Topological resonant cavities and devices including topological resonant cavities

Abstract

This application relates to topological resonators and devices including topological resonators. A topological resonator may include a three-dimensional photonic or phononic crystal having a plurality of supercells modulated at least three independent degrees of freedom to open three-dimensional Dirac points in its band structure, forming a band gap. The intensity parameters of the modulation at the three independent degrees of freedom form a three-dimensional parameter space around an origin. The origin in the three-dimensional parameter space corresponds to a region in real space where the intensity parameters of the modulation at the three independent degrees of freedom are zero, thus the supercell has an equilibrium region of the three-dimensional Dirac point. In real space, with the equilibrium region as the center, the modulation at the three independent degrees of freedom in each direction corresponds to traversing or covering a closed surface around the origin W times continuously or discretely in the three-dimensional parameter space, where W is any positive or negative integer and the absolute value of W corresponds to the number of topological modes of the topological resonator.

Description

Technical Field

[0001] This application generally relates to the field of topological devices, and more particularly to a topological resonant cavity and a device including said topological resonant cavity, such as a laser. Background Technology

[0002] Resonant cavities, also known as resonant arrays, enhance local mode fields or filter frequency modes, and are widely used in devices such as optical lasers, microwave resonators, and acoustic resonant cavities. Examples include commercially available Fabry-Perot (FP) cavities, micro-ring cavities, and distributed Bragg reflector (DBR) cavities in optical lasers. These traditional resonant cavity designs exhibit an inverse relationship between the free spectral range (FSR) and the mode volume. Consequently, in high-power device applications, the larger the cavity volume, the narrower the corresponding free spectral range. As the effective mode volume increases, the single-mode characteristics of the resonant cavity rapidly fail.

[0003] To address this challenge, the same applicant, in their invention patent application 201911035379.9, filed on October 28, 2019, entitled "Two-Dimensional Topological Photonic Crystal Cavity, Its Design Method, and Its Application in Lasers," proposed a Dirac vortex resonator based on a two-dimensional photonic crystal structure. This resonator exhibits a free spectral range inversely proportional to the square root of the mode volume. As the mode volume increases, the decrease in the free spectral range becomes more gradual, thus providing better single-mode cavity design within a certain operating range. Based on this resonator, a topological cavity surface-emitting laser and its monolithic integrated array have been realized, improving the laser's single-mode properties, collimation, and power density.

[0004] To further improve the single-mode free spectrum range of the resonant cavity under a given mode volume, further improvements to the resonant cavity design are needed. Summary of the Invention

[0005] This invention provides a three-dimensional monopole topological resonant cavity whose free spectrum range is inversely proportional to the cube root of the mode volume, thereby enabling better single-mode characteristics.

[0006] According to one embodiment, a topological resonant cavity is provided, comprising a three-dimensional photonic or phononic crystal. The three-dimensional photonic or phononic crystal has multiple supercells, which are modulated in at least three independent degrees of freedom to open three-dimensional Dirac points in their band structure, forming a band gap. The intensity parameters of the modulation in the three independent degrees of freedom form a three-dimensional parameter space around an origin, where the origin corresponds to an equilibrium region in real space where the intensity parameters of the modulation in the three independent degrees of freedom are zero, thus the supercell has a three-dimensional Dirac point. In real space, with the equilibrium region as the center, modulation in the three independent degrees of freedom in each direction corresponds to continuously or discretely traversing or covering a closed surface around the origin W times in the three-dimensional parameter space, where W is any positive or negative integer and the absolute value of W corresponds to the number of topological modes of the topological resonant cavity.

[0007] In one embodiment, the three-dimensional photonic crystal structure includes a first material and a second material with different refractive indices or magnetic permeabilities, and the three-dimensional phononic crystal structure includes a first material and a second material with different sound velocities or densities. The first material forms the unit cell structure of the supercell, and the second material fills the spaces between the unit cells.

[0008] In one embodiment, the modulation on the three independent degrees of freedom includes modulating one or more of the shape, size, position, and material parameters of the unit structures in the supercell.

[0009] In one embodiment, the equilibrium region includes an equilibrium point or a continuous region of arbitrary shape, and the periphery of the equilibrium region is surrounded by a modulation region in which the intensity parameter of the modulation of the three-dimensional photonic or phononic crystal in at least one of the three independent degrees of freedom is not zero.

[0010] In one embodiment, the resonant wavelength of the topological resonant cavity is in the acoustic wave, elastic wave, radio frequency, microwave, terahertz, infrared, or optical bands.

[0011] In one embodiment, the supercell has crystal symmetry represented by space group 230 before modulation, and has symmetry represented by space groups 205, 220, or 214 after modulation on the three independent degrees of freedom.

[0012] According to one embodiment, a device is provided, including the above-described topological resonant cavity.

[0013] In one embodiment, the device includes a laser comprising a topological resonant cavity formed by the three-dimensional photonic crystal.

[0014] The above and other features and advantages of this application will become apparent from the following description of exemplary embodiments. Attached Figure Description

[0015] The above and other objects, features, and advantages of this application will become more apparent from the more detailed description of exemplary embodiments thereof in conjunction with the accompanying drawings. The drawings are provided to further illustrate the embodiments of this application and form part of the specification. They are used together with the embodiments of this application to explain this application and do not constitute a limitation thereof. In the drawings, the same reference numerals generally represent the same components or steps.

[0016] Figure 1A A schematic diagram of the supercell structure of a photonic or phononic crystal according to an embodiment of the present invention is shown.

[0017] Figure 1B Show Figure 1A A schematic diagram of the Brillouin zone of the supercell structure shown.

[0018] Figure 1C Show Figure 1A A schematic diagram of the band structure and density of states of the supercell structure shown.

[0019] Figure 2A This illustrates an embodiment of the invention. Figure 1A The diagram shows the structure of the supercell after modulation in the first independent degree of freedom.

[0020] Figure 2B Show Figure 2A The diagram shows the band structure and density of states of a supercell.

[0021] Figure 3A This illustrates an embodiment of the invention. Figure 1A The diagram shows the structure of the supercell after modulation in the second independent degree of freedom.

[0022] Figure 3B Show Figure 3A The diagram shows the band structure and density of states of a supercell.

[0023] Figure 4A This illustrates an embodiment of the invention. Figure 1A The diagram shows the structure of the supercell after modulation on the third independent degree of freedom.

[0024] Figure 4B Show Figure 4A The diagram shows the band structure and density of states of a supercell.

[0025] Figure 5A This illustrates an embodiment of the invention. Figure 1A The diagram shows the structure of the supercell after modulation on the fourth independent degree of freedom.

[0026] Figure 5B Show Figure 5A The diagram shows the band structure and density of states of a supercell.

[0027] Figure 6 This diagram illustrates the modulation of a supercell of a photonic or phononic crystal in three independent degrees of freedom according to an embodiment of the present invention to form a three-dimensional monopole topological cavity.

[0028] Figure 7A Simulation results are shown showing the free spectrum range of a topological resonant cavity according to an embodiment of the present invention as a function of the radius of the cavity's central potential well.

[0029] Figure 7B Simulation results are shown showing the variation of the free spectral range of a topological resonator with mode volume according to an embodiment of the present invention.

[0030] Figure 8A A photograph of a sample test of a phonon resonator according to an embodiment of the present invention is shown.

[0031] Figure 8B Showing the Figure 8A The spectral response of the sample shown was obtained by measurement.

[0032] Figures 8C to 8E The experimental results of real-space mode field scanning measurements at different frequency locations are shown. Detailed Implementation

[0033] Exemplary embodiments of this application will now be described in detail with reference to the accompanying drawings. It is obvious that the described embodiments are merely some embodiments of this application, and not all embodiments of this application. It should be understood that this application is not limited to the exemplary embodiments described herein.

[0034] An exemplary embodiment of the present invention provides a resonant cavity structure based on a three-dimensional topological photonic / phononic crystal. Based on the topological properties of the energy band and the topological defects of the crystal structure, the topological resonant cavity of the present invention has the following characteristics.

[0035] First, the topological resonant cavity comprises a Dirac photonic / phononic crystal with a three-dimensional double Dirac point in its band structure consisting of eight degenerate bands; or, from the perspective of the system's density of states (DOS), a frequency position with zero density of states exists, and the density of states varies with the square of the frequency interval in the vicinity of this frequency position. It should be understood that photonic and phononic crystals are equally important in the principles of this invention, and the design principles of the monopole topological resonant cavity of this invention, discussed in detail below, are applicable to both photonic and phononic systems. The photonic crystal may comprise two or more different optical media, for example, with different refractive indices or permeabilities, exhibiting dispersion behavior (e.g., frequency-momentum relationship) different from that of light propagation in a homogeneous medium. Examples of such optical media include organic materials (e.g., resins), inorganic materials (e.g., metals, semiconductors, insulators), air, vacuum, etc. The phononic crystal may comprise two or more different acoustic media, for example, with different sound velocities, densities, etc., exhibiting dispersion behavior different from that of sound propagation in a homogeneous medium. Examples of such acoustic media include gases, solids, liquids, etc.

[0036] Second, in embodiments of the present invention, the three-dimensional Dirac point in the band structure of the photonic / phononic crystal is opened by modulation, forming a band gap, and the band gains effective Dirac mass. By modulating the Dirac phonon / photonic crystal structure, a photonic / phononic crystal with complete band gap characteristics (or zero density of states) is obtained near the frequency corresponding to the position of the Dirac point before modulation. Theoretically, for a three-dimensional double Dirac point with eight band degenerates, there are four independent band gap opening modes, and the effective mass terms of the corresponding Hamiltonians are independent of each other (the corresponding representation matrices satisfy anti-reciprocity relations).

[0037] Third, in the embodiments of the present invention, three different independent modulation methods are selected to modulate the photonic / phononic crystal, with the modulation intensity as the three degrees of freedom, and a three-dimensional Dirac mass parameter space is constructed through linear interpolation. Around the cavity center (the position where the modulation intensity of all three degrees of freedom is zero) in the structural space (real space) of the resonant cavity, by mapping the Dirac mass parameters to spatial coordinates, a continuous bandgap within a spherical 4π parameter range enclosing the modulation center is obtained, or a discrete bandgap similarly obtained by discretizing the 4π continuous parameters. Thus, the topological point defect structure obtained by continuous (or discrete) modulation of the Dirac photonic / phononic crystal is a monopole topological cavity.

[0038] Figure 1AA schematic diagram of the supercell structure of a photonic or phononic crystal according to an embodiment of the present invention is shown. It should first be understood that the term "photonic or phononic crystal" in the art does not refer to a material having a periodic repeating structure at the atomic level, but rather to a material that can form a unit structure similar to atoms. This unit structure arrangement forms a periodic repeating structure similar to a crystal, and produces unique properties regarding photons or phonons propagating within it; therefore, it is called a photonic crystal or phononic crystal. Figure 1A An example of a supercell in a photonic or phononic crystal is shown, which is a simple cubic supercell containing two body-centered cubic (BCC) cells. Figure 1A The diagram only shows a unit structure formed by one material; another material could also be used for filling. Figure 1A The space between the unit structures shown, or the other material mentioned, can be air. The supercell structure can also be formed from two or more materials. In short, a three-dimensional photonic crystal can include two or more materials with different refractive indices or magnetic permeabilities, and a three-dimensional phononic crystal can include two or more materials with different sound velocities or densities. One or more materials form the unit structures in the supercell (similar to atoms in a crystal), while another one or more materials fill the spaces between the unit structures. For simplicity, a supercell is described here using two materials as an example.

[0039] A supercell can be used with a region function f D The term is described by (r), where r is the position vector in the supercell. Figure 1A The region function f of the simple cubic supercell shown D (r) is shown in Formula 1 below:

[0040]

[0041] Where a is the lattice constant of the supercell, denoted by ∑ cyc f represents the iterative summation over the spatial location variables (x, y, z). D (r)≥-1.56 indicates the region function f D The portion of (r) greater than or equal to -1.56 is the first material, and the portion less than -1.56 can be the second material with different refractive indices, magnetic permeabilities, or different sound velocities and densities.

[0042] Figure 1AThe geometric symmetry of the supercell shown can be represented by space group #230. For phononic crystals, the structure shown in the figure can be a unit structure composed of acoustic hard material (cured resin was used in the experiment), and the structural region is defined by formula (1). Other regions in the structure can be air, in which sound waves propagate and are reflected at the interface between air and hard material. For photonic crystals, the structure shown in the figure can be a unit structure formed by the first type of photonic crystal material, and the structural region is defined by formula (1). Other regions in the structure can be air or a second type of photonic crystal material with different refractive indices or permeabilities. Photons propagate in the first and second types of photonic crystal materials and are refracted at the interface.

[0043] Figure 1B Show Figure 1A The diagram shows a schematic of the Brillouin zone of a supercell structure, illustrating the two momentum points P and -P in the BCC Brillouin zone, which overlap with the R point of the simple cubic supercell. At these two locations, there are two quadruple degenerate three-dimensional Dirac points, which overlap with the R point to form an octet degenerate three-dimensional bi-Dirac point.

[0044] Figure 1C Show Figure 1A A schematic diagram of the band structure and density of states (DOS) of the supercell structure is shown. Figure 1C As shown, since the simple cubic supercell contains two BCC cells, the number of the first Brillouin bands is doubled. At the position where the reduced frequency (ωa / 2πc) ≈ 1.04, where ω represents the angular frequency, a is the lattice constant of the supercell, and c is the speed of sound in air, an octet degenerate three-dimensional double Dirac point can be observed. The density of states (DOS) at this position is zero, and the density of states in the vicinity varies with the square of the frequency interval. The effective Hamiltonian of the system near the degeneracy point is... The Dirac equation satisfies the massless term, where kx, ky, and kz represent the momentum terms that constitute the coefficient matrix. σ i (i = 1, 2, 3) is the Pauli matrix. It is a matrix direct product.

[0045] It should be understood that the supercell structure of the photonic / phononic crystal described here is only an example, and other supercell structures with different crystal symmetries can also be used, as long as they have three-dimensional double Dirac points.

[0046] Theoretically, for a three-dimensional double Dirac point with eight band degenerate bands, there are four independent bandgap opening modes, and the corresponding effective mass terms of the Hamiltonian are independent of each other. In the embodiments of this application, by selecting any three different independent modulation modes to modulate the size, shape, position, material parameters, etc. of some or all unit structures in the supercell, a photonic / phononic crystal with a complete bandgap and non-zero Dirac mass can be obtained. A three-dimensional parameter space is formed using the modulation intensity of the three modulation modes (or the three obtained Dirac masses) as three dimensions. The three independent modulation degrees of freedom can be one or more of the size, shape, position, or material parameters of the unit structures in the supercell. Material parameters may include material properties, density, permeability, refractive index, dielectric constant, etc. For example, modulating the shape of three different unit structures in the supercell can constitute modulation in three independent degrees of freedom. Similarly, modulating the size of the first unit structure, the shape of the second unit structure, and the position of the third unit structure also constitutes modulation in three independent degrees of freedom. For example, modulating the size, shape, and position of the same structural unit within a supercell also constitutes modulation in three independent degrees of freedom. The degrees of freedom for supercell modulation have been described by the applicant in previous invention patent applications 201911035379.9, 202010494073.6, 202111283095.9 and utility model patent application 202122645165.2, and will not be listed in detail here.

[0047] The following describes some examples of modulating the supercell to open the three-dimensional Dirac points and form a band gap. For ease of description and understanding, these examples do not change the structural material, but only make minor deformation adjustments to the structure, while maintaining certain space group symmetries of the overall structure. Figure 2A This illustrates an embodiment of the invention. Figure 1A The diagram shown illustrates the structure of the supercell after modulation in the first independent degree of freedom. Figure 2B Show Figure 2A The modulated supercell's band structure and density of states are shown. Figure 2A The modulation on the first independent degree of freedom shown can be represented by the following formula 2.

[0048]

[0049] The modulated supercell can be derived from the region function f. D (r)+δf x (r)≥-1.56 indicates that the structural symmetry can be represented by space group #205. Figure 1A The supercell structure f shown before modulation D Compared to (r)≥-1.56, Figure 2A The grayscale value shows the deformation of the structure at the corresponding location relative to the original structure. Figure 2B It can be seen that the three-dimensional Dirac point is opened, forming a complete bandgap of approximately 11.13%. The bandgap range is basically symmetrical about the original Dirac point frequency, and the density of states within the corresponding bandgap range is zero. Near the bandgap frequency, the Hamiltonian of the system changes from H... D (k)+m x Γ 100 Description, where m x This represents the corresponding Dirac mass, which conforms to the Dirac equation containing the mass term.

[0050] Figure 3A This illustrates another embodiment of the invention. Figure 1A The diagram shows the structure of the supercell after modulation in the second independent degree of freedom. Figure 3B Show Figure 3A The modulated supercell's band structure and density of states are shown. Figure 3A The modulation shown can be represented by the following formula 3.

[0051]

[0052] The modulated supercell can be derived from the region function f. D (r)+δf y (r)≥-1.56 indicates that the structural symmetry can also be represented by space group #205. Figure 1A The supercell structure f shown before modulation D Compared to (r)≥-1.56, Figure 3A The grayscale value shows the deformation of the structure at the corresponding location relative to the original structure. Figure 3B It can be seen that the three-dimensional Dirac point is opened, forming a complete bandgap of approximately 11.13%. The bandgap range is basically symmetrical about the original Dirac point frequency, and the density of states within the corresponding bandgap range is zero. Near the bandgap frequency, the Hamiltonian of the system changes from H... D (k)+m y Γ 200 Description, where m y This represents the corresponding Dirac mass, which conforms to the Dirac equation containing the mass term.

[0053] Figure 4A This illustrates another embodiment of the invention. Figure 1A The diagram shows the structure of the supercell after modulation in the third independent degree of freedom. Figure 4B Show Figure 4A The modulated supercell's band structure and density of states are shown. Figure 4A The modulation shown can be represented by the following formula 4.

[0054]

[0055] The modulated supercell can be derived from the region function f. D (r)+δf z (r)≥-1.56 indicates that the structural symmetry can be represented by space group #220. Figure 1A The supercell structure f shown before modulation D Compared to (r)≥-1.56, Figure 4A The grayscale value shows the deformation of the structure at the corresponding location relative to the original structure. Figure 4B It can be seen that the three-dimensional Dirac point is opened, forming a complete bandgap of approximately 12%. The bandgap range is basically symmetrical about the original Dirac point frequency, and the density of states within the corresponding bandgap range is zero. Near the bandgap frequency, the Hamiltonian of the system changes from H... D (k)+m z Γ 320 Description, where m z This represents the corresponding Dirac mass, which conforms to the Dirac equation containing the mass term.

[0056] Figure 5A This illustrates another embodiment of the invention. Figure 1A The diagram shown illustrates the structure of the supercell after modulation on the fourth independent degree of freedom. Figure 5B Show Figure 5A The modulated supercell's band structure and density of states are shown. Figure 5A The modulation shown can be represented by the following formula 5.

[0057]

[0058] The modulated supercell can be derived from the region function f. D (r)+δf′(r)≥-1.56 indicates that the structural symmetry can be represented by the space group #214. Figure 1A The supercell structure f shown before modulation D Compared to (r)≥-1.56, Figure 5A The grayscale value shows the deformation of the structure at the corresponding location relative to the original structure. Figure 5B It can be seen that the three-dimensional Dirac point is opened, forming a complete bandgap of approximately 5.2%. The bandgap range is basically symmetrical about the original Dirac point frequency, and the density of states within the corresponding bandgap range is zero. Near the bandgap frequency, the Hamiltonian of the system changes from H... D (k)+m'Γ 332 The description is as follows, where m' represents the corresponding Dirac mass, which conforms to the Dirac equation containing the mass term.

[0059] The examples above provide four independent modulation degrees of freedom. However, it should be understood that the supercell of a photonic / phononic crystal can also be modulated in other independent degrees of freedom to open the three-dimensional Dirac point and form a band gap, and is not limited to the examples given above. In the examples above, the boundary value of the region function of the modulated cell is -1.56, which means that when the function value is greater than or equal to -1.56, the corresponding region can be the first material; when it is less than -1.56, the corresponding region can be a second material different from the first material. It should be understood that the value "-1.56" here is only an example of a preferred value. When this value is taken, the resulting supercell structure has a large and clean band gap, that is, fewer stray bands near the band gap. Of course, the region function can also have other boundary values.

[0060] Examples of independent modulation degrees of freedom have been described above. In some embodiments of the invention, any three independent degrees of freedom can be selected to modulate a photonic or phononic crystal, as exemplified in [examples omitted]. Figure 6 In. Figure 6 In the example shown, the modulation scheme δf described above was selected to obtain a large bandgap. x (r), δf y (r) and δf z (r), where (a) small figure shows the result after δf z (r) A supercell with space group #220 symmetry after modulation; (b) Inset shows its band structure; (c) and (d) Insets show the band structure after modulation via δf. x (r) and δf y (r) The modulated supercell with space group #205 symmetry; (e) The inset shows its band structure. Taking the intensities of these three modulations as a set of parameters, and assigning each parameter a three-dimensional spatial coordinate basis vector, a three-dimensional parameter space (m) is obtained. x ,m y ,m z Thus, the three axes of the parameter space correspond to three selected modulation schemes, and any other position is obtained by a combination of these three modulation schemes, such as... Figure 6 As shown in small figure (f) in the diagram. In parameter space, any closed sphere enclosing the central singularity is topologically equivalent. Here, we choose a unit sphere, i.e., |(m x ,m y ,m z )|=1. As shown by the arrows in the small figure (f), the parameter vectors on this sphere all point outward, forming a structure similar to a magnetic monopole.

[0061] Mapping the Dirac mass parameter vector on a unit sphere to the real-space coordinates of the photonic / phononic crystal is equivalent to modulating the supercell of the real-space location, so that each spatial point has a specific three-dimensional Dirac mass vector. Except for the center point of the mapping, which is the center of the cavity, the modulation intensity is zero, meaning there is no modulation; therefore, this point is called the equilibrium point. As will be understood from the following description, the equilibrium point in real space can also be a continuous region of any shape, such as a one-dimensional curve, a two-dimensional plane, or a three-dimensional solid region; therefore, it can also be called an equilibrium region. The equilibrium region is surrounded by the modulation region. The modulation at the equilibrium region is zero, and the modulation intensity in at least one independent degree of freedom in the modulation region is not zero; therefore, the equilibrium region in real space corresponds to the origin in parameter space where the modulation intensity is zero. It should be understood that although the equilibrium region or equilibrium point is also called the cavity center or resonant center, it can be any point in the resonant cavity in real space. In this way, a three-dimensional photonic / phononic crystal structure with point topological defects is constructed. In this structure, the band has no effective mass only at the center point; however, in any direction away from the center, the equivalent structural band has an effective mass term. Here, the mapping function between the parameter space and the real space can be expressed by the following formula 6:

[0062]

[0063] Where θ and φ are Figure 6 The spatial angle shown, W θ and W φ For any integer, W = W θ W φ The topological index for mapping also corresponds to the number of modes at the cavity center. It can be any positive or negative integer and is a freely selectable design parameter. For single-mode cavity design, |W θ |=|W φ | = 1, here we choose W θ =W φ =1.

[0064] When the equilibrium region is a single point, the resulting topological cavity structure can be represented by the region function shown in the following formula (7):

[0065]

[0066] Such a resonant cavity (e.g., represented by Equation 7) can be fabricated using techniques such as 3D printing and laser direct writing. For example, in the 3D printing process, by controlling the shape, position, size, or material of the structure formed by the printing material, modulation of the photonic / phononic crystal in three independent degrees of freedom, as represented by Equation 7, can be achieved. Based on the above mapping relationship, it can be understood that, in real space, with the equilibrium region of the formed topological resonant cavity as the center, the Dirac vector formed by the modulation in each direction corresponds to continuously or discretely traversing or covering the closed surface around the origin W times in the three-dimensional parameter space formed by the modulation intensity. Or, in real space, for any closed surface surrounding the equilibrium region and located in the modulation region, the modulation parameters of the three independent degrees of freedom on the closed surface correspond to continuously or discretely traversing or covering the closed surface around the origin W times in the parameter space. W depends on the mapping relationship shown in Equation 6 above, or on the variation of the modulation of the photonic / phononic crystal in the 4π solid angle. When W = ±1, the resulting resonant cavity is a single-mode resonant cavity; when W = ±n and n is a positive integer greater than or equal to 2, the resulting resonant cavity has n modes. In other words, W is the number of topological modes of the resonant cavity.

[0067] As mentioned earlier, the equilibrium region can be of any shape. Here, we assume that the equilibrium region is a spherical region with radius R located at the center of the cavity, which means that a potential well region has been added at the center of the cavity. At the same time, the height of the surrounding barrier can be adjusted by the parameter m0. Decreasing m0 or increasing R can result in a larger mode volume. Thus, the resulting topological cavity structure can be represented by the region function shown in the following formula (8):

[0068]

[0069] As can be seen from Formula 8, in regions with a radius less than or equal to R, the photonic / phononic crystal is not modulated; in regions with a radius greater than R, the photonic / phononic crystal is modulated. Furthermore, in some embodiments, an unmodulated protective region can be provided around the modulation region of the photonic / phononic crystal.

[0070] To verify the single-mode properties of the monopole topological cavity formed above, the inventors conducted numerical simulations. They selected m0 = 1 and increased the value of R from 0 to 10a, where a is the lattice constant. The simulation results are shown below. Figure 7A and 7B middle. Figure 7A The frequency response spectra of topologies with different radius parameters R show that, as the cavity size increases, the single-mode resonant peak at the center of the bandgap remains stable at its frequency position. Meanwhile, nearby modes gradually approach interband modes, and the mode spacing, i.e., the free spectrum range (FSR), is essentially inversely proportional to the radius parameter R. Figure 7B This shows the relationship between FSR and the equivalent mode volume V in the cavity. In the region of large mode volume, the relationship between FSR and V is FSR∝V. -1 / 3 FSR∝V is superior to that of a two-dimensional Dirac vortex topological cavity. -1 / 2 The FSR of most other traditional resonant cavities is ∝ V. -1 .

[0071] Figure 8A Also shown are photographs of a sample test of a phonon resonator according to an embodiment of the present invention. This sample was fabricated using 3D printing technology with a lattice constant α of 3 cm², using a photocurable resin material (as the first material). It is understood that the resin material can also be replaced by any acoustically rigid material (i.e., a material unfavorable to the propagation of sound waves). The resonant characteristics of sound waves in this structure were tested using a sound source and a sound probe. Sound waves propagate in the air (as the second material) within the structural gaps, and are scattered at the interface of the rigid material. Under the multiple scattering effect throughout the structure, a single resonant mode of a specific frequency is formed, with the mode field localized at the center of the structure (the center of monopole modulation). Figure 8B The experimental measurement structure of the frequency response spectrum of this acoustic monopole topological cavity is shown. It can be seen that there is a bandgap in the range of approximately 11 kHz to 12.5 kHz, with a resonance peak at the center of the bandgap at approximately 11.76 kHz. Figures 8C to 8E The results of real-space mode field scanning measurements at three different frequency positions—10.75 kHz, 11.76 kHz, and 12.70 kHz—are presented. It can be seen that only the mode at 11.76 kHz is localized at the center of the cavity, while the modes at the other frequency positions diffuse outwards. Therefore, this sample is a good single-mode phonon resonator.

[0072] The topological monopole resonator formed by the photonic crystal described above can be applied to high-power single-mode (or multi-mode) lasers or microwave resonators. For applications requiring larger resonator mode volumes and a wider free spectral range, the monopole topological resonator of this invention provides the theoretically best design. The corresponding photonic crystal structure can be fabricated using techniques such as three-dimensional laser direct writing and 3D printing. In the embodiments of this invention, the required photonic crystal structure can be directly designed, and the maximum common bandgap range can be selected by using appropriate modulation function coefficients, while retaining considerable design freedom. The mode volume can be arbitrarily increased (decreasing m0 or increasing R), and the number of interband degenerate modes W can be arbitrarily designed, i.e., it can be any positive or negative integer.

[0073] Some embodiments of the present invention also provide devices including the above-described topological resonant cavity, such as optical devices or acoustic devices. Optical devices may include a topological resonant cavity formed by the above-described three-dimensional photonic crystal, and examples of optical devices include, but are not limited to, optical communication devices such as optical fibers, optical modules, etc. Acoustic devices may include a topological resonant cavity formed by the above-described three-dimensional phononic crystal, and examples of acoustic devices include, but are not limited to, acoustic resonant devices that can be directly used in air or underwater (any gaseous or liquid environment), such as sonar and vibration detection devices.

[0074] In one embodiment, the device may include a laser comprising a topological resonant cavity formed of a three-dimensional photonic crystal. The topological resonant cavity of this invention can replace a conventional resonant cavity in a laser; since the structure of lasers is known, it will not be described in detail here. Depending on the design of the number W of interband degenerate modes in the topological resonant cavity, the laser of this invention can be a single-mode laser or a multimode laser. By selecting the lattice parameters of the three-dimensional photonic crystal, the operating wavelength of the laser can be made in the microwave, terahertz, infrared, or optical bands. Of course, it is understood that in other applications, the topological resonant cavity of this invention can also operate in other frequency bands, such as acoustic bands like infrasound (<20Hz), audible sound (20Hz~20kHz), ultrasound (>20kHz), or elastic bands, etc., and this invention does not specifically limit this.

[0075] Some exemplary embodiments of this application also provide an electronic device including the above-described semiconductor laser or laser array. Such electronic device may be, for example, an optical communication device, a lidar device, a detection and sensing device, a laser processing device, a laser medical device, a laser weapon device, a lighting device, or a display device.

[0076] The basic principles of this application have been described above with reference to specific embodiments. However, it should be noted that the advantages, benefits, and effects mentioned in this application are merely examples and not limitations, and should not be considered as essential features of each embodiment of this application. Furthermore, the specific details disclosed above are for illustrative and facilitative purposes only, and are not limitations. These details do not limit the application to the necessity of employing the aforementioned specific details for implementation.

[0077] The above description has been given for purposes of illustration and description. Furthermore, this description is not intended to limit the embodiments of this application to the forms disclosed herein. Although numerous exemplary aspects and embodiments have been discussed above, those skilled in the art will recognize certain variations, modifications, alterations, additions, and sub-combinations thereof.

Claims

1. A topological resonant cavity comprising a three-dimensional photonic or phononic crystal having a plurality of supercells modulated at least three independent degrees of freedom to open three-dimensional Dirac points in its band structure, forming a band gap, wherein the intensity parameters of the modulation at the three independent degrees of freedom form a three-dimensional parameter space around an origin, the origin in the three-dimensional parameter space corresponding to an equilibrium region of the three independent degrees of freedom in real space where the intensity parameters of the modulation at the three independent degrees of freedom are zero, and the supercell having an equilibrium region, wherein modulation at the three independent degrees of freedom in the real space, centered on the equilibrium region, corresponds to traversing or covering a closed surface around the origin W times continuously or discretely in the three-dimensional parameter space, wherein W is any positive or negative integer and the absolute value of W corresponds to the number of topological modes of the topological resonant cavity.

2. The topological resonant cavity as described in claim 1, wherein, The three-dimensional photonic or phononic crystal structure comprises a first material and a second material with different refractive indices, magnetic permeabilities, or different sound velocities and densities. The first material forms the unit structure of the supercell, and the second material fills the space between the unit structures.

3. The topological resonant cavity as described in claim 1, wherein, The modulation on the three independent degrees of freedom includes modulating one or more of the shape, size, position, and material parameters of the unit structures in the supercell.

4. The topological resonant cavity as described in claim 1, wherein, The equilibrium region includes an equilibrium point or a continuous region of arbitrary shape, and the periphery of the equilibrium region is surrounded by a modulation region, in which the intensity parameter of the modulation of the three-dimensional photonic or phononic crystal in at least one of the three independent degrees of freedom is not zero.

5. The topological resonant cavity as described in claim 1, wherein, The resonant wavelength of the topological resonant cavity is in the acoustic wave, elastic wave, radio frequency, microwave, terahertz, infrared or optical bands.

6. The topological resonant cavity as described in claim 1, wherein, The supercell has crystal symmetry represented by space group 230 before modulation, and has symmetry represented by space groups 205, 220, or 214 after modulation on the three independent degrees of freedom.

7. An apparatus comprising the topological resonant cavity according to any one of claims 1-6.

8. The device as claimed in claim 7, wherein, The device includes a laser, which comprises a topological resonant cavity formed by the three-dimensional photonic crystal.

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