2.1.

Deriving Fundamental Limits to Conversion Efficiency

We first formulate the general properties of any passive linear optical device with $N$ total outputs, where an unpolarized light source is coupled to a single input port [Fig. 1(c)]. For maximum conversion efficiency of unpolarized to polarized light, we aim to fully transmit all input power across the outputs. Each output has a predefined target pure polarization state $|{\psi }_{\mathrm{1,2},\dots ,N}⟩$. Mathematically, this transformation to an arbitrary output state $|{\psi }_n⟩$ of the $n^{th}$ output can be defined by a Jones matrix,

The input state $|{\varphi }_n⟩$ is optimized to maximize the transmission efficiency, and $A_n$ are real-valued transmission amplitudes that are bounded for passive devices as $0\le A_n\le 1$. An input unpolarized light source can be represented as a mixed state with the density matrix

The corresponding power transmission to each output is then $P_{\text{unpol},n}=0.5|A_n{|}^2$, meaning that the maximum conversion efficiency for each output is 50% [red marker in Fig. 1(d)]. The total power efficiency is $P_{\text{unpol},\text{total}}=0.5\sum _{n=1}^N|A_n{|}^2$, which should be no more than unity for passive devices. Then, the general research question becomes: can we find a set of input states $|{\varphi }_{\mathrm{1,2},\dots ,N}⟩$ such that $P_{\text{unpol},\text{total}}=1$ for any given set of output states $|{\psi }_{\mathrm{1,2},\dots ,N}⟩$ and arbitrary power splitting portions.

We find that full power efficiency can be achieved for $N\ge 2$, as shown by the blue markers in Fig. 1(d) and in Sec. S1 in the Supplementary Material.³⁵ For $N=2$, the power transmission to each of the two outputs is exactly 50%, meaning that there is no flexibility in the power splitting when realizing 100% total efficiency. Nevertheless, in this scenario, there is still an arbitrary choice of the pure output polarization states, including having the same output polarization. In comparison, previous bulky optics and metasurface-based PBSs have never explored the full potential of a dual-output polarizer.

For three or more outputs, 100% total efficiency can be achieved with arbitrary output states and any power splitting portions, only subject to a condition that each output has a maximum of 50% power, as marked by the red shading in Fig. 1(d). After defining the output states and power splitting portions, analytical forms of the right singular states $|{\varphi }_{\mathrm{1,2},3}⟩$ can be obtained for three outputs, which are derived in Sec. S1.3 in the Supplementary Material.³⁵ For $N\ge 4$ output ports, there are nonunique solutions, since the number of allowed free parameters oversatisfy the necessary conditions. We present a particular analytical solution in Sec. S1.4 in the Supplementary Material.³⁵ We emphasize that this is a general result; a passive optical device with $N\ge 2$ ports may have 100% total efficiency in achieving arbitrary output polarization states. In addition, a device with $N\ge 3$ ports may also have arbitrary power-splitting portions at the outputs, provided that no single port contributes more than 50% to the total efficiency. Replicating the functionality of this device with a conventional optical system would require a series of multiple PBSs, wave plates, and other bulky optical elements (Sec. S2 in the Supplementary Material³⁵).

2.2.

Topology Optimization

We illustrate a particular case of polarization conversion by designing metagratings that split an incoming unpolarized beam into multiple diffraction orders, all having identical output-pure polarizations $|{\psi }_{\mathrm{1,2},\dots ,N}⟩=|\psi ⟩$. This operational functionality achieves spatially uniform polarization that can be beneficial for structured illumination applications.³⁶ Dielectric metasurfaces are commonly and successfully designed in the framework of weakly interacting uncoupled resonators,^28,37–40 where the output fields are determined by the superposition of position-dependent transfer matrices. However, we find that this type of metasurface construction does not enable our desired functionality. Specifically, according to Eq. (6) in Ref. 28 $J_n=A_n|{\varphi }_n^{*}⟩⟨{\varphi }_n|$, conventional nonchiral pillar-based metasurfaces are restricted to the target Jones matrices with $|{\psi }_n⟩=|{\varphi }_n^{*}⟩$. It means that for identical output pure polarizations $|{\psi }_n⟩=|\psi ⟩$, one necessarily has $|{\varphi }_n⟩=|{\psi }^{*}⟩$. This device would filter out one input polarization component with the same output efficiency limit of 50% as a conventional polarizer.

We overcome this apparent roadblock by designing dielectric metasurfaces with a spatially nonlocal response, where the polarization transformations depend on the diffraction order. For this purpose, we perform inverse design with free-form topology optimization^41–44 by adopting the MetaNet codebase⁴⁵ in combination with RETICOLO rigorously coupled wave analysis.⁴⁶ Our figure of merit (FOM) is formulated as the difference between the transmitted target and undesired orthogonal polarization intensities, multiplied over all output diffraction orders,

3.1.

Simulation Results

In the first design, we target the splitting of an incoming unpolarized beam into two outgoing beams ($N=2$) with the same diagonal linear polarization $|D⟩$. This can be achieved with an angled incidence of the input beam such that only two diffraction orders exist in the transmission direction [Fig. 2(a)]. We design the metasurface for equal transmitted diffraction efficiency into the $m=0$ and $m=1$ orders [Fig. 2(b)]. The incident angle is $\alpha =45\mathrm{deg}$ for the unpolarized input light. An oblique angle of incidence is chosen for a first demonstration, which simplifies the optimization by eliminating unwanted higher-order diffraction orders. With 45 deg, the two diffraction channels are reasonably separated spatially to individually measure in an experimental setup.

Fig. 2

(a) Illustration of the dual-output polarizing metasurface. (b) Operating scheme of the metasurface in transmission, with an incident angle of $\alpha =45\mathrm{deg}$ and first-order diffraction angle of $\beta =68\mathrm{deg}$. (c) Final metasurface design of the single unit cell, with a period 950 nm along both directions. The shaded region represents silicon, and the void represents air. (d) Electric field intensity enhancement for incident $|H⟩$ polarization, and (e) $|V⟩$ polarization. These have the same unit cell dimensions as (c). (f) Predicted transmitted power, and (g) extinction ratio of the diagonal linear polarization for each diffraction order.

We run the topology optimization starting with a random distribution of refractive indices in the unit cell. The design is based on a silicon layer ($n=3.48$ at $\lambda =1550\mathrm{nm}$) with a thickness of 1000 nm on a $460\mu \mathrm{m}$ sapphire substrate ($n=1.75$ at $\lambda =1550\mathrm{nm}$), corresponding to our physical sample. The algorithm also incorporates binarization of the perturbation region to either silicon or air with a constraint on minimum feature size.⁴⁷ The optimized complex shape of the nanoresonator is shown in Fig. 2(c). Due to its asymmetric shape, the localized fields induced in the resonator are highly nontrivial. For example, when the incident light is polarized in the $X$-direction ($|H⟩$ polarization) [Fig. 2(d)], most of the field is concentrated on the right tip of the nanoresonator. However, when incident light is polarized in the $Y$-direction ($|V⟩$ polarization) [Fig. 2(e)], the light is instead concentrated along the left arm of the nanoresonator. Therefore, the action of the metasurface on unpolarized light is defined through a complex superposition of induced fields in the nanoresonator.

The modeling predicts highly efficient conversion of unpolarized light to the target diagonal state $|D⟩$ at both outputs [Fig. 2(f)]. The combined total efficiency is beyond 80% over an extended wavelength range of 1520 to 1570 nm. Importantly, this performance fundamentally exceeds the 50% limit of conventional polarizers, with a combined extinction ratio that approaches 100 at 1550 nm [Fig. 2(g)].

While the resulting optimized metasurface geometry delivers the required polarization transmission performance, the underlying mechanism of its operation, and the interplay between local and nonlocal modes, may not be intuitively obvious. We employ the singular value decompositions of the scattering matrix to elucidate the mechanism with which polarizations are split and rotated for different outputs (Sec. S4 in the Supplementary Material³⁵). Then, to obtain physical insight, we perform multipolar decomposition⁴⁸ to identify the predominant local modes of the metasurface under different polarizations (Sec. S5 in the Supplementary Material³⁵). We find that electric and magnetic dipole modes provide the strongest scattering and polarization-filtering response. At the same time, the metasurface has a nonlocal characteristic allowing for the nontrivial dependence of transmission on the diffraction orders beyond the limits of local metasurfaces, as discussed above. We provide performance comparisons to previously demonstrated metasurface polarizers and commercially available polarizers in Sec. S6 in the Supplementary Material (see also Refs. 4950.–51 therein). The metasurface maintains effective performance greater than the threshold 50% efficiency over the entire incident angle ($\alpha$) range from 40 deg to 80 deg. Peak conversion efficiency of $\sim 80\%$ is reached at around 50 deg incidence, which is shown in Sec. S7 in the Supplementary Material.³⁵ While this work focuses on optimization of metasurfaces that are amenable to current planar silicon fabrication platforms, new manufacturing processes in the near future may open up greater possibilities. The total efficiency can likely be increased beyond 90% with multilayer metasurfaces or volumetric metamaterials, which were shown to enhance performance in devices for different applications.^52–55

3.2.

Experimental Results

We successfully fabricate the optimized design using e-beam lithography and standard silicon etching, as shown in Figs. 3(a) and 3(b). The characterization of the metasurface was then performed in free space, where we measured the $m=0$ and $m=1$ diffraction orders, as illustrated in Fig. 3(c). The experiments were performed with the horizontal and vertical input polarization states in two separate measurements, mimicking the unpolarized incident light according to Eq. (2); see Sec. S8 in the Supplementary Material³⁵ for further details. These states were prepared from a continuous-wave tunable laser, operating in the near-infrared within the telecommunications band wavelengths of 1500 to 1575 nm. Calibration measurements against air, described in Sec. S9 in the Supplementary Material,³⁵ are taken to determine the variation in input power across all relevant wavelengths. These values are then used to calculate the absolute metasurface transmission efficiency.

Fig. 3

(a) Top-down and (b) tilted SEM images of the fabricated metasurface. (c) Simplified experimental setup. The source transmits through the metasurface, which diffracts the beam into two orders. Each arm analyses the diagonal polarization before reaching the detector. (d) Measured transmitted power, and (e) extinction ratio of the diagonal polarization for each diffraction order for an incident angle $\alpha =45\mathrm{deg}$.

We experimentally demonstrate the metasurface’s ability to convert unpolarized light to diagonal polarized light with an absolute efficiency close to $\sim 70\%$ [Fig. 3(d)], exceeding the 50% limit of previous approaches. This performance is maintained across a broad wavelength range, from 1540 to 1570 nm. The extinction ratio of desired to undesired output polarization states exceeds 20 at the target wavelength of 1550 nm, see Fig. 3(e). We discuss the fabrication performance tolerances for our metasurface in Sec. S11 in the Supplementary Material.³⁵

For another demonstration, we optimize for and experimentally characterize a two-output circular polarizer; see Sec. S10 in the Supplementary Material.³⁵ The metasurface is able to realize conversion of unpolarized light into circularly polarized light with approximately 60% efficiency, thereby exceeding the 50% limit.

3.3.

Designs for Three and More Outputs

We then extend our method to design metasurfaces that generate three or more outputs. As discussed above and visualized in Fig. 1(d), there is freedom in the distribution of output powers for more than two diffraction orders. We present a metasurface with nonequal power splitting portions for each of the three outputs [Fig. 4(a)], with the optimized design shown in Fig. 4(b). The unit cell is chosen to be 1600 nm by 800 nm. The reason for the rectangular unit cell is to avoid diffraction in air along the $y$ direction, while only allowing first-order diffraction in the $x$ direction. In this configuration, the zeroth order is normal to the plane of the metasurface. This three-output design realizes high combined transmittance of over 80% to the target vertical linear polarization $|V⟩$, and an extinction ratio over $|H⟩$ output approaching 40 at 1550 nm [Figs. 4(c) and 4(d)]. Moreover, in Sec. S12 in the Supplementary Material,³⁵ we show a nontrivial metasurface design where incoming unpolarized light is split into four outputs with the same polarization while achieving similar performance.

Fig. 4

(a) Operating principle of a three-output metasurface polarizer. The incident beam is normal, and $\alpha =75\mathrm{deg}$. (b) The binarized metasurface pattern, with shaded regions representing silicon and the void representing air. (c) Transmittance of desired vertical polarization versus wavelength. (d) Extinction ratio of combined vertical to horizontal polarization.

It is instructive to discuss a question on whether it is possible to recombine the multiple output beams into one. Fundamentally, for two-output splitting, the beams are mutually incoherent and cannot be recombined coherently with any passive device. However, for three of more outputs, the beams can be partially spatially coherent. There is potential in future work to explore different beam recombination schemes and utilize them for further tailoring different structured light and dot projection schemes.