Broadband vectorial ultrathin optics with experimental efficiency up to 99% in the visible region via universal approximators

Integrating conventional optics into compact nanostructured surfaces is the goal of flat optics. Despite the enormous progress in this technology, there are still critical challenges for real-world applications due to the limited operational efficiency in the visible region, on average lower than 60%, which originates from absorption losses in wavelength-thick (≈ 500 nm) structures. Another issue is the realization of on-demand optical components for controlling vectorial light at visible frequencies simultaneously in both reflection and transmission and with a predetermined wavefront shape. In this work, we developed an inverse design approach that allows the realization of highly efficient (up to 99%) ultrathin (down to 50 nm thick) optics for vectorial light control with broadband input–output responses in the visible and near-IR regions with a desired wavefront shape. The approach leverages suitably engineered semiconductor nanostructures, which behave as a neural network that can approximate a user-defined input–output function. Near-unity performance results from the ultrathin nature of these surfaces, which reduces absorption losses to near-negligible values. Experimentally, we discuss polarizing beam splitters, comparing their performance with the best results obtained from both direct and inverse design techniques, and new flat-optics components represented by dichroic mirrors and the basic unit of a flat-optics display that creates full colours by using only two subpixels, overcoming the limitations of conventional LCD/OLED technologies that require three subpixels for each composite colour. Our devices can be manufactured with a complementary metal-oxide-semiconductor (CMOS)-compatible process, making them scalable for mass production at low cost.

withσ = dωσ(ω)e iωt being the inverse Fourier transform of σ and δ(t) the diract delta function, as the reader can verify by direct substitution. The solution of Eq. (1) reads:σ (t) = e At · (σ 0 + K), withσ 0 =σ(t = 0) the initial state and e At being the matrix exponential, with A = iΩ − Γ. The matrix exponential is expressed in closed form by diagonalizing A = QΛQ −1 , with Λ mm = λ m δ mm being a diagonal matrix of complex eigenvalues λ m = iα m − β m : In Eq. (3), Q = Q(Ω, Γ) and Λ = Λ(Ω, Γ) originate from the eigenvectors and eigenvalues of A = iΩ − Γ and as such they are functions of Ω and Γ. The elementsσ mn are calculated by substituting (3) into (2): and expressed as the sum of complex damped exponentials with µ mnh = µ mnh (Ω, Γ) coefficients arising from the matrix product of e At · (σ 0 + K). By applying the Fourier transform on Eq. (4), we obtain the expression of the elements σ mn of σ: represented as a complex rational function with poles ω h + iγ h , and coefficients µ mnh .

SUPPLEMENTARY NOTE II
We considered a general rational function: with N α + 1 and N β + 1 coefficients α n and β n , respectively. Without loss of generality and for the sake of simplicity, we assumed that the function y out is real. If we control N α + N β + 2 coefficients α n and β n , we can then can uniquely determine the behavior of the output function y out (ω m ) at a set of m = 1, ..., M = N α + N β + 1 spectral frequencies ω m . To demonstrate this result, we impose the conditions y out (ω m ) = y m (ω m ) on Eq. (6) and obtain the following algebraic system: with diag being the diagonal matrix for the M function values y(ω m ) at the target points, V i the first i columns of the Vandermonde matrix V generated at the sample points ω m with components V ij = ω j−1 i , α = [α 0 , ..., α Nα ] and β = [β 0 , ..., β N β ] being the rational coefficients sought. The solution to Eq. (6) requires one to find a null vector of the M × (N α + N β + 2) matrix: For M = N α + N β + 1, the matrix (8) has a null space of dimension N α + N β + 2 − M = 1, and each null vector gives an exact solution in (6) [1]. For M < N α + N β + 1, the system is under determined and the null space of (8) has dimension N α + N β + 2 − M > 1. In this case, any null vector among the space of ∞ Nα+N β +2−M provides an exact solution to the problem. For M > N α + N β + 1, the system is over determined and the solution to Eqs. (6) can be found in the least square sense by constrained minimization: with . being the vector norm. The constraint α + β = 1 is introduced to avoid the trivial solution β = α = 0.
In the case of Eq. (1) of the main text, the transfer function H is written as follows: We begin by considering the case C(ω) = C independent of the frequency ω. In this case, Eq. (5) shows that H is a complex rational function with coefficients depending on the resonance frequencies Ω and mode dampings Γ. The control of 2M resonances Ω n implies the control of 2M coefficients in the transfer function H. Equation (7) shows that we can therefore control the system response in amplitude and phase exactly at M − 1 spectral points in both real and imaginary part, or equivalently, amplitude and phase in a single scattering channel. At any number of points ≥ M , the problem can be solved in the least square sense via (9). This result applies also in the case where C(ω) is frequency dependent. In this case every condition H(ω l ) = H l generates a rational function with different constants C l = C(ω l ), maintaining the validity of Eqs. (7)-(9).

SUPPLEMENTARY NOTE III
We considered an optical network with N = 10 output scattering channels, initialized with random scattering C = e iH , with H being a random symmetric matrix with elements drawn from a uniform distribution between zero and 2π. The random couplings K = A + iB are initialized with random matrices A and B, each with elements drawn from a uniform distribution. We then initialized the dampings to Γ = KK † 2 . The transmission T is measured in the first channel s +1 computed at n = 1.

SUPPLEMENTARY NOTE IV
We use the generalized cost function defined in Eq. (6) of the main text to design an 800 nm robust (α = 0.5) polarizer, and compare it to a design found without taking into account tolerance to the fabrication errors (α = 0). To model fabrication tolerances we varied the sizes of individual boxes using a uniform distribution with random values in the range ±25 nm. The main contribution to the errors being the proximity effect at the electron beam lithography stage, which affects the sizes but not the position of the centers of the boxes, nor the overall period.
Supplementary Figure 1. ALFRED performance on 2500 randomly chosen samples from a test dataset a. The colors represent the mean squared error between the ground truth and the predicted spectra for each sample. An example of the output from the predictor part of ALFRED for the random test sample c. The blue and red curves on the figure represent TE and TM spectra respectively, for the true values and dotted lines represents neural network predictions. . The integral of the difference between TE and TM in the interval from 0.6 to 0.8µm was used as the cost function.
Supplementary Figure 5. Panels a-c shows results of a 800 nm polarizer designed α = 0, while d-f present results for α = 0.5. The solid red line in Panels c and f represents the efficiency of the polarizer with no random perturbation. The histograms in panels a and d report the distribution of the whole set of perturbed polarizers. As the probability distribution is not perfectly Gaussian, we split the probability it into a bottom, middle and a top region comprising 16%, 68%, and 100%−16%, respectively, of the total probability area. The central 68% of all values is chosen to be equivalent to one standard deviation of a normal distribution. The dotted lines in panel c shows the minimum efficiency of the polarizers comprising the set at 68%, implying that 68% of the polarizer designs have efficiency higher than that the dotted limit. Panels c and f reports the same analysis for α = 0.5. Panels b and e report the overlay of all perturbed structures used in this analysis, respectfully for α = 0 and α = 0.5.