Polarization is an important variable of electromagnetic waves, typical types of which include linear, circular, and elliptical ones. From the perspective of electromagnetic field, it refers to the trajectory of the electric field vibration, and from the perspective of the photons, it (circular polarization) can also be related to the spin angular momentum [1,2]. Therefore, polarization of light has been widely studied and applied in imaging, quantum optics, and other fields, and polarization manipulation has become an important research content of modern optics and photonics [3–5]. Traditional optical devices based on bulk crystals are used for polarization generation or conversion, such as polarizers and wave plates. The wave plate is based on the linear birefringence effect of the optical crystal, which produces a phase shift between the ordinary and the extraordinary components with an appropriate thickness, and then superimposes them into a new polarization. Such devices are large in size and require high processing accuracy, while only specific materials can be used in each electromagnetic band [6]. More importantly, these devices can hardly realize the simultaneous control of other optical parameters while performing polarization conversion, such as amplitude and phase. This is not conducive to the multi-functional and integrated development of modern optical devices.

Electromagnetic metasurfaces are planar functional devices composed of artificially designed sub-wavelength meta-atoms [7–10]. After more than 10 years of rapid development, significant progress of them has been made in the fields of information optics [11,12], quantum optics [4,5], non-linear optics, and terahertz photonics [13–17]. Electromagnetic functional devices based on metasurfaces are flexible in design and conducive for integration. They not only provide new ideas for the development of new optical devices, but also provide powerful tools for many basic physics researches [18–20]. In particular, metasurfaces also achieve optical linear birefringence and polarization conversion. Different from the material birefringence of traditional optical devices, the meta-atoms in metasurface utilize the shape birefringence [21], which has been realized from microwave to ultraviolet bands using a dielectric or metal [22,23]. For shape-induced linear birefringence, when the incident polarization is consistent with the symmetry axis of the anisotropic meta-atom, the output polarization state remains unchanged, and the incident polarization state is called the eigen-polarization state. When the other linearly polarized wave is incident, its projection in the eigen-polarization state will cause polarization transformation [21], and the output polarization state can be designed by changing the geometric parameters of the structure. The advantage of shape birefringence is that the working frequency can be designed, and other functions can be achieved with polarization conversion, such as non-linear effect [10], wavefront shaping [5,24,25], and polarization-dependent beam energy distribution [26,27]. In particular, researchers set the appropriate geometric dimensions to make the linear polarization conversion efficiency of the meta-atoms be about 50%, controlled the phase difference between the co- and cross-polarized components, and then obtained beam shaping with designed polarization [28–30]. This method is similar to the principle of a quarter-wave plate and can realize the polarization conversion with a simple structure, but the transformation channels are non-independent due to the symmetry of the meta-atoms. It is difficult to achieve independent polarization conversion between different channels. In addition to the linear polarization-based approach, the circular polarization-based one has also been reported, that is, the phase manipulation method of spin decoupling which combines geometric and dynamic phases, while the cross- and co-polarizations can be controlled at the same time [31–33]. However, the optical response of each unit needs to meet several conditions at the same time, which means that their design process is complicated. Therefore, metadevices for polarization conversion with a simple design and two or more independent channels are very necessary.

Here, we propose a new method for terahertz beam shaping with multiple polarization conversion channels. Two sets of spatially interleaved anisotropic units are used to obtain the linearly co- and cross-polarized components, which are superimposed to obtain a new polarization state in beam shaping. We take the focused beam as an example to improve the signal-to-noise ratio (SNR) of the measurement for terahertz electric field. We use two samples to demonstrate the high-efficiency, broadband polarization conversion and the multi-channel function brought about by the incident polarization and the sub-arrays. This method is not only suitable for design of linear polarization-based metasurfaces, but also can be extended to circular polarization-based devices.

The proposed scheme of beam shaping with polarization conversion is shown in Fig. 1, in which Fig. 1(a) is a schematic diagram of the metasurface. We take the incidence of two orthogonal linear polarization states as an example (such as $x$ and $y$ polarizations). In the transmitted wave, we can obtain two independent polarization states, such as different linear or circular polarizations. Here we define the polarization states with an angle of 45° or $-45°$ to the $x$ axis as “$a$” or “$b$” polarization, respectively [28]. The above-mentioned functions are realized by the spatially interleaved structure shown in Fig. 1(b), in which the two kinds of units differ by half a period ($p/2$) in both the $x$ and $y$ directions. When an $x$- or $y$-polarized wave is incident, the transmitted wave of the cross-ellipse cylinder [structure (i)] maintains its polarization state, which is called the eigen-polarization. Moreover, the transmission phases of the two polarization states can be independently controlled. For structure (ii), the long axis of the ellipse forms an angle of 45° or $-45°$ with respect to the $x$ axis, and its eigen-polarization states are $a$ and $b$ polarizations. The transmitted wave can be efficiently converted into cross-polarization when $x$- or $y$-polarized beam is incident. In this way, we have three polarization components to control, and then two independent polarization channels can be obtained after superposition. In addition, other incident polarization states can be regarded as vector superposition of $x$ and $y$ polarizations with a specific phase shift, so more transmission polarization states can also be obtained. For example, when $a/b$-polarized waves are incident, the transmitted waves will be elliptically polarized.

Here, we make a brief theoretical analysis of the working principle for the metasurface in Fig. 1. Without loss of generality, we consider the condition of $x$ or $y$ polarization incidence. Considering that there are two kinds of units in our metasurface, the operational relationship between the transmitted and incident electric field is $$\left(\begin{array}{c}{E}_t^x\\ E_t^y\end{array}\right)=\left(\begin{array}{cc}{T}_{xx}& T_{xy}\\ T_{yx}& T_{yy}\end{array}\right)\left(\begin{array}{c}{E}_i^x\\ E_i^y\end{array}\right),$$(1)where the matrix $\mathit{T}$ is the transmission matrix of the meta-atoms, and its elements $T_{ij}$$(i,j=x,y)$ represent the transmission coefficients. When the polarization conversion efficiency of the second type meta-atoms is high enough, the unconverted part can be ignored, and the matrix $\mathit{T}$ can be decomposed into the sum of the co- and cross-polarization components: $$\mathit{T}={\mathit{T}}_{\mathrm{co}}+{\mathit{T}}_{\mathrm{cross}}=\left(\begin{array}{cc}{T}_{xx}& 0\\ 0& T_{yy}\end{array}\right)+\left(\begin{array}{cc}0& T_{xy}\\ T_{yx}& 0\end{array}\right).$$(2)The transmission coefficient $T_{ij}$ can be written as the product of its modulus and phase, namely, $$\mathit{T}={\mathit{T}}_{\mathrm{co}}+{\mathit{T}}_{\mathrm{cross}}=\left(\begin{array}{cc}{t}_{xx}{\phi }_{xx}& 0\\ 0& t_{yy}{\phi }_{yy}\end{array}\right)+\left(\begin{array}{cc}0& t_{xy}{\phi }_{xy}\\ t_{yx}{\phi }_{yx}& 0\end{array}\right).$$(3)In our structure (ii), the angle between the long axis of the ellipse and the $x$ axis is 45° or $-45°$, so $t_{xy}·{\phi }_{xy}=t_{yx}·{\phi }_{yx}$; then the two coefficients can be unified as $t_{\mathrm{cross}}·{\phi }_{\mathrm{cross}}$.

When the $x$-polarized wave is incident, we have $$\left(\begin{array}{c}{E}_t^x\\ E_t^y\end{array}\right)=\left(\begin{array}{cc}{T}_{xx}& T_{xy}\\ T_{yx}& T_{yy}\end{array}\right)\left(\begin{array}{c}{E}_i^x\\ 0\end{array}\right)=\left(\begin{array}{cc}{T}_{xx}& E_i^x\\ T_{yx}& E_i^x\end{array}\right)=\left(\begin{array}{cc}{t}_{xx}{\phi }_{xx}& E_i^x\\ t_{\mathrm{cross}}{\phi }_{\mathrm{cross}}& E_i^x\end{array}\right).$$(4)Assuming that the phase difference between the transmitted $x$- and $y$-polarized components is $\mathrm{\Delta }{\phi }_x={\phi }_{\mathrm{co}}-{\phi }_{\mathrm{cross}}={\phi }_{xx}-{\phi }_{\mathrm{cross}}$, and the transmission amplitude is 1, then the total transmitted electric field is $${\mathit{E}}_t^{(1)}=({\phi }_{\mathrm{cross}}+\mathrm{\Delta }{\phi }_x)E_i^x·{\mathit{i}}_x+{\phi }_{\mathrm{cross}}{E}_i^x·{\mathit{i}}_y,$$(5)where ${\mathit{i}}_x$ and ${\mathit{i}}_y$ are unit vectors in the $x$ and $y$ directions.

Similarly, when the $y$-polarized wave is incident, the transmitted electric field is $$\left(\begin{array}{c}{E}_t^x\\ E_t^y\end{array}\right)=\left(\begin{array}{cc}{T}_{xx}& T_{xy}\\ T_{yx}& T_{yy}\end{array}\right)\left(\begin{array}{c}0\\ E_i^y\end{array}\right)=\left(\begin{array}{cc}{T}_{xy}& E_i^y\\ T_{yy}& E_i^y\end{array}\right)=\left(\begin{array}{cc}{t}_{\mathrm{cross}}{\phi }_{\mathrm{cross}}& E_i^y\\ t_{yy}{\phi }_{yy}& E_i^y\end{array}\right).$$(6)The phase difference between the transmitted $x$- and $y$-polarized components at this time is $\mathrm{\Delta }{\phi }_y={\phi }_{\mathrm{co}}-{\phi }_{\mathrm{cross}}={\phi }_{yy}-{\phi }_{\mathrm{cross}}$, and the transmission amplitude is 1: $${\mathit{E}}_t^{(2)}={\phi }_{\mathrm{cross}}{E}_i^y·{\mathit{i}}_x+({\phi }_{\mathrm{cross}}+\mathrm{\Delta }{\phi }_y)E_i^y·{\mathit{i}}_y.$$(7)It can be seen from Eqs. (5) and (7) that when we fix the phase of cross-polarization, the total output electric field is determined by $\mathrm{\Delta }{\phi }_y$ and $\mathrm{\Delta }{\phi }_x$, which means that the transmission polarization state is also determined by them.

In order to obtain the above-mentioned metasurface, we use commercial numerical simulation software to obtain parameter libraries of the two kinds of meta-atoms. We choose high-resistance silicon as the constituent material, and process meta-atoms directly on the silicon wafer. High-resistance silicon ($\rho >5000\mathrm{\Omega }·\mathrm{m}$) shows almost no absorption for terahertz wave, and its dielectric constant is set as $\epsilon =11.9$ during the simulation. The two structures are shown in Fig. 2(a), where the period is $p=140\mathrm{\mu m}$, the height of the silicon pillar is $h=200\mathrm{\mu m}$, and the thickness of the substrate is 300 μm. The major and minor axes of the ellipse in structure (ii) are $w_4$ and $w_3$, respectively. The minor axes of the two ellipses in structure (i) are both $D=40\mathrm{\mu m}$, and the major axes are $w_1$ and $w_2$, respectively.

By scanning the parameters of the two structures (the original data can be seen in Appendix B, Fig. 6, the working frequency is $f=1.1\mathrm{THz}$), we first obtained the parameter library of structure (ii), as shown in Fig. 2(b). Since the structure will generate $y$- and $x$-polarized waves that are equal in value when the $x$ and $y$ polarizations are incident, respectively, we use only one component to represent them uniformly ($T_{\mathrm{cross}}$ and ${\phi }_{\mathrm{cross}}$). The phase shifts of the eight elements are changed from 135° to 180°, which is equivalent to covering the interval of [−180°, 180°] with a step of 45°. The first-order transmission coefficients of the eight elements are all around 0.6. The parameter library of structure (i) is shown in Figs. 2(c)–2(f); the values in the figure are obtained by probing the co-polarization components when $x$- or $y$-polarized waves are incident separately. Each unit in the library needs to consider the transmission phase shifts for both $x$- and $y$-polarized waves (eigen-polarizations) at the same time, and the amplitude remains the same. It can be seen from the figure that the selected phases show a good gradient with a step of 45°, and the first-order amplitudes of all units basically remain above 0.6. The transmission efficiency limitation of the two types of structures is mainly due to the reflection at the air/silicon interface.

Based on the above parameter library, we design two metasurfaces to verify the beam shaping with polarization conversion. The samples are processed from a 500 μm thick high-resistance silicon through inductively coupled plasma (ICP) etching technology. Each sample is a square with a side length of 1.4 cm. The specific processing steps can be seen in Appendix A. The scanning electron microscope (SEM) images of the first metasurface are shown in Fig. 3(a), which is called sample 1. Its effective area is a circular array composed of the two kinds of meta-atoms mentioned above. There are 80 units (1.12 cm) of each kind in the radial direction. It can be seen from the figure that the sidewalls of the meta-atoms are very steep, which means the processing error is not obvious. In order to illustrate the polarization conversion function of sample 1 intuitively, we use Poincaré spheres to show the incident and transmission polarization states, which are denoted by $|\mathrm{in}⟩$ and $|\mathrm{out}⟩$, respectively, as shown in Fig. 3(b). To show the independence of the two polarization channels, we hope that when the $x$- and $y$-polarized waves are incident, the transmission waves are $a$-polarized or right circularly polarized (RCP), respectively. In addition, we design the transmitted beam as a focused vortex beam, which can improve the SNR of the measurement results. Therefore, the expression of the phase profile of the three channels in the metasurface should be $${\overline{\phi }}_{xx}=-(\frac{2\pi }{\lambda }\sqrt{x^2+y^2+f^2}-f)+\arctan \left(\frac{y}{x}\right)-\frac{\pi }{2},$$(8)$${\overline{\phi }}_{yy}=-(\frac{2\pi }{\lambda }\sqrt{x^2+y^2+f^2}-f)+\arctan \left(\frac{y}{x}\right),$$(9)$${\overline{\phi }}_{\mathrm{cross}}=-(\frac{2\pi }{\lambda }\sqrt{{\left(x+\frac{p}{2}\right)}^2+{\left(y-\frac{p}{2}\right)}^2+f^2}-f)+\arctan \left(\frac{y}{x}\right),$$(10)where ${\overline{\phi }}_{xx}$ and ${\overline{\phi }}_{yy}$ are responses for the $x$- or $y$-polarized wave of structure (i), and ${\overline{\phi }}_{\mathrm{cross}}$ is the phase response of structure (ii). In order to eliminate the influence of spatial dislocation of meta-atoms on the focusing phase superposition, both $x$ and $y$ coordinates of ${\overline{\phi }}_{\mathrm{cross}}$ are corrected by half a period. In sample 1, the focal length in Eqs. (8)–(10) is $f=7\mathrm{mm}$, and the wavelength is $\lambda =273\mathrm{\mu m}$. The simulated and measured results for electric field intensity of sample 1 at the focal plane are shown in Fig. 3(c), where all the values are normalized, and the design frequency is 1.1 THz. They are obtained from a full-wave simulation of the transmitted electric field under the illumination of $x$- and $y$-polarized waves, and the specific simulation settings are explained in Appendix A. In order to clearly show the polarization state of the transmitted beam, we give the simulation results of two orthogonal polarization components, including a 2D intensity and a 1D curve of the central section. As can be seen in the figure, the RCP and the $b$-polarized components are the main parts, while the orthogonal polarization components are very weak, which is consistent with our design. The corresponding experimental results of sample 1 are shown in the second line of Fig. 3(c), where the actual frequency is 1.08 THz at the plane of $z=6.8\mathrm{mm}$. They are basically in agreement with the simulation results of the first line. The phase distributions which indicate the mode order of the generated vortex beam are shown in Appendix C (Fig. 8). For specific measurement steps, see Appendix A. In addition, we also consider the working bandwidth of the metasurface, taking the incidence of $x$-polarized wave as an example. Figure 3(d) shows the electric field intensity distributions in the $xoz$-plane of several frequencies near 1.1 THz, in which the focusing efficiencies are relatively high. Due to the high refractive index of silicon in the terahertz band ($n=3.45$ around 1 THz), the reflection is up to 30% at the air/silicon interface under normal incidence [34]. Here we only focus on the transmitted part. Figure 3(e) shows the polarization conversion efficiency in the range of 0.7–1.4 THz, which is obtained at the focal plane ($z=7\mathrm{mm}$) by $$\eta =\frac{{|E_{\mathrm{RCP}}|}^2}{{|E_{\mathrm{RCP}}|}^2+{|E_{\mathrm{LCP}}|}^2}.$$(11)

It can be seen that the conversion efficiency at the design frequency of 1.1 THz is up to 89.2%, and it is greater than 60% in the entire survey frequency band. The simulated results show that the efficiency is higher than 80% in the range of 0.8–1.3 THz, with a peak value of 93.8% at 1.1 THz. The measured values are slightly smaller than the simulated ones; after all, there are some noise signals in the edge area of the focal plane in the measurement results.

In fact, we can obtain more channels in one metasurface via area division of the sample, which means designing different phase distributions in different areas [30]. For simplicity, we only consider the condition of dividing the effective area into two sub-arrays. When the $x$- or $y$-polarized wave is incident, the two areas produce different circularly polarized beams. To this end, we design and process the sample 2 shown in Fig. 4(a), where the phase functions of the left- and right-half regions are shown in Appendix B (Fig. 7). The focal length of sample 2 is $f=7\mathrm{mm}$, and the working wavelength is $\lambda =273\mathrm{\mu m}$. The Poincaré sphere in Fig. 4(b) shows the polarization conversion function of sample 2. At this time, one input linear polarization state can generate two output circular polarization states at different positions. Figure 4(c) shows the polarization components of the transmitted electric field when the $x$- or $y$-polarized wave is incident. In addition to the electric field intensity and section curve similar to Fig. 3(c), here we also use the $S_3$ component (normalized) of Stokes parameters to intuitively show the circular polarization distribution of the transmitted wave. It can be seen that both left- and right-handed circular polarizations have been obtained efficiently, and the simulation results are in agreement with the experimental data. In addition, we show the electric field intensity and $S_3$ parameter distributions of the metasurface in the longitudinal section ($xoz$-plane) in Fig. 4(d). It can be found that the division of the regions does not significantly affect the focusing efficiency of the metasurface.

Generally, in the case where linearly polarized waves are incident, all the polarization states we get are derived from the superposition of orthogonal linearly polarized components transmitted by the two types of meta-atoms. With different phase shifts, various states such as linear, elliptical, or circular polarization can be obtained. In particular, when the $x$- or $y$-polarized wave is incident, both of the polarization components appear in the transmitted beam. Assuming that the phase difference between the $x$ and $y$ polarization components in the transmission is $\delta ={\delta }_x-{\delta }_y$, Fig. 5(a) shows the generated polarization states when $\delta$ takes different values. The corresponding positions on the Poincaré sphere are shown in Fig. 5(b). It should be noted that when only the $x$- or $y$-polarized wave is incident, we will never get a purely $x$- or $y$-polarized transmitted wave. However, as shown in Fig. 5(c), when the RCP or LCP wave is incident in sample 2, we can get $x$- or $y$-polarized beams. In this way, we have achieved the coverage of various typical polarizations of the transmitted wave.

In summary, we propose a new method for terahertz beam shaping with multi-channel polarization conversion based on all-silicon metasurfaces. We alternately arrange two kinds of meta-atoms in a single metasurface, eigen-polarizations of which are $x$, $y$, or $\pm 45°$ polarizations, respectively. Therefore, when $x$- or $y$-polarized waves are incident, they will respectively generate co- or cross-polarized transmitted waves while achieving beam shaping. In this way, a new polarization state in transmission can be obtained after far-field interference, and there are two independent channels available. In addition, a single metasurface can be divided into two or even more sub-arrays to achieve more channels for polarization conversion or beam shaping. To this end, we designed and processed two samples to demonstrate the above functions. The simulated and experimental results of the first sample verify the dual-channel polarization conversion with linearly polarized incidence. The second one shows the multi-channel polarization conversion controlled by polarization incidence and region division. We use electric field intensities, Stokes parameters, and Poincaré spheres to show the working performance of the metasurface, which confirm our proposed scheme. The experimental results of the electric field are in agreement with the simulations. This design method is not only easier to implement and can provide independent polarization conversion channels, but can also be extended from a linear polarization base to a circular polarization base. The proposed metasurfaces are expected to be applied for polarization multiplexing imaging, holography, and detection of polarization state of terahertz wave.