A) Theoretical properties of Mikaelian lenses

The Mikaelian lens is a cylindrical lens of thickness T and radius R (see Fig. 2(a)). Its refractive index varies with the radial distance r according to the following equation:

(1)$$n\lpar r\rpar =\displaystyle{{n\lpar 0\rpar } \over {\cosh \left({\displaystyle{\pi \over {2T}}r} \right)}}\quad {\rm with}\quad 0 \leq r \leq R\comma \;$$

where n(0) is the refractive index along the cylinder axis. This lens is also called “hyperbolic cosine lens” in [Reference Lo and Lee9] because of its refractive index distribution or “constant thickness Luneburg lens” in [Reference Jasik10], since the thickness of the lens (the height of the cylinder) is fixed contrary to the classical cylindrical Luneburg lens used for two-dimensional (2D) focusing.

Fig. 2.

Geometrical optics ray tracing inside (a) a Mikaelian lens and (b) a generalized Mikaelian lens with the associated notations.

It is important to note that the ratio T/R has a direct impact on the refractive index dynamic. As plotted in Fig. 4, the thicker the lens with respect to its radius, the smaller the refractive index (and therefore permittivity) variation. Indeed, the ray path length differences are then less important. For a thin lens (T = R), the permittivity inside the dielectric varies from a factor 1 to more than 5, whereas for a thick lens (T = 5R), it goes only from 1 to 1.11.

Fig. 3.

Relative permittivity distribution of a dielectric Mikaelian lens as a function of the normalized radius: influence of the ratio thickness over radius of the lens T = R.

Fig. 4.

Near-electric field mappings computed by CST Microwave Studio of a plane wave impinging on (a) a Mikaelian lens and (b) a generalized Mikaelian lens. The white square shows the location of the lens.

Regarding the properties of the lens, geometrical optics predicts that the Mikaelian lens transforms a point source into a beam of parallel rays as illustrated in Fig. 1(a). The near-electric field mapping of Fig. 3(a) shows that a plane wave impinging on one side of the cylinder is focused on a point.

The behavior of the lens can be adjusted by modifying the refraction index distribution. To have a focal point at a given distance F from the lens, the law of the so-called generalized Mikaelian (given in [Reference Mikaelian5]) is:

(2)$$n\lpar r\rpar =\displaystyle{{n\lpar 0\rpar T - F\left({\displaystyle{1 \over {\cos \gamma }} - 1} \right)} \over {T\cosh \left({\displaystyle{\pi \over {2T}}\lpar r - {r_1}\rpar } \right)}}\comma \;$$

where

$$\displaystyle{\pi \over {2T}}\lpar r - {r_1}\rpar ={\tanh ^{ - 1}}\left({\displaystyle{{T\sin \gamma } \over {n\lpar 0\rpar \, T - F\left({\displaystyle{1 \over {\cos \gamma }} - 1} \right)}}} \right).$$

The angle γ and the radial distance r ₁ are defined in Fig. 2(a). The near-electric field mapping of a plane wave impinging on an ideal generalized Mikaelian lens with F = R = T/2 in Fig. 3(b) illustrates this focusing property. Note that the refraction index distribution (2) is very close to the one of the Kelleher lens [Reference Goatley and Parker11].

B) Gradient index manufacturing

Traditional gradient index lenses have been fabricated using homogeneous dielectric shells to approximate by steps the continuous refractive index distribution [Reference Peeler and Coleman12]. In [Reference Peeler and Archer13], the dielectric distribution of a planar Luneburg lens is achieved by varying the thickness of a dielectric between parallel plate waveguide working in the TE₁₀ mode. The desired permittivity profile of Luneburg lenses can also be obtained by controlling the hole density in a dielectric as done with Teflon in [Reference Rondineau, Himdi and Sorieux14]. All these techniques are expensive in terms of fabrication time and money.

More recently, printed planar Luneburg lenses have been proposed. The refractive index variation is performed by etching holes on a printed circuit board in [Reference Xue and Fusco15]. In [Reference Pfeiffer and Grbic16], the refractive index of the Luneburg lens is controlled through a combination of meandering crossed microstrip lines and varying their widths.

Finally, a planar Luneburg lens has been realized using a variable printed metasurface in [Reference Bosiljevac, Casaletti, Caminita, Sipus and Maci17]. The desired refractive index is obtained by modulating the surface impedance inside a parallel-plate structure.

Recently, an ingenious technological process has been developed to control the permittivity of foam materials [Reference Merlet, Le Bars, Lafond and Himdi6, Reference Bor, Lafond, Merlet, Le Bars and Himdi7]. Foam is composed of a core material into which gas is injected the ratio between the core material and the gas, i.e. by pressing more or less the foam; it is possible to control up to a certain extent the permittivity of the foam. This permittivity can in principle range from the initial foam permittivity to the one of the core material composing the foam. More details on the pressing process and the relation between the foam density and its permittivity are given in [Reference Bor, Lafond, Merlet, Le Bars and Himdi7]. This process is a cheap way to easily manufacture very lightweight gradient index lenses. It has already been successfully implemented for Luneburg lenses [Reference Bor, Lafond, Merlet, Le Bars and Himdi8] and is here applied to Mikaelian lenses.

A) Focusing performances

The directivity and aperture efficiency of ideal Mikaelian and Luneburg lenses are compared for various lens diameters in Fig. 5. Let us recall that the aperture efficiency is the ratio between the directivity of the lens antenna and the one of a constant field circular aperture of the same diameter. Both lenses are composed of 20 homogeneous dielectric shells to approximate the ideal continuous gradient index. The Mikaelian lens has a central relative permittivity ε _r(0) of 2 and a thickness T equal to 2R. The lenses are fed by a circular open-ended waveguide of diameter 2.85 mm whose directivity alone is 8.1 dB. These radiation characteristics show that, roughly speaking, both lenses behave similarly. It seems that the Luneburg lens antenna performs better than the Mikaelian lens for large diameters (above 20 λ), whereas the opposite happens for small diameter (below 15 λ). However, caution must be taken when interpreting these results since the directivity and consequently the aperture efficiency are relatively sensitive to the position of the antenna with respect to the lens.

Fig. 5.

Directivity and aperture efficiency as a function of the lens diameter for the ideal Mikaelian (solid line) and Luneburg (dashed line) lens fed by a circular waveguide.

B) Experimental results

A foam-based planar Mikaelian lens of thickness T = 2R = 56 mm and height 3 mm has been manufactured. The lens diameter (or rather width for the planar version) is equal to 11.2 λ at 60 GHz. The lens is made from the foam material Airex PXc245 [18]. Without being pressed the properties of the foam are at 60 GHz: ε _r = 1.31 and tan δ = 0.008. The permittivity law of the Mikaelian lens is approximated by 10 values given in Table 1 which correspond to the 10 heights before pressing shown in Fig. 6(a). The lens is fed by a WR15 open-ended waveguide operating in the V-band (see Fig. 6(b)).

Fig. 6.

Pictures of the Mikaelian lens: (a) before being pressed and (b) pressed inside its mechanical support, including the feeding waveguide. (c) Full-lens antenna system.

Table 1.

Characteristics of the manufactured Mikaelian lens

The measured and simulated radiation patterns of the Mikaelian lens antenna at 60 GHz are plotted in Figs 7(a) and (b). A fairly good agreement is obtained between simulation and measurements for both E- and H-planes. In the H-plane, the measured half-power beamwidth is equal to 4.4°, sidelobes are below −19 dB and the cross-polarization level is below – 25 dB. Note that the asymmetry in the E-plane is due to the non-symmetry of the mechanical support. The gain patterns in the H-plane are plotted as a function of the frequency in Fig. 7(c). They confirm the expected broadband behavior of the lens antenna and show that the Mikaelian permittivity law is properly respected. The measured gain and loss efficiency (gain over directivity) are plotted as a function of the frequency in Fig. 8. The gain increases from about 14 dB at 57 GHz to 15 dB at 66 GHz, whereas the loss efficiency is between 35 and 39%. These low values of the loss efficiency are due to the relatively high losses of the foam especially when pressed (see the loss tangent in Table 1).

Fig. 7.

Radiation performances of the Mikaelian lens antenna. Normalized far-field pattern at 60 GHz in the (a) H- and (b) E-planes: measurements (solid line), simulation (dashed line). (c) Measured gain as a function of the frequency.

Fig. 8.

Measured gain and loss efficiency of the Mikaelian lens antenna as a function of the frequency.