Unlike bulky traditional optical components, metasurfaces achieve remarkable functionalities in ultrathin and lightweight forms [1–3]. The definition and function of metasurfaces pertain to two-dimensional (2D) artificial structures composed of subwavelength elements, commonly referred to as meta-atoms [4]. These structures are specifically designed to manipulate electromagnetic (EM) waves with a high degree of precision. By introducing spatially varying amplitude, phase, frequency, and polarization responses at the subwavelength scale, metasurfaces facilitate the engineering of wavefronts, thereby allowing for enhanced control over EM wave propagation [5–7]. Consequently, metasurfaces can act as polarization-selective filters, allowing or blocking certain polarization states. Plasmonic metasurfaces rely on metallic materials (e.g., gold, silver), which have significant ohmic (resistive) losses, especially in the THz spectral ranges [8–10]. All-dielectric metasurfaces, on the other hand, use high-refractive-index dielectric materials (e.g., silicon, titanium dioxide) that exhibit negligible absorption losses in the same spectral range. This leads to higher efficiency in wavefront manipulation [11–13]. Moreover, the fabrication of all-dielectric metasurfaces is compatible with existing CMOS (complementary metal-oxide-semiconductor) technologies. This compatibility facilitates large-scale and cost-effective production for practical applications, such as beam shaping, orbital angular momentum (OAM) multiplexing [14–16], nonlinear optics [17–19], and imaging [20–22]. Nevertheless, the advancement of compact and on-chip polarization optics has generated an increasing demand for integrated tunable metasurfaces. While tunable metasurfaces integrated with functional materials have been reported, the constraints of static metasurfaces can be addressed through thermal [23], electrical [24], optical [25], and chemical methods [26]. Scaling these processes for commercial production presents significant challenges and may require sophisticated deposition techniques. Additionally, the functional materials utilized in tunable metasurfaces may experience degradation over time due to repeated cycling through various states or extended exposure to external stimuli.

Recently, Moiré metasurfaces featuring twisted bilayer configurations have emerged as an innovative approach for the dynamic control of the wavefront of EM waves [27–33]. Twisted bilayer metasurfaces allow independent engineering of both the individual layers and their mutual alignment. This decoupled design framework enables highly customizable optical functionalities that are not possible with single-layer metasurfaces. Unlike conventional metasurfaces that require active stimuli, Moiré metasurfaces can achieve tunability passively or via mechanical twisting/sliding. This reduces complexity, power consumption, and potential damage from external stimuli. Tsai et al. designed and fabricated a Moiré meta-platform with a center operating frequency of 532 nm, with variable focal length that can be increased from 10 to 125 mm by tuning mutual angles [29]. Zhang et al. reported a Moiré device consisting of two cascaded metasurfaces that can dynamically manipulate the wavefront of THz waves by mutual rotation between the metasurfaces. Both the order of the resulting Bessel beam and its non-diffractive length can be continuously adjusted [28]. Ogawa et al. proposed a design strategy for dynamic EM focusing using twisted metasurfaces. The method combines transmissive and reflective layers in a cascade fashion and achieves precise manipulation of the beam propagation and scanning area from $-44.7°$ to 44.7° by varying the relative angle between the layers [30]. Apparently, the design strategies for Moiré metasurfaces have been greatly developed. However, conventional meta-atoms, such as dielectric or plasmonic resonators, typically emphasize geometric alignment while neglecting birefringent or polarization-sensitive structures. Thus, significant shortcomings remain in the implementation of solutions tailored to achieve customized polarization filtering characteristics.

In this work, we propose and demonstrate a tunable THz focused beam generator designed for multi-dimensional modulation of the transmitted field. Simultaneous implementation of polarization filtering and adjustable focal length functions is accomplished using two cascaded all-silicon metasurfaces. Polarization-maintaining and polarization-converting meta-atoms were employed to assemble Layer I and Layer II of the proposed design, and labeled as PMMs and PCMs, respectively. By varying the relative twisted angle $p$ between Layer I and Layer II from 90° to 240°, the focal length of the resultant focused beam can be incrementally adjusted from 12 to 4.5 mm, achieving an average focusing efficiency exceeding 35%. Meanwhile, the electric field distribution transmitted by the Moiré meta-device under $x$-LP illumination is concentrated in the right-handed circularly polarized (RCP) component. After switching the incident direction, by further rotating the twisting angle $q$ of Layer II with respect to Layer I, the focusing field generated under the same conditions can be distributed in the left-handed circularly polarized (LCP) and RCP components as desired. Not only that, but the performance of the Moiré meta-device was also quantitatively evaluated by calculating the absolute percentage error (APE), including the focal length and numerical aperture (NA). The experimental results are in good agreement with the theoretical predictions, exhibiting an average APE of less than 10%. Thus, the Moiré meta-platform to tune optical properties by mechanical twisting eliminates the need for external power sources or active materials, simplifying integration into THz polarization systems.

The proposed design strategy corresponding to the Moiré meta-device contains two layers, which can be defined as Layer I and Layer II, as shown in Fig. 1(a). Here, Layer I primarily functions to manipulate the focal length, whereas Layer II is responsible for the polarization filtering of the proposed design. Moreover, the in-plane rotation angles of Layer I and Layer II are predefined as $p$ and $q$, respectively, in degrees. Taking a twist angle of $p=180°$ as an example, the encoded phase distributions utilized in the construction of Layer I and Layer II are depicted in Fig. 1(b). By varying the parameter $p$ from 0° to 360°, the intensity of the transmitted electric field generated by the proposed Moiré meta-device under $x$-LP illumination is concentrated in the RCP component, as illustrated in the first row of Fig. 1(c). By reversing the direction of illumination of the incident $x$-LP, the normalized intensity of the transmitted electric field is distributed in the RCP and LCP components as desired, specifically, $I_{\mathrm{RCP}}=(\cos q+1)/2$ and $I_{\mathrm{LCP}}=(\sin q+1)/2$. This distribution occurs as the relative rotation angle of Layer II with respect to Layer I increases from 0° to 360°, as illustrated in the second row of Fig. 1(c). Meanwhile, the theoretical values of the focal lengths for the focused beams generated within the RCP and LCP channels, as a function of the relative rotation angles $p$ or $q$, are presented in Fig. 1(d). It is evident that the theoretical predictions indicate a gradual decrease in the focal length as the parameter $p$ or $q$ increases, demonstrating polarization-insensitive properties.

This configuration facilitates multi-dimensional joint tuning with tailored focusing characteristics. Thus, a set of polarization-maintaining meta-atoms (PMMs) possessing the same phase delays can be used to assemble Layer I, as illustrated in Fig. 2(a). The silicon pillars, arranged in a cylindrical configuration, exhibit a period $P\hspace{0.17em}=\hspace{0.17em}\mathrm{150}$ μm and a height $H\hspace{0.17em}=\hspace{0.17em}\mathrm{200}$ μm. Moreover, the substrate is also composed of silicon and has a thickness of 300 μm. By systematically varying the radius parameter $R$ in the time-domain solver, the amplitudes of the selected meta-atoms that satisfy the phase encoding criteria are illustrated in Fig. 2(b). The selected units exhibit consistent and relatively uniform transmission amplitudes in both the $x$- and $y$-directions. This observation includes the co-polarization coefficients $t_{xx}$ and $t_{yy}$, as well as a relative amplitude ratio of $t_{xx}/t_{yy}$. The phase distribution at 45° intervals is depicted in Fig. 2(c). Additionally, the phase differences (PDs) calculated along the $x$- and $y$-directions further demonstrate the polarization-insensitive properties of the PMM. Moiré metasurfaces with twisted bilayer configurations introduce a second degree of freedom, which is relative angular displacement, offering distinct advantages over conventional tunable metasurfaces. Theoretically, the encoding phase profile ${\mathrm{\Phi }}_1(r,p_0)$ for generating Layer I can be expressed as [29,34,35] $${\mathrm{\Phi }}_1(r,p_0-p)=-\text{round}\left(\frac{r^2}{\lambda ·f_0}\right)·(p_0-p).$$(1)

Here, $r$ represents the radial coordinate at the metasurface, $p_0$ denotes the relative rotation angle between Layer I and Layer II, $\lambda =375\mathrm{\mu m}$ indicates the central operating wavelength, $f_0=6\mathrm{mm}$ is the initial focal length at $p_0=180°$, and the round ($•$) function is implemented to avoid the fanning effect of the Moiré meta-device during the rotation process. Subsequently, the normalized magnetic field intensity distribution of the PMM was extracted under periodic boundary conditions, as shown in Fig. 2(d). The EM energy carried by the incident plane wave is mainly localized inside the dielectric pillars with a high refractive index ($\sim 3.45$). On the other hand, the meta-atoms for assembling Layer II were selected by engineering rectangular silicon pillars that exhibit shape birefringence properties, as illustrated in Fig. 2(e). Then, such a set of units with EM properties similar to the quarter-wave plates (QWPs) can be defined as polarization-converting meta-atoms (PCMs). The period and pillar height of the PCM with an all-silicon configuration were consistent with that of the PMM. This design choice ensures consistency in the overall metasurface structure and allows for a fair comparison between the two types of pillars in terms of their optical performance. Consequently, the Jones matrix of PCMs in a linearly polarized basis can be represented as [36–39] $$T_0=\left[\begin{array}{cc}|t_{xx}|e^{i{\phi }_{xx}}& 0\\ 0& |t_{yy}|e^{i{\phi }_{yy}}\end{array}\right],$$(2)where $|t_{xx}|$ (${\phi }_{xx}$) and $|t_{yy}|$ (${\phi }_{yy}$) denote the amplitudes (phase delays) under orthogonal linearly polarized illumination, and $T_0$ is the transmitted Jones matrix of a PCM. By introducing an additional rotation factor $\varphi$, the transmission matrix $T_0$ can be further calculated using the standard transformation matrix ${\mathrm{\Lambda }}_1=\left[\begin{array}{cc}\cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \end{array}\right]$ as follows: $$T_1={\mathrm{\Lambda }}_{1}T{\mathrm{\Lambda }}_1^{-1}=\left[\begin{array}{cc}{\cos }^2\varphi ·|t_{xx}|e^{i{\phi }_{xx}}+{\sin }^2\varphi ·|t_{yy}|e^{i{\phi }_{yy}}& \sin \varphi \cos \varphi ·(|t_{xx}|e^{i{\phi }_{xx}}-|t_{yy}|e^{i{\phi }_{yy}})\\ \sin \varphi \cos \varphi ·(|t_{xx}|e^{i{\phi }_{xx}}-|t_{yy}|e^{i{\phi }_{yy}})& {\sin }^2\varphi ·|t_{xx}|e^{i{\phi }_{xx}}+{\cos }^2\varphi ·|t_{yy}|e^{i{\phi }_{\mathrm{yy}}}\end{array}\right],$$(3)where $T_1$ denotes the transmission matrix of the PCM after rotation. Equation (3) can be further simplified by applying the parametric conditions met by the QWP in a linearly polarized basis to Eq. (3), specifically: $\varphi =\pi /4$, $|t_{xx}|=|t_{yy}|=t$, and ${\phi }_{yy}-{\phi }_{xx}=\pi /2$. Equation (3) can be further calculated as $$T_1=\frac{t}{2}{e}^{i{\phi }_{xx}}\left[\begin{array}{cc}1+e^{i\frac{\pi }{2}}& 1-e^{i\frac{\pi }{2}}\\ 1-e^{i\frac{\pi }{2}}& 1+e^{i\frac{\pi }{2}}\end{array}\right]=\frac{\sqrt{2}t}{2}{e}^{i({\phi }_{xx}+\frac{\pi }{4})}\left[\begin{array}{cc}1& -i\\ -i& 1\end{array}\right].$$(4)

In this case, the transmission matrix produced by a PCM with periodic boundary conditions under $x$-LP illumination can be described as $$E_1=\frac{\sqrt{2}t}{2}{e}^{i{\phi }_{xx}}\left[\begin{array}{cc}1& -i\\ -i& 1\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]=\frac{\sqrt{2}t}{2}{e}^{i{\phi }_{xx}}\left[\begin{array}{c}1\\ -i\end{array}\right].$$(5)

Note that PCMs that satisfy specific parametric conditions can efficiently achieve the conversion from a linearly polarized state to a single-handed circularly polarized state. Subsequently, such a PCM can realize advanced wavefront shaping with the assistance of the propagation phase. In the simulation, the $x$- and $y$-directions along the dielectric pillars are set as periodic boundary conditions, while the $z$-direction is set as an open boundary condition. The amplitude and phase delays produced by the selected meta-atoms under $x$-LP illumination are shown in Figs. 2(f) and 2(g). When the in-plane orientation angle of the rectangular pillar is fixed at $\pi /4$, the amplitudes extracted in both the co-polarized and cross-polarized channels are approximately 0.6, i.e., $t_{xx}\approx t_{yy}\approx 0.6$. Furthermore, as the meta-atom numbering increases, the phase delays in the transmission mode not only achieve full-phase modulation of $2\pi$ but also maintain a consistent PD (i.e., ${\phi }_{xx}-{\phi }_{yy}\approx \pi /2$) at the target frequency of 0.9 THz. In addition, Fig. 2(h) illustrates the normalized magnetic field distributions obtained from the time domain solver for a PCM with periodic boundary conditions. In this analysis, we focus on two adjacent pillars as a representative example. The distribution of incident EM energy predominantly occurs within the pillars that possess a high refractive index. This observation suggests that the coupling effects between neighboring dielectric pillars can be considered negligible. Thus, the target phase distribution ${\mathrm{\Phi }}_2(r,p_0)$ for generating Layer II can be described as $${\mathrm{\Phi }}_2(r,p_0)=\mathrm{round}\left(\frac{r^2}{\lambda ·f_0}\right)·p_0.$$(6)

Here, setting $a=1/(\lambda ·f_0)$ serves as a crucial factor for modulating the focal length, and we define a variable, denoted as $p\in [0,2\pi ]$, to facilitate the manipulation of the Moiré metasurface rotation, employing a step size of 30°. Further, the functional relationship between the focal length $f_p$ of the focused beam produced in the transmission mode and the rotation angle $p$ can be expressed as $$f_p=\frac{\pi }{a·p·\lambda }.$$(7)

Obviously, the angle of the designed Moiré metalens can be continuously tuned by varying the relative rotation angle $p$ between Layer I and Layer II. Thus, the numerical aperture (NA) of the designed Moiré meta-device can be calculated from the ideal focal length as [40] $$\mathrm{NA}=\sin \left(\arctan \frac{L}{2f_p}\right),$$(8)where $L$ denotes the diameter of the metasurface. The calculated NA can effectively evaluate the transmission properties of the designed Moiré metasurfaces. The geometric parameters corresponding to PMM and PCM can be found in Appendix A.

The coded phase profiles for generating Layer I are indicated in Fig. 3(a), arranged pixel by pixel from the selected PCMs for performing the function of polarization filtering. Figure 3(b) illustrates the coded phase distributions employed in the construction of Layer II, which correspond to the mutual rotation angles $p=90°$, 120°, 150°, 180°, 210°, and 240°. Therefore, the total modulation phase of the designed Moiré metasurface can be calculated as ${\mathrm{\Phi }}_{\mathrm{tot}}(r,p)={\mathrm{\Phi }}_1(r,p_0-p)+{\mathrm{\Phi }}_2(r,p_0)=a{r}^{2}p$ [29]. Figure 3(c) illustrates the joint phase distribution corresponding to different rotation angles, presenting typical focusing phase profiles with different focal lengths. The electric field distribution of the metasurface under $x$-LP illumination was obtained by using the finite integration technology (FIT) in the time-domain solver, as shown in Fig. 3(d). According to Eq. (5), Layer II is composed of PCMs that adhere to the propagating phase gradient. This design facilitates the conversion of $x$-linearly polarized states into single-handed circularly polarized states. Thus, the electric field intensity distributions extracted from the RCP channels within the $xoz$ plane are shown in Fig. 3(d). As the rotation angle $p$ gradually increases from 90° to 240°, the focal length of the transmitted beam decreases from 9.4 to 3.9 mm. Figures 3(e) and 3(f) present the distribution of the monitored electric field intensity in the $xoy$ plane, including the RCP and LCP components, respectively. Obviously, the Moiré meta-device effectively converts incident $x$-polarization into the RCP component while facilitating zoom capabilities. This indicates that the proposed design strategy utilizing twisted bilayer metasurfaces can concurrently achieve multi-dimensional modulation of both focal length and polarization state. In fact, the proposed design generated by the PCM- and PMM-based assembly strategies, also known as polarization-sensitive Moiré meta-device [16,31,41], exhibits broadband operational characteristics in proximity to the central frequency, as illustrated in Fig. 3(g). Taking $p=120°$ as an example, the focal length of the transmitted beam gradually increases from 6.22 to 7.92 mm as the operating frequency increases from 0.65 to 0.95 THz. Once the frequency is shifted, the focusing efficiency obtained at the corresponding focal plane is reduced, which is shown in the cyan dotted line in Fig. 3(g). Here, the focusing efficiency with quantitative analytical properties is defined as the ratio of the power at the focal spot to the incident power, which can be written as ${\eta }_f=P_f/P_{\mathrm{in}}=\iint E_f(E_f\ge E_{\max }/e)/\iint E_f$ [42]. Here, $P_f$ denotes the integral of the focused field distribution over a range greater than $E_{\max }/e$, and $P_{\mathrm{in}}$ denotes the integral of the field distribution at the entire focal plane. Subsequently, Fig. 3(h) presents a comparison of the intensity of individual polarized components generated by the proposed design under arbitrarily polarized illumination. In this figure, the horizontal axis represents the polarization components, while the vertical axis denotes the incident polarization states. It is not surprising that the polarization conversion behavior is reversible and that the resulting polarization states are distinct.

Such Moiré meta-platforms, including Layer I and Layer II, were prepared using standard ultraviolet (UV) lithography and inductively coupled plasma (ICP) etching techniques. Figure 4(a) depicts the overall appearance of Layer I, which measures $1.4\mathrm{cm}\times 1.4\mathrm{cm}$ and comprises a total of $80\times 80$ PMMs. By analyzing the scanning electron micrographs (SEMs) of Layer I (see the first row of Fig. 4), the fabricated samples exhibited good fabrication accuracy. The SEM images of Layer II were obtained under identical preparation conditions. The included PCMs exhibited smooth surfaces and steep sidewalls, indicating a high level of fabrication accuracy. The performance of the proposed Moiré meta-device was evaluated by employing a probe-based near-field THz imaging system (Model TeraCube Scientific M2) along the longitudinal direction, as shown in Fig. 4(c). The optical path of the THz polarization imaging system is illustrated in Fig. 4(c). This system utilizes a femtosecond laser source characterized by a central wavelength of 780 nm, a pulse width of 100 fs, and a repetition frequency of 80 MHz [43,44]. The microprobe exhibits a spatial resolution of 60 μm and is capable of recording the complex amplitude at the focal plane on a pixel-by-pixel basis. Since the designed twisted bilayer metasurface is mechanically rotated to control the focal length of the transmitted beam, a critical point is the alignment of the two metasurfaces. As shown in Fig. 4(d), strict alignment between Layer I and Layer II is ensured by introducing an optical tool of model GCT-090101. The GCT-090101 tool is equipped with four through holes, each measuring 6 mm in diameter, designed to accommodate the GCT-01 series coaxial stubs. Additionally, this optical component incorporates a built-in 360° rotating dial with an accuracy of 2°. As shown in Fig. 4(d), optical components GCT-090101-1 and GCT-090101-2 with two samples mounted were placed coaxially in the optical path [28]. Moreover, the selection of distance between two layers depends on the phase gradient of the previous one. In other words, a greater phase gradient necessitates a shorter distance. To minimize the coupling between the twisted bilayer metasurfaces, the distance between Layer I and Layer II was maintained at 1.5 wavelengths during this measurement.

The simulation and experimental results produced by the proposed THz Moiré metalens under $x$-LP illumination are represented in Fig. 5, further illustrating the feasibility of this design. As shown in Fig. 5(a), the normalized amplitudes of the electric field distribution were extracted along the longitudinal direction in the $xoz$ plane, corresponding to $p=90°$, 120°, 150°, 180°, 210°, and 240°. Apparently, the peak of the normalized amplitude obtained in the RCP component is gradually red-shifted as the rotation angle is gradually increased. This means that the focal length of the focused beam produced in the transmission mode gradually decreases. Subsequently, the focused field distribution at the corresponding focal plane was obtained using a THz near-field scanning system equipped with a microprobe, as shown in Fig. 5(b). The electric field distribution obtained is basically consistent with the simulation results, as depicted in Fig. 4(e). To quantitatively analyze the performance of the proposed Moiré meta-device, the normalized amplitude curves in the $xoy$ plane were extracted along the horizontal direction, as illustrated in Fig. 5(c). In this figure, the red solid line represents the experimental results, while the cyan solid line denotes the simulation results. Despite a minor discrepancy between the measured and simulated results—potentially attributable to an inaccurate twist angle—the resulting beam quality aligns with the anticipated coding requirements. The focal lengths of the Moiré metasurfaces as a function of the mutual rotation angle $p$ are indicated in Fig. 5(d). The results of the correlational study on focal length and twisted angle demonstrate a negative correlation between these two variables. Another parameter used to evaluate the focusing performance is the numerical aperture (NA), as defined in Eq. (8). Since the numerical aperture is inversely related to focal length, the NA values derived from theoretical, simulating, and experimental methods demonstrate a positive correlation with the rotation angle, as shown in Fig. 5(e). Subsequently, the definition of the absolute percentage error ($\mathrm{APE}=|F_{\exp }-F_{\mathrm{sim}}|/F_{\mathrm{sim}}$) parameter further assesses the errors generated during the rotation process [45], as illustrated in Fig. 5(f). The findings indicate that as the angle of rotation increases, corresponding to the focal length, the maximum error observed in this experiment occurs at $q=150°$, measuring only 9.9%. The maximum APE recorded in the experiment was 6.45%, while the average APE was found to be less than 4%. Based on the calculations, the measured focusing efficiencies related to different rotation angles $q=60°$, 90°, 120°, 150°, 180°, 210°, 240°, and 270° are 48.11%, 46.05%, 46.41%, 41.99%, 39.38%, 33.15%, 27.19%, and 19.37%, respectively. The comparison clearly indicates that the simulation results align closely with the experimental findings.

To further demonstrate the versatility of the proposed Moiré metalens in simultaneously manipulating polarization and focal length, experimental results corresponding to various twist angles were acquired by rotating Layer II. It is worth noting that the generation with a tailored focused field distribution requires a change in the direction of incidence. In other words, by fixing the position of Layer I, the desired function is realized by continuously tuning the parameter $q$. Subsequently, Layer II is considered as a whole and an additional rotation factor $q$ is introduced to simultaneously change the polarization state and the focal length of the transmitted beam. Layer II is composed of PCMs exhibiting specific polarization conversion characteristics. When Layer II is rotated collectively, each PCM will rotate correspondingly. Consequently, the transmitted Jones matrix can be further evaluated as [46] $$T_2={\mathrm{\Lambda }}_2{T}_1{\mathrm{\Lambda }}_2^{-1}=\frac{\sqrt{2}t{e}^{i({\phi }_{xx}+\frac{\pi }{4})}}{2({\cos }^{2}q+{\sin }^{2}q)}(\left[\begin{array}{cc}\sin q(\sin q+i\cos q)& \sin q(\cos q+i\sin q)\\ -\sin q(\cos q-i\sin q)& -\sin q(-\sin q+i\cos q)\end{array}\right]+\begin{array}{c}\left[\begin{array}{cc}\cos q(\cos q+i\sin q)& -\cos q(\sin q+i\cos q)\\ -\cos q(-\sin q+i\cos q)& \cos q(\cos q-i\sin q)\end{array}\right])\end{array},$$(9)where ${\mathrm{\Lambda }}_2=\left[\begin{array}{cc}\cos q& -\sin q\\ \sin q& \cos q\end{array}\right]$ denotes the standard $2\times 2$ rotation matrix. Then, the transmission matrix of the rotated Moiré device under $x$-LP illumination can be described as $$E_2=T_2\left[\begin{array}{c}1\\ 0\end{array}\right]=\frac{\sqrt{2}t{e}^{i({\phi }_{xx}+\frac{\pi }{4})}}{2({\cos }^{2}q+{\sin }^{2}q)}\left(\left[\begin{array}{c}\sin q(\sin q+i\cos q)+\cos q(\cos q+i\sin q)\\ -\sin q(\cos q-i\sin q)-\cos q(-\sin q+i\cos q)\end{array}\right]\right).$$(10)

For simplicity, some eigenvalues are selected to further characterize the trend of the acquired focusing intensity in the LCP and RCP channels with the rotation angle $q$, and detailed calculations are available in Appendix B. The results indicate that the intensity of the RCP component of the transmitted electric field will experience a sequential evolution process of $1\to \sqrt{2}/2\to 0\to \sqrt{2}/2\to 1\to \sqrt{2}/2\to 0$, while the theoretical intensity of the LCP component will also experience a continuous evolutionary process of $0\to \sqrt{2}/2\to 1\to \sqrt{2}/2\to 0\to \sqrt{2}/2\to 1$ [47]. As the rotation angle $q$ was gradually increased from 90° to 240°, the RCP and LCP components of the electric field were recorded pixel by pixel at the corresponding focal planes, as shown in Figs. 6(a) and 6(b). The measured results obtained from the near-field THz scanning system generally align with the simulation results. It is noteworthy that the RCP and LCP components derived from the time-domain solver are detailed in Appendix C. The captured normalized electric field intensity as a function of rotation angle is shown in Fig. 6(c). The comparison reveals a strong alignment among the theoretical analysis, simulations, and experimental results, thereby affirming the consistency and reliability of the findings. The focal length and NA obtained from simulations and experiments are presented in Figs. 6(d) and 6(e), respectively. Consistent with expectations, the focal length exhibits a decreasing trend, while the NA shows an increasing trend as the rotation angle increases. Moreover, the obtained experimental and simulated results are in good agreement. To quantitatively analyze the performance of the transmitted beam, the APE parameters related to the focal length and NA are presented in Fig. 6(f). The average APE observed during the experiment was calculated to be 3.65% for the focal length and 2.15% for the NA.

The fundamental mechanism of the proposed twisted bilayer metasurface utilizes PMMs for Layer I and PCMs for Layer II. This configuration enables simultaneous modulation of polarization state and focal length. In other words, this design strategy is applicable universally, regardless of the operating band or the beam type. The generation of vortex beams with variable focal lengths was further evaluated by changing the encoded phase profile of Layer II, as shown in Fig. 7. The coded phase distribution utilized for generating OAM can be described as $${\mathrm{\Phi }}_3(r,{\theta }_0,\varphi )=\mathrm{round}\left(\frac{r^2}{\lambda ·f_0}\right)·{\theta }_0+l·\varphi ,$$(11)where $l·\varphi$ corresponds to the wavefront of a flat spiral, and $l=1$ denotes the number of topological charges carried by the transmitted beam. Subsequently, the generated Layer II carrying the OAM was replaced with the previous one, and the transmissive vortex field distribution was obtained by rotating Layer I in the time-domain solver. As the rotation angle $p$ is gradually increased from 90° to 180°, the joint phase distribution used to generate the focused vortex beam can be found in Appendix D. As a result, the electric field intensity distribution generated by this Moiré device under $x$-LP illumination is mainly concentrated in the RCP component, as shown in Figs. 7(a) and 7(b). The RCP components extracted in the $xoz$ plane using the field monitor, corresponding to the field distribution of the proposed twisted bilayer metasurface under $x$-LP illumination, are given in Appendix D. The electric field and phase distributions of the LCP components are indicated in Figs. 7(c) and 7(d), further elucidating the working mechanism of this design. Based on the simulations, the focal lengths related to different mutual rotation angles $p=90°$, 120°, 150°, 180°, are 9.0 mm, 6.9 mm, 5.6 mm, and 4.6 mm, respectively. Obviously, the focal lengths are shown as a function of the rotation angles, which is in qualitative agreement with theoretical predictions, as shown in Fig. 7(e). Not only that, but the NA obtained from the numerical calculations also meets the expected design requirements. The quality of the vortex beam is demonstrated through the focusing efficiency, as depicted in Fig. 7(f). As the rotation angle $p$ increases, the focusing efficiency exhibits a declining trend, with an average value of approximately 40%. The comparison shows that the theoretical analysis and simulation results are in good agreement, thus verifying the reliability of the findings.

In conclusion, we propose and experimentally validate a design strategy for a twisted bilayer meta-device that facilitates the simultaneous manipulation of polarization state and focal lengths through cascading techniques. By selecting PMMs and PCMs with distinct polarization properties for the construction of Layer I and Layer II, respectively, the degrees of freedom for modulation of the Moiré meta-device can be significantly enhanced. The proposed design strategy presents a generalized design framework capable of effectively reducing the generation of aberrations, while achieving adjustable focal lengths and wavefront shaping. This is accomplished through a compact and lightweight structure that does not require external interventions. By varying the rotation angle, $p$, of Layer I relative to Layer II within the range of 90° to 240°, the transmitted electric field generated by the designed Moiré meta-device under $x$-LP illumination is predominantly concentrated in the RCP component. Simultaneously, the average focusing efficiency of the transmitted beam exceeded 35%, with experimental errors assessed through the APE parameters. The generation of a varifocal THz focused beam exhibiting customized polarization conversion behavior is further demonstrated through the variation of the incident direction. Experimental results indicate that the intensity and focal length of the transmitted beam in both the LCP and RCP components can be actively modulated by varying the twisting angle $q$ of Layer II with respect to Layer I. Furthermore, these experimental findings are in good agreement with the simulation and theoretical predictions. In addition, the numerical results proved that the focal length and polarization of the generated THz vortex beam can also be flexibly controlled by the designed cascading strategy. This investigation has the potential to offer a comprehensive and dynamic solution for adaptive optics systems, with broad applications in laser processing, long-range optical quantum communications, and high-resolution imaging.