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Abstract
Recently, high-order topological photonic crystals (PhCs) have attracted huge research attention due to their novel physics mechanism and the application potential in integrated photonics. Based on the two-dimensional Su-Schrieffer-Heeger model, we construct and study the mutual coupling between the high-order corner states in 2D dielectric PhCs. Simulation results show that the Q-factor of such corner-localized quasi-bound states in the continuum (BICs) could be enhanced following mutual coupling in finite size. Furthermore, we study the side-coupled structure based on defect-hybrid waveguides and the edge state microring, the quasi-BIC microcavity. The refractive index sensing application based on corner-localized quasi-BICs shows outstanding simulated sensitivity (312.8 nm/RIU) and figure of merit (∼103 1/RIU). The robustness against fabrication errors due to its topologically protected nature makes it competitive compared with other quasi-BICs sensors.
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1. Introduction
The research of high-order topological photonics has received intensive attention in the last few years. The topological phases of matter originate in condensed-matter research. For example, the topological invariants describing the global property of band-structures were proved related to the integer-valued quantity in the Hall conductance. Meanwhile, a series of concepts of solid-state physics like energy-band, lattice potential could find counterparts like eigenfrequency, dielectric constant in photonic crystals (PhCs) system, the topology has been introduced into PhCs naturally [1–3].
PhCs are periodic artificial structures with wavelength-comparable lattice sizes. Since the tunability of geometric configuration provides the controllability of band-structures [4], PhCs become an ideal platform to study topological phenomena like the Su-Schrieffer-Heeger (SSH) model [5,6], the analog quantum spin Hall effect [7,8], the valley Hall effect [9,10]. There are quite a few micro/nanophotonic devices based on topological PhCs emerge because the topologically protected states show remarkable defects-immune and ultra-compact properties [11]. The tightly localized effect of the topological state meets demands for denser photonic integrated circuits (PICs), and the back-scattering free property is crucial to overcome unavoidable fabrication imperfections. The passive topological photonic devices mainly include light transmission waveguides [12,13], optical signal processing devices [14,15], and functional devices [16,17]. By introducing the gain material, more active photonic devices like topological lasers have also been demonstrated [18–20].
A key component in topological PhCs is the boundary between topologically trivial and nontrivial regions by the bulk-edge correspondence [1]. In the example of two-dimensional (2D) systems, when two bulk regions with different topological invariants contact, the band-gap would close somewhere in the edge region, which leads to forming localized states at the interface, obeying the form of a d-dimensional (dD) system and (d-1)D localized states. However, a high-order topological insulator would support (d-2)D states following generalized bulk-edge correspondence [21,22], corner states in sorts of 2D lattices have been designed [23–27]. Besides the advantages mentioned above of topological robustness, the high quality factor and small mode volume of corner states exhibit their potential applying on lasers, fibers and filters [26,28–32]. Furthermore, several works have revealed the relationship between the high-order topological states and Bound states in the continuum (BICs) [33,34], which has been experimentally observed in higher-order waveguides arrays [35]. Considering practical application environments, constructing and analyzing quasi-BICs in finite-size periodic structures is a problem worthy of more attention.
The structure of this article is the following: In Section 2, we construct a high-order topological dielectric disk PhCs structure based on the 2D SSH model. The basic modes for the following design, including bulk, edge, and corner states, are discussed in theory and simulation. Besides, a hybrid waveguide composing defects and edge states is proposed. In Section 3, we simulate and discuss the high-order states in finite-size topological PhCs, which are proved to belong to quasi-BICs. The coupled-mode theory is used to analyze the mutual coupling between corner-localized quasi-BICs. This analytical method could be guided for enhancement of the corner state’s Q factor in a limited footprint. In Section 4, we construct the side-coupled structures with the edge states square ring, and with the corner states microcavity. The sensing application was demonstrated by assuming background refractive index change. Its unique advantages compared with other PhCs sensors, especially the robustness against defects, have been illustrated by simulation. All simulations are performed based on the finite elements method (FEM) in commercial simulation software COMSOL Multiphysics (version 5.6).
2. Topological structure: bulk states, edge states, and corner states
We design a simple square lattice of dielectric disks in air, and the disks refractive index $\epsilon _d = 3.377$, the inherent dispersion and loss of material are ignored. As shown in lower panel in Fig. 1(a), the lattice constant $a$ is 482.6 $\mu$m, the radius of disks $r = 0.275a$. The unit cell could be divided under different definitions (Fig. 1(a)) within the same PhCs. The transverse magnetic (TM) polarization band-structures of four unit cells are obtained using the 2D plane wave expansion method. Fig. 1(b) shows the bulk bands overlap with each other because they share the same overall expanded structure. However, the symmetry of the cell’s electric field $E_z$ component at X and Y high symmetry points are different, as shown in Fig. 1(c). This torus phenomenon of bands could be depicted using the SSH model [36] and Zak phases [37].
Fig. 1. (a) Definitions of the four types of unit cell and geometric configuration of A cell with corresponding Brillouin zone. (b) The band-structures of four unit cells under TM polarization, the 1st and 2nd band under grey light-cone region are marked as “1” and “2”. (c) The electric field $E_z$ of four unit cells, the label “X1” represents the mode distribution of the 1st band at high symmetry X point, other labels are so on.
The 2D Zak phases $\boldsymbol {\theta ^{Zak}} = (\theta _x^{Zak},\theta _y^{Zak})$ are defined as integration of Berry connection over the first Brillouin zone (BZ) [24]:
Fig. 2. (a) The left schematic shows a typical edge constructed by A-cell region and C-cell region, the projected band of A-C type edge (16$a$-length) shows the edge state (red dots) among bulk states (grey dots) gap, the 3D illustration of ES at $\Gamma$ point shows the localized effect. (b) Schematic and projected band for defects-hybrid edge.
The ESs could serve as outstanding on-chip waveguides, owing to backscatter-immune and size-compact features. However, the key limitation of this waveguide is the lack of tunability. Since the topologically protected effect is guaranteed by the difference in bulk topological invariants, and the ESs characteristics are determined by the configuration of entire PhCs, the most effective and direct tuning method is changing the parameter of bulk-region. These bulk-tuning waveguides are hard to design in PICs because the adjustments in bulk regions would inevitably influence the adjacent module. Here we propose a defects-hybrid waveguide that could overcome the tuning problem. A row of disks with expanded radius (r = 0.3575a) could serve as defects, and the projected band-structure of the defects-hybrid supercell (Fig. 2(b)) indicates the defect states and topologically protected ESs both exist. The transmission range of waveguides could be adjusted flexibly with the geometry size of defects. In the example of Fig. 2, the transmission range increases from inherent $1.53\times 10^{14}$ ∼ $1.83\times 10^{14} {\rm Hz}$ to $1.53\times 10^{14}$ ∼ $2.02\times 10^{14} {\rm Hz}$. Although the localized effect of topological ES could be improved in ideal PhCs, the normalized $E$ field of defect states and ESs are comparable under the finite-size condition in Fig. 2(b).
Furthermore, the corner states (CSs) are demonstrated in the structure shown in Fig. 3(a) [29,30]. This high-order topological state could be regarded as the consequence of the “edge-corner” corresponding relationship. The A-B type ES polarization is non-zero because B-cell is topologically non-trivial in $x$ and $y$ directions, the induced polarization at corner $q_{CS} = P_{ES_x} + P_{ES_y} =1$ [38]. The TM and transverse electric (TE) eigenfrequencies of the system are calculated in Fig. 3(b), four CSs are isolated from the ESs and BSs. The normalized $E$ fields of typical states shown in Figs. 3(d),(e) display their characteristics. All the BSs exist in the bulk region, but the mode fields of lower BS is concentrated in the dielectric disk, while the BS in the higher band is concentrated in the air. The ESs in two boundary directions are equivalent due to symmetry. The CSs at the system boundary could also be supported because the surrounding condition could be regarded as a trivial phase, while the degenerated CSs couple with each other. And the TE modes are lossy and continuous in the spectrum.
Fig. 3. (a) The schematic illustration of typical corner state demonstration, the blue boxes mark the edges between region of A-cell and region of B-cell, and the red circle marks the high-order corner. (b) The TM and TE eigenmodes of the entire 16a$\times$16a structure and (c) corresponding Q-factors. (d) The normalized $E$ field of the corner-localized quasi-BIC. (e) The normalized $E$ field of lower band BS, ES, lossy CS, and higher band BS.
3. Mechanism of quasi-BICs tuning
The typical BICs [33,39] in photonics are dark modes without radiation channels. In theory, the “bound” resonant modes would not be coupled to radiation “continuum”, leading to infinite radiative lifetimes. However, the finite extended structure in practice and defects perturbation caused by fabrication would make ideal BICs collapse to quasi-BICs. Previous research has proved the BICs are signatures of the topological phase [34,35]. In the configuration of Fig. 3(a), the “continuum” contains double meaning: the background bulk states and the surrounding scattering boundary. In Figs. 3(b),(d), the protected CS is spectrally isolated from TE bulk modes due to symmetry protection, and the exponentially attenuating mode profile indicates the weak degenerate with TM bulk modes. Besides, the CS’s extremely high quality factor ($Q$) value within 8-units distance to the boundary also indicate nearly vanishing radiation to surroundings.
Furthermore, the schematic illustration of quasi-BICs optimization construction is shown in Fig. 4, the light-blue background B-cell square region located at grey background A-cell square region, whose side-length is $l$ (default $8a$)and $L$ (default $24a$) respectively. The $\Delta x$ and $\Delta y$ represent the distance between two square centers along with orthogonal directions. In the simulation, the complex eigenfrequencies of the entire structure are calculated in the Finite-elements method. The boundary is surrounded by PMLs and scattering boundary conditions to ensure radiative channel to circumstance, where the $Q$ is calculated following the formula : $Q(f) = {\rm Re}[f]/{\rm 2Im}[f]$. As shown in Fig. 4(b), we set $l$ to be half of $L$, and $\Delta x = \Delta y = l/2$, Q of the protected CS (CS 1) increases exponentially with increasing system size $L$, this point would be more intuitive in nearly linearly decreasing imaginary part under logarithmic coordinates. The quasi-BICs would be BICs with totally vanishing radiative pars in the unlimited system size.
Fig. 4. (a) The schematic of CS construction with tuning parameters $L$, $l$, $\Delta x$ and $\Delta y$. (b),(c),(e) show the Q factor relationship with changing parameters. (d) The modes splitting of eigen-CSs at typical configuration. (f) indicates the change of imaginary part of CSs when the topological-nontrivial region approaches the boundary corner.
The previous optimization of corner-localized quasi-BICs was mainly limited to modifying corner structure parameters [28,40], which becomes defects hybridized states essentially. The existence of four degenerated CSs inspires us to achieve adjustment through mutual coupling, which has been discussed in symmetrical condition [29]. Here, we provide qualitative description by coupled-mode Hamiltonian in the following format, and analyze the asymmetrical tuning:
To better understand the mutual coupling process, we keep the B-cell region at the corner of the system ($\Delta x = \Delta y = (L-l)/2$), and the $L$ keeps 24a. In previous works, the CS 2 ∼ 4, whose fields concentrated in PMLs usually be regarded as pseudo modes and ignored. Here, we term these modes as lossy modes with large imaginary parts, while CS 1 is the protected mode. The maximum $Q$ of CS 1 appears near the system center. When $l$ is small, both inherent lossy term $\gamma _1$, which is related to the distance to boundary, and coupling lossy terms $e^{ikl}$ and $e^{ik\sqrt {2}l}$ both contribute to the imaginary component. The $\gamma _1$ becomes the dominant factor under a larger $l$, the slight difference of the line-shape on two sides of the peak shown in Fig. 4(c) confirms the discussion.
Another consequence of mutual coupling is mode splitting. Fig. 4(d) indicates the same eigenfrequency real component under central symmetry, where the coupling coefficient $\kappa$ is the same. When a single side of the topological-nontrivial region approaches the system boundary ($\Delta x = 0$ $\Delta y = 8a$), which means the sharply declining $\kappa _{34}$, the real components of CS 1 and 2 split with CS 3 and 4. Analogously, when nontrivial region approach to corner ($\Delta x = \Delta y = 8a$), the protected quasi-BICs would be totally isolated state causing weak $\kappa _{34}$ and $\kappa _{24}$. The normalized field distribution inserted in Fig. 4(d) exhibits degenerated effect. Besides, the evolution of $Q$ and $\Delta x$, $\Delta y$ (Fig. 4(e)) reveals interesting optimization: the radiation of lossy modes CS 2 ∼ 4 increase when they approach to boundary, but the mutual coupling leads the increase of CS 1 imaginary part abnormally in range 0 ∼ 3a of $\Delta x$ $\Delta y$ (Fig. 4(f)), even the CS 1 is approaching to center actually. When $\Delta x$, $\Delta y$ increase further, the imaginary part of lossy modes increases as the same trend. In contrast, the total lossy term of CS 1 would be suppressed following Eq. (3) and reach the minimum when lossy modes approach system boundaries. The case when a single side of the topological-nontrivial region approaches the system boundary shows similar behavior, which is shown in Fig. S1 from Supplement 1.
Generally speaking, the quasi-BICs in finite-size high-order topological PhCs could be adjusted based on mutual coupling. In order to obtain higher $Q$ quasi-BICs, enlarging the side-length of the topological-nontrivial region and the distance of target CS to boundaries is necessary. Besides, the mode splitting due to asymmetric radiation of other CSs could also improve the characteristic of corner-localized quasi-BICs.
4. Sensing application coupled with edge state ring and corner statemicrocavity
As discussed in Section 2, high-order topological PhCs contain 1D edge states (waveguides or microrings) and 0D corner-localized quasi-BICs (microcavity), which is naturally an excellent platform for constructing high-performance photonic devices. Here, we design and discuss the two types of filter structures: waveguides side coupling with ESs square ring, and with a quasi-BIC microcavity, the refractive index sensing application is demonstrated based on later one.
Firstly, we design a composite structure including a defects-hybrid waveguide and a side-coupling ES square ring. The geometry configuration is shown in Fig. 5(b). In ideal finite PhCs, the equivalent ES could compose the square ring, and the modes should be similar to Whispering Gallery Modes (WGMs). However, the ESs at the ring’s four sides in Fig. 5(b) would split due to asymmetry structure and nonequivalent lossy effect, just like CSs discussed in section 3. Figures 5(a),(c) display resonant dips in two scales, the blue block in Fig. 5(a) indicates main dips group (marked as roman numbers) and zoom-in figure of dips group IV shows four sub-dips marked as circled numbers.
Fig. 5. (a) The transmission in units of dB, the blue background mark the dips group. (b) The schematic illustration of the composite structure including defects-hybrid waveguide and ES ring. (c) The zoom in figure in transmission shows more details of dips group IV and (d) corresponding normalized $E$ field distribution.
With the assist of modes field pattern (Fig. 5(d)), the ESs splitting effect would be easier to understand. The strongest sub-dip ① corresponds to the ES at the lower side of the square, which is the closest to the waveguide, and the second strongest sub-dip ③ corresponds to left-side and right-side ESs because they are equivalent. Since the distance between top-side ES and the waveguide is larger than other sides, the weaker coupling results in a weak sub-dip ④. Moreover, the sub-dip ② could be regarded as the coupled resonant mode between top side ES and other ESs. In addition, the normalized $E$ profiles of sub-dip ① in each group indicate the order of resonant modes. Another interesting phenomenon is the weakening trend of transmission dips from group IV to I, the higher of resonant mode’s order, the poorer localized effect is, which would improve the coupling strength. This mechanism also explains the vanishing of sub-dips ② and ④ in other groups. It should be noted that the resonant dip caused by corner-localized quasi-BIC is not be presented in Fig. 5(a), which will be discussed later within an optimized configuration.
Subsequently, the side-coupling system is constructed between the waveguide and quasi-BICs, the red-dash line box mark the waveguide region, as shown in Fig. 6(a). We simulate and compare the filter characteristic under conventional ES waveguide (W1) and defects-hybrid waveguide (W2). Specially, we adjust the radius of the disks in the C-cell region to ensure a suitable transmission range of W1. The transmission dips could be fitted by Fano resonance following the formula [41]:
where the $B_0$ and $A_0$ are background and amplitude factor respectively, Fano parameter $q$ describes the ratio of transition amplitudes in the scattering process, which relate to the asymmetry of resonance dip, and the reduced energy $\epsilon$ is defined as: where $\lambda _0$ is the resonance center wavelength and $\Delta \lambda$ is the Full Width at Half Maximum (FWHM) of resonance. The inevitable influence caused by C-cell region adjustment in W1 design leads to the resonant dip shift, the $\lambda _0$ is slightly different from the corresponding eigenfrequency. Such shift brings additional difficulties in PICs design. Moreover, the fitting results show that the $q$ is 0.32 (0.03) for the W1 (W2) structure. The smaller $q$ indicates the typical antiresonance (resonant reflection) [42] because the scattering of defects-hybrid W2 is small enough compared to the scattering of the quasi-BIC. The antiresonance could also be observed from the resonant profile close to the reversed Lorentzian lineshape. The quality factor for coupling ($Q_c$) calculating in $Q_c = \lambda _0/\Delta \lambda$ is $1.81\times 10^4$ and $1.58\times 10^5$ for W1 and W2 respectively.Fig. 6. (a) The schematic illustration of the composite structure including defect-hybrid waveguide and CS microcavity, the red arrows indicate the operating path. (b) Corresponding normalized $E$ field distribution in coupling process, where W1 denotes bulk-tuning waveguide and W2 denotes defect-hybrid waveguide. (c),(d) show the transmission simulations and Fano lineshape fitting results of two waveguide configuration, respectively. (e) The transmission dips under different $n_0$ and (f) the linear fitting for dip wavelength-$n_0$ and FOMs at different $n_0$, both (e)(f) are investigated in W2 configuration.
Furthermore, this side-coupled configuration could be designed as an optical refractive index (RI) sensor due to the high Q resonant dip. In a similar construction based on PhCs [43], the resonant peaks or dips in nanostructure would be sensitive to medium RI. The sensitivity is relatively high in such a design because of the strong interaction between the environment and the confined optical field. As shown in Table 1, both in-plane [43–48]and out-of-plane [49–54] sensing configurations have been designed based on various microcavity types. The signals to be detected contain gas, humidity, and biochemical analyte if it is related to effective refractive index variations [55]. Here, we change the background refractive index $n_0$ in the side-coupled structure (W2) for sensing demonstration. The sensitivity ($S$) is defined as:
where the $\Delta \lambda _{dip}$ represents the resonant dip wavelength and corresponding $n_0$ change. The linear fitting result of $\lambda _{dip}$-$n_0$ in Fig. 6(f) shows $S_\lambda$ could reach 312.8 nm/RIU. The Figure of Merit (FOM) value is defined as: and the FOM achieves 31918 1/RIU around $n_0 = 1.00$. It is worthy noting that the FOM is related the Q factor of quasi-BIC microcavity. In practical consideration, the free-standing slab with supporting fins would introduce extra leakage. The discussion in Fig. S2 from Supplement 1 indicates the this change would the Q-factor drop by an order of magnitude. Therefore, the FOM should maintains the order of ${\sim }10^3$ under this estimate. Generally, the FOM value reveals the capability detecting the small perturbation of environment refractive index, because the overlap of transmission dips will limit the lower values of detection. and the FOM achieves 31918 1/RIU around $n_0 = 1.00$. It is worth noting that the FOM is related to the Q factor of the quasi-BIC microcavity. In practical consideration, the free-standing slab with supporting fins would introduce extra leakage. The discussion in Fig. S2 from Supplement 1 indicates that this change would cause the Q-factor to drop by one order of magnitude. Therefore, the FOM should maintain around ${\sim }10^3$ under this estimate. Generally, the FOM value reveals the capability of detecting the small perturbation of the environment refractive index because the overlap of transmission dips will limit the lower detection value.
Table 1. Performance of the representative PhCs sensing applications in recent years
Compared with conventional in-plane PhCs sensor (Table 1), our side-coupled sensing structure owes higher simulated sensitivity and FOM value. The sensitivity is still competitive among quasi-BICs sensing simulations/experiments, and a relatively high FOM value is also critical for detection limitation. Besides, the common quasi-BICs sensing focuses on out-of-plane incident light, which needs a larger footprint with more unit cells, as well as a more complex optical system. Of course, out-of-plane sensing owes unique features, including stronger interaction volume with the matter [49]. Our sensing solution based on corner-localized quasi-BICs is in-plane configuration, which is advantageous for integrated photonics. Furthermore, due to their intrinsic nature, the current sensing with quasi-BICs is extremely sensitive to structure asymmetry factors [53,54] or incident angle [51]. However, these demands are negative for practical application. In contrast, the corner-localized quasi-BICs are robust against fabrication errors owing to the topologically protected effect.
As shown in Fig. 7(a), six defects (including two radii defects, two location defects, and two vacancy defects) are introduced at random sites among the side-coupled structure. The simulation shows that the coupling at resonant dip does not be interrupted by nearby defects (Fig. 7(b)). The dip shift $\Delta \lambda$ caused by defects is around 0.033 nm, which is quite slight. For a more intuitive impression of the robustness of such sensing structure, the relative change of eigenfrequencies and Q-factor with defect size have been compared between corner-localized quasi-BIC and point-defect-localized quasi-BIC, as shown in Fig. S3 from Supplement 1, the corner-localized quasi-BIC is much more stable.
Fig. 7. (a) The schematic for sensing structure with radii defects, location defects and vacancy defects at random sites. (b) The normalized E field with defects at resonant dip and (c) indicates the tiny shift of the resonant dip with or without defects.
5. Conclusion
In this work, we design the 2D dielectric topological PhCs and construct topologically protected edge states and corner states. A waveguide based on defects and edge states hybridized structure is proposed. It is more suitable for integrated optical design. Furthermore, we focus on corner-localized quasi-BICs tuning in finite-size PhCs. The Q factor of the quasi-BIC could be enhanced according to mutual coupling between lossy modes around the scattering boundary and protected modes. Moreover, we construct and simulate the side-coupled filter structure based on defect-hybrid waveguides, the edge state square ring, and the quasi-BIC microcavity. We show the WGMs and modes splitting in the edge states ring caused by asymmetric configuration. Additionally, the ultra-narrow resonant dip of the defects-hybrid waveguide + corner quasi-BIC coupled structure displays the resonant reflection effect. The refractive index sensing application based on such structure shows sensitivity up to 312.8 nm/RIU, and FOM around ${\sim }10^3$ 1/RIU. Further simulations have illustrated that the in-plane quasi-BICs sensing owes unique robustness against defects.
Funding
Strategic Priority Research Program of Chinese Academy of Sciences (XDB43010000); National Natural Science Foundation of China (12075244, 61835011); Key Research Program of Frontier Science, Chinese Academy of Sciences (QYZDY-SSW-JSC004); National Key Research and Development Program of China (2020YFB2206103).
Acknowledgments
We appreciate the editor and the reviewers very much for their constructive comments and suggestions on our manuscript. We are also grateful for discussions about image layout and language polish with Dr. Qu Changming and Miss. Wang Danqing in University of Chinese Academy of Sciences.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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