Polarization-Independent Narrowband Terahertz Filter Based on All-Dielectric Metasurfaces

All-dielectric metasurfaces, which are periodic 2D gratings on slab waveguides, are designed to obtain angular-dependent polarization-insensitive guided-mode resonance filters operating in the terahertz region. The effect of the metasurface grating periods along the <italic>x</italic>- and <italic>y</italic>-axis on the filter's performance is systematically studied via the rigorous coupled-wave analysis method. When the grating periods along the <italic>x</italic>-axis are 380 <inline-formula><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula>m, 400 <inline-formula><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula>m, and 420 <inline-formula><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula>m, the polarization-independent resonances of the split bands occur at 0.517 THz, 0.498 THz, and 0.481 THz, respectively. When the grating periods along the <italic>y</italic>-axis are adjusted to the above values, such polarization-independent resonances show good robustness, which means the polarization-independent resonance frequency of the split bands is mainly determined by the grating periods along the <italic>x</italic>-axis and is in an inverse proportion. Meanwhile, there are additional polarization-independent resonances formed by the intersection of the split and degenerate bands for the two unequal grating periods. This kind of component provides potential applications for narrowband filtering in emerging terahertz devices and systems for communication or sensing.

2D gratings, and provide new solutions of spatially periodic modulations for numerous potential applications: beam deflection [10], multiplexers [11], flat lenses [12] and metamirror [13]. Essentially, it is a resonance phenomenon in photonics that enables strong localization of electromagnetic waves, which is essential to control the amplitude, phase, or polarization [14], [15], [16], [17], [18], [19]. The resonances with high intensity and narrow linewidth are significant for most practical applications, but most of the resonances cannot have both [20]. Guided mode resonance (GMR), which can provide arbitrary resonance intensity and linewidth through structural design [21], refers to incorporating resonant coupling to leaky Bloch modes in the waveguide-grating layer system and was initially proposed for use as a filter in 1992 [22]. Currently, the exploration of GMR filters is mainly limited to operating in the optical and radio frequency regions, such as in color filters [23], [24], bandpass filters [25], [26], and angular-tolerant filters [27], [28]. Because the device size in the THz band is less than in the microwave region but greater than in the optical region, the THz GMR filter components can be relatively easier to manufacture in terms of size. Recently, THz filters using GMR effects have started to be reported [29], [30], [31] due to their narrow-band filtering features, and simple design. Metals show considerable Ohmic loss beyond the microwave frequencies, while dielectric metasurfaces as a platform perform lower levels of dissipation [32]. Bark et al. proposed an all-dielectric THz GMR filter through one-dimensional (1D) grating for the first time [33], and then realized tunable THz filters [34], notch filters [35], and polarization-independent filters [36]. Han et al. experimentally demonstrated high-Q THz GMRs in silicon-based 2D metasurfaces [37].
The independence of the filter regarding the polarization of the incident beam is a vital element not only for optical systems and devices but also for THz applications. The abovementioned notch filters can be implemented as polarizationindependent and polarization-dependent THz filters by controlling the relative rotation angles of cascading two identical 1D grating structures [35]. And the Bark proposed polarizationindependent filters were designed by adjusting the filling factor and grating thickness [36]. These reported GMR filters are based on 1D gratings to realize polarization-insensitive filtering under normal incidence. For oblique incidence, Zhan et al. investigated the characteristics of an all-dielectric polarizationindependent THz filter with 1D grating under different incident planes between classic and fully conical mountings [38]. Up to now, there have been no reports of 2D gratings related This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ to polarization-independent GMR filters operating in the THz region.
In this letter, we propose an angular-dependent polarizationindependent THz filter with 2D all-dielectric metasurfaces built by subwavelength quartz microstructures, which is based on the split features under oblique incidence. We studied the polarization-independence of the designed THz filter according to grating periods and the plane of incidence is set to be in xz plane. The rigorous coupled-wave analysis (RCWA) method is applied to find the polarization-independent GMR filters operating in the THz region. Fig. 1(a) illustrates a schematic structure of the proposed GMR filter, which combinates the 2D metasurface consisting of a subwavelength periodic array of cuboid elements and a slab waveguide of the filter substrate. The unit cell of the metasurface is defined by the grating periods Λ x and Λ y along the x-and y-axis, the corresponding grating widths F · Λ x and F · Λ y (where F is the fill factor and is equal along the x-and y-axis), the grating thickness d g and the homogeneous layer thickness d h . α denotes the azimuthal angle of the incident plane; α = 0 • means the incident wave vector is in the xz plane, while α = 90 • is in the yz plane. The polarization of the E vector is oriented in a polarization (p-s) plane normal to the direction of the wave vector K. The orientation of E is marked by the polarization angle ϕ, where ϕ = 0 • corresponds to s polarization and ϕ = 90 • denotes p polarization. θ is the incident angle measured from the z-axis in the incident plane. The surrounding media of the filter is assumed to be air. Quartz, which has a refractive index of 1.95 in the THz region, is chosen to be the material of the THz GMR filter because of its low loss and low dispersion characteristics at the THz frequency range 0.4 -0.6 THz covered in this paper [39].

II. STRUCTURE AND DESIGN THEORY
When a THz wave from the air is incident vertically (θ = 0 • ) on the surface of the filter and is diffracted by the grating and simultaneously guided by the slab waveguide, the GMR phenomenon occurs, leading to sharp resonant reflection peaks as shown in Fig. 1(b). The incidence is an s-polarize (ϕ = 0 • ) THz wave in the xz plane (α = 0 • ). Based on the RCWA method, the optimized design structure parameters where low sideband and narrow linewidth can be realized are d g = 144 μm, d h = 153 μm, F = 0.38, and Λ x = Λ y = 400 μm. The calculated spectral result by the RCWA method and the other simulated one by the commercial software COMSOL Multiphysics with the frequency-domain finite-element method (FEM) are both illustrated in Fig. 1(b) with the blue and red lines, respectively. These results of the two computational methods are in excellent agreement. The resulting two resonant frequencies are at 0.4655 THz (644 μm) with the full-width half-maximum (FWHM) linewidth of 1.0 GHz and 0.5297 THz (566 μm) with the FWHM of 2.0 GHz, which corresponds to quality factors Q = 466 and 265, respectively. The two calculated zero-order reflectance (R 0 ) peaks are formed due to the double periodicity. In order to reveal more about the physics behind the observed resonant peaks, we perform the electric (E y ) and magnetic (H x ) field distributions via FEM full-wave simulations, at the frequency of 0.4655 THz and 0.5297 THz, respectively. The results are shown in Fig. 1(c) and the high-intensity fields associated with the standing T E 0 and T M 0 modes excitation are strongly confined in the slab waveguide. This also explains the essence of the double periodicity. For the normal incidence, the THz frequency spectral response under an s-polarized incidence manifests a polarization-insensitive feature compared to that for a p-polarized (ϕ = 90 • ) incidence because of the 2D structure symmetry. While for the oblique incidence, the symmetry of the structure is broken, which is the main aim of this work to obtain the polarization-insensitive filter in this case.
The spectral responses of the 2D GMR filter can be considered as a composition of the R 0 maps of two equivalent 1D GMR filters with vertical grating bar orientation [40]. It should be noted that the reflection spectrum of the 2D metasurface structure is still very slightly different from the two 1D grating filters, which is caused by the approximation of the second-order effective medium refractive indices. In previous work [38], we demonstrated that the resonance frequency of the 1D GMR filter would split into two branches for a classic mounting at oblique incidence, and there was a pair of resonance bands for a full conical mounting at oblique incidence, which was degenerate and never split no matter what the θ was. Here, we can analyze the physical mechanism of the 2D all-dielectric GMR filter's spectral responses on the basis of the diffracted wave vectors of 1D GMR filters with classic and fully conical incidence.
For the plane of incidence in the xz plane with α = 0 • as illustrated in Fig. 1(a), the metasurface gratings extending along the y-axis (with the grating period Λ x ) are perpendicular to the incident plane, which can be regarded as the case of classic mounting of the 1D GMR filter. Likewise, the metasurface gratings extending along the x-axis (with the grating period Λ y ) are parallel to the incidence and can be considered full conical mounting. The THz wave vectors associated with m-th order diffraction by the metasurface gratings in Fig. 1(a) obey the following equations [28], [41]: where k x,m and k y,m are the components along the x and y axis, respectively. k 0 = 2πf 0 /c is the free-space wavenumber, c is the speed of light, f 0 is the resonant frequency. In this work, the R 0 peak is established by first-order diffraction (m = ±1). The real part of propagation constant of a leaky mode can be approximated by β m = (k 2 x + k 2 y ) 1 2 [42]. For the metasurface gratings extending along the y-axis, the m = ±1 orders of wave vectors can be expressed as Similarly, for the metasurface gratings extending along the xaxis, we find that the propagation constants with m = ±1 orders are identical and expressible as As for the incident plane in the yz plane, i.e., azimuthal angle α = 90 • , to obtain the m = ±1 orders of propagation constants β ±1 corresponding to the metasurface gratings along the x direction and along the y direction, it is only necessary to switch both x and y. Thanks to the C4 symmetry of the structure (Λ x = Λ y ), the spectral behavior is exactly the same in the incident azimuthal angle α = 0 • and α = 90 • . For Λ x = Λ y , which means that the structure is C2 symmetric, the spectral responses of the yz incident plane can also be analyzed by swapping the x and y of the above formulas in the case of the xz incident plane. Therefore, the subsequent discussion will be primarily established on the basic hypothesis: α = 0 • , also called the plane of incidence in the xz plane.  to the angle symmetry. For the s-polarized incidence, two pairs of resonant bands are observed in Fig. 2(a). One pair of these resonant bands (branches I and II) is split from point A with coordinates (0, 0.4655), whereas the other pair of them (branches III and IV) is degenerate and no longer splits in the R 0 peak at off-normal incidence. Likewise, the calculated R 0 map for the p-polarized incidence in Fig. 2(d) also exhibits two pairs of branches, one of which (branches V and VI) shows a split from point B with coordinates (0, 0.5297), and the other (branches III and IV) is as well degenerate. The two points A and B represent the positions of the R 0 peaks at normal incidence, which are corresponding to those illustrated in Fig. 1(b). As θ increases, the R 0 peak of the split branches (branches I, II, V, and VI) displays a drastic, almost-linear red-or blueshift, while the case of the degenerate branches (branches III, IV) generates a tiny blueshift. The physical mechanism of the 2D all-dielectric metasurface spectral features can be explained by the angular variation of the diffracted wave vector with corresponding quasi-equivalent 1D gratings [40]. From (3), as the incident angle θ increases, the propagation constant of backward-diffracted wave vector (m = −1) decreases but the forward-diffracted wave vector (m = +1) increases, which leads to decreases (corresponding to branches I and V) and increases (corresponding to branches II and VI) in resonance frequency respectively, because β ±1 is proportional to the frequency. As for full conical mounting indicated by (4), the m = ±1 orders of diffracted wave vectors are identical, and thus the resonance branches III and IV are all degenerate and display high angular tolerance.
The calculated angle-dependent R 0 maps of the 2D GMR filter with intermediate polarization angles of 30 • and 60 • are shown in Fig. 2(b) and (c). There are two pairs of split branches (I and II, V and VI) which are composing from the superposition  Fig. 3(b) shows a magnified view of the reflectance spectra near the polarization-insensitive frequency in Fig. 3(a). of Fig. 2(a) and (d), and the two other degenerate branches III and IV are in agreement with Fig. 2(a) and (d). These branches generate a continuously diffraction efficiency transition varying ϕ from 0 • to 90 • , and there is a distinct intersection point labeled P0 (θ P 0 = 7.3 • ) which can be easily obtained from Fig. 2(b) or (c), where the same also exists under s-and p polarizations as shown in Fig. 2(a) and (d). That is, when θ is 7.3 • , the polarization-independent filtering is achieved only in a narrow band around 0.498 THz, and the polarization-dependent GMR filters are obtained at other resonance frequencies which can be used as polarization-controlled tunable filters or switchers.
To elucidate clearly the polarization-insensitivity characteristics of the GMR filter in our metasurface, the reflectance spectra of the designed THz filter with four different polarization angles (ϕ = 0 • , 30 • , 60 • , and 90 • ) for a fixed angle of incidence θ = 7.3 • is shown in Fig. 3(a). Simulation results indicate that the resonant frequency which is insensitive to arbitrary polarization state is clearly observed at f P 0 = 0.498 THz, as shown in the regional magnified view of the R 0 spectrum in Fig. 3(b). The FWHMs of the four different polarized configurations are very small and less than 0.81 GHz.  In addition, it's not uncommon to find that the resonance frequencies of the split bands are related to the grating period along the x-axis Λ x , while the resonance frequencies of the degenerate bands are controlled by the grating period along the y-axis Λ y , according to the analyzation of (3) and (4). In consequence, the resonant frequency of the above polarizationinsensitive crossing point is mainly determined by the grating period Λ x , but not by the grating period Λ y .
For comparison, we show the angle-resolved reflectance spectra of the filter under three different polarization angles (ϕ = 0 • , 45 • , and 90 • ) for different Λ x or Λ y values of 380 μm and 420 μm in Figs. 4 and 5, respectively. As the grating period in the x-axis changes, as shown in Fig. 4, the degenerate resonance bands (branches III and IV) show almost identical spectra compared with those in Fig. 2, because the Λ y is kept unchanged. When the Λ x is 380 μm, as shown in Fig. 4(a)-(c), there also exists a distinct crossing point P1 (6.8, 0.517) located between the split branches, which indicates the incident angle and resonance frequency where the polarization-independence occurs. When  TABLE I  LIST OF THE POLARIZATION-INDEPENDENT POINTS AT OFF-NORMAL INCIDENCE the Λ x increases from 400 μm to 420 μm, the polarizationindependent location is P2 (7.7, 0.481) as shown in Fig. 4(d)-(f). It is clear that there is an inverse proportional relationship between the polarization-independent resonance frequency of the split bands and the period Λ x .
For the case of adjusting Λ y in Fig. 5, there is a negligible variation of the polarization-independent point P3 (7.4, 0.499) and P4 (7.1, 0.497), which is almost the same as the point P1 in Fig. 2. The resonant positions of the split branches display very slight shifts due to the variational strength of the nearest-neighbor magnetic coupling along the y-axis [32], [37], [43]. While the degenerate branches shift to high or low frequency, as the Λ y decreases to 380 μm or increases to 420 μm, corresponding to Fig. 5(a)-(c) or 5(d)-(f), respectively. Under such circumstances (where Λ x and Λ y are not equal), the split and degenerate branches will meet which can also form polarization-independent intersections, such as points Q1 (4.9, 0.466) and Q2 (4.3, 0.466) in Fig. 4, and points Q3 (4.6, 0.486) and Q4 (4.6, 0.448) in Fig. 5. Table I summarizes the above-mentioned polarization-independent points at oblique incidence for different periods. The configuration of changing Λ y provides higher angle and frequency stability relative to changing Λ x for the polarization-independent resonance of the split bands. There, the R0 peak frequency shifts by 1 GHz as Λ y changes from 400 μm to 380 μm or 420 μm. We can easily find that polarization-insensitive locations Q1 and Q2 occur with different θ, corresponding to an approximately identical frequency, which are the opposite of those observed in Q3 and Q4. These results provide a cost-effective solution for realizing efficient on-chip THz communication components, such as wavelength selective polarizers, beam splitters, and wavelength division multiplexers. Moreover, it's worth noting that the small band gaps appear where there should be intersections between those resonant bands in all angle-resolved R 0 maps. Because the interactions of these GMRs lead to the bound states in the continuum (BICs) with the resonance linewidth asymptotically approaching zero at specific angles of incidence [37], [44].

IV. CONCLUSION
In summary, we have provided a detailed analysis of the properties of the angle-dependent polarization-independent filter in all-dielectric metasurfaces consisting of a 2D grating array on a slab waveguide layer by using the RCWA method. Based on the physical mechanism of the 2D all-dielectric GMR filter that can be converted to equivalent 1D gratings with classic and full conical mountings, polarization-insensitivity can be realized due to the features of split resonant bands, and the polarization-independent resonance frequency is only inversely proportional to the grating period Λ x . The computation results show that, when varying Λ x from 380 μm to 400 μm to 420 μm, the polarization-independent frequency shifts from 0.517 THz to 0.498 THz to 0.481 THz; when varying Λ y from 380 μm to 400 μm then to 420 μm, the polarization-independent frequency exhibits a 2 GHz redshift. Furthermore, if Λ x and Λ y are not equal, resulting in the intersection of the split and degenerate bands, the additional polarization-independent resonances will occur, such as the polarization-insensitive frequency are at 0.466 THz under the oblique incidences of 4.9 • (for Λ x = 380 μm, Λ y = 400 μm) and 4.3 • (for Λ x = 420 μm, Λ y = 400 μm), and at 0.486 THz (for Λ x = 400 μm, Λ y = 380 μm) and 0.448 THz (for Λ x = 400 μm, Λ y = 420 μm) under oblique incidences of 4.6 • . The 2D all-dielectric metasurfaces perform high Q-factor, excellent polarization-independence and polarizationdependence, which can be used for realizing wavelength selective polarizers, narrowband polarization-independent filters and polarization-controlled tunable filters in the field of THz communications and sensing.