1. Introduction

Transverse spin (T spin), an interesting component of electromagnetic (EM) field spin, has attracted much attention in recent years [1–4]. In contrast to the conventional longitudinal spin, the helix-independent T spin is perpendicular to the direction of light propagation. In physics, the T spin is related to the inhomogeneity of momentum distribution in the EM field. The direction of the spin in the evanescent waves is locked by momentum, leading to the concept of optical spin-momentum locking [1,2,5–9]. For beams with complex momentum structures, the T spin originates from the longitudinal electric field, as explained by the theory of photonic wheels [10]. As a physically measurable quantity [11–13], T spin sparks research interest in a wide range of fascinating phenomena such as optical manipulation [9,11,13–15], imaging [16], and communications [17] due to its unique physical properties.

Bessel beams are typical nondiffractive beams, and have properties of self-healing and self-accelerating [18–21]. In the conventional methods, Bessel beams can be generated by utilizing diffractive optical elements or axicons with a high numerical aperture (NA) [22,23], which provides the possibility of applying T spin according to the theory of photonic wheels [10]. However, employing axicons result in a complex optical system with limited flexibility and substantial volume. Fortunately, metasurfaces have emerged as potential candidates for manipulating light in compact optical systems due to the outstanding advantages of their thinness, ease of integration, and design flexibility [24–26]. With the capability to flexibly manipulate the wavefront, polarization and dispersion [27,28], the metasurface allows for the generation of Bessel beams with high NA, which also implies that the metasurface can be a powerful platform for the observation and application of T spins. In addition, compared to conventional tightly focused beams (such as Gaussian beams [29]), the nondiffractive beams allow the pure T spin not restricted to the focal plane. In order to obtain the pure T spin in the whole space, we choose nondiffractive Bessel beams to study.

In this work, we analytically and numerically demonstrate the pure T spin existing in Bessel beams, which is oriented along the azimuthal direction and polarization-dependent. Based on the principle of the Pancharatnam-Berry (PB) phase [24,30,31], we designed a single layer dielectric metasurface, in which arrayed $\text {TiO}_2$ nanopillars provide the required geometrical phase distribution for the incident linearly polarized (LP) light and efficiently generate Bessel beams with pure T spin. According to the design, the different linear polarizations of the incident light affect the polarization of the scattered vector Bessel beam, which results in altering the ratio of the electric and magnetic spin components. To be specific, $x$ LP incident light generates a T spin with the pure electric component, and in contrast, $y$ LP incident light generates a pure magnetic T spin. In addition, the T spin allows the local EM field to oscillate around the transverse axis, which is perpendicular to the direction of propagation [29,32]. Upon analyzing the local polarization, we observe that the C point [29,33], symbolizing the transition of the polarization ellipse into a perfect circle, displays a periodic circular distribution within the scattered Bessel beam.

2. Method and structure description

To obtain the pure T spin in the beam, the electric field of the vector Bessel beam is analyzed, which is written in the cylindrical coordinate as [34–37],

As a consequence, we utilize the principle of the PB phase to design the metasurface [24]. Since the adjustment of the meta-atom, the circularly polarized incident light is converted into the scattered cross-polarized field and carries a phase of $2\sigma \theta$ [39], where $\sigma$ characterizes the polarization state of the incident light, $\sigma =\pm 1$ corresponds to the right (RCP) and left (LCP) circularly polarized light, respectively. And $\theta$ is the rotation angle of the meta-atom. Actually, this additional phase is a geometric phase resulting from the evolution of the polarization state of light on the Poincare sphere. The phase imparted to the incident light from an axicon can be expressed as $\varphi _1=-kr\text {sin}\alpha$ [31,39,40], and we consider loading this phase for the meta-atom. Here, the numerical aperture of the axicon is $\text {NA}=\text {sin}\alpha$. For an LP incident light, the meta-atoms individual effect of its LCP and RCP components. As an example, for $x$ LP light $\begin {bmatrix}1 \\0\end {bmatrix}=\frac {1}{2}\left (\begin {bmatrix}1 \\i\end {bmatrix}+\begin {bmatrix}1 \\-i\end {bmatrix} \right )$, to obtain a scattered Bessel beam $\frac {1}{2}\left (\begin {bmatrix}1 \\-i\end {bmatrix}e^{ikr\text {sin}\alpha }+\begin {bmatrix}1 \\i\end {bmatrix}e^{-ikr\text {sin}\alpha } \right )=\begin {bmatrix}1 \\0\end {bmatrix}e^{-ikr\text {sin}\alpha }$, we expect that the RCP and LCP components of the LP incident light are compensated for the phase with a difference of $2m\pi$ [39], i.e., $2kr\text {sin}\alpha =2m\pi$, where $m$ is a positive integer. This implies that the nanopillars are arranged with certain conditions in the radial direction. Thereafter, in order to induce the electric field in the transverse direction as shown in Eq. (3), the phase $\varphi _2=\text {arctan}(y/x)$ is assigned to the meta-atom, which causes different circularly polarized light to acquire opposite topological charges [31,40], and it is the key to generating radial polarization. For $x$ LP incident light, the RCP and LCP components obtain the phase of $\pm \varphi _2$ respectively, and the scattered light can be written as $\frac {1}{2}\left (\begin {bmatrix} 1 \\ -i \end {bmatrix} e^{i \varphi _2}+\begin {bmatrix} 1 \\ i \end {bmatrix} e^{-i \varphi _2}\right ) e^{-i k r\text {sin}\alpha }=\begin {bmatrix} \text {cos}\phi \\ \text {sin}\phi \end {bmatrix}$, which is exactly the radially polarized light. In summary, the characterizations of the designed metasurface can be expressed as

Equation (5) characterizes the angle of rotation that the meta-atoms acquire and the positional conditions that should be satisfied to arrange them.

We designed a metasurface to produce the Bessel beam with a cone angle $\alpha =70^{\circ }$ and pure T spin when illuminated by LP incident light with a wavelength $\lambda =630$ nm. Figures 1(a) and 1(b) indicate that for $x$ and $y$ LP incidence, the metasurface generates pure electric spin $\mathbf {s}_E$ and magnetic spin $\mathbf {s}_H$, respectively. To establish the condition in Eq. (5), the meta-atoms are arranged in cylindrical coordinate $(r,\phi,z=0)$ and the interval between the meta-atoms in the radius direction is calculated to be $\delta r=335$ nm, with 40 arranged rings. In the azimuthal direction, the number of nanopillars arranged on the $m$th circle is 6$m$. As shown in Fig. 1(c), the metasurface consists of $\text {TiO}_2$ nanopillars and a glass substrate (with diameter $D=27$ $\mathrm{\mu}$m), and the relative refractive indexes are set as 2.40 and 1.45, respectively. To obtain high phase response and transmittance, the dimensions of the $\text {TiO}_2$ nanopillars are chosen to be length $W_x=330$ nm, width $W_y=90$ nm, and height $H=600$ nm (black dot in Figs. 1(d) and (e)). Figures 1(f) and (g) demonstrate the additional phases $\varphi _\text {RCP}$ and $\varphi _\text {LCP}$ produced by the metasurface for the RCP and LCP components, respectively.

 

Fig. 1. Schematic illustration of the metasurface. (a) and (b): Incident $x$ and $y$ LP lights generate pure electric spin $\mathbf {s}_E$ and magnetic spin $\mathbf {s}_H$, respectively. (c): Structural illustration of the meta-atoms, which consists of $\text {TiO}_2$ nanopillars and the glass substrate (with diameter $D=27$ $\mathrm{\mu}$m). (d) and (e): For the $x$ LP incident light, calculated phase response $\varphi _x$ and transmittance $T_x$ as functions of nanopillars’ dimensions $W_x$ and $W_y$ at the wavelength 630 nm, and the height is chosen $H=600$ nm. The black dot denotes selected structural parameters of the nanopillar with length $W_x=330$ nm and width $W_y=90$ nm. The nanopillar with a rotation angle of $\theta$ attaches a phase of $2\sigma \theta$ to the plane wave. The phase loaded on the metasurface consists of $\varphi _1$ and $\varphi _2$. And the phase diagrams of the RCP and LCP lights are shown in (f) and (g).

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3. Results and discussions

Firstly, we give the analytical calculations according to Eqs. (3) and (4). Figures 2(a) and 2(b) show the intensity distributions of the electric and magnetic fields in the Bessel beams, respectively. The operating wavelength and the cone angle are also fixed at $\lambda =630$ nm and $\alpha =70^{\circ }$. While the polarization parameters $c_1=0$, $c_2=1$ ($c_1=1$, $c_2=0$) correspond to the radially polarized electric field (magnetic field) in Fig. 2(a) (Fig. 2(b)). The phase difference between the longitudinal and radial fields (see insets of Fig. 2(a) and 2(b)) creates the azimuthal T spin, which is the pure T spin in Eq. (4). Figure 2(c) is the analytical calculation according to Eq. (4) on the $xy$ plane. The intensity of spin density $|\mathbf {s}|^2$, and its components $|s_x|^2$, $|s_y|^2$ and $|s_z|^2$ remain identical for both $c_1=0$, $c_2=1$, and $c_1=1$, $c_2=0$ in Eqs. (4b) and (4c), so we only have to give the intensity of $c_1=0$, $c_2=1$ in Fig. 2(c). The intensities of the transverse components of the spin density ($|s_x|^2$ and $|s_y|^2$) show a two-lobe structure, while the intensity of the longitudinal component ($|s_z|^2$) is strictly zero, indicating the presence of pure T spin in the beam. Simulations with the three dimensional finite difference time domain (FDTD) method (Lumerical Solutions) are performed to demonstrate the designed metasurface. The entire simulation is immersed in a vacuum and selected as a perfectly matched layer (PML) in the $x$, $y$ and $z$ directions, and the incident light is an LP plane wave propagating in the $z$ direction. The radially polarized ($c_1=0$, $c_2=1$) Bessel beam $\begin {bmatrix}\text {cos}\phi \\\text {sin}\phi \end {bmatrix} e^{-ikr\text {sin}\alpha }$ in Fig. 2(a) is induced by the $x$ LP incident plane wave $\begin {bmatrix}1 \\0\end {bmatrix}$. Figure 3 is the numerical results of the metasurface. The electric field intensities of the scattered Bessel beam are exhibited in Fig. 3(a), which correspond to the analytical result of Eq. (3), and the $y$ component of the electric field is neglected in the $xoz$ plane. Hereinafter, we fix the reference plane at $z = 3$ $\mathrm{\mu}$m. Figures 3(b) and 3(c) show the intensity distributions of the spin density, which are normalized by the maximum value of $|\mathbf {s}_E|^2$. As shown in Fig. 3(b), the intensity distributions of the electric spin density $|\mathbf {s}_{E}|^2$, and its components $|s_{E,x}|^2$, $|s_{E,y}|^2$ and $|s_{E,z}|^2$, which are similar to the analytical results in Fig. 2(c). Figure 3(c) shows the intensity distributions of the magnetic spin densities, which are much smaller than the electric spin density. Therefore, the pure T spin originating from electric fields is observed in scattered Bessel beams, which corresponds to the analytical predictions in Fig. 2(b).

Switch the incident light to $y$ LP light $\begin {bmatrix}0 \\1\end {bmatrix}=\frac {i}{2}\left (\begin {bmatrix}1 \\i\end {bmatrix}e^{i\pi }+\begin {bmatrix}1 \\-i\end {bmatrix}\right )$, the polarization of the scattered Bessel beam is changed into $c_1=1$, $c_2=0$, which leads the pure T spin originated from the magnetic spin component $\mathbf {s}_H$. This function, which tunes the polarization and spin of the Bessel beam by merely switching the incident plane wave, is impossible to achieve with traditional optical elements such as axicon. The effect of the metasurface on the $y$ LP light is $\frac {i}{2}\left (\begin {bmatrix} 1 \\ -i \end {bmatrix}e^{i(\pi +\varphi _2)}+\begin {bmatrix} 1 \\ i \end {bmatrix} e^{-i \varphi _2}\right ) e^{-i k r\text {sin}\alpha }=-\begin {bmatrix} -\text {sin}\phi \\ \text {cos}\phi \end {bmatrix} e^{-i k r\text {sin}\alpha }$, which indicates that the transverse component of electric field in the Bessel beam is azimuthally polarized according to Eq. (3), and the transverse magnetic field is radially polarized. The simulation results of $y$ LP incidence are shown in Fig. 4. Figure 4(a) indicates the magnetic field intensities of the scattered Bessel beam in the $xoz$ plane. And the magnetic field component $|H_y|^2$, which corresponds to the Eq. (3) for parameters $c_1=1$ and $c_2=0$, is negligible. The electric spin density $|\mathbf {s}_E|^2$, and the components $|s_{E,x}|^2$, $|s_{E,y}|^2$, $|s_{E,z}|^2$ are negligible in Fig. 4(b). Figure 4(c) illustrates the intensity distribution of the magnetic spin density, which is consistent with the analytical results in Fig. 2(c). Whether the incident light is $x$ or $y$ linear polarization, the designed metasurface can flexibly generate pure T spin as analytically predicted in the demonstration, which further validates that the metasurface can be a new platform for T spin applications.

 

Fig. 2. The analytical results of the EM field and spin density in the Bessel beams. (a): The intensity of two nonzero components of the electric field $|E_r|^2$ and $|E_z|^2$ (normalized by the maximum value of $|\mathbf {E}|^2$), when setting parameters $c_1=0$ and $c_2=1$. (b): The intensity of the magnetic fields $|H_r|^2$ and $|H_z|^2$ (normalized by the maximum value of $|\mathbf {H}|^2$), when setting parameters $c_1=1$ and $c_2=0$. The phases of each EM component are shown as insets. (c): The intensity distribution of the spin density, in which the transverse component shows a two-lobe structure and the longitudinal component is zero.

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Fig. 3. The electric field and spin density in the scattered Bessel beam when the metasurface is illuminated by $x$ LP incident light. (a): The intensity of the electric field in the $xoz$ plane, normalized by the maximum value of $|\mathbf {E}|^2$. (b): The intensity of the electric spin density $|\mathbf {s}_E|^2$, and its components $|s_{E,x}|^2$, $|s_{E,y}|^2$, and $|s_{E,z}|^2$. (c): The negligible intensity of the magnetic spin density. (b) and (c) are normalized by the maximum value of $|\mathbf {s}_E|^2$ and the reference planes are fixed at $xy$ plane ($z=3$ $\mathrm{\mu}$m).

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Fig. 4. The electric magnetic field and spin density in the scattered Bessel beam when the metasurface is illuminated by $y$ LP incident light. (a): The intensity of the magnetic field in the $xoz$ plane, normalized by the maximum value of $|\mathbf {H}|^2$. (b): The negligible intensity of the electric spin density. (c): The intensity of the magnetic spin density $|\mathbf {s}_H|^2$, and its components $|s_{H,x}|^2$, $|s_{H,y}|^2$, and $|s_{H,z}|^2$. The intensities in (b) and (c) are normalized by the maximum value of $|\mathbf {s}_H|^2$ and the reference planes are also fixed at $xy$ plane ($z=3$ $\mathrm{\mu}$m).

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Furthermore, it is worth to notice that the pure T spin in the scattered Bessel beam allows the oscillation axis of the local electric field to be perpendicular to the propagation direction [32,33], thus exhibiting a remarkable polarization structure. To illustrate this phenomenon, we use the concept of C point, which represents the local polarization ellipse of the EM field at that position degenerating into a perfect circle [32], where $\mathbf {s}_E$ is the normalized electric field spin density with the electric field energy density $w_E=\varepsilon (|E_x|^2+|E_y|^2+|E_z|^2)/4$ [5,8], and

 

Fig. 5. The C points and polarization ellipses in a scattered Bessel beam, which is produced by illuminating $x$ LP light on the metasurface. (a): The distribution of $|\mathbf {s}_E|/w_E$ in the $xoz$ plane (upper panel) and $xy$ plane (lower panel and $z=3$ $\mathrm{\mu}$m), normalized by $1/\omega$. (b): Schematic illustration of the polarization ellipse with major axis $\mathbf {A}$, minor axis $\mathbf {B}$ and and normal vector $\mathbf {s}_E$. (c): The polarization ellipse and its enlarged view in the $xoz$ plane, and the background is the distribution of $|\mathbf {s}_E|/w_E$.

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4. Conclusion

In conclusion, we successfully analyze and demonstrate the pure T spin along the azimuthal direction in a Bessel beam, which is related to the polarization parameter. Based on the principle of the PB phase, we have designed a single layer dielectric metasurface that can convert $x$ LP light into a Bessel beam with pure T spin. By switching the incident light to $y$ LP light, the transverse electric field of the scattered Bessel beam is changed from radial polarization to azimuthal polarization, and the ratio of the electric and magnetic spin components is tunable. Moreover, the local EM field oscillates around the transverse axis, creating the C points with the ring profile, and the local polarization also exhibits a periodic distribution associated with these C points. With the method of FDTD, we validated the function of the metasurface and compared it with the results of analytical calculations to ensure the reliability of the results. This work enhances our understanding of spin-related physics and opens a new route for spin-based applications.