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.