## Abstract

Employing artificially structured metamaterials provides a means of circumventing the limits of conventional optical materials. Here, we use transformation optics (TO) combined with nanolithography to produce a planar Luneburg lens with a flat focal surface that operates at telecommunication wavelengths. Whereas previous infrared TO devices have been transformations of free-space, here we implement a transformation of an existing optical element to create a new device with the same optical characteristics but a user-defined geometry.

© 2012 OSA

Materials can be used to guide and manipulate light in a variety of ways. Refraction at the interface between air and glass, for example, can be harnessed to produce high quality lenses by engineering the shape of the interface. Alternatively, the refractive index of a medium can be varied spatially to control the optical path lengths over a region, producing a gradient index (GRIN) optical element. GRIN optics are well known but less common than refractive optics due to the difficulty in producing large, controlled, index variations in conventional materials. Nevertheless, GRIN optics are compelling, since a number of GRIN lens designs exist that correspond to perfect imaging instruments, typically based on spherically or cylindrically symmetric inhomogeneous distributions of refractive index [1,2].

A particular example of this class of optical devices is the Luneburg lens, which has a refractive index profile that varies radially according to [3],

where*r*is the lens radius. The Luneburg takes rays incident from infinity and focuses them perfectly (in the geometrical optic limit) to a point on a spherical surface [1,2]. The Luneburg is an appealing device for wide field-of-view imaging applications; however, its curved focal surface makes it incompatible with standard planar detector arrays. Thus, in addition to achieving the large refractive index variation needed to implement the Luneburg solution, a modification to the lens that would make the lens compatible with planar detectors is also desirable.

_{lens}Over the past decade, artificially structured metamaterials have provoked intense interest for their potential use in realizing unusual or extreme material parameters. At low (sub terahertz) frequencies, structured metals have been used with great success to implement large positive and negative values of the electric permittivity or the magnetic permeability, enabling the experimental demonstrations of exotic media and devices, including negative index materials, “perfect” lenses, hyperlenses, and invisibility cloaks. Though metals are currently being investigated as metamaterial inclusions at infrared and visible wavelengths, the strong absorption inherent to metals at these wavelengths limits their application. Structured dielectrics, however, offer a promising path to GRIN and advanced optical design, since nearly the entire index range from air to that of the bulk value of a given dielectric is available, subject to fabrication constraints. As the available index range of a material becomes larger, more optical design space becomes available. This design space has allowed devices such as carpet cloaks and GRIN lenses to be implemented at IR and optical wavelengths [4–6].

The Luneburg and similar rotationally symmetric GRIN lenses are of practical interest as elements in photonic systems. These types of devices have previously been fabricated in planar waveguides, however they retain their traditional geometries [7,8]. Some variations of the Luneburg footprint, while retaining the aberration-free focusing properties, can lead to more useful devices that can be integrated into photonic systems with greater flexibility and possibly reduced need for additional elements. Transformation optics (TO)—in which coordinate transformations assume the design role—provides just such a tool that allows the determination of an index distribution for a medium corresponding to any modification in the topography of a lens element. Applying a transformation to empty space can produce “cloaking” structures, since the properties of free space are unchanged by the transformation [9]. Likewise, applying a transformation to a region of space containing an optical element leaves all optical properties of that element intact, even though the topology of the element can be changed radically [10].

A TO approach to flattening a Luneburg lens was proposed by Schurig, who showed that an analytic coordinate transformation can be applied to flatten one side of the lens [11]. Unfortunately, this transformation results in a complex medium that is challenging to fabricate at any wavelength, especially IR or visible wavelengths. However, if propagation is restricted to two dimensions (2D) and the magnetic field is restricted to be parallel to the plane of propagation (transverse magnetic, or TM polarization), then quasi-conformal (QC) optimization can be performed to arrive at coordinate transformations that minimize the need for magnetic response as well as anisotropic media [12,13]. While a large index variation is still necessary, the QCTO medium is much more readily realized since it requires only isotropic, dielectric constituents.

While QC transformations are not possible in three dimensions, where anisotropy is always required, they are an enabling tool in 2D [14]. After a QC transformation is applied to a region of space, the permittivity and permeability components out of the plane of the transformation are varied, while the in-plane components are approximately unity [15]. Thus, for TM guided waves, the QC transformation can be exactly implemented with a variation in the isotropic permittivity, while for transverse electric (TE) guided waves, with the electric field restricted to be in the plane of propagation, a variation in the out of plane permeability is required to implement the transformation exactly. Since the index of refraction required by both polarizations is the same, in the geometric-optics limit a transformation can be implemented in a dielectric-only GRIN material for TE as well as TM modes. The transformation is then exact for TM polarized waves but only valid in the geometrical optic limit for TE polarized waves, since a dielectric-only material will give the prescribed index but not the prescribed permeability. Kundtz *et al.* demonstrated a flattened Luneburg at microwave frequencies, designed using QCTO and implemented with metal “cut-wire” metamaterial elements that equated to an isotropic gradient index medium [16]. Recent work at microwave frequencies has implemented such an 'index only' TO Luneburg using patterned dielectric waveguides, a technique suitable for the fabrication of IR and visible wavelength photonic devices [17]. In the device presented here, the only mode supported by the slab waveguide is the fundamental TE_{0} mode.

The index profile for a Luneburg is given by Eq. (1) and shown in Fig. 1(a) . A QC transformation that flattens a section of the lens surface is found by numerically solving the nonlinear partial differential equations given in [15] with Neumann-Dirichlet boundary conditions on the top, left, and right boundaries of Fig. 1(a), which are connected to free-space, and a Dirichlet boundary condition on the bottom bold boundary, which contains the focal surface of the lens to be flattened. The solution of these equations are the QC transformation functions x’(x,y) and y’(x,y), which do not have an analytical form but can be visualized graphically [16]. The permittivity and permeability tensors that implement this transformation are then obtained from the transformation optics equation,

where $\Lambda $ is the Jacobian of the transformation. As a result of applying a QC transformation, only the components out of the plane of the transformation, ${\in}_{zz}$ and_{${\mu}_{zz}$, are significant. Fig. 1 Raytraces through the Luneburg index distributions. (a) The original 2D Luneburg lens, (b) the flattened Luneburg lens, and (c) the fabricated flattened Luneburg lens. The bold black line in (a) and (b) shows the transformed boundary. The range of indexes of the flattened lens, (b), have increased compared to the original lens, (a). In the fabricated lens, (c), indices less than 1.5 have been approximated as 1.5 to maintain waveguiding, and the transformation has been truncated with an index matching region that matches the transformation to the waveguide index. Download Full Size | PPT Slide | PDF Because the device presented here supports only a TE wave, the index of refraction that implements this transformation in the geometrical optic limit is $\sqrt{{\mu}_{zz}^{\text{'}}}$. The index the flattened lens, shown in shown in Fig. 1(b), is then(3)$${n}^{\text{'}}(x,y)=\sqrt{{\mu}_{zz}^{\text{'}}}n(x,y)$$where the index prescribed by the transformation has been multiplied by the index of the untransformed Luneburg lens, n(x,y), given by Eq. (1) [10]. As the degree of flattening increases, the maximum required index also increases. The degree of flattening, and thus the field of view of the lens is determined by the index of the host dielectric.In previous microwave realizations of flattened TO Luneburg lenses, the transformation was truncated at the lens boundary. This truncation was a reasonable approximation of the transformed lens because the index required by the transformation outside the lens, while having some spatial variation, was close to unity everywhere [16,17]. Because the device implemented here is embedded in a slab waveguide—essentially a high index background—a similar truncation is not suitable since it would introduce refraction at the lens interface. For the present work, the transformation is truncated at a distance of 2.5rlens from the center of the lens, so that the flattened lens is surrounded by a circular low index region concentric with the center of the transformed lens. Not only does including this region avoid unwanted refraction at the lens-waveguide interface, but because more of the transformation is included spherical aberrations introduced by truncating the transformation are reduced [17]. At the boundary of the transformation, we must transition to the unpatterned waveguide. This is achieved by matching the index at the boundary of the transformation to the index of the slab waveguide with a low-reflection index matching region. In addition, the transformation also introduces spatial regions where the refractive index assumes values below unity. Values of refractive index less than unity are undesirable, as they imply frequency dispersion and hence introduce bandwidth limitations. Fortunately, approximating these regions by setting their index value to unity has little effect on the focusing behavior of the lens [8]. These modifications to the index profile of the transformed lens are shown in Fig. 1(c).To realize a GRIN device in a slab waveguide, sub-wavelength holes can be etched into silicon on insulator (SOI) to create a dielectric-only metamaterial. This distribution should ideally meet several requirements. First, the maximum lattice constant in the array should be substantially smaller than the wavelength of light in the material so that we are operating in the first band where the material can be described as a homogeneous material rather than a photonic crystal. Secondly, in order for the index to be accurately defined over as small a region as possible and to reduce Rayleigh scattering, the hole-array should have local crystalline symmetry [18]. Third, this crystallinity should be hexagonal, allowing for the maximum range of achievable indices and good isotropy [19]. Because our index is exactly defined over the scale of a single unit cell, requirements two and three also relax fabrication constraints. The unit cells and index gradients can be larger with respect to the wavelength than possible with a statistical or dithered hole distribution, allowing larger holes to be used [7].These requirements are met by using an algorithm that treats the holes as a system of interacting particles where the interaction length of each particle is dependent on its position in such a way that the filling fraction of holes in any location yields the desired index at that location. By allowing the particles to interact, an optimized distribution of holes is achieved with varying spatial density corresponding to the desired index distribution and with local hexagonal symmetry [20]. The transformed lens had a diameter of ten free-space wavelengths or 15.5 μm, and consisted of 84303 holes. Because our waveguide mode is TE0, where the electric field is polarized perpendicular to the axis of the holes, the desired filling fraction was determined by applying the 2D Maxwell-Garnett effective medium (MG) formula for the homogenization of mixtures of cylindrical voids in a dielectric matrix. For the permittivity of the matrix we used the square root of the mode index of our TE0 mode [18]. For 1.55μm wavelength TE0 mode in a 250nm thick SOI waveguide, this mode index is 2.93. The upper limit on the spacing of the holes, determined by the deviation of unit cell simulations from the MG mixing formula and the onset of photonic crystal effects, is 460nm. The lower limit wasdetermined by fabrication constraints and in our case was 94nm for 85nm diameter holes. These limits allow us to vary the mode index from 1.5 to 2.86, as shown in Fig. 2 . Since the minimum index cannot be smaller than the index of the bulk SiO2 cladding layer, this index range covers nearly the entire range of available indices. The dispersion of this hole-array metamaterial is related to the dispersion of the mode index of the SOI slab. Over a wavelength range of 1.45μm-1.65μm the fundamental mode index varies only 1.5% from the design index of 2.93, giving a wide operation bandwidth. Fig. 2 Effective index dispersion. The dashed lines show the index of bulk Si and SiO2 while the solid lines show the effective mode index in the Si slab waveguide vs. frequency. The solid red region is the range of indexes achievable with the hole-array metamaterial used here. This region covers nearly the entire range of accessible indices and exhibits small dispersion. Download Full Size | PPT Slide | PDF The lens was designed to be excited by an array of four waveguides, each at a different location on the focal plane of the flattened lens. When an IR laser is coupled into its corresponding input grating, each 0.8 µm wide input waveguide forms an approximate diffraction limited source on the image plane of the lens which produces a Gaussian plane wave in a different direction (Figs. 3(b) -3(e)). Because the lens is reciprocal, this is equivalent to focusing a plane wave to a point on the flattened focal surface. The output beam transitions from the low index region of the transformed free-space outside the lens to the high index unpatterned slab waveguide through an index matching region concentric with the center of the lens. For characterization purposes, the beam is then coupled out of the waveguide by a curved grating. Fig. 3 Fabricated Luneburg sample. (a) Cut-away view of the lens fabricated in silicon-on-insulator (SOI). Inset shows an SEM image of the same region of the lens where the local-crystallinity of the hole distribution can be seen. (b) Schematic of the fabricated lens. The pattern shown was etched through the Si slab using EBL followed by DRIE, stopping abruptly at SiO2 layer - except for the input/output gratings which only partially penetrate the Si slab. Input waveguides were defined by etching air trenches in the silicon slab-waveguide. (c) SEM image of the fabricated lens. Download Full Size | PPT Slide | PDF Fabrication began with a <100> oriented SOI wafer with a Si top device layer thickness of 340 nm and a buried oxide layer thickness of 2 µm. In order to achieve the designed device layer thickness of 250 nm, the SOI wafer was thermally oxidized such that 90 nm of silicon was consumed; the sacrificial oxide layer was then removed in buffered oxide etch, yielding a silicon device layer thickness of 250 nm. The hole pattern was achieved using electron beam lithography (EBL; Elinonix ELS-7500) followed by deep reactive ion etching (DRIE; SPTS Pegasus).DRIE (Bosch process) is extremely selective with respect to silicon versus silicon dioxide, so the etched holes pass completely through the silicon device layer and terminate abruptly at the buried oxide interface. A second EBL step was performed to define the four input gratings and the semi-circular output grating along the perimeter of the lens; again, DRIE was used to etch the silicon, but for the gratings the etch depth was 120 nm. Finally, a third EBL step followed by DRIE was used to define the four input waveguides. Because the silicon device layer serves as the waveguide material, the waveguides were defined by patterning 0.8 µm wide trenches on both sides of the four waveguides; the trenches were fully etched and terminated on the buried oxide of the SOI, thus providing air cladding on the sides of the waveguides.The lens was characterized by coupling a λ = 1.55μm optical input laser into each waveguide independently with a 20x objective lens and observing the location of the output spot on the output grating with an IR camera. The experimental setup and images of the operating device are shown in Fig. 4 . The output beam direction was determined from the experimental images of the operating lens by measuring the position of the output spot relative to features on the lens structure. Due to the resolution of camera, there is an uncertainty of 3.6° in the measured angles. Table 1 shows the experimentally measured and theoretically predicted angles of the beam outputs from each of the four input waveguides. Fig. 4 Experimental characterization of the sample. (a) The optical circuit of our characterization setup. An amplified spontaneous emission source was used to illuminate the entire lens, while a 1.55 μm laser was focused to one of the four input gratings at a time. A CCD camera was used to image the lens and observe the location of the output beam. The half-wave plate and polarizer were oriented to partially filter the input illumination to reduce saturation of the CCD detector. (b)-(e) Images of the lens with the IR laser coupled to each of the four input gratings. Download Full Size | PPT Slide | PDF Table 1. Experimental Results for Transformed Luneburg Lens Beam AngleView TableAnalysis of SEM images of the fabricated device showed that the input waveguides, which were patterned in a separate step from the lens hole array, were shifted by 170nm from the center line of the lens, which accounts for the offset of the central beam from the optical axis. Taking this shift into account, the experimentally measured beam directions agree very well with theory, and shows focusing over a range of incidence angles of ± 33.5°.In conclusion, we have demonstrated the first transformation optical device based on a transformation of an existing GRIN device operating at IR wavelengths. Such an approach allows new optical elements to be designed that utilize the optical properties of existing devices, but enables the designer to reconfigure the geometry of the device to different more useful geometries, as demonstrated herein by the flattening of the focal surface of a Luneburg lens. By implementing this TO device with a dielectric-only hole-array metamaterial, it is possible to implement this new GRIN device in SOI with a broad bandwidth. The flattened Luneburg lens implemented here shows beam forming (focusing) from a planar focal surface over a wide field of view of 67°, in excellent agreement with the theoretical perfect focusing of the Luneburg lens.References and links1. E. W. Marchland, Gradient Index Optics (Academic Press, 1978).2. R. Luneburg, Mathematical Theory of Optics (Brown Univ. Press, 1944).3. S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. 29(9), 1358–1368 (1958). [CrossRef] 4. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef] [PubMed] 5. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009). [CrossRef] 6. M. 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