CN111209524B - Calculation method for vortex light scattering characteristics of topological insulator particles - Google Patents
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Abstract
The invention discloses a method for calculating vortex light scattering characteristics of topological insulator particles, which specifically comprises the following steps: step 1, laguerre-The Gaussian beams are unfolded through a plane angle spectrometry to obtain the electric field intensity and the magnetic field intensity in the x, y and z directions respectively, and the radial components of the electric field and the magnetic field under a spherical coordinate system are obtained by utilizing a local approximation method; step 2, calculating a beam factor; step 3, the Laguerre-Gaussian beam is incident to the incident field of TI sphere, and the scattering field and the internal field of spherical particles are unfolded by vector spherical harmonic and scattering coefficient to obtain E inc 、E sca 、E int 、H inc 、H sca 、H int The method comprises the steps of carrying out a first treatment on the surface of the Step 4, calculating the scattering coefficient of the Laguerre-Gaussian beam incident TI sphere; step 5, calculating a scattered electric field component of the Laguerre-Gaussian beam incident TI sphere far field region; and 6, calculating a scattering efficiency factor, a extinction efficiency factor and an absorption efficiency factor. The method provided by the invention enables the topological insulator particles to have good application prospects in the aspects of design and production of electronic equipment and optical equipment.
Description
Technical Field
The invention belongs to the technical field of wireless laser communication, and relates to a calculation method of vortex light scattering characteristics of topological insulator particles.
Background
Topological insulator materials have unusual quantum states with an insulator-like band structure between their internal conduction and valence bands and topology-protected, energy-gap-free surface states at their outer edges. In the energy level structure, the surface states are connected with the valence band and the conduction band, are caused by the topological property of the energy band structure, are protected by time inversion symmetry and can exist very stably. The existence of topological magneto-electric coupling items in the topological insulator enables the Maxwell effect to be corrected as follows:wherein epsilon and mu are respectively permittivity and permeability, E and beta are respectively electric field intensity and magnetic induction intensity, and alpha is a fine structureConstant d 3 xdt is a spatiotemporal volume element, Θ is an axion angle describing TI is topological magneto-electric polarizability (TMEP); the constitutive relationship of the topological insulator is modified as: d=ε - (Θα/pi) BETA, and h=BETA/μ+ (Θα/pi) Δζ, where D is the electrical displacement and h is the magnetic field strength.
The scattering problem of spherical particles, from single sphere to multiple spheres, from single-layer to multi-layer dielectric spheres, from plane wave to tangible beam incidence, from common dielectric spheres to chiral dielectric spheres, and the like are intensively studied by expert scholars at home and abroad. However, the scattering study of spherical particles by the ragel gaussian vortex beam is still in the initial stage. Ke Xizheng et al studied the scattering properties of high order LGBs with orbital angular momentum on single sphere particles; zhao Jizhi the scattering properties of spherical and ellipsoidal particles for focused higher order LGBs were studied by army et al; kiselev et al studied the near field scattering properties of isotropic spherical particles on higher order LGB using the T matrix method; qu Tan, wu Zhensen et al systematically investigated the scattering properties of uniaxial anisotropic spheres and bi-spheres and chiral medium spheres on LGB; xu Jiang, han Yiping et al studied the scattering properties of LGB vector far field single sphere particles.
Due to the occurrence of the topological magneto-electric effect, the scattering problem of the topological insulator causes the research hot tide of scholars at home and abroad. Zeng Lunwu et al studied the scattering properties of TI spheres and TI ellipsoidal particles on planar electromagnetic waves, corrected the analytic formulas of the internal and scattered fields according to the constitutive relation and boundary conditions of TI, and utilized the deduction of the scattered electromagnetic field; ge Lixin et al studied the electromagnetic scattering properties of TI cylinders and spherical particles and, under rayleigh approximation, derived an expression that estimates TMEP; akhlesh Lakhtakia et al studied the electromagnetic scattering properties of topologically insulating spheres characterized by a surface admittance γ, and found that for larger values of γ, the scattering properties of TI are close to those of a perfectly conductive sphere, regardless of whether the sphere is composed of dissipative or non-dissipative material, regardless of whether the material supports plane wave transmission at positive or negative phase velocity; faheel Ashraf et al studied the scattering of TI cylinders in chiral media and compared the scattering properties of chiral cylinders of chiral media, and found that the behavior of topologically cylindrical cross-polarized field components in chiral media was more strongly scattered than chiral media cylinders.
Disclosure of Invention
The invention aims to provide a calculation method of vortex light scattering characteristics of topological insulator particles, which utilizes electromagnetic wave theory and generalized Mie scattering theory to research scattering characteristics of topological insulator particles on Laguerre-Gaussian beams, so that the topological insulator particles have good application prospects in the aspects of design and production of electronic equipment and optical equipment.
The technical scheme adopted by the invention is that the method for calculating the vortex light scattering characteristics of the topological insulator particles specifically comprises the following steps:
step 1, spreading Laguerre-Gaussian beams by a plane angle spectrometry to obtain electric field intensities E in x, y and z directions respectively x 、E y 、E z And the magnetic field strength and H x 、H y 、H z And obtaining the electric field E under the spherical coordinate system by using a local approximation method r And the radial component of the magnetic field and H r ；
Step 2, the radial components E of the electric field and the magnetic field under the spherical coordinate system obtained in the step 1 r and Hr Substituting the integrated expression of the beam factor in the transverse magnetic transverse electric mode to obtain the beam factor and />
Step 3, according to the beam factor obtained in step 2 and />Spreading Laguerre-Gaussian beam incident field of TI sphere, scattered field and internal field of spherical particle with vector spherical harmonic and scattering coefficient to obtain E inc 、Ε sca 、Ε int 、H inc 、H sca 、H int ；
Step 4, according to the result obtained in the step 3, calculating the scattering coefficient of the Laguerre-Gaussian beam incident TI sphere;
step 5, based on the scattering coefficient obtained in step 4, obtaining the theta component and the theta component of the scattering electric field of the Laguerre-Gaussian beam incident TI-sphere far field regionA component;
step 6, based on the θ component sum of the scattered electric field of the far field region obtained in step 5Component, find out scattering efficiency factor Q of Laguerre-Gaussian beam incident TI sphere sca Extinction efficiency factor Q ext And absorption efficiency factor Q abs 。
The present invention is also characterized in that,
the specific process of the step 1 is as follows:
in the spherical coordinate system, a beam of LGB linearly polarized in the X-axis direction is incident on a uniform spherical particle along the Z-axis, and the electric field expression of the incident low-order LGB at z=0 is:
wherein E0 The electric field amplitude, x, y are rectangular coordinates, k is wave number, ω is beam waist radius, and l is topological charge number;
based on vector angle spectroscopy, the electromagnetic field distribution of the LGB is unfolded along the x, y and z directions as follows:
wherein ,
E x 、E y 、E z and Hx 、H y 、H z The electric field strength and the magnetic field strength in the x, y and z directions respectively.
Converting the expression in the rectangular coordinate system into the expression of each component in the spherical coordinate system:
(4) The method comprises the steps of carrying out a first treatment on the surface of the Wherein r, theta,Is a spherical coordinate system coordinate.
The specific process of the step 2 is as follows:
substituting the radial components of the electric field and the magnetic field in the spherical coordinate system obtained in the step 1 into an integral expression of the beam factor in the transverse magnetic transverse electric mode, wherein the integral expression is as follows:
wherein, when m=0, the normalization factor is compoundedWhen m is not equal to 0, the composite normalization factor
Substituting formula (4) into formula (5), and calculating a beam factor expression of the on-axis incident LGB by using orthogonality of an exponential function and a trigonometric function:
the beam is incident on the axis, m= ±1-l.
The specific process of the step 3 is as follows:
assuming that the dielectric constant and permeability of TI ball and air are ε, respectively 1 ,μ 1 and ε2 ,μ 2 The method comprises the steps of carrying out a first treatment on the surface of the The radius of the sphere is a; the LGB beam is incident on the TI sphere on axis, and when the incident electromagnetic field is TE (TM) wave, both the scattering electric field and the transmission electric field are TE waves, and in the topological insulator, due to magneto-electric response, the scattering electromagnetic field is cross polarized, the lager-gaussian beam is incident on the incident field of the TI sphere, and the scattering electric field and the internal field of the spherical particle are unfolded by vector spherical harmonics M and N:
wherein ,An ，B n Is the scattering coefficient of TI spherical particles, C n and Dn Is the scattering coefficient inside the TI sphere, and />Is the cross scatter coefficient due to the TME effect of TI spheres.
The specific process of the step 4 is as follows:
combining the tangential component continuous boundary condition of the electromagnetic field on the surface of the spherical particles: r× (E) inc +E sca -E int )＝0；
According to and />Solving for the orthogonality with respect to the scattering coefficient A n 、B n 、C n 、D n 、/>Andthe linear independent equation of (c) yields the scattering coefficient, specifically:
wherein ,
wherein ,jn and hn The first class of ball Bessel functions and Hankel functions are respectively; x=ka is a size parameter;is the relative refractive index of the sphere; beta 1 and β2 Is a function of the axial angle alpha of the spherical particles of the topological insulator, expressed respectively as
When the value of theta is to be 0,a n 、b n 、c n and dn Will degrade into the scattering coefficient and the intra-sphere coefficient of the normal medium sphere.
The specific process of the step 5 is as follows:
the θ component of the scattered electric field of the lager-gaussian beam incident upon the TI sphere far field region was found according to the following formula
Andcomponent->
When (when)Degradation into the scattered field of ordinary dielectric sphere versus LGB;
a n and bn Is the scattering coefficient of spherical particles for plane waves, and />Is a scattering angle function>Is a first order n-th order Legendre function.
The specific process of the step 6 is as follows:
the scattering efficiency factor Q is calculated by the following formula sca Extinction efficiency factor Q ext And absorption efficiency factor Q abs ：
Q abs ＝Q ext -Q sca ；
Wherein lambda is the wavelength, and />Is under the spherical coordinate systemθ component and +.>Component (F)> and />θ component in spherical coordinate system of incident electric field and scattering electric field, +.> and />In the spherical coordinate system of the incident electric field and the scattered electric field, respectively>Component (F)> and />θ component, < > in spherical coordinates of incident magnetic field and scattered magnetic field, respectively> and />In the spherical coordinate system of the incident magnetic field and the scattering magnetic field, respectively>A component.
The method for calculating the vortex light scattering characteristics of the topological insulator particles has the advantages that theoretical basis can be provided for object stealth, electromagnetic object scattering calculation, parameter inversion and the like, quantum communication field based on photon angular momentum can be expanded, and a new thought can be provided for deep study of optical characteristics of topological materials.
Drawings
FIG. 1 is a schematic diagram of a Laguerre-Gaussian vortex beam on-axis incident topological insulator sphere in a method for calculating the vortex light scattering characteristics of topological insulator particles according to the present invention.
Detailed Description
The invention will be described in detail below with reference to the drawings and the detailed description.
The invention discloses a method for calculating vortex light scattering characteristics of topological insulator particles, which specifically comprises the following steps:
step 1, spreading Laguerre-Gaussian beams by a plane angle spectrometry to obtain electric field intensities E in x, y and z directions respectively x 、E y 、E z And the magnetic field strength and H x 、H y 、H z And obtaining the electric field E under the spherical coordinate system by using a local approximation method r And the radial component of the magnetic field and H r ；
The specific process of the step 1 is as follows:
in the spherical coordinate system, a beam of LGB linearly polarized in the X-axis direction is incident on a uniform spherical particle along the Z-axis as shown in fig. 1, and the electric field expression of the incident low-order LGB at z=0 is:
wherein E0 The electric field amplitude, x, y are rectangular coordinates, k is wave number, ω is beam waist radius, l is topological charge number, and the time factor exp (-iωt) is ignored;
based on vector angle spectroscopy, the electromagnetic field distribution of the LGB is unfolded along the x, y and z directions as follows:
wherein ,
E x 、E y 、E z and Hx 、H y 、H z The electric field strength and the magnetic field strength in the x, y and z directions respectively.
Converting the expression in the rectangular coordinate system into the expression of each component in the spherical coordinate system:
(4) The method comprises the steps of carrying out a first treatment on the surface of the Wherein r, theta,Is a spherical coordinate system coordinate.
Step 2, the radial components E of the electric field and the magnetic field under the spherical coordinate system obtained in the step 1 r and Hr Substituting the integrated expression of the beam factor in the transverse magnetic transverse electric mode to obtain the beam factor and />
The specific process of the step 2 is as follows:
wherein, when m=0, the normalization factor is compoundedWhen m is not equal to 0, the composite normalization factor
Substituting formula (4) into formula (5), and calculating a beam factor expression of the on-axis incident LGB by using orthogonality of an exponential function and a trigonometric function:
the beam is incident on the axis, m= ±1-l.
Step 3, according to the beam factor obtained in step 2 and />Spreading Laguerre-Gaussian beam incident field of TI sphere, scattered field and internal field of spherical particle with vector spherical harmonic and scattering coefficient to obtain E inc 、Ε sca 、Ε int 、H inc 、H sca 、H int ；
Assuming that the dielectric constant and permeability of TI ball and air are ε, respectively 1 ,μ 1 and ε2 ,μ 2 The method comprises the steps of carrying out a first treatment on the surface of the The radius of the sphere is a; LGB beam is axially incident to TI ball, and electric field E of incident field inc Electric field of scattered field E sca E of internal field int Magnetic field H of incident field inc Magnetic field H of scattered field sca Magnetic field H of internal field int . In a typical medium, when the incident electromagnetic field is a TE (TM) wave, both the scattered electric field and the transmitted electric field are TE (TM) waves. In topological insulators, due to magneto-electric response, the scattering electromagnetic field is cross polarized, the Laguerre-Gaussian beam is incident to the incident field of the TI sphere, the scattering electric field and the spherical particle internal field are spread out as vector spherical harmonics M and N:
wherein ,An ，B n Is the scattering coefficient of TI spherical particles, C n and Dn Is the scattering coefficient inside the TI sphere. and />Is the cross scatter coefficient due to the TME effect of TI spheres.
Step 4, according to the result obtained in the step 3, calculating the scattering coefficient of the Laguerre-Gaussian beam incident TI sphere;
combining the tangential component continuous boundary condition of the electromagnetic field on the surface of the spherical particles: r× (E) inc +E sca -E int )＝0；r=a, again according to-> and />Solving for the orthogonality with respect to the scattering coefficient A n ，B n ，C n ，D n ，/> and />The linear independent equation of (c) yields the scattering coefficient:
wherein
wherein ,jn and hn The first class of ball Bessel functions and Hankel functions are respectively; x=ka is a size parameter;is the relative refractive index of the sphere; beta 1 and β2 About the axis of spherical particles of topological insulatorFunctions of the sub-angles alpha, which are respectively expressed as
It is apparent that, when Θ=0,a n ，b n ，c n and dn Will degrade into the scattering coefficient and the intra-sphere coefficient of the normal medium sphere.
Step 5, based on the scattering coefficient obtained in step 4, the θ component and the θ component of the scattered electric field of the Laguerre-Gaussian beam incident TI-sphere far field region can be obtainedA component.
When (when)Degradation into the scattered field of ordinary dielectric sphere versus LGB;
a n and bn Is the scattering coefficient of spherical particles for plane waves, and />Is a scattering angle function>Is a first order n-th order Legendre function.
Step 6Based on the sum of the θ component of the scattered electric field of the far field region obtained in step 5The component can obtain the scattering efficiency factor Q of the Laguerre-Gaussian beam incident TI sphere sca Extinction efficiency factor Q ext And absorption efficiency factor Q abs The method comprises the following steps:
Q abs ＝Q ext -Q sca
wherein lambda is the wavelength, and />Is the theta component and +.>Component (F)> and />θ component in spherical coordinate system of incident electric field and scattering electric field, +.> and />In the spherical coordinate system of the incident electric field and the scattered electric field, respectively>Component (F)> and />θ component, < > in spherical coordinates of incident magnetic field and scattered magnetic field, respectively> and />In the spherical coordinate system of the incident magnetic field and the scattering magnetic field, respectively>A component. />
Claims (7)
1. A method for calculating vortex light scattering characteristics of topological insulator particles is characterized by comprising the following steps of: the method specifically comprises the following steps:
step 1, spreading Laguerre-Gaussian beams by a plane angle spectrometry to obtain electric field intensities E in x, y and z directions respectively x 、E y 、E z And the magnetic field strength and H x 、H y 、H z And obtaining the electric field E under the spherical coordinate system by using a local approximation method r And the radial component of the magnetic field and H r ；
Step 2, the radial components E of the electric field and the magnetic field under the spherical coordinate system obtained in the step 1 r and Hr Substituting the integrated expression of the beam factor in the transverse magnetic transverse electric mode to obtain the beam factor and />
Step 3, according to the beam factor obtained in step 2 and />Spreading Laguerre-Gaussian beam incident field of TI sphere, scattered field and internal field of spherical particle with vector spherical harmonic and scattering coefficient to obtain E inc 、E sca 、E int 、H inc 、H sca 、H int ；
Step 4, according to the result obtained in the step 3, calculating the scattering coefficient of the Laguerre-Gaussian beam incident TI sphere;
step 5, based on the scattering coefficient obtained in step 4, obtaining the theta component and the theta component of the scattering electric field of the Laguerre-Gaussian beam incident TI-sphere far field regionA component;
step 6, based on the θ component sum of the scattered electric field of the far field region obtained in step 5Component, find out scattering efficiency factor Q of Laguerre-Gaussian beam incident TI sphere sca Extinction efficiency factor Q ext And absorption efficiency factor Q abs 。
2. The method for calculating the vortex light scattering characteristics of the topological insulator particles according to claim 1, wherein the specific process of the step 1 is as follows:
in the spherical coordinate system, a beam of LGB linearly polarized in the X-axis direction is incident on a uniform spherical particle along the Z-axis, and the electric field expression of the incident low-order LGB at z=0 is:
wherein E0 The electric field amplitude, x, y are rectangular coordinates, k is wave number, ω is beam waist radius, and l is topological charge number;
based on vector angle spectroscopy, the electromagnetic field distribution of the LGB is unfolded along the x, y and z directions as follows:
wherein ,
E x 、E y 、E z and Hx 、H y 、H z The electric field intensity and the magnetic field intensity in the x, y and z directions respectively;
converting the expression in the rectangular coordinate system into the expression of each component in the spherical coordinate system:
wherein r, theta,Is a spherical coordinate system coordinate.
3. The method for calculating the vortex light scattering characteristics of the topological insulator particles according to claim 2, wherein the specific process of the step 2 is as follows:
substituting the radial components of the electric field and the magnetic field in the spherical coordinate system obtained in the step 1 into an integral expression of the beam factor in the transverse magnetic transverse electric mode, wherein the integral expression is as follows:
wherein, when m=0, the normalization factor is compoundedWhen m is not equal to 0, the composite normalization factor
Substituting formula (4) into formula (5), and calculating a beam factor expression of the on-axis incident LGB by using orthogonality of an exponential function and a trigonometric function:
the beam is incident on the axis, m= ±1-l.
4. A method for calculating the vortex light scattering characteristics of a topological insulator particle according to claim 3, wherein the specific process of step 3 is as follows:
assuming that the dielectric constant and permeability of TI ball and air are ε, respectively 1 ,μ 1 and ε2 ,μ 2 The method comprises the steps of carrying out a first treatment on the surface of the The radius of the sphere is a; the LGB beam is incident on the TI sphere on axis, and when the incident electromagnetic field is TE (TM) wave, both the scattering electric field and the transmission electric field are TE waves, and in the topological insulator, due to magneto-electric response, the scattering electromagnetic field is cross polarized, the lager-gaussian beam is incident on the incident field of the TI sphere, and the scattering electric field and the internal field of the spherical particle are unfolded by vector spherical harmonics M and N:
wherein ,An ，B n Is the scattering coefficient of TI spherical particles, C n and Dn Is the scattering coefficient inside the TI sphere,andis the cross scatter coefficient due to the TME effect of TI spheres.
5. The method for calculating the vortex light scattering characteristics of the topological insulator particles according to claim 4, wherein the specific process of the step 4 is as follows:
combining the tangential component continuous boundary condition of the electromagnetic field on the surface of the spherical particles:
r×(E inc +E sca -E int )＝0；r×(μ 0 -1 (B inc +B sca )-(μ r μ 0 ) -1 B int )＝0,r＝a,
according to and />Solving for the orthogonality with respect to the scattering coefficient A n 、B n 、C n 、D n 、/> and />The linear independent equation of (c) yields the scattering coefficient, specifically:
wherein ,
wherein ,jn and hn The first class of ball Bessel functions and Hankel functions are respectively; x=ka is oneA size parameter;is the relative refractive index of the sphere; beta 1 and β2 Is a function of the axial angle alpha of the spherical particles of topological insulator, expressed as +.>
When the value of theta is to be 0,a n 、b n 、c n and dn Will degrade into the scattering coefficient and the intra-sphere coefficient of the normal medium sphere.
6. The method for calculating the vortex light scattering characteristics of the topological insulator particles according to claim 5, wherein the specific process of the step 5 is as follows:
the θ component of the scattered electric field of the lager-gaussian beam incident upon the TI sphere far field region was found according to the following formula and />Component(s)
When (when)Degradation into the scattered field of ordinary dielectric sphere versus LGB;
a n and bn Is the scattering coefficient of spherical particles for plane waves, and />Is a scattering angle function>Is a first order n-th order Legendre function.
7. The method for calculating the vortex light scattering characteristics of the topological insulator particles according to claim 6, wherein the specific process of the step 6 is as follows:
the scattering efficiency factor Q is calculated by the following formula sca Extinction efficiency factor Q ext And absorption efficiency factor Q abs ：
Q abs ＝Q ext -Q sca ；
Wherein lambda is the wavelength, and />Is the theta component and +.>Component (F)> and />θ component in spherical coordinate system of incident electric field and scattering electric field, +.> and />In the spherical coordinate system of the incident electric field and the scattered electric field, respectively>Component (F)> and />θ component, < > in spherical coordinates of incident magnetic field and scattered magnetic field, respectively> and />In the spherical coordinate system of the incident magnetic field and the scattering magnetic field, respectively>A component.
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