Milestone 2 Print Article

Milestones

Maxwell’s equations are as important today as ever. They led to David Pile David the development of special relativity (Milestone 4) and, nowadays, almost every optics problem that can be formulated in terms of dielectric per- mittivity and magnetic permeability (two key constants in Maxwell’s equations), ranging from optical- fibre waveguides (Milestone 13) to metamaterials and transformation optics (Milestone 21), is treated within the framework of these equations or systems of equations derived MILEStONE 2 from them. Their actual solution can, however, be challenging for all but Classical monument the most basic physical geometries. Numerical methods for solving the equations were pioneered by Kane By the middle of the nineteenth only necessary for accuracy but also Yee and Allen Taflove, but went century, a significant body of experi- led to a conceptual breakthrough. unnoticed for many years owing to mental and theoretical knowledge Maxwell predicted an ‘electromag- the limited computing power avail- about electricity and magnetism had netic wave’, which can self-sustain, able at the time. Now, however, these been accumulated. In 1861, James even in a vacuum, in the absence of methods can be easily employed for Clerk Maxwell condensed it into 20 conventional currents. Moreover, solving electromagnetic problems for equations. Maxwell published various he predicted the speed of this wave structures as complex as aircraft. reduced and simplified forms, but to be 310,740,000 m s−1 — within a David Pile, Oliver Heaviside is frequently cred- few percent of the exact value of the Associate Editor, Nature Photonics

ited with simplifying them into the speed of light. ORIGINAL RESEARCH PAPERS Maxwell, J. C. modern set of four partial differential “The agreement of the results On physical lines of force. Phil. Mag. 11, 161–175; equations: Faraday’s law, Ampère’s seems to show that light and magnet- 281–291; 338–348 (1861); ibid. 12, 12–24; 85–95 (1862) | Maxwell, J. C. A dynamical theory of the law, Gauss’ law for magnetism and ism are affections of the same sub- electromagnetic field. Phil. Trans. R. Soc. Lond. Gauss’ law for electricity. stance, and light is an electromagnetic 155, 459–512 (1865) | Maxwell, J. C. A Treatise on Electricity and Magnetism (Clarendon Press, 1873) | One of the most important disturbance propagated through the Yee, K. S. Numerical solution of initial boundary contributions made by Maxwell was field according to electromagnetic value problems involving Maxwell’s equations in actually a correction to Ampère’s law. laws”, wrote Maxwell in 1865. The isotropic media. IEEE Trans. Antenn. Propag. 14, 302–307 (1966) He had realized that magnetic fields concept of light was thus unified with FuRtHER REAdING Taflove, A. Computational can be induced by changing electric electricity and magnetism for the first Electrodynamics: The Finite-Difference Time- fields — an insight that was not time. Domain Method (Artech House, 1995)