### Decoding the Optical Response of Nonlinear Scattering Media: A Leap Towards Highly Scalable Physical Operators.

Is it possible to see through a diffusion medium such as stained glass? Traditionally, this would be considered impossible. When light passes through an opaque substance, the information in the light becomes “jumbled together,” almost as if undergoing complex encryption.

Recently, a remarkable scientific breakthrough by Professor Choi Wonshik’s team from the IBS Center for Molecular Spectroscopy and Dynamics (IBS CMSD) has revealed a method to exploit this phenomenon in optical computing and machine learning.

Since 2010, several previous studies have attempted to exploit information lost due to scattering media, such as biological tissues, using mathematics. This has typically been done by using optical operators such as linear scattering matrices, which can be used to determine the input-output ratio of photons as they undergo scattering.

This topic has been of primary research interest to Professor Choi’s team from IBS CMSD, who published many works combining both hardware- and software-based adaptive optics for tissue imaging. Some of their work was demonstrated in new types of microscopes that can see through high-opacity scattering media, such as mouse skulls, as well as perform deep 3D imaging of tissue.

But things get much more complex when nonlinearity enters the equation. If a scattering medium generates non-linear signals, it can no longer be simply represented by a linear matrix, as the superposition principle is violated. Furthermore, measuring the non-linear input-output characteristics becomes a daunting challenge that sets a demanding stage for research.

#### Unraveling the mystery of nonlinear scattering media

This time, Professor Choi’s team has achieved another scientific breakthrough. They were the first to discover that the optical input-output response of a nonlinear scattering medium can be defined by a third-order tensor as opposed to a linear matrix.

The third-order tensor is a mathematical object used to represent relationships between three sets of data. Simply put, it is a series of numbers arranged in a three-dimensional structure. Tensors are generalizations of scalars (0-order tensors), vectors (1st-order tensors) and matrices (2nd-order tensors) and are commonly used in various fields of mathematics, physics and engineering to describe physical quantities and their relationships.

To demonstrate this, the team used a medium consisting of barium titanate nanoparticles, which generate nonlinear second harmonic generation (SHG) signals due to the intrinsic non-centrosymmetric properties of barium titanate. These SHG signals emerge as a square of the input electric field through the second harmonic process, causing crosstalk when multiple input channels are activated simultaneously, which violates the linear superposition principle. The researchers devised and experimentally validated a new theoretical framework involving these cross terms in a third-order tensor.

To illustrate this, the researchers measured cross-expression by isolating the difference between the electrical output fields produced when two input channels were activated simultaneously and when each channel was activated separately. This required an additional 1,176 measurements set of the possible combinations of two independent input channels, even with only 49 input channels.

“The effort required to detect cross-expression from weak nonlinear signals was significant,” noted Dr. Moon Jungho, the study’s lead author.

#### Real-World Applications Released

The tensor derived from the nonlinear scattering media has a higher rank than matrices of linear scattering media, suggesting its potential as a scalable physical operator. The team demonstrated this through real-world implementation of nonlinear optical encryption and all-optical logic gates.

First, the team successfully demonstrated that nonlinear scattering media can be used for the optical encryption process. When specific image information is loaded into the medium, the output second harmonic wave signals appear as random patterns, akin to a series of encryption processes.

Conversely, by performing an inverse operation of the third-order tensor representation of the second harmonic wave, the original input information can be retrieved through a decryption process. Using the inverse operation of the tensor input-output response, they decoded the original signals from randomly encoded SHG signals, which provides improved security over standard optical encryption that uses linear scattering media.

Furthermore, the integration of digital phase conjugation allowed the researchers to demonstrate all-optical AND logic gates that are activated only when two specific input channels are activated simultaneously. This approach offers potential advantages over silicon-based logic, including reduced power consumption and parallel light-speed processing capabilities.

This research is expected to open up new frontiers in optical computing and machine learning. “In the burgeoning field of optical machine learning, nonlinear optical layers are key to improving model performance. We are currently exploring how our research can be integrated into this field,” said Professor Choi.

Reference: “Measuring the scattering tensor of a disordered non-linear medium” by Jungho Moon, Ye-Chan Cho, Sungsam Kang, Mooseok Jang and Wonshik Choi, 31 July 2023, *Natural physics*.

DOI: 10.1038/s41567-023-02163-8

The study is funded by the Department of Basic Science.

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