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Photonic crystal ring resonators (PhCRR) combine the features of ring resonators with the slow-light effects present in photonic crystal waveguides, resulting in better mode confinement and increased light-matter interaction. When the resonator modes are near the photonic band edge, this enhancement is maximized. However, for this to be useful it is necessary to design the resonator so that these modes are at a desired wavelength. We introduce a design prescription, based on a theoretical analysis of the mode spectrum of PhCRRs, that maximizes these effects at a given wavelength. We test the procedure using numerical simulations, finding a good agreement between the design objectives and the simulated mode structures. We also consider the effects of disorder on the device.
© 2014 Optical Society of America
1. Introduction
Microscale ring resonators are one of the main building blocks for integrated photonics. When the resonator losses are low, the optical confinement in microrings leads to very narrow spectral features, an increase in the local electric field intensity, and an enhancement of the light-matter interaction. The sharp resonant features make ring resonators ideal as narrow-band photonic filters [1], or ultra-sensitive biosensing elements [2]. The stronger intra-cavity light-matter interaction is useful for microscale laser sources with much lower thresholds than those of macroscopic cavities [3]. The combination of these two features (narrow linewidths and stronger light-matter interaction), also results in microrings being good candidates for photonic switches [4]. Finally, the increase in the local field intensity also enhances optical non-linear interactions [5] making microrings a great choice for devices such as frequency combs [6] or mode-locked lasers [7].
Standard ring resonators are fabricated from waveguides with weak but non-zero dispersion, which distorts the mode spectrum slightly compared to that of an ideal ring. Better dispersion control and engineering can be realized by using photonic crystal (PhC) waveguides instead. This control has allowed the demonstration of phenomena like reduced-dispersion and slow light in waveguides [8]. Slow light happens naturally in photonic crystals, due to the flattening of the photonic bands in periodic structures near the band edge, and it enhances the interaction of the confined light with the material and its environment [9, 10].
A photonic crystal ring resonator (PhCRR) is a ring made out of a PhC waveguide. While it bears a resemblance to the standard ring resonator, the slow-light properties of the underlying PhC waveguide alter its behaviour significantly. The slow light not only further enhances light-matter interaction in the PhCRR (very useful for microrings used as sensors, lasers, or non-linear optical devices), it also increases the quality factor of the resonant modes [11] resulting in spectrally narrower resonances (important for filters and switches). The maximum enhancement of the slow-light effect happens at the photonic band edge, where the waveguide group index can become extremely high. In order to maximize this effect, the PhCRR must be designed in such a way that a resonant mode is located at the band edge (or, at least, as close as possible to it). Previous design efforts [11, 12] were able to find resonant modes with high group index, but not exactly located at the photonic band edge. Furthermore, it is important for technological purposes to be able to design the PhCRR so that this band-edge mode has an appropriate resonant wavelength (for instance, 1550 nm for easy interfacing with telecom fiber devices).
In this article, we present a theoretical analysis of PhCRRs based on a straight waveguide approximation. Based on this analysis, we introduce a prescription for the design of PhCRRs with resonant modes at the band edge at any desired wavelength. We then test the procedure using finite-difference time-domain (FDTD) simulations [13], finding good agreement between the design objectives and the simulated results. Finally, we discuss qualitatively how disorder could affect the structure.
2. Theoretical analysis
2.1. General considerations and geometry under study
The theoretical analysis of ring resonators is greatly simplified by their cylindrical symmetry, allowing for an almost completely analytical solution [14]. In the case of the PhCRR this symmetry is broken by the periodic pattern of holes, which makes an analytical solution for the fields and resonances much harder to find. Nevertheless, in the limit of low curvature (that is, a large enough radius) where bending losses can be neglected, it is possible to approximate the ring by a straight waveguide with periodic boundary conditions. The resonant frequencies of the PhCRR can then be found by calculating the waveguide dispersion and applying the boundary conditions.
The geometry we will consider in this article is that of a two-dimensional (2D) high-refractive index (n) ring of radius R and width w with N holes arranged in a periodic lattice, such that the spacing between the center of the holes is a (see Fig. 1 for an illustration). The shape of the holes is not critical (leading to only small changes in the dispersion relations [11]), so we will use circular holes of radius r for simplicity. While the design prescription outlined in this paper can be generalized to PhCRRs of various different geometric parameters, we will examine the case of r = 0.3a, and w = a. We will consider modes with transverse electric (TE) polarization, where the electric field is oriented parallel to the plane of the ring. Even though we restrict ourselves to the 2D case due to computational constraints, the results should correctly illustrate the qualitative behavior of a slab-based 3D ring. We will use a refractive index of n = 2.83 corresponding to the effective index of a 220-nm-thick slab of silicon on a silica substrate, and consider the background dielectric medium to be air (nair = 1).
Fig. 1 a) Geometry of the PhCRR under study. Black corresponds to a high-refractive index material and white to the background medium. b) Geometry of the straight periodic waveguide used to approximate the ring resonator.
2.2. Dispersion relation and boundary conditions
Since we are considering propagating TE modes in a waveguide, we will assume that the ẑ component of the magnetic field can be written as
Under this assumption we can compute the dispersion using the MIT Photonics Bands software [15], which uses a plane wave expansion method to compute the frequencies and field patterns of the periodically patterned waveguide. Thanks to the periodicity, the band structure is uniquely defined by its behavior in the Brillouin zone (for −π/a ≤ kx ≤ π/a). A representative photonic band structure is presented in Fig. 2(a). The different bands flatten as they approach the band edge at kx = π/a, showing the high dispersion of the photonic crystal waveguide modes there. This can be also appreciated in the divergence of the group velocity dispersion and the group index calculated for a lattice constant such that the band edge is located at 1550 nm (Figs. 2(b) and (c), respectively).Fig. 2 (a) Representative dispersion relation for a periodically patterned waveguide.The greyed area above the dashed light-line corresponds to leaky modes. The inset shows three waveguide unit cells with the computed Hz field for the first band. (b) The group velocity dispersion, for the first band of a periodically patterned waveguide with a = 397.3 nm. (c) The group index for the same waveguide as in (b).
Once the dispersion is known, there are two more conditions that need to be satisfied. The first one is the periodic boundary condition on the magnetic field, imposing that the field at x = 2πR is equal to that at x = 0, converting the straight waveguide into a ring. Expressing this condition using Eq. (1), we find it reduced to
which states that an integer number of wavelengths must fit along the ring perimeter. The length of the waveguide is equal to the ring perimeter, but it also needs to have an integer number of lattice periods (otherwise the ring periodicity would be broken). This leads to a geometrical condition that also has to be satisfied: The resonant modes will happen at the propagation constants that satisfy both Eqs. (2) and (3), and the dispersion relation will then determine their frequencies.2.3. Design of modes at the band edge
Our objective is to design a resonator (that is, determine the parameters R, N, a, r, and w) that will have a mode at the band edge of the photonic crystal waveguide with a resonant wavelength of our choice, which we will call λ0. The first insight needed to achieve this goal is to realize that a mode at the band edge will have a wavenumber equal to π/a. The second, and key, insight is understanding that, thanks to the scale invariance of Maxwell’s equations [16], the absolute values in frequency of the dispersion relation (νd) depend strongly on a, the lattice spacing. The r/a ratio will affect the shape of the dispersion relation, but not as strongly. For this reason, the dispersion relationship is usually expressed in units related to the lattice constant. Putting these two ideas together we can devise a prescription (these steps resemble those used to optimize a photonic crystal nanobeam microcavity [17]):
- Set the refractive index to the effective refractive index of the appropriate slab waveguide.
- Fix the waveguide geometry by choosing the filling fraction r/a and the width w. Wider waveguides will reduce losses, but could introduce higher order radial modes into the ring.
- Compute the dispersion relation to find out the frequency as a function of the wavevector (with typical results shown in Fig. 2). From there, find the νd (in units of c/a) corresponding to the mode at the band edge
- Since the calculated frequency is related to the physical frequency ν0 as νd = ν0a/c and λ0ν0 = c, by choosing a = νdλ0 the wavelength of the band-edge mode will match the design wavelength.
While setting the lattice constant alone is enough to place the band edge at the desired wavelength, the PhCRR is not fully specified yet. Applying the periodic boundary conditions (Eq. (2)) restricts the allowed values of kx to integer multiples of R−1. As a result, the mode at the band edge will have a mode number mBE such that mBE/R = π/a. This, combined with the geometrical condition (Eq. (3)), results in
We can immediately see that there will be a mode at the band edge only if the number of holes is even. Now, the only step left is to either fix the number of holes (or an estimated desired radius) and then use Eq. (3) to find the corresponding radius (or corresponding number of holes and a precise value for the radius). It is important to remember that the radius of curvature must be large enough (R >> λ0/n) for the straight waveguide approximation to be valid.At this point we have specified the complete geometry for a PhCRR possessing a resonant mode at the band edge. However, we can perform a more detailed spectral analysis based on the dispersion relation and the boundary conditions. Figure 3 shows graphically where to find the modes. There will be one at the intersection of each allowed wavevector with the dispersion relation, with a resonant wavelength that can be calculated from the value of the dimensionless frequency at that intersection. From this graph we can also appreciate visually the main spectral features of the PhCRR that separate it from a regular ring resonator: (i) its cutoff frequency, appearing due to the bounded nature of the dispersion curve, and (ii) the non-uniformity of the mode spacing arising from the quadratic dispersion relation. In comparison, the linear nature of a standard ring resonator’s dispersion relation results in equidistant spacing between adjacent modes. By carefully designing the periodic waveguide these features can be tuned. It is of particular importance that the mode wavevectors are symmetrically distributed with respect to the band edge. Since the band structure is also symmetric with respect to the band edge, that leads to pairs of degenerate modes separated by an even number of mode numbers (i.e. mBE ± 1, mBE ± 2), etc. This degeneracy is critical to allow for standing wave modes. At first sight, a mode with a field like that in Eq. (1) does not possess N-fold symmetry unless kx = N/(2R), so one might be tempted to conclude that the mode at the band edge is the only one that could form matching standing waves. A linear combination of any two degenerate modes will result in another one, so we can choose any of the the pairs in Fig. 3 where both modes are p/R wavenumbers away from kBE = mBE/R. This combination will result in a standing wave modulated by a spatial beating with an even number of nodes along the ring, which is also of the right form for a system with N-fold symmetry [18]:
Figure 3 also allows one to understand the case of a PhCRR with an odd number of holes. In this case, there is no mode at the band edge, but the allowed wavevectors are still symmetrically distributed around the band edge in pairs with values kx+ = (N − 1)/2+ p, kx− = (N − 1)/2 − p. The result of the superposition is again an N-fold symmetric field with a spatial beating with an odd number of nodes:
Furthermore, as in the case of a regular ring resonator, the dispersion relation is also mirrored around kx = 0, so the mode with mode number −m corresponds exactly (and is degenerate with) the one with number m. Thanks to this we can write the fields as either travelling waves or standing waves without modifying any of the aforementioned characteristics.
Fig. 3 Dispersion relation (plotted over a shifted Brillouin cell) for the lowest band of the periodically patterned waveguide. The vertical lines represent some of the allowed values of the wavevector kx for a ring with an even (panel a) or odd (panel b) number of holes. The horizontal lines illustrate how to graphically find the resonant frequencies of the modes.
3. Example structure
As an illustration of the process, let’s design a structure with the band edge mode at 1550 nm with a ring radius of approximately 2.5 μm. The properties of the design are confirmed through FDTD simulations, utilizing the MIT Electromagnetic Equation Propagation (MEEP) open-source software package [13] with a simulation grid resolution of ≈ 8.3 nm and sub-pixel averaging. We first excite the fields with broadband Gaussian dipole current sources centered on 1550 nm with a spectral width of 200 nm. In order to avoid placing the excitation source in the node of a resonant mode, we use two separate sources: the first is located in an air hole slightly off-center, while the second is placed in the dielectric material between the first hole and its closest neighbor. Once the strength of the source has decayed to negligible values, the spectral distribution of the resonant modes is analyzed using either a harmonic inversion algorithm or Fourier transform.
To compute the Hz field distributions and Q-factor of each individual mode, a narrowband Gaussian dipole source of 5 nm spectral width is centered on each resonant wavelength. The excited fields are allowed to evolve until |Hz|2 has decayed to 90% of its peak value. The decaying exponential fields are then analyzed to extract the Q-factor for each individual mode, and the field patterns are plotted.
3.1. Design
From the dispersion relation in Fig. 3 we find that the calculated frequency at the band edge is νd = 0.2563. Then, a = 397.3 nm will make the resonant wavelength of the band edge mode λ0 = 1550 nm, and the hole radius will be r = 119.2 nm. Now, from Eq. (3), we find that N = 40 will give us a radius of R = 2.529 μm, which is close enough to the desired one.
Figure 4(a) shows the result of a simulation using a broadband source, centered at 1550 nm. It shows a number of resonant modes with the right behavior: The lowest-wavelength one (at the photonic band edge) is found at 1552.8 nm, and the spacing between the following modes increases as the resonant wavelength increases. For comparison, Fig. 4(a) also shows the magnetic field spectrum of an equivalent standard ring resonator (RR). The standard RR clearly does not exhibit any of the previously mentioned spectral features; namely the non-uniformity of mode spacing and cut-off frequency. In Fig. 4(b) we can see the Hz field distributions corresponding to the labeled modes. It is clear that mBE = 20, as expected, and the spatial beating with an even number of nodes can be seen in the other modes.
Fig. 4 (a) Magnetic field spectrum of the resonant modes in the designed PhCRR (black, see text for parameters) and an equivalent standard RR (red) when excited by a broadband pulse. (b) Hz distributions for each one of the labeled modes in panel a).
3.2. Slow light enhancement and Q-factor
As a verification that the slow-light effect induced by the periodic patterning is present, we can compare the results to those of equivalent simulations on a standard RR. We will compare the Q-factors, as they are enhanced by the slow light effect from the photonic crystal waveguide [11, 19]. From the computed fields we have extracted a Q-factor of 1.26×107 for the band-edge mode of the PhCRR. The closest in wavelength RR-mode is the one found at 1566 nm, which has a Q-factor of 3.68 × 106. The patterning, via the slow light effect, increased the Q-factor by a factor slightly larger than 3. The enhancement factor is modest since the ring diameter is small; larger ring diameters should show higher enhancement factors [11].
As further evidence of the slow-light enhancement of the Q-factor, we can look at the different modes in the PhCRR. When the wavelength increases, each successive mode is found farther away from the band edge. Because of the monotonicity of the group index, we would expect the Q-factor to decrease as we move away from the band edge (as the modes lose the slow-light Q-factor enhancement). The Q-factors extracted from the simulations are plotted as crosses in Fig. 5(a) as a function of the resonant wavelength, where the descending trend with increasing wavelength can be clearly observed.
Fig. 5 (a) Q-factor as a function of the mode resonant wavelength. The crosses and circles correspond to the cases without and with surface roughness, respectively. (b) Q-factor versus artificially introduced geometrical disorder.
3.3. The effect of disorder
The results discussed so far involve an ideal structure with a perfect geometrical design and smooth surfaces (up to discretization-induced roughness), so the Q-factors are much higher than those reported in fabricated devices. In real devices the Q-factor will be affected by the presence of disorder, typically due to imperfections introduced during the fabrication process. Two main sources of disorder are roughness on the device surface and perturbations of the periodicity of the lattice.
While state-of-the-art techniques have been able to reduce the values of fabrication-induced surface roughness to less than 2 nm RMS [20], it cannot be avoided. Furthermore, roughness-induced scattering increases quadratically with the group index [21]. This casts doubt about whether the Q-factor for the modes near the band edge will be improved as much as expected in an ideal case. Due to technical limitations, we were not able to fully include surface roughness in our FDTD simulations. We have been able to approximate an average roughness of 3 nm on the waveguide edges by randomly placing a large number of small scatterers at the boundaries. Since in this way we are not considering roughness in the holes the calculated Q-factors will still be higher than those of a real device. This approximation lets us see the trends in a somewhat more realistic case. The resulting Q-factors are plotted in Fig. 5(a), as circles. All the observed modes show a decrease in their Q-factors but this reduction is particularly dramatic in the case of the mode at the band edge (by a factor of almost 30), becoming less significant the farther the mode is from the band edge. This is consistent with an increase of the surface roughness scattering when the group index is very high. While this seems to dampen the usefulness of the band edge mode, it is still possible to find an optimal trade-off by moving slightly away from the band edge (for instance using the mode at 1560 nm, which shows the maximum Q-factor). It is worth noting that our prescription can be modified in a straightforward manner to be able to put modes other than the one at the band edge at a desired wavelength.
Geometrical disorder in the periodic lattice can also introduce extra losses [19, 22]. If the periodicity is not perfect the initial assumption breaks down and, in principle, this analysis would be invalidated. However, as long as the deviation from perfect periodicity is small, we can consider it as a perturbation on a perfect periodic structure which will not qualitatively change the modes much but that will introduce scattering losses. Figure 5(b) shows how the band-edge mode Q-factor evolves as a progressively higher degree of disorder is artificially introduced into the structure. In this case the disorder is given by a random, normally distributed, variation of the hole radius which would be a dominating disorder source in a real device. These results show that the losses are very sensitive to the presence of disorder, with a dispersion of 1% in the radius of the holes already reducing the Q-factor by two orders of magnitude. In principle, this implies that tight fabrication tolerances will be required to obtain large Q-factors. These tolerances are within those achievable by current processing techniques [23]. Nevertheless, it might be possible to engineer the photonic crystal structure to minimize the effect of fabrication-induced losses [24]. Additionally, the presence of a significant amount of disorder can cause Anderson localization of the field [25]. The evolution of the field distributions from the band-edge mode of PhCRRs, illustrated in Fig. 6, clearly show that when the disorder is large enough the mode transitions from a whispering gallery pattern to a localized one. Using suitable modifications to the hole diameters it should be possible to turn the localization into a controlled nanobeam-style nanocavity [17, 26].
Fig. 6 Hz field distributions for the fundamental mode of a ring with a random variation on the hole radius. The label identifies the average deviation from the nominal hole radius.
4. Conclusion
In summary, we have presented a theoretical analysis of photonic crystal ring resonators using a straight waveguide approximation. From this analysis, we have derived a prescription for the design of PhCRRs with a specified wavelength for the mode at the band edge, and thus we have found resonant modes with maximized slow-light effects. By testing this procedure using FDTD simulations we have found a rich mode structure, much more interesting than that of a comparable ring resonator, with non-uniformly spaced standing-wave modes whose field distributions possess spatial beatings. We have also estimated the losses introduced by the presence of disorder in the structure, suggesting that tolerances better than 1% in the fabrication are necessary to maintain low losses and avoid localization. In the presence of surface roughness the slow light also increases the scattering losses very close to the band edge, reducing the effective enhancement there significantly, but this can be compensated for by operating in modes slightly away from it. Due to the slow-light enhancement of the Q-factor, using the right modes in PhCRRs would result in better narrow-band devices than standard rings. In addition, since the modes near the band edge have an increased group index, light-matter interaction in this structure should be greatly enhanced, making it a good candidate for applications requiring this such as a lasers, non-linear devices, or microring-based sensors.
Acknowledgments
This work was supported by the National Science and Engineering Research Council (NSERC) Discovery grant program and Concordia University.
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