# McCormick’s Allen Taflove: 40 Years Solving Maxwell’s Equations

## Professor’s algorithm has made his book the seventh most-cited in physics

Over the past 40 years, Allen Taflove has reshaped the way physicists and electrical engineers solve problems in classical electrodynamics.

Taflove, professor of electrical engineering and computer science at Northwestern University’s McCormick School of Engineering since 1984, has pioneered finite-difference time domain (FDTD) computational solutions of Maxwell’s equations. Overlooked for many years, FDTD methods are now used worldwide. Taflove’s book, *Computational Electrodynamics: The Finite-Difference Time-Domain Method*, ranks seventh on the all-time list of the most-cited books in physics.

Formulated in the 1860s by Scottish physicist James Clerk Maxwell, Maxwell’s equations are fundamental rules of nature that govern nonquantum interactions of electric charges, currents, and electromagnetic waves.

“Maxwell’s equations govern electromagnetic phenomena ranging, literally, from direct current (dc) to daylight,” Taflove said. “Much of the technology that distinguishes our society from that of the year 1800 is based upon Maxwell’s equations. This includes electric power, motors and generators, radio and television transmission, satellite telecommunications, radar, wireless communications, and optical fiber links forming the backbone of the Internet.”

By the time Taflove began his PhD studies at McCormick in 1972 (mentored by Professor Morris Brodwin), researchers had spent decades working with Maxwell’s equations, especially during World War II to advance radar technology. However, because all existing solution techniques assumed that electromagnetic fields vary sinusoidally in time, large amounts of computer memory and running time were required to process the resulting matrices. Available computers could model only simple structures.

Brodwin asked Taflove to look at the problem of assessing microwave exposure levels causing human ocular cataracts, reported during World War II by radar technicians. (At the time, the government was establishing standards for safe microwave exposure.) Attacking this problem required solving Maxwell’s equations to quantify how microwaves penetrate into the eye and its surrounding tissues.

Taflove initially thought that Brodwin’s problem was intractable. “I needed to solve approximately 100,000 electromagnetic field components,” he said. “In 1972, the best available computers could solve for only a few hundred field components using any existing technique.”

Scanning back issues of journals, Taflove spotted a 1966 paper by Kane Yee in *IEEE Transactions on Antennas and Propagation*. Yee reported solving Maxwell’s equations by allowing fields to vary arbitrarily in time, rather than sinusoidally. This yielded an algorithm requiring much less computer storage and running time than anything in the literature.

No one had cited Yee’s paper. It had a major theoretical error and inaccurate means to source and terminate its computation space. But its chief sin was that it had abandoned the sinusoidal paradigm. Taflove realized that, using Yee’s algorithm in concert with Northwestern’s big Control Data CDC-6400 computer, he could indeed implement the human eye model with its 100,000 field unknowns. Still, Yee’s algorithm required much work to remedy its major flaws and understand its theoretical basis.

Subsequently, for his PhD dissertation, Taflove developed Yee’s algorithm into a technique working sufficiently well to implement the human eye model. In 1975, he published his results in two papers in *IEEE Transactions on Microwave Theory and Techniques*. Taflove’s approach emulated how nature works — tracking electromagnetic wave interactions as they evolve in time.

“I was proud of myself, and I expected the world to beat a path,” Taflove said. But it didn’t. “Nobody was interested. The electromagnetics community was wedded to sinusoidal solutions of Maxwell’s equations.”

It wasn’t until 1980 that Taflove was able to go further, working as a research engineer at IIT Research Institute. The U.S. Air Force Rome Air Development Center (RADC) was faced with a problem whereby radar beams emitted by “friendly” aircraft caused its air-to-air missiles to go haywire. After all attempts at modeling this problem had failed, RADC asked Taflove to try his technique.

“Using a Control Data STAR-100 supercomputer to solve 800,000 field unknowns, I determined the failure mechanism and the radar frequency where the problem occurred, without having prior knowledge,” he said. “At this point, I was solving problems 1,000 times more complex than anyone else.”

In a 1980 paper in *IEEE Transactions on Electromagnetic Compatibility* resulting from this work, Taflove coined “finite difference time domain” and its FDTD acronym for his method. Four years later, he joined McCormick’s Department of Electrical Engineering and Computer Science as a faculty member, and subsequently developed supercomputer-based FDTD models for Lawrence-Livermore National Laboratory, Lockheed Missiles & Space Company, and the U.S. Air Force. His supercomputer resource was generously provided by Cray Research.

By 1992, Taflove’s FDTD models routinely solved for tens of millions of electromagnetic field unknowns in a single simulation. And FDTD finally achieved some visibility in the electromagnetics community, with 90 papers presented that year at the IEEE Antennas and Propagation Society International Symposium.

Then Taflove began to write books. In 1995, he authored *Computational Electrodynamics: The Finite-Difference Time-Domain Method*. Three years later, he edited *Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method.* Subsequently, working with Susan Hagness, his former PhD student (and currently a chair professor at the University of Wisconsin-Madison), Taflove expanded and updated the 1995 book in two more editions in 2000 and 2005.

By 2000, hundreds of FDTD-related papers were published annually. But that was only the beginning. Major advances in nanophotonics caused FDTD modeling to “really take off, especially in the physics and physical chemistry communities,” Taflove said. “With FDTD, you can solve Maxwell’s equations to see how light interacts at the nanoscale,” which is necessary for bold photonics goals such as “invisibility cloaks.”

The incredible rise in the popularity of FDTD was recognized in 2010 when, along with Kane Yee, Taflove was cited in an article in the prestigious *Nature Milestones: Photons* as one of the two principal pioneers of computational solutions of Maxwell’s equations. Besides Maxwell himself, only Yee and Taflove were named in this article. And, by 2012, Taflove’s *Computational Electrodynamics* book had attained the rank of seventh on the all-time list of the most-cited books in physics.

Currently, more than two-dozen companies market FDTD software based in large part upon Taflove’s 40 years of work. And while sinusoidal methods of solving Maxwell’s equations have also advanced, Taflove’s FDTD method remains the go-to for the most complicated problems.

“Am I proud of this record? Yes,” said Taflove. “I’m proud that people are using FDTD to solve these fundamental equations of nature, especially for the benefit of human society.”

The latter is well demonstrated at McCormick, where since 2003, Taflove has collaborated with Vadim Backman, professor of biomedical engineering. FDTD modeling has helped to establish the fundamental physics foundation of Backman’s partial-wave spectroscopy (PWS) biophotonics technique for early detection of cancers of the colon, lung, pancreas, and ovaries. In preclinical trials involving hundreds of human subjects, PWS has demonstrated better than 90 percent sensitivity in detecting deeply-seated cancers by scanning cells brushed from accessible body cavities. PWS has the potential to save many lives.

James Clerk Maxwell would be proud, as well.