Goos–Hänchen and Imbert–Fedorov shifts of the Airy beam in dirac metamaterials

We theoretically derive the expression for the Goos–Hänchen (GH) and Imbert–Fedorov (IF) shifts of the Airy beam in Dirac metamaterial. In this work, the large GH and IF shifts can be found when the Airy beam is reflected near the Dirac and Brewster angles. Compared to the Gaussian beam, the GH shifts of the Airy beam are more obvious in the vicinity of the Brewster angle. Interestingly, it is found that the ability to produce an Airy vortex beam at the Dirac point. In addition, the magnitude and the direction of the GH shifts can be controlled by the rotation angles of the Airy beam. We take advantage of this property to design a reflective optical switch based on the rotation angle-controlled GH shifts of the Dirac metamaterial. Our solutions provide the possibility to implement light-tuned optical switches. Moreover, our model can also be used to describe the GH and IF shifts generated by novel beams in other similar photonic systems.


Introduction
In geometric optics, light is reflected and refracted when it is incident from one homogeneous medium to another, and obeys the famous Snell's law. However, longitudinal and transverse shifts will be produced at directions parallel and perpendicular to the incident surface when the wave properties in the actual beam are not to be ignored. The longitudinal shift was first discovered experimentally by Goos and Hänchen in 1947 and was therefore known as the Goos-Hänchen (GH) shift [1]. In 1948, the theoretical explanation of the GH shift was provided by Artmann [2]. The transverse shift was first proposed theoretically by Fedorov in 1955 [3] and verified experimentally by Imbert in 1972 [4], and is therefore known as Imbert- Fedorov (IF) shift. Based on the definition of GH and IF shifts, they refer to the shifts of the reflected beam from the path normally expected by geometrical optics along the x-and y-axis respectively [5]. With the development of novel materials and structures, the GH and IF shifts can be enhanced and modulated effectively in various materials, such as graphene [6][7][8][9][10][11], weakly absorbing dielectric [12][13][14][15], photonic crystal [16][17][18], plasmonics [19,20] and nonoptical systems [21][22][23][24][25]. In addition, the GH and IF shifts have potential applications in optical sensors [26,27], image edge detection [28] and optical switches [29,30].
Lately, topological semimetals are categorized into Dirac semimetals, Weyl semimetals and node-line semimetals by their energy band crossings, which are fourfold degeneracy [31], double degeneracy [32] and the crossover points form loops [33], respectively. The Dirac semimetal is a central gapless topological phase that connects conventional insulators, topological insulators and Weyl semimetals. In photonics, the fourfold degeneracy of the Dirac point is generally obtained by various spatial symmetries in photonic crystals [34,35]. And the special topological photonic states can be achieved through the Dirac point. In addition, a strong photonic spin Hall effect and giant nonspecular effect appear near the Dirac point when a Gaussian beam impinges on the interface of a photonic Dirac metamaterial [36,37]. However, the Gaussian beams are axial symmetry beams. The symmetry of the field intensity distribution of the Airy beam is found to have an important role in the spin Hall effect of light by Li [38]. It is also known that the GH and IF shifts of the Airy beams and the polarized vortex beams have been explored [39,40]. However, the analytical theories of the GH and IF shifts of the Airy beams are poorly developed. We are therefore interested in the GH and IF shifts of the Airy beam in Dirac metamaterials.
In this paper, we construct a model to investigate the GH and IF shifts of the Airy beam in Dirac metamaterials. In our model, the GH and IF shifts of the Gaussian and Airy beams in Dirac metamaterials are contrasted and discussed. The Airy beam was found to have large shifts near both the Brewster and Dirac angles. In addition, the rotation angles of the Airy beam can be used to manipulate the GH shifts around the Brewster angle. Finally, by making use of this property, a reflective optical switch based on rotational angle control of GH shifts has been designed.

Shift equations in dirac metamaterials
As shown in figure 1, we assume that the Airy beam is incident in Dirac metamaterial from a vacuum that the angle of incidence is θ. We set up a Cartesian coordinate system (x, y, z) in the Dirac metamaterial, (x i , y i , z i ) and (x r , y r , z r ) represent the coordinate system of the incident beam and the reflected beam, respectively. Generally, the permittivity and permeability tensors of the medium are the anisotropic form ε = ε 0 ε xxx + ε yŷŷ + ε zẑẑ and µ = µ 0 µ xxx + µ yŷŷ + µ zẑẑ , respectively [28]. For the sake of analysis, we assume ε y = ε z = const, µ y = µ z = const, and where ω 0 is the resonance frequency and ω is the incident frequency. The adjustable constants f 1 and f 2 are determined by the structural parameters. Here, the parameters for the Dirac metamaterial are selected as ε y = µ y = 0.5, then we can give the critical angle of incidence as θ c = 30 • (= arcsin √ ε y µ y ). Additionally, we set the structural coefficients as From the already derived Dirac metamaterial reflection coefficients we can obtain [37]: where k ix and k iy denote the x and y components of the arbitrary vector with respect to the centre of the incident beam, respectively. r pp , r ss and r ps (r sp ) represent the parallel, vertical and crossing polarization of the reflection coefficients, respectively. When ε x = ε y = const and µ x = µ y = 1, the reflection coefficient in equations (2)-(4) can be equated to the isotropic case [41].
In this paper, we only consider the case of the GH shifts of the incident beams for horizontal polarization (HP). The IF shifts take into account only the left circular polarization (LCP) of the incident beam. The analytical expression for the GH shifts of the Gaussian beam in Dirac metamaterial as (see appendix for detailed derivation): where ς pp = ∂rpp ∂θ , R pp = r pp , R sp = r sp . k 0 = 2π/λ indicates the number of waves in the vacuum, and the incident wavelength is λ = 2πc/ω. The analytical expression for the IF shifts of the Gaussian can be written as (see appendix for detailed derivation): where ϑ = 1/(k 0 w 0 ), f p = a p and f s = a s e iη , with FP = F p , FS = |F s |. The analytical expression for the GH shifts (HP) of the Airy beam in Dirac metamaterial can be expressed as (see appendix for detailed derivation): where in which α and β are decay factors. And the analytical expression for the IF shifts (LCP) of the Airy beam can be obtained as (see appendix for detailed deduction):

The GH shift and IF shift
In the following discussion, the beam waist of the beams is set as w 0 = 100λ and the resonance frequency ω 0 = 10 13 Hz. The position of the Dirac point can be determined from the eigenvalues of the Hamiltonian form in the proposed metacrystals [42], k D = k 0 √ ε y µ y , ω D = 1 + f 1 ω 0 . In this paper, we choose (k D , ω D ) to refer to the Dirac point, with ω D being the Dirac frequency. We can obtain the Brewster angles for parallel polarization by First, we discuss the GH shifts of the Airy beam. Figure 2(a) shows the GH shifts of the Airy beam and the Gaussian beam at ω = 1.001ω D . The solid line is the numerical solution of the Airy beams, the dotted line is the analytic solution of the Airy beams, and the dashed line is the analytic solution of the Gaussian beams. As we can see, the numerical simulation and the analytical solution of the GH shifts of the Airy beam in the figure coincide perfectly, which proves that the calculations are accurate. More interestingly, the Airy beam has larger GH shifts and is found to have a special phenomenon near the Brewster angle (θ B = 29.852 • ) compared to the Gaussian beam. The phenomenon can be analyzed theoretically as shown in figure 2(b). It can be visualized that the Airy beam has a significant GH shift at Brewster angle while the Gaussian beam is zero, which corresponds to the analytical expression equation (10) where GH (G) = 0 and GH (Ai) = χ. Hence, the parameter χ determines the values of GH shifts of the Airy beam in this case. Furthermore, both parameters Λ and χ in equation (10) have an effect on the GH shifts of the Airy beam in the vicinity of the Dirac angle which can be seen from figures 2(c) and (d). It can be obtained that the values of the GH shifts are increased by the parameters Λ and χ in the equation (10), and the parameter Λ has more obvious effect on the GH shifts. Simultaneously, the values of the GH shifts near the Dirac angle increase with the incident frequency approaching the Dirac frequency. Consequently, the Airy beam has the large GH shifts not only near the Dirac angles, but also near the Brewster angles.
Next, the IF shifts of the Airy beam are investigated. Figure 3   The impact of σ on the IF shifts is reduced, but the values are relatively small and can be ignored. The enhancement of the IF shifts are greater as the incident frequency approaches the Dirac frequency. Hence, the IF shifts of the Airy beam are larger compared to that of the Gaussian beam. Then, we further investigate other interesting characteristics of the GH and IF shifts of the Airy beam in Dirac metamaterials. The reflection field intensity distributions in the vicinity of a Dirac point (θ = 30 − 10 −10 • , ω = 1 + 10 −12 ωD) are shown in figures 4(a) and (b) with the GH shifts for HP incidence and the IF shifts for LCP incidence, respectively. The generation of a vortex beam in the reflection near the Dirac point is resulted from HP incidence for the GH shift (see figure 4(a)). Furthermore, the IF shifts for LCP incidence are clearly observed to occur in the direction perpendicular (ŷ r ) to the plane of incidence from figure 4(b). Therefore, the Airy vortex beam can be achieved by the GH shift in the vicinity of the Dirac point. This provides a new approach to the generation of vortex beams.

Rotation angle of the Airy beam
The rotation angles of the Airy beam have impact on the shifts owing to the asymmetry of the Airy beam. The angular spectrum of rotation Airy beams can be obtained as follow [14]: where k ixo = k 2 ix + k 2 iy cos arctan k iy /k ix + θ 0 and k iyo = k 2 ix + k 2 iy sin arctan k iy /k ix + θ 0 , θ 0 is the clockwise rotation angle of the initial field.   Figure 5(e) presents the GH shifts for different rotation angles of the Airy beam at the Brewster angle θ B in the polar coordinate system. It can be clearly observed that the GH shifts vary considerably for different rotation angles. Specifically, the maximum values of positive and negative GH shifts are located at the rotation angles of 45 • ( figure 5(a)) and −135 • (figure 5(d)) for the Airy beam, respectively, with the initial field strength symmetric about the x-axis. In addition, the GH shifts change symmetrically with the rotation angle. The IF shifts for different rotation angles of the Airy beam at θ = 29.855 • in the polar coordinate system are given in figure 5(f). The rotation angles of the Airy beam have little influence on the IF shifts. The IF shifts change symmetrically about the rotation angles 135 • ( figure 5(b)) and −45 • (figure 5(c)), where the initial field strengths of the rotation angles are symmetric about the y-axis. Accordingly, the GH shifts can be adjusted optionally by the rotation angle of the Airy beam around the Brewster angle.
As mentioned above, we present the value and sign of the GH shifts which can be effectively controlled by adjusting the rotation angles of the Airy beam without changing the structure or the device. Here, we design a reflected-type optical switching based on the GH shifts controlled by rotation angles in Dirac metamaterial. The schematic diagram of the proposed optical switching is shown in figure 6(a). Then, we take one of schemes as an example for discussion. Figure 6(b) describes the GH shifts near the Brewster angle θ B for the rotation angles of the Airy beam of 45 • , 90 • , −135 • and −90 • at ω = 1.001ω D , respectively. We can observe that the GH shifts are equal and opposite for rotation angles of 45 • and −135 • and for rotation angles of 90 • and −90 • . Corresponding to figure 6(a), a pinhole array with four ports (1,2,3,4) is placed in the propagation path of the reflected beam, with an adjustable rotation angle determining which output port the reflected beam spot would pass through. The truth table for which output port the reflected beam passes through (see figure 6(a)) in response to the incident beam and the rotation angle is listed in figure 6(c). Both the incident beam and the reflected beam are denoted by the logic 0 and the logic 1, which corresponds to the off and the on states, respectively. By optimizing the rotation angle, a one-to-four space-division optical switching can be achieved, in which only one of the output ports 1, 2, 3 and 4 in the reflected beam path is at logic 1 (on state) and their corresponding rotation angles are 90 • , 45 • , −90 • and −135 • , respectively.

Conclusions
To summarize, we have investigated the GH and IF shifts of the Airy beam in Dirac metamaterial. The variation of the GH and IF shifts are analyzed theoretically by deriving analytical solutions for the spatial GH and IF shifts of the Airy beam in Dirac metamaterial and comparing them with the Gaussian beam. Distinct GH shifts are evident around both the Dirac and Brewster angles when the Airy beam incidents in this interface. Furthermore, it is demonstrated that the values of the GH shifts in the case of the Airy beam are markedly higher than that of the incident Gaussian beam by considering the effect of the parameters in the analytical solution of the GH shifts of the Airy beam. Simultaneously the values of the IF shifts around the Dirac point have also increased. More interestingly, we provided a new approach to the generation of vortex beams. Additionally, the values of the GH shifts around the Brewster angle are discovered to be more controllable by changing the rotation angles of the Airy beam. As a result, this property is exploited to design a reflective optical switch based on a rotational angle-controlled GH shifts of the Dirac metamaterial. Our proposal could have potential application in optical signal processing and telecommunications.

Data availability statement
The data generated and/or analysed during the current study are not publicly available for legal/ethical reasons but are available from the corresponding author on reasonable request.
The GH and IF shifts of the Airy beam can be calculated using the equation [14]: