Experimental demonstration of Luneburg waveguides

Transformation optics gives rise to numerous unusual optical devices, such as novel metamaterial lenses and invisibility cloaks. Very recently Mattheakis et al. have suggested theoretical design of an optical waveguide based on a network of Luneburg lenses, which may be useful in sensing and nonlinear optics applications. Here we report the first experimental realization of such Luneburg waveguides. We have studied wavelength and polarization dependent performance of the waveguides.

2 frequency range. The individual Luneburg lenses in the fabricated waveguides are based on lithographically defined metal/dielectric waveguides [12][13][14]. Adiabatic variations of the waveguide shape enable control of the effective refractive index experienced by the TM light propagating inside the waveguide. Our experimental designs appear to be broadband, which has been verified in the 480-633 nm range. These novel optical waveguides considerably extend our ability to control light on sub-micrometer scales.
We have recently demonstrated that metamaterial parameter distribution required for TO-based designs can be emulated by adiabatic changes of shape of a 2D metal-dielectric optical waveguide [9,14]. Devices employed in our experiments have a three-layer waveguide geometry which is shown schematically in the inset in Fig.1(a).
Assuming adiabatic changes of the waveguide thickness, the wave vector k of the guided mode can be calculated as a function of light frequency  and waveguide thickness d for TE and TM polarized light, resulting in the definition of the effective refractive index n eff =k/c for both polarizations [14]: in which the refractive index varies from n(0)= f f / 1 2  to n(a)=1 is easy to realize for TM polarized light based on the comparison of experimental and theoretical data plotted in Fig.1(b,c). Theoretical performance of such a lens for f = 1 is presented in near the lens edge, the same device will operate as a spatial (directional) filter for TE light, as shown in Fig.2(b). This result is natural since most of TE light must experience total reflection from the interface between air (n = 1) and the lens edge (n ~ 0) coming from the medium with higher refractive index.
In our earlier work [14], we have developed a lithography technique which enables precise shape d(r) control of the dielectric photoresist on gold film substrate.
Unlike the traditional lithographic applications which aim for rectangular photoresist 4 edges, we need to create a more gradual adiabatic edge profile. To produce gradual decrease of photoresist thickness (Shieply S1811 photoresist having refractive index n~1.5 was used for device fabrication) several methods have been used. Instead of contact printing (when mask is touching the substrate), we used soft contact mode (with the gap between the mask and the substrate). This allows for the gradient of exposure due to the diffraction at the edges, which leads to a gradual change of thickness of the In these experiments a near-field scanning optical microscope (NSOM) fiber tip was brought in close proximity to the arrays of lithographically formed lenses and used as an illumination source. Almost diffraction-limited (~0.7) focusing of 515 nm light [14] emitted by the fiber tip (seen on the left) clearly demonstrates Luneburg lens-like focusing behavior of the fabricated devices for TM polarized light. Comparison of the theoretical and experimental images performed in Fig. 2(b) demonstrates excellent agreement between theory and experiment for both polarizations (artificial color scheme used to represent experimental images in Fig.2(b) has been chosen to better highlight this close match). 5 Let us now consider the concept of a TO waveguide based on a linear chain of Luneburg lenses, which has been developed in [1]. Its operation is obvious from Fig.   3(d), which shows a ray tracing simulation of the waveguide. A single Luneburg lens focuses incoming parallel rays onto a single spot located on its edge. Therefore, a linear set of Luneburg lenses placed next to each other is able to guide light while exhibiting periodic diffraction-limited foci, which are spaced at twice the lens diameter. The same result has been obtained in our numerical simulations of the straight Luneburg waveguide ( Fig. 3(a,b)) performed using COMSOL Multiphysics using refractive index distribution corresponding to the experimental variation of waveguide thickness.
However, if the Luneburg waveguide is curved, the ideal double periodicity appears to be broken, as shown in Fig. 3(c). Nevertheless, Mattheakis et al. predicted [1] that the Luneburg waveguides may be bent considerably without the loss of guiding. Our experimental results described below confirm these theoretical predictions.
Using the waveguide fabrication technique described above we were able to produce and study both straight and curved Luneburg waveguides, as demonstrated by  Fig. 4 (a,c). This waveguide consists of a linear set of 1.5 m diameter individual Luneburg lenses. The microscopic images of the waveguide taken while 488 nm light was coupled into the waveguide are shown in Fig.   4(b,d). The expected double periodicity of light distribution in the Luneburg waveguide is indicated by arrows in frames (c) and (d). This experiment clearly demonstrates a successful experimental realization of a Luneburg waveguide for the TM polarized light. Moreover, measurements of the polarization-dependent light propagation through 6 the waveguide studied in Fig. 5 also confirm our theoretical model. As expected, TE polarized 488 nm light does not exhibit much propagation along the waveguide.
We have also studied propagation of TM light though curved Luneburg waveguides as illustrated in Fig. 6. As indicated by our theoretical simulations, the ideal double periodicity of light distribution appears to be broken in such waveguides. While some apparent double periodicity is indicated by arrows in Fig. 6(b), the cross-sectional analysis of the measured light distribution along the waveguide (Fig.6(c)) demonstrates that the double periodicity is generally broken. Nevertheless, the FFT analysis of the cross section ( Fig. 6(d)) indicates that the double and the other even periods still dominate light distribution inside the curved waveguide, which confirms its Luneburg nature.
In conclusion, we have reported the first experimental realization of the TObased Luneburg waveguides, which may be useful in sensing and nonlinear optics applications. We have studied wavelength and polarization dependent performance of the waveguides. Our technique opens up an additional ability to manipulate light on submicrometer scale.