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

Method and system for calculating vortex beam electromagnetic scattering field of Debye dispersion plasma sphere Download PDF

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CN112287516B
CN112287516B CN202011004255.7A CN202011004255A CN112287516B CN 112287516 B CN112287516 B CN 112287516B CN 202011004255 A CN202011004255 A CN 202011004255A CN 112287516 B CN112287516 B CN 112287516B
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刘松华
孙明浩
郭立新
朱从宽
史晨鸽
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Abstract

The invention belongs to the field of vortex electromagnetic wave scattered field simulation calculation, and discloses a method and a system for calculating a vortex beam electromagnetic scattered field of a Debye dispersion plasma sphere, wherein the implementation process of the invention comprises the following steps: (1) obtaining a spherical vector wave function expansion of high-order Hermite-Gaussian beam vector potential by using a complex source point method; (2) a spherical vector wave function expansion of a Laguerre-Gaussian vortex beam; (3) the Mie scattering coefficient of the dispersive plasma sphere; (4) spherical vector wave function expansion of an inner field and a scattering field of the Laguerre-Gaussian vortex beam; (5) combining the boundary condition and the Mie scattering coefficient to obtain a scattering field expansion coefficient; (6) obtaining a far-field radar scattering cross section through a scattering field expansion coefficient and a scattering phase function; (7) and (3) repeating the steps (1) to (6) by changing the vortex wave frequency to analyze the dispersion effect of the plasma sphere. The invention obtains the scattering cross section of the dispersion plasma sphere by analyzing and solving the generalized Lorentz Mie of the vortex beam electromagnetic scattering field, and fills the blank of the industry.

Description

Method and system for calculating vortex beam electromagnetic scattering field of Debye dispersion plasma sphere
Technical Field
The invention belongs to the technical field of vortex electromagnetic wave detection, and particularly relates to a method and a system for calculating a vortex beam electromagnetic scattering field of a Debye dispersion plasma sphere.
Background
In the reentry process of the hypersonic aerocraft in the near space, the hypersonic aerocraft can be in violent friction with the surrounding atmosphere, so that the temperature around the aerocraft rises, and the pressure is increased. Under high temperature and high pressure, complex chemical reactions occur between different air components, ionization occurs, and plasma streaming consisting of free electrons, various ions and neutral particles is formed around the aircraft. The plasma on the surface of the target seriously influences the early warning and detection of the radar on the flying target and the communication quality between the monitoring station and the aircraft, even causes the phenomenon of 'black barrier' in communication, and the development of the adjacent space aircraft is seriously restricted by the plasma on the surface of the target. In addition, the plasma is widely existed in the environments such as space ionosphere, microwave laboratory and the like, and is also widely applied to the stealth technology of the environmental target, so that the research on the electromagnetic scattering property of the plasma target has important significance.
The electromagnetic scattering characteristics of the plasma target are mostly researched based on plane wave incidence, and the incident wave type is rarely reported as vortex electromagnetic wave. The study of vortex beams originally originated in the optical field, and the major research is still focused on the optical field at present. The vortex electromagnetic wave is the electromagnetic wave carrying the orbital angular momentum, and the phase wavefront is distributed in a spiral shape, so that compared with the traditional electromagnetic wave, the vortex electromagnetic wave has higher dimensional information modulation freedom degree, and has great application potential in the fields of wireless communication, target detection, imaging and the like. Under the irradiation of vortex electromagnetic waves, radar beams form radiation field excitation with different distribution at different targets, and the scattering characteristics of the targets are greatly different from the characteristics of the targets under the irradiation of traditional plane waves. The eigenstates of different orbital angular momentum are orthogonal to each other and can be theoretically stretched into a Hilbert space with infinite dimensions. Each orbital angular momentum eigenstate is represented by a specific integer, namely the topological charge number, and any other state can be represented as a linear superposition of eigenstates. The property shows that the vortex electromagnetic waves with different mode numbers can be physically separated, and physical and mathematical foundations are laid for the application of the vortex electromagnetic waves in the fields of wireless communication, radar detection and the like. In addition, the traditional plane electromagnetic wave only has one intensity peak value, the wave front distribution is smooth, and the change of angular phase does not exist. The vortex electromagnetic wave carries orbital angular momentum, so that the vortex electromagnetic wave has obvious difference from the traditional electromagnetic wave. First, the beam intensity distribution of the eddy electromagnetic wave is a special hollow ring distribution, and a phase singularity with zero intensity exists on the transmission axis. Secondly, the vortex electromagnetic wave carries a new space degree of freedom-orbital angular momentum, so that the vortex electromagnetic wave has great application prospect in the fields of large-capacity communication, high-performance radar detection and the like.
The research on the scattering characteristics of the vortex electromagnetic waves irradiating the plasma target is expected to solve the problem of difficulty in detecting the target radar of the hypersonic aircraft, and a new choice is provided for the anti-stealth research of the stealth aircraft. The scattering properties of the vortex electromagnetic beam on the plasmon sphere were studied, however, the dispersion properties of the plasmon sphere itself were not considered. The invention provides a method for calculating a dispersion plasma sphere radar scattering cross section based on a Debye dispersion model, and fills up the blank of the industry.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a method and a system for calculating a vortex beam electromagnetic scattering field of a Debye dispersion plasma sphere.
The invention is realized in such a way that a calculation method of vortex beam electromagnetic scattering field of a Debye dispersive plasma sphere 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 to obtain a spherical vector wave function expansion formula of the vector potential of the high-order hermitian-Gaussian wave beam;
step two, through the mathematical relation between the Laguerre-Gaussian beam and the Hermite-Gaussian beam model, obtaining the scalar wave function expansion of the Laguerre-Gaussian vortex beam, substituting the scalar wave function expansion into the vector potential expression of the Laguerre-Gaussian vortex beam carrying orbital angular momentum, and further obtaining the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam;
combining a dielectric parameter expression under a Debye dispersion model, a refractive index-dielectric parameter relation and a Mie theory to obtain a Mie scattering coefficient of the dispersion plasma sphere;
step four, an electromagnetic field incident expression can be obtained through further processing of the spherical vector wave function expansion of the vector potential of the Laguerre-Gaussian vortex beam, and the spherical vector wave function expansion of the inner field and the scattering field of the Laguerre-Gaussian vortex beam can also be obtained 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 characteristics of the vortex beam.
Further, the describing the high-order wave beam by using a complex source point method and a complex source point multipole superposition method to obtain a spherical vector wave function expansion of the vector potential of the high-order hermitian-gaussian wave beam includes: 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:
Figure RE-GDA0002806778480000031
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;
Figure BDA0002695357810000032
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:
Figure RE-GDA0002806778480000033
Figure BDA0002695357810000034
wherein
Figure BDA0002695357810000035
The following iterative relationship is satisfied:
Figure BDA0002695357810000041
Figure BDA0002695357810000042
when u-v-0, the higher order hermi-gaussian beam degenerates into the gaussian fundamental mode,
Figure BDA0002695357810000043
the expression is as follows:
Figure BDA0002695357810000044
the parameters are specifically as follows:
Figure BDA0002695357810000045
cosθ 0 ＝(z 0 +ib)/r 0 ，
Figure BDA0002695357810000046
further, the obtaining of the scalar wave function expansion of the laguerre-gaussian vortex beam through the mathematical relationship between the laguerre-gaussian beam and the hermitian-gaussian beam mode and substituting the scalar wave function expansion into the vector potential expression of the laguerre-gaussian vortex beam carrying the orbital angular momentum further may include:
(1) the Laguerre-Gaussian vortex beam is an approximate solution of paraxial wave equation in a cylindrical coordinate system
Figure BDA0002695357810000047
The following expression is written:
Figure BDA0002695357810000048
let the solution of the equation be:
Figure BDA0002695357810000049
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
Figure BDA00026953578100000410
p(z)＝iln(q 0 +z)， q(z)＝q 0 +z，
Figure BDA00026953578100000411
Which represents the confocal parameters of the image to be scanned,
Figure BDA00026953578100000412
in order to be the azimuth angle,
Figure BDA00026953578100000413
(2) substituting the solution of the equation into a paraxial wave equation, and utilizing a separation variable method:
g＝M(ζ)Z(z)；
wherein
Figure BDA0002695357810000051
The following system of equations is obtained:
Figure BDA0002695357810000052
Figure BDA0002695357810000053
wherein
Figure BDA0002695357810000054
Is a concomitant laguerre polynomial expressed as:
Figure BDA0002695357810000055
(3) will be provided with
Figure BDA0002695357810000056
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:
Figure BDA0002695357810000057
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:
Figure BDA0002695357810000058
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 vector potential of a laguerre-gaussian vortex beam carrying orbital angular momentum is expressed as:
Figure BDA0002695357810000059
wherein:
Figure BDA0002695357810000061
in the formula
Figure BDA0002695357810000062
Respectively corresponding to angle dependence
Figure BDA0002695357810000063
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:
Figure BDA0002695357810000064
Figure BDA0002695357810000065
(6) will be provided with
Figure BDA0002695357810000066
And
Figure BDA0002695357810000067
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:
Figure BDA0002695357810000068
Figure BDA0002695357810000071
then the spherical vector wave function of the vector potential of the laguerre-gaussian vortex beam is expanded as:
Figure RE-GDA0002806778480000072
further, 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 and dielectric parameter relation and a Mie theory;
(1) note a n ，b n The scattering coefficient in Mie theory is expressed by the following specific expression:
Figure BDA0002695357810000073
wherein
Figure BDA0002695357810000074
(2) Based on a Debye dispersion model, the dielectric parameter expression of the dispersion plasma sphere is as follows:
Figure BDA0002695357810000075
μ r ＝1。
further, the electromagnetic field incident expression can be obtained by further processing the spherical vector wave function expansion of the vector potential of the Laguerre-Gaussian vortex beam, and the spherical vector wave function expansion of the inner field and the scattering field of the Laguerre-Gaussian vortex beam can also be obtained according to the same method, and the method comprises the following steps:
(1) the spherical vector wave function development formula of the vector potential of the Laguerre-Gaussian vortex beam is as follows:
Figure RE-GDA0002806778480000076
substituting the above formula into:
Figure RE-GDA0002806778480000081
obtaining an electromagnetic field incidence expression with the propagation direction being + z-axis direction and the electric vector being x-direction polarization:
Figure BDA0002695357810000082
(2) and obtaining spherical vector wave function expansion of the Laguerre-Gaussian vortex beam internal field and the scattering field in the same way.
Further, the step of obtaining the expansion coefficient of the scattering field under the incidence of the laguerre-gaussian vortex beam by combining the Mie scattering coefficient of the dispersive plasma sphere obtained in the step (3) on the surface of the dielectric sphere according to the principle that the electric field and the magnetic field meet the continuity 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 comprises the following steps:
the scatter field expansion coefficient is:
Figure BDA0002695357810000083
wherein
Figure BDA0002695357810000084
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:
Figure BDA0002695357810000085
in the formula
Figure BDA0002695357810000086
And
Figure BDA0002695357810000087
representing the components of the scattered field, the specific expression is as follows:
Figure BDA0002695357810000091
in which the scattering phase function
Figure BDA0002695357810000092
And
Figure BDA0002695357810000093
the method specifically comprises the following steps:
Figure BDA0002695357810000094
Figure BDA0002695357810000095
Figure BDA0002695357810000096
Figure BDA0002695357810000097
is a conjunctive legendre polynomial, having:
Figure BDA0002695357810000098
wherein:
Figure BDA0002695357810000099
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, the vortex beam electromagnetic scattered field calculation system comprising:
the spherical vector wave function expansion acquisition module of the high-order hermite-Gaussian beam vector potential 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 a spherical vector wave function expansion of the high-order hermite-Gaussian beam vector potential;
the acquisition module of the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam is used for acquiring the scalar wave function expansion of the Laguerre-Gaussian vortex beam through the mathematical relationship between the Laguerre-Gaussian beam and the Hermi-Gaussian beam module, substituting the scalar wave function expansion into the vector potential expression of the Laguerre-Gaussian vortex beam carrying the orbital angular momentum, and further acquiring the spherical vector wave function expansion of the Laguerre-Gaussian vortex 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 relational expression and a Mie theory to obtain the Mie scattering coefficient of the dispersion plasma sphere;
the spherical vector wave function expansion module of the Laguerre-Gaussian vortex beam internal field and the scattering field is used for further processing through the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam vector potential to obtain an electromagnetic field incident expression, and the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam internal field and the scattering field 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 an electric field and a magnetic field meet 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 to obtain the scattered field expansion coefficient under the incidence of the Laguerre-Gaussian vortex beam;
and the far-field radar scattering cross section acquisition module is used for acquiring the far-field radar scattering cross section through the scattered field expansion coefficient and the scattered phase function.
And the influence analysis module of the vortex beam electromagnetic scattering property 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 on the vortex beam electromagnetic scattering property under the irradiation of radar beams with different frequencies.
The invention also aims to provide a near space hypersonic aircraft communication method which uses the vortex beam electromagnetic scattered field calculation method.
By combining all the technical schemes, the invention has the advantages and positive effects that: according to the invention, the radar scattering cross section of the dispersive plasma sphere can be obtained by analyzing and solving the generalized Lorentz Mie of the vortex beam electromagnetic scattering field, only the scattering characteristic of the non-dispersive plasma sphere is calculated in the past, the blank of the industry is made up, the problem of difficulty in detecting the target radar of the hypersonic aircraft is expected to be solved, and a new choice is provided for the anti-stealth research of the stealth aircraft.
The radar scattering cross section of the Debye dispersion plasma sphere is calculated by utilizing the characteristics of the unique phase wavefront structure, the special intensity distribution and the like of the vortex electromagnetic wave, the method for calculating the vortex beam electromagnetic scattering field of the Debye dispersion plasma sphere is provided, the scattering characteristic of the Debye dispersion plasma sphere under the irradiation of the vortex electromagnetic wave is researched, the problem that the target radar of the hypersonic aircraft is difficult to detect is hopefully solved, and a new choice is provided for the anti-stealth research of the stealth aircraft.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings used in the embodiments of the present application will be briefly described below, and it is obvious that the drawings described below are only some embodiments of the present application, and it is obvious for those skilled in the art that other drawings can be obtained according to the drawings without creative efforts.
FIG. 1 is a flowchart of a method for calculating an electromagnetic scattered field of a vortex beam according to an embodiment of the present invention.
FIG. 2 is a schematic structural diagram of a vortex beam electromagnetic fringe field calculation system provided by an embodiment of the invention;
in fig. 2: 1. a spherical vector wave function expansion acquisition module of high-order Hermite-Gaussian beam vector potential; 2. a spherical vector wave function expansion acquisition module of the Laguerre-Gaussian vortex wave beam; 3. a Mie scattering coefficient obtaining module; 4. a spherical vector wave function expansion module of the inner field and the scattering field of the Laguerre-Gaussian vortex beam; 5. a scattered field expansion coefficient acquisition module; 6. a far field radar scattering cross section acquisition module; 7. and the influence analysis module is used for analyzing the influence of the electromagnetic scattering characteristics of the vortex beam.
Fig. 3 is a schematic of the geometry of a laguerre-gaussian vortex beam illuminating a Debye dispersive plasmon ball.
Fig. 4 is a block diagram of a scattering cross section of a Debye dispersion plasma sphere under irradiation of a laguerre-gaussian vortex beam calculated by using generalized lorentzian Mie theory according to an embodiment of the present invention.
FIG. 5 is a graph showing the effect of different plasma impact frequencies on the dispersion plasma spherical scattering cross-section at LG00 vortex beam incidence according to an embodiment of the present invention.
FIG. 6 is a schematic diagram illustrating the effect of different plasma collision frequencies on the incident dispersive plasma spherical scattering cross-section of LG01 vortex beam provided by an embodiment of the present invention.
FIG. 7 is a graph illustrating the effect of different plasma collision frequencies on the incident dispersive plasma spherical scattering cross-section of LG02 vortex beam provided by an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Aiming at the problems in the prior art, the invention provides a method and a system for calculating a vortex beam electromagnetic scattering field of a Debye dispersive plasma sphere, and the invention is described in detail below with reference to the attached drawings.
As shown in fig. 1, the method for calculating the electromagnetic scattered field of the vortex beam provided by the invention comprises the following steps:
s101: describing a high-order wave beam by using a complex source point method and a complex source point multipole superposition method to obtain a spherical vector wave function expansion of the vector potential of the high-order Hermite-Gaussian wave beam;
s102: 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 can be obtained, and the scalar wave function expansion is substituted into the vector potential expression of the Laguerre-Gaussian vortex beam carrying orbital angular momentum, so that the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam can be obtained;
s103: combining a dielectric parameter expression under a Debye dispersion model, a refractive index-dielectric parameter relation and a Mie theory to obtain a Mie scattering coefficient of the dispersion plasma sphere;
s104: an electromagnetic field incident expression can be obtained through further processing of a spherical vector wave function expansion of the vector potential of the Laguerre-Gaussian vortex beam, and spherical vector wave function expansions of an inner field and a scattering field of the Laguerre-Gaussian vortex beam can also be obtained according to the same method;
s105: combining the Mie scattering coefficient of the dispersion plasma sphere obtained in the step S103 according to the principle that the electric field and the magnetic field meet the continuous boundary condition on the surface of the medium sphere, and substituting the sphere 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;
s106: and obtaining a far-field radar scattering cross section through a scattering field expansion coefficient and a scattering phase function.
S107: changing the frequency of the Laguerre-Gaussian vortex beam, repeating the step S101 to the step S106 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 electromagnetic scattering characteristics of the vortex beam.
Those skilled in the art can also implement the method for calculating the electromagnetic scattering field of the vortex beam according to the present invention by using other steps, and the method for calculating the electromagnetic scattering field of the vortex beam according to the present invention shown in fig. 1 is only one specific example.
As shown in FIG. 2, the present invention provides a system for calculating an electromagnetic scattered field of a vortex beam, comprising:
the device comprises a high-order hermite-Gaussian beam vector wave function expansion acquisition module 1, a complex source point method and a complex source point multipole superposition method, wherein the high-order hermite-Gaussian beam vector wave function expansion acquisition module is used for describing a high-order beam by using the complex source point method and obtaining a high-order hermite-Gaussian beam vector wave function expansion;
the spherical vector wave function expansion obtaining module 2 of the Laguerre-Gaussian vortex beam is used for obtaining the scalar wave function expansion of the Laguerre-Gaussian vortex beam through the mathematical relation between the Laguerre-Gaussian beam and the Hermite-Gaussian beam module, substituting the scalar wave function expansion into the vector potential expression of the Laguerre-Gaussian vortex beam carrying the orbital angular momentum, and further obtaining the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam;
a Mie scattering coefficient obtaining module 3, configured to combine a dielectric parameter expression under a Debye dispersion model, a refractive index and dielectric parameter relation, and a Mie theory to obtain a Mie scattering coefficient of the dispersive plasma sphere;
the spherical vector wave function expansion module 4 of the Laguerre-Gaussian vortex beam internal field and the scattering field is used for further processing through the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam vector potential to obtain an electromagnetic field incident expression, and the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam internal field and the scattering field can also be obtained according to the same method;
a scattered field expansion coefficient acquisition module 5, configured 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 the magnetic field satisfy the continuity boundary condition, and substitute the spherical vector wave function expansion of the incident field, the internal field, and the scattered field into the boundary condition, so as to obtain the scattered field expansion coefficient under the incidence of the laguerre-gaussian vortex beam;
and the far-field radar scattering cross section acquisition module 6 is used for acquiring the far-field radar scattering cross section through the scattered field expansion coefficient and the scattered phase function.
And the influence analysis module 7 for the electromagnetic scattering property of the vortex beam is used for changing the frequency of the Laguerre-Gaussian vortex beam to obtain the electromagnetic scattering field of the dispersive plasma ball under different frequencies, and further analyzing the influence of the dispersion effect of the plasma ball on the electromagnetic scattering property of the vortex beam under the irradiation of radar beams with different frequencies.
The technical solution of the present invention is further described below with reference to the accompanying drawings.
As shown in FIG. 4, the steps for calculating the electromagnetic scattering field of the vortex beam of the present invention are as follows:
step 1: and describing the high-order wave beam by using a complex source point method and a complex source point multipole superposition method to obtain a spherical vector wave function expansion of the vector potential of the high-order Hermite-Gaussian wave beam.
(1.1) the complex source point method is an equivalent fast algorithm based on the Green function, so in the product classification method, the complex source point method can be introduced.
(1.2) describing the high-order wave beam by using a complex source point method, namely a complex source point multipole superposition method, and obtaining an expansion of a spherical vector wave function of the vector potential of the high-order Hermi-Gaussian wave beam as follows:
Figure RE-GDA0002806778480000141
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;
Figure BDA0002695357810000142
as a function of the first spherical vector wave, 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:
Figure RE-GDA0002806778480000143
Figure BDA0002695357810000144
wherein
Figure BDA0002695357810000145
The following iterative relationship is satisfied:
Figure BDA0002695357810000146
Figure BDA0002695357810000147
when u-v-0, the higher order hermi-gaussian beam degenerates into the gaussian fundamental mode,
Figure BDA0002695357810000148
the expression is as follows:
Figure RE-GDA0002806778480000151
the parameters are specifically as follows:
Figure BDA0002695357810000152
cosθ 0 ＝(z 0 +ib)/r 0 ，
Figure BDA0002695357810000153
step 2: through the mathematical relationship between the Laguerre-Gaussian beam and the Hermite-Gaussian beam model, the scalar wave function expansion of the Laguerre-Gaussian vortex beam can be obtained, and the scalar wave function expansion is substituted into the vector potential expression of the Laguerre-Gaussian vortex beam carrying orbital angular momentum, so that the spherical vector wave function expansion of the Laguerre-Gaussian vortex beam can be obtained;
(2.1) the Laguerre-Gaussian vortex beam is an approximate solution of paraxial wave equation in a cylindrical coordinate system. Paraxial wave equation in cylindrical coordinate system
Figure BDA0002695357810000154
The following expression can be written:
Figure BDA0002695357810000155
let the solution of the equation be:
Figure BDA0002695357810000156
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, and 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
Figure BDA0002695357810000157
p(z)＝iln(q 0 +z)，q(z)＝q 0 +z，
Figure BDA0002695357810000158
Which represents the confocal parameters of the image to be scanned,
Figure BDA0002695357810000159
in order to be the azimuth angle,
Figure BDA00026953578100001510
(2.2) substituting the solution of the equation into the paraxial wave equation by using the separation variable method, assuming that:
g＝M(ζ)Z(z)；
wherein
Figure BDA00026953578100001511
The following system of equations can be obtained:
Figure BDA00026953578100001512
Figure BDA0002695357810000161
wherein
Figure BDA0002695357810000162
To accompany a laguerre polynomial, it can be expressed as:
Figure BDA0002695357810000163
(2.3) mixing
Figure BDA0002695357810000164
And the expression of Z (z) is substituted into the expressions of M (ζ) and g and compared with ψ, a specific expression of Laguerre-Gaussian vortex beam can be obtained:
Figure BDA0002695357810000165
wherein p and l respectively represent the radial module order and the topological charge number of the Laguerre-Gaussian vortex beam
(2.4) normalizing the specific expression of the Laguerre-Gaussian vortex beam to obtain:
Figure BDA0002695357810000166
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 On → infinity, there is degradation from the gaussian beam to the plane wave.
(2.5) the rise of a laguerre-gaussian vortex beam carrying orbital angular momentum can be expressed as:
Figure BDA0002695357810000167
wherein:
Figure BDA0002695357810000168
in the formula
Figure BDA0002695357810000169
Are respectively paired withIn angle dependent relation
Figure BDA00026953578100001610
δ 0l Is a dirac function. According to a mathematical relation between the Laguerre-Gaussian beam and the Hermite-Gaussian beam mode, the scalar wave function expansion of the Laguerre-Gaussian vortex beam can be obtained, and the method specifically comprises the following steps:
Figure BDA0002695357810000171
Figure BDA0002695357810000172
(2.6) mixing
Figure BDA0002695357810000173
And
Figure BDA0002695357810000174
the specific expression of the method 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:
Figure BDA0002695357810000175
Figure BDA0002695357810000176
then the spherical vector wave function of the vector potential of the laguerre-gaussian vortical beam can be expanded as:
Figure RE-GDA0002806778480000177
and step 3: combining a dielectric parameter expression under a Debye dispersion model, a refractive index-dielectric parameter relation and a Mie theory to obtain a Mie scattering coefficient of the dispersion plasma sphere;
(3.1) note a n ，b n For the scattering coefficient in Mie theory, a specific expression can be written as:
Figure BDA0002695357810000181
wherein
Figure BDA0002695357810000182
(3.2) Single layer different parameter plasma coating because the target is uniform and isotropic Debye dispersive plasma sphere, considering the influence of plasma frequency dispersion on the electromagnetic scattering of vortex beam plasma sphere
Figure BDA0002695357810000183
μ r ＝1。
And 4, step 4: an electromagnetic field incident expression can be obtained by further processing the spherical vector wave function expansion of the vector potential of the Laguerre-Gaussian vortex beam, and the spherical vector wave function expansion of the inner field and the scattering field of the Laguerre-Gaussian vortex beam can also be obtained according to the same method;
(4.1) the spherical vector wave function development of the vector potential of the Laguerre-Gaussian vortex beam is as follows:
Figure RE-GDA0002806778480000184
substituting the above formula into:
Figure RE-GDA0002806778480000185
the electromagnetic field incidence expression formula with the propagation direction being + z-axis direction and the electric vector being x-direction polarization can be obtained:
Figure BDA0002695357810000186
and (4.2) similarly obtaining a spherical vector wave function expansion formula of the Laguerre-Gaussian vortex beam internal field and the scattering field.
And 5: combining the Mie scattering coefficient of the dispersion plasma sphere obtained in the step (3) 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;
the scatter field expansion coefficient is:
Figure BDA0002695357810000191
wherein
Figure BDA0002695357810000192
In the formula a n ，b n Is the scattering coefficient in Mie theory.
And 6: obtaining a far-field radar scattering cross section through a scattering field expansion coefficient and a scattering phase function;
the far field radar scattering cross section expression is as follows:
Figure BDA0002695357810000193
in the above formula
Figure BDA0002695357810000194
And
Figure BDA0002695357810000195
representing the components of the scattered field, the specific expression is as follows:
Figure BDA0002695357810000196
in which the scattering phase function
Figure BDA0002695357810000197
And
Figure BDA0002695357810000198
the method specifically comprises the following steps:
Figure BDA0002695357810000199
Figure BDA00026953578100001910
Figure BDA0002695357810000201
Figure BDA0002695357810000202
is a conjunctive legendre polynomial, having:
Figure BDA0002695357810000203
wherein:
Figure BDA0002695357810000204
and 7: changing the frequency of the Laguerre-Gaussian vortex beam, repeating the steps (1) to (6), 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.
The technical effects of the present invention will be described in detail with reference to simulations.
(1) Conditions of test simulation
The relevant calculation parameters are selected as follows: the spherical radius a of the dispersion plasma is 0.05m, and the plasma frequency omega p The beam waist radius is one wavelength, the collision frequency v is 10GHz, v is 20GHz and v is 50GHz respectively, the frequency of the incident vortex electromagnetic wave is f 4GHz and f is 10GHz respectively, the radial mode order p is 0, the angular quantum number is l 0, l is 1 and l is 2 respectively.
(2) Analysis of test simulation results
FIG. 5 calculates the electromagnetic scattering field of the dispersive plasmonic sphere at different impact frequencies for the vortex electromagnetic wave (LG00) frequencies of 4GHz and 10GHz, respectively. It can be seen from the calculation results that the scattering field of the dispersive plasmon ball gradually decreases as the collision frequency increases. This is because the more frequent the collisions, the greater the absorption of the electromagnetic wave by the plasmonic sphere, resulting in a decrease in scattering intensity. In addition, as the frequency of the vortex electromagnetic wave increases, the main lobe width of the scattered field of the dispersive plasma sphere becomes narrower, and the width of the side lobe increases.
FIG. 6 calculates the electromagnetic scattering field of the dispersive plasmonic sphere at different impact frequencies for the vortex electromagnetic wave (LG01) frequencies of 4GHz and 10GHz, respectively. The variation law of the scattered field intensity with the collision frequency is similar to that of fig. 3, i.e. the larger the collision frequency, the smaller the scattered field intensity. Furthermore, as the frequency of the vortex electromagnetic wave increases, the sidelobe intensity of the dispersive plasmon ball fringe field decreases.
FIG. 7 calculates the electromagnetic scattering field of the dispersive plasmonic sphere at different collision frequencies for the vortex electromagnetic wave (LG02) frequencies of 4GHz and 10GHz, respectively. The law of the variation of the scattered field intensity with the collision frequency is similar to that of fig. 3, i.e. the larger the collision frequency, the smaller the scattered field intensity. In addition, as the frequency of the vortex electromagnetic wave increases, the main lobe peak value of the dispersive plasmon polariton decreases, and the side lobe width also decreases.
It should be noted that the embodiments of the present invention can be realized by hardware, software, or a combination of software and hardware. The hardware portion may be implemented using dedicated logic; the software portions may be stored in a memory and executed by a suitable instruction execution system, such as a microprocessor or specially designed hardware. Those skilled 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, such code provided on a carrier medium such as a diskette, CD-or DVD-ROM, a programmable memory such as read-only memory (firmware), or a data carrier such as an optical or electronic signal carrier, for example. The apparatus and its modules of the present invention may be implemented by hardware circuits such as very large scale integrated circuits or gate arrays, semiconductors such as logic chips, transistors, or programmable hardware devices such as field programmable gate arrays, programmable logic devices, etc., or by software executed by various types of processors, or by a combination of hardware circuits and software, e.g., firmware.
The above description is only for the specific embodiments of the present invention, but the scope of the present invention is not limited thereto, and any modification, equivalent replacement, and improvement made by those skilled in the art within the technical scope of the present invention disclosed in the present invention should be covered within the 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:
Figure RE-FDA0002806778470000021
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;
Figure RE-FDA0002806778470000022
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:
Figure RE-FDA0002806778470000023
Figure RE-FDA0002806778470000024
wherein
Figure RE-FDA0002806778470000025
The following iterative relationship is satisfied:
Figure RE-FDA0002806778470000026
Figure RE-FDA0002806778470000027
when u-v-0, the higher order hermi-gaussian beam degenerates into the gaussian fundamental mode,
Figure RE-FDA0002806778470000028
the expression is as follows:
Figure RE-FDA0002806778470000029
the parameters are specifically as follows:
Figure RE-FDA00028067784700000210
cosθ 0 ＝(z 0 +ib)/r 0 ，
Figure RE-FDA00028067784700000211
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
Figure FDA0002695357800000031
The following expression is written:
Figure FDA0002695357800000032
let the solution of the equation be:
Figure FDA0002695357800000033
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
Figure FDA0002695357800000034
p(z)＝iln(q 0 +z)，q(z)＝q 0 +z，
Figure FDA0002695357800000035
Which represents the confocal parameters of the image to be scanned,
Figure FDA0002695357800000036
in order to be the azimuth angle,
Figure FDA0002695357800000037
(2) substituting the solution of the equation into a paraxial wave equation, and utilizing a separation variable method:
g＝M(ζ)Z(z)；
wherein
Figure FDA0002695357800000038
The following system of equations is obtained:
Figure FDA00026953578000000313
Figure FDA0002695357800000039
wherein
Figure FDA00026953578000000312
Is a concomitant laguerre polynomial expressed as:
Figure FDA00026953578000000310
(3) will be provided with
Figure FDA00026953578000000314
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:
Figure FDA00026953578000000311
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:
Figure FDA0002695357800000041
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:
Figure FDA0002695357800000042
wherein:
Figure FDA0002695357800000043
in the formula
Figure FDA0002695357800000044
Respectively corresponding to angle dependence
Figure FDA0002695357800000045
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:
Figure FDA0002695357800000046
Figure FDA0002695357800000047
(6) will be provided with
Figure FDA0002695357800000048
And
Figure FDA0002695357800000049
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:
Figure FDA0002695357800000051
Figure FDA0002695357800000052
then the spherical vector wave function of the vector potential of the laguerre-gaussian vortex beam is expanded as:
Figure FDA0002695357800000053
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:
Figure FDA0002695357800000054
wherein
Figure FDA0002695357800000055
(2) Based on a Debye dispersion model, the dielectric parameter expression of the dispersion plasma sphere is as follows:
Figure FDA0002695357800000056
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:
Figure FDA0002695357800000061
substituting the above formula into:
Figure FDA0002695357800000062
obtaining an electromagnetic field incidence expression with the propagation direction being + z-axis direction and the electric vector being x-direction polarization:
Figure FDA0002695357800000063
(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:
Figure FDA0002695357800000064
wherein
Figure FDA0002695357800000065
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:
Figure FDA0002695357800000071
in the formula
Figure FDA0002695357800000072
And
Figure FDA0002695357800000073
representing the components of the scattered field, the specific expression is as follows:
Figure FDA0002695357800000074
in which the scattering phase function
Figure FDA0002695357800000075
And
Figure FDA0002695357800000076
the method comprises the following specific steps:
Figure FDA0002695357800000077
Figure FDA0002695357800000078
Figure FDA0002695357800000079
Figure FDA00026953578000000710
is a conjunctive legendre polynomial, having:
Figure FDA00026953578000000711
wherein:
Figure FDA00026953578000000712
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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