The spin of a photon is the quantum mechanical counterpart of the polarization of light, and each photon of left-/right-handed circular polarized (LHCP/RHCP) light carries a spin angular momentum of $\pm \hslash$, where $\hslash$ is the reduced Planck constant [1]. In addition to spin angular momentum (SAM), a photon of circular polarization can also have orbital angular momentum (OAM) which is dependent on the spatial degrees of freedom of light [2], including the intrinsic orbital angular momentum due to optical vortex within the beam and external orbital angular momentum due to the beam propagation in the transverse direction [3]. The spin-orbit interactions (SOIs) between these three types of angular momentums underpin a variety of interesting phenomena [4], including optical spin Hall effect [5–7], spin-directional coupling [8–12], and spin-dependent beam shaping [13–16], which have potential applications in quantum communication and spin-based optical elements. The underlying physical mechanism of SOI can be explained by the geometric Berry phase originating from the coupling between the SAM and the local coordinates of the media system [17], that is, rotating the transverse $x\text{-}y$ coordinates of the inhomogeneous media induces opposite phase shifts for LHCP and RHCP incidences. Conventionally, SOIs have been studied in helical fibers [18], gratings [13], and anisotropic crystals [15]. These systems are either of low efficiency, application-limited, or bulky. Recently, metasurfaces comprising artificial subwavelength planar structures have presented unprecedented abilities to manipulate the polarization, amplitude, and phase of light [19], offering a competitive artificial platform for flexible and considerably enhanced SOI effects [4,20,21].

The building element of SOI metasurfaces is generally called a Pancharatnam–Berry optical element (PBOE) [13,22]. A perfect PBOE will completely flip the input circular polarization to its orthogonal state with an extra geometric phase shift that is double the element’s orientation angle. Single layer plasmonic metasurfaces were first utilized to study SOI [16,23]. However, their power efficiency was low due to the compromise between anisotropy and transmittance [23]. To increase the SOI efficiency, multilayer plasmonic, coherently controlled, and dielectric metasurfaces have been adopted [22,24,25]. Specifically, dielectric metasurfaces consisting of high contrast dielectric elements have recently attracted increasing research interest [26,27]. For such a dielectric element, the anisotropy is flexibly tailorable over a large range, and the material loss can be made negligible by properly selecting dielectric materials, endowing it with a significant advantage in the search for a perfect PBOE.

Besides the power efficiency, another challenge of SOI is that the inherent geometric phase shifts of a PBOE over different circularly polarized inputs are equal in magnitude with opposite signs [16]. Therefore, the achieved functionalities are limited only through the geometric phase. To make the imposed phase shifts for RHCP and LHCP inputs independent and thus the spin-control more flexible, further transport phases have been introduced by utilizing PBOEs of various geometries [28]. Through the combination of the transport phase and the geometric phase, independent and complete phase and wavefront control over orthogonal circular polarizations has been demonstrated. Such a method has greatly expanded the applications of PBOEs in polarization optics [29–31], including polarization detection [32], splitting [33], polarization contrast imaging [34], multiplexing, and encryption [35–37].

In this context, we present a design method of spin-dependent all-silicon dielectric metasurfaces by simultaneously involving the geometric phase and transport phase in the constituting PBOEs. The PBOEs are anisotropic rectangular posts distributed in a tetragonal lattice on silicon substrate. Two functional metasurface devices are designed and experimentally characterized in the terahertz regime, including a spin-selective Bessel/half-wave metasurface device and a spin-dependent Bessel/vortex beam generator. The Bessel/half-wave metasurface generates a cross-polarized Bessel beam under the LHCP incidence, whereas it only flips the polarization under the RHCP incidence. The Bessel/vortex beam generator produces Bessel and vortex beams separately under the LHCP and RHCP incidences. Such metasurface devices have the potential to help realize miniature THz systems, with key applications in spin switching, contrast imaging, coding, and encryption.

A perfect PBOE functions as a half-wave plate, i.e., there is a 180° phase difference between the phase shifts along the local fast and slow axes of the PBOE. The corresponding rotated transmission matrix in the Cartesian coordinate system can be expressed as [33] $$\begin{gathered} \mathbf{T}=\left[\begin{array}{cc}{t}_{xx}& t_{xy}\\ t_{yx}& t_{yy}\end{array}\right]=t_0\mathbf{R}(-\alpha )\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\mathbf{R}(\alpha ) \\ =t_0{e}^{i{\phi }_f}\left[\begin{array}{cc}\cos (2\alpha )& \sin (2\alpha )\\ \sin (2\alpha )& -\cos (2\alpha )\end{array}\right]. \end{gathered}$$(1)Here, the former and latter subscripts of the transmission coefficient $t_{ij}(i,j=x,y)$ indicate the polarization directions of the output and input lights, respectively; $t_0$ is the transmission amplitude along the two eigen axes, fast ($f$) and slow ($s$); ${\phi }_f$ is the transport phase shift along the fast axis; and $\mathbf{R}(\alpha )=\left[\begin{array}{cc}\cos (\alpha )& \sin (\alpha )\\ -\sin (\alpha )& \cos (\alpha )\end{array}\right]$ is the rotation matrix, in which $\alpha$ denotes the rotation angle of the PBOE’s local fast axis with respect to the $x$ coordinate. To directly study the spin-dependent response of the PBOE, it is required to further transfer the above matrix into the circular polarization base $$\begin{gathered} {\mathbf{T}}_c=\left[\begin{array}{cc}{t}_{ll}& t_{lr}\\ t_{rl}& t_{rr}\end{array}\right]=t_0\left[\begin{array}{cc}0& e^{i({\phi }_f-2\alpha )}\\ e^{i({\phi }_f+2\alpha )}& 0\end{array}\right] \\ =t_0\left[\begin{array}{cc}0& e^{i{\mathrm{\Phi }}_{lr}}\\ e^{i{\mathrm{\Phi }}_{rl}}& 0\end{array}\right], \end{gathered}$$(2)where the subscripts $l$ and $r$ indicate LHCP and RHCP, respectively. It is easily seen from Eq. (2) that, with a perfect PBOE, the LHCP light can be completely converted to RHCP light and vice versa. The overall phase shift ${\mathrm{\Phi }}_{lr}$ for the RHCP incidence equals ${\phi }_f-2\alpha$, and the overall phase shift ${\mathrm{\Phi }}_{rl}$ for the LHCP incidence equals ${\phi }_f+2\alpha$. In addition to the geometric phase terms $2\alpha$ and $-2\alpha$ determined by the rotation angle, the output phase shifts for the LHCP and RHCP incidences also contain the transport phase term ${\phi }_f$, which is determined by the dimensions of the PBOE element. This provides an effective freedom in decoupling the overall output phases ${\mathrm{\Phi }}_{lr}$ and ${\mathrm{\Phi }}_{rl}$. With the desired ${\mathrm{\Phi }}_{lr}$ and ${\mathrm{\Phi }}_{rl}$, the corresponding required transport phase ${\phi }_f$ and rotation angle $\alpha$ can be obtained as $${\phi }_f=\frac{{\mathrm{\Phi }}_{rl}+{\mathrm{\Phi }}_{lr}}{2},$$(3a)$$\alpha =\frac{{\mathrm{\Phi }}_{rl}-{\mathrm{\Phi }}_{lr}}{4}.$$(3b)Thus, by varying ${\phi }_f$ and $\alpha$ of the PBOEs, complete and independent control over the orthogonal circular polarizations can be achieved. As ${\phi }_f$ is determined by the geometric parameters, PBOEs of different dimensions are needed to achieve independent phase patterns ${\mathrm{\Phi }}_{lr}$ and ${\mathrm{\Phi }}_{rl}$. Therefore, two key criteria should be taken into consideration: (1) a 180° phase difference between the two local axes with equal transmission amplitudes for every single PBOE is required to ensure perfect cross polarization conversion, and (2) the relative transport phase shifts ${\phi }_f$ of the PBOEs need to enable the complete 360° phase coverage of ${\mathrm{\Phi }}_{lr}$ and ${\mathrm{\Phi }}_{rl}$.

To further investigate the phase control and cross-polarization conversion performances of the 8 fundamental PBOEs, the electric field distributions at 1.0 THz are simulated, as shown in Fig. 2(a). It can be seen that the output phases of $E_x$ and $E_y$ are out of phase in each PBOE, and both of them gradually change by 180° from No. 1 to No. 8. As for the last 7 PBOEs (No. 9 to No. 15), the corresponding simulated electric field distributions under $x$- and $y$-polarized incidences are the same as those of the first 7 PBOEs under $y$- and $x$-polarized incidences. Figures 2(b) and 2(c) illustrate the simulated transmission amplitude spectra $|t_{yx}|$ and $|t_{xx}|$ of the 8 fundamental PBOEs, whereas Fig. 2(d) illustrates the corresponding polarization conversion rates (PCRs) calculated by $|t_{yx}{|}^2/({|t_{yx}|}^2+{|t_{xx}|}^2)$. It can be seen that the $y$-polarized (cross-polarized) transmissions are obviously larger than those of the $x$-polarized (co-polarized) transmissions at around 1.0 THz for all PBOEs, where the PCRs all nearly approach 1. To show the cross-polarization conversion performances more clearly, the corresponding $|t_{yx}|,|t_{xx}|$, and PCR values at 1.0 THz marked by the dashed red lines are extracted in Fig. 2(e).

To experimentally verify that the selected PBOEs meet the two criteria, i.e., perfect cross-polarization conversion and complete phase coverage, a polarization conversion deflector (PCD) is first constructed by patterning the PBOEs into a linear phase distribution along the $x$-axis. To enable the complete phase coverage of 360° with a phase increment of 22.5°, the super unit cell comprises 16 PBOEs of which the corresponding phase shifts range from 0° to 337.5° in sequence. Therefore, the periods are $D_x=16p_x=2.4\mathrm{mm}$ along the $x$-direction and $p_x=0.15\mathrm{mm}$ along the $y$-direction. The orientation angles of all the PBOEs are 45°. Deep reactive ion etching process is employed for the fabrication; see Appendix B. The total size of the fabricated PCD sample is $9.6\mathrm{mm}\times 9.6\mathrm{mm}$, which is the same for all devices in this work. Figure 3(a) shows the optical microscope image of the fractional PCD sample.

The PCD was then characterized using our customized all-fiber terahertz time-domain spectrometer (TDS) system [38]. For bandwidth characterization and simplicity, a linearly polarized input is used here. Figures 3(b) and 3(c) show the measured $y$- and $x$-polarized outputs under $x$-polarized input. It can be seen that almost all the $x$-polarized input is converted to $y$-polarized light and deflected to the $+1$st diffraction order, which confirms both the ability to achieve a full phase coverage and the 180° phase difference between ${\phi }_f$ and ${\phi }_s$. Notably, the PCD also supports broadband operation. To illustrate the bandwidth, the power efficiency as a function of frequencies is shown in Fig. 3(d). Here, the power efficiency is defined as the ratio of the output power at the desired polarization and diffraction order to the overall output power. Figure 3(d) highlights that the maximum power efficiency is about 91% and appears at 1.1 THz. The deviation of this operating frequency from the design is likely due to fabrication accuracy. Thus, we focus on the following device performance at 1.1 THz. Figure 3(e) shows the measured normalized intensity curves of the $x$- and $y$-polarized outputs at 1.1 THz. The measured deflection angle is ${\theta }_d=6.5°$, which coincides with the general Snell's law ${\theta }_d=\arcsin (\lambda /D_x)$ [39].

From the measured results of the PCD, it can be concluded that the selected basic PBOEs can well satisfy our requirements in achieving efficient and independent control over the complete phases of the orthogonal circular polarizations at 1.1 THz. Following the above experimental confirmation, we further fabricate two functional devices.

One is a Bessel/half-wave metasurface, which serves as a Bessel generator under the LHCP incidence but as a half-wave plate under the RHCP incidence. To generate the Bessel beam under the LHCP incidence, a linear gradient of ${\mathrm{\Phi }}_{rl}$ along the polar coordinate is required, assembling the conventional axicon phase distribution. Therefore, ${\mathrm{\Phi }}_{rl}=2\pi \rho \arcsin ({\theta }_B)/\lambda$ [40]. Here, $\rho$ is the polar coordinate, and ${\theta }_B$ indicates the deflection angle of the output light with regards to the optical axis and is dependent on the radial span $P$ of the 360° phase coverage along the polar direction by ${\theta }_B=\arcsin (\lambda /P)$.

Meanwhile, attributing to the inherent cross-polarization conversion function of the PBOE metasurface, ${\mathrm{\Phi }}_{lr}$ does not need any localized alteration but keeps a constant $C$. Then, the metasurface can be constructed with the selected PBOEs of the transport phase ${\phi }_f=\pi \rho \arcsin ({\theta }_B)/\lambda +C/2$ and the rotation angle $\alpha =\pi \rho \arcsin ({\theta }_B)/(2\lambda )-C/4$. Figure 4(a) shows the optical microscope image of the fabricated Bessel/half-wave metasurface for which $P$ is set as 1.24 mm. The metasurface was characterized by a fiber-based near-field scanning THz microscopy (FNSTM) [41]. Figure 4(b) illustrates the measured normalized RHCP intensity profile in the longitudinal plane $x=0$ under the LHCP incidence from the substrate side. It is clearly seen that there is a typical diffraction-free feature in a distance longer than $30\lambda$ along the propagation direction. The transverse normalized intensity and phase profiles at 8 mm above the metasurface are also measured, as shown in Figs. 4(c) and 4(d). Both profiles show typical Bessel features: an intense central spot is surrounded by weak concentric rings due to the limited size of the metasurface, and an almost linear phase gradient along the radial direction. Under the RHCP incidence, the measured longitudinal normalized LHCP intensity, and the transverse normalized intensity and phase profiles are shown in Figs. 4(e)–4(g). It can be observed that the beam keeps propagating as a Gaussian-type wave. Thus, under the RHCP incidence, the metasurface only performs the cross-polarization conversion function as a half-wave plate.

The second device is a Bessel/vortex metasurface, which is capable of producing Bessel and vortex beams separately by switching between the LHCP and RHCP incidences. The Bessel beam for the LHCP input is designed to the same specifications as the former device, while the first-order vortex beam for the RHCP input is designed with a topological charge $+1$ by ${\mathrm{\Phi }}_{lr}=\theta$ [42], where $\theta$ is the azimuthal angle. Using the same design principle, we can obtain the ${\phi }_f$ and $\alpha$ required to be patterned on the metasurface. Figure 5(a) shows the optical microscope image of the fabricated sample. Under the LHCP incidence, the measured normalized RHCP longitudinal normalized intensity profile at $x=0$ is illustrated in Fig. 5(b). Figures 5(c) and 5(d) illustrate the transverse intensity and phase profiles at $z=5\mathrm{mm}$. Similar Bessel features are observed as those seen in Figs. 4(b), 4(c), and 4(d). Figure 5(e) illustrates the measured normalized LHCP longitudinal intensity profile at $x=0$, and Figs. 5(f) and 5(g) illustrate the transverse intensity and phase profiles at $z=8\mathrm{mm}$, under the RHCP incidence. Two distinct divergent lobes can be seen from the longitudinal propagation plane, in which the intensity along their central line is almost 0. Meanwhile, the transverse intensity profile in Fig. 5(f) presents a doughnut spot, and the phase profile presents a helical phase of 360° encircling the phase singularity at the center. These features coincide with the $+1$st vortex beam as expected.

Terahertz all-silicon dielectric metasurfaces for spin-dependent and complete wavefront control are realized using PBOEs through introducing extra geometry inhomogeneity besides the rotation inhomogeneity. In this work, a simplified design strategy is utilized to facilitate the metasurface design attributing to the structure symmetry. Based on the simplified design, several exemplar metasurfaces are accomplished using the minimized fundamental constituting elements. One is used to confirm the performance of the selected PBOEs, while the other two metasurfaces demonstrate functional devices—a Bessel/half-wave metasurface and a Bessel/vortex metasurface. The measured results agree well with our design. Such dielectric metasurfaces are easy to fabricate, compatible with semiconductor integration, and have promising applications in ultra-compact and spin-involved optical terahertz imaging system and communication systems for polarization contrast imaging, coding, and encryption.