CN106066925A - Magneto-electric coupled Meta Materials constitutive matrix method of acquiring - Google Patents

Abstract

The present invention relates to magneto-electric coupled Meta Materials constitutive matrix method of acquiring, in theory, by setting up electromagnetism Meta Materials coupling model, obtain the expression formula of 2 × 2 constitutive matrixes based on the degree of coupling, then derive refractive index formula based on the degree of coupling；In emulation, use nonlinear fitting, by observation error of fitting value in error allowed band (less than 0.1), and then the correctness of based on the degree of coupling two-dimentional constitutive matrix expression formula and the refractive index formula proposed in proof theory；Finally, the relation curve of four physical quantitys of constitutive matrix and operating frequency is obtained.The invention have the advantage that the frequency response curve of four constitutive parameters of magneto-electric coupled Meta Materials obtained, this is for analyzing and understanding that magneto-electric coupled Meta Materials electromagnetic property has important function.

Description

Magneto-electric coupled Meta Materials constitutive matrix method of acquiring

Technical field

The present invention relates to the method that magneto-electric coupled Meta Materials constitutive matrix obtains.

Background technology

At present, scattering parameter (S parameter) extracts the method for Meta Materials constitutive parameter because its calculating is simple and feasibility obtains To paying close attention to widely.S parameter method is the refraction that the scattering parameter obtained based on experiment or Electromagnetic Simulation calculates Meta Materials Rate n and normalized natural impedance z, and then extract the effective dielectric constant (ε of material_eff=n/z) and equivalent permeability (μ_eff= nz).If the electric resonance element in LHM and magnetic resonance element do not have cross-couplings or coupling the least in the case of, It is very effective that S parameter method extracts Meta Materials constitutive parameter.Due to magnetic resonance metamaterial (negative magnetoconductivity medium), electric resonance The material with negative refractive index that Meta Materials (negative permittivity medium) and line-split ring resonator are constituted do not has magneto-electric coupled or magnetic Electric coupling is the most weak, and therefore the extraction of constitutive parameter is typically with S parameter method.But the electric resonance element in Meta Materials and magnetic Resonant element often produces cross-couplings phenomenon, shows biisotropy or double anisotropic properties, the most only use etc. It is obvious that effect dielectric constant and equivalent permeability describe this magneto-electric coupled (magnetoelectric coupling) Meta Materials It is inappropriate, it is necessary to introducing constitutive matrix or constitutive tensor express the electromagnetic property of magneto-electric coupled Meta Materials.Therefore, use S parameter method obtains magneto-electric coupled Meta Materials constitutive parameter frequency response curve and naturally there are the biggest limitation.

Summary of the invention

The problem that present invention mainly solves is, the constitutive parameter obtained for existing S parameter method of acquiring can not be fully described The defect of magneto-electric coupled Meta Materials electromagnetic property, it is provided that one relatively can relatively accurate description magnetoelectricity in the presence of being applicable to coupling The method of the electromagnetic property of coupling Meta Materials.

The present invention solves the method that this problem be the technical scheme is that the electromagnetic property of magneto-electric coupled Meta Materials；Its It is characterised by:

In theory, by setting up electromagnetism Meta Materials coupling model, average electric flux density of deriving and magnetic flux density with power up outward Relation between magnetic field, obtains the expression formula of 2 × 2 constitutive matrixes based on the degree of coupling, then derives based on the degree of coupling Refractive index formula；

In emulation, build magnetic resonance and electric resonance Meta Materials coupling model unit, HFSS Yu MATLAB obtain refractive index The extraction of values of emulation experiment；In MATLAB carries out nonlinear fitting, bent by the theoretical value of observation refractive index real part and imaginary part The error of fitting value of line and extraction of values based on emulation experiment data is in error allowed band, and then proposition in proof theory Two-dimentional constitutive matrix expression formula based on the degree of coupling and the correctness of refractive index formula；

Finally, the series of parameters numerical value obtained in nonlinear fitting is substituted in 2 × 2 constitutive matrixes based on the degree of coupling Four physical quantitys in, obtain the relation curve of four physical quantitys of constitutive matrix and operating frequency.

The present invention is used for describe four physical quantitys that magneto-electric coupled Meta Materials electromagnetic property is constitutive matrix, is respectively ε_eff、ξ_eff、ζ_eff、μ_eff, four indispensable.

The present invention not only to obtain out refractive index, also to obtain four constitutive parameters, and complexity is higher, has certain elder generation The property entered；Four elements of constitutive matrix and refractive index formula not only comprise effective dielectric constant and Effective permeability, also comprises Cross coupling parameters and spatial dispersion item.

The present invention needs utilize non-linear fitting technique refractive index theoretical formula to be fitted, obtain the relevant of formula Parameter, then substitute into the constitutive matrix formula of acquisition, controlled the precision of relevant parameter by the precision controlling nonlinear fitting, enter And ensure the precision of extraction of values.

Four physical quantitys ε in magneto-electric coupled Meta Materials constitutive matrix are present invention determine that_eff、ξ_eff、ζ_eff、μ_effWith work frequency The relation of rate, and in order to reflect the implication of curve intuitively, by four physical quantitys divided by four constant ε₀、μ₀Process as vertical coordinate afterwards.

Implement technical scheme, there is beneficial effect: when describing magneto-electric coupled Meta Materials electromagnetic property, Not only need outside effective dielectric constant and two constitutive parameters of Effective permeability, in addition it is also necessary to reflect the effective basis of magneto-electric coupled two Structure parameter ξ_eff、ζ_effThe electromagnetic property of magneto-electric coupled Meta Materials could completely be described；It is magneto-electric coupled super that the present invention finally obtains The frequency response curve of four constitutive parameters of material, this is for analyzing and understanding that magneto-electric coupled Meta Materials electromagnetic property has important work With.

Accompanying drawing explanation

Fig. 1 is that electric resonance element HFSS of the present invention emulates geometric representation.

Fig. 2 is that magnetic resonance element HFSS of the present invention emulates geometric representation.

Fig. 3 is that magnetic of the present invention, electric resonance position of components HFSS emulate schematic diagram.

Fig. 4 is the present invention magneto-electric coupled metamaterial unit HFSS design of Simulation figure.

Fig. 5 is theoretical value and the extraction of values curve chart of the present invention magneto-electric coupled Meta Materials refractive index real part.

Fig. 6 is theoretical value and the extraction of values curve chart of the present invention magneto-electric coupled Meta Materials imaginary index.

Fig. 7 is constitutive matrix parameter ε of the present invention_eff/ε₀Frequency response chart.

Fig. 8 is constitutive matrix parameter of the present inventionFrequency response chart.

Fig. 9 is constitutive matrix parameter of the present inventionFrequency response chart.

Figure 10 is constitutive matrix parameter μ of the present invention_eff/μ₀Frequency response chart.

Figure 11 is invention acquisition methods flow chart.

Detailed description of the invention

1. theoretical derivation:

The Meta Materials formed in spatial arrangements by electric resonance element and magnetic resonance element, due to electric resonance element and magnetic resonance Cross-couplings (cross-couplings between electric field and magnetic field in other words) between element, this magneto-electric coupled Meta Materials shows as double Isotropic characteristics.The present invention is by setting up electromagnetism Meta Materials coupling model, and average electric flux density of deriving and magnetic flux density are with outer Adding the relation between electromagnetic field, the constitutive matrix element mathematic(al) representation obtaining magneto-electric coupled Meta Materials is

$\left(\begin{array}{c}D\\ B\end{array}\right)=\left(\begin{array}{cc}{\mathrm{\ϵ}}_{eff}& {\mathrm{\ξ}}_{eff}\\ {\mathrm{\ζ}}_{eff}& {\mathrm{\μ}}_{eff}\end{array}\right)\left(\begin{array}{c}E\\ H\end{array}\right)---\left(1\right)$

Constitutive matrix element ε in formula (1)_eff, ξ_eff, ζ_eff, μ_effIt is respectively

${\mathrm{\ϵ}}_{eff}=\frac{{\mathrm{\ϵ}}_{av}}{1+{\mathrm{\α}}^2{\mathrm{\ω}}^2{d}^2{\mathrm{\ϵ}}_{av}{\mathrm{\μ}}_{av}}---\left(2\right)$

${\mathrm{\ξ}}_{eff}=-\frac{{\mathrm{\α\ϵ}}_{av}{\mathrm{\μ}}_{av}\mathrm{\ω}d}{1+{\mathrm{\α}}^2{\mathrm{\ω}}^2{d}^2{\mathrm{\ϵ}}_{av}{\mathrm{\μ}}_{av}}---\left(3\right)$

${\mathrm{\ζ}}_{eff}=\frac{{\mathrm{\α\ωd\ϵ}}_{av}{\mathrm{\μ}}_{av}}{1+{\mathrm{\α}}^2{\mathrm{\ω}}^2{d}^2{\mathrm{\ϵ}}_{av}{\mathrm{\μ}}_{av}}---\left(4\right)$

${\mathrm{\μ}}_{eff}=\frac{{\mathrm{\μ}}_{av}}{1+{\mathrm{\α}}^2{\mathrm{\ω}}^2{d}^2{\mathrm{\ϵ}}_{av}{\mathrm{\μ}}_{av}}---\left(5\right)$

And then derive periodic magnetic electric coupling Meta Materials refractive index expression formula and be

$n_{eff}=\frac{2}{k_{0}d}arcsin\left(\frac{1}{\sqrt{{\mathrm{\ϵ}}_0{\mathrm{\μ}}_0}}\sqrt{\frac{{\mathrm{\ϵ}}_{av}{\mathrm{\μ}}_{av}}{1+{\mathrm{\α}}^2{\mathrm{\ϵ}}_{av}{\mathrm{\μ}}_{av}{\mathrm{\ω}}^2{d}^2}}\right(\frac{k_{0}d}{2}\left)\right)---\left(6\right)$

In above formulaH is the thickness of metamaterial unit.

Can be seen that periodic magnetic electric coupling Meta Materials refractive index and the magnetic permeability μ of its magnetic cell, electric device in the present invention DIELECTRIC CONSTANT ε, coefficient of coup α, spatial dispersion item k₀Functional relationship rather than the tradition electromagnetism reason of complexity is there is between d Opinion is thought, refractive index is the Effective permeability evolution with effective dielectric constant of material.Refractive index obtained by the present invention this Kind of complex mathematical relation will provide important theory to depend on for the aspect such as negative index characteristic of people's analysis of magnetic electric coupling material According to.

2. simulating, verifying:

Whether the theoretical formula that the present invention derives is applicable to magneto-electric coupled metamaterial unit, and we select the material of Fig. 1 Unit is verified.Fig. 1 is that electromagnetic coupled metamaterial structure unit designs example, and shown in (a), double annulus SRR are as magnetic resonance unit Part, its structural parameters r₁=0.65mm, r₂=0.85mm, r₃=0.9mm, r₄=1.1mm, g=0.5mm.Choose shown in (b) Single-groove road ELC resonator as electric resonance element, its structural parameters a=0.2mm, b=2.2mm, c=0.4mm, p=0.02mm, E=0.19mm, e are the distances ((c)) between electric device and magnetic cell；C electric device and the face at magnetic cell place in () are mutual Vertically, the conductor material of electric device and magnetic cell is copper, and its thickness is 0.017mm；D () is to use simulation software HFSS design The complete analogous diagram of electromagnetic coupled metamaterial unit, perfect condition is that surrounding fills air, a length of d=of air cartridge subcycle 3mm, the coordinate axes in (d) lower right corner show propagation and the polarised direction of ripple.

In present invention emulation, the dielectric constant Lorentz model equivalents of double slit gap ELC structure is

$\mathrm{\ϵ}={\mathrm{\ϵ}}_0({\mathrm{\ϵ}}_b-\frac{f_{pe}^2}{f^2-f_{0e}^2+{\mathrm{j\γ}}_{e}f})---\left(7\right)$

The equivalents of the Lorentz pcrmeability of magnetic resonance element is

$\mathrm{\μ}={\mathrm{\μ}}_0({\mathrm{\μ}}_b-\frac{f_{pm}^2}{f^2-f_{0m}^2+{\mathrm{j\γ}}_{m}f})---\left(8\right)$

The present invention utilizes high-frequency electromagnetic simulation software HFSS to emulate Fig. 1 (d) coupled structure unit, obtains required S parameter, experiment simulation parameter importing modeling software MATLAB is obtained carrying of Meta Materials refractive index of the present invention by recycling S parameter method Valued curve, then the present invention utilizes non-linear fitting technique to be fitted according to extraction of values real part, obtains Fig. 2.According to formula (2) real part of non-linear fitting technique refractive index of the present invention experiment extraction of values, is utilized to carry out with the real part of refractive index theoretical value Matching (red blue line in Fig. 2), it is determined that parameter ε_b、f_pe、f_0e、γ_e、μ_b、f_pm、f_0m、γ_m, h, α numerical value, see table 1.

The parameter values of table 1 theoretical formula matching

Fig. 2 is theoretical value and the extraction of values results contrast of the present invention magneto-electric coupled Meta Materials refractive index.Solid line is refractive index Real part and the extraction of values of imaginary part, dotted line is real part and the imaginary part of refractive index match value.The error of real part imaginary part is found by comparison The least, it is seen that the refractive index formula of the present invention used by matching is correct.

3. obtain the frequency response curve of constitutive matrix element

Relevant parameter obtained by refractive index matching of the present invention is substituted in theoretical formula (2)-(5) of constitutive matrix, To magnetic electric coupling of the present invention Meta Materials four physical quantitys ε of four constitutive parameters_eff、ξ_eff、ζ_eff、μ_effBent with the relation of operating frequency Line (Fig. 3), in order to reflect the implication of curve intuitively, by four physical quantitys divided by four constant ε₀、μ₀After Process as vertical coordinate.

Above in conjunction with accompanying drawing, the case study on implementation of the present invention is described, but constitutive matrix involved in the present invention obtains Access method, it is not limited to above-mentioned concrete magneto-electric coupled metamaterial unit, above-mentioned detailed description of the invention is only schematically Rather than restrictive, others skilled in the art person, under the enlightenment of the present invention, is wanting without departing from present inventive concept and right Ask under protected ambit, it is also possible to make the magneto-electric coupled metamaterial unit of a lot of other form, and use the present invention The extracting method related to, within these belong to the protection of the present invention.

Claims (6)

1. The most magneto-electric coupled Meta Materials constitutive matrix method of acquiring, it is characterised in that:

   In theory, by setting up electromagnetism Meta Materials coupling model, average electric flux density of deriving and magnetic flux density and additional electromagnetic field Between relation, obtain the expression formula of 2 × 2 constitutive matrixes based on the degree of coupling, then derive refraction based on the degree of coupling Rate formula；

   In emulation, build magnetic resonance and electric resonance Meta Materials coupling model unit, HFSS Yu MATLAB obtain refractive index emulation The extraction of values of experiment；In MATLAB carries out nonlinear fitting, by observation refractive index real part and imaginary part theoretical value curve with The error of fitting value of extraction of values based on emulation experiment data in error allowed band, and then in proof theory propose based on The two-dimentional constitutive matrix expression formula of the degree of coupling and the correctness of refractive index formula；

   Finally, the series of parameters numerical value obtained in nonlinear fitting is substituted into four in 2 × 2 constitutive matrixes based on the degree of coupling In individual physical quantity, obtain the relation curve of four physical quantitys of constitutive matrix and operating frequency.
2. Magneto-electric coupled Meta Materials constitutive matrix method of acquiring the most according to claim 1, is characterized in that: described method of acquiring is necessary The theoretical formula having constitutive matrix and refractive index does basis.
3. Magneto-electric coupled Meta Materials constitutive matrix method of acquiring the most according to claim 1, is characterized in that: it addition, not only to obtain Taking out refractive index, also to obtain four constitutive parameters, complexity is higher, has certain advance.
4. Magneto-electric coupled Meta Materials constitutive matrix method of acquiring the most according to claim 3, is characterized in that: four of constitutive matrix Element and refractive index formula not only comprise effective dielectric constant and Effective permeability, also comprises cross coupling parameters and space color Dissipate item.
5. Magneto-electric coupled Meta Materials constitutive matrix method of acquiring the most according to claim 1, is characterized in that: need to utilize non-linear Fitting technique refractive index theoretical formula is fitted, and obtains the relevant parameter of formula, then substitutes into the constitutive matrix formula of acquisition, Controlled the precision of relevant parameter by the precision controlling nonlinear fitting, and then ensure the precision of extraction of values.
6. Magneto-electric coupled Meta Materials constitutive matrix method of acquiring the most according to claim 1, is characterized in that: determine magneto-electric coupled Four physical quantitys ε in Meta Materials constitutive matrix_eff、ξ_eff、ζ_eff、μ_effWith the relation of operating frequency, and in order to reflect intuitively The implication of curve, by four physical quantitys divided by four constants μ₀Process as vertical coordinate afterwards.