## 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 [1–4], magnetic reflectors [5,6], zero-index materials [7–9], 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 [16–18]. 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 ($\u03f5$) and permeability
($\mu $) of a mixture containing a simple cubic
array of spheres [20]. He employed
the first term in an infinite series of a vector wave function, assuming
that interparticle interactions were negligible due to sufficiently small
particle size and volume fraction. Using Lewin’s model, Holloway
*et al.* showed numerically that a composite
structure containing lossless magnetodielectric spheres exhibits negative
values ($\u03f5<0$ and $\mu <0$) at Mie resonance [21]. Moreover, O’Brien and Pendry found
magnetic activity ($\mu <0$) 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
($\u03f5<0$ or $\mu <0$) 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 [1–4].

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 [1–6,15,24–28]. 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 ($\u03f5$), and permeability ($\mu $) as noted in Fig. 1(a). A standard retrieval algorithm [31–33] then delivers the reflectance of each device studied; it is known that the retrieved parameters may be nonphysical (non-passivity and non-causality) due to neglected spatial dispersion [34–36]. We compare the results of this method to the results of numerical optimization indicated in Fig. 1(b). The MM reflectors provided here annihilate classic Mie resonances by a connective sublayer that prevents individual particle resonance from appearing in the array. We show smooth transitions in the reflectance spectra on morphing from the fully discrete to the connected-particle reflector. Using the standard homogenization method, the connected reflector exhibits negative MM parameter ($\u03f5<0$ or $\mu <0$) spectra as commonly attributed to Mie scattering, even though no Mie resonances can be supported.

## 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
(${\u03f5}_{\mathrm{Si}}=12$) 2D grating where
$\mathrm{\Lambda}$ = period, $H$ = grating height,
$D$ = rod diameter, and
${D}_{h}$ = homogenous layer thickness. We set
$\mathrm{\Lambda}=660\text{\hspace{0.17em}}\mathrm{nm}$, $H=500\text{\hspace{0.17em}}\mathrm{nm}$, and $D=400\text{\hspace{0.17em}}\mathrm{nm}$ as in the reference design [2]. We apply rigorous coupled-wave
analysis (RCWA) [37] to find
the zero-order reflectance (${R}_{0}$) at normal incidence, with results
shown in Fig. 2(b) for
${D}_{h}$ ranging up to 300 nm. A
complex mixture of high-reflection bands appears in the contour map.
The widest bands appear at ${D}_{h}=78$ and ${D}_{h}=110\text{\hspace{0.17em}}\mathrm{nm}$, 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 ${D}_{h}=78\text{\hspace{0.17em}}\mathrm{nm}$ and $\lambda =1.45\text{\hspace{0.17em}}\mathrm{\mu m}$. 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 $\mathrm{\Delta}\u03f5={n}_{H}^{2}-{n}_{c}^{2}$ controls the reflection bandwidth, we
modify this value to investigate the evolution of the reflection
bands. For strong modulation (${n}_{H}={n}_{\mathrm{Si}}$, ${n}_{c}={n}_{\text{air}}$ and $\mathrm{\Delta}\u03f5=11$), as in Fig. 2(e), wideband reflection bands
are achievable for particular grating heights
($H$). For ${R}_{0}>99\%$, the largest bandwidth is
72 nm at $H=500\text{\hspace{0.17em}}\mathrm{nm}$, as previously designed [2]. However, with the same
$H$, it increases to
$\sim 220\text{\hspace{0.17em}}\mathrm{nm}$ at ${D}_{h}=78$ and 110 nm, as seen in
Fig. 2(c). On
decreasing the modulation strength to $\mathrm{\Delta}\u03f5\sim 2.5$, 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].

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 {$\mathrm{\Lambda}=660\text{\hspace{0.17em}}\mathrm{nm}$, $F=0.6$, ${D}_{g}=430\text{\hspace{0.17em}}\mathrm{nm}$, ${D}_{h}$} where ${D}_{g}$ and F are grating depth and fill factor. Here, the input plane wave is TM polarized and at normal incidence. In the range $250\text{\hspace{0.17em}}\mathrm{nm}<{D}_{h}<350\text{\hspace{0.17em}}\mathrm{nm}$, as shown in Fig. 3(b), the high-reflection band is much wider than the band for the reflector consisting of discrete parallel ridges (${D}_{h}=0\text{\hspace{0.17em}}\mathrm{nm}$). 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 (${D}_{h}=267\text{\hspace{0.17em}}\mathrm{nm}$).

#### 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 $\lambda =1.62\text{\hspace{0.17em}}\mathrm{\mu m}$, 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 $\lambda =1.426\text{\hspace{0.17em}}\mathrm{\mu m}$, at which the electric Mie dipole appears.

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 [31–33], 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 $\mu $ and $\epsilon $ from the $S$-parameters (${S}_{11}$ and ${S}_{21}$). The reflection ($r$) and transmission ($t$) coefficients are calculated by RCWA and then ${S}_{11}$ and ${S}_{21}$ are taken by $r$ and $t{\mathrm{e}}^{{ik}_{0}d}$, where $d$ and ${k}_{0}$ 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 ${n}^{\prime \prime}\gg 0$ when ${\mathrm{Z}}^{\prime}=0$, which is consistent with previous reports [1–3]. High reflection also arises when ${Z}^{\prime \prime}\gg {Z}^{\prime}$ because the reflectance at the boundary of the slab (${|\mathrm{\Gamma}|}^{2}=[{({\mathrm{Z}}^{\prime}-1)}^{2}+{Z}^{\prime 2}]/[{({\mathrm{Z}}^{\prime}+1)}^{2}+{Z}^{\prime 2}]$) is high. At 1.347 μm in Fig. 5(e), $Z=2.813\u201314.38i$, with ${|\mathrm{\Gamma}|}^{2}=0.95$. 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 $\mu $ as well as in $\epsilon $, 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.

## 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 ($\u03f5=12$) 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 ${n}_{\mathrm{air}}\le \lambda /\mathrm{\Lambda}<{n}_{\mathrm{Si}}$, 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 $\mathrm{\Lambda}<\lambda $ but not the deep subwavelength condition $\mathrm{\Lambda}\ll \lambda $ demanded by accurate effective-medium theory.

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 [1–3], we show results for discrete and connected reflectors in Figs. 5(b)–5(g) computing effective $Z$, $n$, $\u03f5$ and $\mu $. 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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