AbstractThis paper reviews the Green's function theory and Green's function method in electromagnetism. Firstly, the Green's function corresponding to the scalar wave equation is introduced, and the shock response function in the signal system is analogous. Secondly, the parallel vector Green's function corresponding to the vector wave equation is introduced, including the Green's function in homogeneous and inhomogeneous media, and the relationship between it and the reciprocity theorem and equivalence principle in electromagnetic theory is discussed. Finally, the theory of Green's function considering the motion effect of the medium and the quantum effect of the electromagnetic field is introduced. In addition, the application of electromagnetic Green's function in wireless communication, electromagnetic compatibility and other engineering fields is also discussed.
Key words: Green's function, medium, electromagnetic field, motion effect, quantum effect, wireless communication, electromagnetic compatibility
乔治·格林(George Green，1793—1841)在他1828年出版的论文“An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism”中曾说过：“Although many of the artifices employed in the works before mentioned are remarkable for their elegance, it is easy to see they are adapted only to particular objects, and that some general method, capable of being employed in every case, is still wanting (虽然前述作品中采用的众多技术手段以其优雅而著称，但不难发现这些技术仅适用于特定场合。 因此，我们迫切需要一种更为通用的方法，能够适应各种不同的应用场景)”。 他意识到，虽然当时的实验方法在解释特定现象方面非常成功，但它们无法提供一个统一的理论框架来解释所有的电和磁现象。 格林提出的方法旨在填补这个空白，通过数学分析提供一个更通用的理论框架，从而帮助科学家更全面地理解电磁现象。 格林的工作对后来的科学家，如麦克斯韦等，产生了深远的影响，他们进一步发展了这些理论，形成了现代电磁理论的基础。 格林在他的论文中提出了现在被称为“格林函数”的概念，可用来解决物理学中的各种问题，特别是边界值问题。
The so-called Green's function refers to the "effect" produced by a "point source of unit intensity", and if the distribution source is divided into the superposition of many point sources of different intensities, the effect it produces is the superposition of the effects produced by these point sources, which is the Green's function method. Therefore, the Green's function method is easy to analytically represent, and the formulation of many problems is more concise and elegant, and it is easier to understand physically. It is a general method that has been applied to many fields such as differential equations, electromagnetism, quantum mechanics, seismology, etc., and has important theoretical significance and engineering application value especially in the field of electromagnetism.
1 Green's function of the scalar wave equation
Assuming that there is a continuous, oscillating charge density distribution ρ(r)/ε0 in free space (whose time factor is exp(jωt); here, divided by the dielectric constant ε0 in free space to be consistent with the common Green's function formulation), can be expressed as a linear superposition of point charges:
The oscillation potential generated by the point charge δ (r-r ') at r ' is denoted as g(r, r ') and according to the principle of linear superposition, the oscillation potential resulting from this continuous charge distribution is
Here, g(r, r ') is the scalar Green's function of the scalar wave equation.
For a more rigorous derivation of equation (2), we can use the definition of the scalar wave equation and the scalar Green's function in free space [1,2]:
Here, k0=ω/c is the wavenumber of free space. Multiply Eq. (3) by the potential φ(r ') subtract Eq. (4) by the Green's function g(r, r ') and integrate the source r'; considering that the Laplace operator∇2 is a symmetry operator (area fraction at the boundary at infinity
is 0, i.e., the Sommerfeld radiation condition
), we get equation (2).
In free space, the oscillation potential generated by the point source is g(r, r ')=exp(-jk0R)/4πR, where R=|r-r '| is the distance between the source point and the field point. On the one hand, when the frequency approaches 0, the oscillation term exp(-jk0R) in this potential degenerates to 1, and the Green's function of the scalar wave equation degenerates into the Green's function gs(r, r ')=1/4πR of the electrostatic field. On the other hand, it is not difficult to find that the Green's function g(r, r ') in free space satisfies the spatial translation invariance. Thus, equation (2) can be written as a convolution:
Equation (5) can be compared to the relationship between the input signal x(t) and the output signal y(t) of a linear, time-translationally invariant system:
Here, the impulse response function h of the signal system is analogous to the Green's function g of the scalar wave equation.
It is worth noting that, compared with the Green's function in free space, the Green's function in any linear and inhomogeneous medium does not satisfy the spatial translational symmetry (non-uniformity), but the Green's function formula (2) of the scalar wave equation still holds true (linear). Once the medium satisfies the spatial translational symmetry, the potential can be written in the form of Green's function and charge distribution convolution, which can be efficiently calculated using fast Fourier transform, which is convenient for engineering applications. The same is true for the parallel vector Green's function of electromagnetic fields.
2 The parallel vector Green's function of the electromagnetic field
In order to obtain the Green's function of the electromagnetic field, we first need to answer a question: What kind of electric field will be generated by a current source polarized in the x-direction of free space? The answer is that the electric field in the three polarization directions of x, y, and z will be generated! Therefore, a current source in any polarization direction can be decomposed into x, y, z three polarization directions, and each polarization direction source will produce a field in x, y, z three polarization directions. If the source of any polarization direction is written as a column vector (Jx, Jy, Jz) t with 3 rows ×1 column, the electric field generated by it is also written as a column vector (Ex, Ey, Ez) t with 3 rows × 1 column. Since Maxwell's equations are linear, the Green's function must be a matrix of 3 rows × 3 columns, where columns 1(2, 3) correspond to the vector electric field generated by the x(y, z) polarization current. If the electric field and current are still expressed in vector form, the Green's function of the electromagnetic field is a parallel-vector form, which is called the parallel-vector Green's function [2,3]. where the parallel vector is the second-order tensor, which can be expressed as a matrix or as a tensor product of two vectors.
Consider the parallel vector Green's function of the time-harmonic electromagnetic field
, which is rigorously derived from a scalar field, can be defined by the vector wave equation of free space and the corresponding parallel vector Green's function:
Here, μ0 is the permeability of free space. Using the symmetry of the double-curl operator (the area at the infinity boundary is divided into 0, i.e., the Sommerfeld radiation condition), it can be obtained
The parallel vector Green's function in free space has an analytic form, ie
Here, g(r, r ') is the scalar Green's function corresponding to the scalar wave equation. The parallel-vector Green's function corresponding to the time-harmonic field
, which is expanded in the spherical coordinate system, we can obtain three items containing 1/R3, 1/R2, and 1/R, which correspond to the physical meanings of induced near field, radiated near field (midfield), and far field, respectively. The attenuation law of the induced near-field 1/R3 is essentially the attenuation law of the electric dipole source in the electrostatic field, and the attenuation law of the far-field 1/R satisfies the conservation of power (the spherical area of the Poyinting vector is divided into constants). In addition, the characteristics of electromagnetic waves in the induced near-field region are mainly directly affected by the source, and the wave nature is not obvious, and the main occurrence in the region is the storage and release of energy, rather than the continuous transmission of energy. Therefore, the induced near-field region is more related to virtual or reactive power, representing a reciprocating exchange of energy rather than actual energy transfer. In the far-field region, energy propagates in the form of real power, that is, energy propagates outward from the source in the form of waves, which is the basis for applications such as wireless communications and radar systems.
According to Faraday's theorem of electromagnetic induction (∇×E=-jωμ0H), the magnetic field in free space can be expressed as
Over here
This is called the magnetic and vector Green's function;
can be recorded
, which is called the electrically parallel vector Green's function. According to the symmetry of the operator, i.e., the gradient operator is an antisymmetric operator, and the double gradient operator is a symmetric operator, it can be seen that the electric and vector Green's function is a symmetric operator, while the magnetic and vector Green's function is an anti-symmetric operator. According to the symmetry of the electric parallel vector Green's function operator, the reciprocal theorem of the electromagnetic field [1,2] can be deduced, i.e
。 In the circuit analysis, the voltage can be obtained by the line integral of the electric field, and the current can be obtained by the area fraction of the current density, so the reciprocity theorem V1/I1=V2/I2 of the circuit is obtained, and the conclusion of the mutual impedance of the antenna is also obtained.
For any linear, inhomogeneous medium (non-magnetic), the following electrically parallel vector Green's function is defined:
Then the expression of the electric field (9) and the expression of the magnetic field (11) are still true. thereinto
is the dielectric constant tensor of an anisotropic medium. However, in a non-uniform medium, the spatial translational symmetry of the free-space parallel vector Green's function cannot be maintained, and Equation (12) does not have an analytical solution in general, and can only be calculated numerically.
In the right-angled, cylindrical, and spherical coordinate systems, the plane layering, cylindrical layering, spherical layering media, and the parallel vector Green's function can be analytically expressed, and one of the basic ideas is to extend the parallel vector Green's function into a feature mode, taking the electric parallel vector Green's function as an example, its electric field mode can be solved as follows:
The electric field modes here include a longitudinal mode (zero space, kλ=0) and a transverse mode (kλ≠0). These modes may be continuous or discrete, depending on the boundary conditions of the electromagnetic system (open or closed). According to the orthogonal completeness of the eigenmode, the vector Green's function can be expanded
Here the summation sign represents the contribution of the discrete spectrum and the integral sign represents the contribution of the continuum.
3 Parallel Vector Green's Function and Equivalence Principle
According to the dual principle [1,2], magnetic current is introduced, and the dual forms of equations (9) and (11) are
Combining equations (9) and (16) and (11) and (15), we can obtain the parallel vector Green's function expression of the electric and magnetic fields [1,2]:
Consider the electromagnetic interference problem of RF chips in free space (Figure 1(a)). The chip can be surrounded by a closed surface or an infinitely large plane at the outer boundary of the closed surface (the outer normal direction
) to define the equivalent current
and equivalent magnetic current
(Figure 1(b)). When the spot point r is located at the outer boundary or outer region of the closed surface, the surface integral form of equations (17), (18) still holds (the volume integral becomes the area integral), which is called the equivalence principle or Huygens principle. When the spot point r is located at the inner boundary or inner region of a closed surface, the electromagnetic field to the left of equations (17), (18) is zero, which is called the vanishing theorem. The equivalence principle or Huygens principle has a definite physical meaning (every point on the wavefront can be seen as a new wave source), while the vanishing theorem is a mathematical conclusion or the result of an analytical calculation.
Fig.1 Electromagnetic interference of RF chips in free space (a) Original problem: Analyze the electromagnetic interference generated by RF chips in free space (the antenna effect of RF chips is generated due to unreasonable design, and the conduction current in some traces or connecting lines will radiate electromagnetic waves); (b) Equivalent problem 1: The electromagnetic radiation from the outer region of an arbitrarily closed surface is equivalent to the electromagnetic radiation generated by the equivalent current and the equivalent magnetic current at the outer boundary of the closed surface. (c) Equivalent problem 2: The closed surface is filled with an ideal electrical conductor, and the electromagnetic radiation in the outer region is equivalent to the electromagnetic radiation generated by the equivalent magnetic current on the outer boundary. (d) Equivalent problem 3: The electromagnetic radiation in the outer region is equivalent to the equivalent current on the outer boundary. Calculate the parallel vector Green's function after filling an ideal magnetic conductor with free space
According to the uniqueness theorem of time-harmonic electromagnetic fields [1,2], the solution of the electromagnetic field is unique if a tangential electric field or tangential magnetic field is given on the source and its boundary surface in a region. Therefore, for the electromagnetic interference problem of the above-mentioned RF chips, if only the equivalent current or equivalent magnetic current is given (equivalent to only the tangential magnetic field or tangential electric field), can the radiated field in the external region be found? On the one hand, if the equivalent current on the boundary surface
and equivalent magnetic current
Known, then the external field is actually independent of the internal medium, so it can be regarded as the same as the inside and the outside (both are free space), so the parallel vector Green's function in equations (17), (18) is still the Green's function in free space.
On the other hand, in order to facilitate engineering applications (only the tangential electric field or tangential magnetic field on the boundary surface of a closed surface is measured), the inner region can be filled with an ideal electrical conductor or an ideal magnetic conductor (Fig. 1(c), (d)), so that the equivalent current term or equivalent magnetic current term in equation (17) disappears, but the corresponding parallel vector Green's function needs to be modified or recalculated to satisfy the boundary conditions, respectively
or
。 For an infinite plane, the correction of the parallel vector Green's function is very simple when the inner (lower) region is filled with an ideal electrical conductor or an ideal magnetic conductor, as long as the front of the equation is multiplied by a factor of 2, and the free-space parallel vector Green's function can still be used, which is actually the result of applying the Schelkunoff equivalent and the mirror image method [4]. Therefore, for infinitely large planes, there are two simpler forms of equivalent principles with important engineering applications:
4 Parallel Vector Green's Function and Moving Target
When a single-frequency electromagnetic signal is scattered by a non-uniform moving target in free space, the spectrum expands to form a Doppler spectrum. In free space, a single frequency will be incident on a plane wave
From the laboratory coordinate system K by Lorentz transformation
After switching to the follow-up coordinate system K', which moves at a constant velocity v, it still has
of plane waves. Over here
and
and (t ', r ') and (t, r) satisfy the following Lorentz transform [5]:
thereinto
。 From the above equation, we can see the similarity between the "time and frequency" and "space and wave vector" Lorentz transform forms. The moving target is stationary in its follower coordinate system, so the equivalent current of the moving target under the action of the transformed incident wave can be solved directly by conventional computational electromagnetics method in the follower coordinate system
and equivalent magnetic current
。 Make use of static scalar Green's functions
Calculate the scattered electromagnetic field in the follower coordinate system
and
, and then by the Lorentz inverse transform
, the target scattering field is converted into the laboratory coordinate system, and the scattering electromagnetic field of the moving target in the same laboratory coordinate system as the original incident field is obtained
and
[6]，
Here, the matrix operator
and
It constitutes the static parallel vector Green's function in the follower coordinate system, that is, the form of the parallel vector Green's function in equations (17) and (18). In free space, the Lorentz transform operator of the electromagnetic field is [7]
The operator maps the electromagnetic field (E ', H ') t in the follower coordinate system to the electromagnetic field (E, H)t in the laboratory coordinate system. It is worth noting that since the inhomogeneous moving target is equivalent to the electromagnetic current in the follower coordinate system, the constitutive relation of the electromagnetic field in free space does not change, i.e., D=ε0E, B=μ0H, D '=ε0E ', B '=μ0H '. The scattering field in the laboratory coordinate system K is a time-domain signal, and the Doppler spectrum of the scattered electromagnetic field can be obtained through the Fourier transform, and then the longitudinal and transverse Doppler frequency shift characteristics of the target can be used to realize target detection and recognition. If the main focus is on the Doppler spectrum of the scattered electromagnetic field, the computational efficiency of equation (21) can be improved by switching the operator order.
5 Parallel Vector Green's Function and Wireless Communication Channel
Massive multiple-input, multiple-output (MIMO) antenna arrays are an important part of 5G/6G wireless communication systems. Current research hotspots include holographic MIMO, continuous aperture MIMO, and near-field MIMO. To analyze the communication degree of freedom limit of finite physical aperture MIMO antenna arrays, the parallel vector Green's function method can be used [8—11]. The channel matrix that connects the transmitting antenna array and the receiving antenna array is the parallel vector Green's function matrix. Firstly, the electric field at the receiving antenna can be calculated by the parallel vector Green's function and the source or equivalent source of the transmitting antenna, and secondly, the power density at the receiving antenna is proportional to the electric field strength. Assuming that the number of transmitting and receiving antennas is Nt and Nr respectively, the discrete matrix form of the above physical process can be written as
Here, Jd is the column vector of 3Nt×1, which represents the (equivalent) current source at the Nt transmit points (for approximating the transmitting source at the Nt transmitting antennas), and Ed is the column vector of 3Nr×1, which represents the vector electric field at the Nr receiving points (for approximating the electric field at the Nr receiving antennas),
is a matrix of 3Nr×3Nt, which represents the parallel vector Green's function matrix corresponding to the complex inhomogeneous channel.
is the correlation matrix corresponding to the transmitting antenna, which is Ermitic and can be diagonalized, i.e
。 After diagonalization, equation (23) is written as an inner product of quantum mechanics:
It can be seen that the information propagates in each independently orthogonal eigenmode channel, and the amount of information transmitted by each mode depends on the magnitude of the corresponding eigenvalue and the projection of the emission current in the eigenmode. The diagonalization method here is essentially consistent with the precoding technique in wireless communication [11]. If the eigenvalue elements in the eigenvalue matrix λ are arranged from largest to smallest, the limit of the degree of freedom of the channel can be analyzed, that is, the number of significant eigenvalues (the mode channel corresponding to the small eigenvalue will be submerged in the noise). One question left to the reader: known correlation matrices
The eigenmode of can be used as the optimal transmission mode for the transmitting antenna array; if the channel is reciprocal (
，
), then the optimal receiving mode of the receiving antenna array should be the eigenmode of another correlation matrix, please give the mathematical expression of the correlation matrix.
6 Parallel Vector Green's Function and Quantum Electrodynamics
One of the most important applications of the parallel vector Green's function in quantum electrodynamics is to analyze the quantum vacuum fluctuations of electromagnetic fields. The quantum vacuum fluctuation of electromagnetic fields is related to physical phenomena such as spontaneous radiation, Lamb shift, Casimir force, etc. Due to the dissipative nature of the medium, it will absorb the fluctuating electromagnetic waves in the vacuum and generate a fluctuating noise current, which in turn will radiate the fluctuating electromagnetic waves. The noise current (also known as Langzhiwanyuan) gives the medium the characteristics of gain, compensates for the loss of the medium, so that the process of interaction between the field and the matter satisfies the conservation of energy in the sense of the overall average, and reaches a statistical equilibrium between the absorption and radiation of electromagnetic waves. According to the fluctuation dissipation theorem, the fluctuation (correlative) characteristics of the noise current are as follows [12]:
Over here
is the average photon state energy. When the temperature approaches absolute zero, the average photon energy is the zero energy Hω/2 (the average number of photons is 0, no photons), and the zero energy is the root of the quantum vacuum fluctuation, and when the temperature is high enough, the average photon state energy is approximately the classical kBT form. In addition, the fluctuation of the noise current is proportional to the imaginary part of the dielectric constant of the medium, which is related to the dissipation characteristics of the medium. With the fluctuation of electric current, how to obtain the fluctuation of the electromagnetic field? Or use the parallel vector Green's function in a non-uniform medium! According to equations (9) and (25), we can obtain:
Using Equation (12) and its conjugated form, it is possible to deduce:
(27)
Finally, the vacuum fluctuation of the electric field is proportional to the imaginary part of the parallel vector Green's function in a non-uniform medium [12—14], and its value is regular finite (no singularity).
The trace of the imaginary part of the vector Green's function is related to the photon local density of states and the spontaneous emissivity of the atom at the spatial position (r = r ') of the atom, resulting in the broadening of the atomic spectral line. Different from the imaginary part of the parallel vector Green's function, its real part is singular (r = r '), which is related to the Lamb shift phenomenon (the movement of atomic spectral lines), and the imaginary part of the parallel vector Green's function can be numerically transformed by Hilbert to obtain the real part of the parallel vector Green's function.
Interestingly, if the spatially uncorrelated noise current is analogized to an uncorrelated random bit stream sent by different users, then equation (28) directly gives the electric field correlation function in wireless communication. For example, for a two-dimensional electromagnetic problem, the free-space Green's function (
) is a Bézier function of order 0
This is the famous Clarke channel model [15]. The model assumes that the phases of the plane wave field arriving from different directions are random and uniformly distributed, and that the probability of the occurrence of the plane wave field is equal at each angle, which happens to be closely related to the statistical distribution characteristics of noise sources in free space. Quantum electrodynamics and wireless communication theory are actually linked, and this is the surprise that the Green function brings us!
The author finds that in the development of the theory and method of electromagnetic Green's function, Chinese scholars such as Kong Jinou, Dai Zhenduo, Zeng Liang, Zhou Yongzu, and Li Lewei, as well as domestic scholars such as Lin Weigan, Fang Gangnan, Liang Changhong, Nie Zaiping, and Gong Shuxi, have made important contributions. The author also sincerely hopes that more young scholars in China will make innovative work in this field.
Acknowledgments We would like to thank Professor Chen Xudong of the National University of Singapore and Professor Xia Mingyao of Peking University for their valuable comments. Thanks to Yijia Cheng from Zhejiang University for helping to draw the illustrations.
Author: Sha Wei Ying Lei Xiao Gaobiao
School of Information and Electronic Engineering, Zhejiang University
School of Physics, Zhejiang University
School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University
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