CN113406730B - Method for realizing multiband topological angular state by using 2D S-T photonic crystal - Google Patents

Method for realizing multiband topological angular state by using 2D S-T photonic crystal Download PDF

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CN113406730B
CN113406730B CN202110830838.3A CN202110830838A CN113406730B CN 113406730 B CN113406730 B CN 113406730B CN 202110830838 A CN202110830838 A CN 202110830838A CN 113406730 B CN113406730 B CN 113406730B
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刘建军
彭宇宸
颜贝
谢建斓
史奥芊
高峰
彭鹏
蒋家培
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Abstract

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.

Description

Method for realizing multiband topological angular state by using 2D S-T photonic crystal
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 the d-1 dimension. Among them, the 2D 2-order topological insulator produces zero-dimensional topological boundary states, also called topological corner states. The appearance of the optical waveguide widens the non-trivial topological isolated 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 angular states has mainly focused 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, at present, the multi-frequency topological angular state is not realized in the photonic crystal, which greatly limits the application of the photonic crystal in the multi-band photonic device, and therefore, how to simultaneously realize the topological angular states with different frequencies in the same structure is urgently needed to be 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, compare with other C6The symmetrical structure, namely the 2D S-T photonic crystal energy band structure, is very 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 angular state is further adjusted in a larger space, and the possibility is provided for realizing the topological angular state with variable position.
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 7 scatterers are greater than 12 scatterers, interior outer unanimous 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;
adjusting the diameters of Type I and Type III photonic crystals until the two types of photonic crystals have a common band gap, and then 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 and 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. 2 is a 2D S-T photonic crystal with Stampfli type photonic quasicrystal as basic structural unit and arranged in 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: d is a radical of1＝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 at 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 pointzDot tableThe space between the signs is parity and the space between the triangles is 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) 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 2D S-T photonic crystal composite structure, the projection energy bands of the strip-shaped super cell at 360-420THz and 480-550 THz; (c) e at A, B point in the projection bandZField distribution; (d) e at C, D point in the projection bandZAnd (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 at THz angle; (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 (upper panel) and 385.72THz (lower panel) 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 the linear current wave source to overcome the defect includes a region a: impurity addition, B region: part of the scatterers are removed.
Fig. 7 shows the topological angular states in a parallelogram box structure: (a) adding 509.23THz line current; (b) adding 385.72THz line current; (c) adding 123.48THz left-handed polarized wave source; (d) a source of 123.48THz right-hand polarized waves 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. C at the interface by adjusting the diameters of internal and external scatterers of the photonic crystal6The 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 quasicrystals and 2D S-T photonic crystal structures, as shown in FIG. 1. The basic structural unit of the Stampfli type photonic 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, thereby meeting the requirement of 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 figure 2 (only part of which is shown). Assuming that the lattice constant a is 1 μm, the lattice vector is:
Figure BDA0003175490210000061
Figure BDA0003175490210000062
distance between adjacent scatterers
Figure BDA0003175490210000063
(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,Py) A topology invariant is defined. Polarization P in i-directioniThe expression is as follows:
Figure BDA0003175490210000071
wherein BZ represents a first Brillouin zone,
Figure BDA0003175490210000072
indicating the Zak phase in the i direction,
Figure BDA0003175490210000073
representing the belief, psi 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, based on the formula (1), a judgment polarization value P can be obtainediThe simpler method is to solve by judging the signs of the parity at the high symmetry point of the Brillouin zone, as follows:
Figure BDA0003175490210000074
wherein eta isn(Mi) And etan(Γ) represents the n-th band M in the first Brillouin zoneiAnd the parity size at the gamma point,
Figure BDA0003175490210000075
to judge the symbol. In addition, for satisfying C6The symmetrical structure has a relation P because it also satisfies the mirror symmetryx＝PyWhen the obtained polarization is 1/2, the quadrupole state Qxy＝PxPy1/4, 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 the three scatterer diameter cases was calculated as shown in FIG. 3.
3(a), 3(b) and 3(c), when the Type I of the 2D S-T photonic crystal is changed into the Type III, the evolution process of opening to degeneracy and reopening of multiple Dirac points exists in the energy band structure. According to the conclusion obtained by previous research, the 2D S-T photonic crystal can realize the photon spin Hall effect at a low frequency band, so that 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 number of band calculations and rule summaries, it is found that only the trivisual (left) and the trivisual (right) situations shown in fig. 3(d) and the trivisual (left) and the trivisual (right) situations shown in fig. 3(e) occur in the common band gap of the Type I and Type III photonic crystal high-band dislocation as long as the low-band is subjected to 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, verifying the appearance of topological boundary states with band gaps, calculating the projected energy bands of a super-cell consisting of a topological mediocre state and a topological non-mediocre photonic crystal with a common band gap having a dislocation before and after 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 have found that two Type I or two Type III photonic crystals can also produce topological boundary states in the projected energy bands, as long as the nearest neighbor energy bands under their dislocated bandgaps are a combination of Nontrivial and Trivial, as well as those with C6Compared with the energy band structure of the symmetrical honeycomb lattice photonic crystal, the topological boundary state of the 2D S-T photonic crystal is easier to have larger band gap frequency difference with the bulk state. 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 the 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 in different directions, while the boundary state of FIG. 4(b) is generated by the non-trivial Zak phase only, and can also be transmitted in the boundary but not in the boundaryIt 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 truncated 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 identical to that in fig. 3, but independent solutions (in a frame line) which do not belong to the projected energy band of fig. 3 appear in the bandgap, and the eigenmode field distributions corresponding to three representative points in the bandgap are analyzed, and as shown in fig. 5(b) and 5(d), the electric fields are concentrated at six corners inside the box-shaped structure, demonstrating the feasibility of realizing topological corner states of two different mechanisms. In order to further verify whether the topological angular state can be realized in reality and overcome the defects, waveguides are constructed in the simulation in the same arrangement mode as in fig. 5, wave sources of the waveguides are located in the center of the lower boundary of the hexagon of the box-shaped structure, and the topological characteristics of the 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. Moreover, the invention also analyzes the topological angular state of the parallelogram box-shaped structure to show the universality and stability of the method for realizing the topological angular 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 causes band inversion in the band structure to generate topological phase change, and the common band gap dislocated outside the band inversion of the twoThere is also a process of opening, degenerating and reopening the Dirac point, calculating the projected energy band of the two combined super cell structures results in boundary states with band gaps, compared to other structures 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.

Claims (2)

1. A method for implementing a multiband topological corner state by using a 2D S-T photonic crystal, the method comprising the steps of:
step one, based on a 2D S-T photonic crystal, the diameter of a scatterer is changed, and three different types of basic structural units are generated:
inside big outer small-size Type I: inner layer 7 scatterer diametersd 1Greater than 12 scatterer diameters of the outer layerd 2，d 1=0.3R，d 2=0.1R，RIs the distance between adjacent scatterers;
type II is unanimous outside and inside: inner layer 7 scatterer diametersd 1Equal to 12 scatterer diameters of the outer layerd 2，d 1=d 2=0.6R；
Inside small and outside large Type III: inner layer 7 scatterer diametersd 1Less than 12 scatterer diameters of the outer layerd 2，d 1=0.1R，d 2=0.34R；
The basic structure unit is formed by arranging germanium medium columns in the air;
and step two, adjusting the diameters of the Type I photonic crystals and the Type III photonic crystals to the extent that the two types of photonic crystals have a common band gap, then combining the Type I photonic crystals and the Type III photonic crystals in a left-right array mode by taking a lattice constant as a space, generating a topological boundary state with a band gap in a projection energy band, designing a box-shaped structure of the inner layer Type III photonic crystals surrounded by the outer layer Type I photonic crystals, solving the truncation energy band of the box-shaped structure, generating independent solutions different from the projection energy band in the band gap, taking a plurality of representative points of the solutions, analyzing the electric fields of the solutions, finding that the electric fields are all gathered at six corners inside the box-shaped structure, and proving feasibility of realizing the topological angle states of two different mechanisms.
2. The method for achieving multiband topological angular states using 2D S-T photonic crystals as claimed in claim 1, wherein the second step is replaced with: the nearest energy bands under the band gaps of the two Type I photonic crystals or the two Type III photonic crystals meeting dislocation are the combined situation of Nontrivial and Trivisual, the diameter parameters are adjusted to have a common band gap, and a topological boundary state is generated in the projection energy band, namely a topological angular state can be further generated.
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