Dynamically Tunable Singular States Through Air-Slit Control in Asymmetric Resonant Metamaterials

Abstract

This study presents a novel method for dynamically tuning singular states in one-dimensional (1D) photonic lattices (PLs) using air-slit-based structural modifications. Singular states, arising from symmetry-breaking-induced resonance radiation, generate diverse spectral features through interactions between resonance modes and background radiation. By strategically incorporating air slits to break symmetry in 1D PLs, we demonstrated effective control of resonance positions, enabling dual functionalities including narrowband band pass and notch filtering. These singular states originate from asymmetric guided-mode resonances (aGMRs), which can be interpreted by analytical modeling of the equivalent slab waveguide. Moreover, the introduction of multiple air slits significantly enhances spectral tunability by inducing multiple folding behaviors in the resonance bands. This approach allows for effective manipulation of optical properties through simple adjustments of air-slit displacements. This work provides great potential for designing multifunctional photonic devices with advanced metamaterial technologies.

1. Introduction

Metamaterials and metasurfaces have been widely recognized as transformative platforms for controlling electromagnetic waves [1,2,3]. These advanced materials utilize subwavelength structures to achieve remarkable electromagnetic properties, such as negative refraction, flat lensing and perfect light absorption [4,5,6]. By engineering their composition at the nanoscale, these systems enable accurate control over the amplitude, phase, and polarization of light [7]. Depending on their dimensionality (i.e., 1D, 2D or 3D), these periodic structures provide compact solutions for manipulating light–matter interactions [8,9,10].

The realization of such advanced materials relies on several physical concepts such as effective medium theory (EMT) [11], lattice resonances, particle scattering resonances [12], and surface plasmon effects [13]. EMT uses macroscopic effects from complex subwavelength interactions, which generates extraordinary properties such as negative refractive indices or anisotropic behavior [11,14]. Lattice resonances, namely guided-mode resonances (GMRs), arise from the coupling of Bloch eigenmodes with diffracted waves in periodic arrays [15,16]. These periodic structures exhibit diverse spectral responses with high quality factor (high-Q) resonances, which enable them highly desirable for wavelength-selective photonic devices [17]. Additionally, particle scattering resonances driven by electric or magnetic dipoles in individual structures facilitate strong light–matter interactions and related phenomena [12,18].

Recently, among various mechanisms, the concept of bound state in the continuum (BIC) has attracted significant attention in metasurface research, which are non-radiative states embedded within resonance modes under perfect symmetry condition [19,20]. By controlling symmetrical breaking such as off-normal incidence or structural asymmetry, BICs can transition into radiative modes known as quasi-BICs, which shows a great potential for innovative devices including narrowband filtering, optical switching, and biosensing and plasmonic devices [21,22,23,24]. Recent advancements have revealed a new type of resonance in photonic lattices (PLs), where strong resonances radiate at isolated spectral positions from adjacent bands when symmetry is broken. This phenomenon, termed the singular state [25], represents an exotic spectral response of quasi-bound states in the continuum (quasi-BICs). These states emerge through asymmetric guided-mode resonances (aGMRs) where diffracted light couples to leaky modes in perturbed waveguide systems [26].

In this work, we present a novel method for dynamically tuning singular states by employing PLs with air-slit-based structural modifications. This approach enables effective control over resonance positions and dual functionalities, including narrowband reflection and notch filtering. By incorporating multiple air-slits, the resonance bands exhibit enhanced spectral tunability through the induction of multiple folding behaviors, highlighting the significant potential of these tunable singular states for the development of versatile photonic devices.

2. Air-Slit-Induced Singular States

To investigate tunable singular states, we broke the symmetry of a simple one-dimensional (1D) photonic lattice (PL) by incorporating an air slit. Figure 1a illustrates the 1D resonant structure, defined by grating parameters such as period (Λ), fill factor (F), and grating height (H) with the refractive index (n). Continuing the study of singular states, we employed the previous design of the 1D PL [25], where the grating parameter set is {$\Lambda$ = 1 μm, $F$ = 0.5, $H$ = 0.5 μm} with a lossless material ($n$ = 2). In this study, we characterized the zeroth-order reflectance ($R_0$) using the rigorous coupled-wave analysis (RCWA) [27] method implemented in commercial software [28]. The analysis is performed using polarized light, with the electric field parallel to the grating grooves (TE polarization), illuminating the 1D photonic lattice (PL) in the air ($n_{air}$ = 1). As shown in calculated $R_0$ map of Figure 1a, wideband perfect reflection is observed in the spectral region due to the resonant radiation, which has been understood by the guided-mode resonance (GMR) phenomenon [15]. As the height ($H$) increases, additional resonance bands are generated by excitation of higher GMR modes. Figure 1b presents the case where an air-slit with a width of 0.05 μm is introduced at the center of the grating, preserving the structural symmetry. The $R_0$ map shows resonance bands like those observed without the air slit (i.e., Figure 1a) but only a slight narrowing observed. This narrowing can be attributed to the effective refractive index reduction due to the presence of air. In contrast, the resonance band is significantly changed when the symmetry is broken by shifting the air slit. Figure 1c shows the $R_0$ map of the 1D PL with air slit displaced 0.05 μm from the center. The asymmetry of the 1D PL leads to the formation of a sharp band, which can be interpreted as an off-BIC (bound state in continuum) [19] or asymmetric GMR [25]. Observed in the asymmetric system, these resonance bands can be classified as asymmetric guided-mode resonance (aGMR) modes owing to their origin in asymmetric excitation of GMR modes. Notably, shown in the $R_0$ map, the fundamental aGMR mode manifests as a singular state, distinctly separated from the other resonance bands. This singular state will be shown to be tunable through manipulation of the air slit.

3. Interpretation of Singular States

Singular states can be interpreted by analyzing the slab waveguide modes of an equivalent 1D PL. As depicted in Figure 2a, the 1D PL can be approximated as a slab waveguide where electromagnetic fields are confined through the coupling of diffracted waves to a propagation constant ($\beta$). For this equivalent slab waveguide, the 1D grating layer is homogenized using effective medium theory (EMT) with the Rytov’s formalism [29]. The effective refractive indices ($n_m^{TE}$ and $v_m^{TE}$) are determined by the mth order solutions from the following equations [29,30]:

$${\displaystyle \frac{\sqrt{n^2-{(n_m^{TE} )}^2} }{\sqrt{1-{(n_m^{TE} )}^2 }}}={\displaystyle \frac{tan\left\{\frac{\pi \Lambda }{\lambda }(1-F)\sqrt{1-{(n_m^{TE} )}^2 }\right\}}{tan\left\{\frac{\pi \Lambda }{\lambda }F \sqrt{n^2-{(n_m^{TE} )}^2 }\right\}}} (\mathrm{symmetric}),$$

(1)

$${\displaystyle \frac{\sqrt{n^2-{(v_m^{TE} )}^2} }{\sqrt{1-{(v_m^{TE} )}^2 }}}={\displaystyle \frac{tan\left\{\frac{\pi \Lambda }{\lambda }F \sqrt{n^2-{(v_m^{TE} )}^2 }\right\}}{tan\left[\frac{\pi \Lambda }{\lambda }\right(1-F\left)\sqrt{1-{(v_m^{TE} )}^2 }\right]}} (\mathrm{asymmetric}).$$

(2)

Here, the $n_m^{TE}$ and $v_m^{TE}$ represent the symmetric and asymmetric field EMT solutions from each Equations (1) and (2). In case of TM polarization, the EMT solutions are also given by previous works [29,30]. By using the asymmetric solutions of $v_m^{TE}$, the EMT model can estimate the quasi-BIC wavelength at which resonant radiation begins when symmetry is gently broken. In this analysis, we focus on the first-order diffraction because the singular states primarily couple with the first-order diffraction modes in this 1D PL. Considering the first diffraction waves, we apply the first solution $n_1^{TE}$ and $v_1^{TE}$ for slab waveguide analysis. The eigenvalue problem of the lth TE mode of the slab waveguide can be expressed as [26,31]:

$$tan\left\{q_1{\displaystyle \frac{H}{2}}-l{\displaystyle \frac{\pi }{2}}\right\}={\displaystyle \frac{\sqrt{{\left(\beta \right)}^2-{\left(k_0\right)}^2}}{q_1}}l=0, 1,\dots ,$$

(3)

where the $q_1$ and $k_0$ are vertical propagation vectors of first-order diffracted waves and propagation constant ($2\pi /\lambda$) in vacuum, respectively. Based on the different EMT solutions ($n_1^{TE}$ and $v_1^{TE}$), the $q_1$ is given by $n_1^{TE}{k}_0$ or $v_1^{TE}{k}_0$. The $\beta$ represents the propagation constant of the guided mode, assuming coupled from the first-order grating vector ($K=2\pi /\Lambda$). Consequently, the lth TE guided modes (${TE}_l$ or $a{TE}_l$) are obtained as eigen-solutions using symmetric and asymmetric effective refractive indices. Figure 2b displays the calculated ${TE}_l$ or $a{TE}_l$ modes, which matches the resonance bands observed in the asymmetric air-slit 1D PL in Figure 1c. Indeed, the singular state originates from the $a{TE}_0$, indicating the fundamental guided mode of asymmetric EMT slab coupled by first-order diffracted waves.

4. Analysis of Air Slit-Induced Singular States

To characterize the ${aTE}_0$ mode of the singular state, we analyze the electric field profiles at specific points on the $R_0$ spectra of both the symmetric and asymmetric air-slit 1D PLs. Figure 3a shows points (i)–(iv) on the $R_0$ spectra for the air-slit 1D PLs. In contrast to point (i) on the spectrum of the symmetric air-slit 1D PL (black dash line), the point (ii) on the asymmetric air-slit 1D PL (red solid line) spectrum exhibits a sharp resonance peak at λ = 1.4266 µm, associated with a singular state. In the shorter-wavelength region, the point (iii) corresponds to high reflection while point (iv) shows a sharp dip at $\lambda$ = 1.0734 µm. The electric field distributions ($E_y$, as out of plane) corresponding to these points are presented in Figure 3b–e. To differentiate between symmetry and asymmetric field distribution, we included the real values of the $E_y$ scale in these plots. As shown in Figure 3b, the light propagates through the 1D PL without resonance. In Figure 3c, however, the light is strongly confined to the lattice with asymmetric air slit. As this asymmetric structure transitions toward symmetry, the aGMR generates a high-Q resonance exhibiting a highly confined field. This $E_y$ profile represents the singular state, formed by the fundamental guided modes (i.e., ${aTE}_0$). In Figure 3d, the light is highly reflected from the symmetric air-slit 1D PL. Conversely, in Figure 3e, it passes through in the asymmetric air-slit 1D PL due to interaction with enhanced $E_y$ field, forming the first guided modes (i.e., ${aTE}_1$).

5. Tunable Singular States by Asymmetric Air-Slit 1D Lattice

Prior to designing tunable singular states, we examine the refractive index ($n$) effects on the singular state in the asymmetric air-slit 1D PL. Figure 4a presents the $R_0$ map for the symmetric air-slit 1D PL, where the spectral response is calculated by varying the $n$ from 1.5 to 3.5. As observed, two resonance bands shift toward longer wavelengths as $n$ increases. This red shift is a characteristic behavior of GMRs as higher $n$ increases a propagation constant of guided mode. Figure 4b shows the corresponding $R_0$ map for the asymmetric air-slit case under the same refractive index variations. As expected, sharp resonance bands appear and exhibit a red shift. Notably, the singular state becomes increasingly isolated from the other resonance bands for higher-refractive-index materials such as silicon nitride (Si₃N₄, n ≈ 2) and silicon (Si, n ≈ 3.5).

The aGMR enables the generation of tunable singular state by varying asymmetry of the air-slit 1D PL. Figure 5 shows the tunable singular state in an air-slit 1D PL based on a dielectric material ($n$ = 2), where the grating parameter set is {$\Lambda$ = 1 µm, $F$ = 0.5, $H$ = 0.5 µm}. As illustrated in the schematic of Figure 5a, a single air-slit with a width of 0.05 µm is displaced by a distance $d$ from the symmetric line. To estimate the quantity of asymmetry, we derived the asymmetry index (AI) by

$$AI={\displaystyle \frac{{\int }_0^{\Lambda /2}\left|x-x_0\right|·\left|n\left(x,d\right)-n\left(-x,d\right)\right|dx}{{\int }_0^{\Lambda /2}\left|x-x_0\right|dx}}$$

(4)

where the $n\left(x,d\right)$ denotes the refractive index profile as a function of spatial coordinate $x$ and parameter $d$. The $\left|x-x_0\right|$ term assigns greater weight to asymmetries occurring farther from the symmetry line $x_0$. As shown in the plot, the AI steadily increases, reaching its maximum value at $d$ = 0.225 μm in the single-air-slit 1D PL. The AI then gradually decreases to zero as the structure becomes symmetric. In the $R_0$ map of the asymmetric air-slit 1D PL, two sharp resonances are observed due to the aGMRs. The lower peak corresponds to the ${aTE}_0$ mode, while the upper dip represents the ${aTE}_1$ mode. These modes arise from the interference between leaky waveguide modes and background radiation, as a phenomenon characteristic of Fano resonances. At the low reflection region, the $R_0$ peak can be used as a notch filter. In contrast, the $R_0$ dip around the high reflection band for band-pass filter. When the $d$ gradually increases from zero to ±0.15 µm, these dual functionalities can be simultaneously tuned by varying the $d$. Compared to the AI plot, the tuned wavelength shows good agreement with the quantitative asymmetry of the single air slit.

This dual functionality becomes even more dynamic with the introduction of multiple air-slits. As shown in the schematic of Figure 6a, two air-slits are aligned with a pitch width of $F\Lambda /2$ = 0.25 μm. By varying the displacement $d$ from zero to ±0.35 μm, as seen in $R_0$ map, the two sharp aGMRs are rapidly tuned. This behavior corresponds to the one-fold radiation observed in the single-air-slit case of Figure 5. Furthermore, the AI profile for two air-slits closely matches the folded resonant wavelengths, where the AI is calculated using Equation (4) with $x_0=\pm F\Lambda /2$. Using triple and quadruple air-slits, the aGMRs exhibit even faster tuning due to two- and three-fold radiation effects as presented in Figure 6b,c. Herein, the triple and quadruple air-slits are equally distributed within the same pitch width ($F\Lambda /2$ = 0.25 μm). These additional slits introduce more degrees of asymmetry as represented corresponding AI profiles, thus leading to more change in the spectral position of the aGMRs. Meanwhile, these resonances shift toward shorter wavelengths (blue-shift) as the number of air-slits increases because the effective refractive index is reduced by more air gaps. This ability to dynamically tune singular states by simply adjusting the multiple air-slit promise for applications in metamaterials and metasurfaces.

6. Discussion

We present a novel design and analysis of tunable singular states in 1D PLs through air-slit-based structural modifications. While previous studies primarily examined singular states as exotic phenomena arising from isolated resonance modes [25], this approach employs controlled asymmetry to dynamically tune these states. By introducing air-slit modifications, we demonstrate tunable bandpass and notch filtering in the singular states. Analytical modeling of equivalent slab waveguides validates that air-slit-induced singular states originate from asymmetric guided-mode resonance (aGMR). When incident light is diffracted by the broken symmetry, it couples to an equivalent slab waveguide undergoing an asymmetric effective medium, as formulated by Rytov’s EMT [26,29].

Air-slit PLs effectively control structural asymmetry, serving as a dynamically tunable platform for optical filtering. Notably, the multiple air-slits significantly enhance spectral tunability by inducing multiple folding behavior in the resonance bands. The analysis of the asymmetry index (AI) of these multiple air-slits clearly reveals the relationship between displacement-induced symmetry breaking and the dynamic tuning of singular states. This insight provides a clear understanding of the optimization potential for these structures in advanced filtering applications.

Compared to existing photonic devices, our tunable singular states in 1D photonic lattices (PLs) provide a compact and efficient solution for manipulating metamaterials. Notably, we achieve compatible tuning ranges of up to 70 nm with high Q-factors (i.e., λ/∆λ) ranging from ~10² to ~10³ in both notch and bandpass filtering applications. For instance, surface lattice resonances of gold arrays have achieved a Q-factor of up to 2340 but with limited tunability [32]. In another study, a 1D photonic crystal structure composed of 12 layers of Nb₂O₅ and SiO₂ films achieved a transmission peak of 91.37% with a Q-factor of 419 [33]. Theoretically, this structure demonstrated a wide tuning range from approximately 1245 nm to 1850 nm by varying the thicknesses of each multilayer film. In a different approach, by varying the angle of incidence from 0° to 5.3°, two resonance peaks are tuned as their separation increases from 23 nm to 78 nm in switchable graphene-based photonic lattice structures [34]. While these results are promising, there are other photonic structures that offer higher performance. For example, topological photonic crystals have demonstrated ultra-high Q-factors > 10⁸ with high transmission, although these require highly specific conjugated structures and optimization for tunable designs [35]. Recently, silicon-based guided-mode resonance (GMR) structures have demonstrated Q-factors as high as 2.39 × 10⁵, with tunability from 1525 nm to 1570 nm by varying the grating period [16].

However, fabrication for these air-slit 1D PLs is still challenging, primarily due to the requirement for sub-nanometer pitch deep etching processes. For silicon material, the Bosch DRIE (deep reactive ion etching) process achieves aspect ratios exceeding 120:1 for 35 nm-wide trenches via cryogenic etching [36]. To achieve widths comparable to or larger than 35 nm for the air-slits, the dimensions of the air-slit 1D PLs need to be scaled up by at least seven times. This scaling would allow the structure to operate in the long-wave infrared region (λ = 8–14 μm), where dynamic tunable filtering can be expected. Notably, these air-slit PLs provide tolerant tunability to variations in the void factor (${VF}^{\prime }=VF\left(1+\Delta VF\right)$) of air-slits. The effects of $\Delta VF$ deviations on the quadruple air-slit PL are illustrated in Figure 7, which presents $R_0$ maps for $\Delta VF$ deviations of (i) −40%, (ii) −20%, (iii) 20% and (iv) 40%, with all other parameters consistent with those in Figure 6c. Notably, even when $\Delta VF$ is reduced to −40%, both filtering channels maintain their functionality, but the operational tuning range decreases by 35%. This reduction is less degraded at $\Delta VF$ = −20%, where the tuning range decreases by only 8%. Positive $\Delta VF$ deviations exhibit distinct effects, allowing only the bandpass filtering channel. At $\Delta VF$ = 20%, this channel benefits from an increased tuning range. However, at $\Delta VF$ = 40%, the bandpass channel undergoes significant spectral broadening.

7. Conclusions

In conclusion, the asymmetric guided-mode resonance (aGMR) mechanism enables the radiation of tunable singular states by introducing structural asymmetry in photonic lattices (PLs) through air-slit modifications. Our analysis demonstrates that the air-slit-induced singular state can effectively control the spectra position of strong resonance interfering with background radiation. Through systematic investigation, we have shown that the refractive index plays a critical role in determining the spectral position and isolation of singular states. Dielectric materials such as silicon nitride (Si₃N₄) and silicon (Si) are proper for realizing tunable singular states due to their high refractive indices and compatible fabrication technologies. Additionally, the incorporation of multiple air-slits significantly enhances spectral tunability by inducing folding behaviors in resonance bands, enabling dynamic control over resonance positions and achieving dual functionalities such as narrowband band-pass filtering and notch filtering. These results will be useful for versatile metasurface and metamaterial applications in optical filters, switches and photonic devices. Despite potential fabrication challenges such as precise air-slit alignment and symmetry control, these structures are feasible with advanced nanofabrication techniques. Furthermore, the proposed concepts are extendable to other spectral regions and polarization-independent designs using two-dimensional metastructures, which will offer substantial promise for developing highly tunable optical devices for advanced photonic applications.