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# Metamaterial computer solves integral equations encoded in electromagnetic waves

22 Mar 2019

Metamaterials have been used by researchers in the US to solve mathematical problems by transforming data that are encoded into electromagnetic waves. The researchers believe their new analogue computing paradigm offers several advantages over conventional digital computers and are now working to make it compatible with traditional silicon photonics devices.

Metamaterials are synthetic, compound materials that are structured in ways that give them specific properties — such as a negative refractive index – that are rare or absent in natural materials. To design optical metamaterials, researchers often rely on a branch of mathematics called transformation optics, which transforms the coordinates of space to control the path of light through a material. A famous example is the invisibility cloak whereby transformation optics is used to control the refraction of light in the cloak so that incident light travels smoothly around the cloaked object rather than scattering off it. The result is that an observer will conclude that the cloaked object is not present.

In 2014, researchers led by Nader Engheta of the University of Pennsylvania proposed another possible use for transformation optics. They pointed out that electromagnetic waves encode mathematical functions in their amplitudes and phases – both of which can be transformed by metamaterials. This led the team to suggest that metamaterials could perform mathematical operations on these functions.

### Integral equations

Now, Engheta and colleagues have designed a metamaterial that not only performs mathematical operations but can also find solutions to an important class of equations called integral equations.

“In almost any field of science and engineering you can describe the numerical values of the phenomena that you are after using integral equations,” explains Engheta. Solving these equations is therefore vital to modelling a wide range of phenomena. Algebraic solutions are often impossible, however, so researchers often must rely on computational analysis. This involves rearranging the equation so that the unknown solution appears on both sides. Starting from an arbitrary point, the calculation is then run repeatedly in a feedback loop until the correct solution is reached. At this point, performing the mathematical operation described by the equation does not change the value, so the solution remains stable.

“That takes time,” explains Engheta, which is why finding numerical simulations can often require significant computational resources.

### Speed of light

The researchers believed metamaterials could offer several important advantages over this conventional digital process. One benefit is that the computational process could be extremely fast because electromagnetic waves pass through metamaterials at the speed of light. Also, the same metamaterial can process multiple waves simultaneously: “Waves can pass through each other, giving you a parallel system,” explains Engheta.

To test their ideas, the researchers designed metamaterials from carefully-patterned dielectrics to perform mathematical transformations related to three different integral equations. Computational modelling of how electromagnetic waves interact with the metamaterials suggests that the solutions provided by the hypothetical systems should agree very well the solutions obtained from traditional numerical methods. Furthermore, the computational modelling suggests that the metamaterial systems can reach the correct solutions very quickly.

The team also created a metamaterial in the lab for one of the integral equations (see figure). It was made from patterned low-loss polystyrene and is designed for use with microwaves. The team found that its performance was in very good agreement with computational predictions.

In future, the researchers aim to build their metamaterials from a silica dielectric, which would make integration with standard silicon photonics devices easier. A silica dielectric metamaterial would also allow infrared light at telecom wavelengths to be used to perform calculations. This means that future devices could be much smaller than the microwave prototype.

The team also hopes that in the future reconfigurable metameterials could be developed, effectively creating a kind of reprogrammable analogue computer. Nevertheless, stresses Engheta, the present platform does not offer the prospect of an alternative to the conditional logic of a true computer, in which one computation depends on the outcome of another: “We don’t have any optical logic here,” he says.

Andrea Alù at the City University of New York was involved in the 2014 research and continues to work independently on computing based electromagnetic waves. He praises Engheta and colleagues for turning the original idea into reality. “I find it interesting because it’s not at all trivial that this can be worked out, especially given all the tolerances present.”

The research is described in Science.

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