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Abstract
In this study, we shed light on the properties of a photonic ring resonator made up of a closed array of circular dielectric nanorods arranged periodically in a background material. This type of resonator can reach high-quality factors (Q-factor) for specific transverse-magnetic (TM)-like modes, while maintaining a small footprint. We validate this by full 3D finite difference time domain simulations. The properties of the mode most interesting for applications are determined for various parameters of the resonator for the material parameters of GaN. This study provides design guidelines for the realization of this type of photonic nano-resonator and proposes and analyses two practical implementations.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical resonators with high quality-factor, Q, and small modal volume, V, are of great interest for various optical applications, including low-threshold lasers [1], filters [2], nonlinear optical devices [3], and devices to study cavity quantum electrodynamics [4]. The most commonly studied high-Q, small V resonators are based on a defect mode in a photonic crystal (PhC) [5], or whispering gallery modes (WGMs) that occur when light waves circulate in a microstructure due to total internal reflection [6,7]. Generally, PhCs consist of dielectric rods in air (high dielectric pillars in low dielectric medium) or air holes in a dielectric region (usually air holes in a dielectric slab). Rods in air typically favor a photonic bandgap (PBG) for transverse magnetic (TM) modes, with the E field polarized in the length direction of the rods [8]. Due to the bending of bands near the PBG in the photonic bandstructure of PhCs, guided modes at these locations have a small group velocity. This slow-light effect can be used in combination with WGMs in a waveguide ring resonator to significantly increase the Q-factor in PhC ring resonators [9–11]. While PhC ring resonators are commonly formed by air holes in a dielectric waveguide on a substrate containing TE modes [12,13] or in a microdisk [14], the opposite type (dielectric rods in air) is also theoretically viable [15,16], and nanolasers based on rods in a linear configuration have been demonstrated [17]. This type supports TM-like polarized modes, which could be beneficial for specific applications like PhC quantum cascade lasers, as the intersubband transitions in quantum wells are TM-polarized [18,19] or, for example, be used to selectively couple transitions of rare-earth dopants having a specific dipole moment with respect to the crystal structure of the host material [20]. These structures can be applied to a wide range of materials, and could offer opportunities for materials for which current fabrication potential is limited. For example, epitaxial grown GaN containing optical active materials, like quantum wells or rare-earth dopants, require a relatively thick buffer layer in order to improve the degraded crystal quality resulting from the lattice mismatch with commonly used substrates. This precludes the direct formation of optical active GaN based materials on a low-refractive index material, and thus the use of conventional waveguide-based PhC ring-resonators on a suitable substrate.
In 2007 Nojima proposed this type of (quasi-)1D system, consisting of nanostructures arranged in a closed loop, and named it “photonic-atoll resonator” (PA) [21]. Evaluation of a closed array of cylindrical rods in 2D by the analytic multiple-scattering theory generally described in Ref. [22,23], demonstrated extremely high Q-values (∼1015) for a specific mode for a PA consisting of 50 GaAs rods. Later studies also revealed that the modification of the PA structure into elliptical shapes could lift the degeneracy of the optical eigen-modes [24]. Although these works did not consider the vertical Q that comes into play in a real system, it was concluded that PAs could potentially have better confinement of light propagating in 2D space as compared to similar sized 2D PhCs, which typically require a large periodic medium. Despite these promising characteristics there have neither been follow-up studies into 3 dimensions, nor any experimental realizations of this type of structures.
In this study we analyze this type of structures by a finite difference time domain (FDTD) method, and determine the influence of various parameters on the optical properties. In the first part we show how the FDTD method qualitatively reproduces the results of the analytic multiple scattering theory in 2 dimensions. In the second part we extend the simulations to 3 dimensions and include the influence of rod height. In the final part we propose and analyze two structures which could be realized in practice. While in this work we focus on the material properties of specifically GaN, the concepts developed here can be extended to any transparent dielectric material.
2. Simulations
2.1 Design
All FDTD simulations were performed using Lumerical inc. commercial software. Figure 1 shows a schematic structure of the resonator. It consists of N periodically arranged rods in a circle in the background material air. The fill factor is defined by r/a, with r the rod radius and a the array period. The full structure is defined by N, r, r/a and rod height h. For all simulations where the filling factor is not explicitly mentioned it is fixed as r/a = 0.45, following the results of Ref. [21], and which is also the optimum value according to our 3D simulations (See Fig. S1 in Supplement 1) The system is excited by an electric dipole oriented parallel to the rod-length, placed inside one of the rods to excite TM-like-modes, while the electric field is recorded by a monitor placed in a different rod.
Fig. 1. a) 3D schematic of a ring resonator with N = 16 rods. b) Top-view of the PA with indicated the PA radius d, and angle Θ. c) Side-view of 2 rods with indicated the parameters defining the structure.
2.2 2D
2.2.1 Electric field distribution
2D FDTD simulations are performed in order to evaluate if these reproduce the results of Ref. [21]. Furthermore, these simulations can be performed in a reasonable simulation time to get qualitative information about, for example, the influence of the number of rods. We use the refractive index of GaN for the dielectric material, where the imaginary part is ignored as it is close to 0 [25], and optimize the structure in such a way that the mode with the electric field distribution having antinodes inside all rods is around 620 nm, coinciding with the peak of Eu3+ emission in GaN [26]. Figure 2(a) depicts the spectrum of the electric field intensity for a rod radius r = 71 nm, and N = 30. A number of narrow peaks can be observed in the wavelength range 450 to 700 nm, while there are no modes between 520 and 610 nm. The spatial electric field distributions of the 4 largest peaks are shown in Fig. 2(b). The complex behavior of different modes having varying amount of nodes around the ring, resulting from Bragg reflections, has also been observed in Ref. [21], and it is possible to distinguish similar modes here. The modes can be allocated to the first two branches in the photonic band structure, with those on the short-wavelength side of the PBG originating from the “air” branch, having high field intensities between the rods, and those at the long-wavelength side originating from the “dielectric” bands (see Fig. S2 and S3 in Supplement 1). The most interesting mode for applications where light-matter interactions are involved is around 617 nm, located at the edge of the dielectric branch; it has high field intensities located inside all of the rods, allowing for optimum light-matter coupling, and has a high Q-factor. Therefore, we will focus on this mode for the remaining part of this study.
Fig. 2. Left: Simulated spectrum of the electric field intensity for a resonator consisting of N = 30 rods with a radius r = 71 nm. The 3 colored regions indicate the location of the “air” and “dielectric” branches in the photonic band structure, and the location of the PBG. Right: The electric field distribution for 4 largest peaks, the black dotted circles indicate the location of the rods.
2.2.2 Effect of number of rods and background material
In order to determine the effect of the number of rods and background material, the Q-factor was determined by fitting the slope of the envelope of the decaying electric field. The obtained values for structures consisting of different numbers of rods N are shown by the black circles in Fig. 3(a). The Q-factor increases exponentially as a function of the number or rods. Due to limitations in the simulation time, values for N > 40 cannot be accurately determined and are extrapolated by the dashed line. It can be seen that for N = 50 it should reach a value of ∼109, which is about 4 orders lower than that found for this mode in Ref. [21]. This difference can be assigned to be due to the smaller refractive index material used here, nGaN = 2.38, while that in Ref. [21] is nGaAs = 3.63, leading to larger bending losses.
Fig. 3. The dependence of the Q-factor on the number of rods, N, for same-sized rods with r = 71 nm (black circles) and with a radius distribution, σ = 0.03 (red circles). The blue squares depict the peak wavelength with values indicated on the right-axis. The black dashed line serves as a guide to the eye. a) rods in air (n = 1), b) rods in PDMS (n = 1.42).
Although the theoretical Q-factor can reach these very high values, practical values are limited by the quality of the fabrication process. For example, in a top-down fabrication procedure, electron beam lithography can produce accurate rod-to-rod spacing, but the precise rod radius and etch roughness are more difficult to control. In order to evaluate the influence of variation on attainable Q-factors, we have determined the values for rods with a normally distributed dispersion of the rod radius σ = 0.03 (Fig. 3). Generally, the obtained Q-factor can be expressed by the following equation,
where Qi the intrinsic Q-factor of the structure (σ = 0) and Qd represents the radiation losses due to the disorder. Q is thus dominated by the smallest factor, which can be observed in the behavior in Fig. 3. For small N the value of Q is dominated by Qi and follows a similar trend as that of σ = 0, while for larger N, Qd starts to dominate and limits Q to approximately 105 for N > 40.Since proper optical confinement is complicated to realize in the vertical direction when the rods are standing on a substrate (see 3D simulations), we foresee one implementation by embedding them in a low-refractive index polymer like PDMS, that is commonly used for lift-off processes [27,28]. Figure 3(b) shows the results of simulations for the same structures embedded in PDMS with a refractive index nPDMS = 1.42. Larger bending losses occur due to the smaller refractive index contrast with the dielectric material, and Q-factors are considerably smaller. However, by increasing the number of rods to 80, and introducing the same size-dispersion as before, a value of the Q-factor of ∼4 × 104 is obtained.
2.3 3D
In the 2D simulations the rod height is implicitly assumed to be infinite. We extend the simulations to 3 dimensions using the same layout and adding a height to the rods. Given that the size of the structures becomes critical in the 3D simulations due to limitations in processing power and memory size, we analyzed relatively small structures composed of N = 20 GaN rods in air to determine the general influence of the rod height.
2.3.1 Rod height-dependence
As the resonance wavelength is dependent on the amount of dielectric material, the resonance wavelength shifts to shorter wavelengths for smaller rod heights h. The rod height dependence was determined by varying only r in order to get a resonance wavelength around 620 nm. Figure 4 shows the dependence of the Q-factor on the rod height for 300 < h < 1000 nm. The value increases for larger h and saturates at a value slightly larger than obtained from the 2D simulation. We note that the mode volume V increases approximately linearly with the rod height (NB V ≈ 0.2N(λ/n)3 for a rod height of 1000 nm). The height dependence of the figure of merit Q/V, which determines the strength of cavity interactions, is depicted in the inset of Fig. 4 and has a maximum for a rod height of ∼500 nm. We note that for rod heights > 600 nm also higher order vertical modes appear, although with considerably smaller Q-factor. These could offer a broader spectral range than the fundamental modes for specific applications requiring a large bandwidth.
Fig. 4. Q-factor as function of rod height for a structure consisting of N = 20 rods with a resonance wavelength of 620 nm. The 2D value is indicated in red at the top. The inset shows the normalized values of Q/V as a function of rod height.
2.4 Practical implementations
Figure 5 depicts the components of the electric field of the resonant mode in radial coordinates for a structure with r/a = 0.35 and h = 500 nm. It can be seen that inside the rods the mode is purely oriented in the z-direction. There is also a strong field between the rods in the azimuthal direction, which are a result of the bending of the quasi-1D structure in the circular structure. The intensity of this inter-rod field increases for shorter rods and larger filler fraction r/a. The resonant mode profile precludes the direct formation of these structures on a substrate with small refractive index contrast, as optical energy can easily dissipate into the substrate via the fields between the rods. One of the solutions to this limitation is embedding into another low-refractive index material, like PDMS, which was analyzed above in the 2D case. Although this will lower the refractive index contrast with the surrounding material, and thus the Q-factor, this method is relatively easy to apply.
Fig. 5. Mode profiles for different components in radial coordinates, all intensities are on the same scale. The black lines indicate the rods.
A second solution is using a barrier layer with lower refractive index underneath the ring resonator to confine the light inside the rods. However, it is necessary to raise the structure on the barrier material to isolate the inter-rod fields from the substrate. Figure 6 shows the suggested structure, in which the rods underneath the resonator have a sufficient height (650 nm here, about 1 wavelength) to prevent energy from dissipating into the substrate. For the barrier layer in this simulation we use a refractive index nbar = 2.2, a typical value for lattice-matched AlInN [29]. The electric field profiles in Fig. 6 show that the mode extends slightly into the barrier layer, and also that the field between the rods extends relatively far down, indicating the importance of the height of the barrier rods. For structures with N = 20 the values of the Q-factor for these raised structures are similar to the ideal structures in air where the resonance wavelength is slightly red-shifted due to the vertical extension of the mode into the barrier
Fig. 6. Schematic of barrier structure with radial and azimuthal cross-sections of the electric field of the resonance mode for a structure with r/a = 0.45 and h = 700 nm. The dashed lines indicate the boundaries of the structures.
There are some specific benefits for such type of rod-based resonators as compared to conventional ones. Since the high-intensity regions of the mode profiles are well confined inside the rods, this type of nanocavity offers clear advantages for experimental characterization. In the commonly used free-space excitation and probing layout, there is usually a considerable background contribution to the detected signal from emitters that are randomly distributed in the dielectric material and not in the same physical location as the resonant modes. This occurs in microdisks and slab-based PhC cavities, but is less of an issue for rod-based cavities due to the lack of surrounding dielectric material. We note that direct coupling of light from a waveguide injected by a fiber into a rod-based 1D waveguide has also been demonstrated in the GaAs platform [30]. However, while such a technique is possible on a substrate, it will be challenging to incorporate and likely be detrimental to device performance.
Although this work considers GaN, the general design can be applied to any material, and larger refractive index materials, like GaAs, will benefit from smaller bending losses, especially when the structure is embedded in a polymer. Rods with the dimensions indicated in the last paragraph could potentially be produced, as high quality GaN nanorods with vertical sidewalls and rods with a large aspect-ratio up to ∼12 have been shown before [31,32]. Especially, the narrow separation between the rods is difficult to fabricate. Therefore, smaller values of r/a are more realistic for most material systems, but this will come at the cost of a reduction of the Q-factor resulting from a decrease in average amount of dielectric material, and thus refractive index, and associated bending losses. For a more realistically realizable structure with h = 500 nm, r/a = 0.35 and N = 48 on a barrier, we obtained a Q-factor of 25.000, demonstrating that high Q-factors could potentially be achieved on the GaN platform.
Finally, we note again the importance of fabrication variations on the obtainable Q-factors. While we analyzed only variations in rod diameter, other features like roughness and sidewall taper, will contribute to this limitation. Further experimental exploration is required to determine if the structures suggested in this manuscript can be successfully created.
3. Conclusion
We have employed FDTD simulations on photonic ring cavities made up of circular dielectric nanorods arranged periodically in a background material. These structures can support TM-like modes with high Q-factors, while maintaining a small footprint. For the resonance mode that has high field intensities inside all rods, which is most interesting for applications where light-matter interactions are involved, the theoretical Q-factor in 2D increased exponentially as function of the number of rods. The rod height dependence in 3D indicated that the optimum value for Q/V was found for a rod height slightly shorter than the resonance wavelength in air. We proposed and analyzed two practical implementations for these structures: 1) embedding it in a low-refractive index polymer and lift-off from the substrate; 2) using a lower refractive index material underneath the structure with enough height to isolate it from the substrate. Our results demonstrate the general behavior of this type structure and its potential for the realization of high-quality resonators.
Funding
Japan Society for the Promotion of Science (18H05212).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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