Phase-shifting surfaces bend the rules of ray optics

One way to arrive at Snell’s law of refraction is to assume, as Pierre de Fermat did nearly four centuries ago, that light rays travel the fastest path between two points. If the points lie on opposing sides of the interface between optically different materials, then that path likely isn’t a straight line.

Richard Feynman likened the problem to that of a lifeguard—presumably a faster runner than swimmer—tasked with rescuing a drowning swimmer. The astute lifeguard, rather than make a beeline to the swimmer, would run toward a point on the shoreline that extends the length of the run in exchange for shortening the length of the swim. Likewise, when a light ray passes from a medium with a low refractive index n_i to one with a high refractive index n_t, its angle of incidence θ_i is larger than its angle of refraction θ_t, as formalized by Snell’s law: n_tsinθ_t − n_isinθ_i = 0. The fastest-path assumption also leads to the law governing the angle of reflection θ_r: sinθ_r − sinθ_i = 0.

Now, Harvard University researchers led by Federico Capasso have posed a question that Fermat, Feynman, and Willebrord Snell likely never considered: How would a light ray’s trajectory change if, at the surface of reflection and refraction, it experienced a position-dependent phase shift?

Combining theory and experiments, they’ve arrived at a conclusive answer: A carefully constructed phase-shifting surface—implementable with plasmonic antennas or other small optical resonators—can bend light in ways that defy the traditional laws of reflection and refraction. ¹

Rethinking Snell’s law

Fermat’s principle works not because light is in a particular hurry to get from one place to the next, but because the fastest path lies at an extremum, where the derivative of the optical path length with respect to small deviations in trajectory is zero. Light waves that hew closely to the fastest path arrive at their destination at nearly the same time and with nearly the same phase and thus interfere constructively. Away from the extremum, neighboring trajectories of light arrive out of phase and cancel each other.

Fermat’s principle can therefore be recast as a principle of stationary phase. A ray like that depicted in figure 1 reflects and refracts in such a way that light originating at point A arrives at points B and C with phases ϕ_B and ϕ_C that are constant with respect to small perturbations of the point of incidence x′. That is, light travels the path for which dϕ_B/dx′ and dϕ_C/dx′ are zero.

Adopting that more general interpretation of Fermat’s principle, Capasso and his colleagues showed theoretically that a position-dependent phase shift Φ(x) imposed at the reflecting and refracting surface can alter the location of the path of stationary phase and, along with it, the usual rules of refraction and reflection. They could accommodate the effect with straightforward modifications to Snell’s law, n_tsinθ_t − n_isinθ_i = (λ/2π)dΦ/dx, and to the law of reflection, sinθ_r − sinθ_i = (λ/2π)dΦ/dx, where λ is the wavelength of light.

“Snell’s law assumes that the interface between two media is just a mathematical boundary,” lead author Nanfang Yu explains. “By introducing a structured phase delay with a gradient along the surface, it becomes much more than that.”

In fact, the Harvard team’s equations suggest that reflection and refraction can be altered at will. Negative refraction, multiple critical angles for total internal reflection, and other oddities become possible, as depicted on the cover.

The effect is different from that of metamaterials, which bend light due to anomalous bulk permittivities. (See the article by Martin Wegener and Stefan Linden, PHYSICS TODAY, October 2010, page 32 .) Metamaterials obey Snell’s law; they just have unusual indices of refraction. What Capasso and company envisioned are more aptly thought of as metasurfaces.

Plasmonic shifts

Adopting strategies previously used to shape RF wavefronts, the researchers sought to create a metasurface from plasmonic antennas, which scatter light with an amplitude and phase delay that depend on the antennas’ resonance properties. (See the article by Lukas Novotny, PHYSICS TODAY, July 2011, page 47 .) However, the simplest plasmonic antenna—a straight rod—offers limited control over phase; its phase delay can range only from 0 to π, and the scattering amplitude is appreciable over just half that range.

Coauthor Zeno Gaburro provided a crucial insight—a V-shaped antenna, with its two orthogonally oriented resonance modes, can be designed to scatter light with a phase delay anywhere in the 0 to 2π range. Applying Maxwell’s equations, he and his colleagues identified four shapes of V antennas that, along with their mirror images, yield phase delays spanning 0 to 2π in π/4 increments.

Arrayed in the repeating pattern shown in figure 2, those antennas constitute a discrete approximation to the phase-shifting surface Capasso and company had envisioned with their generalization of Snell’s law. The gradient in the phase shift, dΦ/dx, could be adjusted by simply altering the spacing of the antennas. As long as the interantenna distance and the antennas themselves are smaller than the wavelength of light, the phase grating behaves like an effective medium.

In a proof-of-principle experiment, the researchers aimed a mid-IR quantum cascade laser at various angles of incidence to their phase-grated wafer. For normal incidence, Snell’s law predicts that the angle of refraction should be zero. Indeed, portions of a normally oriented beam passed straight through the wafer, but the portion scattered by the plasmonic antennas refracted with an angle of 40°, in near-perfect agreement with the group’s theory. The metasurface proved to be broadband, deflecting light with wavelengths from 5 to 10 µm.

Subwavelength optics

Capasso and his coworkers are already busy dreaming up ways to put their new ideas about reflection and refraction to work. “We have a treasure trove of things to explore—low-aberration lenses, birefringent polarizers, phase plates, you name it,” says Gaburro. Among that trove is the phase grating shown in figure 3a, which the researchers created by arranging V-shaped antennas in order of clockwise-increasing phase delay. The subwavelength-thin plate successfully converted a plane wave of IR light into the vortex beam shown in figure 3b.

Nader Engheta of the University of Pennsylvania thinks the work bodes well for low-profile optical devices. “The ability to control the phase of light at will, to shape simple beams into very complex beams, all over subwavelength distances, is a big breakthrough,” he says. “It’s extremely elegant and beautiful work.”