## Abstract

Luneburg lens with flat focal surface has been developed to work together with planar antenna feeds for beam steering applications. According to our analysis of the conventional flattened Luneburg lens, it cannot accommodate enough feeding elements which can cover its whole scan range with half power beamwidths (HPBWs). In this paper, a novel Luneburg lens with extended flat focal surface is proposed based on the theory of Quasi-Conformal Transformation Optics (QCTO), with its beam steering features reserved. To demonstrate this design, a three-dimensional (3D) prototype of this novel extend-flattened Luneburg lens working at Ku band is fabricated based on 3D printing techniques, whose flat focal surface is attached to a 9-element microstrip antenna array to achieve different scan angles. Our measured results show that, with different antenna elements being fed, the HPBWs can cover the whole scan range.

© 2016 Optical Society of America

## 1. Introduction

Covering the whole scan range with HPBWs is the basic requirement for a beam scanning array [1]. To realize multi-direction beams or beam controlling, phase-shift networks are needed for a conventional array, which are always complex to design and expensive to fabricate. Compared with conventional beam scanning array, Luneburg lens, being an attractive gradient index lens, can focus plane wave incident from arbitrary direction to generate/receive high gain and multi-direction beams without any phase-shift networks [2–7]. Thus Luneburg lens or any similar lens has great potential in the applications of beam scanning. Recently, flattened Luneburg lens has been developed with QCTO procedures and applied in microwave and millimeter wave bands [8–18]. So far, due to the limited size of Luneburg lens’ flat focal surface, in most developed flattened Luneburg lens beam steering systems, a single antenna is used as the source and moved along the flat focal surface to realize HPBW-covered beam scanning [6–16]. So the beam scanning speed is mainly determined by the moving velocity of the single source, which limits its applications.

In the present work, firstly, we analyze the planar feeding elements capacity of the conventional flattened Luneburg lens, which indicates that the finite size of its flat focal surface is insufficient to accommodate enough feeding elements for HPBW-covered electronic scan. Then, we propose a novel extend-flattened Luneburg lens based on QCTO method, whose flat focal surface is enlarged to satisfy HPBW-covered scanning without deteriorating its beam steering features dramatically. Also, with this focal plane extension, the maximum permittivity of the proposed lens is lower than conventional flattened design, which makes it easy to be fabricated with 3D-printed materials. As a verification of this design, a 148-mm-diameter extend-flattened Luneburg lens prototype with 4 × 4 × 4 mm^{3} unit cell size is manufactured using 3D printing techniques. By controlling the polymer/air filling ratio of each unit cell, the required permittivity ranging from 1.1 to 2.26 has been achieved. This prototype is then fed by a 9-element planar microstrip antenna array working at 15.4 GHz to test its beam scanning performance. The experimental results demonstrate ± 42° HPBW-covered scanning capability of the proposed lens with different antenna elements being fed, which means the proposed lens can provide a simple and low-cost approach for electronic scan applications.

## 2. Analysis of the conventional flattened Luneburg lens

For the two-dimensional (2D) circular Luneburg lens, its permittivity and permeability distribution can be expressed as [2]:

Where*R*is the lens radius. In order to realize beam steering with planar antenna sources, flattened Luneburg lenses have been developed with QCTO procedures [9–13]. In this conventional flattening design, only a part of the circular Luneburg lens with an open angle

*θ*(0 <

*θ*<

*π*) is directly compressed to the flat focal plane of the transformed Luneburg lens without any further deformation, as shown in Fig. 1. Their permittivity and permeability transformation can be performed based on Transformation Optics (TO) formulas [9]:

**is the Jacobian matrix for spatial transformation. For plane wave (TEM) or transverse-electric (TE) wave with vertical polarization to the 2D flattened lens, isotropic and dielectric-only material distribution is enough to realize the propagating features of the lens [19]:**

*A**S*is the unit cell area in the original circular lens domain and Δ

*S*′ is the transformed unit cell area in the flattened lens domain. Since the lens is a bulk effect device while the flattened Luneburg lens can preserve the beam steering features of the original lens, the aperture size

*D*

_{2}and beam scanning range

*θ*

_{scanning}of the flattened Luneburg lens can be approximated as Eq. (4), and the focal plane size

*L*(

*ε*

_{r}≥ 1 region) can be obtained based on the permittivity distribution shown in Fig. 1.

With this 2D QCTO based design, a 3D flattened Luneburg lens can be obtained by rotating the 2D flattened lens symmetrically. While it is fed by a source antenna on the focal plane, its HPBW can be estimated as [1]:

Where*λ*is the wavelength in free space. To achieve electronic scan, a planar antenna array can be attached to the focal plane as the source with only one element fed at each scan step instead of moving a single source antenna along the focal plane, as shown in Fig. 2.

In order to cover the whole scan range efficiently with the above structure, the scanning beam granularity Δ*φ* between two adjacent beams, radiated by two adjacent antennas, should be approximately HPBW or smaller. Thus the total number of feeding elements *N* should satisfy

*N*is limited by the size of focal plane

*L*and the distance

*d*between adjacent Combining Eq. (5)-(7), we can obtain that, to achieve HPBW-covered scan, the distance

*d*

Figure 3 shows the values of *θ*_{scanning}/*π*, *L*/*D*_{2} of the 2D flattened lens obtained based on its permittivity distribution and the maximum values of *d*/*λ* for HPBW-covering calculated using Eq. (8) for various open angle *θ*. It can be seen that with *θ* increasing from 0 to *π*, the maximum *d* for HPBW-covered scanning varies from 0.55*λ* to 0.26*λ*.

Generally, the dimensions of common planar feeding elements, such as microstrip antennas or waveguides, are approximately *λ*_{g}/2 (*λ*_{g} is the guide wavelength) [1]. Although using high permittivity materials as substrates for microstrip antennas or as fillers for waveguides may reduce the size of feeding elements, enough separation between adjacent elements is still required to minimize the mutual coupling. According to our simulation shown in Fig. 4, for a typical planar microstrip antennas array on the focal plane of a flattened Luneburg lens, the distance *d* with coupling coefficients less than −15dB between adjacent elements is more than 0.4*λ*, which is not small enough to meet HPBW-covered scanning once the open angle *θ* ≥ 90°. Although printed dipole antennas can be arranged with much smaller *d* in one dimension, they cannot guarantee HPBW coverage in 2D scanning. Thus for the conventional flattened Luneburg lens with open angle *θ* ≥ 90°, its focal plane size has to be enlarged to accommodate enough feeding elements for HPBW-covered scanning.

## 3. Design of the extend-flattened Luneburg lens

To meet HPBW-covered scanning requirement, we propose a modified flattened Luneburg lens with an extended focal plane based on QCTO method. Figure 5 shows the spatial transformation from conventional 2D flattened Luneburg lens to the extend-flattened lens, with their outlines shown in black lines. The lower part of the conventional flattened lens is enlarged to achieve an extended focal plane. Neumann-Dirichlet boundary (shown in red lines) conditions are applied to this part to relax the grids in the quasi-conformal procedure. Meanwhile, Dirichlet boundary (shown in blue lines) conditions are used to truncate the upper part of transformation domains [9]. To guarantee the beam steering performance, the truncation cannot be close to the lens. The final conformal grids are obtained by iteration algorithm [20] and visualized graphically in Fig. 5.

Here the transformation based on the conventional flattened lens with aperture size *D*_{2} = 2.2 and open angle *θ* = 120° is illustrated as an example. In order to produce a smoother permittivity distribution after the transformation, Neumann-Dirichlet boundaries (red lines) have been applied at a small distance to the left/right side outlines of the lens (black lines). The Neumann-Dirichlet boundaries for the conventional lens domain are:

The lines *PQ* and *P*′*Q*′ shown in Fig. 5 represent the focal planes of conventional lens and the extended lens, and their lengths are 1.24 and 2.04, respectively. By the above transformation, the focal plane size is extended about 64.5%. Meanwhile, the aperture sizes of both lenses are almost the same.

The relative permittivity distribution in the extended lens domain are calculated using Eq. (3) and compared with that in the conventional lens domain, as shown in Fig. 6. Since the unit cell areas Δ*S*′ around the focal plane region in the conventional lens domain are larger than the corresponding cell areas Δ*S* in the extended lens domain, the permittivity around the focal plane region decrease after the present transformation. The relative permittivity of the extend-flattened Luneburg lens after truncation on its boundary (shown in black lines) ranges from 0.43 to 2.26.

To verify that the novel extend-flattened Luneburg lens preserves the beam steering features as the conventional flattened lens, both are integrated with a vertical polarized source on the focal plane to form radiators, respectively. The aperture sizes of both 2D lens models are 148 mm, while the focal plane size is 84 mm for the conventional lens and 136 mm for the extended lens. A line current, serving as a feeding source, is perpendicular to the 2D lens and moving along the focal plane of respective lens to check the beam scanning features. Figure 7 shows the E-field distributions of the two radiators with the same feeding amplitude at 15GHz simulated using COMSOL Multiphysics. Then the 3D models of both lenses are generated by rotating the 2D lens models symmetrically. Both 3D lens models are fed by a WR-62 rectangular waveguide moving along the focal plane. The radiation patterns and directivities of both lens models for different scan angles at 15GHz are also simulated and shown in Fig. 8. From the simulation results we clearly observe that both lenses possess almost the same beam scanning angles. Meanwhile, the extended lens exhibits higher radiation directivity and slightly narrower beamwidth compared to the conventional lens. The physical and performance parameters of both lenses are listed in Table 1. According to Eq. (8), the above transformation has successfully loosened the HPBW-covered scanning requirement from *d* ≤ 0.33*λ* (conventional flattened Luneburg lens) to *d* ≤ 0.54*λ* (present extend-flattened Luneburg lens), which consequently make it possible to place enough feeding elements on the focal plane for HPBW-covered scanning.

## 4. Lens realization and electronic scan application

The 3D extend-flattened Luneburg lens can be generated by rotating above 2D extend-flattened Luneburg lens along its symmetric axis. Metamaterials can be employed to realize the prescribed permittivity distribution ranging from 0.43 to 2.26 [21, 22]. For simplicity, we have only fabricated the 1 ≤ *ε*_{r} ≤ 2.26 part of the 3D extend-flattened Luneburg lens with periodic polymer structure, which can be easily realized utilizing advanced 3D printing technique [23, 24]. Each unit cell of the periodic polymer structure consists of a polymer cube and six polymer connecting rods. The inset in Fig. 9 shows the cubic unit cell, whose dimension is set to 4 × 4 × 4 mm^{3} to guarantee the lens working at Ku-band (12-18 GHz). The polymer material used here is VisiJet M3 Crystal, whose relative permittivity is about 2.9 and lost tangent is about 0.04 at Ku-band. According to the effective medium theory [25], the effective permittivity *ε*_{eff} of each unit cell is determined by its polymer filling ratio. The width *w* of polymer connecting rods is 0.7 mm, and the side-length *b* of the polymer cube varies from 0.7 to 3.7 mm to guarantee the required permittivity distribution. Figure 9 also gives the relationship between the effective permittivity *ε*_{eff} and the side-length *b* calculated with S-parameter retrieval method [26].

The 3D extend-flattened Luneburg lens prototype is fabricated using Projet 3500 HDMax 3D printer with a maximum solution of 34 × 34 × 16 μm^{3}. Figure 10(a) shows the manufactured lens whose aperture diameter *D* is 148 mm and focal plane diameter is 100 mm. To check the HPBW-covered scanning feature, a 9-element microstrip antenna linear array is attached on the focal plane. The separation distance *d* between the centers of adjacent elements is 10 mm (0.51*λ* at 15.4 GHz), as shown in Fig. 10(b). The substrate of the microstrip antenna array is F4B-2 material with *ε*_{r} = 2.65. Because of the effective permittivity variation around the focal plane region of the lens, the dimensions of antenna elements are slightly different (ranging from 5.08 mm to 5.28 mm) to make sure they can work at the same frequency.

Due to the symmetry of the lens and array structure, only elements 1 to 5 [Fig. 10(b)] are fed in turns and tested to verify its scanning performance. The return losses of elements 1 to 5 and coupling coefficients between adjacent elements are measured using Vector Network Analyzer (Rohde& Schwarz EVM) and plotted in Fig. 11. It can be seen that all these elements can work at 15.1-15.7 GHz with return losses less than −10 dB and coupling coefficients less than −22dB. The radiation patterns at 15.4 GHz with only one element fed at each scan step are shown in Fig. 12, which indicate that HPBW-covered scanning within ± 42° range can be achieved once all the 9 elements are fed in turns.

## 5. Conclusion

Based on the analysis of the conventional flattened Luneburg lens, we firstly demonstrate that its flat focal surface is insufficient to accommodate enough feeding elements for HPBW-covered electronic scan. Then, a novel extend-flattened Luneburg lens is proposed with QCTO procedures. Compared to the conventional flattened Luneburg lens, the extended focal plane of the present lens can accommodate enough feeding elements for HPBW-covered scanning. Finally, a 3D lens prototype of the present design working at Ku band is fabricated using 3D printing techniques as an example, while a 9-element microstrip antenna linear array is attached to its focal plane as the source. With different antenna elements being fed in turns, ± 42° HPBW-covered scanning can be achieved. Moreover, the present lens can be used to realize 2D HPBW-covered electronic scan while integrating with a 2D feeding network, which means the proposed lens can provide a simple and low-cost approach for electronic scan applications.

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