1. Introduction

When finite waist width beams totally reflect at the interface of two different media, longitudinal and transverse shifts will generate at the incident plane, as shown in Fig. 1. The longitudinal shift was first discovered by F. Goos and H. Hänchen in 1947 experimentally and named Goos-Hänchen(GH) shift [1]. The existence of the GH shift was proved by Artmann in 1948 [2]. And the transverse shift was proved by Fedorov and Imbert using energy-flow argument and named Imbert-Fedorov(IF) shift [3,4]. With the further study of these shifts, angular Goos-Hänchen shift(GHA) and angular Imbert-Fedorov shift(IFA) were proposed and the existence of these shifts has extended from the situation of total reflection to partial reflection and refraction, and it was found that the reflection coefficient, the transmission coefficient, and the polarization state are closely related to these spatial and angular shifts [5–7]. Thus, shifts in different kinds of material have been deeply studied in the past few years, such as graphene [8–11], photonic crystal [9,12,13], metasurfaces [10,14] and weakly absorbing dielectric [15–18]. Meanwhile, the application of the angular spectrum theory in this area makes researchers pay attention to the kinds of incident beams, such as Gaussian beams [8,19–21], Laguerre-Gaussian beams [22], Hermite-Gaussian beams [23,24], paraxial X-waves [25], and nondiffracting Bessel beams [26]. While most of them are axial symmetry beams. Chamorro-Posada [27] numerically studied the Goos-Hänchen shift of Airy beams at nonlinear interfaces, Cisowski and Correia [28] investigated the orbital Hall effect of asymmetrical vortex beams in free space, Li [29] found that the symmetry of the field distribution of Airy beams plays an important role in the Spin Hall effect of light. And Ornigotti [30] presented the analytical theory for the GH and IF shifts of Airy beams impinging on a dielectric surface. So far, these effects apply to lots of optic fields such as uniaxial crystal [31], measurement of graphene layers [32,33], dichroic polarizer [34], nanophotonics [35], sensing [11,36] and trapping [37].

 

Fig. 1. Schematic diagram of GH and IF shifts reflecting from the surface between air and weakly absorbing medium. $\varepsilon _0$ and $\varepsilon _1$ represent the dielectric constant of air and weakly absorbing medium respectively.

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In this paper, we recalculate the expressions of spatial GH shift(GHS) and spatial IF shift(IFS) of 2-D finite energy Airy beams and show the influences of the second-order terms of the reflection coefficient on GHS and IFS for Airy beams. It can be deduced that the properties of the axial symmetry of finite energy Airy beams have huge influences on GHS and IFS. Furthermore, shifts of rotational finite energy Airy beams are numerically studied and it is found that the rotation angle has huge influences on both GHS and IFS.

2. Theoretical analysis

It is well-known that the electric field intensity of position $\vec {r}$ for the incident and the reflected beams can be both written in the angular spectrum representation as [38]

For the finite energy Airy beams, the spatial expression is $A(X,Y)=\mathrm {Ai}(\vartheta {X})\mathrm {Ai}(\vartheta {Y})\mathrm {exp}(\alpha \vartheta {X}+\beta \vartheta {Y})$, where $\alpha$ and $\beta$ are the decay factor, $\vartheta =1/(k_0w_0)$ and $w_0$ is the length parameter of Airy beams. And the angular spectrum is

When the decay factor $\alpha =\beta$, finite energy Airy beams can be axial symmetry about $x$ or $y$ axes by rotating around the origin. For this circumstance, the spatial expression can be easily obtained by the expression $A(x,y)$ in polar coordinates with $A(r,\theta +\theta _0)$, where $\theta _0$ is the clockwise rotation angle of the initial field. Thus, based on the zeroth-order Hankel transformation,

3. Results and discussions

In the next discussion, the wavelength of beams is set as $\lambda =633nm$, the length parameter of Airy beams is set as $w_0=1mm$, and the dielectric constant of the weakly absorbing medium is $\varepsilon =2+0.02i$ [16]. In the process of the reflection, the sudden change of the phase of the p-polarization Fresnel reflection coefficient at Brewster angle in such medium leads to a large shift near the Brewster angle, which is convenient for us to discuss the phenomenon of GH and IF shifts for finite energy Airy beams. For the parameters of finite energy Airy beams, the decay factor is set as $\alpha =\beta =0.1$, therefore, $\Lambda$ approximately equals to 1. However, this term can not be ignored at all times. As shown in Fig. 2, the solid curve, the dashed line and the dotted line are the value of $1/\Lambda$ when $\alpha$ and $\beta$ are 0.1, 0.01 and 0.001 respectively. Obviously, $1/\Lambda$ almost equals to 1 when $\alpha =\beta =0.1$, and $1/\Lambda$ reaches a minimum value $0.95$ near the Brewster angle when $\alpha =\beta =0.01$, which has little effect on GH and IF shifts. While $1/\Lambda$ reaches 0.65 at Brewster angle when $\alpha =\beta =0.001$, thus the effect on GH and IF shifts should be considered when the decay factor is less than the order of $10^{-2}$.

 

Fig. 2. The influence of high order Taylor expansion terms in denominator on GH and IF shifts for $\alpha =\beta = 0.1$(solid line), $0.01$(dashed line) and $0.001$(dotted line).

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Since $1/\Lambda$ has been ignored in the next discussion, it’s easy to compare GH and IF shifts of finite energy Airy beams with those of Gaussian beams. Thus, the first part of Eq. (19), promptly Eq. (16), is equal to GHS of Gaussian beams, which is calculated from the partial differential of $r(\vec {k}_I)$ to $U$. While the second part is distinguished from the Gaussian solution which comes from the third-order term of the phase part of $\widetilde {A}(U,V)$. Figure 3(a) shows the GHS of finite energy Airy beams and Gaussian beams. The solid line is the analytic solution of finite energy Airy beams, and the symbol ’$\times$’ is the numerical solution of finite energy Airy beams. These two results are completely matched, which proves that the calculation is satisfied with the accuracy. Noticing the GHS of finite energy Airy beams is much bigger than that of Gaussian beams (dashed line) at Brewster angle, thus changing the incident beam might be a feasible way to enhance GH shifts. The same phenomenon also happens in the IFS shift, as shown in Fig. 3(b), solid curve and symbol ’$\times$’ represent analytic solutions and numerical solutions of IFS of finite energy Airy beams for p-polarization respectively, these two results are completely matched as well. The interesting point is that the reflected beam will divide into a pair left-handed and right-handed circular polarization beams symmetrically at the incident plane when the polarization state of the incident beam is linearly polarized, which is called spin hall effect in optics [13,31,32,36]. Thus, $\Delta _{\mathrm {IF}}$ should equal to zero for linearly polarization incident beams. Such an explanation suits for IFS of the p-polarization Gaussian beams compatibly as we can see from Eqs. (22) and (23) that $\Delta _{\mathrm {IF}}^g$ equals to 0 for $a_s=0$. While for the case of IFS of p-polarization finite energy Airy beams, as shown in Fig. 3(b), IFS reaches $-45\lambda$ at the Brewster angle. The same phenomenon also happens in [43], they found the spin Hall effect generates a transverse translation of the beam as a whole(i.e. IFS). Such a phenomenon here might cause by a pair of huge asymmetry displacements of the spin Hall effect, which is initially caused by the odd-order terms of the phase part of $\widetilde {A}(U,V)$ and the second-order terms of the reflection coefficient.

 

Fig. 3. (a) Analytic GHS of finite energy Airy beams(solid line), numerical GHS of finite energy Airy beams(’$\times$’ symbol) and analytic GHS of Gaussian beams(dashed line). (b) Analytic IFS of finite energy Airy beams(solid line) and numerical IFS of finite energy Airy beams (’$\times$’ symbol).

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From the theoretical analysis of GHS and IFS of finite energy Airy beams, we can obtain that the asymmetry about $x$ and $y$ axes of 2-D finite energy Airy beams’ field distribution, as shown in Fig. 4(a), is the main reason for huge GH and IF shifts comparing to Gaussian beams. When the decay factor $\alpha$ equals to $\beta$, the initial field distribution of the finite energy Airy beam can be symmetric about $x$ axis by clockwise rotating $-45$ degree around the origin point, as shown in Fig. 4(b). From the perspective of Fourier Optics, the spectrum of the x-axial symmetry beam is an even function of $U$. Thus, odds terms of the phase part about $U$ parameters are non-existent, therefore, GH shifts of finite energy Airy beams rotating $-45^\circ$ are the same with these of Gaussian beams, exactly as the dotted curve shown in Fig. 4(e). When the rotation angle is $45^\circ$, as shown in Fig. 4(c), the initial field distribution is symmetric about $y$ axis but asymmetric about $x$ axis, for this circumstance, the GHS is still larger than that of Gaussian beams, while it is smaller than that of normal finite energy Airy beams, as shown in the dashed line of Fig. 4(e). However, when the rotation angle is $-90^\circ$, the trail of finite energy Airy beams changes from negative to positive of $x$ axis, as shown in Fig. 4(d), and the GHS changes from a huge increase to a huge decrease comparing to GHS of Gaussian beams, as shown in the dash-dotted line of Fig. 4(e).

 

Fig. 4. (a)-(d) Intensity of finite energy Airy beams at the initial incident plane rotated around the origin for $0^\circ$, $45^\circ$, $-45^\circ$ and $-90^\circ$, respectively. (e) Dependence of GH shifts on the incident angle for different rotation angles of $0^\circ$(solid line), $45^\circ$(dash line), $-45^\circ$(dotted line) and $-90^\circ$(dash-dotted line) respectively.

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Figures 5(a)–5(d) show the initial intensity when rotation angles are $0^\circ$, $-45^\circ$, $90^\circ$ and $135^\circ$, respectively. Figure 5(e) is the IFS for finite energy Airy beams of different rotation angles. When the rotation angle is $-45^\circ$, the initial intensity is symmetric about $y$ while asymmetric about $x$, such circumstance is similar to GH shifts of $45^\circ$ rotation, while IFS of $-45^\circ$ rotation, as shown in the solid line of Fig. 5(e), reaches $-190\lambda$ far larger than IFS of normal finite energy Airy beams (dashed line). When the rotation angle is $90^\circ$, the trail of finite energy Airy beams changes from negative to positive of $y$ axis, as shown in Fig. 5(d). In this circumstance, the IFS of this rotation angle is just the opposite with no rotation, as shown in the dotted line of Fig. 5(e). And IF shifts for $135^\circ$ rotation angle, as shown in the dash-dotted line of Fig. 5(e), are opposite with IF shifts of $-45^\circ$ rotation angle.

 

Fig. 5. (a)-(d) Intensity of finite energy Airy beams at the initial incident plane rotated around the origin for $0^\circ$, $-45^\circ$, $90^\circ$ and $135^\circ$ respectively. (e) Dependence of GH shifts on the incident angle for different rotation angles of $0^\circ$(solid line), $-45^\circ$(dashed line), $90^\circ$(dotted line) and $135^\circ$(dash-dotted line) respectively.

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Furthermore, when the incident angle is the Brewster angle, the dependence of GHS on the rotation angle is shown in Fig. 6(a). When rotation angles are $0^\circ$ and $90^\circ$, these two GHSs are equal and reach the maximum. The reason for these two equal GHSs is that both of the expressions of the amplitude along the $x$ direction are $Ai(\vartheta X)\mathrm {exp}(\alpha \vartheta X)$ due to the decay factor $\alpha =\beta$. When the rotation angles are $-45^\circ$ and $135^\circ$, these two GHSs are equal to that of Gaussian beams, because that both of the amplitude expressions are symmetrical about $y$ axis. While GHSs for $-180^\circ$ and $90^\circ$ rotation angles are equal and reach the minimum because both of the amplitude expressions are $Ai(-\vartheta X)\mathrm {exp}(-\alpha \vartheta X)$ along $x$ axis. Similarly, Fig. 6(b) is the dependence of IFS on the rotation angle when the incident angle is the Brewster angle. When the amplitude expression is symmetrical about $x$ axis, namely when the rotation angles are $-135^\circ$ and $45^\circ$, IFSs of finite energy Airy beams are equal to those of Gaussian beams. And the IFSs are almost odd symmetrical about each rotation angle. Different from GHS, which reaches the maximum and minimum when the tail of 2-D finite energy Airy beams points to the positive and negative orientation of $x$ axis, IFS reaches the maximum and minimum when the rotation angles are $-45^\circ$ and $135^\circ$, namely the amplitude expression is symmetrical about $y$ axis.

 

Fig. 6. The GHS (a) and IFS (b) as a function of the rotation angle when the incident angle is the Brewster angle.

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4. Conclusion

In conclusion, we introduce the theoretical calculation of GHS and IFS for 2-D finite energy Airy beams. When the decay factors of 2-D finite energy Airy beams are smaller than the order of $10^{-2}$, the influence of the second-order terms of the reflection coefficient in the denominator should be considered to precisely calculate the shifts. While decay factors are larger than the order of $10^{-2}$, only the calculation of differential of the amplitude function should keep the second-order terms of the reflection coefficient. Furthermore, it is found that the second-order terms of the reflection coefficient have huge influences on GHS and IFS because of the odd-order terms of the phase part of the angular spectrum of finite energy Airy beams. And we notice that the information of the initial angular spectrum can be easily changed by rotating the initial incident field, thus both GHS and IFS can be greatly increased or decreased by adjusting the rotation angle.