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 [5
]. Quasi-conformal optimization is applied to the transformation, such that any anisotropy is minimized and can be neglected in the final structure. The degree of flattening of the lens, and thus its field-of-view, is ultimately determined by the dielectric constant of the host dielectric material.
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 [14
]. Third, this crystallinity should be hexagonal, allowing for the maximum range of achievable indices and good isotropy [15
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 [11
]. An alternative technique would be to evenly space holes along lines of constant index, but this approach requires special symmetries to produce good crystallinity, absent from the present design [16
]. The algorithm we use treats the holes as a system of interacting particles where the interaction length of each particle is dependent on its position. By allowing the particles to interact, an optimized distribution of holes is achieved with varying spatial density corresponding to the desired index at any location and with local hexagonal symmetry [17
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
and loss tangent 0.002. The selected design of the refractive Luneburg called for an index of
at the outer boundary. The host index allows the lens to be compressed by 13.4 percent of the radius, giving a field of view of ±30° and an f-number of 0.433. The lens comprises 7,979 holes. To achieve the aspect ratio required it was necessary for the lens to be drilled halfway through from both sides of the slab using a computer controlled milling machine. Holes of diameter 1/8th
inch were drilled into the ceramic composite material.
shows the optimized relative permittivity distribution for the flattened lens, the hole distribution computed by the particle interaction approach described above, and the final fabricated lens. Plots of the simulated and measured electric field distributions are shown in . The field distribution was measured using a 2D electric field mapping apparatus previously reported [18
]. For both the simulations and the experimental measurements, a source was placed at the focal plane of the lens, with a roughly collimated beam expected as the output. In the experimental setup, the source consisted of a dielectric rod waveguide with square cross-section coupled to the focal surface of the lens. Simulations were performed using COMSOL Multiphysics, a commercial finite element solver. The agreement between the simulated and measured field patterns was found to be excellent.
Figure 1. (a) The continuous permittivity map for the 2D TE compressed refractive luneburg lens (note the permittivity of two at the outer boundary), (b) a distribution of holes with the same effective permittivity and (c) the fabricated lens. The hole diameter (more ...)
Figure 2. 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 (a) simulated continuous index lens and (b) the measured, fabricated lens. A 10 GHz source was used in both the (more ...)