1. Introduction

The mathematics of transformation optics (TO) [1] has introduced a powerful tool for lens design. In principle, analytic [2] and numerical [2–4] TO techniques allow the lens designer to bend light to almost any specification. Of particular interest for immediate applications, the spatial-warping techniques of TO permit modification of the physical geometry [5] of an existing lens while retaining its optical function. This approach conceptually facilitates conformal lens and antenna design, quite possibly eliminating obtrusions or forming the optical element to its surroundings. The trade-off for this geometric flexibility is increased complexity of the optical parameters (electric permittivity ε and magnetic permeability μ); the optical parameters resulting from TO designs are typically inhomogeneous and anisotropic [6, 7]. Since such materials are nearly absent in nature and are difficult to synthesize, artificially structured metamaterials [8–10] have been proposed as a possible solution [11–13]. Even with the potential that metamaterials offer, most TO designs require material parameters beyond the current technical capabilities of metamaterial fabrication; although, in principle no barriers exist given herculean effort. The present realization of TO designs has thus become one of trade-offs, sacrificing performance in exchange for simplification of the optical parameters and realistic fabrication as is also true of the implementation reported in this work.

Many early examples of using TO have focused modification of pre-existing lens designs drawn from the family of gradient-index lenses [14]. In particular, using TO to modify the classical solution for the Lüneburg lens [15] has seen significant attention. A nearly-perfect [16] imaging and beam-forming device, the Lüneburg lens is a spherical gradient lens with isotropic permittivity that varies with radius as ε(r) = (2 − r²). The primary drawback to the Lüneburg lens is that its image surface is spherical [17], making it incompatible with traditionally planar detector and source arrays. TO presents a simple solution - planarize some portion of the image surface by applying a coordinate transform that reshapes the spherical surface to a planar surface. This approach, first proposed by Schurig [17], was realized by Kundtz and Smith in a two-dimensional geometry. In that work, the use of quasi-conformal transformation optics (QCTO) was applied in an optimization step to achieve a dielectric only implementation. While no three-dimensional (3D) QCTO transformation exists that would easily allow the design of a flattened 3D Lüneburg lens, the rotational symmetry of a 3D Lüneburg lens suggests an approximate 3D solution might be obtained by rotating the 2D QCTO solution about the optical axis. This approach was attempted by Ma et al. [18], who experimentally demonstrated a 3D flattened Lüneburg lens at microwave frequencies. However, in the work of [18], the known detrimental influence of both the TO and metamaterial approaches were not addressed in any detail, and comparison was not made to a conventional Lüneburg lens. In the present work, we continue the exploration into TO design and consider in detail the implications of the material simplifications necessary to realize the TO Lens. In particular, we quantitatively compare the beam-forming directivity of the flattened 3D QCTO Lüneburg lens with a conventional Lüneburg lens.

2. Design

In Lüneburg lens flattening, typically, only a portion of the 2π steradian hemisphere is flattened - creating a fixed field of view (FOV). The greater the FOV is, the more extreme the resultant optical parameters are [3]. The TO procedure can be summarized in the four-vector notation [3]:

As discussed in previous literature [2], the QCTO procedure breaks down when applied to 3D geometries. To create a 3D flattened Lüneburg lens with FOV = π steradian, we follow the procedure of Landy et al. [19], applying a QC transform in 2D and rotating this two-dimensional material-matrix around the optical axis to obtain a three-dimensional axisymmetric profile.

This axisymmetric assumption is equivalent to restricting wave propagation to transverse-electric (TE) polarization for an isotropic, dielectric-only lens. However, even after the simplifying approximations of QC and axisymmetric polarization, fabrication of the resultant optical parameter matrix [3] is still beyond the technical scope of any metamaterial created to date. To obtain a design within closer reach, we impose three further material simplifications: nonmagnetic (μ_θ,ϕ,z = 1), isotropic (ε_θ =ε_ρ,z), and nonmetallic (all ε > 1). Each of these simplifications has negative repercussions in lens performance. The resultant isotropic dielectric profile after these simplifications is shown in Fig. 1(a).

 

Fig. 1 (a) Dielectric prescription for a QCTO Lüneburg lens flattened via QCTO to FOV π steradian, and with the isotropic and nonmagnetic assumptions mentioned in text. (b) Effective medium mapping for a metamaterial consisting of a cylindrical hole matrix of fixed spacing but variable diameter. The square and circular points show finite element solutions for TE and TM polarizations, respectively. (c) Photograph of completed lens, with representative slice showing HDPE ring and FR4 core.

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To realize the prescribed material parameters for the simplified Lüneburg lens structure, we create a cylindrical-hole-array metamaterial in two materials, FR4 fiberglass and High Density Polyethylene (HDPE) plastic. Removal of material reduces the effective dielectric ε_eff from the bulk value - in agreement with effective medium theory [20]. Figure 1(b) thus gives us the effective-medium-to-geometry mapping necessary for creating a metamaterial implementation of the TO lens. The anisotropy of cylindrical holes results in differences between ε_eff for polarizations perpendicular and parallel to the cylinders, and later the effects of this choice and its anisotropy will become apparent. To be thorough we perform finite element simulations to obtain exact solutions for the effective parameters of the unit cells. Scattering-parameter results from the finite element simulations are inverted using the procedure of [21] to find the effective medium permittivity and permeability, though the permeability remains almost exactly unity. The revolved ε(ρ,z) prescription of Fig. 1(a) is broken up into planar sheets, each of which contains holes in FR4 and HDPE mapped from Fig. 1(b). Physically, these holes are made via CNC waterjet - which results in a total-lens cost competitive with existing conventional Lüneburg lenses (often made of nested spherical shells of different density polystyrene) even for this first-generation device. The sheets are stacked to make the complete lens as shown in Fig. 1(c).

3. Experiment

We begin our characterization of the 3D flattened Lüneburg lens by first performing beam-forming measurements of a conventional spherical Lüneburg lens [22] of the same diameter as that of our flattened QCTO Lüneburg lens. This important step is done using a rotational stage and anechoic-chamber to obtain far-field directivity. Figure 2(a) shows this calibration directivity map, taken at 12 GHz. The measured directivity of +26dBi is in good agreement with manufacturers specifications of ∼+27dBi at 12GHz. Because of the inherent imaging and beam-forming properties of the Lüneburg lens, this directivity provides a good indication of the maximum achievable performance we can expect for any lens of the same diameter.

 

Fig. 2 Beamforming measurements for the QCTO Lüneburg lens. Electric-field amplitude as a function of the spherical angles ϕ and θ taken at 12GHz in the Far-field (160cm) for (a) A conventional spherical 9” Lüneburg lens [22]. (b) Our QCTO-Lüneburg lens with the source feed at the center of the flattened plane. Off-center source which directs beam in θ or ϕ for (c) source feed at X+15mm (d) Y-15mm (e) X+43mm (f) Y-43mm. Note the axes are zoomed in (a) to reveal the tight symmetrical beam of the conventional Lüneburg lens. To enable quantitative performance evaluation, the directivity of each is given. (g) illustrates the capability of broadband operation by showing directivity for the conventional Lüneburg lens and our TO Lüneburg lens (center feed) across the entire X-band.

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Far-field directivity measurements for our QCTO-flattened Lüneburg lens are shown in Fig. 2(b). The overall directivity the QCTO-Lüneburg lens is reduced by nearly 6dBi. The defocussing is visually apparent as blurring, with the introduction of side lobes around the central peak. A significant portion of this defocusing is from the material simplifications made to realize the metamaterial, particularly the minimum value of the index at the edges. Although the full-parameter-set QCTO solution for Lüneburg lens flattening is understood to perform focusing nearly equivalent to the original, all deviations made from that ideal design can only degrade performance. Given the extent of the deviations made to realize our design, even comparable performance would be unexpected. In an imperfect realization of a TO design such as ours (as well as previous work [18]), these degradations are inevitable and expected.

We next consider off-center feeds along the flattened surface, which will direct the beam in either azimuthal (± X) or elevation (± Y) angles. Figures 2(c) and 2(d) show these measurements for source locations of ΔX+15mm and ΔY-15mm. The total beam steering is about 15°. For these modest angles of +−15°, the center beam-waist becomes slightly elliptical, although overall directivity is approximately that of the center-feed beam. We notice the side-lobe pattern is distinctly different for azimuthal and elevation steering - an asymmetry which is likely related to the isotropy of the material parameters. The cylinder effective-medium mapping (Fig. 1(b)) was designed for TE polarization; and the TM polarization components inevitably present in elevation-steering result in aberrations that degrade the beam quality.

In Figs. 2(e) and 2(d) we show off-center-feed measurements for more extreme source positions, X+43mm and Y-43mm. Immediately we notice a very aberrant non-gaussian beam. The angular steering of the beam is also only ∼20°, barely more than in Figs. 2(c) and 2(d). Some of this poor-performance may be due to the same isotropy issues mentioned above. However, at extreme angles an additional complication comes into play. The QCTO designs calls for regions of ε < 1, which we have omitted to simplify the material requirements and to enable broadband operation (such regions must be resonant). It is know that when these regions are omitted, the effective FOV becomes constricted [23]. In a QCTO flattened lens with ε > 1, not only can rays not be directed beyond ∼30°, attempts to do so result in additional aberrations. The measured directivity for these extreme angle beams is also reduced - although directivity (calculated from peak angular-intensity) becomes a less meaningful measurement for such a deformed beam.

The use of non-resonant dielectric materials in this design allows operation over a wide bandwidth. The bandwidth will be bound by low-frequencies where the diameter of the lens becomes comparable to the wavelength, and at high-frequencies by breakdown of the effective-medium approximations made to design the unit-cells. To evaluate bandwidth in this design, in Fig. 2(g) we plot the beam-forming directivity across the entire X-band for both the conventional Lüneburg lens and our TO flattened Lüneburg lens (center fed). We see good directivity across this entire range with higher frequencies performing better on both lenses. The TO Lüneburg lens has spectral features not seen in the conventional Lüneburg lens, and a spectrally-broad dip in the directivity centered around 10.5 GHz detracts somewhat from its performance. However the design is still significantly broadband in comparison to resonant metamaterial designs which usually have an operational fractional bandwidth of only ∼10%, and the entire upper quarter of the X-band performs well with little apparent dispersion.

As one further beam-forming measurement, we also perform near-field measurements of the output beam using an open waveguide detector on an XYZ translation stage. Figure 3(c) shows these measurements, plotting XZ and YZ slices of the phased Electric-field Re[Ẽ(x,y,z)] (also at 12 GHz).

 

Fig. 3 XZ and YZ planar slices of the beam Re[Ẽ(x,y,z)], displaying both phase and beam-amplitude information. As measured from the lens front, the scan Z-range is +60cm to +70cm along the optical axis.

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The real utility of flattening a Lüneburg lens exists in compatibility with traditionally planar imaging arrays. Pursuing this, we recorded focal-plane images on the back (flat) side of the lens for incoming paraxial beams. The combination of such images at many angles comprise the impulse response, or Point-Spread-Function (PSF) (a core metric for any imaging device) of our QCTO Lüneburg lens. We create incoming paraxial rays using our conventional Lüneburg lens at various angles of incidence, and measure |E(x,y)| on the back plane (using a XY scanner and tapered-waveguide nearfield-probe detector).

Figure 4(a) plots the raw PSF for a pariaxial beam incident along the optical axis. For an ideal Lüneburg lens in the Eikonal limit, paraxial rays reduce to a single point on the image plane. Considering wave optics, the best focus one can obtain is a diffraction-limited spot [24], and we include these ”ideal-image” spots on each of our PSF graphs as a Abbe-limit sized white circle. Immediately, we notice in Fig. 4(a) that our experimental image spot is broader than the ideal Abbe-spot, and there is considerable structure beyond the central peak.

 

Fig. 4 Imaging performance of the QCTO Lüneburg lens revealed through the Point Spread Function. Collimated beams, using a horn-fed conventional Lüneburg lens, are incident from various azimuthal (θ) and elevation (ϕ) angles. (a) Raw data for normal incidence θ = 0°, ϕ = 0°. (b) Data for θ = 0°, ϕ = 0° after deconvolution of the detector’s transfer function. (c–f) Deconvolved data for off-normal-azimuth beams (TE polarization) for (c) θ = 12°, (d) θ = 29°. Deconvolved data for off-normal-elevation beams (TM polarization) for (e) ϕ = 12°, (f) ϕ = 29°. On each, the white circle shows the expected position and size of a perfect diffraction-limited focus.

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The PSF for the QCTO Lüneburg lens is blurred by the same material imperfections as discussed above, and also convoluted with the transfer function of the detector. It is possible to remove this detector blurring using standard image processing techniques. Deconvolution of the detector transfer function [25] produces a notably improved PSF, shown for normal incidence in Fig. 4(b).

We also plot the PSF for off-normal incidence angles of 12° and 29° in both azimuthal Figs. 4(c) and 4(d) and elevation in Figs. 4(e) and 4(f). As discussed above, azimuthal angles maintain the TE polarization relative to our cylindrical holes. In TE, we see reasonable adherence to the PSF expectations of our isotropic design. At elevation angles, we increasingly couple to TM, thus deviating from our prescription in Fig. 1(a). In the PSF of Figs. 4(e) and 4(f) most of the intensity entirely misses the white ray-tracing prediction points.

Much of the Airy-disc-like side lobe structure seen in the PSF’s may also be the result of refraction at the lens boundaries. Although the designs for both conventional and QCTO Lüneburg lens go to ε_eff = 1 at the boundary (thus eliminating refraction and reflection), our chosen metamaterial only reaches a minimum of ε_eff = 1.4. This produces aberrations in focus as the refractive effect is not accounted for by TO design machinery. The edge-refraction due to non-unity minimum-dielectric can be eliminated by using a refracting-Lüneburg lens design [26], but such designs require larger maximum dielectric values - which creates additional material complications.

We are able to acquire a better understanding of the effect our material simplifications have on the Lüneburg lens performance through Eikonal ray-tracing. Using the beam-forming configuration (a point-source at the image surface of the Lüneburg lens is guided into paraxial rays) we explore the differences between conventional, full-parameter TO, and materials-simplified TO Lüneburg lens designs. Figure 5(a) shows a ray-tracing cross-section for a conventional Lüneburg lens, alongside a 2D directivity plot. Although in the Eikonal limit the directivity for a conventional Lüneburg lens is infinite (an aberration-free imaging device), we can avoid this unphysical metric (without resorting to full-wave simulation) by giving rays a minimum Gaussian width of 1-degree. This imparts a finite width to the far-field directivity, allowing comparisons to be made with TO designs. For a conventional Lüneburg lens, there is complete degeneracy between TE and TM polarizations; we plot only TE polarization in red.

 

Fig. 5 2D Eikonal ray-tracing and directivity results for different Lüneburg lens designs (a) a A conventional spherical Lüneburg lens. (b) Isotropic dielectric-only QCTO design, without any material simplifications. (c) Our fabricated Lüneburg lens, which includes a minimum dielectric cutoff, and anisotropy.

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Figure 5(b) shows the same ray-tracing and directivity plots for an isotropic, dielectric-only TO-flattened Lüneburg lens. This profile, shown in 1a, is prescribed by the QCTO transformation. When the source is considered to be exactly on (or just inside) the flattened image plane, this design produces a high quality beam of directivity nearly equal to that of the conventional Lüneburg lens. In this configuration with a rotationally symmetric isotropic dielectric, there is little difference between TE and TM polarizations (TM polarization plotted in blue).

The situation changes greatly when we look at the Lüneburg lens that was actually fabricated. Here, we have to add in the unintentional anisotropy and minimum dielectric cutoff-value. Ray-tracing and directivity for this lens are shown in Fig. 5(c). We see that the discontinuous dielectric at the lens boundary induces refraction, bending the rays as they exit the front plane. For TE polarization (shown in red on the left half), the resultant distortion is not excessive. The net directivity for the fabricated lens is nearly that of the complete QCTO lens shown in Fig. 5(b). The refraction serves to take the place of the ”missing” section of material gradient - effectively helping to collimate the beam.

TE polarized waves do not see any anisotropy in the dielectric - at least in this 2D cross section. In contrast, when we look at TM polarization (shown in blue on the right half of Fig. 5(c)), significant issues become apparent; refraction at the front of the lens is still an issue. A larger issue, is the discontinuity in dielectric where we have joined our FR4 and HDPE material sections. Because restriction to cylindrical holes does not grant us adequate control over the dielectric tensor, we can only choose to match two of the three (in a non-gyrotropic media) dielectric tensor components at the boundary, and the dielectric constant in the reaming direction is thus discontinuous. This fact is evident upon inspection of Fig. 1(b), and the difference in dielectric constant between TE and TM polarizations. In the ray-tracing, we can observe refraction at the interface in TM polarization, absent in TE. The result of this unintentional internal refraction is a noticeably distorted directivity with prominent side lobes that extend almost ±15°. From an imaging standpoint, the directivity metric underestimates the severity of these aberrations, as they do not distinguish between plus and minus in that ±15°. Looking at the rays, it’s apparent the lens has become partially ”cross-eyed”, with beams that should ideally be paraxial on the right side appearing as errant side-lobes directed to the left.

It is worth also noting that the machinery of TO does not make any consideration for matching the flattened surface to the external world. These ray-tracing results have considered a point-source located exactly on the back - quite dissimilar from the actual experimental setup. Beyond the back plane, any rays from external sources would undergo additional refraction upon entering the lens. In anisotropic media, this refraction also lifts the path degeneracy between TE and TM polarizations, and the added issue of birefringence makes the situation even more complicated. Finding ways to include systems-level design into TO, such as source-impedance-matching of the image plane, is certain to be a significant part of creating real-world useful transformation optical devices.

4. Conclusion

To summarize, we have demonstrated a QCTO-flattened Lüneburg lens with an effective FOV of 30° and directivity of ∼20dBi. Our investigation reveals the performance of our QCTO-flattened metamaterial-fabricated Lüneburg lens suffers relative to a conventional Lüneburg lens: directivity is down 6dB and beam and focusing profiles are notably aberrant. Discussion of factors which cause these problems, such as the material isotropy and dielectric-extrema, is a key first step in finding routes to alleviate the handicaps. We also must bear in mind that the added functionality of conformal geometry presently has no rival. A pertinent example is our experimental evaluation of the PSF of the QCTO Lüneburg lens (Fig. 4). Such a procedure is difficult in a conventional Lüneburg lens, as it requires either a two-axis mechanical rotation stage, or a spherical detector array - which are both well beyond the scope of our investigation. For many applications, the flattened (or any other desired shape) surface enables applications which are practically impossible for spherical Lüneburg lenses.

Thus, the question is not necessarily one of performance metrics in existing applications, but one of creating the best design possible for new applications. Of the material issues mentioned in this work which are detrimental to performance (polarization, anisotropy, unit-cell size, attainable dielectric extrema, etc), there are potential routes to minimize or eliminate most. For instance, a comprehensive design could either avoid geometric anisotropy (using spherical voids, for instance) or alternatively factor the known permittivity anisotropy into the initial design. Similarly, refraction at the edges could be accounted for and the TO prescription modified to maintain imaging performance. More advanced - although not necessarily more expensive - fabrication methods can extend the range of attainable dielectric values.

Use of metamaterials in the implementation of TO leverages rapidly growing fabrication capabilities in metamaterials, as well inviting advanced possibilities such as reconfigurable [27, 28] TO devices. Learning to design around limitations, or use optimization routines to mitigate them, will be a critical push in the advancement of TO metamaterials into real-world applications. It is our belief that a core part of this push has to be honest metric-based evaluation [29] of metamaterial designs and devices as they emerge. However, as is often the case when previously impossible devices become reality, new metrics may also be needed which fairly evaluate metamaterial Transformation Optic designs based on their strengths as well as weaknesses.