Generating a superdirective beam from metamaterials using Fourier optics

When a material is coupled with a metamaterial exhibiting opposite optical properties, they can annihilate. A portion of space is optically removed as a result, allowing light to propagate through the whole structure without diffraction.¹ Optical antimatter behavior is a generalization of the superlens effect, where metamaterial properties are exploited to overcome the diffraction limit. These extraordinary electromagnetic media are also expected to give rise to other interesting phenomena. In particular, an extremely directive beam appears when a divergent beam, or an internal point source, impinges on a medium with a refractive index close to zero.

Metamaterials are artificial materials with structures specifically engineered to have optical properties that are not found in nature, such as a negative refractive index. We use thousands of coupled slabs of equal or almost equal thickness comprising air, which has a refractive index of 1, and metamaterial ‘anti-air’ with an effective refractive index equal to –1. The average refractive index is zero: a quasi-zero (QZAI) medium. The total effect of such complementary media is light transmission in the plane without diffraction, preserving the input source profile.

We have experimentally verified these theoretical predictions using a macroscopic sample with a length of 4mm.² In our sample the metamaterial antimatter slabs are composed of a photonic crystal (PhC) made from silicon (n=3.45 for incident light at wavelength λ=1.55μm) with air holes arranged in a hexagonal lattice in the (x; z) plane. The holes have a radius r, lattice parameter a, and ratio r/a=0.38. This particular PhC shows an almost isotropic effective index n_e =−1 for transverse magnetic (TM) polarized incident light at ω_n=0.305, where ω_n is the frequency normalized to a. For this polarization the electric field is directed along an axis that follows a line of the holes in the array. For λ=1.55μm, the parameters that render an effective index of –1 are r=180nm and a=472nm. The QZAI structure is obtained by alternating slabs of air with slabs of this PhC, and the side edges of the PhC are cut³ in such a way that the zero average index condition is broken slightly.² The grating period is , given by the distance between two air layers (see Figure 1), and acts as a virtual waveguide without a physical structure for the lateral and vertical confinement. A direct quantification of the propagation length and the angular spread of the mode are determined by measuring the angular and spectral width of the diffracted peaks.

Figure 1. A schematic of an optical antimatter structure composed of air and anti-air. The scattered light recorded out of plane² is collimated for 4mm as it propagates through the structure, shown superposed on the top surface of the structure schematic. The figure is not to scale.

High-accuracy experimental results have shown that the angular dispersion of the diffracted beam is as small as Δθ=0.06° as it propagates along the direction of the zeroth order of the diffraction grating composed by coupled slabs of air and anti-air metamaterial. This material exhibits low angular dispersion despite the high divergence of the input beam resulting from the strong focusing by the lensed fiber from which it is emitted. We have also experimentally verified with great accuracy that the wavenumber of the beam propagating in the QZAI plane is the wavenumber in air 2π/λ, i.e., the beam propagates without experiencing diffraction.

Figure 2. The angular profile peak of diffracted order m=1.

The directivity of the scattered beam is determined from a fine angular scan around the m^th-diffracted order of the grating (see Figure 2). This measurement yields an extraordinary value of Δθ_m=(0.061±0.003)° for m = –1.⁴

To measure the spectral dispersion Δλ, we vary the input wavelength and fix the observation angle of the diffracted beam to θ₋₁=21.55°, corresponding to the m=−1 peak at the wavelength λ=1.55μm. We obtain a Gaussian spectral distribution with a full width at half-maximum, Δλ=(3.08±0.06)nm. Considering the spectral resolution (or resolving power) R of the m^th-diffracted order from the grating, this is directly proportional to the number of periods N that comprise the grating:

From (1) we can directly derive the propagation length, the maximum length the beam can travel inside the grating without diffraction, as at least N=516 periods, i.e., a length of NΛ=1.265mm, where Λ=2.453μm is the grating period. This propagation length is underestimated. Experimental limitations aside, the formula in Equation (1) is derived within the fundamental assumption that the propagating wave is a plane wave. In our case, the incident light is focused by the lensed fiber and is far from meeting the conditions of a plane wave.

Spread of the incident wave corresponds to a spread of the k-wavevector interacting with the grating and finally a spread of the diffracted peaks. This is not observed in our experiments: the spectral and angular peaks are extremely narrow, and for this reason we define the radiation as extraordinarily collimated. These properties have potential application in very sensitive integrated spectrometers. We are now working on a hybrid silicon-based platform that integrates microfluidic capability while realizing a real on-chip spectrometer that fully exploits the extreme sensitivity of such a metamaterial structure.

Portions of this work were performed as a user project at the Molecular Foundry, Lawrence Berkeley National Laboratory, which is supported by the US Department of Energy under contract DE-AC02-05CH11231.