We study circular grating resonators (CGRs) which are formed by a central defect surrounded by concentric rings composing a grating and which display perfect azimuthal modal-symmetry. Because of their radial symmetry they exhibit a complete band gap for a minimal index contrast. However, as is the case for all 2D resonators their quality factors are limited by vertical losses. To reduce the vertical losses we introduce a chirp of the grating period by reducing it towards the central defect. The chirped CGRs exhibit drastically improved quality factors of up to tens of millions with a modal volume of a few cubic wavelengths.
© 2009 Optical Society of America
For future integrated optical components one of the most critical parameters is the footprint. Smaller footprints can be achieved by employing resonant structures that lead to longer effective propagation paths. Applications for such resonant structures are filters , delay-lines , electro-optical modulators , all-optical switches , lasers , as well as sensing  and spectroscopy applications . Furthermore, beyond classical optics, resonant structures with a very high quality factor and small mode volume could pave the way towards integrated devices that harness cavity quantum electrodynamics effects . In this letter we focus on one type of resonator structure, namely circular grating resonators (CGR) [9, 10, 11]. Similar cavity structures have shown much promise and already theoretical quality factors of a couple ten thousand [12, 13, 14]. Such structures and CGRs enable cavities with an ultra-small footprint and at the same time a high quality factor. Compared with linear structures, they offer enhanced photonic confinement due to the photonic band gap in the two lateral dimensions. To achieve a complete band gap for radial wavevectors, CGRs require only a minimal refractive index contrast of the materials used to fabricate the structure. This is an inherent advantage over photonic crystal cavities where the photonic band gap is induced by a lattice with high index contrast, and thus, the choice of suitable materials is very limited. Furthermore, because their radial symmetry they enable mode patterns with perfect azimuthal symmetry, e.g. dipole-like resonances. Hence, CGRs could lead to highly integrated resonant devices with an extended choice of materials and mode patterns while exhibiting extremely small optically active volumes.
A CGR consisting of a circular Bragg mirror which is formed by concentric rings around the central defect is shown in Fig. 1. Here we focus on oxide-cladded silicon-on-insulator (SOI) devices because of their immediate relevance for silicon photonic applications although the same concept can be realized with many other material systems. The rings and the defect consist of Si and have a refractive index of 3.44 at the telecom wavelength of 1550 nm. The CGR is surrounded by SiO2 with an index of 1.46. The Bragg mirror imposes a photonic band gap, i.e. a stop band, for a range of radial wavevectors, which prevents lateral leakage of light from the cavity. The radius of the central defect rc determines the resonance frequency ν of the resonator. For a given grating geometry, the reflectivity of the Bragg mirror can be increased by adding more rings. Here, we choose 40 rings corresponding to a nearly infinite system. For small central defect sizes, only few nodes of the resonant optical field are located in the center, and the field exponentially decays radially in the Bragg rings. Hence, the effective modal volume can be strongly reduced to a few cubic wavelengths.
2. Circular grating resonators with periodic rings
To explore the limits of CGRs without a chirp, i.e. with a constant grating period, we search for the geometry with maximum quality factor. To minimize the spatial extension of the cavity mode, we maximize the band gap of the circular Bragg rings. We approximate the circular Bragg rings by their corresponding linear grating. The relevant geometric parameters are the grating period a, the height h and the duty cycle D. We compute the lower and upper band edges by solving Maxwell’s equation in a plane-wave basis using the MPB code . Without restricting the general applicability for other polarizations, we focus in this paper on TE polarization, where the electric field is in the r,φ plane and the magnetic field is in the z-direction (Er, Eφ, Hz). The geometry with maximum band gap of Δν/ν=41% (D=0.62 and h=0.69a) shows a lower band edge of 0.207c/a and an upper band edge of 0.314c/a, where c is the speed of light.
As the quality factor is largely determined by losses in the vertical direction, the configuration with the largest band gap that provides the strongest photonic confinement is not necessarily the one with the highest quality factor. Starting from this circular Bragg geometry we simulate a CGR and maximize its quality factor by varying its three free geometric parameters: its defect radius rc, its duty cycle D, and its height h. The quality factors are calculated using the finite-difference time-domain (FDTD) method using the MEEP code . The cylindrical symmetry was exploited in such a way that the calculations were reduced to 2D. A uniform mesh with 40 grid points per grating period a was used and perfectly matched layer boundary conditions were employed on a calculation cell size of 44a×13a, in the r- and z-directions, respectively. To generalize the results, they are expressed in units normalized to the grating period a. Frequencies are reported in units of c/a.
We maximize the quality factor by varying the three geometric parameters, whereby we find that CGRs exhibit various local maxima. We choose geometries that have rather small defect radii, a small height, and a high quality factor. For the azimuthal order m=0 we determine a CGR geometry with a defect radius of rc=1.96a, a duty cycle D=0.34 and a height h=0.85a exhibiting a maximum quality factor of 17800 and a resonance frequency of 0.182c/a. The band gap for this configuration is Δν/ν=24%. The quality factor Q, resonance frequency ν, and modal volume Vm=∫ε|E|2 dV/max(ε|E|2) are plotted as function of the defect radius in Fig. 2. With increasing defect radius, the resonance frequency decreases and approaches the lower band edge of the circular Bragg rings. As the resonance frequency of the mode approaches the band edge, it becomes more delocalized and like the mode of the band edge, which is a guided mode, and its quality factor therefore increases. This is corroborated by an increase of the modal volume with increasing defect radius. At the maximum quality factor, the modal volume is 9.6a 3 [18.8 (λ/2n)3], where n denotes the refractive index of the high index material. As the resonance frequency of the mode crosses the lower band edge its modal volume increases notedly and becomes more and more delocalized. After reaching its maximum the quality factor decreases again due to increasing radial losses because the resonance frequency of the mode lies outside the band gap.
3. Chirped circular grating resonators with high quality factors
For many applications a resonator with a quality factor on the order of 10000 is insufficient. Therefore, in a next step we modify the geometry of the CGR to improve its quality factor. By relaxing the spatial mode confinement slightly it is possible to achieve a much larger quality factor. We introduce a chirp in the grating period of the concentric rings. The grating period is reduced towards the central defect and the concentric rings are shifted by ΔN with N being the number of the ring. The width of the trenches qN increases for trenches further away from the defect and the shift ΔN is reduced from ring to ring by the factor Γ:
In the limit of large N the trench width approaches the trench width without a chirp.
We introduced two additional geometric parameters to describe the chirped CGRs: the shift of the first ring Δ1 and the decay constant Γ. We now vary all five geometric parameters to maximize the quality factor. For three different azimuthal orders m=0, m=1, and m=3 the results are shown in Table 1. For the azimuthal order m=0 we obtain a maximum quality factor of approx. 6.6×107. Increasing the azimuthal order lowers the maximum quality factor to 5.6×106 for m=1 and to 2.2×106 for m=3. The defect radius rc decreases from 1.60a for m=0 to 1.53a for m=1 and to 1.10a for m=3. It turns out that all three azimuthal orders have the same duty cycle of D=0.34, same height of h=0.80a, and same shift of the first ring
of Δ1=-0.29a. Only for the azimuthal order m=0 is the decay constant is larger (Γ=0.88) compared to the other two configurations. All three resonance frequencies are located well within the band gap which extends from 0.182c/a to 0.239c/a. This corresponds to a band gap of Δν/ν=27%.
In Fig. 3 we plot the quality factor Q, the frequency ν, and the modal volume Vm as a function of the shift of the first ring Δ1 for azimuthal order of m=0. Without chirp (Δ1=0) the CGR only exhibits a quality factor of 780. With decreasing shift of the first ring Δ1 (increasing chirp) the quality factor, the frequency, and the modal volume increase. While the quality factor increases from 780 to 6.6×107 by five orders of magnitude the modal volume only increases to 6.2a 3 [21.8 (λ/2n)3] by a factor of 1.4 compared with the configuration without chirp.
2D photonic-crystal resonators described in the literature yield a quality factor of 7×108 with modal volume of 10.4 (λ/2n)3  or a quality factor of 2×108 with a modal volume of 11.2 (λ/2n)3 . Our maximum quality factor of 6.6×107 is comparable. Because of the azimuthal symmetry of the mode of the CGR, its modal volume is slightly - by a factor of two - larger. Although the CGRs perform slightly worse than 2D photonic-crystal resonators, CGRs have two major advantages: They require only a minimal index contrast and their modes exhibit azimuthal symmetry. Just as 2D photonic-crystal resonators with such high index-contrast CGRs suffer of additional losses due surface roughness when actually fabricated. However, for 2D photonic-crystal cavities such losses reduce only the quality factor from 1.5×107 to 2.5×106 . We expect a similar reduction of the quality factor for fabricated CGRs.
The mechanism of the chirp can be understood by plotting the effective band gap as function of the ring number N as shown in Fig. 4. This is analogous to the local bands used to describe multiheterostructure 2D photonic-crystal resonators . For chirped CGR, the local band edges are modified by the smaller effective lattice constant and the larger effective duty cycle towards the central defect. The band gap therefore decreases towards the central defect. In the inner part of the CGR (N≤2) no band gap is present. Starting from the resonance frequency, a band gap gradually opens up for N>2 and increases towards the outer rings of the CGR. Therefore the resonant mode can adapt from a very delocalized, high-quality-factor, band-edge-like mode in the center of the CGR to the large band-gap region in the outer parts of the CGR. This enables a very high quality factor at only a small compromise on the strong confinement. In contrast, CGR with periodic gratings achieve high quality factors by approaching the resonance frequency to the band edge.
The functionality of the chirp strongly depends on the decay constant Γ. In Fig. 4 the upper band edge increases and lower band edge decreases exponentially with the ring number. The decay constant defines the rate the band gap decreases towards the central defect. Using a too small decay constant would lead to a too fast decrease of the band gap, whereas using a too large decay constant would lead to a too slow decrease. The quality factor shows a sharp maximum at the optimum decay constant. All in all, the quality factor varies with the decay constant in very similar fashion as it varies with the shift of the first ring.
To further explore the nature of the extraordinarily high quality factor of the chirped CGR we investigate the resonant field distributions. In Fig. 5 we plot the cross section of the electric field component Eφ for three different shifts of the first ring Δ1. The color scales are used in such a way that even smaller electric field values further away from the center defect are visible and they are almost everywhere supersaturated. The near-field patterns are visually indistinguishable in the three cases. However, the far-fields pattern differ drastically. An indication of nodal planes appear precisely at the maximum quality factor configuration (Δ1=-0.29a), indicating the cancelation of the lowest-order multi-pole moment (i. e. dipole) . The two other configurations show dipole-like radial far-field patterns which are an indication of large vertical losses and therefore low quality factors. This mechanism offers an additional explanation for the high quality factors.
Finally, in order to assess the quality of our chirp relation (Eq. 1) we start from this optimal configuration and we varied the positions of all Si/SiO2 interfaces fully independently. We found, however, that the assumed chirp relation for this configuration seems to yield the maximum quality factor. For other configurations than m=0 small shifts in the positions of the Si/SiO2 interfaces lead to slightly larger quality factors. However, these shifts are on the order of the calculation grid resolution and therefore at the limits of numerical significance.
In summary, we studied CGRs which are formed by a central defect surrounded by concentric rings composing a grating. However, for periodic gratings their quality factor is limited by vertical losses when the mode is strongly confined. To reduce the vertical losses we introduce a chirp of the grating period by reducing it towards the central defect. This chirp introduces a gradual opening of the photonic band gap in the radial direction which allows the mode to transform from a delocalized band-edge-like mode in the center to a mode within a wide band gap in the outer regions. The radiated field pattern suggests that cancelation of the lowest-order multi-pole moment plays an important role to achieve the high quality factor. These chirped CGRs show drastically improved quality factors up to 6.6×107 with only slightly increasing the modal volume by a factor of 1.4. Whereas similar figures for quality factor and modal volume are achieved for photonic crystal cavities, chirped CGR in contrast only require a minimal index contrast and their modes exhibit perfect azimuthal symmetry.
The authors gratefully acknowledge financial support from EU within the Circles of Light project (FP6-034883).
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