Examples of qTO-Derived Lenses Inspired by Classical Designs
This section will describe several qTO-derived inhomogeneous metamaterials that are either explicit transformations of classical lenses or inspired by classical ideas. First, broadband wide-angle designs explicitly derived from the Maxwell, Luneburg, and Fresnel lenses will be described. After that, an inhomogeneous multibeam lens antenna and a GRIN anti-reflection coating (ARC) design procedure will be detailed by studying how these designs were done in the past and using qTO to generate the appropriate material profile.
Broadband Wide-Angle Lenses Derived from Refractive Lenses
In 1854, James Clerk Maxwell analytically determined the refractive index of a spherical lens  such that any point source on the spherical surface will focus to the opposite point on the sphere, as shown in Fig. 6.5a. The refractive index distribution of this so-called Maxwell lens for a sphere of radius R is given by:
As can be seen in Fig. 6.5b, a point source radiating spherical waves on one surface transforms into a plane wave in the center of the lens. This fact can be exploited by using only half of the Maxwell fish-eye (commonly referred to as an HMFE) to create highly directive lens antennas, as first discovered by Fuchs et al.  and shown in Fig. 6.5c. Fuchs et al. physically realized the HMFE through a series of concentric homogeneous dielectric shells and showed very good agreement for the co-polarized E- and H-plane radiated far-field patterns at a single frequency. The spherical surfaces make it difficult to fabricate, but qTO can be used to flatten the lens. To illustrate the process, the analytical study by Aghanejad et al.  will be briefly described. As shown in Fig. 6.6b, a direct implementation of numerically solving Laplace’s equations to flatten the HMFE leads to refractive indices less than 1, requiring metamaterials that are typically highly resonant and lossy (we will call this design the dispersive design). To overcome this, the regions with refractive indices less than 1 are simply replaced by 1, as shown in Fig. 6.6b (this is referred to as the non-dispersive design). Figure 6.7 shows the fields from a COMSOL MultiPhysics simulation for the dispersive design, the non-dispersive design, and a benchmark horn antenna, all operating at 18 GHz. Figure 6.8 compares the far-field patterns. As can be seen, both the dispersive and non-dispersive designs are highly directional. The non- dispersive design has higher sidelobes but maintains approximately the same gain and half-power beamwidth as the dispersive design. To illustrate the broadband performance of this proposed design, the variations in gains, halfpower beamwidths, and over a frequency range of 15-21 GHz are shown in Fig. 6.9. As can be seen, the gain monotonically increases while the half-power beamwidth stays constant and 511 remains below -9 dB over the frequency range of interest. This design shows the theoretical performance of a broadband, directive, qTO-derived flat lens. The device was not manufactured but could be physically realized through non-resonant metamaterials or dielectric rods with spatially varying radii .
Figure 6.5 Maxwell's fish-eye lens. (a) Ray traces. Image by Catslash from Wikimedia Commons. Used under CC-BY-SA 3.0. https://upload.wikimedia. org/wikipedia/commons/d/d5/Maxwellsfish-eyelens.svg (b) Field distribution. Figure (b) reprinted with permission from Ref. 1, Copyright 2012, IEEE. (c) Half Maxwell's fish-eye.
Figure 6.6 Flattening the half Maxwell's fish-eye (a) original HMFE, (b) flattened dispersive lens, (c) flattened non-dispersive lens. Reprinted with permission from Ref. 1, Copyright 2012, IEEE.
Figure 6.7 COMSOL field distribution comparison for the (a) dispersive, (b) non-dispersive, and (c) horn designs. Reprinted with permission from Ref. 1, Copyright 2012, IEEE.
Figure 6.8 Far-field pattern comparisons for the non-dispersive, dispersive, and horn designs. Reprinted with permission from Ref. 1, Copyright 2012, IEEE.
Figure 6.9 Broadband performance of the half Maxwell's fish-eye, (a) gain versus frequency, (b) half-power beamwidth versus frequency, and (c) 5n versus frequency. Reprinted with permission from Ref. 1, Copyright 2012, IEEE.
Another classic lens design was discovered by Rudolf Luneburg in 1944. The refractive index profile of this spherical lens was designed such that an incident plane wave from any direction will focus to a point on the surface of the lens , as shown in Fig. 6.10.
Figure 6.10 Luneburg lens. Image by Catslash from Wikimedia Commons. Used under CC-BY-SA 3.0. https://upload.wikimedia.org/wikipedia/ commons/2/2f/Luneburg_lens.svg.
The refractive index for this so-called Luneburg lens of radius R is given by:
Despite its incredible imaging properties, the Luneburg lens is not commonly used in optical applications since it requires a detector that conforms to the surface of the sphere, while most detectors in practice are planar. In 2008, D. Schurig proposed an analytical transformation that flattens the Luneburg lens . The mathematical description of the transformation is given as follows:
A resulting ray trace is depicted in Fig. 6.11a. The required permittivity tensor is anisotropic and includes offdiagonal terms. This tensor can be expressed in an orthogonal diagonalizing basis by simply rotating the coordinate system, and the resulting diagonal permittivity tensor (which is the same as the permeability tensor) is:
As expected, the final material properties are diagonalized, but still anisotropic. The resulting anisotropic permittivity distribution is provided in Fig. 6.11b.
Such a device could only be physically realized with anisotropic metamaterials. In 2009, Kundtz and Smith used qTO to derive a GRIN lens that can be implemented with simple dielectrics . The numerical transformation performed is shown in Fig. 6.12 for both the virtual space (a) and physical space (b). The dashed yellow line
Figure 6.11 Flattening of Luneburg lens. (a) Coordinate transformation, (b) permittivity distribution. Reprinted from Ref. 52, Copyright 2008, IOP Publishing under Creative Commons Attribution license.
represents the focal surface, which gets mapped from a spherical surface to a flat surface via slipping Neumann boundary conditions. The red line represents the front surface of the flat lens; Dirichlet boundary conditions are employed to ensure the final design does not have reflections. The choice of Y = 1.4 for the upper boundary of the lens is relatively arbitrary, but it needs to be far enough away from the yellow line to ensure the conformal module is approximately one over the entire map, thereby minimizing anisotropy. The resulting refractive index distribution is shown in Fig. 6.13a. In the physical realization of this design, refractive indices less than 1 are set to 1.08, resulting in an index range of 1.08-4.0. Because this range is so large, the device was divided into two regions, as shown in Fig. 6.13b, a center region where n < 2 and an outer region where n > 2. In the center region, the distribution was broken up into small squares with a constant bulk effective permittivity. In each square region, the permittivity is realized by a metallic I-beam lithographically patterned on an FR4 substrate, as shown in Fig. 6.13d . The dimensions required for a given permittivity at a certain point were found through a standard electromagnetic parameter retrieval method . Over the frequency range of interest (7-15 GHz), I-beam metamaterials with refractive indices greater than two would be very large and yield significant spatial dispersion. Therefore, those regions where n > 2 are physically realized by patterning copper strips on an FR4 substrate, as shown in Fig. 6.13c. The resulting index changes over the frequency range by up to 10% in the outer regions and up to 3% in the inner region. To measure the device, a dielectric waveguide was used to approximate a point source at various points of the focal surface. This should result in plane waves directed at different angles. Figure 6.14 shows the experimental results for a variety of angles and frequencies, showcasing the broadband performance of the device.
Figure 6.12 qTO solution for the flattened Luneburg lens in (a) virtual space and (b) physical space. Reprinted by permission from Macmillan Publishers Ltd: Nature Publishing Group, Ref. 26, Copyright 2010.
Figure 6.13 Flattened Luneburg (a) refractive index distribution; (b) physical device showing two regions, a center region where n > 2 and two outer regions where n < 2; (c) unit cell used to realize the index distribution where n > 2; (d) unit cell used to realize the index distribution where n < 2. Reprinted by permission from Macmillan Publishers Ltd: Nature Publishing Group, Ref. 26, Copyright 2010.
Figure 6.14 Experimental results for flattened Luneburg lens. Results are shown at 10 GHz for beams directed at incident angles of (a) 0°, (b) 35°, (c) 50°, and (d) off-axis at 50°. The broadband performance of the device is showcased by including off-axis 50° results at (e) 7 GHz and (f ) 15 GHz. Reprinted by permission from Macmillan Publishers Ltd: Nature Publishing Group, Ref. 26, Copyright 2010.
While these results are extremely impressive at microwave frequencies, they likely would not scale well to optical frequencies where metals begin to behave like lossy dielectrics and less like conductors. For better scalability, an all-dielectric implementation would be preferred. This was done by Hunt et al. in 2011 by drilling sub-wavelength holes in a dielectric slab where the density of holes dictates the desired refractive index at a given location . The same qTO transformation as  was employed, and regions where n < 1 were approximated as 1. The resulting refractive index profile of the inhomogeneous metamaterial is given in Fig. 6.15a, the density of holes in Fig. 6.15b, and the final manufactured design is shown in Fig. 6.15c. The device theoretically has the potential to scale well to optical frequencies, but it was constructed at microwave frequencies for ease of manufacturing and testing. A comparison between simulated results in COMSOL and measured results for a beam directed to 30° at 10 GHz is shown in Fig. 6.16. Similar to the previous designs, a point source at different positions of the flattened focal surface will yield perfectly planar wavefronts at different angles on planes containing the optical axis. On planes not containing the optical axis, there will be some distortion. To illustrate the FOV of the lens, ray traces with 33 rays were simulated through both the complete lens where no approximations have been made to the qTO-derived refractive index, and the approximated lens where n < 1 regions are replaced by n = 1. As can be seen in Fig. 6.17, the two designs agree very closely for on-axis and an incident angle of 10. At 20° and 30°, some of the rays travel through the n < 1 region, leading to a few stray rays. Still, the majority of the rays all focus to an extremely tight spot. Significant aberrations begin to appear at angles greater than 40°. This demonstrates that the exact qTO- derived index has an FOV of ±40°, while the approximated dielectric- only design has an FOV of ±30°. These results show the tremendous potential of obtaining manufacturable broadband lenses with a wide FOV by applying qTO to classic refractive lens designs such as the Maxwell and Luneburg lenses.
Figure 6.15 Dielectric-only flattened Luneburg lens. (a) Refractive index profile, (b) distribution of holes, and (c) manufactured design. Reprinted from Ref. 21, Copyright 2011, under Creative Commons Attribution license.
Figure 6.16 Results for dielectric-only flattened Luneburg lens: (a) simulation and (b) measurement. Reprinted from Ref. 21, Copyright 2011, under Creative Commons Attribution license.
Figure 6.17 Spot diagram comparison between complete and approximated (where n < 1 regions are approximated by n = 1) at different incident angles. Reprinted from Ref. 21, Copyright 2011, under Creative Commons Attribution license.