CN107562981B - Electric field calculation method and device in conductor rough surface scattering problem - Google Patents
Electric field calculation method and device in conductor rough surface scattering problem Download PDFInfo
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Abstract
The application provides an electric field calculation method and device in the problem of conductor rough surface scattering. The method comprises the following steps: selecting M discrete coordinate points with equal intervals on the rough surface of the conductor with the length of L, wherein M is 2NAnd N is a positive integer, 0-order scattered field to N-order scattered field are respectively calculated according to the abscissa and the ordinate of the M discrete coordinate points, N is a target order of the electric field to be calculated, the first order partial derivative in the z-axis direction of each obtained scattered field to the N-order partial derivative in the z-axis direction are calculated, and the N-order electric field of the conductor rough surface is calculated according to the incident field, the obtained first order partial derivative in the z-axis direction of each scattered field to the N-order partial derivative in the z-axis direction and the N-order electric field expression. Therefore, the electric field of any order can be calculated, the calculation precision is high, the time complexity is reduced, and the calculation efficiency is improved.
Description
Technical Field
The application relates to the technical field of communication, in particular to an electric field calculation method and device in the problem of conductor rough surface scattering.
Background
The electromagnetic scattering of rough surfaces has wide application fields, such as remote sensing, marine geography, communication, material science, medical imaging and the like. In general, the electric field of the rough surface is the sum of the incident field and the scattered field, the incident field is a known function, and the calculation formula is as follows
Wherein,
where k is the wave number of the electromagnetic wave in vacuum, g is the cone factor, θiIs the incident angle of the electromagnetic wave.
The calculation methods of the scattering field of the rough surface are mainly divided into two categories: analytic methods, which are computational methods based on explicit physical concepts and accurate mathematical derivations, and numerical methods. The perturbation method is a classical analytic method, and the application field of the perturbation method is a rough surface with a slightly rough surface.
The perturbation method comprises a traditional perturbation method and a classical perturbation method, and in the prior art, the calculation formula for calculating the scattering field of the rough surface by the classical perturbation method is as follows:
wherein,and
respectively a 0 order scattered field, a 1 order scattered field and a 2 order scattered field; r is a position vector and r is a position vector,
kix＝ksinθi，kiz＝kcosθi，θik is the wave number of the electromagnetic wave in vacuum; f (k)x) Is the Fourier transform pair of (x); w (k)ix-kx) Is a spectral function of the gaussian spectrum,
h is the root mean square height, l is the correlation length; k is a radical ofxIs the x-direction component of k, kzIs the z-direction component of k, kxAnd kzThe relationship between is
It can be seen from the above calculation process that the dimension of the classical perturbation method for solving the integral increases with the increase of the order of the scattered field, and although a solution of a higher-order scattered field can be obtained in theory, the actual calculation is often limited by the calculation difficulty and calculation error of multiple integrals, so that the order of solving the scattered field by the classical perturbation method mostly stays on a second-order or third-order solution, a small part extends to a fourth-order solution, and the higher-order scattered field above the fourth order cannot be solved. Therefore, a high-order electric field with a rough surface cannot be obtained.
Disclosure of Invention
The application provides an electric field calculation method in the problem of conductor rough surface scattering, which can calculate an electric field of any order.
In a first aspect, the present application provides a method for calculating an electric field in a conductor rough surface scattering problem, including:
selecting M discrete coordinate points with equal intervals on the rough surface of the conductor with the length of L, wherein M is 2NN is a positive integer;
respectively calculating a scattering field from 0 order to n order according to the abscissa and the ordinate of the M discrete coordinate points, wherein n is a target order of the electric field to be calculated;
calculating a first order partial derivative of each obtained order of scattering field in the z-axis direction to an n order partial derivative of each obtained order of scattering field in the z-axis direction;
calculating an n-order electric field psi of the conductor rough surface according to the incident field and the n-order electric field expression from the first order partial derivative in the z-axis direction to the z-axis direction of each obtained scattering fieldn(r)；
The n-order electric field expression is:
wherein psii(r) is the incident field and,
is a scattering field of order l, belongs to {0,1,2,. n }, r is a position vector,
x is the abscissa of the discrete coordinate point, and z, f are the ordinate of the discrete coordinate point.
Optionally, the calculating the 0 th-order scattered field to the n th-order scattered field according to the abscissa and the ordinate of the M discrete coordinate points includes:
calculating an n-order scattering field according to the abscissa and the ordinate of the M discrete coordinate points and the following formula
Optionally, the step of calculating a first order partial derivative of each order of scattered field in the z-axis direction to an n-order partial derivative of each order of scattered field in the z-axis direction includes:
according to the calculated n-order scattered field
Calculating the spectral Density function An(kx)：
According to the calculated An(kx) And m-order partial derivatives in the z-axis direction of each order of the scattering field calculated by the following formula
Wherein x is the abscissa of the discrete coordinate point, k is the wave number of the electromagnetic wave in vacuum, and k isxIs the x-direction component of k, kzIs the z-direction component of k, and i is in imaginary units.
Optionally, the method further includes:
according to said spectral density function An(kx) The first order partial derivative in the x-axis direction of the resulting scattered field of each order is calculated by equation (1):
calculating the n-order electric field psi by formula (2)n(r) partial derivatives in the x-axis direction:
calculating the n-order electric field psi by formula (3)n(r) partial derivatives in the z-axis direction:
according to said spectral density function An(kx) Calculating the n-order scattering field by formula (4)
Mixed partial derivatives in the x-axis direction and in the z-axis direction:
according to
And
calculating the normal direction n of any point on the rough surface of the conductor by the formula (5)sCorresponding directional derivative
According to
Calculating a bistatic scattering coefficient sigma of the conductor asperity by equations (6) and (7)n(tapered wave)(θi,θs)：
Wherein, thetaiIs the angle of incidence, θsIs the scattering angle, L is the matte surface length, and g is the pyramid factor.
In a second aspect, the present application provides a rough-surface scattered field calculation apparatus, comprising:
a selecting module for selecting M with equal interval on the rough surface of the conductor with the length of LA discrete coordinate point, M2NN is a positive integer;
the first calculation module is used for calculating a scattering field from 0 order to n order according to the abscissa and the ordinate of the M discrete coordinate points, wherein n is a target order of the electric field to be calculated;
the second calculation module is used for calculating the first order partial derivative of each obtained order of scattering field in the z-axis direction to the n-order partial derivative of each obtained order of scattering field in the z-axis direction;
a third calculating module for calculating the n-order electric field psi of the conductor rough surface according to the incident field and the n-order electric field expression and the first-order partial derivative in the z-axis direction to the z-axis direction of each obtained scattering fieldn(r)；
The n-order electric field expression is:
wherein psii(r) is the incident field and,
is a scattering field of order l, belongs to {0,1,2,. n }, r is a position vector,
x is the abscissa of the discrete coordinate point, and z, f are the ordinate of the discrete coordinate point.
Optionally, the first calculating module is configured to:
calculating an n-order scattering field according to the abscissa and the ordinate of the M discrete coordinate points and the following formula
Optionally, the second calculating module is configured to:
according to the calculated n-order scattered field
Calculating the spectral Density function An(kx)：
According to the calculated An(kx) And m-order partial derivatives in the z-axis direction of each order of the scattering field calculated by the following formula
Wherein x is the abscissa of the discrete coordinate point, k is the wave number of the electromagnetic wave in vacuum, and k isxIs the x-direction component of k, kzIs the z-direction component of k, and i is in imaginary units.
Optionally, the method further includes:
a fourth calculation module to:
according to said spectral density function An(kx) The first order partial derivative in the x-axis direction of the resulting scattered field of each order is calculated by equation (1):
calculating the n-order electric field psi by formula (2)n(r) partial derivatives in the x-axis direction:
calculating the n-order electric field psi by formula (3)n(r) partial derivatives in the z-axis direction:
according to whatThe spectral density function An(kx) Calculating the n-order scattering field by formula (4)Mixed partial derivatives in the x-axis direction and in the z-axis direction:
according toAnd
calculating the normal direction n of any point on the rough surface of the conductor by the formula (5)sCorresponding directional derivative
According to
Calculating a bistatic scattering coefficient sigma of the conductor asperity by equations (6) and (7)n(tapered wave)(θi,θs)：
Wherein, thetaiIs the angle of incidence, θsIs the scattering angle, L is the matte surface length, and g is the pyramid factor.
The method and the device for calculating the electric field in the scattering problem of the rough surface of the conductor are characterized in that M discrete coordinate points with equal intervals on the rough surface of the conductor and with the length of L are selected, a 0-order scattered field to an n-order scattered field are respectively calculated according to the abscissa and the ordinate of the M discrete coordinate points, n is a target order of the electric field to be calculated, then a first-order partial derivative in the z-axis direction of each obtained scattered field to an n-order partial derivative in the z-axis direction are calculated, and finally the n-order electric field of the rough surface of the conductor is calculated according to an incident field, the obtained first-order partial derivative in the z-axis direction of each scattered field, and an n-order electric field expression. Therefore, the electric field of any order can be calculated, the calculation precision is high, the time complexity is reduced, and the calculation efficiency is improved.
Drawings
In order to more clearly illustrate the technical solutions in the present application or the prior art, the drawings needed to be used in the description of the embodiments or the prior art will be briefly introduced below, and it is obvious that the drawings in the following description are some embodiments of the present application, and it is obvious for those skilled in the art to obtain other drawings based on these drawings without inventive exercise.
FIG. 1 is a flowchart illustrating a first embodiment of a method for calculating an electric field in the present application for the problem of scattering from a rough surface of a conductor;
FIG. 2 is a schematic illustration of a one-dimensional conductor asperity;
FIG. 3 is a flowchart of a second embodiment of a method for calculating an electric field in the present application for the problem of scattering from a rough surface of a conductor;
FIG. 4a is a schematic diagram illustrating the simulation and comparison of the calculation accuracy of the 2 nd order perturbation method and the moment method;
FIG. 4b is a schematic diagram illustrating the simulation comparison of the calculation accuracy of the 4 th order perturbation method and the moment method;
FIG. 4c is a schematic diagram illustrating the simulation comparison of the calculation accuracy of the 8 th order perturbation method and the moment method;
FIG. 5 is a schematic diagram illustrating simulation comparison of computation time of the infinite perturbation method and the moment method according to the present application;
FIG. 6 is a schematic diagram of an electric field calculation apparatus according to a first embodiment of the present application for solving the problem of scattering from rough surfaces of conductors;
FIG. 7 is a schematic diagram illustrating a second embodiment of an electric field calculation apparatus for the problem of scattering from a rough surface of a conductor according to the present application.
Detailed Description
To make the purpose, technical solutions and advantages of the present application clearer, the technical solutions in the present application will be clearly and completely described below with reference to the drawings in the present application, and it is obvious that the described embodiments are some, but not all embodiments of the present application. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present application.
The method for calculating the electric field in the problem of scattering of the rough surface of the conductor in the application can also be called an infinite order perturbation method, which is hereinafter referred to as an infinite order perturbation method for short. The expansion form of the scattered field calculation formula can be infinite order series, the scattered field of any order can be calculated, the electric field is equal to the sum of the incident field and the scattered field, and therefore the electric field of any order can be calculated. For example, the method can be used for solving the electric field of the incident one-dimensional conductor rough surface of Transverse electric wave (TE), and has higher precision and calculation efficiency. The following describes in detail an electric field calculation method and apparatus in the scattering problem of conductor rough surface provided by the present application with reference to the accompanying drawings.
Fig. 1 is a flowchart of a first embodiment of a method for calculating an electric field in the problem of scattering from a rough surface of a conductor according to the present application, as shown in fig. 1, the method according to the present embodiment may include:
s101, selecting M discrete coordinate points with equal intervals on the rough surface of the conductor with the length of L, wherein M is 2NAnd N is a positive integer.
S102, respectively calculating a 0-order scattered field to an n-order scattered field according to the abscissa and the ordinate of the M discrete coordinate points, wherein n is the target order of the electric field to be calculated.
Specifically, the n-order scattering field is calculated according to the abscissa and the ordinate of the M discrete coordinate points and the following formula
And obtaining a scattering field of a corresponding order according to the value of n, and if n is equal to 8, obtaining an 8-order scattering field of the rough surface, and when n is equal to 8, calculating a 0-order scattering field, a 1-order scattering field and a 2-order scattering field … 8 according to the abscissa and the ordinate of the M discrete coordinate points.
Wherein psii(r) is the incident field, which is a known function, and the calculation formula of the incident field is as follows:
wherein, thetaiIs the angle of incidence, r is the position vector,
x is the abscissa of the discrete coordinate point and z is the ordinate of the discrete coordinate point.
S103, calculating a first order partial derivative of each obtained order of scattering field in the z-axis direction to an n-order partial derivative of each obtained order of scattering field in the z-axis direction.
S103 may specifically be calculating a 0 th order scattered field
Calculating the first order partial derivatives in the axial direction, the second order partial derivatives in the z-axis direction, … and the n order partial derivatives in the z-axis direction to obtain a 1 order scattering field
Calculating the first order partial derivatives in the axial direction, the second order partial derivatives in the z-axis direction, … and the n-order partial derivatives in the z-axis direction to obtain a 2 nd order scattering field
Axle squareCalculating the first order partial derivatives upwards, the second order partial derivatives in the z-axis direction, … and the n-order partial derivatives in the z-axis direction, …, and calculating the n-order scattered field
First order partial derivatives in the axial direction, second order partial derivatives in the z-axis direction, …, and n-order partial derivatives in the z-axis direction.
In particular, from the calculated n-order scattered field
Calculating the spectral Density function An(kx)：
According to the calculated An(kx) And m-order partial derivatives in the z-axis direction of each order of the scattering field calculated by the following formula
Wherein x is the abscissa of the discrete coordinate point, k is the wave number of the electromagnetic wave in vacuum, and k isxIs the x-direction component of k, kzIs the z-direction component of k, and i is in imaginary units.
S104, calculating an n-order electric field psi of the rough surface of the conductor according to the incident field and the obtained first-order partial derivatives in the z-axis direction of each-order scattered field to n-order partial derivatives in the z-axis direction and n-order electric field expressionsn(r)；
The n-order electric field expression is:
wherein psii(r) is the incident field and,is a scattering field of order l, belongs to {0,1,2,. n }, r is a position vector,x is the abscissa of the discrete coordinate point, and z, f are the ordinate of the discrete coordinate point.
By the operations of S101-S104, an electric field of an arbitrary order can be obtained.
The electric field calculation method in the conductor rough surface scattering problem provided in this embodiment includes selecting M discrete coordinate points with equal intervals on a conductor rough surface having a length L, calculating a 0 th order scattering field to an n th order scattering field according to abscissa and ordinate of the M discrete coordinate points, where n is a target order of an electric field to be calculated, then calculating an n th order partial derivative from a first order partial derivative in a z-axis direction of each obtained order scattering field to the z-axis direction, and finally calculating an n th order electric field of the conductor rough surface according to an incident field, the n th order partial derivative from the first order partial derivative in the z-axis direction of each obtained order scattering field to the z-axis direction, and an n th order electric field expression. Therefore, the electric field of any order can be calculated, the calculation precision is high, the time complexity is reduced, and the calculation efficiency is improved. The dimension of the integral is controlled in one dimension, the purpose of accurate calculation is achieved, and due to the fact that the dimension of the integral is low, the expandability is strong, and the problem of two-dimensional and three-dimensional scattering can be solved.
Generally, the two-station scattering coefficient of the conductor asperity is a parameter that characterizes the computational accuracy of the electrical field across the asperity, and further wherein the method comprises:
s105, according to the spectral density function An(kx) The first order partial derivative in the x-axis direction of the resulting scattered field of each order is calculated by equation (1):
s106, calculating the n-order electric field psi through the formula (2)n(r) partial derivatives in the x-axis direction:
s107, calculating the n-order electric field psi through the formula (3)n(r) partial derivatives in the z-axis direction:
s108, according to the spectral density function Al(kx) Calculating the n-order scattering field by formula (4)
Mixed partial derivatives in the x-axis direction and in the z-axis direction:
s109, according to
And
calculating the normal direction n of any point on the rough surface of the conductor by the formula (5)sCorresponding directional derivative
S110, according to
Calculating the double-station scattering coefficient sigma of the conductor rough surface through the formulas (6) and (7)n(tapered wave)(θi,θs)：
Wherein, thetaiIs the angle of incidence, θsIs the scattering angle, L is the matte surface length, and g is the pyramid factor.
The double-station scattering coefficient sigma of the rough surface of the conductor is calculatedn(tapered wave)(θi,θs) The calculation accuracy of the electric field can be obtained.
The following describes the technical solution of the embodiment of the method shown in fig. 1 in detail by using a specific embodiment.
FIG. 2 is a schematic view of a one-dimensional conductor asperity having a length L and x-z axes, θiIs the angle of incidence, θsIs the scattering angle, nsIs the normal direction of any point on the rough surface,. psiiIs an incident field,. phisIs the scattered field. The following description will be given by taking an example of calculating an 8-step electric field of the one-dimensional conductor rough surface.
Fig. 3 is a flowchart of a second embodiment of the electric field calculation method in the problem of scattering from a rough surface of a conductor according to the present application, and as shown in fig. 3, the method according to the present embodiment may include:
s201, selecting M discrete coordinate points with equal intervals on the rough surface of the conductor with the length L, wherein M is 2NAnd N is a positive integer.
S202, calculating a 0-order scattered field to an 8-order scattered field according to the abscissa and the ordinate of the M discrete coordinate points.
The specific calculation formula can be referred to as the calculation formula in S102, and details are not repeated here.
And S203, calculating a first order partial derivative of each obtained order scattered field in the z-axis direction to an 8 th order partial derivative in the z-axis direction.
The specific calculation process and calculation formula can be referred to as S103, which is not described herein again.
S204, calculating an 8-order electric field psi of the rough surface of the conductor according to the incident field and the obtained first-order partial derivatives in the z-axis direction of each-order scattered field to 8-order partial derivatives in the z-axis direction and an n-order electric field expressionn(r)；
n order of electricityThe field expression is:
and S205, calculating the double-station scattering coefficient of the conductor rough surface according to the calculated 8-order electric field.
The specific calculation process can be seen in S105 to S110, which are not described herein again.
The performance simulation diagram is adopted to compare the calculation accuracy of the infinite perturbation method and the moment method, the moment method is a standard calculation method in the field and is a classical numerical method, the basic idea is that firstly, a proper electric field or magnetic field integral equation is selected, then an expansion function is used for expanding unknown quantity, and finally, a weight function is used for carrying out inner product on two sides of the integral equation, so that the integral equation is converted into a matrix equation to be solved. Fig. 4a is a schematic diagram showing comparison of simulation of calculation accuracy of a 2-order perturbation method and a moment method, fig. 4b is a schematic diagram showing comparison of simulation of calculation accuracy of a 4-order perturbation method and a moment method, fig. 4c is a schematic diagram showing comparison of simulation of calculation accuracy of an 8-order perturbation method and a moment method, in fig. 4a to 4c, HOSPM refers to an infinite perturbation method and MOM refers to a moment method, and it can be seen from fig. 4a to 4c that the calculation accuracy of the infinite perturbation method is consistent with the moment method, especially, the calculation accuracy is highest at a mirror image, i.e., a plane reflection, which is an angle of a scattering coefficient of a single station, and is an angle most concerned in radar detection.
FIG. 5 is a schematic diagram illustrating simulation of computation time of the infinite perturbation method and the moment method, in FIG. 5, HOSPM indicates the infinite perturbation method, MOM indicates the moment method, the abscissa indicates the number of points of discrete coordinate points, also called the number, and the ordinate indicates the computation time, as shown in FIG. 5, the infinite perturbation method is more advantageous than the moment method in computation time, and theoretically, the computation time complexity of the moment method is O (N) (N is a unit of time)3) The infinite order perturbation rule is O (N), the time complexity is reduced, and the calculation efficiency is improved.
The electric field calculation method in the problem of conductor rough surface scattering provided by the application has the advantages that the physical concept is more definite, the universality of a mathematical formula is stronger, the electric field of any order can be theoretically calculated, the calculation precision is higher, the time complexity is reduced, and the calculation efficiency is improved.
Fig. 6 is a schematic structural diagram of a first embodiment of an electric field calculation apparatus for the problem of scattering from a rough surface of a conductor according to the present application, as shown in fig. 6, the apparatus of the present embodiment includes: a selection module 11, a first calculation module 12, a second calculation module 13 and a third calculation module 14, wherein,
the selecting module 11 is configured to select M discrete coordinate points with equal intervals on a conductor rough surface with a length L, where M is 2NAnd N is a positive integer.
The first calculating module 12 is configured to calculate a scattering field from 0 th order to n th order according to the abscissa and the ordinate of the M discrete coordinate points, where n is a target order of the electric field to be calculated.
The second calculating module 13 is configured to calculate a first order partial derivative in the z-axis direction to an n-order partial derivative in the z-axis direction of each obtained order of scattered field.
The third calculating module 14 is configured to calculate an n-order electric field ψ of the rough surface of the conductor according to the incident field, the first-order partial derivative in the z-axis direction of each obtained scattering field, the n-order partial derivative in the z-axis direction, and the n-order electric field expressionn(r)；
The n-order electric field expression is:
wherein psii(r) is the incident field and,is a scattering field of order l, belongs to {0,1,2,. n }, r is a position vector,x is the abscissa of the discrete coordinate point, and z, f are the ordinate of the discrete coordinate point.
Further, the first calculation module 12 is configured to:
calculating an n-order scattering field according to the abscissa and the ordinate of the M discrete coordinate points and the following formula
The second calculation module 13 is configured to:
according to the calculated n-order scattered field
Calculating the spectral Density function An(kx)：
According to the calculated An(kx) And m-order partial derivatives in the z-axis direction of each order of the scattering field calculated by the following formula
Wherein x is the abscissa of the discrete coordinate point, k is the wave number of the electromagnetic wave in vacuum, and k isxIs the x-direction component of k, kzIs the z-direction component of k, and i is in imaginary units.
The apparatus of this embodiment may be used to implement the technical solution of the method embodiment shown in fig. 1, and the implementation principle and the technical effect are similar, which are not described herein again.
Generally, the two-station scattering coefficient of the conductor rough surface is a parameter for characterizing the calculation accuracy of the electric field of the rough surface, fig. 7 is a schematic structural diagram of an embodiment of an electric field calculation apparatus in the present application for the scattering problem of the conductor rough surface, as shown in fig. 7, the apparatus of this embodiment further includes, on the basis of the embodiment shown in fig. 6: a fourth calculation module 15, the fourth calculation module 15 being configured to:
according to the spectral density function An(kx) Calculating the x-axis direction of each order of the obtained scattered field by the formula (1)First order partial derivatives of (1):
calculating the n-order electric field psi by the formula (2)n(r) partial derivatives in the x-axis direction:
calculating the n-order electric field psi by formula (3)n(r) partial derivatives in the z-axis direction:
according to the spectral density function An(kx) Calculating the n-order scattering field by formula (4)
Mixed partial derivatives in the x-axis direction and in the z-axis direction:
according to
And
calculating the normal direction n of any point on the rough surface of the conductor by the formula (5)sCorresponding directional derivative
According to
Calculating the double-station scattering coefficient sigma of the conductor rough surface through the formulas (6) and (7)n(tapered wave)(θi,θs)：
Wherein, thetaiIs the angle of incidence, θsIs the scattering angle, L is the matte surface length, and g is the pyramid factor.
The double-station scattering coefficient sigma of the rough surface of the conductor is calculatedn(tapered wave)(θi,θs) The calculation accuracy of the electric field can be obtained.
Those of ordinary skill in the art will understand that: all or a portion of the steps of implementing the above-described method embodiments may be performed by hardware associated with program instructions. The program may be stored in a computer-readable storage medium. When executed, the program performs steps comprising the method embodiments described above; and the aforementioned storage medium includes: various media that can store program codes, such as ROM, RAM, magnetic or optical disks.
Finally, it should be noted that: the above embodiments are only used for illustrating the technical solutions of the present application, and not for limiting the same; although the present application has been described in detail with reference to the foregoing embodiments, it should be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some or all of the technical features may be equivalently replaced; and the modifications or the substitutions do not make the essence of the corresponding technical solutions depart from the scope of the technical solutions of the embodiments of the present application.
Claims (8)
1. A method for calculating an electric field in a conductor rough surface scattering problem is characterized by comprising the following steps:
selecting the conductor with the length L on the rough surface and the likeM discrete coordinate points, M being 2NN is a positive integer;
respectively calculating a scattering field from 0 order to n order according to the abscissa and the ordinate of the M discrete coordinate points, wherein n is a target order of the electric field to be calculated;
calculating a first order partial derivative of each obtained order of scattering field in the z-axis direction to an n order partial derivative of each obtained order of scattering field in the z-axis direction;
calculating an n-order electric field psi of the conductor rough surface according to the incident field and the n-order electric field expression from the first order partial derivative in the z-axis direction to the z-axis direction of each obtained scattering fieldn(r)；
The n-order electric field expression is:
wherein psii(r) is the incident field and,
is a scattering field of order l, belongs to {0,1,2,. n }, r is a position vector,
x is the abscissa of the discrete coordinate point, and z, f are the ordinate of the discrete coordinate point.
2. The method of claim 1, wherein calculating the 0 th-nth order fringe field from the abscissa and the ordinate of the M discrete coordinate points comprises:
calculating an n-order scattering field according to the abscissa and the ordinate of the M discrete coordinate points and the following formula
3. The method of claim 2, wherein the calculating a first order partial derivative in a z-axis direction to an n-order partial derivative in the z-axis direction of each order of the fringe field comprises:
according to the calculated n-order scattered fieldCalculating the spectral Density function An(kx)：
According to the calculated An(kx) And m-order partial derivatives in the z-axis direction of each order of the scattering field calculated by the following formula
Wherein x is the abscissa of the discrete coordinate point, k is the wave number of the electromagnetic wave in vacuum, and k isxIs the x-direction component of k, kzIs the z-direction component of k, and i is in imaginary units.
4. The method of claim 3, further comprising:
according to said spectral density function An(kx) The first order partial derivative in the x-axis direction of the resulting scattered field of each order is calculated by equation (1):
calculating the n-order electric field psi by formula (2)n(r) partial derivatives in the x-axis direction:
calculating the n-order electric field psi by formula (3)n(r) partial derivatives in the z-axis direction:
according to said spectral density function An(kx) Calculating the n-order scattering field by formula (4)
Mixed partial derivatives in the x-axis direction and in the z-axis direction:
according to
And
calculating the normal direction n of any point on the rough surface of the conductor by the formula (5)sCorresponding directional derivative
According toCalculating a bistatic scattering coefficient sigma of the conductor asperity by equations (6) and (7)n(tapered wave)(θi,θs)：
Wherein, thetaiIs the angle of incidence, θsIs the scattering angle, L is the matte surface length, and g is the pyramid factor.
5. An electric field calculation apparatus for use in the problem of scattering from conductor asperities, comprising:
a selecting module for selecting M discrete coordinate points with equal interval on the rough surface of the conductor with the length of L, wherein M is 2NN is a positive integer;
the first calculation module is used for calculating a scattering field from 0 order to n order according to the abscissa and the ordinate of the M discrete coordinate points, wherein n is a target order of the electric field to be calculated;
the second calculation module is used for calculating the first order partial derivative of each obtained order of scattering field in the z-axis direction to the n-order partial derivative of each obtained order of scattering field in the z-axis direction;
a third calculating module for calculating the n-order electric field psi of the conductor rough surface according to the incident field and the n-order electric field expression and the first-order partial derivative in the z-axis direction to the z-axis direction of each obtained scattering fieldn(r)；
The n-order electric field expression is:
wherein psii(r) is the incident field and,
is a scattering field of order l, belongs to {0,1,2,. n }, r is a position vector,
x is the abscissa of the discrete coordinate point, and z, f are the ordinates of the discrete coordinate point。
6. The apparatus of claim 5, wherein the first computing module is configured to:
calculating an n-order scattering field according to the abscissa and the ordinate of the M discrete coordinate points and the following formula
7. The apparatus of claim 6, wherein the second computing module is configured to:
according to the calculated n-order scattered field
Calculating the spectral Density function An(kx)：
According to the calculated An(kx) And calculating the n-order partial derivatives of each order of scattering field in the z-axis direction by the following formula
Wherein x is the abscissa of the discrete coordinate point, k is the wave number of the electromagnetic wave in vacuum, and k isxIs the x-direction component of k, kzIs the z-direction component of k, and i is in imaginary units.
8. The apparatus of claim 7, further comprising:
a fourth calculation module to:
according to said spectral density function An(kx) The first order partial derivative in the x-axis direction of the resulting scattered field of each order is calculated by equation (1):
calculating the n-order electric field psi by formula (2)n(r) partial derivatives in the x-axis direction:
calculating the n-order electric field psi by formula (3)n(r) partial derivatives in the z-axis direction:
according to said spectral density function An(kx) Calculating the n-order scattering field by formula (4)Mixed partial derivatives in the x-axis direction and in the z-axis direction:
according to
Andcalculating the normal direction n of any point on the rough surface of the conductor by the formula (5)sCorresponding directional derivative
According to
Calculating a bistatic scattering coefficient sigma of the conductor asperity by equations (6) and (7)n(tapered wave)(θi,θs)：
Wherein, thetaiIs the angle of incidence, θsIs the scattering angle, L is the matte surface length, and g is the pyramid factor.
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