Novel Electromagnetic Cavities: Bound States in the Continuum
Authors: Thomas Lepetit and Boubacar Kanté
Affiliation: Department of Electrical and Computer Engineering, University of California San Diego, USA.
In the last 10 years, an intense research effort has been devoted to bringing all-optical signal generation and processing on chip to realize true photonic integrated circuits (PICs). PICs are at their core made of waveguides, which transfer signals to different devices on the circuit, and cavities, which process signals for different functionalities . First, linear devices such as couplers, splitters, and add-drop filters were developed and, more recently, nonlinear devices such as frequency combs, nanolasers, and optical rams have been demonstrated [2-4]. Overall, progress has resulted in devices with increased functionalities that work at lower power and are more compact.
Cavities are an essential building block of PICs because they provide enhanced light-matter interaction. Currently, the most mature technology is based on a silicon on insulator platform and ring resonators. Typically, these dielectric resonators are several microns in diameter . However, due to the difficulty of integration with much smaller electronic components, other technologies such as plasmonics have started to be investigated. One of the main advantages of plasmonic devices, which are made of noble metals such as gold and silver, is that their size is not limited by the wavelength. For example, plasmonic ring resonators of only several hundred nanometers in diameter have been demonstrated . Generally, cavities are characterized by their quality factor Q, which is a measure of their capacity to store signals for a long time. At present, dielectric cavities have reached quality factors of 106, which are only limited by radiation losses coming from sidewall roughness, but typically have a footprint of 80 μm2. In contrast, plasmonic cavities can have a footprint as low as 1.25 μm2 but their quality factors are usually below 102, being limited by thermal losses coming from conduction electrons. Therefore, there is a need for novel cavity designs that can simultaneously achieve high quality factors and low footprints.
Figure 1: Cross-section of the electric field magnitude for the coupled resonators system (Half of it). Resonator 1, whose inner radius is zero, is on top and resonator 2, with a non-zero inner radius, is at the bottom. The symmetry plane on the right of each plot denotes the symmetry plane of interest. Odd modes are mostly confined in resonator 1 and even modes mostly in resonator 2.
Recently, we have demonstrated the possibility of making electromagnetic cavities using a different concept, namely bound states in the continuum (BICs) . BICs were first proposed in 1929 in the context of quantum mechanics by Von Neumann and Wigner . They surprisingly showed that bound states can exist above the continuum threshold, i.e., there are states that do not decay even in the presence of open decay channels. However, due to the theoretical nature of the first proposal, BICs did not become fully appreciated until 1985 when Friedrich and Wintgen showed that they could be interpreted as resulting from the interference of two distinct resonances . In this picture, one resonance traps the other and thus one quality factor decreases while the other one tends to infinity. Since BICs are essentially a wave phenomenon they also appear in electromagnetics where they translate for lossless dielectrics into an infinite quality factor. As a proof of concept, we have designed and measured a BIC in the microwave range using a periodic metasurface [10-11].
BICs are intrinsically sensitive to perturbations as they only exist at a single point in phase space. This is very useful for sensing applications but detrimental for most others. To obtain an extended BIC, we designed a system with two quasi-degenerate BICs. We achieved this by considering a unit cell with two resonators, a disk and a ring (see Figure 1). Odd modes of the disk resonator interfere and lead to one BIC and even modes of the ring resonator interfere and lead to another BIC. We use ceramic resonators of high-permittivity (εr=43±0.75) and they are thus only slightly coupled. Experimentally, to limit the fabrication dispersion inherent to a large array, we made the measurements in a rectangular metallic waveguide (X-band, 8.2-12.4 GHz). It is possible because such a guided setup is equivalent to an infinite array at oblique incidence as shown by image theory.
Figure 2: Modes of two dielectric resonators (εr=43) in a rectangular metallic waveguide (X-band). Both resonators are cylindrical (r=3.5 mm, h1=2.25 mm, h2=3.0 mm) and the second has a non-zero inner radius. a) Resonance frequencies vs. inner radius for even and odd modes. b) Quality factor vs. inner radius for even and odd modes for lossless and lossy resonators.
We explored phase space along a line, by varying the inner radius of the ring resonator, and showed the presence of two avoided resonance crossings (see Figure 2a), which are typical of BICs . As a result, there is an extended region of phase space where the quality factor tends to infinity (see Figure 2b). BICs only serve to cancel radiation losses and in the presence of thermal losses these are the limiting factor. At present, this scheme is therefore practical only for dielectrics but it could be extended to plasmonics by introducing gain materials to achieve loss-compensation.
Beyond the fundamental interest on the limit of quality-factors given a certain volume, there is a sustained interest in reducing the footprint of many cavity-based devices for future PICs. Tailoring the optical potential further, for example by moving away from perfectly periodic structures , opens the possibility improving the field confinement and thus shrink devices. Our work is a first step in this promising direction.
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