CN112287516B - Method and system for calculating electromagnetic scattering field of vortex beam in Debye dispersive plasma sphere - Google Patents

Abstract

The invention belongs to the field of vortex electromagnetic wave scattering field simulation calculation, and discloses a method and system for calculating a vortex beam electromagnetic scattering field of a Debye dispersion plasma sphere. Spherical vector wave function expansion of Hermit-Gaussian beam vector potential; (2) Spherical vector wave function expansion of Laguerre-Gaussian vortex beam; (3) Mie scattering coefficient of dispersive plasma sphere; (4) Laguerre-Gaussian vortex beam expansion Spherical vector wave function expansions of Gal-Gaussian vortex beam inner and scattered fields; (5) Combining boundary conditions and Mie scattering coefficients to obtain scattering field expansion coefficients; (6) Obtaining scattering field expansion coefficients and scattering phase functions Far-field radar scattering cross section; (7) Change the frequency of the vortex wave and repeat (1) to (6) to analyze the dispersion effect of the plasma sphere. The invention obtains the scattering cross section of the dispersive plasma sphere through the analytical solution of the generalized Lorentz Mie of the electromagnetic scattering field of the vortex beam, which fills the gap in the industry.

Description

Method and system for calculating electromagnetic scattering field of vortex beam in Debye dispersive plasma sphere

technical field

The invention belongs to the technical field of vortex electromagnetic wave detection, and in particular relates to a method and system for calculating a vortex beam electromagnetic scattering field of a Debye dispersion plasma sphere.

Background technique

During the re-entry of a hypersonic vehicle in near space, it will rub violently with the surrounding atmosphere, causing the temperature and pressure around the vehicle to rise. Under high temperature and high pressure, complex chemical reactions will occur between different air components, and ionization will occur, and then a plasma flow composed of free electrons, various ions and neutral particles will be formed around the aircraft. Plasma on the surface of the target seriously affects the radar's early warning and detection of flying targets, as well as the quality of communication between the monitoring station and the aircraft, and even leads to the "black barrier" phenomenon in communication. The existence of plasma on the target surface seriously restricts the near space. The development of aircraft. In addition, plasma widely exists in space ionosphere, microwave laboratory and other environments, and is also widely used in stealth technology of environmental targets, so it is of great significance to study the electromagnetic scattering characteristics of plasma targets.

The previous studies on the electromagnetic scattering characteristics of plasma targets are mostly based on the incident plane wave, and there are few reports that the incident wave type is vortex electromagnetic wave. The study of vortex beams originated in the field of optics, and the main research is still in the field of optics. Vortex electromagnetic waves are electromagnetic waves that carry orbital angular momentum. Compared with traditional electromagnetic waves, they have a higher degree of freedom of information modulation due to their helical phase wavefront distribution. great application potential. Under the irradiation of vortex electromagnetic waves, the radar beam forms radiation field excitations with different distributions at different targets, and the target scattering characteristics are quite different from those under the traditional plane wave irradiation. The eigenstates of different orbital angular momentum are orthogonal to each other, and can theoretically be expanded into an infinite-dimensional Hilbert space. Each orbital angular momentum eigenstate is represented by a specific integer, namely the topological charge, and any other state can be represented as a linear superposition of eigenstates. This property shows that vortex electromagnetic waves with different mode numbers can be physically separated, which lays a physical and mathematical foundation for their applications in wireless communication, radar detection and other fields. In addition, the traditional plane electromagnetic wave has only one intensity peak and the wavefront distribution is smooth, and there is no angular phase change. Vortex electromagnetic waves are significantly different from traditional electromagnetic waves because they carry orbital angular momentum. First, the beam intensity distribution of the vortex electromagnetic wave is a special hollow annular distribution, and there is a phase singularity with zero intensity on its transmission axis. Secondly, the vortex electromagnetic wave carries a new space degree of freedom—orbital angular momentum, which makes it have great application prospects in the fields of large-capacity communication and high-performance radar detection.

The research on the scattering characteristics of plasma targets irradiated by vortex electromagnetic waves is expected to solve the difficult problem of radar detection of hypersonic aircraft targets, and provide a new option for the anti-stealth research of stealth aircraft. Previous studies have studied the scattering characteristics of vortex electromagnetic beams to plasma spheres, but the dispersion characteristics of plasma spheres themselves have not been considered. Based on the Debye dispersion model, the invention provides a method for calculating the scattering cross section of the dispersion plasma sphere radar, which fills the gap in the industry.

SUMMARY OF THE INVENTION

Aiming at the problems existing in the prior art, the present invention provides a method and system for calculating the electromagnetic scattering field of the vortex beam of the Debye dispersion plasma sphere.

The present invention is realized like this, a kind of vortex beam electromagnetic scattered field calculation method of Debye dispersion plasma sphere, described vortex beam electromagnetic scattered field calculation method comprises:

Step 1, using the complex source point method and the complex source point multipole superposition method to describe the high-order beam, and obtaining the spherical vector wave function expansion of the high-order Hermitian-Gaussian beam vector potential;

Step 2, through the mathematical relationship between the Laguerre-Gaussian beam and the Hermitian-Gaussian beam mode, the scalar wave function expansion of the Laguerre-Gaussian vortex beam is obtained, and it is substituted into the Laguerre carrying the orbital angular momentum. - The vector potential expression of the Gaussian vortex beam, and then the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam can be obtained;

Step 3, in conjunction with the dielectric parameter expression, the relationship between the refractive index and the dielectric parameter and the Mie theory under the Debye dispersion model, the Mie scattering coefficient of the dispersive plasma sphere can be obtained;

In step 4, the electromagnetic field incident expression can be obtained by further processing the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam vector potential. Ball vector wave function expansion;

Step 5: On the surface of the dielectric sphere, according to the principle that the electric field and magnetic field satisfy the continuity boundary condition, combine the Mie scattering coefficient of the dispersive plasma sphere obtained in step 3, and substitute the spherical vector wave function expansions of the incident field, inner field and scattering field into the boundary conditions , the expansion coefficient of the scattered field under the incidence of the Laguerre-Gaussian vortex beam can be obtained;

Step 6: Obtain the far-field radar scattering cross section through the scattering field expansion coefficient and the scattering phase function;

Step 7: Change the frequency of the Laguerre-Gaussian vortex beam, and repeat steps 1 to 6 to obtain the electromagnetic scattering field of the dispersive plasma sphere at different frequencies, and then analyze the dispersion effect of the plasma sphere under the irradiation of radar beams of different frequencies. Effects on electromagnetic scattering properties of vortex beams.

Further, using the complex source point method to describe the high-order beam by using the complex source point multipole superposition method to obtain the spherical vector wave function expansion of the high-order Hermitian-Gaussian beam vector potential includes: using the complex source point method, The high-order beam is described by the method of complex source point multipole superposition, and the expansion of the spherical vector wave function of the high-order Hermitian-Gaussian beam vector potential is obtained as:

为第一类球矢量波函数，系数α⁽¹⁾(u,v,n,m)、β⁽¹⁾(u,v,n,m)表示HG波束的球矢量 波函数展开系数，其数学表达式为：where the subscripts u and v represent the modulo order of the Hermitian-Gaussian beam along the x and y directions, respectively;

is the first type of spherical vector wave function, the coefficients α ⁽¹⁾ (u,v,n,m), β ⁽¹⁾ (u,v,n,m) represent the expansion coefficient of the spherical vector wave function of the HG beam, and its mathematical The expression is:

满足以下迭代关系：in

The following iterative relations are satisfied:

表达式如下：When u=v=0, the high-order Hermitian-Gaussian beam degenerates into a Gaussian fundamental mode,

The expression is as follows:

cosθ₀＝(z₀+ib)/r₀，

The parameters are specifically:

cosθ ₀ =(z ₀ +ib)/r ₀ ,

Further, through the mathematical relationship between the Laguerre-Gaussian beam and the Hermitian-Gaussian beam mode, the scalar wave function expansion of the Laguerre-Gaussian vortex beam can be obtained, and it is substituted into the Laguerre-Gaussian vortex beam carrying the orbital angular momentum. The vector potential expression of the Gale-Gaussian vortex beam, and then the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam can be obtained, including:

下的表达式写作：(1) The Laguerre-Gaussian vortex beam is an approximate solution of the paraxial wave equation in the cylindrical coordinate system, and the paraxial wave equation in the cylindrical coordinate system

The following expression is written:

Let the solution of the equation be:

p(z)＝iln(q₀+z)， q(z)＝q₀+z，

表示共焦参数，

为方位角，

where w(z) is the beam width at z, p(z) is the complex phase shift, which is related to the transmission of the beam, and q(z) is the parameter of the complex beam, which is used to describe the intensity of the beam with the transmission The Gaussian variation of the axial distance, the phase wavefront curvature is spherical at the paraxial; the specific expression is

p(z)=iln(q ₀ +z), q(z)=q ₀ +z,

is the confocal parameter,

is the azimuth,

(2) Substitute the solution of the equation into the paraxial wave equation, and use the separation of variables method:

g=M(ζ)Z(z);

得到如下方程组：in

The following equations are obtained:

为伴随拉盖尔多项式，表示为：in

is the adjoint Laguerre polynomial, expressed as:

和Z(z)的表达式代入到M(ζ)和g的表达式中并且与ψ进行对 比，得到拉盖尔-高斯涡旋波束具体表达式：(3) will

The expressions of and Z(z) are substituted into the expressions of M(ζ) and g and compared with ψ to obtain the specific expression of the Laguerre-Gaussian vortex beam:

where p and l represent the radial mode order and topological charge of the Laguerre-Gaussian vortex beam, respectively;

(4) Normalize the specific expression of Laguerre-Gaussian vortex beam to get:

When p=l=0, the Laguerre-Gaussian vortex beam will degenerate to the incident Gaussian beam, and when the beam waist radius w ₀ →∞, it will degenerate from the Gaussian beam to the plane wave;

(5) The vector potential of a Laguerre-Gaussian vortex beam carrying orbital angular momentum is expressed as:

in:

分别对应角度依赖关系中的

δ_0l为狄拉克函数，根据拉盖 尔-高斯波束与厄米-高斯波束模之间的数学关系式，得到拉盖尔-高斯涡旋波束 的标量波函数展开，具体如下：in the formula

Corresponding to the angle dependencies in the

δ _0l is the Dirac function. According to the mathematical relationship between the Laguerre-Gaussian beam and the Hermitian-Gaussian beam mode, the scalar wave function expansion of the Laguerre-Gaussian vortex beam is obtained, as follows:

和

的具体表达式代入到拉盖尔-高斯涡旋波束 矢势表达式当中，得到拉盖尔-高斯涡旋波束的球矢量波函数的展开系数 χ⁽¹⁾(p,l,n,m)和κ⁽¹⁾(p,l,n,m)的表达式如下：(6) will

and

Substitute the specific expression into the Laguerre-Gaussian vortex beam vector potential expression to obtain the expansion coefficient χ ⁽¹⁾ (p,l,n,m) of the spherical vector wave function of the Laguerre-Gaussian vortex beam. and κ ⁽¹⁾ (p,l,n,m) are expressed as:

Then the spherical vector wave function of the Laguerre-Gaussian vortex beam vector potential expands as:

Further, the Mie scattering coefficient of the dispersive plasma sphere can be obtained in combination with the dielectric parameter expression, the refractive index and the dielectric parameter relational formula and the Mie theory under the Debye dispersion model;

(1) Denote a _n , b _n is the scattering coefficient in Mie theory, and its specific expression is written as:

in

(2) Based on the Debye dispersion model, the dielectric parameter expression of the dispersive plasma sphere is:

μ_r＝1。

μ _r =1.

Further, the electromagnetic field incident expression can be obtained by further processing the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam vector potential, and the Laguerre-Gaussian vortex beam inner field and scattering field can also be obtained according to the same method. The spherical vector wave function expansion of includes:

(1) The expansion of the spherical vector wave function of the Laguerre-Gaussian vortex beam vector potential is:

Substitute the above formula into:

The electromagnetic field incident expression with the propagation direction as the +z-axis direction and the electric vector as the polarization in the x-direction is obtained:

(2) In the same way, the spherical vector wave function expansions of the inner and scattered fields of the Laguerre-Gaussian vortex beam are obtained.

Further, according to the principle that the electric field and the magnetic field satisfy the continuity boundary condition on the surface of the dielectric sphere, combined with the Mie scattering coefficient of the dispersive plasma sphere obtained in step (3), the spherical vector wave functions of the incident field, the inner field and the scattered field are expanded to the formula By substituting the boundary conditions, the expansion coefficients of the scattered field under the incidence of the Laguerre-Gaussian vortex beam can be obtained, including:

The scattering field expansion coefficient is:

in

_{where an and bn are} the scattering coefficients in the Mie theory;

The far-field radar scattering cross section obtained by the scattered field expansion coefficient and the scattered phase function includes:

The far-field radar cross section expression is as follows:

和

表示散射场的分量，具体表达式如下：in the formula

and

represents the component of the scattered field, and the specific expression is as follows:

和

具体为：where the scattering phase function

and

Specifically:

为连带勒让德多项式，有：

For the associated Legendre polynomial, we have:

in:

Another object of the present invention is to provide a vortex beam electromagnetic scattered field calculation system for implementing the vortex beam electromagnetic scattered field calculation method, and the vortex beam electromagnetic scattered field calculation system includes:

The spherical vector wave function expansion acquisition module of the high-order Hermitian-Gaussian beam vector potential is used to describe high-order beams by using the complex source point method and the method of complex source point multipole superposition to obtain high-order Hermitian-Gaussian beams The spherical vector wave function expansion of the vector potential;

The spherical vector wave function expansion acquisition module of the Laguerre-Gaussian vortex beam, which is used to obtain the Laguerre-Gaussian vortex beam through the mathematical relationship between the Laguerre-Gaussian beam and the Hermitian-Gaussian beam mode The scalar wave function expansion of , and substitute it into the vector potential expression of the Laguerre-Gaussian vortex beam carrying the orbital angular momentum, and then the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam can be obtained;

The Mie scattering coefficient acquisition module is used to obtain the Mie scattering coefficient of the dispersive plasma sphere by combining the expressions of dielectric parameters under the Debye dispersion model, the relationship between the refractive index and the dielectric parameters, and the Mie theory;

Spherical vector wave function expansion module of Laguerre-Gaussian vortex beam internal and scattered fields, which can be used to obtain electromagnetic field incidence expressions by further processing the spherical vector wave function expansion of Laguerre-Gaussian vortex beam vector potential , the spherical vector wave function expansions of the Laguerre-Gaussian vortex beam inner field and scattered field can also be obtained according to the same method;

Scattering field expansion coefficient acquisition module, which is used to expand the spherical vector wave functions of the incident field, internal field and scattered field by combining the Mie scattering coefficient of the obtained dispersive plasma sphere according to the principle that the electric and magnetic fields satisfy the continuity boundary condition on the surface of the dielectric sphere. By substituting the boundary conditions into the formula, the expansion coefficient of the scattered field under the incidence of the Laguerre-Gaussian vortex beam can be obtained;

The far-field radar cross section acquisition module is used to obtain the far-field radar cross section through the scattered field expansion coefficient and the scattered phase function.

The influence analysis module of electromagnetic scattering characteristics of vortex beams is used to change the frequency of Laguerre-Gaussian vortex beams to obtain the electromagnetic scattering fields of dispersive plasma spheres at different frequencies, and then analyze the plasma spheres irradiated by radar beams of different frequencies. Influence of dispersion effects on electromagnetic scattering properties of vortex beams.

Another object of the present invention is to provide a near-space hypersonic vehicle communication method, which uses the vortex beam electromagnetic scattering field calculation method.

Combined with all the above technical solutions, the advantages and positive effects of the present invention are as follows: the present invention can obtain the radar scattering cross section of the dispersive plasma sphere by solving the generalized Lorentz Mie analytical solution of the electromagnetic scattering field of the vortex beam. The scattering properties of non-dispersive plasma spheres are calculated, which fills the gap in the industry, is expected to solve the difficult problem of radar detection of hypersonic aircraft targets, and provides a new option for the anti-stealth research of stealth aircraft.

The invention utilizes the unique phase wave front structure and special intensity distribution of the vortex electromagnetic wave to calculate the radar scattering cross section of the Debye dispersion plasma sphere, and proposes a method for calculating the vortex beam electromagnetic scattering field of the Debye dispersion plasma sphere , to study the scattering characteristics of Debye dispersive plasma spheres under the irradiation of vortex electromagnetic beams, which is expected to solve the difficult problem of radar detection of hypersonic aircraft targets, and provides a new option for anti-stealth research of stealth aircraft.

Description of drawings

In order to explain the technical solutions of the embodiments of the present application more clearly, the following will briefly introduce the drawings that need to be used in the embodiments of the present application. Obviously, the drawings described below are only some embodiments of the present application. For those of ordinary skill in the art, other drawings can also be obtained from these drawings without creative effort.

FIG. 1 is a flowchart of a method for calculating a vortex beam electromagnetic scattering field provided by an embodiment of the present invention.

2 is a schematic structural diagram of a vortex beam electromagnetic scattered field calculation system provided by an embodiment of the present invention;

In Figure 2: 1. The acquisition module of the spherical vector wave function expansion of the high-order Hermitian-Gaussian beam vector potential; 2. The acquisition module of the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam; 3. The Mie scattering coefficient Acquisition module; 4. Spherical vector wave function expansion module of Laguerre-Gaussian vortex beam inner field and scattered field; 5. Scattered field expansion coefficient acquisition module; 6. Far-field radar cross section acquisition module; 7. Vortex A module for analyzing the effects of beam electromagnetic scattering properties.

Figure 3 is a geometric schematic diagram of the Laguerre-Gaussian vortex beam irradiating the Debye dispersive plasma sphere.

Fig. 4 is a block diagram of the Debye dispersion plasma sphere scattering cross section under the Laguerre-Gaussian vortex beam irradiation provided by the generalized Lorentz Mie theory calculation provided by the embodiment of the present invention.

Fig. 5 is the influence of the plasma collision frequency on the scattering cross section of the dispersive plasma sphere under the incidence of the LG00 vortex beam provided by the embodiment of the present invention.

Fig. 6 is a schematic diagram showing the influence of different plasma collision frequencies on the scattering cross section of the LG01 vortex beam incident dispersive plasma sphere according to an embodiment of the present invention.

Fig. 7 is a schematic diagram showing the influence of different plasma collision frequencies on the scattering cross section of the incident dispersive plasma sphere of the LG02 vortex beam provided by the embodiment of the present invention.

Detailed ways

In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention, but not to limit the present invention.

In view of the existing problems in the prior art, the present invention provides a method and system for calculating the electromagnetic scattering field of a vortex beam of a Debye dispersion plasma sphere. The present invention is described in detail below with reference to the accompanying drawings.

As shown in FIG. 1 , the method for calculating the vortex beam electromagnetic scattering field provided by the present invention includes the following steps:

S101: Using the complex source point method and using the complex source point multipole superposition method to describe the high-order beam, obtain the spherical vector wave function expansion of the high-order Hermitian-Gaussian beam vector potential;

S102: Through the mathematical relationship between the Laguerre-Gaussian beam and the Hermitian-Gaussian beam mode, the scalar wave function expansion of the Laguerre-Gaussian vortex beam can be obtained, and it is substituted into the Laguerre carrying the orbital angular momentum - The vector potential expression of the Gaussian vortex beam, and then the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam can be obtained;

S103: The Mie scattering coefficient of the dispersive plasma sphere can be obtained by combining the expressions of dielectric parameters under the Debye dispersion model, the relationship between refractive index and dielectric parameters, and the Mie theory;

S104: The electromagnetic field incident expression can be obtained by further processing the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam vector potential. According to the same method, the spherical inner field and scattering field of the Laguerre-Gaussian vortex beam can also be obtained. Vector wave function expansion;

S105: According to the principle that the electric field and the magnetic field satisfy the continuity boundary condition on the surface of the dielectric sphere, combine the Mie scattering coefficient of the dispersive plasma sphere obtained in step S103, and substitute the spherical vector wave function expansions of the incident field, the inner field and the scattering field into the boundary conditions, Then the expansion coefficient of the scattered field under the incidence of the Laguerre-Gaussian vortex beam can be obtained;

S106: The far-field radar scattering cross section can be obtained by the scattering field expansion coefficient and the scattering phase function.

S107: Change the frequency of the Laguerre-Gaussian vortex beam, and repeat steps S101 to S106 to obtain the electromagnetic scattering field of the dispersive plasma sphere at different frequencies, and then analyze the effect of the plasma sphere on the dispersion effect of the plasma sphere under the irradiation of radar beams of different frequencies. Influence of electromagnetic scattering properties of vortex beams.

Those skilled in the art of the vortex beam electromagnetic scattered field calculation method provided by the present invention can also use other steps to implement, and the vortex beam electromagnetic scattered field calculation method provided by the present invention in FIG. 1 is only a specific embodiment.

As shown in Figure 2, the vortex beam electromagnetic scattered field calculation system provided by the present invention includes:

The spherical vector wave function expansion acquisition module 1 of the high-order Hermitian-Gaussian beam vector potential is used to describe the high-order beam by using the complex source point method and the complex source point multipole superposition method to obtain the high-order Hermitian-Gaussian beam. The spherical vector wave function expansion of the beam vector potential;

The spherical vector wave function expansion acquisition module 2 of the Laguerre-Gaussian vortex beam is used to obtain the Laguerre-Gaussian vortex through the mathematical relationship between the Laguerre-Gaussian beam and the Hermitian-Gaussian beam mode Expand the scalar wave function of the beam, and substitute it into the vector potential expression of the Laguerre-Gaussian vortex beam carrying the orbital angular momentum, and then the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam can be obtained;

The Mie scattering coefficient acquisition module 3 is used to obtain the Mie scattering coefficient of the dispersive plasma sphere by combining the dielectric parameter expression, the relationship between the refractive index and the dielectric parameter and the Mie theory under the Debye dispersion model;

The spherical vector wave function expansion module 4 of the Laguerre-Gaussian vortex beam internal and scattered fields is used to further process the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam vector potential to obtain the electromagnetic field incident expression According to the same method, the spherical vector wave function expansions of the inner and scattered fields of the Laguerre-Gaussian vortex beam can also be obtained;

Scattering field expansion coefficient acquisition module 5 is used to combine the Mie scattering coefficient of the obtained dispersive plasma sphere on the surface of the dielectric sphere according to the principle that the electric field and magnetic field satisfy the continuity boundary condition, and calculate the spherical vector wave function of the incident field, inner field and scattered field. By substituting the expansion into the boundary conditions, the expansion coefficient of the scattered field under the incidence of the Laguerre-Gaussian vortex beam can be obtained;

The far-field radar scattering cross section acquisition module 6 is used to obtain the far-field radar scattering cross section through the scattering field expansion coefficient and the scattering phase function.

The influence analysis module 7 of the electromagnetic scattering characteristics of the vortex beam is used to change the frequency of the Laguerre-Gaussian vortex beam to obtain the electromagnetic scattering field of the dispersive plasma sphere at different frequencies, and then analyze the plasma irradiated by the radar beam of different frequencies. Influence of spherical dispersion effect on electromagnetic scattering properties of vortex beams.

The technical solutions of the present invention will be further described below with reference to the accompanying drawings.

As shown in Figure 4, the specific implementation steps of the vortex beam electromagnetic scattering field calculation of the present invention are as follows:

Step 1: Using the complex source point method, the high-order beam is described by the method of complex source point multipole superposition, and the spherical vector wave function expansion of the high-order Hermitian-Gaussian beam vector potential is obtained.

(1.1) The complex source point method is an equivalent fast algorithm based on Green's function, so the complex source point method can be introduced into the integral method.

(1.2) Using the complex source point method, that is, using the complex source point multipole superposition method to describe the high-order beam, the expansion of the spherical vector wave function of the high-order Hermitian-Gaussian beam vector potential can be obtained as:

为第一类球矢量波函数，系数α⁽¹⁾(u,v,n,m)、β⁽¹⁾(u,v,n,m)表示HG波束的球矢量 波函数展开系数，其数学表达式为：where the subscripts u and v represent the modulo order of the Hermitian-Gaussian beam along the x and y directions, respectively;

is the first type of spherical vector wave function, the coefficients α ⁽¹⁾ (u,v,n,m), β ⁽¹⁾ (u,v,n,m) represent the expansion coefficient of the spherical vector wave function of the HG beam, and its mathematical The expression is:

满足以下迭代关系：in

The following iterative relations are satisfied:

表达式如下：When u=v=0, the high-order Hermitian-Gaussian beam degenerates into a Gaussian fundamental mode,

The expression is as follows:

cosθ₀＝(z₀+ib)/r₀，

The parameters are specifically:

cosθ ₀ =(z ₀ +ib)/r ₀ ,

Step 2: Through the mathematical relationship between the Laguerre-Gaussian beam and the Hermitian-Gaussian beam mode, the scalar wave function expansion of the Laguerre-Gaussian vortex beam can be obtained and substituted into the Laguerre carrying the orbital angular momentum The vector potential expression of the Laguerre-Gaussian vortex beam, and then the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam can be obtained;

下的表达式可以写作：(2.1) The Laguerre-Gaussian vortex beam is an approximate solution of the paraxial wave equation in the cylindrical coordinate system. Paraxial Wave Equation in Cylindrical Coordinate System

The following expression can be written as:

Let the solution of the equation be:

p(z)＝iln(q₀+z)，q(z)＝q₀+z，

表示共焦参数，

为方位角，

where w(z) is the beam width at z, p(z) is the complex phase shift, which is related to the transmission of the beam, and q(z) is the parameter of the complex beam, which is used to describe the intensity of the beam with the transmission Gaussian variation of the axial distance, the phase wavefront curvature is spherical in the paraxial direction. Its specific expression is

p(z)=iln(q ₀ +z), q(z)=q ₀ +z,

is the confocal parameter,

is the azimuth,

(2.2) Substitute the solution of the equation into the paraxial wave equation and use the separation of variables method, assuming:

g=M(ζ)Z(z);

可以得到如下方程组：in

The following equations can be obtained:

为伴随拉盖尔多项式，可以表示为：in

is the adjoint Laguerre polynomial, which can be expressed as:

和Z(z)的表达式代入到M(ζ)和g的表达式中并且与ψ进行 对比，可以得到拉盖尔-高斯涡旋波束具体表达式：(2.3) will

The expressions of and Z(z) are substituted into the expressions of M(ζ) and g and compared with ψ, the specific expression of the Laguerre-Gaussian vortex beam can be obtained:

where p and l represent the radial mode order and topological charge of the Laguerre-Gaussian vortex beam, respectively

(2.4) Normalize the specific expression of Laguerre-Gaussian vortex beam, we can get:

When p=l=0, the Laguerre-Gaussian vortex beam will degenerate to the incident Gaussian beam, and when the beam waist radius w ₀ →∞, it will degenerate from the Gaussian beam to the plane wave.

(2.5) The vector potential of a Laguerre-Gaussian vortex beam carrying orbital angular momentum can be expressed as:

in:

分别对应角度依赖关系中的

δ_0l为狄拉克函数。根据拉盖 尔-高斯波束与厄米-高斯波束模之间的数学关系式，可以得到拉盖尔-高斯涡旋 波束的标量波函数展开，具体如下：in the formula

Corresponding to the angle dependencies in the

δ _0l is the Dirac function. According to the mathematical relationship between the Laguerre-Gaussian beam and the Hermitian-Gaussian beam mode, the scalar wave function expansion of the Laguerre-Gaussian vortex beam can be obtained, as follows:

和

的具体表达式代入到拉盖尔-高斯涡旋波束 矢势表达式当中，可以得到拉盖尔-高斯涡旋波束的球矢量波函数的展开系数 χ⁽¹⁾(p,l,n,m)和κ⁽¹⁾(p,l,n,m)的表达式如下：(2.6) will

and

Substitute the specific expression into the Laguerre-Gaussian vortex beam vector potential expression to obtain the expansion coefficient χ ⁽¹⁾ (p,l,n,m) of the spherical vector wave function of the Laguerre-Gaussian vortex beam. ) and κ ⁽¹⁾ (p,l,n,m) are expressed as follows:

Then the spherical vector wave function of the Laguerre-Gaussian vortex beam vector potential can be expanded as:

Step 3: The Mie scattering coefficient of the dispersive plasma sphere can be obtained by combining the expressions of dielectric parameters under the Debye dispersion model, the relationship between the refractive index and the dielectric parameters, and the Mie theory;

(3.1) Denote a _n , b _n is the scattering coefficient in Mie theory, and its specific expression can be written as:

in

μ_r＝1。(3.2) Since the target is a uniform and isotropic Debye dispersion plasma sphere, considering the effect of the plasma frequency dispersion on the electromagnetic scattering of the vortex beam plasma sphere, the single-layer plasma cladding with different parameters

μ _r =1.

Step 4: The electromagnetic field incident expression can be obtained by further processing the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam vector potential. Ball vector wave function expansion;

(4.1) The spherical vector wave function expansion of the Laguerre-Gaussian vortex beam vector potential is:

Substitute the above formula into:

Then we can obtain the electromagnetic field incident expression with the propagation direction as the +z-axis direction and the electric vector as the polarization in the x-direction:

(4.2) Similarly, the spherical vector wave function expansions of the Laguerre-Gaussian vortex beam inner field and scattered field can be obtained.

Step 5: On the surface of the dielectric sphere, according to the principle that the electric field and magnetic field satisfy the continuity boundary condition, combine the Mie scattering coefficient of the dispersive plasma sphere obtained in step (3), and substitute the spherical vector wave function expansions of the incident field, inner field and scattered field into Boundary conditions, the expansion coefficient of the scattered field under the incidence of the Laguerre-Gaussian vortex beam can be obtained;

The scattering field expansion coefficient is:

in

_{where an and bn are} the scattering coefficients in the Mie theory.

Step 6: The far-field radar scattering cross section can be obtained by the scattering field expansion coefficient and the scattering phase function;

The far-field radar cross section expression is as follows:

和

表示散射场的分量，具体表达式如下：In the above formula

and

represents the component of the scattered field, and the specific expression is as follows:

和

具体为：where the scattering phase function

and

Specifically:

为连带勒让德多项式，有：

For the associated Legendre polynomial, we have:

in:

Step 7: Change the frequency of the Laguerre-Gaussian vortex beam, and repeat steps (1) to (6) to obtain the electromagnetic scattering field of the dispersive plasma sphere at different frequencies, and then analyze the plasma irradiated by radar beams of different frequencies. Influence of spherical dispersion effect on electromagnetic scattering properties of vortex beams.

The technical effects of the present invention will be described in detail below in conjunction with simulation.

(1) Test simulation conditions

The relevant calculation parameters are selected as follows: the radius of the dispersive plasma sphere a=0.05m, the plasma frequency _ωp =10GHz, the beam waist radius is one wavelength, the collision frequencies are ν=10GHz, ν=20GHz, ν=50GHz, the incident vortex electromagnetic wave is The frequencies are f=4GHz and f=10GHz, the radial mode order p=0, and the angular quantum numbers are l=0, l=1, and l=2, respectively.

(2) Analysis of test simulation results

Figure 5 calculates the electromagnetic scattering fields of the dispersive plasma spheres at different collision frequencies when the frequency of the vortex electromagnetic wave (LG00) is 4 GHz and 10 GHz, respectively. It can be seen from the calculation results that with the increase of the collision frequency, the scattered field of the dispersive plasma sphere gradually decreases. This is because the more frequent the collision, the greater the absorption of electromagnetic waves by the plasma sphere, resulting in a decrease in the scattering intensity. In addition, as the frequency of the vortex electromagnetic wave increases, the width of the main lobe of the scattered field of the dispersive plasma sphere narrows, and the width of the side lobe increases.

Figure 6 calculates the electromagnetic scattering fields of the dispersive plasma spheres at different collision frequencies when the frequency of the vortex electromagnetic wave (LG01) is 4 GHz and 10 GHz, respectively. The variation law of scattering field intensity with collision frequency is similar to Fig. 3, that is, the larger the collision frequency, the smaller the scattering field intensity. In addition, as the frequency of the vortex electromagnetic wave increases, the side lobe intensity of the scattered field of the dispersive plasma sphere decreases.

Figure 7 calculates the electromagnetic scattering fields of the dispersive plasma spheres at different collision frequencies when the frequency of the vortex electromagnetic wave (LG02) is 4 GHz and 10 GHz, respectively. The variation law of scattering field intensity with collision frequency is similar to Fig. 3, that is, the larger the collision frequency, the smaller the scattering field intensity. In addition, as the frequency of the vortex electromagnetic wave increases, the peak value of the main lobe of the scattered field of the dispersive plasma sphere decreases, and the width of the side lobe also decreases.

It should be noted that embodiments of the present invention may be implemented by hardware, software, or a combination of software and hardware. The hardware portion may be implemented using special purpose logic; the software portion may be stored in memory and executed by a suitable instruction execution system, such as a microprocessor or specially designed hardware. Those of ordinary skill in the art will appreciate that the apparatus and methods described above may be implemented using computer-executable instructions and/or embodied in processor control code, for example on a carrier medium such as a disk, CD or DVD-ROM, such as a read-only memory Such code is provided on a programmable memory (firmware) or a data carrier such as an optical or electronic signal carrier. The device of the present invention and its modules can be implemented by hardware circuits such as very large scale integrated circuits or gate arrays, semiconductors such as logic chips, transistors, etc., or programmable hardware devices such as field programmable gate arrays, programmable logic devices, etc., It can also be implemented by software executed by various types of processors, or by a combination of the above-mentioned hardware circuits and software, such as firmware.

The above are only specific embodiments of the present invention, but the protection scope of the present invention is not limited to this. Any person skilled in the art is within the technical scope disclosed by the present invention, and all within the spirit and principle of the present invention Any modifications, equivalent replacements and improvements made within the scope of the present invention should be included within the protection scope of the present invention.

Claims (8)

1. A vortex beam electromagnetic scattering field calculation method of a Debye dispersive plasma sphere is characterized by comprising the following steps:

describing a high-order wave beam by using a multi-source point method and a multi-pole superposition method to obtain a spherical vector wave function expansion of a high-order Hermite-Gaussian wave beam vector potential;

step two, through the mathematical relation between the Laguerre-Gaussian beam and the Hermite-Gaussian beam model, the scalar wave function expansion of the Laguerre-Gaussian vortex beam is obtained, and the scalar wave function expansion is substituted into the vector potential expression of the Laguerre-Gaussian vortex beam carrying the orbital angular momentum, so that the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam can be obtained;

combining a dielectric parameter expression under a Debye dispersion model, a refractive index-dielectric parameter relational expression and a Mie theory to obtain a Mie scattering coefficient of the dispersion plasma sphere;

step four, further processing the Laguerre-Gaussian vortex beam vector potential spherical vector wave function expansion to obtain an electromagnetic field incident expression, and obtaining the Laguerre-Gaussian vortex beam internal field and scattering field spherical vector wave function expansion according to the same method;

step five, combining the Mie scattering coefficient of the dispersion plasma sphere obtained in the step three on the surface of the medium sphere according to the principle that the electric field and the magnetic field meet the continuous boundary condition, and substituting the spherical vector wave function expansion of the incident field, the internal field and the scattering field into the boundary condition to obtain the scattering field expansion coefficient under the incidence of the Laguerre-Gaussian vortex beam;

step six, obtaining a far-field radar scattering cross section through a scattering field expansion coefficient and a scattering phase function;

and step seven, changing the frequency of the Laguerre-Gaussian vortex beam, repeating the step one to the step six, obtaining the electromagnetic scattering field of the dispersion plasma ball under different frequencies, and further analyzing the influence of the dispersion effect of the plasma ball under the irradiation of radar beams with different frequencies on the electromagnetic scattering characteristic of the vortex beam.

2. The method for computing an electromagnetic scattered field of a vortex beam according to claim 1, wherein the describing the high-order beam by a multi-source point method and a multi-pole superposition method by using a multi-source point, and obtaining a spherical vector wave function expansion of the vector potential of the high-order Hermi-Gaussian beam comprises the following steps: describing a high-order wave beam by using a complex source point method and a complex source point multipole superposition method, and obtaining an expansion formula of a spherical vector wave function of the vector potential of the high-order Hermite-Gaussian wave beam as follows:

subscripts u and v in the formula respectively represent the mode orders of Hermitian-Gaussian beams changing along the x direction and the y direction;

for the first spherical vector wave function, coefficient α ⁽¹⁾ (u,v,n,m)、β ⁽¹⁾ (u, v, n, m) represents a spherical vector wave function expansion coefficient of the HG beam, and its mathematical expression is:

wherein

The following iterative relationship is satisfied:

when u-v-0, the higher order hermi-gaussian beam degenerates into the gaussian fundamental mode,

the expression is as follows:

the parameters are specifically as follows:

cosθ ₀ ＝(z ₀ +ib)/r ₀ ，

3. the method of calculating an electromagnetic scattered field of a vortex beam of claim 1, wherein the step of obtaining a scalar wavefunction expansion of a Laguerre-Gaussian vortex beam by a mathematical relationship between the Laguerre-Gaussian beam and the Hermite-Gaussian beam mode, and substituting the scalar wavefunction expansion into a vector expression of the Laguerre-Gaussian vortex beam carrying orbital angular momentum, thereby obtaining a spherical vector wavefunction expansion of the Laguerre-Gaussian vortex beam comprises:

(1) the Laguerre-Gaussian vortex beam is an approximate solution of paraxial wave equation in a cylindrical coordinate system

The following expression is written:

let the solution of the equation be:

where w (z) denotes the beam width of the beam at z, p (z) denotes the complex phase shift associated with the transmission of the beam, q (z) is a parameter of the complex beam used to describe the gaussian variation of the intensity of the beam with distance from the transmission axis, the curvature of the phase wavefront being spherical at the proximal axis; the specific expression is

p(z)＝iln(q ₀ +z)，q(z)＝q ₀ +z，

Which represents the confocal parameters of the image to be scanned,

in order to be the azimuth angle,

(2) substituting the solution of the equation into a paraxial wave equation, and utilizing a separation variable method:

g＝M(ζ)Z(z)；

wherein

The following system of equations is obtained:

wherein

Is a concomitant laguerre polynomial expressed as:

(3) will be provided with

And the expression of Z (z) is substituted into the expressions of M (ζ) and g and compared with ψ, resulting in a Laguerre-Gaussian vortex beam specific expression:

in the formula, p and l respectively represent the radial module order and the topological charge number of the Laguerre-Gaussian vortex beam;

(4) normalizing the specific expression of the Laguerre-Gaussian vortex beam to obtain:

when p is 0, the Laguerre-Gaussian vortex beam is degraded to the incident situation of the Gaussian beam, and when the beam waist radius w is equal to ₀ Will degenerate from a gaussian beam to a plane wave → ∞;

(5) the rise of a laguerre-gaussian vortex beam carrying orbital angular momentum is expressed as:

wherein:

in the formula

Respectively corresponding to angle dependence

δ _0l Obtaining a scalar wave function expansion of the Laguerre-Gaussian vortex beam according to a mathematical relation between the Laguerre-Gaussian beam and the Hermite beam mode as a Dirac function, wherein the scalar wave function expansion comprises the following specific steps:

(6) will be provided with

And

the specific expression is substituted into the vector potential expression of the Laguerre-Gaussian vortex beam to obtain the expansion coefficient chi of the spherical vector wave function of the Laguerre-Gaussian vortex beam ⁽¹⁾ (p, l, n, m) and κ ⁽¹⁾ The expression of (p, l, n, m) is as follows:

then the spherical vector wave function of the vector potential of the laguerre-gaussian vortex beam is expanded as:

4. the method for calculating the vortex beam electromagnetic scattering field according to claim 1, wherein the Mie scattering coefficient of the dispersive plasma sphere can be obtained by combining a dielectric parameter expression under a Debye dispersion model, a refractive index-dielectric parameter relational expression and a Mie theory;

(1) note a _n ，b _n The scattering coefficient in Mie theory is expressed by the following specific expression:

wherein

(2) Based on a Debye dispersion model, the dielectric parameter expression of the dispersion plasma sphere is as follows:

μ _r ＝1。

5. the method of claim 1, wherein the further processing by spherical vector wave function expansion of the vector potential of the Laguerre-Gaussian vortex beam to obtain an electromagnetic field incident expression with a propagation direction of + z and an electric vector of x-direction polarization, and obtaining spherical vector wave function expansion of the inner field and the scattered field of the Laguerre-Gaussian vortex beam according to the same method comprises:

(1) the spherical vector wave function development formula of the vector potential of the Laguerre-Gaussian vortex beam is as follows:

substituting the above formula into:

obtaining an electromagnetic field incidence expression with the propagation direction being + z-axis direction and the electric vector being x-direction polarization:

(2) and obtaining spherical vector wave function expansion of the Laguerre-Gaussian vortex beam internal field and the scattering field in the same way.

6. The method for calculating an electromagnetic scattered field of a vortex beam according to claim 1, wherein the scattered field expansion coefficient under the incidence of a Laguerre-Gaussian vortex beam can be obtained by combining the Mie scattering coefficient of the dispersive plasma sphere obtained in step (3) on the surface of the medium sphere according to the principle that the electric field and the magnetic field satisfy the continuity boundary condition and substituting the spherical vector wave function expansion of the incident field, the internal field and the scattered field into the boundary condition:

the scatter field expansion coefficient is:

wherein

In the formula a _n ，b _n Is the scattering coefficient in the Mie theory;

obtaining a far-field radar scattering cross section through a scattering field expansion coefficient and a scattering phase function comprises the following steps:

the far-field radar scattering cross-section expression is as follows:

in the formula

And

representing the components of the scattered field, the specific expression is as follows:

in which the scattering phase function

And

the method comprises the following specific steps:

is a conjunctive legendre polynomial, having:

wherein:

7. a vortex beam electromagnetic scattered field calculation system for implementing the vortex beam electromagnetic scattered field calculation method according to any one of claims 1 to 6, wherein the vortex beam electromagnetic scattered field calculation system comprises:

the device comprises a spherical vector wave function expansion acquisition module of the vector potential of the high-order hermitian-Gaussian beam, a vector wave function expansion acquisition module and a vector wave function expansion module, wherein the spherical vector wave function expansion acquisition module is used for describing the high-order beam by using a complex source point method and a complex source point multipole superposition method to obtain the vector wave function expansion of the high-order hermitian-Gaussian beam vector potential;

the acquisition module is used for obtaining the scalar wave function expansion of the Laguerre-Gaussian vortex wave beam through the mathematical relationship between the Laguerre-Gaussian wave beam and the Hermite-Gaussian wave beam module, substituting the scalar wave function expansion into the vector expression of the Laguerre-Gaussian vortex wave beam carrying the orbital angular momentum, and further obtaining the spherical vector wave function expansion of the Laguerre-Gaussian vortex wave beam;

the Mie scattering coefficient acquisition module is used for combining a dielectric parameter expression under a Debye dispersion model, a refractive index and dielectric parameter relation and a Mie theory to obtain the Mie scattering coefficient of the dispersion plasma sphere;

the Laguerre-Gaussian vortex beam inner field and scattering field spherical vector wave function expansion module is used for further processing through the Laguerre-Gaussian vortex beam vector potential spherical vector wave function expansion to obtain an electromagnetic field incident expression, and the Laguerre-Gaussian vortex beam inner field and scattering field spherical vector wave function expansion can also be obtained according to the same method;

the scattered field expansion coefficient acquisition module is used for combining the Mie scattering coefficient of the obtained dispersion plasma sphere on the surface of the medium sphere according to the principle that the electric field and the magnetic field meet the continuous boundary condition, and substituting the spherical vector wave function expansion of the incident field, the internal field and the scattered field into the boundary condition to obtain the scattered field expansion coefficient under the incidence of the Laguerre-Gaussian vortex beam;

the far-field radar cross section acquisition module is used for obtaining a far-field radar cross section through a scattered field expansion coefficient and a scattered phase function;

and the influence analysis module of the vortex beam electromagnetic scattering characteristic is used for changing the frequency of the Laguerre-Gaussian vortex beam to obtain the electromagnetic scattering field of the dispersion plasma ball under different frequencies, and further analyzing the influence of the plasma ball dispersion effect under the irradiation of radar beams with different frequencies on the vortex beam electromagnetic scattering characteristic.

8. A near space hypersonic aircraft communication method is characterized in that the method for calculating the vortex beam electromagnetic scattering field is used as claimed in any one of claims 1 to 6.

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