High-Q supercavity modes in subwavelength dielectric resonators

TL;DR

This paper introduces a new mechanism to achieve high-Q supercavity modes in subwavelength dielectric resonators by utilizing bound states in the continuum, enabling advanced nanophotonic applications.

Contribution

It uncovers a novel way to realize giant Q-factors in nanoscale dielectric resonators through mode coupling and Fano resonances, expanding nanophotonics capabilities.

Findings

High-Q modes achieved in dielectric nanorods

Strong mode coupling and Fano resonances observed

Bound states in the continuum enable supercavity modes

Abstract

Recent progress in nanoscale optical physics is associated with the development of a new branch of nanophotonics exploring strong Mie resonances in dielectric nanoparticles with high refractive index. The high-index resonant dielectric nanostructures form building blocks for novel photonic metadevices with low losses and advanced functionalities. However, unlike extensively studied cavities in photonic crystals, such dielectric resonators demonstrate low quality factors (Q-factors). Here, we uncover a novel mechanism for achieving giant Q-factors of subwavelength nanoscale resonators by realizing the regime of bound states in the continuum. We reveal strong mode coupling and Fano resonances in high-index dielectric finite-length nanorods resulting in high-Q factors at the nanoscale. Thus, high-index dielectric resonators represent the simplest example of nanophotonic supercavities, expanding substantially the range of applications of all-dielectric resonant nanophotonics and meta-optics.

Full text

Trapping of light in localized modes is extremely important for various applications in optics and photonics including lasing Kodigala et al. (2017), sensing Gather and Yun (2011); Foreman et al. (2015), harmonic generation Dominguez-Juarez et al. (2011); Shorokhov et al. (2016), Raman scattering Li et al. (2014), and photovoltaics Spinelli et al. (2012); Brongersma et al. (2014). For many optical devices, it becomes critical to localize electromagnetic fields in small subwavelength volumes. Plasmonic structures based on metals allow subwavelength localization of light by means of surface plasmon polaritons Hess et al. (2012). However, metals impose inevitable losses and heating, which limit the device performance and efficiency. In contrast, dielectric nanoparticles with high refractive index offer a novel way for the subwavelength localization of light by employing the Mie resonances being limited only by the radiation damping Kuznetsov et al. (2016). Unlike metallic nanoscale structures, dielectric nanoparticles support both electric and magnetic Mie modes that expand substantially the applications of meta-optics opn_2017 . Also, dielectric materials with high refractive index are available in a broad spectral range. At the same time, the standard Mie theory predicts relatively low values of the quality factor ($Q\approx 10$) for nanoparticles made of conventional optical materials such as Si, Ge, and AlGaAs, both in the visible and near-infrared spectral ranges.

However, for many applications of all-dielectric nanophotonics it is very desirable to achieve higher values of the $Q$ factor. One way to enhance the $Q$ factor is to increase the size of the resonator, for example by confining waves by cavities and defects in photonic crystals Asano et al. (2017) or by exploiting modes with high angular momentum known as whispering gallery modes (WGM) Righini et al. (2011). Another way is to arrange several resonators in space and excite collective modes Zywietz et al. (2015); Shalaev et al. (2015). An alternative approach for enhancing the $Q$ factors is to use the so-called anapole mode with the spectrally overlapped electric and toroidal dipole modes Basharin et al. (2015); Miroshnichenko et al. (2015). As a result, the $Q$ factor of the anapole mode realized in a dielectric resonator may exceed 30 Grinblat et al. (2017). Here we suggest a novel approach based on bound states in the continuum.

The physics of bound states in the continuum (BIC) with potentially infinite lifetime attracted a lot of attentions in photonics Marinica et al. (2008); Hsu et al. (2013). Albeit true BIC can exist in structures that are infinitely extended at least at one of spatial dimensions Hsu et al. (2016), finite-size samples can support their analogue in the form of supercavity modes Rybin and Kivshar (2017), with the $Q$ factor growing rapidly before reaching a maximum value owing to the finite-size effects. In accord with the Fridrich-Wintgen theory of BIC Friedrich and Wintgen (1985), the radiating tails of leaky modes can cancel out each other via destructive interference resulting in the rapid growth of $Q$ factor. The earlier studies of high-$Q$ microcavities in the regime of avoided resonance crossing Wiersig (2006) can also be linked to the BIC concept Friedrich and Wintgen (1985). In that way, a pair of strongly coupled modes of a microcavity may interfere destructively outside the resonator Wiersig (2006) realizing the BIC conditions. Recently, it was shown Gomis-Bresco et al. (2017) that even a simple waveguiding system can support BIC by employing an anisotropic material and vectorial nature of electromagnetic fields. It is crucially important that the modes with mixed polarizations may allow the substantial reduction of resonator’s dimensions, and they can be engineering to realize the supercavity regime Kivshar and Rybin (2017).

In this Letter, we reveal that subwavelength dielectric resonators with high refractive index can support the supercavity modes with unexpectedly high $Q$ factors. For an illustrative case of a single dielectric cylindrical resonator, this occurs when two eigenmodes with different polarizations, associated with the Mie resonances and Fabry-Perot resonances of a finite-length nanorod, strongly interact near the avoided crossing regime (Fig. 1). Because these modes are approximately orthogonal inside the resonator, they interfere predominantly outside, realising the so-called bound states in the continuum (BIC) and supercavity regime Rybin and Kivshar (2017) when the subwavelength all-dielectric resonators demonstrate extremely high values of the $Q$ factor.

We calculate the scattering cross-section of the dielectric cylindrical resonator as a function of its aspect ratio $r/l$. For generality, we use the normalized size parameter $x=kr$ being a product of the wavenumber $k$ and resonator radius $r$. To reveal the strong mode coupling effects, we start by setting a high value of the resonator permittivity $\varepsilon=80$. The color map shown in Fig. 2(a) exhibits two families of resonances with the linear dependencies on the aspect ratio $r/l$. This includes the slow-varying TE-polarized (Mie-like) modes mapped to the Mie resonances of an infinitely long cylinder [see Fig. 2(b)], and rapidly-varying (Fabry-Perot-like) modes with the TM polarization. Here, we employ the standard mode classification Zhang and Li (2008), so that both TE and TM modes are defined by the predominant polarization; the first index $m$ corresponds to the azimuthal number of the mode, whereas other two indices $k$ and $n$ enumerate the Mie and Fabry-Perot resonances, respectively. Due to different spectral shifts of the Mie and Fabry-Perot modes when the aspect ratio varies, the modes can intersect at certain points [as shown by red and green circles in Fig. 2(a)]. The modes with the same azimuthal index $m$ interact stronger, and they undergo coupling each other with the avoid crossing scenario at special values of the aspect ratio parameter $r/l$ with nearly complete disappearance of spectral lines (see the SM point in Fig. 1). We notice that the similar spectral feature was observed in the reflectance spectra of a photonic-crystal membrane supporting BIC Hsu et al. (2013).

Figure 3 shows the spectra corresponding to the avoided crossing that occurs for TE020 (Mie-like) and TM021 (Fabry-Perot-like) modes of the dielectric nanoresonator. In the scattering spectra [see Fig. 3(a)], each mode is manifested by an asymmetrical Fano-line shape. We study the spectra in detail through the line characteristics such as the central position, damping (proportional to the inverse value of the $Q$ factor), and Fano asymmetry parameter calculated with the help of the classical Fano formula Rybin et al. (2009). Outside the avoided crossing regime, the frequency shifts of both Mie and Fabry-Perot modes are described by linear relations [see Figs. 2(a) and 3(b). For shorter resonators, when $r/l>0.72$, the spectrum exhibits a pair of asymmetric resonances. The lower-frequency feature corresponds to the excitation of the TE020 (Mie-like) mode at $x=0.66$, and the higher-frequency mode is the TM021 (Fabry-Perot type) mode at $x=0.73$. For longer resonators, when $r/l<0.65$, the sequence of the modes is reversed: the Fabry-Perot mode appears at $x=0.60$, while the Mie-type mode appears at almost the same position $x=0.64$. For $0.65<r/l<0.72$, the modes undergo a strong coupling accompanied by a typical avoided crossing.

Since the modes correspond to asymmetric Fano-like lineshapes, the damping parameter cannot be determined directly from the linewidth, but it can be calculated together with the Fano parameter $q$ by fitting to the Fano formula. Dependence of these two parameters on the resonator aspect ration $r/l$ is shown in Figs. 3(c,d). For small values of the aspect ratio ($r/l<0.62$), both modes are characterized by positive $q$ and comparable damping parameters. When the aspect ratio decreases, the characteristics of the lower-frequency mode do not vary substantially. In contrast, for the high-frequency mode both the parameters change dramatically demonstrating the resonant features in a narrow range of the aspect ratio values, $0.69<r/l<0.71$. At $r/l=0.703$, the asymmetry parameter tends to infinity, and then it becomes negative. Insets in Fig. 3(c) show a change of the scattering spectra in a small range of the aspect ratio values, $0.698<r/l<0.703$. At the same time, the $Q$ factor exhibits a typical dependence of the BIC state Hsu et al. (2016); Rybin and Kivshar (2017), with the maximum value $Q=65000$ limited by finite-size effects.

We notice that the extreme values of both damping and Fano parameters are realized at $r/l=0.703$. In the weak-coupling regime, the Fano asymmetry parameter is known to correspond to the damping rate Rybin et al. (2016), and the case $|q|\to\infty$ corresponds to uncoupling of the resonance from the continuum Riffe (2011). Hence, we expect similar features also in the case of the strong coupling regime, and that suggests that the Fano parameter may be employed for the analysis of specific properties of the BIC states. Indeed, the relation between the Fano resonance and bound states in the continuum was discussed earlier in the context of electronic transport Kim et al. (1999), and the limit $q\to\infty$ corresponding to the BIC condition was termed as “collapse of Fano resonance”. Also, the asymmetric Fano lineshape was identified in the reflection spectra from a photonic-crystal membrane sustaining BIC Hsu et al. (2013). Here, we show that an infinite value of the Fano parameter corresponds to BIC or supercavity mode.

To verify that the effective damping of the higher-frequency mode indeed vanishes, we calculate eigenfrequencies of the resonator. We employ a rigorous method of resonant-state expansion Doost et al. (2014). The real parts of calculated eigenfrequencies are added to Fig. 2(a). Their positions are in an excellent agreement with the features in the scattering spectra. The analysis of the imaginary parts of the resonator eigenfrequencies confirms that in the vicinity of the avoided crossing point the radiation losses of one mode decreases substantially revealing its BIC nature. In particular, for $\varepsilon=80$ the $Q$ factor of the supercavity mode is found to be 63700, being in an agreement with the results obtained from a fitting of the scattering spectra.

Finally, we discuss the conditions of the supercavity regime and BIC states in dielectric resonators. For the values of permittivity from the interval $4\leq\varepsilon\leq 100$, we calculate the resonator eigenvalues by means of the resonant-state expansion method Doost et al. (2014) as functions of the aspect ratio $r/l$. We focus on two anti-crossing regions marked by circles in Fig. 2(a) for the azimuthal numbers $m=0$ and $m=1$, respectively. For each value of permittivity, the $Q$ factor behaves similar to the dependence presented in Fig. 3(d). The maximum value of $Q$ (in the regime of the supercavity mode) vs. $\varepsilon$ is shown in Fig. 4(a). For $\varepsilon>10$, this dependence is well approximated by a power law, $Q=0.04\cdot\varepsilon^{3.2}$, for $m=0$, and $Q=0.1\cdot\varepsilon^{2.3}$, for $m=1$. In particular, for a Si nanoresonator with $\varepsilon=13$, the lower-frequency supercavity mode with $m=1$ demonstrates $Q$ about 50 that is at least an order of magnitude higher than the $Q$ factor of a Si sphere $Q=13.4$, for the magnetic dipole mode, and $Q=6$, for the electric dipole mode. In contrast, the supercavity mode with $m=0$ has $Q$ about 200, that is sufficient for strong nonlinear effects at the nanoscale Shorokhov et al. (2016).

Figure 4(b) demonstrates the sizes (diameter and height) of a cylindrical resonator (normalized by the wavelength) as functions of the material permittivity. Four curves correspond to two diameters (circles) and two heights (squares) for the regime of the supercavity mode with azimuthal numbers $m=0$ (red) and $m=1$ (cyan), respectively. Resonators operate in the subwavelength regime at moderate values of $\varepsilon>3.5$. The Si resonators are shown in the insets. Both dimensions are approximately half-wavelength, being much smaller than WGM resonators exploiting high-order $m$ modes.

The mode interference associated with the BIC state can be described in terms of the Friedrich-Wintgen theory Friedrich and Wintgen (1985) developed earlier for electronic BIC states. This theory assumes that a pair of discrete states interact through the continuum of delocalized states. This interaction shifts the energy levels of the discrete states, and at certain conditions the states undergo a strong mode coupling. As a result, the damping rate of one state increases while the damping rate of the other state decreases until it vanishes and BIC appears. In photonic systems, a pair of resonator’s modes can be described by the Friedrich-Wintgen theory provided the size of the optical modes is negligible relative to the wavelength in the media. The calculated damping rates shown in Fig. 3(d) are in an excellent agreement with the predictions of the Friedrich-Wintgen theory. When the dielectric permittivity increases, one achieves higher $Q$ factors [see Fig. 4(a)] because the modes become close to the theoretical idealization. This theory suggests how to engineer the resonator properties for achieving really high $Q$ factors for simple geometries.

In summary, we have revealed that subwavelength High-index dielectric resonators can support the supercavity modes with high $Q$ factors, provided their parameters are engineered to match the conditions of bound states in the continuum. These results were predicted by analysing the scattering spectra of resonators and also calculating the mode eigenvalues with the resonant-state expansion method for open electrodynamic systems. For Si resonators, our approach allows engineering supercavity modes to achieve $Q\sim 200$, that is sufficient for nonlinear optics and quantum nanophotonics. We believe that our finding will open up new horizons for active and passive nanoscale metadevices including low-threshold nanolasers, biosensors, on-chip parametric amplifiers, and nanophotonics quantum circuits.

Acknowledgments. The authors acknowledge discussions with R.W. Boyd, C. De Angelis, A. Fratalocchi, J.S. Totero Gongora, A.A. Kaplyanskii, and J. Sipe. This work has been supported by the Ministry of Education and Science of the Russian Federation (3.1500.2017/4.6), the Russian Foundation for Basic Research (16-02-00461), Russian Science Foundation (17-12-01581), and the Australian Research Council.