Goos-H&#x00E4;nchen and Imbert-Fedorov shifts of off-axis Airy vortex beams

Mingsheng Gao; Guanghui Wang; Xiangbo Yang; Hongzhan Liu; Dongmei Deng

doi:10.1364/OE.404831

1. Introduction

In 1947, F. Goos and H. Hänchen found that a longitudinal shift will generate between the actual reflection path and the geometric optical reflection path when a total reflection of finite waist width beams occurs at the interface of two different media [1]. Soon after, Fedorov and Imbert replenished the theory of reflection and proved that there exists a lateral shift along the vertical direction of the incident plane besides a longitudinal shift [2,3]. The longitudinal shift is called Goos-Hänchen(GH) shift and the lateral shift is named Imbert-Fedorov(IF) shift, which are also found in the phenomenon of refraction and partial reflection [4,5]. In recent years, the GH and the IF shifts for various kinds of incident beams and the influences of their parameters on the GH and the IF shifts are deeply researched. Aiello and Ornigotti et al. theoretically studied the GH and the IF shifts for “nondiffracting” Bessel beam [6], paraxial X-waves [7], and the propagating Gaussian light beam [8]. Golla [9] numerically studied the reflected beam profiles and demonstrated the GH shift for fundamental higher-order Hermite-Gaussian beams. On the other hand, Cisowski [10] studied the photonic orbital Hall effect of asymmetrical vortex beams during propagation, which gives a new perspective in beam shifts. Ornigotti [11] theoretically studied the GH and the IF shifts of Airy beams soon after. Li [12] studied the spin Hall effect of Airy beams in an inhomogeneous medium and found the symmetry distribution of light beams has huge influences on the spin Hall effect. In the recent work of our group, it is found that the axial symmetry and the phase of a finite energy Airy beam have huge influences on the GH and the IF shifts [13]. On the other hand, the replenishment of the weak measurement theory for the GH and the IF shifts enables the observation and the detection [14–19]. Thus, applications based on beam shifts such as sensors [20–22], measurement of refractive index [23] and graphene layers [24], and optical differential operation and image edge detection [25] are proposed in recent years.

A vortex beam is a special kind of beam carrying orbital angular momentum and phase singularity which has a continuous spiral phase and a phase singularity at the center of the vortex [26]. It becomes a hot spot in recent researches leaning on the wide applications in optical communications [27], trapping [28], and optical microscope [29]. Our group also has some researches in this field such as rapidly auto-focused ring Pearcey Gaussian vortex beam [30] and ring Airy Gaussian vortex beams [31]. In the field of beam shifts, Bliokh [32] researched the GH and the IF shifts of linearly polarized Laguerre-Gaussian vortex beams theoretically and numerically, and found the value and the sign of the vortex-induced shifts can be controlled by the topological charges. However, the GH and the IF shifts of the finite energy off-axis Airy vortex beams (AiVBs) have not been reported. Here, the GH and the IF shifts of the finite energy off-axis AiVBs will be numerically calculated.

In this paper, the angular spectrum of off-axis AiVBs is analytically derived and the GH and the IF shifts near the Brewster angle for different topological charges are numerically calculated. The influences of the topological charges and the off-axis positions on the GH and the IF shifts are analyzed detailly. The results show that the further the off-axis position of the vortex is, the less influences of the vortex on GH and IF shifts become.

2. Theoretical analysis

Based on the geometric optics, it is assumed that the off-axis AiVBs go from the media 0 into the media 1 along the $z$ axis with the incident angle $\theta$ whose dielectric constants are $\varepsilon _0$ and $\varepsilon _1$ respectively. The incident coordinate system ($x_i$-$y_i$-$z_i$) is established with the path of the incident beams and the reflect coordinate system ($x_r$-$y_r$-$z_r$) is established with the path of the geometric optical reflected beams as shown in Fig. 1. Thus, the incident electric field at the position $\vec {r}$ can be expressed in the angular spectrum as [33,34]

(1)$$E_i(\vec{r}_i)=\frac{1}{2\pi}{\iint}\widetilde{E}_i(\vec{k}_i)e^{i(\vec{k}_i{\cdot}\vec{r}_i)}dk_{x}dk_{y},$$

where $\vec {k_i}=\hat {x}_ik_x+\hat {y}_ik_y+\hat {z}_ik_z$ is the wave vector of the incident beam, and $\widetilde {E}_i$ is the angular spectrum of the incident beam

(2)$$\widetilde{E}_i(\vec{k}_i)=\sum_\lambda^{p,s}\hat{e}_\lambda(\vec{k}_i)\alpha(\vec{k}_i)\widetilde{A}(\vec{k}),$$

where $\lambda$ denotes p-polarization or s-polarization, $\hat {e}_p=(\hat {e}_s\times \vec {k})/|\hat {e}_s\times \vec {k}|$ and $\hat {e}_s=(\hat {z}\times \vec {k})/|\hat {z}\times \vec {k}|$ denote basic vectors of p-polarization and s-polarization respectively. $\alpha _\lambda (\vec {k})=\hat {e}_\lambda \cdot \hat {f}$ contains the amplitude information of p- and s-polarization $\hat {f}$, where $\hat {f}=a_p\hat {x}+a_se^{i\eta }\hat {y}$, $a_p, a_s$ and $\eta$ are p-polarization information, s-polarization information and the phase difference between p- and s-polarization respectively and $|a_p|^2+|a_s|^2=1$. $\widetilde {A}(\vec {k})$ is the angular spectrum of the initial incident plane. In this paper, the GH and the IF shifts of finite energy off-axis AiVBs are discussed, and the amplitude $A(x, y)$ of off-axis AiVBs can be expressed as [35]

(3)$$A(x,y)=Ai(\frac{x}{w_0})Ai(\frac{y}{w_0})\mathrm{exp}[\alpha\frac{x}{w_0}+\beta\frac{y}{w_0}](\frac{x-x_0}{w_0}+i\frac{y-y_0}{w_0})^m,$$

where $Ai(\cdot )$ is the Airy function, $w_0$ is the length parameter of off-axis AiVBs, $\alpha$ and $\beta$ are the decay factors of the Airy part along with x and y directions separately. $(x_0, y_0)$ is the vortex’s position, and $m$ is the topological charge of the vortex. Using Fourier transform we can easily calculate the angular spectrum of off-axis AiVBs as

(4)$$\begin{aligned} \widetilde{A}(\vec{k}_i)&=w_0^2e^{[\frac{1}{3}(\alpha^3+\beta^3)-w_0^2(\alpha k_x^2+\beta k_y^2)]}e^{\{\frac{i}{3}[w_0^3(k_x^3+k_y^3)-3w_0(\alpha^2k_x+\beta^2k_y)]\}} \\ &\times\{[(\alpha-iw_0k_x)^2-\frac{x_0}{w_0}]+i[(\beta-iw_0k_y)^2-\frac{y_0}{w_0}]\}^m. \end{aligned}$$

According to Fresnel’s law, the angular spectrum of the reflected electric field can be expressed as

(5)$$\widetilde{E}_r(\vec{k}_r)=\sum_\lambda^{p,s}\hat{e}_\lambda(\vec{k}_r)\alpha_\lambda(\vec{k}_r)r_\lambda(\vec{k}_i)\widetilde{A}(\vec{k}_i;Z_i),$$

where $\vec {k}_r$ is the wave vector of reflected beams which can be calculated by the geometrical relationship between incident and reflected beams of the geometrical reflection law $\vec {k}_r=\vec {k}_i-2\hat {z}(\hat {z}\cdot \vec {k}_i)$. $r_\lambda (\vec {k}_i)$ for p and s are the Fresnel reflection coefficients of each polarization status, and they can be described that [33,34]

(6)$$r_p(\vec{k}_i)=\frac{\varepsilon k_z-\sqrt{\varepsilon k_0^2-k_x^2-k_y^2}}{\varepsilon k_z+\sqrt{\varepsilon k_0^2-k_x^2-k_y^2}}, r_s(\vec{k}_i)=\frac{k_z-\sqrt{\varepsilon k_0^2-k_x^2-k_y^2}}{k_z+\sqrt{\varepsilon k_0^2-k_x^2-k_y^2}}.$$

where $\varepsilon =\varepsilon _1/\varepsilon _0$. Thus, it is easy to numerically calculate the GH and the IF shifts of finite energy off-axis AiVBs using the expression for asymmetrical beams in the frequency domain [13,36]

(7)$$\Delta_\mathrm{GH}=\frac{\iint\mathrm{Im}[\widetilde{E}_r^*\frac{\partial}{\partial k_x}\widetilde{E}_r]dk_xdk_y}{\iint|\widetilde{E}_r|^2dk_xdk_y}+\frac{\iint\mathrm{Im}[\widetilde{E}_i^*\frac{\partial}{\partial k_x}\widetilde{E}_i]dk_xdk_y}{\iint|\widetilde{E}_i|^2dk_xdk_y},$$

(8)$$\Delta_\mathrm{IF}=-\frac{\iint\mathrm{Im}[\widetilde{E}_r^*\frac{\partial}{\partial k_y}\widetilde{E}_r]dk_xdk_y}{\iint|\widetilde{E}_r|^2dk_xdk_y}+\frac{\iint\mathrm{Im}[\widetilde{E}_i^*\frac{\partial}{\partial k_y}\widetilde{E}_i]dk_xdk_y}{\iint|\widetilde{E}_i|^2dk_xdk_y}.$$

3. Results and discussions

To illustrate the influences of vortex on beam shifts for off-axis AiVBs, the structure and parameters of Airy parts are chosen the same as those in Ref. [13]. As shown in Fig. 1, the off-axis AiVBs go from air into the surface of a weakly absorbing media 1 whose dielectric constant is $\varepsilon =2+0.02i$, the wavelength of the incident beam $\lambda =633\mathrm {nm}$, the decay factors of off-axis AiVBs $\alpha$ and $\beta$ are set as $0.1$, and the length parameter $w_0=1\mathrm {mm}$.

Fig. 1. Schematic of Cartesian ($x$-$y$-$z$), incident ($x_i$-$y_i$-$z_i$) and the reflect ($x_r$-$y_r$-$z_r$) coordinate system, the GH shift ($\Delta _{\mathrm {GH}}$) and IF shift ($\Delta _{\mathrm {IF}}$) of reflected off-axis AiVBs. Where $\varepsilon _0$ and $\varepsilon _1$ are the dielectric constant of the media 0 and the media 1 separately.

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First, the influences of topological charges $m$ on GH and IF shifts are discussed with the center of vortex locating at the origin point of the incident frame. Figures 2(a), 2(b), and 2(c) are the intensity distribution of the finite energy off-axis AiVBs for $m=0$, $m=1$, and $m=2$ separately. Clearly, with the increase of the topological charge, the intensity at $(0, 0)$ changes to zero and the intensity nearby is reduced. The quantities of side lobes are also significantly increased and the peaks of the intensity on the x-axis and the y-axis transfer along the negative direction of the x-axis and the y-axis. Figure 2(d) presents GH shifts for different topological charges near the Brewster angle. The black solid line is the dependence of GH shifts ($\Delta _{\mathrm {GH}}$) on the incident angle for m=0 which is discussed in Ref. [13]. The GH shift increases first and then decreases with the increase of the incident angle, and gets the maximum 103$\lambda$ at the Brewster angle. When the topological charges are 1 and 2, as the red dotted line and the blue dashed line shown in Fig. 2(d), the trends of the GH shifts are similar to those for $m=0$. While their peaks sharply increase to $230\lambda$ for $m=1$ and to $430\lambda$ for $m=2$ at the Brewster angle. The IF shifts for different topological charges near the Brewster angle are shown in Fig. 2(e). The influences of the topological charges on the IF shift have some differences comparing with those on GH shifts. The black solid line is the IF shifts for Airy beams whose entire symbol is negative. While the IF shifts for $m=1$ and $m=2$ are both positive as the red dotted line and the blue dashed line shown in Fig. 2(e). Similarly, these IF shifts increase first and then decrease with the increase of the incident angle, and reach the peaks of $-7\lambda$, $37.1\lambda$ and $170\lambda$ for $m=0, 1$ and $2$ at the Brewster angle respectively.

Fig. 2. (a-c) Intensity distributions of the incident finite energy off-axis AiVBs for topological charges $m=0, m=1$ and $m=2$. (d-e) GH shifts and IF shifts for different topological charges near the Brewster angle.

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Then, we choose the first-order topological charge to discuss the GH and the IF shifts for off-axis AiVBs. Figure 3(a) shows the influences of the vortex’s position at the y-axis on GH shifts near the Brewster angle. The black dotted line is the GH shifts when the vortex is located at the origin point $(0, 0)$, which is the same as the red dotted line in Fig. 2(d). The red dash-dotted line, the blue solid line, and the green dashed line in Fig. 3(a) are the GH shifts near the Brewster angle where the vortex locates at $(x_0, y_0)=(0, 1)$, $(0, 0.01)$ and $(0, -0.01)$ respectively. Each of the GH shifts for different vortex’s positions increases first and then decreases with the change of the incident angle, and reaches a peak at the Brewster angle. The biggest peak value is the GH shift with the vortex locating at the origin point $(0, 0)$, while the smallest peak value is the GH shift with the vortex locating at the point $(0, 0.1)$. Figure 3(b) is the IF shifts of the off-axis AiVBs for the first-order topological charge. The black dotted line, the red solid line, the blue dashed line and the green dash-dotted line are IF shifts for the vortex’s positions at $(x_0, y_0)=(0, 0), (0,1), (0, 0.01)$ and $(0, -0.01)$ near the Brewster angle respectively. Similarly, all IF shifts increase to a peak near the Brewster angle and then decrease with the increase of the incident angle, and the peak value of IF shifts with the vortex’s position at $(0, 1)$ is the smallest. While only the sign of IF shifts with the vortex locating at the origin point $(0,0)$ is positive, the signs of the others are negative. Interestingly, the peak values of all GH shifts with different vortex’s positions appear at the Brewster angle, while the peak values of IF shifts with the vortex locating at $(0, 0.01)$ and $(0, -0.01)$ have a little offset from the Brewster angle.

Fig. 3. (a) GH shifts near the Brewster angle when the vortex locates at $(0, 0)$ (black dotted line), $(0, 10^{-0})$ (red dash-dotted line), $(0, 10^{-2})$ (blue solid line) and $(0, -10^{-2})$ (green dashed line). (b) IF shifts near the Brewster angle when the vortex locates at $(0, 0)$ (black dotted line), $(0, 10^{-0})$ (red solid line), $(0, 10^{-2})$ (blue dashed line) and $(0, -10^{-2})$ (green dash-dotted line).

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To illustrate the relationship between GH shifts and the vortex’s position, Fig. 4(a) is the pseudo-color picture of the GH shifts at the Brewster angle for different vortex’s positions, and Fig. 4(b) is the two-dimensional relationship curve of the GH shift at the Brewster angle for the x off-axis AiVBs (black solid line) and the y off-axis AiVBs (red dashed line). Overall, the movement of the vortex’s position along the positive and negative directions of y-axis entails symmetrical decline of the GH shifts. Noticing the movement of the vortex’s position along the negative direction has a sudden change near $x_0=-0.01$ in Fig. 4(a). Combining with Fig. 4(b), it can be found that when the vortex’s position moves along the negative directions of the x-axis or the y-axis, the GH shifts increase in a tiny distance, and then both decrease sharply with the increase of distance between the vortex’s position and the origin point. It is worth noting that the negative directions are also the direction of the tails of the off-axis AiVBs. Thus, the tiny increase is owing to the non-axis symmetry of the off-axis AiVBs. If we compare the curves in Fig. 4(b) with the GH shift for $m=0$, which is the blue dash-dotted line in Fig. 2(d), it can be found that the curves of GH shifts at the Brewster angle for the first-order topological charge off-axis AiVBs approach to the GH shifts at the Brewster angle for $m=0$ with the increase of the vortex’s position. In other words, the more further the vortex’s position is, the tinier the influences of the vortex on GH shifts are.

Fig. 4. (a) Pseudo-color picture of the GH shift of the off-axis AiVBs at the Brewster angle for different vortex positions. (b) GH shifts of the x off-axis AiVBs (black solid line) and GH shifts of the y off-axis AiVBs (red dashed line) at the Brewster angle.

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Using the same method as Fig. 4, the relationship between the IF shifts and the vortex’s positions is depicted in Fig. 5. Figure 5(a) is the pseudo-color picture of the IF shifts at the Brewster angle for different vortex’s positions. When the vortex’s position is far from the origin point, the IF shift at the Brewster angle almost changes symmetrically with the vortex’s position moving along the positive and the negative directions of the x-axis. While when the vortex’s position is near the origin point, the changes of the IF shifts at the Brewster angle are more complicate comparing with the GH shifts. Figure 5(b) is the IF shifts at the Brewster angle for the x off-axis AiVBs (black solid line) and the y off-axis AiVBs (red dashed line). When the vortex’s position moves along the negative directions of the x-axis or the y-axis, the IF shift at the Brewster angle also increases in a tiny distance. While both of the IF shifts at the Brewster angle decline rapidly from a positive big IF shift to a negative big IF shift at $x=\pm 0.004$ and $y=\pm 0.004$. Similarly, when the vortex position keeps moving along the x- and the y-axis further, IF shifts gradually approach to the IF shift at the Brewster angle for the topological charge $m=0$ as the black solid line shows in Fig. 2(e).

Fig. 5. (a) Pseudo-color picture of the IF shifts at the Brewster angle for different positions of vortex. (b) IF shifts of the x off-axis AiVBs (black solid line) and of the y off-axis AiVBs (red dashed line) at the Brewster angle.

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4. Conclusion

In conclusion, the GH and the IF shifts for off-axis AiVBs at the air-weakly absorbing media interface have been studied. It is found that both the GH and the IF shifts rapidly increase with the increase of topological charges. Furthermore, with the distance between the origin point and the vortex’s position increasing, the modulation of the vortex’s position of the off-axis AiVBs leads to the GH and the IF shifts decrease to those of the Airy beam. In other words, the further the vortex’s positions are, the less the vortex influences on the GH and the IF shifts for off-axis AiVBs are. Our results might give a new perspective for controlling the GH and the IF shifts and have potential applications in optical switch and sensing.

Funding

National Natural Science Foundation of China (11374108, 11674107, 11775083, 61875057); Natural Science Foundation of Guangdong Province, China (2018A030313480).

Disclosures

The authors declare no conflicts of interest.

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Goos-Hänchen and Imbert-Fedorov shifts of off-axis Airy vortex beams

Abstract

1. Introduction

2. Theoretical analysis

3. Results and discussions

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (5)

Equations (8)

Optics Express