Infrared all-dielectric Kerker metasurfaces: supplement

This document provides supplementary information to “Infrared all-dielectric Kerker meta-surfaces”. We provide further details on the following topics discussed in the main paper:(i) Transmission and Reﬂection Through Subwavelength of 2D Arrayed Dipole-like Scatters; (ii) Multipole Expansion of Scattered Field for Single Particle and 2D Array; (iii) Kerker Conditions; (iv) Perfect Absorber and Invisible Metasurface; (v) Temporal-coupled Mode Theory of Perfect Absorbers; and (vi) Fabrication and Experimental Characterization. , ,


TRANSMISSION AND REFLECTION THROUGH SUBWAVELENGTH OF 2D ARRAYED DIPOLE-LIKE SCATTERS
(α e f f +χ e f f ) 2 (S11) where t is the transmission coefficient. For the backscattering, given the viewing direction with n z = (0, 0, −1), we have n z × (s × E inc ) = E inc . Then, similarly, we can obtain the reflectance (R) from the structure where r is the reflection coefficient. Our forms for r and t are similar to those given in prior studies. [1] We find that the reflectance is zero when, The transmittance may be zero if we have, Equation S14 is a necessary but not sufficient condition to obtain T = 0. If we also have the case that R = 0, then we find, or in terms of normalized polarizabilities, T = 0 ← α 2,n = χ 2,n = V −1 k 0 ρ (S16)

MULTIPOLE EXPANSION OF SCATTERED FIELD FOR SINGLE PARTICLE AND 2D ARRAY
To compare the array induced absorption to that from a single-particle, we employ multipole expansion method to analyze the absorption and scattering cross section arising from the unit cell of an array and a single silicon disk. There are two major methods used for determination of multipoles: expansion of the fields external to the particle with spherical multipoles using orthogonal spherical harmonics as basis functions, [3] and using induced currents inside the particle in Cartesian coordinates based on the discrete dipole approximation. [4] Here, we adopt the later method since we would like to identify the contribution of scattering cross section from dipoles, quadrupoles, and toroidal dipoles, respectively. [5] In the Cartesian multipole expansion method, the scattering object consists of closely packed cubic lattice of electric dipoles with the same polarizability α p for each dipole. Considering the multipoles located at the Cartesian origin, which is also the center of mass of the disk, the induced electric dipole (ED) moment, and induced polarization current density J are calculated as, where V is the volume of the disk, r is the position vector, p(d) is the relative permittivity of the dielectric disk (surrounding dielectric). Here we assume that our disk resonators are embedded in air with d = 1. Then the magnetic dipole (MD) moment (m), electric quadrupole (EQ) tensor ( Q), magnetic quadrupole (MQ) tensor ( M) and the toroidal dipole (TD) moment (T) can be expressed as, where ⊗ denotes the dyadic product of vectors, U is the 3 × 3 unit tensor. Then the corresponding scattering field for multipoles is given as [6] E(r) sca (ED) = k 2 0 4π 0 where k 0 , c, are the wavenumber and light velocity in vacuum, n = (sinθcosφ, sinθsinφ, cosθ) T is the unit vector along the radius vector r, θ is the polar angle and φ is the azimuthal angle. Then the corresponding scattering cross section for each multipole can be evaluated as, [4] σ sca (ED) = Given the symmetry of the cylindrical shape of disks and the incident field polarized in x direction E inc = E 0,x e ik 0 r , the induced polarization current J p = (J x , 0, 0) T , leading to non-zero multipoles being p x , m y , Q zx = Q xz , M yz = M zy , and T x . Then the scattered field in the forward direction (θ = 0) and backward direction (θ = π) can be described as, Further, we can describe the transmitted and reflected field in the multipole polarizabilities. The the total transmitted and reflected field from periodically arrayed particles can be described as where z → ∞ is the propagation distance from the center of the cylinder in the z direction, ρ is the density of the multipoles per unit area. Then the transmission and reflection coefficient can be given as Therefore to achieve zero reflection r = 0 and zero transmission simultaneously, we can obtain the relations with the assumption of E 0,x = 1 V/m: If only electric dipole and magnetic dipole are considered, we can get the same conditions as shown in Eqs. S14-S16. Given by the polarizabilities [7] : Specifically, for an x-polarized plane wave propagating in +z direction, E inc = (E x , E y , E z ) = (E 0,x , 0, 0)exp ik 0 z . Then the electric dipole moment at the center of the scatter (z=0) in x direction is given as As to the electric quadrupoles, the component ∇E inc in Eq. S40 is given as and similarly, Therefore, the quadrupole in Eq.S40 is shown as Then the electric quadrupole moment component is Similarly, the magnetic quadrupole moment component is described as where η 0 is the free-space impedance. With these polarizabilities, we can further derive Eqs. S36 and S37 as where the electric dipole polarizability is calculated from the induced polarization current density in Eq. S17 and p in Eq. S40 and is given as, and the magnetic polarizability is calculated from the induced polarization current density in Eq. S18 and m in Eq. S40 and is given as, For determination of α p and χ m through Eqs. S48 and S49 from simulation, the electric field is polarized in the x-direction and has the form E inc = (E x , E y , E z ) = (E 0,x , 0, 0)exp ik 0 z .

KERKER CONDITIONS
Generally, we may describe the differential scattering cross section as, [8] here dΩ = sin θdθdφ, and S || (θ) and S ⊥ (θ) are polarized scattering waves parallel and perpendicular (respectively) to the scattering plane, and given as, 2n + 1 n(n + 1) a n dP where P (1) n (cos θ) is the associated Legendre polynomial, and a n and b n are the complex scattering coefficient for TM and TE modes, respectively.
In order to achieve zero forward scattering and zero back scattering, Kerker only considered contributions from dipoles, i.e. only a 1 and b 1 . If we also assume only dipoles contribute, then we find, It is important to note that the S π = 0 condition (a 1 = b 1 ) implies that the electric (α) and magnetic (χ) polarizabilities are equal, i.e. α = χ, and the S 0 = 0 condition (a 1 = −b 1 ) implies α = −χ.

PERFECT ABSORBER AND INVISIBLE METASURFACE
The perfect absorber operating at 25 THz consists of an arrayed silicon disks with periodicity of p= 9.8 µm, radius r = 2.62 µm, and height h = 2.22 µm. In the numerical model, silicon material was set as a Drude model with plasma frequency of ω p = 36.65 THz and collision frequency γ= 7.16 THz. The cylindrical disk array supports two resonant eigenmodes of opposite symmetryan even mode ( Fig. 1 (c)) described by an induced electric polarizability (α), and and odd mode ( Fig. 1 (d)), described by an induced magnetic polarizability (χ). Indeed the induced α n (red curves) and induced χ n (blue curves) shown in Fig. S1(a) and (c), exhibit resonant behavior. Although α n achieves an additional offset due to ∞ -compared to χ n -its oscillator strength is sufficient such that the real part crosses zero near ω 0 -see red curve of Fig. S1 (a). In-fact both α 1,n and χ 1,n cross zero at the same frequency near the ω 0 , shown as the dashed grey line in Fig. S1 (a). We also find that the electric and magnetic imaginary polarizabilities (α 2,n , χ 2,n ) are approximately equal at ω 0 , as shown by the dashed vertical grey line in Fig S1 (c). Thus dissipation of electric and magnetic energy due to the incident external electromagnetic wave is balanced. To achieve invisibility, we simply decreased the disk height from 2.22 µm to 0.6 µm. The transmission and reflection are shown in Fig. 2 (c) and the calculated polarizabilities are shown in Fig. S1 (b) and (d). In our numerical simulation, all the tensor polarizability components polarization vectors were calculated in Cartesian coordinates. Many tensor polarizability components were negligible and thus not shown.
To evaluate the scattering properties, we adopted the asymmetric parameter g, which is defined as the average cosine of the scattering angle, given as [9] where σ s is the scattering cross section. From the definition, it is obvious that g goes to zero when the scattering is isotropic to all direction; and g also vanishes if the scattering is symmetric about the scattering angle of 90 o , which is in the horizontal direction. If the the forward scattering (θ < 90 o ) is stronger than the backward scattering (θ > 90 o ), g is positive, and vice versa. For dipolar scatters, g can be further described as where electric polarizability α = α 1 + iα 2 and χ = χ 1 + iχ 2 . According to Eq.S14 and S15, when t = 0, g = α 2,n ξ − (α 2 1,n + α 2 2,n ) 2(α 2 1,n + α 2 2,n ) + (ξ 2 − 2ξα 2,n ) where α 1,n and α 2,n are real and imaginary parts of electric polarizability per volume, ξ = 2 kρV , V = πr 2 h, r is the radius of the disk, and h is the disk height. The g values for other conditions can be easily obtained from Eq. S57 and shown in the paper. Figure S1 (e) and (f) shows the calculated polarizabilities and the asymmetry value of g for the perfect absorber and invisible metasurface, respectively. At the absorption peak, g is close to 0.5, which matches very well to the theoretical value shown in Table. 1, while g crosses zero at the anti-reflective point. Both of the calculated g values match very well the theoretical values shown in Table. 1.
With the numerically calculated electric and magnetic polarizabilities of the perfect absorber, we can also calculate the transmission and reflection based on Eq. S15 and Eq. S16. The calculated spectra are very close to the numerical results as shown in Fig. S2, with some deviation at high frequencies attributed to high-order multipoles.
In addition, in Fig. S3 (a) we show the phase difference of the induced polarizability resonances, α n and χ n , as a function of frequency. We find there is a slight phase difference of approximately 2 • , (χ n lags α n ), but that they are approximately in-phase up until the resonance frequency, after which they are out-of-phase (shown as the shaded grey area) for higher frequencies, achieving a max phase difference of 141 • at 29.3 THz. In Fig. S3 (b) we show the ratio of the magnitudes of χ n to α n and we find at ω 0 the ratio is unity. Thus in contrast to other studies, we find that induced electric and magnetic polarizabilities are not only equal in magnitude, but also in-phase at ω 0 . Thus we do not achieve the second Kerker condition of zero forward scattering, but rather achieve a modified second Kerker condition of zero total forward scattering, due to destructive interference of S 0 produced by α and χ with the incident wave. Figure S4 (a) shows the calculated transmission, reflection and absorption of the optimized absorber operating at 25 THz, with disk radius r = 2.62 µm, disk height h = 2.22 µm, and periodicity of 9.8 µm. All S-parameters simulations are at normal incidence, whereas experiments were carried out with a FTIR microscope using a 15x Cassegrain lens. Thus, the range of incident angles in experiment spanned from 12 to 23.6 degrees, leading to the difference between simulated and experimental absorptivity. Fig. S4 (c) shows the calculated corresponding scattering cross sections (σ sca ) of the arrayed disk array attributed from electric dipole (ED), magnetic dipole (MD), electric quadrupole (EQ), and magnetic quadrupole (MQ). Clearly, the σ sca of high-order multipoles barely contribute to the total SCS at low frequencies till the absorption peak. The σ sca of electric dipoles and magnetic dipoles intersect at 24.7 THz, which is slightly lower than the resonant frequency of 25 THz. As the frequency is higher than 29.7THz, the σ sca of electric and magnetic quadrupoles surpass that of magnetic dipoles, thus providing additional scattering to the total transmission and reflection as shown in Fig. S2 (c).
As a comparison, we also investigated the scattering properties of a single disk resonator as shown in Fig. S4 (b) and (d). Similarly, the scattering properties are dominated by the electric and magnetic dipole with frequency lower than 25 THz. However, significantly different from the scattering of the disk array, in which the σ sca is only about 28% larger than the absorption cross section (σ abs ), the σ sca of a single disk is 2.5 times larger than the σ abs , which also exhibits a much broader resonance. According to the σ abs of multipole analysis, such a difference is mainly associated with the drastic change of the σ abs ) of electric and magnetic dipoles, while the σ abs of high-order multipoles only change slightly because of their much weaker neighbor coupling as a result of the small σ abs compared to the unit-cell area of 83 µm 2 . Therefore, the neighbor coupling of electric and magnetic dipoles plays an important role to achieve perfect absorption for the disk array. Figure S5 shows the powerflow of time-averaged Poynting vector with incidence from left to right for a unit cell of the Kerker absorber. It is clear that the incident power is totally dissipated inside the disk. Compared the powerflow showing in Fig.5, the powerflow of the Poynting vector for a single disk is shown in Fig. S5. Additionally, some of the power is depleted inside the disk, however most of the power is forward scattered into free space, which is consistent with the scattering cross-section simulation in Fig.S4 (b), where the absorption cross-section is much smaller than the scattering cross-section.
Next, we also show that an invisible metasurface can be achieved via only changing the height of the absorber design, but keeping the radius, periodicity and silicon properties unchanged. Further, such an AR metasurface nearly achieves both Kerker conditions simultaneously. As shown in Fig. 2 (c), the response exhibits a zero reflection, and transmittance as high as 94.8% at 33.25 THz, where the loss tangent of silicon is still as high as 0.024, 40% of the loss tangent for a perfect absorber design at 25 THz. It has been shown that the perfect absorption of all-dielectric absorbers is achieved by critically coupling of degenerated hybrid electric (EH 111 ) and magnetic (HE 111 ) modes. [10] And the dielectric waveguide theory shows that the hybrid HE 111 mode is inversely proportional to the height of disk. [11] Therefore, the decreased disk height results in a lift the degeneracy of the electric and magnetic modes with the the magnetic resonance moving to a much higher frequency. [11,12] The resonance property at 31 THz can be further verified from the calculated polarizabilities. The normalized electric polarizability shown in Fig. S6 (b) exhibits a strong resonance, indicating that the resonance at 31 THz is the result of the induced electric dipole mode. However, the real and imaginary parts of magnetic polarizabilities are nearly zero in the range between 25 and 40 THz, i.e. χ 1,n ≈ χ 2,n ≈ 0. At 33.25 THz, the real part of the electric polarizability α 1,n crosses zero such that χ 1,n ≈ α 1,n = 0, while the imaginary part of the electric polarizability α 2,n reaches a minimum value of 0.5. As a result the transmission must be less than unity, as given by Eq. 6. Although the multipole scattering cross sections are small near 33.25 THz, although not zero (see inset to magnetic quadrupole (purple) destructively interfere such that there is no contribution to the total forward scattering. Meanwhile, we notice that normalized spatial volume factor k 0 ρV = 0.014 1. According to Eqs. S11 and S12, the reflectance R ≈ 0, and transmittance T is very close to unity. Because of these near-zero polarizabilities, the calculated scattering cross sections shown in Fig.S6 for electric and magnetic dipoles are also asymptotic to zero, which means achieving both of the first and second Kerker conditions, i.e. zero backward and forward scatterings, respectively. The reduced transmission at 33.25 THz is attributed from the absorption. We also noticed that the σ scat of the induced electric dipole equals to the σ scat of the toroidal dipole at 34.25 THz. And the total far-field scattering of the electric dipole together with the induced toroidal dipole is suppressed to zero at this frequency as shown in dashed blacked curve in Fig. S6. We conclude that the invisible metasurface achieves an anapole mode at 34.25 THz, where the radiation of the electric dipole destructively interferes with the induced toroidal dipole. Because of an extra weak scattering contribution from the magnetic quadrupole, the anapole mode excitation does not overlap the anti-reflection point exactly.

FABRICATION AND EXPERIMENTAL CHARACTERIZATION
The metasurface array was patterned by deep reactive ion etching (DRIE), resulting in a free standing sample area of 1cm × 1cm. The etching mask of hexagonal disk array was first obtained via the electron-beam lithography of a 400-nm ZEP520A resist coated on an SOI (silicon on insulator) sample with device layer about 2.2 µm. Then the hexagonal silicon disk array was obtained through the DRIE process to remove the extra silicon between disks. Next, to achieve a free-standing structure, the backside of the sample with thickness around 500 µm was again etched in the DRIE system. And the 1-µm buried oxide layer was removed in buffered oxide etch (BOE) solution for about 10 mins. Finally, the free-standing silicon disk array was cleaned using acetone and isopropyl alcohol. The metasurface was characterized in a Fourier transform infrared spectrometer, using a globar (silicon carbide) source, potassium bromide (KBr) beam splitter, and a mercury cadmium telluride (MCT) liquid nitrogen cooled detector. Measurements were collected over a range in the mid infrared in both near normal reflectance R(ω) and normal transmittance T(ω), each normalized by a suitable reference. In the reflection measurement, a gold mirrow was used as the reference, while in the transmission measurement, the free-space was used as the reference give the free-standing of the structure. In both measurements, the incidence angles are between 12 to 23.6 degrees. The frequency dependent absorptivity was calculated as A(ω) = 1-T(ω)-R(ω).