CN111125900B - Method and system for calculating Casimir acting force of three-layer structure of anisotropic ferrite - Google Patents
Method and system for calculating Casimir acting force of three-layer structure of anisotropic ferrite Download PDFInfo
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Abstract
The invention discloses a method and a system for calculating Casimir action of an anisotropic ferrite three-layer structure, and relates to a system for calculating Casimir action of the anisotropic ferrite three-layer structure, which comprises the following steps: the building module is used for building a model of the anisotropic ferrite three-layer structure; a determination module for determining electromagnetic properties of the anisotropic ferrite; the first calculation module is used for calculating a transmission matrix of the uniaxial anisotropic material layer; the second calculation module is used for calculating the reflection coefficient matrix of the three-layer structure of the anisotropic material; and the third calculation module is used for calculating Casimir acting force of the anisotropic ferrite three-layer structure model. According to the invention, the Casimir acting force of the three-layer structure of the anisotropic ferrite is calculated according to the scattering theory, and the Casimir acting force effect of the three-layer structure of the anisotropic ferrite can be accurately analyzed and calculated according to the scattering theory.
Description
Technical Field
The invention relates to the technical field of quantum optics, in particular to a method and a system for calculating an anisotropic ferrite three-layer structure Casimir acting force.
Background
From the viewpoint of quantum mechanics, not all vacuum is available, and the change of vacuum fluctuation zero energy generates macroscopic interaction force, namely Casimir effect. The Casimir acting force is the effect which is most easily measured on quantum vacuum fluctuation in the macroscopic world, and the accurate understanding of Casimir force is also the key point of accurate force measurement in a range of distances from nanometer to millimeter. Casimir calculates the force between a pair of plates parallel to each other, and he found the force F at zero temperature and the limit of perfect reflection Cas The force depends only on two basic constants of plate spacing, namely the speed of light and the planck constant. In particular, it is independent of the fine structure constants that appear in the expression of atomic van der waals forces. This property of popularity is related to the perfect reflection assumption used by Casimir in its derivation. More accurate experiments were performed with metal mirrors that only show perfect reflection at frequencies less than the characteristic plasma frequency, which depends on the nature of the conduction electrons in the metal.
Due to the non-trivial permeability of ferrite, there are conditions under which repulsive Casimir forces can be obtained, and the polarity of the external magnetic field control forces can be adjusted, the anisotropic ferrite trilayer structure must be significantly different from other material structures. Through searching, the research on the acting force of the three-layer structure Casimir of the anisotropic ferrite is not related in the prior art. The invention will therefore propose how to calculate the Casimir forces of the three-layer structure of the anisotropic ferrite.
Disclosure of Invention
The invention aims at overcoming the defects of the prior art, and provides a method and a system for calculating the acting force of an anisotropic ferrite three-layer structure Casimir, which can accurately analyze the scattering theory and calculate the acting force effect of the anisotropic ferrite three-layer structure Casimir.
In order to achieve the above purpose, the present invention adopts the following technical scheme:
a system for calculating a Casimir force for an anisotropic ferrite trilayer structure, comprising:
the building module is used for building a model of the anisotropic ferrite three-layer structure;
a determination module for determining electromagnetic properties of the anisotropic ferrite;
the first calculation module is used for calculating a transmission matrix of the uniaxial anisotropic material layer;
the second calculation module is used for calculating the reflection coefficient matrix of the three-layer structure of the anisotropic material;
and the third calculation module is used for calculating Casimir acting force of the anisotropic ferrite three-layer structure model.
Further, the model for establishing the anisotropic ferrite three-layer structure in the establishment module comprises a first ferrite material plate, a second ferrite material plate and other material plates except for ferrite materials.
Further, the determining module specifically includes determining a permeability of the anisotropic ferrite material; the magnetic permeability is mu when perpendicular to the external magnetic field ⊥ The magnetic permeability is mu when parallel to the external magnetic field z The expressions are respectively:
μ z ＝1
wherein omega ex ＝γH ex ；ω m ＝4πM s The method comprises the steps of carrying out a first treatment on the surface of the Gamma represents gyromagnetic ratio; m is M s Representing the saturation magnetization of ferrite; h ex Represents the external magnetic field strength; omega ex Representing the frequency of the external magnetic field; omega m Representing the ion volume frequency of the material plate.
Further, the first computing module specifically includes:
the optical axis is perpendicular to the interface, the interface is used as an x-y plane, the anisotropic medium is a half space with z >0, and the magnetic permeability tensor is:
wherein mu ⊥ Represents permeability perpendicular to the external magnetic field, mu z Represents the permeability parallel to the external magnetic field;
the influence of a plurality of interfaces is calculated by adopting a transmission matrix method, and for a jth layer of uniaxial anisotropic medium in a multilayer structure, a quaternary vector formed by x and y components of electromagnetic field quantity in the medium is defined, wherein the structural relation of the uniaxial anisotropic material is as follows:
solution of electromagnetic field wave equation
Wherein k is y A wave vector representing the y-direction; q represents a wave vector in the z direction;representing the electric field strength; />Indicating the magnetic field strength.
Further, the first computing module further includes:
equations for TE and TM waves are obtained using maxwell's equations:
wherein c represents the speed of light; ω represents the fluctuation frequency; q TM A z-direction wave vector representing the TE wave; q TM The z-direction wave vector representing the TM wave.
Further, the transmission matrix of the uniaxial anisotropic material layer in the first calculation module is:
wherein M represents a transmission matrix of the uniaxial anisotropic material layer.
Further, the second calculation module calculates the reflection coefficient matrix of the three-layer structure of the anisotropic material, specifically, obtains the total reflection coefficient matrix of the multi-layer structure by using a transmission matrix of each layer of the multi-layer material structure corresponding to the electromagnetic field quaternary vector;
wherein, the electromagnetic field boundary conditions at each interface are adopted, and the obtained reflection coefficient matrix R is:
wherein,
wherein,
wherein θ 0 Represents the angle of incidence, k ∥ Representing wave vectors parallel to the interface; x is x mn Represents a total transmission matrix, m, n=1, 2, 3, 4;
let ω=iζ, virtual frequency ζ and parallel wave vector k ∥ The constituted coordinate system takes the form of polar coordinates:
the three-layer structure reflection coefficient matrix of the anisotropic material is as follows:
wherein X is N Representing a three-layer structure reflection coefficient matrix of the anisotropic material; m represents the transmission matrix of the uniaxial anisotropic material layer.
Further, the third computing module includes:
under the conditions of plane and parallel mirror surfaces, the first ferrite material plate and the second ferrite material plate form a Fabry-Perot cavity with the length of a, the frequency of a field mode of omega and the transverse wave vector of k; the loop function of the cavity optical response to the input field is defined as:
wherein,and->Representing open-loop and closed-loop functions corresponding to one round trip in the cavity, respectively;is an analytical function of frequency ω; the sum of transverse wave vectors is expressed as quantized eigenvector k in the x, y-direction limited interval x ＝2πq x /a x And k y ＝2πq y /a y Is the sum of a when x ,a y At →infinity, the summation transition is integral:
the Airy function defined in classical optics is taken as the ratio of intra-cavity energy to extra-cavity energy in a given mode:
and->Depending on the reflection amplitude of the slab seen from the inside, the Casimir effort is obtained as:
in which the z-direction wave vector k is defined z ＝ik[ω]Factor (F)Representing the difference between medial and lateral;
by utilizing the analytic property of the function, the Casimir acting force is expressed as follows when the Casimir acting force is applied to the frequency complex plane area surrounding the Re omega 0 and the Imomega 0 quadrants:
wherein F is Cas Representing the Casimir force.
Further, the third computing module further includes:
in the case of flat-plate non-specular reflection, the expression between the first ferrite material plate, the second ferrite material plate as the mode sum Casimir energy represented by the marks ζ and m≡k, p is:
wherein,
wherein,representing the phase difference obtained by the field mode when scattering on the cavity wall; the phase difference is rewritten by quantum field theory as a trace of a matrix defined over each mode of the mode sum m≡k, p:
casimir force for anisotropic ferrite three-layer structural model:
correspondingly, the method for calculating the Casimir acting force of the three-layer structure of the anisotropic ferrite comprises the following steps:
s1, establishing a model of an anisotropic ferrite three-layer structure;
s2, determining electromagnetic characteristics of the anisotropic ferrite;
s3, calculating a transmission matrix of the uniaxial anisotropic material layer;
s4, calculating a reflection coefficient matrix of the three-layer structure of the anisotropic material;
s5, calculating Casimir acting force of the anisotropic ferrite three-layer structure model.
Compared with the prior art, the invention has the following advantages:
1. according to the invention, the Casimir acting force of the three-layer structure of the anisotropic ferrite is calculated according to the scattering theory, and the Casimir acting force effect of the three-layer structure of the anisotropic ferrite can be accurately analyzed and calculated according to the scattering theory.
2. The invention can accurately reflect the influence of the nontrivial magnetic permeability of the externally applied static magnetic fields with different intensities and the anisotropic ferrite material on the Casimir acting force.
3. The invention can accurately obtain a stable sandwich structure and a stable balance point, and can adjust the balance stay position of the middle plate by regulating and controlling the intensity of the externally applied electrostatic magnetic field.
4. The invention can accurately reflect the influence of the static magnetic fields in different directions on Casimir repulsive force.
Drawings
FIG. 1 is a system configuration diagram for calculating a Casimir acting force of an anisotropic ferrite three-layer structure according to the first embodiment;
FIG. 2 is a schematic diagram of a three-layer structure model of an anisotropic ferrite provided in the first embodiment;
FIG. 3 is a schematic diagram of system I/O provided in the second embodiment;
FIG. 4 is a schematic view of the propagation direction of a field mode in each material layer according to the first embodiment;
FIG. 5 is a schematic diagram of a space flat panel system according to a first embodiment;
FIG. 6 is a diagram showing normalized Casimir force of the ferrite plate 1 (A plate) in the ferrite-silver plate-ferrite sandwich structure under the condition of applying static magnetic fields of different intensities generated by the simulation provided in the second embodiment;
FIG. 7 is a diagram showing normalized Casimir force of the ferrite plate 1 (A plate) in the ferrite-dielectric plate-ferrite sandwich structure under the application of static magnetic fields of different intensities generated by the simulation provided in the first embodiment;
fig. 8 is a diagram showing normalized Casimir net force of a silver plate in a ferrite-silver plate-ferrite sandwich structure under the condition of applying static magnetic fields in different directions, which is generated by simulation provided in the first embodiment.
Detailed Description
Other advantages and effects of the present invention will become apparent to those skilled in the art from the following disclosure, which describes the embodiments of the present invention with reference to specific examples. The invention may be practiced or carried out in other embodiments that depart from the specific details, and the details of the present description may be modified or varied from the spirit and scope of the present invention. It should be noted that the following embodiments and features in the embodiments may be combined with each other without conflict.
The invention aims at overcoming the defects of the prior art and provides a method and a system for calculating the acting force of a three-layer structure Casimir of an anisotropic ferrite.
Example 1
The present embodiment provides a system for calculating a Casimir acting force of an anisotropic ferrite three-layer structure, as shown in fig. 1, including:
a building module 11 for building a model of an anisotropic ferrite three-layer structure;
a determining module 12 for determining electromagnetic properties of the anisotropic ferrite;
a first calculation module 13 for calculating a transmission matrix of the uniaxial anisotropic material layer;
a second calculation module 14, configured to calculate a reflection coefficient matrix of the three-layer structure of the anisotropic material;
and a third calculation module 15, configured to calculate a Casimir acting force of the anisotropic ferrite three-layer structure model.
In the build-up module 11, a model of the anisotropic ferrite three-layer structure is built up.
As shown in fig. 2, the former for establishing the anisotropic ferrite three-layer structure includes a first ferrite material plate 1, a second ferrite material plate 3, and other material plates 2 except for ferrite materials; wherein a plate 2 of a material other than ferrite material, such as a metallic silver plate. The thicknesses of the three plates of the first ferrite material plate 1, the second ferrite material plate 3 and the other material plates 2 except for ferrite material are d respectively 1 、d 3 、d 2 . In the present embodiment, the first ferrite material plate 1 and the second ferrite material plate 3 are ferrite materials, namely epsilon 1 ＝ε 3 The other material plates 2 except ferrite material have magnetic permeability mu 2 Dielectric constant of epsilon 2 。
In the determination module 12, electromagnetic properties of the anisotropic ferrite are determined.
The model of the anisotropic ferrite three-layer structure used in this example is shown in fig. 2, wherein the plates 1 and 3 are ferrite material plates, and the plate 2 is a metallic silver plate.
Firstly, the magnetic permeability of ferrite material is defined, and when the magnetic permeability is perpendicular to external magnetic field, the magnetic permeability is mu ⊥ Permeability is mu when parallel to external magnetic field z The expressions are respectively:
μ z ＝1
wherein omega ex ＝γH ex ；ω m ＝4πM s The method comprises the steps of carrying out a first treatment on the surface of the Typical ferrite materials have dielectric constants in the range of 12-16; m is M s Representing the saturation magnetization of ferrite, and the value is 135G-239G; gamma represents gyromagnetic ratio and gamma is 1.8X10 7 S -1 G -1 Corresponding distance unitsIn the cm or mm range; h ex Represents the external magnetic field strength; omega ex Representing the frequency of the external magnetic field; omega m Representing the ion volume frequency of the material plate.
In the first calculation module 13, a transmission matrix of the uniaxial anisotropic material layer is calculated.
For the case where the optical axis is perpendicular to the interface, the present embodiment selects the interface as the x-y plane, and the anisotropic medium selects the half-space defined by z >0, as shown in fig. 4, the permeability tensor is given by:
μ ⊥ represents permeability perpendicular to the external magnetic field, mu z Represents the permeability parallel to the external magnetic field;
the influence of a plurality of interfaces is calculated by adopting a transmission matrix method, and for a jth layer of uniaxial anisotropic medium in a multilayer structure, a quaternary vector formed by x and y components of electromagnetic field quantity in the medium is defined, wherein the structural relation of the uniaxial anisotropic material is as follows:
solution of electromagnetic field wave equation
Wherein k is y A wave vector representing the y-direction; q represents a wave vector in the z direction;representing the electric field strength; />Indicating the magnetic field strength.
Combining maxwell's equations, we get equations for TE and TM waves:
wherein c represents the speed of light; ω represents the fluctuation frequency; q TE A z-direction wave vector representing the TE wave; q TM The z-direction wave vector representing the TM wave.
Wherein TE wave (i.e., s wave): there is a magnetic field component but no electric field component in the propagation direction, called a transverse wave. In planar optical waveguides (closed cavity structures), electromagnetic field components have Ey, hx, hz, and the propagation direction is the z direction.
TM wave (i.e. p wave): there is an electric field component in the propagation direction and no magnetic field component, called transverse magnetic wave. In the planar optical waveguide (closed cavity structure), electromagnetic field components include Hy, ex, and Ez, and the propagation direction is the z direction.
The quaternary vectors at (x, y, z) and (x, y, z+z) inside the material are then used to derive the specific form of the transmission matrix of the uniaxial anisotropic medium material layer in relation to the transmission matrix M of the material:
wherein M represents a transmission matrix of the uniaxial anisotropic material layer.
In the second calculation module 14, a reflection coefficient matrix of the three-layer structure of the anisotropic material is calculated.
In the embodiment, the transmission matrix of each layer of the multi-layer material structure corresponding to the electromagnetic field quaternary vector is utilized to obtain the total reflection coefficient matrix of the multi-layer structure. Substituting each layer of transmission matrix into a field quantity quaternary vector for calculating the incidence of electromagnetic waves to the multilayer structure, wherein the electromagnetic field boundary conditions at each interface are applied to obtain each matrix element of the reflection coefficient matrix:
wherein, each parameter is defined as:
wherein,
wherein θ 0 Represents the angle of incidence, k ∥ Representing wave vectors parallel to the interface; x is x mn Represents a total transmission matrix, m, n=1, 2, 3, 4;
let ω=iζ, virtual frequency ζ and parallel wave vector k ∥ The constituted coordinate system takes the form of polar coordinates:
total transmission matrix x mn Is obtained by the continuous multiplication of transmission matrixes of all layers, so that the reflection coefficient matrix of the three-layer structure of the anisotropic material is as follows:
wherein X is N Representing a three-layer structure reflection coefficient matrix of the anisotropic material; m represents the transmission matrix of the uniaxial anisotropic material layer.
In a third calculation module 15, the Casimir force of the anisotropic ferrite three-layer structure model is calculated.
This embodiment first extends in the x and y directions in the case of the original Casimir effect spatial structure with perfect planes and parallel mirrors, as shown in fig. 5. The first ferrite material plate 1 (A plate) and the second ferrite material plate 3 (B plate) are formed into a Fabry-Perot cavity with the length of a. The field mode is characterized by its frequency ω, transverse wave vector k (which has components kx, ky in the plane of the slab) and polarization state p. In the system of fig. 5 the frequency ω, transverse wave vector k≡ (kx, ky) and polarization p=te, TM remain unchanged throughout the scattering process on the mirror or cavity. The scattering process is only specific to kz symbols having opposite wave vector longitudinal componentsNumber free vacuum mode. By usingRepresenting the reflection amplitude of the wall j=a, B seen from the inside of the cavity.
The loop function characterizing the optical response of the cavity to the input field is defined as:
wherein,and->Representing open-loop and closed-loop functions corresponding to one round trip in the cavity, respectively; />Is an analytical function of frequency ω; the sum of transverse wave vectors is expressed as quantized eigenvector k in the x, y-direction limited interval x ＝2πq x /a x And k y ＝2πq y /a y Is the sum of a when x ,a y At →infinity, the summation transition is integral:
/>
the Airy function defined in classical optics is taken as the ratio of intra-cavity energy to extra-cavity energy in a given mode:
and->Depending only on the reflection amplitude of the slab seen from the inside, the Casimir force is obtained as:
in which the z-direction wave vector k is defined z ＝ik[ω]Factor (F)Representing the difference between medial and lateral;
using the analytic nature of the function, applying the cauchy-law on the complex frequency plane area surrounding the Re omega 0, im omega 0 quadrants, ignoring the contribution of higher frequencies considering the application of high frequency transparency characteristics, the expression of Casimir force is:
wherein F is Cas Representing the Casimir force. I.e. expressed as an integral over the complex frequency ω=iζ.
In the case of flat-plate non-specular reflection, the expression between the first ferrite material plate 1 (a plate), the second ferrite material plate 3 (B plate) as the mode sum Casimir energy denoted by the marks ζ and m≡k, p is:
wherein,
wherein,representing the phase difference obtained by the field mode when scattering on the cavity wall; the phase difference is rewritten by quantum field theory as a trace of a matrix defined over each mode of the mode sum m≡k, p:
where R is A And R is B Giving the reflective properties of the two cavity walls and thus a more general formula for the corresponding Casimir force. Thus, casimir force for the anisotropic ferrite three-layer structure model:
wherein the transverse wave vector component k of the parallel interface is explicitly written as k || . And finally substituting the related parameters of the anisotropic ferrite into a formula to obtain the acting force of the anisotropic ferrite Casimir.
Compared with the prior art, the embodiment has the following advantages:
1. according to the invention, the Casimir acting force of the three-layer structure of the anisotropic ferrite is calculated according to the scattering theory, and the Casimir acting force effect of the three-layer structure of the anisotropic ferrite can be accurately analyzed and calculated according to the scattering theory.
2. The invention can accurately reflect the influence of the nontrivial magnetic permeability of the externally applied static magnetic fields with different intensities and the anisotropic ferrite material on the Casimir acting force.
3. The invention can accurately obtain a stable sandwich structure and a stable balance point, and can adjust the balance stay position of the middle plate by regulating and controlling the intensity of the externally applied electrostatic magnetic field.
4. The invention can accurately reflect the influence of the static magnetic fields in different directions on Casimir repulsive force.
Correspondingly, the method for calculating the Casimir acting force of the three-layer structure of the anisotropic ferrite comprises the following steps:
s11, establishing a model of an anisotropic ferrite three-layer structure;
s12, determining electromagnetic characteristics of the anisotropic ferrite;
s13, calculating a transmission matrix of the uniaxial anisotropic material layer;
s14, calculating a reflection coefficient matrix of the three-layer structure of the anisotropic material;
s15, calculating Casimir acting force of the anisotropic ferrite three-layer structure model.
Example two
The system for calculating the acting force of the three-layer structure Casimir of the anisotropic ferrite is different from the first embodiment in that:
this example further illustrates a system for calculating the Casimir force of an anisotropic ferrite three-layer structure.
In this embodiment, as shown in fig. 3. The thicknesses of three plates of the first ferrite material plate 1 (a plate), the second ferrite material plate 3 (B plate), and the other material plate 2 (C plate) except for ferrite material are input to the a port. Inputting dielectric constant and omega of anisotropic ferrite material at B port ex The permittivity, permeability and plasma frequency of the other material plates are input at the C port. The Casimir force of the ferrite plate is output at the D port and the Casimir net force of the other material plates is output at the E port.
In this embodiment, fig. 6 shows normalized Casimir force diagram of the first ferrite plate 1 (a plate) in the ferrite-silver plate-ferrite sandwich structure when static magnetic fields of different intensities are applied. In the figure, a first ferrite material plate 1 (A plate) and a second ferrite material plate 3 (B plate) are ferrite material plates, the other material plates 2 (C plate) except ferrite material are silver plates, and the thickness of the three plates is d 1 ＝d 3 ＝d 2 ＝1.0λ m Dielectric constant epsilon of anisotropic ferrite material 1 ＝ε 3 ＝12，ω ex1 ＝ω ex2 Permeability μ of silver plate 2 (C plate) 2 Dielectric constant =1Wherein the plasma frequency omega m ＝1.37×10 6 rad/s. As can be seen from fig. 6, this example can analyze the characteristic that the electromagnetic response characteristic of the ferrite material plate gradually increases under the condition that the applied magnetic field gradually decreases, in the adjustable range.
In this example, fig. 7 shows normalized Casimir force diagram of the ferrite plate 1 (a plate) in the ferrite-dielectric-ferrite sandwich structure when static magnetic fields of different intensities are applied. The thickness of the three plates in this figure is d 1 ＝d 3 ＝d 2 ＝1.0λ m Dielectric constant epsilon of anisotropic ferrite material 1 ＝ε 3 ＝10，ω ex1 ＝ω ex2 Permeability μ of silver plate 2 (C plate) 2 =1, dielectric constant ε 2 =10. As can be seen from fig. 7, this example can analyze the characteristic that the electromagnetic response characteristic of the ferrite material plate gradually increases under the condition that the applied magnetic field gradually decreases, in the adjustable range.
In this embodiment, fig. 8 shows normalized Casimir net force diagram of silver plate in ferrite-silver plate-ferrite sandwich structure for static magnetic fields with different applied intensities when the applied magnetic field direction is x-axis direction. In the figure, the plates 1 (A plate) and 3 (B plate) are ferrite plates, the other plates 2 (C plate) are silver plates, and the thickness of the three plates is d 1 ＝d 3 ＝d 2 ＝1.0λ m Dielectric constant epsilon of anisotropic ferrite material 1 ＝ε 3 ＝12，ω ex1 ＝ω ex2 Magnetic permeability mu of silver plate 2 Dielectric constant =1Wherein the plasma frequency omega m ＝1.37×10 6 rad/s. The embodiment can analyze that under the condition that the direction of the externally applied magnetic field is the x-axis direction, the magnetic response intensity of the ferrite material is weakened, the Casimir repulsive force can not be obtained, and further the characteristic of a stable sandwich structure can not be obtained. />
According to the invention, the Casimir acting force of the three-layer structure of the anisotropic ferrite is calculated based on the scattering theory, the influence of the applied static magnetic fields with different intensities and the nontrivial magnetic permeability of the anisotropic ferrite material on the Casimir acting force can be analyzed, a stable sandwich structure and a stable balance point can be obtained, and meanwhile, the stable balance position of the middle plate can be regulated and controlled by analyzing and adjusting the intensity parameters of the applied magnetic field. Because the material cost is higher in practice, the theoretical model used in the invention is relatively close to the actual anisotropic ferrite material, and has relatively high application value as a test model. The system can calculate the needed Casimir acting force, can provide a new application for the anisotropic ferrite material, and also provides a new way for researching the Casimir acting force.
Note that the above is only a preferred embodiment of the present invention and the technical principle applied. It will be understood by those skilled in the art that the present invention is not limited to the particular embodiments described herein, but is capable of various obvious changes, rearrangements and substitutions as will now become apparent to those skilled in the art without departing from the scope of the invention. Therefore, while the invention has been described in connection with the above embodiments, the invention is not limited to the embodiments, but may be embodied in many other equivalent forms without departing from the spirit or scope of the invention, which is set forth in the following claims.
Claims (7)
1. A system for calculating a Casimir force for an anisotropic ferrite trilayer structure, comprising:
the building module is used for building a model of the anisotropic ferrite three-layer structure;
a determination module for determining electromagnetic properties of the anisotropic ferrite;
the first calculation module is used for calculating a transmission matrix of the uniaxial anisotropic material layer;
the second calculation module is used for calculating the reflection coefficient matrix of the three-layer structure of the anisotropic material;
the third calculation module is used for calculating Casimir acting force of the anisotropic ferrite three-layer structure model;
the model for establishing the anisotropic ferrite three-layer structure in the establishment module comprises a first ferrite material plate, a second ferrite material plate and other material plates except ferrite materials;
the determination module specifically comprises determining the magnetic permeability of the anisotropic ferrite material; the magnetic permeability is mu when perpendicular to the external magnetic field ⊥ The magnetic permeability is mu when parallel to the external magnetic field z The expressions are respectively:
μ z ＝1
wherein omega ex ＝γH ex ；ω m ＝4πM s The method comprises the steps of carrying out a first treatment on the surface of the Gamma represents gyromagnetic ratio; m is M s Representing the saturation magnetization of ferrite; h ex Represents the external magnetic field strength; omega ex Representing the frequency of the external magnetic field; omega m Representing the ion body frequency of the material plate;
the first computing module specifically comprises:
the optical axis is perpendicular to the interface, the interface is used as an x-y plane, the anisotropic medium is a half space with z >0, and the magnetic permeability tensor is:
wherein mu ⊥ Represents permeability perpendicular to the external magnetic field, mu z Represents the permeability parallel to the external magnetic field;
the influence of a plurality of interfaces is calculated by adopting a transmission matrix method, and for a jth layer of uniaxial anisotropic medium in a multilayer structure, a quaternary vector formed by x and y components of electromagnetic field quantity in the medium is defined, wherein the structural relation of the uniaxial anisotropic material is as follows:
the solution of the electromagnetic field wave equation is:
wherein k is y A wave vector representing the y-direction; q represents a wave vector in the z direction;representing the electric field strength; />Indicating the magnetic field strength.
2. The system for calculating a Casimir force of an anisotropic ferrite three-layer structure according to claim 1, wherein the first calculation module further comprises:
equations for TE and TM waves are obtained using maxwell's equations:
wherein c represents the speed of light; ω represents the fluctuation frequency; q TE A z-direction wave vector representing the TE wave; q TM Representing TM wavesIs a z-direction wave vector of (c).
3. The system for calculating the Casimir acting force of the three-layer structure of the anisotropic ferrite according to claim 2, wherein the transmission matrix of the uniaxial anisotropic material layer in the first calculation module is:
wherein M represents a transmission matrix of the uniaxial anisotropic material layer.
4. The system for calculating the three-layer structure Casimir acting force of the anisotropic ferrite according to claim 3, wherein the second calculation module calculates the three-layer structure reflection coefficient matrix of the anisotropic material, specifically, obtains the total reflection coefficient matrix of the multilayer structure by using a transmission matrix of each layer of corresponding electromagnetic field quaternary vector of the multilayer material structure;
wherein, the electromagnetic field boundary conditions at each interface are adopted, and the obtained reflection coefficient matrix R is:
wherein,
wherein,
wherein θ 0 Represents the angle of incidence, k ∥ Representing wave vectors parallel to the interface; x is x mn Represents a total transmission matrix, m, n=1, 2, 3, 4;
let ω=iζ, virtual frequency ζ and parallel wave vector k ∥ The constituted coordinate system takes the form of polar coordinates:
the three-layer structure reflection coefficient matrix of the anisotropic material is as follows:
wherein X is N Representing a three-layer structure reflection coefficient matrix of the anisotropic material; m represents the transmission matrix of the uniaxial anisotropic material layer.
5. The system for calculating the Casimir force of the three-layer structure of the anisotropic ferrite of claim 4, wherein the third calculation module comprises:
under the conditions of plane and parallel mirror surfaces, the first ferrite material plate and the second ferrite material plate form a Fabry-Perot cavity with the length of a, the frequency of a field mode of omega and the transverse wave vector of k; the loop function of the cavity optical response to the input field is defined as:
wherein,and->Representing open-loop and closed-loop functions corresponding to one round trip in the cavity, respectively; />Is an analytical function of frequency ω; the sum of transverse wave vectors k is expressed as quantized eigenvector k of the x, y-direction limited interval x ＝2πq x /a x And k y ＝2πq y /a y Is the sum of a when x ,a y At →infinity, the summation transition is integral:
the Airy function defined in classical optics is taken as the ratio of intra-cavity energy to extra-cavity energy in a given mode:
and->Depending on the reflection amplitude of the slab seen from the inside, the Casimir effort is obtained as:
in which the z-direction wave vector k is defined z ＝ik[ω]Factor (F)Representing the difference between medial and lateral;
by utilizing the analytic property of the function, the Casimir acting force is expressed as follows when the Casimir acting force is applied to the frequency complex plane area surrounding the Re omega 0 and the Imomega 0 quadrants:
wherein F is Cas Representing the Casimir force.
6. The system for calculating the Casimir force of the three-layer structure of the anisotropic ferrite of claim 5, wherein said third calculation module further comprises:
in the case of flat-plate non-specular reflection, the expression between the first ferrite material plate, the second ferrite material plate as the mode sum Casimir energy represented by the marks ζ and m≡k, p is:
wherein,
wherein,representing the phase difference obtained by the field mode when scattering on the cavity wall; the phase difference is rewritten by quantum field theory as a trace of a matrix defined over each mode of the mode sum m≡k, p:
casimir force for anisotropic ferrite three-layer structural model:
7. a method for calculating the Casimir acting force of an anisotropic ferrite three-layer structure, comprising the steps of:
s1, establishing a model of an anisotropic ferrite three-layer structure;
s2, determining electromagnetic characteristics of the anisotropic ferrite;
s3, calculating a transmission matrix of the uniaxial anisotropic material layer;
s4, calculating a reflection coefficient matrix of the three-layer structure of the anisotropic material;
s5, calculating Casimir acting force of the anisotropic ferrite three-layer structure model;
in the step S1, a model for establishing an anisotropic ferrite three-layer structure comprises a first ferrite material plate, a second ferrite material plate and other material plates except ferrite materials;
in step S2, determining the magnetic permeability of the anisotropic ferrite material; the magnetic permeability is mu when perpendicular to the external magnetic field ⊥ The magnetic flux guideRate is mu when parallel to external magnetic field z The expressions are respectively:
μ z ＝1
wherein omega ex ＝γH ex ；ω m ＝4πM s The method comprises the steps of carrying out a first treatment on the surface of the Gamma represents gyromagnetic ratio; m is M s Representing the saturation magnetization of ferrite; h ex Represents the external magnetic field strength; omega ex Representing the frequency of the external magnetic field; omega m Representing the ion body frequency of the material plate;
the step S3 is specifically as follows:
the optical axis is perpendicular to the interface, the interface is used as an x-y plane, the anisotropic medium is a half space with z >0, and the magnetic permeability tensor is:
wherein mu ⊥ Represents permeability perpendicular to the external magnetic field, mu z Represents the permeability parallel to the external magnetic field;
the influence of a plurality of interfaces is calculated by adopting a transmission matrix method, and for a jth layer of uniaxial anisotropic medium in a multilayer structure, a quaternary vector formed by x and y components of electromagnetic field quantity in the medium is defined, wherein the structural relation of the uniaxial anisotropic material is as follows:
the solution of the electromagnetic field wave equation is:
wherein k is y A wave vector representing the y-direction; q represents a wave vector in the z direction;representing the electric field strength; />Indicating the magnetic field strength.
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