## Abstract

Conventional optical components rely on gradual phase shifts accumulated during light propagation to shape light beams. New degrees of freedom are attained by introducing abrupt phase changes over the scale of the wavelength. A two-dimensional array of optical resonators with spatially varying phase response and subwavelength separation can imprint such phase discontinuities on propagating light as it traverses the interface between two media. Anomalous reflection and refraction phenomena are observed in this regime in optically thin arrays of metallic antennas on silicon with a linear phase variation along the interface, which are in excellent agreement with generalized laws derived from Fermat’s principle. Phase discontinuities provide great flexibility in the design of light beams, as illustrated by the generation of optical vortices through use of planar designer metallic interfaces.

The shaping of the wavefront of light with optical components such as lenses and prisms, as well as diffractive elements such as gratings and holograms, relies on gradual phase changes accumulated along the optical path. This approach is generalized in transformation optics (*1*, *2*), which uses metamaterials to bend light in unusual ways, achieving such phenomena as negative refraction, subwavelength-focusing, and cloaking (*3*, *4*) and even to explore unusual geometries of space-time in the early universe (*5*). A new degree of freedom of controlling wavefronts can be attained by introducing abrupt phase shifts over the scale of the wavelength along the optical path, with the propagation of light governed by Fermat’s principle. The latter states that the trajectory taken between two points A and B by a ray of light is that of the least optical path, *6*–*8*); that is, the derivative of the phase

**Generalized laws of reflection and refraction.** The introduction of an abrupt phase shift, denoted as phase discontinuity, at the interface between two media allows us to revisit the laws of reflection and refraction by applying Fermat’s principle. Consider an incident plane wave at an angle θ_{i}. Assuming that the two paths are infinitesimally close to the actual light path (Fig. 1), then the phase difference between them is zero
_{t} is the angle of refraction; Φ and Φ+*d*Φ are, respectively, the phase discontinuities at the locations where the two paths cross the interface; *dx* is the distance between the crossing points; *n*_{i} and *n*_{t} are the refractive indices of the two media; and *k*_{o} = 2π/λ_{o}, where λ_{o} is the vacuum wavelength. If the phase gradient along the interface is designed to be constant, the previous equation leads to the generalized Snell’s law of refraction*d*Φ/*dx*) is introduced. Because of the nonzero phase gradient in this modified Snell’s law, the two angles of incidence ±θ_{i} lead to different values for the angle of refraction. As a consequence, there are two possible critical angles for total internal reflection, provided that *n*_{t} < *n*_{i}:_{r} is the angle of reflection. There is a nonlinear relation between θ_{r} and θ_{i}, which is markedly different from conventional specular reflection. Equation 4 predicts that there is always a critical angle of incidence

In the above derivation, we have assumed that Φ is a continuous function of the position along the interface; thus, all the incident energy is transferred into the anomalous reflection and refraction. However, because experimentally we use an array of optically thin resonators with subwavelength separation to achieve the phase change along the interface, this discreteness implies that there are also regularly reflected and refracted beams, which follow conventional laws of reflection and refraction (*d*Φ/*dx* = 0 in Eqs. 2 and 4). The separation between the resonators controls the amount of energy in the anomalously reflected and refracted beams. We have also assumed that the amplitudes of the scattered radiation by each resonator are identical, so that the reflected and refracted beams are plane waves. In the next section, we will show with simulations—which represent numerical solutions of Maxwell’s equations—how, indeed, one can achieve the equal-amplitude condition and the constant phase gradient along the interface through suitable design of the resonators.

There is a fundamental difference between the anomalous refraction phenomena caused by phase discontinuities and those found in bulk designer metamaterials, which are caused by either negative dielectric permittivity and negative magnetic permeability or anisotropic dielectric permittivity with different signs of permittivity tensor components along and transverse to the surface (*3*, *4*).

**Phase response of optical antennas.** The phase shift between the emitted and the incident radiation of an optical resonator changes appreciably across a resonance. By spatially tailoring the geometry of the resonators in an array and hence their frequency response, one can design the phase discontinuity along the interface and mold the wavefront of the reflected and refracted beams in nearly arbitrary ways. The choice of the resonators is potentially wide-ranging, from electromagnetic cavities (*9*, *10*) to nanoparticle clusters (*11*, *12*) and plasmonic antennas (*13*, *14*). We concentrated on the latter because of their widely tailorable optical properties (*15*–*19*) and the ease of fabricating planar antennas of nanoscale thickness. The resonant nature of a rod antenna made of a perfect electric conductor is shown in Fig. 2A (*20*).

Phase shifts covering the 0-to-2π range are needed to provide full control of the wavefront. To achieve the required phase coverage while maintaining large scattering amplitudes, we used the double-resonance properties of V-shaped antennas, which consist of two arms of equal length *h* connected at one end at an angle Δ (Fig. 2B). We define two unit vectors to describe the orientation of a V-antenna: ŝ along the symmetry axis of the antenna and â perpendicular to ŝ (Fig. 2B). V-antennas support “symmetric” and “antisymmetric” modes (Fig. 2B, middle and right), which are excited by electric-field components along ŝ and â axes, respectively. In the symmetric mode, the current distribution in each arm approximates that of an individual straight antenna of length *h* (Fig. 2B, middle), and therefore the first-order antenna resonance occurs at *h* ≈ λ_{eff}/2, where λ_{eff} is the effective wavelength (*14*). In the antisymmetric mode, the current distribution in each arm approximates that of one half of a straight antenna of length 2*h* (Fig. 2B, right), and the condition for the first-order resonance of this mode is 2*h* ≈ λ_{eff}/2.

The polarization of the scattered radiation is the same as that of the incident light when the latter is polarized along ŝ or â. For an arbitrary incident polarization, both antenna modes are excited but with substantially different amplitude and phase because of their distinctive resonance conditions. As a result, the scattered light can have a polarization different from that of the incident light. These modal properties of the V-antennas allow one to design the amplitude, phase, and polarization state of the scattered light. We chose the incident polarization to be at 45° with respect to ŝ and â so that both the symmetric and antisymmetric modes can be excited and the scattered light has a substantial component polarized orthogonal to that of the incident light. Experimentally, this allows us to use a polarizer to decouple the scattered light from the excitation.

As a result of the modal properties of the V-antennas and the degrees of freedom in choosing antenna geometry (*h* and Δ), the cross-polarized scattered light can have a large range of phases and amplitudes for a given wavelength λ_{o}; analytical calculations of the amplitude and phase response of V-antennas assumed to be made of gold rods are shown in Fig. 2, D and E. In Fig. 2D, the blue and red dashed curves correspond to the resonance peaks of the symmetric and antisymmetric modes, respectively. We chose four antennas detuned from the resonance peaks, as indicated by circles in Fig. 2, D and E, which provide an incremental phase of π/4 from left to right for the cross-polarized scattered light. By simply taking the mirror structure (Fig. 2C) of an existing V-antenna (Fig. 2B), one creates a new antenna whose cross-polarized radiation has an additional π phase shift. This is evident by observing that the currents leading to cross-polarized radiation are π out of phase in Fig. 2, B and C. A set of eight antennas were thus created from the initial four antennas, as shown in Fig. 2F. Full-wave simulations confirm that the amplitudes of the cross-polarized radiation scattered by the eight antennas are nearly equal, with phases in π/4 increments (Fig. 2G).

A large phase coverage (~300°) can also be achieved by using arrays of straight antennas (fig. S3). However, to obtain the same range of phase shift their scattering amplitudes will be substantially smaller than those of V-antennas (fig. S3). As a consequence of its double resonances, the V-antenna instead allows one to design an array with phase coverage of 2π and equal, yet high, scattering amplitudes for all of the array elements, leading to anomalously reflected and refracted beams of substantially higher intensities.

**Experiments on anomalous reflection and refraction.** We demonstrated experimentally the generalized laws of reflection and refraction using plasmonic interfaces constructed by periodically arranging the eight constituent antennas as explained in the caption of Fig. 2F. The spacing between the antennas should be subwavelength so as to provide efficient scattering and to prevent the occurrence of grating diffraction. However, it should not be too small; otherwise, the strong near-field coupling between neighboring antennas would perturb the designed scattering amplitudes and phases. A representative sample with the densest packing of antennas, Γ = 11 μm, is shown in Fig. 3A, where Γ is the lateral period of the antenna array. In the schematic of the experimental setup (Fig. 3B), we assume that the cross-polarized scattered light from the antennas on the left side is phase-delayed as compared with the ones on the right. By substituting into Eq. 2 –2π/Γ for *d*Φ/*dx* and the refractive indices of silicon and air (*n*_{Si} and 1) for *n*_{i} and *n*_{t}, we obtain the angle of refraction for the cross-polarized beam
θ_{t,⊥} = arcsin[*n*_{Si}sin(θ_{i}) – λ_{o}/Γ]
(6)Figure 3C summarizes the experimental results of the ordinary and the anomalous refraction for six samples with different Γ at normal incidence. The incident polarization is along the *y* axis in Fig. 3A. The sample with the smallest Γ corresponds to the largest phase gradient and the most efficient light scattering into the cross-polarized beams. We observed that the angles of anomalous refraction agree well with theoretical predictions of Eq. 6 (Fig. 3C). The same peak positions were observed for normal incidence, with polarization along the *x* axis in Fig. 3A (Fig. 3D). To a good approximation, we expect that the V-antennas were operating independently at the packing density used in experiments (*20*). The purpose of using a large antenna array (~230 μm by 230 μm) is solely to accommodate the size of the plane-wave–like excitation (beam radius ~ 100 μm). The periodic antenna arrangement is used here for convenience but is not necessary to satisfy the generalized laws of reflection and refraction. It is only necessary that the phase gradient is constant along the plasmonic interface and that the scattering amplitudes of the antennas are all equal. The phase increments between nearest neighbors do not need to be constant, if one relaxes the unnecessary constraint of equal spacing between nearest antennas.

The angles of refraction and reflection are shown in Fig. 4, A and B, respectively, as a function of θ_{i} for both the silicon-air interface (black curves and symbols) and the plasmonic interface (red curves and symbols) (*20*). In the range of θ_{i} = 0° to 9°, the plasmonic interface exhibits “negative” refraction and reflection for the cross-polarized scattered light (schematics are shown in the bottom right insets of Fig. 4, A and B). The critical angle for total internal reflection is modified to ~–8° and +27° (Fig. 4A, blue arrows) for the plasmonic interface in accordance with Eq. 3, compared with ±17° for the silicon-air interface; the anomalous reflection does not exist beyond θ_{i} = –57° (Fig. 4B, blue arrow).

At normal incidence, the ratio of intensity *R* between the anomalously and ordinarily refracted beams is ~0.32 for the sample with Γ = 15 μm (Fig. 3C). *R* rises for increasing antenna packing densities (Fig. 3, C and D) and increasing angles of incidence [up to *R* ≈ 0.97 at θ_{i} = 14° (fig. S1B)]. Because of the experimental configuration, we are not able to determine the ratio of intensity between the reflected beams (*20*), but we expect comparable values.

**Vortex beams created by plasmonic interfaces.** To demonstrate the versatility of the concept of interfacial phase discontinuities, we fabricated a plasmonic interface that is able to create a vortex beam (*21*, *22*) upon illumination by normally incident linearly polarized light. A vortex beam has a helicoidal (or “corkscrew-shaped”) equal-phase wavefront. Specifically, the beam has an azimuthal phase dependence exp(i*l*ϕ) and carries an orbital angular momentum of *23*). Here, the topological charge *l* is an integer, indicating the number of twists of the wavefront within one wavelength; ϕ is the azimuthal angle with respect to the beam axis; and *24*) or a computer-generated hologram (*25*) and can be used to rotate particles (*26*) or to encode information in optical communication systems (*27*).

The plasmonic interface was created by arranging the eight constituent antennas as shown in Fig. 5, A and B. The interface introduces a spiral-like phase shift with respect to the planar wavefront of the incident light, creating a vortex beam with *l* = 1. The vortex beam has an annular intensity distribution in the cross section, as viewed in a mid-infrared camera (Fig. 5C); the dark region at the center corresponds to a phase singularity (*22*). The spiral wavefront of the vortex beam can be revealed by interfering the beam with a co-propagating Gaussian beam (*25*), producing a spiral interference pattern (Fig. 5E). The latter rotates when the path length of the Gaussian beam was changed continuously relative to that of the vortex beam (movie S1). Alternatively, the topological charge *l* = 1 can be identified by a dislocated interference fringe when the vortex and Gaussian beams interfere with a small angle (Fig. 5G) (*25*). The annular intensity distribution and the interference patterns were well reproduced in simulations (Fig. 5, D, F, and H) by using the calculated amplitude and phase responses of the V-antennas (Fig. 2, D and E).

**Concluding remarks.** Our plasmonic interfaces, consisting of an array of V-antennas, impart abrupt phase shifts in the optical path, thus providing great flexibility in molding of the optical wavefront. This breaks the constraint of standard optical components, which rely on gradual phase accumulation along the optical path to change the wavefront of propagating light. We have derived and experimentally confirmed generalized reflection and refraction laws and studied a series of intriguing anomalous reflection and refraction phenomena that descend from the latter: arbitrary reflection and refraction angles that depend on the phase gradient along the interface, two different critical angles for total internal reflection that depend on the relative direction of the incident light with respect to the phase gradient, and critical angle for the reflected beam to be evanescent. We have also used a plasmonic interface to generate optical vortices that have a helicoidal wavefront and carry orbital angular momentum, thus demonstrating the power of phase discontinuities as a design tool of complex beams. The design strategies presented in this article allow one to tailor in an almost arbitrary way the phase and amplitude of an optical wavefront, which should have major implications for transformation optics and integrated optics. We expect that a variety of novel planar optical components such as phased antenna arrays in the optical domain, planar lenses, polarization converters, perfect absorbers, and spatial phase modulators will emerge from this approach.

Antenna arrays in the microwave and millimeter-wave regions have been used for the shaping of reflected and transmitted beams in the so-called “reflectarrays” and “transmitarrays” (*28*–*31*). These typically consist of a double-layer structure comprising a planar array of antennas and a ground plane (in the case of reflectarrays) or another array (in the case of transmitarrays), separated by a dielectric spacer of finite thickness. Reflectarrays and transmitarrays cannot be treated as a single interface for which one can write down the generalized laws because they rely on both antenna resonances and the propagation of waves in the spacer to achieve the desired phase control. The generalization of the laws of reflection and refraction we present is made possible by the deeply subwavelength thickness of our optical antenna arrays and their associated abrupt phase changes, with no contribution from propagation effects. These generalized laws apply to the whole optical spectrum for suitable designer interfaces and can be a guide for the design of new photonic devices.

## Supporting Online Material

www.sciencemag.org/cgi/content/full/science.1210713/DC1

Materials and Methods

SOM Text

Figs. S1 to S6

References (*32–39*)

Movie S1

## References and Notes

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**Acknowledgments**: The authors acknowledge helpful discussion with J. Lin, R. Blanchard, and A. Belyanin. The authors acknowledge support from the National Science Foundation (NSF), Harvard Nanoscale Science and Engineering Center (NSEC) under contract NSF/PHY 06-46094, and the Center for Nanoscale Systems (CNS) at Harvard University. This work was supported in part by the Defense Advanced Research Projects Agency (DARPA) N/MEMS S&T Fundamentals program under grant N66001-10-1-4008 issued by the Space and Naval Warfare Systems Center Pacific (SPAWAR). Z.G. acknowledges funding from the European Communities Seventh Framework Programme (FP7/2007-2013) under grant agreement PIOF-GA-2009-235860. M.A.K. is supported by NSF through a Graduate Research Fellowship. Harvard CNS is a member of the National Nanotechnology Infrastructure Network. The Lumerical (Vancouver, BC, Canada) FDTD simulations in this Research Article were run on the Odyssey cluster supported by the Harvard Faculty of Arts and Sciences Sciences Division Research Computing Group.