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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The Luneburg lens is a powerful imaging device, exhibiting aberration free focusing for parallel rays incident from any direction. However, its advantages are offset by a focal surface that is spherical and thus difficult to integrate with standard planar detector and emitter arrays. Using the recently developed technique of transformation optics, it is possible to transform the curved focal surface to a flat plane while maintaining the perfect focusing behavior of the Luneburg over a wide field of view. Here we apply these techniques to a lesser-known refractive Luneburg lens and implement the design with a metamaterial composed of a semi-crystalline distribution of holes drilled in a dielectric. In addition, we investigate the aberrations introduced by various approximations made in the implementation of the lens. The resulting design approach has improved mechanical strength with small aberrations and is ideally suited to implementation at infrared and visible wavelengths.

A Luneburg lens is a gradient index (GRIN) lens with a spherically or cylindrically symmetric index distribution that exhibits aberration-free focusing of light incident from any direction onto a spherical or cylindrical surface [

In the transformation optical design methodology, a desired distortion of the space that an electromagnetic wave travels through is achieved by introducing a spatial variation of the electric permittivity and magnetic permeability tensors over some region [

In the flattened Luneburg lens reported by Kundtz

The index profile for a standard Luneburg lens of unit radius is given by [_{0}, the index, _{0}.

In the present refractive Luneburg design, the index value at the lens boundary is determined by the minimum allowed separation of the holes, which is in turn determined by fabrication and mechanical strength constraints.

Once a profile for the untransformed refractive Luneburg lens is obtained, a coordinate transformation is applied that flattens one surface of the lens [

Once a profile for the untransformed lens is obtained, the degree of flattening, and thus the field of view, of the lens is determined by the index of the host dielectric. As the degree of flattening increases, the maximum required index also increases. Since the maximum index cannot be larger than the host dielectric’s index, a second constraint on the index variation of the medium is introduced. The achievable degree of flattening may be increased by introducing multiple material regions so that a low index substrate is used at the outer boundary of the lens and successively higher index materials are used as the prescribed index exceeds those of the outer materials. As is typical for the transformations used to develop transformation optical devices, the transformation applied here extends slightly beyond the boundary of the lens, resulting in some variation of the free space index outside the lens. In addition, the transformation also introduces spatial regions where the refractive index takes 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, as will be discussed below.

Once a continuous index profile has been determined, a method for translating that index profile into a distribution of holes must be employed. The resulting distribution should meet several requirements. First, the holes should be uniformly sized, as this greatly simplifies fabrication where drilling or lithography techniques are used. Second, in order for the index to be accurately defined over as small a region as possible and to reduce Rayleigh scattering, the hole distribution should have crystalline symmetry [

Several techniques that meet these requirements have been introduced previously. Note that the present flattened Luneburg design is a transformation of an existing gradient index profile, and so the transformation itself cannot be used to arrive at a regular crystalline distribution, as has been done in other transformation optics designs [

To investigate the performance of the flattened refractive Luneburg lens, we chose to fabricate and characterize an implementation designed to operate over a broad range of frequencies in the microwave range (at least spanning the 8–12 GHz band measurable in our apparatus). The untransformed lens had a diameter of ten free space wavelengths at our central frequency. The transformation was truncated at the transformed lens boundary. The lens was machined from a slab of Emerson&Cuming ECCOSTOCK HiK polymer-ceramic with index of
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In this implementation of the flattened Luneburg, two approximations of the index distribution prescribed by the transformation were made. Because the transformation ideally extends through all space, the free space index outside the untransformed lens is also modified. In order to perfectly preserve the behavior of the Luneburg, the entire transformation should be implemented, but to make a reasonably sized lens the transformation should be truncated close to the transformed lens’s boundary. The deviation from unity index decreases with distance from the lens, but at some point the transformation must be truncated and the index set to one. Thus the first approximation is to set the transformation index to one beyond some radius, as shown in

The second approximation has a slightly more severe effect. For regions at the edges of the flattened region of the lens, the transformation indices take values less than one. To avoid using resonant metamaterials, these indices must be approximated as one, as shown in

The effect of these approximations can be better quantified by performing ray traces through the transformed lens and calculating the optical path difference (OPD) and spot diagrams. Similar ray tracing analysis has been used to compare three dimensional quasi-conformal transformations, which require anisotropic index of refraction, and their isotropic approximations [

When the

In addition to aberrations introduced by these two approximations, distortion aberrations are also present. This is an interesting effect caused by the transformation itself, and indeed is present in the full-transformation lens as well. In order to preserve orthogonality in the quasi-conformal transformation, the x-coordinates are allowed to ‘slip’ on the flattened boundary,

The performance of the lens was quantified by analyzing the far-field patterns of the lens, shown in

In conclusion, we designed and fabricated a lens which exhibits focusing over a field of view of ±30° with very little aberration. The design utilizes a refractive Luneburg which has been flattened using a quasi-conformal transformation to produce a flat focusing surface appropriate for planar detector/transmitter arrays. Using the refractive Luneburg as a starting point for our design allowed the lens to be implemented with a semi-crystalline distribution of holes drilled in a dielectric. The design and fabrication approaches are well suited for scaling transformation optical structures to IR and visible wavelengths. The two approximations made in this implementation—the approximation of indices outside the lens and indices less than one as one—have been investigated using ray tracing and were found to introduce only small aberrations.

This work was supported through a Multidisciplinary University Research Initiative, sponsored by the Army Research Office (Contract No. W911NF-09-1-0539). N. Kundtz would like to acknowledge the IC postdoctoral fellowship program.

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Out of plane electric field for a source located on the flattened focal plane that produces a plane wave propagating at 30 degrees for the

The two approximations made to the transformed Luneburg lens index profile.

The OPD of rays focused through a transformed Luneburg lens for different size transformation regions. The lens has an aperture size of ten wavelengths and rays are incident along the optical axis of the lens. The legend indicates the radius of a circle, concentric with the lens, beyond which the transformation index is set to one as shown in

Spot diagrams of 33 rays focused through a complete and an approximate transformed Luneburg lens for different angles of incidence. In the approximated lens, all indices less than one have been approximated as one. The lens has an aperture size of ten wavelengths and angles of incidence are measured from the optical axis. Though the lens has a field of view of ±40°, in the approximated lens rays travel through the approximated

Far-field radiation patterns for the continuous permittivity lens and the experimental lens at