A method for realizing nonlinear equivalence

Abstract

The present invention discloses a method for realizing nonlinear equivalence, belonging to the technical field of composite materials. The method comprises: a plurality of nanoparticles exist in a background medium to form a composite material, and the relative dielectric constant of the nanoparticles and the background medium in the composite material is obtained; the nonlinear effective dielectric function of the composite material is obtained according to the relative dielectric constant; the first and second formulas of the effective nonlinear magnetic susceptibility of the composite material are obtained according to the nonlinear effective dielectric function and the relative dielectric constant of the nanoparticles and the background medium; and the effective nonlinear magnetic susceptibility correction formula of the composite material is obtained. The present invention accurately predicts the nonlinear optical effect of the composite material caused by the completely different local fields experienced by tiny nonlinear impurities by correcting the effective nonlinear magnetic susceptibility of the composite material. For weak nonlinear composite materials with short range and large particle size difference, the correction scheme of the nonlinear effective medium theory proposed by the present invention is reliable and accurate.

Description

Technical Field

[0001] The invention relates to the technical field of composite materials, in particular to a method for realizing nonlinear equivalence. Background Art

[0002] With the rise of the discipline of micro-electromechanical systems based on micro-mechanical and micro-electronic technologies, micro-motors, i.e. motors with a diameter of less than 160mm or a power of less than 750W, as the core of micro-electromechanical systems, are widely used in various scientific research and national defense fields such as biomedicine, aerospace, precision instruments, and communication engineering. Among them, the medical field is one of the most representative fields of micro-motor application, which can realize functions such as targeted drug delivery, precision surgery, and biosensing. In recent years, relevant researchers have successively developed nano-motors with a size of only a few nanometers. This type of micro-motor can convert chemical energy into mechanical energy, can be integrated into ultra-small machines, and is widely used in medical fields such as local micro-drug delivery in the body, precision microsurgery, and micro-traumatic endoscopic treatment. In order to meet the high requirements of the measurement accuracy of micro-motor speed measurement systems for the rapid development of modern medical technology. People have proposed many methods for measuring the speed of molecular motors suitable for this type of medical use. Among them, the prior art proposes to utilize the highly sensitive characteristics of the second and third harmonic generation (Second / Third-harmonic generation, referred to as THG / SHG) to the structure and the environment, and designs a nonlinear optical metamaterial as a motor speed measurement system. This molecular motor speed measurement system is suitable for the speed measurement of micro-motors at the nanometer level, thereby breaking the limitation of motor speed measurement caused by small volume. At the same time, the speed measurement method of molecular / nano motors is also suitable for the accurate measurement of high-speed and low-speed motor speeds, and can effectively avoid the measurement error caused by time delay. However, this nonlinear metamaterial system has a complex structure, and has too high requirements on the size and shape of the composite structure, and the size is micro-nanometer level, so it is challenging in processing and preparation. At the same time, the complexity of the structure greatly increases the difficulty of nonlinear analysis. Therefore, it is urgent to explore a nonlinear equivalent method to equate the complex nonlinear composite structure to a material system with uniform material distribution, thereby greatly reducing the complexity of these composite structures.

[0003] R.Blumenfeld and DJBergman proposed a perturbation method for finding the effective response of the potential field, which is only applicable to the case of small nonlinear magnetic susceptibility distribution density. P.Ponte Castaneda et al. developed a dual variational method for the nonlinear homogeneous equivalent of nonlinear metamaterials for the HS model, and estimated the effective constitutive behavior of nonlinear composite media through more mathematical analysis. The HS model refers to these composite structures in which the size of the composite particles can be different, but the volume proportion of each material is the same. Subsequently, relevant researchers used this variational method to obtain the effective response suitable for sparse distribution and inhomogeneous dispersion, and expanded the corresponding mathematical expression to a large volume fraction, so that it is applicable to the mathematical analysis of the nonlinear effects of random mixed particle models with two different nonlinear responses. L.Gao et al. further generalized the above nonlinear effective medium theory. They considered the influence of the distribution characteristics of inclusions, temperature, shape, electrostrictive effect and nonlocal response of materials on the effective nonlinear equivalent of nonlinear composite structures, thereby further enriching the theoretical analytical framework of the nonlinear effective medium theory of composite nonlinear structures. In addition to the nonlinear composite medium formed by the combination of nanoparticles and dielectric materials, layered nonlinear composite structures also have corresponding homogenized nonlinear effective medium theories. L. Gao and N. Daryakar proposed a homogenized nonlinear effective medium method suitable for layered nonlinear material composite structures, including the second-order and third-order nonlinear effective medium methods of composite gradient films and the second-order and third-order nonlinear effective medium methods of alternating layer nanostructures. At the same time, Q. Ren and S. Kovalev proposed the effective medium method of second-order and third-order nonlinear metasurfaces.

[0004] The nonlinear equivalent method of the above composite structure is to homogenize the composite into a continuous effective medium with uniform properties. However, in the linear case, local homogenization is not enough to correctly describe the wave behavior in special composite structures involving maximum wave vectors or surface wave resonances. In this case, the equivalent parameters cannot capture the microscopic evanescent and propagating waves and tunneling effects in the composite structure, resulting in a large difference between the nonlinear effects produced by the actual structure and its effective medium model. The nonlinear effects of composite materials are closely related to their linear effects, so the nonlinear effective parameters of composite materials are no longer accurate when involving maximum wave vectors or surface wave resonances. Summary of the invention

[0005] The object of the present invention is to provide a method for realizing nonlinear equivalence to solve the problems raised in the above background technology.

[0006] In order to solve the above technical problems, the present invention provides the following technical solutions:

[0007] A method for realizing nonlinear equivalence comprises the following steps:

[0008] Step S100: a plurality of nanoparticles exist in a background medium to form a composite material, and the relative dielectric constants of the nanoparticles in the composite material and the background medium are obtained;

[0009] Step S200: obtaining a nonlinear effective dielectric function of the composite material according to the relative dielectric constant;

[0010] Step S300: obtaining a first formula of effective nonlinear magnetic susceptibility of the composite material according to the nonlinear effective dielectric function of the composite material and the relative dielectric constant of the nanoparticles;

[0011] Step S400: by analyzing the relative dielectric constants of the nanoparticles and the background medium, and the relationship between the nonlinear effective dielectric function, a second formula for the effective nonlinear magnetic susceptibility of the composite material is obtained;

[0012] Step S500: obtaining a correction formula for the effective nonlinear magnetic susceptibility of the composite material according to the first formula for the effective nonlinear magnetic susceptibility and the second formula for the effective nonlinear magnetic susceptibility.

[0013] Furthermore, a number of nanoparticles c and nanoparticles a exist in a background medium h to form a composite material, and the relative dielectric constants of the nanoparticles c, nanoparticles a and the background medium h are ε c , ε a and ε h .

[0014] Furthermore, according to the volume occupied by nanoparticles c and nanoparticles a in the composite material, the volume fraction f of nanoparticles c and nanoparticles a is obtained. c and f a , and then according to the relative dielectric constant ε c , ε a and ε h , the nonlinear effective dielectric function of the composite material is obtained: ε eff =F(ε h ,ε c ,ε a ,f c ,f a ), where F is the nonlinear effective dielectric function, and the calculated result of the nonlinear effective dielectric function ε eff is the effective dielectric constant of the composite material.

[0015] Furthermore, step S300 includes:

[0016] Step S310: Assume that nanoparticle a is a nonlinear dielectric nanoparticle and nanoparticle c is a linear dielectric nanoparticle. Relative dielectric constant εa The linear part of , then there is the nonlinear dielectric formula of nanoparticle a: Among them, |E a | is the fundamental frequency field inside the nonlinear particle of nanoparticle a in the linear case, <|E a | 2 > represents the average value of the square of the spatial electric field norm, is the third-order nonlinear magnetic susceptibility of nanoparticle a;

[0017] Step S320: Substitute the nonlinear dielectric formula of nanoparticle a into the nonlinear effective dielectric function of the composite material to obtain the dielectric constant ε eff The expansion formula is: Where F' is the relative dielectric constant ε a The partial derivative of in, represents the partial derivative operation of the nonlinear effective dielectric function F, Relative dielectric constant ε a Find the partial derivative, <|E a |> represents the average value of the spatial electric field norm;

[0018] Step S330: Express the partial derivative F' as the average electric field of nanoparticle a in the linear limit, and obtain the electric field formula of nanoparticle a: Where E0 is the intensity of the incident electric field, Represents the effective dielectric constant ε eff Perform partial derivative operations;

[0019] Step S340: Substitute the electric field formula of the nanoparticle a into the dielectric constant ε eff In the expansion formula of represents the effective dielectric constant ε eff The linear part of

[0020] Step S350: Obtain the first formula of the effective nonlinear magnetic susceptibility of the composite material:

[0021] Furthermore, the step of analyzing the relationship between the relative dielectric constant of the nanoparticles and the background medium, and the nonlinear effective dielectric function in step S400 includes:

[0022] Step S410: According to the effective dielectric constant ε eff , as well as the relative dielectric constants of the nanoparticles and the background medium, the effective medium theory formula of the composite material is obtained: Where d represents the dimension of the composite material, ε i and f irepresent the relative dielectric constant and volume fraction of the i-th nanoparticle, respectively;

[0023] Step S420: Taking the assumption in step S310 as an example, if nanoparticle a has a third-order nonlinear effect and the size of nanoparticle a is much smaller than the size of nanoparticle c, then is the first-order polarizability of nanoparticle a, and according to the effective medium theory formula, the effective dielectric constant of the composite material is obtained:

[0024]

[0025] Where N represents the number of nanoparticles c in the composite material, ε i represents the relative dielectric constant of the ith nanoparticle c, M is the number of nanoparticles a in the composite material, β j represents the correction coefficient of the jth nanoparticle a, ε j represents the relative dielectric constant of the jth nanoparticle a.

[0026] Furthermore, the step of obtaining the second formula of the effective nonlinear magnetic susceptibility of the composite material in step S400 includes:

[0027] Step S430: If the nanoparticles are in a low density state, there is ε eff =ε h , and if the composite material dimension d = 3, the effective dielectric constant of the composite material can be simplified to:

[0028] Step S440: Obtain the second formula of the effective nonlinear magnetic susceptibility of the composite material by calculating the partial derivative of the effective dielectric constant of the composite material simplified in step S430:

[0029] Further, step S500 includes: obtaining a correction formula for the effective nonlinear magnetic susceptibility of the composite material according to the first formula for the effective nonlinear magnetic susceptibility of the composite material and the second formula for the effective nonlinear magnetic susceptibility:

[0030] Compared with the prior art, the beneficial effects achieved by the present invention are as follows: the present invention provides a method for realizing nonlinear equivalence, including: a plurality of nanoparticles exist in a background medium to form a composite material, and the relative dielectric constant of the nanoparticles and the background medium in the composite material is obtained; the nonlinear effective dielectric function of the composite material is obtained according to the relative dielectric constant; the first and second formulas of the effective nonlinear magnetic susceptibility of the composite material are obtained according to the nonlinear effective dielectric function and the relative dielectric constant of the nanoparticles and the background medium; and the effective nonlinear magnetic susceptibility correction formula of the composite material is obtained. The present invention accurately predicts the nonlinear optical effect of the composite material caused by the completely different local fields experienced by tiny nonlinear impurities by correcting the effective nonlinear magnetic susceptibility of the composite material. For weak nonlinear composite materials with short range and large particle size differences, the correction scheme of the nonlinear effective medium theory proposed by the present invention is reliable and accurate. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] The accompanying drawings are used to provide a further understanding of the present invention and constitute a part of the specification. Together with the embodiments of the present invention, they are used to explain the present invention and do not constitute a limitation of the present invention. In the accompanying drawings:

[0032] Figure 1 It is a flow chart of a method for realizing nonlinear equivalence of the present invention;

[0033] Figure 2 It is a schematic diagram of a composite material structure of a method for realizing nonlinear equivalence of the present invention;

[0034] Figure 3 It is a schematic diagram of a three-dimensional three-component periodic composite structure of a composite material for realizing a method of nonlinear equivalence of the present invention;

[0035] Among them, Figure 2 Middle: (a) is a schematic diagram of the structure of linear dielectric nanoparticles c and tiny nonlinear nanoparticles a in the background medium h;

[0036] (b) is caused by Figure 2 - Schematic diagram of the structure of a composite material composed of nanoparticles c and nanoparticles a in (a) present in a background medium h;

[0037] exist Figure 3 Middle: (a) is a schematic diagram of the three-dimensional three-component periodic composite structure in the composite material;

[0038] (b) is an enlarged image of each periodic structure;

[0039] (c) Reason Figure 3 - Schematic diagram of the structure of a composite material composed of nanoparticles c and nanoparticles a in (a) present in a background medium h;

[0040] (d) is the distribution of the normalized fundamental frequency electric field amplitude near nanoparticle c in the absence of nanoparticle a;

[0041] (e) As the nanoparticle a moves along Figure 3 -(b) Trajectory motion, normalized THG transmittance corrected using the nonlinear effective medium theory correction scheme (solid line Realcomposite) and calculated using the traditional Br effective nonlinear medium theory (dashed line Br), and the actual THG transmittance of the original composite material (dashed line Correction);

[0042] (f) is the normalized THG transmittance as the relative dielectric constant ε of nanoparticle a when nanoparticle a is located at position 5. a function. DETAILED DESCRIPTION

[0043] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0044] See also Figure 1-Figure 3 The present invention provides a method for realizing nonlinear equivalence, and a corresponding flowchart of the method for realizing nonlinear equivalence is shown in FIG. Figure 1 As shown. The following steps are included:

[0045] It should be noted that, in the present invention, taking the third harmonic generation (abbreviated as THG) as an example, two-dimensional and three-dimensional all-dielectric nonlinear composite material structures are studied on a deep subwavelength scale. Based on the Maxwell Garnett theory and the Bruggeman (Br) effective nonlinear medium theory, a theoretical description of the effective nonlinear susceptibility correction of THG in composite materials under weak nonlinear and rare limits is proposed. The corrected effective nonlinear susceptibility can accurately predict the nonlinear optical effects of composite materials caused by completely different local fields experienced by tiny nonlinear impurities.

[0046] Step S100: A plurality of nanoparticles exist in a background medium to form a composite material, and the relative dielectric constants of the nanoparticles in the composite material and the background medium are obtained.

[0047] Take d-dimensional composite materials as an example, as shown in the attached Figure 2As shown in the figure, a composite material is formed by a number of nanoparticles c and nanoparticles a in a background medium h. The relative dielectric constants of nanoparticles c, nanoparticles a and background medium h are ε c , ε a and ε h .

[0048] exist Figure 2 (a) is a schematic diagram of the structure of a linear dielectric nanoparticle c and a tiny nonlinear nanoparticle a in the background medium h; where ε c , ε a and ε h are the relative dielectric constants corresponding to nanoparticle c, nanoparticle a and background medium h, is the third-order nonlinear magnetic susceptibility of nanoparticle a, and the radius r of nanoparticle c c and the radius r of the nanoparticle a a The relationship between them is: a =0.1r c (b) is Figure 2 -(a) Schematic diagram of the structure of a composite material composed of nanoparticles c and nanoparticles a in a background medium h; eff and are the effective dielectric constant and effective nonlinear susceptibility of the composite material, respectively.

[0049] Step S200: obtaining the nonlinear effective dielectric function of the composite material according to the relative dielectric constant.

[0050] According to the volume occupied by nanoparticles c and nanoparticles a in the composite material, the volume fraction f of nanoparticles c and nanoparticles a is obtained. c and f a , and then according to the relative dielectric constant ε c , ε a and ε h , the nonlinear effective dielectric function of the composite material is obtained: ε eff =F(ε h ,ε c ,ε a ,f c ,f a ), where F is the nonlinear effective dielectric function, and the calculated result of the nonlinear effective dielectric function ε eff is the effective dielectric constant of the composite material.

[0051] Among them, if you use and Relative dielectric constant ε eff , ε c , ε a and ε hThe linear part of

[0052] Step S300: obtaining a first formula of effective nonlinear magnetic susceptibility of the composite material according to the nonlinear effective dielectric function of the composite material and the relative dielectric constant of the nanoparticles.

[0053] Step S310: Assume that nanoparticle a is a nonlinear dielectric nanoparticle and nanoparticle c is a linear dielectric nanoparticle. Relative dielectric constant ε a The linear part of , then there is the nonlinear dielectric formula of nanoparticle a: Among them, |E a | is the fundamental frequency field inside the nonlinear particle of nanoparticle a in the linear case, <|E a | 2 > represents the average value of the square of the spatial electric field norm, is the third-order nonlinear magnetic susceptibility of nanoparticle a.

[0054] Step S320: Substitute the nonlinear dielectric formula of nanoparticle a into the nonlinear effective dielectric function of the composite material to obtain the dielectric constant ε eff The expansion formula is: Where F' is the relative dielectric constant ε a The partial derivative of in, represents the partial derivative operation of the nonlinear effective dielectric function F, Relative dielectric constant ε a Find the partial derivative, <|E a |> represents the average value of the spatial electric field norm:

[0055] The dielectric constant ε in this step eff The expansion formula is based on ε eff The Taylor series expansion of .

[0056] Step S330: Express the partial derivative F' as the average electric field of nanoparticle a in the linear limit, and obtain the electric field formula of nanoparticle a: Where E0 is the intensity of the incident electric field, Represents the effective dielectric constant ε eff Perform partial derivative operation.

[0057] The electric field formula of the nanoparticle a is obtained by substituting the relevant values ​​of the nanoparticles a and c and the background medium h in this embodiment into the existing formula.

[0058] Step S340: Substitute the electric field formula of the nanoparticle a into the dielectric constant ε eff In the expansion formula of represents the effective dielectric constant ε eff The linear part of

[0059] Step S350: Obtain the first formula of the effective nonlinear magnetic susceptibility of the composite material:

[0060] The first formula of the effective nonlinear susceptibility is obtained according to the definition and can be directly applied.

[0061] Step S400: by analyzing the relative dielectric constants of the nanoparticles and the background medium, and the relationship between the nonlinear effective dielectric function, a second formula for the effective nonlinear magnetic susceptibility of the composite material is obtained.

[0062] Step S410: According to the effective dielectric constant ε eff , as well as the relative dielectric constants of the nanoparticles and the background medium, the effective medium theory formula of the composite material is obtained: Where d represents the dimension of the composite material, ε i and f i represent the relative dielectric constant and volume fraction of the i-th nanoparticle, respectively.

[0063] The effective medium theory formula of the composite material is obtained by substituting the relevant values ​​of the nanoparticles a and c and the background medium h in the present embodiment into the existing formula. This step is based on the classic Maxwell Garnett effective medium theory to obtain the effective dielectric constant ε eff Relationship with nanoparticles and background media in composite materials.

[0064] Step S420: Taking the assumption in step S310 as an example, if nanoparticle a has a third-order nonlinear effect and the size of nanoparticle a is much smaller than the size of nanoparticle c, then is the first-order polarizability of nanoparticle a, and according to the effective medium theory formula, the effective dielectric constant of the composite material is obtained:

[0065]

[0066] Where N represents the number of nanoparticles c in the composite material, ε i represents the relative dielectric constant of the ith nanoparticle c, M is the number of nanoparticles a in the composite material, β j represents the correction coefficient of the jth nanoparticle a, ε j represents the relative dielectric constant of the jth nanoparticle a.

[0067] like Figure 2-(a) shows that when an additional tiny nonlinear particle is placed near a dielectric nanoparticle, the electric field near it changes dramatically because the scattered wave of the dielectric nanoparticle contains a large number of evanescent waves. Therefore, when the additional tiny nonlinear particle is in different positions, it will experience different fundamental frequency fields, which may be much larger or smaller than the average field in the entire composite material. In this case, the classical Maxwell Garnett theory needs to be modified according to the change of the local field. Among them, β j It can be calculated by the ratio of the electric field strength inside the jth tiny nonlinear particle to the average electric field of the entire composite material without nonlinear particles.

[0068] Step S430: If the nanoparticles are in a low density state, there is ε eff =ε h , and if the composite material dimension d = 3, the effective dielectric constant of the composite material can be simplified to:

[0069] For simplicity, we consider a three-dimensional three-component composite material structure, that is, d = 3, then:

[0070]

[0071] Nanoparticles have ε in a low-density state eff =ε h ,but

[0072] Step S440: Obtain the second formula of the effective nonlinear magnetic susceptibility of the composite material by calculating the partial derivative of the effective dielectric constant of the composite material simplified in step S430:

[0073] Step S500: obtaining a correction formula for the effective nonlinear magnetic susceptibility of the composite material according to the first formula for the effective nonlinear magnetic susceptibility and the second formula for the effective nonlinear magnetic susceptibility.

[0074] According to the first formula and the second formula of the effective nonlinear magnetic susceptibility of the composite material, the corrected formula of the effective nonlinear magnetic susceptibility of the composite material is obtained:

[0075] Embodiment 1: This embodiment takes a three-dimensional three-component periodic composite structure as an example. By conducting experiments on this three-component periodic composite structure, the results obtained can illustrate that the correction scheme of the nonlinear effective medium theory proposed by the present invention for weak nonlinear composite materials with short range and large particle size difference is reliable and accurate, as follows:

[0076] Attach Figure 3 As shown, Figure 3 -(a) is a schematic diagram of the three-dimensional three-component periodic composite structure in the composite material. Figure 3 -(a), establish a three-dimensional coordinate system of x, y and z axes. If an incident wave is a TM wave, the incident wavelength is λ0, the propagation direction of the incident wave is the z direction, and the incident electric field E ω The polarization direction is the x direction;

[0077] Figure 3 - (b) is a magnified view of each periodic structural unit, whose lattice constant is a=λ0 / 100, where the size of the nanoparticle c is r c =0.2a, relative dielectric constant is ε c , the relative magnetic permeability is μ c =1; the size of nanoparticle a is r a =0.1r c , the relative dielectric constant is ε a , the relative magnetic permeability is μ a =1; the relative dielectric constant of the background medium h is ε h , the relative magnetic permeability is μ h = 1; the gap (i.e., edge-to-edge distance) between nanoparticle c and nanoparticle a is d. The numbers 1-5 correspond to Figure 3 - Five different positions of nanoparticle a in (b), where the curves between 1-5 represent the movement trajectory of nanoparticle a.

[0078] Figure 3 -(c) is formed based on the presence of nanoparticles c and nanoparticles a in the background medium h, with an effective relative dielectric constant ε eff and effective nonlinear susceptibility of composite materials.

[0079] In order to better understand the effect of the evanescent field on the accuracy of the traditional Br nonlinear effective medium theory, we Figure 3 -(c) shows the distribution of the fundamental frequency local field around the dielectric nanoparticles. In this embodiment, the relative dielectric constant ε c =3.

[0080] Figure 3 -(d) is the distribution of the normalized fundamental frequency electric field amplitude near nanoparticle c in the absence of nanoparticle a. Figure 3-(d) shows that the field strength at the two poles of nanoparticle c (positions 1 and 4) is greatly enhanced, while it is greatly weakened at the equator (positions 2 and 3). At position 5, which is farther away, the influence of nanoparticle c on nanoparticle a gradually weakens, and the electric field strength there tends to the incident electric field strength. The generation of this drastic change in the evanescent field originates from the continuity boundary condition of the dielectric nanoparticle-air interface. Since nanoparticle c is at a deep subwavelength scale, the basic electric field inside it can be considered to be uniform and can be approximately expressed as (obtained by the existing formula). Therefore, the continuity of the electric displacement field at the two poles of the dielectric nanoparticle can be obtained in is the electric field at the two poles on the air side. In addition, the electric field at the equator of the dielectric nanoparticle on the air side is: Due to the continuity of the electric field, when ε c >ε h When, there is, When a tiny nonlinear nanoparticle is placed close to a dielectric nanoparticle, it will experience different fundamental electric fields at different locations, thus affecting the position-dependent THG transmittance.

[0081] Figure 3 -(e) is the time when the nanoparticle a moves along Figure 3 -(b) When the trajectory moves, the normalized THG transmittance calculated using the nonlinear effective medium theory correction scheme (solid line Realcomposite) and the traditional Br effective nonlinear medium theory (dashed line Br), while the actual THG transmittance of the original composite material is: dashed line Correction. Let the relative dielectric constant ε of nanoparticle a be a =1, so that the normalized THG transmittance is normalized to 1.

[0082] Figure 3 -(f) is the normalized THG transmittance as the relative dielectric constant ε of nanoparticle a when nanoparticle a is located at position 5 a (ε c =3 and ε h =1).

[0083] exist Figure 3 -(e) and Figure 3 -(f), we show the difference between the normalized THG transmittance calculated using the modified nonlinear effective medium theory (solid line Real composite) and the conventional Br effective nonlinear medium theory (dashed line Br). Figure 3-(e) shows the THG transmittance when nanoparticle a moves from position 1 to position 5 along the trajectory 1→2→3→4→5. When nanoparticle a moves along the trajectory 1→4, the gap L between the two nanoparticles remains unchanged. Here we assume that the nonlinear magnetic susceptibility of nanoparticle a is several orders of magnitude larger than that of nanoparticle c, so the nonlinear response of nanoparticle c can be ignored. The solid line Realcomposite and the dotted line Br in the figure represent the normalized THG transmittance calculated using the nonlinear effective medium theory correction scheme and the traditional Br effective nonlinear medium theory, respectively. Obviously, there is a significant difference between the two results. However, the correction result (solid line Realcomposite) proposed by our nonlinear effective medium theory correction scheme is consistent with the actual THG transmittance of the original composite material (dotted line Correction). This proves the accuracy of the correction scheme using the nonlinear effective medium theory.

[0084] exist Figure 3 -(f), we demonstrate that when nanoparticle a is located at position 5, the THG transmission of the composite structure varies with the relative dielectric constant ε of nanoparticle a. a Here we let ε c =3 and ε h = 1. In the figure we can clearly observe that as ε a →ε h , the revised result of the Br effect nonlinear medium theory will be closer to the traditional Br effect nonlinear medium theory. This is because ε a and ε h The larger the difference between them, the larger the local field inside the nanoparticle a. The greater the difference between the uniform field and the composite structure, the more accurate the correction scheme of the nonlinear effective medium theory proposed by us is.

[0085] It should be noted that, in this article, relational terms such as first and second, etc. are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Moreover, the terms "include", "comprise" or any other variants thereof are intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or device.

[0086] Finally, it should be noted that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art can still modify the technical solutions described in the aforementioned embodiments or replace some of the technical features therein by equivalents. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A method for realizing nonlinear equivalence, characterized in that: The following steps are involved: Step S100: a plurality of nanoparticles exist in a background medium to form a composite material, and the relative dielectric constants of the nanoparticles in the composite material and the background medium are obtained; Step S200: obtaining a nonlinear effective dielectric function of the composite material according to the relative dielectric constant; Step S300: obtaining a first formula of effective nonlinear magnetic susceptibility of the composite material according to the nonlinear effective dielectric function of the composite material and the relative dielectric constant of the nanoparticles; Step S400: by analyzing the relative dielectric constants of the nanoparticles and the background medium, and the relationship between the nonlinear effective dielectric function, a second formula for the effective nonlinear magnetic susceptibility of the composite material is obtained; Step S500: obtaining a correction formula for the effective nonlinear magnetic susceptibility of the composite material according to the first formula for the effective nonlinear magnetic susceptibility and the second formula for the effective nonlinear magnetic susceptibility; Step S310: Assume that nanoparticle a is a nonlinear dielectric nanoparticle and nanoparticle c is a linear dielectric nanoparticle. Relative dielectric constant ε a The linear part of , then there is the nonlinear dielectric formula of nanoparticle a: Among them, |E a | is the fundamental frequency field inside the nonlinear particle of nanoparticle a in the linear case, <E a | 2 > represents the average value of the square of the spatial electric field norm, is the third-order nonlinear magnetic susceptibility of nanoparticle a; Step S320: Substitute the nonlinear dielectric formula of nanoparticle a into the nonlinear effective dielectric function of the composite material to obtain the dielectric constant ε eff The expansion formula is: Where F' is the relative dielectric constant ε a The partial derivative of in, represents the partial derivative operation of the nonlinear effective dielectric function F, Relative dielectric constant ε a Find the partial derivative, <|E a |> represents the average value of the spatial electric field norm; Step S330: Express the partial derivative F' as the average electric field of nanoparticle a in the linear limit, and obtain the electric field formula of nanoparticle a: Where E0 is the intensity of the incident electric field, Represents the effective dielectric constant ε eff Perform partial derivative operations; Step S340: Substitute the electric field formula of the nanoparticle a into the dielectric constant ε eff In the expansion formula of represents the effective dielectric constant ε eff The linear part of Step S350: Obtain the first formula of the effective nonlinear magnetic susceptibility of the composite material: Step S410: According to the effective dielectric constant ε eff , as well as the relative dielectric constants of the nanoparticles and the background medium, the effective medium theory formula of the composite material is obtained: Where d represents the dimension of the composite material, ε i and f i represent the relative dielectric constant and volume fraction of the i-th nanoparticle, respectively; Step S420: Taking the assumption in step S310 as an example, if nanoparticle a has a third-order nonlinear effect and the size of nanoparticle a is much smaller than the size of nanoparticle c, then is the first-order polarizability of nanoparticle a, and according to the effective medium theory formula, the effective dielectric constant of the composite material is obtained: Where N represents the number of nanoparticles c in the composite material, ε i represents the relative dielectric constant of the ith nanoparticle c, M is the number of nanoparticles a in the composite material, β j represents the correction coefficient of the jth nanoparticle a, ε j represents the relative dielectric constant of the jth nanoparticle a; Step S430: If the nanoparticles are in a low density state, there is ε eff =ε h , and if the composite material dimension d = 3, the effective dielectric constant of the composite material can be simplified to: Step S440: Obtain the second formula of the effective nonlinear magnetic susceptibility of the composite material by calculating the partial derivative of the effective dielectric constant of the composite material simplified in step S430:

2. A method for realizing nonlinear equivalence according to claim 1, characterized in that: Step S100 includes: a plurality of nanoparticles c and nanoparticles a exist in a background medium h to form a composite material, and the relative dielectric constants of the nanoparticles c, the nanoparticles a and the background medium h are ε c , ε a and ε h .

3. A method for realizing nonlinear equivalence according to claim 2, characterized in that: Step S200 includes: obtaining the volume fraction f of the nanoparticles c and the nanoparticles a according to the volumes occupied by the nanoparticles c and the nanoparticles a in the composite material. c and f a , and then according to the relative dielectric constant ε c , ε a and ε h , the nonlinear effective dielectric function of the composite material is obtained: eff =F(ε h ,ε c ,ε a ,f c ,f a ), where F is the nonlinear effective dielectric function, and the calculated result of the nonlinear effective dielectric function ε eff is the effective dielectric constant of the composite material.

4. A method for realizing nonlinear equivalence according to claim 3, characterized in that: Step S500 includes: obtaining a correction formula for the effective nonlinear magnetic susceptibility of the composite material according to the first formula for the effective nonlinear magnetic susceptibility of the composite material and the second formula for the effective nonlinear magnetic susceptibility:

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