Single-layer spatial analog meta-processor for imaging processing

Computational meta-optics brings a twist on the accelerating hardware with the benefits of ultrafast speed, ultra-low power consumption, and parallel information processing in versatile applications. Recent advent of metasurfaces have enabled the full manipulation of electromagnetic waves within subwavelength scales, promising the multifunctional, high-throughput, compact and flat optical processors. In this trend, metasurfaces with nonlocality or multi-layer structures are proposed to perform analog optical computations based on Green’s function or Fourier transform, intrinsically constrained by limited operations or large footprints/volume. Here, we showcase a Fourier-based metaprocessor to impart customized highly flexible transfer functions for analog computing upon our single-layer Huygens’ metasurface. Basic mathematical operations, including differentiation and cross-correlation, are performed by directly modulating complex wavefronts in spatial Fourier domain, facilitating edge detection and pattern recognition of various image processing. Our work substantiates an ultracompact and powerful kernel processor, which could find important applications for optical analog computing and image processing.

1. Particularly, according to the digital throughput measured in bits per second (bit/s) , the throughput of analog processor here is evaluated by the processing range of input data per unit of time.
Firstly, both the GF kernel and our proposed processor revel a distinct advantage in device integration for the single-layer spectrum modulation within sub-wavelength scale. Secondly, compared with the increase of waveguide number with more complicated inverse design of metastructure and the device optimization of GF kernel for wider modulation range of incident angle, the computing throughput can boost enormously by simply increasing the metaatom number of 4f Fourier system and our proposed metasurface. Finally, compared with other three methods, the angle-dependent response of GF kernel is suitable for limited Fourier-domain operations (such as derivative and integral). For instance, the differentiation is transformed into the easier multiplication in angular-scattering spectrum, as , which can be implemented by the mechanism of Fano resonance or surface plasmon polariton (SPP). However, for the mathematical function with drastic fluctuations, such as the cross-correlation in our work as ( ) , sinc sinc , the difficulties in angle-dependent metaatom design will be greatly increased. Overall, our proposed meta-processor renders a high level of aggregation with respect to the integration, throughput, function diversity.

Supplementary
should be eliminated by introducing the concave-lens phase factor ( ) on the Huygens' metasurface aperture, where x y y f f = =   and  represents two-dimensional convolution operation. Via applying the convolution theorem,, described as the output image in Fourier spectrum can be obtained by multiplying input signals with repeated Fourier transforms of EH, as . Hence, as the ratio of the output signal and input signal in Fourier spectrum, EH acts as transfer function and x′ and y′ indicate the spatial Fourier frequencies (wavevectors). Huygens' metasurface can directly modulate the spatial Fourier spectrum through EH for predesigned analog processing. For direct expression of formula in the main text, Equation (3) can be described as: , where the additional phase factor ( ) ( ) Overall, by superimposing the specific phase factor related with the input and output focal length on the transfer function EH algorithmically, the proposed Huygens' metasurface can directly manipulate spatial Fourier frequencies for the target outputs with single-layer structure.

Supplementary Note 3 Design of Huygens' meta-atom structure
Huygens' metasurface is a two-dimensional array composed of crossed magnetic and electric dipoles which can tailor the scattered electric and magnetic field distribution. Based on Huygens' principle and boundary conditions, the relationship between the surface impedance of electric, magnetic dipoles and the value of reflection, transmission coefficient is derived as [1,2]: Huygens' metasurface, Equation (5) and (6) where the reflection coefficient remains 0 for the minimization of reflected energy. t φ represents transmission phase ranging from 0 to 2π, while t denotes transmission amplitude ranging from 0 to 1.
η is set to be 377Ω as the impedance of the free space. Hence, Equation (7) and (8) for 2D edge detection.

Supplementary Note 5 The resolution of the first-order derivative and cross-correlation operation
As an indicator of the proposed analog processor, the system resolution of edge detection and crosscorrelation operation is analyzed respectively. As shown in Supplementary Fig. 3a and 3b, the 2D edges of rectangles with different sizes are clearly demonstrated along both horizontal and vertical directions When the minimal edge length decreases to 20mm, the detected edge revels intermittent and a dramatic increase of sidelobe intensity influences the edge extraction. Hence, the resolution of the proposed first-order derivative operator resolution can be considered to be roughly 25mm.

Supplementary Note 7 The wavefront profiles on Huygens' metasurfaces for cross-correlation on one-dimensional sequence
For the validation of the proposed cross-operator, three reference sequences ①, ② and ③ are built by Huygens' metasurfaces respectively to test the similarity with the input two-square-pulse image. With the same method as the first cross-correlation experiment, the features of three reference sequences are built by designing the transfer function first, as ( ) 2 2 , ' sinc

Supplementary Note 8 Application of Huygens' metasurface processor at optical frequencies
To enhance the practicability and maneuverability of the Huygens metasurface processor, the proposed working mechanism has been applied to the optical-spectrum edge detection to improve the image quality, and the Laplace operation is implemented to increase function diversity, respectively.
Firstly, compared with microwave differentiator, the first-order derivative operation performed at optical frequencies can extract micrometer-scale edges and enable detection of sophisticated images.
To detect the details of input image 'H' at λ= 532 nm, the input and output focal length are set to be f1 = f2 = 80λ and the number of meta-atoms is set to be 350 × 350 with periods of 300 nm. As shown in Supplementary Fig. 8 a- , E x y denotes the input electric field), can be performed by our proposed metaprocessor. According to Equation (2) and (4) , Huygens metasurface can extract the edge details of the 2D object. As shown in Supplementary Fig. 8 e-h, the output numerical results demonstrate the prominent edge features, which indicate the feasibility and flexibility of the proposed Huygens' metasurface processor in various scenarios.