Metasurfaces, the two-dimensional version of artificially engineered metal/dielectric structures that control the amplitude, phase, and polarization of light by varying the spatial dimensions and orientation of meta-elements, are promising candidates to mold the outgoing wavefront^[1–10]. Polarization of the incident light plays a key role in light–matter interaction for wavefront manipulation; however, geometry of meta-elements makes the metasurfaces sensitive to the incident polarization of the light. There have been several efforts to realize single-layered polarization-insensitive metasurfaces for wavefront engineering functionalities, e.g., meta-holograms, metalenses, and beam steering for linear and circular polarization^[11–16]. For circularly polarized (CP) wavefront manipulation, these metasurfaces either use propagation phase or compound geometric and propagation phase to obtain the desired phase profiles on CP orthogonal polarization. Polarization-insensitive metasurfaces involving compound phase modulation merge the propagation phase with the polarization-insensitive geometric phase^[17–20]. However, such metasurfaces cannot be considered spin-insensitive because they do not undergo any wavefront change for circularly co-polarized light.

For CP wavefront manipulation, the geometric phase can be imposed on the transmitted light independently as well as collectively with the propagation phase for various metasurfaces-based photonic applications^[21–32]. In compound phase modulation, the energy in the CP transmitted state as the input cannot be eliminated totally^[33]; therefore, researchers have started to explore both co- and cross-polarization states by partially distributing the output energy^[34,35]. By this approach, in a single-layered metasurface, the transmitted phase on three out of four combinations of input and output polarization states can be controlled flexibly for different applications. However, to the best of our knowledge, none of the work has been reported that encodes the same information on all four transmitted polarization states. Here, to realize spin-insensitive functionality through metasurfaces, we use compound phase modulation by merging the propagation phase with the polarization-insensitive geometric phase and utilize both co- and cross-CP channels by partially distributing the energy among them. As a proof of concept, we encode a hologram on the single-layered non-interleaved metasurface, which displays a holographic image for any input and output combination of the CP state. Furthermore, we also numerically realize a spin-free beam deflector, which remains insensitive to the spin of the incident light. The proposed design principle is expected to pave a route to realizing next-generation spin-insensitive integrated photonic systems for communication, imaging, and information processing.

The mathematical description of the desired functionality can be understood through the Jones transmission matrix for the CP input $E_{\text{in}}$ and output $E^{\mathrm{out}}$ electric field as^[33,34,36]$$\begin{gathered} J=R(\theta )\left[\begin{array}{cc}{e}^{i{\varphi }_{\mathrm{RR}}}& e^{i{\varphi }_{\mathrm{RL}}}\\ e^{i{\varphi }_{\mathrm{LR}}}& e^{i{\varphi }_{\mathrm{LL}}}\end{array}\right]R(-\theta ) \\ =\left[\begin{array}{cc}{e}^{i{\varphi }_{\mathrm{RR}}}& e^{i({\varphi }_{\mathrm{RL}}\pm 2\sigma \theta +\frac{\pi }{2})}\\ e^{i({\varphi }_{\mathrm{LR}}\mp 2\sigma \theta +\frac{\pi }{2})}& e^{i{\varphi }_{\mathrm{LL}}}\end{array}\right], \end{gathered}$$(1)where $R(\theta )$ is the rotation matrix, and $\pm 2\sigma \theta$ or $\mp 2\sigma \theta$ are geometric phases (rotation-dependent equal and opposite phases for CP orthogonal polarizations), with $\pm \sigma$ or $\mp \sigma$ indicating the handedness of the CP light. Herein, $+\sigma$ and $-\sigma$ denote the right circular polarization (RCP) and the left circular polarization (LCP) light, respectively. ${\varphi }_{\mathrm{RR}}$, ${\varphi }_{\mathrm{LR}}$, ${\varphi }_{\mathrm{RL}}$, and ${\varphi }_{\mathrm{LL}}$ are the propagation phases, which can be tailored through the physical dimensions of the nanobrick, and it can be determined through the phase average of the linearly orthogonal polarizations as $\frac{{\varphi }_{xx}+{\varphi }_{yy}}{2}$. The left/right subscripts represent the output/input CP state and L/R define the LCP/RCP state. For $+\sigma$ incident light, the transmitted electric field can be determined as $$\begin{gathered} E_{+\sigma }^{\mathrm{out}}=J{E}_{\text{in}}=\left[\begin{array}{cc}{e}^{i{\varphi }_{\mathrm{RR}}}& e^{i({\varphi }_{\mathrm{RL}}+2\sigma \theta +\frac{\pi }{2})}\\ e^{i({\varphi }_{\mathrm{LR}}-2\sigma \theta +\frac{\pi }{2})}& e^{i{\varphi }_{\mathrm{LL}}}\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right] \\ =\left[\begin{array}{c}{e}^{i{\varphi }_{\mathrm{RR}}}\\ e^{i({\varphi }_{\mathrm{LR}}-2\sigma \theta +\frac{\pi }{2})}\end{array}\right]. \end{gathered}$$(2)

Similarly, for the $-\sigma$ incident light, the transmitted electric field can be determined as $$\begin{gathered} E_{-\sigma }^{\mathrm{out}}=J{E}_{\text{in}}=\left[\begin{array}{cc}{e}^{i{\varphi }_{\mathrm{RR}}}& e^{i({\varphi }_{\mathrm{RL}}+2\sigma \theta +\frac{\pi }{2})}\\ e^{i({\varphi }_{\mathrm{LR}}-2\sigma \theta +\frac{\pi }{2})}& e^{i{\varphi }_{\mathrm{LL}}}\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right] \\ =\left[\begin{array}{c}{e}^{i({\varphi }_{\mathrm{RL}}+2\sigma \theta +\frac{\pi }{2})}\\ e^{i{\varphi }_{\mathrm{LL}}}\end{array}\right]. \end{gathered}$$(3)

In Eqs. (2) and (3), the terms with propagation phase only can be referred as co-polarization transmission channels, whereas the expression with both propagation and geometric phase can be referred as cross-polarization channels. These expressions indicate that both (co- and cross-) transmission channels can be manipulated through the suitable phase profile selection.

In our work, to design spin-insensitive metasurfaces, the inherent spin-locked geometric phase restriction is lifted up by fixing the rotation angle at “$\theta {=0}^{°}$” or “‘$\theta {=90}^{°}$.” In this way, exponential expressions for the geometric phase (cross-polarization channels) provide the same phase profiles, indicating that circular orthogonal polarization channels experience the same phase shift; thus, the geometric phase for these two rotation angles can be considered as polarization-insensitive phase shift. In single-layered metasurfaces, the phase shift in CP co-polarization is always same. It is worth noting that, in order to distribute the energy equally in co- and cross-polarization channels, the phase difference between two linearly orthogonal polarizations should be equal to $\mathrm{\Delta }\varphi ={\varphi }_{xx}-{\varphi }_{yy}=\pi /2$^[34]. If the co- and cross-polarization channels exhibit a phase relationship such that $E_{+\sigma }^{\mathrm{out}}=\left[\begin{array}{c}{\varphi }_1(\text{Co})\\ {\varphi }_2={\varphi }_1+\frac{\pi }{2}(\text{Cross})\end{array}\right]$ and $E_{-\sigma }^{\mathrm{out}}=\left[\begin{array}{c}{\varphi }_3={\varphi }_1+\frac{\pi }{2}(\text{Cross})\\ {\varphi }_4={\varphi }_1(\text{Co})\end{array}\right]$, then both transmission channels modulate the incoming wavefront in the same way for LCP and RCP light illumination, and the design can be regarded as spin-insensitive. For single-layered non-interleaved metasurfaces, such a kind of phase connection can be conveniently established by fixing the geometric phase value and modulating the propagation phase.

The illustration in Fig. 1(a) presents the unit cell, which consists of silicon nanobricks with the height of $H=420\mathrm{nm}$ placed on the glass substrate. The physical dimensions of the nanobrick in the $x$ and $y$ directions “$L_x=L_y$” determine the propagation phase, and the orientation angle $\theta$ between the $x$ axis and local coordinate “$u$” determines the geometric phase. Here, we obtain the propagation phase by modifying the side dimensions and merging it with fixed geometric phase by rotating the nanobrick at $\theta {=0}^{°}$ or $\theta {=90}^{°}$ depending upon the fast axis [Fig. 1(a)]. Hence, the $2\pi$ phase control can be achieved to realize the proposed functionality. It is worth mentioning that the choice of nanobrick instead of isotropic structure facilitates partial distribution of output energy in CP co- and cross-polarization channels. To obtain the $2\pi$ phase control, the side dimensions of the nanobrick are swept from 50 to 250 nm at $\theta {=0}^{°}$ by using the finite difference time domain (FDTD) simulation tool (Lumerical) under the normally incident linearly orthogonal polarizations. The wavelength of incident light is $\lambda =780\mathrm{nm}$. The corresponding numerically simulated amplitude and phase delays are shown in Fig. 1(b). A library search approach is used to pick the nanobricks that satisfy the “$\mathrm{\Delta }\varphi \approx \pi /2$” phase difference [Fig. 1(c)] between the fast and slow axes, which guarantees almost equal amplitude distribution in CP co- and cross-transmission channels and fulfills the 0−2π phase span. The optimized parametric values of nanobricks are presented in Table 1. Figure 1(d) shows that, for RCP/LCP incident light, these basic nanobricks partially distribute the amplitude between the CP co- and cross- [($\text{Amp\_RR}$ and $\text{Amp\_LR}$) or ($\text{Amp\_LL}$ and $\text{Amp\_RL}$)] transmission channels. The amplitude distribution is the same for CP co-polarization ($\text{Amp\_RR}=\text{Amp\_LL}$) and cross-polarization ($\text{Amp\_LR}=\text{Amp\_RL}$) channels. The optimized nanobricks provide the discrete phase coverage of $2\pi$ with a phase interval of $2\pi /7$, as presented in Fig. 1(e). It is shown that the pair of co-polarization channels (RR and LL) has the same profiles, and the pair of cross-polarization channels (LR and RL) experiences the additional $\pi /2$ phase shift. Based on optimized parametric values, in the following, we design and numerically realize two types of spin-insensitive metasurface designs: (1) spin-insensitive meta-hologram and (2) spin-insensitive beam deflector possessing power splitting functionality.

Holography, the technique of recording and reconstructing wavefronts, has seen the evolution of three-dimensional (3D) displays from its beginnings. Conventional holographic techniques record the interference between the coherent reference beam and target object and reconstruct the virtual 3D image through reference beam illumination^[37,38]. In recent years, computer-generated holography (CGH) is replacing conventional techniques by calculating the amplitude and phase information of the target’s object through a computer without any complex recording process^[39]. In general, CGH techniques use the spatial light modulator (SLM) for encoding and decoding; however, large pixel size limits its functionality to obtain holographic images with high resolution^[40,41].

The subwavelength-scaled feature size of meta-elements overcomes the SLM’s pixel size limit and opens new frontiers for optical and holographic devices. Metasurface holography can be achieved by precisely mapping the encoding image’s configuration (phase and/or amplitude) with the meta-elements at the interface according to their position and local scattering properties. Metasurface holography can be categorized into three major types: phase-only holography, amplitude-only holography, and complex (combined amplitude and phase) holography^[42]. To design a metasurface holographic device (phase only or complex), discrete phase shifts are the crucial information that can be tailored by engineering the spatial dimensions and/or orientation of the meta-elements depending upon the input/output polarization of the light. Several polarization-dependent and independent metasurface holograms have been reported for linearly polarized light^{[11,14,43–47]}. For CP light, most meta-holograms involving the spin of light display the holographic image on the cross polarization, making them sensitive to handedness of the incident light^[48–52]. In this work, we propose the spin-insensitive metasurface hologram, which displays the identical holographic image irrespective of the spin of incident/output polarization through compound phase modulation by controlling the spatial dimension and orientation of meta-elements.

The schematic of the proposed spin-insensitive metasurfaces hologram is shown in Fig. 2. It is presented that the light at the output with the LCP or RCP state displays an identical holographic image regardless of the spin of incident light. To design the meta-hologram, phase information of the target image with the letters “NRC” [Fig. 3(a)] is extracted through the Gerchberg–Saxton algorithm^[53]. The reconstructed hologram is shown in Fig. 3(b). Then, the mapping of the nanobrick’s discrete phase levels is performed with the hologram’s phase mask, and two-dimensional (2D) matrices containing the physical dimensions of the nanobricks in the spatial domain are obtained. These spatially distributed parametric values of nanobricks are then imported onto the interface of the glass substrate. When the RCP light is normally incident on the metasurface, the spatially distributed nanobricks interact with the incoming light and reproduce the holographic image of the letters “NRC” on both transmitted channels, i.e., “RR and LR.” Similarly, by switching the spin of the incident light to LCP, the spatially distributed nanobricks reproduce the same image on the “RL and LL” channels; the far-field results are shown in Figs. 3(c). The transmission channels “RR” and “LL” reconstruct the same image due to “${\varphi }_1={\varphi }_4$,” and, because of the relative additional phase shift “$\pi /2$,” the same holographic image appears on the “LR” and “RL” channels.

The proposed design principle can also be implied to realize other polarization-insensitive metasurfaces devices. Similar to the meta-hologram, previously reported metasurfaces-based CP beam steering devices (beam deflectors and splitters) only utilize the cross-polarization channels^[26,54]. In the following, for RCP/LCP incident light, a spin-insensitive beam deflector possessing the beam splitting functionality in CP orthogonal channels at output is designed. Such beam deflectors can refract the transmitted CP orthogonal beams (LCP and RCP) at a specific angle with an energy distribution ratio of 50:50. In order to realize the proposed beam deflector design, the phase function for the normally incident light can be designed as follows: $\mathrm{d}\varphi (x,\lambda )=\frac{2\pi }{\lambda }{n}_t\mathrm{d}x\sin {\theta }_{\text{trans}}$, in which $\lambda$ is the operating wavelength, $x$ is the lateral position on the metasurface, $n_t$ is the refractive index of transmission media, and $\mathrm{d}x$ is the period size. At operating frequency, the transmission/refraction angle ${\theta }_{\text{trans}}$ can be determined through the size of the super cell composed of unit cells covering the 0−2π discrete phase span.

The schematic of the proposed spin-insensitive beam deflector is shown in Fig. 4(a). It is shown that under CP (LCP/RCP), both co- and cross-polarized lights deflect at ${\theta }_{\text{trans}}$. To numerically realize the proposed functionality, three super cells composed of seven basic nanobricks are distributed in four rows. The size of each super cell in the $x$ direction is ${\mathrm{\Lambda }}_x=400\mathrm{nm}\times 7=2800\mathrm{nm}$. Hence, the transmission angle in glass ($n_t=1.46$) can be calculated as ${\theta }_{\text{trans}}=\arcsin (\frac{780\mathrm{nm}\times 2\pi }{2\pi \times 2800\times 1.46\mathrm{nm}})=10.99°$. The numerically calculated near-field and far-field results are shown in Figs. 4(b) and 4(c), which validate the analytical results and show that both CP co- and cross-polarization beams are deflecting at the same angle. Furthermore, the power of incident CP light is partially splitting in co- and cross-polarization channels (approximately 48% in each channel) under RCP/LCP light. It is worth noting here that the additional relative phase shift “$\pi /2$” does not significantly change the refraction angle and power in CP cross-polarization channels. The proposed meta-hologram and spin-insensitive designs can be experimentally validated through standard electron beam lithography by following the process reported in Refs. [12,55].

In summary, we proposed and numerically realized spin-independent metasurfaces for CP transmission channels using compound propagation and polarization-insensitive geometric phase modulation. By exploring both the co- and cross-transmission channels, the compound propagation and polarization-insensitive geometric phase modulation impose indistinct phase, i.e., ${\varphi }_1={\varphi }_4$, on CP co-polarization transmission channels, i.e., “RR” and “LL.” Meanwhile, the transmitted light experiences an additional relative phase shift of “$\pi /2$” on the cross-polarization channels, i.e., “LR” and “RL.” Such phase distribution can achieve spin-insensitive wavefront engineering functionality for CP outgoing light. This paper incorporates spin-insensitive functionality in meta-hologram and metasurface-based beam deflectors by establishing the desired phase relationships on all CP transmission channels. The numerically simulated results of the meta-hologram reconstruct the same image of the letters “NRC” on all four CP transmission channels regardless of the spin of the incident light. Similarly, the beam deflector design refracts the light at 11° for all four CP transmission channels with equal power distribution (approximately 48% into each co- and cross-transmission channel for RCP/LCP incidence). It is expected that the proposed design principle may pave a route to realizing next-generation spin-insensitive integrated photonic systems for communication, imaging, and information processing.