The metamaterial device gives the output g = inv( I nxn − K) e, where I nxn represents the identity matrix of the same order as the kernel. In a physical realization of the device based on N input and N output waveguides, this equation can be discretized into matrix form, g = e + Kg, where g represents a N × 1 vectorial solution obtained as the complex-valued signal at each waveguide, e is the N × 1 vector input signal and K is an N × N discretized version of the kernel K( u,v). Which constitutes a Fredholm integral equation of the second kind.
In this work, we embark on a more ambitious approach to fulfill two important goals: (1) To design a single metamaterial structure that can simultaneously solve several independent equations with different kernels each with arbitrary input signals encoded on a wave with different frequency and (2) to include more general, i.e., nonsymmetric and symmetric, kernels, using the transmissive cavity and feedback mechanism. In an earlier work from our group, this concept was experimentally demonstrated with a reflective cavity, which necessitates symmetric kernels 17. Other researchers have exploited known resonant geometries to achieve differential equation solvers for time-envelope signals 41, and an example of parallel photonic differentiation has also been recently proposed 42.īy implementing feedback paths into linear systems the output and input modes can be linked together, resulting in platforms that, in addition to performing mathematical operations, can also compute the solution of mathematical problems requiring matrix inversion 17. Similar to other photonic devices, in the linear regime these structures can be characterized by a scattering matrix connecting a set of complex-valued input modes to a set of complex-valued output modes, thus capturing the operation of matrix multiplication 40. Inverse-design techniques have been largely exploited for the design of complex nanophotonic structures, such as couplers for multiplexing, 37 devices involving nonlinear phenomena 38, 39, and devices for sound classification 35. While earlier efforts using optical signal processing required extensive free-space propagation to do mathematical operations in a Fourier-transformed space 29, 30, 31, 32, more recent approaches have benefitted from the use of compact inverse-designed structures that can reliably reproduce the desired optical functionality in a more limited footprint 17, 33, 34, 35, 36.
Researchers have managed to exploit the laws of physics in such a way that the response of well-known physical systems can be utilized to solve more abstract mathematical problems 4. As traditional electronic computers find it more difficult to increase their capabilities by achieving further miniaturization 3, novel approaches based on different platforms have become closer to being functional.
We design, build, and test a computing structure at microwave frequencies that solves two independent integral equations with any two arbitrary inputs and also provide numerical results for the calculation of the inverse of four 5 x 5 matrices.įrom nonlinear image processing 1 to cryptography 2, the ability to quickly solve mathematical problems involving linear algebra is the key for handling the increasing volume of information generated every day. Here we demonstrate that a transmissive cavity filled with a judiciously tailored dielectric distribution and embedded in a multi-frequency feedback loop can calculate the solutions of a number of mathematical problems simultaneously. However, previous devices have not fully exploited the linearity of the wave equation, which as we show here, allows for the simultaneous parallel solution of several independent mathematical problems within the same device.
Therefore, researchers have explored wave systems, such as photonic or quantum devices, for solving mathematical problems at higher speeds and larger capacities. Technical specifications and characteristics Key Features Number of Channels: 5.In the search for improved computational capabilities, conventional microelectronic computers are facing various problems arising from the miniaturization and concentration of active electronics.