Surface integral self-consistent coupling analysis method, device, equipment and storage medium

Abstract

The embodiment of the present invention discloses a surface integral self-consistent coupling analysis method, device, equipment and storage medium, wherein the surface integral self-consistent coupling analysis method includes: establishing a unified PMCHWT surface integral equation for scattering problems at fundamental frequency and second harmonic frequency; reducing the unknowns to be calculated by the unified surface integral equation by half; comprehensively considering the mutual coupling between the fundamental field and the second harmonic field in a self-consistent coupling manner; calculating the second harmonic field of nanoparticles from the current coefficient and the magnetic flux coefficient. In the embodiment of the present invention, the unknowns calculated by the unified surface integral equation are reduced by half, and at the same time, the impedance matrix distribution is improved, two groups of highly sparse matrix elements are eliminated, the condition number of the impedance matrix is ​​improved, the convergence of the equation is improved, the equation behavior in the surface integral method is improved, and the efficiency of numerical simulation when considering the mutual coupling between the second harmonic field and the fundamental field is improved.

Description

Technical Field

[0001] The present invention relates to the technical field of harmonic analysis, and in particular to a surface integral self-consistent coupling analysis method, device, equipment and storage medium. Background Art

[0002] Nanoparticles have made great progress in plasmonics and energy conversion due to their excellent optical and catalytic properties. Second harmonic generation (SH) of nanoparticles is a nonlinear optical process in which two photons of the same frequency interact to produce a photon with a frequency twice that of the fundamental frequency. Second harmonic generation (SHG) of nanoparticles is mainly contributed by local surface SH sources and has high radiation directionality and interface sensitivity. Therefore, it has important practical value in material interface characterization, directional antenna design, biological directional detection, biological microscopy imaging, etc.

[0003] Since SHG is extremely sensitive to the symmetry of nanoparticles, the modeling and analysis of SHG is quite strict. In practical applications, the generation of harmonics can be decomposed into coupled linear problems, so the measured parameters of the material can be directly used, making the frequency domain method popular in the modeling of nonlinear optical processes. Nanoparticles are homogeneous media, and the surface integral method only needs to be divided on the surface, which can reduce the amount of calculation, as shown in the existing technical literature " J, Suuriniemi S, Kauranen M. Boundary element method for surface nonlinear optics of nanoparticles [J]. Optics express, 2011, 19 (23): 23386-23399” is applicable to solving the second harmonic generated by the local surface SH source. However, due to the introduction of two sets of excessively high unknowns on the inner and outer surfaces, the properties of the equations in the surface integral method become worse, and the efficiency of the numerical simulation is further reduced when considering the mutual coupling between the second harmonic field and the fundamental field. Summary of the invention

[0004] The embodiments of the present invention provide a surface integral self-consistent coupling analysis method, device, equipment and storage medium, which can reduce the unknown quantity calculated by the unified surface integral equation by half, improve the impedance matrix distribution, eliminate two groups of highly sparse matrix elements, improve the condition number of the impedance matrix, improve the convergence of the equation, make the equation state in the surface integral method better, and improve the efficiency of numerical simulation when considering the mutual coupling between the second harmonic field and the fundamental field.

[0005] The embodiment of the present invention provides a surface integral self-consistent coupling analysis method, including:

[0006] A uniform plane wave is incident on the nanoparticles, and a triangular mesh is used to divide the nanoparticles to obtain a discrete model of the nanoparticles, and the number, number and geometric position information of the edges in the nanoparticle model are obtained;

[0007] Taking the equivalent induced current and magnetic current on the inner and outer surfaces of the nanoparticles as unknown quantities, a unified PMCHWT surface integral equation for the scattering problem at the fundamental frequency and the second harmonic frequency is established according to the boundary conditions of the electric field and the magnetic field.

[0008] According to the equivalent relationship between the field and the source on the boundary, the second-order polarization source is equivalent to the incident source at the second harmonic frequency, which reduces the unknown quantity to be calculated by the unified surface integral equation by half and improves its impedance matrix distribution.

[0009] The mutual coupling between the fundamental field and the second harmonic field is comprehensively considered in a self-consistent coupling manner;

[0010] The surface integral equation is discretely solved by the moment method to obtain the current coefficient and the magnetic flux coefficient, and then the second harmonic field of the nanoparticles is calculated based on the current coefficient and the magnetic flux coefficient.

[0011] Furthermore, the segmentation size is Where λ is the electrical wavelength of free space, μ r and ε r are the relative dielectric constant and the magnetic permeability of the nanoparticles, respectively. The electric field intensity and the magnetic field intensity of the uniform plane wave as the incident field are respectively and The solution area of ​​the nanoparticle is divided into the medium outer area V as the outer domain e and the inner region V of the medium as the inner domain i , inc represents a uniform plane wave as the incident wave, S is the area outside the medium V e and the dielectric interior area V i The interface, n represents the area V pointing to the outside of the medium e The unit normal vector is introduced as l = i, e, where i and e are used as subscripts to represent the internal area V of the medium. i Physical quantity and medium external area V e The physical quantity of the nanoparticles is introduced as ν = ω, 2ω, where ω and 2ω are used as superscripts to represent the physical quantity of the fundamental frequency ω and the physical quantity of the second harmonic frequency 2ω, respectively. The dielectric constant and magnetic permeability of the background space medium of the nanoparticles are and The dielectric constant and magnetic permeability of the material of the nanoparticles are and

[0012] Furthermore, the method for establishing a unified PMCHWT surface integral equation for the fundamental frequency and the second harmonic frequency includes:

[0013] Polarization source of second harmonic generation on the nanoparticles Determined by the fundamental field components and material parameters shown in formula (1):

[0014]

[0015] where ε0 is the dielectric constant in vacuum, is the second-order surface nonlinear polarizability, is the total field on the inner surface of the nanoparticle at the fundamental frequency; then formula (1) can be further expressed by formula (2):

[0016]

[0017] Among them, (t1, t2, n) is a set of mutually perpendicular vectors defined on the surface of the nanoparticle, t1 and t2 are unit tangential vectors, and n is the unit normal vector; the second harmonic on the nanoparticle has a tangential component and the normal component

[0018] The outer domain and the inner domain are respectively equivalent by using the surface integral method, which includes: by combining the tangential electric field integral equation T-EFIE and the tangential magnetic field integral equation T-MFIE of the inner surface and the outer surface of the nanoparticle, a unified PMCHWT equation for solving the scattering problem under two linear frequencies, the fundamental frequency and the second harmonic frequency, as shown in formula (3) is obtained:

[0019]

[0020] in Corresponding to the equivalent current and magnetic current on the inner surface at frequencies ν=ω and ν=2ω respectively, n is the unit normal vector, η i and η e are the wave impedance of the inner domain and the wave impedance of the outer domain respectively, I is the unit matrix, and are the current source and magnetic current source on the interface at the corresponding frequencies ν=ω and ν=2ω, respectively, and under different frequencies they are:

[0021]

[0022] in and are the electric field intensity and magnetic field intensity of the uniform plane wave, respectively.

[0023] and are the tangential and normal components of the second harmonic of the nanoparticles, j is an imaginary unit, and ε′ is the dielectric constant of the interface.

[0024] As a linear integral operator and The definitions of are shown in formula (5) and formula (6):

[0025]

[0026]

[0027] Where S is the integral area of ​​the nanoparticle surface, r and r′ represent the position vector of the observation point and the position vector of the source point in the integral area, respectively. is the scalar Green's function in three-dimensional space, k l is the wave number of the corresponding space, μ l is the magnetic permeability line of the corresponding area, i is the imaginary unit, i and j remain perpendicular, and X(r′) is the vector representation of the source point.

[0028] Furthermore, the method of equivalently treating the second-order polarization source as an incident source at the second harmonic frequency according to the equivalent relationship between the field and the source on the boundary, reducing the unknown quantity to be calculated by the unified surface integral equation by half, and improving the impedance matrix distribution includes:

[0029] According to the equivalent relationship between the field and the source on the boundary, as shown in formula (7), the second-order polarization source and are equivalent to the incident source at the second harmonic frequency and

[0030]

[0031] The subscript r represents that the equivalent incident field is non-uniformly distributed;

[0032] Formula (3) is improved to formula (8):

[0033]

[0034] Among them, J (ν) and M (ν) are the new equivalent current and magnetic current on the outer surface of the nanoparticle, E (ν,inc) and H (ν,inc) They represent the electric field intensity and magnetic field intensity of the incident wave irradiating the outer surface of the nanoparticle, respectively.

[0035] Furthermore, the method of comprehensively considering the mutual coupling between the fundamental field and the second harmonic field in a self-consistent coupling manner includes:

[0036] Update the right vector matrix of the basis field in formula (8) to formula (9):

[0037]

[0038] The coupling of the electric field and the coupling of the magnetic field are shown in formula (10) and formula (11) respectively:

[0039]

[0040]

[0041] in and Represents the fundamental polarization source The tangential component and fundamental polarization source Normal component of; fundamental polarization source It is determined by the fundamental field and the second harmonic field as shown in formula (12):

[0042]

[0043] By continuously updating the incident source, the self-consistent coupling process of the fundamental field and the second harmonic field is completed using formula (8).

[0044] Furthermore, the method of completing the self-consistent coupling process of the fundamental field and the second harmonic field by continuously updating the incident source using formula (8) includes:

[0045] Initialize the primary polarization source is zero, then

[0046] Solution stage 1: This stage uses the surface integral method to solve the basis field through formula (8), at this time ν = ω;

[0047] The second harmonic polarization source is obtained by formula (1): And further use formula (7) to calculate the equivalent incident field at the second harmonic frequency: and

[0048] Solution stage 2: This stage uses the surface integral method to solve the second harmonic field through formula (8), at this time ν = 2ω;

[0049] The rotation phase is to return to the solution phase 1. and return to the solution calculated in the second stage In this way, the fundamental polarization source is updated through formula (12): The base incident field is further updated by formula (9).

[0050] Repeat the process from solution phase 1 to rotation phase until the self-consistent iteration converges.

[0051] Furthermore, the method of using the moment method to discretely solve the surface integral equation to obtain the current coefficient and the magnetic current coefficient, and then calculating the second harmonic field of the nanoparticles from the current coefficient and the magnetic current coefficient includes:

[0052] The equivalent current J and magnetic current M induced on the surface of the nanoparticles are expanded using the RWG basis function as shown in formula (13):

[0053]

[0054] where α n and β n are the corresponding current coefficient and magnetic flux coefficient respectively, N is the number of sides of the triangle, f n (r) is the nth RWG basis function, n is a natural number, and formula (13) is substituted into formula (8), and then the RWG basis function f is used on both sides of formula (8) n (r) Performing the Galerkin test, the matrix form of the surface integral equation is shown in formula (14):

[0055]

[0056] Where Z EJ The matrix representing the electric field generated by the current, Z EM The matrix representing the magnetic field generated by the current, Z HJ The element matrix representing the electric field generated by magnetic current, Z HM Represents the element matrix of the magnetic field generated by the magnetic current, V H The right-hand side vector matrix, V, represents the electric field generated by the incident wave. H The right-hand side vector matrix representing the magnetic field produced by the incident wave.

[0057] The far-field scattering field E of the nanoparticles can be obtained by self-consistent iteration to obtain the current coefficient and magnetic current coefficient. sca , then the scattering cross-sectional area RCS of the nanoparticles is expressed as:

[0058]

[0059] The embodiment of the present invention further provides a device for surface integral self-consistent coupling analysis, comprising:

[0060] A profile module is used to inject uniform plane waves into nanoparticles and use triangular meshes to divide the nanoparticles to obtain a nanoparticle model by discrete modeling, and to obtain the number, number and geometric position information of the edges in the nanoparticle model;

[0061] Establish a module for establishing the unified PMCHWT surface integral equation for scattering problems at fundamental frequency and second harmonic frequency based on the boundary conditions of electric field and magnetic field, taking the equivalent induced current and magnetic current on the inner and outer surfaces of nanoparticles as unknown quantities;

[0062] A reduction module is used to convert the second-order polarization source into an incident source at the second harmonic frequency according to the equivalent relationship between the field and the source on the boundary, thereby halving the unknown quantities to be calculated by the unified surface integral equation and improving its impedance matrix distribution;

[0063] A coupling module is used to comprehensively consider the mutual coupling between the fundamental field and the second harmonic field in a self-consistent coupling manner;

[0064] The calculation module is used to discretely solve the surface integral equation by the moment method to obtain the current coefficient and the magnetic flux coefficient, and then calculate the second harmonic field of the nanoparticles by the current coefficient and the magnetic flux coefficient.

[0065] An embodiment of the present invention further provides a device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the surface integral self-consistent coupling analysis method is implemented when the processor executes the program.

[0066] An embodiment of the present invention further provides a computer-readable storage medium storing computer-executable instructions, wherein the computer-executable instructions are used to execute the surface integral self-consistent coupling analysis method.

[0067] In the fields of material interface characterization, directional antenna design, biological directional detection or biological microscopic imaging, the embodiments of the present invention unify the surface integral equations at the fundamental frequency and the second harmonic frequency, and according to the equivalent relationship between the field and the source on the boundary, the second-order polarization source is equivalent to the incident source at the second harmonic frequency, and the unknown quantity calculated by the unified surface integral equation is halved. At the same time, its impedance matrix distribution is improved, two groups of highly sparse matrix elements are eliminated, the condition number of the impedance matrix is ​​improved, and the convergence of the equation is improved; by continuously updating the fundamental polarization source jointly determined by the fundamental field and the second harmonic field, the self-consistent iterative coupling of the fundamental field and the second harmonic field is realized.

[0068] Other features and advantages of the present invention will be described in the following description, and partly become apparent from the description, or understood by practicing the present invention. The purpose and other advantages of the present invention can be realized and obtained by the structures particularly pointed out in the description, claims and drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] Figure 1 is a schematic diagram of mesh division of nanoparticles according to an embodiment of the present invention;

[0070] Figure 2Schematic diagram of the local coordinate system and rectangular coordinate system of the surface of the nanoparticles according to an embodiment of the present invention;

[0071] Figure 3 is a comparison diagram of the fundamental frequency RCS of the metal nano-gold spheres according to the embodiment of the present invention;

[0072] Figure 4 This is a comparison chart of the second harmonic frequency RCS of the nano-gold spheres in the embodiment of the present invention. Figure 4 The normal component of (a) is (χ ⊥⊥⊥ ,x ||⊥|| )=(1,0); Figure 4 The tangential component of (b) is (χ ⊥⊥⊥ ,x ||⊥|| )=(0,1);

[0073] Figure 5 Schematic diagram of the self-consistent convergence of the surface integral self-consistent coupling analysis method according to an embodiment of the present invention;

[0074] Figure 6 A flow chart of a surface integral self-consistent coupling analysis method according to an embodiment of the present invention;

[0075] Figure 7 A structural diagram of a device for surface integral self-consistent coupling analysis according to an embodiment of the present invention;

[0076] Figure 8 The figure is a structural diagram of a device according to an embodiment of the present invention. DETAILED DESCRIPTION

[0077] The embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0078] The steps shown in the flowchart of the accompanying drawings can be executed in a computer system such as a set of computer executable instructions. Also, although a logical sequence is shown in the flowchart, in some cases, the steps shown or described can be performed in a sequence different from that shown here.

[0079] The embodiment of the present invention belongs to the field of harmonic analysis, also belongs to the field of second harmonic characteristic analysis of nanoparticles, and also belongs to a hybrid surface integral method, device, equipment and storage medium, and is particularly suitable for a surface integral self-consistent coupling analysis method, device, equipment and storage medium of the second harmonic of nanoparticles. The embodiment of the present invention reduces the unknown quantities to be solved, and can improve the calculation efficiency by about 7.5 times, and saves memory consumption by about 6 times. It is suitable for high-precision and high-efficiency analysis of second harmonic generation of nanoparticles of arbitrary shapes.

[0080] like Figure 6 As shown, the surface integral self-consistent coupling analysis method of the embodiment of the present invention includes:

[0081] Step 101, consider using a uniform plane wave to impinge on the nanoparticles, the frequency of the uniform plane wave as the incident wave is f inc , considering the sensitivity of the second harmonic at the boundary, such as Figure 1 As shown, a triangular mesh is used to divide the nanoparticles to discretely model the nanoparticles to obtain a nanoparticle model, and the number, serial number and geometric position information of the edges in the nanoparticle model are obtained.

[0082] In one embodiment, the segmentation size is Where λ is the electrical wavelength of free space, μ r and ε r are the relative dielectric constant and the magnetic permeability of the nanoparticles, respectively. The electric field intensity and the magnetic field intensity of the uniform plane wave as the incident field are respectively and The solution area of ​​the nanoparticle is divided into the medium outer area V as the outer domain e and the inner region V of the medium as the inner domain i , where the medium is the nanoparticle. inc represents a uniform plane wave as the incident wave, and S is the area outside the medium V e and the dielectric interior area V i The interface, n represents the area V pointing to the outside of the medium e The unit normal vector is introduced as l = i, e, where i and e are used as subscripts to represent the internal area V of the medium. i Physical quantity and medium external area V e The physical quantity of the nanoparticles is introduced as ν = ω, 2ω, where ω and 2ω are used as superscripts to represent the physical quantity of the fundamental frequency ω and the physical quantity of the second harmonic frequency 2ω, respectively. The dielectric constant and magnetic permeability of the background space medium of the nanoparticles are and The dielectric constant and magnetic permeability of the material of the nanoparticles are and

[0083] Step 102, taking the equivalent induced current and magnetic current on the inner and outer surfaces of the nanoparticles as unknown quantities, and according to the boundary conditions of the electric field and the magnetic field, a unified PMCHWT surface integral equation for the scattering problem under the fundamental frequency and the second harmonic frequency is established. The fundamental frequency is the frequency of the fundamental wave of the incident wave. The second harmonic frequency is the frequency of the second harmonic. The fundamental is the fundamental wave of the incident wave.

[0084] In one embodiment, the method of establishing a unified PMCHWT surface integral equation of the fundamental frequency and the second harmonic frequency includes:

[0085] Polarization source of second harmonic generation on the nanoparticles Determined by the fundamental field components and material parameters shown in formula (1):

[0086]

[0087] where ε0 is the dielectric constant in vacuum, is the second-order surface nonlinear polarizability, is the total field on the inner surface of the nanoparticle at the fundamental frequency; usually, the surface magnetic susceptibility tensor has three independent and mutually perpendicular components, then formula (1) can be further expressed as formula (2):

[0088]

[0089] Among them, Figure 2 As shown. (t1, t2, n) is a set of mutually perpendicular vectors defined on the surface of the nanoparticle, t1 and t2 are unit tangent vectors, and n is a unit normal vector; the second harmonic on the nanoparticle has a tangential component and the normal component

[0090] The surface integral method is used to respectively perform equivalence on the external domain and the internal domain, which includes: for the external domain problem, its equivalent current and magnetic current In the outer domain V e The scattered field is generated, while in the inner domain V i No scattered field is generated; for the internal domain problem, its equivalent current and magnetic current In the internal domain V i Produce scattered fields, while in the external domain V e No scattered field is generated. By combining the tangential electric field integral equation T-EFIE and the tangential magnetic field integral equation T-MFIE on the inner and outer surfaces of the nanoparticles, the unified PMCHWT equation for solving the scattering problem at two linear frequencies, the fundamental frequency and the second harmonic frequency, is obtained as shown in formula (3):

[0091]

[0092] in and Corresponding to the equivalent current and magnetic current on the inner surface at frequencies ν=ω and ν=2ω respectively, n is the unit normal vector, η i and η e are the wave impedance of the inner domain and the wave impedance of the outer domain respectively, I is the unit matrix, and are the current source and magnetic current source on the interface at the corresponding frequencies ν=ω and ν=2ω, respectively, and under different frequencies they are:

[0093]

[0094] in and are the electric field intensity and magnetic field intensity of the uniform plane wave, respectively. and are the tangential component of the nanoparticle second harmonic and the normal component of the nanoparticle second harmonic, respectively, j is an imaginary unit, and ε′ is the dielectric constant of the interface;

[0095] As a linear integral operator and The definitions of are shown in formula (5) and formula (6):

[0096]

[0097]

[0098] Where S is the integral area of ​​the nanoparticle surface, r and r′ represent the position vector of the observation point and the position vector of the source point in the integral area, respectively. is the scalar Green's function in three-dimensional space, k l is the wave number of the corresponding space, μ l is the magnetic permeability line of the corresponding area, i is the imaginary unit, i and j remain perpendicular, and X(r′) is the vector representation of the source point.

[0099] Step 103, based on the equivalent relationship between the field and the source on the boundary, the second-order polarization source is equivalent to the incident source at the second harmonic frequency, the unknown quantity to be calculated by the unified surface integral equation is halved, and the impedance matrix distribution is improved.

[0100] In one embodiment, the method of equivalently treating the second-order polarization source as an incident source at the second harmonic frequency according to the equivalent relationship between the field and the source on the boundary, reducing the unknown quantity to be calculated by the unified surface integral equation by half, and improving the impedance matrix distribution includes:

[0101] According to the equivalent relationship between the field and the source on the boundary, as shown in formula (7), the second-order polarization source and are equivalent to the incident source at the second harmonic frequency and

[0102]

[0103] The subscript r represents that the equivalent incident field is non-uniformly distributed;

[0104] Substituting the relationship between current and magnetic current into the electric field integral equation and the magnetic field integral equation, formula (3) is improved to formula (8):

[0105]

[0106] Among them, J (v) and M (v) are the new equivalent current and magnetic current on the outer surface of the nanoparticle, E (v,inc) and H (v,inc) They represent the electric field intensity and magnetic field intensity of the incident wave irradiating the outer surface of the nanoparticle, respectively;

[0107] Comparing formula (8) with formula (4), the equivalent relationship between the field and the source on the boundary is used to reduce the number of unknowns in the surface integral equation by half. At the same time, the element distribution of the impedance matrix is ​​readjusted, two groups of highly sparse matrix elements are eliminated, the condition number of the impedance matrix is ​​improved, and the convergence of the equation is improved.

[0108] Step 104, determine the fundamental polarization source by using the second harmonic field and the fundamental field It represents the feedback of the second harmonic field to the fundamental field, and comprehensively considers the mutual coupling between the fundamental field and the second harmonic field in a self-consistent coupling manner, making it more consistent with the experimental physical model.

[0109] In one embodiment, the fundamental polarization source is determined by the second harmonic field and the fundamental field. Representing the feedback of the second harmonic field to the fundamental field, the mutual coupling between the fundamental field and the second harmonic field is comprehensively considered in a self-consistent coupling manner to make it more consistent with the experimental physical model, including:

[0110] The interaction between the fundamental field and the second harmonic field is comprehensively considered by means of self-consistent coupling. When considering the mutual coupling between the second harmonic field and the fundamental field, the incident source of the fundamental field includes not only the incident plane wave, but also the coupling of the second harmonic field to the fundamental field. Then, the right-hand side vector matrix of the fundamental field in formula (8) is updated to formula (9):

[0111]

[0112] The coupling of the electric field and the coupling of the magnetic field are shown in formula (10) and formula (11) respectively:

[0113]

[0114]

[0115] in and Represents the fundamental polarization source The tangential component and fundamental polarization source Normal component of; fundamental polarization source It is determined by the fundamental field and the second harmonic field as shown in formula (12):

[0116]

[0117] The surface integral method of the fundamental field-second harmonic field coupling is considered, that is, by continuously updating the incident source, the self-consistent coupling process of the fundamental field and the second harmonic field is completed using formula (8), where: represents the total field at the inner surface of the nanoparticle at the second harmonic frequency.

[0118] In one embodiment, the method of completing the self-consistent coupling process of the fundamental field and the second harmonic field by continuously updating the incident source using formula (8) includes:

[0119] Step 201, initializing the primary polarization source is zero, then

[0120] Step 202, solution stage 1, this stage is to solve the basis field by using the surface integral method through formula (8), at this time ν = ω.

[0121] Step 203, calculate the second harmonic polarization source by formula (1): And further use formula (7) to calculate the equivalent incident field at the second harmonic frequency: and

[0122] Step 204, solving stage 2, this stage is to solve the second harmonic field by using the surface integral method through formula (8), at this time ν = 2ω.

[0123] Step 205, the rotation phase, which returns to the solution phase 1 calculated using step 202. and return the result calculated by the second solution phase of step 204 In this way, the fundamental polarization source is updated through formula (12): The base incident field is further updated by formula (9).

[0124] Step 206, repeat the process from step 202 to step 205 from the solution phase 1 to the rotation phase until the self-consistent iteration converges.

[0125] Step 105, using the moment method to discretely solve the surface integral equation to obtain the current coefficient and the magnetic current coefficient, and then calculating the second harmonic field of the nanoparticles based on the current coefficient and the magnetic current coefficient.

[0126] In one embodiment, the method of using the moment method to discretely solve the surface integral equation to obtain the current coefficient and the magnetic current coefficient, and then calculating the second harmonic field of the nanoparticles from the current coefficient and the magnetic current coefficient includes:

[0127] The equivalent current J and magnetic current M induced on the surface of the nanoparticles are expanded using the RWG basis function as shown in formula (13):

[0128]

[0129] where α n and β n are the corresponding current coefficient and magnetic flux coefficient respectively, N is the number of sides of the triangle, f n (r) is the nth RWG basis function, n is a natural number, and formula (13) is substituted into formula (8), and then the RWG basis function f is used on both sides of formula (8) n (r) Performing the Galerkin test, the matrix form of the surface integral equation is shown in formula (14):

[0130]

[0131] Where Z EJ The matrix representing the electric field generated by the current, Z EM The matrix representing the magnetic field generated by the current, Z HJ The element matrix representing the electric field generated by magnetic current, Z HM The element matrix representing the magnetic field generated by the magnetic current, V H The right-hand side vector matrix, V, represents the electric field generated by the incident wave. H The right-hand side vector matrix representing the magnetic field produced by the incident wave.

[0132] The far-field scattering field E of the nanoparticles can be obtained by self-consistent iteration to obtain the current coefficient and magnetic current coefficient. sca , then the radar cross-sectional area RCS of nanoparticles is expressed as:

[0133]

[0134] In order to verify the correctness and efficiency of the surface integral self-consistent coupling analysis method of the embodiment of the present invention, a nano-gold sphere with a radius of r = 50nm as a nanoparticle is analyzed. The incident wave is a uniform plane wave with a wavelength of λ = 520nm and an incident direction of The polarization direction is The observation angle of the dual-station RCS is The relative dielectric parameters of gold are ε(ω)=(-3.95,2.58), ε(Ω)=(-1.20,4.67). Figure 3 It is a comparison diagram of the fundamental frequency RCS of the nano-gold spheres of the embodiments of the present invention. The Mie series of the method of the present invention is well consistent with that of the analytical method, which verifies the correctness of the method of the present invention in calculating the linear scattering of nanoparticles. Figure 4The comparison of the second harmonic frequency RCS comparison diagram of the nano-gold sphere of the present invention is made by considering only the normal component and tangential component of the second-order nonlinear polarizability. Figure 4 It can be seen that the surface integral self-consistent coupling analysis method of the embodiment of the present invention is in good agreement with the analytical method SH-Mie series, which verifies the correctness of this method in calculating the second-order nonlinear response of nanoparticles. Figure 5 The self-consistent convergence of the surface integral self-consistent coupling analysis method of the embodiment shows that the relative error can reach 10 within 10 steps. -4 Table 1 is a comparison of the computational resources of the surface integral self-consistent coupling analysis method of the embodiment of the present invention and the conventional surface integral method:

[0135] Table 1

[0136]

[0137] The surface integral self-consistent coupling analysis method of the embodiment of the present invention reduces the unknown quantities to be solved, can improve the computational efficiency by about 7.5 times, and saves memory consumption by about 6 times, and is suitable for high-precision and high-efficiency analysis of second harmonic generation of nanoparticles of arbitrary shapes.

[0138] like Figure 7 As shown, an embodiment of the present invention further provides a device for surface integral self-consistent coupling analysis, comprising:

[0139] The profile module 71 is used to inject a uniform plane wave into the nanoparticles, and adopt a triangular mesh to divide the nanoparticles to discretely model the nanoparticles to obtain a nanoparticle model, and obtain the number, number and geometric position information of the edges in the nanoparticle model;

[0140] Establishing module 72, for establishing a unified PMCHWT surface integral equation for scattering problems at fundamental frequency and second harmonic frequency, taking equivalent induced current and magnetic current on the inner surface and outer surface of the nanoparticle as unknown quantities and according to the boundary conditions of the electric field and the magnetic field;

[0141] A reduction module 73 is used to, based on the equivalent relationship between the field and the source on the boundary, equate the second-order polarization source to an incident source at the second harmonic frequency, reduce the unknown quantity to be calculated by the unified surface integral equation by half, and improve its impedance matrix distribution;

[0142] A coupling module 74, for comprehensively considering the mutual coupling between the fundamental field and the second harmonic field in a self-consistent coupling manner;

[0143] The calculation module 75 is used to discretely solve the surface integral equation using the moment method to obtain the current coefficient and the magnetic current coefficient, and then calculate the second harmonic field of the nanoparticles based on the current coefficient and the magnetic current coefficient.

[0144] like Figure 8As shown, an embodiment of the present invention further provides a device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the surface integral self-consistent coupling analysis method is implemented when the processor executes the program.

[0145] An embodiment of the present invention further provides a computer-readable storage medium storing computer-executable instructions, wherein the computer-executable instructions are used to execute the surface integral self-consistent coupling analysis method.

[0146] In this embodiment, the above-mentioned storage medium may include but is not limited to: a USB flash drive, a read-only memory (ROM), a random access memory (RAM), a mobile hard disk, a magnetic disk or an optical disk, and other media that can store program codes.

[0147] It will be appreciated by those skilled in the art that all or some of the steps, systems, and functional modules / units in the methods disclosed above may be implemented as software, firmware, hardware, and appropriate combinations thereof. In hardware implementations, the division between the functional modules / units mentioned in the above description does not necessarily correspond to the division of physical components; for example, a physical component may have multiple functions, or a function or step may be performed by several physical components in cooperation. Some or all components may be implemented as software executed by a processor, such as a digital signal processor or a microprocessor, or implemented as hardware, or implemented as an integrated circuit, such as an application-specific integrated circuit. Such software may be distributed on a computer-readable medium, which may include a computer storage medium (or non-transitory medium) and a communication medium (or temporary medium). As known to those skilled in the art, the term computer storage medium includes volatile and non-volatile, removable and non-removable media implemented in any method or technology for storing information (such as computer-readable instructions, data structures, program modules, or other data). Computer storage media include, but are not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disk (DVD) or other optical disk storage, magnetic cassettes, magnetic tapes, magnetic disk storage or other magnetic storage devices, or any other medium that can be used to store the desired information and can be accessed by a computer. In addition, it is well known to those of ordinary skill in the art that communication media typically include computer-readable instructions, data structures, program modules, or modulated data signals such as carrier waves or other transport mechanisms.

Claims

1. A surface integral self-consistent coupling analysis method, characterized in that: include: A uniform plane wave is incident on the nanoparticles, and a triangular mesh is used to discretely model the nanoparticles by segmentation, thereby obtaining a nanoparticle model, and obtaining the number, serial number and geometric position information of the edges in the nanoparticle model; The split size is Where λ is the electrical wavelength of free space, μ r and ε r are the relative dielectric constant of the nanoparticles and the magnetic permeability of the nanoparticles, respectively; Taking the equivalent induced current and magnetic current on the inner and outer surfaces of the nanoparticles as unknown quantities, a unified PMCHWT surface integral equation for the scattering problem at the fundamental frequency and the second harmonic frequency is established according to the boundary conditions of the electric field and the magnetic field; specifically, by combining the tangential electric field integral equation T-EFIE and the tangential magnetic field integral equation T-MFIE on the inner and outer surfaces of the nanoparticles, a unified PMCHWT surface integral equation for solving the scattering problem at two linear frequencies, the fundamental frequency and the second harmonic frequency, is obtained; According to the equivalent relationship between the field and the source on the boundary, the second-order polarization source is equivalent to the incident source at the second harmonic frequency, the unknown quantity to be calculated by the unified surface integral equation is halved, and its impedance matrix distribution is improved; specifically, according to the equivalent relationship between the field and the source on the boundary, the second-order polarization source is equivalent to the incident source at the second harmonic frequency; and the unified PMCHWT surface integral equation is improved to the fundamental field equation; The mutual coupling between the fundamental field and the second harmonic field is comprehensively considered in a self-consistent coupling manner; specifically, the vector matrix in the fundamental field equation is updated to a matrix about the coupling of the electric field and the magnetic field, and the fundamental polarization source is jointly determined by the fundamental field and the second harmonic field; the self-consistent coupling process of the fundamental field and the second harmonic field is completed by continuously updating the incident source; The surface integral equation is discretely solved by the moment method to obtain the current coefficient and the magnetic flux coefficient, and then the second harmonic field of the nanoparticles is calculated based on the current coefficient and the magnetic flux coefficient.

2. The surface integral self-consistent coupling analysis method according to claim 1, characterized in that: The electric field intensity of the uniform plane wave as the incident field and the magnetic field intensity of the uniform plane wave are respectively and The solution area of ​​the nanoparticle is divided into the medium outer area V as the outer domain e and the inner region V of the medium as the inner domain i , inc represents a uniform plane wave as the incident wave, S is the area outside the medium V e and the dielectric interior area V i The interface, n represents the area V pointing to the outside of the medium e The unit normal vector is introduced as l = i, e, where i and e are used as subscripts to represent the internal area V of the medium. i Physical quantity and medium external area V e The physical quantity of the nanoparticles is introduced as ν = ω, 2ω, where ω and 2ω are used as superscripts to represent the physical quantity of the fundamental frequency ω and the physical quantity of the second harmonic frequency 2ω, respectively. The dielectric constant and magnetic permeability of the background space medium of the nanoparticles are and The dielectric constant and magnetic permeability of the material of the nanoparticles are and 3. The surface integral self-consistent coupling analysis method according to claim 2, characterized in that: The method for establishing a unified PMCHWT surface integral equation for the fundamental frequency and the second harmonic frequency comprises: Polarization source of second harmonic generation on the nanoparticles Determined by the fundamental field components and material parameters shown in formula (1): where ε0 is the dielectric constant in vacuum, is the second-order surface nonlinear polarizability, is the total field on the inner surface of the nanoparticle at the fundamental frequency; then formula (1) can be further expressed by formula (2): Among them, (t1, t2, n) is a set of vectors defined on the surface of the nanoparticles and perpendicular to each other, t1 and t2 are unit tangent vectors, and n is the region V pointing to the outside of the medium. e The unit normal vector of the nanoparticle has a tangential component. and the normal component The outer domain and the inner domain are respectively equivalent by using the surface integral method, which includes: by combining the tangential electric field integral equation T-EFIE and the tangential magnetic field integral equation T-MFIE of the inner surface and the outer surface of the nanoparticle, a unified PMCHWT surface integral equation for solving the scattering problem under two linear frequencies, the fundamental frequency and the second harmonic frequency, as shown in formula (3) is obtained: in and Corresponding to the equivalent current and magnetic current on the inner surface at frequencies ν=ω and ν=2ω respectively, n is the unit normal vector, η i and η e are the wave impedance of the inner domain and the wave impedance of the outer domain respectively, I is the unit matrix, and are the current source and magnetic current source on the interface at the corresponding frequencies ν=ω and ν=2ω, respectively, and under different frequencies they are: in and are the electric field intensity and magnetic field intensity of the uniform plane wave, respectively. and are the tangential component of the nanoparticle second harmonic and the normal component of the nanoparticle second harmonic, respectively, j is an imaginary unit, and ε′ is the dielectric constant of the interface; As a linear integral operator and The definitions of are shown in formula (5) and formula (6): Where S is the integral area of ​​the nanoparticle surface, r and r′ represent the position vector of the observation point and the position vector of the source point in the integral area, respectively. is the scalar Green's function in three-dimensional space, k l is the wave number of the corresponding space, μ l is the magnetic permeability of the corresponding area, i is the imaginary unit, i and j remain perpendicular, and X(r′) is the vector representation of the source point.

4. The surface integral self-consistent coupling analysis method according to claim 2, characterized in that: The method of equivalently treating the second-order polarization source as an incident source at the second harmonic frequency according to the equivalent relationship between the field and the source on the boundary, reducing the unknown quantity to be calculated by the unified surface integral equation by half, and improving the impedance matrix distribution, comprises: According to the equivalent relationship between the field and the source on the boundary, as shown in formula (7), the second-order polarization source and are equivalent to the incident source at the second harmonic frequency and The subscript r represents that the equivalent incident field is non-uniformly distributed; Formula (3) is improved to formula (8): Among them, J (ν) and M (ν) are the new equivalent current and magnetic current on the outer surface of the nanoparticle, E (ν,inc) and H (ν,inc) They represent the electric field intensity and magnetic field intensity of the incident wave irradiating the outer surface of the nanoparticle, respectively.

5. The surface integral self-consistent coupling analysis method according to claim 2, characterized in that: The method of comprehensively considering the mutual coupling between the fundamental field and the second harmonic field in a self-consistent coupling manner includes: Update the right vector matrix of the basis field in formula (8) to formula (9): The coupling of the electric field and the coupling of the magnetic field are shown in formula (10) and formula (11) respectively: in and Represents the fundamental polarization source The tangential component and fundamental polarization source The normal component of ; Fundamental polarization source It is determined by the fundamental field and the second harmonic field as shown in formula (12): By continuously updating the incident source, the self-consistent coupling process of the fundamental field and the second harmonic field is completed using formula (8), where: represents the total field at the inner surface of the nanoparticle at the second harmonic frequency.

6. The surface integral self-consistent coupling analysis method according to claim 5, characterized in that: The method of completing the self-consistent coupling process of the fundamental field and the second harmonic field by continuously updating the incident source using formula (8) includes: Initialize the primary polarization source is zero, then Solution stage 1: This stage uses the surface integral method to solve the basis field through formula (8), at this time ν = ω; The second harmonic polarization source is obtained by formula (1): And further use formula (7) to calculate the equivalent incident field at the second harmonic frequency: and Solution stage 2: This stage uses the surface integral method to solve the second harmonic field through formula (8), at this time ν = 2ω; The rotation phase is to return to the solution phase 1. and return to the solution calculated in the second stage In this way, the fundamental polarization source is updated through formula (12): The base incident field is further updated through formula (9); Repeat the process from solution phase 1 to rotation phase until the self-consistent iteration converges.

7. The surface integral self-consistent coupling analysis method according to claim 2, characterized in that: The method of using the moment method to discretely solve the surface integral equation to obtain the current coefficient and the magnetic current coefficient, and then calculating the second harmonic field of the nanoparticles from the current coefficient and the magnetic current coefficient, comprises: The equivalent current J and magnetic current M induced on the surface of the nanoparticles are expanded using the RWG basis function as shown in formula (13): where α n and β n are the corresponding current coefficient and magnetic flux coefficient respectively, N is the number of sides of the triangle, f n (r) is the nth RWG basis function, n is a natural number, and formula (13) is substituted into formula (8), and then the RWG basis function f is used on both sides of formula (8) n (r) Performing the Galerkin test, the matrix form of the surface integral equation is shown in formula (14): Where Z EJ The matrix representing the electric field generated by the current, Z EM The matrix representing the magnetic field generated by the current, Z HJ The element matrix representing the electric field generated by magnetic current, Z HM The element matrix representing the magnetic field generated by the magnetic current, V E The right-hand side vector matrix, V, represents the electric field generated by the incident wave. H The right-hand side vector matrix representing the magnetic field generated by the incident wave; The far-field scattering field E of the nanoparticles can be obtained by self-consistent iteration to obtain the current coefficient and magnetic current coefficient. sca , then the scattering cross-sectional area RCS of the nanoparticles is expressed as:

8. A device for surface integral self-consistent coupling analysis, characterized in that: include: The profile module is used to model the nanoparticles discretely by using a triangular mesh by incident on the nanoparticles with a uniform plane wave, thereby obtaining a nanoparticle model and obtaining the number, numbering and geometric position information of the edges in the nanoparticle model; The split size is Where λ is the electrical wavelength of free space, μ r and ε r are the relative dielectric constant of the nanoparticles and the magnetic permeability of the nanoparticles, respectively; A module is established for establishing a unified PMCHWT surface integral equation for scattering problems at fundamental frequency and second harmonic frequency, taking the equivalent induced current and magnetic current on the inner and outer surfaces of the nanoparticles as unknown quantities and according to the boundary conditions of the electric field and the magnetic field; specifically, by combining the tangential electric field integral equation T-EFIE and the tangential magnetic field integral equation T-MFIE on the inner and outer surfaces of the nanoparticles, a unified PMCHWT surface integral equation for solving the scattering problems at two linear frequencies, the fundamental frequency and the second harmonic frequency, is obtained; The reduction module is used to treat the second-order polarization source as an incident source at the second harmonic frequency according to the equivalent relationship between the field and the source on the boundary, reduce the unknown quantity to be calculated by the unified surface integral equation by half, and improve its impedance matrix distribution; specifically, according to the equivalent relationship between the field and the source on the boundary, treat the second-order polarization source as an incident source at the second harmonic frequency; and improve the unified PMCHWT surface integral equation into a fundamental field equation; The coupling module is used to comprehensively consider the mutual coupling between the fundamental field and the second harmonic field in a self-consistent coupling manner; specifically, the vector matrix in the fundamental field equation is updated to a matrix about the coupling of the electric field and the magnetic field, and the fundamental polarization source is jointly determined by the fundamental field and the second harmonic field; the self-consistent coupling process of the fundamental field and the second harmonic field is completed by continuously updating the incident source; The calculation module is used to discretely solve the surface integral equation by the moment method to obtain the current coefficient and the magnetic flux coefficient, and then calculate the second harmonic field of the nanoparticles by the current coefficient and the magnetic flux coefficient.

9. A device, characterized in that: The invention comprises a memory, a processor and a computer program stored in the memory and executable on the processor, wherein when the processor executes the program, the surface integral self-consistent coupling analysis method described in any one of claims 1 to 7 is implemented.

10. A computer-readable storage medium, characterized in that: Computer executable instructions are stored, and the computer executable instructions are used to execute the surface integral self-consistent coupling analysis method described in any one of claims 1 to 7.

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