Article Cite This: ACS Photonics XXXX, XXX, XXX−XXX
Loss-Free Negative-Index Metamaterials Using Forward Light Scattering in Dielectric Meta-Atoms SeokJae Yoo,‡ Suyeon Lee,‡ and Q-Han Park* Department of Physics, Korea University, Seoul, Korea ABSTRACT: Negative-index metamaterials are significantly limited by their inherent losses. We demonstrate that a negative index without dissipative and reflective losses can be achieved in subwavelength dielectric cuboid arrays. We can avoid reflection by making dielectric cuboids exhibit the resonant forward scattering of light, while dissipative loss is inherently absent in dielectric materials. A joint study combining theoretical and experimental research demonstrates that loss-free negative-index metamaterials can be realized. They fulfill the criteria of negative refraction based on causality, while conventional active loss compensation schemes lead to the disappearance of negative refraction by causality. KEYWORDS: metamaterials, effective medium theory, refraction, reflection, optical resonators
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exhibit spectrally overlapping electric and magnetic dipole resonances and result in resonant forward scattering. We experimentally realize loss-free negative-index metamaterials by fabricating high-index dielectric meta-atoms, working in the microwave spectral band. At the frequency of the resonant forward scattering of light, our metamaterial shows a refractive index up to Re(n) = −3 with loss-free perfect transmission. We also demonstrate that loss-free negativity within the refractive index can be maintained for multiple stacks of meta-atom layers at wavelength-order metamaterial thicknesses. Our strategy for creating loss-free negative-index metamaterials is not restricted by the fundamental limit of loss compensation in negativeindex metamaterials because our strategy obeys causality.
egative refraction of light in metamaterials promises the intriguing ability to control light in an unnatural manner, causing it to travel with opposite phase and energy velocities.1,2 Although there have been numerous demonstrations of negative-index metamaterials, their performance is significantly limited by losses and reflection.3 Resonant metallic inclusions, namely, meta-atoms, present in negative-index metamaterials not only absorb light energy through dissipative ohmic losses but also scatter incident light in unwanted directions, e.g., backward or perpendicularly. Therefore, the major challenge for metamaterial researchers and engineers has been to passively minimize and actively compensate for these losses in metamaterials. Such passive loss minimization strategies include geometric tailoring,4 the use of low-loss materials,5 the use of plasmonically induced transparency-based metamaterials,6,7 and the plasmon injection scheme.8 However, even after adopting these passive strategies, losses in the negative-index metamaterials cannot be significantly reduced.8 The active loss compensation strategy is achieved by incorporating gain media into negative-index metamaterials.3,9 However, by causality, it has been proved in principle that the loss compensation of metamaterials leads to the disappearance of their negative refraction properties.10 On the other hand, the approach of tailoring directional scattering by dielectric nanoantennae has come into focus recently. It has been demonstrated that, due to the interplay between electric and magnetic modes in dielectric nanoantennae, perfect control of scattering direction is possible and that they can even play a role as Huygens sources (perfect forward scattering) and reflectors (perfect backward scattering).11,12 Another advantage of dielectric nanoantennae is due to dielectric materials being inherently free from dissipative loss, unlike plasmonic metals, which feature significant ohmic losses. Here, we propose an alternative loss-free negative-index metamaterial design strategy that uses resonant forward light scattering. Our inclusions, namely, meta-atoms, are designed to © XXXX American Chemical Society
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RESULTS AND DISCUSSION
Design Guideline for Loss-Free Negative-Index Metamaterials. We assume that meta-atoms with the electric dipole moment p = ε0αeE and the magnetic dipole moment m = αmH are arranged in low number density N. αe (αm) is the electric (magnetic) polarizability of the meta-atom. For individual meta-atoms, the differential scattering cross section at the observation angle θ is given by13 dσsca k4 (θ ) = {(|αe|2 + |αm|2 )(1 + cos2 θ) dΩ 2 + 4μ Re(αeαm*)cos θ } 0
(1)
with the solid angle Ω and wavenumber k. At the low density of the meta-atoms, the coupling between them is negligible, and the differential scattering cross section can be written as in eq 1. Then, the backward scattering cross section is given by Received: November 21, 2017 Published: February 7, 2018 A
DOI: 10.1021/acsphotonics.7b01400 ACS Photonics XXXX, XXX, XXX−XXX
Article
ACS Photonics dσsca (180°) = k 4 |αe − αm|2 dΩ
(2)
When the resonant forward scattering condition, αe = αm, is satisfied, we find that the backward scattering is perfectly suppressed.13 The resonant forward scattering condition requires magnetoelectric particles, and thus its realization has remained obscure. However, it has recently been reported that the resonant forward scattering condition can be realized in high-index dielectric particles at resonance.11 On the other hand, the effective permittivity and permeability are given by ε ≈ ε0(1 + Nαe) and μ ≈ μ0(1 + Nαm), respectively.14,15 For the metamaterial with the effective parameters ε and μ, the negative-index condition is given by ε′μ″ + μ′ε″ < 0. The condition can be expressed in terms of αe and αm as follows: ′ + αM ″ )} ε′μ″ + μ′ε″ ≈ 2ε0μ0 + Nε0μ0 {(αE′ + αE″) + (αM 0 and μ″ < 0 (ε″ < 0 and μ″ > 0) become possible at the same frequency. In our dielectric cuboid metamaterials, which exhibit resonant forward scattering, electric and magnetic dipole resonances overlap and the antiresonance can be found with the effective parameters ε″ > 0 and μ″ < 0 (Figure 5b,c). We also emphasize that our metamaterials are in the metamaterial phase, rather than the photonic crystal phase. Although both metamaterials and photonic crystals are composed of periodic structures, their structures differ in terms of periodicity.26 When the periodicity of the structure is comparable to the light wavelength, the periodic structure belongs to the photonic crystal phase, and thus the optical properties of the periodic structure are governed according to the Bragg scattering of periodic elements. On the other hand, the periodic structure is in the metamaterial phase when each inclusion determines the optical properties of the periodic structure and the periodicity cannot support Bragg scattering. The literature has established that the homogenization approach to metamaterials is justified when the Mie gap opens up below the lowest Bragg band gap in the photonic crystal phase.26 In our metamaterials, the negativity of effective indices originates from the Mie resonances of each meta-atom, rather than Bragg scattering by periodic structures, because we use the simultaneous overlap of the electric and magnetic dipole Mie resonances of each dielectric cuboid.
2 ε″(ω1)μ′(ω1) + μ″(ω1)ε′(ω1) 3 ω1 dω1 π (ω12 − ω 2)2 (5)
It was pointed out that loss compensation by active gain materials cannot provide negative refraction. 10 In loss compensation methods, gain materials incorporated into metamaterials cause the imaginary parts of permittivity and permeability to vanish, i.e., ε″ = μ″ = 0. Therefore, since the integral in eq 5 becomes null, such metamaterials incorporating gain materials violate the criterion of negative refraction. Here, we show that our strategy of using the resonant forward scattering in dielectric cuboid meta-atoms can satisfy the criterion of negative refraction, ξ(ω) ≤ −1. In Figure 5a, we plot the ξ(ω) spectrum of the single-layered dielectric cuboid metamaterial whose optical responses are given in Figure 3. We
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CONCLUSIONS To sum up, we demonstrate that dielectric cuboid meta-atoms can realize loss-free metamaterials. The radiative and dissipative losses in metamaterials can be circumvented by resonant D
DOI: 10.1021/acsphotonics.7b01400 ACS Photonics XXXX, XXX, XXX−XXX
Article
ACS Photonics
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forward scattering and the inherent low dissipation characteristic of dielectric materials, respectively. We also prove that the realization of loss-free metamaterials obeys the criterion of negative refraction and does not violate causality or energy conservation. Loss-free metamaterials of wavelength-order thickness are also presented. We expect that our strategy for designing loss-free metamaterials will foster the application of negative refraction.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
SeokJae Yoo: 0000-0002-6438-7123 Q-Han Park: 0000-0002-6301-2645 Author Contributions ‡
S. Yoo and S. Lee contributed equally.
Author Contributions
S.J.Y. established a theory and designed the metamaterials; S.J.Y. and S.Y.L. performed numerical calculations and the microwave experiment; S.J.Y., S.Y.L., and Q.H.P. analyzed data and wrote the manuscript. All authors have given approval to the final version of the manuscript. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the Center for Advanced MetaMaterials (CAMM) funded by the Ministry of Science, ICT and Future Planning as a Global Frontier Project (CAMM2014M3A6B3063710). SeokJae Yoo was also supported by a Korea University Grant, and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2017R1A6A3A11034238).
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REFERENCES
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DOI: 10.1021/acsphotonics.7b01400 ACS Photonics XXXX, XXX, XXX−XXX
