Hyperbolic metamaterial based on anisotropic Mie-type resonance Chuwen Lan,1,5 Ke Bi,2 Bo Li,1,2 Xiaohan Cui,3 Ji Zhou,2,* and Qian Zhao4,6 2
1 Advanced Materials Institute, Shenzhen Graduate School, Tsinghua University, Shenzhen, China State Key Lab of New Ceramics and Fine Processing, School of Materials Science and Engineering, Tsinghua University, Beijing, China 3 School of Material Science and Engineering, University of Science and Technology Beijing, Beijing, China 4 State Key Lab of Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing, China 5 [email protected] 6 [email protected] * [email protected]
Abstract: A hyperbolic metamaterial (MM) based on anisotropic Mie-type resonance is theoretically and experimentally demonstrated in microwave range. Based on the shape-dependent Mie-type resonance, metamaterials with indefinite permeability or permittivity parameters are designed by tailoring the isotropic particle into an anisotropic one. The flat lens consisting of anisotropic dielectric resonators has been designed, fabricated and tested. The experimental observation of refocusing and a plane wave with ominidirectional radiation directly verify the predicted properties, which confirm the potential application in negative index material and superlens. This work will also help to develop all-dielectric anisotropic MM devices such as 3D spatial power combination, cloak, and electromagnetic wave converter, etc. ©2013 Optical Society of America OCIS codes: (160.3918) Metamaterials; (160.1190) Anisotropic optical materials.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
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#189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29592
16. L. Kang, Q. Zhao, B. Li, J. Zhou, and H. Zhu, “Experimental verification of a tunable optical negative refraction in nematic liquid crystals,” Appl. Phys. Lett. 90(18), 181931 (2007). 17. Q. Zhao, L. Kang, B. Li, J. Zhou, H. Tang, and B. Zhang, “Tunable negative refraction in nematic liquid crystals,” Appl. Phys. Lett. 89(22), 221918 (2006). 18. J. Sun, J. Zhou, B. Li, and F. Kang, “Indefinite permittivity and negative refraction in natural material: Graphite,” Appl. Phys. Lett. 98(10), 101901 (2011). 19. R. Wang, J. Sun, and J. Zhou, “Indefinite permittivity in uniaxial single crystal at infrared frequency,” Appl. Phys. Lett. 97(3), 031912 (2010). 20. L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of lefthanded behavior in an array of standard dielectric resonators,” Phys. Rev. Lett. 98(15), 157403 (2007). 21. Q. Zhao, L. Kang, B. Du, H. Zhao, Q. Xie, X. Huang, B. Li, J. Zhou, and L. Li, “Experimental demonstration of isotropic negative permeability in a three-dimensional dielectric composite,” Phys. Rev. Lett. 101(2), 027402 (2008). 22. H. Němec, P. Kužel, F. Kadlec, C. Kadlec, R. Yahiaoui, and P. Mounaix, “Tunable terahertz metamaterials with negative permeability,” Phys. Rev. B 79(24), 241108 (2009). 23. J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99(10), 107401 (2007). 24. J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012). 25. D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(3 3 Pt 2B), 036617 (2005). 26. T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 065602 (2003). 27. Q. Cheng, W. Jiang, and T. Cui, “Multi-beam generations at pre-designed directions based on anisotropic zeroindex metamaterials,” Appl. Phys. Lett. 99(13), 131913 (2011). 28. Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012). 29. A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett. 12(7), 3749–3755 (2012). 30. A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci Rep 2, 492 (2012). 31. A. E. Miroshnichenko and Y. S. Kivshar, “Fano resonances in all-dielectric oligomers,” Nano Lett. 12(12), 6459–6463 (2012). 32. Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nat Commun 4, 1527 (2013).
1. Introduction The term metamaterial (MM) refers to engineered composite materials with carefully tailored subwavelength building blocks. MM with exotic properties such as negative refraction index, negative permeability, and near zero permeability or permittivity makes it possible to realize new functional devices like superlens, cloaking, and perfect absorber [1–5]. In particular, great progress has been made in microwave area, which the novel antenna, filter, power divider, and bend waveguide have been realized [4,5]. Within the rapidly growing field of MM, hyperbolic MM (or indefinite medium), in which not all elements of the principal permittivity and permeability tensors have the same sign, is currently attracting much attention [6–13]. Unlike conventional anisotropic media such as YVO4, calcite and liquid crystal [14–17], hyperbolic MM can enable all-angle negative refractive because the dispersion relation in such medium is hyperbolic. It is also demonstrated to support evanescent modes propagation, thus allowing the potential application in superlens [9,10]. Recently, researchers find that such medium have a broadband singularity in the density of photonic states, which can be used to produce enhanced and highly directional spontaneous emission [12,13]. Up to now, almost all researchers focus on the hyperbolic MM with strong anisotropy of permittivity whose principal dielectric tensor elements have different signs. To achieve such medium, researchers have proposed metallic nanowires and metal-dielectric multi-layer structures [8–11]. These structures become the two most popular hyperbolic MM structures operating at optical wavelengths. Interestingly, some all-dielectric natural materials have also been demonstrated to generate indefinite permittivity in several certain spectral ranges, such as graphite in ultraviolet and strong anisotropic uniaxial crystals in the infrared [18,19].
#189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29593
However, realizing hyperbolic medium with strong anisotropy of permeability seems more difficult than one with strong anisotropy of permittivity, especially in optical ranges. So far, tailored metallic structure such as SRRs has been extensively studied to achieve anisotropic permeability. In microwave ranges, it is quite easy to realize SRRs-based hyperbolic MM, which has been theoretically and experimentally demonstrated [7]. However, this structure suffers from serious metal loss and complex micromachining at high frequencies. All-dielectric MM based on Mie-type resonance with advantages such as low intrinsic loss and tunable features provides a simple route to achieving high-performance MM operating in a broad spectrum [20–24]. As a matter of fact, considerable attention is focused on the isotropy of all-dielectric MM, while the feasibility of realizing anisotropic electromagnetic response is less studied. In addition, the corresponding experiment has not been carried out so far. In this work, we theoretically and experimentally propose a hyperbolic MM based on Mie-type resonance. It is shown that hyperbolic MM can be easily obtained by breaking the geometrical symmetry of dielectric particles. Due to the high tunability and multi-resonance response of Mie-based MM, we believe that such hyperbolic MM provides a convenient route to achieving superlens and negative index material. This study also helps to develop other anisotropic MM such as 3D spatial power combination, 3D invisibility cloak, and EM-wave converter in a broad spectrum. 2. Design hyperbolic MM
Fig. 1. (a) The unit cell for all-dielectric MM based on Mie-type resonance: a ceramic particle with permittivity 245 and loss tangent 0.004 is embedd in the air cubic box. The geometrical parameters are set as follows: p = 4mm, a = 2mm. (b), (c), (d) show the calculated components of the permeability tensor near the first Mie-type resonance when the dz is set as 2mm, 1.5mm, 1mm respectively. The ranges where indefinite permeability properties can be obtained are marked in blue and purple respectively.
In this section, we demonstrate that hyperbolic MM can be obtained easily by breaking the geometrical symmetry of dielectric resonators. Let us consider a MM with unit cell composed of a single ceramic particle embedded in an air cubic box (see in Fig. 1(a)). In this work, the particle is made of dielectric ceramic BST (Ba0.5Sr0.5TiO3) doped with 10 wt% MgO, owing to its high permittivity and low loss (with permittivity 245 and loss tangent 0.004). The
#189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29594
geometrical parameters for the unit cell are set as follows: p = 4mm, a = 2mm. At the first stage, numerical simulations based on the commercial finite integration time domain package (CST Microwave Studio) have been carried out to investigate the impact of dz (i.e. the length along the z direction) on the components of the permeability tensor near the first Mie-type resonance. It should be noted that we focus only on the first resonance in our study, although the high-order resonances can also be available for hyperbolic MM. In the calculation, the electric field is set along the y direction, and two different simulations have been carried out to obtain the S parameters when the magnetic field is set along the x and z directions respectively. Based on the S parameters, the components of the permeability tensor in the x and z directions are calculated by well-developed retrieval method [25]. All the retrieved permeability tensors are showed in Figs. 1(b)-1(d). As plotted in Fig. 1(b), the Lorentz-type permeability dispersion with dz = 2mm appears at 7.31 GHz, where a strong magnetic Mie-type resonance is induced. Obviously, the components of the permeability tensor in the x and z directions are always the same due to the isotropic configuration. As dz decreases to 1.5mm, they are obviously different. As one can see in Fig. 1(c), the Lorentz-type permeability tensors are displayed at 7.76 GHz and 8.39 GHz respectively. As a result, one can obtain indefinite permeability in two narrow frequency ranges, which are marked in blue and purple. The components of permeability tensor for particle with dz decreasing to 1mm are plotted in Fig. 1(d). As for μz, the Mie-type resonance is induced in 8.58 GHz and followed negative permeability values from 8.58 GHz to 8.92 GHz. The magnetic plasmas frequency, with effective permeability equal to zero, locates at 8.92 GHz. In Fig. 1(c), the negative permeability values for μx can be obtained from 10.77 GHz to 11.05 GHz, while the magnetic plasmas frequency can be evaluated at 11.05 GHz. We have also calculated the corresponding component of the permittivity tensor εy when the magnetic field is set in x and y directions respectively (see Figs. 2(a)-2(b)). It is obvious that εy is always positive in the frequency range considered. As a result, indefinite permeability can be obtained in range marked in blue and range marked in purple. Obviously, anisotropic zero-index can be generated around the magnetic plasmas frequencies, namely near 8.92 GHz and 11.05 GHz. It should be noted that anti-resonance behaviors can be observed for the component of the permittivity tensor around the magnetic resonance in Fig. 2(a) and Fig. 2(b), where the imaginary parts of the effective permittivity and permeability are opposite in sign. This is a consequence of periodicity effects [26]. The Lorentz-type dispersion for the effective permittivity appearing around 11.7 GHz is originating from the 2nd Mie resonance, i.e. electric resonance.
Fig. 2. The calculated permittivity εy for two simulations: (a) the magnetic field is set in x direction. (b) the magnetic field is set in z direction.
#189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29595
Fig. 3. Schematic of hyperbolic MM based on Mie-type resonance
At this point, we have seen that the EM properties of Mie-type resonance are strongly affected by the geometrical parameters. This gives us hope that hyperbolic MM can be easily realized by breaking the geometric symmetry of resonators. In order to demonstrate the properties predicted, we have designed a flat lens, which is schematically showed in Fig. 3. Here, we consider a TE wave with electric field along y propagating in such flat lens. From Maxwell’s equations, we can easily obtain the dispersion relation, which is given by k x2
k z2
ω
= ( )2 ε y (1) μz μx c It’s obviously that when μz and μx have opposite sign, the curve for such equation becomes a hyperbola. Hence, the group velocity of incident ray will undergo negative refraction, while the phase velocity will undergo positive refraction. As a result, a source located on one side of such slab lens will refocus inside or outside of the lens. If we make μz be zero, the hyperbola will be squeezed into a single line. As a result, the MM becomes a special indefinite medium, which is called anisotropic zero-index MM. In such case, the cylindrical waves produced by the source will be converted into a plane one, and ominidirectional radiation can also be obtained [27,28]. Based on the analysis above, it is clear that the key to demonstrate the indefinite properties of our metamaterial is observing the refocusing phenomena in the range marked in blue and ominidirectional radiation phenomena at 8.92 GHz respectively when the source ray passes through such flat lens. It should be mentioned that the range marked in purple or higher modes are also available for achieving hyperbolic MM. +
3. Experiment and analysis Employing the tape casting technique, the green tapes doped with MgO were fabricated and then sintered at 1400 degrees to get ceramic slabs with thickness of 1mm. The dielectric ceramic slabs were then cut into cuboids with size of 2mm × 2mm × 1mm. The manufactured particles are showed in Fig. 4(a).
#189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29596
Fig. 4. (a) The photograph of fabricated ceramic particles. (b) The photograph of fabricated flat lens.
A flat lens with sizes of 100mm × 55mm × 8mm has been made (see shown in Fig. 4(b)). The corresponding geometry is illustrated in Fig. 3. As matter of fact, the flat metmaterial is composed of two layers with thickness of 4mm to simplify the fabrication. For each layer, a 0.3mm thin Teflon slab with hole arrays was used to support the fabricated particles. The scotch tape was also employed to stick smoothly to the bottom surface of such slab for the purpose of reinforcing. In order to make the results more accurate and practical, we have calculated the EM parameters of bilayer MM with Teflon slab, shown in Figs. 5(a)-5(d). In the calculation, two-layer MM was employed to take the coupling effects between adjacent cells along the wave vector into consideration. In addition, the Teflon slab with permittivity 2.1 and loss tangent 0.001 was also considered. As one can see, indefinite permeability can be obtained from 8.52 GHz to 8.89 GHz, where μz and μx have opposite sign and εy keeps positive. The anisotropic zero-index can be obtained at 8.89 GHz.
Fig. 5. The modified EM parameters for the hyperbolic MM. (a), (b) show the calculated components of the permeability tensor in x direction and in z direction, respectively. The calculated permittivity εy for two simulations: (c) the magnetic field is set in x direction. (d) the magnetic field is set in z direction.
#189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29597
The sample was tested by the 2D near-field microwave scanning apparatus, which is a parallel-plate waveguide with operating frequencies from 8.0 GHz to 12.0 GHz. As showed in Fig. 6(a), there is a dipole source fixed on the lower plate, while a detector probe is mounted in the upper plate to obtain the electric field within the chamber. The lower plate can move in x and z directions by using two step motors. As a result, we can obtain spatially sampling in a finite region. In the experiment, the flat lens was placed on the front of source, and the measurement region was selected as 150mm × 200mm. The vector network analyzer (Agilent ENA5071C) was employed to collect the measure data, which was then sent to a computer for subsequent processing.
Fig. 6. (a) The photograph of the 2D near-field microwave scanning apparatus; (b) The corresponding simulation model using HFSS.
Fig. 7. The simulated electric field distributions in x-z plane at different frequencies: (a) 8.6 GHz, (b) 8.7GHz (c) 8.8 GHz, (d) 8.89GHz. (e), (f), (g), (h) show the measured electric field distributions at corresponding frequencies. The region marked by black frame represents the flat lens.
As a comparison, numerical simulations have also been carried out by employing the software package HFSS, a finite-element based frequency-domain electromagnetic solver. The simulation model is showed in the Fig. 6(b), where computational domain is actually an air box with size of 150mm × 200mm × 11mm. As plotted in this picture, the hyperbolic MM is inserted to the air box, where top and bottom are set as perfectly electric conducting (PEC)
#189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29598
while others are radiation boundary condition. A dipole source is placed before the hyperbolic MM to excite a cylindrical wave. In the simulation, four frequencies (8.6 GHz, 8.7 GHz, 8.8 GHz and 8.89 GHz) have been chosen for observation. The corresponding components of the permeability tensor and permittivity tensor for the aforementioned frequencies are set as the same to the calculated parameters above (see Fig. 5). The simulation results are shown in Figs. 7(a)-7(d), where electric fields are plotted for different frequencies in x-z plane. As plotted in Figs. 7(a)-7(c), excellent refocusing phenomena are obviously observed inside or outside the slab, verifying the extremely anisotropic properties. Note that the frequency dependent focus is attributed to the dispersion of permeability tensor in z direction. As we can in Fig. 7(d), the cylindrical wave produced by the source is converted into a plane one, and ominidirectional radiation can also be obtained. The phenomena can be easily understood by plotting the isofrequency surface of hyperbolic metamaterial, as shown in Figs. 8(a)-8(b). It can be seen that, when μz 0, the isofrequency surface for such medium is hyperbolic in the (kx, kz) plane. In such case, the group velocity of incident ray will undergo negative refraction, and a source located on one side of slab lens will refocus inside or outside of the lens. It is worth mentioning that when the value of μz increases gradually, the hyperbola is squeezed (see Figs. 8(a)-8(b)). As a result, the position of focus outside the lens is changed gradually. When the μz tend to zero, the hyperbola will be squeezed into a single line, and any waves with kz≠0 cannot transmit through the medium. Hence, only the wave with kz = 0 can be supported in such medium. So the cylindrical waves produced by the source will be converted into a plane one.
Fig. 8. Isofrequency surface of hyperbolic metamaterial with component of the permeability tensor in z direction for three cases: (a) μz1 (b) μz2 (c) μz3, where μz1 < μz2 < μz3 and μz3 →0. Si and ki are the Poynting vector and the wave vector of incident wave, respectively. kr and Sr are the refracted wave vector and Poynting vector, respectively.
The corresponding experimental results for electric field distribution at aforementioned frequencies are shown in Figs. 7(e)-7(h), respectively. As shown in Figs. 7(e)-7(g), the refocusing occurs obviously inside and outside the lens. The experimental results show a good agreement with the simulated ones. It’s worth noting that the electric field intensities are not consistent because the sources in simulation and experiment are not the same. As plotted in Fig. 7(h), approximate plane wave with direction of propagation perpendicular to the surfaces of the lens is obtained. We can also find that the produced plane wave splits into two beams, which are slightly different from the simulation result. The slight departure can be attributed to the fabrication. Even though, the agreement between experiment results and simulated one is quite good. Although the feasibility of hyperbolic MM based on Mie-type resonance is demonstrated at microwave frequencies, it is possible to extend this property to THz or even higher frequencies by choosing suitable materials and dimensions. For example, the dielectric metamaterials based on Mie resonances of dielectric rod or cube arrays have been #189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29599
experimentally demonstrated in THz ranges [22]. Optical magnetism in mid-infrared range was realized by arrays high-index tellurium dielectric structure [24]. Very recently, alldielectric structures with a finite magnetic response have been demonstrated in visible ranges [29–32]. Those results tell us that such kind of hyperbolic MM can be easily achieved in optical ranges. It’s important to note that anisotropic Mie-type resonances can also be employed to design other anisotropic MM like 3-D spatial power combination, cloak, and EM-wave converter. In contrast to the anisotropic MM consisting of SRRs structure, which is actually planer structure that can only produce magnetic response in one direction, the dielectric anisotropic MM based on Mie-type resonance which is a body phenomenon can provide more freedom in the design of novel metamaterial. For example, we can independently design the components of the permeability tensor and permittivity tensor. This is the main advantage respect to the metallic structure based anisotropic MM. In addition, multi-band anisotropic metamaterial with either anisotropic permeability or permittivity can be obtained due to the fact that the excitations of various Mie magnetic and electric resonances. Furthermore, the all-dielectric structure offers more convenient methods to get a metamaterial with tunable operation frequency. 4. Conclusion We have demonstrated theoretically and experimentally a hyperbolic MM based on anisotropic Mie-type resonance. The flat lens consisting of dielectric resonators has been designed, fabricated and tested. The hyperbolic properties are verified by experimental observation of refocusing and ominidirectional radiation phenomena of cylindrical electromagnetic wave propagation through such medium. The simulation results show a good agreement with the experimental ones. This study is believed to provide a simple route toward developing negative index material and superlens. It also helps to develop anisotropic MM such as 3-D spatial power combination, cloak, and EM-wave converter, etc. Acknowledgment This work was supported by the National High Technology Research and Development Program of China under Grant No. 2012AA030403, the National Natural Science Foundation of China under Grant Nos. 51032003, 11274198, 61275176, 61077029 and 51221291, and, the Shenzhen Science and technology projects under Grant Nos. XCL201110009, JCY201110096, and JSE201007200050A, and the China Postdoctoral Research Foundation under Grant No. 2013M530042.
#189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29600
1 Advanced Materials Institute, Shenzhen Graduate School, Tsinghua University, Shenzhen, China State Key Lab of New Ceramics and Fine Processing, School of Materials Science and Engineering, Tsinghua University, Beijing, China 3 School of Material Science and Engineering, University of Science and Technology Beijing, Beijing, China 4 State Key Lab of Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing, China 5 [email protected] 6 [email protected] * [email protected]
Abstract: A hyperbolic metamaterial (MM) based on anisotropic Mie-type resonance is theoretically and experimentally demonstrated in microwave range. Based on the shape-dependent Mie-type resonance, metamaterials with indefinite permeability or permittivity parameters are designed by tailoring the isotropic particle into an anisotropic one. The flat lens consisting of anisotropic dielectric resonators has been designed, fabricated and tested. The experimental observation of refocusing and a plane wave with ominidirectional radiation directly verify the predicted properties, which confirm the potential application in negative index material and superlens. This work will also help to develop all-dielectric anisotropic MM devices such as 3D spatial power combination, cloak, and electromagnetic wave converter, etc. ©2013 Optical Society of America OCIS codes: (160.3918) Metamaterials; (160.1190) Anisotropic optical materials.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004). D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008). N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (Wiley Interscience, 2006). T. Cui, D. Smith, and R. Liu, Metamaterials: Theory, Design, and Applications (Springer, 2009). D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with Indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405 (2003). D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004). J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008). Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express 16(20), 15439–15448 (2008). A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B 79(24), 245127 (2009). A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007). Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012). I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105(6), 067402 (2010). X. L. Chen, M. He, Y. X. Du, W. Y. Wang, and D. F. Zhang, “Negative refraction: an intrinsic property of uniaxial crystals,” Phys. Rev. B 72(11), 113111 (2005). Y. Zhang, B. Fluegel, and A. Mascarenhas, “Total negative refraction in real crystals for ballistic electrons and light,” Phys. Rev. Lett. 91(15), 157404 (2003).
#189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29592
16. L. Kang, Q. Zhao, B. Li, J. Zhou, and H. Zhu, “Experimental verification of a tunable optical negative refraction in nematic liquid crystals,” Appl. Phys. Lett. 90(18), 181931 (2007). 17. Q. Zhao, L. Kang, B. Li, J. Zhou, H. Tang, and B. Zhang, “Tunable negative refraction in nematic liquid crystals,” Appl. Phys. Lett. 89(22), 221918 (2006). 18. J. Sun, J. Zhou, B. Li, and F. Kang, “Indefinite permittivity and negative refraction in natural material: Graphite,” Appl. Phys. Lett. 98(10), 101901 (2011). 19. R. Wang, J. Sun, and J. Zhou, “Indefinite permittivity in uniaxial single crystal at infrared frequency,” Appl. Phys. Lett. 97(3), 031912 (2010). 20. L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of lefthanded behavior in an array of standard dielectric resonators,” Phys. Rev. Lett. 98(15), 157403 (2007). 21. Q. Zhao, L. Kang, B. Du, H. Zhao, Q. Xie, X. Huang, B. Li, J. Zhou, and L. Li, “Experimental demonstration of isotropic negative permeability in a three-dimensional dielectric composite,” Phys. Rev. Lett. 101(2), 027402 (2008). 22. H. Němec, P. Kužel, F. Kadlec, C. Kadlec, R. Yahiaoui, and P. Mounaix, “Tunable terahertz metamaterials with negative permeability,” Phys. Rev. B 79(24), 241108 (2009). 23. J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99(10), 107401 (2007). 24. J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012). 25. D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(3 3 Pt 2B), 036617 (2005). 26. T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 065602 (2003). 27. Q. Cheng, W. Jiang, and T. Cui, “Multi-beam generations at pre-designed directions based on anisotropic zeroindex metamaterials,” Appl. Phys. Lett. 99(13), 131913 (2011). 28. Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108(21), 213903 (2012). 29. A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett. 12(7), 3749–3755 (2012). 30. A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci Rep 2, 492 (2012). 31. A. E. Miroshnichenko and Y. S. Kivshar, “Fano resonances in all-dielectric oligomers,” Nano Lett. 12(12), 6459–6463 (2012). 32. Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nat Commun 4, 1527 (2013).
1. Introduction The term metamaterial (MM) refers to engineered composite materials with carefully tailored subwavelength building blocks. MM with exotic properties such as negative refraction index, negative permeability, and near zero permeability or permittivity makes it possible to realize new functional devices like superlens, cloaking, and perfect absorber [1–5]. In particular, great progress has been made in microwave area, which the novel antenna, filter, power divider, and bend waveguide have been realized [4,5]. Within the rapidly growing field of MM, hyperbolic MM (or indefinite medium), in which not all elements of the principal permittivity and permeability tensors have the same sign, is currently attracting much attention [6–13]. Unlike conventional anisotropic media such as YVO4, calcite and liquid crystal [14–17], hyperbolic MM can enable all-angle negative refractive because the dispersion relation in such medium is hyperbolic. It is also demonstrated to support evanescent modes propagation, thus allowing the potential application in superlens [9,10]. Recently, researchers find that such medium have a broadband singularity in the density of photonic states, which can be used to produce enhanced and highly directional spontaneous emission [12,13]. Up to now, almost all researchers focus on the hyperbolic MM with strong anisotropy of permittivity whose principal dielectric tensor elements have different signs. To achieve such medium, researchers have proposed metallic nanowires and metal-dielectric multi-layer structures [8–11]. These structures become the two most popular hyperbolic MM structures operating at optical wavelengths. Interestingly, some all-dielectric natural materials have also been demonstrated to generate indefinite permittivity in several certain spectral ranges, such as graphite in ultraviolet and strong anisotropic uniaxial crystals in the infrared [18,19].
#189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29593
However, realizing hyperbolic medium with strong anisotropy of permeability seems more difficult than one with strong anisotropy of permittivity, especially in optical ranges. So far, tailored metallic structure such as SRRs has been extensively studied to achieve anisotropic permeability. In microwave ranges, it is quite easy to realize SRRs-based hyperbolic MM, which has been theoretically and experimentally demonstrated [7]. However, this structure suffers from serious metal loss and complex micromachining at high frequencies. All-dielectric MM based on Mie-type resonance with advantages such as low intrinsic loss and tunable features provides a simple route to achieving high-performance MM operating in a broad spectrum [20–24]. As a matter of fact, considerable attention is focused on the isotropy of all-dielectric MM, while the feasibility of realizing anisotropic electromagnetic response is less studied. In addition, the corresponding experiment has not been carried out so far. In this work, we theoretically and experimentally propose a hyperbolic MM based on Mie-type resonance. It is shown that hyperbolic MM can be easily obtained by breaking the geometrical symmetry of dielectric particles. Due to the high tunability and multi-resonance response of Mie-based MM, we believe that such hyperbolic MM provides a convenient route to achieving superlens and negative index material. This study also helps to develop other anisotropic MM such as 3D spatial power combination, 3D invisibility cloak, and EM-wave converter in a broad spectrum. 2. Design hyperbolic MM
Fig. 1. (a) The unit cell for all-dielectric MM based on Mie-type resonance: a ceramic particle with permittivity 245 and loss tangent 0.004 is embedd in the air cubic box. The geometrical parameters are set as follows: p = 4mm, a = 2mm. (b), (c), (d) show the calculated components of the permeability tensor near the first Mie-type resonance when the dz is set as 2mm, 1.5mm, 1mm respectively. The ranges where indefinite permeability properties can be obtained are marked in blue and purple respectively.
In this section, we demonstrate that hyperbolic MM can be obtained easily by breaking the geometrical symmetry of dielectric resonators. Let us consider a MM with unit cell composed of a single ceramic particle embedded in an air cubic box (see in Fig. 1(a)). In this work, the particle is made of dielectric ceramic BST (Ba0.5Sr0.5TiO3) doped with 10 wt% MgO, owing to its high permittivity and low loss (with permittivity 245 and loss tangent 0.004). The
#189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29594
geometrical parameters for the unit cell are set as follows: p = 4mm, a = 2mm. At the first stage, numerical simulations based on the commercial finite integration time domain package (CST Microwave Studio) have been carried out to investigate the impact of dz (i.e. the length along the z direction) on the components of the permeability tensor near the first Mie-type resonance. It should be noted that we focus only on the first resonance in our study, although the high-order resonances can also be available for hyperbolic MM. In the calculation, the electric field is set along the y direction, and two different simulations have been carried out to obtain the S parameters when the magnetic field is set along the x and z directions respectively. Based on the S parameters, the components of the permeability tensor in the x and z directions are calculated by well-developed retrieval method [25]. All the retrieved permeability tensors are showed in Figs. 1(b)-1(d). As plotted in Fig. 1(b), the Lorentz-type permeability dispersion with dz = 2mm appears at 7.31 GHz, where a strong magnetic Mie-type resonance is induced. Obviously, the components of the permeability tensor in the x and z directions are always the same due to the isotropic configuration. As dz decreases to 1.5mm, they are obviously different. As one can see in Fig. 1(c), the Lorentz-type permeability tensors are displayed at 7.76 GHz and 8.39 GHz respectively. As a result, one can obtain indefinite permeability in two narrow frequency ranges, which are marked in blue and purple. The components of permeability tensor for particle with dz decreasing to 1mm are plotted in Fig. 1(d). As for μz, the Mie-type resonance is induced in 8.58 GHz and followed negative permeability values from 8.58 GHz to 8.92 GHz. The magnetic plasmas frequency, with effective permeability equal to zero, locates at 8.92 GHz. In Fig. 1(c), the negative permeability values for μx can be obtained from 10.77 GHz to 11.05 GHz, while the magnetic plasmas frequency can be evaluated at 11.05 GHz. We have also calculated the corresponding component of the permittivity tensor εy when the magnetic field is set in x and y directions respectively (see Figs. 2(a)-2(b)). It is obvious that εy is always positive in the frequency range considered. As a result, indefinite permeability can be obtained in range marked in blue and range marked in purple. Obviously, anisotropic zero-index can be generated around the magnetic plasmas frequencies, namely near 8.92 GHz and 11.05 GHz. It should be noted that anti-resonance behaviors can be observed for the component of the permittivity tensor around the magnetic resonance in Fig. 2(a) and Fig. 2(b), where the imaginary parts of the effective permittivity and permeability are opposite in sign. This is a consequence of periodicity effects [26]. The Lorentz-type dispersion for the effective permittivity appearing around 11.7 GHz is originating from the 2nd Mie resonance, i.e. electric resonance.
Fig. 2. The calculated permittivity εy for two simulations: (a) the magnetic field is set in x direction. (b) the magnetic field is set in z direction.
#189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29595
Fig. 3. Schematic of hyperbolic MM based on Mie-type resonance
At this point, we have seen that the EM properties of Mie-type resonance are strongly affected by the geometrical parameters. This gives us hope that hyperbolic MM can be easily realized by breaking the geometric symmetry of resonators. In order to demonstrate the properties predicted, we have designed a flat lens, which is schematically showed in Fig. 3. Here, we consider a TE wave with electric field along y propagating in such flat lens. From Maxwell’s equations, we can easily obtain the dispersion relation, which is given by k x2
k z2
ω
= ( )2 ε y (1) μz μx c It’s obviously that when μz and μx have opposite sign, the curve for such equation becomes a hyperbola. Hence, the group velocity of incident ray will undergo negative refraction, while the phase velocity will undergo positive refraction. As a result, a source located on one side of such slab lens will refocus inside or outside of the lens. If we make μz be zero, the hyperbola will be squeezed into a single line. As a result, the MM becomes a special indefinite medium, which is called anisotropic zero-index MM. In such case, the cylindrical waves produced by the source will be converted into a plane one, and ominidirectional radiation can also be obtained [27,28]. Based on the analysis above, it is clear that the key to demonstrate the indefinite properties of our metamaterial is observing the refocusing phenomena in the range marked in blue and ominidirectional radiation phenomena at 8.92 GHz respectively when the source ray passes through such flat lens. It should be mentioned that the range marked in purple or higher modes are also available for achieving hyperbolic MM. +
3. Experiment and analysis Employing the tape casting technique, the green tapes doped with MgO were fabricated and then sintered at 1400 degrees to get ceramic slabs with thickness of 1mm. The dielectric ceramic slabs were then cut into cuboids with size of 2mm × 2mm × 1mm. The manufactured particles are showed in Fig. 4(a).
#189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29596
Fig. 4. (a) The photograph of fabricated ceramic particles. (b) The photograph of fabricated flat lens.
A flat lens with sizes of 100mm × 55mm × 8mm has been made (see shown in Fig. 4(b)). The corresponding geometry is illustrated in Fig. 3. As matter of fact, the flat metmaterial is composed of two layers with thickness of 4mm to simplify the fabrication. For each layer, a 0.3mm thin Teflon slab with hole arrays was used to support the fabricated particles. The scotch tape was also employed to stick smoothly to the bottom surface of such slab for the purpose of reinforcing. In order to make the results more accurate and practical, we have calculated the EM parameters of bilayer MM with Teflon slab, shown in Figs. 5(a)-5(d). In the calculation, two-layer MM was employed to take the coupling effects between adjacent cells along the wave vector into consideration. In addition, the Teflon slab with permittivity 2.1 and loss tangent 0.001 was also considered. As one can see, indefinite permeability can be obtained from 8.52 GHz to 8.89 GHz, where μz and μx have opposite sign and εy keeps positive. The anisotropic zero-index can be obtained at 8.89 GHz.
Fig. 5. The modified EM parameters for the hyperbolic MM. (a), (b) show the calculated components of the permeability tensor in x direction and in z direction, respectively. The calculated permittivity εy for two simulations: (c) the magnetic field is set in x direction. (d) the magnetic field is set in z direction.
#189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29597
The sample was tested by the 2D near-field microwave scanning apparatus, which is a parallel-plate waveguide with operating frequencies from 8.0 GHz to 12.0 GHz. As showed in Fig. 6(a), there is a dipole source fixed on the lower plate, while a detector probe is mounted in the upper plate to obtain the electric field within the chamber. The lower plate can move in x and z directions by using two step motors. As a result, we can obtain spatially sampling in a finite region. In the experiment, the flat lens was placed on the front of source, and the measurement region was selected as 150mm × 200mm. The vector network analyzer (Agilent ENA5071C) was employed to collect the measure data, which was then sent to a computer for subsequent processing.
Fig. 6. (a) The photograph of the 2D near-field microwave scanning apparatus; (b) The corresponding simulation model using HFSS.
Fig. 7. The simulated electric field distributions in x-z plane at different frequencies: (a) 8.6 GHz, (b) 8.7GHz (c) 8.8 GHz, (d) 8.89GHz. (e), (f), (g), (h) show the measured electric field distributions at corresponding frequencies. The region marked by black frame represents the flat lens.
As a comparison, numerical simulations have also been carried out by employing the software package HFSS, a finite-element based frequency-domain electromagnetic solver. The simulation model is showed in the Fig. 6(b), where computational domain is actually an air box with size of 150mm × 200mm × 11mm. As plotted in this picture, the hyperbolic MM is inserted to the air box, where top and bottom are set as perfectly electric conducting (PEC)
#189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29598
while others are radiation boundary condition. A dipole source is placed before the hyperbolic MM to excite a cylindrical wave. In the simulation, four frequencies (8.6 GHz, 8.7 GHz, 8.8 GHz and 8.89 GHz) have been chosen for observation. The corresponding components of the permeability tensor and permittivity tensor for the aforementioned frequencies are set as the same to the calculated parameters above (see Fig. 5). The simulation results are shown in Figs. 7(a)-7(d), where electric fields are plotted for different frequencies in x-z plane. As plotted in Figs. 7(a)-7(c), excellent refocusing phenomena are obviously observed inside or outside the slab, verifying the extremely anisotropic properties. Note that the frequency dependent focus is attributed to the dispersion of permeability tensor in z direction. As we can in Fig. 7(d), the cylindrical wave produced by the source is converted into a plane one, and ominidirectional radiation can also be obtained. The phenomena can be easily understood by plotting the isofrequency surface of hyperbolic metamaterial, as shown in Figs. 8(a)-8(b). It can be seen that, when μz 0, the isofrequency surface for such medium is hyperbolic in the (kx, kz) plane. In such case, the group velocity of incident ray will undergo negative refraction, and a source located on one side of slab lens will refocus inside or outside of the lens. It is worth mentioning that when the value of μz increases gradually, the hyperbola is squeezed (see Figs. 8(a)-8(b)). As a result, the position of focus outside the lens is changed gradually. When the μz tend to zero, the hyperbola will be squeezed into a single line, and any waves with kz≠0 cannot transmit through the medium. Hence, only the wave with kz = 0 can be supported in such medium. So the cylindrical waves produced by the source will be converted into a plane one.
Fig. 8. Isofrequency surface of hyperbolic metamaterial with component of the permeability tensor in z direction for three cases: (a) μz1 (b) μz2 (c) μz3, where μz1 < μz2 < μz3 and μz3 →0. Si and ki are the Poynting vector and the wave vector of incident wave, respectively. kr and Sr are the refracted wave vector and Poynting vector, respectively.
The corresponding experimental results for electric field distribution at aforementioned frequencies are shown in Figs. 7(e)-7(h), respectively. As shown in Figs. 7(e)-7(g), the refocusing occurs obviously inside and outside the lens. The experimental results show a good agreement with the simulated ones. It’s worth noting that the electric field intensities are not consistent because the sources in simulation and experiment are not the same. As plotted in Fig. 7(h), approximate plane wave with direction of propagation perpendicular to the surfaces of the lens is obtained. We can also find that the produced plane wave splits into two beams, which are slightly different from the simulation result. The slight departure can be attributed to the fabrication. Even though, the agreement between experiment results and simulated one is quite good. Although the feasibility of hyperbolic MM based on Mie-type resonance is demonstrated at microwave frequencies, it is possible to extend this property to THz or even higher frequencies by choosing suitable materials and dimensions. For example, the dielectric metamaterials based on Mie resonances of dielectric rod or cube arrays have been #189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29599
experimentally demonstrated in THz ranges [22]. Optical magnetism in mid-infrared range was realized by arrays high-index tellurium dielectric structure [24]. Very recently, alldielectric structures with a finite magnetic response have been demonstrated in visible ranges [29–32]. Those results tell us that such kind of hyperbolic MM can be easily achieved in optical ranges. It’s important to note that anisotropic Mie-type resonances can also be employed to design other anisotropic MM like 3-D spatial power combination, cloak, and EM-wave converter. In contrast to the anisotropic MM consisting of SRRs structure, which is actually planer structure that can only produce magnetic response in one direction, the dielectric anisotropic MM based on Mie-type resonance which is a body phenomenon can provide more freedom in the design of novel metamaterial. For example, we can independently design the components of the permeability tensor and permittivity tensor. This is the main advantage respect to the metallic structure based anisotropic MM. In addition, multi-band anisotropic metamaterial with either anisotropic permeability or permittivity can be obtained due to the fact that the excitations of various Mie magnetic and electric resonances. Furthermore, the all-dielectric structure offers more convenient methods to get a metamaterial with tunable operation frequency. 4. Conclusion We have demonstrated theoretically and experimentally a hyperbolic MM based on anisotropic Mie-type resonance. The flat lens consisting of dielectric resonators has been designed, fabricated and tested. The hyperbolic properties are verified by experimental observation of refocusing and ominidirectional radiation phenomena of cylindrical electromagnetic wave propagation through such medium. The simulation results show a good agreement with the experimental ones. This study is believed to provide a simple route toward developing negative index material and superlens. It also helps to develop anisotropic MM such as 3-D spatial power combination, cloak, and EM-wave converter, etc. Acknowledgment This work was supported by the National High Technology Research and Development Program of China under Grant No. 2012AA030403, the National Natural Science Foundation of China under Grant Nos. 51032003, 11274198, 61275176, 61077029 and 51221291, and, the Shenzhen Science and technology projects under Grant Nos. XCL201110009, JCY201110096, and JSE201007200050A, and the China Postdoctoral Research Foundation under Grant No. 2013M530042.
#189675 - $15.00 USD Received 14 Aug 2013; revised 25 Sep 2013; accepted 27 Sep 2013; published 22 Nov 2013 (C) 2013 OSA 2 December 2013 | Vol. 21, No. 24 | DOI:10.1364/OE.21.029592 | OPTICS EXPRESS 29600
