A model once thought to be nearly impossible for quickly and accurately designing radio frequency (RF) waves needed to fire up doughnut-shaped tokamak fusion facilities has been developed by a graduate student at the U.S. Department of Energy’s (DOE) Princeton Plasma Physics Laboratory (PPPL). The student, Nick Lopez, has innovated a fast and accurate way to calculate the energy and path of RF waves that are distorted by roadblocks called “caustics” that make the behavior of the waves highly complex.

“Caustics are extraordinarily difficult to describe mathematically in order to simulate — and we need to be able to simulate them to optimize the delivery of RF waves,” said Lopez, whose findings are reported in a paper in the *Journal of Optics*. “The trick is to simulate caustics accurately and we’ve now found a way to do that.”

**Causing bounce-backs**

Caustics arise in fusion research when researchers beam RF waves into the plasma that fuels fusion reactions to help raise it to the required million-degree temperatures. “The wave can be propagating through the plasma and hit a region that’s too difficult to propagate in and bounces back and you have a caustic there,” Lopez said.

The bounce-back causes difficulties for scientists who seek to harvest on Earth the fusion reactions that power the sun and stars to produce safe and clean energy to generate electricity. Fusion reactions combine light elements in the form of plasma — the state of matter composed of free electrons and atomic nuclei — to create massive amounts of energy.

Scientists use what are called “full-wave codes” to accurately calculate the impact of caustics on RF waves. But such calculations can take days or weeks of costly runs on supercomputers. “This is a problem if we’re trying to develop a tokamak heating system by iterating through a bunch of design ideas and it’s not a good thing if it takes a long time for each iteration,” Lopez said.

An alternative method called “ray tracing” can speedily determine the impact. “Ray tracing is really fast but doesn’t calculate caustics correctly,” Lopez said. “My paper tries to find a way between these two extremes to give us a framework that’s fast and accurate.”

His model applies a geometrical technique called a “metaplectic transform” to recast the wave equation that governs how the wave travels through the plasma. The transformation eliminates the term in the equation that describes the wave folding upon itself when it meets a reflection. “Now there is no caustic in the equation and it can be simulated very easily,” Lopez said. “This is sort of a math trick based on geometrical insights into the equations.”

**Superior alternative**

The trick provides a superior alternative for dealing with caustics, said PPPL physicist Ilya Dodin, advisor to Lopez and co-author of the paper. The new study, he notes, is one of five that Lopez has written on caustics and the one that amplifies a key open access paper published in the *New Journal of Physics* last year. “I would put Nick’s theory in textbooks already for the beauty of it,” Dodin said. “But there is also a practical aspect: This approach can lead to faster codes that would allow for improved optimization of tokamak fusion devices.”

The theory could also benefit researchers at the Lawrence Livermore National Laboratory (LLNL) in California who are developing a fusion-creation technique called inertial confinement fusion (ICF) that implodes plasma capsules with the world’s most powerful laser beams. “Caustics occur all the time in ICF experiments,” said Lopez, who gave a talk at LLNL earlier this year and will test how well his model works there as an intern there this summer.

Scientists at the National Ignition Facility, the LLNL unit that conducts ICF research, look forward to his arrival. “Our group has been extremely impressed with Nick’s work on the modeling of caustics and his analytical and mathematical abilities,” said LLNL physicist Pierre Michel, a leader of the ICF team. “His work has direct implications for laser-plasma interaction and improving caustic modeling should improve the predictive capability of our codes.”

For Lopez, such improvement would complement the more accurate calculation of caustics in tokamaks that his model aims to create. “It’s really important for researchers to know how much wave energy they have at a certain point,” he said, “and that’s required for all types of fusion research.”

Support for this work comes from the DOE Office of Science.

PPPL, on Princeton University's Forrestal Campus in Plainsboro, N.J., is devoted to creating new knowledge about the physics of plasmas — ultra-hot, charged gases — and to developing practical solutions for the creation of fusion energy. The Laboratory is managed by the University for the U.S. Department of Energy’s Office of Science, which is the single largest supporter of basic research in the physical sciences in the United States and is working to address some of the most pressing challenges of our time. For more information, visit energy.gov/science.