Angle-selective all-dielectric Huygens' metasurfaces

Silicon nanodisk metasurfaces were fabricated by structuring hydrogenated amorphous silicon (a-Si:H) thin-films on a fused silica (${\rm SiO}_{2}$) substrate via an electron-beam lithography (EBL) based process.

Figure 1(a) shows a sketch of the metasurface design. The nanodisks are arranged in a periodic square array with lattice constant $a=860~{\rm nm}$ and covered by an embedding ${\rm SiO}_{2}$ layer of thickness t. In order to allow for Huygens' metasurfaces with spectrally overlapping electric and magnetic dipolar Mie-type resonances in the telecommunication spectral range around $1.55~\mu{\rm m}$, the nanodisk height was fixed to $h=248~{\rm nm}$ and several metasurface samples, each with a unique nanodisk radius r between $230~{\rm nm}$ and $380~{\rm nm}$, were fabricated. All arrays were realised with a lateral size of $100\times100~\mu{\rm m}^2$, which is sufficient for boundary effects to be negligible.

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A scanning-electron micrograph (SEM) of a typical metasurface sample is displayed in figure 1(b). For the fabrication of all samples, $248~{\rm nm}$ of a-Si:H were deposited onto a ${\rm SiO}_{2}$ microscope cover slip with a thickness of $0.16~{\rm mm}$ by plasma-enhanced chemical vapour deposition (PECVD) at $250~^\circ{\rm C}$. Then, the silicon film was spin-coated with hexamethyldisilazane (HMDS) as an adhesion promoter ($1500~{\rm rpm}$, $30~{\rm s}$), followed by the negative-tone electron-beam resist maN-2403 ($3000~{\rm rpm}$, $30~{\rm s}$), resulting in a resist thickness of about $300~{\rm nm}$. Afterwards, a pre-exposure bake was performed at $90~^\circ{\rm C}$ for $1~{\rm min}$. A conductive polymer (ESPACER) was spin-coated ($1500~{\rm rpm}$, $30~{\rm s}$) on top of the resist in order to enhance the electrical conductivity of the sample for EBL.

Next, the EBL exposure was performed at an acceleration voltage of $20~{\rm kV}$ and a current of $13~{\rm pA}$. After exposure, the sample was developed in ma-D 525 for $105~{\rm s}$, followed by a rinse in deionised water for several minutes. The sample then underwent a fluorine based, inductively-coupled plasma—reactive-ion etching (ICP-RIE) process in a Plasmalab System 100 from Oxford Instruments by utilizing the obtained electron-beam resist pattern as an etch mask. The etching was performed with ${\rm SF}_{6}$ $(1.8~{\rm sccm})$ and ${\rm CHF}_{3}$ $(50~{\rm sccm})$ at $20~^\circ{\rm C}$ with $10~{\rm mT}$orr at $500~{\rm W}$ induction power and $15~{\rm W}$ bias power. The etch rate was monitored in situ with a D-205 Digilem laser interferometer from Jobin Yvon-Sofie.

Finally, after removing the residual resist and organic solvent residue left on the sample using an oxygen plasma, the nanodisks were embedded in an additional ${\rm SiO}_{2}$ layer with a target thickness t between $500~{\rm nm}$ and $600~{\rm nm}$ using PECVD at $250~^\circ{\rm C}$.

Angularly resolved transmittance spectra of the metasurface samples were measured with a custom-built white-light spectroscopy setup. In order to ensure a homogeneous intensity profile on the sample, we employed Köhler illumination in conjunction with an intensity-stabilised tungsten halogen lamp. The divergence angle was reduced to less than $1^\circ$ by the condenser diaphragm in the Köhler illumination path. A linear polariser was introduced to control the polarisation of the incident light field. The light transmitted by a selected metasurface was collected with a large-working-distance near-infrared microscope objective (Mitutoyo M Plan Apo NIR $20\times$, $0.40~{\rm NA}$), imaged onto a rectangular aperture and finally coupled into an optical spectrum analyser (Yokogawa AQ6370B). Tilting of the sample holder allowed for a controlled variation of the incidence angle. The size of the rectangular aperture was set to the smallest projected size of a single metasurface sample. In order to determine the relative transmittance $T=I/I_0$, each intensity spectrum I was referenced to a spectrum I₀ obtained from a measurement through the substrate and embedding layer directly next to the selected metasurface. Note that due to this referencing procedure, T may exceed unity.

To compare our experimental results with theory, we numerically calculate the relative transmittance $T=\left\vert t/t_0\right\vert ^2$ and the relative phase $\phi={\rm arg}({t/t_0})$ for a systematic variation of the incidence angle by using a finite element solver (Comsol Multiphysics). Here, t denotes the zeroth-order transmission coefficient of the full model and t₀ that of an equivalent model in which the material of the nanodisks (a-Si:H) is replaced by that of the embedding layer (${\rm SiO}_{2}$), thus mimicking the referencing procedure performed in the experiments. Note that the phase is given with respect to the $\newcommand{\e}{{\rm e}} {\exp}({-{\rm i}\omega t})$ time convention. Dispersive refractive index data for ${\rm SiO}_{2}$ and a-Si:H were taken from [29] and measured via ellipsometry, respectively. For reference, the average refractive indices are $1.444$ (${\rm SiO}_{2}$) and $3.698$ (a-Si:H) for $1.4~\mu{\rm m} \leqslant \lambda \leqslant 1.65~\mu{\rm m}$. In the calculations, we assume an infinitely thick ${\rm SiO}_{2}$ substrate; the upper half-space is taken to be vacuum.

Since our numerical model does not account for sample imperfections, deviations from the assumed refractive index data and measurement inaccuracies, we determined optimised geometrical parameters by performing a least-square fit on a representative subset of the experimental transmittance data. In particular, by including angle-dependent data in the set of fitted points, we were able to obtain a very good overall agreement between the numerical and the experimental transmittance spectra for all angles of incidence. Furthermore, we simultaneously optimised the nanodisk height, the embedding layer thickness and the nanodisk radii of the two metasurfaces characterised under oblique incidence in order to obtain a consistent set of geometrical parameters. The lattice spacing was kept constant as it can be controlled with high precision in the experiment. Initial values for the parameters were taken from SEMs and experimental target values. We used the gradient-based Levenberg-Marquardt algorithm [30–32] to minimise the sum of the squared differences between the measured and the simulated transmittance for those combinations of wavelengths and incidence angles which were empirically identified to be most sensitive to parameter changes. After an optimal set of parameters was found, the transmittance and phase spectra were then calculated for the full spectral and angular ranges studied in the experiment.

Furthermore, to unambiguously determine the multipolar order of the observed resonances, we computed the scattering contribution $\rho_{cn}=\sum_{m=-n}^{n} \left\vert c_{nm}\right\vert ^2$ on the basis of a multipole decomposition of the induced polarization density [33] within an individual nanodisk. Here, $c_{nm}$ stands for either the electric or the magnetic multipole coefficient of the decomposition with polar order n and azimuthal order m. The rigorous simulation of the induced polarization density was performed in CST Microwave Studio. The multipole coefficients of the octupolar order were found to result in insignificantly small scattering contributions and were, together with all higher orders, neglected in the further analysis. Note that the computed multipole coefficients do not only describe the field scattered by one nanodisk, but include the superposition of the fields scattered by all other nanodisks and the interface of the embedding layer.

First, in order to identify suitable metasurfaces for the oblique incidence study, we experimentally measure the relative transmittance spectra of a range of fabricated metasurfaces with different nanodisk radii at normal incidence and determine the corresponding radii from SEMs. These results are presented in figure 2(a). Figures 2(b) and (c) present optimised simulated spectra for comparison.

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While a good qualitative agreement with experimental data is achieved for a nanodisk height of $h_{\rm opt}=224~{\rm nm}$ and an embedding layer thickness of $t_{\rm opt}=870~{\rm nm}$, note that all nanodisks appear to be effectively smaller in the simulations than in the SEMs by about $r-r_{\rm opt}=30~{\rm nm}$ and $h-h_{\rm opt}=24~{\rm nm}$. This deviation is partly due to the formation of an oxide layer at the nanodisk surfaces during nanofabrication. Furthermore, differences between the actual silicon refractive index and the refractive index data used in the model may also play a role, considering that ellipsometric data was obtained from an unstructured a-Si:H layer with a much lower surface-to-volume ratio compared to the nanodisk array.

Both the experimental and the simulated spectra clearly reveal the tuning of the dipole resonance positions as a function of the nanodisk radius as well as the characteristic transparency effect for spectrally overlapping resonances in accord with previous work [2, 6]. In particular, at an exemplary wavelength of $1.45~\mu{\rm m}$, a full $2\pi$ phase coverage is obtained while the transmittance is bounded by $0.83$ and $1.02$ (experiment) or $0.83$ and $0.97$ (simulation) as the radius is varied over a range of $150~{\rm nm}$. For large radii, the electric and magnetic dipole resonances separate spectrally into two individual resonances with their characteristic near-zero transmittance and a phase shift of π at resonance. Note that in this case, the electric dipole resonance (ED) is located at longer wavelengths than the magnetic dipole resonance (MD). Also note that the transparency effect in the Huygens' regime occurs despite a strong resonant interaction of the incident light with the silicon nanoresonators, as evidenced by the strong spectral dispersion in the transmitted phase. This is in stark contrast to the off-resonant transparent regions in the spectrum, which show weak spectral dispersion in the transmitted phase.

Figure 3 illustrates the near-field profiles of the electric and magnetic field inside one nanodisk of the metasurface ($r_{\rm opt}=300~{\rm nm}$) for a resonant excitation of the ED ($\lambda=1.595~\mu{\rm m}$) and MD ($\lambda=1.525~\mu{\rm m}$) at normal incidence.

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In the case of the ED, the electric field shows a linear orientation of its field vectors and a field enhancement around the centre of the nanodisk (see figure 3(a)), while the magnetic field is circulating in a plane normal to the electric field vectors (see figure 3(b)). In the case of the MD, the situation is reversed; the electric field is circulating and the magnetic field vectors are oriented linearly (see figures 3(c) and (d)). Consequently, the near-field patterns of the two resonances closely resemble those of the electric and magnetic dipole resonances known from Mie theory, and we label them ED and MD, respectively. This is further underpinned by the multipolar decomposition of the induced polarization densities associated with these resonances as discussed below.

In the following, we will concentrate on the angle-dependent transmission properties of two particular metasurfaces which are representative for the two cases of (i) spectrally overlapping (see figures 4 and 5) and (ii) spectrally separate (see figures 6 and 7) electric and magnetic dipole resonances at normal incidence. The radii of the selected metasurfaces with (i) $r=287~{\rm nm}$ ($r_{\rm opt}=257~{\rm nm}$) and (ii) $r=330~{\rm nm}$ ($r_{\rm opt}=300~{\rm nm}$) are marked by the vertical lines in figure 2. Note that in all presented results, both the wavelength λ and the incidence angle θ are measured in vacuum.

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Here, we consider the case (i) of a metasurface ($r=287~{\rm nm}$) with spectrally overlapping dipole resonances for normal incidence. Figures 4(a) and (d) show the experimentally measured relative transmittance spectra for TE and TM polarisation, respectively. While the metasurface is almost transparent for normal incidence over the entire spectral range of interest, the gradual reappearance of a resonance dip is observed for both polarisations as the incidence angle is increased. In the case of TE polarisation, however, the transmittance immediately decreases as the incidence angle increases, whereas for TM polarisation, high transmittance is preserved up to about $\theta<5^\circ$. The ED and MD remain in a partial spectral overlap around $\lambda=1.485~\mu{\rm m}$ for incidence angles up to about $\theta<10^\circ$, then the dip splits into two distinct resonance minima. For a further increase of the incidence angle, the spectral separation between these two resonance minima is continuously growing. Figures 4(b) and (e) show the corresponding simulated transmittance spectra. Good qualitative agreement with experimental data is obtained, allowing for a further analysis of the observed resonance shift based on the numerical model.

An important point which deserves our attention for dielectric metasurfaces at oblique incidence is the onset of the first diffraction order. In our case, no higher diffraction orders can be excited for $\lambda>\left(n+{\sin}{\theta}\right)a$, where $n=1.444$ is the averaged refractive index of the substrate. The limit of this condition is marked by dashed lines in figures 4(b) and (e). Consequently, at an exemplary wavelength of $\lambda=1.6~\mu{\rm m}$ and for incidence angles up to about $\theta<25^\circ$ it is possible to evade scattering loss into higher diffraction orders. The simulated relative phase is displayed in figures 4(c) and (f), indicating that full $2\pi$ phase coverage is preserved up to the incidence angle where the splitting of the resonance peaks occurs.

Furthermore, to identify the dominant multipolar order of the resonances observed under normal and oblique incidence, we plot the scattering contributions in figure 5. As seen in figure 5(a), at normal incidence, the spectral positions of the ED and MD are very close, as required for Huygens' metasurfaces [2]. At an incidence angle of $15^\circ$ and for TE polarisation, the ED shifts towards longer wavelengths, whereas the MD remains mostly stationary; the spectral overlap of the ED and MD reduces continuously. In the case of TM polarisation, the shifting directions of the ED and MD are reversed with the MD dominated mode shifting towards longer wavelengths. Thus, although the angular dependence of the transmittance spectra of the Huygens' metasurface appears very similar for the two orthogonal polarisations, the multipole decomposition reveals that at oblique incidence, the ED is the fundamental resonance of the metasurface for TE polarisation while for TM polarisation the MD contribution is the strongest at the fundamental resonance. The slight detuning of the spectral positions of the ED and MD for normal incidence in combination with the reversed direction of the resonance shifts also explains the difference in the onsets of the low transmittance regime in the TE and TM case.

Next, we consider the case (ii) of a metasurface ($r=330~{\rm nm}$) with spectrally separate dipole resonances for normal incidence. Figure 6 summarises the experimentally measured and numerically calculated angle-dependent relative transmittance and phase spectra; figure 7 presents the scattering contributions for incidence angles of $0^\circ$ and $15^\circ$. At normal incidence, the MD ($\lambda=1.525~\mu{\rm m}$) is located at a wavelength much shorter than the ED ($\lambda=1.595~\mu{\rm m}$). As already seen in case (i), for TE polarisation we observe a red shift of the ED and a slight blue shift of the MD as the incidence angle is increased, leading to a further spectral separation of the two resonances at oblique incidence (see figures 6(a)–(c)). For TM polarisation, an increase in the incidence angle leads to a red shift of the MD, while the ED remains almost stationary (see figures 6(d)–(f)). As a consequence, the ED and MD are shifted into spectral overlap for an incidence angle of $\theta\approx 10^\circ$ ($\theta\approx 13^\circ$) in experiment (simulation) at a wavelength of $\lambda=1.595~\mu{\rm m}$, and for a small angular range of few degrees, the Huygens' regime of high transmittance and full $2\pi$ phase coverage is restored. The small discrepancy of about $3^\circ$ between the measured and simulated results in the position of this angular range can be explained by the alignment accuracy of the optical setup in combination with deviations of the experimentally realised sample geometry from the ideal model system. Again, the onset of the first diffraction order is indicated by the dashed lines in figures 6(b) and (d), showing that the Huygens' regime appears at wavelengths well below that of the first diffraction order at the corresponding angle.

Importantly, due to the stationary behaviour of the ED in TM polarisation, the metasurface reflects all TM light of the corresponding resonance wavelength which is arriving at incidence angles other than the particular angle for which the Huygens' regime is established. Thus, this metasurfaces superimposes a Huygens' metasurface response with an angular filtering function. This also holds for the metasurface of case (i), where the Huygens' regime is established at near-normal incidence. As such, case (i) represents one special case of a larger class of angle-selective Huygens' metasurfaces.