CN108614911B - Material interface light beam IF displacement system based on finite surface band gap topological insulator - Google Patents

Material interface light beam IF displacement system based on finite surface band gap topological insulator Download PDF

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CN108614911B
CN108614911B CN201810195567.7A CN201810195567A CN108614911B CN 108614911 B CN108614911 B CN 108614911B CN 201810195567 A CN201810195567 A CN 201810195567A CN 108614911 B CN108614911 B CN 108614911B
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topological insulator
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曾然
唐龙
王驰
侯金鑫
胡淼
李齐良
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Hangzhou Dianzi University
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Abstract

The invention discloses a method for calculating the Imbert-Fedorov displacement of a reflected light beam on a topological insulator with a finite surface band gap based on a fluence method, which comprises the following steps: the first step is as follows: establishing a single interface model of a common isotropic medium and an isotropic finite band gap topological insulator; the second step is that: determining boundaries and initial conditions; the third step: calculating a transmission coefficient and a reflection coefficient on a single interface of a common medium and a chiral medium; the fourth step: solving the energy flow in each direction by using the corrected energy flow method; the fifth step: the Imbert-Fedorov (IF) shift is found. The invention can accurately analyze the transverse displacement characteristics of the single interface model of the common medium and the finite surface band gap topological insulator, and the topological insulator with the finite surface band gap is closer to the theoretical model of the actual topological insulator material and has application value when being used as a test model; meanwhile, a new way is provided for regulating IF displacement, and an optical method is provided for measuring the topological magnetic polarization property.

Description

Material interface light beam IF displacement system based on finite surface band gap topological insulator
Technical Field
The invention belongs to the technical field of optical information, and particularly relates to an IF (Imbert-Fedorov) displacement system on a topological insulator interface of a common medium-finite surface band gap obtained by a modified energy flow method.
Background
When the electromagnetic wave is totally reflected on the interface of two uniform media, the center of the reflected wave will be displaced relative to the center of the incident wave, i.e. the Goos parallel to the incident surface
Figure GDA0003565432710000011
The (GH) lateral displacement and the Imbert-Fedorov (IF) transverse displacement perpendicular to an incident surface, theoretical and experimental researches on GH and IF displacement are always important means for exploring the interaction between a light wave and a physical medium, and the influence of an incident mode, a polarization state and medium characteristic parameters of an incident wave on a reflected wave and a transmitted wave can be known through researches on displacement characteristics. In 1948, Goos-
Figure GDA0003565432710000012
Theoretical explanation of the effects, Renard subsequently proposed the fluence theory, which states that part of the fluence of the incident beam enters the optically hydrophobic medium as evanescent waves and is perpendicular to the interfaceAfter traveling a certain distance, the GH is reflected to an optically dense medium, but GH displacement obtained by the energy flow method is inconsistent with the result of the fixed phase method used by Artmann. Yasumoto et al increase the influence term of the mutual interference of the incident beam and the reflected beam on the energy flow on the basis of the energy flow method to ensure the correctness of the energy flow method.
Current research on IF shifts has been extended to a variety of structural and material media, such as layered structural media, periodic media, absorbing and non-absorbing materials, photonic crystals, chiral media, and the like. Despite such material media and structures, few IF effects have been addressed at the surface of the topological insulator, and in particular few studies have been addressed at the surface band gap-limited topological insulator, so the present system will address the lateral displacement characteristics of the reflected beam at the surface band gap-limited topological insulator interface.
Topological insulators protected by time-reversal symmetry are a completely new material state discovered in recent years by the condensed state physics community. The material has peculiar physical properties, the surface of the material is in a metal state without energy system, and the material has energy gaps in the body, and is similar to a common insulator. When studying the lateral displacement characteristics of the surface of the topological insulator, the surface of the topological insulator needs to be covered by the magnetic coating to open the surface band gap, and the limited surface band gap will also influence the lateral displacement characteristics. With the increasing research on topological insulators, people also begin to combine the unusual topological magnetoelectric properties of the surface of the topological insulator with the displacement properties of the reflected beam. Xu et al discusses the IF shift produced by emission at the anisotropic topological insulator interface; liu et al studied IF shifts at the interface of common media and topological insulators using the fluence method, and found that IF shifts also exist with linear polarization.
Although many people have studied the special electromagnetic properties of the topological insulator, the research on the finite surface band gap of the topological insulator is very rare, Chen studies the influence of the finite surface band gap topological insulator on the Casimir force, and the research on the characteristic displacement of the surface of the finite surface band gap topological insulator is not. The Imbert-Fedorov (IF) shift has become the research focus of many researchers, and with the emergence of various special material media, the research on the IF shift is more and more abundant.
Disclosure of Invention
The invention provides a method for calculating the Imbert-Fedorov displacement of a reflected light beam on a topological insulator with a finite surface band gap based on a fluence method. The topological insulator with the limited surface band gap is closer to a theoretical model of an actual topological insulator material, and has application value when being used as a test model; meanwhile, a new way is provided for regulating IF displacement, and an optical method is provided for measuring the topological magnetic polarization property.
In order to achieve the purpose of the invention, the invention adopts the following technical scheme:
a method for analyzing the Imbert-Fedorov displacement of a reflected light beam on a topological insulator with a limited surface band gap based on a fluence method is carried out according to the following steps:
the first step is as follows: establishing a topological insulator single interface model of a common isotropic medium and an isotropic finite surface band gap;
the second step is that: determining boundaries and initial conditions;
the third step: obtaining a transmission coefficient and a reflection coefficient on a single interface of a topological insulator with a common medium and a limited surface band gap;
the fourth step: solving the energy flow in each direction by using the corrected energy flow method;
the fifth step: the Imbert-Fedorov (IF) shift is found.
Further, the first step of the method is specifically:
the bulk of the topological insulator is an insulating state protected by time-reversal symmetry, alpha/(4 pi)2)∫d3xdt Theta E.B topological nontrivial term can be rewritten as a spin momentum locking Fermi on the interface of a topological insulator and a common insulator, and the system only considers the condition that the topological magnetic susceptibility theta is pi or-pi and can be directly popularized to multiple Fermi. The effect of Dirac fermi on the surface of the topological insulator can be expressed as:
Figure GDA0003565432710000031
a＝0，x，y，γ0＝σz，γx＝ivFσy，γy＝-ivFσx。σx、σy、σzare the three pauli spin matrices. v. ofFIs the Fermi velocity of the surface Fermi, for different materials vFWill be different in value of Bi2Te3Surface, vF＝1.3×10-3c, in Bi2Se3Surface vF＝1.7×10-3c, for convenience of calculation, take v hereF＝1.0×10-3c. m denotes the surface band gap by applying a magnetic coating, m ═ m |, corresponds to Θ ═ pi, typically Θ ═ 2n +1) pi denotes the presence of multiple fermi-ons on the surface of the topological insulator. A. theaAre the first three terms of electromagnetic potential, the corresponding electromagnetic field can be described as:
Figure GDA0003565432710000032
where E and B are the electric and magnetic fields, respectively,. epsilon2And mu2It is the permittivity and permeability inside the topological insulator. The effective effect on a (2+1) -dimensional external electromagnetic field can be obtained by standard quantum field theory
Figure GDA0003565432710000033
Introducing a Feynman parameter and integrating the fermi field into a single cycle correction, the following form can be obtained:
Figure GDA0003565432710000041
the dimensionless parameters φ and φ can be obtained by integration:
Figure GDA0003565432710000042
Figure GDA0003565432710000043
where sign (m) denotes the sign of the finite surface bandgap, i.e. the sign corresponding to different topological magnetic susceptibility. α represents a fine structure constant, and
Figure GDA0003565432710000044
k here0、kPRespectively representing the total wave vector and the plane and plane of incidence wave vectors within the topological insulator. As can be understood from the above expression, in the case of a large surface band gap (| m | → ∞), Φ (λ) → sign (m) α, Φ (λ) ═ α/6.
Substituting equation (3) into the standard electromagnetic field equation (2) can obtain the corresponding Maxwell equation:
Figure GDA0003565432710000045
in the case of | m | → ∞ according to an constitutive relation that can be corrected:
Figure GDA0003565432710000046
assuming a finite beam from a semi-infinite isotropic medium 1 (. epsilon.)1，μ1) Topological insulator 2 (epsilon) with oblique incidence to semi-infinite thick uniform finite surface band gap2，μ2) As shown in fig. 2 (a). From these corresponding Maxwell equations, the modified constitutive relations and the corresponding boundary conditions can be obtained:
Figure GDA0003565432710000047
Figure GDA0003565432710000051
the expressions of the reflection electric field and the transmission electric field on the surface of the topological insulator with the limited surface band gap can be deduced according to the equations (6-9):
Figure GDA0003565432710000052
Figure GDA0003565432710000053
Figure GDA0003565432710000054
Kr＝k1(-cosθix+sinθiz)、Kt＝k2(cosθix+sinθiz) are each independently of the other
Figure GDA0003565432710000055
Figure GDA0003565432710000056
λ0Denotes the wavelength of the incident wave in vacuum, Kx、PxThe x-axis components of the wavevectors in medium 1 and medium 2, respectively. According to the corresponding electric field Er、EtAnd the constitutive relation (7) can obtain a corresponding magnetic field.
The light beam coming from an optically-dense medium
Figure GDA0003565432710000057
Incident on an optically thinner medium
Figure GDA0003565432710000058
At an incident angle satisfying thetai＞θc＝arcsin(n1/n2) Total reflection will occur and the energy flow transmitted by the beam will be reflected at the boundary between the two media. The electromagnetic wave propagating in the optically thinner medium is oneNon-uniform and in the form of an exponential decay, commonly referred to as an evanescent wave. Renard states that the fluence of an incident beam will enter the optically thinner medium as an evanescent wave and travel some distance along the interface before returning to the optically denser medium. Considering a finite plane wave incident on the surface of a topological insulator with a finite surface bandgap, the resulting IF shift will be in the y-axis direction. As shown in FIG. 2(b), first consider a time-averaged Poynting vector S due to the coherence of the incident beam and the reflected beamirThe average power flow P can be calculated as shown by the shaded portionir(ξ＝z,y)。
Figure GDA0003565432710000061
In the figure PtThe total energy flow representing the evanescent wave can be found by equation (14), where SrTime-averaged Poynting vector representing evanescent wave:
Figure GDA0003565432710000062
and the component of the time-averaged Poynting vector of the reflected wave on the x-axis
Figure GDA0003565432710000063
Can be expressed as:
Figure GDA0003565432710000064
the IF displacement expression can be obtained by the fluence method:
Figure GDA0003565432710000065
under total reflection, i.e. angle of incidence θi＞θcIn this case, the reflection coefficient becomes complex, the phase also changes with the change of the incident angle, and the IF shift occurs in the reflected wave.
The invention has the following characteristics:
the invention uses Yasumoto and Oisihi improved energy flow method to calculate the IF displacement of the light beam from the common dielectric medium to the surface of the topological insulator with limited surface band gap, can accurately analyze the transverse displacement characteristics of the single interface model of the common dielectric medium and the topological insulator with limited band gap, and the analyzed displacement characteristics can accurately reflect the incident mode, polarization state and influence of the characteristic parameters of the medium of the incident wave on the reflected wave and the transmitted wave.
Because the actual manufacturing cost of the topological insulator is high, the expected IF beam deviation can be obtained by firstly calculating and testing the topological insulator material with any parameter, and the topological insulator with the limited surface band gap is a theoretical model which is closer to the actual topological insulator material and has a use value as a test model. When the width of the band gap of the finite surface is changed during linear polarization, the IF displacement of TE incident waves is reduced along with the reduction of the band gap width; IF displacement of TM incident wave decreases the width of the finite surface bandgap of the topological insulator, the displacement of the beam will decrease, and the value of the displacement will increase slightly as the width of the finite surface bandgap continues to decrease. To study the effect of the reflected light beams of different polarization states on the displacement characteristics, we also studied the IF displacement of the elliptically polarized incident light beam, and from the results, it can be seen that the IF displacement of the elliptically polarized incident light beam is significantly enhanced, and when the band gap width is reduced, the value of the IF displacement also starts to be reduced, and when the band gap width is further reduced, the IF displacement will be increased by a small extent.
Drawings
FIG. 1 is an analytical flow chart of the present invention.
FIG. 2(a) is a schematic diagram of a single interface model composed of a common medium and a topological insulator with a finite surface bandgap in the present invention.
FIG. 2(b) a model of the energy flow of a finite plane wave over a topological insulator of finite surface bandgap.
Fig. 3 is an input-output diagram of the system.
Fig. 4 is a graph of IF shift produced at linearly polarized incidence.
Fig. 5(a) is a graph of IF shift as a function of polarization angle γ when determining an angle of incidence.
Fig. 5(b) is a graph of IF shift at different finite surface bandgaps for left and right hand polarizations incident on the interface of a right hand bandgap topological insulator.
Detailed Description
The following describes embodiments of the present invention in detail with reference to the accompanying drawings.
FIG. 1 is an analytical flow chart of the present invention. The method for analyzing the Imbert-Fedorov displacement of a reflected light beam on a topological insulator with a limited surface band gap based on a power flow method comprises the following steps:
the first step is as follows: establishing a single-interface model of a topological insulator of a common isotropic medium and an isotropic finite surface band gap;
the second step is that: determining boundaries and initial conditions;
the third step: obtaining a transmission coefficient and a reflection coefficient on a single interface of a topological insulator with a common medium and a limited surface band gap;
the fourth step: solving the energy flow in each direction by using the corrected energy flow method;
the fifth step: the Imbert-Fedorov (IF) shift is found.
The bulk of the topological insulator is an insulating state protected by time-reversal symmetry, alpha/(4 pi)2)∫d3xdt Theta E.B topological nontrivial term can be rewritten as a spin momentum locking Fermi on the interface of a topological insulator and a common insulator, and the system only considers the condition that the topological magnetic susceptibility theta is pi or-pi and can be directly popularized to multiple Fermi. The effect of Dirac fermi on the surface of the topological insulator can be expressed as:
Figure GDA0003565432710000081
a＝0，x，y，γ0＝σz，γx＝ivFσy，γy＝-ivFσx。σx、σy、σzare the three pauli spin matrices. v. ofFIs the Fermi velocity of the surface Fermi, forMaterials v of the sameFWill be different in value of Bi2Te3Surface, vF＝1.3×10-3c, in Bi2Se3Surface vF＝1.7×10-3c, for convenience of calculation, take v hereF＝1.0×10-3c. m denotes the surface band gap by applying a magnetic coating, m ═ m |, corresponds to Θ ═ pi, typically Θ ═ 2n +1) pi denotes the presence of multiple fermi-ons on the surface of the topological insulator. A. theaAre the first three terms of electromagnetic potential, the corresponding electromagnetic field can be described as:
Figure GDA0003565432710000082
where E and B are the electric and magnetic fields, respectively,. epsilon2And mu2It is the permittivity and permeability inside the topological insulator. The effective effect on a (2+1) -dimensional external electromagnetic field can be obtained by standard quantum field theory
Figure GDA0003565432710000083
Introducing a Feynman parameter and integrating the fermi field into a single cycle correction, the following form can be obtained:
Figure GDA0003565432710000084
the dimensionless parameters φ and φ can be obtained by integration:
Figure GDA0003565432710000091
Figure GDA0003565432710000092
where sign (m) denotes the sign of the finite surface bandgap, i.e. the sign corresponding to different topological magnetic susceptibility. α represents a fine structure constant, and
Figure GDA0003565432710000093
k here0、kPRespectively representing the total wave vector and the plane and plane of incidence wave vectors within the topological insulator. As can be understood from the above expression, in the case of a large surface band gap (| m | → ∞), Φ (λ) → sign (m) α, Φ (λ) ═ α/6.
Substituting equation (3) into the standard electromagnetic field equation (2) can obtain the corresponding Maxwell equation:
Figure GDA0003565432710000094
in the case of | m | → ∞ according to an constitutive relation that can be corrected:
Figure GDA0003565432710000095
assuming a finite beam from a semi-infinite isotropic medium 1 (. epsilon.)1，μ1) Topological insulator 2 (epsilon) with oblique incidence to semi-infinite thick uniform finite surface band gap2，μ2) As shown in fig. 2 (a). From these corresponding Maxwell equations, the modified constitutive relations and the corresponding boundary conditions can be obtained:
Figure GDA0003565432710000096
Figure GDA0003565432710000097
the expressions of the reflection electric field and the transmission electric field on the surface of the topological insulator with the finite surface band gap can be deduced according to the equations (6-9):
Figure GDA0003565432710000101
Figure GDA0003565432710000102
Figure GDA0003565432710000103
Kr＝k1(-cosθix+sinθiz)、Kt＝k2(cosθix+sinθiz) are each independently of the other
Figure GDA0003565432710000104
(j＝1,2)，λ0Denotes the wavelength of the incident wave in vacuum, Kx、PxThe x-axis components of the wavevectors in medium 1 and medium 2, respectively. According to the corresponding electric field Er、EtAnd the constitutive relation (7) can obtain a corresponding magnetic field.
The light beam coming from an optically-dense medium
Figure GDA0003565432710000105
Incident on an optically thinner medium
Figure GDA0003565432710000106
At an incident angle satisfying thetai＞θc＝arcsin(n1/n2) Total reflection will occur and the energy flow transmitted by the beam will be reflected at the boundary between the two media. The electromagnetic wave propagating in the optically thinner medium is then non-uniform and in an exponentially decaying form, commonly referred to as an evanescent wave. Renard states that the fluence of an incident beam will enter the optically thinner medium as an evanescent wave and travel some distance along the interface before returning to the optically denser medium. Considering a finite plane wave incident on a topological insulator surface with a finite surface bandgap, the resulting IF shift will be in the y-direction. As shown in FIG. 2(b), first consider a time-averaged Poynting vector S due to the coherence of the incident beam and the reflected beamirE.g. hatched in the figure, can be calculatedMean power flow Pir(ξ＝z,y)。
Figure GDA0003565432710000111
In the figure PtThe total energy flow representing the evanescent wave can be found by equation (14), where SrTime-averaged Poynting vector representing evanescent wave:
Figure GDA0003565432710000112
and the component of the time-averaged Poynting vector of the reflected wave on the x-axis
Figure GDA0003565432710000113
Can be expressed as:
Figure GDA0003565432710000114
the IF displacement expression can be obtained by the fluence method:
Figure GDA0003565432710000115
under total reflection, i.e. angle of incidence θi＞θcIn this case, the reflection coefficient becomes complex, the phase also changes with the change of the incident angle, and the IF shift occurs in the reflected wave.
In the present embodiment, as shown in fig. 3, assuming that light is input only to the a port by the width of the finite surface band gap, the dielectric constants and magnetic permeabilities of the media 1 and 2 are input to the B port, and the polarization types (1. represents linear polarization, 2. represents elliptical polarization) of the incident light beam are input to the port C. Suppose that μ is input at port B1＝μ2＝1，ε1＝5，ε24, and 100W is input at port aR、200WR、300WR、400WR、500WR、10000WRW hereinRRepresenting the resonant frequency. Assuming that port C is identified as 1, in linear polarization, the phase difference δ is 0, where the electric field is called TE polarization parallel to the interface and the magnetic field is called TM polarization parallel to the interface. And changing the surface band gap m and the incidence angle to obtain different polarized wave patterns. I.e. linear polarization, an IF shift can be obtained as shown in fig. 4.
On the other hand, when the input value at port C is 2, that is, when elliptically polarized light enters, δ at this time is ± pi/2 (indicating right-handed polarization and left-handed polarization, respectively). Based on the data input from ports A and B, an angle of incidence θ is determinediChanging the polarization angle γ at 65 ° results in an IF shift as shown in fig. 5 (a). A larger IF shift can be obtained at the corresponding polarization state and gamma angle at which the maximum minima occur as shown in fig. 5 (b).
The invention researches the IF displacement of a light beam with limited width at the topological insulator interface with limited surface band gap based on a corrected energy flow method, and analyzes the influence of incident light beams with different polarization states and the width m of different limited surface band gaps on the IF displacement. Because the actual manufacturing cost of the topological insulator is high, the system can calculate and test which parameter of the topological insulator material obtains the wanted IF beam deviation, and the topological insulator with limited surface band gap is a theoretical model which is closer to the actual topological insulator material, and has application value as a test model. Meanwhile, the research on IF displacement of the interface of a common medium and a finite band gap topological insulator provides a new way for regulating the IF displacement and also provides an optical method for measuring the topological magnetic polarization property.
While the preferred embodiments and principles of this invention have been described in detail, it will be apparent to those skilled in the art that variations may be made in the embodiments based on the teachings of the invention and such variations are considered to be within the scope of the invention.

Claims (1)

1. A method for calculating IF shift of a reflected beam on a topological insulator with a finite surface bandgap based on a fluence method, the method comprising the steps of:
the first step is as follows: establishing a single-interface model of a topological insulator of a common isotropic medium and an isotropic finite surface band gap;
the second step is that: determining boundaries and initial conditions;
the third step: obtaining a transmission coefficient and a reflection coefficient on a single interface of a topological insulator of a common medium and a finite surface band gap;
the fourth step: solving energy flows in all directions by adopting a corrected energy flow method;
the fifth step: obtaining IF displacement;
the first step is as follows:
rewriting topological non-trivial terms as spin momentum locked fermions on the interface of the topological insulator and the common insulator; the effect of Dirac fermi on the surface of the topological insulator is expressed as:
Figure FDA0003516970880000011
wherein a is 0, x, y; gamma ray0＝σz，γx＝ivFσy，γy＝-ivFσx，
Figure FDA0003516970880000012
And sigmax、σy、σzAre three pauli spin matrices; v. ofFIs the Fermi velocity of the surface FermiF＝1.0×10-3c; m represents a surface band gap by applying a magnetic coating, m ═ m |, corresponds to Θ ═ pi, Θ ═ 2n +1) pi represents the presence of multiple fermi seeds on the surface of the topological insulator; a. theaAre the first three terms of electromagnetic potential; the corresponding electromagnetic field equation is:
Figure FDA0003516970880000013
e and B are electric and magnetic fields, respectively, epsilon2And mu2Respectively, the dielectric constant and the magnetic permeability inside the topological insulator;
effective action of (2+1) -dimensional external electromagnetic field obtained in standard quantum field theory
Figure FDA0003516970880000014
Introducing the Feynman parameters and integrating the fermi field into a single cycle correction to obtain the following form:
Figure FDA0003516970880000021
dimensionless parameters phi and phi are found by integration:
Figure FDA0003516970880000022
Figure FDA0003516970880000023
wherein sign (m) represents a sign of finite surface band gap, i.e., a sign corresponding to magnetic susceptibility of different topologies, α represents a fine structure constant, and
Figure FDA0003516970880000024
k of (a)0、kPRespectively representing the total wave vector in the topological insulator and the wave vectors of the plane and the incident plane; f denotes the electromagnetic field tensor;
substituting equation (3) into the standard electromagnetic field equation (2) results in the corresponding Maxwell equation:
Figure FDA0003516970880000025
at | m | → ∞ according to the constitutive relation corrected:
Figure FDA0003516970880000026
the second step is as follows:
assuming a finite beam from a semi-infinite isotropic medium 1 (. epsilon.)1，μ1) Topological insulator 2 (epsilon) with oblique incidence to semi-infinite thick uniform finite surface band gap2，μ2) The modified constitutive relation and the corresponding boundary conditions are obtained from the corresponding Maxwell equation (6):
Figure FDA0003516970880000031
Figure FDA0003516970880000032
and thirdly, deducing expressions of a reflection electric field and a transmission electric field on the surface of the topological insulator with the limited surface band gap according to equations (6-9), wherein the expressions are as follows:
Figure FDA0003516970880000033
Figure FDA0003516970880000034
Figure FDA0003516970880000035
Kr＝k1(-cosθix+sinθiz)、Kt＝k2(cosθix+sinθiz) are each independently of the other
Figure FDA0003516970880000036
j＝1,2，λ0Denotes the wavelength of the incident wave in vacuum, Kx、PxThe x-axis components of the wavevectors in medium 1 and medium 2, respectively; according to the corresponding electric field Er、EtObtaining a corresponding magnetic field; n represents the refractive index of the medium, c represents the speed of light, KxWave vector representing the x-axis component in medium 1, and PxA wave vector representing the x-axis component in medium 2;
the fourth step is as follows:
calculating the mean energy flow Pir(ξ＝z,y)，
Figure FDA0003516970880000041
Figure FDA0003516970880000042
And the component of the time-averaged Poynting vector of the reflected wave on the x-axis
Figure FDA0003516970880000043
Expressed as:
Figure FDA0003516970880000044
the fifth step is as follows:
Figure FDA0003516970880000045
the IF shift of the reflected beam on the topological insulator with finite surface band gap is thus obtained.
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