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# Metamaterials solve integral equations

23 April 2019

Analog computers made with dielectric structures process electromagnetic signals.

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As computers approach the physical limits of electronics, scientists are developing different technologies to further improve speed and capacity. Although signal processing is generally still done electronically, in 2014 Nader Engheta at the University of Pennsylvania and his collaborators proposed using metamaterials to process signals encoded in light. The device would have a spatially varying dielectric constant and would inhomogeneously scatter an incoming signal in such a way as to transform it into, say, the signal’s spatial derivative. Inspired by the idea, Engheta’s group has now used metastructures as part of a basic analog optical computer, a schematic of which is shown in the figure, that can solve integral equations.

The devices are designed to solve integral equations that take the form

$gu = Iinu + ∫abKuvgvdv.$

Known as Fredholm integral equations of the second kind, they come up in many areas of science and engineering, including antenna theory and perturbation theory in quantum mechanics. For a given kernel K, the device takes an input signal Iin(u) and outputs the function g(u) that is the solution to the equation. The patterned block in the figure represents the metastructure itself, which plays the role of K in the integral. As with all numerical integration, the domain of the continuous variable u is discretized. Then the integral equation can be recast as a matrix relation in which the integral over K is a matrix and an algorithm translates the matrix’s elements into a spatial distribution of dielectric material for the metastructure.

Each of the N values of u at which the equation is evaluated has a corresponding waveguide (loops in the figure) that allows the signal to flow around the network and through a coupling element where Iin is introduced (orange block). Eventually the signal reaches a steady state in which the signal entering the kernel (green wave) matches the output signal (red wave). That is the solution g. The light is sampled at the coupling elements to track the device’s output.

Unlike some other designs for optical signal processors, Engheta’s is not yet reconfigurable. Each individual metastructure serves only as the kernel function for which it was designed. However, the group is working to develop the capability to reconfigure. Their devices also have a much smaller footprint than other systems. The group’s analysis shows that a steady state can be reached in fewer than 300 periods of the input light, which means that at optical frequencies the computer could evaluate the solutions to integral equations in picoseconds. (N. M. Estakhri, B. Edwards, N. Engheta, Science 363, 1333, 2019.)

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