1. Introduction

The Q-factor of a photonic resonance in the far-field spectrum is related with the extended lifetime of the photon or the effective optical path of light circulating within the structure, and is thus an important attribute in the characterization of enhanced light-matter interactions. Consequently a high-Q factor optical resonance is intensively pursued in nanophotonic researches where wavelength or even subwavelength scale nanostructures can help achieve some optical effects which are comparable to those in bulk optics [1–3]. Optical resonances with high Q factors, ranging from several thousands to millions, can significantly enhance the interaction between light and matter, and they have shown important applications across many areas including optical sensing [4], active optical devices [5], amplification of photon-matter coupling [6], and light sources [7]. To date, researchers have explored different physics and device design to achieve resonances with ultra-high Q factors, aiming to realize stronger light-matter interactions. In the meanwhile, metasurfaces, due to their optically thin thickness and multifunctional light control capabilities, are considered a new platform for achieving unique functionalities and novel photonic devices. Traditional metasurface structures rely on the gradient of scattering phases from individual unit cells to manipulate the wavefront of incident light, and thus usually cannot benefit from the advantages of high-Q resonances. A combination of metasurface and high-Q photonics can be achieved resulting from the interactions between a large number of unit cells in the nonlocal metasurface structures [8,9].Although this can open new horizons and offer new possibilities for innovative optical devices, the exploring of new physics is still required.

Recently, we first proposed a new type of leaky modes [10] by utilizing the period-doubling perturbation in photonic lattices to transform the infinite-Q guided modes (GMs) into the quasi-guided modes (QGMs) whose dispersion band are folded into the continuum. The main characteristic of the QGMs lies in the ultra-high Q factors, which can be controlled between an intermediate value and the infinity by the magnitude of the perturbation. Furthermore, the GMs in subwavelength photonic lattices usually exhibit a continuous dispersion extending over a large spectral band. This dispersion, although folded at the boundary of the first Brillouin Zone (FBZ) in the new lattice, will be inherited by the QGMs. As a result, the QGMs can exhibit robust Q factors over a large spectral band, and the resonance frequency can be controlled by the incident angle of external radiations. Compared to another type of high-Q resonances of the quasi-bound states in the continuum (Q-BIC), the QGMs provide a new and superior way to realize robust and high-Q leaky modes in a wide range of the energy-momentum space.

In this work, we combine the Brillouin zone folding effect with moiré physics to achieve similar robust and ultra-high Q resonance with more flexibility of the period increasing. Moiré physics and related applications have become an emerging and exciting research topic, since Cao et al. investigated in 2018 the generation of superconducting phases in bilayer graphene structures at magic angles [11]. In most cases, moiré superlattices are produced by single or multi-layer periodic structures undergoing relative displacement [12] or relative rotation [13], and thus the period of the resulted moiré's superlattice strongly depends on the relative displacement or rotation. The initial breakthrough was made in condensed matter systems with subsequently discovered exotic phenomena in moiré superlattices, including unconventional superconductivity [11], moiré excitons [14], and anomalous Hall ferromagnetism [15], to name just a few. Recently, the concepts of moiré physics have been introduced into the field of optics. For example, Wang et al. designed a moiré lattice by using a 2D photonic bilayer sublattice in order to explore the light localization and delocalization controlled by changing the ratio of the sublattice amplitude [16], and Huang et al. realized moiré Q-BIC [17] by controlling the interlayer coupling and interlayer torsion angle on a bilayer photonic crystal plate, which provided a new avenue to study novel optical effects.

We demonstrate in this work that the original subwavelength lattice can be transformed into waveguide moiré gratings (WMGs) by rotating a one-dimensional (1D) ridge grating on a slab waveguide by angle θ in both the clockwise and counterclockwise directions. The new period of the WMGs is mainly determined by the angle θ, which is significantly different from the period-doubling perturbation achieved by changing the size or moving the position of every second column of a 1D or 2D structure [10,18]. We numerically calculate the dispersion relation and Q-factor of the QGMs supported by the WMGs with θ = 60°, and the results demonstrate that the WMGs can achieve ultra-high and robust Q-values in a wide region of the energy-momentum space. We further use the example of optical-trapping of polystyrene (PS) spheres to effectively demonstrate that the QGMs-based WMGs can generate quite high optical gradient forces on PS spheres at different wavelengths or wavenumbers. The results confirm the high potential of WGMs in enhanced light-matter interactions.

2. Structure and results

Figure 1(a) shows the schematic structure of the WMG, which consists of a crossed silicon (index 3.5) grating separated from a quartz (index 1.5) substrate by a silicon slab waveguide layer. The left panel of Fig. 1(b) presents the top view of the original 1D ridge grating. After the rotation operation by an angle θ along both the clockwise and counterclockwise directions, WGMs are formed with the top view shown in the right panel of Fig. 1(b), where the blue solid line represents the primary cell of the WMG. The cladding material above the WMGs is assumed to be water (index 1.31), and the geometrical parameter are described in the caption of Fig. 1. The area surrounded by the blue solid lines illustrates the primitive cell of the WMGs. The period along the x-direction and y-direction of the WGMs are changed to be a₁ = a/cosθ and a₂ = a/sinθ, respectively, where a is the period of the original 1D grating. Since the period increase in the y-direction for a certain θ is equivalent to the increase of period in the x-direction at the angle of 90°-θ, we only pay attention to the dispersion and Q factor evolution as a function of k_x in this work. The dispersion relation and Q factor for the modes supported by both the 1D grating and the WMGs are numerically calculated by employing the eigen-frequency analysis implemented in COMSOL Multiphysics, a commercial software based on the finite element method (FEM). The transverse Floquet periodic boundary condition (PBC) is used to characterize the wavenumber in the xy directions. All the results are for the transverse magnetic (TM) polarization, where the magnetic field is assumed to be along y direction. That is to say, along the length direction of the 1D grating. For the WMGs shown on the right panel of Fig. 1(b), the magnetic field is along the a₂ direction of the primitive cell. Figure 2(a) presents the calculated dispersion diagrams for the first three orders of modes supported by the original 1D grating, whose unit cell is schematically shown in the upper inset. The TM₁ mode is all located within the blue shaded area (below the light line). So it represents the dispersion for a well-confined GM mode. Due to the presence of the ridge gratings on top of the slab waveguide to provide a periodic modulation of the refractive index, a band-folding of TM₁ is seen giving rise to TM₂ and TM₃. TM₃ is the well-known guided mode resonances (GMR), which can be easily accessed by free space radiations. In contrast, TM₂ is hybrid GMR/GM mode. For wavevectors close to the Г point, its dispersion band is in the continuum, manifesting itself the GMR. Close to the boundary of the FBZ where the dispersion band is below the light line, it becomes the well-confined GM mode. The top silicon grating layer with a low height provides a weak refractive index modulation, manifested by two small spectral gaps at k_x= 0 between TM₂ and TM₃, and at k_x=π/a between TM₁ and TM₂. The inset shows two close-ups of the spectral gaps at these two high-symmetry points. Noteworthy, the Q factor at the Г point for the TM₃ band reaches infinity, as shown by the results of the dependence of the Q-factor on the wavevector in Fig. 2(b). It corresponds to the regular symmetry-protected BIC, at which the field of incident plane wave has a zero overlap with the supported mode. For wavenumbers away from the Г point, i.e. k_x ≠ 0, the overlap is non-zero and the mode sees a significant drop of the Q factor as larger wavenumber. So with the BIC effect, leaky modes with ultra-high Q values can only be obtained within a very small region of wavenumbers around the Г point, while Q factors in other regions of the momentum space are significantly deteriorated. The high-dependence of the Q factors with the Q-BIC modes posses strong limitations for many optical applications.

 

Fig. 1. (a) Schematic illustration of the Si WMGs on a quartz substrate consisting of crossed gratings on a slab waveguide layer. The geometric parameters are: slab thickness t = 220 nm and grating thickness h = 60 nm.(b) The left panel is a top view of the 1D ridge grating, whose period is a = 270 nm and ridge width w = 50 nm.The rotations by an angle θ in both the clockwise and counterclockwise directions, the 1D grating is transformed in the WMG shown in the right panel.

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Fig. 2. (a) Dispersion diagrams of the modes supported by the 1D ridge grating. TM₁, TM₂, TM₃ denote the first three TM bands, and the light blue shaded area marks the non-radiating bound modes (below the light line). The two close-ups at the Г and Х points highlight the spectral gaps between TM₂/TM₃ and TM₂/TM₃ bands. The upper inset shows a top view of the 1D ridge grating. (b)The Q factor as a function of k_x along the TM₃ band, and it tends to infinity at k_x = 0, indicating the occurrence of a symmetry-protected BIC here. (c) Dispersions of the QGMs supported by WMGs resulting from different values of θ. The inset shows a top view of the WMG unit cell.

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We further show that ultra-high and robust Q-factors in a wide region of energy-momentum space can be realized by using the concept of WMGs. Since the period of the WGMs are dependent on the value of θ, different WGMs can be achieved by simply choosing the proper θ. Figure 2(c) shows the dispersions of the QGMs in the WMGs at four different rotation angles θ (25°,30°,35°,40°). The dispersion band of the TM₁ mode in the original 1D grating (i.e. θ=0°) is also shown for comparison. Since the period of the WMGs in the x direction becomes larger, the FBZ for the WMGs along the k_x direction shrinks, leading to the folding of the TM₁ dispersion band at the boundary of the new FBZ, i.e. at different wavevectors of π/a₁. It is also observed that as θ increases, the dispersion of the QGMs are shifted downward compared to the original TM₁ band. This phenomenon is attributed to an increase in the filling ratio of the ridges in the unit cell, as θ increases from 0 to 40°, and a corresponding increase of the effective index of the cladding ridge layer [19,20].

We use the simplest case of θ equaling to 60° as an example to demonstrate the evolution from GMs to QGMs in the WMGs. In that case, the period in the k_x direction is doubled from a in the original lattice to a₁ (a₁= 2a) in the new WMGs. As a result, the FBZ shrinks from (-π/a, π/a) to (-π/a₁, π/a₁) in the k_x space and all the bands in the original lattice are folded at k_x=±π/a₁. For simplicity, only the TM₁ and TM₂ modes are considered. The dispersion diagrams for the resulted QGMs in the WMGs as well as TM₁/TM₂ modes in the 1D ridge grating, are shown in Fig. 3(a), where the hollow orange circles extending across the wavenumber range of (-π/a, π/a) denote the TM₁/TM₂ modes in the original lattice. Due to the halving of the FBZ for the WMGs, they will be folded to be above the light line for k_x in the region of (-π/a₁, π/a₁). The GM part of TM₂ mode originally below the light line appears in the continuum close to the Г point now. The two new QGMs resulting from the folding of the two GMs of TM₁ and TM₂ are labelled as TM₁₁ and TM₂₁, respectively. Both TM₁₁ and TM₂₁ inherit the same steep dispersion of TM₁ and TM₂, respectively, suggesting that one can rely on the use of different wavenumber to control the QGM resonance. The Q-factors of QGMs along TM₁₁ and TM₂₁ bands are shown in Fig. 3(b). It is seen from the results in Figs. 3(a) and (b) that with the increase in wavenumber, both bands exhibit ultrahigh Q-factors over large bandwidth. However, slight difference in the Q-factor evolutions between the two modes is also found. For the TM₁₁ band, as k_x becomes larger, the Q factor increases slightly and then stabilizes. Beyond the wavenumber k_x = ±0.82π/a₁, the Q factor increases dramatically to infinity. This is because the TM₁₁ band has extended below the light line (see Fig. 3(a)), so that the QGMs in the WMGs switches to GMs which cannot be coupled to free space. For the TM₂₁ band, one signature is the existence of infinity Q-factor at k_x = 0. This is due to the mirror symmetry of the WMG with respect to the center in the x direction, and thus enables the support of ideal symmetry-protected BIC. Further away from the Г point, the Q-factor along the TM₂₁ band maintains robust and ultra-high Q-factors as k_x increases. The Q factor suddenly drops from wavenumbers beyond k_x=±0.56π/a₁. This is because the predecessor TM₂ band is a hybrid GMR/GM mode and only part of it is below the light line (see Fig. 3(a)). So when it is folded, only part of the TM₂₁ band can maintain ultra-high and robust Q factor resonance. To see straightforwardly the resonance associated with the QGM, we calculated the transmission spectrum using normally incident plane waves with the magnetic field aligned in the y-direction. The results in Fig. 3(c) clearly show a Fano-profiled resonance around 223.2 THz with a narrow linewidth. The Fano-profile is attributed to the interference between the QGM and the Fabry-Perot resonance in the silicon slab. The linewidth of the resonance is consistent with the Q-factor presented in Fig. 3(b). Further results not shown here confirm the dependence of the QGM resonance on the wavevector by changing the light incident angle and the robustness of the Q factor against the wavevector. Figure 3(d) shows the real part of the magnetic field distribution at the resonance wavelength in Fig. 3(c) across the xz plane of the WMGs sampled in the center of the primitive cell along the y-direction. It can be seen that the field distribution is mainly concentrated within the waveguide layer. A out-of-phase distribution of the magnetic field is also found, which is a signature of the QGM which is folded from the predecessor GM at the boundary of the original FBZ.

 

Fig. 3. (a) Dispersion bands of the QGMs supported by the WMGs (solid lines) and the GMR/GM by the 1D ridge gratings (orange hollow circles), where the light blue shaded areas represent the confined modes. (b) Calculated results of the QGM Q factor as a function of wavenumber. (c) Transmission spectrum of the QGMs in the WMGs for TM-polarized plane waves at normal incidence. (d) On-resonance distribution of the real part of the magnetic field across the central xz plane of the WMG primitive cell.

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To demonstrate the potential application of the designed WMG in enhanced light-matter interactions and also to show the spectral tunability of the QGM resonance by the wavenumber, we used the optical trapping of nanoparticle as an example. The PS (refractive index 1.59) sphere with radius r = 15 nm is assumed to be at a distance of d = 120 nm from the top surface of the slab waveguide layer. The same WMG with the rotation angle θ=60° is used for the numerical calculations. The water cladding above the WMGs is assumed to mimic the real suspending conditions of the PS nanospheres. The optical force exerted on the PS sphere was calculated based on the Maxwell's stress tensor method (MST) [21]. Within the framework of classical electrodynamics, the component of the total time-averaged force F acting on the irradiated object can be calculated by surface integration:

Figure 4(a) and (c) presents the transmission spectrum calculated in the presence of PS sphere for two different incident angles, normal incidence in Fig. 4(a) and at k_x = 0.4 π/a₁ in Fig. 4(c). The latter corresponds to an incident angle around 34.3°. In both calculations, the magnetic field of the incident plane waves is along the y direction. For the normal incidence in Fig. 4(a), a strong resonance close to 223.27 THz is seen with the Q-factor found about 1.28 × 10⁷. The slight spectral shift here compared to the results in Fig. 3(d) is attributed to the presence of the small PS sphere introduced into the system. At an inclinded incidence condition of k_x = 0.4 π/a₁, the results in Fig. 4(c) shows a strong shift of the resonance to 197.18THz, with a Q-factor found to be 1.43 × 10⁷. This provides a clear evidence that the QGM resonance can be controlled by the incident angle. The inset in Figs. 4(a) and (c) shows the magnetic field distribution across the xz cross-section of the primitive cell at the resonance frequency. A similar out-of-phase distribution as in Fig. 3(d) is seen.

 

Fig. 4. (a) and (c) Transmission spectra through the WMGs calculated at normal incidence and at the wavenumber k_x = 0.4π/a₁, respectively. The left inset presents the top view of magnetic field H_x at the resonance peak across the center of the ridge layer while the right inset shows cross sectional view of H_yit across the xz plane. (b) and (d) Calculated optical force spectrum on the PS sphere for two incident conditions in (a) and (c).

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We assume an input power intensity of 1 mW/µm² and then use Eq. (1) to calculate the optical force exerted onto the PS sphere. Figure 4(b) and (d) present the calculated optical forces for the two incident conditions. From the results, it can be seen that the peak of the optical force is realized at the QGM resonance, and the component of F_z is much higher than the those in the other two directions. The peak value of F_x on the PS sphere is about 0.01 pN/mW, F_y = -48 pN/mW, and F_z is about -1236 pN/mW. Even though the structure is symmetric along the x and y directions, there are counter-intuitively non-zero F_x and F_y values, which are related to the slight asymmetric field distribution at the peak of the Fano resonance [22]. This indicates that the PS sphere is not stably trapped in a fixed position above the surface of the WMGs, but is pulled towards the center of the WMGs in the negative z direction. For the inclinced incidence in Fig. 4(c), the optical force applied to the PS sphere has the peak values of F_x around -0. 25 pN/mW, F_y around 12 pN/mW, and F_z about -2148.5 pN/mW. Compared to the optical forces along the x and y axes in Fig. 4(b), the optical forces along these axes in Fig. 4(d) are in the opposite directions. This effect is attributed to the different field distributions at those two Fano resonances [22] for two incident angles. This difference, although hard to observe in the cross sectional views of the H_y field (see the right inset of Figs. 4(a) and (c)), can be easily seen in the top-views across the ridge layer (see the left inset of Figs. 4(a) and (c)).

The results from normal and inclinded incidences suggest that the PS sphere is subjected to a large optical force at a larger wave number (k_x = 0.4 π/a₁₎. This is consistent with the trend of the Q-factor as a function of the wavenumber. This spectral tuning by the incident angle is important in the optical trappling applications because many commerical setups use lasers at a single wavelength. Thus one can tune the resonance of the QGM to match the wavelength of the available lasers to achieve the highest trapping effect.

3. Conclusion

In conclusion, we have proposed and numerically demonstrated that the QGMs-based WMGs are capable of supporting ultra-high and robust Q resonances in a wide range of energy-momentum space regions. The period perturbation of the WMGs is mainly determined by the rotation angle θ, which provides more flexibility from the period perturbation brought about by changing the size or moving the position of every second element of a 1D or 2D structure. By applying the WMGs to the optical capturing of PS nanospheres, our work show that it is possible to provide large optical forces at the QGM resonance. The spectrally tunability of the QGM resonance by the incident angle facilitates the practical applications with a single optical structure. Our results in this work opens up a novel avenue for the design of photonic devices with ultrahigh and robust Q resonance, where the realization of ultra-sharp spectral features in a very wide region of momentum-energy space may enhance the performance of photonic devices such as biosensors, ultra-narrowband thermal radiation, and filters.