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We present the first realized three-dimensional (3D) practical implementation of the so called “optical black hole” in microwave frequencies, an electromagnetic (EM) concentrator. The 3D EM wave concentrator was designed with non-resonant gradient index (GRIN) 3D woodpile photonic crystals (PCs) structure in metamaterial regime, and fabricated by Stereolithography (SL) process. Omnidirectional EM wave capture and absorbing ability of the device in a broad bandwidth (12GHz-15GHz) were validated by full-wave simulation and experiments. Such devices may have applications in microwave energy harvesting and radiation detector.
©2013 Optical Society of America
1. Introduction
Artificial metamaterials made scientists magicians in the EM wave world with Transformation Optics (TO) [1,2], the magic wands in their hands. TO theory has demonstrated its magic power in the design of novel EM devices with specific material parameters distribution profile, which offered a way to manipulate the propagation of EM waves. “Tricks” like cloak [3–5], field rotators [6] and novel lens [7,8] have been realized with metamaterials. Motivated by imitating celestial mechanics, researchers have proposed devices as the optical analogues of cosmic phenomena [8–10]. One example is termed as the effective “optical black hole” or “EM black hole”. Different approaches were suggested to realize the interesting issue [8,11–13]. The isotropic and non-resonant approach proposed by Narimanov and Kildishev [12] was further demonstrated theoretically [14–17] and implemented in microwave regime [18–20]. The so called “black hole” devices behave more like omnidirectional and broadband EM wave concentrators. The scheme has also inspired analogy work in other realm, such as the acoustic and the elastic wave “black hole” [21–23]. However, the implementations mentioned above were all cylindrical versions of the “black hole”, which were limited to 2D situation with waveguide geometry. 3D implementation of such devices would broaden the potential applications, but it also increases the difficulty in design and fabrication.
In this article, we present the first practical implementation of a 3D spherical effective “EM black hole”. The device consists of an inner core and an outer shell. The lossy core was realized with liquid dielectric medium. The shell with radially varying permittivity profile was designed with non-resonant 3D gradient index (GRIN) woodpile photonic crystals (PCs) structures in metamaterial regime. The highly integrated complex 3D GRIN structure, which was used to manipulate the local EM field propagation in the device, was fabricated by SL process using photosensitive resin as raw material. 3D full-wave simulation and experimental results validated the broadband performance of the device. The 3D EM wave concentrator proposed may find potential applications in energy harvesting and radiation detector.
2. Design of the 3D EM wave concentrator
The permittivity distribution profile of the non-magnetic 3D EM wave concentrator derived from the equation in [12]:
where defines a GRIN shell and a lossy core, and R represent the radius of the inner core and the outer radius of the spherical shell respectively, represents the complex permittivity of the lossy core and stands for the permittivity of the background medium. was taken as 1 in the research, meaning the system was matched to free-space (air). The radius of the core was defined as according to the impedance-matched condition at the core-shell interface. Integer power index leads to incident wave falling into the core [12,14,15]. Higher orders n results in stronger and more efficient beam trap ability of the system. But it also increases the difficulty to realize, for the correspondingly growing values and gradients of the permittivity index. Thus, which constitutes the criterion for beam capture was chosen in the research.To fulfill the inhomogeneous isotropic permittivity distribution, woodpile PCs structure [24] in metmaterial regime was utilized as 3D GRIN medium in this letter. In long-wavelength limit, PCs structure can be homogenized and served as GRIN medium [4,25]. Diamond-structured woodpile PCs in metamaterial regime with highly symmetrical face-centered cubic lattice can be used to realize 3D GRIN designs. Its approximate spherical equifrequency surface satisfies the requirement of homogenization under effective medium approximation in a broad bandwidth, which results in 3D isotropic EM properties [4,26]. The effective medium approximation would work well when the wavelength of the incident wave is much longer than the rod spacing of the woodpile PCs unit cells. But as demonstrated in [4], the effective medium limit is more forgiving than that. In order to simplify the analysis process to determine the effective permittivity of the GRIN PCs, effective medium theory was employed in the research. The obtained effective permittivity show good conformity with those derived from the dispersion curves of a PC based on photonic band gap calculations [4,25]. In fact, by modifying the dispersion properties and engineering the equifrequency contours or surfaces of PCs, precise control of the EM waves propagation can be achieved [27–29]. But in our research, we only exploited the local isotropic properties and the relationship between volume fraction of the constituent material and the local effective permittivity to fulfill the required discrete permittivity distribution. For practical engineering realization, the continuous gradient permittivity profile of the shell was divided into multiple layers. More non-uniform layers could improve the performance of the “EM black hole”. Considering the design and fabrication factors, 12 layers with equal thickness was adopted in the present research, which has been proved adequate to provide absorption efficiency comparable with the theoretical ideal case [14,17]. Concerning the fabrication capability and the desired working frequency range of the device which satisfies the long wavelength limit, the rod spacing of the PCs was set to be a = 5mm. The thickness of each spherical layer of the EM wave concentrator shell was also accordingly set as 5mm. A nearly non-dispersive photosensitive resin with a permittivity of 3.0, which was measured using a well-established waveguide-based retrieval method [30] from 8 GHz to 18 GHz, was used as the structure material of the GRIN shell. The effective permittivity can be varied from 1.0 (full air) to 3.0 (full resin) and the real part of the complex permittivity of the lossy core was set to be 3.0. Then, from Eq. (1), R and Rc can be determined as 142.0mm and 82.0mm respectively. As previously described, employing effective medium theory, the effective permittivity of the PCs can be calculated as , where is the volume filling fraction of the photosensitive resin. The local effective permittivity is controlled by altering the logs width ω of the woodpile PCs unit cells within each spherical layer area while the rod spacing of the unit cells keeps unchanged as 5mm. The relationship between local effective permittivity and logs width is shown in Fig. 1. With rods remaining connected, the integrated structure is self-supporting and presents considerable mechanical strength. The 3D EM wave concentrator was fabricated on a SL process based 3D printing machine (SPS450B, Institute of Advanced Manufacturing Technology, Xi’an Jiaotong University), as shown in Figs. 1(a) and 1(b). Comparing with the Fused Deposition Modeling 3D printing process employed in [5], the precision of the SL process is relatively higher. But it also comes at higher cost because of the use of laser.
Fig. 1 Fabricated sample of the 3D EM wave concentrator shell with GRIN woodpile PCs structure. (a) The sample of the EM wave concentrator. (b) Partial view of the cross-section along the axis of the spherical shell. (c) The relation between the local effective permittivity and log width of woodpile PCs with a rod spacing of 5mm. (d) The schematic of woodpile PCs structure.
3. Simulation results
The use of isotropic 3D GRIN PCs structure ensures the device to be independent on polarization of the incident EM waves. To examine the broadband and omnidirectional EM wave capturing performance, the near-field wave propagating in the device was simulated in CST MICROWAVE STUDIO. A beam at a side position from the axis was incident on the model. The power flow in space and inside the device at 10 GHz was examined. As shown in Fig. 2, the power flow rapidly bends toward the core, indicating that the incoming field powerwas effectively concentrated. To further visualize the simulation results, we investigated the field intensity at different frequencies in the cross section of the system along the axis with incident beam on centre [Figs. 3(a)-3(c)] and off centre [Figs. 3(e)-3(g)]. This is equivalent to simulating incident beam from different direction because of the circular symmetry. The incident wave followed a bending path when entering the shell. The broad bandwidth performance was also demonstrated in simulation covering X and Ku band as Fig. 3 shows. For comparison, the gradient index shell was substituted with homogenous woodpile PCs structure as a reference sample. The unit cell lattice constant and sample dimensions kept the same. From Figs. 3(d) and 3(h), it can be seen that for the reference sample, the incident beam was severely scattered at the core-shell and shell-air interface in comparison with Fig. 3(b) and 3(e), respectively. Figure 4 demonstrates the field intensity distribution under the incidence of plane waves at the frequency of 13 GHz. From Fig. 4(a), it’s clear that nearly all incident waves hitting the “black hole” were trapped at the core and did not emerge. While for the same reference sample used in Figs. 3(d) and 3(h), the incident waves severely emerged [shown in Fig. 4(b)]. The simulation results demonstrate omnidirectional and broadband wave trapping ability of the proposed 3D EM wave concentrator.
Fig. 2 Power flow in space and in the 3D EM wave concentrator” when an off-centre beam is incident on the structure. (a) Perspective view. (b) Front view (c) Side view.
Fig. 3 Simulation results of the electric field intensity distributions in the cross section of the system. (a-c) Incident beam on centre at 8 GHz, 13 GHz and 18GHz. (d) Incident beam on centre to the reference sample at 13 GHz. (e-g) Incident beam off centre at 8 GHz, 13 GHz and 18 GHz. (h) Incident beam off centre to the reference sample at 13 GHz.
Fig. 4 Simulation results of the electric field intensity distributions in the cross section under the incidence of a plane wave at 13GHz. (a) The EM wave concentrator. (b) Reference sample.
4. Realization and experimental performance of the 3D EM wave concentrator
Two hemispheres of the woodpile shell were fabricated by SL process and assembled with liquid medium core. High shaping precision of the process ensures the continuous changing of the unit cell parameters and smooth GRIN to realize the designed permittivity profile accurately, which reduces the impedance-mismatch and the EM wave scattering. The lossy core was realized via a compound liquid medium approach [31]. Mixture of ethanol and oleic acid was chosen in the research, for ethanol is a damping medium with high dielectric constant and oleic acid is a lossless medium with relatively low dielectric constant. The required complex dielectric constant was fulfilled, when volume ratio between ethanol and oleic acid reaches 60:40, as demonstrates in Fig. 5. Broadband performance is ensured by the weak frequency dependence of the dielectric constants. A spherical shell with a radius of82.0 mm was used to contain the compound liquid medium. The shell was also made by SL with photosensitive resin and was only 200 μm thick, and would not affect the impedance-match condition at the core-shell interface, which was demonstrated with simulation.
Fig. 5 Dielectric properties of ethanol, oleic acid and the compound medium. (a) Dielectric constants of ethanol, oleic acid and their mixture at a volume ratio of 60:40 in the frequency range of 5 GHz – 20 GHz. The oleic acid has nearly no loss (b) Dielectric constants of the mixture with different volume ratio at the frequency of 12 GHz, 13 GHz, 14 GHz and 15 GHz.
The cross-sectional slice along the axis of the spherical EM wave concentrator was measured in a 2D field mapping device [5,32] to determine its electric field intensity distribution with a beam obliquely incident on the structure. A slice of 3D EM wave concentrator with a height of 8.8mm can be considered as a quasi 2D “EM black hole”. In comparison with the simulation, similar wave trapping behavior was observed in a relatively narrower bandwidth from 12 GHz to 15 GHz, as illustrated in Fig. 6. The reduced operationalfrequency range for the quasi 2D sample was mainly restricted by experiment limitations and the frequency dispersion characteristic of the core material. Standing wave pattern appears, when the frequency of the incident wave exceeds 15 GHz, which is the cutoff frequency of the higher order modes for the test device. Coexistence of the high-order modes and the main mode will affect the proper measurement. Also, the absorbing effect begins to deteriorate due to the smaller dielectric loss of the core material in higher frequencies. For incident wave below 12 GHz, distortion in electric field map introduced by the hardware flaws (e.g. the sag and the unevenness of the chamber plates in the test device) becomes significant. Thus, the incident beam scattered severely in propagation. The 2D near-field experiment results reflect an overall good wave trapping performance of the device, though it exhibits limitations compared with the simulation.
Fig. 6 Measured electric field intensity distributions for the cross-sectional slice along the axis of the device. (a-d) At 12 GHz, 13 GHz, 14 GHz and 15GHz, respectively.
The far-field scattering pattern for the 3D sample was also measured in an anechoic chamber with a pair of Ku-band horn antennas. One served as a feeding source generating incident wave on the device along the axis, which is the representative situation, while the other one acted as a detector of far-field scattering waves. For comparison, we also measured the far-field pattern when the spherical outer shell was removed. The measured far-field pattern at 12, 13, 14 and 15 GHz are illustrated in Fig. 7, respectively. Compared with the bare core situation, scattering fields are significantly reduced with the existence of the shell, reflecting a good EM wave absorbing performance. The reduced backscattering also suggests a relatively small monostatic RCS. The absorbing behavior of the 3D EM wave concentrator was verified by the contrast experiments.
Fig. 7 Measured far-field scattering pattern of EM waves for the 3D “EM black hole” and the bare core. Feeding source is along the axis. (a-d) at 12 GHz, 13 GHz, 14 GHz and 15 GHz, respectively.
5. Conclusion
In conclusion, a 3D EM wave concentrator was designed and realized with GRIN 3D PCs structure in metamaterial regime and a liquid medium approach. The complicated structure was fabricated by SL. Simulations and experimental results validated the capability for the concentrator in capturing and absorbing the broadband and omnidirectional EM wave. The proposed device may find applications in energy harvesting and radiation detector for 3D situation. Furthermore, by changing the photosensitive material used in SL process, and the composition in the compound liquid medium, more flexible effective index range can be realized. Also, the proposed structure is all dielectric and non-resonant, which can be extended to optical regime. The approach in this report presents promising prospects for practical implementation of novel 3D EM GRIN designs.
Acknowledgments
This work is supported by National Natural Science Foundation of China (51105300), Ph.D. Program Foundation of Ministry of Education of China (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.
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