阅读说明：本技术 利用2d s-t光子晶体实现多频段拓扑角态的方法 (Method for realizing multiband topological angular state by utilizing 2D S-T photonic crystal ) 是由 刘建军 彭宇宸 颜贝 谢建斓 史奥芊 高峰 彭鹏 蒋家培 于 2021-07-22 设计创作，主要内容包括：本发明公开了一种利用2DS-T光子晶体实现多频段拓扑角态的方法,所述方法包括如下步骤：步骤一、基于2DS-T光子晶体,改变其散射子直径大小,产生里大外小型TypeI、里外一致型TypeII、里小外大型TypeIII三种不同类型的基本结构单元；步骤二、将TypeI光子晶体与TypeIII光子晶体以晶格常数为间距左右阵列组合,或者两种TypeI光子晶体或两种TypeIII光子晶体满足错位的带隙下最近邻的能带是Nontrivial与Trivial的组合情形,在投影能带中产生拓扑边界态。该方法不仅丰富了高阶拓扑态的研究,也为实现拓扑角态提供了更简易的方法。(The invention discloses a method for realizing a multiband topological angular state by utilizing a 2DS-T photonic crystal, which comprises the following steps: changing the diameter of a scatterer of a 2DS-T photonic crystal to generate three basic structural units of different types, namely a small type I with a large inside and a small outside, a uniform type II with a large inside and a small outside, and a large type III with a large inside and a small outside; and step two, combining the TypeI photonic crystals and the TypeIII photonic crystals in a left-right array mode by taking the lattice constant as the interval, or combining the Nontrivial and Trivial nearest neighbor energy bands of the two TypeI photonic crystals or the two TypeIII photonic crystals under the condition that dislocation band gaps are met, and generating a topological boundary state in a projection energy band. The method not only enriches the research of high-order topological states, but also provides a simpler method for realizing topological angular states.)

1. A method for realizing multiband topological angular states by utilizing a 2DS-T photonic crystal is characterized by comprising the following steps:

step one, based on a 2DS-T photonic crystal, changing the diameter of a scatterer to generate a TypeI with a large inside and a small outside: the diameter of 7 scatterers in the inner layer is larger than that of 12 scatterers in the outer layer, and the inner and outer consistent type TypeII: the diameter of 7 scatterers in the inner layer is equal to the diameter of 12 scatterers in the outer layer, and the inner layer is small and the outer layer is large type III: the diameter of 7 scatterers in the inner layer is smaller than that of 12 scatterers in the outer layer;

and secondly, adjusting the diameters of the TypeI photonic crystals and the TypeIII photonic crystals until the two types of photonic crystals have a common band gap, and combining the TypeI photonic crystals and the TypeIII photonic crystals in a left-right array mode by taking the lattice constant as the interval, so that a topological boundary state with the band gap is generated in a projection energy band.

2. The method for implementing multiband topological corner states by using 2DS-T photonic crystals as claimed in claim 1, wherein the step two is replaced by: the two TypeI photonic crystals or the two TypeIII photonic crystals satisfy the combined case of nonotrivia and trivia for the nearest neighbor band under the bandgap of the dislocation and adjust the diameter parameters to have a common bandgap, creating topological boundary states in the projected energy bands.

Technical Field

The invention relates to a simple method for realizing a topological angular state, in particular to a method for realizing a multiband topological angular state by using a two-dimensional Stampfli-Triangle (2D S-T) photonic crystal.

Background

For conventional topological insulators, d-dimensional topological insulators typically have d-1-dimensional topological boundary states. However, the discovery of high-order topological insulators suggests an unconventional body-edge correspondence: the d-dimensional topological insulator presents a gapless topological boundary state smaller than d-1 dimension. Wherein the 2D 2-order topological insulator produces zero-dimensional topological boundary states, also referred to as topological corner states. The appearance of the optical waveguide widens the non-trivial topological insulating phase family, and provides a new idea for optical imaging, photon local area and control of optical waveguide transmission in the optical field. Meanwhile, the optical topological angular state provides a simple method for researching the optical microcavity because artificial construction defects are not needed, and new power is further provided for researching related devices such as a novel laser.

The study of topological angle states mainly focuses on tetragonal, kagome and honeycomb lattice structures based on the 2D Su-Schrieffer-Heeger (SSH) model. Unlike the conventional lattices described above, which have translational symmetry, 2D photonic quasicrystals have rotational symmetry and long-range order. The existing research shows that the photonic quasicrystal has the characteristics of rich energy band structure, local mode, low dielectric constant threshold value for generating complete band gap and the like, is superior to the periodic photonic crystal, but the energy band structure of the photonic quasicrystal cannot be effectively calculated, so that the photonic crystal formed by periodically cycling the basic structural unit of the 2D photonic quasicrystal can not only keep the advantages of the photonic quasicrystal, but also obtain an accurate energy band structure so as to research the high-order topological state therein. In a photonic system, the development of a high-order topological state is from the initially considered difficult realization to the realization of a similar phenomenon of a topological angular state, and then, stretching and compressing a photonic crystal primitive cell lattice to construct a 2D SSH model and theoretically and experimentally realize a second-order photonic topological insulator becomes a common method. However, not currently implemented in photonic crystalsThe frequency topological angle state greatly limits the application of the frequency topological angle state in a multiband photonic device, so how to simultaneously realize the topological angle states with different frequencies in the same structure is urgently researched. The 2D S-T photonic crystal formed by the periodicity of the triangular lattice of the basic structural unit of the Stampfli type photonic quasicrystal satisfies C₆Symmetry, can realize the photon spin Hall effect, compared with other C₆The symmetrical structure, 2D S-T photonic crystal energy band structure is easy to have broadband characteristics, so that the band gap between the topological boundary state and the bulk state is larger, the generation of the topological angle state is further adjusted, and the possibility is provided for realizing the position-variable topological angle state.

Disclosure of Invention

The invention aims to provide a method for realizing a multiband topological angular state by utilizing a 2D S-T photonic crystal, which not only enriches the research of high-order topological states, but also provides a simpler method for realizing the topological angular state.

The purpose of the invention is realized by the following technical scheme:

a method for realizing multiband topological angular states by using a 2D S-T photonic crystal comprises the following steps:

step one, changing the diameter of a scatterer of the 2D S-T photonic crystal to generate a Type I with a large inside and a small outside: the internal layer 7 scatterers are greater than the external layer 12 scatterers, and the internal and external consistent Type II: the diameter of 7 scatterers in the inner layer is equal to the diameter of 12 scatterers in the outer layer, and the inner layer, the small layer and the outer layer are large Type III: the diameter of 7 scatterers in the inner layer is smaller than that of 12 scatterers in the outer layer;

secondly, adjusting the diameters of the Type I photonic crystals and the Type III photonic crystals until the two types of photonic crystals have a common band gap, and combining the 16 Type I photonic crystals and the 16 Type III photonic crystals in a left-right array mode by taking a lattice constant as a distance, wherein the two types of photonic crystals have different polarization values under the common band gap to generate a topological boundary state with the band gap in a projection energy band; or the two Type I photonic crystals or the two Type III photonic crystals meet the combined situation of Nontrivial and Trivial of the nearest neighbor energy bands under the dislocation band gaps, the diameter parameters are adjusted to have a common band gap, and a topological boundary state can also be generated in the projection energy band;

thirdly, in order to further explore whether the topological angular state can be generated by the polarization of the topological boundary state obtained in the second step, a hexagonal box-shaped structure of the outer-layer Type I photonic crystal surrounding the inner-layer Type III photonic crystal is designed, the cutoff energy band of the hexagonal box-shaped structure is solved, independent solutions which do not belong to the projection energy band in the second step appear in the band gap, the solutions are taken from a plurality of representative points to analyze the electric field of the solutions, the electric field is found to be gathered at six corners inside the box-shaped structure, and the feasibility of realizing the topological angular state of two different mechanisms is proved;

and step four, verifying whether the topological corner state can be realized in reality and whether the defects can be overcome, constructing the waveguide in the simulation in the same arrangement mode of the step three, positioning a wave source of the waveguide at the center of the lower boundary of the hexagon of the box-shaped structure, simultaneously introducing the defects to verify the topological characteristics of the corner state, finding that electric fields still exist at six corners and the defects near the corners have immunity, and proving that the obtained corner state is protected by the topology.

Compared with the prior art, the invention has the following advantages:

the invention researches the high-order topological state and mechanism of the 2D S-T photonic crystal, finds that if partial energy bands of the photonic crystal are subjected to order exchange, all common band gaps have topological boundary states and topological angle states with gaps, and then constructs a waveguide to introduce defects to verify the topological characteristics of the photonic crystal. The invention is a method for realizing topological angular state based on two physical mechanisms in the same photonic crystal for the first time, wherein the method is initiated by photon spin Hall effect and by topological interface state, the former can change the position distribution of the topological angular state, and the latter can realize the topological angular state of different frequency bands in the same structure. The discovery enriches the research of high-order topological states and also provides a simpler method for realizing topological angular states. The research result of the invention has guiding significance on the design of optical integrated devices such as optical microcavities, high-quality factor lasers and the like.

Drawings

FIG. 1 is a Stampfli type photonic quasicrystal.

FIG. 2The photonic crystal is a 2D S-T photonic crystal which takes a Stampfli type photonic quasicrystal as a basic structural unit and is arranged according to a triangular lattice; comprises three different types of basic structural units, wherein Type I is large inside and small outside (d)₂/d₁<1) (ii) a Type II is of inside-outside coincidence Type (d)₂/d₁1); type III is small inside and large outside (d)₂/d₁>1)。

FIG. 3 is an energy band structure of a 2D S-T photonic crystal: (a) type I: d₁＝0.9R，d₂＝0.4R；(b)Type II：d₁＝d₂＝0.6R；(c)Type III：d₁＝0.1R，d₂＝0.8R；Type I(d₁＝0.3R，d₂0.1R) and Type III (d)₁＝0.1R，d₂0.34R) band structure in different frequency bands (common band gap but band gap between different order bands, i.e. dislocated common band gap): (d)300-450 THz; (e)400-550THz, the inset is Brillouin zone gamma and corresponding E at M point_zThe field, dots, represents the even parity and the triangles represent the odd parity.

FIG. 4 shows the projected energy bands and boundary states: (a) by Type I (d)₁＝0.9R，d₂0.4R) and Type III (d)₁＝0.1R，d₂0.8R) projection energy band of the elongated super cell in the 2D S-T photonic crystal composite structure; (b) by Type I (d)₁＝0.3R，d₂0.1R) and Type III (d)₁＝0.1R，d₂0.34R) in the projection energy bands of the strip-shaped super cell in the 2D S-T photonic crystal composite structure at 360-420THz and 480-550 THz; (c) e corresponding to point A, B in the projection band_ZField distribution; (d) e corresponding to point C, D in the projection band_ZAnd (6) field distribution.

Fig. 5 shows the mode field distribution of the truncated energy band and topological angular state of the box structure: (a) type I (d)₁＝0.9R，d₂0.4R) and Type III (d)₁＝0.1R，d₂0.8R) solution number relationship of the box-shaped structure of the 2D S-T photonic crystal combination; (b)124.68 mode field distribution of THz angular states; (c) type I (d)₁＝0.3R，d₂0.1R) and Type III (d)₁＝0.1R，d₂0.34R) the solution number relationship of two frequency bands in the box-shaped structure of the 2D S-T photonic crystal combination; (d)509.23THz (top)Graph) and 385.72THz (lower graph) angular mode field distributions.

FIG. 6 is a topological property of the angular state: (a) FIG. 4(a) shows a defect-free mode field distribution of the structure with the addition of a left-handed polarized wave source (left) and a right-handed polarized wave source (right); (b) the mode field distribution of the structure of fig. 4(b) after adding a linear current wave source to overcome the defect includes a region a: impurity addition, B region: part of the scatterers are removed.

Fig. 7 is a topological angular state in a parallelogram box structure: (a) adding 509.23THz line current; (b) adding 385.72THz line current; (c) adding 123.48THz left-hand polarization wave source; (d) 123.48THz right-handed polarized wave source was added.

Detailed Description

The technical solution of the present invention is further described below with reference to the accompanying drawings, but not limited thereto, and any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention shall be covered by the protection scope of the present invention.

The invention provides a method for realizing multiband topological angle states by utilizing a 2D S-T photonic crystal, which is characterized in that the spatial inversion symmetry is broken under the condition of keeping time inversion symmetry, so that the order exchange of partial energy bands of the photonic crystal is realized, the photonic crystal is combined with a photonic crystal which does not generate the order exchange of the energy bands and has a common photonic band gap, a boundary state with the band gap can be generated in a projection energy band, the phenomenon meets the performance characteristics of a high-order topological insulator, topological angle states with two physical mechanisms can be generated in the 2D S-T photonic crystal, one is triggered by a quantum spin Hall effect, and the other is triggered by a topological interface state. Adjusting the diameter of internal and external scatterers of photonic crystal to make C at interface₆The symmetry breaking into C₃Symmetry to produce topology can be considered as a generalization of the 2D SSH model. The specific contents are as follows:

first, model and theory

Stampfli type photonic quasicrystal and 2D S-T photonic crystal structure, as shown in FIG. 1. The basic structural unit of the Stampfli type photon quasicrystal is formed by splicing and combining a triangle and a square edge to edge, and rotating for 6 times by taking the vertex which does not participate in combination as a rotating center, becauseThis satisfies C₆Symmetry, as shown in the blue lattice in fig. 1, the lattice structure of the Stampfli type photonic quasicrystal can be formed by reducing the blue lattice to 1/σ 0.2680 times according to the self-similarity factor σ of the Stampfli type photonic quasicrystal, which is 2+2cos (2 pi/12) ═ 3.7320, and placing the lattice point at the lattice point of the blue lattice. The Stampfli type photonic quasicrystal basic structural unit is periodically arranged according to a triangular lattice to form the 2D S-T photonic crystal, as shown in FIG. 2 (only part of which is shown). Assuming that the lattice constant a is 1 μm, the lattice vector is: distance between adjacent scatterers(also the lattice constant of a Stampfli type photonic quasicrystal) and the diameter of the internal 7 scatterers is d₁The outer 12 scatterers have a diameter d₂The basic structural unit is composed of germanium dielectric column (epsilon)_ra16) arranged in air (epsilon)_rb1) above.

For Type I and Type III photonic crystals, the coupling coefficient between scatterers differs according to the difference in scatterer diameter, and further energy bands with topologically non-mediocre Zak phases can be obtained to cause the appearance of topological boundary states, which are also called topological interface states, if non-mediocre Zak phases exist at the x-direction and y-direction boundaries at the same time, polarization of the boundaries can be caused, so that topological angular states are generated between photonic crystals with topologically mediocre states and topologically non-mediocre Zak phases. Therefore, in order to characterize the topological properties of photonic crystals, the present invention derives from the 2D polarization vector P ═ (P)_x,P_y) A topology invariant is defined. Polarization P in i-direction_iThe expression is as follows:

wherein BZ represents a first Brillouin zone,indicating the Zak phase in the i direction,representing the belief, ψ is the periodic bloch function of the energy band. It can be deduced that the Hamilton quantity of the 2D S-T photonic crystal satisfies H (-k) ═ H^*(k) And it satisfies the time-reversal symmetry so that the sum of the total Berry curvatures is zero. Under the condition that the system is in zero Berry curvature, a judgment polarization value P can be obtained based on the formula (1)_iThe simpler method is to solve by judging the signs of the parity at the high symmetry point of the Brillouin zone, as follows:

wherein eta is_n(M_i) And eta_n(Γ) represents the n-th band M in the first Brillouin zone_iAnd the parity size at the Γ point,to judge the symbol. In addition, for satisfying C₆The symmetrical structure has a relation P because it also satisfies mirror symmetry_x＝P_yWhen the obtained polarization is 1/2, the quadrupole state Q_xy＝P_xP_y1/4, the topological boundary states and topological corner states are generated.

Second, result and discussion

The band structure of the 2D S-T photonic crystal for three scatterer diameter cases was calculated as shown in FIG. 3.

As can be seen from fig. 3(a), 3(b) and 3(c), when the Type I of the 2D S-T photonic crystal is changed to Type III, there is an evolution process from open to degenerate to reopen of multiple Dirac points in the band structure. According to the conclusion obtained by previous research, the 2D S-T photonic crystal can realize the photon spin Hall effect in a low frequency band, and therefore, the degree of freedom of pseudo spin is introduced. However, in the high frequency range, compared with the previous research, the p-d band inversion phenomenon with regularity in the low frequency band does not appear in the band, and the defect process of multiple Dirac points still exists. Further, by analyzing the parity at the Γ and M points of the brillouin zone, it can be seen from equation (2) that the polarization value of the band gap nearest neighbor energy band in the left graph of fig. 3(d) and the right graph of fig. 3(e) is 0, and the polarization value of the band gap nearest neighbor energy band in the right graph of fig. 3(d) and the left graph of fig. 3(e) is 1/2, and therefore, the energy bands at this time have topological characteristics and accordingly, a topological angular state can be generated. For the sake of simplicity, the case with different parity in the band gap nearest to the band gap Γ and M point is denoted as nintrivia, and the case with the same parity is denoted as trivia. Through a large amount of energy band calculation and rule summarization, it is found that only Trivisual (left) and Nontrivial (right) situations shown in FIG. 3(d) and Nontrivial (left) and Trivisual (right) situations shown in FIG. 3(e) occur under the common band gap of the Type I and Type III photonic crystal high-frequency band dislocation as long as the low-frequency band is subjected to energy band inversion. Conversely, if no band inversion occurs in the low band, the band nearest to the common band gap with the high band offset can only occur in Type I photonic crystals of different materials.

Further exploring the topological characteristics of the 2D S-T photonic crystal at different frequency bands and verifying the appearance of topological boundary states with band gaps, calculating the projected energy bands of the super cell consisting of the topological mediocre state and the topological non-mediocre state photonic crystal with the common band gap being staggered before and after the energy band inversion, as shown in FIG. 4.

As shown in fig. 4(a), the 2D S-T photonic crystal composite structure of Type I and Type III is a topological boundary state generated by the photon spin hall effect, and there is a gap of a certain frequency band between the topological boundary state and the bulk state. As can be seen from FIG. 4(b), when a Type I photonic crystal is combined with a Type III photonic crystal having a common photonic bandgap without band inversion and having a dislocation, topological boundary states having bandgaps are generated due to different polarization values at the bandgaps, and the bandgap widths of the three frequency bands shown in FIGS. 4(a) and 4(b) are Δ f₁＝5.8THz、Δf₂20.4THz and Δ f₃20.2 THz. In addition, we found that two types I orTwo Type III photonic crystals can also generate topological boundary states in a projection energy band as long as the nearest neighbor energy band under the dislocation band gap of the two Type III photonic crystals is the combined situation of Nontrivial and Trivisual₆Compared with the energy band structure of the symmetrical honeycomb lattice photonic crystal, the topological boundary state and the bulk state of the 2D S-T photonic crystal are more prone to have larger band gap frequency difference. Therefore, the 2D S-T photonic crystal can provide abundant conditions for the appearance of topological angular states. As can be seen from fig. 4(c) and 4(D), the mode fields corresponding to A, B, C and D on the dispersion curve of the boundary state are mainly distributed at the boundary between Type I and Type III photonic crystals and attenuate to both sides, which is in accordance with the characteristics of the boundary state. The 2D S-T photonic crystal composite structure related to FIG. 4(a) can realize unidirectional transmission effect with spin-direction locking due to the introduction of pseudo spin freedom, i.e. wave sources with different circular polarization directions can generate unidirectional transmission with different directions, while the boundary state of FIG. 4(b) is generated by non-trivial Zak phase only, and can also be transmitted at the boundary but has no unidirectionality. Due to different physical mechanisms of two topological boundary states, topological angular states of different physical mechanisms are possible to realize for the 2D S-T photonic crystal. To further explore whether the topological angular state can be generated by boundary state polarization, a box structure is designed and its truncation energy band is solved, as shown in fig. 5.

The frequency range of the topological boundary states in fig. 5(a) and 5(c) is the same as that in fig. 3, but independent solutions (inside the frame line) which do not belong to the projected energy band in fig. 3 appear in the band gap, and the distribution of the eigenmode fields corresponding to three representative points is analyzed, as shown in fig. 5(b) and 5(d), the electric fields are all concentrated at six corners inside the box-shaped structure, and the feasibility of realizing topological angular states of two different mechanisms is proved. In order to further verify whether the topological angular state can be realized in reality and overcome the defects, the waveguides are constructed in the simulation in the same arrangement mode as in fig. 5, the wave source of the waveguides is positioned at the center of the lower boundary of the hexagon of the box-shaped structure, and the topological characteristics of the topological angular state are verified by introducing the defects, as shown in fig. 6.

As shown in fig. 6(a), the topological angular state generated by the topological boundary state of the quantum spin hall effect has a spin-direction locking relationship, and in accordance with the generation mechanism of the topological boundary state, the topological angular state with gradually weakened strength in the counterclockwise direction or the clockwise direction can be excited by the wave source with different circular polarization directions. For the topological corner state generated by the topological interface state, two different defect modes are introduced, and as can be seen from fig. 6(b), electric fields still exist at six corners and immunity is provided for the defects near the corners, which proves that the obtained corner state is protected by the topology. In addition, the invention also analyzes the topological angle state of the parallelogram box-shaped structure to show the universality and stability of the method for realizing the topological angle state, as shown in fig. 7.

Third, conclusion

The invention is based on 2D S-T photonic crystal, changes the diameter of scatterer to generate three different types of basic structural units of small type inside and outside, consistent type inside and outside and large type inside and outside, and destroys C at the boundary₆The symmetry enables the energy band structure to generate energy band inversion to generate topological phase change, at the moment, the opening, degenerating and reopening processes of Dirac points also exist in the common band gap dislocated outside the energy band inversion of the two energy band structures, the two combined super cell structures are calculated, the projection energy band is calculated to obtain the boundary state with the band gap, and the boundary state is compared with other boundary states with C₆The symmetrical structure is easier to generate a wide band gap by the 2D S-T photonic crystal. Meanwhile, as long as the nearest neighbor energy bands under the band gap have different polarization values, the topological boundary state with the band gap can exist to generate a topological angle state, which is a simple method for realizing the topological angle state and also provides clues for realizing the multiband topological angle state.