# Reflection Characteristics of Airy Beams Impinging on Graphene-Substrate Surfaces

Authors: Fuping Wu; Yuanfei Hui; Zhiwei Cui; Ju Wang
Journal: International Journal of Optics
 (2022)

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## Abstract

In this work, we analytically and numerically investigate the reflection characteristics of the airy beams impinging on graphene-substrate surfaces. The explicit analytical expressions for the electric and magnetic field components of the airy beams reflected from a graphene-substrate interface are derived. The local-field amplitude, Poynting vector, and spin and orbital angular momentum of the reflected airy beams with different graphene structure and beam parameters are presented and discussed. The results show that the reflection properties of the airy beams can be flexibly tuned by modulating the Fermi energy of the graphene and have a strong dependence on the incident angle and polarization state. These results may have potential applications in the modulation of airy beams and precise measurement of graphene structure parameters.


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## Body

## 1. Introduction

Airy beams, which were predicted theoretically in 1979 by Berry and Balazs [1] and first observed experimentally in 2007 by Siviloglou et al. [2, 3], are the solutions of scalar wave equation under paraxial approximation or its quantum-mechanics analog, i.e., the Schrödinger equation. It has been shown that an airy beam tends to accelerate in transverse space during propagation and maintains its shape unchanged over long distances [4, 5]. Moreover, the airy beam has the ability to self-reconstruct or self-heal during propagation [6]. Such unique properties of the airy beam have led to many applications in a variety of domains, such as optical micromanipulation, optical communications, light-sheet microscopy, plasma channel generation, and light bullets [7–15]. Since most of these applications involve the airy beams pass through an interface between two media, a deep understanding of the properties of airy beams upon reflection has important significance. The reflection properties of airy beams have been usually investigated by the conventional air-glass interface [16]. In recent years, various material media, such as atomic medium [17], nonlinear medium [18], epsilon-near-zero metamaterial [19], weakly absorbing medium [20], and graphene metamaterial [21], have been introduced into the reflection of airy beams to understand more of their properties. Among these material media, it is worth mentioning that graphene can be used to flexibly tune the reflection properties of the incident beams.It is known that graphene, a one-atom-thick layer of carbon atoms arranged in a hexagonal honeycomb lattice [22], has attracted wide attention due to its striking electronic and optical properties [23–27]. One of the unique optical properties of the graphene is that its reflectance is determined by the fine structure constant and intrinsic parameters [28, 29], which can be tuned by altering the Fermi level through electrostatic doping without changing the device structure [30–32]. Such interesting feature makes graphene a good candidate for tuning the reflection properties of airy beams. Compared to the case of conventional air-glass interface, graphene-substrate surface may reveal many other important and interesting features of the airy beams upon reflection, which will pave the way to many unprecedented applications, such as nanoparticles trapping or manipulation and design of new integrated devices. Recently, the reflection of airy beams impinging on graphene-based structures has been demonstrated [21, 33–35]. Yang et al. first studied the deflection and propagation distance of airy beams reflected by a single graphene sheet with different external biased voltage [33]. Guan et al. further reported the generation of achromatic airy plasmons on graphene surfaces in the infrared range [34]. Later, Imran et al. investigated the propagation lengths and transverse displacements of airy beams in bilayer graphene with tunable surface conductivity [35]. Subsequently, Zhen and Deng analyzed the spatial and angular Goos–Hänchen (GH) shifts for the airy beams impinging onto graphene-substrate surfaces [21]. Despite all these works, few attention has been paid to the local reflection characteristics, including the field distribution, Poynting vector, spin angular momentum (SAM), and orbital angular momentum (OAM), of the airy beams impinging on graphene-substrate surfaces. These reflection characteristics are helpful not only to understand the modulation of airy beams but also to examine the optical properties of graphene. Compared to the case of conventional fundamental Gaussian beams, airy beams may reveal more interesting properties of the graphene. It is known that the fundamental Gaussian beams are typical cylindrical symmetric beams, while the airy beams possess an asymmetrical field distribution. The existing study has shown that the symmetry of the field distribution of airy beams have huge influences on GH shifts [36] which involve the reflection of airy beams at the interface of two different media. Airy beams can also raise the magnitude of GH shifts compared to the case of fundamental Gaussian beams [37]. In addition, the airy beams exhibit some intriguing propagation properties as mentioned above, i.e., self-accelerating, self-healing, and nondiffracting. Predictably, the asymmetrical field distribution and unique properties of Airy beams are of great importance in understanding the optical properties of grapheme. In this work, we analytically and numerically investigate the reflection characteristics of the airy beams impinging on graphene-substrate surfaces.This work is arranged as follows. In Section2, we present explicit analytical expressions for the electromagnetic field components of Airy beams reflected at a graphene-substrate surface. The numerical simulation results are performed and discussed in Section 3, and Section 4 is a summary of this paper.


## 2. Theoretical Formulae

To determine the reflected fields, we start with a short derivation of the angular spectrum amplitude of the airy beams. Considering the finite-power airy beams that propagate along the direction of+z axis, the complex scalar function of the beams in the source plane z=0 is given as [3, 6](1)ux,y,z=0=Aixx0Aiyy0expa0xx0expa0yy0,in which x0 and y0 are the arbitrary transverse scales in the x and y directions, respectively, a0 represents exponential factor which determines the beam propagation distance, and Ai⋅ denotes the airy function [38]. Using the 2D Fourier transform of equation (1), the angular spectrum amplitude of the airy beams in the source plane can be expressed as [39](2)u˜kx,ky=x0y04π2exp−a0kx2x02+ky2y02+23a03×expi3kx3x03+ky3y03−3a02kxx0+kyy0.Now, we consider the reflection of an airy beam illuminating from air onto the surface of a graphene-substrate system, as illustrated in Figure1. For simplicity, the air is characterized by the refractive index n0, and the substrate is characterized by n1. The global coordinate system x,y,z is established at the graphene-substrate surface, where the z-axis of the laboratory frame is normal to the interface pointing to the graphene-substrate system. The monolayer graphene layer is located on the top of the substrate lying at z=0. The incident and reflected beam coordinates are represented by the local coordinate systems xi,yi,zi and xr,yr,zr, respectively. The angles θi and θr indicate the incident angle and reflection angle of the central-wave component, respectively.Figure 1
Illustration of the reflection of an airy beam illuminating from air onto the surface of a graphene-substrate system.As well known, the Fresnel reflection coefficients are the crucial parameters to determine the reflected fields. Several methods, such as the transfer matrix method [40] and the classical macroscopic method [41], have been developed to calculate the Fresnel reflection coefficients in a structure containing graphene. For the graphene-substrate system considered in this paper, we employ the transfer matrix method to obtain the Fresnel reflection coefficients. By utilizing the boundary conditions at z=0 and Ohm’s law J=σE, the transfer matrix of graphene surface for the light beams with parallel polarization state can be represented as [42](3)Tp0⟶1=121+ηp+ξp1−ηp−ξp1−ηp+ξp1+ηp−ξp,in which ηp=n02k1z/n12k0z and ξp=σk1z/ε0n12ω. Similarly, the transfer matrix of graphene surface for the beams with perpendicular polarization state is given in [42](4)Ts0⟶1=121+ηs+ξs1−ηs+ξs1−ηs−ξs1+ηs−ξs,where ηs=k1z/k0z, ξs=σμ0ω/k0z, ε0 and μ0 are the permittivity and permeability of free space, respectively, ω=2πc/λ0 is the angular frequency with c and λ0 being the speed and wavelength of the incident beams in free space, and k0z=k0cosθi and k1z=k0n12−n02sin2θi with k0=2π/λ0 being the wave vector of the beams in free space and θi being the incident angle. The electromagnetic properties of graphene monolayer are characterized by the optical conductivity σ, which can be determined by the semiconductor theory as [43, 44](5)σω,Ef=e2Efπℏ2iω+i/τ+e24ℏHℏω−2Ef+iπlnℏω−2Efℏω+2Ef,where the elementary charge e=1.6×10−19C and the reduced Plank’s constant ℏ=1.05×10−34J⋅s. Also, H⋅ indicates the Heaviside step function, Ef is the graphene Fermi energy, which can be modulated by external gate voltage via electric doping, τ=μEf/evf2 denotes the electron-phonon relaxation time with μ being the mobility, and vf is the Fermi velocity. In addition, the propagation matrix of this substrate can be expressed as(6)P=1001.Then, for the graphene-substrate system, the transfer matrix of the system is given by(7)M=Tp,s0⟶1P=Tp,s0⟶1.With the help of the transfer matrix, the Fresnel reflection coefficients for the beams with parallel and perpendicular polarization states can be written as [40](8)rp=M21M11=n12/k1z−n02/k0z+σ/ε0ωn12/k1z+n02/k0z+σ/ε0ω,rs=M21M11=k0z−k1z−k0σ/ε0ck0z+k1z+k0σ/ε0c.To reveal the local reflection characteristics of the airy beams impinging on graphene-substrate surfaces, it is necessary to carry out a full vector wave analysis of airy beams upon reflection. Following the work of reference [39], the vector potential of the reflected beams can be expressed as(9)Ar=urHxr,yr,zrx^r+urVxr,yr,zry^rexpikrzr,where x^r and y^r are the unit vector in the positive xr and yr axis directions, respectively, kr=k0, and urHxr,yr,zr and urVxr,yr,zr are the complex amplitude of the reflected airy beams with horizontal and vertical polarizations, which can be evaluated via the inverse Fourier transformations of the corresponding angular spectrums. In fact, the reflected angular spectrums u˜rH,u˜rV related to the angular spectrums u˜iH,u˜iV of the incident beam can be written as(10)u˜rHu˜rV=rp1−krxk0∂lnrp∂θikryrp+rscotθik0−kryrp+rscotθik0rs1−krxk0∂lnrs∂θiu˜iHu˜iV,in which u˜iHV=αβu˜r denotes the angular spectrum of the incident airy beam with horizontal (vertical) polarization, where α,β=1,±i/2 corresponds to the left-handed circular polarization (L-CP) and right-handed circular polarization (R-CP) states, respectively. By applying the reflection boundary conditions krx=−kix and kry=kiy, we can express the angular spectrum of reflected airy beams by(11)u˜rkrx,kry=u˜−kx,ky=x0y04π2exp−a0krx2x02+kry2y02+23a03×expi3−krx3x03+kry3y03−3a02−krxx0+kryy0.Combining equations (10)–(12), after some algebra, and simplifying, we obtain(12)urHxr,yr,zrurVxr,yr,zr=ArHxr,yr,zrArVxr,yr,zrur,where(13)ArH=αrp−αrp1k0∂lnrp∂θii2krx02a0+izr2krx03+ix0Ai′TrxAiTrx−βrp+rscotθik0i2kry02a0+izr2kry03+iy0Ai′TryAiTry,ArH=αrp−αrp1k0∂lnrp∂θii2krx02a0+izr2krx03+ix0Ai′TrxAiTrx−βrp+rscotθik0i2kry02a0+izr2kry03+iy0Ai′TryAiTry,ur=AiTrxAiTryexpMrxexpMry,in which(14)Trx=−xrx0−zr24kr2x04+ia0zrkrx02,Try=yry0−zr24kr2y04+ia0zrkry02,Mrx=−a0xrx0−a0zr22kr2x04−izr312kr3x06+ia02zr2krx02−ixrzr2krx03,Mry=a0yry0−a0zr22kr2y04−izr312kr3y06+ia02zr2kry02+iyrzr2kry03.Within the paraxial approximation, the electric and magnetic field vectors of the reflected airy beams can then be expressed in terms of the vector potentialAr as(15)Er=ikrZrurHx^r+urVy^r+ikr∂urH∂xr+∂urV∂yrz^rexpikrzr,Hr=ikr−urVx^r+urHy^r−ikr∂urV∂xr−∂urH∂yrz^rexpikrzr,where Zr=μ0/ε0 denotes the wave impedance and the explicit expressions of ∂urH/∂xr, ∂urV/∂xr, ∂urH/∂yr, and ∂urV/∂yr are given in [39].Having written explicitly the analytical expressions of the electric and magnetic fields of the reflected airy beams, we can study the local-field amplitude, Poynting vector, SAM, and OAM of the reflected airy beams with different graphene structure and beam parameters. The Poynting vector, namely, the energy flux density, can be written in the form:(16)S=12ReEr∗×Hr.Base on the Poynting vector, the spin and orbital currents can be defined to describe the SAM and OAM in terms of the electric and magnetic field vectors [45]. However, such an established formalism produces fundamental difficulties when applied to structured optical fields [46]. Here, we adopt a canonical approach to describe the SAM and OAM densities of the airy beams during the reflection process. With such an approach, the SAM density is proportional to the local ellipticity of the field polarization, and the OAM density is defined via the canonical momentum density, which corresponds to the local gradient of the phase of the field [46]. Specifically, the SAM and OAM densities can be expressed as(17)S=14ωImε0Er∗×Er+μ0Hr∗×Hr,L=r×P,in which(18)P=14ωImε0Er∗⋅∇Er+μ0Hr∗⋅∇Hr,where ω is the angular frequency, Im⋅ denotes the imaginary parts, the superscript “∗” denotes the complex conjugate, and the notation A⋅∇B=Ax∇Bx+Ay∇By+Az∇Bz is used.


## 3. Numerical Results and Discussion

In this section, we performed some numerical calculations to explore the reflection properties of airy beams impinging on graphene-substrate surfaces. Unless otherwise mentioned, the adopted parameters are the wavelength of the beamλ0=1550nm, the scaled parameters x0=y0=1.5λ0, the decay parameter a0=0.1, the incident angle θi=30°, the refractive index of the substrate n1=1.517, the mobility μ=1m2V/s, the Fermi velocity vf=106m/s, and the Fermi energy of grapheme Ef=0.5eV. In addition, we choose the plane zr=λ0 as the observation plane, and the transverse coordinates are set as x=1.5λ0 and y=−1.5λ0.As mentioned before, the graphene exhibits Fermi level modulated electromagnetic characteristic. To investigate such a characteristic, the dependence of the electric field magnitude on the incident angleθi for the reflected airy beams with L-CP and R-CP states at different Fermi energy Ef is demonstrated in Figure 2. From this figure, one can see that a magnitude peak of airy beams with L-CP appears initially, and then a valley is emerged near Brewster angle θB, while the variation trend is opposite for the beams with R-CP. Moreover, the pair of maximum magnitude peak and minimum valley for all cases moves to a smaller incident angle θi when the Fermi energy Ef switches from Ef<0.4eV to Ef>0.4eV. This is due to the fact that the Brewster angle θB is 57.18° at Ef<0.4eV and decreases to 56.61° at Ef>0.4eV. It is also found that, for a bigger value of the Fermi energy EfEf<0.4eV, a larger enhancement of the magnitude for reflected Airy beams with circular polarization is obtained.Figure 2
Variations of the electric field magnitude for reflected airy beams versus incident anglesθi with different fermi energy Ef. (a) L-CP and (b) R-CP.
(a)(b)Figure3 describes the magnitude of electric fields as a function of the Fermi energy Ef for reflected airy beams with L-CP and R-CP. As we can see, the magnitude of electric fields reaches its minimum valleys and maximum peaks near the Fermi energy Ef=0.4eV. The reason is based on the fact that the real part of graphene conductivity σ is 0.6×10−4s/m at Ef<0.4eV and becomes 0 at Ef>0.4eV, thus leading to the Brewster angle θB changes from 57.18° to 56.61°. Besides, the electric field magnitude of airy beams with R-CP is smaller than that of the beams with L-CP.Figure 3
Dependence of the electric field magnitude on the Fermi energyEf for reflected airy beams with the state of circular polarization: (a) L-CP and (b) R-CP.
(a)(b)Next, we explore the effect of beam wavelength and refractive index of substrate on electric magnitude distribution of reflected airy beams. Figure4 gives the electric field magnitude of airy beams with L-CP changing with incident angle θi for different beam wavelengths λ0 and refractive index of substrate n1. For each situation, there are a peak and a valley in the pattern of electric field magnitude. Specifically, the magnitude will increase suddenly and reach peak value. After that, it decreases sharply until reaching a valley. Then, it will saturate to different levels depending on the beam wavelength λ0 and substrate refractive index n1. As seen from Figure 4(a), the pair of peak and valley for beam wavelengths λ0≤850nm (488nm, 633nm, and 850nm) and λ0=1550nm are located around θi=57.18° and 56.61°, respectively. This phenomenon can be explained as follows: the real part of the graphene conductivity is 0.6×10−4s/m at beam wavelengths λ0=488nm, 633nm, and 850nm and decreases to 0 at λ0=1550nm, thus leading to the Brewster angle θB decreasing to 56.61°. It also can be seen that the incident angle of a pair of peak and valley of the magnitude decreases with the decrease of n1, as shown in Figure 4(b). Moreover, the magnitudes of electric field become larger with the increase of the substrate refractive n1 or the decrease of the beam wavelength λ0.Figure 4
Magnitude variations of electric field as a function of incident anglesθi for reflected airy beams with (a) different beam wavelengths λ0 and (b) different substrate refractive index n1.
(a)(b)We now turn our attention to the analysis of the influence of polarization states on the Poynting vector of the reflected airy beams. The corresponding results are depicted in Figure5, where the transverse Poynting vector is represented by white arrows and the components of the magnitude of electric fields are presented by means of color shading. As we can see, the change of states of circular polarization has almost no effect on the profile of electric field magnitude in the xr−yr plane, but leads to a significant effect on the orientations of transverse Poynting vector. Meanwhile, we also note that the orientations of the main lobe of the transverse Poynting vector for the reflected airy beams with L-CP and R-CP are opposite when the incident angle θi<56.61°, and both of them are reversed when θi>56.61°. Specifically, when the incident angle θi<56.61°θi=30°, the main lobe of the Poynting vector exhibits a clockwise spiral for reflected airy beams with L-CP as depicted in Figure 5 (a1), while it exhibits an anticlockwise spiral for the beams with R-CP as plotted in Figure 5 (b1). Furthermore, the orientations of the main lobe of the transverse Poynting vector for the airy beams with L-CP (R-CP) with incident angle θi=30°θi<56.61° will present the same fashion as that for the beams with R-CP (L-CP) at incident angle θi=75°θi>56.61°. This implies that the change of the orientations of transverse Poynting vector occurs at the incident angle θi=56.61° since the GH shifts with graphene at Ef>0.4eVEf=0.5eV are maximal at the Brewster angle θB=56.61°.Figure 5
Distributions of electric field magnitude (color shading) and Poynting vector (white arrows) of reflected airy beams with circular polarization states: (a1–a3) L-CP and (b1–b3) R-CP. An enlarged view onto the main lobe of reflected airy beams is shown as insets in the top left corners. The first column to the last column corresponds to the incident anglesθi=30°, 56.61°, and 75°.To gain further insight into the reflection characteristics of airy beams, we analyze the SAM and OAM of the airy beams after reflection. Figure6 shows the transverse and longitudinal SAM density distributions of the reflected airy beams with different circular polarization states and incident angles at the graphene-substrate interface. As shown in Figure 6 (a1), the transverse SAM density increases with the incident angle θi increasing, and the transverse SAM density of L-CP airy beams is smaller than that of the R-CP beams. When the incident angle θi=50°, the main lobe of the transverse SAM density exhibits crescent pattern and the change of the state of circular polarization (from L-CP to R-CP, or vice versa) only leads to a slight change of the main lobe of the reflected beams, as shown schematically in Figures 6(b1) and 6(c1). Upon further observation, with the increase of the incident angle θi, we find that the longitudinal SAM density of the R-CP beam decreases, while that of the L-CP beam increases gradually. Notably, the longitudinal SAM densities with L-CP and R-CP have equal magnitude but opposite signs when the incident angle θi≠56.61° (Brewster angle). In addition, comparing the longitudinal SAM density components in Figures 6(b2) and 6(c2), we see that the change of the state of circular polarization induces the reversal of the longitudinal SAM density.Figure 6
Transverse and longitudinal SAM density distributions of the circular-polarization Airy beams during the reflection process. (a1) and (a2) are the transverse and longitudinal SAM density distributions of the reflected beam changing with incident angleθi for beams with L-CP and R-CP. (b1), (c1) and (b2), (c2) are the transverse and longitudinal SAM density distributions in the xr−yr plane for incident angles θi=50°, respectively. (b1, b2) L-CP and (c1, c2 R-CP. Here, the transverse and longitudinal components of the SAM density in the plane zr=λ0 are plotted in normalized units with respect to the maximum value.Figure7 shows the distributions of the OAM density for reflected airy beams with L-CP and R-CP states. From Figure 7(a1), it is observed that the OAM density of the reflected airy beams with L-CP initially exhibits a valley and then emerges a peak when the Fermi energy Ef approaches the value of 0.4eV, namely, the total OAM density will decrease suddenly and reach a valley value. After that, it increases until reaching a maximum value. Then, it will decrease suddenly. Opposite variation trend occurs for the airy beams with R-CP, as illustrated in Figure 7(b). It can be found from Figures 7(a1) and 7(b1) that the OAM density undergoes a slight increase when the incident angle θi is increased, but the OAM density near Ef=0.4eV shows different properties. It is also interesting to find that the change of the sign of circular polarization state has almost no effect on the transverse OAM density, but induces a reversal of the longitudinal OAM density, as shown in Figures 7(b1), 7(b2), 7(c1), and 7(c2).Figure 7
Distributions of the OAM density for reflected airy beams with L-CP and R-CP states. (a1) and (a2) show variations of the OAM density as a function of the Fermi energy for the L-CP and R-CP beams with different incident anglesθi. (b1), (b2) and (c1), (c2) show the normalized transverse and longitudinal OAM density distributions in the xr−yr plane when the incident angle θi=30°, respectively. (a1–c1) L-CP and (a2–c2) R-CP.


## 4. Conclusions

In this paper, a theoretical study on the reflection characteristics of the airy beams impinging on graphene-substrate interfaces is investigated. The explicit expressions for the electric and magnetic fields of the airy beams reflected at a graphene-substrate interface are derived using a hybrid method based on angular spectrum representation and vector potential. The influences of the graphene-substrate structure and incident beam parameters on the pattern distributions of the local field amplitude, Poynting vector, SAM, and OAM of the reflected airy beams are examined in detail. The numerical results show that the position of the Brewster angle is closely related to the optical conductivity of the graphene. The incident wavelength and Fermi energy can change the optical conductivity of the graphene, which controls the angular momentum inversion near the Brewster angle in the specific tuning region. In addition, the profiles of electric amplitude, Poynting vector, SAM, and OAM densities strongly depend on the incident angle and polarization states of the airy beams. Specifically, the orientation of transverse Poynting vector reverses at the Brewster angleθi=56.61°Ef=0.5eV. Actually, the orientation of the energy flow of the reflected airy beams with L-CP exhibits a clockwise rotation at incident angle θi=30° and becomes an anticlockwise rotation at θi=75°. Furthermore, the change of the sign of circular polarization state induces the reversal of the longitudinal SAM and OAM densities when the incident angle θi=30°. These findings may provide useful insights into the development of precision metrology and manipulation of the structured beams.



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