Generalized Brewster effect in high-refractive-index nanorod-based metasurfaces

Diego Romero Abujetas; José A. Sánchez-gil; Juan José Sáenz

doi:10.1364/OE.26.031523

1. Introduction

In recent years, high-refractive index (HRI) nanophotonic structures are attracting a widespread interest [1], mostly due to the strong magnetic dipole resonances found in the visible and telecomm spectral ranges for HRI nanoparticles [2–5], which make them especially appealing to tailor light at the nanoscale, as an alternative to plasmonic nanostructures. Actually, resonant dielectric nanostructures are nearly lossless precluding energy dissipation into heat [6,7], which undermines plasmonics due to their high losses. The appearance of such magnetic dipole resonances, fully predicted by Mie theory for spherical or cylindrical HRI subwavelength particles [8], has been experimentally verified in different spectral regimes [9–12]. HRI dielectric nanostructures are thus emerging as a new field in nanophotonics in light of their unique optically induced magnetic and (also) electric resonances, leading to a wealth of fascinating phenomenology such as nanoantenna directionality [9,13–18], Raman scattering [7], enhanced nonlinear effects [19,20] and optoelectronic devices [21].

HRI nanostructures are being used as building blocks of metamaterials [22–24] and especially of metasurfaces [25–28] which can be seen as ultrathin gratings whose thickness and periodicity are small compared to a wavelength in the surrounding media. Metasurfaces have been shown to be relevant in a variety of applications as perfect reflectors [29, 30, 36], beam deflectors or metalenses [31–35, 37]. Most of these applications rely on the interference between electric and magnetic dipolar fields and the associated asymmetric angular distributions of the scattered intensity [4,5].

As an example of particular interest, it has been recently shown that properly designed all-dielectric metasurfaces, based on HRI spheres, exhibit zero reflectivity, a generalized Brewster’s effect, potentially for any angle, wavelength and polarization of choice [38]. This effect is a direct consequence of the angle-suppressed radiation/scattering from small HRI spheres or disks. At normal incidence, the effect is related to the absence of backscattering at the, so-called, first Kerker condition which provides an interesting connection between two apparently unrelated phenomena such as Brewster’s angle and general Kerker’s conditions [38]. Opposite, homogeneous subwavelength HRI cylinders do not fulfil the first Kerker condition due to the mismatch between the local electric and magnetic density of states.

The unconventional optical properties of metasurfaces, in particular those related to arrays of non-touching resonant particles (metafilms [26]), have been often associated to the resonant properties of the building blocks themselves, while disregarding multiple scattering effects, e.g. the coupling with leaky-mode resonances (also known as guided-mode or lattice resonances) [39]. Understanding the complex interplay between particle and leaky-mode or lattice resonances in metafilms from full wave numerical calculations is not always simple and quasi-analytical methods can shed much light onto the underlying physics. Multiple scattering theory [40] is particularly well suited to analyse this complex interplay in metafilms (guided wave approaches provide equivalent insights for metasurfaces and resonant gratings based on subwavelength-structured continuous films [41,42]). Introduced in the pioneering work of Twersky to explain the Wood anomalies in one-dimensional gratings [43], this approach has been subsequently been applied to understand the nature of the resonant coupling in one-dimensional arrays of non-resonant subwavelength dielectric wires [44–47].

In this work we shall focus on the complex interference, scattering and resonant phenomena arising in simple geometrical arrangements like one-dimensional arrays of parallel, free-standing, subwavelength HRI dielectric cylinders. We analyze analytically the reflection and transmission of all-dielectric metasurface consisting of HRI cylinders whose scattering properties can be described by their two lowest-order Mie resonances. To this aim, we derived a coupled electric and magnetic dipole (CEMD) formulation that unravels the interference and coupling between electric and magnetic dipolar fields. Based on the multiple scattering approach, we discuss the nature of reflection resonances and generalized Brewster conditions induced by the coupling of electric and magnetic dipoles with the lattice periodicity. Although a zero back-scattering condition can never be achieved for individual HRI cylinders, here we will show that the mutual (multiple scattering) interaction, when they are arranged in a nanorod-based metasurface, leads to a rich optical response including an effective Kerker condition and an associated generalized Brewster's effect. It should be mentioned explicitly that the cylinder sizes and spectral regions considered below ensure that no higher-order resonances are needed, unlike in [36, 37] where the role of multipole resonances is crucial, to fully describe reflection and transmission both qualitative-and quantitatively. As a matter of fact, we show that the results derived from our analytical theoretical approach are in full agreement with numerical results obtained from commercial full wave electromagnetic solvers.

The manuscript is structured as follows: The scattering properties of a single cylinder are summarized in Section II where we also discuss the absence of the first Kerker condition in HRI cylinders. The multiple scattering approach, described in Sec. III, leads to analytical expressions for the transmitted and reflected fields; the optical theorem is explicitly derived too. In Sec. IV the specular reflection amplitude is introduced, and the Brewster’s effect are formally connected with the Kerker condition. The formulation is exploited to study an array of Si nanocylinders in Section V, analyzing the interplay between longitudinal and transverse leaky-mode resonances: Brewster conditions are obtained in both polarizations. Concluding remarks are presented in Sec. VI.

2. Scattering properties of a single HRI cylinder

Let us consider a dielectric cylinder with its axis along the x-axis, radius R, electric permittivity ε and centred at r′ = (y′, z′) in an otherwise homogeneous medium (which, for simplicity, we assume to be vacuum or air). The infinitely long cylinder is illuminated by an external plane wave with a wave vector k₀(with |k₀| = k = 2π/λ, λ being the wavelength of the incident field in the surrounding media) perpendicular to the cylinder axis. We describe the electromagnetic field as a 6-dimensional vector field Ψ(r) as

Ψ (r) = [\begin{matrix} E (r) \\ Z H (r) \end{matrix}], E (r) [\begin{matrix} E_{x} (r) \\ E_{y} (r) \\ E_{z} (r) \end{matrix}], H (r) [\begin{matrix} H_{x} (r) \\ H_{y} (r) \\ H_{z} (r) \end{matrix}],

E(r) and H(r) being the electric and magnetic vector fields defined in Cartesian coordinates and Z = (µ₀/ϵ₀)^1/2the vacuum impedance (ZH(r) and E(r) have the same units). We assume that the cylinder’s electrodynamics response can be fully described by its lowest electric and magnetic multipoles in terms of a 6 × 6 polarizability tensor [48–50], $\overset{\leftrightarrow}{α}$ ,

\overset{\leftrightarrow}{α} = (\begin{matrix} α^{(E)} & 0 \\ 0 & α^{(M)} \end{matrix}), k^{2} α^{(E)} = (\begin{matrix} 4 i b_{0} & 0 & 0 \\ 0 & 8 i a_{1} & 0 \\ 0 & 0 & 8 i a_{1} \end{matrix}), k^{2} α^{(M)} = (\begin{matrix} 4 i a_{0} & 0 & 0 \\ 0 & 8 i b_{1} & 0 \\ 0 & 0 & 8 i b_{1} \end{matrix}),

where α⁽^E⁾and α⁽^M⁾are the electric and magnetic polarizabilities and a_nand b_nare the (dimensionless) Mie coefficients [8]. For homogeneous and isotropic cylinders, the polarizability is diagonal and, to this end, we define:

α_{x} \equiv α_{x x}, α_{y} = α_{z} = α_{y, z} \equiv α_{y y} = α_{z z}

The multipoles associated to $α_{x}^{(M)}$ and $α_{x}^{(E)}$ (i.e. a₀and b₀, respectively) can be seen as magnetic and electric dipoles along the cylinder axis. We will refer to them as longitudinal dipoles. Strictly speaking, a dipole along the axis is meaningful and it can be considered as a (polarization) current line. The multipoles $α_{y, z}^{(E)}$ and $α_{y, z}^{(M)}$ (i.e. a₁and b₁, respectively) are dipoles in the transversal yz plane, and we will call them transverse dipoles.

In Fig. 1 we plot the first Mie coefficients of an isolated HRI dielectric cylinder with real permittivity ε = 12:25 and radius R = 50 nm as a function of the frequency of the incoming field (ω = ck = 2πc/λ, c the speed of light in the surrounding media). As the incident field is normal to the cylinder axis, the TE and TM waves are decoupled and can be treated independently. Notice that, in contrast with the Mie theory for spheres, for cylinders at normal incidence (in general for 2D objects) the coefficients a_n are related to the response to TE (p-polarized) waves -with the magnetic field parallel to the cylinder axis-, while b_n described the response upon TM (s-polarized) waves -with the electric field parallel to the cylinder axis-.

Fig. 1 Contributions of the first terms in Mie theory to the scattering efficiency, Q_sca, for (a) p-polarized TE and (b) s-polarized TM waves of a dielectric cylinder of radius R = 50 nm and relative permittivity ε = 12.25 as a function of the energy ω. The second order terms a₂, b₂are also shown to indicate the limit of validity of the dipolar approximation.

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The field scattered by the cylinder, Ψ_sca, at the position r = (y, z) can then be expressed as a function of the polarizing (incoming) field, Ψ_inc(r′) at the cylinder position, r′ = (y′, z′), as

Ψ_{sca} (r) = k^{2} (\begin{matrix} G (r, r^{'}) & G_{E M} (r, r^{'}) \\ - G_{E M} (r, r^{'}) & G (r, r^{'}) \end{matrix}) \overset{\leftrightarrow}{α} Ψ_{inc} (r^{'}) = k^{2} \overset{\leftrightarrow}{G} (r, r^{'}) \overset{\leftrightarrow}{α} Ψ_{inc} (r^{'})

where G(r, r′) and G_EM (r, r′) are the Green functions of the cylinder (see Appendix A)

G (r, r^{'}) = {I \frac{\nabla \nabla}{k^{2}}} g (r - r^{'})

G_{E M} (r, r^{'}) = i \frac{\nabla}{k} \times (g (r - r^{'}) I)

where g(r − r′) ≡ (i/4)H₀(r −r′) is the two-dimensional scalar Green function (with H₀ the Hankel function of order zero) and I the (3 × 3) unit dyadic. For a single, isolated, cylinder the incoming field is simply given by the external plane wave Ψ⁰(r′) (see below).

2.1. Radiation intensity patterns and absence of first Kerker condition

In the far field, the intensity I_ffof the light scattered is, in general, asymmetric due to the interference between electric and magnetic contributions [8]

\begin{array}{l} I_{ff}^{(T E)} (ω, ϕ) \propto {| a_{0} + 2 a_{1} \cos ϕ |}^{2} \propto {| α_{x}^{(M)} + α_{y, z}^{(E)} \cos ϕ |}^{2}, \\ I_{ff}^{(T E)} (ω, ϕ) \propto {| b_{0} + 2 b_{1} \cos ϕ |}^{2} \propto {| α_{x}^{(E)} + α_{y, z}^{(M)} \cos ϕ |}^{2}, \end{array}

where the scattering angle ϕ, is the angle between the incident and the scattered wave. Figure 2 illustrates typical radiation patterns at different frequencies and polarizations (see also Fig. 3). Although at specific wavelengths the scattering backwards can be almost suppressed, the Optical Theorem precludes the existence of zero backscattering: The conservation of energy for 2D lossless objects implies [50]

Re [1 / a_{n}] = Re [1 / b_{n}] = 1 \to {Im [1 / (k^{2} α_{x})] = - 1 / 4, Im [1 / (k^{2} α_{y, z})] = - 1 / 8}

or, equivalently,

Im {{\overset{\leftrightarrow}{α}}^{- 1}} = - k^{2} Im {\overset{\leftrightarrow}{G} (r_{0})},

so Eq. (7) can not vanish for the backward wave (ϕ = 180°). Unlike spheres, in cylinders the degrees of freedom of the electric the magnetic fields are different and there is a mismatch between the local density of states of each multipole. Nonetheless, when 2α_x= α_y_,_z, Eq. (7) can be zero at ϕ = ±120°.

Fig. 2 Sketch of the metafilm based on an array of parallel nanorods and scattering geometry for both s and p polarizations of the incoming light. Continuous blue lines correspond to radiation intensity diagrams of a single cylinder HRI cylinder for different frequencies and polarizations (same parameters as in Fig. 1).

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Fig. 3 The two-dimensional maps correspond to the metafilm normalized specular reflection versus angle of incidence θ₀and frequency ω (or a/λ = ω/(2πc)). (a,c) Maps calculated from single scattering and neglecting interference effects corresponding to the asymmetric and resonant scattering from a single cylinder. (b) Schematic of the single scattering geometry. (d,f) Results of a (COMSOL) full wave numerical solution. Black solid line delimit the onset of diffraction. Notice that we use two different color palettes to emphasise both transparent and total reflection regions. (e) Illustration of the metafilm and the scattering configuration.

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3. Periodic array of HRI nanorods: Formal Scattering theory and general results

Let us now consider an infinite set of parallel cylinders with the same radius R placed at r_n= (y_n= na, z_n= 0), n = 0, ±1, ±2, … where a is the lattice constant, illuminated by an external plane wave, Ψ⁽⁰⁾(r) normal to the cylinder axis and incident angle θ₀, as shown in Fig. 2. The spatial dependence of the plane wave can be taken as exp(ik₀· r) = exp (iK₀y) exp (−iq₀z) where

\begin{matrix} K_{0} = k \sin θ_{0}, & q_{0} = k \cos θ_{0}, & k^{2} = K_{0}^{2} + q_{0}^{2} \end{matrix},

with k = |k₀|, while the time dependence exp (−iωt) will be assumed for all the fields. Therefore, the external plane wave travels from the upper to the lower half plane. The upper half plane (z > 0) is considered as the reflected region, while the lower half plane (z < 0) is the transmitted region.

3.1. Single scattering versus full wave numerical calculations

In order to emphasize the relevance of interference and multiple scattering effects (even for subwavelength gratings), in Fig. 3(a,c) we have ploted the reflected intensity in the specular direction in a ‘a/λ vs. θ₀’ (or ω vs. θ₀) map assuming non-interacting cylinders, i.e. a plot of I_ff(ω, π − θ₀) normalized to its maximum value. The Q_scaplots on top serve to identify frequency regions where only the first two Mie coefficients are relevant and do not overlap with higher order multipoles. The results for I_ffinclude all relevant multipoles and reduces to Eq. (7) with ϕ_s= π − θ₀when only the two first terms are relevant. These results summarise the different intensity radiation patterns of a single cylinder as a function of the ratio a/λ (each pattern in Fig. 2 corresponds to a vertical line in the I_ff(ω, π – θ₀) map).

The maps in Fig. 3(d,f) correspond to the full wave numerical calculation of the specular reflection R₀(ω, θ₀) obtained from a well established electromagnetic solver (COMSOL Multiphysics). The regions with

0 \leq | \sin θ_{0} | \leq \frac{λ}{a} - 1

correspond to a subwavelength, ‘zero order’, grating (black solid line delimits the onset of diffraction in the bottom half of Fig. 3). Upon comparing upper and lower graphs in Fig. 3, It becomes evident that most relevant features in the full wave numerical simulations cannot be reproduced by the scattering from single, non-interacting cylinders.

3.2. Formal Scattering theory

Formal scattering theory can help us to understand the strong influence of the lattice in the optical properties of the metafilm observed in Fig. 3. The discussion below follows the notation introduced in Ref. [47] which is now extended to include electric and magnetic responses.

Let us consider an external plane wave, Ψ⁽⁰⁾(with wave vector $k_{0} = K_{0} \hat{y} - q_{0} \hat{z}$ ), as:

E^{(0)} (r) = (E_{s_{0}} \hat{x} + E_{p_{0}} [(K_{0} / k) \hat{z} + (q_{0} / k) \hat{y}]) e^{i k_{0} \cdot r}

Z H^{(0)} (r) = (- E_{s_{0}} [(K_{0} / k) \hat{z} + (q_{0} / k) \hat{y}] + E_{p_{0}} \hat{x}) e^{i k_{0} \cdot r}

where Ψ⁽′⁾is divided into s (E_s₀) and p (E_p₀) wave contributions.

Each nanorod is excited by the external plane wave plus the waves scattered from the rest of the grating [43]. The self-consistent incident field, on the n = 0 rod (|r₀| = 0), is then given by the solution of

Ψ_{inc} = (r_{0}) = Ψ^{(0)} (r_{0}) + \sum_{n \neq 0} k^{2} \overset{\leftrightarrow}{G} (r_{n}) \overset{\leftrightarrow}{α} Ψ_{inc} (r_{n}) .

For a periodic array and plane wave illumination the Bloch’s theorem holds, Ψ_inc(r_n) = Ψ_inc(r₀) exp(iK₀na), and the self-consistent incident field can be written as

Ψ_{inc} (r_{0}) = Ψ^{(0)} (r_{0}) + k^{2} {\sum_{n \neq 0} \overset{\leftrightarrow}{G} (r_{n}) e^{i K_{0} n a}} \overset{\leftrightarrow}{α} Ψ_{inc} (r_{0}) \equiv Ψ^{(0)} (r_{0}) + k^{2} {\overset{\leftrightarrow}{G}}_{b} \overset{\leftrightarrow}{α} Ψ_{inc} (r_{0})

where

{\overset{\leftrightarrow}{G}}_{b}

is the lattice ‘depolarization’ dyadic (or return Green function),

{\overset{\leftrightarrow}{G}}_{b} \equiv \sum_{n \neq} \overset{\leftrightarrow}{G} (r_{n}) e^{i K_{0} n a} .

${\overset{\leftrightarrow}{G}}_{b}$ tell us about the coupling strength between the resonators, and is crucial to determine all the scattering properties, as shown below. Explicit expressions of the different Green dyadics in Cartesian coordinates are given in Appendix B.

The solution of the self-consistent equation can be formally written as a function of the external plane wave:

Ψ_{inc} (r_{0}) = {[\overset{\leftrightarrow}{I} - k^{2} {\overset{\leftrightarrow}{G}}_{b} \overset{\leftrightarrow}{α}]}^{- 1} Ψ^{(0)} (r_{0}),

where

\overset{\leftrightarrow}{I}

is the unit (6 × 6) dyadic.

Once we know the self-consistent incoming field, Eq. (17), the field scattered by the nanorod placed at (y_n= na, z_n= 0) is given by

Ψ_{scat} (r) = k^{2} \overset{\leftrightarrow}{G} (r - r_{n}) \overset{\leftrightarrow}{α} Ψ_{inc} (r_{n}),

and the total scattered field can be written as

Ψ_{scat-tot} (r) = k^{2} {\sum_{n = - \infty}^{\infty} \overset{\leftrightarrow}{G} (r - r_{n}) e^{i K_{0} a n}} \overset{\leftrightarrow}{α} Ψ_{inc} (r_{0}) \equiv k^{2} {\overset{\leftrightarrow}{G}}^{\pm} (r) \overset{\leftrightarrow}{α} Ψ_{inc} (r_{0}),

where the tensor lattice sum

{\overset{\leftrightarrow}{G}}^{\pm} (r)

can be written as a sum over all diffracted spectral orders (m = ⋯, −2, −1, 0, 1, 2 ⋯) as:

{\overset{\leftrightarrow}{G}}^{\pm} (r) \equiv \sum_{n = - \infty}^{\infty} \overset{\leftrightarrow}{G} (r - r_{n}) e^{i K_{0} a n} = \sum_{m = - \infty}^{\infty} {\overset{\leftrightarrow}{G}}_{m}^{\pm} e^{i K_{m} y} e^{\pm i q_{m} z},

where, “−” (“+”) corresponds to downward scattered waves in the region z < 0 (upwards reflected waves in the region z > 0). K_mand q_mare the wavevectors of the diffracted orders, Eq. (47). See Eq. (52) in Appendix C for an explicit expression in Cartesian coordinates

Finally, using Eq. (17) in Eq. (19), the reflected and transmitted fields are then given by

Ψ_{r} (r) = k^{2} {\overset{\leftrightarrow}{G}}^{+} (r) \tilde{\overset{\leftrightarrow}{α}} Ψ^{(0)} (r_{0}),

Ψ_{t} (r) = Ψ^{(0)} (r) k^{2} {\overset{\leftrightarrow}{G}}^{-} (r) \tilde{\overset{\leftrightarrow}{α}} Ψ^{(0)} (r_{0}),

where

\tilde{\overset{\leftrightarrow}{α}}

is the renormalized (dressed) polarizability,

\tilde{\overset{\leftrightarrow}{α}} = \overset{\leftrightarrow}{α} {[I - k^{2} {\overset{\leftrightarrow}{G}}_{b} \overset{\leftrightarrow}{α}]}^{- 1} = {[{\overset{\leftrightarrow}{α}}^{- 1} - k^{2} {\overset{\leftrightarrow}{G}}_{b}]}^{- 1} .

3.3. Optical Theorem

The incoming power density can be written as

S^{(0)} = - \frac{1}{2} Re {E^{(0)} \times H^{(0) *}} \cdot \hat{z} = \frac{q_{0}}{4 k Z} {| Ψ^{(0)} |}^{2}

Integrating the Poynting vector in the far field, the reflected and transmitted power densities can be shown to be given by a sum over all N_pdiffracted, propagating (i.e. q_mreal, Eq. (47)), beams

S_{R} = \frac{k^{4}}{4 k Z} Ψ^{(0) †} (r_{0}) {\tilde{\overset{\leftrightarrow}{α}}}^{†} (\sum_{m = 0}^{N_{p}} q_{m} {\overset{\leftrightarrow}{G}}_{m}^{+}^{†} {\overset{\leftrightarrow}{G}}_{m}^{+}) \tilde{\overset{\leftrightarrow}{α}} Ψ^{(0)} (r_{0}),

\begin{array}{l} S_{T} = \frac{k^{4}}{4 k Z} Ψ^{(0) †} (r_{0}) {\tilde{\overset{\leftrightarrow}{α}}}^{†} (\sum_{m = 0}^{N_{p}} q_{m} {\overset{\leftrightarrow}{G}}_{m}^{-}^{†} {\overset{\leftrightarrow}{G}}_{m}^{-}) \tilde{\overset{\leftrightarrow}{α}} Ψ^{(0)} (r_{0}) \\ + \frac{q_{0}}{4 k Z} Ψ^{(0) †} (r_{0}) (\overset{\leftrightarrow}{I} + i \frac{k^{2}}{a q_{0}} {\tilde{\overset{\leftrightarrow}{α}} - {\tilde{\overset{\leftrightarrow}{α}}}^{†}}) Ψ^{(0)} (r_{0}), \end{array}

Energy conservation (in absence of absorption) implies that S⁽⁰⁾= S_R+ S_Tand, taking into account that

\sum_{m}^{N_{p}} q_{m} [{\overset{\leftrightarrow}{G}}_{m}^{+}^{†} {\overset{\leftrightarrow}{G}}_{m}^{-} + {\overset{\leftrightarrow}{G}}_{m}^{-}^{†} {\overset{\leftrightarrow}{G}}_{m}^{-}] = \frac{2}{a} Im {\overset{\leftrightarrow}{G} (r_{0})},

we obtain

2 i {\tilde{\overset{\leftrightarrow}{α}} - {\tilde{\overset{\leftrightarrow}{α}}}^{†}} = k^{2} {\tilde{\overset{\leftrightarrow}{α}}}^{†} Im {\overset{\leftrightarrow}{G} (r_{0})} \tilde{\overset{\leftrightarrow}{α}} .

This is a general condition for the dressed polarizability tensor $\tilde{\overset{\leftrightarrow}{α}}$ imposed by energy conservation (valid for periodic arrays of arbitrary dipolar particles) and is similar to the one obtained for electrically small particles [48–50]. Assuming ${\tilde{\overset{\leftrightarrow}{α}}}^{- 1}$ exists, then we have

{\tilde{\overset{\leftrightarrow}{α}}}^{- 1} = {\tilde{\overset{\leftrightarrow}{α}}}_{0}^{- 1} - i k^{2} Im {\overset{\leftrightarrow}{G} (r_{0})},

where, in absence of absorption,

{\tilde{\overset{\leftrightarrow}{α}}}_{0}^{- 1}

must be Hermitian. Taking into account Eqs. (9) and (23), it is easy to find that our grating of homogeneous isotropic cylinders

{\tilde{\overset{\leftrightarrow}{α}}}^{- 1} = Re {{\overset{\leftrightarrow}{α}}^{- 1} - k^{2} {\overset{\leftrightarrow}{G}}_{b}} - i k^{2} Im {\overset{\leftrightarrow}{G} (r_{0})},

fulfil the general condition given by Eq. (28).

4. Fresnel reflection amplitudes for nanorod-based metafilms

We shall restrict our discussion to subwavelength gratings (metafilms) with period a < λ below the onset of the first diffracted beam (this takes place at the Rayleigh condition |2π/a ±K₀| = k). This limits the study to angles of incidence verifying Eq. (11), such that only the zero (specular, m = 0) order can propagate in the far field (|z| ≫ λ) input and output regions, being higher order diffracted beams evanescent and confined in the grating region. The specular reflectance, R₀, can then be derived from Eq. (24) and (25) for both p- and s-polarized fields

\begin{array}{l} R_{0}^{(p)} = {(\frac{k^{2}}{2 k a \cos θ_{0}})}^{2} {| γ^{(p)} ({\tilde{α}}_{x}^{(M)} + \sin^{2} θ_{0} {\tilde{α}}_{z}^{(E)} + 2 k^{2} \sin θ_{0} G_{b z x} {\tilde{α}}_{x}^{(M)} {\tilde{α}}_{z}^{(E)}) - \cos^{2} θ_{0} {\tilde{α}}_{y}^{(E)} |}^{2} \\ R_{0}^{(s)} = {(\frac{k^{2}}{2 k a \cos θ_{0}})}^{2} {| γ^{(s)} ({\tilde{α}}_{x}^{(E)} + \sin^{2} θ_{0} {\tilde{α}}_{z}^{(M)} + 2 k^{2} \sin θ_{0} G_{b z x} {\tilde{α}}_{x}^{(E)} {\tilde{α}}_{z}^{(M)}) - \cos^{2} θ_{0} {\tilde{α}}_{y}^{(M)} |}^{2}, \end{array}

where

k^{2} {\tilde{α}}_{i}^{(E)} = {(\frac{1}{k^{2} α_{i}^{(E)}} - G_{b i i})}^{- 1}, k^{2} {\tilde{α}}_{i}^{(M)} = {(\frac{1}{k^{2} α_{i}^{(M)}} - G_{b i i})}^{- 1}

and

γ^{(p)} = \frac{1}{1 - k^{4} G_{b z x}^{2} {\tilde{α}}_{x}^{(M)} {\tilde{α}}_{x}^{(E)}}, γ^{(s)} = \frac{1}{1 - k^{4} G_{b z x}^{2} {\tilde{α}}_{x}^{(E)} {\tilde{α}}_{z}^{(M)}},

being G_biiand G_bzxthe matrix elements of

{\overset{\leftrightarrow}{G}}_{b}

(see Appendix B). Since the particles are lossless, the specular transmittance, T₀, is simply given by T₀= 1 − R₀. For completeness, the specular transmittance and the reflectance for any diffracted order is shown in Appendix D.

4.1. Kerker induced Brewster’s effect (KiBs)

When the cylinders form a periodic array, the response by fields along the y axis is different from fields along the z axis. Furthermore, the lattice breaks the azimuthal symmetry and evens the local density of states for electric and magnetic fields, allowing the balance between electric and magnetic multipoles. It is interesting to note that the balance is due to the cancellation of the scattering effects of the individual resonators by the array. Formally, the imaginary parts of the lattice Green functions compensate the imaginary parts of the polarizabilities, being possible to totally suppress the reflected wave at any angle of incidence. The suppression comes from the interference between the electric and the magnetic lowest-multipoles, assisted by the lattice, calling this condition as Kerker induced Brewster’s effect (KiBs). For example, at normal incidence the KiBs condition is easily read as

R_{0}^{(p)} = {(\frac{k^{2}}{2 k a})}^{2} {| {\tilde{α}}_{x}^{(M)} - {\tilde{α}}_{y}^{(E)} |}^{2} = 0, R_{0}^{(s)} = {(\frac{k^{2}}{2 k a})}^{2} {| {\tilde{α}}_{x}^{(E)} - {\tilde{α}}_{y}^{(M)} |}^{2} = 0,

\begin{array}{l} ℜ [1 / α_{x}^{(M)}] - ℜ [1 / α_{y, z}^{(E)}] - k^{2} ℜ (G_{b x x} - G_{b y y}) = 0 for p waves, \\ ℜ [1 / α_{x}^{(E)}] - ℜ [1 / α_{y, z}^{(M)}] - k^{2} ℜ (G_{b x x} - G_{b y y}) = 0 for s waves . \end{array}

Hence, the lattice plays a crucial role, being possible to have a zero in reflexion potentially at any angle of incidence.

5. Reflection and transmission by a Silicon nanorod metafilm

Let us study the scattering properties of an array of non-absorbing dielectric cylinders with a ratio between the lattice parameter and the radius a/R = 4 and relative permittivity ε = 12.25; this is a typical semiconductor permittivity in the optical domain above the gap (e.g. crystalline Si [51]). We will first consider the role of each multipole separately and later the interplay between them; in this manner the interference among all them will become clearer. Although for semiconductor cylinders in the optical domain all multipoles somewhat overlap, recall that for very HRIs in the THz and microwave domain, all resonant multipoles of a single cylinders would be isolated.

5.1. Longitudinal electric/magnetic dipoles

Figure 4 shows the total reflectance for both polarizations when the cylinder array is only described by their lower-order resonant multipoles ( ${\tilde{α}}_{x}^{(E)}$ and ${\tilde{α}}_{x}^{(M)}$ for s and p waves, respectively); the orientation of the equivalent, longitudinal, dipole configuration is also depicted to illustrate the discussion (recall however that, in infinitely long cylinders, a dipole along the cylinder axis is not strictly speaking meaningful, but rather a current line). The black, vertical dashed lines mark the position of the resonance of the polarizabilities. The black solid line delimits the diffractive region at which the first diffraction channel is opened.

Fig. 4 (a,b) Total reflectance calculated only considering ${\tilde{α}}_{x}^{(E)}$ and ${\tilde{α}}_{x}^{(M)}$ on Eq. (31) for a/R = 4 and n = 3.5 for (a) p and (b) s waves as a function of the energy ω of the incident wave and the angle of incidence θ₀. The black, vertical dashed lines marks the position of the resonance of the polarizabilities, while the black solid lines delimit the diffractive region. Middle panel: Configuration of the equivalent longitudinal dipole multipoles (bear in mind that, in infinitely long cylinders, a dipole along the axis is not strictly speaking meaningful, but rather a current line).

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The reflectance for the specular wave can be written as

R_{0}^{(p, s)} = {(\frac{1}{2 k a \cos θ_{0}})}^{2} {| {(ℜ {\frac{1}{k^{2} α_{x}^{(M, E)}} - G_{b x x}} - i \frac{1}{2 k a \cos θ_{0}})}^{- 1} |}^{2} .

It is known that the scattered wave and the incident one are 180° out of phase at resonance, so that they interfere destructively in transmission: Thus the reflectance tends to unity at resonance, namely, the system behaves as a perfect mirror [44]. From Eq. (36), $R_{0}^{(p, s)} = 1$ if $ℜ [1 / {\tilde{α}}_{x}^{(M, E)} - k^{2} G_{b x x}] = 0$ . Therefore, the resonance of the array is shifted from the resonance of the cylinders by the action of the lattice [by G_xx, see Eq. (32)]. For example, for p waves the reflectance is maximum near the resonances of the polarizability, so G_xxslightly modifies this condition. However, for s waves the maximum is placed at much higher energies: indeed shifted and broadened by the action of the lattice.

Furthermore, when only the longitudinal dipole is considered the system always behaves as a thin dielectric film; the scattered wave is the same both in transmission and in reflection (the scattering is symmetric and t = 1 + r) even if the cylinders are not thin compared to the incident wavelength. This means that the cylinder array can be described as a thin dielectric film even at wavelengths of the order of the cylinder size (2R/λ > 0.1). A retrieval of the effective parameter of the system based on the reflectance and the transmittance properties, considering a reasonable width of 2R, leads to an unexpected non-zero magnetic susceptibility (µ_eff, 1) in order to keep a symmetric scattering. Finally, note that the reflectance for p waves vanishes at low frequencies, due to its magnetic character.

5.2. Transverse electric/magnetic dipoles

The phenomenology of the transverse dipole becomes richer because the lattice breaks the degeneracy (G_byy, ≠ G_bzz) and interference issues arise. Figure 5 shows the reflectance of the array for both polarizations when the lower-order resonant multipoles are neglected ( ${\tilde{α}}_{x}^{(E)}$ and ${\tilde{α}}_{x}^{(M)}$ set to zero); the insets depicts dipole orientations and far-field patterns. The black vertical dashed lines, the blue dashed curves, and the black solid curves mark the resonances of the polarizabilities, the zero of reflectance, and the first diffraction condition, respectively. Although this configuration in truth only occurs for TE waves at low frequencies (or for extremely HRI materials for both polarizations), we find it very illustrative to characterize the self-interference of these dipoles alone before addressing below the more complex situation in which the two lowest-order multipoles contribute to the total response.

Fig. 5 (a,b) Reflectance calculated with Eq. (31) setting ${\tilde{a}}_{x}^{(E)} = {\tilde{a}}_{x}^{(M)} = 0$ for a/R = 4 and n = 3.5: (a) p and (b) s waves as a function of the energy ω of the incident wave and the angle of incidence θ₀. The black, vertical dashed lines marks the position of the resonance of the polarizabilities, while the blue dashed lines show the zero reflectance (total transmission) bands. The black solid lines delimit the diffractive region. Middle panel: Configuration of the equivalent transverse dipole multipoles.

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Close to normal incidence and near to the resonance of the cylinders (shown in black dashed lines) the reflectance goes to zero, surrounded by two high reflectivity lobes. This spectrally narrow zero comes from the destructive interference between the multipoles, sort of an electromagnetic-induced transparency (EIT) window; the narrow EIT band disappears exactly at normal incidence and the reflectivity becomes large again ( $R_{0}^{(p, s)} ~ 1$ ). In addition, there is a total transmission band at low frequencies for both polarizations (around θ₀~ 45°) that correspond to the Brewster condition achieved for p and s waves in electric and magnetic media, respectively. Also, for s waves there is another total transmission band at higher frequencies above the resonance of the polarizabilities (ω = 2.85 eV). Although this band does not appear for the other polarization, the analysis of cases for different a/R ratio reveals that this characteristic depends on the spectral position of the resonance and not on the polarization (not shown here). The only difference that comes from the polarization itself is the extension of the reflection lobes towards lower or higher frequencies. This is explained by the different behaviour of the a₁, b₁resonances: for p waves, a₁ goes to zero above resonance (at higher frequencies), whereas for s waves b₁remains non-zero.

All this phenomenology can be understood from the analytic form of the reflectance. In the absence of the first multipole $(α_{x}^{(E, M)} = 0)$ , Eq. (31) can be written as

R_{0}^{(p, s)} = {(\frac{1}{2 k a \cos θ_{0}})}^{2} {| \sin^{2} θ_{0} {(ℜ {\frac{1}{k^{2} α_{y, z}^{(E, M)}} - G_{b z z}} - i \frac{\sin^{2} θ_{0}}{2 k a \cos θ_{0}})}^{- 1} - \cos^{2} θ_{0} {(ℜ {\frac{1}{k^{2} α_{y, z}^{(E, M)}} - G_{b y y}} - i \frac{\cos^{2} θ_{0}}{2 k a \cos θ_{0}})}^{- 1} |}^{2},

that it can be shown that is proportional to

R_{0}^{(p, s)} \propto {| ℜ [\frac{1}{k^{2} α_{y, z}^{(E, M)}}] (\cos^{2} θ_{0} - \sin^{2} θ_{0}) - (ℜ [G_{b z z}] \cos^{2} θ_{0} - ℜ [G_{b y y}] \sin^{2} θ_{0}) |}^{2} .

It is seen that the reflection must be large at resonance due to the cancellation of the incident wave in transmission. However, as the first term of Eq. (38) is zero at resonance, it can be compensated by the second term at some frequency, leading to a zero of reflection surrounded by a large (close to one) and broad reflectivity band. Overall, the reflectance is large, except for a specific frequency near resonance where this narrow EIT condition is met and reflectance vanishes. Furthermore, this phenomenology can be better understood if the transverse contributions are separated into the contributions along the y and z axes, namely, parallel and perpendicular to the metasurface; this is done in Fig. 6(a-d). The width of each multipole is proportional to the angle of incidence

J [1 / {\tilde{α}}_{y}^{(E, M)}] \propto k^{2} \cos^{2} θ_{0}, J [1 / {\tilde{α}}_{z}^{(E, M)}] \propto k^{2} \sin^{2} θ_{0},

Thus, close to normal incidence the dipole along the z axis is extremely narrow, while the other dipole is broader. The interference of these multipoles lead to a narrow EIT window, that becomes broader as the angle of incidence increases. Interestingly, this behaviour is also observed at oblique incidence. Now the dipole along the y axis plays the role of the narrow resonance and a narrow EIT window appears close to 90°. This can be seen in Figure 5(b) for s waves, where the reflectivity goes to zero at oblique incidence. However, it is not met for p waves because the resonance falls in the diffraction region (it can be tuned by modifying the lattice parameter). The latter condition (narrow EIT window at grazing incidence with high transmission for specific frequencies) is worth mentioning since all surfaces behave as mirrors at grazing incidence.

Fig. 6 (a,b) Contribution to the p total reflectance in Fig. 5(a) from the projection of the transverse dipole multipoles along the y (a) and z axis (b). (c,d) Contribution to the s total reflectance in Fig. 5(b) from the projection of the transverse dipole multipoles along the y (c) and z axis (d). Top panel: Configuration of the equivalent transverse dipole multipoles.

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The first term of Eq (38) dominates the reflectance out of resonance ( $ℜ [1 / (k^{2} α_{y, z}^{(E, M)})] ≫ 1$ out of resonance), although at the diffraction edge $ℜ [G_{b z z}]$ diverges and both terms should be considered. However, the first term is always zero at θ₀= 45°. Then, at low energies there is a nearly dispersionless, total transmission band at fixed angle close to θ₀= 45° (horizontal blue dashed curves in Fig. 5). This band is the zero in transmission associated with the Brewster effect, in which a dipole cannot radiate along its axis. As the energy increases, this band becomes dispersive due to the lattice and/or to the polarizabilities, presenting different features. If there is no resonance of the polarizability in the non-diffraction region, the band bends toward smaller angles θ₀following the frequencies at which $ℜ [G_{b z z}]$ compensates the first term, up to the diffraction line at normal incidence [45]. On the other hand, if the Mie resonance lies in the spectral range of frequencies with no diffraction, the broad (Brewster-like) transmission band is connected with lowest-energy EIT window described above, as shown in Fig. 5(a-b). Depending on the spectral position of the resonances, the broad transmission band bends either toward normal or toward grazing incidence, for p waves or s waves, respectively. Therefore, the lattice presents a phase transition, depending of the spectral position of the resonance, that would be interesting to study. Finally, from Eq (38), the reflectance is always zero at ω = 3.3 eV and θ₀= 45° ( $ℜ [G_{b y y} - G_{b z z}] = 0$ ) regardless of the properties of the particles. The zeros of reflections form closed bands, so this point must be included within a total transmission band. If the first narrow EIT window is located above this zero-reflection point, as for p waves, both bands appear as hybridized (see the lower-energy, blue dashed curve in Fig. 5a). If the narrow EIT band is otherwise placed below the zero-reflection point, as is the case for s waves, the broad transmission band exhibits a sort of splitting, as shown in Fig. 5(b) for s waves with the two blue dashed curves: interestingly, note that in this case the transmission bands behave as in a strong coupling phenomena, where the cylinder resonance separates the broad Brewster band and the narrow EIT windows.

5.3. Longitudinal and transverse electric/magnetic multipoles

After studying the phenomenology of each resonance separately, we now consider all of them together. In Fig. 7 the reflectance of the array is shown when both dipoles (electric and magnetic) are considered. The position of the resonant frequencies are displayed in black, dashed curves, while the zeros of reflectance is shown in blue, dashed curves. In addition, all these curves are labeled. The diffraction lines is marked in black, solid lines. In this regard, it follows from Eq. (31) that, apart from the direct interference between the different multipoles, there is also an additional term arising from the coupling between the dipoles in the x- and z-axis. This term is zero for normal incidence (G_bzx= 0 at normal incidence) and increases toward higher angles. It is important to note, upon considering all terms, the reflectance from the exact numerical calculation (bottom half Fig. 3) is in full agreement with the CEMD analytical formulation shown in Fig. 7, ensuring the validity of our method.

Fig. 7 (a,b) Reflectance calculated considering all terms on Eq. (31) for a/R = 4 and n = 3.5: (a) p and (b) s waves as a function of the energy ω of the incident wave and the angle of incidence θ₀. The black, vertical dashed lines mark the position of the resonance of the polarizabilities, labeled for a better identification. The blue dashed lines show the zero reflectance (total transmission) bands, also labeled. The black solid line delimits the diffractive region. Middle panel: Configuration of the equivalent dipole multipoles (bear in mind that, in infinitely long cylinders, a dipole along the axis is not strictly speaking meaningful).

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As predicted by the KiBs condition at normal incidence [Eq. (35)] there are frequencies at which the reflectance is zero (zero back-scattering): These frequencies are at ω = 2.6, 4.1 eV for p waves and ω = 2.4, 4.5 eV for s waves. Furthermore, there is another zero in reflection close to normal incidence at ω = 4.0 eV for p waves and at ω = 2.9 eV for s waves. For p waves the minimum mainly arises from the interference of the in-plane dipoles (EIT window) as in Fig. 5. Nonetheless, for s waves the zero in reflection comes from the (Fano asymmetric) interference between the longitudinal dipole and the transverse dipole in the y axis.

The interference between multipoles is easily appreciable in both cases. For p waves, the first resonance $(α_{x}^{(M)})$ bends the first zero-reflection (Brewster-like) band, that appears at θ₀~45° at low frequencies, toward the first KiBs condition (ω = 2.6 eV) and a maximum of reflectance covering all angles emerges. However, above the first resonance the interference leads to a second total transmission band that starts at θ₀= 90° and disappears when it crosses the diffraction line. The low frequency (first) band is due to the self- interference between the transverse dipoles, but the inter-resonance (second) band is induced by the coupling between the different multipoles described by γ⁽^p⁾in Eqs. (33). This coupling would lead to a Fano interference, although both multipoles are qualitative narrow and separated in energies, so the interference is not very pronounced. Finally, there is another narrow minimum of reflectance near to the resonance of $α_{y, z}^{(E)}$ (ω = 4.0 eV), as in Fig. 5. This minimum is connected to the second zero predicted by the KiBs condition (ω = 4.1 eV), forming the third transmission band. In addition, the high reflectance band that covers all angles of incidences above the resonance in Fig. 5(a) disappears, result of the interference with the longitudinal dipole.

For s waves, the existence of electric response at low frequencies precludes the emergence of the low frequency zero-reflection band associated to the Brewster effect. In this case, multipole coupling originates a narrow high reflective band in the proximity of the second resonance that separates two regions with zero-reflection bands, the first at normal and the second at grazing incidences. However, this phenomenology changes if the resonances shift to lower frequencies (i.e, increasing the radii R): the narrow high reflective band becomes a zero-reflexion band (not shown here). The narrow band comes from to the Fano interference between the multipoles connected by γ⁽^s⁾. For this polarization the longitudinal dipole is broad (see Fig. 4) and the interaction with the perpendicular traversal dipole leads into a strong Fano interference. Nevertheless, the normal interference with the y traversal dipole determines the background and the sign of the Fano parameter. Regarding the zero-reflection bands in Fig. 7(b), the first corresponds to the closed band defined by the first KiBs condition (ω = 2.4 eV) and the minimum of reflectance related to the Fano interference (ω = 2.9 eV). At this energy there is a EIT window for the transverse dipoles (Fig. 5(b)), but the coupling to the longitudinal dipole modifies the profile. The second zero in reflection is placed at grazing incidence, connecting the another EIT windows with a zero coming from the Fano interference, also at oblique incidence. Lastly, the second KiBs condition (ω = 4.5 eV) arises at higher frequencies, forming the third band that covers all the diffraction-free region.

6. Concluding remarks

To summarize, we have theoretically investigated the reflection and transmission through an array of all-dielectric metasurface consisting of HRI cylinders. To this aim, a coupled electric and magnetic dipole (CEMD) formulation has been developed where the electric and magnetic polarizabilities are given by the two lowest-order Mie cylinder resonances in both polarizations. General analytical expressions are explicitly derived for reflection and transmission, the optical theorem and the Kerker conditions of the cylinder array. The formulation could be applied to any array of HRI subwavelength wires across the electromagnetic spectrum, as long as the wire response can be reasonably described by the lowest order longitudinal/transverse electric/magnetic polarizibilities.

As a case study, we analyzed a metasurface consisting of an array of Si nanocylinders of interest in nanophotonics. By unraveling the role of each dipole resonance separately, we are able to determine the impact of the lattice in all cases; then the interplay between them is fully accounted for. Despite being transparent, total reflection from the HRI cylinder array is observed on resonance. Brewster conditions are obtained in both polarizations, predicted indeed analytically through lattice Kerker conditions; recall that Kerker conditions are forbidden for isolated cylinders. Our analytical results are shown to agree with full numerical simulations, not only qualitatively but also quantitatively.

In this manner, we have analytically described the physics underlying most phenomenology of these all-dielectric metasurfaces, which indeed hold potential for light control and sensing devices at the nanoscale, through their polarization-dependent total reflection/transmission and Fano interference. Furthermore, the coupled electric-magnetic dipole formulation and associated intrinsic phenomenology can be easily extrapolated to lower frequencies, wherein dielectric wires exist with much larger HRI.

Appendix

A. Green tensors of a single cylinder

The scalar Green function, g(r – r_n), is the solution of the Helmholtz equation with a point source located at r = r_n: ∇²g + k²g = −δ(r – r_n),

g (r - r_{n}) = \frac{i}{4} H_{0} (k | r - r_{n} |) = \int_{- \infty}^{+ \infty} \frac{d K}{2 π} e^{i K (y - y_{n})} {\frac{i}{2 q} e^{i q | z - z_{n} |}}, q \sqrt{k^{2} - K^{2}} .

We then have

\begin{array}{l} G (r, r_{n}) = {I + \frac{\nabla \nabla}{k^{2}}} g (r - r_{n}) \\ = (\begin{matrix} G_{x x} & 0 & 0 \\ 0 & G_{y y} & G_{y z} \\ 0 & G_{y z} & G_{z z} \end{matrix}) = \frac{1}{k^{2}} (\begin{matrix} k^{2} & 0 & 0 \\ 0 & k^{2} + \partial^{2} / \partial y^{2} & \partial^{2} / \partial y \partial z \\ 0 & \partial^{2} / \partial y \partial z & k^{2} + \partial^{2} / \partial z^{2} \end{matrix}) g (r - r_{n}), \end{array}

\begin{array}{l} G_{E M} (r, r_{n}) = i \frac{\nabla}{k} \times (g (r - r_{n}) I) \\ = (\begin{matrix} 0 & G_{E M x y} & G_{E M x z} \\ - G_{E M x y} & 0 & 0 \\ - G_{E M x z} & 0 & 0 \end{matrix}) = \frac{i}{k} (\begin{matrix} 0 & - \partial / \partial z & \partial / \partial y \\ \partial / \partial z & 0 & 0 \\ - \partial / \partial y & 0 & 0 \end{matrix}) g (r - r_{n}) . \end{array}

B. Lattice depolarization dyadic

The lattice depolarization function can be written as

{\overset{\leftrightarrow}{G}}_{b} \equiv \sum_{n \neq 0} \overset{\leftrightarrow}{G} (r_{n}) e^{i K_{0} n a} = \lim_{r \to r_{0}} {\overset{\leftrightarrow}{G} (r) - \overset{\leftrightarrow}{G} (r)} = (\begin{matrix} G_{b} & G_{b}^{(E M)} \\ - G_{b}^{(E M)} & G_{b} \end{matrix})

where

G_{b}

is a matrix whose elements, formally equivalent to those given in Ref. [47], can be written as:

G_{b x x} \equiv {(G_{b})}_{x x} = i [\frac{1}{2 q_{0} a} - \frac{1}{4}] + \frac{1}{2 a} \sum_{m = 1}^{\infty} (\frac{i}{q_{m}} + \frac{i}{q_{- m}} - \frac{i}{k_{m}}) + \frac{1}{2 π} [\ln \frac{k a}{4 π} + γ_{e}],

\begin{array}{l} G_{b y y} {(G_{b})}_{y y} = \frac{i}{4} [\frac{2 q_{0}}{k^{2} a} - = \frac{1}{2}] + \frac{1}{2 k^{2} a} \sum_{m = 1}^{\infty} (i q_{m} + i q_{- m} + 2 k_{m} - \frac{k^{2}}{k_{m}}) \\ + \frac{1}{4 π} [\ln \frac{k a}{4 π} + γ_{e} - \frac{1}{2}] - \frac{q_{0}^{2}}{4 k^{2} π} + \frac{1}{6} \frac{π}{k^{2} a^{2}}, \end{array}

\begin{array}{l} G_{b z z} {(G_{b})}_{z z} = \frac{i}{4} [\frac{2 K_{0}^{2}}{k^{2} q_{0} a} - \frac{1}{2}] + \frac{1}{2 k^{2} a} \sum_{m = 1}^{\infty} (i \frac{K_{m}^{2}}{q_{m}} + \frac{K_{- m}^{2}}{q_{- m}} - 2 k_{m} - \frac{k^{2}}{k_{m}}) \\ + \frac{1}{4 π} [\ln \frac{k a}{4 π} + γ_{e} - \frac{1}{2}] - \frac{K_{0}^{2}}{4 k^{2} π} - \frac{1}{6} \frac{π}{k^{2} a^{2}}, \end{array}

where

k_{m} = \frac{2 m π}{a}, K_{m} \equiv K_{0} - k_{m}, q_{m} = \sqrt{k^{2} - K_{m}^{2}} .

and γ_eis the Euler-Mascheroni constant. In addition, G_bxx= G_byy+ G_bzz.

The non-vanishing components of $G_{b}^{(E M)}$ are given by

\begin{array}{l} G_{b z x} \equiv {(G_{b}^{(E M)})}_{z x} = - {(G_{b}^{(E M)})}_{x z} \\ = i \frac{K_{0}}{2 k q_{0} a} + \frac{i}{2 k a} \sum_{m = 1}^{\infty} (\frac{K_{m}}{q_{m}} + \frac{K_{- m}}{q_{- m}}) - \frac{1}{2 π} \frac{K_{0}}{k} \end{array}

C. Lattice Green dyadic

Combining the Poisson summation formula:

\frac{2 π}{a} \sum_{m = - \infty}^{+ \infty} δ (K - \frac{2 π m}{a}) = \sum_{n = - \infty}^{+ \infty} e^{i K a n}

with the Weyl expansion of the Green function, Eq. (40), and for r_n= (y_n= na, z_n= 0), we have

\begin{array}{l} \sum_{n = - \infty}^{\infty} e^{i K_{0} a n} g (r - r_{n}) = \int_{- \infty}^{+ \infty} \frac{d K}{2 π} {\sum_{n = - \infty}^{\infty} e^{i (K_{0} - K) a n}} e^{i K y} {\frac{1}{2 q} e^{\pm i q z}} \\ = {\sum_{m = - \infty}^{\infty} e^{i K_{m} y} e^{\pm i q_{m} z} \frac{i}{2 a q_{m}}} . \end{array}

The lattice Green dyadic can then be easily written as

\begin{array}{l} \overset{\leftrightarrow}{G} (r) \equiv \sum_{n = - \infty}^{\infty} e^{i K_{0} a n} \overset{\leftrightarrow}{G} (r - r_{n}) = \sum_{n = - \infty}^{\infty} e^{i K_{0} a n} (\begin{matrix} G (r - r_{n}) & G_{E M} (r - r_{n}) \\ - G_{E M} (r - r_{n}) & G (r - r_{n}) \end{matrix}) \\ = \sum_{m = - \infty}^{\infty} e^{i K_{m} y} e^{\pm i q_{m} z} {\overset{\leftrightarrow}{G}}_{m}^{\pm} = \sum_{m = - \infty}^{\infty} e^{i K_{m} y} e^{\pm i q_{m} z} (\begin{matrix} G_{m}^{\pm} & G_{E M m}^{\pm} \\ - G_{E M m}^{\pm} & G_{m}^{\pm} \end{matrix}) \end{array}

where

G_{m}^{\pm} = \frac{i}{2 q_{m} a} \frac{1}{k^{2}} (\begin{matrix} k^{2} & 0 & 0 \\ 0 & k^{2} - K_{m}^{2} & \mp q_{m} K_{m} \\ 0 & \mp q_{m} K_{m} & k^{2} - q_{m}^{2} \end{matrix}), G_{E M m}^{\pm} = \frac{i}{2 q_{m} a} \frac{1}{k} (\begin{matrix} 0 & \pm q_{m} & - K_{m} \\ \mp q_{m} & 0 & 0 \\ K_{m} & 0 & 0 \end{matrix}),

For a subwavelength grating, the far-field (|z| → ∞) Green dyadic is simply given by

\overset{\leftrightarrow}{G} (r) ~ e^{i K_{0} y} e^{\pm i q_{0} z} (\begin{matrix} G_{0}^{\pm} & G_{E M 0}^{\pm} \\ - G_{E M 0}^{\pm} & G_{0}^{\pm} \end{matrix}) .

where

G_{0}^{\pm} = \frac{i}{2 q_{0} a} \frac{1}{k^{2}} (\begin{matrix} k^{2} & 0 & 0 \\ 0 & q_{0}^{2} & \mp q_{0} K_{0} \\ 0 & \mp q_{0} K_{0} & K_{0}^{2} \end{matrix}), G_{E M m}^{\pm} = \frac{i}{2 q_{0} a} \frac{1}{k} (\begin{matrix} 0 & \pm q_{0} & - K_{0} \\ \mp q_{0} & 0 & 0 \\ K_{0} & 0 & 0 \end{matrix}),

C.1. Imaginary part of the lattice Green dyadic

\begin{array}{l} Im {\overset{\leftrightarrow}{G} (0)} \equiv \frac{1}{2} Im {{\overset{\leftrightarrow}{G}}^{+} (0) + {\overset{\leftrightarrow}{G}}^{-} (0)} \\ = Im {(\begin{matrix} G_{0} & G_{E M 0} \\ - G_{E M 0} & G_{0} \end{matrix}) .} + \sum_{m \neq 0}^{diffracted} Im {(\begin{matrix} G_{m} & G_{E M 0} \\ - G_{E M 0} & G_{m} \end{matrix}) .} \end{array}

where the sum runs over the propagating diffracted waves (with q_m‘real’). For a subwavelength grating:

Im {G_{0}} = \frac{1}{2 q_{0} a} \frac{1}{k^{2}} (\begin{matrix} k^{2} & 0 & 0 \\ 0 & q_{0}^{2} & 0 \\ 0 & 0 & K_{0}^{2} \end{matrix}), Im {G_{E M 0}} = \frac{1}{2 q_{0} a} \frac{K_{0}}{k} (\begin{matrix} 0 & 0 & - 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix}),

D. Fresnel amplitudes

The specular transmittance, T₀, of the lattice is

\begin{array}{l} T_{0}^{(p)} = {| 1 + \frac{i}{2 k a \cos θ} Z H_{x 0}^{(T)} / E_{p_{0}} |}^{2}, \\ T_{0}^{(s)} = {| 1 + \frac{i}{2 k a \cos θ} E_{x 0}^{(T)} / E_{s_{0}} |}^{2}, \end{array}

where

H_{x 0}^{(T)}

and

E_{x 0}^{(T)}

are the fields scattered by the lattice to the transmitted region z < 0

\begin{array}{l} Z H_{x 0}^{(T)} / E_{p_{0}} = k^{2} γ^{(p)} ({\tilde{α}}_{x}^{(M)} + \sin^{2} θ_{0} {\tilde{α}}_{z}^{(E)} + 2 k^{2} \sin θ_{0} G_{b z x} {\tilde{α}}_{x}^{(M)} {\tilde{α}}_{z}^{(E)}) + k^{2} \cos^{2} θ_{0} {\tilde{α}}_{y}^{(E)}, \\ E_{x 0}^{(T)} / E_{s_{0}} = k^{2} γ^{(s)} ({\tilde{α}}_{x}^{(E)} + \sin^{2} θ_{0} {\tilde{α}}_{z}^{(M)} + 2 k^{2} \sin θ_{0} G_{b z x} {\tilde{α}}_{x}^{(E)} {\tilde{α}}_{z}^{(M)}) + k^{2} \cos^{2} θ_{0} {\tilde{α}}_{y}^{(M)} . \end{array}

Finally, the reflectance for the diffracted order m (for = [q_m] = 0) is given by

\begin{array}{l} R_{m}^{(p)} = {(\frac{1}{2 k a \cos θ_{0}})}^{2} {| γ^{(p)} (k^{2} {\tilde{α}}_{x}^{(M)} + K_{0} K_{m} {\tilde{α}}_{z}^{(E)} + k^{3} (K_{0} + K_{m}) G_{b z x} {\tilde{α}}_{x}^{(M)} {\tilde{α}}_{z}^{(E)}) - q_{0} q_{m} {\tilde{α}}_{y}^{(E)} |}^{2}, \\ R_{m}^{(s)} = {(\frac{1}{2 k a \cos θ_{0}})}^{2} {| γ^{(s)} (k^{2} {\tilde{α}}_{x}^{(E)} + K_{0} K_{m} {\tilde{α}}_{z}^{(M)} + k^{3} (K_{0} + K_{m}) G_{b z x} {\tilde{α}}_{x}^{(E)} {\tilde{α}}_{z}^{(M)}) - q_{0} q_{m} {\tilde{α}}_{y}^{(M)} |}^{2} . \end{array}

The transmittance follows the same expression, replacing −q₀q_mby +q₀q_m.

Funding

Spanish Ministerio de Economía, Industria y Competitividad (LENSBEAM FIS2015-69295-C3-2-P and FIS2015-69295-C3-3-P and FPU PhD Fellowship FPU15/03566).

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Generalized Brewster effect in high-refractive-index nanorod-based metasurfaces

Abstract

1. Introduction

2. Scattering properties of a single HRI cylinder

2.1. Radiation intensity patterns and absence of first Kerker condition

3. Periodic array of HRI nanorods: Formal Scattering theory and general results

3.1. Single scattering versus full wave numerical calculations

3.2. Formal Scattering theory

3.3. Optical Theorem

4. Fresnel reflection amplitudes for nanorod-based metafilms

4.1. Kerker induced Brewster’s effect (KiBs)

5. Reflection and transmission by a Silicon nanorod metafilm

5.1. Longitudinal electric/magnetic dipoles

5.2. Transverse electric/magnetic dipoles

5.3. Longitudinal and transverse electric/magnetic multipoles

6. Concluding remarks

Appendix

A. Green tensors of a single cylinder

B. Lattice depolarization dyadic

C. Lattice Green dyadic

D. Fresnel amplitudes

Funding

References and links

Cited By

Figures (7)

Equations (59)

Optics Express