Abstract

Metamaterials are important, as they possess properties not found in simple materials. Photonic device technology applying metamaterials supports many new and useful applications. Here, we address the fundamental physics of wideband metamaterial reflectors. We show that these devices operate because of resonant leaky Bloch modes propagating in the periodic lattice. Moreover, in contrast to published literature, we demonstrate that Mie scattering in individual array particles is not a causal effect. In particular, by connecting the constituent particles by a matched sublayer and thereby destroying the Mie cavity, we find that the resonance bandwidth actually expands even though localized Mie resonances have been extinguished. There is no abrupt change in the reflection characteristics on addition of a sublayer to any metamaterial array consisting of discrete particles. Thus, the physics of the discrete and connected arrays is the same. The resonant Bloch mode picture is supported by numerous additional examples and analyses presented herein.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

All-dielectric metamaterials (MMs) are of interest because of certain unique properties that are not available with natural materials. Dielectric media are generally low-loss, and thus do not exhibit the levels of dissipation typically found in metallic MMs. Extensive studies of theoretical and experimental electromagnetics of MMs are motivated by numerous potential applications: perfect mirrors [14], magnetic reflectors [5,6], zero-index materials [79], negative-index media [10,11], Huygens metasurfaces [12], achromatic metasurfaces [13], and flat lenses [14] have been suggested. Thus far, most MMs have been developed by tuning the geometry of an assembly of particles including cylinders, rods, and spheres that are embedded in, or laid on, lower-index media [15]. Therefore, theoretical modeling aimed at quantifying the pertinent spectral response has focused on resonant interactions between light and particles [1618]. A seminal work by Mie led to analytical solutions for the scattering of an electromagnetic plane wave by a homogenous sphere [19]. Indeed, Mie scattering by individual particles is considered in most of the literature modeling MM response.

From a historical perspective, in the 1940s, Lewin derived a formula for the effective permittivity (ϵ) and permeability (μ) of a mixture containing a simple cubic array of spheres [20]. He employed the first term in an infinite series of a vector wave function, assuming that interparticle interactions were negligible due to sufficiently small particle size and volume fraction. Using Lewin’s model, Holloway et al. showed numerically that a composite structure containing lossless magnetodielectric spheres exhibits negative values (ϵ<0 and μ<0) at Mie resonance [21]. Moreover, O’Brien and Pendry found magnetic activity (μ<0) in multilayer arrays of dielectric cylinders exploiting Mie resonances [22]. For practical realization, the composite effective-medium structure should be simple, preferably a single layer. Thus, Ginn and Brener predicted magnetic activity in a single-layer dielectric cubic structure by homogenization [23]. Slovick et al. found that single-negative MMs (ϵ<0 or μ<0) led to high reflection in a semi-infinite model medium and also theoretically illustrated a wideband reflector by a monolayer of silicon (Si) microcubes [1]. Subsequently, several research groups proposed various perfect dielectric metamaterial reflectors using cylinder or rod arrays [14].

Here, we address the fundamental physics of MM reflectors. It is widely assumed that Mie resonance in the constituent unit cell medium is foundational to wideband MM reflection. Indeed, there is a substantial and growing body of literature citing particle-based Mie resonance as the fundamental effect enabling MM reflectors [16,15,2428]. We provide an alternate view by modeling two-dimensional (2D) and one-dimensional (1D) reflectors with all Mie resonances extinguished. We show that MM reflectors operate in lattice resonance dominated by lateral leaky Bloch modes [29,30]. This lattice resonance is one type of Fano resonance referred to as guided-mode or leaky-mode resonance. Figure 1 compares the classic Mie resonance approach and the leaky Bloch mode approach. The usual procedure is to cast the periodic MM reflectors into an equivalent homogeneous slab and then derive the effective material parameters of impedance (Z), refractive index (n), permittivity (ϵ), and permeability (μ) as noted in Fig. 1(a). A standard retrieval algorithm [3133] then delivers the reflectance of each device studied; it is known that the retrieved parameters may be nonphysical (non-passivity and non-causality) due to neglected spatial dispersion [3436]. We compare the results of this method to the results of numerical optimization indicated in Fig. 1(b). The MM reflectors provided here annihilate classic Mie resonances by a connective sublayer that prevents individual particle resonance from appearing in the array. We show smooth transitions in the reflectance spectra on morphing from the fully discrete to the connected-particle reflector. Using the standard homogenization method, the connected reflector exhibits negative MM parameter (ϵ<0 or μ<0) spectra as commonly attributed to Mie scattering, even though no Mie resonances can be supported.

 figure: Fig. 1.

Fig. 1. Fundamental theories to model dielectric MM reflectors. (a) Assembled from single particles into periodic arrays, Mie scattering and effective medium theory (EMT) are widely used to predict the spectral response of a reflector. This methodology engineers the electric and magnetic dipoles of isolated elements to design dielectric MM reflectors. (b) Lattice resonance grounded in lateral leaky Bloch modes provides a unified approach that applies to arrays of discrete and connected elemental particles. With numerical optimization, both 1D and 2D periodic reflectors are designed.

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2. RESULTS

A. Resonance Effects in Discrete/Connected Dielectric Gratings

To connect with current literature on the subject, we build our first example on a design provided by Moitra et al. [2]. Figure 2(a) illustrates a Si-based (ϵSi=12) 2D grating where Λ = period, H = grating height, D = rod diameter, and Dh = homogenous layer thickness. We set Λ=660nm, H=500nm, and D=400nm as in the reference design [2]. We apply rigorous coupled-wave analysis (RCWA) [37] to find the zero-order reflectance (R0) at normal incidence, with results shown in Fig. 2(b) for Dh ranging up to 300 nm. A complex mixture of high-reflection bands appears in the contour map. The widest bands appear at Dh=78 and Dh=110nm, as detailed in Fig. 2(c). This enhancement is attributed to guided-mode resonant lateral Bloch modes that locate primarily in the sublayer [33]. This is clearly signified by the view of the internal electric field in the inset of Fig. 2(c), where Dh=78nm and λ=1.45μm. The field distribution shows that the resonance drives counterpropagating lateral modes forming standing waves, as noted schematically in Fig. 1(b). Similarly, lateral Bloch modes exist in the discrete-element grating displayed in Fig. 2(d). Since the modulation strength of the dielectric constant Δϵ=nH2nc2 controls the reflection bandwidth, we modify this value to investigate the evolution of the reflection bands. For strong modulation (nH=nSi, nc=nair and Δϵ=11), as in Fig. 2(e), wideband reflection bands are achievable for particular grating heights (H). For R0>99%, the largest bandwidth is 72 nm at H=500nm, as previously designed [2]. However, with the same H, it increases to 220nm at Dh=78 and 110 nm, as seen in Fig. 2(c). On decreasing the modulation strength to Δϵ2.5, narrower reflection signatures appear as in Fig. 2(f). Again, these sharp resonance loci originate from the lateral leaky modes effectively propagating in the discrete grating. Figure 2(g) shows classical modal curves in the equivalent homogenous slab waveguide, where the effective refractive indices are estimated with second-order effective medium theory for TM and TE modes. The details of the treatment and calculations are explained in Supplement 1 and summarized in Fig. S1. There is good qualitative agreement between Figs. 2(f) and 2(g). Moreover, we can explain the complex modal processes in both discrete and connected 2D reflectors by decomposition into quasi-equivalent 1D grating components; this is provided in Supplement 1 in Fig. S2. It is clear that the major reflectance features of the 2D grating reflector originate in a blend of TM and TE resonant leaky modes derivable from the 1D equivalents [38]. We note that results similar to those in Figs. 2(f) and 2(g) were found before for 1D gratings without a sublayer [39]; here, this is extended to the 2D case pertinent to most metamaterial reflectors. We remark that Mie scattering was not specifically treated in [39].

 figure: Fig. 2.

Fig. 2. Wideband metamaterial reflector examples. (a) Structure of a silicon (Si) reflector with a homogenous sublayer. (b) Calculated reflectance (R0) map as a function of sublayer thickness Dh. (c) Reflectance spectra for optimal Dh=78 and 110 nm as compared with the spectrum without a sublayer. The electric-field distribution in the device is shown in the inset. (d) Grating reflector without a sublayer with a high-index (nH) dielectric array in a low-index (nC) background. (e) R0 map for the reflector in (d) under strong index modulation (nH=3.464, nC=nL=1). As the refractive-index modulation is weakened, the reflection bands narrow and take on the signature of discrete waveguide modes in an effective-medium slab corresponding to the reflector. Thus, in (f) we show an R0 map for weak index modulation (nH=2.794, nL=2.3, nC=1) and in (g) we display the modal curves in the slab waveguide where the observed modes are driven by the first evanescent diffraction orders of the grating. The agreement between (f) and (g) is undeniable and strongly supports the leaky-mode resonance picture of this device class.

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To compare 1D and 2D reflectors, we modify the 2D device to obtain a Si-based 1D reflector with comparable parameters as shown in Fig. 3(a). The 1D grating structure is identified by a parameter set {Λ=660nm, F=0.6, Dg=430nm, Dh} where Dg and F are grating depth and fill factor. Here, the input plane wave is TM polarized and at normal incidence. In the range 250nm<Dh<350nm, as shown in Fig. 3(b), the high-reflection band is much wider than the band for the reflector consisting of discrete parallel ridges (Dh=0nm). The local field again derives form interference between couterpropagating leaky modes, resulting in a standing-wave configuration. In Fig. 3(c), the field is localized in the isolated rods whereas in Fig. 3(d) the guided lateral Bloch modes locate largely in the sublayer of the constituent grating (Dh=267nm).

 figure: Fig. 3.

Fig. 3. TM-polarized resonant reflector based on a 1D Si grating. The grating excites resonant leaky modes, providing wide reflection bands. (a) Reflector designed with traditional grating parameters. (b) Calculated R0 map as a function of the sublayer thickness. Cross-sectional views of the magnetic-field amplitude distribution appear in (c) for the grating with discrete ridges (Dh=0nm) at λ=1.62μm and in (d) for the connected grating with a sublayer Dh=267nm at λ=1.55μm.

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B. Local Mode Signatures in Isolated/Periodic Nanorods

To investigate the influence of any Mie resonances in the lattice, we treat the 1D Si-grating reflector in Fig. 3(a) further. Thus, we apply the finite-difference time-domain (FDTD) method to calculate the total scattering cross section (TSCS) spectrum of an isolated infinite Si rod with the same geometry as shown in Figs. 4(a) and 4(b). At wavelengths (i) 1.426 and (ii) 2.1 μm, the TSCS peaks mark the appearance of the electric and magnetic dipoles, respectively, whose field maps are shown in Fig. 4(c). In between these resonances, for example at λ=1.62μm, the amplitude distribution maintains features of the dipole fields as seen in Fig. 4(d). Whereas the single rod does not effectively suppress forward scattering, light transmission is totally negated by the periodic array as seen in Fig. 4(e). There, the amplitude distribution takes on a shape reminiscent of the Mie magnetic dipole that is located nearly 500 nm away in wavelength. Moreover, Figs. 4(d) and 4(e) illustrate the distinctly different phase distribution of the individual and arrayed rods. This point is elaborated further in Supplement 1 in Fig. S3 at λ=1.426μm, at which the electric Mie dipole appears.

 figure: Fig. 4.

Fig. 4. Mie scattering and guided-mode resonance in relation to the 1D Si grating in Fig. 3(c). (a) Schematic of Mie scattering with a single infinite Si rod. (b) Calculated total scattering cross-section spectrum for the rod with R0 for the corresponding periodic array measured on the right-hand scale. (c) Field magnitude profiles at principal wavelengths (i) 1.426 and (ii) 2.1 μm. Magnitude and phase of the field distribution in (d) a single rod and (e) Si grating at the same wavelength of (iii) 1.62 μm.

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To demonstrate the electrodynamics of light propagation in the reflector, we provide video demonstration simulated with FDTD computations. In an infinite grating with periodic boundary conditions (Visualization 1 and Visualization 2), the field forms a standing wave by counterpropagating Bloch modes for gratings with and without a sublayer. Thus, the field appears to be laterally stationary. To eliminate the standing waves numerically, we annul reflection at the sides of the simulation boundary using perfectly matched layers. In Visualization 3 and Visualization 4, lateral propagation, characteristic of resonant leaky modes, is seen. Further explanations of the movie simulations are given under Supplement 1 in Fig. S4.

C. Effective Parameters of Homogenized MM Reflectors

We now apply the standard methods used to find reflectance in the Mie resonance domain to our non-Mie resonators. Following the literature [3133], the effective material parameters are derived by homogenization of the 1D or 2D constituent gratings. The procedure is summarized in Fig. 5(a). To retrieve the effective optical parameters, we treat an example periodic array as a single homogenous slab and obtain the complex μ and ε from the S-parameters (S11 and S21). The reflection (r) and transmission (t) coefficients are calculated by RCWA and then S11 and S21 are taken by r and teik0d, where d and k0 denote the thickness of the slab and the wavenumber of the incident wave in air. Figures 5(b) and 5(c) show the numerical RCWA reflectance for both the discrete and non-discrete grating reflectors. The device parameters are as in Figs. 3(c) and 3(d). In Figs. 5(d) and 5(e), the high-reflection band matches the region for n0 when Z=0, which is consistent with previous reports [13]. High reflection also arises when ZZ because the reflectance at the boundary of the slab (|Γ|2=[(Z1)2+Z2]/[(Z+1)2+Z2]) is high. At 1.347 μm in Fig. 5(e), Z=2.81314.38i, with |Γ|2=0.95. Compared to the discrete grating, the connected grating provides a wider spectral range while meeting these conditions. In Fig. 5(g), it is noted that the non-discrete grating also provides negative values in μ as well as in ε, even though the interconnecting sublayer mutes any local Mie resonances. Therefore it is not reasonable to attribute the dispersion in the fundamental parameters shown in Figs. 5(f) and 5(g) to Mie scattering, which is the main point of this example.

 figure: Fig. 5.

Fig. 5. Effective optical properties of 1D discrete and connected grating reflectors. (a) Algorithmic procedure for determination of effective electromagnetic parameters. Retrieved and RCWA-computed reflectance for the (b) 1D discrete reflector and (c) 1D non-discrete reflector. The effective material constants (Z, n) and (μ+jμ and ε+jε) are calculated by homogenization for these (d), (f) discrete and (e), (g) non-discrete reflectors. The grating parameter sets are {Λ=660nm, F=0.6, Dg=430nm, Dh=0nm} and {Λ=660nm, F=0.6, Dg=430nm, Dh=267nm}.

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3. DISCUSSION

We have addressed the fundamental physics behind wideband dielectric metamaterial reflectors. We have shown that this device class operates under guided-mode resonance effects grounded in lateral leaky Bloch modes in both 1D and 2D periodic lattices. The results presented herein contradict many published reports on wideband metamaterial reflectors that more or less uniformly declare Mie scattering, characteristic of the individual constituent particles, as the fundamental causal effect. We have presented many arguments to support our claims. In particular, we have shown that the addition of a sublayer of the same material to the collection of particles (pillars or cylinders) actually greatly expands the reflectance bandwidth. As the matched sublayer annihilates any localized resonances (Mie or Fabry–Perot) by destroying the resonance cavity, a Mie-resonance-based reflector should cease to function under this modification. From another point of view, connecting the individual particles with the sublayer should adversely alter the resonance dynamics. This is not the case, as evidenced by numerous computed results. For example, in Fig. 2(b) for 2D pillars and Fig. 3(b) for rectangular cylinders, there is a smooth change in the reflectance as the sublayer thickness rises. Indeed, these devices work very well under conditions where most of the resonant fields reside nearly entirely in the sublayer. These conclusions are consistent with prior work showing that high-contrast interfaces and local Fabry–Perot resonances are not the root cause of wideband reflection observed in high-contrast gratings [40].

Another convincing argument supporting the guided-mode resonance picture is provided in Figs. 2(d)2(g), pertaining to the isolated 2D pillar array. In Fig. 2(e), there appears a reflectance map for the fully modulated array with pillar refractive index of 3.464 (ϵ=12) located in air. Reducing the modulation strength as in Fig. 2(f) retains key features including the mode-line curvatures of Fig. 2(e), albeit with reduced reflectance zone widths. Then, we simplify the reduced reflector in Fig. 2(f) to an equivalent slab using classic Rytov-type effective-medium theory [41]. Solving the attendant eigenvalue problem yields Fig. 2(g), which has clear qualitative connection to the periodic device and reveals the identity of the various classic waveguide modes behind the resonance signatures. In the periodic reflector, these classic modes morph into the lateral leaky Bloch modes while retaining some features of the originating mode shape. Under increasing modulation strength, such as in Fig. 2(e), the leaky modes become abstractions of the original parent modes. Additionally, the numerical movies provided demonstrate lateral mode propagation along the lattice vectors.

Further treating the guided-mode resonance regime, in Fig. 6 we provide a reflectance map as a function of period for the Si cylinder array in Fig. 2(d). A first point to note is that the high-reflectance regions fall within the subwavelength guided-mode resonance regime nairλ/Λ<nSi, as expected for an efficient lattice-resonance-based process. For the larger periods within this region, there first appear discrete-line modal signatures as noted in the figure. As the period falls and the pillar density of the reflector increases, the effective refractive index increases and the mode loci merge, providing a substantially wide band. We note that this region, delimited by dashed lines in Fig. 6, denotes a subwavelength region Λ<λ but not the deep subwavelength condition Λλ demanded by accurate effective-medium theory.

 figure: Fig. 6.

Fig. 6. Variation of the reflectance of the 2D grating with period. Reflectance as a function of the period for the 2D Si discrete-particle reflector where the rod diameter is D=400nm with height H=500nm.

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The methodology used to derive the effective parameters of periodic arrays of Mie scatterers is illustrated schematically in Fig. 1. A main point is homogenization, whereby the periodic array is cast into a single “equivalent” slab. The parametric retrieval methods that have been developed are robust in that parameters are retrieved as summarized in the introduction. Nevertheless, it has been pointed out that the extracted parameters frequently violate conventional passivity and causality near material or lattice resonances [23,35,36]. For that reason, Alù suggested first-principles homogenization considering spatial dispersion in periodic metamaterials yielding physically meaningful parameters [35]. Applying the numerical steps outlined in Fig. 5(a) often used for dielectric MM reflectors [13], we show results for discrete and connected reflectors in Figs. 5(b)5(g) computing effective Z, n, ϵ and μ. Spectral signatures widely attributed to Mie-scattering-based resonances appear even though the connected device is imbued with a 267 nm thick sublayer and supports no Mie resonances.

As a final point, works by Gomez–Medina et al. [42] and Ghenuche et al. [43] provide additional support for the main conclusions of the present paper. Treating arrays of subwavelength cylinders with small radii (nanorods), narrowband unity reflection was found even though the geometric and materials parameters were such that no Mie resonances were supported [42]. Experimental verification was subsequently reported [43]. The narrow “geometrical resonances” reported in these works are fundamentally the same as the guided-mode resonances that are the main focus here.

4. CONCLUSION

In summary, wideband MM reflectors operate under guided-mode resonance consistent with propagation of Bloch modes in the periodic lattice. Their physics cannot be explained with the localized Mie of Fabry–Perot resonant modes. Progress in application of periodic or aperiodic nanostructures necessitates clear understanding of the correct fundamentals. The explanations set forth in this paper aid in charting a progressive path forward as this field of study matures.

Funding

National Science Foundation (NSF) (ECCS-1549851, ECCS-1606898, IIP-1444922); Texas Instruments Distinguished University Chair in Nanoelectronics Endowment.

 

See Supplement 1 for supporting content.

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References

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  1. B. Slovick, Z. G. Yu, M. Berding, and S. Krishnamurthy, “Perfect dielectric-metamaterial reflector,” Phys. Rev. B 88, 165114 (2013).
    [Crossref]
  2. P. Moritra, B. A. Slovick, Z. G. Yu, S. Krishnamurthy, and J. Valentine, “Experimental demonstration of a broadband all-dielectric metamaterial perfect reflector,” Appl. Phys. Lett. 104, 171102 (2014).
    [Crossref]
  3. P. Moritra, B. A. Slovick, W. Li, I. Kravchencko, D. P. Briggs, S. Krishnamurthy, and J. Valentine, “Large-scale all-dielectric metamaterial perfect reflectors,” ACS Photon. 2, 692–698 (2015).
    [Crossref]
  4. Z. Liu, X. Liu, Y. Wang, and P. Pan, “High-index dielectric meta-materials for near-perfect broadband reflectors,” J. Phys. D 49, 195101 (2016).
    [Crossref]
  5. J. Z. Hao, Y. Seokho, L. Lan, D. Brocker, D. H. Werner, and T. S. Mayer, “Experimental demonstration of an optical artificial perfect magnetic mirror using dielectric resonators,” in IEEE Antennas and Propagation Society International Symposium (2012), pp. 1–2.
  6. S. Liu, M. B. Sinclair, T. S. Mahony, Y. C. Jun, S. Campione, J. Ginn, D. A. Bender, J. R. Wendt, J. F. Ihlefeld, P. G. Clem, J. B. Wright, and I. Grener, “Optical magnetic mirrors without metals,” Optica 1, 250–256 (2014).
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  8. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582–586 (2011).
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  9. P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7, 791–795 (2013).
    [Crossref]
  10. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438, 335–338 (2005).
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  11. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007).
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  12. M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3, 813–820 (2015).
    [Crossref]
  13. F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
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  14. F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
    [Crossref]
  15. S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotechnol. 11, 23–36 (2016).
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  16. P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface Mie resonators,” Nat. Commun. 3, 692 (2012).
    [Crossref]
  17. I. Popa and S. A. Cummer, “Compact dielectric particles as a building block for low-loss magnetic metamaterials,” Phys. Rev. Lett. 100, 207401 (2008).
    [Crossref]
  18. 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, 027402 (2008).
    [Crossref]
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  21. C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
    [Crossref]
  22. S. O’Brien and J. B. Pendry, “Magnetic activity at infrared frequencies in structured metallic photonic crystals,” J. Phys. Condens. Matter 14, 6383–6394 (2002).
    [Crossref]
  23. J. C. Ginn and I. Brener, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
    [Crossref]
  24. A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354, aag2472 (2016).
    [Crossref]
  25. S. J. Corbitt, M. Francoeur, and B. Raeymaekers, “Implementation of optical dielectric metamaterials: a review,” J. Quant. Spectrosc. Radiat. Transfer 158, 3–16 (2015).
    [Crossref]
  26. I. Staude and J. Schilling, “Metamaterial-inspired silicon nanophotonics,” Nat. Photonics 11, 274–284 (2017).
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  27. M. Decker and I. Staude, “Resonant dielectric nanostructures: a low-loss platform for functional nanophotonics,” J. Opt. 18, 103001 (2016).
    [Crossref]
  28. L. Li, J. Wang, J. Wang, H. Du, H. Huang, J. Zhang, S. Qu, and Z. Xu, “All-dielectric metamaterial frequency selective surfaces based on high-permittivity ceramic resonators,” Appl. Phys. Lett. 106, 212904 (2015).
    [Crossref]
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    [Crossref]
  32. D. R. Smith and S. Schultz, “Determination of effective permittvity and permeability of metamaterials from reflection and transmittance coefficients,” Phys. Rev. B 65, 195104 (2002).
    [Crossref]
  33. X. Chen, T. M. Grzegorczyk, B. I. Wu, and J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004).
    [Crossref]
  34. C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. 13, 013001 (2011).
    [Crossref]
  35. A. Alù, “Restoring the physical meaning of metamaterial constitutive parameters,” Phys. Rev. B 83, 081102 (2011).
    [Crossref]
  36. V. Grigoriev, G. Demésy, J. Wenger, and N. Bonod, “Singular analysis to homogenize planar metamaterials as nonlocal effective media,” Phys. Rev. B 89, 245102 (2014).
    [Crossref]
  37. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
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  38. Y. H. Ko, M. Shokooh-Saremi, and R. Magnusson, “Modal processes in two-dimensional resonant reflector and their correlation with spectra of one-dimensional equivalents,” IEEE Photon. J. 7, 4900210 (2015).
    [Crossref]
  39. R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16, 3456–3462 (2008).
    [Crossref]
  40. R. Magnusson, “Wideband reflectors with zero-contrast gratings,” Opt. Lett. 39, 4337–4340 (2014).
    [Crossref]
  41. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).
  42. R. Gomez-Medina, M. Laroche, and J. J. Saenz, “Extraordinary optical reflection from sub-wavelength cylinder arrays,” Opt. Express 14, 3730–3737 (2006).
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  43. P. Ghenuche, G. Vincent, M. Larcoche, N. Bardou, R. Hadïar, J. Pelouard, and S. Collin, “Optical extinction in a single layer of nanorods,” Phys. Rev. Lett. 109, 143903 (2012).
    [Crossref]

2017 (1)

I. Staude and J. Schilling, “Metamaterial-inspired silicon nanophotonics,” Nat. Photonics 11, 274–284 (2017).
[Crossref]

2016 (4)

M. Decker and I. Staude, “Resonant dielectric nanostructures: a low-loss platform for functional nanophotonics,” J. Opt. 18, 103001 (2016).
[Crossref]

A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354, aag2472 (2016).
[Crossref]

Z. Liu, X. Liu, Y. Wang, and P. Pan, “High-index dielectric meta-materials for near-perfect broadband reflectors,” J. Phys. D 49, 195101 (2016).
[Crossref]

S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotechnol. 11, 23–36 (2016).
[Crossref]

2015 (6)

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3, 813–820 (2015).
[Crossref]

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
[Crossref]

P. Moritra, B. A. Slovick, W. Li, I. Kravchencko, D. P. Briggs, S. Krishnamurthy, and J. Valentine, “Large-scale all-dielectric metamaterial perfect reflectors,” ACS Photon. 2, 692–698 (2015).
[Crossref]

S. J. Corbitt, M. Francoeur, and B. Raeymaekers, “Implementation of optical dielectric metamaterials: a review,” J. Quant. Spectrosc. Radiat. Transfer 158, 3–16 (2015).
[Crossref]

L. Li, J. Wang, J. Wang, H. Du, H. Huang, J. Zhang, S. Qu, and Z. Xu, “All-dielectric metamaterial frequency selective surfaces based on high-permittivity ceramic resonators,” Appl. Phys. Lett. 106, 212904 (2015).
[Crossref]

Y. H. Ko, M. Shokooh-Saremi, and R. Magnusson, “Modal processes in two-dimensional resonant reflector and their correlation with spectra of one-dimensional equivalents,” IEEE Photon. J. 7, 4900210 (2015).
[Crossref]

2014 (4)

R. Magnusson, “Wideband reflectors with zero-contrast gratings,” Opt. Lett. 39, 4337–4340 (2014).
[Crossref]

V. Grigoriev, G. Demésy, J. Wenger, and N. Bonod, “Singular analysis to homogenize planar metamaterials as nonlocal effective media,” Phys. Rev. B 89, 245102 (2014).
[Crossref]

S. Liu, M. B. Sinclair, T. S. Mahony, Y. C. Jun, S. Campione, J. Ginn, D. A. Bender, J. R. Wendt, J. F. Ihlefeld, P. G. Clem, J. B. Wright, and I. Grener, “Optical magnetic mirrors without metals,” Optica 1, 250–256 (2014).
[Crossref]

P. Moritra, B. A. Slovick, Z. G. Yu, S. Krishnamurthy, and J. Valentine, “Experimental demonstration of a broadband all-dielectric metamaterial perfect reflector,” Appl. Phys. Lett. 104, 171102 (2014).
[Crossref]

2013 (2)

B. Slovick, Z. G. Yu, M. Berding, and S. Krishnamurthy, “Perfect dielectric-metamaterial reflector,” Phys. Rev. B 88, 165114 (2013).
[Crossref]

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7, 791–795 (2013).
[Crossref]

2012 (4)

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref]

P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface Mie resonators,” Nat. Commun. 3, 692 (2012).
[Crossref]

J. C. Ginn and I. Brener, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
[Crossref]

P. Ghenuche, G. Vincent, M. Larcoche, N. Bardou, R. Hadïar, J. Pelouard, and S. Collin, “Optical extinction in a single layer of nanorods,” Phys. Rev. Lett. 109, 143903 (2012).
[Crossref]

2011 (3)

C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. 13, 013001 (2011).
[Crossref]

A. Alù, “Restoring the physical meaning of metamaterial constitutive parameters,” Phys. Rev. B 83, 081102 (2011).
[Crossref]

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582–586 (2011).
[Crossref]

2010 (1)

Z. Szabó, G. H. Park, R. Hedge, and E. P. Li, “A unique extraction of metamaterial parameters based on Kramer–Kronig relationship,” IEEE Trans. Microw. Theory Tech. 58, 2646–2653 (2010).
[Crossref]

2008 (3)

I. Popa and S. A. Cummer, “Compact dielectric particles as a building block for low-loss magnetic metamaterials,” Phys. Rev. Lett. 100, 207401 (2008).
[Crossref]

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, 027402 (2008).
[Crossref]

R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16, 3456–3462 (2008).
[Crossref]

2007 (2)

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007).
[Crossref]

M. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B 75, 075119 (2007).
[Crossref]

2006 (1)

2005 (1)

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438, 335–338 (2005).
[Crossref]

2004 (2)

Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express 12, 5661–5674 (2004).
[Crossref]

X. Chen, T. M. Grzegorczyk, B. I. Wu, and J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004).
[Crossref]

X. Chen, T. M. Grzegorczyk, B. I. Wu, and J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004).
[Crossref]

2003 (1)

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[Crossref]

2002 (2)

S. O’Brien and J. B. Pendry, “Magnetic activity at infrared frequencies in structured metallic photonic crystals,” J. Phys. Condens. Matter 14, 6383–6394 (2002).
[Crossref]

D. R. Smith and S. Schultz, “Determination of effective permittvity and permeability of metamaterials from reflection and transmittance coefficients,” Phys. Rev. B 65, 195104 (2002).
[Crossref]

1995 (1)

1990 (1)

1956 (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

1947 (1)

L. Lewin, “The electrical constants of a material loaded with spherical particles,” J. Inst. Electr. Eng. Part III 94, 65–68 (1947).
[Crossref]

1908 (1)

G. Mie, “Beiträge zur Optic truber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[Crossref]

Aieta, F.

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
[Crossref]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref]

Alù, A.

A. Alù, “Restoring the physical meaning of metamaterial constitutive parameters,” Phys. Rev. B 83, 081102 (2011).
[Crossref]

Anderson, Z.

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7, 791–795 (2013).
[Crossref]

Bagby, J. S.

Baker-Jarvis, J.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[Crossref]

Bardou, N.

P. Ghenuche, G. Vincent, M. Larcoche, N. Bardou, R. Hadïar, J. Pelouard, and S. Collin, “Optical extinction in a single layer of nanorods,” Phys. Rev. Lett. 109, 143903 (2012).
[Crossref]

Bender, D. A.

Berding, M.

B. Slovick, Z. G. Yu, M. Berding, and S. Krishnamurthy, “Perfect dielectric-metamaterial reflector,” Phys. Rev. B 88, 165114 (2013).
[Crossref]

Blanchard, R.

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref]

Bonod, N.

V. Grigoriev, G. Demésy, J. Wenger, and N. Bonod, “Singular analysis to homogenize planar metamaterials as nonlocal effective media,” Phys. Rev. B 89, 245102 (2014).
[Crossref]

Brener, I.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3, 813–820 (2015).
[Crossref]

J. C. Ginn and I. Brener, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108, 097402 (2012).
[Crossref]

Briggs, D. P.

P. Moritra, B. A. Slovick, W. Li, I. Kravchencko, D. P. Briggs, S. Krishnamurthy, and J. Valentine, “Large-scale all-dielectric metamaterial perfect reflectors,” ACS Photon. 2, 692–698 (2015).
[Crossref]

P. Moitra, Y. Yang, Z. Anderson, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nat. Photonics 7, 791–795 (2013).
[Crossref]

Brocker, D.

J. Z. Hao, Y. Seokho, L. Lan, D. Brocker, D. H. Werner, and T. S. Mayer, “Experimental demonstration of an optical artificial perfect magnetic mirror using dielectric resonators,” in IEEE Antennas and Propagation Society International Symposium (2012), pp. 1–2.

Brongersma, M. L.

A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354, aag2472 (2016).
[Crossref]

Campione, S.

Capasso, F.

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
[Crossref]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref]

Chan, C. T.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582–586 (2011).
[Crossref]

Chen, X.

X. Chen, T. M. Grzegorczyk, B. I. Wu, and J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004).
[Crossref]

Clem, P. G.

Collin, S.

P. Ghenuche, G. Vincent, M. Larcoche, N. Bardou, R. Hadïar, J. Pelouard, and S. Collin, “Optical extinction in a single layer of nanorods,” Phys. Rev. Lett. 109, 143903 (2012).
[Crossref]

Corbitt, S. J.

S. J. Corbitt, M. Francoeur, and B. Raeymaekers, “Implementation of optical dielectric metamaterials: a review,” J. Quant. Spectrosc. Radiat. Transfer 158, 3–16 (2015).
[Crossref]

Cummer, S. A.

I. Popa and S. A. Cummer, “Compact dielectric particles as a building block for low-loss magnetic metamaterials,” Phys. Rev. Lett. 100, 207401 (2008).
[Crossref]

Decker, M.

M. Decker and I. Staude, “Resonant dielectric nanostructures: a low-loss platform for functional nanophotonics,” J. Opt. 18, 103001 (2016).
[Crossref]

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3, 813–820 (2015).
[Crossref]

Demésy, G.

V. Grigoriev, G. Demésy, J. Wenger, and N. Bonod, “Singular analysis to homogenize planar metamaterials as nonlocal effective media,” Phys. Rev. B 89, 245102 (2014).
[Crossref]

Ding, Y.

Dominguez, J.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3, 813–820 (2015).
[Crossref]

Du, B.

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, 027402 (2008).
[Crossref]

Du, H.

L. Li, J. Wang, J. Wang, H. Du, H. Huang, J. Zhang, S. Qu, and Z. Xu, “All-dielectric metamaterial frequency selective surfaces based on high-permittivity ceramic resonators,” Appl. Phys. Lett. 106, 212904 (2015).
[Crossref]

Engheta, N.

M. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B 75, 075119 (2007).
[Crossref]

Falkner, M.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3, 813–820 (2015).
[Crossref]

Firsov, A. A.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438, 335–338 (2005).
[Crossref]

Francoeur, M.

S. J. Corbitt, M. Francoeur, and B. Raeymaekers, “Implementation of optical dielectric metamaterials: a review,” J. Quant. Spectrosc. Radiat. Transfer 158, 3–16 (2015).
[Crossref]

Gaburro, Z.

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref]

Gaylord, T. K.

Geim, A. K.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438, 335–338 (2005).
[Crossref]

Genevet, P.

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
[Crossref]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref]

Ghenuche, P.

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P. Moritra, B. A. Slovick, Z. G. Yu, S. Krishnamurthy, and J. Valentine, “Experimental demonstration of a broadband all-dielectric metamaterial perfect reflector,” Appl. Phys. Lett. 104, 171102 (2014).
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P. Moritra, B. A. Slovick, Z. G. Yu, S. Krishnamurthy, and J. Valentine, “Experimental demonstration of a broadband all-dielectric metamaterial perfect reflector,” Appl. Phys. Lett. 104, 171102 (2014).
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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, 027402 (2008).
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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, 027402 (2008).
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ACS Photon. (1)

P. Moritra, B. A. Slovick, W. Li, I. Kravchencko, D. P. Briggs, S. Krishnamurthy, and J. Valentine, “Large-scale all-dielectric metamaterial perfect reflectors,” ACS Photon. 2, 692–698 (2015).
[Crossref]

Adv. Opt. Mater. (1)

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3, 813–820 (2015).
[Crossref]

Ann. Phys. (1)

G. Mie, “Beiträge zur Optic truber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
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Supplementary Material (5)

NameDescription
» Supplement 1       Supplementary Materials
» Visualization 1       Infinite grating with discrete ridges without sublayer.
» Visualization 2       Nondiscrete infinite grating with a sublayer.
» Visualization 3       Discrete finite grating without sublayer.
» Visualization 4       Nondiscrete finite grating with a sublayer.

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Figures (6)

Fig. 1.
Fig. 1. Fundamental theories to model dielectric MM reflectors. (a) Assembled from single particles into periodic arrays, Mie scattering and effective medium theory (EMT) are widely used to predict the spectral response of a reflector. This methodology engineers the electric and magnetic dipoles of isolated elements to design dielectric MM reflectors. (b) Lattice resonance grounded in lateral leaky Bloch modes provides a unified approach that applies to arrays of discrete and connected elemental particles. With numerical optimization, both 1D and 2D periodic reflectors are designed.
Fig. 2.
Fig. 2. Wideband metamaterial reflector examples. (a) Structure of a silicon (Si) reflector with a homogenous sublayer. (b) Calculated reflectance ( R 0 ) map as a function of sublayer thickness D h . (c) Reflectance spectra for optimal D h = 78 and 110 nm as compared with the spectrum without a sublayer. The electric-field distribution in the device is shown in the inset. (d) Grating reflector without a sublayer with a high-index ( n H ) dielectric array in a low-index ( n C ) background. (e)  R 0 map for the reflector in (d) under strong index modulation ( n H = 3.464 , n C = n L = 1 ). As the refractive-index modulation is weakened, the reflection bands narrow and take on the signature of discrete waveguide modes in an effective-medium slab corresponding to the reflector. Thus, in (f) we show an R 0 map for weak index modulation ( n H = 2.794 , n L = 2.3 , n C = 1 ) and in (g) we display the modal curves in the slab waveguide where the observed modes are driven by the first evanescent diffraction orders of the grating. The agreement between (f) and (g) is undeniable and strongly supports the leaky-mode resonance picture of this device class.
Fig. 3.
Fig. 3. TM-polarized resonant reflector based on a 1D Si grating. The grating excites resonant leaky modes, providing wide reflection bands. (a) Reflector designed with traditional grating parameters. (b) Calculated R 0 map as a function of the sublayer thickness. Cross-sectional views of the magnetic-field amplitude distribution appear in (c) for the grating with discrete ridges ( D h = 0 nm ) at λ = 1.62 μm and in (d) for the connected grating with a sublayer D h = 267 nm at λ = 1.55 μm .
Fig. 4.
Fig. 4. Mie scattering and guided-mode resonance in relation to the 1D Si grating in Fig. 3(c). (a) Schematic of Mie scattering with a single infinite Si rod. (b) Calculated total scattering cross-section spectrum for the rod with R 0 for the corresponding periodic array measured on the right-hand scale. (c) Field magnitude profiles at principal wavelengths (i) 1.426 and (ii) 2.1 μm. Magnitude and phase of the field distribution in (d) a single rod and (e) Si grating at the same wavelength of (iii) 1.62 μm.
Fig. 5.
Fig. 5. Effective optical properties of 1D discrete and connected grating reflectors. (a) Algorithmic procedure for determination of effective electromagnetic parameters. Retrieved and RCWA-computed reflectance for the (b) 1D discrete reflector and (c) 1D non-discrete reflector. The effective material constants ( Z , n ) and ( μ + j μ and ε + j ε ) are calculated by homogenization for these (d), (f) discrete and (e), (g) non-discrete reflectors. The grating parameter sets are { Λ = 660 nm , F = 0.6 , D g = 430 nm , D h = 0 nm } and { Λ = 660 nm , F = 0.6 , D g = 430 nm , D h = 267 nm }.
Fig. 6.
Fig. 6. Variation of the reflectance of the 2D grating with period. Reflectance as a function of the period for the 2D Si discrete-particle reflector where the rod diameter is D = 400 nm with height H = 500 nm .

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