New theory describes how waves carry information from the environment

Waves capture information from their environment in which they propagate. A theory of information carried by waves has been developed at TU Wien, with astonishing results that can be used for technical applications.

Ultrasound is used to analyze the body, radar systems to study airspace or seismic waves to study the interior of our planet. Many areas of research focus on waves deflected, scattered or reflected by their environment. As a result, these waves carry a certain amount of information about their environment, which must then be extracted as completely and precisely as possible.

The search for the best way to achieve this has been the subject of research around the world for many years. TU Wien has now succeeded in describing with mathematical precision the information conveyed by a wave about its environment. This made it possible to show how waves capture information about an object and then transport it to a measuring device.

This can now be used to generate custom waves to extract the maximum information from the environment, for example for more precise imaging processes. This theory was confirmed by microwave experiments. The results were published in the journal Natural physics.

Where exactly is the information located?

“The basic idea is quite simple: you send a wave at an object and the part of the wave scattered by the object is measured by a detector,” explains Professor Stefan Rotter from the Institute for Theoretical Physics of the TU Vienna.

“The data can then be used to learn something about the object, for example its precise position, speed or size.” This information about the environment that this wave carries with it is known as “Fisher information”.

However, it is often not possible to capture the entire wave. Usually only part of the wave reaches the detector. This begs the question: where exactly is this information in the wave? Are there parts of the wave that can be safely ignored? Would perhaps a different waveform provide more information to the detector?

“To get to the bottom of these questions, we took a closer look at the mathematical properties of this Fisher information and came up with some surprising results,” says Rotter.

“The information fulfills what is called the continuity equation: the information contained in the wave is preserved as it moves through space, according to laws that are very similar to those of conservation energy, for example.

An understandable information path

Using the newly developed formalism, the research team was now able to calculate exactly at which point in space the wave actually carries how much information about the object. It turns out that information about different properties of the object (such as position, speed and size) can be hidden in completely different parts of the wave.

As theoretical calculations show, the information content of the wave depends precisely on the extent to which the wave is influenced by certain properties of the studied object.

“For example, if we want to measure whether an object is a little more to the left or a little more to the right, then the Fisher information is conveyed precisely by the part of the wave which comes into contact with the right edges and left of the object”, explains Jakob Hüpfl, the doctoral student who played a key role in the study.

“This information then spreads, and the more this information reaches the detector, the more precisely the position of the object can be read there.”

Microwave experiments confirm theory

In Ulrich Kuhl’s group at the University of Côte d’Azur in Nice, experiments were carried out by Felix Russo as part of his master’s thesis: a disordered environment was created in a microwave chamber at the using randomly positioned Teflon objects. Between these objects was placed a metal rectangle whose position remained to be determined.

The microwaves were sent through the system and then captured by a detector. The question now was: to what extent can the position of the metal rectangle be deduced from the waves picked up by the detector in such a complex physical situation and how does the information flow from the rectangle to the detector?

By precisely measuring the microwave field, it was possible to show exactly how information about the horizontal and vertical position of the rectangle propagates: it emanates from the respective edges of the rectangle and then moves with the wave, without any information is lost, just as the newly developed theory predicts.

Possible applications in many areas

“This new mathematical description of Fisher information has the potential to improve the quality of various imaging methods,” says Rotter. If it is possible to quantify where the desired information is located and how it propagates, it also becomes possible, for example, to position the detector more appropriately or to calculate custom waves that carry the maximum amount of information to the detector.

“We tested our theory with microwaves, but it is also valid for a wide variety of waves of different wavelengths,” emphasizes Rotter. “We provide simple formulas that can be used to improve microscopy methods as well as quantum sensors.”