Construction of magneto-optical photonic crystal waveguide and robust unidirectional transmission method of nontrivial bandgap

Abstract

The application discloses a construction of a magneto-optical photonic crystal waveguide and a nontrivial one-way bulk state robust transmission method. The waveguide can excite a reverse-handed one-way boundary state and a one-way bulk state with reverse transmission simultaneously by parameter design of a square magneto-optical photonic crystal waveguide. The reverse-handed one-way boundary state is absorbed by upper and lower boundary conditions with scattering boundary conditions. Under the strong light transmission restriction of the square magneto-optical photonic crystal, the one-way bulk state is limited in the strip to transmit in one direction, thereby avoiding being absorbed, so that the waveguide forms a nontrivial one-way bulk state. The structure is topologically protected, and the nontrivial one-way bulk state robust transmission with complete one-way conductivity of the waveguide channel is realized. Further, the controllability of the nontrivial one-way bulk state transmission frequency can be realized by adjusting the magnetic field strength. The application has wide application prospects in the fields of robust energy transmission, optical communication technology and integrated photonic circuit.

Description

Technical Field

[0001] This invention belongs to the fields of microwave optics, topological photonics, and robust energy transfer, and specifically relates to the construction of a magneto-optical photonic crystal waveguide and a non-trivial unidirectional bulk robust transmission method. Background Technology

[0002] Microwave technology refers to the technology of using electromagnetic waves in the microwave frequency band (generally ranging from 300MHz to 300GHz, corresponding to wavelengths of 1 meter to 1 millimeter) for information transmission, signal processing, energy transfer, and other related applications. Research and applications of microwave technology are constantly expanding, with cutting-edge fields such as millimeter-wave communication and quantum communication showing broad development potential. This invention proposes a method for achieving nontrivial unidirectional bulk robust transmission in the microwave frequency band.

[0003] Robust energy transfer refers to a technology that ensures signal transmission in a specific direction, maintaining stable and efficient transmission even in the face of interference, noise, or external disturbances. Compared to traditional bidirectional or multidirectional transmission methods, robust energy transfer emphasizes the stability and anti-interference capability of unidirectional transmission. The initial inspiration for robust energy transfer came from the study of topological states, particularly the discovery of phenomena such as topological insulators and the quantum Hall effect. Topological insulators are materials that are insulating internally but support electron conductivity at their surfaces or boundaries. These surface states form topologically protected states at the material's edges, exhibiting unidirectional propagation and robustness against interference from defects and impurities. The quantum Hall effect is a topological phenomenon caused by a strong magnetic field when electrons move in a two-dimensional material. The quantum Hall effect demonstrates that at the material's boundaries, electrons can propagate in a single direction without reflection and are robust to impurities and defects.

[0004] Furthermore, in the classical Haldane model, the time-reversal symmetry of the system is broken by giving the electrons undergoing second-nearest neighbor transitions in the two sets of sublattices opposite phases, thus realizing chiral topological boundary states similar to the quantum Hall effect without the need for an external magnetic field. Therefore, the Haldane model proves that the quantum Hall effect does not depend on a strong magnetic field or Landau levels; it can be achieved simply by breaking the time-reversal symmetry of the system. The modified Haldane model, by giving the electrons undergoing second-nearest neighbor transitions in the two sets of sublattices the same phase, allows the chiral boundary states, which originally had opposite propagation directions, to acquire the same dispersion, thus transforming them into antichiral boundary states propagating in the same direction. However, the band structure of this antichiral state is covered by trivial bulk states. The heterogeneous Haldane model is composed of two stacked, opposite modified Haldane models. This model not only moves the unidirectional antichiral boundary states into the band gap but also generates a new unidirectional bulk state within the frequency range of the antichiral state, propagating inside the model and with the opposite propagation direction. However, these two states typically coexist in the band gap as a trivial band formed by both.

[0005] With the deepening research on topological insulators and the quantum Hall effect, topological photonics has gradually emerged as a new field. Researchers have begun to explore the transmission characteristics of light waves in materials and structures with topological properties, and photonic crystals have become a structure of interest. Photonic crystals refer to structures with periodic dielectric arrangements on the wavelength scale, capable of generating photonic passbands and photonic band gaps, and having the ability to effectively manipulate electromagnetic wave transmission. Similar to the periodic potential energy modulation of electron transmission behavior in crystalline materials in solid-state physics, the transmission behavior of photons in photonic crystal materials is also regulated by the periodic medium, and its global characteristics can be described through the photonic band structure. Therefore, the concept of topological structure can be naturally extended to photonic crystals. Secondly, inspired by topological phases and topological phase transitions in condensed matter physics, it has been realized that topological phases are a common phenomenon in periodic structures, thus predicting that topologically protected chiral unidirectional boundary states can be generated in time-reversal symmetry-broken photonic crystals.

[0006] Subsequently, by analogy with the Haldane model, researchers constructed a two-dimensional microwave cellular magneto-optical photonic crystal using yttrium iron garnet (YIG) material with magnetic response; and by applying an external magnetic field in the same direction to each YIG to break the time reversal symmetry of the structure, they successfully generated topological chiral unidirectional boundary states with opposite transmission directions at the upper and lower boundaries of the two-dimensional microwave cellular magneto-optical photonic crystal.

[0007] Secondly, by analogy with the improved Haldane model, researchers applied external magnetic fields in opposite directions to the two nested sublattices within a two-dimensional microwave cellular magneto-optical photonic crystal. This not only breaks the time-reversal symmetry of the structure but also realizes the concept of equal next-nearest neighbor transition coefficients in the improved Haldane model, thus successfully observing anti-chiral unidirectional boundary states propagating in the same direction at the upper and lower boundaries in this structure.

[0008] Furthermore, by analogy with the heterogeneous Haldane model, researchers have constructed two-dimensional microwave cellular magneto-optical photonic crystal waveguides by periodically stacking two magneto-optical photonic crystals of opposite analogous modified Haldane models. This allows anti-chiral unidirectional boundary states and their oppositely propagating unidirectional bulk states to be generated within the bandgap, and the robust propagation of unidirectional bulk states within the strip has been successfully observed. However, based on the law of conservation of energy, at the same frequency, the anti-chiral unidirectional boundary states need to achieve propagation balance with the oppositely propagating unidirectional bulk states. Therefore, this waveguide supports the propagation of anti-chiral unidirectional boundary states and the propagation of oppositely propagating unidirectional bulk states at the boundary and within the strip, respectively. (J. Chen and ZYLi, “Prediction and Observation of Robust One-Way Bulk States in a Gyromagnetic Photonic Crystal,” Physical Review Letters 128(25), 257401(2022)). Current state-of-the-art waveguide designs cannot yet achieve fully unidirectional conduction characteristics that only support unidirectional bulk state propagation. Even though antichiral unidirectional boundary states and unidirectional bulk states have relatively low spatial interference, they still weaken the robust transmission performance of unidirectional bulk states. Furthermore, existing methods typically utilize cellular magneto-optical photonic crystal waveguides to achieve unidirectional bulk state transmission, which involve complex waveguide structures and magnetic field configurations; they also neglect the tunability of the unidirectional bulk state transmission frequency. These are all problems that urgently need to be solved.

[0009] The main reason why existing cellular magneto-optical photonic crystal waveguides cannot achieve fully unidirectional bulk state conduction is that the high-frequency band of the cellular magneto-optical photonic crystal with serrated boundaries is located below the optical cone, thus limiting the leakage of anti-chiral boundary states outside the serrated boundary. This invention provides a square magneto-optical photonic crystal waveguide composed of a square magneto-optical photonic crystal with straight boundaries. This waveguide can also generate unidirectional anti-chiral boundary states and reverse-propagating unidirectional bulk states. However, because the high-frequency band of the square magneto-optical photonic crystal with straight boundaries is located above the optical cone, anti-chiral boundary states can leak outside the straight boundary. Simultaneously, by utilizing the upper and lower boundaries of the waveguide with scattering boundary conditions to absorb the leaked anti-chiral boundary states, the propagation path supporting the anti-chiral boundary states is blocked, thereby effectively reducing the impact on the propagation of unidirectional bulk states within the structure and improving its transmission efficiency. Building upon this, the powerful optical transmission confinement capability of the square magneto-optical photonic crystal makes it difficult for unidirectional bulk states in the strip to propagate to the upper and lower boundaries, thus avoiding absorption. This enables robust transmission of nontrivial unidirectional bulk states with completely unidirectional conductivity in the transmission channel. Furthermore, compared to cellular magneto-optical photonic crystal waveguides, the square magneto-optical photonic crystal waveguide used in this invention has a simpler structure and magnetic field configuration. Moreover, based on the influence of magnetic field changes on the frequency of nontrivial unidirectional bulk state generation, this invention achieves tunability of the nontrivial unidirectional bulk state transmission frequency. Summary of the Invention

[0010] To overcome the shortcomings and deficiencies of existing unidirectional bulk transmission technologies, one objective of this invention is to provide a robust method for nontrivial unidirectional bulk transmission based on the construction of a square magneto-optical photonic crystal waveguide. This method ensures that the waveguide transmitting the unidirectional bulk state possesses complete unidirectional conductivity and exhibits strong robustness against obstacles, defects, and interference. Furthermore, by adjusting the magnetic field strength, the transmission frequency of the nontrivial unidirectional bulk state supported by the square magneto-optical photonic crystal waveguide can be flexibly controlled. This square magneto-optical photonic crystal waveguide provides a new platform for studying robust nontrivial unidirectional bulk transmission, and its rich characteristics will play a role in applications such as robust energy transmission, optical communication technology, and integrated photonic circuits.

[0011] The objective of this invention is achieved by at least one of the following technical solutions.

[0012] The method for constructing a magneto-optical photonic crystal waveguide includes the following steps:

[0013] S1. Construct a square magneto-optical photonic crystal with yttrium iron garnet (YIG) pillars as the dielectric pillars;

[0014] S2. Using the plane of periodic YIG pillars in the square magneto-optical photonic crystal as the coordinate plane, apply a magnetic field to the square magneto-optical photonic crystal in the z direction on the xoy plane.

[0015] S3. Set boundaries and boundary conditions around the square magneto-optical photonic crystal to construct a square magneto-optical photonic crystal waveguide.

[0016] Further, in step S1, YIG dielectric pillars with radius r are used as the magneto-optical material of the square magneto-optical photonic crystal. Multiple YIG pillars are arranged in a square lattice to construct a square magneto-optical photonic crystal with a lattice constant a. The square magneto-optical photonic crystal contains m rows and n columns of YIG pillars.

[0017] Furthermore, in step S2, external magnetic fields of equal magnitude and opposite direction are applied to the odd-numbered and even-numbered rows of dielectric pillars of the square magneto-optical photonic crystal, wherein the magnitude of the magnetic field H of the odd-numbered rows is k and the direction is along +z.

[0018] Further, in step S3, a boundary with scattering boundary conditions is set at 1.7a in the +y direction of the first row of dielectric pillars of the square magneto-optical photonic crystal, which serves as the upper boundary of the waveguide of the square magneto-optical photonic crystal; a boundary with scattering boundary conditions is set at 1.7a in the -y direction of the last row of dielectric pillars of the square magneto-optical photonic crystal, which serves as the lower boundary of the waveguide; a is the lattice constant.

[0019] Furthermore, a boundary with both incident and outgoing functions is set at 3.5a in the -x direction of the leftmost column of dielectric pillars in the square magneto-optical photonic crystal, which serves as the left boundary of the waveguide; a boundary with both incident and outgoing functions is set at 3.5a in the +x direction of the rightmost column of dielectric pillars in the square magneto-optical photonic crystal, which serves as the right boundary of the waveguide, wherein the left and right boundaries are the incident and outgoing ends of the waveguide, respectively.

[0020] Furthermore, the background medium within the boundary is air.

[0021] Furthermore, in step S3, the frequency range of the projected bandgap of the constructed square magneto-optical photonic crystal waveguide is within the microwave frequency band.

[0022] The nontrivial unidirectional bulk robust transmission method based on the aforementioned magneto-optical photonic crystal waveguide construction method includes the following steps:

[0023] S1. Calculate and analyze the projected band structure and intrinsic mode field of the square magneto-optical photonic crystal waveguide under the current conditions using the finite element method. Determine the group velocity direction of the non-trivial unidirectional bulk state by calculating the slope of the band of the non-trivial unidirectional bulk state, and then determine the completely unidirectional conduction direction of the non-trivial unidirectional bulk state in the square magneto-optical photonic crystal waveguide.

[0024] S2. Plane electromagnetic waves within the frequency range of the non-trivial unidirectional bulk state energy band are sequentially incident from the conducting and non-conducting ends of the square magneto-optical photonic crystal waveguide. The electromagnetic wave intensity at the corresponding transmitting end of the square magneto-optical photonic crystal waveguide and the electric field distribution within the corresponding waveguide are detected to verify the unidirectional bulk state transmission and the complete unidirectional conductivity of the waveguide.

[0025] S3. After inserting an obstacle into the center of the square magneto-optical photonic crystal waveguide or creating a defect by removing the dielectric pillar in the center of the square magneto-optical photonic crystal waveguide, the same plane electromagnetic wave as in step S2 is incident from the conducting end of the waveguide to detect the electromagnetic wave intensity at the output end and the electric field distribution inside the waveguide.

[0026] S4. Compare the electromagnetic wave intensity at the output end and the electric field distribution inside the waveguide when the plane electromagnetic wave is incident from the conducting end of the waveguide in steps S3 and S2, and analyze the robust transmission of the non-trivial unidirectional bulk state when facing obstacles and defects.

[0027] Furthermore, the finite element method was used to calculate and analyze the data in the RF-frequency domain module of the COMSOL MULTIPHYSICS software.

[0028] Furthermore, by continuously changing the magnetic field strength applied to the square magneto-optical photonic crystal waveguide and adjusting the magnetic field magnitude, the transmission frequency of the nontrivial unidirectional bulk state is changed, thereby achieving tunability of the nontrivial unidirectional bulk state transmission frequency, as detailed below:

[0029] While keeping other parameters of the square magneto-optical photonic crystal waveguide constant, by continuously varying the magnetic field strength applied to it, the corresponding band structure can be calculated. Since the nontrivial unidirectional bulk state band within the waveguide exhibits a certain tolerance to magnetic field variations, as the magnetic field strength increases, the band gradually shifts towards higher energy regions with slow structural changes. Within a magnetic field range of 2500 Gs to 11000 Gs, this band remains constant, and its center frequency exhibits a wide shift range of 9 GHz to 13 GHz. Therefore, by simply adjusting the magnetic field strength, the transmission frequency of the nontrivial unidirectional bulk state supported by the square magneto-optical photonic crystal waveguide can be adjusted, thus achieving tunability of the nontrivial unidirectional bulk state transmission frequency.

[0030] The principle of this invention is as follows: This invention uses a square magneto-optical photonic crystal waveguide to achieve robust propagation of nontrivial unidirectional bulk states. This waveguide is composed of a square lattice magneto-optical photonic crystal. First, by applying a co-directional external magnetic field to the dielectric pillars of the square magneto-optical photonic crystal to break the time-reversal symmetry of the structure, the square magneto-optical photonic crystal can generate chiral boundary states with topological protection in the bandgap. However, since the high-frequency band of the square magneto-optical photonic crystal with straight boundaries is located above the optical cone, the unidirectional chiral boundary states will leak outside the straight boundaries. Therefore, it is necessary to set metal cladding at the upper and lower boundaries of the square magneto-optical photonic crystal waveguide to prevent the leakage of chiral boundary states, thereby achieving stable robust propagation of chiral unidirectional boundary states at the upper and lower boundaries. Furthermore, when external magnetic fields with opposite directions are applied to the odd-numbered and even-numbered rows of dielectric pillars of the square magneto-optical photonic crystal, and the parameters are adjusted, due to the reversal of some magnetic field directions, the original chiral unidirectional boundary states will acquire the same dispersion, thus transforming into co-directional propagating anti-chiral unidirectional boundary states. Based on the law of conservation of energy, the antichiral unidirectional boundary state needs to achieve propagation balance with the corresponding reverse-propagating bulk state. Therefore, the waveguide generates both antichiral unidirectional boundary states and unidirectional bulk states within the same bandgap. These two states propagate in opposite directions at the upper and lower boundaries of the metallic-clad square magneto-optical photonic crystal waveguide and inside the waveguide structure, respectively. Thus, the waveguide supporting these two states is not completely unidirectional. Then, when the metallic cladding is removed and the upper and lower boundaries of the square magneto-optical photonic crystal waveguide are subjected to scattering boundary conditions, the antichiral unidirectional boundary state leaks outward and is absorbed by the upper and lower boundaries with scattering boundary conditions. This blocks the propagation path supporting the antichiral unidirectional boundary state. Simultaneously, due to the strong optical transmission confinement capability of the square magneto-optical photonic crystal, the bulk state is effectively confined within the band and propagates unidirectionally, making it difficult to diffuse to the upper and lower boundaries and avoiding absorption, thereby ensuring the integrity of the unidirectional bulk state. This mechanism enables the generation of nontrivial unidirectional bulk states in the bandgap of the square magneto-optical photonic crystal waveguide, thereby achieving nontrivial unidirectional bulk state transmission with completely unidirectional conduction in the waveguide channel. Furthermore, due to the topological protection of the structure, the nontrivial unidirectional bulk state exhibits resistance to backscattering and strong transmission robustness against obstacles, defects, and interference. Finally, because the energy band of the nontrivial unidirectional bulk state has a high tolerance to magnetic field variations, as the external magnetic field changes within a certain range, the energy band of the nontrivial unidirectional bulk state shifts to higher or lower energy regions, and its structure possesses a certain degree of stability. Therefore, by adjusting the magnitude of the external magnetic field, the transmission frequency of the nontrivial unidirectional bulk state supported by the square magneto-optical photonic crystal waveguide can be flexibly controlled.

[0031] Compared with the prior art, the present invention has the following advantages and superior effects:

[0032] (1) Mechanism of unidirectional bulk robust transmission in this invention: Existing unidirectional bulk robust transmission technology usually utilizes a unidirectional bulk state located below the light cone generated by a honeycomb magneto-optical photonic crystal with serrated boundaries, while this invention utilizes a non-trivial unidirectional bulk state located above the light cone generated by a square magneto-optical photonic crystal with straight boundaries.

[0033] (2) Complete unidirectional conductivity: Existing cellular magneto-optical photonic crystal waveguides that support unidirectional bulk robust transmission often simultaneously support anti-chiral unidirectional boundary state transmission at the boundary, which is opposite to the bulk transmission direction. Therefore, these waveguides do not possess complete unidirectional conductivity. In contrast, the square magneto-optical photonic crystal waveguide of this invention only supports non-trivial unidirectional bulk robust transmission, and the channel accompanying the reverse bulk transmission is prohibited, thus possessing complete unidirectional conductivity.

[0034] (3) Robust transmission performance: Existing cellular magneto-optical photonic crystal waveguides transmit anti-chiral unidirectional boundary states and reverse-propagating unidirectional bulk states at the boundaries and stripes, respectively. Although these two states have less spatial interference, they still weaken the robust transmission performance of the unidirectional bulk state. However, the square magneto-optical photonic crystal waveguide of the present invention does not have a transmission channel opposite to the transmission direction of the unidirectional bulk state, so the unidirectional bulk state has better transmission robustness.

[0035] (4) Transmission frequency tunability: No existing waveguides supporting tunable transmission frequency of unidirectional bulk states have been reported. In this invention, as the external magnetic field increases, the energy band of the nontrivial unidirectional bulk state shifts to the high-energy region, while its structure changes only slightly. Therefore, by simply changing the magnitude of the magnetic field, the transmission frequency of the nontrivial unidirectional bulk state supported by the square magneto-optical photonic crystal waveguide can be flexibly adjusted, and the adjustable frequency range is relatively wide.

[0036] (5) Simple structure and magnetic field configuration: Most existing magneto-optical photonic crystals that support unidirectional bulk robust transmission adopt a honeycomb lattice structure and require the application of external magnetic fields with opposite directions to the two sets of sublattices in the honeycomb lattice. In contrast, the present invention adopts a square lattice magneto-optical photonic crystal, and only requires the application of external magnetic fields with opposite directions to the odd-numbered rows and even-numbered rows of dielectric pillars in the square lattice, making its structure and magnetic field configuration much simpler. Attached Figure Description

[0037] Figure 1 This is a schematic diagram of the waveguide structure based on a square magneto-optical photonic crystal according to the present invention;

[0038] Figure 2 The image shows the projected band structure and the eigenmode field diagram at the corresponding frequency of the waveguide in Embodiment 1 of the present invention.

[0039] Figure 3 The electric field intensity distribution and electric field intensity curves of Embodiments 1 and 2 of the present invention are shown below;

[0040] Figure 4 This is a graph showing the changes in the width and center frequency of the non-trivial unidirectional bulk state energy band as the magnetic field strength increases, as shown in Embodiment 3 of the present invention. Detailed Implementation

[0041] The present invention will be further described in detail below with reference to the embodiments, but the implementation of the present invention is not limited thereto.

[0042] Example 1

[0043] The method for constructing a magneto-optical photonic crystal waveguide in this embodiment includes the following steps:

[0044] S1. Construct a square magneto-optical photonic crystal with yttrium iron garnet (YIG) pillars as the dielectric pillars; in this invention, "square" refers to the photonic crystal having a square lattice.

[0045] Using YIG dielectric pillars with a radius of r = 1.83 mm as the magneto-optical material for a square magneto-optical photonic crystal, multiple YIG pillars are arranged in a square lattice to construct a square magneto-optical photonic crystal with a lattice constant of a = 1.16 cm. The square magneto-optical photonic crystal contains 14 rows and 30 columns of YIG pillars.

[0046] S2, the xoy plane with the plane of periodic YIG pillars in the square magneto-optical photonic crystal as the coordinate plane, such as Figure 1 As shown on the coordinate axis, a magnetic field is applied to the square magneto-optical photonic crystal in the z-direction;

[0047] External magnetic fields of equal magnitude and opposite direction are applied to the odd-numbered and even-numbered rows of dielectric pillars of a square magneto-optical photonic crystal. The magnetic field H of the odd-numbered rows has a magnitude of 4800 Gs and a direction along +z.

[0048] S3. Set boundaries and boundary conditions around the square magneto-optical photonic crystal to construct a square magneto-optical photonic crystal waveguide, such as... Figure 1 As shown;

[0049] A boundary with scattering boundary conditions is set at 1.7a in the +y direction of the first row of dielectric pillars of the square magneto-optical photonic crystal, which serves as the upper boundary of the waveguide of the square magneto-optical photonic crystal; a boundary with scattering boundary conditions is set at 1.7a in the -y direction of the last row of dielectric pillars of the square magneto-optical photonic crystal, which serves as the lower boundary of the waveguide.

[0050] A boundary with both incident and outgoing functions is set at 3.5a in the -x direction of the leftmost column of dielectric pillars in the square magneto-optical photonic crystal, which serves as the left boundary of the waveguide; a boundary with both incident and outgoing functions is set at 3.5a in the +x direction of the rightmost column of dielectric pillars in the square magneto-optical photonic crystal, which serves as the right boundary of the waveguide, wherein the left and right boundaries are the incident and outgoing ends of the waveguide, respectively.

[0051] The background medium within the boundary is air.

[0052] The frequency range of the projected bandgap of the constructed square magneto-optical photonic crystal waveguide is within the microwave band.

[0053] Because the upper and lower boundaries of the square magneto-optical photonic crystal waveguide with scattering boundary conditions absorb the anti-chiral unidirectional boundary states, the anti-chiral unidirectional boundary states cannot propagate in the waveguide.

[0054] Due to the strong optical transmission confinement capability of the square magneto-optical photonic crystal, the unidirectional bulk state located in the waveguide is difficult to transmit to the upper and lower boundaries and is avoided from being absorbed, thus enabling the bulk state to maintain unidirectional transmission in the waveguide.

[0055] In this embodiment, the method for implementing non-trivial unidirectional volume transfer includes the following steps:

[0056] Step 1, take Figure 1 The supercell of the square magneto-optical photonic crystal waveguide shown is used to calculate the projected band structure and intrinsic mode field along the x-direction using the finite element method. It can be found that only nontrivial unidirectional bulk state bands exist in the band gap. By calculating the slope of the nontrivial unidirectional bulk state bands, the group velocity direction of the nontrivial unidirectional bulk state is determined, and thus the completely unidirectional conduction direction of the nontrivial unidirectional bulk state in the square magneto-optical photonic crystal waveguide is determined. The design and simulation calculation of this invention are both completed with the support of the radio frequency-frequency domain module in the COMSOLMULTIPHYSICS software.

[0057] like Figure 2 As shown, within the photonic bandgap of frequency f from 11.34 GHz to 11.62 GHz, there are two dispersion curves connecting the upper and lower trivial bulk state energy bands. The slope of the dispersion curves characterizes the group velocity of the electromagnetic wave, and the slopes of both curves are negative. Furthermore, the eigenmode fields of the dispersion curves at points M1 and M2 are distributed in bulk form within the supercell, which means that the waveguide only supports non-trivial unidirectional bulk states propagating along the -x direction.

[0058] Step 2: Constructing as follows Figure 1 The normal waveguide structure model shown is selected. Figure 2The frequency of 11.55 GHz, marked by a dashed line, was used as the excitation frequency to verify the nontrivial unidirectional bulk state transmission characteristics of the square magneto-optical photonic crystal waveguide. When a plane wave of 11.55 GHz is incident from the left end of the waveguide, the electric field intensity distribution within the waveguide is as follows. Figure 3 As shown in (a), it can be seen from the figure that when a plane wave propagates along the +x direction, it attenuates rapidly, and the waveguide propagation path in the +x direction is completely blocked; while when a plane wave of the same frequency is incident from the right end of the waveguide, the electric field intensity distribution in the waveguide is as follows. Figure 3 As shown in (b), the plane wave propagates in a bulk form along the -x direction, and the unidirectional bulk waveguide propagation channel in the -x direction is open. These results are consistent with the conclusions drawn from the first step through band structure analysis.

[0059] Step 3: Collection Figure 3 The electric field intensity along marker line 1 near the right boundary of the waveguide shown in (a) and the data collected... Figure 3 The electric field intensity along marked line 2 near the left boundary of the waveguide shown in (b) is as follows: Figure 3 As shown in curves 1 and 2 in (e), it can be seen from the figure that the electric field strength near the right boundary is almost 0, basically below 10. -5 The electric field strength is on the order of magnitude greater than 10, while the electric field strength near the left boundary is generally greater than 10. -2 This indicates that the waveguide exhibits strong non-reciprocal transmission characteristics, supports only non-trivial unidirectional bulk states propagating along the -x direction, and possesses complete unidirectional conductivity.

[0060] Example 2

[0061] In this embodiment, the method for achieving non-trivial unidirectional robust volume transfer is the same as that in Embodiment 1, except for the features described below.

[0062] Step 1: Constructing Barrier Waveguides and Defect Waveguides. Since perfect electrical conductors exhibit perfect reflection and scattering of electromagnetic waves, this embodiment uses perfect electrical conductors as barriers to electromagnetic wave transmission. By inserting perfect electrical conductors with lengths of 2a and widths of 0.1a into the center of the waveguide, a barrier waveguide can be constructed as follows: Figure 3 The barrier waveguide shown in (c) can be constructed by removing the two YIG dielectric pillars at the center of the waveguide. Figure 3 The defective waveguide shown in (d) is an example.

[0063] The second step involves using the same frequency of 11.55 GHz as the excitation frequency to verify the robust transmission characteristics of the non-trivial unidirectional bulk state. For example... Figure 3 As shown, Figure 3The diagrams show the electric field intensity distributions of plane waves incident at the left end of a normal waveguide and at the right end of a normal waveguide, a barrier waveguide, and a defective waveguide, respectively, as well as the electric field intensity curves along the dashed lines near the output terminals, for examples 1 and 2. When a plane wave with a frequency of 11.55 GHz is incident from the right end of a barrier waveguide and a defective waveguide, the electric field intensity distributions are as follows: Figure 3 As shown in (c) and (d). By Figure 3 As can be seen in (c), a plane wave can recover to its normal state after bypassing an obstacle. Figure 3 The electric field distribution is similar to that shown in (b) of the normal transmission case, and continues to propagate unidirectionally along the -x direction; by Figure 3 As can be seen in (d), a plane wave can recover to its original state after passing through a defect. Figure 3 The electric field distribution is similar to that shown in (b) of the normal transmission case, and continues to transmit unidirectionally along the -x direction.

[0064] Step 3: Collection Figure 3 (c) shows the electric field intensity along marker line 3 near the left boundary of the barrier waveguide, and the data collected... Figure 3 The electric field intensity along marked line 4 near the left boundary of the defective waveguide shown in (d) is as follows: Figure 3 As shown in curves 3 and 4 in (e), curves 3 and 4 are consistent with... Figure 3 Compared to curve 2 of the electric field intensity along marked line 2 near the left boundary under normal transmission conditions shown in (b), they are almost identical due to the topological protection of the structure. That is, the nontrivial unidirectional bulk state propagating in the waveguide can propagate robustly in the face of obstacles and defects. This achieves robust transmission of nontrivial unidirectional bulk states.

[0065] Example 3

[0066] In this embodiment, the method for achieving the tunability of the non-trivial unidirectional bulk transmission frequency is the same as that in Embodiment 2, except for the features described below.

[0067] By continuously changing the magnetic field strength applied to the square magneto-optical photonic crystal waveguide and adjusting the magnetic field magnitude, the transmission frequency of the nontrivial unidirectional bulk state can be changed, thus achieving tunability of the nontrivial unidirectional bulk state transmission frequency, as detailed below:

[0068] Step 1, take Figure 1 The supercell of the square magneto-optical photonic crystal waveguide shown is used to calculate the band structure under different magnetic field strengths, keeping other structural parameters constant. Analysis of the calculation results reveals the range of magnetic field strengths required for the square magneto-optical photonic crystal waveguide to generate non-trivial unidirectional bulk bands within the band gap.

[0069] Step 2: Within the above-mentioned magnetic field strength range, calculate the variation law of the frequency range and center frequency of the non-trivial unidirectional bulk state energy band as the magnetic field strength continuously increases; plot the relationship between magnetic field strength and the width of the non-trivial unidirectional bulk state energy band. Figure 4 (a) and plotting the relationship between magnetic field strength and the center frequency of the band structure of nontrivial unidirectional bulk states. Figure 4 (b) of. Figure 4 As can be seen in (a), the nontrivial unidirectional bulk band remains present as the magnetic field strength increases from 2500 Gs to 11000 Gs; its frequency range first increases and then decreases, reaching a maximum of 0.281 GHz at 5000 Gs. Figure 4 As can be seen in (b), as the magnetic field strength increases from 2500 Gs to 11000 Gs, the center frequency of the nontrivial unidirectional bulk state band shifts from 9 GHz to around 13 GHz, a shift range of approximately 4 GHz, with a center frequency of 11.59 GHz at 5000 Gs. Therefore, by adjusting the strength of the external magnetic field, the transmission frequency of the nontrivial unidirectional bulk state supported by the square magneto-optical photonic crystal waveguide can be flexibly controlled over a wide controllable frequency range. This achieves the tunability of the nontrivial unidirectional bulk state transmission frequency. This characteristic significantly improves the usability and practicality of the square magneto-optical photonic crystal waveguide of this invention.

[0070] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the embodiments described above. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A method for constructing a square magneto-optical photonic crystal waveguide supporting robust bulk transmission in nontrivial unidirectional states, characterized in that, Includes the following steps: S1. Construct a square magneto-optical photonic crystal dielectric pillar using a yttrium iron garnet pillar as the dielectric pillar; utilizing a radius of... r The dielectric pillars are used as the magneto-optical material of a square magneto-optical photonic crystal. Multiple dielectric pillars are arranged in a square lattice, forming a lattice constant of . a A square magneto-optical photonic crystal, wherein the square magneto-optical photonic crystal comprises m rows and n columns of dielectric pillars; r =1.83mm, a =1.16cm; S2, with the plane of periodically arranged dielectric pillars in a square magneto-optical photonic crystal as the coordinate plane. xoy noodles, in z A magnetic field is applied to the square magneto-optical photonic crystal in the direction to break the time reversal symmetry of the structure, so as to generate a unidirectional bulk state and an anti-chiral unidirectional boundary state that satisfy the energy conservation condition and the left-right transport balance. S3. Set boundaries and boundary conditions around the square magneto-optical photonic crystal to construct a square magneto-optical photonic crystal waveguide. The upper and lower boundaries are set with scattering boundary conditions to absorb the anti-chiral unidirectional boundary states propagating near the upper and lower boundaries, so that the square magneto-optical crystal waveguide only supports this non-trivial unidirectional bulk state propagation. In the first row of dielectric pillars of the square magneto-optical photonic crystal, + y Direction 1.7 a A boundary with scattering boundary conditions is set at a certain point, which serves as the upper boundary of the square magneto-optical photonic crystal waveguide; at the last row of dielectric pillars of the square magneto-optical photonic crystal... y Direction 1.7 a A boundary with scattering boundary conditions is set at the location, which serves as the lower boundary of the waveguide; a It is the lattice constant; In the leftmost column of dielectric pillars of the square magneto-optical photonic crystal - x Direction 3.5 a A boundary with both incident and exit functions is set at a certain point, serving as the left boundary of the waveguide; in the rightmost column of the dielectric pillars of the square magneto-optical photonic crystal, the + x Direction 3.5 a A boundary with both incident and outgoing functions is set at a point, which serves as the right boundary of the waveguide, where the left and right boundaries are the incident and outgoing ends of the waveguide, respectively.

2. The method for constructing a square magneto-optical photonic crystal waveguide supporting robust bulk transmission in nontrivial unidirectional states according to claim 1, characterized in that, In step S2, external magnetic fields of equal magnitude and opposite direction are applied to the odd-numbered and even-numbered dielectric pillars of the square magneto-optical photonic crystal.

3. The method for constructing a square magneto-optical photonic crystal waveguide supporting robust bulk transmission in nontrivial unidirectional states according to claim 1, characterized in that, The background medium within the boundary is air.

4. The method for constructing a square magneto-optical photonic crystal waveguide supporting robust bulk transmission in nontrivial unidirectional states according to claim 1, characterized in that, In step S3, the frequency range of the projected bandgap of the constructed square magneto-optical photonic crystal waveguide is within the microwave frequency band.

5. A nontrivial unidirectional bulk robust transmission method based on the construction method of a square magneto-optical photonic crystal waveguide supporting nontrivial unidirectional bulk robust transmission according to any one of claims 1 to 4, characterized in that, Includes the following steps: S1. Calculate and analyze the projected band structure and intrinsic mode field of the square magneto-optical photonic crystal waveguide under the current conditions using the finite element method to determine the generation of the nontrivial unidirectional bulk state. Calculate the slope of the nontrivial unidirectional bulk state band to determine the group velocity direction of the nontrivial unidirectional bulk state, and then determine the propagation direction of the nontrivial unidirectional bulk state in the square magneto-optical photonic crystal waveguide, i.e., the completely unidirectional conduction direction. S2. Plane electromagnetic waves within the frequency range of the non-trivial unidirectional bulk state energy band are sequentially incident from the conducting and non-conducting ends of the square magneto-optical photonic crystal waveguide. The electromagnetic wave intensity at the corresponding transmitting end of the square magneto-optical photonic crystal waveguide and the electric field distribution within the corresponding waveguide are detected to verify the unidirectional bulk state transmission and the complete unidirectional conductivity of the waveguide. S3. After inserting an obstacle into the center of the square magneto-optical photonic crystal waveguide or creating a defect by removing the dielectric pillar in the center of the square magneto-optical photonic crystal waveguide, the same plane electromagnetic wave as in step S2 is incident from the conducting end of the waveguide to detect the intensity of the electromagnetic wave at the output end and the electric field distribution inside the waveguide. S4. Compare the electromagnetic wave intensity at the output end and the electric field distribution inside the waveguide when the plane electromagnetic wave is incident from the conducting end of the waveguide in steps S3 and S2, and analyze the robust transmission of the non-trivial unidirectional bulk state when facing obstacles and defects.

6. The non-trivial unidirectional bulk robust transmission method according to claim 5, characterized in that, The finite element method was used to calculate and analyze the data in the RF-Frequency Domain module of COMSOLMULTIPHYSICS software.

7. The non-trivial unidirectional bulk robust transmission method according to claim 6, characterized in that, The transmission frequency of the nontrivial unidirectional bulk state can be adjusted by continuously changing the magnetic field strength applied to the square magneto-optical photonic crystal waveguide.

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