same parameters as in Fig. 1 and w=100 nm as a function of the bend angle. Oscillating behavior of modal wavelengths and Q-factors can be observed. The oscillation pattern is however somewhat distorted from the periodical picture with the period of 15 degrees expected from the application of the above-mentioned rule-of-thumb. As the results plotted in Fig. 2(b) indicate, low-loss transmission along this CROW section can be expected at bend angles around 0, 45, and 60 degrees. Figs. 2(c) and 2(d) present a visualization of the WG-mode near-field distributions of the two modes in the vicinity of the bend region for the bend angle of 62 degrees, where one of them (EE) reaches its highest Q-value (c) and the other (OO) is suppressed (d).
4. Switching in branched CROW sections
We have previously shown  that CROW bend losses can be minimized for any value of the bend angle by tuning the size of the resonator positioned at the bend. Here, a possibility of post-fabrication tuning of bent and branched CROW sections and of designing switchable optical elements by changing the material parameters of the central disk rather than its size is explored. To keep the following discussion general, no specific physical effect causing the refractive index change Δn is considered, though each of them has a specific index perturbation signature .
In Figs. 3 and 5 two possible designs of switchable CROW-based devices that enable splitting of waves or pulses into two output waveguide ports or routing all the energy into a single port are considered. In both cases, the branches bend angle values (β 1 and β 2) were chosen such that the bend losses are minimized. Fig. 3 shows how the change of the effective refractive index of the central microdisk affects the resonant wavelengths and the Q-factors of the anti-bonding WG12, 1 CROW modes. In can be seen that the two modes couple with the avoided crossing scenario (real parts of their complex eigenfrequencies repel each other and the imaginary parts cross at the coupling point ). An additional point of avoided level crossing with one of the red-shifted low-Q CROW modes is observed in Fig. 5. Note that in this section the term “modes coupling” refers to the interactions among complex eigenfrequencies of the spectrally-resolved CROW modes under the change of the CROW structural parameters rather than to electromagnetic coupling between WG-modes of individual resonators. Coupling of eigenstates of optical microcavities, photonic molecules, and coupled-resonator waveguides can lead to undesired effects, such as degradation of the working mode Q-factor . In properly configured deformed-cavity or coupled-cavity configurations, however, this effect can be exploited either to enhance useful mode features, e.g., to increase the working mode quality factor [16–18, 37], or to realize optical switching functionality [18, 29].
In the structures shown in Figs. 3–6 switching can be realized by exploiting a fundamental feature of complex eigenstates coupling with the avoided crossing scenario. The two modes interchange their identities (i.e., Q-factors and fields distributions) upon passing the anti-crossing point, and at this point their modal eigenvectors represent a symmetric and an anti-symmetric superposition of the eigenvectors of the uncoupled system [35, 37]. In the simplest case of a double-microdisk asymmetric photonic molecule this effect causes formation of bonding (the symmetric superposition) and anti-bonding (the anti-symmetric one) photonic molecule modes at the point where both cavities are identical [18, 29]. As demonstrated in Figs. 4 and 6, switching between two high-Q supermodes of the branched CROW structures can be achieved by moving the natural frequency of the high-Q mode to/from the anticrossing (AC) point by tuning the permittivity value of the central microdisk (e.g., thermo-optically or electro-optically). Movies in Figs. 4 and 6 show the dynamics of the spatial near-fields distributions in branched CROW sections with the change of the central disk effective index. The movies demonstrate that if the eigenfrequency of the higher-Q CROW mode is pushed away from the anti-crossing point (blue line to the left of the AC point and red line to the right of the AC point), only the lower branch of the CROW is switched on. In the vicinity of the anti-crossing point, coupling of the eigenstates yields high intensity of WG-modes in both lower and upper branches of the coupled-cavity structure. Thus, by modulating the refractive index of only one resonator in the CROW section, all the energy can be either routed to the lower channel or split between two channels. Furthermore, it is also possible to entirely transfer the power from the lower to the upper arm of the branched CROW section. The movie in Fig. 7 shows the near-field pattern variation of the lower-Q mode (red line to the left of the AC point in Fig. 5). For the values of the mode eigenfrequency away from the AC point, the field intensity of this mode is mostly localized in the CROW upper arm. If the eigenfrequency of this mode is pushed closer to the AC point, a situation is possible when the mode near-field intensity is redistributed in such a way that the left arm and the upper right arm of the CROW section are switched on (all the energy is routed to the upper channel).
The tuning capability demonstrated above may potentially be useful for realization of tunable CROW-based filters, switches and routers. Our preliminary studies indicate that similar switching effect can be achieved if other microdisks located in the branch region are tuned (e.g., disks 2, 4 or 5). Furthermore, every microdisk can be controlled separately (e.g., via an individual heater ) thus adding extra tuning capability to the system. The study of CROWs composed of microcavities of various sizes indicates that the higher the CROW mode Q-factor and the smaller the inter-cavity coupling coefficient, the smaller the effective index change is required to perform the switching . However, even for the case of small cavity sizes and rather strong inter-cavity electromagnetic coupling considered here, required shifts of the refractive index are easily achievable in practice. Change of refractive index of microdisk and microring resonators fabricated in SOI platform can be achieved all-optically with central microdisk refractive index being modulated by photo-injection. The index variations as large as Δn eff=1.07×10-3(Δn Si=-0.97×10-3 ) and Δn eff=-1.45×10-3 (Δn Si=-1.6×10-3, see ) have been induced by a photo-generated carrier densities of ΔN=2.7×1017 cm-3 and ΔN=4.8×1017 cm-3, respectively. Similar tunable and switchable CROW structures can be realized in other popular material platforms, where even more significant detuning of the microdisk refractive index can be achieved. For example, free-carrier-injection-induced changes of the refractive index of Δn=-6×10-3 (InGaAsP/InP microdisk ) and of Δn=-4×10-3 (InP microdisk ) have recently been reported. Large thermo-optical effect of silicon (∂n/∂T=1.86×10-4 K -1) and some polymers also makes possible tuning of resonant wavelengths of the microdisks by the change of their temperature.
As already mentioned, each of the physical mechanisms that may potentially be used for realizing switching in WG-mode microcavity structures has a unique “r-signature” . Parameter r relates the material refractive index perturbation Δn to the extinction coefficient perturbation Δκ as follows: r=Δκ/Δn (κ=αλ/4π, where α is the material absorption coefficient). Previous studies of tunable microdisk resonators fabricated in various material systems show that in the near infrared region of interest Δn has much greater effect on the eigenfrequencies of microdisks than Δκ, which in many cases can be neglected. For example, in the case of Si microdisks tuned via the thermo-optic effect, the r-value is negligibly small . Free-carrier plasma dispersion (FCPD) effect increases resonator material losses, however, the microcavity-based devices can still be tuned without substantially degrading their Q-factors. For FCPD-tuned Si microresonators operating at 1.55 µm (assuming equal accumulation of electrons and holes in the microcavity) the r-value has been estimated as r=0.0533 (Δκ=5.33×10-5 for Δn=-1×10-3) . Experimental study of InP microdisks in the same frequency range revealed that the refractive index perturbation of Δn=-4×10-3 caused by the free carriers injection was accompanied by the Δα=3.5 cm-1 change in the absorption coefficient (Δκ=4.317×10-5) . Taking into account the finite values of Δκ in the numerical experiments with branched CROW structures did not produce any noticeable effect on the eigenfrequencies of the CROW modes and did not affect the switching sensitivity. FCPD-induced losses, however, cause slight degradation of the CROW mode Q-factors. For example, the quality factor of the higher-Q mode in Fig. 5 for n 3=2.895 (Δn=-5×10-3) reduces from Q=5.117×104 (r=0) to Q=4.979×104 (r=0.0533, Δκ=2.665×10-4), while the mode resonant frequency does not shift (λ ′=1543.64 nm).
Finally, it should be noted that switching between two optical states in a branched CROW section can be used not only for energy routing and splitting but also for realization of a dynamical control of coupling between two spatially separated quantum dots (QDs) placed in two microcavities belonging to different CROW branches. The interaction between two QDs can be achieved by switching the upper branch of the CROW section in Figs. 5, 6 on and off. The proposed structure has several advantages over the previously explored for this purpose photonic molecule composed of two coupled micropillar cavities of different diameters . First, both CROW modes (one having field localized in a lower CROW arm and the other with the whole CROW section switched on) have high Q-factors and almost identical resonant frequencies. Second, the area to which external control impacts are to be applied to achieve switching between two optical states (namely, disk 3) can be spatially separated from the areas used for the readout process, where emitters are located (e.g., disks 4 and 5 adjacent to disk 3 and located in the lower and the upper CROW branches, respectively).
Using rigorous boundary-integral-equations theory, spectral properties of coupled-resonator optical waveguide sections with bends and branches composed of high-index-contrast thin-disk resonators of ~3 µm in diameter were simulated at λ=1.55 µm. Through computer modeling, optimal CROW configurations were designed that minimize bend radiation losses and enable modulation of the electromagnetic energy flow along the structure by a change in the microcavity refractive index in the 1÷5×10-3 range. Switching functionality in such structures is achieved by exploiting the features of the complex eigenmodes energy levels coupling with the avoided crossing scenario under the change of the CROW structural and material parameters. Spectrally-designed tunable structures based on coupled microdisk resonators are expected to find applications as dynamic low-power routers and splitters, optical memory elements, and switchable structures for manipulation of optical coupling between two or more spatially separated nano-emitters.
The author is grateful to Trevor Benson and Ana Vukovic of the University of Nottingham, Sergei Tarapov and Vadim Derkach of Kharkov Institute of Radiophysics and Electronics NASU, and Andrea Melloni of Politecnico di Milano for useful discussions and advice.
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