1. Introduction

Multifunctional artificial layered materials with sub-wavelength thickness have become the current research hotspot, the so-called “metasurfaces” [1]. Since metasurfaces can modulate the characteristics of incident light waves, including their amplitude, phase, dispersion, and polarization characteristics, they have recently attracted great attention [2–5]. For traditional optics, wavefront control is achieved through optical elements (for example, prisms and lenses), spatial light modulators, or phase plates, all of which rely on the gradual phase accumulation effect that occurs during the propagation of light waves. On the contrary, the metasurfaces composed of sub-wavelength units could abruptly change the phase of the incident light wave by adjusting its optical and geometric characteristics. Therefore, metasurfaces have a wide range of applications, such as beam deflection [4,6], beam focusing [7,8], optical vortex generation [9,10], and filters [11].

The all-dielectric metasurfaces have much lower intrinsic losses compared with the plasmonic metallic metasurfaces, so it has broad prospects in practical applications [12]. Silicon materials are often used in all-dielectric metasurfaces due to their high refractive index and low loss in the near-infrared. It is well known that all-dielectric metasurfaces composed of low-loss dielectric materials have high-Q resonance modes, including the guided mode resonances (GMR) [13–15] and bound state in the continuum (BIC) [16,17]. Since GMR can be easily coupled with free space light, much work has focused on how GMR can be used for various optical components, such as polarizers, filters, and biosensors [13–15]. Recently, the BIC has aroused great interest among researchers. Although the BIC resonances exist in the energy spectrum of the continuum [16,17], the use of symmetry breaking to form a quasi-BIC can achieve precise control of the radiation lifetime, and thus providing a feasible solution for various applications, such as lasers [18,19], optical modulators [20–23], nonlinear light generation [24,25] and sensors [26]. Such planar optics provide a versatile and highly compact approach for free-space light manipulation, which usually meets or exceeds the performance of bulk optics. However, the quality factor of the metasurfaces is limited to a few tens so far, which greatly limits its applications [27–29].

Here, we designed a high-quality factor (high-Q) pass-band filter with dynamic adjustment of the central wavelength. By calculating the diffraction spectra of a rationally designing silicon nano-antennas that simultaneously support GMR and Mie resonances [30], we found that over the wavelength range from 1370 nm to 1500 nm only a quasi-BIC resonance at ∼1400 nm with a Q factor of up to 36922 could transmit along the zeroth order direction. With varying the density of the free-carriers and thus the refractive index of the ITO substrate in the simulations, we showed that the resonance wavelength of the high-Q resonance could be electrically tuned. Therefore, our work provides a feasible way for the design of ultra-narrow band-pass filter with a low-voltage adjustable central wavelength [31].

2. Methods

The finite element software COMSOL Multiphysics was employed to calculate the diffraction spectra of the proposed structures. The designed structure consists of an array of silicon nanowires supported on the Al₂O₃ substrate. In the calculations, the width of the silicon nanowire and the periodicity of the array are set to be w = 220 nm and p = 2120 nm, respectively. The geometrical parameters of the grooves that are introduced into the silicon nanowires are assumed to be: groove length l = 80nm, depth d = 30nm, and groove period Λ = 654nm. The refractive indices of Al₂O₃ substrate, silicon nanowires, and surrounding medium are set to be n_Al2O3 = 1.74, n_Si = 3.48, and n_air = 1, respectively. Periodic boundary conditions are applied to the four side boundaries of the calculation domain, with the top and bottom set as perfectly matched layers to absorb reflected and transmitted light. Different mesh densities inside and around the silicon nanowires have been used in the simulations and the deviation of the results is found to be less than 1%, which indicates that converged solutions are achieved. The Q-factor of the GMR is calculated as a ratio of its resonance wavelength (λ) to its full width at half maximum (FWHM, Δλ), i.e., Q = λ/Δλ.

3. Results and discussion

Figure 1(a) schematically shows a periodic silicon array metasurfaces supported on an Al₂O₃ substrate. The metasurface is composed of two groups of equidistant structures. The first group is composed of three silicon nanorods with the same width and distance. The second group is a blank structure, and the two groups of structures are selected to correspond to a constant relative phase delay between adjacent elements is equal to π. By adjusting the height t and width w of Si nanorods, we make their electric dipole and magnetic dipole resonances overlap in the spectrum, which can achieve high transmittance [30]. The phase of transmitted light can also be changed by adjusting the size of Si nanorods. As shown in Figs. 1(b) and 1(c), at a fixed wavelength of 1400 nm, the transmittance |T_cross|² and the two-dimensional color map of phase φ_cross of metasurfaces with different Si nanorod sizes are numerically simulated. It can be observed that there is a high transmittance area (transmittance over 90%), in which two silicon nanorod sizes (circles in Fig. 1(b), (c)) are selected to provide an incremental phase shift of π for transmission deflection. In the chosen two parameters 1 and 2, the width w of silicon is 0 nm and 220 nm, respectively, and the height t is 600 nm.

 

Fig. 1. Concept and numerical design of beam deflection realized by all-dielectric metasurfaces. (a) Schematic illustrating that the metasurfaces deflect the beam to first-order diffraction at an angle of about 45 degrees. On the left, the width and height of a single silicon nanorod are w = 220 nm and t = 600 nm, respectively. The black dashed line box on the right represents a “blank structure” without silicon nanorods, and the unit cell period p is 2120 nm. (b) Simulated transmission amplitude |T_cross|² for Si metasurfaces with different geometries at a fixed wavelength of 1400 nm. (c) Simulated transmission phase φ_cross of Si metasurfaces with different geometries. The tables inset in the figure are our parameters corresponding to the green circles 1 and 2 in the (b) and (c) plots. (d) Simulated diffraction spectroscopy of all-dielectric metasurfaces. Inset: Schematic illustration of device incidence and diffraction, with k_inc representing the incident wave vector. Diffraction arrow colors correspond to curve colors in the spectrum.

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Next, we use the above silicon nanorod parameters to create periodic metasurfaces to simulate the transmission beam deflection characteristics. Figure 1(d) shows the diffraction efficiency of the allowable order within the wavelength of 1350 nm - 1550 nm. A transmittance close to 45% is observed in both the designed +1st and -1st diffraction orders, and the transmittance of the 0th-order diffraction is close to zero in the entire range. According to the generalized Snell's law [32] sinθ = λ/n_T p, where θ is the deflection angle of the transmitted light, λ is the free-space wavelength, n_T is the refractive index of the transmitted environment (in our design, the refractive index of air), and p is the period of the metasurfaces. In our structure, λ = 1350 nm - 1550 nm, p = 2120 nm, n_T= 1, so the deflection angle of transmitted light θ = 40° - 47°. Therefore, when the incident light passes through our beam deflection metasurfaces, it will be completely turned to the θ and -θ directions, and the transmitted energy along the wave vector direction approaches 0.

In order to produce high-Q, we are considering introducing GMR. It should be noted that silicon nanowires in our design could also act as waveguides [16]. Figure 2(a) shows the dispersion curves of the five lowest-order guided modes of the metasurfaces within the wavelength range of 1350 nm - 1550 nm. Figure 2(b) shows the normalized electric field distribution of the five guided modes in this structure. As can be seen, these modes are usually confined within or between nanorods. We choose mode 3 and mode 4 to study, in which the fields of the Mode 3 are mostly localized in the gap regions of the nanowires, while the fields of the mode 4 are mostly distributed inside the nanowires. In order to excite GMR, a periodic arrangement of grooves with a periodicity of Λ is introduced along the silicon nanowires to provide reciprocal vector 2π/Λ for compensating the momentum mismatch between the free-space wave and the guided modes, i.e., 2π/Λ = n_eff2π/λ₀, where n_eff is the effective refractive index of the guided mode and λ₀/n_eff represents the guided mode wavelength. It could be seen from Fig. 2(a) that the mode 3 and mode 4 (blue and red lines) will leak into the free space within the free-space wavelength range of 1350 nm - 1550 nm provided that the groove periodicity of Λ is chosen between 622 nm and 667 nm. As indicated by a horizontal dashed line in Fig. 2(a), the periodicity of Λ is assumed to be Λ = 654 nm in the following discussion, which ensures that the mode 3 and mode 4 could resonant around 1400 nm and 1500 nm, respectively.

 

Fig. 2. Concept and parameter design of introducing BIC effect and quasi-BIC effect into all-dielectric metasurfaces. (a) Waveguide dispersion in all-dielectric metasurfaces. The ordinate represents the BIC wavelength, the abscissa corresponds to the free-space wavelength (λ), and the dashed line represents Λ = 2π/k = 654 nm. (b) The electric field distributions correspond to the five waveguide modes, color-coded to match the dispersion map. |E|/|E₀| represents the magnitude of the electric field E normalized by the incident field magnitude. (c) Enlarged top view of the metasurfaces with the BIC effect introduced, the width of a single silicon nanorod is w = 220 nm, the groove length and depth are l = 80 nm and d = 30 nm, respectively, and the period is Λ = 654 nm. (d)Front view of Si nanorods with grooves and schematic diagram of the BIC effect. The red dotted line corresponds to the periodic grooves with a groove depth d = 30 nm. In Si nanorods, the radiation mode outside the groove and the scattering mode inside the Si nanorod form an “accidental” BIC effect through intra-waveguide interference. (e) Simulated diffraction spectra of a metasurface introducing the BIC effect. (f) Enlarged top view of metasurfaces with broken groove symmetry. (g) Simulated first-order diffraction spectra of metasurfaces with broken groove symmetry. The modes at wavelengths λ = 1400 nm and 1503 nm correspond to the intersections of the grey dashed lines with modes 4 and 3 in (a). (h)Simulated 0th order diffraction spectra of metasurfaces with broken groove symmetry. The mode Q values for wavelengths λ = 1400 nm and 1503 nm are 4786 and 2679, respectively.

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Figure 2(c) shows a schematic diagram of the structure with the groove periodicity of Λ = 654 nm, the groove depth of d = 30 nm, and the groove length of l = 80 nm. Figure 2(e) shows the front view of the grooved silicon nanorods and a schematic diagram of the BIC effect. The symmetry of the groove structure introduces the internal interference between the different diffraction channels, which can significantly suppress the sidewall diffraction and produce the “accidental” BIC effect that limits the diffraction [33]. This effect is reflected in the diffraction spectrum that the resonance peak linewidth tends to 0, as shown in Fig. 2(d). When we break this symmetry, it will trigger the transition from BIC to quasi-BIC. Figure 2(f) shows a schematic diagram of a Si waveguide with broken symmetry. The GMR supports measurable far-field resonance, that is, it is a quasi-bound state. What is important is that the excitation of GMR only occurs in a very narrow range of wavelengths, resulting in a strong resonance field and thus a high-Q.

The first-order diffraction spectral response of the groove symmetry-breaking structure is shown in Fig. 2(g). It is clearly seen that the introduction of grooves only causes quasi-BIC at wavelengths of 1400 nm and 1503 nm, which are mode-matched with Mode 4 and Mode 3, and will not affect the diffraction efficiency of other wavelengths. As shown in Fig. 2 (h), the 0th-order diffraction has high-Q resonance at wavelength λ = 1400 nm and 1503 nm, and the Q factors are 4786 and 2679, respectively. Therefore, when the incident light passes through the metasurfaces, only the light less than 1nm near the working wavelength can propagate along the direction of the wave vector, and the light of other wavelengths will be deflected to the first-order diffraction in the direction of the angle θ, achieving extremely narrow wavelength filtering.

We conduct further research on the symmetry breaking of the designed groove symmetry metasurfaces and understand the transition process from BIC to quasi-BIC in more detail. Figure 3(a) shows two design ideas. The first is to change the depth d of the groove on the right, and the second is to change the position y of the groove on the right. Firstly, we set the groove depth d₂ on the right side to 0, 10, 20, 30, 40, 50, 60 nm, and the groove depth d₁ on the left side is fixed to 30 nm, Fig. 3(b) shows the 0th-order diffraction efficiency map of these structures; secondly, we fix the left groove position y₁ = 0, and adjust the right groove position y₂ in the S direction, Fig. 3(c) shows the 0th-order diffraction efficiency map corresponding to this scheme. The symmetry breaking can be achieved through the above two schemes. The sharp peaks in the spectra of Fig. 3(b) and Fig. 3(c) are features of quasi-BIC. They can fit Lorentzian or Fano shapes well without any obvious background or side peaks. As the degree of groove symmetry becomes higher and higher, the spectral linewidth of the quasi-BIC decreases significantly with the red shift or blue shift of the resonance wavelength, until (d₁, d₂) (30, 30) or (y₁, y₂) (0, 0), the resonance peak disappears in the spectrum, and the quasi-BIC changes to BIC. We further extracted the Q values of several structures in Fig. 3(b) and Fig. 3(c), and plotted them in Fig. 3(d). The maximum Q value reached an astonishing 36922 (d₁, d₂ = 30, 20), which provides a theoretical basis for achieving ultra-narrow linewidth filtering.

 

Fig. 3. The effect of the degree of groove symmetry breaking on the BIC effect. (a) Enlarged top view of a metasurface with two grooves. The width of a single silicon nanorod is w = 220 nm, the groove length is l = 80 nm, the left groove depth d₁ = 30 nm, the right groove depth d₂ is a variable, and the period is Λ = 654 nm. The red axis coordinates are the scale bar of the position y₁ and y₂ of the groove in the S direction. (b) and (c) are the simulated diffraction spectra corresponding to changing the groove depth d₂ on the right side of the device and the position y₂ in the S direction, respectively. (d) Change the Q-value of the formant corresponding to groove depth (black pentagram) or position (red circle).

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Traditional filter designs are static, and their working wavelengths are fixed once they are manufactured. Therefore, it is of great significance to design a dynamically tunable filter. We adopted an electro-optical modulation scheme, adding a layer of 5 nm thick ITO (SnO₂:In₂O₃) between the sapphire substrate and the silicon nanorods. ITO thin film is a transparent conductive oxide material with high carrier concentration and low resistivity, which has become an ideal material for tuning surface plasmon resonance in the near-infrared range [34]. Changing the carrier concentration of ITO can realize the dynamic regulation of the optical mode at the ITO interface [35], and the regulation of the material carrier concentration is one of the important methods to realize electro-optic modulation. There are many ways to adjust the ITO carrier concentration, such as changing the ITO applied voltage [36], changing the SnO₂ doping concentration [35], changing the annealing temperature and annealing gas [37], and changing the deposition temperature [38].

We use voltage regulation to change the carrier concentration at the Si/ITO interface, resulting in a dramatic change in the refractive index n_ITO of ITO (n_ITO varies from 1.96 to 0.57) [34]. In the simulation, we set n_ITO to 1.96, 1.265, and 0.57, respectively. Figure 4(a) shows the guided mode dispersion of mode 3 (dashed line) and mode 4 (solid line) under different refractive indices of ITO. Figure 4(b) shows that when no voltage is applied (n_ITO = 1.96), mode 3 is confined in the silicon nanorods, and mode 4 partially leaks into the substrate ITO. Based on the difference between mode 3 and mode 4, we have reason to believe that when the refractive index of the substrate changes, mode 4 will be greatly affected while mode 3 will not [33]. We set Λ = 654nm, Fig. 4(c) shows the resonance curves of the two modes. The resonance peak corresponding to mode 4 undergoes a significant blue shift with the decrease of n_ITO, from 1410 nm to 1391 nm. The resonant peak of mode 3 is not sensitive to the change of n_ITO.

 

Fig. 4. Dynamic adjustment of the position of the 0th-order diffraction resonance peak by changing the refractive index of the substrate ITO. (a) Waveguide dispersion of all-dielectric metasurfaces introduced into ITO substrates. The solid line corresponds to mode 4 and the dashed line corresponds to mode 3. The refractive indices of ITO are 1.96, 1.265, and 0.57 corresponding to black, red, and blue lines. (b) 0th-order simulated diffraction spectra of all-dielectric metasurfaces varying the refractive index of the substrate ITO. The long-wavelength and short-wavelength regions correspond to modes 3 and 4, respectively. (c) The electric field distributions corresponding to waveguide modes 3 and 4, from top to small, ITO refractive indices are 1.96, 1.265, and 0.57, respectively.

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4. Conclusion

Using finite element software for simulation, we first proved that the all-dielectric metasurfaces achieved beam deflection through a reasonable phase gradient design, and the transmittance along the wave vector approached zero; then, by introducing the quasi-BIC effect, light of a specific wavelength can continue to propagate in the direction of the wave vector through the metasurfaces, realizing the function of ultra-narrow linewidth filtering. Moreover, we introduce an electro-optical modulation scheme, which enables dynamic control of the device's operating wavelength range up to 19 nm according to our calculations. It is worth noting that the Q value of the transmission window with the center-wavelength of ∼1397 nm in our proposed band-pass filter is up to 36922, and the FWHM is only about 0.05 nm, which is two orders of magnitude smaller than the typical FWHM (from a few to several tens of nanometers) in the commercially available near-infrared band-pass filters. However, the spectral range of blocked light (defined as the transmission less than 1%) on either side of the transmission window in our proposed filters is from 1370 nm to 1410 nm, which is smaller than the background suppression band (typically several hundreds of nanometers) of the commercial bandpass filters. To broaden the blocking range, materials with broad blocking ranges could be used for or coated onto the substrate, which has already been utilized in the commercial filters. Our proposed design represents a general method for sub-wavelength metasurfaces filtering and could have applications in the design of narrow-band, lightweight, reconfigurable, and efficient filtering devices.