## Abstract

Birefringent metamaterial shows fantastic properties in controlling the propagation of electromagnetic wave. Based on the birefringent theory, a Ku-band birefringent metamaterial lens is proposed, which can radiate high gain TE wave with a TE wave feed and deflect incident TM wave. To realize bi-functional high gain radiation or deflection, the required permittivity distribution is analyzed first. Then, the unit cells of metamaterial lens are carefully designed based on effective permittivity equations to achieve the desired permittivity distribution. To demonstrate the present design, a cylindrical birefringent lens prototype is fabricated using 3D printing techniques. The experiments verify that, with a rectangular waveguide feeding on the lens surface or a horn antenna illuminated in the far field, the Ku-band birefringent metamaterial lens performs as a high-gain broadband radiator for the TE wave feed or a deflector for the incident TM wave.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The metamaterial is a kind of artificial material engineered to have fantastic electromagnetic properties that are hardly achieved by natural materials [1,2]. It has played an important role in precisely controlling the propagation characteristics of EM waves, such as negative refraction, high directivity transmission, arbitrary bending, invisible cloaking and absorbing [3–10]. In addition to the single functional metamaterial above, birefringent metamaterial has attracted more and more scientists and engineers recently. This kind of metamaterial possess anisotropic electromagnetic parameters that enable it to manipulate the propagation characteristics of differently-polarized EM waves independently [11]. Such metamaterial devices include bi-functional Luneburg-fisheye lens, planar lens with arbitrary beam deflections, bi-functional focusing/deflection metasurface and so on [12–16]. Nevertheless, the existing bi-functional metamaterial devices are mainly composed of metal-dielectric periodic structures with resonant characteristics. The inherent drawbacks of resonant metamaterials, such as limited bandwidth and complicated fabrication process have limited the applications of bifunctional metamaterial devices.

In this paper, in order to achieve high-gain radiation for TE wave as well as deflection for incident TM wave, an all-dielectric birefringent metamaterial lens is proposed. The desired permittivity distribution of the lens to realize bi-functional high-gain radiation and deflection is analyzed first based on the birefringent theory. To realize the desired permittivity distribution, the design formula of the birefringent lens with metamaterials is presented. The all-dielectric periodic structure is non-resonant that guarantees broad working bandwidth. As a verification of this design, a 124-mm-diameter cylindrical birefringent lens prototype working at Ku band is manufactured using 3D printing techniques. This prototype is fed by a rectangular waveguide to verify its high-gain radiation for TE wave. On the other hand, its deflection performance is tested by illuminating TM wave with a horn antenna in the far field. The experimental results demonstrate that the gain of the lens prototype with feeding waveguide is around 12 dB at Ku band, which is about 4.8 ~6.5 dB higher than the waveguide itself. Also, the E-field distribution near the lens region shows a beam deflection angle *θ* of 19° ~22° for TM wave. Therefore, the proposed birefringent lens can be used to achieve high gain radiation for TE wave with a feed and deflection for incident TM wave in broad band.

## 2. Permittivity distribution analysis for birefringent lens

Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light [17]. For example, when incident upon a birefringent material with optic axis in *z*-direction, the EM waves propagating along the *xy*-plane can be separated to two rays with different paths, one is TM wave (ordinary ray) with E-field polarization perpendicular to *z*-axis and the other is TE wave (extraordinary ray) with E-field parallel to *z*-axis, as shown in Fig. 1. The propagation of TM and TE waves are governed by refractive index *n _{o}* and

*n*(

_{e}*n*≠

_{o}*n*), respectively.

_{e}Assuming the birefringent material is linear and non-magnetic, then its relative permittivity tensor ${\overline{\overline{\epsilon}}}_{r}$ and relative permeability matrix ${\overline{\overline{\mu}}}_{r}$ should take the following forms,

To realize a cylindrical birefringent lens with high gain radiation for TE wave as well as deflection for TM wave incident from arbitrary azimuth directions, the permittivity components *ε _{zz}* and

*ε*=

_{xx}*ε*should be designed for high gain radiation and deflection properties, respectively.

_{yy}The high gain radiation property of the cylindrical lens can be realized by focusing the incident TE wave with a focal length *F*, which means the refractive index *n _{e}*(

*r*) should satisfy the following equation [18],

*R*is the lens radius,

*ρ = n*(

_{e}*r*)

*r*and

*F*≥

*R*. In particular, when the focal length is

*F*=

*R*,

*n*(

_{e}*r*) follows the refractive index of a Luneburg lens [18], in which the permittivity component

*ε*is preferred to be

_{zz}Meanwhile, to deflect the incident TM wave with a certain angle *θ*, the refractive index *n _{o}*(

*r*) of a cylindrical lens should satisfy the following equation [19],

*ε*=

_{xx}*ε*of the birefringent lens are preferred to be,whose distribution will depend on the deflection angle

_{yy}*θ*.

For example, to focus an incident TE wave with *F* = *R* while deflect an incident TM wave with *θ* = 30°, the permittivity distribution of the birefringent lens can be calculated and shown in Fig. 2.

Figure 3 shows the simulated E-field distributions with TE and TM plane waves illuminating the cylindrical birefringent lens at 15 GHz. It can be seen that the incident TE wave can be focused on the opposite side of the lens’ surface. Meanwhile, the TM wave incident on the right part of the lens can be deflected with *θ* = 30°. Due to the circularly symmetric permittivity distribution, the cylindrical birefringent lens can perform the same function for TE or TM wave incident from arbitrary azimuth directions.

## 3. Design principle of the birefringent lens with metamaterials

Based on the anisotropic permittivity formulas given above, an appropriate metamaterial periodic dielectric structure has been proposed in this section to realize the required anisotropic permittivity distribution for the birefringent lens.

A kind of air and dielectric filled cubic unit cell has been employed to form the periodic structure in the present design, as shown in Figs. 4(c) and 4(f). Each cubic unit cell with the dimension of *a* × *a* × *a* is composed of three intersect dielectric cuboids with the long edges in *x*-, *y*- and *z*-directions, respectively. And the rest region of the unit cell is filled with air. The dimension of the *z*-directional cuboid is *w*_{1} × *w*_{1} × *a*, while the *x*- and *y*-directional cuboids have a dimension of *w*_{2} × *w*_{2} × *a*. With this unit cell structure, the air/dielectric filling ratio in *z*-direction is different from that in *x*- and *y*-directions, which means the anisotropic effective permittivity *ε _{xx}* =

*ε*≠

_{yy}*ε*can be achieved.

_{zz}The anisotropic effective permittivity of the proposed periodic structure can be analyzed based on the effective medium theory [20]. Here the relative permittivities of air and dielectric are denoted as *ε _{a}* and

*ε*. According to the sizes of

_{d}*w*

_{1}and

*w*

_{2}, the proposed unit cell can be divided into 4 types of sub-structures I ~IV, whose permittivity tensors are ${\overline{\overline{\epsilon}}}_{1}$ ~${\overline{\overline{\epsilon}}}_{4}$, as shown in Figs. 4(a) and 4(c). Sub-structure I is filled with air. Sub-structures II and III are anisotropic sandwiched structures. If

*w*

_{1}≤

*w*

_{2}, sub-structure IV is an isotropic dielectric cuboid with the dimension of

*w*

_{2}×

*w*

_{2}×

*a*; or else it is a complicated sandwiched structure. Since the anisotropic permittivity of the sandwiched structure have been derived from effective medium theory [21], the permittivity components of sub-structures I ~IV in

*y*- and

*z*-directions are as follows,

These 4 types of sub-structures can form two sandwiched structures V and VI in each unit cell, as shown in Figs. 4(b) and 4(e). Their effective permittivity components in *y*- and *z*-directions can be expressed as,

Then the proposed unit cell can be regarded as a sandwiched structure composed of structure V and VI. In this way, the effective permittivity components of the periodic structure can be calculated as,

Based on the above analysis, the anisotropic permittivity of the proposed periodic structure with different internal sizes has been calculated. The dimension of each unit cell is a = 4mm, which ensures the periodic structure working at Ku band [22]. The value range of *w*_{1} and *w*_{2} is set to 0.7 ~4 mm, which guarantees that the nearby unit cells can be firmly joint together by the dielectric cuboids during the fabrication process. The dielectric material used in the unit cell has the relative permittivity of *ε _{d}* = 2.9 and loss tangent of 0.02. Figures 5(a) and 5(b) shows the anisotropic permittivity components

*ε*,

_{xx}*ε*and

_{yy}*ε*versus different

_{zz}*w*

_{1}and

*w*

_{2}calculated according to Eq. (11), while Figs. 5(c) and 5(d) shows the effective permittivity results obtained with S-parameter retrieval method [23]. It can be seen that the results derived from both methods are close to each other, which demonstrate the above formulas can be used to approximate the anisotropic permittivity components of the prescribed periodic structure.

Furthermore, to evaluate the anisotropy range which can be achieved using the proposed periodic structure, the anisotropic factor Δ*ε* has been defined as,

The variation of Δ*ε* with different *w*_{1} and *w*_{2} is shown in Fig. 6, whose maximum available range is −0.32 ≤ Δ*ε* ≤ 0.34.

According to the available range of anisotropic permittivity and anisotropic factor Δ*ε* with the proposed periodic structure, the unit cell internal sizes and realized permittivity distribution of three birefringent lenses with *F* = *R* and different *θ* are studied. Their realizable permittivity distributions is compared to the ideal permittivity distribution in Figs. 7(a)–7(c), while the corresponding Δ*ε* is shown in Fig. 7(d). It can be seen that with the increasing of *θ*, the ideal permittivity distribution of the birefringent lenses can be precisely realized with the proposed periodic structure at large radius region, but is difficult to be realized at small radius region. Taking this into account, the birefringent lens with an appropriate permittivity distribution (for example, the permittivity distribution for *θ* = 30°) is designed and fabricated with the prescribed periodic structure to realize the high gain radiation for TE wave and deflection for TM wave at Ku band.

## 4. Birefringent lens prototype design and experiment validation

Based on the above analysis, a cylindrical birefringent lens with a diameter of 124mm and a height of 24 mm has been designed, as shown in Fig. 8. The lens model is composed of 16 layers of unit cells along the radius and 6 layers along the axis, with the dimension of 4 × 4 × 4 mm^{3} for each unit cell. The internal sizes *w*_{1} and *w*_{2} of different unit cell layers along the radius and their permittivity distribution have been shown in Fig. 7(b), while the unit cells along the axis have the same dimension. It can be seen that except for the innermost region, the realized permittivity distribution agree well with the ideal permittivity distribution.

To validate the high-gain radiation and deflection performance of the birefringent lens, a prototype of the lens model is fabricated based on 3D printing techniques [24–26], as shown in Fig. 9. The periodic structure of the lens prototype is constructed using Projet 3500 HDMax 3D printer with a maximum solution of 34 × 34 × 16 μm^{3}. The polymer material used for 3D printing is VisiJet M3 Crystal, whose relative permittivity and loss tangent are the same as the dielectric material parameters in the design procedure.

The high gain performance of the proposed lens for TE wave can be demonstrated by feeding the lens with a WR62 waveguide polarized along the z-axis, as shown in Fig. 10. The measured radiation patterns at Ku band are presented in Figs. 11(a)–11(c). Then, the measured gain is compared with that of a single waveguide in Fig. 11(d). It can be seen that the gain of the lens prototype with a waveguide feed is about 4.8 ~6.5 dB higher than that of the single waveguide across Ku band.

On the other hand, the deflecting function of the proposed lens for incident TM wave is also checked. The lens prototype is placed in the far field of a Ku-band horn antenna, which is used to generate TM wave incoming to the lens from one side, as shown in Fig. 12. The E-field distribution on the other side of the lens is tested using a WR62 waveguide. This is similar with the experimental setup for the anisotropic metamaterial device [27]. Figure 13 shows the simulated and measured E-field distribution in the testing region. It can be found that the incident wave is deflected from normal direction with an approximate angle of *θ* = 22°, 20° and 19° at 12.4, 15 and 18 GHz, respectively, which is similar with the HFSS simulation results. Due to the limited layers of unit cells and the approximation of permittivity distribution in the present lens prototype, both the simulated and measured deflection angles are smaller than the ideal deflection angle *θ* = 30°.

## 5. Conclusion

Based on the birefringent theory, a broadband birefringent metamaterial lens with high gain for TE wave and deflection for TM wave is proposed. At first, the desired permittivity distribution to realize bifunctional high gain radiation and deflection is analyzed. Then, the birefringent lens is designed with all-dielectric metamaterials, which guarantees broad working bandwidth. The effective permittivity equations for the present periodic structure are developed to calculate its permittivity distribution, which is close to the results retrieved from simulated S-parameters. In this way, the unit cell internal sizes of an appropriate cylindrical birefringent lens are given to realize the desired permittivity distribution. To validate this design, a birefringent metamaterial lens prototype is fabricated using 3D printing techniques. The broadband high-gain radiation performance for TE wave is verified by feeding the lens prototype with a rectangular waveguide on its cylindrical surface. Further, when the lens prototype is illuminated by a Ku-band horn antenna in the far field region, the E-field distributions near the lens show expected beam deflection for TM wave. Both simulated and measured results indicate that, the birefringent metamaterial lens prototype can exhibit high gain radiation for TE wave as well as prospective deflection for TM wave across Ku band.

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