A method for calculating the transmission spectrum of multilayer anisotropic chiral media

Abstract

The present invention relates to a method for calculating the transmission spectrum of a multilayer anisotropic chiral medium, comprising the following steps: S1, establishing a multilayer periodic structure model of a multilayer anisotropic chiral medium and a common medium; S2, calculating the electric field and magnetic field expressions in the anisotropic chiral medium layer and the common medium layer; S3, writing the boundary conditions at the interface of the two material layers; S4, calculating the transmission matrix of the multilayer periodic structure; S5, obtaining the transmission spectrum of the multilayer periodic structure of the anisotropic chiral medium and the common medium; the multilayer structure of the multilayer anisotropic chiral metamaterial and the common medium of the present invention is more valuable as a test model; at the same time, the calculation process involves a calculation method based on the transmission matrix, which provides an optical method for analyzing multilayer metamaterial structures with anisotropy; and produces beneficial technical effects.

Description

Technical Field

[0001] The invention belongs to the field of optical information technology, and in particular relates to a method for calculating the transmission spectrum of a multi-layer anisotropic chiral medium. Background Art

[0002] In recent years, chiral media have attracted the attention of many researchers due to their unique electromagnetic properties. Since negative refraction was first realized in 2004 using a chiral metamaterial structure, the study of chiral metamaterials has become a new hotspot in the field of electromagnetics. After that, many chiral media with complex structures were proposed in the microwave band, THz band, and even in the optical band. In 2006, asymmetric transmission characteristics were first discovered in planar chiral media. This new characteristic once again aroused people's enthusiasm for the study of chiral metamaterials, and many meaningful results were quickly obtained. Not only did it realize the asymmetric transmission of linearly polarized waves and circularly polarized waves, but it could also realize the asymmetric transmission of linearly polarized waves and circularly polarized waves at the same time by one structure. In addition to negative refraction and asymmetric transmission characteristics, chiral metamaterials also have other exotic properties, such as huge optical activity, circular dichroism, polarization deflection characteristics, etc. Chiral metamaterials have the important properties of causing cross-coupling of electric and magnetic fields and realizing negative refraction. The propagation characteristics of electromagnetic waves in chiral metamaterials can be changed by adjusting the chiral parameters. In addition, chiral exotic media have different responses to electromagnetic waves of different frequencies, resulting in special transmission spectra.

[0003] In view of the above technical problems, it is necessary to improve them. Summary of the invention

[0004] Based on the above-mentioned deficiencies in the prior art, the present invention provides a method for calculating the transmission spectrum of a multilayer anisotropic chiral medium.

[0005] The multilayer structure of the multilayer anisotropic chiral metamaterial of the present invention and the common medium is more valuable as a test model; at the same time, the calculation process involves a calculation method based on the transmission matrix, which provides an optical method for analyzing multilayer anisotropic metamaterial structures.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is: a method for calculating the transmission spectrum of a multilayer anisotropic chiral medium, comprising the following steps:

[0007] S1. Establish a multi-layer periodic structure model composed of multi-layer anisotropic chiral media and ordinary media;

[0008] The structure of the model is a multi-layer periodic structure constructed with anisotropic chiral media and ordinary media. In order to facilitate the explanation of this method, the ordinary medium layer is arranged periodically from left to right, taking the vacuum layer as an example, but the method is still general.

[0009] The constitutive equation of anisotropic chiral media is:

[0010] The constitutive equation of anisotropic chiral media is:

[0011]

[0012]

[0013]

[0014] when When the longitudinal component is included, when When the lateral component is included, For the sake of convenience, the present invention takes the case of including the longitudinal component as an example for detailed description, and directly gives the results for other cases;

[0015] Constitutive equation in vacuum:

[0016] D=ε·E

[0017] B=μ·H

[0018] ε t (μ t )ε z (μ z ) is the relative dielectric constant (permeability) of the uniaxial chiral medium perpendicular and parallel to the optical axis; κ is the chiral parameter, which describes the electromagnetic coupling and has Gaussian units. The dielectric constant and permeability in vacuum are ε=1 and μ=1.

[0019] S2. Calculate the electric and magnetic field expressions in the anisotropic chiral medium layer and the ordinary medium layer; write the expressions of the incident electric field, reflected electric field, and transmitted electric field of the electromagnetic wave on the interface of the anisotropic chiral medium layer;

[0020] Electric and magnetic fields of incident electromagnetic waves:

[0021]

[0022]

[0023] The electric and magnetic fields of reflected electromagnetic waves:

[0024]

[0025]

[0026] The electric and magnetic fields of right-handed transmitted electromagnetic waves:

[0027]

[0028]

[0029] The electric and magnetic fields of left-handed transmitted electromagnetic waves:

[0030]

[0031]

[0032] The electric and magnetic fields reflected back from another interface after right-hand transmission:

[0033]

[0034]

[0035] The electric and magnetic fields reflected from another interface after left-handed transmission:

[0036]

[0037]

[0038] The electric and magnetic fields of the electromagnetic waves transmitted from the second interface:

[0039]

[0040]

[0041] The electric and magnetic fields of the electromagnetic wave reflected from the third interface to the second interface:

[0042]

[0043]

[0044] Where "+" and "-" represent the direction of the component of the electromagnetic wave propagation direction on the z-axis. When the component direction is the same as the z-direction, it is "+", and the opposite is "-". θ is the angle between the incident electromagnetic wave and the normal. R is the angle between the right-handed transmitted wave and the normal, θ L is the angle between the left-handed transmitted wave and the normal. In vacuum, the wave number k0 = ω / c, ω = 2πf, and f is the frequency of the electromagnetic wave. In a uniaxial chiral medium, there are two propagation modes: R The right circularly polarized (RCP) wave with wave number k L Left circularly polarized (LCP) wave, k R and k L The expression is as follows:

[0045]

[0046] in The "+" and "-" in the "±" symbol correspond to the RCP wave and LCP wave respectively. On the other hand, the continuity of electric and magnetic fields at the dielectric interface requires k0sinθ i =k R sinθ R =k L sinθ L .

[0047] S3. Write down the boundary conditions at the interface between the two material layers. From the electromagnetic field boundary conditions, we know that at the interface, the tangential components of the electric and magnetic fields are conserved, so at the first interface:

[0048]

[0049]

[0050]

[0051]

[0052] The second interface has:

[0053]

[0054]

[0055]

[0056]

[0057] E x represents the electric field component in the x-axis direction, H x represents the magnetic field component in the x-axis direction, E y represents the electric field component in the y-axis direction, H y Represents the magnetic field component in the y-axis direction.

[0058] S4. Calculate the transmission matrix of the multilayer periodic structure; the transmission matrix for linear polarization incidence is written according to the boundary conditions. The boundary conditions at the first interface are written in matrix form:

[0059]

[0060]

[0061]

[0062] Write the boundary conditions at the second interface in matrix form:

[0063]

[0064]

[0065]

[0066] The intermediate variable Eliminate and get E is 、E ip 、E rs 、E rp and relation:

[0067]

[0068] The transfer matrix expression is obtained:

[0069]

[0070] Similarly, when circular polarization is incident, the transmission matrix is ​​obtained:

[0071]

[0072]

[0073]

[0074] E iL refers to the incident left-handed electric field, E iR refers to the incident right-handed electric field, E rL refers to the reflected left-handed electric field, E rR It refers to the reflected right-handed electric field;

[0075]

[0076] Similarly, when the chiral parameter has only the transverse component, the following changes occur:

[0077]

[0078]

[0079] k0 sinθ=k' R sinθ' R =k' L sinθ' L

[0080]

[0081] The above matrix θ R(L) →θ'R(L) , other matrix forms are the same as above, and the transmission matrix is ​​obtained:

[0082]

[0083] S5. Calculate the transmission spectrum of the multilayer periodic structure constructed by anisotropic chiral medium and ordinary medium; take the transmission matrix at linear polarization incidence as an example, the transmission matrix is ​​known:

[0084]

[0085] Get the transmission coefficient matrix:

[0086]

[0087] Where Δ = t 11 t 22 -t 12 t 21 After the transmission coefficient matrix is ​​obtained, simulation can be performed to obtain the transmission spectrum of the multi-layer periodic structure constructed by anisotropic chiral medium and ordinary medium;

[0088] Similarly, the transmission matrix of circular polarization incident is known as:

[0089]

[0090] Get the transmission coefficient matrix:

[0091]

[0092] Where Δ'=t' 11 t' 22 -t′ 12 t' 21 .

[0093] The beneficial effects of the present invention are:

[0094] The multilayer structure of the multilayer anisotropic chiral metamaterial of the present invention and the common medium is more valuable as a test model; at the same time, the calculation process involves a calculation method based on the transmission matrix, which provides an optical method for analyzing multilayer anisotropic metamaterial structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0095] Figure 1 is a flow chart of analysis and calculation of an embodiment of the present invention;

[0096] Figure 2 A multilayer structure model of anisotropic chiral medium and vacuum layer according to an embodiment of the present invention;

[0097] Figure 3 is a system input and output diagram of an embodiment of the present invention;

[0098] FIG4( a ) is a graph showing the direct transmission coefficient versus frequency under different chiral parameters according to an embodiment of the present invention;

[0099] FIG4( b ) is a graph showing the transformation of the cross transmission coefficient with frequency under different chiral parameters according to an embodiment of the present invention;

[0100] FIG5( a ) is a graph showing the direct transmission coefficient of an embodiment of the present invention with different numbers of layers as a function of frequency;

[0101] FIG5( b ) is a graph showing the cross transmission coefficient at different numbers of layers according to an embodiment of the present invention as a function of frequency;

[0102] FIG6( a ) is a graph showing the direct transmission coefficient at different incident angles as a function of frequency according to an embodiment of the present invention;

[0103] FIG. 6( b ) is a graph showing the cross transmission coefficient at different incident angles as a function of frequency according to an embodiment of the present invention. DETAILED DESCRIPTION

[0104] The technical solution of the present invention will be described clearly and completely below in conjunction with the accompanying drawings. 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.

[0105] In the description of the present invention, it should be noted that, unless otherwise clearly specified and limited, the terms "installed", "connected", and "connected" should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection, or it can be indirectly connected through an intermediate medium, or it can be the internal communication of two components. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to specific circumstances.

[0106] Example: Figure 1 As shown, the analysis and calculation flow chart of the embodiment of the present invention; the analysis and calculation of the optical topological transformation based on the double-layer periodic structure of anisotropic material and dielectric structure includes the following steps:

[0107] S1. Establish a multi-layer periodic structure model constructed by multi-layer anisotropic chiral media and ordinary media; specifically, the structure of the model is a multi-layer periodic structure constructed by multi-layer anisotropic chiral media and ordinary media; in order to facilitate the description of the method, the ordinary medium layer takes the vacuum layer as an example and is arranged periodically from left to right, but the method is still general;

[0108] The constitutive equation of anisotropic chiral media is:

[0109]

[0110]

[0111]

[0112] when When the longitudinal component is included, when When the lateral component is included, For the sake of convenience, the present invention takes the case of including the longitudinal component as an example for detailed description, and directly gives the results for other cases;

[0113] Constitutive equation in vacuum:

[0114] D=ε·E

[0115] B=μ·H

[0116] ε t (μ t )ε z (μ z ) is the relative dielectric constant (permeability) of the uniaxial chiral medium perpendicular and parallel to the optical axis; κ is the chiral parameter, which describes the electromagnetic coupling and has Gaussian units. The dielectric constant and permeability in vacuum are ε=1 and μ=1.

[0117] S2, calculating the electric field and magnetic field expressions in the anisotropic chiral medium layer and the ordinary medium layer; specifically, step S2 specifically includes: writing the expressions of the incident electric field, reflected electric field, and transmitted electric field of the electromagnetic wave on the interface of the anisotropic chiral medium layer as;

[0118] The electric and magnetic fields of the incident electromagnetic wave:

[0119]

[0120]

[0121] The electric and magnetic fields of reflected electromagnetic waves:

[0122]

[0123]

[0124] The electric and magnetic fields of right-handed transmitted electromagnetic waves:

[0125]

[0126]

[0127] The electric and magnetic fields of left-handed transmitted electromagnetic waves:

[0128]

[0129]

[0130] The electric and magnetic fields reflected back from another interface after right-hand transmission:

[0131]

[0132]

[0133] The electric and magnetic fields reflected from another interface after left-handed transmission:

[0134]

[0135]

[0136] The electric and magnetic fields of the electromagnetic waves transmitted from the second interface:

[0137]

[0138]

[0139] The electric and magnetic fields of the electromagnetic wave reflected from the third interface to the second interface:

[0140]

[0141]

[0142] Where "+" and "-" represent the direction of the component of the electromagnetic wave propagation direction on the z-axis. When the component direction is the same as the z-direction, it is "+", and the opposite is "-". θ is the angle between the incident electromagnetic wave and the normal. R is the angle between the right-handed transmitted wave and the normal, θ L is the angle between the left-handed transmitted wave and the normal. In vacuum, the wave number k0 = ω / c, ω = 2πf, and f is the frequency of the electromagnetic wave. In a uniaxial chiral medium, there are two propagation modes: R The right circularly polarized (RCP) wave with wave number k L Left circularly polarized (LCP) wave, k R and k L The expression is as follows:

[0143]

[0144] in The "+" and "-" in the "±" symbol correspond to the RCP wave and LCP wave respectively. On the other hand, the continuity of electric and magnetic fields at the dielectric interface requires k0sinθ i =k R sinθ R =k L sinθ L .

[0145] S3. Write out the boundary conditions at the interface between the two material layers; specifically, step S3 specifically includes: From the electromagnetic field boundary conditions, it can be seen that at the interface junction, the tangential components of the electric field and the magnetic field are conserved, so at the first interface:

[0146]

[0147]

[0148]

[0149]

[0150] The second interface has:

[0151]

[0152]

[0153]

[0154]

[0155] E x represents the electric field component in the x-axis direction, H x represents the magnetic field component in the x-axis direction, E y represents the electric field component in the y-axis direction, H y Represents the magnetic field component in the y-axis direction.

[0156] S4, calculating the transmission matrix of the multilayer periodic structure; specifically, step S4 specifically includes: writing the transmission matrix when the linear polarization is incident according to the boundary conditions, and writing the boundary conditions at the first interface into a matrix form:

[0157]

[0158]

[0159]

[0160] Write the boundary conditions at the second interface in matrix form:

[0161]

[0162]

[0163]

[0164] The intermediate variable Eliminate and get E is 、E ip 、E rs 、E rp and relation:

[0165]

[0166] The transfer matrix expression is obtained:

[0167]

[0168] Similarly, when circular polarization is incident, the transmission matrix is ​​obtained:

[0169]

[0170]

[0171]

[0172] E iL refers to the incident left-handed electric field, E iR refers to the incident right-handed electric field, E rL refers to the reflected left-handed electric field, E rR It refers to the reflected right-handed electric field;

[0173]

[0174] Similarly, when the chiral parameter has only the transverse component, the following changes occur:

[0175]

[0176]

[0177] k0sinθ=k' R sinθ' R =k' L sinθ' L

[0178]

[0179] The above matrix θ R(L) →θ'R(L) , other matrix forms are the same as above, and the transmission matrix is ​​obtained:

[0180]

[0181] S5, obtaining the transmission spectrum of the multilayer periodic structure constructed by the anisotropic chiral medium and the ordinary medium; specifically, step S5 specifically includes: taking the transmission matrix at the time of linear polarization incidence as an example, the known transmission matrix is:

[0182]

[0183] Get the transmission coefficient matrix:

[0184]

[0185] Where Δ = t 11 t 22 -t 12 t 21 After the transmission coefficient matrix is ​​obtained, simulation can be performed to obtain the transmission spectrum of the multi-layer periodic structure constructed by anisotropic chiral medium and ordinary medium;

[0186] Similarly, the transmission matrix of circular polarization incident is known as:

[0187]

[0188] Get the transmission coefficient matrix:

[0189]

[0190] Where Δ'=t' 11 t' 22 -t′ 12 t' 21 .

[0191] In this embodiment, if Figure 3 As shown in the system, the electromagnetic wave frequency parameter is input at the A end, the number of layers parameter is input at the B end, the incident angle parameter is input at the C end, the chiral parameter of the chiral medium is input at the D end, the thickness of the material is input at the E end, the relationship between the direct transmission coefficient and each parameter is output at the F end, and the relationship between the cross transmission coefficient and each parameter is output at the G end.

[0192] In this embodiment, the anisotropic chiral medium parameter is set to ε t =3+0.01i,ε z =2+0.01i, dielectric constant in vacuum ε=1, μ=1, thickness of anisotropic chiral medium d=10mm.

[0193] In Figure 4(a) and Figure 4(b), the incident angle is first fixed, the incident angle is set to 15°, the number of layers is 3, and the chiral parameter is set to 0.3, 0.6, 3, and 6. The direct transmission coefficient (|T ss | 2 ) and the cross transmission coefficient (|T ps | 2 ) change curve. It can be seen from the figure that different frequencies have different effects on transmission. At some frequencies, there can be almost no loss, or very low loss through the material. Under different chiral parameter conditions, the change trend is almost the same. It can also be seen that the cross transmission coefficient has an obvious increase at some frequencies, which means that a more obvious polarization deflection occurs at this frequency.

[0194] In Figure 5(a) and Figure 5(b), the incident angle is set to 15°, the chiral parameter is set to 0.3, and the number of layers is set to 1, 3, 5, and 7, respectively. The influence of the number of layers on the direct transmission coefficient and the cross transmission coefficient is obtained, and the direct transmission spectrum and cross transmission spectrum under different numbers of layers are obtained. As shown in the figure, it can be seen that due to the increase in the number of layers, some frequencies with high transmission coefficients in a single layer are suppressed in a multilayer; for the cross transmission coefficient, it is found that there is a significant increase with the increase in the number of layers.

[0195] In Figure 6(a) and Figure 6(b), the number of layers is set to 3, the chiral parameter is set to 0.3, and the incident angles are set to π / 6, 2π / 6, 3π / 6, and 4π / 6, respectively. The peak value of the direct transmission spectrum at different incident angles begins to decrease, and the frequency corresponding to the peak value increases. The cross transmission spectrum also has an obvious change. When the incident angle increases, the cross transmission component has a significant increase.

[0196] The present invention provides a method for calculating the transmission spectrum of a multilayer anisotropic chiral medium based on a transmission matrix. The multilayer anisotropic chiral metamaterial of the present invention is used to construct a multilayer structure with ordinary media, which has application value as a test model; at the same time, the calculation process involves a calculation method based on a transmission matrix, which provides an optical method for analyzing multilayer anisotropic metamaterial structures.

[0197] The above description is only a detailed description of the preferred embodiments and analytical calculations of the present invention. For ordinary technicians in this field, according to the ideas provided by the present invention, there will be changes in the specific implementation methods, and these changes should also be regarded as the protection scope of the present invention.

Claims

1. A method for calculating the transmission spectrum of a multilayer anisotropic chiral medium, characterized in that: The following steps are involved: S1. Establish a multi-layer periodic structure model composed of multi-layer anisotropic chiral media and ordinary media; S2. Calculate the electric and magnetic field expressions in anisotropic chiral medium layer and ordinary medium layer; S3. Write the boundary conditions at the interface between the two material layers; S4, calculating the transmission matrix of the multi-layer periodic structure; S5. Calculate the transmission spectrum of the multilayer periodic structure constructed by anisotropic chiral medium and ordinary medium; Step S1 specifically includes: the structure of the model is a multi-layer periodic structure constructed by anisotropic chiral medium and ordinary medium; the ordinary medium layer is arranged periodically from left to right, taking the vacuum layer as an example; Among them, the constitutive equation of anisotropic chiral media is: when When the longitudinal component is included, when When the lateral component is included, Constitutive equation in vacuum: D=ε·E B=μ·H ε t (μ t ), ε z (μ z ) are the relative permittivities of the uniaxial chiral medium perpendicular to and parallel to the optical axis, respectively; κ is the chiral parameter, describing the electromagnetic coupling, and the permittivity and permeability in vacuum are Gaussian units ε=1, μ=1; Step S2 specifically includes: writing down the expressions of the incident electric field, reflected electric field and transmitted electric field of the electromagnetic wave on the interface of the anisotropic chiral medium layer as follows; The electric and magnetic fields of the incident electromagnetic wave: The electric and magnetic fields of reflected electromagnetic waves: The electric and magnetic fields of right-handed transmitted electromagnetic waves: The electric and magnetic fields of the left-handed transmitted electromagnetic wave: The electric and magnetic fields reflected back from another interface after right-hand transmission: The electric and magnetic fields reflected from another interface after left-handed transmission: The electric and magnetic fields of the electromagnetic waves transmitted from the second interface: The electric and magnetic fields of the electromagnetic wave reflected from the third interface to the second interface: Among them, + and - respectively represent the direction of the component of the electromagnetic wave propagation direction on the z-axis. When the component direction is the same as the z-direction, it is +, and the opposite is -. θ is the angle between the incident electromagnetic wave and the normal, θ R is the angle between the right-handed transmitted wave and the normal, θ L is the angle between the left-handed transmitted wave and the normal. In vacuum, the wave number k0 = ω / c, ω = 2πf, and f is the frequency of the electromagnetic wave. In a uniaxial chiral medium, there are two propagation modes: R The right circularly polarized wave with wave number k L The left circularly polarized wave, k R and k L The expression is as follows: in, The + and - in the symbol ± correspond to the RCP wave and LCP wave respectively. On the other hand, the continuity of the electric and magnetic fields at the dielectric interface requires k0 sinθ i =k R sinθ R =k L sinθ L ; Step S3 specifically includes: It is known from the electromagnetic field boundary conditions that at the interface junction, the tangential components of the electric field and the magnetic field are conserved, so at the first interface: The second interface has: Among them, E x represents the electric field component in the x-axis direction, H x represents the magnetic field component in the x-axis direction, E y represents the electric field component in the y-axis direction, H y represents the magnetic field component in the y-axis direction; Step S4 specifically includes: the transmission matrix at the time of linear polarization incidence is written out according to the boundary conditions, and the boundary conditions at the first interface are written into a matrix form: Write the boundary conditions at the second interface in matrix form: The intermediate variable Eliminate and get E is 、E ip 、E rs 、E rp and relation: The transfer matrix expression is obtained: Similarly, when circular polarization is incident, the transmission matrix is ​​obtained: E iL refers to the incident left-handed electric field, E iR refers to the incident right-handed electric field, E rL refers to the reflected left-handed electric field, E rR It refers to the reflected right-handed electric field; Similarly, when the chiral parameters have only the transverse component, the following changes occur: k0sinθ=k' R you' R =k' L you' L The above matrix θ R(L) →θ' R(L) , other matrix forms are the same as above, and the transmission matrix is ​​obtained: Step S5 specifically includes: taking the transmission matrix when linear polarization is incident as an example, the known transmission matrix is: Get the transmission coefficient matrix: Where Δ = t 11 t 22 -t 12 t 21 After the transmission coefficient matrix is ​​obtained, simulation is performed to obtain the transmission spectrum of the multi-layer periodic structure constructed by anisotropic chiral medium and ordinary medium; Similarly, the transmission matrix of circular polarization incident is known as: Get the transmission coefficient matrix: Where, Δ'=t1'1t' 22 -t1'2t' 21 .

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