Method for chiral manipulation of a vortex beam based on graphene

Abstract

The application discloses a graphene-based vortex beam chirality control method, and specific steps are as follows: S1, a graphene-substrate system surface vortex beam reflection schematic diagram is established; S2, a graphene-substrate system Fresnel reflection coefficient is described; S3, a scalar angular spectrum description of an incident vortex beam is performed; S4, a graphene surface vortex beam reflection full-vector theoretical model is established; and S5, a graphene surface reflected vortex beam chirality density is calculated. The method utilizes the unique tunability of graphene, and realizes flexible control on the vortex beam chirality by changing the Fermi energy of graphene and the substrate refractive index.

Description

Technical Field

[0001] This invention belongs to the field of optical chirality control, specifically relating to a method for chiral control of vortex beams based on graphene. Background Technology

[0002] Chirality refers to the property of a structure or system that it cannot be superimposed on its mirror image regardless of rotation or movement; it is a universal characteristic of nature. When light exhibits a helical wavefront or a distorted phase during propagation, it possesses chiral characteristics; this type of light is called chiral light. Vortex beams are a type of chiral light with a helical phase wavefront, and their complex amplitude expression contains a phase factor. Where l is the topological charge number. It's the azimuth angle. The phase factor of a vortex beam. Related to orbital angular momentum (OAM), each photon in a vortex beam with topological charge l carries ±|l|. OAM (Optical Amplification) studies have shown that the interaction between vortex beams and chiral matter can produce an optical rotation effect similar to that between circularly polarized light and chiral matter. Compared to traditional circularly polarized chiral light fields, vortex beams with unique amplitude, phase, and polarization state distributions offer new possibilities for obtaining richer information about chiral matter. Effective manipulation of the chirality of vortex beams has significant application value in chiral molecule recognition, chiral structure detection, and chiral optical micromanipulation.

[0003] Currently, there are two methods for controlling the chirality of vortex beams. One method is to use a high numerical aperture lens to focus the vortex beam. [1] However, this approach only enhances the chirality of vortex beams under specific conditions. Another method involves using artificial metamaterials such as chiral media. [2]The dispersive properties of vortex beams are entirely determined by the structural dimensions of their periodic units, resulting in fixed optical characteristics after design, making flexible control of the chirality of vortex beams impossible. However, combining materials for controlling the chirality of vortex beams with active excitation allows for more flexible control. Currently, ferroelectric materials, temperature-controlled materials, and two-dimensional materials have been used for dynamic beam control. Graphene, as a typical two-dimensional material, is a promising source for dynamic beam control. Graphene is a flat, monolayer graphite composed of a single atom-thick layer of carbon atoms arranged in a hexagonal honeycomb lattice. Its Fermi energy determines its electrical conductivity, which can be tuned over a wide range by applying a bias voltage or an external electric field. A unique property of graphene is that its reflection characteristics are determined by its fine-structure constant and intrinsic parameters, which can be modulated by changing the Fermi level through electrostatic doping. Therefore, we propose a method for controlling the chirality of vortex beams based on graphene, which achieves flexible control of the chirality of vortex beams by changing the graphene Fermi energy, substrate refractive index, incident angle, and topological charge number.

[0004] Wenguo Zhu et al. reported on the manipulation of spin splitting of vortex beams based on graphene, and presented a theoretical model for vortex beam propagation in graphene. [3] However, this theoretical model does not involve vector analysis of the vortex beam reflected from the graphene surface, and therefore cannot be used to calculate the chiral density of the vortex beam reflected from the graphene surface, thus failing to provide a theoretical basis for controlling the chiral density of the vortex beam based on graphene. The purpose of this invention is to address the shortcomings and deficiencies of the above methods by establishing a full-vector theoretical model for the reflection of vortex beams from the graphene surface, numerically calculating the influence of graphene parameters on the chirality of the vortex beam, and demonstrating that flexible control of the chirality of the vortex beam can be achieved based on graphene.

[0005] References:

[0006] [1] Guo Shenyan, Cui Zhiwei, Wang Ju, Wu Fuping. Local optical chirality analysis of tightly focused vortex beams. Acta Photonica Sinica, 50:1026002(2021).

[0007] [2] Fuping Wu, Zhiwei Cui, Shenyan Guo, Wanqi Ma, and Ju Wang. Chirality of optical vortex beams reflected from an air-chiral medium interface. Optics Express, 30: 21678-21697 (2022).

[0008] [3]WGZhu,MJJiang,HYGuan,JHYu,HHLu,J.Zhang,Z.Chen, Tunablespin splitting of Laguerre-Gaussian beams in graphene metamaterials,PhotonicsRes.5:684-688(2017). Summary of the Invention

[0009] The purpose of this invention is to provide a graphene-based method for controlling the chirality of vortex beams. This method utilizes the unique tunability of graphene to achieve flexible control of the chirality of vortex beams by changing the graphene Fermi energy and the substrate refractive index.

[0010] The technical solution adopted in this invention is a graphene-based method for controlling the chirality of vortex beams, and the specific steps are as follows:

[0011] S1. Establishment of a schematic diagram of vortex beam reflection on the surface of the graphene-substrate system;

[0012] S2, Description of the Fresnel reflectance coefficient of the graphene-substrate system;

[0013] S3. Scalar angular spectrum description of the incident vortex beam;

[0014] S4. Establishment of a full-vector theoretical model for reflection of vortex beams on graphene surfaces;

[0015] S5. Calculation of the chiral density of reflected vortex beams on graphene surfaces.

[0016] The invention is further characterized in that,

[0017] In step S1, the specific method for establishing the schematic diagram of the vortex beam reflection on the surface of the graphene-substrate system is as follows: Assume the refractive index of air is n0, and the refractive index of the substrate is n1; wherein, the graphene-substrate interface is located in the global coordinate system (x, y, z), and the z-axis of the global coordinate system (x, y, z) is perpendicular to the graphene-substrate interface and points towards the substrate; the monolayer graphene located on top of the substrate is at position z = 0; (x i ,y i ,z i ) and (x r ,y r ,z r ) represent the coordinate system of the incident beam and the coordinate system of the reflected beam, respectively; θ i and θ r These represent the incident angle and reflection angle of the central wave vector, respectively.

[0018] In step S2, the Fresnel reflectance of the graphene-substrate system is expressed as:

[0019]

[0020]

[0021] Where, r p Let r be the Fresnel reflection coefficient for the horizontal polarization case. s Let n be the Fresnel reflection coefficient for vertical polarization, n1 be the refractive index of the substrate, n0 be the refractive index of air, ε0 be the dielectric constant in vacuum, and ω be the angular frequency. k0 is the wavenumber of the light beam in air, θ i σ(ω,E) is the incident angle of the light beam, c is the light beam in vacuum, and σ(ω,E) is the incident angle of the light beam. f Let be the optical conductivity of graphene, expressed as follows:

[0022]

[0023] Where i represents an imaginary number, and e = 1.6 × 10 -19 C is the electron charge, E f The Fermi energy of graphene. To reduce Planck's constant, ω is the angular frequency. For relaxation time, v f denoted as Fermi rate, μ as electron mobility, and H(·) as Heaviside step function, i.e., unit step function.

[0024] The specific implementation of step S3 is as follows: in the incident beam coordinate system (x i ,y i ,z i In ), z i The complex amplitude expression for the vortex beam on the initial plane with = 0 is:

[0025]

[0026] Where w0 is the waist radius of the vortex beam, l is the topological charge number, and sign(·) is the sign function.

[0027] Substitute equation (4) into equation (5) below and perform a Fourier transform;

[0028]

[0029] The scalar angular spectrum of the incident vortex beam is obtained. The expression is:

[0030]

[0031] Where, k ix and k iy Let k be the incident beam wave vector.i The horizontal component.

[0032] The specific implementation method of step S4 is as follows:

[0033] First, after the vortex beam is reflected off the graphene surface, its angular spectrum expression is determined as follows:

[0034]

[0035] in, and These represent the horizontal and vertical components of the angular spectrum of the reflected vortex beam, respectively, k rx and k ry Let k be the wave vector of the reflected beam. r The transverse component, α and β are the polarization parameters of the incident vortex beam. To determine the scalar angular spectrum of the reflected beam, a boundary condition k is applied to the scalar angular spectrum of the incident vortex beam. rx =-k ix and k ry =k iy The scalar angular spectrum of the reflected vortex was obtained. Right now

[0036]

[0037] Then, in the incident beam coordinate system (x r ,y r ,z r In this context, a vector potential A is introduced. r The expression is:

[0038]

[0039] Where, k r Let k be the wave vector of the reflected beam. r The modulus, and respectively along x r and y r The unit vector of the axis, and These are the horizontal and vertical components of the complex amplitude of the reflected vortex beam, respectively, obtained through the inverse Fourier transform of the following equation:

[0040]

[0041] Substituting equation (7) into equation (10) and integrating, we get...

[0042]

[0043]

[0044] in

[0045] I r1 =u r (13)

[0046]

[0047]

[0048]

[0049] In the formula, The Rayleigh distance is the distance at which the reflected vortex beam is measured.

[0050] Finally, under the Lorentz gauge condition, the electric field E of the reflected vortex beam r and magnetic field H r Using vector potential A r Represented as:

[0051]

[0052]

[0053] Among them, Z r The wave impedance for reflecting a vortex beam. and respectively along x r y r and z r The unit vector of the axis, and The specific expression is:

[0054]

[0055]

[0056]

[0057]

[0058] In the formula

[0059]

[0060]

[0061]

[0062]

[0063]

[0064]

[0065] The specific implementation method of step S5 is as follows:

[0066] The electric field E after the vortex beam derived in step S4 is reflected on the graphene surface r and magnetic field H r Substituting into the expression for optical chirality density, where the optical chirality density C is defined as:

[0067]

[0068] In the formula, ω is the angular frequency, c is the speed of light in a vacuum, Im[·] represents the imaginary part, and "*" represents the complex conjugate.

[0069] The beneficial effects of this invention are:

[0070] This invention proposes a graphene-based method for controlling the chirality of vortex beams. This method leverages the dynamic tunability of graphene to achieve more flexible control over the chirality of vortex beams. A unique property of graphene is that its reflection characteristics are determined by its fine structure constant and intrinsic parameters, which can be modulated by altering the Fermi level through electrostatic doping. The Fermi energy determines the electrical conductivity of graphene, which can be tuned over a wide range by applying a bias voltage or an external electric field. Based on this invention, the chirality of vortex beams can be controlled by changing the Fermi energy of graphene and the refractive index of the substrate, as well as by changing the incident angle, topological charge, and polarization state of the vortex beam. Attached Figure Description

[0071] Figure 1 This is a schematic diagram of the reflection of the vortex beam incident from air onto the surface of the graphene-substrate system in the method of the present invention;

[0072] Figure 2 This is a graph showing the variation of the graphene Fresnel reflection coefficient with the incident angle according to an embodiment of the present invention;

[0073] Figure 3 This is a graph showing the variation of graphene conductivity with Fermi energy in an embodiment of the present invention.

[0074] Figure 4 This is a graph showing the variation of the normalized chiral density of the reflected beam as a function of the incident angle when the topological charge number takes different values, according to an embodiment of the present invention.

[0075] Figure 5The following are normalized chiral density distribution diagrams of the reflected beam of the embodiment of the present invention when the topological charge number takes different values: (a) is the chiral density distribution diagram of the reflected beam of the embodiment when the topological charge number l = -1, (b) is the chiral density distribution diagram of the reflected beam of the embodiment when the topological charge number l = -2, (c) is the chiral density distribution diagram of the reflected beam of the embodiment when the topological charge number l = 1, and (d) is the chiral density distribution diagram of the reflected beam of the embodiment when the topological charge number l = 2.

[0076] Figure 6 This is a graph showing the variation of the normalized chiral density of the reflected beam as a function of the incident angle when the Fermi energy takes different values, according to an embodiment of the present invention.

[0077] Figure 7 This is a graph showing the variation of the normalized chiral density of the reflected beam with the incident angle when the refractive index takes different values, according to an embodiment of the present invention.

[0078] Figure 8 The normalized chiral density of the reflected beam in this embodiment of the invention is at the incident angle θ. i The graph showing the change in Fermi energy at 35°.

[0079] Figure 9 The normalized chiral density of the reflected beam in this embodiment of the invention is at the incident angle θ. i The graph showing the change in Fermi energy at 57.5°. Detailed Implementation

[0080] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0081] This invention provides a method for controlling the chirality of vortex beams based on graphene, the specific steps of which are as follows:

[0082] S1. Establishment of a schematic diagram of vortex beam reflection on the surface of the graphene-substrate system. (See diagram below.) Figure 1 As shown, for simplicity, let the refractive index of air be n0 and the refractive index of the substrate be n1; where the graphene-substrate interface is located in the global coordinate system (x,y,z), and the z-axis of the global coordinate system (x,y,z) is perpendicular to the graphene-substrate interface and points towards the substrate; the monolayer of graphene on top of the substrate is located at z=0; (x i ,y i ,z i ) and (x r ,y r ,z r ) represent the coordinate system of the incident beam and the coordinate system of the reflected beam, respectively; θ i and θ r These represent the incident angle and reflection angle of the central wave vector, respectively.

[0083] S2, Description of the Fresnel reflectance coefficient of the graphene-substrate system. Based on the boundary conditions, Figure 1 The Fresnel reflectance coefficient of the graphene-substrate system is given as follows:

[0084]

[0085]

[0086] Where, r p Let r be the Fresnel reflection coefficient for the horizontal polarization case. s Let n be the Fresnel reflection coefficient for vertical polarization, n1 be the refractive index of the substrate, n0 be the refractive index of air, ε0 be the dielectric constant in vacuum, ω be the angular frequency, and k be the refractive index of air. 0z =k0cosθ i , k0 is the wavenumber of the light beam in air, θ i σ(ω,E) is the incident angle of the light beam, c is the light beam in vacuum, and σ(ω,E) is the incident angle of the light beam. f Let be the optical conductivity of graphene, expressed as follows:

[0087]

[0088] Where i represents an imaginary number, and e = 1.6 × 10 -19 C is the electron charge, E f The Fermi energy of graphene. To reduce Planck's constant, ω is the angular frequency. For relaxation time, v f denoted as Fermi rate, μ as electron mobility, and H(·) as Heaviside step function, i.e., unit step function.

[0089] S3. Scalar angular spectrum description of the incident vortex beam. In the incident beam coordinate system (x... i ,y i ,z i In ), z i The complex amplitude expression for the vortex beam on the initial plane with = 0 is:

[0090]

[0091] Where w0 is the waist radius of the vortex beam, l is the topological charge number, and sign(·) is the sign function.

[0092] Substitute equation (4) into equation (5) below and perform a Fourier transform;

[0093]

[0094] The scalar angular spectrum of the incident vortex beam is obtained. The expression is:

[0095]

[0096] Where, k ix and k iy Let k be the incident beam wave vector. i The horizontal component.

[0097] S4. Establishment of a full-vector theoretical model for reflection of vortex beams on graphene surfaces;

[0098] First, after the vortex beam is reflected off the graphene surface, its angular spectrum expression is determined as follows:

[0099]

[0100] in, and These represent the horizontal and vertical components of the angular spectrum of the reflected vortex beam, respectively, k rx and k ry Let k be the wave vector of the reflected beam. r The transverse component, α and β are the polarization parameters of the incident vortex beam. To determine the scalar angular spectrum of the reflected beam, a boundary condition k is applied to the scalar angular spectrum of the incident vortex beam. rx =-k ix and k ry =k iy The scalar angular spectrum of the reflected vortex was obtained. Right now:

[0101]

[0102] Furthermore, in the incident beam coordinate system (x r ,y r ,z r In this context, a vector potential A is introduced. r The expression is:

[0103]

[0104] Where, k r Let k be the wave vector of the reflected beam. r The modulus, and respectively along x r and y r The unit vector of the axis, and These are the horizontal and vertical components of the complex amplitude of the reflected vortex beam, respectively, obtained through the inverse Fourier transform of the following equation:

[0105]

[0106] Substituting equation (7) into equation (10) and integrating, we get...

[0107]

[0108]

[0109] in

[0110] I r1 =u r (13)

[0111]

[0112]

[0113]

[0114] In the formula, The Rayleigh distance is the distance at which the reflected vortex beam is measured.

[0115] Under the Lorentz gauge conditions, the electric field E of the reflected vortex beam r and magnetic field H r Using vector potential A r Represented as:

[0116]

[0117]

[0118] Among them, Z r The wave impedance for reflecting a vortex beam. and respectively along x r y r and z r The unit vector of the axis, and The specific expression is

[0119]

[0120]

[0121]

[0122]

[0123] In the formula

[0124]

[0125]

[0126]

[0127]

[0128]

[0129]

[0130] S5. Calculation of the chiral density of the vortex beam reflected from the graphene surface. The electric field E of the vortex beam after reflection from the graphene surface, derived in step S4, is calculated. r and magnetic field H r Substituting these parameters into the expression for optical chirality density allows us to analyze the effect of graphene parameters on the chirality of vortex beams. The optical chirality density C is defined as:

[0131]

[0132] In the formula, ω is the angular frequency, c is the speed of light in a vacuum, Im[·] represents the imaginary part, and "*" represents the complex conjugate.

[0133] The key point of this invention is the establishment of a full-vector theoretical model for calculating the chiral density of a reflected vortex beam on a graphene surface. It cleverly combines the angular spectrum expansion method and the vector potential method to numerically calculate the influence of graphene parameters on the chirality of the vortex beam. The method first uses the angular spectrum expansion method to establish the relationship between the angular spectrum of the incident vortex beam and the angular spectrum of the reflected vortex beam on the graphene surface. Then, it performs a Fourier transform and uses the vector potential method to derive the explicit expression of the electromagnetic field components under the condition of reflection of the vortex beam on the graphene surface, which can facilitate the calculation of the chiral density of the reflected vortex beam.

[0134] Example:

[0135] Using the method of the present invention, the parameters are set as follows in the embodiments: wavelength λ of the vortex beam = 1550 nm, beam waist radius w0 = 2.0λ, refractive index of the substrate n1 = 1.428, and Fermi rate v f =10 6 m / s, Fermi energy E f =0.3eV, electron mobility μ =0.5m 2 V / s.

[0136] exist Figure 2 and Figure 3 In the simulation, we simulated the incident angle θ of the vortex beam. i Fresnel reflectance r of graphene-substrate system s r p The effects and Fermi energy E f The effect on the optical conductivity σ of graphene. From Figure 2 As can be seen from rs With the incident angle θ i The increase is due to the increase of r p With the incident angle θ i The increase of r first decreases and then increases. p The minimum value occurs at the incident angle θ i = 57.5°, which is the Brewster angle of graphene. The Fresnel reflectance coefficient is also closely related to the optical conductivity σ of graphene, and the conductivity of graphene is highly dependent on the Fermi energy. Figure 3 As shown, when E f At <0.4 eV, the real part of the optical conductivity σ of graphene is Re(σ) = 0.6 × 10⁻⁶. -14 S / m. When E f At a voltage greater than 0.4 eV, the real part of the optical conductivity σ of graphene is Re(σ) = 0. Simultaneously, with the increase in Fermi energy E... f As the α-coefficient of graphene increases, the imaginary part of its optical conductivity σ first decreases, reaches its minimum, and then increases; that is, the real and imaginary parts increase in a step at the critical point (E0). f It has a minimum value at (=0.4eV).

[0137] Depend on Figure 4 It can be seen that for a fundamental Gaussian beam (l = 0), the chiral density of the reflected beam is zero and does not change with the increase of the incident angle. When l ≠ 0, the peak value of the chiral density of the reflected beam changes with the incident angle θ. i The chiral density increases with the increase of the topological charge number l, and changing the sign of the topological charge number l will reverse the peak value of the chiral density, but the value of the peak value will not change. Figure 5 In the figure, (a) shows the chiral density distribution of the reflected beam in the embodiment when the topological charge number l = -1; (b) shows the chiral density distribution of the reflected beam in the embodiment when the topological charge number l = -2; (c) shows the chiral density distribution of the reflected beam in the embodiment when the topological charge number l = 1; and (d) shows the chiral density distribution of the reflected beam in the embodiment when the topological charge number l = 2. Figure 5 As shown, the incident angle is set to θ. i =45°, we can again observe that the chiral density reverses as the sign of the topological charge l changes.

[0138] from Figure 6 It can be seen that the chiral density of the reflected vortex beam gradually increases with the increase of the incident angle. When the Fermi energy E f At 0.4 eV, the phenomenon is obvious and the Fermi energy is E f =0.3eV and E f The situation is different when the incident angle θ is 0.5 eV. i At <31°, the Fermi energy E f=0.4eV corresponds to the minimum peak value of chiral density; when the incident angle θ i At >43°, the Fermi energy E f The peak chiral density is highest at 0.4 eV. Figure 7 In this context, the Fermi energy is set to E. f =0.3eV, which shows that the peak chiral density of the reflected vortex beam increases with the increase of the refractive index.

[0139] exist Figure 8 In the case, the incident angle θ i Set to θ i =35°; in Figure 9 In the case, the incident angle θ i Set to θ i = 57.5°. When the angle of incidence θ i At 35°, the peak chiral density increases with the Fermi energy E f The increase first decreases and then increases, in E f It reaches its minimum value when the incident angle θ is 0.4 eV; i At 57.5°, the peak chiral density first increases and then decreases with increasing Fermi energy, at E f It reaches its maximum value at 0.4 eV.

[0140] The above examples demonstrate that the Fresnel reflection coefficient of the graphene-substrate system is closely related to the incident angle, and the graphene conductivity is highly dependent on the Fermi energy. By changing the Fermi energy and refractive index of the graphene-substrate system, as well as the incident angle and topological charge of the vortex beam, the chirality of the vortex beam can be flexibly controlled. The method of this invention for effectively controlling the chirality of vortex beams has significant application value in fields such as chiral molecule recognition, chiral structure detection, and chiral optical micromanipulation.

Claims

1. A method for controlling the chirality of vortex beams based on graphene, characterized in that, The specific steps are as follows: S1. Establishment of a schematic diagram of vortex beam reflection on the surface of the graphene-substrate system; In step S1, the specific method for establishing the schematic diagram of vortex beam reflection on the surface of the graphene-substrate system is as follows: Assume the refractive index of air is... The refractive index of the substrate is The graphene-substrate interface is located in the global coordinate system. In the global coordinate system In The axis is perpendicular to the graphene-substrate interface and points towards the substrate; the monolayer graphene located on top of the substrate... Location; and These represent the coordinate system of the incident beam and the coordinate system of the reflected beam, respectively. and These represent the angle of incidence and the angle of reflection of the center wave vector, respectively. S2, Description of the Fresnel reflectance coefficient of the graphene-substrate system; In step S2, the Fresnel reflectance of the graphene-substrate system is expressed as: (1) (2) in, Here is the Fresnel reflection coefficient for the horizontal polarization case. Here is the Fresnel reflection coefficient for vertical polarization. The refractive index of the substrate, The refractive index of air, It is the dielectric constant in a vacuum. Angular frequency, , , Let be the wavenumber of the light beam in air. It is the angle of incidence of the light beam. A beam of light in a vacuum. Let be the optical conductivity of graphene, expressed as follows: (3) in, represents an imaginary number, The amount of electron charge. The Fermi energy of graphene. To reduce Planck's constant, Angular frequency, For relaxation time, For Fermi rate, It is electron mobility. This is the Heaviside step function, i.e., the unit step function; S3. Scalar angular spectrum description of the incident vortex beam; The specific implementation of step S3 is as follows: in the incident beam coordinate system middle, The complex amplitude expression of the vortex beam on the initial plane is: (4) in, Let be the waist radius of the vortex beam. For topological load number, It is a symbolic function; Substitute equation (4) into equation (5) below and perform a Fourier transform; (5) The scalar angular spectrum of the incident vortex beam is obtained. The expression is: (6) in, and The incident beam wave vector The horizontal component; S4. Establishment of a full-vector theoretical model for reflection of vortex beams on graphene surfaces; The specific implementation method of step S4 is as follows: First, after the vortex beam is reflected off the graphene surface, its angular spectrum expression is determined as follows: (7) in, and These are the horizontal and vertical components of the angular spectrum of the reflected vortex beam, respectively. and The wave vector of the reflected beam The horizontal component, and The polarization parameters of the incident vortex beam are denoted as . To obtain the scalar angular spectrum of the reflected beam, boundary conditions are applied to the scalar angular spectrum of the incident vortex beam. and The scalar angular spectrum of the reflected vortex was obtained. ,Right now (8) Then, in the incident beam coordinate system In this context, a vector potential is introduced. The expression is: (9) in, The wave vector of the reflected beam The modulus, and respectively along and The unit vector of the axis, and These are the horizontal and vertical components of the complex amplitude of the reflected vortex beam, respectively, obtained through the inverse Fourier transform of the following equation: (10) Substituting equation (7) into equation (10) and integrating, we get... (11) (12) in (13) (14) (15) (16) In the formula, The Rayleigh distance for the reflected vortex beam; Finally, under the Lorentz gauge condition, the electric field of the reflected vortex beam and magnetic field Using vector potential Represented as: (17) (18) in, The wave impedance for reflecting a vortex beam. , and respectively along , and The unit vector of the axis, , , and The specific expression is: (19) (20) (21) (22) In the formula (23) (24) (25) (26) (27) (28)； S5. Calculation of the chiral density of reflected vortex beams on graphene surfaces; The specific implementation method of step S5 is as follows: The electric field after the vortex beam derived in step S4 is reflected on the graphene surface and magnetic field Substituting into the expression for optical chirality density, where optical chirality density... Defined as: (29) In the formula, Angular frequency, The speed of light in a vacuum. "*" indicates the imaginary part, and "*" indicates complex conjugation.

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