In recent years there has been significant interest in broadband, omnidirectional antireflection (AR) nanostructures that minimize Fresnel reflection due to index (impedance) mismatch at an optical interface ^{[1] [2]} .  The reflection can be suppressed by using adiabatic impedance matching implemented as an intermediate material with gradually varying index in the direction of surface normal. Subwavelength patterning is an effective method to implement such a gradient index (GRIN) surface.  However, these recent studies have been mostly restricted to planar surfaces.  Diffractive optical elements such as diffraction gratings, Fresnel zone plates, and holographic optics also suffer from Fresnel reflection losses evidenced as undesirable reflection orders.  Recently, we proposed a new class of GRIN diffractive optics that is capable of suppressing such reflection losses ^[3] .  Using the same GRIN principles, we can demonstrate diffractive elements where the reflected energy can be suppressed.

The proposed concept of the nanostructured GRIN grating is illustrated in Figure 1, where subwavelength tapered nanostructures with period p are integrated on both the ridge and groove of the grating.  Top-view and cross-section micrographs of the fabricated GRIN grating in silicon substrate are depicted in Figure 2. The grating has a period Λ of 5 mm, and the subwavelength cone-shaped pillars have nominal base diameter of 150 nm. The fabricated structure resembles a grating with nano-engineered surfaces. A cross-sectional micrograph is shown in Figure 2(b), where the cone heights on the ridge and groove are 650 and 600 nm, respectively. Some point defects characteristic of nanosphere self-assembly used in the fabrication process can be observed.  Broadband characterization of the fabricated structure indicated suppression by at least two orders of magnitude in the reflected orders of the GRIN grating over a large range of incident angles up to 60º.

1. P. Lalanne and G. M. Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnology, vol. 8, pp. 53-56, Oct. 1997.
2. Y. Kanamori, M. Sasaki, and K. Hane, “Broadband antireflection gratings fabricated upon silicon substrates,” Opt. Lett., vol. 24, no. 20, pp. 1422-1424, Oct. 1999.
3. C.-H. Chang, L. Waller, and G. Barbastathis, “Design and optimization of broadband wide-angle antireflection structures for binary diffractive optics,” Opt. Lett., vol. 35, no. 7, pp. 907-909, April 2010.

Luneburg lens is a gradient index (GRIN) element ^[1] known to produce diffraction-limited focus at the lens edge opposite to an incident plane wave. Despite its usefulness in applications such as radar systems, omnireflectors, or integrated optics, implementing the Luneburg lens in optical frequencies due to the difficulty in producing the desired GRIN profile. In this work, we describe the design and fabrication of a Luneburg lens for operation at near infrared optical frequencies using subwavelength aperiodic nanostructures.

The Luneburg lens is designed using a dielectric periodic square lattice of circular silicon rods with spatially varying parameters and subwavelength features. If the variation in the structure is gradual enough to be considered periodic within the adiabatic length scale, local dispersion relations can be analyzed through established photonic crystal theory ^[2] , and Hamiltonian optics can be used to analyze and design the propagation of light with adequate accuracy ^{[3] [4]} .  Using the developed algorithm less computational power is needed, and it allows for convenient structure optimization.

We designed a Luneburg lens with lattice constant of λ/6 at operating wavelength λ = 1.55 µm and fabricated a planar (2D) implementation using silicon-on-insulator (SOI) substrate.  The structure was patterned using electron-beam lithography and transferred into the device layer using reactive ion etching.  Although this structure suffers slightly from structure anisotropy, through design and optimization we were able to obtain a geometrical waist diameter calculated as λ/3 at the lens focus, as depicted in Figure 1. The fabricated structure, shown in Figure 2, has minimum feature size of around 90 nm, which can be readily achieved by the resolution limits of our lithographic approach. The fabricated lens is being characterized using scanning near-field optical microscope, and initial results demonstrate tight focusing of light.

1. R. K. Luneburg, Mathematical Theory of Optics, Berkeley, CA: University of California Press, 1964.
2. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, Princeton, NJ: Princeton University Press, 2008.
3. P. S. J. Russell and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. Lightwave Technol., vol. 17, pp. 1982-1988, Nov. 1999.
4. Y. Jiao, S. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method,” Phys Rev E, vol. 70, pp. 036612, Sep. 1999.