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Abstract
Resonances play a key role in the practical application of polaritons because they can be easily detected from the far field and integrated with various devices. Although the mechanism and implementation of polariton resonances have been studied extensively, the resonant behaviors of in-plane anisotropic polaritons and their tuning strategies remain largely unexplored so far. Here, using in-plane hyperbolic phonon polaritons in α-MoO3 as an example, we analyze their resonant absorption upon multiple conditions theoretically. We unveil the collective resonant mode originating from synergistic Fabry−Pérot cavity resonances and Bragg resonances. Beyond controlling polarizations and geometries, active tuning of resonant absorption is further achieved by rotating resonators or changing polariton topologies. Our results bridge the gap between polariton dispersions and resonant absorption, and provide a bottom-up strategy for the design of polariton-based resonators, photonic crystals, and metasurfaces. The resonant structures proposed here could serve as versatile building blocks for infrared absorbers, polarization detectors, sensors, modulators, and other photonic devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Polaritons as part−light, part−matter quasiparticles are an important branch of nanophotonics, owing to their subdiffractional scales and tight field compression [1–4]. They provide a powerful platform for light manipulation in the subwavelength regime [5]. In practice, however, it is challenging to excite polaritons directly by free-space light because of momentum mismatch. To circumvent this limitation, fabricating polaritonic nanostructures with characteristic scales comparable to or less than polariton wavelengths is a simple yet efficient method [6]. These subwavelength structures can not only launch polaritons directly but scatter polaritons generated from adjacent ones, leading to vibrant polariton modes. Compared with polaritons supported by infinite slabs or films which are usually studied at the near field, the resonant modes with enhanced signals can be recorded and recognized by relatively lower-cost far-field spectra, which can be easily integrated into molecular sensors [7–9], photodetectors [10], modulators [11], among others.
Prior works on polariton resonances mainly focused on in-plane isotropic polaritons, for example, plasmon polaritons (PPs) on metals or graphene [12,13], phonon polaritons (PhPs) hosted by silicon carbide or hexanol boron nitride [14,15], and exciton polaritons (EPs) in transition metal dichalcogenides [16]. The wavevectors (kj, j = x, y, or z) of these polaritons are isotropic in the x–y plane, namely, kx = ky. Recently discovered polaritons in metasurfaces or low-symmetry crystals possess in-plane hyperbolic polaritons with kx × ky < 0 at certain frequencies [17–19]. This unique property not only leads to appealing reflection [20,21], refraction [22,23], and diffraction [24], but also bears great potential in polariton canalization [25], twist polaritonics [26–29], and polaritonic crystals [30]. The resonances of periodically arranged structures that accommodate in-plane hyperbolic polaritons have been studied theoretically [31,32] and experimentally [33,34]. However, the tuning strategies of such resonances, especially stemming from in-plane hyperbolicity, are by far less explored.
Here, we study theoretically the tunable resonant absorption of periodic structures made from α-MoO3 that intrinsically hosts in-plane hyperbolic PhPs. Thanks to the low-symmetry lattice structure of α-MoO3, its absorption exhibits high polarization and incidence sensitivity. Cavity resonances and Bragg resonances together enhance the resonant absorption of α-MoO3 resonators, yielding high quality factors and strong geometry and rotation dependence. Besides, based on the topological transition of in-plane hyperbolic polaritons, we also realize actively configurable resonant absorption by building twisted α-MoO3 bilayers and graphene/α-MoO3 heterostructures. Our findings reveal the collective resonances of in-plane hyperbolic polaritons as well as their tuning mechanisms, thus paving the way to the design of polariton-based detectors, modulators, sensors, or other photonic elements and devices at the subdiffractional scales.
2. Methods
2.1 Full-wave simulations
Numerical simulations were performed by finite-element methods using Comsol Multiphysics. To visualize the wavefronts of polaritons, a vertically polarized dipole was used to excite polaritons. The absorption spectra of resonant structures were calculated from a unit cell under periodic boundary conditions. The light sources were linearly polarized plane waves. Using a similar setup, the electric field distributions at certain frequencies were captured. The field distributions over plain flakes were recorded at the height of 200 nm above flake surfaces, while the fields over resonators (except for the twisted ones) were captured at the height of 2 nm. The thickness of α-MoO3 slabs was set as 50 nm for all the simulations. And the periodicity of the unit cell was 1 µm. The permittivity of plain α-MoO3 was described by a Lorentzian model [35]
2.2 Dispersion relations of polaritons
Under the limit of large wave vectors, the momentum of PhPs (q) in α-MoO3 was given by
3. Results and discussion
3.1 Phonon resonances in α-MoO3
Due to the low-symmetry lattice structure of α-MoO3, its complex-valued optical permittivity (εj, j = x, y, or z) varies at different crystalline axes, as shown in Fig. 1(a), which can be described by a Lorentzian model in Eq. (1). In the long-wavelength infrared range, there are three frequency bands between the transverse (TO) and longitudinal (LO) optical phonon frequencies, known as Reststrahlen bands (RBs), in which εj is negative along one of the crystalline axes. Here we focus on RB1 (16.3−25.5 THz) and RB2 (24.5−29.2 THz) with negative εy and εx, respectively. A branch of linearly polarized light dictated by the polarization angle (αp) and incident angle (αi) is employed to study the phonon absorption of an α-MoO3 slab with the thickness (d) of 50 nm, as sketched in Fig. 1(b). The calculated absorption spectra against αp and αi are plotted respectively in Figs. 1(c) and 1(d), where two dominant absorption peaks locate at the almost same positions coincident with TO1 and TO2 of α-MoO3. Polarization directions determine the relative absorption strengths at TO1 and TO2, leading to only one absorption peak when αp = 0° and 90° and otherwise two absorption peaks. Note that the phonon resonance is maximized when the polarization direction is in line with crystalline axes (αp = 0° and 90°), as confirmed by the simulated extinction spectra in Fig. S1 in the Supplement 1. This polarization-dependent property renders α-MoO3 a natural platform for polarization detection [39]. The absorption intensities both decrease gradually with the reduction of αi from 90° to 45°. However, neither αp nor αi change peak position distinctly. For convenience, hereafter, we mainly consider normal-incident light sources (αi = 90°) with x- or y- polarizations.
Fig. 1. Phonon resonances and in-plane hyperbolic PhPs in α-MoO3. (a) Optical permittivity of α-MoO3 along different crystalline axes. The blue and red shaded areas indicate the first and second Reststrahlen bands. (b) Schematic of the simulation setup, where k0 and E are the wavevector and energy of the linearly polarized incident light. (c) Calculated absorption spectra of α-MoO3 illuminated by the normal-incident light with different αp. (d) Absorption spectra as a function of αi. (e) In-plane hyperbolic electric field distributions of PhPs launched by a vertically polarized dipole (gray dots). (f) Corresponding distributions of the real parts of electric fields at the z-direction, Re(Ez).
Free-space light cannot excite PhPs directly in a plain α-MoO3 slab due to large momentum mismatch. To visualize the wavefronts of PhPs, we use a vertically polarized dipole to launch PhPs. The captured electric fields in the x-y plane are shown in Fig. 1(e), exhibiting hyperbolic distributions with orthogonal open directions in different RBs. Such hyperbolic wavefronts in Fig. 1(f) are significantly different from the circular ones sustained by in-plane isotropic polaritons. This unique property favors the tuning of resonant absorption which will be discussed in detail below.
3.2 High-Q polariton resonances
Tailoring slabs into certain geometries with a feature size comparable to or less than polariton wavelengths is a useful method to excite polaritons from the far field, which is hardly attainable by infinite slabs or films. We first consider linear ribbon arrays with a square unit cell. As sketched in Fig. 2(a), the unit cell has a fixed pitch (p = 1 µm) but varied widths along the x-direction (wx). Simulated field distributions and absorption spectra are displayed in Figs. 2(b) and 2(c), respectively. The absorption peak at 16.3 THz is relatively insensitive to scale variance and should assign to the phonon absorption at TO1. By contrast, when the polarization direction is in line with the arrangement direction (αp = 0°), the absorption peak in RB2 emerges and redshifts with the increase of wx. We attribute this effect to the resonant absorption of PhPs. The greatly enhanced intensity of the electric field in the right panel of Fig. 2(b) confirms the generation of polariton resonances. For the arrays packed along the y-direction, polaritons are excited at αp = 90° and subsequent absorption peaks follow a similar shift tendency to the case with the x-polarization, as seen from Figs. 2(c)−2(e).
Fig. 2. Collective resonances in α-MoO3 linear ribbon arrays. (a) Schematic of the unit cell with p = 1 µm and wx = 0.5 µm. (b) Corresponding electric field distributions. (c) Absorption spectra of the ribbon arrays with different widths along the x- (top) and y-directions (bottom). (d) Schematic of the unit cell with p = 1 µm and wy = 0.5 µm. (e) Corresponding electric field distributions. (f) Analytically calculated dispersion relations of polariton resonances. Symbols represent extracted values from simulations in (c). (g) IFCs at the frequencies of 26.2 THz (left) and 20.2 THz (right) marked by arrows in (c). Gray curves relate to IFCs in plain α-MoO3. Background color maps represent the normalized amplitude of FT harmonics of the corresponding linear arrays. (h) Calculated propagation lengths of polaritons. Horizontal dashed line indicates the pitch of arrays.
A single α-MoO3 ribbon performs like a Fabry−Pérot cavity, due to the waveguide-like propagation of PhPs within the α-MoO3 thin slab. The momentum of longitudinal cavity resonances ($k_j^r$) satisfy $k_j^r$wj + φ = πn [40], where φ is the reflection phase shift at the sidewalls of the slab, and n = 0, 1, 2… is the waveguide mode number. Previous studies have demonstrated that φ is −π at the α-MoO3/air interface [20]. For the fundamental waveguide mode (n = 0), its dispersion relation follows $k_j^r$ = π/wj. The extracted results from the strongest peaks are pointed out by symbols in Fig. 2(f). Besides, the general dispersion relations of PhPs in undecorated α-MoO3 slabs can be deduced analytically by Eq. (2), as plotted in Fig. 2(f) (gray curves). In periodic ribbon arrays, however, momenta should be modified because of the resonant nature. We thus double the wavevectors obtained from plain α-MoO3 slabs [41]. As shown by black curves in Fig. 2(f), the calculated results match well with the extracted values from simulations for ribbons with varied widths.
For tightly arranged ribbon arrays, polariton scattering should also be considered. The interaction between polaritons and periodic structures can be interpreted by the intersections between the isofrequency contours (IFCs) of polaritons and the Fourier transform (FT) of periodic arrays [30]. IFCs represent the dispersion of polaritons in the frequency domain and can be derived from dispersion relations at a given frequency. Note that here we also use $k_j^r$ as the input. The IFCs at 26.2 and 20.2 THz are plotted in Fig. 2(g), where the background color maps relate to the normalized amplitude of the FTs of arrays in Figs. 2(a) and 2(d). Intersections indicate that the first-order harmonics play a dominant role in the resonance, corresponding to $k_j^s$ = 2π/p, where $k_j^s$ represents Bragg resonance condition. The propagation length of polaritons estimated by $L_j^r = 1/\textrm{Im}({k_j^r} )$ is plotted in Fig. 2(h), which is larger than p over the frequencies studied (except the case with wj = 0.3 µm) and thus fulfills the prerequisite of Bragg resonances [42]. $k_j^s$ approximately equals to $k_j^r$ when wj = p/2. At this condition, intra-ribbon cavity resonances couple with inter-ribbon Bragg resonances. Such superposition gives rise to the maximum absorption strength in our system at 26.2 THz, corresponding to a quality factor (Q) of 74. The further enhancement of Q can be achieved by reducing ribbon widths or controlling the polarization directions (see Fig. S2 in the Supplement 1).
3.3 Broadband and tunable resonant absorption
Beyond high Q, broadband absorption is a prerequisite for many other photonic applications, such as photodetection. Bearing in mind that absorption peaks shift with ribbon widths and the in-plane hyperbolic dispersion of PhPs in α-MoO3, we propose in Fig. 3(a) doubly periodic structures with a quad-triangular resonator composed of four common-vertex triangles with the same base (b). As shown in Figs. 3(b) and 3(c), such resonators sustain polaritons in both RB1 and RB2, and, more importantly, can expand absorption ranges significantly, thanks to their quasi-hyperbolic edges along both the x- and y-directions which can greatly enhance the resonance of in-plain hyperbolic PhPs through negative reflection [21]. The resonant peaks in absorption spectra relate to different diffraction orders, indicating an increased contribution of Bragg resonances. Comparing with linear ribbons and circular resonators, the quad-triangular structure with b = 0.8 µm exhibits the highest absorption coefficients over the whole frequency ranges (10−30 THz), indicating that the quad-triangular structures can indeed broaden the resonant absorption based on in-plane hyperbolic polaritons (see Fig. S3 and Table S2 in the Supplement 1). The absorption can be further enhanced and reach nearly 100% using a metal back reflector (see Fig. S4 in the Supplement 1), finding potentials in perfect absorbers.
Fig. 3. Doubly periodic resonators with broad absorption ranges and rotation tunability. (a) Schematic of quad-triangular resonant unit cells rotated by θr. (b) Electric field distributions over the unit cells with b = 0.8 µm at 18.2 (top) and 25.4 THz (bottom). (c) Absorption spectra as a function of b. Arrows indicate 18.2 and 25.4 THz. Electric field profiles are extracted in the plane indicated by pink and green dashed lines in (b). (d) Absorption maps against θr in RB1 (top) and RB2 (bottom).
Except for geometry dependence, in-plane hyperbolic polaritons are also sensitive to the alignment directions of unit cells [43]. As shown in Fig. 3(a), we fix crystalline axes and rotate resonators anticlockwise by the angle of θr. The snapshots of field distributions in Fig. 3(b) indicate that rotation gradually concentrates energies into the gaps normal to the polarization directions and enhances the in-plane anisotropy of field distributions, giving rise to relatively narrower absorption ranges. A clearer tendency can be seen from the absorption maps in Fig. 3(d). Note that rotation changes the relative position between polarization and resonator edges as polarization direction does, but the relative position between polarization and crystalline axes remains unchanged in the former thus leading to different phenomena.
In the above cases the topology of IFCs remains hyperbolic because the crystalline axes of α-MoO3 are fixed. Recent studies have reported that the topological transition of IFCs can be realized by building twisted bilayers [26–29] or heterostructures [38,44–46]. We first consider in the left panel of Fig. 4(a) the topological transitions in twisted bilayer α-MoO3 with different twist angles (θt). Figure 4(b) shows representatively the electric field distributions of dipole-launched PhPs in RB1 upon twist. Results in RB2 are provided in the Fig. S5 of Supplement 1. They both see topological transitions from hyperbolas to quasi-circles, during which polaritons propagate larger distance. Close to the transition angle, polaritons become more canalized and collimated due to the flattened IFCs and highly directive group velocity of polaritons. In far-field spectra, such transitions weaken the polarization dependence of absorption and finally, the absorption under the x- and y-polarizations become equal when θt = 90°, as shown in Fig. 4(c). A periodic structure (p = 1 µm) with an aforementioned quad-triangular geometry (b = 0.8 µm) is then patterned in these twisted bilayers, as shown in the right panel of Fig. 4(a). The electric fields in the cross-section between the two layers are displayed in Fig. 4(d). At the same frequency (18.2 THz), the untwisted α-MoO3 double layers are almost unresponsive to the x-polarized light. With the increase of θt, the field intensities under the x-polarized illuminations grow gradually. Strong field intensities can be seen under both the x- and y-polarizations when θt = 90°. The evolution of absorption spectra is summarized in Fig. 4(e), following a similar tendency to those of twisted plain slabs but exhibiting broader absorption ranges.
Fig. 4. Tuning resonant absorption via topological transitions in twisted bilayers. (a) Schematic of twisted bilayer α-MoO3 slabs (left) and the relevant resonant unit cell (right). Green and blue dashed frames indicate the top and bottom α-MoO3 slabs. (b) Dipole-launched electric field distributions over twisted bilayers with different θt. Gray dots indicate dipoles. (c) Absorption spectra of twisted bilayers as a function of θt. (d) Electric field distributions over the twisted resonators. (e) Same to (c) but for quad-triangular resonators.
The topological transition of in-plane hyperbolic polaritons can also be realized in heterostructures. We sketch a heterostructure composed of graphene on the top of α-MoO3 in the top panel of Fig. 5(a) and then change the fermi energies (EF) of graphene. The simulated polariton distributions in RB1 and RB2 are shown in the Fig. S6 of Supplement 1. Polariton wavefronts change from open hyperbolas to closed quasi-ellipses with the increase of EF. The IFCs of hybrid polaritons in Fig. 5(b) and Fig. S5 reveal such evolution more explicitly. Note that polaritons within hyperbolic sectors are the volume-confined mode, namely, the electric field is mainly concentrated within the α-MoO3 thin slab. By contrast, polaritons out of hyperbolic sectors are the surface mode with their energy confined on the graphene monolayer [46]. This difference can be seen from the field distributions in the x-z and y-z planes shown in Fig. 5(c).
Fig. 5. Gate-tunable resonant absorption of hybrid polaritons. (a) Schematic of a graphene/α-MoO3 heterostructure (top) and the relevant quad-triangular resonant unit cell (bottom). (b) IFCs of graphene/α-MoO3 heterostructures with EF = 0 (gray), 0.3 (red), 0.6 (blue), and 0.9 eV (orange). (c) Dipole-launched electric field distributions at the y-z and x-z planes. (d) Absorption spectra as a function of EF. (e) Electric field distributions over graphene/α-MoO3 resonators at 18.2 THz.
Such topological transition offers a new degree of freedom for resonant absorption control. Using the quad-triangular unit cell in the bottom panel of Fig. 5(a) with b = 0.8 µm as an example, we plot its absorption spectra as a function of EF in Fig. 5(c). New absorption peaks appear at frequencies below TO1 and blueshift with the increase of EF, which should be attributed to the resonant absorption of PPs hosted by graphene (see Fig. S7 in the Supplement 1), whereas α-MoO3 serves purely as the substrate without polaritonic response. Within RB1 and RB2, hybrid plasmon-phonon polaritons form and consequent absorption ranges expand with the increase of EF. Although slightly attenuated polarization sensitivity is observed, its effect is not as remarkable as the twisted structure does, because of the quasi-elliptic IFCs in graphene/α-MoO3 heterostructures and the aforementioned different polariton modes in and out of the hyperbolic sectors. Such difference can be visualized more clearly from different field strengths over graphene/α-MoO3 resonators upon the x- and y-polarizations in Fig. 5(d) and Fig. S5.
4. Conclusion
In summary, we have analyzed theoretically the resonant absorption of in-plane hyperbolic polaritons and their tuning strategies. The scaling law of polariton resonances is established in terms of the synergistic Fabry−Pérot resonances and Bragg resonances. High-Q, broadband, and actively tunable resonant absorption are achieved by adjusting feature scales, modifying geometries, and controlling alignment directions or polariton topologies, respectively. We envision that other tuning methodologies used in metasurfaces and polaritonic crystals, for example, lattice arrangement and periodicity, could readily be transplanted into our systems. Given that polariton dispersion relations also play a dominant role in resonances, other strategies for the modification of dispersion relations can also be employed as tuning knobs, such as controlling the dielectric environment, slab thicknesses, and temperature.
Funding
National Natural Science Foundation of China (61735017, 61822510, 62020106002); National Key Research and Development Program of China (2021YFC2401403); Major Scientific Research Project of Zhejiang Laboratory (2019MC0AD02); Zhejiang University Education Foundation Global Partnership Fund.
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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