Physicists have solved a mathematical equation at the speed of light by shining microwaves onto a metamaterial – a cluster of unusual shapes laid out in a carefully arranged pattern.

Nader Engheta of the University of Pennsylvania in the US and his colleagues came up with the metamaterial, and used it to solve a class of equation known as “Fredholm integral equations of the second kind”.

The collection of shapes, which the team nicknamed Swiss cheese, processes microwave radiation into which specific parameters have been encoded. It then spits out a solution to an ore-set integral equation, also encoded onto microwave properties.

And because it happens at the speed of light, it is orders of magnitude faster than conventional computer-based methods says Engheta, whose work is published in the journal *Science*.

Fredholm integral equations have many uses. For example, they can describe how wifi strength varies through a building, or calculate perturbation theory in quantum mechanics, or be applied to acoustics.

“If you were trying to plan the acoustics of a concert hall, you could write an integral equation where the inputs represent the sources of the sound, such as the position of speakers or instruments, as well as how loudly they play,” says Engheta.

“Other parts of the equation would represent the geometry of the room and the material its walls are made of. Solving that equation would give you the volume at different points in the concert hall.”

Solutions for different situations, such as musicians in different locations or playing at different volumes, can be found by changing the characteristics of the input microwaves; for example, their phase and amplitude.

Each metamaterial configuration corresponds to a single equation only: in the concert hall analogy, for a differently-shaped concert hall you would need a new configuration.

For ease of manufacture in this proof-of-principle experiment, Engheta chose to work with microwaves. Metamaterials are built from shapes and holes that are of similar scale to the wavelength of the electromagnetic radiation. In the case of microwaves, that equates to about a few centimetres for each component, a scale easily manufactured out of polystyrene with a milling machine.

Now that the team has succeeded with microwaves they plan to scale down to infrared wavelengths, which fibre-optic telecommunications use. The move to micron-scale components will make manufacture more difficult, but will allow the components to be included on a microchip.

The 10,000-fold decrease in size will also slash the time it takes for the system to solve equations, from the current nanosecond processing time to mere picoseconds.

As well as its high speed, light has the advantage that electromagnetic waves pass through each other without interacting, so many solutions can be processed in parallel.

Engheta and his team previously showed that metamaterials can operate as electrical components such as resistors and capacitors, and can also perform mathematical operations such as differentiation and integration

It was a natural step, therefore, to use them to solve equations, a task team member Brian Edwards calls “photonic calculus”.

“This structure was calculated through a computational process known as inverse design, which can be used to find shapes that no human would think of trying,” he says.

The team hopes to incorporate metamaterials which can be dynamically changed to solve a range of equations.

“Perhaps in the future this can also lead to some form of materials that can learn, based on training from the input and output functions on these metamaterials,” says Engheta.

**Related reading:** Equations show how order falls into chaos