APPLIED PHYSICS LETTERS 104, 094101 (2014)
All-dielectric three-dimensional broadband Eaton lens with large refractive index range Ming Yin, Xiao Yong Tian,a) Ling Ling Wu, and Di Chen Li State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China
(Received 23 December 2013; accepted 21 February 2014; published online 4 March 2014) We proposed a method to realize three-dimensional (3D) gradient index (GRIN) devices requiring large refractive index (RI) range with broadband performance. By combining non-resonant GRIN woodpile photonic crystals structure in the metamaterial regime with a compound liquid medium, a wide RI range (1–6.32) was fulfilled flexibly. As a proof-of-principle for the low-loss and nondispersive method, a 3D Eaton lens was designed and fabricated based on 3D printing process. Full-wave simulation and experiment validated its omnidirectional wave bending effects C 2014 AIP Publishing LLC. in a broad bandwidth covering Ku band (12 GHz–18 GHz). V [http://dx.doi.org/10.1063/1.4867704] GRIN medium plays an important role in devices with gradient material property distribution profile designed for controlling electromagnetic (EM) waves. Lenses based on GRIN medium, such as Maxwell fisheye lens,1 Luneburg lens,2 and Eaton lens,3 are typical GRIN devices. They have important engineering applications in imaging, radar, and antenna. Recent development of transformation optics4,5 and metamaterials lights a new sight on these GRIN lenses. For instance, transmutation of singularities in the RI profile of Eaton lens facilitates its implementation;6,7 compression of the RI profile of a lens reduces its volume;8–10 flattened Luneburg lens with a flat focal locus makes it matched to conventional planar antenna arrays.11–13 However, practical applications of these devices are still restricted due to challenges in the implementation of three-dimensional gradient index (3D GRIN) medium with large RI range and broadband property. The use of resonant metamaterials will result in loss and bandwidth limitation.7 Non-resonant metamaterials exhibit a relatively broad operational bandwidth, yet they are still difficult to be extended to 3D.11,14 Drilling subwavelength holes in stacked dielectric plates12,15–17 is a way to realize 3D GRIN materials. However, the RI range available is relatively small (1.08–2.10), which is mainly restricted by the dielectric materials matrix; Moreover, the inherent 2D cylindrical symmetry of the holes potentially harms the 3D isotropic property. An all dielectric way based on titanates dispersed into polymeric material9 can also be used to realize 3D GRIN. But the RI range realized by this method is still limited (1.70–3.80), especially when impedance match to air is needed; meanwhile the complex implementation process is not facile to adjust the RI to the exact target values. In this paper, we propose a method to realize a broadband 3D GRIN lens with a large variation of RI. The woodpile photonic crystals (PCs) structure in metamaterial regime and a compound liquid medium were utilized as 3D GRIN medium to achieve a wide range of refractive index (RI) a)
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(1–6.32). The complex 3D structure involved was fabricated by stereolithography (SL)-based 3D printing process. To demonstrate the feasibility of the proposed method, a 3D Eaton lens was designed and fabricated. By full-wave simulation and far-field experimental results, the right-angle bending effect of EM waves was verified from 12 to 18 GHz. The RI distribution of the 90 Eaton lens18,19 is described by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a a 2 1 for r a; and n ¼ 1 for r > a; n ¼ þ nr nr (1) where n represents the RI and a is the radius of the Eaton lens. n ¼ 1 at the boundary of the device which ensures the impedance match to air background. The RI increases when r goes to zero and tends to infinity at the centre of the sphere. pffiffiffiffi For non-magnetic approach (n ¼ er ), the equation describes a radially varying distribution profile for the permittivity. To fulfill the gradient profile with large permittivity range, we adopted an all dielectric approach involving nonresonant gradient periodic structure and a compound liquid medium. 3D PCs with high symmetry like diamondstructured woodpile PCs possess approximately spherical equifrequency surface in the long-wavelength limit. They can be homogenized and lead to nearly 3D isotropic EM properties under effective medium approximation.20–23 Such periodic structures are suited as 3D GRIN materials. The local effective permittivity can be controlled by changing the volume filling fraction of the constituent material. In our implementation, the volume filling fraction was controlled by changing the logs width of the woodpile structure, and the structural material used was photo-curable resin with a permittivity of 3.0. Thus, the effective permittivity can be varied from 1.0 (air) to 3.0 (photo-curable resin). To realize larger permittivity, a compound liquid medium composed of benzene and acetonitrile was employed.13,24 By adjusting the composition of the liquid mixture, permittivity range from 2.5 to 40 can be achieved. Combining the GRIN woodpile
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structure and the compound liquid medium, permittivity within the range from 1.0 (air) to 40 can be realized flexibly. The resin and the liquids used in the research are nearly non-dispersive from 12 GHz to 18 GHz; in practical implementation, the effective medium limit requiring small period-to-wavelength ratio is demonstrated to be forgiving for non-resonant woodpile PCs structure.22 These two features provide the basis of the broadband property for this method. In our implementation of the 3D Eaton lens (as shown in Fig. 1), a was set as 87 mm. The continuous permittivity distribution was divided into multiple spherical layers. The permittivity increases sharply when r approaches zero, so the layer thickness at the central area of the system decreases accordingly to better define the permittivity gradient. The corresponding parameters of the 3D Eaton lens multi-layer system are listed in Ref. 25. In fact, more layers could improve the performance at the cost of increasing difficulty in design and fabrication, whereas, the current setting in our implementation is adequate as demonstrated by simulation and experiment. When 0:345a < r a, the permittivity ranges from 1.0 to 3.0. Woodpile PCs structure was utilized in this area to realize the gradient permittivity profile. Considering the fabrication capability and the designed working frequency range covering Ku band, the rod spacing of the woodpile PCs was set as 5 mm which satisfies the effective medium limit. By changing the logs width of the woodpile PCs within each layer while keeping the rod spacing unchanged, the required local permittivity was achieved.23 As shown in Fig. 1(a), from the innermost layer to the outermost layer, the self-supporting integrated structure varied from pure resin to air. Impedance-match condition to air at the boundary was also ensured. When 0:0079a r 0:345a, the permittivity varies from 3.0 to 40.0. The target permittivity in this area of the proposed 3D Eaton lens was realized by the compound liquid medium approach.13 The permittivity at the central part (r 0:0079a) goes beyond 40.0, which is larger than the maximum value provided by the mixture of benzene and acetonitrile. For practical realization, extreme value was truncated and the permittivity in this very small region was set as 40.0 (pure acetonitrile). A multilayer framework was introduced in order to contain the liquid mixture with different
FIG. 1. Photographs of the fabricated sample: (a) half part of the fabricated 3D Eaton lens with pinholes in the framework for liquid injection; (b) the internal structure of the framework for containing liquid; (c) the final sample of the 3D Eaton lens.
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composition, as illustrated in Fig. 1(b). The designed model was divided into two hemispherical systems [as shown in Fig. 1(a)] in order to facilitate the injection of the compound liquid and the fabrication. Such complex structures were fabricated based on 3D printing process with a SL machine (SPS450B, Shaanxi Hengtong Intelligent Machine Co., Ltd, China). The final sample was then assembled after encapsulation of the injected liquids with adhesive, as shown in Fig. 1(c). The corresponding parameters of the woodpile PCs structure and the compound liquid medium are listed in Ref. 25. To illustrate the wave bending effects of the proposed 3D Eaton lens, full-wave simulations were generated using CST MICROWAVE STUDIO. A beam was incident on the 3D Eaton lens model along x direction and the wave propagation was examined at 15 GHz (centre frequency of Ku band). As shown in Fig. 2(a), the power flow indicates that the incoming waves experienced a 90 bend around the origin of the system. The far-filed scattering pattern plotted in Fig. 2(b) verifies the 90 bending angle. To further study the wave bending effects, the electric field distribution in the middle cross-section of the 3D Eaton lens was investigated. From Fig. 3(a), it can be seen that when an off-centre beam was incident on the Eaton lens along x direction, very few rays passed through the simplified central region while propagating inside the lens along a 90 bending path. When an on-centre beam was incident on the device along x-axis, the proposed Eaton lens behaved like a T-branch waveguide, as shown in Fig. 3(b). The narrow beam can actually be divided into two halves separated by x-axis, and each experienced a sharp twist near the central region. In this case, a slight scattering was observed due to the simplification of the permittivity profile in the central region. The far-field scattering pattern of these two cases were also calculated and plotted in Figs. 3(c) and 3(d). In Fig. 3(c), the bending effects of the Eaton lens resulted in a main lobe at 90 in scattering field, while in Fig. 3(d) two main lobes at 90 and 270 were observed. Nearly perfect 90 bending angle was demonstrated, even though a slight scattering existed in Fig. 3(d). The impact of the simplification on performance is small, in general. In practical application, it could be further reduced by avoiding beam incident directly towards the very small central region.
FIG. 2. Full-wave simulation results at 15 GHz when an off-centre beam is incident on the 3D Eaton lens: (a) the power flow distribution and (b) the calculated far-field scattering pattern.
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FIG. 3. Full-wave simulations results at 15 GHz in the middle cross-section of the 3D Eaton lens: the electric field distribution when the beam is incident (a) off-centre and (b) on-centre; the calculated far-field scattering pattern when the beam is incident (a) off-centre and (b) on-centre.
To verify the broadband performance of the fabricated 3D Eaton lens, the scattering behavior of the 3D sample was measured in an anechoic chamber. A Ku-band waveguide connected to a vector network analyzer (Agilent, E8363B) was utilized as the feeding source producing a narrow-beam incident wave in the near-field region. The experiment scheme is illustrated in Fig. 4(a). The spherical symmetry ensures a 3D Eaton lens works for incident waves from all direction, so we arbitrary chose to fix the feeding source in x-y plane generating incident wave along x direction. The far-field scattering wave was detected as a function of azimuth angle and frequency. The measured scattering field is plotted in Fig. 4(b), normalized by the largest value. Similar to the simulation results, the 90 wave bending effects was
FIG. 4. The measurement of the radiation behavior of the 3D Eaton lens in Ku-band: (a) the experiment scheme and (b) the measured results of the scattering field as a function of azimuth and frequency.
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verified in a broad bandwidth covering Ku band (12 GHz–18 GHz). The slight scattering is due to the simplification of the centre area. And the framework used for containing liquid might also introduce small impedance mismatch, though the wall thickness (0.3 mm) is thin. It should be noted that the model simplification and engineering approximation involved in our implementation include: (1) the multilayer approximation; (2) the homogenization of the woodpile PCs structure and the effective medium approximation; (3) the introduction of the framework for containing liquid; (4) the simplification in the small centre region. As shown in the simulation and experimental results, the influence can be considered negligible in the microwave regime, which demonstrates the feasibility of these simplifications and approximations in practical implementation. In conclusion, to realize a 3D broadband lens requiring large RI range and flexible GRIN control, an approach combining gradient woodpile structure in the long-wavelength limit and a compound liquid medium was proposed. As a proof-of-principle, a 3D Eaton lens was designed and fabricated using SL process. Broadband performance of the 3D Eaton lens was demonstrated by simulation and experiment. This paper provides a practical approach to realize 3D broadband GRIN lens. Such 3D GRIN devices may find applications in imaging, radar, and antenna. The proposed approach is non-resonant and all-dielectric, which also makes it promising to be extended to optical regime. This work was supported by National Natural Science Foundation of China (No. 51105300), Ph.D. Program Foundation of Ministry of Education of China (No. 20110201120075), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Ministry of Education of China, and the Fundamental Research Funds for the Central Universities of China. 1
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M. Yin, X. Y. Tian, L. L. Wu, and D. C. Li, Opt. Express 21(16), 19082 (2013). H. X. Han, L. L. Wu, X. Y. Tian, D. C. Li, M. Yin, and Y. Wang, J. Appl. Phys. 112(11), 114913 (2012). 25 See supplementary material at http://dx.doi.org/10.1063/1.4867704 for the calculated corresponding parameters of the 3D Eaton lens, the woodpile PCs structure, and the compound liquid medium. 24
