High Quality Factor Unidirectional Guided Resonances in Etchless Lithium Niobate Metagratings for Polarization Modulation

Abstract

Unidirectional guided resonances (UGRs), as distinctive resonant eigenstates in planar photonic lattices, exhibit unique capability of emitting light in a single direction. In this work, UGRs with high-Q factor and infinite proximity to the $\mathsf{\Gamma }$-point infinitely using etchless lithium niobate (LN) metagratings are proposed and investigated numerically. By adjusting the parameters of metagraings, the Q-factor and asymmetric radiation ratio of UGRs can be flexibly tuned, and the wavelength center of UGRs respect will move with respect to the wave vector along the $\mathsf{\Gamma }$-X direction. Accompanied by the optimizing of asymmetric radiation ratio, the evolution of two dispersion curves from avoided crossing to crossing can be observed. Furthermore, leveraging the polarization sensitivity of UGRs, we achieve a broadband linear-to-circular polarization conversion with a high polarization extinction ratio. This work advances the fundamental understanding of UGRs while potentially offering promising applications in metagratings-based surface-emitting lasers, beam steering, and refractive index sensors.

1. Introduction

Resonances in slab-type photonic structures have attracted significant interest for designing optoelectronic devices over the past decade [1]. Bound states in the continuum (BICs) are exotic electromagnetic states that lie inside the continuum and coexist with radiating waves but remain perfectly confined without any radiation [2]. Through destructive interference or symmetry protection, BICs prohibit resonances from radiating into free space, allowing for infinite quality (Q) factor and topological protection based on polarization singularity [2,3,4]. When perturbations are introduced to break the symmetry, BICs evolve into quasi-BICs, which are coupled to radiating states, and possess a Fano resonance line shape with a finite but ultrahigh-Q factor [3,4]. According to their different formation mechanisms, BICs are generally split into two categories: symmetry-protected BICs at $\mathsf{\Gamma }$ point and accidental BICs at off-$\mathsf{\Gamma }$ point [5,6,7]. In the past two decades, enormous efforts have been devoted to exploring their topological physical phenomena as well as practical applications of BICs, such as lasers, sensors and so on [8,9,10,11].

Recently, the concept of unidirectional guided resonances (UGRs) has been proposed, and ultrahigh unidirectional radiation has been realized [12,13,14]. When a UGR is assisted in the photonic crystal slabs, the radiation on one side of the slab is totally canceled and the radiation on the other side is at a particular direction [15,16]. As peculiar eigenmodes of the photonic systems, UGRs are closely related to the concept of BICs, and hence UGRs are also referred to as unidirectional BICs [17,18,19]. From the perspective of topology, UGRs and BICs are fascinating topological phenomena observed in planar photonic structures [17]. BICs have been demonstrated as polarization vector singularities carrying integer topological charges in the momentum, where far-field radiative coefficients are strictly equal to zero [17,20,21]. In contrast, UGRs, also acting as vortex centers, carry topological charges but with finite Q factors, enabling their unidirectional radiation [13,22]. In another word, the UGRs have radiative loss to only the down or up side and can induce large local field enhancement. This distinctive directional radiation stems from the inherent topological characteristics of these resonances [23,24]. Unlike traditional methods, these topologically enabled UGRs obviate the need for additional reflectors, presenting a streamlined approach for unidirectional light emission [25,26]. It has been proposed theoretically or experimentally in various type of optoelectronic devices via UGRs, such as all-pass phase shifters, perfect absorbers and dynamic switch [14,23,27]. However, the quality (Q) factors and polarization characteristics of UGRs are crucial for practical applications. Though high-Q UGR mode empowered by a photonic crystal slab with tilted sidewalls has been experimentally demonstrated [12], for most of the previously reported UGRs, their Q factors remain at the level of a few hundred, which is one of the main obstacles for UGRs.Nevertheless, constrained by the in-plane symmetry protection, UGRs usually take place at off-$\mathsf{\Gamma }$ points locations in momentum space randomly and the resonance can only be excited by obliquely incident light. As a result, it not only leads to limited Q factors but also brings a technical challenge to practical applications. Very recently, the work of confining UGR states at the $\mathsf{\Gamma }$ point through designing tilted sidewalls, i.e., breaking the symmetries of structure in photonic crystal slabs, has been reported [24]. However, the Q factors of UGRs will decline rapidly once UGRs deviate the $\mathsf{\Gamma }$ point so that is dependent on the tilt angle of sidewalls profoundly. To attain wider range and high-Q factors, it becomes imperative to enable the migration of UGRs around the $\mathsf{\Gamma }$ point by means of designing structural parameters without symmetry-breaking. Lithium niobate (LN), a type of material exhibiting strong Pockels electro-optical (EO) effects, has been used for tunable metadevices [28,29,30,31]. Combining the LN with the high-Q property of UGR, electrically tunable UGR based on LN slab-type devices can be expected.

In this work, we investigate the Q factors and radiation of UGRs supported by etchless LN vertical gratings in air, which can be easily fabricated using current nanofabrication technologies. The symmetric protected BIC is found at the $\mathsf{\Gamma }$ point, and this BIC will be transformed into UGR modes. By varying the lattice constant of silicon bars, the UGR mode can remain at the area around the $\mathsf{\Gamma }$ point within a certain range rather than moving away from the $\mathsf{\Gamma }$-point immediately. Therefore, the Q factors of UGRs can maintain a high value corresponding to the range. Meanwhile, the Q factor and asymmetric radiation ratio have been improved to ${10}^5$ and above 80 dB by adjusting the lattice constant. After optimizing the parameters, we select the structure that both its transverse electric (TE) mode and transverse magnetic (TM) mode of UGRs near the $\mathsf{\Gamma }$ point at the same time. A broadband polarization conversion from linearly polarized light to circularly polarized light is realized. Compared with the migration of UGRs to the $\mathsf{\Gamma }$ point by means of designing sidewalls, this proposed vertical gratings may reduce some fabricating difficulties in practical applications.

2. Results and Discussion

The schematic of our proposed etchless LN metagratings in air (n_air = 1) is displayed in Figure 1a. The heteronanostructure consists of an x-cut thick LN thin film bonding on a quartz substrate (n_SiO₂ = 1.46). The periodic silicon grating with a lattice constant a is deposited on the LN layer along the y direction, and h₂ donates the fixed thickness of the LN film. Owing to the high index contrast between the LN layer and Si ribbon resonators, together with the noticeable leaky channels for the electromagnetic fields to the EO material, BIC and UGR resonant modes can be designed. The specific parameters are shown in Figure 1b, where w and h₁ refer to the fixed width and thickness of the silicon bars with a refractive index n_Si = 3.48. We begin by considering the Si-on-LN metagratings (w = 500 nm, a = 780 nm, h₁ = 430 nm and h₂ = 150 nm) in the sub-micron scale. The refractive indexs of dielectric LN films are set as n_o = 2.21 and n_e = 2.14, respectively. We use the finite element method (Comsol Multiphysic) to perform the numerical simulation. Based on the Comsol eigenfrequency solver, a unit cell of proposed gratings is built with Floquet periodic boundary conditions along the x direction. The numerical results can be obtained, such as the dispersions, Q factors, and asymmetric radiation ratio. To ensure the convergence, the mesh of the unit cell is set as a physics-controlled and extremely fine element size in the wave optics module of Comsol. The corresponding band diagram of the TE mode along the $\mathsf{\Gamma }$-X direction is presented in Figure 1c. Here, the dispersion relations are plotted within the Brillouin zone corresponding to the grating with its period for convenience. We focus on the lowest dispersion relation (lower band); the BIC mode always takes place at the $\mathsf{\Gamma }$-point when wave vector k_x(2$\pi /a$) = 0, marked by a red triangle. The eigenstates in the case of off-$\mathsf{\Gamma }$ point are all UGR modes for this band. Figure 1d displays the corresponding Q factor, which indicates a UGR state whose high-Q factor of ${10}^5$ is located at the position closest to the $\mathsf{\Gamma }$ point approximately. Figure 1e shows the eigen electric field profile of the BIC state and the UGR mode nearest to the $\mathsf{\Gamma }$ point. Obviously, they are well localized in the grating and substrate without radiative loss for the BIC mode. On the other hand, the existence of the substrate will result in breaking up–down mirror symmetry and thus enable a potential way to achieve UGRs. In contrast, the downward radiation is lower compared with the unidirectional upward radiation in terms of the UGR mode.

Though its asymmetric radiation ratio ($\eta$) is about 50 dB, it still greatly needs to be improved as shown in Figure 2a. The asymmetric radiation ratio is defined as $\eta ={\gamma }_t/{\gamma }_b$ and that at the $\mathsf{\Gamma }$ point is nonessential, where ${\gamma }_t$ and ${\gamma }_b$ are the radiation decay rates toward the top and bottom, respectively. To confirm this BIC, a polarization vector of far-field radiation projected onto the xy plane is thus introduced in Figure 2b. The red/blue colors represent the right-handed/left-handed polarization states, and a circular-polarized state marked by a blue dot appears at the $\mathsf{\Gamma }$-point. The topological charge q of the polarization singularity is defined as follows [5]:

$$q={\displaystyle \frac{1}{2\pi }}{\oint }_L\mathrm{dk}·{\nabla }_{\mathbf{k}}\varphi \left(\mathbf{k}\right),$$

(1)

where L is a closed path in the k space that goes around the polarization singularity in the counterclockwise direction. dk is the polarization vector projected into the xoy plane, $\varphi \left(k\right)$ is the angle between polarization major axis and the x axis. From Equation (1), we can immediately draw the conclusion that BIC carries topological charges $q=-1$, further confirming the topological nature of this class of BIC mode.

After obtaining the UGR mode with a low asymmetric radiation ratio ($\eta$), we optimize it by adjusting the lattice constant. Lattice constants are easier to control compared to some other structure parameters in practical fabrication including height, the tilted angle of the side walls, and so on. The band structure in parameter space (k_x, a) is plotted as displayed in Figure 3a. We can see that the band undergoes a transition from the crossing-type into the avoided crossing-type when the lattice constant increases from 700 to 800 nm because the two leaky modes of up–down bands possess different transverse parities [32]. The black solid line refers to the path of evolution for the crossed point. Figure 3b shows the crossed band when the lattice constant a = 700 nm. The Q factor of UGRs with increasing lattice constant is depicted in Figure 3c. The Q factors can be up to ${10}^{11}$ near the $\mathsf{\Gamma }$-point and decay quadratically with respect to the wave vector k_x moving in the direction away from the $\mathsf{\Gamma }$-point. On the other hand, the lattice constant has a slight influence on the Q factor, relatively. It is clear that the Q factors of UGRs are insensitive to the lattice constant of gratings and keep high values over the area around the $\mathsf{\Gamma }$-point. Therefore, we have a larger scope to choose the modes that are close to the $\mathsf{\Gamma }$-point to obtain high-Q factors. Meanwhile, the asymmetric radiation ratio of UGRs can be modified by changing the lattice constant of gratings. Figure 3d shows the asymmetric radiation ratio of the structure within the range of lattice constant a from 700 to 800 nm. To begin with, the maximum value of the asymmetric radiation ratio increases gradually and stays at the position that infinitely approaches the $\mathsf{\Gamma }$-point with the reducing of the lattice constant until a = 760 nm approximately. Next, the lattice constant a decreases sequentially, and the central wavelength of the asymmetric radiation ratio will move towards the $\mathsf{\Gamma }$-X direction. Moreover, the peak value of the asymmetric radiation ratio will decrease accompanied by the process. As a result, it is not satisfied with the tendency to modulate for high-Q UGR modes. It is noteworthy that the wave vector k_x of the crossed point for band structure is not completely consistent with that for the peak value of the asymmetric radiation ratio, which can be found in Ref. [32] as well.

Considering both high Q and the asymmetric radiation ratio simultaneously, we select the case of lattice constant a = 753 nm. The corresponding dispersion relations of TE mode are shown in Figure 4a, where the normalized frequency $\omega =$0.474 (upper band) of UGR with high-Q factor is closest to the $\mathsf{\Gamma }$ point but completely different from it. The right part of Figure 4a shows a nearly perfect upward UGR electrical mode profile with poor downward radiation whose asymmetric radiation ratio $\eta =90$ dB given in Figure 3d. Since experimental challenges are associated with vertical incidence of light and fabrication difficulties for tilted sidewalls of the silicon bars, the influence of an extremely small angle oblique incidence on its wavelength can be almost ignored. We adapt the case of normal incidence excitation for simulation. The corresponding transmittance and reflectance are given in Figure 4b. Next, we consider the polarization sensitivity of UGRs. Figure 4c plots the simulation energy bands of the transverse magnetic (TM) mode when the lattice constant a = 753 nm, and we investigate the dispersion relation (red line) mainly. Specifically, a downward UGR ($\omega =$0.471) with an asymmetric radiation ratio of −45 dB appears close to $\mathsf{\Gamma }$ point on the TM energy band, and the mode profile of the magnetic field distribution is shown at the right panel of Figure 4c. The transmittance and reflectance of the TM mode are given in Figure 4d. It is difficult to ensure that both the UGRs of TM and TE modes close to $\mathsf{\Gamma }$ point have a large asymmetric radiation ratio simultaneously.

Benefiting from the high-Q UGR resonances, the etchless LN gratings can be used for the application of EO modulation. For taking advantage of the ${\gamma }_{13}$ element of LN, the EO coefficient tensor of x-cut LN, the applied electrode configuration can refer to Refs. [28,33]. The EO effect of LN is expressed as $\mathsf{\Delta }{n}_e=-n_e^3{\gamma }_{33}E/2$, where E denotes the electrostatic field caused by the applied voltage along the z direction, and the linear EO coefficient is ${\gamma }_{33}=33$ pm/V. No resonances can be found in the transmittance and reflectance spectra in Figure 4b,d. Figure 5a provides the phase retardation under the incidence of TE and TM plane polarization wave for the aforementioned UGR gratings, respectively. Clearly, the phase-abruption takes place at 1588.5 and 1595.4 nm of TE and TM modes, suggesting strong phase resonance with unitary amplitude. which is consistent with the normalized frequency in Figure 4a,c as well. As a result of the phase resonance, high group delay and shift of resonant wavelength could be achieved as shown in Figure 5b, and the related group delay is given by $-darg\left(r\right)/d\omega$ [34]. It can be seen that the maximum group delay is 29 ps around the resonance wavelength 1588.5 nm. With the refractive index $n_e$ of LN changing from 2.139 to 2.141, the resonance peak shifts from 1588.3 to 1558.7 nm. Under such a configuration, the change of refractive index can be obtained by driving the voltage below 10 V/$\mathsf{\mu }$m. The LN film is helpful to break the up–down mirror symmetry for inducing UGRs and offers the tunability of UGR for active applications.

Metadevices have become powerful tools for achieving polarization conversion due to their anisotropic phase-abruption at the interface [35,36,37]. Despite there not being much difference (6.9 nm) between the central wavelengths of the TE and TM modes for UGR resonances of Figure 5a, dominated by the feature of high-Q narrow line width for this UGR phase resonance, our proposed gratings can be exploited for polarization conversion. The transmittance and reflectance under the exciting of 45 degree of electric field polarization angle wave are shown in Figure 6a, which indicates the property of independent electric field polarization angle for this grating. In general, this device provides high reflectances within a certain wavelength range around the UGR resonances for two orthogonal polarizations and 45 degree of electric field polarization angle. We retrieve this grating structure operating under normal incidence for linear-to-circular polarization conversion. The left-handed circular polarization (LCP) and right-handed circular polarization (RCP) conversion efficiency are defined as [35]

$$\begin{array}{cc}\hfill & {\eta }_{RCP}=r_{RCP}^2={\displaystyle \frac{1}{4}}\left[R_{\Vert }+R_{\perp }+2\sqrt{R_{\Vert }{R}_{\perp }}sin\left(\mathsf{\Delta }\phi \right)\right]\hfill \\ \hfill & {\eta }_{LCP}=r_{LCP}^2={\displaystyle \frac{1}{4}}\left[R_{\Vert }+R_{\perp }+2\sqrt{R_{\Vert }{R}_{\perp }}sin\left(\mathsf{\Delta }\phi \right)\right],\hfill \end{array}$$

(2)

where r denotes the amplitude, R refers to the reflectance, ‖, ⊥ is the incident electric field parallel and perpendicular to the gratings, and $\mathsf{\Delta }\phi ={\varphi }_{\perp }-{\varphi }_{\Vert }$ is the relative phase delay, reflectivity. The retrieved linear-to-circular polarization conversion efficiency is shown in Figure 6b. The conversion efficiency of RCP reaches $50\%$ across a successive wavelength range from 1554.2 to 1627.7 nm (73.5 nm bandwidth). The polarization extinction ratio (PER) is another important index from an engineering point of view and is widely used in optical devices, which is approximately equal to [35,37,38]

$$PER=20log\left({\displaystyle \frac{C_y}{C_x}}\right),$$

(3)

where $C_x$ and $C_y$ denote the major and minor axes of the ellipse for the projected polarization vector ($C_y$, $C_x$). These two parameters give us information about the plate only from the amplitude, but some parameters connect both the amplitude and the phase deviation. The related Stokes parameters are used to write their expressions [5,36,37]: $S_0={\left|C_x\right|}^2+{\left|C_y\right|}^2$, $S_1={\left|C_x\right|}^2-{\left|C_y\right|}^2$, $S_2=R\left(C_x{C}_y\right)$ and $S_3=-R\left(C_x{C}_y\right)$. Figure 6c demonstrates that the PER, remaining below 3 $dB$, spans from 1532.2 to 1621.1 nm, producing a bandwidth of approximately 88.9 nm. Compared with these reported works for polarization conversion in the visible-near-infrared region, the conversion efficiency of our proposed UGR gratings is normal, and the bandwidth of PER is a wide and attractive performance [35,36,37,38]. Not that, except the shift of phase resonant wavelength and group delay, the minor refractive index change of the LN layer induced by a small driving voltage has a very slight impact on the polarization conversion in transmittance and reflectance spectra without sharp resonances.

3. Conclusions

In conclusion, a periodic Si grating on LN is proposed. Introducing the LN film aims to break up–down mirror symmetry for inducing the tunable high-Q UGR resonances. A BIC state with a topological charge $q=-1$ always exists at the $\mathsf{\Gamma }$-point and the UGR mode appears close to the $\mathsf{\Gamma }$ point in momentum space infinitely. By varying the lattice constant of silicon bars, the UGR mode can remain at the area near the $\mathsf{\Gamma }$ point within a certain range and move along the direction of $\mathsf{\Gamma }$-X. Meanwhile, the Q factor and asymmetric radiation ratio have been improved to ${10}^5$ and above 80 dB by optimizing the lattice constant. This class of vertical grating without tilted sidewalls may reduce some fabricating difficulties in practical applications. Finally, the class of etchless Si-on-LN grating is utilized for the application of linear-to-circular polarization conversion. The results indicate that we can achieve a broadband polarization conversion with high polarization extinction ratio enabled by the grating with high-Q UGR resonance. Our work establishes a foundation for lossless polarization conversion, offering potential applications in circular dichroism spectroscopy, beam steering, and refractive index sensors.