Metamaterials that do math

A new study by researchers in Penn’s School of Engineering and Applied Science shows that metamaterials can be designed to do “photonic calculus” by working like an analog computer.

The predecessors of modern digital computers, ushered in by the University's own ENIAC, analog computers were essentially purpose-built calculators. Rather than breaking information down into bits, these devices transformed input to output all at once using physical elements, such as complex arrays of gears and drive shafts, and later, electrical components, like resistors, capacitors, inductors, and amplifiers, to produce results.

By swapping analog computers’ mechanical gears and electrical circuits for optical materials that operate on light waves, researchers are applying these analog computing principles at the micro- and nanoscale.

“Compared to digital computers, analog computers were bulky, power hungry, and slow,“ says senior author Nader Engheta, the H. Nedwill Ramsey Professor in the Department of Electrical and Systems Engineering. “But by applying the concepts behind them to optical metamaterials, one day we might be able to make [analog computers] at micro- and nanoscale sizes, and operate them at nearly the speed of light using little power.”

Metamaterials are composites of natural materials, but are designed so they manipulate electromagnetic waves in ways that are more than the simple sum of their parts. Multiple manipulations can be combined or performed in sequence, allowing metamaterial researchers to change the shape of waves in complex ways.

A light wave, when described in terms of space and time, can be thought of as a curve on a Cartesian plane; that curve is known as its profile. The researchers have simulated a material that can perform a specific mathematical operation on that wave’s profile, such as finding its first or second derivative.

Their metamaterial gradually changes light waves as they pass through it, such that the exiting light waves' profiles are the derivatives of the incoming light waves' profiles. The researchers found that metamaterials capable of other calculus operations, such as integration and convolution, could also be produced.

Viewing and manipulating this type of light wave profile is an everyday occurrence for applications like image processing, though it is typically done after the light wave has been converted to electronic signals in the form of digital information. The researchers’ proposed computational metamaterials could almost instantly perform such operations on the original wave, such as the light coming in through the lens of a camera, without conversion to electronic signals. 

The researchers’ experiments began with a computer simulation of an ideal metamaterial, one that could perfectly change the shape of the incoming wave profile into that of its derivative. With the ideal metamaterial as a guide, the researchers then constrained their simulations to specific materials suitable for existing fabrication techniques, such as silicon and aluminum-doped zinc oxide.

“The simulation results of the two were almost identical, so we’re hopeful we’ll be able to do photonic calculus in a physical experiment in the future,” Engheta says.