#### 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.

#### 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 Section 2, 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 axis, the complex scalar function of the beams in the source plane is given as [3, 6]in which and are the arbitrary transverse scales in the and directions, respectively, represents exponential factor which determines the beam propagation distance, and 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]

Now, we consider the reflection of an airy beam illuminating from air onto the surface of a graphene-substrate system, as illustrated in Figure 1. For simplicity, the air is characterized by the refractive index , and the substrate is characterized by . The global coordinate system is established at the graphene-substrate surface, where the -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 . The incident and reflected beam coordinates are represented by the local coordinate systems and , respectively. The angles and indicate the incident angle and reflection angle of the central-wave component, respectively.

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 and Ohm’s law , the transfer matrix of graphene surface for the light beams with parallel polarization state can be represented as [42]in which and . Similarly, the transfer matrix of graphene surface for the beams with perpendicular polarization state is given in [42]where , , and are the permittivity and permeability of free space, respectively, is the angular frequency with and being the speed and wavelength of the incident beams in free space, and and with being the wave vector of the beams in free space and being the incident angle. The electromagnetic properties of graphene monolayer are characterized by the optical conductivity , which can be determined by the semiconductor theory as [43, 44]where the elementary charge and the reduced Plank’s constant . Also, indicates the Heaviside step function, is the graphene Fermi energy, which can be modulated by external gate voltage via electric doping, denotes the electron-phonon relaxation time with being the mobility, and is the Fermi velocity. In addition, the propagation matrix of this substrate can be expressed as

Then, for the graphene-substrate system, the transfer matrix of the system is given by

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]

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 aswhere and are the unit vector in the positive and axis directions, respectively, , and and 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 related to the angular spectrums of the incident beam can be written asin which denotes the angular spectrum of the incident airy beam with horizontal (vertical) polarization, where corresponds to the left-handed circular polarization (L-CP) and right-handed circular polarization (R-CP) states, respectively. By applying the reflection boundary conditions and , we can express the angular spectrum of reflected airy beams by

Combining equations (10)–(12), after some algebra, and simplifying, we obtainwherein which

Within the paraxial approximation, the electric and magnetic field vectors of the reflected airy beams can then be expressed in terms of the vector potential aswhere denotes the wave impedance and the explicit expressions of , , , and 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:

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 asin whichwhere is the angular frequency, denotes the imaginary parts, the superscript “” denotes the complex conjugate, and the notation 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 beam , the scaled parameters , the decay parameter , the incident angle , the refractive index of the substrate , the mobility , the Fermi velocity , and the Fermi energy of grapheme . In addition, we choose the plane as the observation plane, and the transverse coordinates are set as and .

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 angle for the reflected airy beams with L-CP and R-CP states at different Fermi energy 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 , 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 when the Fermi energy switches from to . This is due to the fact that the Brewster angle is at and decreases to at . It is also found that, for a bigger value of the Fermi energy , a larger enhancement of the magnitude for reflected Airy beams with circular polarization is obtained.

**(a)**

**(b)**

Figure 3 describes the magnitude of electric fields as a function of the Fermi energy 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 . The reason is based on the fact that the real part of graphene conductivity is at and becomes 0 at , thus leading to the Brewster angle changes from to . Besides, the electric field magnitude of airy beams with R-CP is smaller than that of the beams with L-CP.

**(a)**

**(b)**

Next, we explore the effect of beam wavelength and refractive index of substrate on electric magnitude distribution of reflected airy beams. Figure 4 gives the electric field magnitude of airy beams with L-CP changing with incident angle for different beam wavelengths and refractive index of substrate . 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 and substrate refractive index . As seen from Figure 4(a), the pair of peak and valley for beam wavelengths (, , and ) and are located around and , respectively. This phenomenon can be explained as follows: the real part of the graphene conductivity is at beam wavelengths , , and and decreases to 0 at , thus leading to the Brewster angle decreasing to . It also can be seen that the incident angle of a pair of peak and valley of the magnitude decreases with the decrease of , as shown in Figure 4(b). Moreover, the magnitudes of electric field become larger with the increase of the substrate refractive or the decrease of the beam wavelength .

**(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 Figure 5, 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 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 , and both of them are reversed when . Specifically, when the incident angle , 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 will present the same fashion as that for the beams with R-CP (L-CP) at incident angle . This implies that the change of the orientations of transverse Poynting vector occurs at the incident angle since the GH shifts with graphene at are maximal at the Brewster angle .

To gain further insight into the reflection characteristics of airy beams, we analyze the SAM and OAM of the airy beams after reflection. Figure 6 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 increasing, and the transverse SAM density of L-CP airy beams is smaller than that of the R-CP beams. When the incident angle , 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 , 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 (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 7 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 approaches the value of , 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 is increased, but the OAM density near 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).

#### 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 angle . Actually, the orientation of the energy flow of the reflected airy beams with L-CP exhibits a clockwise rotation at incident angle and becomes an anticlockwise rotation at . Furthermore, the change of the sign of circular polarization state induces the reversal of the longitudinal SAM and OAM densities when the incident angle . These findings may provide useful insights into the development of precision metrology and manipulation of the structured beams.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The work was supported by the Natural Science Foundation of Shaanxi Province (2020JM-210); Key Research and Development Projects of Shaanxi Province (2019GY-146); and Xi’an Municipal Science and Technology Project (201805029YD7CG13(4)).