CN118940515A - A method for realizing nonlinear equivalence - Google Patents
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Abstract
The invention discloses a method for realizing nonlinear equivalence, and belongs to the technical field of composite materials. Comprising the following steps: a plurality of nano particles exist in a background medium to form a composite material, and the relative dielectric constants of the nano particles in the composite material and the background medium are obtained; obtaining a nonlinear effective dielectric function of the composite material according to the relative dielectric constant; obtaining first and second formulas of effective nonlinear susceptibility of the composite material according to the nonlinear effective dielectric function and relative dielectric constants of the nano particles and the background medium; and obtaining an effective nonlinear susceptibility correction formula of the composite material. The invention accurately predicts the nonlinear optical effect of the composite material caused by completely different local fields experienced by tiny nonlinear impurities by correcting the effective nonlinear magnetic susceptibility of the composite material, and the correction scheme of the nonlinear effective medium theory provided by the invention is reliable and accurate for weak nonlinear composite materials with short ranges and large particle size differences.
Description
Technical Field
The invention relates to the technical field of composite materials, in particular to a method for realizing nonlinear equivalence.
Background
With the rising subjects of the micro-electromechanical systems based on micro-mechanical and micro-electronic technology, the micro-motor, namely, the motor with the diameter smaller than 160mm or the power smaller than 750W, is used as the core of the micro-electromechanical system and is widely applied to the fields of various scientific research national defense such as biomedical, aerospace, precision instruments, communication engineering and the like. The medical field is one of the most representative fields of micro-motor application, and can realize the functions of directional drug delivery, precise surgery, biological sensing and the like. In recent years, related researchers have successively developed nanomotors having a size of only a few nanometers. The micro-motor can convert chemical energy into mechanical energy, can be integrated on a microminiature machine, and is widely applied to the medical fields of local micro-drug delivery in vivo, precise microsurgery, micro-wound endoscopic treatment and the like. In order to meet the high requirements of the rapid development of modern medical technology on the measurement precision of a micro-motor rotating speed measurement system. A number of methods for measuring the rotational speed of molecular motors suitable for such medical applications have been proposed. The non-linear optical metamaterial is designed to be used as a motor rotation speed measuring system by utilizing the high sensitivity characteristic of Second and Third harmonic generation (THG/SHG) to structures and environments, and the molecular motor rotation speed measuring system is suitable for the rotation speed measurement of a nano-level micro motor, so that the limitation of motor rotation speed measurement caused by small volume is broken. Meanwhile, the speed measuring method of the molecular/nano motor micro motor is simultaneously suitable for accurate measurement of the rotating speeds of the high-speed motor and the low-speed motor, and measurement errors caused by time delay can be effectively avoided. However, the nonlinear metamaterial system structure is complex, the size and shape of the composite structure are excessively high, and the size is micro-nano, so that the nonlinear metamaterial system structure has challenges in processing and preparation. Meanwhile, the complexity of the structure greatly increases the difficulty of nonlinear analysis. Therefore, it is highly desirable to explore a nonlinear equivalent method to equivalent complex nonlinear composite structures to material systems with uniform material distribution, thereby greatly reducing the complexity of these composite structures.
Blumenofeld and d.j. Bergman propose a perturbation method for finding the effective response of the potential field, which is only suitable for treating the situation that the nonlinear susceptibility distribution density is small. Ponte Castanada et al developed a method of non-linear homogeneous equivalent dual variation of a non-linear metamaterial for H-S models and estimated the effective constitutive behavior of a non-linear composite medium by more mathematical analysis. The H-S model refers to these composite structures in which the composite particles may vary in size, but the volume fraction of each material is the same. Then, the relevant researchers also adopt the variation method to obtain effective response suitable for thin distribution and non-uniform dispersion, and expand the corresponding mathematical expression with large volume fraction, so that the method is suitable for mathematical analysis of nonlinear effects of a random mixed particle model with two different nonlinear responses. Gao et al further generalized the nonlinear effective medium theory described above. They consider the effects of the distribution characteristics, temperature, shape, electrostriction effect of inclusions and the nonlinear equivalent of the material in response to the effective nonlinearity of the nonlinear composite structure, thereby further enriching the theoretical resolution framework of the nonlinear effective medium theory of the composite nonlinear structure. In addition to the nonlinear composite medium formed by combining the nanoparticles with the dielectric material, the layered nonlinear composite structure also has a corresponding homogenized nonlinear effective medium theory. Gao and n.daryakar propose a method of homogenizing nonlinear effective media suitable for layered nonlinear material composite structures, including second and third order nonlinear effective media methods of composite gradient films and second and third order nonlinear effective media methods of alternating layer nanostructures. Meanwhile, Q.ren and S.Kovalev propose effective medium methods for second and third order nonlinear supersurfaces.
The nonlinear equivalent method of the composite structure described above is to homogenize the composite into a continuous effective medium with uniform properties. But in the linear case, local homogenization is not sufficient to correctly describe the wave behavior in special composite structures involving very large wave vectors or surface wave resonances. In this case, the equivalent parameters cannot capture microscopic evanescent and propagating waves and tunneling effects in the composite structure, resulting in a large difference between the nonlinear effect generated by the actual structure and its effective medium model. The nonlinear effect of the composite material is closely related to the linear effect of the composite material, so that the nonlinear effective parameters of the composite material are not accurate when the condition of extremely large wave vector or surface wave resonance is involved.
Disclosure of Invention
The present invention is directed to a method for achieving nonlinear equivalence, so as to solve the problems set forth in the background art.
In order to solve the technical problems, the invention provides the following technical scheme:
A method of achieving nonlinear equivalence comprising the steps of:
step S100: a plurality of nano particles exist in a background medium to form a composite material, and the relative dielectric constants of the nano particles 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 nano particles;
Step S400: obtaining an effective nonlinear susceptibility second formula of the composite material by analyzing the relation between the relative dielectric constants of the nano particles and the background medium and the nonlinear effective dielectric function;
step S500: and obtaining an effective nonlinear magnetic susceptibility correction formula of the composite material according to the effective nonlinear magnetic susceptibility first formula and the effective nonlinear magnetic susceptibility second formula.
Further, a plurality of nano particles c and nano particles a exist in the background medium h to form a composite material, and the relative dielectric constants of the nano particles c, the nano particles a and the background medium h are epsilon c、εa and epsilon h respectively.
Further, according to the volume occupied by the nano particles c and the nano particles a in the composite material, the volume fractions f c and f a of the nano particles c and the nano particles a are obtained, and further according to the relative dielectric constants epsilon c、εa and epsilon h, the nonlinear effective dielectric function of the composite material is obtained: epsilon eff＝F(εh,εc,εa,fc,fa), wherein F is a nonlinear effective dielectric function, and epsilon eff is the effective dielectric constant of the composite material.
Further, step S300 includes:
step S310: let nanoparticle a be a nonlinear dielectric nanoparticle and nanoparticle c be a linear dielectric nanoparticle, using The linear part representing the relative permittivity epsilon a has a nonlinear dielectric formula for the nanoparticle a: Where, |E a | is the fundamental frequency field inside the nonlinear particle of nanoparticle a in the linear case, <|E a|2 > represents the average of the squares of the spatial electric field modes, Is the third order nonlinear susceptibility of nanoparticle a;
Step S320: substituting the nonlinear dielectric formula of the nano particle a into the nonlinear effective dielectric function of the composite material to obtain an expansion formula of dielectric constant epsilon eff: wherein F' is the partial derivative of the relative permittivity epsilon a, Wherein, Representing a biasing operation on the nonlinear effective dielectric function F,Represents the partial derivative of the relative permittivity epsilon a of the independent variable, and < |E a | > represents the average value of the spatial electric field modes;
Step S330: the partial derivative F' is expressed as the average electric field of the nano particle a under the linear limit, and the electric field formula of the nano particle a is obtained: wherein E 0 is the intensity of the incident electric field, Representing a partial derivative operation of the effective dielectric constant epsilon eff;
step S340: substituting the electric field formula of the nano particle a into the expansion formula of dielectric constant epsilon eff to obtain A linear portion representing the effective dielectric constant ε eff;
Step S350: obtaining a first formula of effective nonlinear susceptibility of the composite material:
Further, the step of analyzing the relation between the relative dielectric constants of the nanoparticle and the background medium and the nonlinear effective dielectric function in step S400 includes:
Step S410: according to the effective dielectric constant epsilon eff and the relative dielectric constant of the nano particles and the background medium, the effective medium theoretical formula of the composite material is obtained: Wherein d represents the composite dimension, ε i and f i represent the relative permittivity and volume fraction, respectively, of the ith nanoparticle;
Step S420: taking the assumption in step S310 as an example, if the nanoparticle a has a third-order nonlinear effect and the size of the nanoparticle a is much smaller than that of the nanoparticle c, the nonlinear dielectric equation The first-order polarizability of the nano particles a is obtained according to an effective medium theoretical formula, and the effective dielectric constant of the composite material is obtained:
Where N represents the number of nanoparticles c in the composite material, epsilon i represents the relative permittivity of the ith nanoparticle c, M represents the number of nanoparticles a in the composite material, beta j represents the correction factor of the jth nanoparticle a, and epsilon j represents the relative permittivity of the jth nanoparticle a.
Further, the step of obtaining the second formula of the effective nonlinear susceptibility of the composite material in step S400 includes:
Step S430: if the nanoparticle is in a low density state, there is epsilon eff＝εh, and if the composite dimension d=3, the effective dielectric constant of the composite can be reduced to:
step S440: obtaining a second formula of effective nonlinear magnetic susceptibility of the composite material according to the partial derivative of the effective dielectric constant of the composite material simplified in the step S430:
Further, step S500 includes: according to the first formula of the effective nonlinear magnetic susceptibility of the composite material and the second formula of the effective nonlinear magnetic susceptibility, an effective nonlinear magnetic susceptibility correction formula of the composite material is obtained:
Compared with the prior art, the invention has the following beneficial effects: the invention provides a method for realizing nonlinear equivalence, which comprises the following steps: a plurality of nano particles exist in a background medium to form a composite material, and the relative dielectric constants of the nano particles in the composite material and the background medium are obtained; obtaining a nonlinear effective dielectric function of the composite material according to the relative dielectric constant; obtaining first and second formulas of effective nonlinear susceptibility of the composite material according to the nonlinear effective dielectric function and relative dielectric constants of the nano particles and the background medium; and obtaining an effective nonlinear susceptibility correction formula of the composite material. The invention accurately predicts the nonlinear optical effect of the composite material caused by completely different local fields experienced by tiny nonlinear impurities by correcting the effective nonlinear magnetic susceptibility of the composite material, and the correction scheme of the nonlinear effective medium theory provided by the invention is reliable and accurate for weak nonlinear composite materials with short ranges and large particle size differences.
Drawings
The accompanying drawings are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate the invention and together with the embodiments of the invention, serve to explain the invention. In the drawings:
FIG. 1 is a schematic flow chart of a method of implementing nonlinear equivalence in accordance with the present invention;
FIG. 2 is a schematic illustration of a composite structure for implementing a nonlinear equivalent method of the present invention;
FIG. 3 is a schematic view of a three-dimensional, three-component periodic composite structure of a composite material for implementing a nonlinear equivalent method of the present invention;
Wherein in fig. 2: (a) A schematic structural diagram of linear dielectric nano particles c and tiny nonlinear nano particles a in a background medium h;
(b) Is a schematic structural diagram of a composite material composed of nanoparticles c and nanoparticles a in fig. 2- (a) present in a background medium h;
in fig. 3: (a) Is a schematic diagram of a three-dimensional three-component periodic composite structure in the composite material;
(b) An enlarged view for each periodic structure;
(c) Is a schematic structural diagram of a composite material composed of nanoparticles c and nanoparticles a in fig. 3- (a) present in a background medium h;
(d) In order to normalize the distribution of the amplitude of the fundamental electric field in the vicinity of nanoparticle c in the absence of nanoparticle a;
(e) To correct for and calculate the normalized THG transmittance using the nonlinear effective medium theory Correction scheme (solid line Realcomposite) and using the traditional Br effective nonlinear medium theory (dashed line Br) along with the trajectory of nanoparticle a in fig. 3- (b), as well as the actual THG transmittance of the original composite material (dashed line Correction);
(f) To normalize the THG transmission as a function of the relative permittivity epsilon a of nanoparticle a when nanoparticle a is at position 5.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
Referring to fig. 1-3, the present invention provides a method for implementing nonlinear equivalence, and a flow chart of a corresponding method for implementing nonlinear equivalence is shown in fig. 1. The method comprises the following steps:
In the present invention, the generation of Third Harmonic (THG) is taken as an example, and two-dimensional and three-dimensional all-dielectric nonlinear composite material structures are studied on the deep sub-wavelength scale. Based on Maxwell Garnett theory and Bruggeman (Br) effective nonlinear medium theory, theoretical description of effective nonlinear susceptibility correction of the composite material THG under weak nonlinearity and a weak limit is provided. The corrected effective nonlinear susceptibility can accurately predict nonlinear optical effects of the composite material caused by the disparate local fields experienced by the tiny nonlinear impurities.
Step S100: and a plurality of nano particles exist in the background medium to form a composite material, and the relative dielectric constants of the nano particles in the composite material and the background medium are obtained.
Taking d-dimensional composite material as an example, as shown in fig. 2, the composite material is formed by a plurality of nano particles c and nano particles a existing in a background medium h, and the relative dielectric constants of the nano particles c, the nano particles a and the background medium h are epsilon c、εa and epsilon h respectively.
In fig. 2: (a) A schematic structural diagram of linear dielectric nano particles c and tiny nonlinear nano particles a in a background medium h; wherein epsilon c、εa and epsilon h are respectively the relative dielectric constants corresponding to the nano particle c, the nano particle a and the background medium h,Is the third order nonlinear susceptibility of nanoparticle a, and the relationship between radius r c of nanoparticle c and radius r a of nanoparticle a is: r a＝0.1rc. (b) Is a schematic structural diagram of a composite material composed of nanoparticles c and nanoparticles a in fig. 2- (a) present in a background medium h; epsilon eff The effective dielectric constant and the effective nonlinear magnetic susceptibility of the composite material are respectively.
Step S200: and obtaining the nonlinear effective dielectric function of the composite material according to the relative dielectric constant.
According to the volume occupied by the nano particles c and the nano particles a in the composite material, the volume fractions f c and f a of the nano particles c and the nano particles a are obtained, and further according to the relative dielectric constants epsilon c、εa and epsilon h, the nonlinear effective dielectric function of the composite material is obtained: epsilon eff＝F(εh,εc,εa,fc,fa), wherein F is a nonlinear effective dielectric function, and epsilon eff is the effective dielectric constant of the composite material.
Wherein if it is usedAndThe linear portions representing the relative dielectric constants ε eff、εc、εa and ε h are then present
Step S300: and obtaining a first formula of the 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 nano particles.
Step S310: let nanoparticle a be a nonlinear dielectric nanoparticle and nanoparticle c be a linear dielectric nanoparticle, usingThe linear part representing the relative permittivity epsilon a has a nonlinear dielectric formula for the nanoparticle a: Where, |E a | is the fundamental frequency field inside the nonlinear particle of nanoparticle a in the linear case, <|E a|2 > represents the average of the squares of the spatial electric field modes, Is the third order nonlinear susceptibility of nanoparticle a.
Step S320: substituting the nonlinear dielectric formula of the nano particle a into the nonlinear effective dielectric function of the composite material to obtain an expansion formula of dielectric constant epsilon eff: wherein F' is the partial derivative of the relative permittivity epsilon a, Wherein, Representing a biasing operation on the nonlinear effective dielectric function F,Represents the partial derivative of the relative permittivity epsilon a of the independent variable, <|E a | > represents the average of the spatial electric field modes:
The expansion formula for dielectric constant ε eff in this step is expanded according to the Taylor series of ε eff.
Step S330: the partial derivative F' is expressed as the average electric field of the nano particle a under the linear limit, and the electric field formula of the nano particle a is obtained: wherein E 0 is the intensity of the incident electric field, Indicating that the effective dielectric constant epsilon eff is biased.
The electric field formula of the nanoparticle a is obtained by substituting and converting the relevant values of the nanoparticles a and c and the background medium h in the embodiment according to the existing formula.
Step S340: substituting the electric field formula of the nano particle a into the expansion formula of dielectric constant epsilon eff to obtainA linear portion representing the effective dielectric constant ε eff;
Step S350: obtaining a first formula of effective nonlinear susceptibility of the composite material:
the first formula of the effective nonlinear susceptibility is obtained by definition and can be directly applied.
Step S400: and obtaining the second formula of the effective nonlinear susceptibility of the composite material by analyzing the relation between the relative dielectric constants of the nano particles and the background medium and the nonlinear effective dielectric function.
Step S410: according to the effective dielectric constant epsilon eff and the relative dielectric constant of the nano particles and the background medium, the effective medium theoretical formula of the composite material is obtained: Where d represents the composite dimensions, ε i and f i represent the relative permittivity and volume fraction, respectively, of the ith nanoparticle.
The effective medium theoretical formula of the composite material is obtained by substituting and converting the related numerical values of the nano particles a and c and the background medium h in the embodiment according to the existing formula. The step is to obtain the relation between the effective dielectric constant epsilon eff and the nano particles and the background medium in the composite material according to the classical MaxwellGarnett effective medium theory.
Step S420: taking the assumption in step S310 as an example, if the nanoparticle a has a third-order nonlinear effect and the size of the nanoparticle a is much smaller than that of the nanoparticle c, the nonlinear dielectric equationThe first-order polarizability of the nano particles a is obtained according to an effective medium theoretical formula, and the effective dielectric constant of the composite material is obtained:
Where N represents the number of nanoparticles c in the composite material, epsilon i represents the relative permittivity of the ith nanoparticle c, M represents the number of nanoparticles a in the composite material, beta j represents the correction factor of the jth nanoparticle a, and epsilon j represents the relative permittivity of the jth nanoparticle a.
When an additional tiny nonlinear particle is placed near the dielectric nanoparticle, the electric field near the dielectric nanoparticle is greatly changed due to the large amount of evanescent wave contained in the scattered wave of the dielectric nanoparticle, as shown in fig. 2- (a). Thus, when the additional tiny nonlinear particles are in different positions, they will experience different fundamental fields, which may be much larger or smaller than the average field in the overall composite. In this case, the classical Maxwell gamnett theory requires correction according to local field variations. Wherein β j can be calculated from the ratio of the electric field strength within the j-th tiny nonlinear particle to the average electric field of the entire composite without nonlinear particles.
Step S430: if the nanoparticle is in a low density state, there is epsilon eff＝εh, and if the composite dimension d=3, the effective dielectric constant of the composite can be reduced to:
for simplicity we consider a three-dimensional three-component composite structure, i.e. d=3, then there are:
the nano particles have epsilon eff＝εh under the low density state, then
Step S440: obtaining a second formula of effective nonlinear magnetic susceptibility of the composite material according to the partial derivative of the effective dielectric constant of the composite material simplified in the step S430:
step S500: and obtaining an effective nonlinear magnetic susceptibility correction formula of the composite material according to the effective nonlinear magnetic susceptibility first formula and the effective nonlinear magnetic susceptibility second formula.
According to the first formula of the effective nonlinear magnetic susceptibility of the composite material and the second formula of the effective nonlinear magnetic susceptibility, an effective nonlinear magnetic susceptibility correction formula of the composite material is obtained:
embodiment one: in this embodiment, taking a three-dimensional three-component periodic composite structure as an example, by performing experiments on the three-component periodic composite structure, the obtained result can indicate that the proposed correction scheme of the nonlinear effective medium theory is reliable and accurate for weak nonlinear composite materials with short ranges and large particle size differences, and is specifically as follows:
In fig. 3, a three-dimensional three-component periodic composite structure schematic diagram of the composite material is shown in fig. 3- (a), a three-dimensional coordinate system of x, y and z axes is established, if an incident wave is a TM wave, the incident wavelength is lambda 0, the propagation direction of the incident wave is the z direction, and the polarization direction of the incident electric field E ω is the x direction;
Fig. 3- (b) is an enlarged view of each periodic structural unit, having a lattice constant of a=λ 0/100, wherein the size r c =0.2a of the nanoparticle c, a relative permittivity of ε c, and a relative permeability of μ c =1; the size r a＝0.1rc of the nanoparticle a has a relative permittivity of epsilon a and a relative permeability of μ a =1; the relative permittivity of the background medium h is epsilon h, and the relative permeability is mu h =1; the gap (i.e., edge-to-edge distance) between nanoparticle c and nanoparticle a is d. Wherein the numbers 1-5 correspond to five different positions of nanoparticle a in fig. 3- (b), wherein the curve between 1-5 represents the movement trajectory of nanoparticle a.
FIG. 3- (c) shows the effective relative permittivity ε eff and the effective nonlinear susceptibility according to the presence of nanoparticle c and nanoparticle a in the background medium hIs a composite material of (a).
To better understand the effect of the evanescent field on the theoretical accuracy of a conventional Br nonlinear effective medium, we present the distribution of the fundamental frequency local field around the dielectric nanoparticle in fig. 3- (c), in this example, with a relative permittivity epsilon c =3.
Fig. 3- (d) shows the normalized distribution of the amplitude of the fundamental electric field in the vicinity of the nanoparticle c in the absence of the nanoparticle a, and it is apparent from fig. 3- (d) that the field strength is greatly increased at the two poles (position 1 and position 4) of the nanoparticle c, and is greatly decreased at the equator (position 2 and position 3). At a position 5 further away, the influence of the nanoparticle c on the nanoparticle a gradually diminishes, where the electric field strength tends to the incident electric field strength. The creation of this strongly varying evanescent field results from the continuous boundary conditions of the dielectric nanoparticle-air interface. Since the nanoparticle c is in a deep sub-wavelength scale, the basic electric field inside it can be considered to be uniform and can be expressed approximately as(Obtained by the existing formula). Thus, the continuity of the electric displacement field at both poles of the dielectric nanoparticle is availableWherein the method comprises the steps ofIs the electric field of the two poles on the air side. In addition, the electric field on the equator of the air-side dielectric nanoparticle is: due to the continuity of the electric field, when epsilon c＞εh, then there is, When a tiny nonlinear nanoparticle is placed close to a dielectric nanoparticle, it will experience different fundamental electric fields at different locations, affecting the location-dependent THG transmission.
Fig. 3- (e) is the normalized THG transmittance calculated with the nonlinear effective medium theory correction scheme (solid line Realcomposite) and with the conventional Br effective nonlinear medium theory (dashed line Br) as the nanoparticle a moves along the trajectory in fig. 3- (b), while the actual THG transmittance of the original composite is: and (5) dotted line Correction. Wherein the relative dielectric constant epsilon a =1 of the nanoparticle a is made such that the normalized THG transmittance is normalized to 1.
Fig. 3- (f) is the normalized THG transmission as a function of the relative dielectric constants epsilon a(εc =3 and epsilon h =1 of nanoparticle a when nanoparticle a is at position 5.
In fig. 3- (e) and 3- (f), we show the difference between normalized THG transmittance calculated using the nonlinear effective medium theory correction scheme (solid line Real composition) and using the conventional Br effective nonlinear medium theory (dashed line Br). We show in fig. 3- (e) the THG transmittance of nanoparticle a as it moves from position 1 to position 5 in sequence along the trajectory 1→2→3→4→5. When the nanoparticle a moves along the trajectory 1→4, the gap between the two nanoparticles remains unchanged. Here we assume that the nonlinear susceptibility of nanoparticle a is several orders of magnitude greater than that of nanoparticle c, so that the nonlinear response of nanoparticle c is negligible. The solid line Realcomposite and the dashed line Br in the figure represent the normalized THG transmission calculated using the nonlinear effective medium theory correction scheme, and using the conventional Br effective nonlinear medium theory, respectively. Clearly, there is a significant difference between the two results. However, the proposed Correction result (solid line Realcomposite) using the nonlinear effective medium theory Correction scheme is better matched with the actual THG transmittance (dashed line Correction) of the original composite material. Thereby proving the accuracy of the correction scheme by using the nonlinear effective medium theory.
In fig. 3- (f), we demonstrate that the THG transport of the composite structure varies with the relative permittivity epsilon a of nanoparticle a when nanoparticle a is in position 5, where we let epsilon c =3 and epsilon h =1. In the figure, it can be clearly observed that as epsilon a→εh, the correction result of the Br effect nonlinear medium theory is closer to the conventional Br effect nonlinear medium theory. This is because the larger the difference between ε a and ε h, the more the local field inside nanoparticle aThe greater the difference from the uniform field of the composite structure. Thus, the correction scheme of the nonlinear effective medium theory is reliable and accurate for the weak nonlinear composite materials with short distance and large particle size difference.
It is noted that relational terms such as first and second, and the like are used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. Moreover, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus.
Finally, it should be noted that: the foregoing description is only a preferred embodiment of the present invention, and the present invention is not limited thereto, but it is to be understood that modifications and equivalents of some of the technical features described in the foregoing embodiments may be made by those skilled in the art, although the present invention has been described in detail with reference to the foregoing embodiments. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (7)
1. A method for achieving nonlinear equivalence, comprising the steps of:
step S100: a plurality of nano particles exist in a background medium to form a composite material, and the relative dielectric constants of the nano particles 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 nano particles;
Step S400: obtaining an effective nonlinear susceptibility second formula of the composite material by analyzing the relation between the relative dielectric constants of the nano particles and the background medium and the nonlinear effective dielectric function;
step S500: and obtaining an effective nonlinear magnetic susceptibility correction formula of the composite material according to the effective nonlinear magnetic susceptibility first formula and the effective nonlinear magnetic susceptibility second formula.
2. A method of achieving nonlinear equivalence according to claim 1, wherein step S100 comprises: the nano particles c and the nano particles a exist in the background medium h to form a composite material, and the relative dielectric constants of the nano particles c, the nano particles a and the background medium h are epsilon c、εa and epsilon h respectively.
3. A method of achieving nonlinear equivalence according to claim 2, wherein step S200 includes: according to the volume occupied by the nano particles c and the nano particles a in the composite material, the volume fractions f c and f a of the nano particles c and the nano particles a are obtained, and further according to the relative dielectric constants epsilon c、εa and epsilon h, the nonlinear effective dielectric function of the composite material is obtained: epsilon eff＝F(εh,εc,εa,fc,fa), wherein F is a nonlinear effective dielectric function, and epsilon eff is the effective dielectric constant of the composite material.
4. A method of achieving nonlinear equivalence according to claim 3, wherein step S300 comprises:
step S310: let nanoparticle a be a nonlinear dielectric nanoparticle and nanoparticle c be a linear dielectric nanoparticle, using The linear part representing the relative permittivity epsilon a has a nonlinear dielectric formula for the nanoparticle a: Where, |E a | is the fundamental frequency field inside the nanoparticle a, which is nonlinear in the linear case, E a|2 > represents the average of the squares of the spatial electric field modes, Is the third order nonlinear susceptibility of nanoparticle a;
Step S320: substituting the nonlinear dielectric formula of the nano particle a into the nonlinear effective dielectric function of the composite material to obtain an expansion formula of dielectric constant epsilon eff: wherein F' is the partial derivative of the relative permittivity epsilon a, Wherein, Representing a biasing operation on the nonlinear effective dielectric function F,Represents the partial derivative of the relative permittivity epsilon a of the independent variable, and < |E a | > represents the average value of the spatial electric field modes;
Step S330: the partial derivative F' is expressed as the average electric field of the nano particle a under the linear limit, and the electric field formula of the nano particle a is obtained: wherein E 0 is the intensity of the incident electric field, Representing a partial derivative operation of the effective dielectric constant epsilon eff;
step S340: substituting the electric field formula of the nano particle a into the expansion formula of dielectric constant epsilon eff to obtain A linear portion representing the effective dielectric constant ε eff;
Step S350: obtaining a first formula of effective nonlinear susceptibility of the composite material:
5. the method of claim 4, wherein the step of analyzing the relative dielectric constants of the nanoparticle and the background medium and the relationship between the nonlinear effective dielectric functions in step S400 comprises:
Step S410: according to the effective dielectric constant epsilon eff and the relative dielectric constant of the nano particles and the background medium, the effective medium theoretical formula of the composite material is obtained: Wherein d represents the composite dimension, ε i and f i represent the relative permittivity and volume fraction, respectively, of the ith nanoparticle;
Step S420: taking the assumption in step S310 as an example, if the nanoparticle a has a third-order nonlinear effect and the size of the nanoparticle a is much smaller than that of the nanoparticle c, the nonlinear dielectric equation The first-order polarizability of the nano particles a is obtained according to an effective medium theoretical formula, and the effective dielectric constant of the composite material is obtained:
Where N represents the number of nanoparticles c in the composite material, epsilon i represents the relative permittivity of the ith nanoparticle c, M represents the number of nanoparticles a in the composite material, beta j represents the correction factor of the jth nanoparticle a, and epsilon j represents the relative permittivity of the jth nanoparticle a.
6. The method of claim 5, wherein the step of obtaining the second formula for the effective nonlinear susceptibility of the composite material in step S400 comprises:
Step S430: if the nanoparticle is in a low density state, there is epsilon eff＝εh, and if the composite dimension d=3, the effective dielectric constant of the composite can be reduced to:
step S440: obtaining a second formula of effective nonlinear magnetic susceptibility of the composite material according to the partial derivative of the effective dielectric constant of the composite material simplified in the step S430:
7. The method for achieving nonlinear equivalence according to claim 6, wherein step S500 includes: according to the first formula of the effective nonlinear magnetic susceptibility of the composite material and the second formula of the effective nonlinear magnetic susceptibility, an effective nonlinear magnetic susceptibility correction formula of the composite material is obtained:
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| Publication number | Priority date | Publication date | Assignee | Title |
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| US5023139A (en) * | 1989-04-04 | 1991-06-11 | Research Corporation Technologies, Inc. | Nonlinear optical materials |
| US20110267617A1 (en) * | 2008-03-18 | 2011-11-03 | The Board Of Trustees Of The University Of Illinois | Second-Order Nonlinear Susceptibility of a Nanoparticle Using Coherent Confocal Microscopy |
| CN106972278A (en) * | 2017-03-16 | 2017-07-21 | 浙江大学 | A kind of electromagnetism stealth material of the scattering particles of all dielectric zero and its composition |
| CN112016225A (en) * | 2020-08-12 | 2020-12-01 | 网络通信与安全紫金山实验室 | Area point self-consistent coupling analysis method, device, equipment and storage medium |
| CN116660108A (en) * | 2023-05-31 | 2023-08-29 | 郑州轻工业大学 | A Method for Estimating the Size Distribution Information of Magnetic Nanoparticles in a Weak Diamagnetic Environment |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| US5023139A (en) * | 1989-04-04 | 1991-06-11 | Research Corporation Technologies, Inc. | Nonlinear optical materials |
| US20110267617A1 (en) * | 2008-03-18 | 2011-11-03 | The Board Of Trustees Of The University Of Illinois | Second-Order Nonlinear Susceptibility of a Nanoparticle Using Coherent Confocal Microscopy |
| CN106972278A (en) * | 2017-03-16 | 2017-07-21 | 浙江大学 | A kind of electromagnetism stealth material of the scattering particles of all dielectric zero and its composition |
| CN112016225A (en) * | 2020-08-12 | 2020-12-01 | 网络通信与安全紫金山实验室 | Area point self-consistent coupling analysis method, device, equipment and storage medium |
| CN116660108A (en) * | 2023-05-31 | 2023-08-29 | 郑州轻工业大学 | A Method for Estimating the Size Distribution Information of Magnetic Nanoparticles in a Weak Diamagnetic Environment |
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