1 Introduction

Recently, higher-order topological states have been widely researched due to its unique properties in manipulating the propagation of light [1, 2]. Photonic crystal (PC) [3,4,5] is an excellent platform for realizing topological edge state (TES) [6,7,8,9] and topological corner state (TCS) [1, 6, 10, 11] since its energy band structure can be designed artificially. At present, these topological states have been successfully realized in photonic crystals with different lattices such as hexagonal lattice [12, 13], Kagome lattice [14, 15], square lattice [4, 16]. The band gap of photonic crystals at point of high symmetry can be opened by lattice transformation, and PCs with different topologies can be obtained through energy band inversion and topological phase transition [17,18,19,20]. TESs at the interface of combination of PCs with different topological properties [6, 21,22,23] can be realized, which have unique properties of topologically protection and strong robustness to detects due to lattice symmetry and backscattering suppression, respectively. Additionally, researchers have found that TCSs can be generated at the intersection of interfaces with two non-zero edge polarization [5, 24, 25] values based on the principle of polarization. TCSs have extremely strong optical localization property, but it cannot transmit optical signals and extremely tiny frequency difference of TCSs limiting its practical applications.

In this paper, we propose a square lattice PC composed of rhombic dielectric columns, and the topological phase transition is realized by lattice transformation. The formation mechanism and transmission characteristics of TESs are investigated and the robustness to impurities and disorder of TESs is verified. Based on the topological polarization principle, the TCSs with strong localization are successfully realized in the box-type structure consisting of with non-trivial and trivial PCs. We combine the TCS cavities with the PC waveguide to construct the topological photonic crystal (TPC) waveguide–cavity coupling system. In this system, the energy transmitted in the PC waveguide can be coupled to the TCS cavities at a specific frequency, resulting in the interruption of the energy transmission and realizing the multifunction of topological optical switch and controllabled optical storage. Overall, our research highlights the potential of higher-order topological states and paves a new way for the design of optical communication devices.

2 Model and analysis

A large photonic energy band gap can be obtained by reducing the symmetry of dielectric columns [26, 27]. Figure 1a–c shows the corresponding photonic band gap (PBG) of the square lattice PCs with circular, square, and rhombic dielectric columns when their distances from the center of the dielectric column to the center of the unit cell are equal. It is clear that the PBG (71.30 THz) of the square lattice PCs with rhombic dielectric columns is larger than that (68.66 THz or 70.00 THz) of the square lattice PCs with circular or square dielectric columns due to the lower symmetry of rhombic dielectric column. Herein, we propose a kind of square lattice PCs composed of rhombic dielectric columns with \({\varepsilon }_{r}\)=11.7 surrounded in the air background with \({\varepsilon }_{0}\)=1. Its unit cell (UC) consists of four rhombic dielectric columns with the minor axis b1 = 0.20a and major axis b2 = 0.28a as shown in Fig. 2a, \(a\)=0.5 µm is the lattice constant, and R = 0.35a is the distance between the unit cell centers. The first Brillouin zone of the square lattice is displayed in Fig. 2d. We calculate the variation of the PBG of UC at high symmetric point X(Y) of the first Brillouin zone with R/a by the COMSOL Multiphysics—a finite element analysis software, as shown in Fig. 2e. It is clear that the degenerate point can appear when R = 0.35a and disappears when R < 0.35a or R > 0.35a. The UC1 (R = 0.56a) and the UC2 (R = 0.14a) are obtained by expanding the UC and shrinking the UC as shown in Fig. 2b and c, respectively. We scan the first Brillouin zone of square lattice and the obtained energy band structures of the three unit cells (UC, UC1, and UC2) are shown in Fig. 2f–h, respectively. From PBGs of UC, UC1, and UC2, we can find that the degenerate point appears at point X(Y) for the case of UC, while the PBGs open in the expanded and shrunken cases of UC1 and UC2, and there is the same PBG with the frequency range from 182.18 to 253.48 THz. The insets at the bottoms of Fig. 2g and h display the electric field distributions along the x(y) direction of UC1 and UC2 for the first band. It can be seen that the electric field distribution of UC1 at the X(Y) point along the x(y) direction is mirror anti-symmetrical, while that of UC2 at the X(Y) point along the x(y) direction is mirror symmetrical. This indicates the occurrence of the topological phase transition between UC1 and UC2.

Fig. 1
figure 1

a The PBG of the square lattice PCs with circular dielectric columns. b The PBG of the square lattice PCs with square dielectric columns. c The PBG of the square lattice PCs with rhombic dielectric columns

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Fig. 2
figure 2

a UC (R = 0.35a) consisting of four rhombic dielectric columns. b UC1 (R = 0.56a) obtained by expanding the UC. c UC2 (R = 0.14a) obtained by shrinking the UC. d The first Brillouin zone of the square lattice. e The variation of the PBG of UC at high symmetric point X(Y) of the first Brillouin zone with R/\(a\). f The band structure of UC in first Brillouin zone. g The band structure of UC1 in first Brillouin zone. The eigenmodes of UC1 are inset at the bottom. h The band structure of UC2 in first Brillouin zone. The eigenmodes of UC2 are inset at the bottom

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The topological properties of UC1 and UC2 can be characterized by extended Zak phase [24, 28, 29]

$$\theta_{j}^{Zak} = \smallint dk_{x} dk_{y} Tr[A_{j} (k_{x} ,k_{y} )] = 2\pi P_{j} ,j = x,y,$$
(1)

where \(\theta_{j}^{Zak}\) represents the value of Zak phase in the j direction, the integration range is the first Brillouin zone, \(A_{j} (k_{x} ,k_{y} ) = i\langle \psi \left| {\partial k_{j} } \right.\left| \psi \right.\rangle\) is the Berry connection, \(\psi\) denotes the periodic part of Bloch function, and \(P_{j}\) represents the quantized value of two-dimensional polarization along the j direction in UC. For C4 symmetry, \(P_{x}\) is equal to \(P_{y}\) and the value of two-dimensional polarization \(P_{j}\) can be calculated by the following formula [24]

$$P_{j} = \frac{1}{2}(\sum\limits_{n} {q_{j}^{n} {\text{modulo 2), }}j = x,y,( - 1)}^{{q_{j}^{n} }} = \frac{{\eta (X_{j} )}}{\eta (\Gamma )},$$
(2)

where \(\sum\limits_{n} {q_{j}^{n} }\) is for all the occupied bands and \(\eta\) represents the parity of the band at the high symmetric points X(Y) and \(\Gamma\), \(X_{j}\) denotes the \(X\) or \(Y\) point of the first Brillouin zone. For UC1, the value of polarization is \(P{ = }(P_{x} , \, P_{y} )\) = (1/2, 1/2) and the corresponding value of Zak phase is (\({\uppi }\), \({\uppi }\)), which means that UC1 corresponds to topologically non-trivial case. While, the value of polarization of UC2 is \(P{ = }(P_{x} , \, P_{y} )\) = (0, 0) and the corresponding value of the Zak phase is (0, 0). This indicates that UC2 is topologically trivial case. TES can be obtained at the interface between two PCs with different topological properties [7, 9, 30,31,32,33].

3 Topological edge states and corner states in the combined structure

To realize TESs, we construct a combined structure composed of two PCs (PC1 and PC2) with different Zak phases as shown in Fig. 3a and choose a supercell shown in Fig. 3b to calculate the energy band structure using COMSOL Multiphysics—a finite element analysis software. The calculated dispersion relation of the configuration is depicted in Fig. 3c, where TESs (highlighted by red dot line) with the frequency range from 184.11 to 205.89 THz appear in the band gap of 182.18–253.48 THz. The electric field distribution of the combined structure at the frequency of 189.50 THz is shown in Fig. 3d, which indicates that the TESs appear at the interface between PC1 and PC2. For verifying the robustness of TESs to defects, we randomly introduce impurities and disorders into the combined structure as shown in the left side of Fig. 3e. The electric field distribution at frequency of 189.50 THz as shown in the right side of Fig. 3e exhibits that there is no apparent scattering loss in the propagation process of TES. Transmission efficiency is an important index to measure the robustness of TES. We design a harmonic source (indicated by the red arrow) composed of four line sources with sequential phase changes and place it on the left side of the combined structure. According to the detected transmission power values of the input and output ports, the transmission efficiency of TES is calculated and defined as

$$T = \frac{{P_{{{\text{out}}}} }}{{P_{{{\text{in}}}} }} = \frac{{\iint {p_{{{\text{out}}}} ds}}}{{\iint {p_{{{\text{in}}}} ds}}},$$
(3)

where \(P_{{{\text{in}}}}\) and \(P_{{{\text{out}}}}\) denote the total power values of input and output harmonic electromagnetic wave. \(p_{{{\text{in}}}}\) and \(p_{{{\text{out}}}}\) denote the measured time-average power values of harmonic electromagnetic wave propagating at the input and output ports. The transmission efficiencies of TESs with and without defects in the combined structures are calculated and plotted in Fig. 3f and g, respectively, we find that the transmission efficiencies of TESs in the structure without defects are above 96%, and transmission efficiencies with different defects (such as disorders and impurities represented by red dotted line boxes on the left side of Fig. 3e) are also above 96% in the frequency range from 184.11 to 205.89 THz. However, when these defects are introduced into the W1 PC waveguide as shown in Fig. 3h, the electric field distribution as shown in Fig. 3i at frequency of 189.50 THz indicates that the transmission energy is reflected seriously by impurity and disorder.

Fig. 3
figure 3figure 3

a The combined structure composed of PC1 and PC2. b The supercell of the structure and the electric field distributions when kx = 0 and kx = π/\(a\). c The dispersion curves of the supercell. The TESs are highlighted by the red dots and the bulk states are by the black dots. d The electric field distribution of the proposed combined structure without defects at the frequency of 189.5 THz. e Diagrams of three combined structures with defects such as impurities and disorder (the left panel), and the corresponding electric field distributions at the frequency of 189.5 THz (the right panel). f The transmission efficiencies of TES in the combined structure without defects. g The transmission efficiencies of TES in the combined structure with defects. h The W1 PC waveguide structure with defects. i The electric field distribution of the W1 PC waveguide in (h) at the frequency of 189.5 THz

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The principle of bulk–edge–corner correspondence manifests that the existence of TCSs is determined by the edge polarization [6, 10, 25, 34]. \(P_{{\text{j}}}^{{{\text{edge}}}}\) represents the value of edge polarization along the j direction, which depends on the symmetry of the electric field distribution in the supercell. It can be seen from the middle diagrams of Fig. 3b that the electric field distributions in the supercell are mirror symmetrical and mirror anti-symmetrical when kx = 0 and kx = π/\(a\), respectively; therefore, the value of the edge polarization along the x direction is \(P_{x}^{{{\text{edge}}}} = 1/2\). Similarly, the value of the edge polarization along the y direction is \(P_{y}^{{{\text{edge}}}} = 1/2\). The relationship between TCSs and edge polarization is expressed by the following equation:

$$Q^{{{\text{corner}}}} = P_{{\text{x}}}^{{{\text{edge}}}} + P_{y}^{{{\text{edge}}}} .$$
(4)

We can judge the presence of TCSs according to Eq. (4). When \(P_{j}^{{{\text{edge}}}} = 1/2\)(j = x, y), \(Q^{{{\text{corner}}}} = 1\); therefore, TCSs can be generated in the structure composed of PC1 and PC2. To demonstrate the existence of TCSs, a box-type combined structure with inside PC1 and outside PC2 is constructed as shown in Fig. 4a. Figure 4b shows a series of eigenstates (bulk states, TESs, TCSs) by calculating the eigenmodes of the box-type combined structure, it can be observed that there are TESs and four nearly degenerate TCSs in the band gap, the corresponding frequencies of TCSs are C1 = 230.26 THz, C2 = 230.38 THz, C3 = 230.38 THz, and C4 = 230.49 THz. Figure 4c–f shows the electric field distributions of four TCSs, which demonstrates that four TCSs are strong localization at the intersections of interfaces between PC1 and PC2 in the box-type combined structure. Alternatively, from the C4 symmetry of the square lattice, the topological index of TCSs can be expressed as

$$Q^{c} = \frac{1}{4}([X_{1} ] + 2[M_{1} ] + 3[M_{2} ]),$$
(5)

where \(\left[ {\Pi_{p} } \right] = \# \Pi_{p} - \# \Gamma_{p}\), \(\# \Pi_{p}\) represents the number of bands underneath the PBG and rotation eigenvalues \(\Pi_{p} = e^{[2\pi i(p - 1)]/4}\)(p = 1, 2, 3, 4). \(\Pi\) represents high symmetric points X(Y), M and \(\Gamma\) at the first Brillouin zone. Since PC1 is non-trivial case, \([X_{1} ] = - 1, \, [M_{1} ] = 1\), and \([M_{2} ] = {0}\). Therefore, the topological index of TCSs is \(Q^{{\text{c}}} = 1/4\), indicating 1/4 fractionalized TCSs at each of the four intersections of interfaces between PC1 and PC2 in the box-type combined structure [25, 33, 35,36,37,38].

Fig. 4
figure 4

a The box-type combined structure consisting of inside PC1 and outside PC2. b The eigenmodes of the box-type combined structure. c–f The electric field distributions of TCSs at frequencies C1 = 230.26 THz, C2 = 230.38 THz, C3 = 230.38 THz, and C4 = 230.49 THz, respectively

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To make four nearly degenerate TCSs having larger frequency difference, we explore a new method of changing the sizes of dielectric columns A, B, C, and D (b1 = 0.20\(a\), b2 = 0.28\(a\)) in the intersections of interfaces between PC1 and PC2 in the box-type combined structure. The corresponding minor axis b2 of the A, B, C, and D dielectric column is changed to be 0.336\(a\), 0.280\(a\), 0.224\(a\), and 0.168\(a\), keeping the major axis b1 the same as shown in Fig. 5a. According to the eigenmodes of the box-type combined structure shown in Fig. 5b, we can find that the frequency difference of TCSs is larger and the frequencies of TCSs are C1 = 224.71 THz, C2 = 230.38 THz, C3 = 237.28 THz, and C4 = 245.05 THz. Figure 5c–f indicates the corresponding electric field distributions of the four TCSs have the strong localization property.

Fig. 5
figure 5

a The box-type combined structure for changing the sizes of the corner dielectric columns. b The eigenmodes of the box-type combined structure. c–f The electric field distributions of TCSs at C1 = 224.71THz, C2 = 230.38THz, C3 = 237.28 THz, and C4 = 245.05 THz, respectively

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4 Waveguide–cavity coupling system based on topological photonic crystals

Based on the above analysis, we design a TPC waveguide–cavity coupling system with a Y-type PC waveguide and four TCS cavities in the box-type combination structure. As shown in Fig. 6a, the Y-type PC waveguide is constructed in PC2 and widths of the input waveguide and the output waveguides are 2 \(a\) and \(a\), respectively. The four TCS cavities T1, T2, T3, and T4 are marked with blue dotted boxes. We detect the average output power of the output waveguide 1 in the Y-type PC waveguide and the electric field intensities of cavity T1 and cavity T2. The results are shown in Fig. 6c, the blue solid line represents the average output power of the output waveguide 1, and we find that its values drop to almost zero at the frequencies 225.70 THz and 231.70 THz. The red solid line (purple solid line) depicts the electric field intensity of cavity T1 (cavity T2) reaching maximum at the corresponding frequency. It indicates the energy can be coupled to TCS cavities T1, T2 from the output waveguide 1 of the Y-type PC waveguide at specific frequencies. Similarly, the output waveguide 2 of the Y-type PC waveguide couples with the cavity T3 and the cavity T4, and the corresponding coupling frequencies are 238.70 THz and 246.40 THz, respectively. The average output power of the output waveguide 2, the electric field intensities of the cavity T3 and the cavity T4 are shown in Fig. 6d. Figure 6e–h exhibits the electric field distributions of TPC waveguide–cavity coupling system at 225.70 THz, 231.70 THz, 238.70 THz, and 246.40 THz, respectively. Obviously, due to the strong localization of TCS, the transmission energy in one waveguide is coupled into the TCS cavities at the coupling frequencies, while the transmission energy in another waveguide can flow out from the output. When the TPC waveguide–cavity system is operating at the uncoupled frequencies, both output waveguides of the Y-type PC waveguide can transmit energy simultaneously, as shown in Fig. 6b. Therefore, the TPC waveguide–cavity coupling system can realize the functions of topological optical switch [39, 40] and controllable optical storage [41, 42] at a specific frequency.

Fig. 6
figure 6

a TPC waveguide–cavity coupling system. b The electric field distribution of the TPC waveguide–cavity coupling system at f = 200.00 THz. c The average output power of the output waveguide 1 (blue solid line) in the Y-type PC waveguide, the electric field intensities of the cavity T1 (red solid line) and the cavity T2 (purple solid line). d The average output power of the output waveguide 2 (blue solid line) in the Y-type PC waveguide and the electric field intensities of cavity T3 (yellow solid line) and cavity T4 (green solid line). \({\varvec{e}}\)–h The electric field distributions corresponding to f = 225.70 THz, 231.70 THz, 238.70 THz, and 246.40 THz in the TPC waveguide–cavity coupling system

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Finally, we investigate the robustness of TCSs to defects in the TPC waveguide–cavity coupling system by randomly introducing impurities and air defects in the regions represented by the green dotted line box, as shown in Fig. 7a. Figure 7b–d shows the waveguide–cavity coupled system with three different random defects, and impurities and air defect are labeled by the red dotted line boxes. We detect the corresponding average out power of the two output waveguides in the Y-type PC waveguide and the electric field intensity of the four TCS cavities after introducing defects into the TPC waveguide–cavity coupling system, as shown in Fig. 7e–g. Compared with the TPC waveguide–cavity coupling system without introducing defects, the coupling effect is almost unchanged. Therefore, TCSs in the proposed TPC waveguide–cavity coupling system have strong robustness to defects.

Fig. 7
figure 7

a The green dashed region where defects are randomly introduced in the TPC waveguide–cavity coupled system. b–d The waveguide–cavity coupled system with three different random defects, the impurities, and air defect are labeled by the red dotted line boxes. e–g The average out power of the two output waveguides in the Y-type PC waveguide and the electric field intensity of the four TCS cavities after introducing defects into the TPC waveguide–cavity coupling system

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5 Conclusion

In this paper, we propose a kind of square lattice PCs composed of rhombic dielectric columns, and achieve topological phase transition by expanding and shrinking of UC. TESs are implemented at the interface PCs with different Zak phases, which have high transmission efficiency and robustness to impurities and disorder. Based on the principle of topological polarization, four nearly degenerate TCSs are realized in the box-type structure. We find that the frequency difference of the four nearly degenerate TCSs is larger after changing the size of dielectric columns located at the intersections of interfaces between PC1 and PC2 in the box-type combined structure. On this basis, we combine the conventional PC waveguide and TCS cavities to construct a TPC waveguide–cavity coupling system. This system has the characteristic of strongly coupling and can realize the multifunction of topological optical switch and controllable optical storage at the specific frequency. Meanwhile, the robustness of TCSs in this system is proved by introducing impurities and air defects. These work provides a new idea for the applications of TESs and TCSs in optical communication devices.