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Abstract
The study of topological photonics has gained significant attention due to its potential application for robust and efficient light manipulation. In this work, we theoretically investigate a two-dimensional photonics crystal that exhibits a topological edge state (TES) and a topological corner state (TCS). Furthermore, we also achieve a coupling between a topological corner state and a trivial cavity (TC), resulting in a phenomenon similar to the electromagnetically induced transparency (EIT) effect. To verify the stability of the EIT-like effect, disorders around TES and TCS are introduced, and the theoretical results show that this structure is immune to the disorders. The achievement of the coupling between topological states can have potential applications in the areas of waveguiding, sensing, and logic gates. It is hoped that this work will contribute to the ongoing efforts in the exploration and utilization of topological photonics.
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1. Introduction
The ability to control light-matter interactions is vital in many applications, including lasers, medical imaging, and quantum computing [1–3]. The cavity-waveguide coupling system is a crucial optical coupling system with vast application prospects and significant potential for further development. By adjusting the structure, material and size parameters of the cavity and waveguide, researchers can achieve various optical performances [4–6] and applications [7–12]. However, the performance of cavity-waveguide coupling systems is highly susceptible to structural factors such as slight disorder in the geometry, surface roughness and material inhomogeneity [13,14]. This highlights the need for careful consideration of the system’s design and fabrication processes in order to achieve optimal performance.
Photonic crystals are excellent artificial microstructures for controlling the flow of light, as electromagnetic waves can exhibit similar behavior to that of electrons in solid-state materials. In recent years, the advantages of topological photonic crystals have become increasingly evident, owing to the rapid progress in topology research in condensed matter physics and topological photonics [15–19]. This has led to a surge of interest in exploring the potential of topological photonic crystals for various applications, including robust and efficient information processing and communication technologies. The importance of topological photonics lies in its insensitivity to disorder [20–23]. In an ordinary optical waveguide, unwanted feedback and loss are primarily caused by back-reflection [14]. However, by utilizing topological corner state (TCS) as a cavity and topological edge states (TES) as a waveguide, the system becomes highly stable against disorders, owing to the lattice symmetric protection [4,24]. Substituting the photonic crystal cavity and waveguide with TCS and TES, respectively, is therefore a significant improvement [25,26]. While some studies have investigated the stability of cavities and waveguides, a robust sensing system must possess both stability and sensitivity. Achieving this balance is crucial for developing high-performance sensors.
In this study, we propose a theoretical realization of the analog of electromagnetically induced transparency (EIT) [27,28] in a coupled cavity-waveguide system based on a two-dimensional photonic crystal. The EIT-like effect is created through the interaction between a TC, which is a photonic crystal cavity composed of point defects [29–31], and a TCS. This results in a transparency window in the transmission spectrum. Moreover, the incorporation of TCS and TES helps to reduce the impact of structural disorders on the EIT-like effect. The proposed design strikes a balance between trivial photonic crystals and nontrivial photonic crystals, rendering the coupled cavity-waveguide system insensitive to disorder. Notably, the EIT-like effect is highly sensitive in the presence of TC, even a small change in the background refractive index can cause it to change. These findings have significant implications for designing optical sensors, offering a novel approach to enhance their sensitivity and stability.
2. Structure
Here, we introduce the structure of the proposed two-dimensional (2D) photonic crystal. Figure 1(a) illustrates the square lattice photonic crystal with four inversion centers, thereby allowing it to be classified into four unit cell (UC) types, namely, UC1, UC2, UC3, and UC4. Each UC consists of two dielectric circles with different diameters, namely, $r_1=0.4a$ and $r_2=0.2a$. The yellow region denotes the background refractive index $n_b=1$, and the blue region represents the dielectric refractive index $n_d=3.4$. The lattice constant of each UC is $a$, as depicted in Fig. 1(b). The dispersion bands in reciprocal space are used to define the topologies of material systems in photonics, which is depicted in Fig. 1(c). The energy band structures of all UCs are identical. Notably, there exists a bandgap between the first and second energy bands, indicating that the photonic crystal is a gapped 2D material.
Fig. 1. (a) The proposed 2D photonic crystal, four types of the unit cell are marked with red dashed boxes. (b) Unit cell. (c) The band structure of the unit cell under TM polarization. (d) The electric $E_z$ of the unit cells at the high symmetry point in the first Brillouin zone. (e-f) The band structures of the combinations of UC1 and UC3, UC2 and UC3, respectively. The insets show the electric fields $E_z$ of the two different combinations at high symmetry points (0,0) (left) and (0, $\pi$ ) (right) along the $k_x$ direction.
To enhance our comprehension of the topological properties of the proposed photonic crystal, it is imperative to ascertain the Zak phases of the 2D photonic crystal. There is a simple connection between Zak phase and the electric fields $E_z$. The electric fields $E_z$ at $X(Y)$ point in the first Brillouin zone of the UCs are shown in Fig. 1(b). The Zak phase of a 2D photonic crystal can be described as [32,33]: $\theta _{x(y)}^{Zak} =\int dk_{x}dk_{y}Tr\left [ A_{x(y)}\left ( k_x,k_y \right ) \right ]$, here the first Brillouin zone is the integral range, $A_{x(y)}\left ( k_x,k_y \right ) =i\left\langle\psi \left |\delta _{k_{x(y)}} \right | \psi \right\rangle$, is the Berry connection, $\psi$ is the period of Bloch function. Especially in this paper, all the UCs are $C_4$ symmetry, and the values of the Zak phases can only be 0 or $\pi$[32]. In Fig. 1(b), the electric field $E_z$ of UC1 at $X$ point along the $X$ direction is symmetric, so the Zak phase in this direction is $0$. On the contrary, at $Y$ point along the $Y$ direction the electric field is antisymmetric, so the Zak phase in the direction is $\pi$. For the same reason, the Zak phases of UC2 and UC3 are $(\pi,\pi )$ and $(0,0)$, respectively. The relationship between Zak phase and two-dimensional polarization along the $X(Y)$ direction is [32]: $\theta _{x(y)}^{Zak} =2\pi P_{x(y)}$. And the two-dimensional polarizations correspond to UC1, UC2 and UC3 are $(0,1/2)$, $(1/2,1/2)$ and $(0,0)$, respectively.
According to the body-edge correspondence principle, the combination of two UCs along the $X(Y)$ direction with different polarizations can form a TES [34]. The Zak phase of UC1 and UC3 are $(0,\pi )$ and $(0,0)$, respectively. Therefore, there will be a TES named TES1 along the $X$ direction ranging from $0.2812(a/\lambda )$ to $0.3433(a/\lambda )$ in the bandgap, As is shown in Fig. 1(e). Likewise, in Fig. 1(f), the combination of UC2 and UC3 along the $X$ direction can form a TES named TES2 ranging from $0.2773(a/\lambda )$ to $0.3078(a/\lambda )$. But the two TESs are different, the reason is that their edge polarization $(p_{x}^{edge})$ are not the same. In Fig. 1(e-f),the electric fields $E_z$ at high symmetric point in the first Brillouin zone are plotted. For TES1, at $(0,0)$ and $(\pi /a,0)$, the electric fields $E_z$ are mirror symmetric. For TES2, at $(0,0)$, the electric field $E_z$ is mirror symmetric but mirror antisymmetric at $(\pi /a,0)$. Therefore, the two edge polarizations along the $x$ direction are $p_{x(TES1)}^{edge}=0$ and $p_{x(TES2)}^{edge}=1/2$, respectively. In addition, there is a TES when UC2 and UC3 are combined along the $Y$ direction with $p_{y(TES2)}^{edge}=1/2$.
The existence of TCS depends on the edge polarization: $Q^{corner}=p_{x}^{edge}+p_{y}^{edge}=1$[35]. Combining UC2 with UC3 along the $x$ and $y$ directions results in $Q^{corner}=1$, which indicates the occurrence of a TCS. As shown in Fig. 2(a), PC2 is enclosed by PC3, and the electromagnetic field of frequency $0.3295(a/\lambda )$ is confined to the right-angled corner where PC2 and PC3 are in contact. We have identified that the frequency of the TCS lies within the frequency range of the TES1. Therefore, we can achieve both of these states simultaneously in a photonic crystal. As illustrated in Fig. 2(b), the PC1 and PC3 are in contact along the $X$ direction, forming the TES, and the TCS is formed at the right angle junction between PC3 and PC2. When a beam of electromagnetic waves with a frequency range from $0.3285(a/\lambda )$ to $0.3325(a/\lambda )$ is incident from one side and exits from the other side of the photonic crystal at the junction of PC1 and PC3, we can observe that the transmission rate of this structure drops to a minimum at frequency $0.3307(a/\lambda )$. This phenomenon occurs due to the presence of the TCS, which localizes the electromagnetic wave at frequency $0.3307(a/\lambda )$.
Fig. 2. (a) The eigenmodes of the combination of UC2 and UC3. The insets show the combination of UC2 and UC3 (upper), and the electric field of TCS (lower). (b) The transmission spectra of the combination of TES and TCS. The inset show the combination. (c) The eigenmodes of the TC. The insets show the TC in PC3 (upper) and its electric field (lower). (d) The transmission spectra of the combination of TES and TC. The inset show the combination.
To realize the phenomenon of EIT-like, it is necessary to introduce another localized state within the photonic crystal. As depicted in Fig. 2(c), we have successfully introduced a point defect in a photonic crystal consisting of UC3, which serves as a TC and is delineated by a red dashed box and shown enlarged. The point defect was generated by reducing the diameter of the central circular to $r=0.168r_1$, leading to the localization of electromagnetic wave with frequency $0.3292(a/\lambda )$ around it. In both cases we can see that TC and TCS have an eigenmode around $0.3290(a/\lambda )$. Upon combining PC1 and PC3 and examining the transmittance of the composite structure, we observed that the minimum transmittance occurs at the frequency of $0.3307(a/\lambda )$, as illustrated in Fig. 2(d).
3. Theoretical model
It is essential to understand the photonic crystal architecture to comprehend the coupling between TC and TCS, as well as the impact of EIT-like. As depicted in Fig. 3(a), the pale yellow region signifies PC1, the blue region corresponds to PC2, and the deep yellow region represents PC3. The position of the point defect is indicated by the red dashed rectangle. As discussed earlier, this structure is capable of realizing TES, TCS, and TC. Electromagnetic waves with frequency range from $0.3290(a/\lambda )$ to $0.3320(a/\lambda )$ are incident from the left side of the photonic crystal. The waves propagate exclusively along the interface between PC1 and PC3, and this phenomenon can be attributed to the TES. Thus, it can be regarded as a waveguide in this case. As the TCS and TC can localize electromagnetic waves at specific frequencies, they can be considered as optical cavities. When the TCS and TC coexist on a photonic crystal, they exhibit direct coupling interactions, as well as indirect coupling effects mediated by the TES waveguide. Figure 3(c) presents the transmission spectrum of this coupled system. By comparing with Fig. 2(b) and Fig. 2(d), we can clearly observe that the transmission becomes extremely high at frequency $0.3307(a/\lambda )$ due to the splitting of the small eigenfrequencies of the two cavities. To further illustrate the coupling phenomenon between them, Fig. 3(d) displays the distribution of electric fields in this photonic crystal. It is evident that TC and TCS do not localize electromagnetic waves at the frequency of $0.3307(a/\lambda )$, but instead localize them at the frequencies of $0.3300(a/\lambda )$ and $0.3315(a/\lambda )$.
Fig. 3. (a) The proposed structure that can realize EIT-like. (b) Illustration of TC and TCS coupling system. (c) Calculated(black) and CMT fitted(red) transmission spectra of the proposed coupling system. (d) Electric fields of $E_z$ at $0.3300(a/\lambda )$, $0.3307(a/\lambda )$ and $0.3315(a/\lambda )$, respectively. (e) Frequency diagram for coupling of TC and TCS.
CMT is introduced to explain the coupling between TC and TCS, as is shown in Fig. 3(b). The electromagnetic wave with the form of $e^{i\omega t}$ enters along TES from the port close to TC, so $S_{(+4)}=0$. The coupling between the TC and TCS can be expressed as:
4. Simulation and discussions
To verify the robustness of the cavity-waveguide coupling, we introduced structural disorder around the TCS and TES as marked by the red dashed box in Fig. 4. In Fig. 4(a), the dielectric circle marked in red is randomly displaced by a certain distance (10% of the diagram, with no preferred direction). From the transmission spectrum shown in Fig. 4(b), we can clearly see that the coupling between TC and TCS still exists, and the energy level is still splitted, as confirmed by the electric field profile in Fig. 4(a). Next, we introduced the structural disorders around the TES waveguide, as shown in Fig. 4(b), where the red-marked dielectric circle are also randomly displaced (10% of the diagram, with no preferred direction) while keeping the structural disorder at TCS unchanged. Similarly, we can demonstrate from the electric field profile and transmission spectrum that this coupling phenomenon still exists and is not affected by the structural disorders.
Fig. 4. Coupling between TC and TCS when disturbances are introduced. The red dotted box marks the disturbances (left). The electric field of $E_z$ (right) at the frequency corresponding to the lowest transmittance. (a) Disorder at TCS. (b) The corresponding transmission spectrum when the disorder is at the TCS. (c) Disorder at TCS and TES. (d) The corresponding transmission spectrum when the disorder is at the TCS and TES.
Let us focus on the change in the coupling phenomenon after altering the background refractive. Figure 5 provides a detailed description of the variation trends of the two transmission valleys with respect to changes in the background refractive index of the photonic crystal from 1.01 to 1.05. As can be seen from Fig. 5(b), The change of frequency is approximately linear with the change of background refractive index, which can be quantified by the sensor sensitivity $S=\Delta \omega /\Delta n$. The left transmission valley ($-0.1014 a/\lambda RIU$) is more sensitive than the right transmission valley ($-0.0786 a/\lambda RIU$). Since the two transmission valleys have different sensitivities to the change of background refractive index, they can be used to verify each other. When one of the transmission valley changes, the other should also correspondingly change, and through this mutual verification of changes, we can more accurately determine the change of the background refractive index. Optical sensors have been widely used due to their real-time monitoring, label-free, and non-destructive characteristics. However, the unstable environment, such as fluctuations in the light source intensity and changes in the surroundings, makes it difficult to evaluate the analytes by detecting a single detection signal. Therefore, the study of self-calibrated refractive index sensors is of great significance. The EIT-like effect we proposed exhibits two resonance valleys, which can serve as two detection channels for self-referencing sensors, paving a way for the realization of self-calibrated refractive index sensor optical sensors.
Fig. 5. (a) The transmission spectrum when the background refractive index varied from 1.01 to 1.05. (b) Relationship between the resonant frequency and the background refractive index.
5. Conclusion
In conclusion, we have achieved the coupling phenomenon of cavity-waveguide on the basis of a 2D topological photonic crystal. With the presence of TES and TCS, the entire photonic crystal is topologically protected. In addition, the presence of TC gives this structure a certain level of sensitivity. More importantly, the EIT-like coupling effect between TCS and TC can effectively combine robustness and sensitivity, providing new ideas for the design of future optical sensors.
Funding
National Natural Science Foundation of China (61275174); Hunan Provincial Innovation Foundation for Postgraduate (No. 2022ZZTS0170).
Acknowledgments
We are grateful for resources from the High Performance Computing Center of Central South University.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
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