Topological rainbow optical waveguide based on valley photonic crystal

Abstract

The invention provides a topological rainbow optical waveguide based on a valley photonic crystal, which is used for localizing and constraining topological edge states with different frequencies on a specific interface and providing a feasible platform for the interface as a valley degree of freedom to regulate and control nanoscale electromagnetic waves. Valley photonic crystals VPC-1 and VPC1 are constructed through full-dielectric aluminum oxide cylinders, a U-shaped waveguide structure composed of a beard-type interface 1, a beard-type interface 2 and a handrail-type interface is constructed based on the two types of valley photonic crystals, and the structure is obtained through theoretical calculation and numerical simulation. It is proved that the valley Hall topology state in the U-shaped waveguide can be localized and bound to a specified interface depending on the frequency. Compared with the design that different frequency edge states need to be separated and bound to different spatial positions in a traditional topological rainbow system, the waveguide achieved through the method does not need to modulate an external magnetic field or design a complex structure with gradually-changed parameters, and photon integration development is greatly promoted.

Description

Technical Field

[0001] This invention relates to the technical fields of topological photonics, optical quantum communication and integrated photonic devices, and in particular to a topological rainbow waveguide based on valley photonic crystals, which promotes the application and development of valley electronics in the field of integrated optics, especially in the research and development of novel functional photonic devices. Background Technology

[0002] In two-dimensional topological photonic systems, by breaking the parity-reversal symmetry (PRS) of the photonic system, valley-dependent edge states can be generated at the interface of a stack of two mirror-inverted valley photonic crystals (VPCs). This mechanism is analogous to the quantum valley Hall effect (QVHE). These valley-dependent edge states are protected against inter-valley scattering caused by defects that preserve the symmetry of the C3 crystal. Generally speaking, compared with the quantum Hall effect (QHE) and quantum spin Hall effect (QSHE) based on special magnetic field conditions or complex material structures, the quantum valley Hall effect is easier to fabricate using all-dielectric materials and combined with traditional fabrication techniques to create nanophotonic devices. This is mainly based on two factors: first, the parity-reversal symmetry breaking generates valley-dependent spin-splitting bulk state energy bands; second, the valley, as a degree of freedom (DOF), becomes a superior information carrier. To date, researchers have made significant progress in the field of terahertz on-chip communication.

[0003] Based on the quantum valley Hall effect, the laboratory has successfully developed and fabricated micro-nano photonic chips, with continuously improving wireless communication efficiency. For example, transmission rates of 13 Gbps (Terahertz topological photonics for on-chip communication), 100 Gbps (Valley-conserved topological integrated antenna for 100-Gbps THz 6G wireless), 160 Gbps (Phototunable chip-scale topological photonics: 160 Gbps waveguide and demultiplexer for THz 6G communication), and 320 Gbps (On-chip topological beamformer for multi-linkterahertz 6G to XG wireless) have been achieved. In addition to studying the transmission characteristics of terahertz topological photonic chips, researchers have also explored the localization and trapping phenomena of topological edge states. For example, the topological rainbow effect is a phenomenon in which topological edge states are robustly localized and trapped at specific spatial locations at different frequencies, providing a novel platform for manipulating integrated photonic electromagnetic waves (EM waves). In 2021, the research group of Cuicui Lu at Beijing Institute of Technology proposed a topological rainbow based on synthetic dimension, realizing the trapping of electromagnetic waves of different frequencies at different spatial locations (Topological rainbow concentrator based on synthetic dimension), and subsequently observed this topological rainbow phenomenon in the laboratory (On-chip nanophotonic topological rainbow). Another important application is the localization of chiral edge states by the non-Hermitian skin effect proposed by Guigeng Liu at Westlake University in 2024—chiral edge states are localized at specific volume boundaries based on the non-Hermitian skin effect. However, no research has yet reported the case of localizing topological edge states to specific interfaces through frequency dependence.

[0004] To our knowledge, the design and development of topological rainbows primarily stem from three principles and methods: dispersion engineering, synthesis dimension, and Landau levels, with dispersion engineering being the most widely used. Topological rainbow phenomena can be achieved by combining dispersion engineering with topological properties, such as modulating external magnetic field strength, designing device structures with gradually varying parameters, and introducing gradient losses. However, in practical applications, the fabrication processes of these methods are complex, hindering the integration of photonic devices. On the other hand, numerous studies have shown that interface coupling distance and interface configuration, or interface stacking type (such as sawtooth interfaces, armrest interfaces, and mustache interfaces), are key factors for achieving precise control of electromagnetic wave frequencies. Summary of the Invention

[0005] To address the complex technical issues of traditional topological rainbow waveguide design, this invention proposes a topological rainbow waveguide based on valley photonic crystals. By studying the dispersion curves of valley Hall edge states at different interfaces, this invention can also construct topological rainbow structures composed of different interfaces, and the interfaces themselves hold promise as novel degrees of freedom for electromagnetic wave manipulation. Compared to previous research, this approach does not require special technical means, which is of positive significance for the development of integrated photonic devices. This invention proposes a graphene-like lattice valley photonic crystal topological rainbow waveguide state, which can realize the localization of valley Hall boundary states of different frequencies on different interfaces. Compared with existing topological rainbow designs, the structure is simpler, the development cost is lower, and it possesses waveguide transmission characteristics. These research results provide a potentially feasible technical path and platform for manipulating electromagnetic waves at the nanoscale using interfaces as degrees of freedom, demonstrating significant value, particularly in multi-frequency routing and ultra-compact topological rainbow nanolasers in the field of integrated photonics.

[0006] To achieve the above objectives, the technical solution of this invention is implemented as follows: A topological rainbow waveguide based on valley photonic crystal is a U-shaped waveguide structure, including a U-shaped waveguide interface, the region enclosed by the U-shaped waveguide interface is the inner region, and the outer side of the U-shaped waveguide interface is the outer region; the outer region is composed of valley photonic crystal VPC1, and the inner region is composed of valley photonic crystal VPC-1, with valley photonic crystal VPC1 and valley photonic crystal VPC-1 being spatially antisymmetric; the U-shaped waveguide interface includes a beard-shaped interface-1, a handrail-shaped interface, and a beard-shaped interface-2, with beard-shaped interface-1 and beard-shaped interface-2 respectively connected to both sides of the handrail-shaped interface, and beard-shaped interface-1 and beard-shaped interface-2 being mirror-symmetric; a chiral polarization source is provided on beard-shaped interface-2.

[0007] Preferably, both the valley photonic crystal VPC1 and the valley photonic crystal VPC-1 are arranged in a graphene-like lattice; the valley Hall boundary states propagate along the wave vector kx direction on the beard-shaped interface-1 and the beard-shaped interface-2, and along the wave vector ky direction on the armrest-shaped interface.

[0008] Preferably, both the valley photonic crystal VPC1 and the valley photonic crystal VPC-1 are composed of alumina dielectric pillars, which are all dielectric materials. The rhombic lattice unit contains dielectric pillar A and dielectric pillar B, and the lattice constant a = 16 mm.

[0009] The diameters of dielectric pillars A and B in the valley photonic crystal VPC1 are d and d, respectively. A =6 / 16a,d B =5 / 16a; Cylinder r of dielectric column A and dielectric column B A =r0(1+ )=3mm, r B =r0(1- =2.5mm, Here, r0 represents the radius of the dielectric pillar when no valley deformation occurs, and r0 is the valley deformation parameter. The diameters of dielectric pillars A and B in the valley photonic crystal VPC-1 are d and d, respectively. A =5 / 16a,d B =6 / 16a; where, d A d B These represent the cylinder diameters of medium column A and medium column B, respectively.

[0010] Preferably, the valley photonic crystal VPC0 exhibits degeneracy of photonic states at points K and K′ in the photonic bandgap, forming a Dirac cone, as shown by the green arrows. In the valley photonic crystal VPC-1, photonic states at points K and K′ separate, creating a photonic bandgap. Furthermore, the polarization of the photonic states at points K and K′ exhibits valley Hall locking. At point K, the valley spin of the photonic states in the first photonic bandgap is upward, and the valley spin of the photonic states in the second photonic bandgap is downward. At point K′, the corresponding valley polarization directions of the photonic states are opposite. All cases for the valley photonic crystal VPC-1 are reversed.

[0011] Preferably, based on the calculation of the photonic band Chern number of the valley photonic crystal VPC1, the Chern number at points K and K′ is obtained as follows: The Chern number of the valley photonic crystal VPC-1 at points K and K' is The valley number of the photonic bandgap at points K and K′ is: .

[0012] Preferably, the valley-shaped variable parameter of the beard-shaped interface-1 and Valley-shaped variable parameters of the beard-shaped interface-2 And the value is This allows for the reversal of the two types of valley Hall boundary dispersion curves with respect to valleys K and K′.

[0013] Preferably, the eigenmodes of the Valley Hall boundary state are antisymmetric about the half-horizontal plane and belong to odd modes; when propagating on the beard-shaped interface-1, regardless of whether the waveguide structure is linear or Z-shaped, the Valley Hall boundary state exhibits extremely high unidirectional propagation performance within the bandwidth range; when excited by a left-handed circularly polarized chiral polarization source, the Valley Hall edge state propagates unidirectionally along the beard interface-1 and does not backscatter even when encountering sharp corners.

[0014] Compared with the whisker-type interface-1, the dispersion curve of the valley Hall edge state formed at the whisker-type interface-2 is reversed about the K and K′ valleys. The eigenmode of the valley Hall edge state is symmetrical about the half-horizontal plane and belongs to the even mode. Its transmission spectrum in the band gap has a high unidirectional transmittance in the bandwidth range.

[0015] Preferably, the valley-Hall boundary states on the handrail-shaped interface have electric fields concentrated on the handrail-shaped interface and decay exponentially as they move away from the handrail-shaped interface; within the bandwidth range, the valley-Hall boundary states also have extremely high unidirectional transmission characteristics; the valley-Hall boundary states propagate unidirectionally along the handrail-shaped interface at a frequency of 0.3686 (c / a); the valley-Hall boundary states can bypass intentionally placed obstacles and maintain unidirectional propagation along the handrail-shaped interface without generating back reflection, exhibiting excellent transmission robustness, where c is the speed of light and a is the lattice constant.

[0016] Preferably, at frequency f1 = 0.3493 (c / a), the valley Hall edge states at the U-shaped waveguide interface will propagate unidirectionally along the beard-shaped interface-2, the armrest-shaped interface, and the beard-shaped interface-1 in sequence; at frequency f2 = 0.3639 (c / a), the valley Hall edge states propagate unidirectionally along the beard-shaped interface-1 and the beard-shaped interface-2, and are absent on the armrest-shaped interface, thus becoming localized and confined to the beard-shaped interface-2; when the frequency is f3 = 0.3731 (c / a), the valley Hall edge states propagate unidirectionally along the beard-shaped interface-2 and the armrest-shaped interface, and are absent on the beard-shaped interface-1, thus similarly localized at the armrest-shaped interface.

[0017] Preferably, by changing the polarization direction of the chiral polarization source, based on the valley spin-locking mechanism, the valley Hall edge state can propagate unidirectionally to the right along the whisker-shaped interface -2 under excitation at frequencies f1, f2, and f3.

[0018] This invention addresses the problem of localizing and confining topological edge states of different frequencies to specific interfaces, providing a practical platform for using interfaces as valley degrees of freedom to manipulate nanoscale electromagnetic waves. The invention constructs valley photonic crystals VPC-1 and VPC1 from all-dielectric alumina cylinders. Based on these two types of valley photonic crystals, a U-shaped waveguide structure is built, consisting of a whisker-shaped interface-1, a whisker-shaped interface-2, and a handrail-shaped interface. Theoretical calculations and numerical simulations demonstrate that valley Hall topological states in this U-shaped waveguide can be frequency-dependently localized and confined to designated interfaces. Compared to the design of traditional topological rainbow systems where edge states of different frequencies need to be separated and confined to different spatial locations, the waveguide achieved by this invention does not require complex structures involving modulated external magnetic fields or gradually changing design parameters, greatly promoting the development of photonic integration.

[0019] The beneficial effects of this invention are as follows: This invention designs a valley photonic crystal topological rainbow waveguide based on alumina pillars arranged in a honeycomb lattice. The waveguide structure is U-shaped and mainly consists of valley photonic crystals VPC1 and VPC-1. Valley photonic crystals VPC1 and VPC-1 are spatially antisymmetric. The U-shaped waveguide structure is composed of three different types of interfaces: whisker-type interfaces-1 / -2 and armrest-type interfaces. Whisker-type interfaces-1 and-2 are mirror-symmetric. Based on the analysis of electromagnetic wave transmission characteristics, the topological rainbow waveguide can not only achieve electromagnetic wave transmission but also achieve energy localization and confinement of specific interfaces through frequency modulation. Compared with previous designs, the fabrication process of this invention is simpler and more efficient, requiring no complex technical means. These results provide a potentially feasible research platform and development opportunity for controlling electromagnetic waves at the nanoscale using interfaces as degrees of freedom. In future work, if experimental research on topological rainbow waveguides can be carried out in higher frequency bands such as terahertz, infrared, and optical frequencies, it is expected to promote the development of more compact on-chip integrated applications. Attached Figure Description

[0020] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0021] Figure 1 This is a schematic diagram of the overall structure of the present invention. The inset on the right shows the boundary type of the graphene-like valley photonic crystal.

[0022] Figure 2 This diagram shows the photon band distribution and corresponding photon eigenmodes of the first Brillouin zone in the lattice unit cell of the graphene-like valley photonic crystal of this invention. (a) is a schematic diagram of the graphene-like valley photonic crystal structure. (b1) and (b3) represent the transmission vectors, with a lattice constant of a = 16 mm; (b1)-(b3) are schematic diagrams of three valley photonic crystal units, where d represents the diameter of the valley photonic crystal VPC1. A =6 / 16a,d B =5 / 16a; Diameter d of Valley Photonic Crystal VPC0 A =2r0=11 / 32a,d B =2r0=11 / 32a; Diameter d of Valley Photonic Crystal VPC-1 A =5 / 16a,d B =6 / 16a (d) A d B Representing the diameter of the cylinder, r A r B (These represent the radii of the cylinders, respectively). In the valley photonic crystal VPC1, r A =r0(1+ =3mm, r B =r0(1- =2.5mm, ,in (c1) represents the valley shape variable parameters; (c3) corresponds to the photonic band diagrams of the first Brillouin zone of valley photonic crystals VPC1, VPC0, and VPC-1, respectively. The diagram on the right corresponds to the electric field amplitude and phase distribution of the eigenmodes of the valley Hall photonic states at points K and K'; (d) represents the valley photonic crystal stacking interface, where the blue lines indicate the interface type, which consists of beard-shaped, serrated, and armrest-shaped interfaces, respectively.

[0023] Figure 3 The diagram shows the valley Hall edge states and transmission spectral lines at the beard-shaped interface. (a) and (f) are schematic diagrams of the structures of beard-shaped interfaces-1 and-2, respectively; (b) and (g) are the photonic band diagrams projected onto the wave vector kx direction of (a) and (f), respectively, showing a valley Hall boundary state appearing in the photonic band gap, with the right-hand diagram representing one of the boundary eigenmodes; (c) and (h) are the transmission spectral lines of the valley Hall boundary state along the straight waveguides and Z-type waveguides of beard-shaped interfaces-1 and-2, respectively; (d) and (e), (i) and (j) are the electric field amplitude distributions of the valley Hall boundary state along the straight waveguides and Z-type waveguides of beard-shaped interfaces-1 and-2, respectively.

[0024] Figure 4The diagram shows the valley-Hall edge states and transmission spectrum in the handrail-type interface. (a) is a schematic diagram of the handrail-type interface structure; (b) is a photonic band diagram projected onto the wave vector ky direction of (a), showing two valley-Hall boundary states within the photonic bandgap, with the right side showing the boundary state eigenmodes and Poynting vector distribution; (c) shows the electric field amplitude distribution of the valley-Hall boundary states in the straight waveguide of the handrail-type interface; (d) shows the electric field amplitude distribution of the valley-Hall boundary states in the straight waveguide of the handrail-type interface when a defect exists; and (e) is a schematic diagram of the defect structure in the straight waveguide of the handrail-type interface.

[0025] Figure 5 This is a diagram of the topological rainbow photonic waveguide of the present invention. (a) shows the Valley Hall boundary state dispersion curves of the whisker-type interface-1, whisker-type interface-2, and armrest-type interface, marked with pink, green, and red curves respectively; (b1)-(b4) are schematic diagrams of the topological rainbow waveguide, with black arrows representing the propagation direction of the topological waveguide states; "This indicates that the interface cannot support the transmission of the Valley Hall edge state; the pink, green and red lines represent the beard-type interface-1, the beard-type interface-2 and the armrest-type interface, respectively; the dashed rectangle represents the localized region of the topological rainbow waveguide; (c1)-(c4) correspond to the electric field amplitude distribution of the topological rainbow waveguides (b1)-(b4), respectively."

[0026] In the figure, 1 is the unit cell of valley photonic crystal VPC1, 2 is the outer region composed of valley photonic crystal VPC1, i.e., the outer region of the U-shaped waveguide structure, 3 is the unit cell of valley photonic crystal VPC-1, 4 is the inner region composed of valley photonic crystal VPC-1, i.e., the inner region of the U-shaped waveguide structure, 5 is bearded interface-1, 6 is bearded interface-2, 7 is armchair interface, 8 is a chiral polarization source, and 9 is the graphene-like photonic crystal interface type, which includes bearded, armchair, and sawtooth types. Detailed Implementation

[0027] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0028] like Figure 1As shown, a topological rainbow waveguide based on valley photonic crystal is a U-shaped waveguide structure, including a U-shaped waveguide interface. The region enclosed by the U-shaped waveguide interface is the inner region 4, and the area outside the U-shaped waveguide interface is the outer region 2. This invention includes the outer region 2 of the U-shaped waveguide structure, which is composed of valley photonic crystal VPC1 arranged in a graphene-like lattice, as shown... Figure 1 As shown, the inner region 4 of the U-shaped waveguide structure is composed of valley photonic crystals VPC-1, arranged in a graphene-like lattice. Valley photonic crystals VPC1 and VPC-1 are mirror- or spatially antisymmetric. Both valley photonic crystals VPC1 and VPC-1 break parity symmetry, and these two types of valley photonic crystals satisfy mirror antisymmetry. The U-shaped waveguide interface consists of a mustache-type interface (labeled 5), a mustache-type interface (labeled 6), and a handrail-type interface (labeled 7), represented by magenta, green, and red solid lines, respectively. The mustache-type interfaces of this invention are divided into two categories, mainly based on the valley shape variable parameters. Decide, Get the beard-shaped interface -1, Beard-type interface-2 was obtained. The three interface types disclosed in the literature [Yonatan Plotnik, Mikael C. Rechtsman, Daohong Song, et al. Observation of unconventional edge states in 'photonic graphene', Nature Materials, 13(1):57-62, 2014]—beard-type, armrest-type, and serrated—are readily known to those skilled in the art. In this invention, a chiral polarization source 8 is placed on the beard-type interface-2. Reference numeral 9 indicates the three boundary types in graphene-like valley photonic crystals: beard-type, armrest-type, and serrated. Beard-type interface-1 and beard-type interface-2 are mirror-symmetric, and the two types of valley Hall boundary dispersion curves on beard-type interface-1 and beard-type interface-2 are inverted about valleys K and K′, respectively. From Figure 1 As can be seen, the Valley Hall boundary states propagate along the wave vector kx on the beard-shaped interfaces-1 and-2, and along the wave vector ky on the handrail-shaped interface. The kx direction corresponds to the momentum space. The direction, ky, is the momentum space. direction.

[0029] like Figure 2As shown, (a) is a schematic diagram of a two-dimensional valley photonic crystal structure. The valley photonic crystal designed in this invention is composed of alumina dielectric pillars, which are all dielectric materials with a relative permittivity of 7.5. The rhombus and regular hexagon in the figure can both be represented as unit cells of the valley photonic crystal. In the rhombus lattice unit, there are two dielectric pillars A and B, with a lattice constant a = 16 mm. The vector represents the lattice transport vector, which is the definition of a Bravais lattice. Any two-dimensional lattice can have two non-perpendicular unit vectors representing its lattice transport direction, and any position of the lattice can be represented by a transport vector. (b1)-(b3) represent the unit cells of three types of valley photonic crystals, respectively. The diameters of dielectric pillars A and B in the valley photonic crystal VPC1 are d and d, respectively. A =6 / 16a,d B =5 / 16a; Diameter d of Valley Photonic Crystal VPC0 A =2r0=11 / 32a,d B =2r0=11 / 32a, valley-shaped variable parameter The valley photonic crystal VPC0 was obtained, and it can be seen that the photonic band structure at points K and K′ is degenerate, serving as a reference for valley photonic crystals VPC-1 and VPC1. The diameter d of valley photonic crystal VPC-1 is... A =5 / 16a,d B =6 / 16a, where d A d B Let r represent the cylinder diameters of dielectric column A and dielectric column B, respectively. A r B Let r represent the cylindrical radii of dielectric pillars A and B, respectively. In the valley photonic crystal VPC1, the cylindrical radius r of dielectric pillar A is... A =r0(1+ )=3mm, r B =r0(1- =2.5mm, ,in, Here, r0 represents the radius of the dielectric pillar (2.75 cm) without valley deformation, and r0 represents the valley deformation parameter. (c1)-(c3) correspond to the photonic band diagrams of the first Brillouin zone of valley photonic crystals VPC1, VPC0, and VPC-1, respectively. The diagram on the right corresponds to the amplitude and phase distribution of the eigenmodes of the valley Hall photonic states at points K and K′. As can be seen from the diagram, for valley photonic crystal VPC0, the photonic states at points K and K′ degenerate to form a Dirac cone, as shown by the green arrows; for valley photonic crystal VPC-1, the photonic states at points K and K′ separate, producing a photonic band gap, and the polarization of the photonic states at points K and K′ exhibits valley Hall locking. For example, at point K, the valley spin of the photonic states in the first photonic band is upward, and the valley spin of the photonic states in the second photonic band is downward; at point K′, the valley polarization directions of the corresponding photonic states are opposite. This result is confirmed by the phase distribution of the eigenmodes of the photonic states on the right. Correspondingly, for the valley photonic crystal VPC-1, all cases are reversed. (d) shows the valley photonic crystal stacking interface, where the blue lines indicate the interface type, consisting of a mustache-shaped interface, a serrated interface, and a handrail-shaped interface. Based on the calculation of the photonic band Chern number of the valley photonic crystal VPC1, the Chern number at points K and K′ is obtained as follows: The Chern number of the valley photonic crystal VPC-1 at points K and K' is Therefore, the valley Chern number of the photonic bandgap at points K and K′ is: The valley-Hall boundary state number in the photonic bandgap can be calculated by determining the number of boundary states in the bandgap based on the volume-edge correspondence. According to the volume-edge correspondence, there is one valley-Hall boundary state in the photonic bandgap, and the propagation group velocities of the valley-Hall boundary state at points K and K′ are opposite.

[0030] like Figure 3 As shown, (a) represents a schematic diagram of the structure of the beard-shaped interface-1, and the photonic band structure along the wave vector kx is shown in Figure (b). The photonic band structure was calculated using COMSOL multiphysics simulation software based on the finite element method. In the band gap region marked by the shade, there are valley Hall edge states located in the K / K′ energy valley, which propagate towards each other along the beard-shaped interface-1. At the same time, there are two pairs of trivial topological edge states (marked by pink curves), which are separated from the bulk band due to the breaking of crystal symmetry. The right side of Figure (b) shows the eigenmode of the valley Hall edge state, which is antisymmetric relative to the half-horizontal plane and belongs to the odd mode (mainly to illustrate the mode type). As shown in Figures (d) and (e), regardless of whether the waveguide structure is linear or zigzag, the valley Hall boundary states exhibit extremely high unidirectional transmission performance within the bandwidth range marked by the light shade. Through full-wave simulation, the electric field amplitude (|E) of the valley Hall boundary states in linear and zigzag waveguides is given respectively. Z|) Distribution. Under chiral left-handed circular polarization (LCP) excitation, the valley-Hall edge states propagate unidirectionally along the whisker interface-1, exhibiting no backscattering even at sharp corners, demonstrating the propagation robustness brought by topological protection. Similarly, Figure (f) shows a schematic diagram of the structure of the whisker interface-2, which is obtained by mirror symmetry of the lattice interface-1. Therefore, the energy band of the valley-Hall edge states formed at the whisker interface-2 is flipped at the K and K′ valleys, as shown in Figure (g). In this case, the eigenmodes of the valley-Hall edge states are symmetric about the half-horizontal plane and belong to even modes, as seen on the right side of Figure (g). The transmission spectrum within the photonic bandgap shows a similar trend with frequency as described above, exhibiting high unidirectional transmittance over the bandwidth. Therefore, it is concluded that in valley photonic crystals, the whisker interface is beneficial for the application of waveguide-based integrated photonic devices and circuits.

[0031] like Figure 4 As shown, (a) is a schematic diagram of the armchair-type interface structure, and its photonic band structure is shown in Figure (b). From the eigenmodes on the right, we can obtain the electric field E... Z The energy concentration occurs at the armchair interface and decreases exponentially with distance from the interface; this is a characteristic of boundary states, concentrated at the boundary and absent in the bulk state. The energy flow vortex phenomenon here can be further elucidated by the Poynting vector (marked by the magenta arrow). For example, the energy flow indicated by the pink arrow in the figure rotates counterclockwise (clockwise) as shown by the black circular arrow, corresponding to the valley spin-up (downward) states, respectively. Figure (c) shows the transmission spectrum of the Valley Hall edge state at the armchair interface, indicating that within the bandwidth marked by the light shading, the Valley Hall edge state also exhibits extremely high unidirectional transmission characteristics, while this phenomenon is absent at the interface with the small band gap. This result can be further verified by the inset of Figure (c), which depicts the forward and reverse energy flows in the vertical plane. As can be seen from the figure, both the forward and reverse energy flows approach zero within the small band gap, which is a complete band gap and does not support the Valley Hall boundary state. The Valley Hall edge state propagates unidirectionally along the armchair interface at a frequency of 0.3686 (c / a), as shown in Figure (d). Furthermore, the Valley Hall edge states are able to bypass intentionally placed obstacles and maintain unidirectional propagation along the armchair interface without generating back reflections, demonstrating their robustness to the transmission of defects or disordered disturbances, as shown in Figures (e) and (f).

[0032] like Figure 5 As shown, based on previous discussions and analyses of the dispersion relations of valley-Hall edge states at different interfaces, this invention designs a U-shaped waveguide structure composed of a beard-shaped interface-1, a beard-shaped interface-2, and an armchair interface. To analyze and predict the transmission characteristics of the valley-Hall edge states in this U-shaped waveguide structure, by integrating... Figure 3 of (b), 3 of (g) and Figure 4 The dispersion curve shown in (b) was plotted. Figure 5 The edge state dispersion spectrum is shown in (a). A schematic diagram of the U-shaped waveguide structure is presented in Figures (b1)-(b4), where the magenta, green, and red lines represent the beard-shaped interface-1, beard-shaped interface-2, and armrest-shaped interface, respectively, and the black arrows indicate the propagation direction of the valley-Hall edge state. Based on the dispersion relationship in Figure (a), it can be inferred that at frequency f1 = 0.3493 (c / a), the valley-Hall edge state will propagate unidirectionally along the beard-shaped interface-2, armrest-shaped interface, and beard-shaped interface-1 in sequence, as shown in Figure (b1). At frequency f2 = 0.3639 (c / a), in principle, the valley-Hall edge state will propagate unidirectionally along the beard-shaped interface-1 and beard-shaped interface-2, and since there is no armrest-shaped interface, it becomes localized and confined to the beard-shaped interface-2 marked with a dashed box, as shown in Figure (b2). This is because the frequency f2 within the tiny bandgap cannot support the transmission of the valley Hall edge state in the armchair interface, resulting in the electromagnetic wave being completely reflected at the interface. When the frequency is f3 = 0.3731 (c / a), the valley Hall edge state propagates unidirectionally along the mustache-shaped interface-2 and the armchair interface. Since the mustache-shaped interface-1 does not exist, it will propagate along the mustache-shaped interface-2 and the armchair interface, and similarly localize at the armchair interface, as shown in Figure (b3). By referring to the figure and using the idea and method of wavelength division multiplexing, three frequencies f1, f2, and f3 can be given, where c is the speed of light and a is the lattice constant. By changing the polarization direction of the chiral source, based on the valley spin-locking mechanism, the valley Hall edge state can propagate unidirectionally to the right along the mustache-shaped interface-2 at a frequency of f1 = 0.3493 (c / a) (in fact, all three frequencies can be achieved), as shown in Figure (b4). In a U-shaped waveguide structure, the valley-Hall edge states can not only be conducted, but also confined or localized at specific interfaces through frequency modulation. Therefore, this U-shaped waveguide is called a topological rainbow waveguide (TRW). The simulation results of the full-band topological rainbow waveguide at different frequencies are shown in Figures (c1) and (-c4). As can be seen from the figures, the amplitude distribution of the topological rainbow waveguide's electric field at each frequency is in complete agreement with our previous analysis and predictions. Figure 5 (b1-b4) represents a schematic diagram of the principle, for analysis and prediction. Figure 5 In the diagram (c1-c4), numerical simulation is used. The outstanding performance of topological rainbow waveguides lies in their ability to localize different frequency topological waveguide states at different spatial interfaces without adjusting external magnetic field strength, designing structures with gradually varying parameters, or introducing gradient losses. Topological rainbow waveguides have enormous application potential in the field of nanophotonics, such as in robust light trapping and ultracompact topological rainbow nanolasers.

[0033] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A topological rainbow optical waveguide based on valley photonic crystal, characterized in that, It is a U-shaped waveguide structure, including a U-shaped waveguide interface, the area enclosed by the U-shaped waveguide interface is the inner region (4), and the outer side of the U-shaped waveguide interface is the outer region (2); the outer region (2) is composed of valley photonic crystal VPC1, and the inner region (4) is composed of valley photonic crystal VPC-1. Valley photonic crystal VPC1 and valley photonic crystal VPC-1 are spatially antisymmetric; the U-shaped waveguide interface includes a beard-shaped interface-1, a handrail-shaped interface and a beard-shaped interface-2. The beard-shaped interface-1 and the beard-shaped interface-2 are respectively connected to the two sides of the handrail-shaped interface. The beard-shaped interface-1 and the beard-shaped interface-2 are mirror-symmetric; a chiral polarization source (8) is provided on the beard-shaped interface-2.

2. The topological rainbow waveguide based on valley photonic crystal according to claim 1, characterized in that, Both the valley photonic crystal VPC1 and the valley photonic crystal VPC-1 are arranged according to a graphene-like lattice; the valley Hall boundary states propagate along the wave vector kx direction on the beard-shaped interface-1 and the beard-shaped interface-2, and along the wave vector ky direction on the armrest-shaped interface.

3. The topological rainbow waveguide based on valley photonic crystal according to claim 2, characterized in that, Both the valley photonic crystal VPC1 and the valley photonic crystal VPC-1 are composed of alumina dielectric pillars, which are all dielectric materials. The rhombic lattice unit contains dielectric pillar A and dielectric pillar B, and the lattice constant a = 16 mm. The diameters of dielectric pillars A and B in the valley photonic crystal VPC1 are d and d, respectively. A =6 / 16a,d B =5 / 16a; Cylinder r of dielectric column A and dielectric column B A =r0(1+ )=3mm, r B =r0(1- =2.5mm, Here, r0 represents the radius of the dielectric pillar when no valley deformation occurs, and r0 is the valley deformation parameter. The diameters of dielectric pillars A and B in the valley photonic crystal VPC-1 are d and d, respectively. A =5 / 16a,d B =6 / 16a; Where, d A d B These represent the cylinder diameters of medium column A and medium column B, respectively.

4. The topological rainbow waveguide based on valley photonic crystal according to claim 3, characterized in that, The valley photonic crystal VPC0 exhibits degeneracy of photonic states at points K and K′ in the photonic bandgap, forming Dirac cones, as shown by the green arrows. The valley photonic crystal VPC-1 shows photonic state separation at points K and K′, creating a photonic bandgap. Furthermore, the polarization of the photonic states at points K and K′ exhibits valley Hall locking. At point K, the valley spin of the photonic states in the first photonic bandgap is upward, while the valley spin of the photonic states in the second photonic bandgap is downward. At point K′, the corresponding valley polarization directions of the photonic states are opposite. All cases for the valley photonic crystal VPC-1 are reversed.

5. The topological rainbow waveguide based on valley photonic crystal according to claim 3 or 4, characterized in that, Based on the calculation of the photonic band Chern number of the valley photonic crystal VPC1, the Chern number at points K and K′ is obtained as follows: The Chern number of the valley photonic crystal VPC-1 at points K and K' is The valley number of the photonic bandgap at points K and K′ is: .

6. The topological rainbow waveguide based on valley photonic crystal according to claim 5, characterized in that, The valley-shaped variable parameters of the beard-shaped interface-1 and Valley-shaped variable parameters of the beard-shaped interface-2 And the value is This allows for the reversal of the two types of valley Hall boundary dispersion curves with respect to valleys K and K′.

7. The topological rainbow waveguide based on valley photonic crystal according to claim 6, characterized in that, The eigenmodes of the Valley Hall boundary state are antisymmetric about the half-horizontal plane and belong to odd modes. When propagating on the beard-shaped interface-1, regardless of whether the waveguide structure is straight or Z-shaped, the Valley Hall boundary state exhibits extremely high unidirectional propagation performance within the bandwidth range. Under the excitation of the left-hand circularly polarized chiral polarization source (8), the Valley Hall edge state propagates unidirectionally along the beard interface-1 and does not backscatter even when encountering sharp corners. Compared with the whisker-type interface-1, the dispersion curve of the valley Hall edge state formed at the whisker-type interface-2 is reversed about the K and K′ valleys. The eigenmode of the valley Hall edge state is symmetrical about the half-horizontal plane and belongs to the even mode. Its transmission spectrum in the band gap has a high unidirectional transmittance in the bandwidth range.

8. The topological rainbow waveguide based on valley photonic crystal according to claim 6 or 7, characterized in that, The valley-Hall boundary states on the handrail-shaped interface have electric fields concentrated on the handrail-shaped interface and decay exponentially as they move away from the interface. Within the bandwidth, the valley-Hall boundary states also have extremely high unidirectional transmission characteristics. The valley-Hall boundary states propagate unidirectionally along the handrail-shaped interface at a frequency of 0.3686 (c / a). The valley-Hall boundary states can bypass intentionally placed obstacles and maintain unidirectional propagation along the handrail-shaped interface without back reflection, exhibiting excellent transmission robustness, where c is the speed of light and a is the lattice constant.

9. The topological rainbow waveguide based on valley photonic crystal according to claim 8, characterized in that, At frequency f1 = 0.3493 (c / a), the valley-Hall edge states of the U-shaped waveguide interface will propagate unidirectionally along the beard-shaped interface-2, the armrest-shaped interface, and the beard-shaped interface-1 in sequence. At frequency f2 = 0.3639 (c / a), the valley-Hall edge states propagate unidirectionally along the beard-shaped interface-1 and the beard-shaped interface-2, and are absent on the armrest-shaped interface, thus becoming localized and confined to the beard-shaped interface-2. When the frequency is f3 = 0.3731 (c / a), the valley-Hall edge states propagate unidirectionally along the beard-shaped interface-2 and the armrest-shaped interface, and are absent on the beard-shaped interface-1, thus similarly localized at the armrest-shaped interface.

10. The topological rainbow waveguide based on valley photonic crystal according to claim 9, characterized in that, By changing the polarization direction of the chiral polarization source (8), based on the valley spin-locking mechanism, the valley Hall edge state can propagate unidirectionally to the right along the whisker-shaped interface -2 under excitation at frequencies f1, f2 and f3.

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