1.

Introduction

An optical whispering gallery mode (WGM) microdisk cavity is a structure that confines light to small volumes via resonant recirculation.¹ Optical microcavities offer a high confinement of the electromagnetic field in a small volume, with dimensions comparable to the wavelength of light, which provides a unique system for the enhancement of light–matter interactions on the nanoscale. Optical devices based on WGM microcavities, such as lasers, LEDs, and sensors, are already in use for a wide variety of applications.¹ Other photonic structures, such as photonic crystal cavities, can also enhance the light–matter interaction by orders of magnitude, enabling the device’s optical properties to be controlled externally, both optically² and electrically.³

Optical metamaterials present a unique opportunity for light–mater interaction enhancement and attaining special electromagnetic properties that are not found in nature.⁴ An intriguing class of metamaterials, which exhibit an optical topological transition (OTT) from a closed ellipsoid to opened hyperboloid dispersion, has been recently studied.^5–7 The iso-frequency surface (IFS) is enclosed by

Fig. 1

A visualization of the changes in the isofrequency contour for the different types of dispersion. The values for the real part of the optical permittivity are given for a specific material (see Sec. 2).

In isotropic materials, e.g., silicon or ${\mathrm{SiO}}_2$, reducing the size of the optical cavity (from microcavity to nanocavity) is limited to the fundamental diffraction limit.⁸ In HMMs, large wavevectors can be achieved along the unbounded IFS. Therefore, optical WGM cavities, created from HMMs, allow the confinement of the electromagnetic field to an extremely small volume,^9,10 which opens new opportunities for creating infinitely small three-dimensional (3-D) nanocavities with hyperbolic dispersion and dimensions significantly smaller than the wavelength of light for applications in ultrafast LEDs.

In a microcavity, light is emitted into a cavity mode and acquires an enhanced efficiency or emission rate. The Purcell factor ($F$)¹¹ quantifies the enhancement of an atomic emission rate in a cavity versus its rate in free space

Such enhancement is desired for both future quantum information processing and visible light communications. For example, a rate-equation model revealed that when a small (subwavelength) mode volume is combined with a $Q$ (order of 100), the spontaneous emission rate for an otherwise conventional LED can be enhanced, resulting in a 200-GHz modulation bandwidth.¹²

Here, we perform finite-difference element numerical simulations for the electromagnetic field confinement and study optical properties of the WGMs in such deep-subwavelength $\mathrm{Al}/{\mathrm{SiO}}_2$ disk nanocavities. This study aims to investigate the fundamental limits for the dimensions of this class of nanocavities, optical mode confinement, and the properties of the nanocavities’ resonance frequencies. The choice of $\mathrm{Al}/{\mathrm{SiO}}_2$ HMMs nanolayered material and silicon substrate material will allow the future incorporation of such nanocavities to the well-established silicon photonics optical communications networks.¹³

2.

Nanolayered Metamaterial Permittivity Models

When designing 3-D nanocavities, the scale of each layer and dispersion of HMM is vital to increasing the bandwidth of such devices, which calls for a detailed model for the dispersion properties of multilayered HMMs. Using aluminum in metal-dielectric structures offers several advantages over these materials. In particular, the plasma frequency of the aluminum is higher than that of gold or silver, which allows for a significant surface plasmon response to occur in the ultraviolet (UV) portion of the spectrum. As a result, aluminum exhibits metallic characteristics over a broader spectral range than gold and silver. In addition, ${\mathrm{SiO}}_2$ is used as the dielectric for this HMM because it could be easily integrated into current CMOS technology and has near-zero losses in the UV region.

In our recent work,¹⁴ we used an effective medium approximation (EMA) with nonlocal corrections to describe $\mathrm{Al}/{\mathrm{SiO}}_2$ nanolayered structures with hyperbolic dispersion. Based on our previously published study (Ref. 14), we found that when the Al fill fraction is reduced to 1%, the number of layers does not affect the dielectric function of HMM (for $N>25$). In addition, we found that the permittivity of the HMM does not change (Al fraction $\sim 0.3\%$) when the thickness of an individual layer is varied from 10 to 2 nm. For the simulations done in this paper, the disk nanocavity is assumed to be made of the $\mathrm{Al}/{\mathrm{SiO}}_2$ nanolayered HMM with 250 nm thickness. The number of the layers ($N=25$) and the thickness of the individual layer ($d=10\mathrm{nm}$) were constant. The $\mathrm{Al}/{\mathrm{SiO}}_2$ nanolayered structure is assumed to be a uniaxial material with different values for the dielectric permittivity in the parallel direction, ${ϵ}_{\parallel }$, and the perpendicular direction, ${ϵ}_{\perp }$. The permittivity is modeled using EMA.^15,16 According to EMA, in-plane permittivity, ${ϵ}_{\parallel }$, and out-of-plane permittivity, ${ϵ}_{\perp }$, can be described by

Manipulating the Al fill fraction ($f_{\mathrm{AL}}$) controls the spectral position of the OTT. The optical permittivity predicted by EMA (Fig. 2) varies with wavelength and fill fraction, which allows for the designing of the material to fit the optical permittivity parameters in desired resonance range. The out-of-plane permittivity, ${ϵ}_{\perp }$, stays relatively constant while the in-plane permittivity, ${ϵ}_{\parallel }$, varies with small aluminum concentration. The optical permittivity values used for nanocavity studies are chosen near the OTT for a given frequency in an attempt to reduce the effects of the absorption.

Fig. 2

(a) Nanolayered HMM, (b) real and (c) imaginary parts of the out-of-plane optical permittivity (solid line), ${ϵ}_{\perp }$, and the in-plane optical permittivity (dashed line), ${ϵ}_{\parallel }$, for $\mathrm{Al}/{\mathrm{SiO}}_2$ metamaterials with aluminum fill fractions $f_{\mathrm{AL}}$ of 0.3% to 1.5%.

3.

Computational Methods

COMSOL’s multiphysics module was used to create a finite element method (FEM) study that produces a resonance frequency and optical mode intensity distribution for the disk nanocavities with different radii. FEM solves a set of weak partial differential Maxwell equations to find the relevant modes.¹⁹ The optical mode volume and quality factor were deduced from the intensity distribution. The FEM model is axisymmetric, which is computationally efficient as it allows for symmetric modes to be extrapolated. Over a two-dimensional space, from the edge to the center of the disk, the electric field, electric displacement, and magnetic field were calculated as though they are mirrored around the disk in an azimuthal order ($m$) number of times (Fig. 3). Perfectly matched layers, which are not depicted here, were used to simulate infinite space beyond the computation space.

Fig. 3

(a) $m=7$, radial electric intensity. (b) Diagram of computation space. The axisymmetric model displays one symmetric node of the resonance.

A free triangular mesh was used in the simulation. To ensure triangular mesh properties did not affect the solution, both a Delaunay mesh and a free quadrature mesh were implemented. Changing the mesh type had a minimal effect on the overall simulation (difference $<0.1\%$). The smallest mesh settings offered by the COMSOL program are “extra fine” and “extremely fine.” The respective maximum mesh size elements are 32.6 and 16.3 nm. The “extremely fine” mesh was used on disks with radii of 200 nm and lower. Finally, this proved to be adequate as the solutions converged with a tolerance of $1\times {10}^{-6}$ and retained resolution.

While this model allows for exceptional computation times, the axisymmetric nature of this model does not always provide a clear picture of the modes. When reducing mode order to $m=1$, higher order modes are hard to distinguish from the lowest order mode. FEM models also create spurious mode solutions that solve the required weak form partial differential equations, but have no physical counterpart. These factors must be taken into account for this numerical study.

4.

Optical Mode Confinement: Results and Discussion

First, the ${\mathrm{SiO}}_2$ microdisks with various radii (microdisk’s height is 250 nm) were studied using the FEM model (Fig. 4). The free space wavelength ${\lambda }_0$ of the cavity in Fig. 4 (radius $R=1\mu \mathrm{m}$) is ${\lambda }_0\sim 2828\mathrm{nm}$. As the radius decreases, the optical power is leaked around the circumference of the disk. As expected, by $R=0.4\mu \mathrm{m}$, the majority of the optical intensity is outside the cavity and the $Q$ factor is significantly reduced indicating that the mode is no longer confined within the cavity. Based on the wavelength of the lowest order mode, the diffraction limit for this microdisk optical cavity is around $R_d\sim 0.9\mu \mathrm{m}$. It indicates (Fig. 4) that no optical mode is likely in microdisks with dimensions smaller than $R_d$. This result is consistent with previously published experimental and numerical results for silicon microcavities.⁸

Fig. 4

${|E_{\mathrm{rad}}|}^2$ for the lowest order optical mode for the ${\mathrm{SiO}}_2$ microdisks with different radii $R$. The resonance frequencies are (a) 105.98 THz ($R=1\mu \mathrm{m}$), (b) 107.14 THz ($R=0.8\mu \mathrm{m}$), (c) 109.72 THz ($R=0.4\mu \mathrm{m}$), and (d) 111.06 THz ($R=0.2\mu \mathrm{m}$).

The results of the numerical FEM study for disk nanocavities with different radii $R$ (disk nanocavity’s height is 250 nm) show that with hyperbolic dispersion present in the nanolayered $\mathrm{Al}/{\mathrm{SiO}}_2$ material, the optical mode is highly confined in the disk considerably beyond the diffraction limit (Fig. 5). The EMA was used (see Sec. 2) to obtain the dielectric permittivity in the parallel direction, ${ϵ}_{\parallel }$, and in the perpendicular direction, ${ϵ}_{\perp }$ for nanodisks with a different aluminum fill fraction $f_{\mathrm{Al}}$. The wavelength of the optical lower order mode for a disk nanocavity with radius $R=1\mu \mathrm{m}$ and various aluminum fill fractions was around $\sim 2810\mathrm{nm}$ [Fig. 6(c)]. Figure 5 shows the intensity of the lowest order mode (Al fill fraction is varied between 0.1% and 3.35%) as the radius of the disk is decreased. For all radii, the highest confinement for the optical mode is achieved at the positive side of the OTT with a fill fraction of aluminum between 0.25% and 0.225%. Figure 5 shows that the optical mode is well confined in a disk nanocavity with a radius of up to an order of magnitude smaller than the free-space resonant wavelength ${\lambda }_0$. Our recent experimental study²⁰ shows the feasibility of fabrication of the $\mathrm{Al}/{\mathrm{SiO}}_2$ nanolayered films with a low aluminum fill fraction using the RF sputtering technique.

Fig. 5

${|E_{\mathrm{rad}}|}^2$ for the lowest order optical mode for disk nanocavities with different radii $R$ and varying aluminum fill fractions $f_{\mathrm{Al}}$. The optical constants were defined in Sec. 2.

Fig. 6

(a) Quality factor $Q$ and (b) Purcell factor $F$ numerically calculated for the lowest optical mode of the disk nanocavities with different radii (between 60 nm and $1.08\mu \mathrm{m}$) and aluminum fill fractions at wavelength $\lambda \approx 2800\mathrm{nm}$. The diffraction limit for ${\mathrm{SiO}}_2$ nanocavity is indicated by a vertical dotted line ($R_d\sim 0.9\mu \mathrm{m}$). (c) Real part of the in-plane permittivity (${ϵ}_{\parallel }$) and spectral position of the optical resonant wavelength for nanocavity with radius $R=1\mu \mathrm{m}$ and various aluminum fill fractions.

As expected, the $Q$-factor diminishes rapidly with decreasing the microdisk’s dimensions [Fig. 6(a)]. In addition, as the fill fraction is increased and the dielectric permittivity in the parallel direction becomes $|{ϵ}_{\parallel }|>0$, the $Q$-factor is reduced due to presence of higher optical losses for HMMs [Fig. 6(a)]. However, the Purcell factor exhibits a different type of behavior [Fig. 6(b)]. For the lowest order mode ($m=1$), there is an increase in the Purcell factor for the disk nanocavity. It can be seen that the Purcell factor $F$ is an order of magnitude higher for the disk HMM nanocavity compared to the isotropic case (${\mathrm{SiO}}_2$ microdisk) (Fig. 4). Further reducing the nanocavity size ($R<60\mathrm{nm}$) leads to the drop in the $F$ factor due to the reducing of the $Q$-factor ($Q<10$) and, as a result, loss of the optical mode.

The disk nanocavity numerical study presented in this paper provides a valuable insight into the enhancement of the optical mode confinement due to hyperbolic dispersion. The strong optical mode confinement results in low radiation loss and, as a result, a higher $Q$ factor for nanocavities from anisotropic nanolayered metamaterials compared to the nanocavities with isotropic dispersion. Circular disk geometry of a nanocavity results in a lower scattering loss for high-$k$ wavevectors compared to the rectangular geometry studied in Ref. 21. The upper bound on the enhancement of the photonic density of states is imposed by the characteristic metamaterial patterning scale.²² Using nanolayered metamaterial with the thickness of the individual layer approaching the atomic layer²³ could provide greater flexibility in designing disk nanocavities with highly confined optical modes. Possible incorporation of the laser gain material²⁴ will allow using this type of nanocavity for laser devices and ultrafast LEDs.

5.

Conclusions

The numerical study of the $\mathrm{Al}/{\mathrm{SiO}}_2$ nanocavities with hyperbolic dispersion provides an opportunity to study the unique properties of nanocavities with increased photon density of states and enhanced light–matter interactions. Results of the numerical simulations show that nanocavities with radius of up to an order of magnitude smaller than ${\lambda }_0$ and Purcell factor reaching $2\times {10}^3$ are possible to create. Future incorporation of the laser gain material will allow using this type of nanocavity for laser devices and ultrafast LEDs for a variety of applications, from optical communications to biological nanoparticle sensing.