Putting Order in Disorder

Image Courtesy of Noora Said.

What is scattering?

“Why is the sky blue?” is perhaps one of the first, most vexing questions a kid can ask their parents. After all, how do you explain to a three-year-old that the answer lies in the scattering of light in opaque diffusive systems?

In our day-to-day lives, scattering occurs when particles pass through a medium—such as air or water, for example—and collide with other particles, resulting in a change in their trajectory. “Scattering of light is very common,” said Hui Cao, a Professor of Applied Physics at Yale University, focusing on mesoscopic physics, nanophotonics, and biophotonics. “The sky looks blue because blue light is scattered [more] strongly than red light, and the reason why we look opaque is that there’s strong scattering by the cells in biological tissues. That’s why we cannot see through most biological tissues.” We encounter many things in everyday life that we cannot see through or send information through—all because of this strong scattering of light. 

However, scattering can cause challenges in applications such as medicine, where the opacity—or lack of transparency—of human tissues can limit their visualization and manipulation. 

In recent years, researchers have explored different ways to transmit energy through diffusive systems. In medical applications, for example, it is critical to focus on delivering and depositing energy inside systems such as human tissue instead of simply sending energy through the system. Medical applications include procedures like photothermal therapy, which uses heat generated by near-infrared light to treat cancer, or deep tissue imaging, which allows researchers to image whole tissues without dividing them into thinner sections. 

Can’t we just send light into the system?

One of the greatest challenges in depth-targeted delivery of light—in other words, sending light into a specific spot in the system—is that energy scatters in multiple directions and diffuses upon entering the system. 

Previous research relied on controlling the wavefront of the input energy wave to limit diffusion, which allowed researchers to focus light on a specific spot in a scattering medium. A wavefront is an imaginary surface where all the points are at the same phase in their wave cycle (think of the ripples you see when you drop something in water). Shaping the wavefront involves controlling the distribution of wave intensities and phases in the input beam.

However, this method was less practical for real-world applications because medical targets such as tumors or neurons often sprawl over a region rather than remaining in a single focal spot. The upper limit on energy deposition in a region at a certain depth in a diffusive system, which is important to know for practical applications of this technique, was also unclear using this method. 

Another difficulty lays in observing the delivery of light into the system. “Scientifically, it’s really easy to study sending something into a system and measure something coming out of a system—you just put a camera on one end and input on the other,” said Nicholas Bender, formerly a Yale doctoral student in Cao’s lab and now a postdoctoral researcher at Cornell University. “What’s hard to do is to understand what this light is doing inside a system because by observing the system, you may interfere with it.” Simply put, trying to see what was going on in the system could mean inadvertently altering whatever process was underway within it. 

Lasers and Math

To confront this issue of targeted energy delivery into diffusive systems, researchers at Yale University performed a comprehensive series of experiments, numerical simulations, and theory. “I like to call it ‘creating and controlling disorder, randomness, and chaos with lasers,’” Bender said. 

The team began by defining a matrix that mathematically described the relationship between the laser beam input into a diffusive system and the way the light was distributed across a region of specific depth in the system. By running repeated simulations of virtual disordered systems, the research team plotted what different input beams would look like at various points within the system.

The maximum energy that could be delivered to the target region corresponded to the largest eigenvalue of the deposition matrix. Eigenvalues are factors representing the scales of eigenvectors, which are characteristic vectors in linear algebra. The input wavefront could be found from the eigenvector associated with that largest eigenvalue. 

Causing Chaos (On Purpose)

One unique feature of this study was the experimental setup. The researchers devised an experimental platform consisting of a two-dimensional disordered structure (picture a rectangular slab with holes that randomly let light through) where the optical field could be analyzed by the researcher looking down at the platform from above (from the third dimension). “​​This is new,” Cao said. “Before, people usually made a three-dimensional sample. With three-dimensional scattering, when you send in the light, you cannot see it, so you don’t know [which wavefront is best] to deliver light. But by using this system, we’re able to peek in, to take a look from the third dimension, and say, ‘Oh, we see! This is where we can deposit and how much.’” For example, if you rolled a marble into a non-transparent box, you wouldn’t be able to see its path or where it stopped. If you rolled that marble onto a sheet of paper, you would be able to look down onto the paper and track the marble’s progress. 

According to Cao, creating this experimental setup was not an easy process. “It’s absurd how much effort we had to put into making this disordered system just the way we want it,” he said. “I mean, calibrating and controlling disorder is ridiculous…it’s crazy and great and horrible to do,” Bender said. This breakthrough experimental technique allowed the researchers to observe scattering and light delivery with a degree of control that had never before been possible. 

Using a spatial light modulator, a device that controls the intensity and phase of light emitted, the researchers could shape the wavefront of a laser beam in one dimension. They found the two-dimensional field distribution inside the system—the field of randomly scattered light in the platform. 

The two-dimensional sample consisted of a silicon-on-insulator wafer with photonic-crystal sidewalls to keep light inside the system. The team added random optical scattering to this system by etching a random array of holes in the wafer. When the laser beam was sent in, some of the scattered light would come out of the holes and into the path of a reference beam. A camera then recorded these interference patterns. “Basically, by shaping the incident wavefront using a spatial light modulator, we can control how we’re going to send the light in,” Cao said. “By finding the correct wavefront, we can deposit light into different target areas deep inside.”

This experimental platform allowed the researchers to directly map the diffusive system at any depth. “Once we have this system that allows us to see what light is doing inside a random disordered system, we can essentially say, ‘Okay, instead of just describing the relationship from the input to the output, we can describe the relationship of the light from the input to anywhere inside,’” Bender said.

This experimental setup allowed the researchers to create a system where the disorder could be controlled, precisely tuned, and analyzed. By optimizing the laser input wavefront, they were then able to maximize energy delivery. 

This is a unique and novel set up—one that has fundamentally changed the ways light can interact with opaque systems. “We’re the only people who can actually do this study,” Bender said. “We can control the input with very, very good precision using the spatial light modulators we have available. We can make the waveguides any way we want just because of the technical capabilities we have in the facilities at Yale.” 

The Theory

The research team also built a theoretical model to predict the maximum amount of energy that could be delivered to a certain depth in the system. Theoretical modeling was important in showing the researchers the limits of their new technology. “What’s the fundamental limit? How well can we reach it, and what determines this limit?” Cao said. 

Through mathematical calculations, the team found that energy enhancement depended on the sample thickness and depth of the region. Energy enhancement was also affected by the transport mean free path of the system, which is the distance light can travel in a random system before it loses all of the information about the initial direction of propagation. They then experimentally measured the internal field distribution at different depths to find the deposition matrix for regions in a diffusive system. They found that the highest possible energy enhancement occurs at three-fourths of the system’s thickness. 

Moment of Discovery

This research differs from previous studies in the field because it focuses on deposition instead of energy transmission. “Everybody doing wavefront shaping was looking at the transmission matrix or some version of the transmission matrix,” Bender said. Compared to transmission, having disorder after the region of light deposition can result in light traveling backward, interfering with the light going forwards, leading to a greater energy enhancement than in the case of transmission. 

Controlling Randomness 

The ability to control random wave scattering allows for energy deposition into specific regions of opaque systems. “This is interesting for medical applications or anything dealing with a real system because most systems are disordered to some extent,” Bender said. Research into this type of targeted energy delivery could be used in applications ranging from the optogenetic control of neurons to tissue imaging. “There are people in [the] community trying to do imaging through the skull. They try to send the laser beams through the skull for both a diagnosis and also to try to simulate neurons,” Cao said. 

This study pushes the boundaries of what was previously thought to be possible. “Traditionally, people [think that] if something looks white, then you just cannot see through it,” Cao said. “I wish more people knew that actually, that is not true. Random scattering is not something just impossible to control.”

About the Author: Eunsoo Hyun is a junior in Berkeley College majoring in Biomedical Engineering. Outside of writing for YSM, Eunsoo enjoys painting, learning new languages, and dancing with the Yale Jashan Bhangra team. 

Acknowledgments: The author would like to thank Professor Hui Cao and Dr. Nicholas Bender for their time and enthusiasm for their research.

Extra reading:

Bender, N., Yamilov, A., Goetschy, A., Yılmaz, H., Hsu, C.W., and Cao, H. (2022). Depth-targeted energy delivery deep inside scattering media. Nat. Phys. 18, 309–315.