Control of Imbert-Fedorov shifts by the optical properties of rotating elliptical Gaussian vortex beams

Guoqi Fan; Dongmei Deng

doi:10.1364/OE.439156

1. Introduction

A long time ago, Fedorov and Imbert proved the lateral shift by using the energy flow parameter, and named it Imbert Fedorov (IF) shift [1,2]. Soon after, the study of the IF shift was further deepened, and the angular Imbert Fedorov shift (IFA) was proposed. The existence of the IF shift was extended from the total reflection to the partial reflection and the refraction. It was found that the reflection coefficient, the transmission coefficient and the polarization state were closely related to the spatial and the angular shift [3,4]. With the development of the angular spectrum theory, people begin to pay attention to the influence of the beam type on the IF shift, such as Gaussian beams [5–11], Laguerre-Gaussian beams [12–14], Hermite Gaussian beams [15], "nondiffracting" Bessel beams [16], Airy beams [17–19]. On the one hand, the rotating beams in a nonlocal medium can be obtained by the interaction of moving solitons [20]. In general, rotating elliptical Gaussian beams are obtained by Gaussian beams passing through lens groups [21]. Harrigan [22] explained in detail how a circular Gaussian beam will be converted into a rotating elliptical beam after passing through crossed cylinder tenses. Chowdhury [23] directly observed the existence of the lateral heat confinement and its corresponding effects on the unusual temperature rise during the Heat Trap effect by using a rotating elliptical laser beam. Zhou [24] proposed a multi head laser deformation equipment using rotating polygon as scanning beam splitter and modulator. In experiments, different ellipticity ratios of beams can be designed by controlling defocus and other deformation parameters. Ye [25] proposed a theoretical model of the rotating elliptical Gaussian optical coherent lattice in the ocean turbulence, and proved that the lattice constant can be used to adjust the evolution mode of the rotating elliptical Gaussian optical coherent lattice. Based on the extended Huygens Fresnel principle and the spatial power spectrum of the ocean turbulence, Sun [26] derived the analytical expression of partially coherent rotating elliptical Gaussian beams passing through the anisotropic ocean turbulence. Liu [11] studied the GH and IF shifts of rotating elliptical Gaussian beams in a weak absorbing medium. It is proved that the spatial GH and IF shifts are closely related to the optical parameters. On the other hand, due to the continuous spiral phase and the phase singularity of the vortex center [27], the vortex beams have been deeply studied in the optical field in recent years [28–30]. The feasibility of optical vortex generation in the system of uniaxial crystal with tilted Gaussian beam was theoretically and experimentally investigated by Sokolenko [31]. Masajada [32] proposed a cheap, accurate and stable method to introduce optical vortices into Gaussian beams, and carried out numerical and experimental analysis. Plociniczak [33] studied the optical system of focused Gaussian beam carrying high-order optical vortex, proposed the optical vortex motion in the beam, and carried out experimental verification. Liu [34] generated the second harmonic Airy beams and Airy vortex beams in a switchable way through the cooperation of the binary mask engineering nonlinear photonic crystal and the space variable liquid crystal geometric phase element. Sharma [35] proposed a high average power picosecond near-infrared tunable optical vortex source, which uses an antiresonance interferometer inside the optical parametric oscillator and an external cylindrical lens for the astigmatism mode conversion. Gao [19] studied the GH and IF shifts of finite energy off-axis Airy vortex beams, and the effects of the topological charge and the off-axis position on the GH and IF shifts are analyzed in detail by numerical calculation. However, the IF shift of rotating elliptical Gaussian vortex beams (REGVBs) has not been reported. In this paper, the IF shift of REGVBs will be calculated analytically.

In this paper, the angular spectrum of REGVBs is derived analytically. The IF shift of REGVBs near the Brewster angle is calculated analytically and numerically when the topological charge is 1. The influence of the topological charge and the off-axis position of REGVBs on the IF shift is analyzed in detail. The results show that with the increase of the topological charge, the value of the IF shift increases gradually, and the farther away the vortex is from the axis, the smaller the influence of the vortex on the IF shift. We also find that the special properties of REGVBs can be used to control the direction of the IF shift.

2. Theoretical model

According to the theory of geometric optics, we consider the REGVBs which come from media 0 impinging on media 1 along the z axis. Therefore, it is useful to define two reference frames. The first reference frame represents the incident coordinate system (x$_i$, y$_i$, z$_i$) of the incident beam propagating in medium 0, and the second reference coordinate system represents the reflection coordinate system (x$_r$, y$_r$, z$_r$) of the reflected beam propagating in medium 1.

Based on the Fourier representation, the incident electric field, in the incident and the reflect frames, can be written as [36,37]

(1)$$E_{I}\left(\vec{r}_{I}\right)=\iint_{-\infty}^{+\infty} \widetilde{E}_{I}\left(\vec{k}_{I}\right) e^{2 \pi i\left(\vec{k}_{I} \cdot \vec{r}_{I}\right)} d k_{x} d k_{y},$$

where the wave vector can be expressed as $\vec {k}_{i}=\hat {x}_{i} k_{x}+\hat {y}_{i} k_{y}+\hat {z}_{i} k_{z}$, $\widetilde {E}_{I}$ can be obtained from the following expression

(2)$$\tilde{E}_{I}\left(\vec{k}_{I}\right)=\sum_{\gamma}^{p, s} \hat{e}_{\gamma}\left(\vec{k}_{I}\right) \alpha\left(\vec{k}_{I}\right) \tilde{A}(\vec{k}),$$

where $\gamma$ indicates p-polarization or s-polarization. To understand $\gamma$, instead, one should note that when denoting p-polarization or s-polarization, basis vectors can be characterized by $\hat {e}_{p}=\left (\hat {e}_{s} \times \vec {k}\right ) /\left |\hat {e}_{s} \times \vec {k}\right |$ and $\hat {e}_{s}=(\hat {z} \times \vec {k}) /|\hat {z} \times \vec {k}|$. Moreover, $a_{\gamma }(\vec {k})$ which is defined by $a_{\gamma }(\vec {k})=\hat {e}_{\gamma } \cdot \hat {f}$ is polarized vector spectral amplitudes, with $\hat {f}=a_{p} \hat {x}+a_{s} e^{i \eta } \hat {y}$ being the unit vector in the incident frame. The different polarization states are respectively shown by $a_{p}$ , $a_{s}$ which are linked by relation $\left |a_{p}\right |^{2}+\left |a_{s}\right |^{2}=1$, and the phase difference information between the p-polarization and the s-polarization is shown by $\eta$. $\tilde {A}(\vec {k})$ is the angular spectrum of the incident field.

In the spatial domain, it is assumed that the initial field of the REGVBs can be shown in the following form

(3)$$E(\mathrm{x}, \mathrm{y}, 0)=e^{-\frac{x^{2}}{a^{2} w_{0}^{2}}-\frac{y^{2}}{b^{2} w_{0}^{2}}-\frac{i x y}{c^{2} w_{0}^{2}}}\left(\frac{x-x_{0}}{w_{0}}+i \frac{y-y_{0}}{w_{0}}\right)^{m}$$

where $w_{0}$ is the beam waist of REGVBs, $a$ and $b$ are the elliptical parameters of the REGVBs separately, $c$ is the rotating parameter, ($x_{0}$, $y_{0}$) is the vortex’s position, and $m$ is the topological charge of the vortex.

In this case, the impinging electric field can be written, in the incident frame, by the Fourier transform

(4)$$\tilde{A}(U, V)=\left(A_{1} U i+A_{2} V\right) e^{B_{1} V^{2}+B_{2} U V i+B_{3} U^{2}},$$

where U,V are the frequency domain coordinates,

(5)$$\begin{aligned} & A_{1}={-}a^{3} w_{0}^{2} \pi^{2}\left(\frac{4 c^{4}+a^{2} b^{2}}{4 c^{4} b^{2} w_{0}^{2}}\right)^{-\frac{1}{2}}+\left(\frac{a^{5} \pi^{2}-2 a^{3} c^{2} \pi^{2}}{4 c^{4}}\right)\left(\frac{4 c^{4}+a^{2} b^{2}}{4 c^{4} b^{2} w_{0}^{2}}\right)^{-\frac{3}{2}}, \\ & A_{2}=\left(a \pi^{2}-\frac{1}{2 c^{2}} a^{3} \pi^{2}\right)\left(\frac{4 c^{4}+a^{2} b^{2}}{4 c^{4} b^{2} w_{0}^{2}}\right)^{-\frac{3}{2}}, \\ & B_{1}=\frac{-4 c^{4} \pi^{2} b^{2} w_{0}^{2}}{4 c^{4}+a^{2} b^{2}}, \quad B_{2}=\frac{4 c^{2} \pi^{2} a^{2} b^{2} w_{0}^{2}}{4 c^{4}+a^{2} b^{2}}, \quad B_{3}=\frac{\pi^{2} a^{4} b^{2} w_{0}^{2}}{4 c^{4}+a^{2} b^{2}}-a^{2} \pi^{2} w_{0}^{2}. \end{aligned}$$

The information of the reflected electric field and the angular spectrum of REGVBs is included in

(6)$$\tilde{E}_{r}\left(\vec{k}_{r}\right)=\sum_{\gamma}^{p, s} \hat{e}_{\gamma}\left(\vec{k}_{r}\right) a_{\gamma}\left(\vec{k}_{r}\right) \tilde{A}\left(\vec{k}_{i} ; Z_{i}\right).$$

In the expression above, the $\vec {k}_{r}$ which is the wave vector of reflected beams can be obtained from the geometrical relationship law $\vec {k}_{r}=\vec {k}_{i}-2 \hat {z}\left (\hat {z} \cdot \vec {k}_{i}\right )$ between $\vec {k}_{i}$ and $\vec {k}_{r}$. When the Fresnel reflection coefficients are taken into account, the $r_{\gamma }\left (\vec {k}_{i}\right )$ should be expressed as [36,37]

(7)$$r_{p}\left(\vec{k}_{i}\right)=\frac{\varepsilon k_{z}-\sqrt{\varepsilon k_{0}^{2}-k_{x}^{2}-k_{y}^{2}}}{\varepsilon k_{z}+\sqrt{\varepsilon k_{0}^{2}-k_{x}^{2}-k_{y}^{2}}}, \quad r_{s}\left(\vec{k}_{i}\right)=\frac{k_{z}-\sqrt{\varepsilon k_{0}^{2}-k_{x}^{2}-k_{y}^{2}}}{k_{z}+\sqrt{\varepsilon k_{0}^{2}-k_{x}^{2}-k_{y}^{2}}},$$

where $\varepsilon =\frac {\varepsilon _{1}}{\varepsilon _{0}}$. In addition, to calculate the IF shifts of REGVBs, it is useful to recall the following well-known expressions [17,19]

(8)$$k_{0} \Delta_{I F}={-}\frac{\iint_{-\infty}^{+\infty} \operatorname{Im}\left[\tilde{E}_{R}^{*} \frac{\partial}{\partial V} \tilde{E}_{R}\right] d U d V}{\iint_{-\infty}^{+\infty}\left|\tilde{E}_{R}\right|^{2} d U d V}+\frac{\iint_{-\infty}^{+\infty} \operatorname{Im}\left[\tilde{E}_{I}^{*} \frac{\partial}{\partial V} \tilde{E}_{I}\right] d U d V}{\iint_{-\infty}^{+\infty}\left|\tilde{E}_{\mathrm{I}}\right|^{2} d U d V}.$$

Then, the angular spectrum given by Eq. (4) can be used to calculate the beam shifts of REGVBs, we obtain the following result

(9)$$\begin{aligned} & k_{0} \Delta_{I F}^{R E G V B}=\frac{k_{0} \Delta_{I F}^{G}-C_{1}}{C_{2}}, \\ & C_{1}=\frac{3 \pi}{4} A_{1} A_{1} B_{2}\left(2 B_{3}\right)^{-\frac{5}{2}}\left(2 B_{1}\right)^{-\frac{1}{2}}+\left(\frac{\pi}{32} A_{2} B_{2} B_{1}^{{-}1}+\frac{\pi}{8} A_{1}\right) A_{2} B_{1}^{-\frac{1}{2}} B_{3}^{-\frac{3}{2}}, \\ & C_{2}={-}\frac{\pi}{8}\left(B_{1} B_{3}\right)^{-\frac{1}{2}}\left(A_{1} A_{1} B_{3}^{{-}1}+A_{2} A_{2} B_{1}^{{-}1}\right), \end{aligned}$$

where $k_{0} \Delta _{I F}^{G}$ is the IF shift of Gaussian beams. It is worth noting that the effect of the vortex position on the IF shift is ignored in the expression of the analytical solution. And the IFA can be easily calculated by using the same method as the IFA of Gaussian beams [5]. Thus,

(10)$$\begin{aligned} & \Theta_{I F}^{R E G V B}=2 \frac{C_{3}}{C_{2}} \cot \theta \cos \eta \frac{a_{s}^{2} w_{p}-a_{p}^{2} w_{s}}{a_{p} a_{s}}, \\ & C_{3}=\left(\frac{3 \pi}{4} A_{2} A_{2}\left({-}2 B_{1}\right)^{-\frac{5}{2}}\left({-}2 B_{3}\right)^{-\frac{1}{2}}+\frac{\pi}{32} A_{1} A_{1}\left(B_{1} B_{3}\right)^{-\frac{3}{2}}\right). \end{aligned}$$

where $w_{\gamma }=\frac {a_{\gamma }^{2} R_{\gamma }^{2}}{a_{s}^{2} R_{s}^{2}+a_{p}^{2} R_{p}^{2}}$ [17].

3. Results and discussions

Before numerically calculating, it is worth building a model which is used to explain the IF-shift phenomenon of the reflective REGVB at the surface dividing the vacuum and the weakly absorbing medium with dielectric constant $\varepsilon =2+0.02 i$ as shown in Fig. 1. For later convenience, we can choose related parameters: $\lambda =633 \mathrm {~nm}$, $w_{0}=1 \mathrm {~mm}$, $\mathrm {a}=0.1, \mathrm {~b}=0.1$ and $\mathrm {c}=0.1$, where $\lambda$ means the wavelength of the REGVBs.

Fig. 1. The diagram of the IF shift reflected from the surface between the air and the weak absorbing medium. $\varepsilon _{0}$ and $\varepsilon _{1}$ represent the dielectric constant of the air and the weak absorbing medium respectively.

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To start our analysis, Fig. 2(a) shows the IF shift of REGVBs, the solid line is the numerical solution of REGVBs numerically simulated by angular spectrum method [36], and the symbol "x" is the analytical solution of REGVBs. From Fig. 2(a), we can see that the analytical solution and the numerical solution are basically equal, which proves that the calculation accuracy is satisfactory. Figure 2(b) compares the IF shift of REGVBs, Gaussian beams and finite energy Airy beams. We find that changing the incident beam may be a feasible method to adjust the IF shift. Interestingly, when the polarization state of the incident beam is linear, the reflected beam is symmetrically divided into a pair of the left-handed and the right-handed circularly polarized beams on the incident plane, which is optically called the spin Hall effect [38–41]. Therefore, for linearly polarized incident beams, the IF shifts should be equal to zero. This explanation is applicable to the IF shift of p-polarized Gaussian beams. In Eq. (9), since $a_{s}$= 0, the IF shift of Gaussian beams is equal to 0. As shown in Fig. 2(b), the IF shift of p-polarized finite energy Airy beams and p-polarized REGVBs are similar to even symmetry and odd symmetry near the Brewster angle, respectively. This phenomenon is caused by the odd order term of the phase part of $\tilde {A}(U, V)$ and the second order term of the reflection coefficient.

Fig. 2. (a) The analytic IF shift of REGVBs (’x’ symbol) and numerical IF shift of REGVBs when m=1 (solid line). (b) The numerical IF shift of REGVBs when m=0 (solid line), the numerical IF shift of Gaussian beams (dash line) and the numerical IF shift of finite energy Airy beams (dotted line).

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Next, the effect of the topological charge m on the IF shift is discussed when the vortex center is located at the origin of the incident coordinate system. Figures 3(a), (b) and (c) are the intensity distribution of REGVBs when m = 0, m= 1 and m = 2, respectively. Obviously, with the increase of the topological charge, the intensity at (0,0) changes to zero, the intensity near (0,0) decreases, and the intensity peaks on x-axis and y-axis diffuse to all sides. Figure 3(d) shows the IF shift of different topological charges near the Brewster angle. We numerically calculate and compare the IF shift when the topological load m = 0, 1, 2. The red solid line shows the relationship between the IF shift and the incident angle when m = 0, which is discussed by Liu [11]. The IF shift increases first and then decreases with the increase of the incident angle, showing a similar to the odd symmetry around the Brewster angle, with a peak value of 105$\lambda$. When the topological charges are 1 and 2, as shown by the green dashed line and the blue dotted line in Fig. 3(d), the trend of the IF shift is similar to that of m = 0, while the peaks of m = 1 and m= 2 increase sharply to 420 $\lambda$ and 730 $\lambda$, respectively.

Fig. 3. (a-c) Intensity distributions of the incident REGVBs for topological charges m = 0, m = 1 and m = 2. (d) IF shifts for different topological charges near the Brewster angle.

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In the following discussion, we select the first-order topological charge to study the IF shift of REGVBs, discuss the influence of the vortex position on the IF shift, and propose two peculiar properties of REGVBs in the control of the IF shift. In order to illustrate the relationship between the IF shift and the vortex position, Fig. 4(a) shows pseudo-color picture of the IF shift at different vortex positions with $\theta = 54.6^{\circ }$, and Fig. 4(b) is the two-dimensional relation curve of the IF shifts of the off-axis REGVBs at $\theta = 54.6^{\circ }$ and $\theta = 54.87^{\circ }$, respectively. Generally speaking, the movement of the vortex position along the positive and the negative directions of the x-axis leads to the decrease of the symmetry of the IF shift. It is noticed that in Fig. 4(a), the value of the IF shift has a depression on both sides of the peak along the positive and negative directions of the x-axis of the vortex position. Combining with Fig. 4(b), it can be found that when the position of the vortex moves along the positive and the negative direction of the x-axis, the IF shift will have a slight mutation within a small distance, and then change sharply in reverse with the increase of the distance between the vortex position and the origin. It can be found that for the first-order topological charge of off-axis REGVBs, the farther the position of the vortex is, the smaller the influence of the vortex on the IF shift is. However, through the analysis of Fig. 4(b), it can also be found that no matter the vortex position moves to the x-axis or the y-axis, the IF shifts of $\theta = 54.6^{\circ }$ and $\theta = 54.87^{\circ }$ are the same, but the direction is opposite. This shows that the direction of the IF shift can be well controlled by adjusting the incident angle, which is the first peculiar property of REGVBs in the control of the IF shift.

Fig. 4. (a) Pseudo-color picture of the IF shift of the REGVBs at $\theta = 54.6^{\circ }$ for different vortex positions. (b) IF shifts for REGVBs of the x-offaxis at $\theta = 54.6^{\circ }$ (red solid line), the y-offaxis at $\theta = 54.6^{\circ }$ (black dotted line), the x-offaxis at $\theta = 54.87^{\circ }$ (green dash line), the y-offaxis at $\theta = 54.87^{\circ }$ (blue dash-dotted line).

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Next, we introduce another peculiar property of REVGBs in the control of the IF shift, as shown in Fig. 5(a). The black dotted line and the green dash-dotted line respectively indicate the influence of ellipse parameters a and b on the IF shift. It can be found that the influence of ellipse parameters a and b on the IF shift is mainly reflected in the numerical value. However, as shown in the red solid line of Fig. 5(a), when the value of the rotation parameter c is less than about 0.02, The direction of the IF shift is negative, but when the rotation parameter c is greater than about 0.02, the direction of the IF shift becomes positive. This shows that the value of the rotation parameter c can not only change the value of the IF shift, but also change the direction of the IF shift, which provides us with an idea to control the direction of the IF shift. As shown in Fig. 5(b), by changing the value of the rotation parameter c, two groups of the IF shift with the same value and the opposite direction are obtained.

Fig. 5. (a) The two-dimensional relationship between the IF shift and optical parameters at $\theta = 54.6^{\circ }$. (b) IF shifts for different rotating parameters near the Brewster angle.

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4. Conclusion

In a word, this paper introduces the theoretical calculation of the IF shift of REGVBs with the topological charge of 1, and studies the influence of the topological charge and the vortex position on the IF shift of REVGBs. The results show that the IF shift can be effectively changed by adjusting the topological charge and the vortex position. By comparing the IF shifts of Gaussian beams, Airy beams with finite energy and REVGBs, we find that the IF shifts of REVGBs have the properties similar to the odd symmetry near the Brewster angle, and the rotation parameter $c$ can specially regulate the value and the direction of IF shifts at the same time. At present, the direction control of the IF shift mostly depends on the reflective material. However, material based control often requires frequent replacement of materials and is not easy to obtain continuous IF shift values. Adjusting the IF shift based on the optical properties of REVGBs can change the way of changing reflective materials into some more convenient and feasible ways, such as adjusting incident angle, adding vortex lens and adjusting optical parameters. Our research opens up a new way to control the direction of the IF shift through the properties of the beam, which has a potential application prospect in the field of optical switches and sensors.

Funding

National Natural Science Foundation of China (11374108, 11775083, 12174122); Guangzhou Science and Technology Program key projects (2019050001).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Control of Imbert-Fedorov shifts by the optical properties of rotating elliptical Gaussian vortex beams

Abstract

1. Introduction

2. Theoretical model

3. Results and discussions

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (10)

Optics Express