CN103235888B - The method of a kind of accurate Calculation Bianisotropic medium ball electromagnetic scattering - Google Patents

Abstract

The present invention proposes the method for a kind of accurate Calculation Bianisotropic medium ball electromagnetic scattering.Step of the present invention is as follows: 1. the differential equation utilizing the eigen[value derivation magnetic induction density <b>B</bGreatT.Gr eaT.GT of passive maxwell equation group and two anisotropy medium; 2. the factor relevant with <b>B</bGreatT.Gr eaT.GT in the differential equation is expressed with the form of spherical vector wave functions, then spherical vector wave functions <b>M is utilized, the property of orthogonality of N</b> draws a matrix equation containing ginseng, the condition first utilizing matrix equation to meet untrivialo solution calculates the parameter of this matrix equation, then parameter generation is got back to the untrivialo solution obtaining matrix equation in the matrix equation of containing parameter; Does 3. the function that structure one is new, use new function again magnetic induction density <b>B is represented, </b> and then obtain the electromagnetic field of medium ball inside, then the electromagnetic field in medium ball and the incident electromagnetic field outside ball, scattering field are updated in boundary condition, draw scattering matrix.The present invention is applicable to the electromagnetic scattering solving the less Bianisotropic medium ball of electric size.

Description

A Method for Accurately Computing Electromagnetic Scattering of Double Anisotropic Dielectric Spheres

technical field

The invention belongs to the field of theoretical calculation of electromagnetic scattering, and in particular relates to a method for accurately calculating electromagnetic scattering of a bi-anisotropic medium sphere.

Background technique

The more traditional research method for solving electromagnetic scattering is the analytical method, which has always been the goal pursued by scientists. The so-called analytical research method is a method of closed-form mathematical solution, which is to directly find various mathematical equations derived from Maxwell's equations. Very effective for electromagnetic problems with certain boundary condition rules. Analytical solutions can provide more effective data for other numerical calculations, verify the correctness of numerical calculation results, and give clear physical concepts, so they have very important guiding significance.

For media such as plasma and ferrite under the action of a constant magnetic field, their electromagnetic properties need to use the tensor dielectric constant and tensor permeability to describe, that is , they have anisotropic properties, called anisotropic coal quality.

The research on the analytical solution of the electromagnetic scattering of the anisotropic medium sphere is widely carried out now, and the radar cross section of the sphere can also be calculated by using different analytical methods.

The doubly anisotropic medium makes the cross-coupling between the provided electric field and magnetic field difficult to solve analytically for such a medium sphere. Therefore, there are few researches on the analytical solutions of electromagnetic scattering by bi-anisotropic dielectric spheres.

Contents of the invention

The purpose of the present invention is to address the deficiencies of the prior art, to propose a method for accurately calculating the electromagnetic scattering of a bi-anisotropic medium sphere, and to prove that the spherical vector wave function is suitable for a bi-anisotropic medium.

The technical solution adopted by the present invention to solve its technical problems is as follows:

Step 1. Utilize the passive Maxwell's equations and the eigenequation of the double anisotropic medium to derive the differential equation about the magnetic induction B ;

Step 2. Express the factors related to B in the differential equation in the form of spherical vector wave functions, and then use the orthogonal properties of spherical vector wave functions M and N to obtain a matrix equation with parameters. First, use the matrix equation to satisfy non The condition of zero solution calculates the parameters of the matrix equation, and then substitutes the parameters back into the matrix equation with parameters to obtain the non-zero solution of the matrix equation;

Step 3. Construct a new function, use the new function Re-express the magnetic induction intensity B, and then obtain the electromagnetic field inside the dielectric sphere, and then substitute the electromagnetic field inside the dielectric sphere, the incident electromagnetic field outside the sphere, and the scattered electromagnetic field into the boundary conditions to obtain the scattering matrix.

As described in step 1, add an item to the eigenequation of the anisotropic medium to become a double anisotropic medium, and the eigenequation of the double anisotropic medium is as follows:

(1)

Among them, the electric displacement vector D , the electric field strength E , the magnetic field strength H and the magnetic induction strength B are all vectors, and the vectors are represented in black bold; represent imaginary units; It is a parameter used to measure the electromagnetic characteristics of the medium;

Passive Maxwell's equations are as follows:

(2a)

(2b)

(2c)

(2d)

Substituting Equation 1 into Equations 2a, 2b, 2c, and 2d, the differential equation of magnetic induction B is derived as follows:

(3)

Among them, the symbol ▽× represents the curl of a vector; ω is the frequency of the electromagnetic wave; for inverse of ;

As described in step 2, the formula 3 Written in the form of a spherical vector wave function, the details are as follows:

(4)

(5)

(6)

in, and The same means the spherical vector wave function, the superscript (1) means that the vector wave function is composed of the first kind of spherical Bessel function, the subscript Represents the parameters in the spherical vector wave function; represents a pending quantity, Represents a vector in the spherical coordinate system; the coefficients in front of the wave function of the spherical vector are determined by the tensor in the eigenequation of the medium, and , Indicates the field strength of the incident electric field.

The coefficients in front of the vector wave function are defined as follows:

, (7)

, (8)

(9)

(10)

(11)

(12)

(13)

in: ,when hour, ;when hour, , ;

, (14)

, (15)

(16)

, (17)

(18)

Among them, in the present invention, Indicates the same quantity, and the difference in subscripts is used to distinguish between different expressions; similarly , , Both represent the same quantity.

Substitute equations 4, 5, and 6 into equation 3 to get:

(19)

Using the properties of the spherical vector wave function, we can get:

Expressed in matrix form as follows:

(20)

Equation 20 transforms into the following form:

(twenty one)

Among them , I is the identity matrix, and the meaning expressed in formula 21 is: there is such a parameter k that the equation has a non-zero solution, and the matrix knowledge only needs to make the determinant of formula 21 zero to solve the parameter ,parameter recorded as , then use Substitute into equation 20 to find the solution that the equation is not zero ,remember ,

Construct a new vector function in the described step 3 ,details as follows:

in is an undetermined coefficient, which is determined by the boundary conditions on the surface of the medium sphere;

make (twenty two)

(twenty three)

(twenty four)

In formulas 22, 23, and 24, the extra subscript l in front of the coefficients of the vector wave function is to convert the solution Caused by substituting into formulas 7, 8, 9, and 16; , , are non-zero numbers, and only become zero when curling them.

The parameters are as follows:

, ;

The scattered field and the incident field outside the sphere are respectively defined as: E _I , H _I and Es , Hs , the expressions are as follows (see: ZF Lin and S.T. Chui. "Electromagneticscatteringbyopticallyanisotropicmagneticparticle."PhysicalReviewE,vol.69,pp.056624-2- 056624-24, 2004)

(25)

(26)

(27)

(28)

in, , is the dielectric constant in vacuum, is the magnetic permeability in vacuum; Indicates the incident wave direction, polarization characteristics, etc.; the superscript (3) in Equations 27 and 28 indicates that the spherical vector wave function is composed of Bessel functions of the third kind; both the internal medium and the external medium of the sphere are ideal media, so There is no surface charge and surface current on the surface of the sphere, so the tangential components of the electric field and magnetic field at any point on the surface of the sphere are continuous, that is:

(29)

(30)

Substituting formulas 23-28 into the above formulas 29 and 30 to simplify:

in, , is the radius of the sphere; , is a spherical Bessel function, is a spherical Hankel function of the first kind. express yes Find the derivative, similarly .

Write the above formula in matrix form:

(31)

(32)

Solving equations 31 and 32 gives:

The radar cross section (radar cross section. RCS) is used to characterize the characteristics of the scattering rate of the reflected radar wave of the target, and it is used as the most basic parameter for evaluating the electromagnetic scattering characteristics of the target. The details are as follows (see: Z.F.LinandS.T.Chui. vol.69, pp.056624-2-056624-24, 2004):

(33)

(34)

(35)

Substitute Equation 27 into Equation 35 to get:

(36)

The beneficial effects of the present invention are as follows:

Based on the spherical vector wave function, an analytical solution for the electromagnetic scattering of a doubly anisotropic dielectric sphere is proposed. The calculation result is given for the first time. It is necessary to solve the determinant of the matrix equation with parameters, solve the parameters from the determinant, and then obtain the non-zero solution of the matrix equation. Since a high-order matrix is needed to calculate a sphere with a large electrical size, the computer is not easy to calculate the parameters of the high-order matrix equation containing parameters. Therefore, this method is more suitable for solving the electromagnetic scattering of bi-anisotropic dielectric spheres with small electric size.

Description of drawings

Fig. 1 is the corresponding relationship between the radar scattering cross section and the scattering angle of the dielectric sphere given in Embodiment 1 of the present invention

Fig. 2 is the corresponding relationship between the radar scattering cross section and the scattering angle of the dielectric sphere given in Embodiment 2 of the present invention

Figure 3 is the corresponding relationship between the radar scattering cross section and the scattering angle of the dielectric sphere given in Embodiment 3 of the present invention Figure 4 is the corresponding relationship between the radar scattering cross section and the scattering angle of the dielectric sphere given in Embodiment 4 of the present invention

Fig. 5 is that the main parameters of embodiment 5 of the present invention are the same as in example 4, research The influence of the change on the radar cross section of the E surface:

Fig. 6 is that the main parameters of embodiment 6 of the present invention are the same as in example 4, research The influence of the change on the radar cross section of the H surface:

detailed description

The present invention will be further described below in conjunction with drawings and embodiments.

A method for accurately calculating electromagnetic scattering of a double anisotropic dielectric sphere, comprising the following steps:

Assign values to each tensor of the medium eigenequation:

Calculated by fortran , and output these two matrices to MATLAB

Calculate determinant in MATLAB

. Obtain an equation about k, and solve the unknown quantity k of this equation. The obtained solution is denoted as ,Will Substitute into the equation to find . then pass and calculate .

To sum up, the inside of the dielectric sphere can be calculated, and the incident field is known, the incident field and the scattering field can be calculated through the boundary conditions to calculate the scattering field of the dielectric sphere, and then the radar cross section (RCS) can be obtained by the scattering field

.

As shown in Figure 1, the figure shows the correspondence between the radar scattering cross section and the scattering angle of the dielectric sphere for Embodiment 1 of the present invention, and its parameters are , , , , , , The compared data comes from the literature (You-Lin Geng. "Scattering of a plane wave by ananisotropic ferrite-coated conductingsphere" IET Microw. Antennas Propag, 2008, 2, (2), pp.158-162.).

As shown in Figure 2, the figure shows the correspondence between the radar cross section of the dielectric sphere and the scattering angle for Embodiment 2 of the present invention, and its parameters are , , , , .

As shown in Figure 3, the figure shows the corresponding relationship between the radar scattering cross section and the scattering angle of the dielectric sphere for Embodiment 3 of the present invention, and its parameters are .

As shown in Figure 4, the figure shows the corresponding relationship between the radar scattering cross section and the scattering angle of the dielectric sphere for Embodiment 4 of the present invention, and its parameters are .

As shown in Figure 5, the main parameters of embodiment 5 of the present invention are the same as in example 4 among the figure, research The influence of the change on the radar cross section of the E surface:

As shown in Figure 6, the main parameters of embodiment 6 of the present invention are the same as in example 4 among the figure, research The influence of the change on the H-surface radar cross section.

Claims (1)

1. A method for accurately calculating the electromagnetic scattering of a bi-anisotropic medium ball is characterized by comprising the following steps:

step 1, deducing a differential equation related to magnetic induction intensity B by utilizing a passive Maxwell equation set and an intrinsic equation of a double-anisotropic medium;

step 2, expressing factors related to B in the differential equation in a spherical vector wave function mode, then obtaining a matrix equation containing parameters by utilizing the orthogonal property of the spherical vector wave functions M and N, firstly calculating the parameters of the matrix equation by utilizing the condition that the matrix equation meets the non-zero solution, and then replacing the parameters back to the matrix equation containing the parameters to obtain the non-zero solution of the matrix equation;

step 3, constructing a new function, and using the new function V_lRepresenting the magnetic induction intensity B again, further solving an electromagnetic field inside the medium ball, and then substituting the electromagnetic field inside the medium ball, an incident electromagnetic field outside the medium ball and a scattering electromagnetic field into boundary conditions to obtain a scattering matrix;

in the step 1, one term is added to the intrinsic equation of the anisotropic medium to be changed into the double anisotropic medium, wherein the intrinsic equation of the double anisotropic medium is as follows:

$\begin{array}{cc}D=\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}\&CenterDot;E+\stackrel{\&OverBar;}{\mathrm{\&xi;}}\&CenterDot;H& B=\stackrel{\&OverBar;}{\mathrm{\&mu;}}\&CenterDot;H---\end{array}\left(1\right)$

$\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}={\mathrm{\&epsiv;}}_s\left[\begin{array}{ccc}{\mathrm{\&epsiv;}}_t& -{\mathrm{i\&epsiv;}}_g& 0\\ {\mathrm{i\&epsiv;}}_g& {\mathrm{\&epsiv;}}_t& 0\\ 0& 0& 1\end{array}\right],\stackrel{\&OverBar;}{\mathrm{\&xi;}}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& \mathrm{\&xi;}\end{array}\right],\stackrel{\&OverBar;}{\mathrm{\&mu;}}={\mathrm{\&mu;}}_s\left[\begin{array}{ccc}{\mathrm{\&mu;}}_t& -{\mathrm{i\&mu;}}_g& 0\\ {\mathrm{i\&mu;}}_g& {\mathrm{\&mu;}}_t& 0\\ 0& 0& 1\end{array}\right]$

wherein, the electric displacement vector D, the electric field intensity E, the magnetic field intensity H and the magnetic induction intensity B are vectors, and the vectors are represented by black bold; i represents an imaginary unit;_s,_t,_g,μ_s,μ_t,μ_gis a parameter for measuring the electromagnetic property of the medium;

the passive maxwell equations are specifically as follows:

$\&dtri;\&times;E=i\mathrm{\&omega;}B---\left(2a\right)$

$\&dtri;\&times;H=-i\mathrm{\&omega;}D---\left(2b\right)$

$\&dtri;\&CenterDot;B=0---\left(2c\right)$

$\&dtri;\&CenterDot;D=0---\left(2d\right)$

substituting equation 1 into equations 2a, 2B, 2c, and 2d, the differential equation for deriving the magnetic induction B is as follows:

$\&dtri;\&times;\&lsqb;{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}^{-1}{\mathrm{\&epsiv;}}_s\&CenterDot;(\&dtri;\&times;{\mathrm{\&mu;}}_s{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}^{-1}\&CenterDot;B)\&rsqb;+i\mathrm{\&omega;}\&dtri;\&times;\&lsqb;\stackrel{\&OverBar;}{\mathrm{\&xi;}}\&CenterDot;B\&rsqb;-k_s^{2}B=0---\left(3\right)$

wherein, the symbol×, calculating the rotation of a vector, wherein omega is the frequency of the electromagnetic wave;is composed ofThe inverse of (1); $k_s^2={\mathrm{\&omega;}}^2{\mathrm{\&epsiv;}}_s{\mathrm{\&mu;}}_s;$

in the step 2, the compound shown in the formula 3 B is written in the form of a spherical vector wave function, as follows:

$B=\underset{n=1}{\overset{+\mathrm{\&infin;}}{\&Sigma;}}\underset{m=-n}{\overset{+n}{\&Sigma;}}{\stackrel{\&OverBar;}{E}}_{mn}\&lsqb;d_{mn}{M}_{mn}^{\left(1\right)}(k,r)+c_{mn}{N}_{mn}^{\left(1\right)}(k,r)\&rsqb;---\left(4\right)$

${\mathrm{\&epsiv;}}_s{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}^{-1}\&CenterDot;(\&dtri;\&times;{\mathrm{\&mu;}}_s{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}^{-1}\&CenterDot;B)=k\underset{n=0}{\overset{+\mathrm{\&infin;}}{\&Sigma;}}\underset{m=-n}{\overset{+n}{\&Sigma;}}{\stackrel{\&OverBar;}{E}}_{mn}({\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{c}}}_{mn}{M}_{mn}^{\left(1\right)}+{\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{d}}}_{mn}{N}_{mn}^{\left(1\right)}+{\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{w}}}_{mn}{L}_{mn}^{\left(1\right)})---\left(5\right)$

$\stackrel{\&OverBar;}{\mathrm{\&xi;}}\&CenterDot;B=\underset{n=0}{\overset{+\mathrm{\&infin;}}{\&Sigma;}}\underset{m=-n}{\overset{+n}{\&Sigma;}}{\stackrel{\&OverBar;}{E}}_{mn}({\hat{d}}_{mn}{M}_{mn}^{\left(1\right)}+{\hat{c}}_{mn}{N}_{mn}^{\left(1\right)}+{\hat{w}}_{mn}{L}_{mn}^{\left(1\right)})---\left(6\right)$

(4-6) the expansion coefficients in the formulas are respectively:

$\begin{array}{cc}{\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{d}}}_{mn}=\underset{q,p}{\mathrm{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{pq}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\stackrel{\&OverBar;}{p}}_{mn}^{pq}{\stackrel{\&OverBar;}{d}}_{pq}+{\tilde{p}}_{mn}^{pq}{\stackrel{\&OverBar;}{c}}_{pq}),& {\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{c}}}_{mn}=\underset{q,p}{\mathrm{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{pq}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\stackrel{\&OverBar;}{O}}_{mn}^{pq}{\stackrel{\&OverBar;}{d}}_{pq}+{\tilde{O}}_{mn}^{pq}{\stackrel{\&OverBar;}{c}}_{pq})\end{array}---\left(7\right)$

$\begin{array}{cc}{\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{w}}}_{mn}=\underset{q,p}{\mathrm{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{pq}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\stackrel{\&OverBar;}{q}}_{mn}^{pq}{\stackrel{\&OverBar;}{d}}_{pq}+{\tilde{q}}_{mn}^{pq}{\stackrel{\&OverBar;}{c}}_{pq}),& {\stackrel{\&OverBar;}{d}}_{mn}=\underset{v=1}{\overset{+\mathrm{\&infin;}}{\&Sigma;}}\underset{u=-v}{\overset{+v}{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\tilde{g}}_{mn}^{uv}{d}_{uv}+{\stackrel{\&OverBar;}{g}}_{mn}^{uv}{c}_{uv})\end{array}---\left(8\right)$

${\stackrel{\&OverBar;}{c}}_{mn}=\underset{v=1}{\overset{+\mathrm{\&infin;}}{\&Sigma;}}\underset{u=-v}{\overset{+v}{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\tilde{e}}_{mn}^{uv}{d}_{uv}+{\stackrel{\&OverBar;}{e}}_{mn}^{uv}{c}_{uv})---\left(9\right)$

${\tilde{g}}_{pq}^{mn}={\mathrm{\&sigma;}}_{nq}{\mathrm{\&sigma;}}_{mp}+\&lsqb;\frac{(n^2+n-m^2){\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;}+{\mathrm{m\&mu;}}_g^{\&prime;}}{n(n+1)}\&rsqb;{\mathrm{\&sigma;}}_{nq}{\mathrm{\&sigma;}}_{mp}---\left(10\right)$

${\tilde{e}}_{pq}^{mn}=\frac{i(n+m)\&lsqb;m{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;}-(n+1){\mathrm{\&mu;}}_g^{\&prime;}\&rsqb;{\mathrm{\&sigma;}}_{n-1,q}{\mathrm{\&sigma;}}_{mp}}{n(2n+1)}+\frac{i(n-m+1)\&lsqb;m{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;}+{\mathrm{n\&mu;}}_g^{\&prime;}\&rsqb;{\mathrm{\&sigma;}}_{n+1,q}{\mathrm{\&sigma;}}_{mp}}{(n+1)(2n+1)}---\left(11\right)$

${\stackrel{\&OverBar;}{g}}_{pq}^{mn}=-\frac{i(n+m)(n+1)\&lsqb;m{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;}+(n-1){\mathrm{\&mu;}}_g^{\&prime;}\&rsqb;{\mathrm{\&sigma;}}_{n-1,q}{\mathrm{\&sigma;}}_{mp}}{n(n-1)(2n-1)}-\frac{in(n-m+1)\&lsqb;m{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;}-(n+2){\mathrm{\&mu;}}_g^{\&prime;}\&rsqb;{\mathrm{\&sigma;}}_{n+1,q}{\mathrm{\&sigma;}}_{mp}}{(n+1)(n+2)(2n+1)}---\left(12\right)$

$\begin{array}{c}{\stackrel{\&OverBar;}{e}}_{pq}^{mn}=\&lsqb;1+\frac{(4n^2+4n-3){\mathrm{m\&mu;}}_g^{\&prime;}}{n(n+1)(2n-1)(2n+3)}\&rsqb;{\mathrm{\&sigma;}}_{nq}{\mathrm{\&sigma;}}_{mp}+\frac{\&lsqb;(2n^2+2n+3)m^2+(2n^2+2n-3)n(n+1)\&rsqb;{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;}}{n(n+1)(2n-1)(2n+3)}{\mathrm{\&sigma;}}_{nq}{\mathrm{\&sigma;}}_{mp}\\ -\frac{(n+1)(n+m)(n+m-1)}{(n-1)(2n-1)(2n+1)}{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;}{\mathrm{\&sigma;}}_{n-2,q}{\mathrm{\&sigma;}}_{mp}-\frac{n(n-m+2)(n-m+1)}{(n+2)(2n+1)(2n+3)}{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;}{\mathrm{\&sigma;}}_{n+2,q}{\mathrm{\&sigma;}}_{mp}\end{array}---\left(13\right)$

wherein:when n is equal to q, the compound is,_nq1 is ═ 1; when n ≠ q, it is,_nq＝0，

${\mathrm{\&mu;}}_t^{\&prime;}=\frac{{\mathrm{\&mu;}}_t}{{\mathrm{\&mu;}}_t^2-{\mathrm{\&mu;}}_g^2},{\mathrm{\&mu;}}_g^{\&prime;}=-\frac{{\mathrm{\&mu;}}_g}{{\mathrm{\&mu;}}_t^2-{\mathrm{\&mu;}}_g^2},{\mathrm{\&epsiv;}}_t^{\&prime;}=\frac{{\mathrm{\&epsiv;}}_t}{{\mathrm{\&epsiv;}}_t^2-{\mathrm{\&epsiv;}}_g^2},{\mathrm{\&epsiv;}}_g^{\&prime;}=-\frac{{\mathrm{\&epsiv;}}_g}{{\mathrm{\&epsiv;}}_t^2-{\mathrm{\&epsiv;}}_g^2}$

$\begin{array}{cc}{\stackrel{\&OverBar;}{p}}_{pq}^{mn}={\stackrel{\&OverBar;}{e}}_{pq}^{mn}{|}_{{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_t^{\&prime;}={\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;},{\mathrm{\&epsiv;}}_g={\mathrm{\&mu;}}_g},& {\tilde{p}}_{pq}^{mn}={\tilde{e}}_{pq}^{mn}{|}_{{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_t^{\&prime;}={\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;},{\mathrm{\&epsiv;}}_g={\mathrm{\&mu;}}_g}\end{array}---\left(14\right)$

$\begin{array}{cc}{\stackrel{\&OverBar;}{O}}_{pq}^{mn}={\stackrel{\&OverBar;}{g}}_{pq}^{mn}{|}_{{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_t^{\&prime;}={\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;},{\mathrm{\&epsiv;}}_g={\mathrm{\&mu;}}_g},& {\tilde{O}}_{pq}^{mn}={\tilde{g}}_{pq}^{mn}{|}_{{\stackrel{\&OverBar;}{\mathrm{\&epsiv;}}}_t^{\&prime;}={\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_t^{\&prime;},{\mathrm{\&epsiv;}}_g={\mathrm{\&mu;}}_g}\end{array}---\left(15\right)$

${\hat{d}}_{mn}=\underset{v=1}{\overset{+\mathrm{\&infin;}}{\&Sigma;}}\underset{u=-v}{\overset{+v}{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\tilde{r}}_{mn}^{uv}{d}_{uv}+{\stackrel{\&OverBar;}{r}}_{mn}^{uv}{c}_{uv}),{\hat{c}}_{mn}=\underset{v=1}{\overset{+\mathrm{\&infin;}}{\&Sigma;}}\underset{u=-v}{\overset{+v}{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\tilde{s}}_{mn}^{uv}{d}_{uv}+{\stackrel{\&OverBar;}{s}}_{mn}^{uv}{c}_{uv})---\left(16\right)$

$\begin{array}{cc}{\tilde{r}}_{pq}^{mn}={\tilde{g}}_{pq}^{mn}\&CenterDot;\mathrm{\&xi;}{|}_{{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_r^{\&prime;}=-1,{\mathrm{\&mu;}}_k^{\&prime;}=0},& {\tilde{s}}_{pq}^{mn}={\tilde{e}}_{pq}^{mn}\&CenterDot;\mathrm{\&xi;}{|}_{{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_r^{\&prime;}=-1,{\mathrm{\&mu;}}_k^{\&prime;}=0}\end{array}---\left(17\right)$

${\stackrel{\&OverBar;}{r}}_{pq}^{mn}={\stackrel{\&OverBar;}{g}}_{pq}^{mn}\&CenterDot;\mathrm{\&xi;}{|}_{{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_r^{\&prime;}=-1,{\mathrm{\&mu;}}_k^{\&prime;}=0},{\stackrel{\&OverBar;}{s}}_{pq}^{mn}={\stackrel{\&OverBar;}{e}}_{pq}^{mn}\&CenterDot;\mathrm{\&xi;}{|}_{{\stackrel{\&OverBar;}{\mathrm{\&mu;}}}_r^{\&prime;}=-1,{\mathrm{\&mu;}}_k^{\&prime;}=0}---\left(18\right)$

wherein,andthe same expression is shown, namely a spherical vector wave function, an upper mark (1) shows that the vector wave function is composed of a first type of spherical Bessel function, and a lower mark mn shows parameters in the spherical vector wave function; (4) in the formula d_mn、c_mnThe expansion coefficient of the magnetic flux density B in the bi-anisotropic medium is a quantity to be solved, k is a quantity to be quantified, and r represents a vector in a spherical coordinate system; the coefficient in front of the spherical vector wave function is determined by the eigen equation of the mediumIs determined by the tensor of (1), andE₀representing the field strength of the incident electric field;

$C_{mn}={\&lsqb;\frac{2n+1}{n(n+1)}\frac{(n-m)!}{(n+m)!}\&rsqb;}^{\frac{1}{2}}$

the properties of the spherical vector wave function are utilized to obtain:

$k^2{\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{d}}}_{mn}-k_s^2{d}_{mn}+i\mathrm{\&omega;}\hat{c}=0$

$k^2{\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{c}}}_{mn}-k_s^2{c}_{mn}+i\mathrm{\&omega;}\hat{d}=0$

expressed in matrix form as follows:

$\left[\begin{array}{cc}{k}^2& 0\\ 0& k^2\end{array}\right]\left[\begin{array}{c}\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{d}}\\ \stackrel{\&OverBar;}{\stackrel{\&OverBar;}{c}}\end{array}\right]+\left[\begin{array}{cc}0& i\mathrm{\&omega;}k\\ i\mathrm{\&omega;}k& 0\end{array}\right]\left[\begin{array}{c}\hat{d}\\ \hat{c}\end{array}\right]-k_s^2\left[\begin{array}{c}d\\ c\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]---\left(20\right)$

wherein,i is an identity matrix;to representOther similarities; d, c is expressed as a waiting quantity d_mn,c_mnM, n, u, v, p, q represent integers;the expression is as follows,

$\begin{array}{cc}\left[\begin{array}{c}\hat{d}\\ \hat{c}\end{array}\right]=\left[\begin{array}{cc}\tilde{R}& \stackrel{\&OverBar;}{R}\\ \tilde{S}& \stackrel{\&OverBar;}{S}\end{array}\right]\left[\begin{array}{c}d\\ c\end{array}\right]& \left[\begin{array}{c}\stackrel{\&OverBar;}{\stackrel{\&OverBar;}{d}}\\ \stackrel{\&OverBar;}{\stackrel{\&OverBar;}{c}}\end{array}\right]=\left[\begin{array}{cc}\tilde{p}& \stackrel{\&OverBar;}{p}\\ \widetilde{\mathrm{\&theta;}}& \stackrel{\&OverBar;}{\mathrm{\&theta;}}\end{array}\right]\left[\begin{array}{c}d\\ c\end{array}\right]\end{array}$

$\begin{array}{cc}{\tilde{R}}_{mn,uv}=\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}{\tilde{r}}_{mn}^{uv}& {\tilde{S}}_{mn,uv}=\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}{\tilde{S}}_{mn}^{uv}\end{array}$

$\begin{array}{cc}{\stackrel{\&OverBar;}{R}}_{mn,uv}=\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}{\stackrel{\&OverBar;}{r}}_{mn}^{uv}& {\stackrel{\&OverBar;}{S}}_{mn,uv}=\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}{\stackrel{\&OverBar;}{s}}_{mn}^{uv}\end{array}$

${\tilde{p}}_{mn,uv}=\underset{p,q}{\mathrm{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\stackrel{\&OverBar;}{p}}_{mn}^{pq}{\tilde{g}}_{pq}^{uv}+{\tilde{p}}_{mn}^{pq}{\tilde{e}}_{pq}^{uv}),{\stackrel{\&OverBar;}{p}}_{mn,uv}=\underset{p,q}{\mathrm{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\stackrel{\&OverBar;}{p}}_{mn}^{pq}{\stackrel{\&OverBar;}{g}}_{pq}^{uv}+{\tilde{p}}_{mn}^{pq}{\stackrel{\&OverBar;}{e}}_{pq}^{uv})$

${\widetilde{\mathrm{\&theta;}}}_{mn,uv}=\underset{p,q}{\mathrm{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\stackrel{\&OverBar;}{o}}_{mn}^{pq}{\tilde{g}}_{pq}^{uv}+{\tilde{o}}_{mn}^{pq}{\tilde{e}}_{pq}^{uv}),{\widetilde{\mathrm{\&theta;}}}_{mn,uv}=\underset{p,q}{\mathrm{\&Sigma;}}\frac{{\stackrel{\&OverBar;}{E}}_{uv}}{{\stackrel{\&OverBar;}{E}}_{mn}}({\stackrel{\&OverBar;}{o}}_{mn}^{pq}{\stackrel{\&OverBar;}{g}}_{pq}^{uv}+{\tilde{o}}_{mn}^{pq}{\stackrel{\&OverBar;}{e}}_{pq}^{uv})$

formula 20 transforms into the following form:

$(\left[\begin{array}{cc}{k}^2& 0\\ 0& k^2\end{array}\right]\left[\begin{array}{cc}\tilde{p}& \stackrel{\&OverBar;}{p}\\ \widetilde{\mathrm{\&theta;}}& \stackrel{\&OverBar;}{\mathrm{\&theta;}}\end{array}\right]+\left[\begin{array}{cc}0& i\mathrm{\&omega;}k\\ i\mathrm{\&omega;}k& 0\end{array}\right]\left[\begin{array}{cc}\tilde{R}& \stackrel{\&OverBar;}{R}\\ \tilde{S}& \stackrel{\&OverBar;}{S}\end{array}\right]-k_s^{2}I)\left[\begin{array}{c}d\\ c\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]---\left(21\right)$

the meaning of expression of formula (21) is: the existence of the parameter k leads the equation to have non-zero solution, only the determinant of the formula (21) needs to be zero through the knowledge of the matrix, the parameter k is solved, and the parameter k is recorded as k_l(l ═ 1,2,3 …), and k again_lSolving for a solution [ d ] for non-zero equations by substituting 21_mn,lc_mn,l]^-1，

Constructing a new function V in the step 3_lThe method comprises the following specific steps:

$V_l=-\frac{k_l}{\mathrm{\&omega;}}\underset{n=1}{\overset{+\mathrm{\&infin;}}{\&Sigma;}}\underset{m=-n}{\overset{+n}{\&Sigma;}}{\stackrel{\&OverBar;}{E}}_{mn}\&lsqb;d_{mn,l}{M}_{mn}^{\left(1\right)}(k_l,r)+c_{mn,l}{N}_{mn}^{\left(1\right)}(k_l,r)\&rsqb;$

α therein_lThe undetermined coefficient is determined by the boundary condition of the surface of the medium sphere;

order toWherein a is_lIs represented by V_lThe weight of (c); correspondingly obtaining the magnetic field inside the ballElectric fieldIncident electric field E outside the ball_IMagnetic field H_IScattering electric field E_sMagnetic field H_sSubstituting into the following boundary conditions:

[E_I+E_s]×e_r＝E_l×e_r[H_I+H_s]×e_r＝H_l×e_r

wherein e_rIs the direction vector of electromagnetic wave propagation;

and simplifying and sorting to obtain a scattering matrix (21) formula, so that the radar scattering cross section is calculated.