CN110568524A - Zero-refractive-index metamaterial with low loss and design method - Google Patents
Zero-refractive-index metamaterial with low loss and design method Download PDFInfo
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Abstract
The invention provides a zero-refractive-index metamaterial with low loss and a design method thereof. The zero-refractive-index metamaterial comprises a plurality of periodically arranged two-dimensional crystal units, and the height of a first dielectric column in each crystal unit meets the condition of a photon bound state, so that the quality factor of a magnetic dipole at the center of a Brillouin area of the zero-refractive-index metamaterial is the highest, and the loss of the zero-refractive-index metamaterial is the lowest. The design method comprises the following steps: establishing a photonic crystal model for simulating the metamaterial; keeping the heights of all the crystal units unchanged, and adjusting the periods and the structural parameters along the x direction and the y direction of the crystal units to obtain a photonic crystal model meeting the zero refractive index; establishing a relation between the height of the first dielectric column in the crystal unit and the eigenmode quality factor of the photonic crystal model; the height of the first dielectric column in the crystal unit when the quality factor of the magnetic dipole reaches the maximum is screened out by utilizing the relation. The invention makes large-scale commercial use of zero-refractive-index metamaterial possible.
Description
Technical Field
The invention belongs to the field of photonic crystals, and particularly relates to a low-loss zero-refractive-index metamaterial and a design method thereof.
Background
the refractive index is a basic property parameter of all substances in nature, and the refractive index of most natural substances is greater than one. In 2011, the group of subjects taught by chen zi pavilion, hong kong science and technology university, for the first time, related dirac cones to zero refractive index. The Chen Zi proves that after the center of the Brillouin zone of the photonic crystal is induced to generate the Dirac-like cone dispersion, the equivalent refractive index of the photonic crystal is zero, so that a theoretical basis is provided for the experiment to realize the zero-refractive-index material, and a gate is opened for researching the zero-refractive-index material in an optical band.
Based on the theory of zero-index photonic crystals, many scientists have invested in experimental studies of zero-index photonic crystals. In 2015, littleleaf first designed and fabricated photonic crystals with equivalent zero refractive index consisting of an array of low aspect ratio silicon pillars embedded in a polymer matrix coated with a gold film, and this structure could be fabricated in arbitrary shapes over large areas using standard planar processes. Subsequently in 2017, Vulis experiment realized a zero-refractive-index metamaterial based on 220-nanometer thick silicon-on-insulator, and the structure is composed of a square matrix of air columns in silicon and is compatible with CMOS. Meanwhile, Kita designs and experiments realize the zero-refractive-index metamaterial on the all-dielectric sheet with certain robustness on processing defects.
In addition to the processing research on zero-index photonic crystals, there have been many researchers in recent years who focus on studying the properties and application expansion of zero-index materials. Zhai investigated the theoretical feasibility of laser ignition using zero index metamaterial. Fang demonstrates that at the dirac point, the curved wavefront of the zero index metamaterial region is transformed into a planar wavefront in the dihedral region.
The loss of materials is one of very important factors which puzzles the commercial industry, and the reduction of the loss of materials is the subject of cumin-less research of many scholars. Li provides a method for designing a low-loss photonic crystal in a visible light frequency band by utilizing a silver array, and the simulation method mainly aims at metallic silver, so that the use cost of the metallic silver is high, and the value of practical industrial application is low.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a method for enabling large-scale commercial application of a zero-refractive-index metamaterial. The invention directly utilizes the photon bound state to design the low-loss zero-refractive-index metamaterial, and can reduce the loss of the zero-refractive-index metamaterial to the maximum degree by establishing the relation between the physical structure parameter of the crystal unit, namely the height, and the quality factor of the eigenmode without depending on the selected material of the crystal unit. In addition, the invention can be based on low-cost and mature processing technology of the insulating silicon, has lower industrial use cost and has great advantages when being used for large-scale industrial production.
In order to achieve the purpose, the technical scheme provided by the invention is as follows:
The invention provides a zero-refractive-index metamaterial with low loss, which comprises a plurality of periodically arranged two-dimensional crystal units, wherein each crystal unit comprises a first medium column and a second medium column wrapping the first medium column; the zero-refractive-index metamaterial is characterized in that the height of the first dielectric column in each crystal unit meets the condition of a photon bound state, namely, the optical wave can be completely bound in the height range of the first dielectric column of the corresponding crystal unit, so that the quality factor of a magnetic dipole at the center of the Brillouin zone of the zero-refractive-index metamaterial is highest, and the loss of the zero-refractive-index metamaterial is lowest.
The invention also provides a design method of the zero-refractive-index metamaterial, which is characterized by comprising the following steps of:
1) establishing a photonic crystal model for simulating the metamaterial by using a finite element analysis method based on Maxwell equations, wherein the photonic crystal model is composed of a plurality of two-dimensional crystal units which are periodically arranged in the x direction and the y direction of a three-dimensional coordinate system xyz, and the structural parameters, namely the heights, of the crystal units along the z direction are equal; the parameters set in the photonic crystal model include: the excitation light wavelength and the polarization direction of the excitation light of the photonic crystal model, the initial values of the structural parameters and the periods of the crystal units set according to the materials of the first dielectric column and the second dielectric column, and the boundary conditions of the photonic crystal model;
2) Keeping the heights of all the crystal units unchanged, and adjusting the period of each crystal unit and the structural parameters along the x and y directions to enable the whole photonic crystal model to adjust a Dirac cone at the center of the Brillouin zone to obtain a photonic crystal model meeting the zero refractive index;
3) Keeping the period of the photonic crystal model meeting the zero refractive index in the step 2) and the structural parameters along the x and y directions unchanged, adjusting the height of the first dielectric column in each crystal unit, and establishing the relationship between the height of the first dielectric column in each crystal unit and the eigenmode quality factor of the photonic crystal model;
4) Screening the height of the first dielectric column in the crystal unit when the quality factor of the magnetic dipole reaches the maximum according to the relationship between the height of the first dielectric column in the crystal unit and the quality factor of the eigenmode established in the step 3) near the set wavelength of the excitation light and the set polarization direction of the excitation light, wherein the loss corresponding to the whole photonic crystal model is the lowest.
Further, when the relationship between the height of the first dielectric column in the crystal unit and the eigenmode quality factor of the photonic crystal model is established in the step 3), only the central point k of the brillouin zone of the photonic crystal model is solved for the height value of any single crystal silicon columnx＝kyAll eigenmodes are 0, and several eigenmodes closest to the absolute value of the optical wave frequency corresponding to the set excitation light wavelength are selected from the eigenmodes.
The invention has the characteristics and beneficial effects that:
The invention creatively provides for the first time that the low-loss zero-refractive-index photonic crystal is designed in a mode of adjusting the height of the photonic crystal to meet the optical constraint state, the loss is creatively connected with the quality factor of the magnetic dipole at the center of the Brillouin zone, the loss of all the zero-refractive-index photonic crystals distributed in a two-dimensional periodic array can be reduced as far as possible without depending on the materials selected by the crystal units, and the large-scale commercial use of the zero-refractive-index super-structure material in the aspects of interconnection of integrated optics, beam scanning and holography becomes possible.
drawings
FIG. 1 is an overall flow diagram of the method of the present invention.
FIG. 2 is a schematic of the model and parameters for a single crystal unit established in the method of the present invention;
FIG. 3 is a schematic view of a Dirac cone adjusted in the process of the present invention;
FIG. 4 is a graph of crystal unit height versus quality factor for all modes established by the method of the present invention;
FIG. 5 is a graph of crystal unit height versus electric monopole and magnetic dipole quality factor established by the method of the present invention;
FIG. 6 is a schematic diagram of the method of the present invention.
Detailed Description
The technical solution of the present invention will be further described in detail with reference to the accompanying drawings and embodiments.
The invention provides a low-loss zero-refractive-index metamaterial, which comprises a plurality of periodically arranged two-dimensional crystal units, wherein each crystal unit comprises a first dielectric column and a second dielectric column wrapping the first dielectric column, the first dielectric column adopts a monocrystalline silicon column, the second dielectric column adopts a silicon dioxide column (the invention is also applicable to other dielectric columns with mature preparation processes in the field), and the photonic crystal is convenient to adjust a Dirac cone so as to realize the zero-refractive-index metamaterial. The height of the first dielectric columns in each crystal unit meets the condition of a photon bound state, namely, the light waves can be completely bound in the height range of the first dielectric columns of the corresponding crystal units, so that the quality factor of the magnetic dipole at the center of the Brillouin zone of the zero-refractive-index metamaterial is the highest, and the loss of the zero-refractive-index metamaterial is the lowest.
The invention also provides a design method of the low-loss zero-refractive-index metamaterial, the overall process is shown in figure 1, and the design method comprises the following steps:
1) establishing a photonic crystal model for simulating a super-structure material by using a finite element analysis method based on Maxwell equations, wherein the photonic crystal model is composed of a plurality of two-dimensional crystal units which are periodically arranged in the x and y directions of a three-dimensional coordinate system xyz, and the structural parameters, namely the heights, of the crystal units along the z direction are equal; the parameters set in the photonic crystal model include: the excitation light wavelength and the polarization direction of the excitation light of the photonic crystal, the initial values of the structural parameters and the periods of the crystal units set according to the materials of the first dielectric column and the second dielectric column, and the boundary conditions of the photonic crystal model;
this implementationIn the example, optical simulation software COMSOL (or FDTD and the like) is adopted to establish a two-dimensional photonic crystal model for simulating the metamaterial. Referring to fig. 2, each crystal unit is composed of a single crystal Silicon column (Silicon in the COMSOL software is selected) nested in a Silicon dioxide column (Silicon in the COMSOL software is selected), two adjacent crystal units are arranged in close proximity, and the parameters set in the established photonic crystal model include: according to the application background (i.e. integrated optics) of the photonic crystal, the wavelength of excitation light of the photonic crystal model is set to be 1550 nanometers (optical communication waveband), and the polarization direction of the excitation light is selected to be TM polarized light in consideration of the larger aspect ratio of the structure; setting the relative dielectric constant and the relative magnetic permeability of each dielectric column according to the material of each dielectric column, wherein the relative dielectric constant of the single crystal silicon column is 11.7, the relative magnetic permeability of the single crystal silicon column is 1, the relative dielectric constant of the silicon dioxide column is 2.09, and the relative magnetic permeability of the silicon dioxide column is 1; setting the initial value of the period of the whole photonic crystal model (namely the diameter of the silicon dioxide column) as pitch, setting the initial value of the radius of the single crystal silicon column in each crystal unit as radius and the initial value of the height as height, and setting the initial values of the distance between the top end and the bottom end of the single crystal silicon column and the top end and the bottom end of the silicon dioxide column as h1and h2Wherein, the initial values of the period and the structural parameters are set according to the material of the selected dielectric column, the embodiment is in the range of hundred nanometers, the initial height of the crystal unit set according to the material of the dielectric column, the pitch and radius are in the range of hundred nanometers, the height is in the range of thousand nanometers, and h is1and h2Equal and half the wavelength of the excitation light, i.e. 775 nm; setting boundary conditions of a crystal unit, wherein the boundary conditions in the z direction select a self-contained PML (perfect matching layer) in COMSOL software, the boundary conditions in the x and y directions select a self-contained Floquet (FloQuit) periodic boundary condition in the COMSOL software, and components of a corresponding k vector (namely a space wave vector of a Brillouin zone) in the x, y and z directions are respectively set as:
In the formula, kx，ky，kzRespectively, empty of Brillouin zoneThe components of the inter-wave vector in the x, y, z directions; k is a radical ofFx，kFy，kFzthe components of the space wave vector of the Brillouin zone in the x, y and z directions under the Floquet periodic boundary condition are respectively.
2) Keeping the heights of all the crystal units unchanged, adjusting the period of each crystal unit and the structural parameters (the diameter when a circular column is adopted and the width when other shapes are adopted) along the x direction and the y direction to enable the whole photonic crystal model to adjust a Dirac cone at the center of the Brillouin zone, obtaining a photonic crystal model meeting the zero refractive index, and obtaining the photonic crystal model corresponding to the zero refractive index metamaterial. The specific implementation process of adjusting the dirac cone in this embodiment is as follows:
Setting the wave vector of the exciting light at the central point of the Brillouin zone, namely kx＝kyat 0, for all periods of each crystal unit and all structural parameters along the x and y directions, the eigenfrequency corresponding to all eigenmodes is solved through an eigenfrequency solver in finite element analysis software. For any crystal unit period value or any structure parameter along the x direction and the y direction, screening a plurality of eigenmodes which are closest to the set excitation light wavelength (1550 nanometers) and the light wave frequency corresponding to the polarization direction from the characteristic frequency of the eigenmodes through MATLAB data processing software (the number of the eigenmodes is selected to at least ensure that the central point k of the Brillouin zone is at least selectedx＝kyThe number of the eigenmodes is 10, namely 10 eigenmodes are screened out for any period value or any structural parameter value along the x and y directions, and the corresponding wavelength is between 1500 nanometers and 1600 nanometers.
In the scanning process, the period and the diameter of the photonic crystal are gradually changed, and the rule between the characteristic frequencies of the eigenmodes of the photonic crystal and the characteristic frequencies of the eigenmodes of the photonic crystal is as follows:
When the radius of the photonic crystal is not changed, the wavelengths corresponding to the characteristic frequencies of the three modes are gradually increased along with the increase of the crystal period, but the speed of the electric monopole is increased faster than that of the magnetic dipole, and the electric monopole and the magnetic dipole can be degenerated at a certain specific frequency.
secondly, when the period of the photonic crystal is not changed, the wavelengths corresponding to the characteristic frequencies of the three modes are increased along with the increase of the radius of the crystal, but the magnetic dipole is increased faster than the electric monopole, and the triple degeneracy can be realized at a certain specific frequency.
according to the two rules, the values of the radius and the period are continuously adjusted, and finally the dirac cone is adjusted at the center of the brillouin area, as shown in fig. 3.
3) Keeping the period of the photonic crystal model satisfying the zero refractive index in the step 2) and the structural parameters along the x and y directions unchanged, scanning the height of the first dielectric column in each crystal unit, and establishing the relationship between the height and the eigenmode quality factor of the photonic crystal modelx＝kyAll eigenmodes 0, the figure of merit value corresponding to each eigenmode that has been set (i.e., 10 eigenmodes selected hereinafter) is calculated at the same time. For any single crystal silicon column height value, 10 eigenmodes closest to the optical wave frequency absolute value corresponding to the set excitation light wavelength 1550 nanometers are still solved at the central point of the Brillouin zone of the photonic crystal model. Since the appropriate height range is not known, a larger height scan range is initially set: 10-1200 nm at an interval of 10 nm, and 2400 eigenmodes are finally obtained (the parameter selection basis is that the peak value appears on the quality factor curve corresponding to the magnetic dipole mode). The data were exported to MATLAB for processing, and the results are shown in fig. 4. In fig. 4, the abscissa is the scan height and the ordinate is the logarithm of the figure of merit for the eigenmodes, each solved eigenmode being represented by a circle in the figure.
4) Screening the height of the first dielectric column in the crystal unit when the quality factor of the magnetic dipole reaches the maximum according to the relationship between the height of the first dielectric column in the crystal unit and the quality factor of the eigenmode established in the step 3) near the set wavelength of the excitation light and the set polarization direction of the excitation light, wherein the loss corresponding to the whole photonic crystal model is the lowest.
and selecting the electric monopole and the magnetic dipole in all the eigenmodes, and selecting the height of the corresponding first dielectric column when the quality factor of the magnetic dipole is highest, so as to obtain the low-loss zero-refractive-index material. The eigenmode selection is mainly chosen around the set excitation light wavelength and polarization direction. The specific selection is described herein by way of example as silicon dioxide nested in monocrystalline silicon. Since this example uses TM polarized light incident as an excitation source, dirac cones are induced. The TM polarization requires that the ratio between the z-direction electric field component of the eigenmode and all the electric field intensities is large, and the range set by this embodiment is that the ratio between the z-direction electric field component and all the electric field intensities needs to be larger than 50%; the eigen frequency corresponding to the eigen mode solved by the degeneracy requirement at the frequency corresponding to the incident light wave is not much different from the frequency corresponding to the incident light wave, and since the excitation light wave set in this embodiment is in the 1550 nm optical communication band, the wavelength corresponding to the eigen mode solved is set between 1500 nm and 1600 nm. Under the above two conditions, the relationship between the height of the obtained single crystal silicon pillar and the eigenmode quality factor is shown in fig. 5. In fig. 5, the top row of circles are all the quality factors of the electric monopole, and the other circles are the quality factors of the magnetic dipole resolved by different heights of the monocrystalline silicon column. The corresponding quality factors of the magnetic dipoles resolved by different heights of the monocrystalline silicon columns are different. When the height of the single crystal silicon column is 1090 nanometers, the quality factor of the magnetic dipole reaches the maximum value, and the sum of the quality factors of the electric monopole and the magnetic dipole reaches the maximum value at the moment, namely the design with the lowest overall loss of the whole photonic crystal is realized. 1090 nm, the design that maximizes the quality factor of the magnetic dipole is the high design parameter of the desired low-loss zero-refraction metamaterial.
The design method of nested single crystal silicon in silicon dioxide is described in the above examples, and the design parameters of the low-loss zero-refractive-index metamaterial with the structure finally obtained are radius 171.3 nm, pitch 851 nm and height 1090 nm.
The principle of the invention is as follows:
The invention provides a low-loss zero-refractive-index metamaterial which is realized by adjusting the height of a photonic crystal to meet the condition of a photon bound state. The concrete expression is as follows: when light waves propagate in the photonic crystal, they radiate outward from the upper and lower boundaries of the photonic crystal. In this case, each interface can be regarded as a semi-transparent and semi-reflective mirror, and a part of the radiated light will radiate outward through the interface, and another part of the light will be reflected back to the photonic crystal. Light reflected back into the photonic crystal will undergo multiple such semi-reflections. When the height of the first dielectric column in the photonic crystal is controlled so that the path difference of all the outward radiated light is just an odd multiple of half the wavelength of the incident light, all the outward radiated light will be coherently cancelled, which is equivalent to no more light being radiated out of the photonic crystal, so that the optical confinement state in the zero-refractive-index medium can be realized, as shown in fig. 6.
For each propagation mode of the photonic crystal, the degree of coupling between each eigenmode and the outside during propagation is different due to different coupling conditions with the outside, and macroscopically, the loss of propagation of each eigenmode in the photonic crystal is completely different. In the actual simulation process, the loss-related parameter is a quality factor, also called Q value. The quality factor is mainly used for measuring the capability of the crystal to store energy and select frequency, and the Q value is defined as:
Where ν corresponds to the eigenfrequency to be solved, E1Energy per second lost for photonic crystals, E2is the energy stored in the photonic crystal. By definition, the larger the Q value, the corresponding E2/E1the larger the ratio of (a) is, the smaller the energy loss per second is. Therefore, when the Q value of the photonic crystal corresponding to the eigenmode is larger, the loss of the corresponding eigenmode is smaller, and if a mode which meets the zero-refractive-index condition (degeneracy of an electric monopole mode and two magnetic dipole modes) and has a high Q value can be designed, the low-loss zero-refractive-index metamaterial can be obtained.
The chen zigzi teaches for the first time that in the center of the brillouin zone, as long as the photonic crystal has dirac cone dispersion, the photonic crystal has an equivalent zero refractive index, and the equivalent dielectric constant and the equivalent magnetic permeability of the metamaterial both cross zero at the dirac point frequency, and externally show impedance matching zero refractive index. For the photonic crystal of the silicon column array, the dirac cone at the central point of the brillouin zone can be designed as long as the period and the diameter are adjusted to have a proper ratio, for the photonic crystal plate, the period and the radius are two parameters, the period and the radius are also an important parameter of the height of the dielectric column, different heights are necessarily corresponding to different boundary conditions, so that different eigen mode solutions are obtained, the eigen modes may correspond to different Q values, and if the corresponding height of the crystal column when the Q value is maximum can be found, the low-loss zero-refraction metamaterial can be obtained.
the closer to the triple degenerate state, the higher the quality factor of the magnetic dipole, the closer the quality factor of the two magnetic dipoles degenerated at the Γ point remains, while the quality factor of the electric monopole remains continuously higher. When the equivalent zero refractive index is realized, the magnetic dipole and the electric monopole which play the most important roles are the gamma point, and the quality factor of the electric monopole is always very high, so that the corresponding height value when the quality factor of the magnetic dipole is the highest can be found by scanning the height of the crystal column, and the low-loss zero-refractive-index metamaterial can be obtained.
While the above is a preferred embodiment of the present method based on silicon dioxide nested silicon, the scope of the present invention is not limited thereto, and any person skilled in the art should be within the scope of the present invention when designing a low-loss two-dimensional zero-index photonic crystal. Therefore, the protection scope of the present invention shall be subject to the protection scope defined by the claims.
Claims (4)
1. a zero-refractive-index metamaterial with low loss comprises a plurality of periodically arranged two-dimensional crystal units, wherein each crystal unit comprises a first dielectric column and a second dielectric column wrapping the first dielectric column; the method is characterized in that the height of the first dielectric column in each crystal unit meets the condition of a photon bound state, namely, the optical wave can be completely bound in the height range of the first dielectric column of the corresponding crystal unit, so that the quality factor of a magnetic dipole at the center of the Brillouin zone of the zero-refractive-index metamaterial is the highest, and the loss of the zero-refractive-index metamaterial is the lowest.
2. the zero index metamaterial according to claim 1, wherein the first dielectric pillars are silicon pillars; the second dielectric column is a silicon dioxide column.
3. A method of designing a zero index metamaterial according to claim 1 or 2, comprising the steps of:
1) Establishing a photonic crystal model for simulating the metamaterial by using a finite element analysis method based on Maxwell equations, wherein the photonic crystal model is composed of a plurality of two-dimensional crystal units which are periodically arranged in the x direction and the y direction of a three-dimensional coordinate system xyz, and the structural parameters, namely the heights, of the crystal units along the z direction are equal; the parameters set in the photonic crystal model include: the excitation light wavelength and the polarization direction of the excitation light of the photonic crystal model, the initial values of the structural parameters and the periods of the crystal units set according to the materials of the first dielectric column and the second dielectric column, and the boundary conditions of the photonic crystal model;
2) Keeping the heights of all the crystal units unchanged, and adjusting the period of each crystal unit and the structural parameters along the x and y directions to enable the whole photonic crystal model to adjust a Dirac cone at the center of the Brillouin zone to obtain a photonic crystal model meeting the zero refractive index;
3) Keeping the period of the photonic crystal model meeting the zero refractive index in the step 2) and the structural parameters along the x and y directions unchanged, adjusting the height of the first dielectric column in each crystal unit, and establishing the relationship between the height of the first dielectric column in each crystal unit and the eigenmode quality factor of the photonic crystal model;
4) screening the height of the first dielectric column in the crystal unit when the quality factor of the magnetic dipole reaches the maximum according to the relationship between the height of the first dielectric column in the crystal unit and the quality factor of the eigenmode established in the step 3) near the set wavelength of the excitation light and the set polarization direction of the excitation light, wherein the loss corresponding to the whole photonic crystal model is the lowest.
4. The design method according to claim 3, wherein in the step 3), when the relationship between the height of the first dielectric column in the crystal unit and the eigenmode quality factor of the photonic crystal model is established, only the central point k of the Brillouin zone of the photonic crystal model is solved for any single-crystal silicon column height valuex＝kyAll eigenmodes are 0, and several eigenmodes closest to the absolute value of the optical wave frequency corresponding to the set excitation light wavelength are selected from the eigenmodes.
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| CN114609775A (en) * | 2022-02-14 | 2022-06-10 | 江苏大学 | Rapid modeling design method of super-structured lens based on object-oriented idea |
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| CN112185487B (en) * | 2020-09-28 | 2023-05-16 | 北京环境特性研究所 | Target excitation frequency searching method and device for photonic crystal and readable medium |
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