Abstract

Expressions of Goos-Hänchen and Imbert-Fedorov shifts of rotational 2-D finite energy Airy beams are introduced in this paper. The influences of the second-order terms of the reflection coefficient on the spatial Goos-Hänchen shift (GHS) and spatial Imbert-Fedorov shift (IFS) of rotational 2-D finite energy Airy beams are theoretically and numerically investigated at the surface between air and weakly absorbing medium for the first time. It is found that the axial symmetry of the initial field of beams has huge influences on GHS and IFS and both of the GHS and IFS can be controlled by adjusting the rotation angle of the initial field distribution.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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 [57]. Thus, shifts in different kinds of material have been deeply studied in the past few years, such as graphene [811], photonic crystal [9,12,13], metasurfaces [10,14] and weakly absorbing dielectric [1518]. 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 [8,1921], 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].

 figure: Fig. 1.

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]

$$\vec{E}_\mu(\vec{r}_\mu)=\frac{1}{2\pi}{\iint}\widetilde{E}_\mu(\vec{k}_\mu)e^{i(\vec{k}_\mu{\cdot}\vec{r}_\mu)}dk_{x_{\mu}}dk_{y_{\mu}},$$
where the subscripts $\mu \in \{I,R\}$ represent the incident or the reflected coordinate system; $\vec {k}$ denotes the wave vector, in the incident coordinate system, the wave vector can be decomposed into $k_{x_I}\hat {x}_I+k_{y_I}\hat {y}_I+k_{z_I}\hat {z}_I$. To simplify the computation in this work, let $k_{x_I}=k_0U$, $k_{y_I}=k_0V$, $k_{z_I}=k_0W$, where $k_0=|\vec {k}|=2\pi /\lambda$ is the wave number, and let $k_0\hat {x}_I=\hat {X}_I$, $k_0\hat {y}_I=\hat {Y}_I$, $k_0\hat {z}_I=\hat {Z}_I$. According to the geometrical relationship between $\vec {k}_I$ and $\vec {k}_R$, $\vec {k}_R=\vec {k}_I-2\hat {z}(\hat {z}{\cdot }\vec {k}_I)$, thus $\vec {k}_R=k_0(-U\hat {x}_R+V\hat {y}_R+W\hat {z}_R)$. Therefore, the electric field of the incident and the reflected beams can be expressed as
$$\vec{E}_I(k_0\vec{r}_I)=\frac{k_0^2}{2\pi}{\iint}\widetilde{E}_I(U,V;\theta)e^{i(U\hat{X}_I+V\hat{Y}_I+W\hat{Z}_I)}dUdV,$$
$$\vec{E}_R(k_0\vec{r}_R)=\frac{k_0^2}{2\pi}{\iint}\widetilde{E}_R(U,V;\theta)e^{i(-U\hat{X}_R+V\hat{Y}_R+W\hat{Z}_R)}dUdV,$$
the incident and the reflected angular spectrums can be expressed as
$$\widetilde{E}_\mu(U,V;\theta)=\sum_{\lambda}^{p,s}\hat{e}_\lambda(\vec{k}_\mu)\alpha_\lambda(U,V;\theta)\widetilde{A}_{\mu}(U,V;\theta),$$
where $\lambda =p,s$ represent p- or s-polarization, $\hat {e}_\lambda (\vec {k}_\mu )$ are polarization unit basis vectors can be characterized by
$$\hat{e}_p(\vec{k}_\mu)=\frac{\hat{e}_s(\vec{k}_\mu){\times}\vec{k}_\mu}{|\hat{e}_s(\vec{k}_\mu){\times}\vec{k}_\mu|}, \hat{e}_s(\vec{k}_\mu)=\frac{\hat{z}{\times}\vec{k}_\mu}{|\hat{z}{\times}\vec{k}_\mu|}.$$
Polarized vector spectral amplitudes $\alpha _\lambda (U,V;\theta )=\hat {e}_\lambda (k_I)\cdot \hat {f}$, where $\hat {f}=f_p\hat {x}_I+f_s\hat {y}_I$, and $f_p=a_p$ contains the amplitude information for p-polarization $a_p$, $f_s=a_se^{i\eta }$ contains the amplitude information $a_s$ and the phase difference information between p- and s-polarization $\eta$. The initial incident angular spectrum $\widetilde {A}_I(U,V;\theta )=\widetilde {A}(U,V;0)$ is the amplitude of the initial beam field $A(x_I,y_I,z_I)|_{z_I=0}$ in the Fourier space. And $\widetilde {A}_{R}(U,V;\theta )=r_\lambda (U,V;\theta )\widetilde {A}(U,V;Z^0_I)$ is the initial reflected angular spectrum, where $Z^0_I$ is the transmission distance before reflecting along $\hat {Z}_I$. $r_\lambda (U,V;\theta )$ is the Fresnel reflection coefficients. When the incident beam is considered as a paraxial beam with rich information of angular spectrum, and the Fresnel reflection coefficient is sensitive to the incident angle, high order terms of the reflection coefficient should be maintained to obtain a precise result [7,13,25,30]. Thus, the second order terms are maintained in the calculation with the high order terms being neglected. Considering a beam reflecting at the surface between vacuum and medium whose dielectric constant is $\varepsilon _1$ as shown in Fig. 1, the Fresnel reflection coefficients of p- and s-polarization here can be characterized by the ordinary Fresnel reflection coefficient using Taylor expansion method at $U=0, V=0$ as [39]
$$r_\lambda(U,V;\theta){\simeq}r_\lambda+Ur'_\lambda+\frac{1}{2}U^{2}{r''_\lambda}+\frac{1}{2}V^{2}{r'_\lambda},$$
here $r'_\lambda$ and $r''_\lambda$ refer to the first and the second order differentials of ordinary reflection coefficients to the incident angle respectively. According to the definition of GH and IF shifts, these shifts refer to the shifts of a reflected beam from the path usually expected from geometrical optics along x- and y-axis respectively [40,41]. Due to the transmission trajectory of Airy beam is a parabolic line [27,42], establishing a coordinate system with the centroid of Airy beams at the origin point of the incident plane would entail the calculation more complex. Hence, choosing the incident frame with the usual expression of Airy function [29,30], these two shifts can be characterized by
$$\bar{X}=\frac{{\iint}X_R|\vec{E}_R(k_0\vec{r}_R)|^2dX_RdY_R}{{\iint}|\vec{E}_R(k_0\vec{r}_R)|^2dX_RdY_R}-\frac{{\iint}X_I|\vec{E}_I(k_0\vec{r}_I)|^2dX_IdY_I}{{\iint}|\vec{E}_I(k_0\vec{r}_I)|^2dX_IdY_I},$$
$$\bar{Y}=\frac{{\iint}Y_R|\vec{E}_R(k_0\vec{r}_R)|^2dX_RdY_R}{{\iint}|\vec{E}_R(k_0\vec{r}_R)|^2dX_RdY_R}-\frac{{\iint}Y_I|\vec{E}_I(k_0\vec{r}_I)|^2dX_IdY_I}{{\iint}|\vec{E}_I(k_0\vec{r}_I)|^2dX_IdY_I}.$$
Regarding the first and the second terms of Eq. (7) as $\bar X_R$ and $\bar X_I$, terms of Eq. (8) note as $\bar Y_R$ and $\bar Y_I$ as well, we can easily obtain these expressions of incident and reflected beams in the Fourier form
$$\bar X_R=\frac{{\iint}\mathrm{Im}[\widetilde{E}_R^*\frac{\partial}{\partial U}\widetilde{E}_R]dUdV}{{\iint}|\widetilde{E}_R|^2dUdV}-(Z_R+Z_I)\frac{{\iint}\frac{U}{W}{|\widetilde{E}_R|^2dUdV}}{{\iint}|\widetilde{E}_R|^2dUdV},$$
$$\bar X_I=\frac{-{\iint}\mathrm{Im}[\widetilde{E}_I^*\frac{\partial}{\partial U}\widetilde{E}_I]dUdV}{{\iint}|\widetilde{E}_I|^2dUdV}-Z_I\frac{{\iint}\frac{U}{W}{|\widetilde{E}_I|^2dUdV}}{{\iint}|\widetilde{E}_I|^2dUdV},$$
$$\bar Y_R=\frac{-{\iint}\mathrm{Im}[\widetilde{E}_R^*\frac{\partial}{\partial V}\widetilde{E}_R]dUdV}{{\iint}|\widetilde{E}_R|^2dUdV}+(Z_R+Z_I)\frac{{\iint}\frac{V}{W}{|\widetilde{E}_R|^2dUdV}}{{\iint}|\widetilde{E}_R|^2dUdV},$$
$$\bar Y_I=\frac{-{\iint}\mathrm{Im}[\widetilde{E}_I^*\frac{\partial}{\partial V}\widetilde{E}_I]dUdV}{{\iint}|\widetilde{E}_I|^2dUdV}+Z_I\frac{{\iint}\frac{V}{W}{|\widetilde{E}_I|^2dUdV}}{{\iint}|\widetilde{E}_I|^2dUdV}.$$
Therefore, GHS and IFS of non-axial symmetric beams should be modified as
$$k_0\Delta_\mathrm{GH}=\frac{{\iint}\mathrm{Im}[\widetilde{E}_R^*\frac{\partial}{\partial{U}}\widetilde{E}_R]dUdV}{{\iint}|\widetilde{E}_R|^2dUdV}+\frac{{\iint}\mathrm{Im}[\widetilde{E}_I^*\frac{\partial}{\partial V}\widetilde{E}_I]dUdV}{{\iint}|\widetilde{E}_I|^2dUdV},$$
$$k_0\Delta_\mathrm{IF}=-\frac{{\iint}\mathrm{Im}[\widetilde{E}_R^*\frac{\partial}{\partial{V}}\widetilde{E}_R]dUdV}{{\iint}|\widetilde{E}_R|^2dUdV}+\frac{{\iint}\mathrm{Im}[\widetilde{E}_I^*\frac{\partial}{\partial V}\widetilde{E}_I]dUdV}{{\iint}|\widetilde{E}_I|^2dUdV}.$$
While GHA and IFA of non-axial symmetric beams are $\partial \bar {X}/\partial {Z_R}$ and $\partial \bar {Y}/\partial {Z_R}$ respectively, which are the same as GHA and IFA of Gaussian beams [21].

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

$$\begin{aligned} \widetilde{A}(U,V)&=\frac{w_0^2}{2\pi}\mathrm{exp}(\frac{\alpha^3+\beta^3}{3})\mathrm{exp}(-\frac{\alpha U^2+\beta V^2}{\vartheta^2}) \\ &\times\mathrm{exp}[i(\frac{U^3+V^3}{3\vartheta^3}-\frac{\alpha^2U+\beta^2V}{\vartheta})]. \end{aligned}$$
By substituting the angular spectrum into Eq. (13), we can obtain $k_0\Delta _{\mathrm {GH}}=\bar x_{Rr}+\bar x_{RA}+\bar x_{IA}$. Where the three terms refer to solutions of the partial derivative of the Fresnel reflection coefficient $r_\lambda (\vec {k_I})$($\bar x_{Rr}$), the angular spectrum $\widetilde A(U,V)$ of the reflected beams($\bar x_{RA}$), and the incident beams($\bar x_{IA}$) respectively. The results are as follows
$$\bar x_{Rr}=\frac{1}{\Lambda}k_0\Delta_\mathrm{GH}^g,$$
$$\bar x_{IA}=\frac{\alpha^2}{\vartheta}-\frac{1}{4\alpha\vartheta},$$
$$\bar x_{RA}=-\bar x_{IA}+\frac{1}{\Lambda}\frac{\vartheta}{8\alpha^2}\sum_{\lambda}^{p,s}\omega_{\lambda_{1}},$$
where $\Lambda =1+\frac {\vartheta ^2}{4\alpha }\sum _{\lambda }^{p,s}\omega _{\lambda _{1}}+\frac {\vartheta ^2}{4\beta }\sum _{\lambda }^{p,s}\omega _{\lambda _{2}}$, and $\Delta _{\mathrm {GH}}^g=\omega _p\partial \phi _p/\partial \theta +\omega _s\partial \phi _s/\partial \theta$ is the GHS for Gaussian beams whose initial beam width is $w_0$, $\omega _\lambda$ inside is $a_\lambda ^2R_\lambda ^2/(a_p^2R_p^2+a_s^2R_s^2)$, and $\omega _{\lambda _{1}}=a_\lambda ^2(R_\lambda ^2+\mathrm {Re}[r''_\lambda r^*_\lambda ])/(a_p^2R_p^2+a_s^2R_s^2)$. Thus,
$$k_0\Delta_\mathrm{GH}^\mathrm{Airy}=\frac{1}{\Lambda}(k_0\Delta_\mathrm{GH}^g+\frac{\vartheta}{8\alpha^2}\sum_{\lambda}^{p,s}\omega_{\lambda_{1}}).$$
And the GHA can be easily calculated that
$$\Theta_\mathrm{GH}^\mathrm{Airy}=\frac{1}{4\alpha\Lambda}\Theta_\mathrm{GH}^g,$$
where $\Theta _{\mathrm {GH}}^g=-2\vartheta ^2(\omega _p\frac {\partial }{\partial \theta }\mathrm {ln}R_p+\omega _s\frac {\partial }{\partial \theta }\mathrm {ln}R_s)$ is GHA of Gaussian beams, Eq. (20) agrees well with that in [30]. By the similar calculation process, the IF shift of Airy beams can be described as $k_0\Delta _{\mathrm {IF}}=\bar y_{Rp}+\bar y_{Rr}+\bar y_{RA}+\bar y_{Re}-\bar y_{Ip}-\bar y_{IA}-\bar y_{Ie}$, these terms correspond to the partial derivative of the reflected polarized vector($\bar y_{Rp}$), the reflection coefficient($\bar y_{Rr}$), the reflected angular spectrum($\bar y_{RA}$), the reflected polarization unit basis vectors ($\bar y_{Re}$), the partial derivative of the incident polarized vector($\bar y_{Ip}$), the incident angular spectrum($\bar y_{IA}$), and the incident polarization unit basis vectors($\bar y_{Ie}$) respectively. The results are as follows
$$\bar y_{Rp}=-\frac{1}{\Lambda}\frac{2a_pa_s\mathrm{sin}\eta\mathrm{cot}\theta}{a_p^2R_p^2+a_s^2R_s^2}(R_p^2+R_s^2),$$
$$\bar y_{Re}=\frac{1}{\Lambda}\frac{2a_pa_s\mathrm{cot}\theta}{a_p^2R_p^2+a_s^2R_s^2}R_pR_s\mathrm{sin}(\phi_p-\phi_s-\eta),$$
$$\bar y_{IA}=\frac{\beta^2}{\vartheta}-\frac{1}{4\beta\vartheta},$$
$$\bar y_{RA}=\bar y_{IA}+\frac{\vartheta}{8\beta^2\Lambda}\sum_{\lambda}^{p,s}\omega_{\lambda_{2}},$$
$$\bar y_{Rr}=-\frac{1}{\Lambda}\frac{\vartheta^2}{4\beta}\frac{2a_pa_s\mathrm{sin}\eta\mathrm{cot^2}\theta}{a_p^2R_p^2+a_s^2R_s^2}(R_p^2\phi'_p-R_s^2\phi'_s),$$
$$\bar y_{Ip}=-\bar y_{Ie}=-\frac{2a_pa_s\mathrm{sin}\eta\mathrm{cot}\theta}{a_p^2+a_s^2}.$$
$\bar y_{Rr}\ll \bar y_{Rp}$, $\bar y_{Re}$, $\bar y_{IA}$ and $\bar y_{RA}$, thus,
$$k_0\Delta_\mathrm{IF}^\mathrm{Airy}=\frac{1}{\Lambda}(k_0\Delta_\mathrm{IF}^g+\frac{\vartheta}{8\beta^2}\sum_{\lambda}^{p,s}\omega_{\lambda_{2}}),$$
where $k_0\Delta _{\mathrm {IF}}^g=\bar y_{Rp}+\bar y_{Re}$ is the IFS of Gaussian beams [21], and $\omega _{\lambda _2}=a_\lambda ^2(\mathrm {cot}\theta \mathrm {Re}[r'_\lambda r^*_\lambda ]+\mathrm {cot}^2\theta R^2_{\bar \lambda })/(a_p^2R_p^2+a_s^2R_s^2)$, $\bar \lambda =s,p$ when $\lambda =p,s$. In the same way, the IFA for finite energy Airy beams, agree with [30], is
$$\Theta_\mathrm{IF}^\mathrm{Airy}=\frac{1}{4\beta\Lambda}\Theta_\mathrm{IF}^g,$$
where $\Theta ^g_{IF}=2\vartheta ^2(w_pa_s^2-w_sa_p^2)\mathrm {cot}{\theta }\mathrm {cos}{\eta }/(a_pa_s)$. It’s worth stressing that the expressions of $\hat {e}_\lambda (\vec {k}_\mu )\alpha _\lambda (U,V;\theta )$ $r_\lambda (U,V;\theta )$ and its differential need to be expressed with Taylor expansion, $\widetilde {A}(U,V)$ and its differential should all be remained. The second-order terms in numerators as well as higher order terms in numerators and denominators are ignored except in Eq. (18) and Eq. (24), while the second-order terms in denominators are all remained. Besides, other terms such as the differential of the polarization base vector and the polarization vector amplitude are zero. Leaving out the coefficient error of [30], the results of GHA in Eq. (20) and IFA in Eq. (28), we obtained are the same with those of [30] for decay factor $\alpha =\beta\;>\;0.01$.

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,

$$\widetilde{A}(\rho,\phi+\theta_0)=\iint A(r,\theta+\theta_0)\mathrm{exp}\{-i\rho r\mathrm{cos}[(\theta+\theta_0)-(\phi+\theta_0)]\}rdrd\theta,$$
when the initial field rotates $\theta _0$, the expression of the angular spectrum rotates $\theta _0$ as well. where $r=\sqrt {x^2+y^2}$, $\theta =\mathrm {arctan}(y/x)$, $\rho =\sqrt {k_x^2+k_y^2}$ and $\phi =\mathrm {arctan}(k_y/k_x)$. Thus, the angular spectrum of rotation finite energy Airy beams can be easily obtained as follow
$$\widetilde{A}_o(U,V)=\frac{w_0^2}{2\pi}\mathrm{exp}(\frac{\alpha^3+\beta^3}{3})\mathrm{exp}(-\frac{\alpha U_o^2+\beta V_o^2}{\vartheta^2}) \mathrm{exp}[i(\frac{U_o^3+V_o^3}{3\vartheta^3}-\frac{\alpha^2U_o+\beta^2V_o}{\vartheta})],$$
where $U_o=\sqrt {U^2+V^2}\mathrm {cos}[\mathrm {arctan}(V/U)+\theta _0]$ and $V_o=\sqrt {U^2+V^2}\mathrm {sin}[\mathrm {arctan}(V/U)+\theta _0]$. Numerical solutions of GH and IF shifts of normal 2-D finite energy Airy beams and rotational 2-D finite energy Airy beams are obtained in Matlab by calculating Eq. (13) and Eq. (14) with the angular spectrum we obtained analytically.

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 $10^{-2}$.

 figure: Fig. 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.

 figure: Fig. 3.

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).

 figure: Fig. 4.

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.

 figure: Fig. 5.

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.

 figure: Fig. 6.

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.

Funding

National Natural Science Foundation of China (11374108, 11775083).

Disclosures

The authors declare no conflicts of interest.

References

1. F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947). [CrossRef]  

2. K. Artmann, “Berechnung der seitenversetzung des totalreflektierten strahles,” Ann. Phys. 437(1-2), 87–102 (1948). [CrossRef]  

3. F. I. Fedorov, “K teorii polnogo otrazheniya,” Doklady Akademii Nauk SSSR 105, 465–468 (1955).

4. O. C. de Beauregard and C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. Lett. 28(18), 1211–1213 (1972). [CrossRef]  

5. M. Player, “Angular momentum balance and transverse shifts on reflection of light,” J. Phys. A: Math. Gen. 20(12), 3667–3678 (1987). [CrossRef]  

6. K. Y. Bliokh and Y. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75(6), 066609 (2007). [CrossRef]  

7. A. Aiello and J. Woerdman, “Theory of angular goos-hänchen shift near brewster incidence,” arXiv preprint arXiv:0903.3730 (2009).

8. S. Grosche, M. Ornigotti, and A. Szameit, “Goos-hänchen and imbert-fedorov shifts for gaussian beams impinging on graphene-coated surfaces,” Opt. Express 23(23), 30195–30203 (2015). [CrossRef]  

9. L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017). [CrossRef]  

10. A. Farmani, M. Miri, and M. H. Sheikhi, “Tunable resonant goos-hänchen and imbert-fedorov shifts in total reflection of terahertz beams from graphene plasmonic metasurfaces,” J. Opt. Soc. Am. B 34(6), 1097–1106 (2017). [CrossRef]  

11. X. Zhou, S. Liu, Y. Ding, L. Min, and Z. Luo, “Precise control of positive and negative goos-hänchen shifts in graphene,” Carbon 149, 604–608 (2019). [CrossRef]  

12. J. He, J. Yi, and S. He, “Giant negative goos-hänchen shifts for a photonic crystal with a negative effective index,” Opt. Express 14(7), 3024–3029 (2006). [CrossRef]  

13. X.-H. Ling, H.-L. Luo, M. Tang, and S.-C. Wen, “Enhanced and tunable spin hall effect of light upon reflection of one-dimensional photonic crystal with a defect layer,” Chin. Phys. Lett. 29(7), 074209 (2012). [CrossRef]  

14. V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant goos-hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6(1), 19319 (2016). [CrossRef]  

15. H. M. Lai and S. W. Chan, “Large and negative goos-hänchen shift near the brewster dip on reflection from weakly absorbing media,” Opt. Lett. 27(9), 680–682 (2002). [CrossRef]  

16. L. Wang, H. Chen, and S. Zhu, “Large negative goos-hänchen shift from a weakly absorbing dielectric slab,” Opt. Lett. 30(21), 2936–2938 (2005). [CrossRef]  

17. R.-L. Chern, “Effect of damping on goos-hänchen shift from weakly absorbing anisotropic metamaterials,” J. Opt. Soc. Am. B 31(5), 1174–1181 (2014). [CrossRef]  

18. C. Wang, F. Wang, R. Liang, Z. Wei, H. Meng, H. Dong, H. Cen, and N. Lin, “Electrically tunable goos-hänchen shifts in weakly absorbing epsilon-near-zero slab,” Opt. Mater. Express 8(4), 718–726 (2018). [CrossRef]  

19. O. Emile, T. Galstyan, F. A. Le, and F. Bretenaker, “Measurement of the nonlinear goos-hänchen effect for gaussian optical beams,” Phys. Rev. Lett. 75(8), 1511–1513 (1995). [CrossRef]  

20. A. Aiello and H. Woerdman, “The reflection of a maxwell-gaussian beam by a planar surface,” arXiv preprint arXiv:0710.1643 (2007).

21. O. Marco and A. Andrea, “Goos-hänchen and imbert-fedorov shifts for bounded wavepackets of light,” J. Opt. 15(1), 014004 (2013). [CrossRef]  

22. H. Lin, M. Jiang, L. Zhuo, W. Zhu, H. Guan, J. Yu, H. Lu, J. Tan, J. Zhang, and Z. Chen, “Enhanced imbert-fedorov shifts of higher-order laguerre-gaussian beams by lossy mode resonance,” Opt. Commun. 431, 136–141 (2019). [CrossRef]  

23. D. Golla and S. D. Gupta, “Goos-hänchen shift for higher-order hermite-gaussian beams,” Pramana 76(4), 603–612 (2011). [CrossRef]  

24. C. Prajapati and D. Ranganathan, “Goos-hänchen and imbert-fedorov shifts for hermite-gauss beams,” J. Opt. Soc. Am. A 29(7), 1377–1382 (2012). [CrossRef]  

25. M. Ornigotti, A. Aiello, and C. Conti, “Goos-hänchen and imbert-fedorov shifts for paraxial x-waves,” Opt. Lett. 40(4), 558–561 (2015). [CrossRef]  

26. A. Aiello and J. P. Woerdman, “Goos-hänchen and imbert-fedorov shifts of a nondiffracting bessel beam,” Opt. Lett. 36(4), 543–545 (2011). [CrossRef]  

27. P. Chamorro-Posada, J. Sánchez-Curto, A. B. Aceves, and G. S. McDonald, “Widely varying giant goos-hänchen shifts from airy beams at nonlinear interfaces,” Opt. Lett. 39(6), 1378–1381 (2014). [CrossRef]  

28. C. M. Cisowski and R. R. B. Correia, “Splitting an optical vortex beam to study photonic orbit-orbit interactions,” Opt. Lett. 43(3), 499–502 (2018). [CrossRef]  

29. H. Li, M. Tang, J. Wang, J. Cao, and X. Li, “Spin hall effect of airy beam in inhomogeneous medium,” Appl. Phys. B 125(3), 51 (2019). [CrossRef]  

30. M. Ornigotti, “Goos-hänchen and imbert-fedorov shifts for airy beams,” Opt. Lett. 43(6), 1411–1414 (2018). [CrossRef]  

31. K. Y. Bliokh, C. T. Samlan, C. Prajapati, G. Puentes, N. K. Viswanathan, and F. Nori, “Spin-hall effect and circular birefringence of a uniaxial crystal plate,” Optica 3(10), 1039–1047 (2016). [CrossRef]  

32. X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012). [CrossRef]  

33. L. Jiang, J. Wu, X. Dai, and Y. Xiang, “Comparison of goos-hänchen shifts of the reflected beam from graphene on dielectrics and metals,” Optik 125(23), 7025–7029 (2014). [CrossRef]  

34. K. Y. Bliokh, C. Prajapati, C. T. Samlan, N. K. Viswanathan, and F. Nori, “Spin-hall effect of light at a tilted polarizer,” Opt. Lett. 44(19), 4781–4784 (2019). [CrossRef]  

35. F. I. Baida, D. Van Labeke, and J.-M. Vigoureux, “Numerical study of the displacement of a three-dimensional gaussian beam transmitted at total internal reflection. near-field applications,” J. Opt. Soc. Am. A 17(5), 858–866 (2000). [CrossRef]  

36. X. Zhou, L. Sheng, and X. Ling, “Photonic spin hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8(1), 1221 (2018). [CrossRef]  

37. M. C. Beeler, R. A. Williams, K. Jiménez-García, L. J. LeBlanc, A. R. Perry, and I. B. Spielman, “The spin hall effect in a quantum gas,” Nature 498(7453), 201–204 (2013). [CrossRef]  

38. L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).

39. M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

40. C.-F. Li, “Unified theory for goos-hänchen and imbert-fedorov effects,” Phys. Rev. A 76(1), 013811 (2007). [CrossRef]  

41. A. Aiello and J. P. Woerdman, “Role of beam propagation in goos-hänchen and imbert-fedorov shifts,” Opt. Lett. 33(13), 1437–1439 (2008). [CrossRef]  

42. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of airy beams,” Opt. Lett. 33(3), 207–209 (2008). [CrossRef]  

43. K. Y. Bliokh and A. S. Desyatnikov, “Spin and orbital hall effects for diffracting optical beams in gradient-index media,” Phys. Rev. A 79(1), 011807 (2009). [CrossRef]  

References

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  • |

  1. F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
    [Crossref]
  2. K. Artmann, “Berechnung der seitenversetzung des totalreflektierten strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
    [Crossref]
  3. F. I. Fedorov, “K teorii polnogo otrazheniya,” Doklady Akademii Nauk SSSR 105, 465–468 (1955).
  4. O. C. de Beauregard and C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. Lett. 28(18), 1211–1213 (1972).
    [Crossref]
  5. M. Player, “Angular momentum balance and transverse shifts on reflection of light,” J. Phys. A: Math. Gen. 20(12), 3667–3678 (1987).
    [Crossref]
  6. K. Y. Bliokh and Y. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75(6), 066609 (2007).
    [Crossref]
  7. A. Aiello and J. Woerdman, “Theory of angular goos-hänchen shift near brewster incidence,” arXiv preprint arXiv:0903.3730 (2009).
  8. S. Grosche, M. Ornigotti, and A. Szameit, “Goos-hänchen and imbert-fedorov shifts for gaussian beams impinging on graphene-coated surfaces,” Opt. Express 23(23), 30195–30203 (2015).
    [Crossref]
  9. L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
    [Crossref]
  10. A. Farmani, M. Miri, and M. H. Sheikhi, “Tunable resonant goos-hänchen and imbert-fedorov shifts in total reflection of terahertz beams from graphene plasmonic metasurfaces,” J. Opt. Soc. Am. B 34(6), 1097–1106 (2017).
    [Crossref]
  11. X. Zhou, S. Liu, Y. Ding, L. Min, and Z. Luo, “Precise control of positive and negative goos-hänchen shifts in graphene,” Carbon 149, 604–608 (2019).
    [Crossref]
  12. J. He, J. Yi, and S. He, “Giant negative goos-hänchen shifts for a photonic crystal with a negative effective index,” Opt. Express 14(7), 3024–3029 (2006).
    [Crossref]
  13. X.-H. Ling, H.-L. Luo, M. Tang, and S.-C. Wen, “Enhanced and tunable spin hall effect of light upon reflection of one-dimensional photonic crystal with a defect layer,” Chin. Phys. Lett. 29(7), 074209 (2012).
    [Crossref]
  14. V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant goos-hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6(1), 19319 (2016).
    [Crossref]
  15. H. M. Lai and S. W. Chan, “Large and negative goos-hänchen shift near the brewster dip on reflection from weakly absorbing media,” Opt. Lett. 27(9), 680–682 (2002).
    [Crossref]
  16. L. Wang, H. Chen, and S. Zhu, “Large negative goos-hänchen shift from a weakly absorbing dielectric slab,” Opt. Lett. 30(21), 2936–2938 (2005).
    [Crossref]
  17. R.-L. Chern, “Effect of damping on goos-hänchen shift from weakly absorbing anisotropic metamaterials,” J. Opt. Soc. Am. B 31(5), 1174–1181 (2014).
    [Crossref]
  18. C. Wang, F. Wang, R. Liang, Z. Wei, H. Meng, H. Dong, H. Cen, and N. Lin, “Electrically tunable goos-hänchen shifts in weakly absorbing epsilon-near-zero slab,” Opt. Mater. Express 8(4), 718–726 (2018).
    [Crossref]
  19. O. Emile, T. Galstyan, F. A. Le, and F. Bretenaker, “Measurement of the nonlinear goos-hänchen effect for gaussian optical beams,” Phys. Rev. Lett. 75(8), 1511–1513 (1995).
    [Crossref]
  20. A. Aiello and H. Woerdman, “The reflection of a maxwell-gaussian beam by a planar surface,” arXiv preprint arXiv:0710.1643 (2007).
  21. O. Marco and A. Andrea, “Goos-hänchen and imbert-fedorov shifts for bounded wavepackets of light,” J. Opt. 15(1), 014004 (2013).
    [Crossref]
  22. H. Lin, M. Jiang, L. Zhuo, W. Zhu, H. Guan, J. Yu, H. Lu, J. Tan, J. Zhang, and Z. Chen, “Enhanced imbert-fedorov shifts of higher-order laguerre-gaussian beams by lossy mode resonance,” Opt. Commun. 431, 136–141 (2019).
    [Crossref]
  23. D. Golla and S. D. Gupta, “Goos-hänchen shift for higher-order hermite-gaussian beams,” Pramana 76(4), 603–612 (2011).
    [Crossref]
  24. C. Prajapati and D. Ranganathan, “Goos-hänchen and imbert-fedorov shifts for hermite-gauss beams,” J. Opt. Soc. Am. A 29(7), 1377–1382 (2012).
    [Crossref]
  25. M. Ornigotti, A. Aiello, and C. Conti, “Goos-hänchen and imbert-fedorov shifts for paraxial x-waves,” Opt. Lett. 40(4), 558–561 (2015).
    [Crossref]
  26. A. Aiello and J. P. Woerdman, “Goos-hänchen and imbert-fedorov shifts of a nondiffracting bessel beam,” Opt. Lett. 36(4), 543–545 (2011).
    [Crossref]
  27. P. Chamorro-Posada, J. Sánchez-Curto, A. B. Aceves, and G. S. McDonald, “Widely varying giant goos-hänchen shifts from airy beams at nonlinear interfaces,” Opt. Lett. 39(6), 1378–1381 (2014).
    [Crossref]
  28. C. M. Cisowski and R. R. B. Correia, “Splitting an optical vortex beam to study photonic orbit-orbit interactions,” Opt. Lett. 43(3), 499–502 (2018).
    [Crossref]
  29. H. Li, M. Tang, J. Wang, J. Cao, and X. Li, “Spin hall effect of airy beam in inhomogeneous medium,” Appl. Phys. B 125(3), 51 (2019).
    [Crossref]
  30. M. Ornigotti, “Goos-hänchen and imbert-fedorov shifts for airy beams,” Opt. Lett. 43(6), 1411–1414 (2018).
    [Crossref]
  31. K. Y. Bliokh, C. T. Samlan, C. Prajapati, G. Puentes, N. K. Viswanathan, and F. Nori, “Spin-hall effect and circular birefringence of a uniaxial crystal plate,” Optica 3(10), 1039–1047 (2016).
    [Crossref]
  32. X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012).
    [Crossref]
  33. L. Jiang, J. Wu, X. Dai, and Y. Xiang, “Comparison of goos-hänchen shifts of the reflected beam from graphene on dielectrics and metals,” Optik 125(23), 7025–7029 (2014).
    [Crossref]
  34. K. Y. Bliokh, C. Prajapati, C. T. Samlan, N. K. Viswanathan, and F. Nori, “Spin-hall effect of light at a tilted polarizer,” Opt. Lett. 44(19), 4781–4784 (2019).
    [Crossref]
  35. F. I. Baida, D. Van Labeke, and J.-M. Vigoureux, “Numerical study of the displacement of a three-dimensional gaussian beam transmitted at total internal reflection. near-field applications,” J. Opt. Soc. Am. A 17(5), 858–866 (2000).
    [Crossref]
  36. X. Zhou, L. Sheng, and X. Ling, “Photonic spin hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8(1), 1221 (2018).
    [Crossref]
  37. M. C. Beeler, R. A. Williams, K. Jiménez-García, L. J. LeBlanc, A. R. Perry, and I. B. Spielman, “The spin hall effect in a quantum gas,” Nature 498(7453), 201–204 (2013).
    [Crossref]
  38. L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).
  39. M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).
  40. C.-F. Li, “Unified theory for goos-hänchen and imbert-fedorov effects,” Phys. Rev. A 76(1), 013811 (2007).
    [Crossref]
  41. A. Aiello and J. P. Woerdman, “Role of beam propagation in goos-hänchen and imbert-fedorov shifts,” Opt. Lett. 33(13), 1437–1439 (2008).
    [Crossref]
  42. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of airy beams,” Opt. Lett. 33(3), 207–209 (2008).
    [Crossref]
  43. K. Y. Bliokh and A. S. Desyatnikov, “Spin and orbital hall effects for diffracting optical beams in gradient-index media,” Phys. Rev. A 79(1), 011807 (2009).
    [Crossref]

2019 (4)

X. Zhou, S. Liu, Y. Ding, L. Min, and Z. Luo, “Precise control of positive and negative goos-hänchen shifts in graphene,” Carbon 149, 604–608 (2019).
[Crossref]

H. Lin, M. Jiang, L. Zhuo, W. Zhu, H. Guan, J. Yu, H. Lu, J. Tan, J. Zhang, and Z. Chen, “Enhanced imbert-fedorov shifts of higher-order laguerre-gaussian beams by lossy mode resonance,” Opt. Commun. 431, 136–141 (2019).
[Crossref]

H. Li, M. Tang, J. Wang, J. Cao, and X. Li, “Spin hall effect of airy beam in inhomogeneous medium,” Appl. Phys. B 125(3), 51 (2019).
[Crossref]

K. Y. Bliokh, C. Prajapati, C. T. Samlan, N. K. Viswanathan, and F. Nori, “Spin-hall effect of light at a tilted polarizer,” Opt. Lett. 44(19), 4781–4784 (2019).
[Crossref]

2018 (4)

2017 (2)

2016 (2)

V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant goos-hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6(1), 19319 (2016).
[Crossref]

K. Y. Bliokh, C. T. Samlan, C. Prajapati, G. Puentes, N. K. Viswanathan, and F. Nori, “Spin-hall effect and circular birefringence of a uniaxial crystal plate,” Optica 3(10), 1039–1047 (2016).
[Crossref]

2015 (2)

2014 (3)

2013 (2)

O. Marco and A. Andrea, “Goos-hänchen and imbert-fedorov shifts for bounded wavepackets of light,” J. Opt. 15(1), 014004 (2013).
[Crossref]

M. C. Beeler, R. A. Williams, K. Jiménez-García, L. J. LeBlanc, A. R. Perry, and I. B. Spielman, “The spin hall effect in a quantum gas,” Nature 498(7453), 201–204 (2013).
[Crossref]

2012 (3)

C. Prajapati and D. Ranganathan, “Goos-hänchen and imbert-fedorov shifts for hermite-gauss beams,” J. Opt. Soc. Am. A 29(7), 1377–1382 (2012).
[Crossref]

X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012).
[Crossref]

X.-H. Ling, H.-L. Luo, M. Tang, and S.-C. Wen, “Enhanced and tunable spin hall effect of light upon reflection of one-dimensional photonic crystal with a defect layer,” Chin. Phys. Lett. 29(7), 074209 (2012).
[Crossref]

2011 (2)

A. Aiello and J. P. Woerdman, “Goos-hänchen and imbert-fedorov shifts of a nondiffracting bessel beam,” Opt. Lett. 36(4), 543–545 (2011).
[Crossref]

D. Golla and S. D. Gupta, “Goos-hänchen shift for higher-order hermite-gaussian beams,” Pramana 76(4), 603–612 (2011).
[Crossref]

2009 (1)

K. Y. Bliokh and A. S. Desyatnikov, “Spin and orbital hall effects for diffracting optical beams in gradient-index media,” Phys. Rev. A 79(1), 011807 (2009).
[Crossref]

2008 (2)

2007 (2)

C.-F. Li, “Unified theory for goos-hänchen and imbert-fedorov effects,” Phys. Rev. A 76(1), 013811 (2007).
[Crossref]

K. Y. Bliokh and Y. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75(6), 066609 (2007).
[Crossref]

2006 (1)

2005 (1)

2002 (1)

2000 (1)

1995 (1)

O. Emile, T. Galstyan, F. A. Le, and F. Bretenaker, “Measurement of the nonlinear goos-hänchen effect for gaussian optical beams,” Phys. Rev. Lett. 75(8), 1511–1513 (1995).
[Crossref]

1987 (1)

M. Player, “Angular momentum balance and transverse shifts on reflection of light,” J. Phys. A: Math. Gen. 20(12), 3667–3678 (1987).
[Crossref]

1972 (1)

O. C. de Beauregard and C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. Lett. 28(18), 1211–1213 (1972).
[Crossref]

1948 (1)

K. Artmann, “Berechnung der seitenversetzung des totalreflektierten strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
[Crossref]

1947 (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
[Crossref]

Aceves, A. B.

Achanta, V. G.

V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant goos-hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6(1), 19319 (2016).
[Crossref]

Agarwal, G. S.

V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant goos-hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6(1), 19319 (2016).
[Crossref]

Aiello, A.

Andrea, A.

O. Marco and A. Andrea, “Goos-hänchen and imbert-fedorov shifts for bounded wavepackets of light,” J. Opt. 15(1), 014004 (2013).
[Crossref]

Artmann, K.

K. Artmann, “Berechnung der seitenversetzung des totalreflektierten strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
[Crossref]

Baida, F. I.

Beeler, M. C.

M. C. Beeler, R. A. Williams, K. Jiménez-García, L. J. LeBlanc, A. R. Perry, and I. B. Spielman, “The spin hall effect in a quantum gas,” Nature 498(7453), 201–204 (2013).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, C. Prajapati, C. T. Samlan, N. K. Viswanathan, and F. Nori, “Spin-hall effect of light at a tilted polarizer,” Opt. Lett. 44(19), 4781–4784 (2019).
[Crossref]

K. Y. Bliokh, C. T. Samlan, C. Prajapati, G. Puentes, N. K. Viswanathan, and F. Nori, “Spin-hall effect and circular birefringence of a uniaxial crystal plate,” Optica 3(10), 1039–1047 (2016).
[Crossref]

K. Y. Bliokh and A. S. Desyatnikov, “Spin and orbital hall effects for diffracting optical beams in gradient-index media,” Phys. Rev. A 79(1), 011807 (2009).
[Crossref]

K. Y. Bliokh and Y. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75(6), 066609 (2007).
[Crossref]

Bliokh, Y. P.

K. Y. Bliokh and Y. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75(6), 066609 (2007).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

Bretenaker, F.

O. Emile, T. Galstyan, F. A. Le, and F. Bretenaker, “Measurement of the nonlinear goos-hänchen effect for gaussian optical beams,” Phys. Rev. Lett. 75(8), 1511–1513 (1995).
[Crossref]

Broky, J.

Cai, L.

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
[Crossref]

Cao, J.

H. Li, M. Tang, J. Wang, J. Cao, and X. Li, “Spin hall effect of airy beam in inhomogeneous medium,” Appl. Phys. B 125(3), 51 (2019).
[Crossref]

Cen, H.

Chamorro-Posada, P.

Chan, S. W.

Chen, H.

Chen, S.

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
[Crossref]

Chen, Z.

H. Lin, M. Jiang, L. Zhuo, W. Zhu, H. Guan, J. Yu, H. Lu, J. Tan, J. Zhang, and Z. Chen, “Enhanced imbert-fedorov shifts of higher-order laguerre-gaussian beams by lossy mode resonance,” Opt. Commun. 431, 136–141 (2019).
[Crossref]

Chern, R.-L.

Christodoulides, D. N.

Cisowski, C. M.

Conti, C.

Correia, R. R. B.

Dai, X.

L. Jiang, J. Wu, X. Dai, and Y. Xiang, “Comparison of goos-hänchen shifts of the reflected beam from graphene on dielectrics and metals,” Optik 125(23), 7025–7029 (2014).
[Crossref]

de Beauregard, O. C.

O. C. de Beauregard and C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. Lett. 28(18), 1211–1213 (1972).
[Crossref]

Desyatnikov, A. S.

K. Y. Bliokh and A. S. Desyatnikov, “Spin and orbital hall effects for diffracting optical beams in gradient-index media,” Phys. Rev. A 79(1), 011807 (2009).
[Crossref]

Ding, Y.

X. Zhou, S. Liu, Y. Ding, L. Min, and Z. Luo, “Precise control of positive and negative goos-hänchen shifts in graphene,” Carbon 149, 604–608 (2019).
[Crossref]

Dogariu, A.

Dong, H.

Emile, O.

O. Emile, T. Galstyan, F. A. Le, and F. Bretenaker, “Measurement of the nonlinear goos-hänchen effect for gaussian optical beams,” Phys. Rev. Lett. 75(8), 1511–1513 (1995).
[Crossref]

Farmani, A.

Fedorov, F. I.

F. I. Fedorov, “K teorii polnogo otrazheniya,” Doklady Akademii Nauk SSSR 105, 465–468 (1955).

Galstyan, T.

O. Emile, T. Galstyan, F. A. Le, and F. Bretenaker, “Measurement of the nonlinear goos-hänchen effect for gaussian optical beams,” Phys. Rev. Lett. 75(8), 1511–1513 (1995).
[Crossref]

Golla, D.

D. Golla and S. D. Gupta, “Goos-hänchen shift for higher-order hermite-gaussian beams,” Pramana 76(4), 603–612 (2011).
[Crossref]

Goos, F.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
[Crossref]

Grosche, S.

Guan, H.

H. Lin, M. Jiang, L. Zhuo, W. Zhu, H. Guan, J. Yu, H. Lu, J. Tan, J. Zhang, and Z. Chen, “Enhanced imbert-fedorov shifts of higher-order laguerre-gaussian beams by lossy mode resonance,” Opt. Commun. 431, 136–141 (2019).
[Crossref]

Gupta, S. D.

D. Golla and S. D. Gupta, “Goos-hänchen shift for higher-order hermite-gaussian beams,” Pramana 76(4), 603–612 (2011).
[Crossref]

Hänchen, H.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
[Crossref]

He, J.

He, S.

Imbert, C.

O. C. de Beauregard and C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. Lett. 28(18), 1211–1213 (1972).
[Crossref]

Jiang, L.

L. Jiang, J. Wu, X. Dai, and Y. Xiang, “Comparison of goos-hänchen shifts of the reflected beam from graphene on dielectrics and metals,” Optik 125(23), 7025–7029 (2014).
[Crossref]

Jiang, M.

H. Lin, M. Jiang, L. Zhuo, W. Zhu, H. Guan, J. Yu, H. Lu, J. Tan, J. Zhang, and Z. Chen, “Enhanced imbert-fedorov shifts of higher-order laguerre-gaussian beams by lossy mode resonance,” Opt. Commun. 431, 136–141 (2019).
[Crossref]

Jiménez-García, K.

M. C. Beeler, R. A. Williams, K. Jiménez-García, L. J. LeBlanc, A. R. Perry, and I. B. Spielman, “The spin hall effect in a quantum gas,” Nature 498(7453), 201–204 (2013).
[Crossref]

Lai, H. M.

Le, F. A.

O. Emile, T. Galstyan, F. A. Le, and F. Bretenaker, “Measurement of the nonlinear goos-hänchen effect for gaussian optical beams,” Phys. Rev. Lett. 75(8), 1511–1513 (1995).
[Crossref]

LeBlanc, L. J.

M. C. Beeler, R. A. Williams, K. Jiménez-García, L. J. LeBlanc, A. R. Perry, and I. B. Spielman, “The spin hall effect in a quantum gas,” Nature 498(7453), 201–204 (2013).
[Crossref]

Li, C.-F.

C.-F. Li, “Unified theory for goos-hänchen and imbert-fedorov effects,” Phys. Rev. A 76(1), 013811 (2007).
[Crossref]

Li, H.

H. Li, M. Tang, J. Wang, J. Cao, and X. Li, “Spin hall effect of airy beam in inhomogeneous medium,” Appl. Phys. B 125(3), 51 (2019).
[Crossref]

Li, X.

H. Li, M. Tang, J. Wang, J. Cao, and X. Li, “Spin hall effect of airy beam in inhomogeneous medium,” Appl. Phys. B 125(3), 51 (2019).
[Crossref]

Liang, R.

Lin, H.

H. Lin, M. Jiang, L. Zhuo, W. Zhu, H. Guan, J. Yu, H. Lu, J. Tan, J. Zhang, and Z. Chen, “Enhanced imbert-fedorov shifts of higher-order laguerre-gaussian beams by lossy mode resonance,” Opt. Commun. 431, 136–141 (2019).
[Crossref]

Lin, N.

Ling, X.

X. Zhou, L. Sheng, and X. Ling, “Photonic spin hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8(1), 1221 (2018).
[Crossref]

X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012).
[Crossref]

Ling, X.-H.

X.-H. Ling, H.-L. Luo, M. Tang, and S.-C. Wen, “Enhanced and tunable spin hall effect of light upon reflection of one-dimensional photonic crystal with a defect layer,” Chin. Phys. Lett. 29(7), 074209 (2012).
[Crossref]

Liu, M.

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
[Crossref]

Liu, S.

X. Zhou, S. Liu, Y. Ding, L. Min, and Z. Luo, “Precise control of positive and negative goos-hänchen shifts in graphene,” Carbon 149, 604–608 (2019).
[Crossref]

Liu, Y.

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
[Crossref]

Lu, H.

H. Lin, M. Jiang, L. Zhuo, W. Zhu, H. Guan, J. Yu, H. Lu, J. Tan, J. Zhang, and Z. Chen, “Enhanced imbert-fedorov shifts of higher-order laguerre-gaussian beams by lossy mode resonance,” Opt. Commun. 431, 136–141 (2019).
[Crossref]

Luo, H.

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
[Crossref]

X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012).
[Crossref]

Luo, H.-L.

X.-H. Ling, H.-L. Luo, M. Tang, and S.-C. Wen, “Enhanced and tunable spin hall effect of light upon reflection of one-dimensional photonic crystal with a defect layer,” Chin. Phys. Lett. 29(7), 074209 (2012).
[Crossref]

Luo, Z.

X. Zhou, S. Liu, Y. Ding, L. Min, and Z. Luo, “Precise control of positive and negative goos-hänchen shifts in graphene,” Carbon 149, 604–608 (2019).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).

Marco, O.

O. Marco and A. Andrea, “Goos-hänchen and imbert-fedorov shifts for bounded wavepackets of light,” J. Opt. 15(1), 014004 (2013).
[Crossref]

McDonald, G. S.

Meng, H.

Min, L.

X. Zhou, S. Liu, Y. Ding, L. Min, and Z. Luo, “Precise control of positive and negative goos-hänchen shifts in graphene,” Carbon 149, 604–608 (2019).
[Crossref]

Miri, M.

Mulay, G. L.

V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant goos-hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6(1), 19319 (2016).
[Crossref]

Nori, F.

Ornigotti, M.

Perry, A. R.

M. C. Beeler, R. A. Williams, K. Jiménez-García, L. J. LeBlanc, A. R. Perry, and I. B. Spielman, “The spin hall effect in a quantum gas,” Nature 498(7453), 201–204 (2013).
[Crossref]

Player, M.

M. Player, “Angular momentum balance and transverse shifts on reflection of light,” J. Phys. A: Math. Gen. 20(12), 3667–3678 (1987).
[Crossref]

Prajapati, C.

Puentes, G.

Ranganathan, D.

Ravishankar, A. P.

V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant goos-hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6(1), 19319 (2016).
[Crossref]

Samlan, C. T.

Sánchez-Curto, J.

Sheikhi, M. H.

Sheng, L.

X. Zhou, L. Sheng, and X. Ling, “Photonic spin hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8(1), 1221 (2018).
[Crossref]

Shu, W.

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
[Crossref]

Siviloglou, G. A.

Spielman, I. B.

M. C. Beeler, R. A. Williams, K. Jiménez-García, L. J. LeBlanc, A. R. Perry, and I. B. Spielman, “The spin hall effect in a quantum gas,” Nature 498(7453), 201–204 (2013).
[Crossref]

Szameit, A.

Tan, J.

H. Lin, M. Jiang, L. Zhuo, W. Zhu, H. Guan, J. Yu, H. Lu, J. Tan, J. Zhang, and Z. Chen, “Enhanced imbert-fedorov shifts of higher-order laguerre-gaussian beams by lossy mode resonance,” Opt. Commun. 431, 136–141 (2019).
[Crossref]

Tang, M.

H. Li, M. Tang, J. Wang, J. Cao, and X. Li, “Spin hall effect of airy beam in inhomogeneous medium,” Appl. Phys. B 125(3), 51 (2019).
[Crossref]

X.-H. Ling, H.-L. Luo, M. Tang, and S.-C. Wen, “Enhanced and tunable spin hall effect of light upon reflection of one-dimensional photonic crystal with a defect layer,” Chin. Phys. Lett. 29(7), 074209 (2012).
[Crossref]

Van Labeke, D.

Vigoureux, J.-M.

Viswanathan, N. K.

Wang, C.

Wang, F.

Wang, J.

H. Li, M. Tang, J. Wang, J. Cao, and X. Li, “Spin hall effect of airy beam in inhomogeneous medium,” Appl. Phys. B 125(3), 51 (2019).
[Crossref]

Wang, L.

Wei, Z.

Wen, S.

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
[Crossref]

X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012).
[Crossref]

Wen, S.-C.

X.-H. Ling, H.-L. Luo, M. Tang, and S.-C. Wen, “Enhanced and tunable spin hall effect of light upon reflection of one-dimensional photonic crystal with a defect layer,” Chin. Phys. Lett. 29(7), 074209 (2012).
[Crossref]

Williams, R. A.

M. C. Beeler, R. A. Williams, K. Jiménez-García, L. J. LeBlanc, A. R. Perry, and I. B. Spielman, “The spin hall effect in a quantum gas,” Nature 498(7453), 201–204 (2013).
[Crossref]

Woerdman, H.

A. Aiello and H. Woerdman, “The reflection of a maxwell-gaussian beam by a planar surface,” arXiv preprint arXiv:0710.1643 (2007).

Woerdman, J.

A. Aiello and J. Woerdman, “Theory of angular goos-hänchen shift near brewster incidence,” arXiv preprint arXiv:0903.3730 (2009).

Woerdman, J. P.

Wolf, E.

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

Wu, J.

L. Jiang, J. Wu, X. Dai, and Y. Xiang, “Comparison of goos-hänchen shifts of the reflected beam from graphene on dielectrics and metals,” Optik 125(23), 7025–7029 (2014).
[Crossref]

Xiang, Y.

L. Jiang, J. Wu, X. Dai, and Y. Xiang, “Comparison of goos-hänchen shifts of the reflected beam from graphene on dielectrics and metals,” Optik 125(23), 7025–7029 (2014).
[Crossref]

Yallapragada, V. J.

V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant goos-hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6(1), 19319 (2016).
[Crossref]

Yi, J.

Yu, J.

H. Lin, M. Jiang, L. Zhuo, W. Zhu, H. Guan, J. Yu, H. Lu, J. Tan, J. Zhang, and Z. Chen, “Enhanced imbert-fedorov shifts of higher-order laguerre-gaussian beams by lossy mode resonance,” Opt. Commun. 431, 136–141 (2019).
[Crossref]

Zhang, J.

H. Lin, M. Jiang, L. Zhuo, W. Zhu, H. Guan, J. Yu, H. Lu, J. Tan, J. Zhang, and Z. Chen, “Enhanced imbert-fedorov shifts of higher-order laguerre-gaussian beams by lossy mode resonance,” Opt. Commun. 431, 136–141 (2019).
[Crossref]

Zhou, X.

X. Zhou, S. Liu, Y. Ding, L. Min, and Z. Luo, “Precise control of positive and negative goos-hänchen shifts in graphene,” Carbon 149, 604–608 (2019).
[Crossref]

X. Zhou, L. Sheng, and X. Ling, “Photonic spin hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8(1), 1221 (2018).
[Crossref]

X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012).
[Crossref]

Zhu, S.

Zhu, W.

H. Lin, M. Jiang, L. Zhuo, W. Zhu, H. Guan, J. Yu, H. Lu, J. Tan, J. Zhang, and Z. Chen, “Enhanced imbert-fedorov shifts of higher-order laguerre-gaussian beams by lossy mode resonance,” Opt. Commun. 431, 136–141 (2019).
[Crossref]

Zhuo, L.

H. Lin, M. Jiang, L. Zhuo, W. Zhu, H. Guan, J. Yu, H. Lu, J. Tan, J. Zhang, and Z. Chen, “Enhanced imbert-fedorov shifts of higher-order laguerre-gaussian beams by lossy mode resonance,” Opt. Commun. 431, 136–141 (2019).
[Crossref]

Ann. Phys. (2)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. 436(7-8), 333–346 (1947).
[Crossref]

K. Artmann, “Berechnung der seitenversetzung des totalreflektierten strahles,” Ann. Phys. 437(1-2), 87–102 (1948).
[Crossref]

Appl. Phys. B (1)

H. Li, M. Tang, J. Wang, J. Cao, and X. Li, “Spin hall effect of airy beam in inhomogeneous medium,” Appl. Phys. B 125(3), 51 (2019).
[Crossref]

Appl. Phys. Lett. (1)

X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012).
[Crossref]

Carbon (1)

X. Zhou, S. Liu, Y. Ding, L. Min, and Z. Luo, “Precise control of positive and negative goos-hänchen shifts in graphene,” Carbon 149, 604–608 (2019).
[Crossref]

Chin. Phys. Lett. (1)

X.-H. Ling, H.-L. Luo, M. Tang, and S.-C. Wen, “Enhanced and tunable spin hall effect of light upon reflection of one-dimensional photonic crystal with a defect layer,” Chin. Phys. Lett. 29(7), 074209 (2012).
[Crossref]

J. Opt. (1)

O. Marco and A. Andrea, “Goos-hänchen and imbert-fedorov shifts for bounded wavepackets of light,” J. Opt. 15(1), 014004 (2013).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (2)

J. Phys. A: Math. Gen. (1)

M. Player, “Angular momentum balance and transverse shifts on reflection of light,” J. Phys. A: Math. Gen. 20(12), 3667–3678 (1987).
[Crossref]

Nature (1)

M. C. Beeler, R. A. Williams, K. Jiménez-García, L. J. LeBlanc, A. R. Perry, and I. B. Spielman, “The spin hall effect in a quantum gas,” Nature 498(7453), 201–204 (2013).
[Crossref]

Opt. Commun. (1)

H. Lin, M. Jiang, L. Zhuo, W. Zhu, H. Guan, J. Yu, H. Lu, J. Tan, J. Zhang, and Z. Chen, “Enhanced imbert-fedorov shifts of higher-order laguerre-gaussian beams by lossy mode resonance,” Opt. Commun. 431, 136–141 (2019).
[Crossref]

Opt. Express (2)

Opt. Lett. (10)

H. M. Lai and S. W. Chan, “Large and negative goos-hänchen shift near the brewster dip on reflection from weakly absorbing media,” Opt. Lett. 27(9), 680–682 (2002).
[Crossref]

L. Wang, H. Chen, and S. Zhu, “Large negative goos-hänchen shift from a weakly absorbing dielectric slab,” Opt. Lett. 30(21), 2936–2938 (2005).
[Crossref]

M. Ornigotti, A. Aiello, and C. Conti, “Goos-hänchen and imbert-fedorov shifts for paraxial x-waves,” Opt. Lett. 40(4), 558–561 (2015).
[Crossref]

A. Aiello and J. P. Woerdman, “Goos-hänchen and imbert-fedorov shifts of a nondiffracting bessel beam,” Opt. Lett. 36(4), 543–545 (2011).
[Crossref]

P. Chamorro-Posada, J. Sánchez-Curto, A. B. Aceves, and G. S. McDonald, “Widely varying giant goos-hänchen shifts from airy beams at nonlinear interfaces,” Opt. Lett. 39(6), 1378–1381 (2014).
[Crossref]

C. M. Cisowski and R. R. B. Correia, “Splitting an optical vortex beam to study photonic orbit-orbit interactions,” Opt. Lett. 43(3), 499–502 (2018).
[Crossref]

M. Ornigotti, “Goos-hänchen and imbert-fedorov shifts for airy beams,” Opt. Lett. 43(6), 1411–1414 (2018).
[Crossref]

K. Y. Bliokh, C. Prajapati, C. T. Samlan, N. K. Viswanathan, and F. Nori, “Spin-hall effect of light at a tilted polarizer,” Opt. Lett. 44(19), 4781–4784 (2019).
[Crossref]

A. Aiello and J. P. Woerdman, “Role of beam propagation in goos-hänchen and imbert-fedorov shifts,” Opt. Lett. 33(13), 1437–1439 (2008).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of airy beams,” Opt. Lett. 33(3), 207–209 (2008).
[Crossref]

Opt. Mater. Express (1)

Optica (1)

Optik (1)

L. Jiang, J. Wu, X. Dai, and Y. Xiang, “Comparison of goos-hänchen shifts of the reflected beam from graphene on dielectrics and metals,” Optik 125(23), 7025–7029 (2014).
[Crossref]

Phys. Rev. A (3)

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin hall effect in graphene,” Phys. Rev. A 95(1), 013809 (2017).
[Crossref]

K. Y. Bliokh and A. S. Desyatnikov, “Spin and orbital hall effects for diffracting optical beams in gradient-index media,” Phys. Rev. A 79(1), 011807 (2009).
[Crossref]

C.-F. Li, “Unified theory for goos-hänchen and imbert-fedorov effects,” Phys. Rev. A 76(1), 013811 (2007).
[Crossref]

Phys. Rev. E (1)

K. Y. Bliokh and Y. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75(6), 066609 (2007).
[Crossref]

Phys. Rev. Lett. (2)

O. C. de Beauregard and C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. Lett. 28(18), 1211–1213 (1972).
[Crossref]

O. Emile, T. Galstyan, F. A. Le, and F. Bretenaker, “Measurement of the nonlinear goos-hänchen effect for gaussian optical beams,” Phys. Rev. Lett. 75(8), 1511–1513 (1995).
[Crossref]

Pramana (1)

D. Golla and S. D. Gupta, “Goos-hänchen shift for higher-order hermite-gaussian beams,” Pramana 76(4), 603–612 (2011).
[Crossref]

Sci. Rep. (2)

X. Zhou, L. Sheng, and X. Ling, “Photonic spin hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8(1), 1221 (2018).
[Crossref]

V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant goos-hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6(1), 19319 (2016).
[Crossref]

Other (5)

A. Aiello and H. Woerdman, “The reflection of a maxwell-gaussian beam by a planar surface,” arXiv preprint arXiv:0710.1643 (2007).

A. Aiello and J. Woerdman, “Theory of angular goos-hänchen shift near brewster incidence,” arXiv preprint arXiv:0903.3730 (2009).

F. I. Fedorov, “K teorii polnogo otrazheniya,” Doklady Akademii Nauk SSSR 105, 465–468 (1955).

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

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Figures (6)

Fig. 1.
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.
Fig. 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).
Fig. 3.
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).
Fig. 4.
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.
Fig. 5.
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.
Fig. 6.
Fig. 6. The GHS (a) and IFS (b) as a function of the rotation angle when the incident angle is the Brewster angle.

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

E μ ( r μ ) = 1 2 π E ~ μ ( k μ ) e i ( k μ r μ ) d k x μ d k y μ ,
E I ( k 0 r I ) = k 0 2 2 π E ~ I ( U , V ; θ ) e i ( U X ^ I + V Y ^ I + W Z ^ I ) d U d V ,
E R ( k 0 r R ) = k 0 2 2 π E ~ R ( U , V ; θ ) e i ( U X ^ R + V Y ^ R + W Z ^ R ) d U d V ,
E ~ μ ( U , V ; θ ) = λ p , s e ^ λ ( k μ ) α λ ( U , V ; θ ) A ~ μ ( U , V ; θ ) ,
e ^ p ( k μ ) = e ^ s ( k μ ) × k μ | e ^ s ( k μ ) × k μ | , e ^ s ( k μ ) = z ^ × k μ | z ^ × k μ | .
r λ ( U , V ; θ ) r λ + U r λ + 1 2 U 2 r λ + 1 2 V 2 r λ ,
X ¯ = X R | E R ( k 0 r R ) | 2 d X R d Y R | E R ( k 0 r R ) | 2 d X R d Y R X I | E I ( k 0 r I ) | 2 d X I d Y I | E I ( k 0 r I ) | 2 d X I d Y I ,
Y ¯ = Y R | E R ( k 0 r R ) | 2 d X R d Y R | E R ( k 0 r R ) | 2 d X R d Y R Y I | E I ( k 0 r I ) | 2 d X I d Y I | E I ( k 0 r I ) | 2 d X I d Y I .
X ¯ R = I m [ E ~ R U E ~ R ] d U d V | E ~ R | 2 d U d V ( Z R + Z I ) U W | E ~ R | 2 d U d V | E ~ R | 2 d U d V ,
X ¯ I = I m [ E ~ I U E ~ I ] d U d V | E ~ I | 2 d U d V Z I U W | E ~ I | 2 d U d V | E ~ I | 2 d U d V ,
Y ¯ R = I m [ E ~ R V E ~ R ] d U d V | E ~ R | 2 d U d V + ( Z R + Z I ) V W | E ~ R | 2 d U d V | E ~ R | 2 d U d V ,
Y ¯ I = I m [ E ~ I V E ~ I ] d U d V | E ~ I | 2 d U d V + Z I V W | E ~ I | 2 d U d V | E ~ I | 2 d U d V .
k 0 Δ G H = I m [ E ~ R U E ~ R ] d U d V | E ~ R | 2 d U d V + I m [ E ~ I V E ~ I ] d U d V | E ~ I | 2 d U d V ,
k 0 Δ I F = I m [ E ~ R V E ~ R ] d U d V | E ~ R | 2 d U d V + I m [ E ~ I V E ~ I ] d U d V | E ~ I | 2 d U d V .
A ~ ( U , V ) = w 0 2 2 π e x p ( α 3 + β 3 3 ) e x p ( α U 2 + β V 2 ϑ 2 ) × e x p [ i ( U 3 + V 3 3 ϑ 3 α 2 U + β 2 V ϑ ) ] .
x ¯ R r = 1 Λ k 0 Δ G H g ,
x ¯ I A = α 2 ϑ 1 4 α ϑ ,
x ¯ R A = x ¯ I A + 1 Λ ϑ 8 α 2 λ p , s ω λ 1 ,
k 0 Δ G H A i r y = 1 Λ ( k 0 Δ G H g + ϑ 8 α 2 λ p , s ω λ 1 ) .
Θ G H A i r y = 1 4 α Λ Θ G H g ,
y ¯ R p = 1 Λ 2 a p a s s i n η c o t θ a p 2 R p 2 + a s 2 R s 2 ( R p 2 + R s 2 ) ,
y ¯ R e = 1 Λ 2 a p a s c o t θ a p 2 R p 2 + a s 2 R s 2 R p R s s i n ( ϕ p ϕ s η ) ,
y ¯ I A = β 2 ϑ 1 4 β ϑ ,
y ¯ R A = y ¯ I A + ϑ 8 β 2 Λ λ p , s ω λ 2 ,
y ¯ R r = 1 Λ ϑ 2 4 β 2 a p a s s i n η c o t 2 θ a p 2 R p 2 + a s 2 R s 2 ( R p 2 ϕ p R s 2 ϕ s ) ,
y ¯ I p = y ¯ I e = 2 a p a s s i n η c o t θ a p 2 + a s 2 .
k 0 Δ I F A i r y = 1 Λ ( k 0 Δ I F g + ϑ 8 β 2 λ p , s ω λ 2 ) ,
Θ I F A i r y = 1 4 β Λ Θ I F g ,
A ~ ( ρ , ϕ + θ 0 ) = A ( r , θ + θ 0 ) e x p { i ρ r c o s [ ( θ + θ 0 ) ( ϕ + θ 0 ) ] } r d r d θ ,
A ~ o ( U , V ) = w 0 2 2 π e x p ( α 3 + β 3 3 ) e x p ( α U o 2 + β V o 2 ϑ 2 ) e x p [ i ( U o 3 + V o 3 3 ϑ 3 α 2 U o + β 2 V o ϑ ) ] ,

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