Broadband angular spectrum differentiation using dielectric metasurfaces

Abstract

Signal processing is of critical importance for various science and technology fields. Analog optical processing can provide an effective solution to perform large-scale and real-time data processing, superior to its digital counterparts, which have the disadvantages of low operation speed and large energy consumption. As an important branch of modern optics, Fourier optics exhibits great potential for analog optical image processing, for instance for edge detection. While these operations have been commonly explored to manipulate the spatial content of an image, mathematical operations that act directly over the angular spectrum of an image have not been pursued. Here, we demonstrate manipulation of the angular spectrum of an image, and in particular its differentiation, using dielectric metasurfaces operating across the whole visible spectrum. We experimentally show that this technique can be used to enhance desired portions of the angular spectrum of an image. Our approach can be extended to develop more general angular spectrum analog meta-processors, and may open opportunities for optical analog data processing and biological imaging.

Fig. 1. Angular spectrum meta-differentiator.

a–c Angular spectrum analog meta-processors for optical analog differentiation processing in the angular spectrum domain, as light propagates through them. The corresponding angular spectrum operation, Ĥ, and complex real-space transfer function, t(x, y), provided by the silicon metasurfaces are, respectively, $\propto \frac{{\partial }^2}{\partial k_x\partial k_y}$ and ∝ xy (a), $\propto \frac{\partial }{\partial k_x}$ and ∝ x (b), and $\propto \frac{\partial }{\partial k_x}+\frac{\partial }{\partial k_y}$ and ∝ x + y (c). It is assumed that C₁ is equal to C₂ in (c). d Schematic of the rotated silicon nanopillar on a silica substrate with W_u = 200 nm, W_v = 100 nm, h = 220 nm, and P = 280 nm. e, f Amplitude (e) and phase (f) of σ versus orientation angle θ. The refractive index and extinction coefficient of silicon were measured by an ellipsometer (see Supplementary Fig. S1 of Supplementary Note 2), and the refractive index of silica is extracted from Ref. . g–l Schematics of the top view presenting the distribution of the orientation angles (g–i) and distributions of the real parts of the required σ(x, y)/|σ(x, y)|_max of the three meta-differentiators (j–l) with $\frac{{\partial }^2}{\partial k_x\partial k_y}$ (g, j), $\frac{\partial }{\partial k_x}$ (h, k) and $\frac{\partial }{\partial k_x}+\frac{\partial }{\partial k_y}$ (i, l), respectively.

Fig. 2. Angular spectrum intensity distributions for three types of differentiations.

a, b Field intensity ${∣E_x∣}^2/{∣E_x∣}_{\max }^2$ (a) and angular spectrum intensity ${∣A_x∣}^2/{∣A_x∣}_{\max }^2$ (b) of the input x-polarized Gaussian beam with a waist radius of w = 3.5 μm at 685.5 nm. c, d Output angular spectrum intensity distributions for the three types of differentiations: theory (c) and simulation (d). k₀ is the wavenumber in the air.

Fig. 3. Measured transmission field for three types of differentiations.

a–c SEM images of partial samples for angular spectrum differentiation with $\frac{{\partial }^2}{\partial k_x\partial k_y}$ (a) $\frac{\partial }{\partial k_x}$ (b) and $\frac{\partial }{\partial k_x}+\frac{\partial }{\partial k_y}$ (c). The scale bars are 1 μm. d Experimental setup for measurement. The meta-differentiator and the CCD camera are on the object and the image planes of objective 2, respectively. e–g Transmission field intensity distributions recorded by CCD with red light (685.5 nm) (e), green light (532 nm) (f) and blue light (450 nm) (g). The scale bars are 100 μm.

Fig. 4. Measured angular spectrum differentiations of three parallel rectangular holes.

a Experimental setup for performing angular spectrum operation of an object. The object and the meta-differentiator are on the object and the image planes of objective 1, respectively. The back focal plane of objective 1 is also the object plane of objective 2 and the CCD camera is on the image plane of the objective 2. b Theoretical (the first panel) and experimental (the second to fourth panels) intensity distributions of three parallel rectangular holes drilled in a stainless steel plate (1 mm thick). The geometrical parameters of the holes are set at W_x = 200 μm and W_y = 1 mm. c, d Angular spectrum intensity distributions (c) and normalized intensity profiles along lines A-B in (c) (d). e–j Angular spectrum intensity distributions with $\frac{{\partial }^2}{\partial k_x\partial k_y}$ (e), $\frac{\partial }{\partial k_x}$ (g), and $\frac{\partial }{\partial k_x}+\frac{\partial }{\partial k_y}$ (i), and their normalized intensity profiles along lines A-B (f, h, j).

Fig. 5. Angular spectrum isolation experiment.

a Experimental setup for performing angular spectrum isolation at 685.5 nm. b–e Angular spectrum intensity distributions of the mixed light without (b) and with (c) the meta-differentiator with $\frac{\partial }{\partial k_x}$. Their normalized angular spectrum intensity profiles along the lines A-B are shown in (d) and (e), respectively.