One mode-model in nanostructures with inclined sidewalls applied to nano Fabry-Perot structures

. Metasurfaces are engineered with specific shapes and sizes to interact with light in a unique way. By manipulating the design of the metasurface, it is possible to control optical properties of the surface such as its thermal emissivity. However, thin layers patterning techniques can lead to side-wall angles, thus modifying light propagation within the structure. Here, we introduce a one-mode model that fairly describes the propagation of light in structures with inclined sidewalls. We applied this method to two families of plasmonics resonators: nano Fabry-Perot and coupled nano Fabry-Perot with refractory materials ZrC and tungsten.


Introduction
Metasurfaces are arrays of subwavelength-scale structures that interacts with incident light providing unique optical properties such as being able to manipulate the amplitude, phase, and polarization of light at the surface.They offer a new way to control the properties of light in a wide range of applications such as telecommunications, imaging, sensing, and energy conversion.Most patterning techniques such as lift-off or, ion beam etching can result in angled side-walls up to 20° [1].Such patterning techniques are mandatory for refractory materials such as ZrC, an alternative to tungsten for high temperature applications (T>1000°C), as for example thermo-photovoltaic applications [2].
Here, we introduce a one-mode model that fairly describes the propagation of light in structures with inclined sidewalls.We applied this method to two families of plasmonics resonators: nano Fabry-Perot and coupled nano Fabry-Perot with refractory materials ZrC and tungsten.The method agrees well with electromagnetic solvers simulations

One-mode model for simulations of inclined slits
Modal simulation involves modelling the metasurface as an array of layers (structured or not) and calculating the eigenmodes (both propagating and evanescent), of the structure, providing physical information on the resonance.With modal simulations, by calculating the eigenmodes, it is possible to determine the metasurface response to specific wavelengths of light, like its absorption properties.It can also be used to study the behaviour of metasurfaces under different conditions, such as at varying angles of incidence or polarization [3].
The typical structure considered in the following is depicted in Fig. 1.It consists in a periodic array of metallic slits with inclined sidewalls.The slit with vertical sidewalls is known to act as a vertical Fabry-Perot resonator with the height h determining the cavity propagation length and its width control the effective index of the propagating gap plasmon.With inclined sidewalls, the gap plasmon is modified continuously along its propagation and the resonance behaviour is modified.To model such a structure with classical modal methods, the inclined slit can be divided into a superposition of smaller layers of decreasing width, but it would lengthen drastically the calculus time (more layers to solve and extremely meshed horizontal interfaces).
To circumvolute this issue and calculate the propagation of the propagating mode in the slit, we use in the following a one-mode model [4].In that case, the inclined layer can be simulated using two regular structured layers, one with the width at the top of the inclined layer and one with the dimensions at the bottom.By retrieving the respective scattering matrices, the interfaces of the inclined layer can be simulated.Then the gap plasmon is computed analytically along its propagation.
In Ping Sheng's article [5], the mode wavevector kz in the slit (in our case gap plasmon), obeys Eq. 1: for an incident light inclined of , a slit of period P, a relative width of metal and by posing and (with the complex wave vector of the incident light and the complex dielectric constant of the metal).
By dividing the inclined layer into infinitesimal ones of progressive width, we can calculate recursively the complex eigenvalue for with Gauss-Newton algorithm.By calculating the propagation thought every individual slit of height (with Eq.2), the propagation of the propagative mode in the inclined layer is obtained as: Figure 2 shows the reflectivity spectra (orange line) computed with this method for an inclined slit in a gold layer (top aperture 90 nm with 5° inclined sidewalls, slit depth is 0.38μm, bottom width of 20 nm).For the sake of comparison, the reflectivity spectra of regular vertical slits with widths 20 (green) and 90 nm (blue) are also plotted.As expected, the smaller width, in green, has a higher resonant wavelength compared to the larger one in blue, as the gap plasmon effective index is higher for smaller gaps.Moreover, the response of the slit with inclined sidewalls is as expected exhibiting a resonance that is spectrally between the two extreme cases.
More refined geometry can also be considered with the one-mode model introduced before and extending its applications to a one-mode per slit model.By adding a second slit, with a geometry slightly different than the first one (either wider or deeper), it is possible to use wider slits that would be otherwise be undercoupled [6].This is the principle of the coupled nano Fabry-Perot resonator that relies on a three-waves interference to reach critical coupling.This kind of geometry can achieve very thin resonance (more than 100 of q-factor) even in the near IR while being reasonable to fabricate.By using the previously described method, on this geometry, it is possible to find optimal resonance with for example refractory materials for high temperature emissivity control.As shown in Fig. 3, we studied this coupled nanoFabry-Perot resonator with refractory materials ZrC [7] and tungsten.We obtained respective Q-factors of 25 and 4 for W and ZrC.W having better metallic properties than ZrC, optimal geometry of W yields better Q-factors.

Conclusion
Modal simulation is a fast and simple way to simulate metamaterials.We have introduced a one-mode model that accounts for inclined sidewalls in nanostructures and have demonstrated its use on nano Fabry-Perot and coupled nano Fabry-Perot resonators.Experimental investigations of these resonators are ongoing with the materials aforementioned.

Fig. 2 .
Fig. 2. Modal simulation of a simple slit of height h = 0.38 μm repeated at period P = 2.65μm for different width and for an inclined slit with respective dimensions at the top and bottom in Au metal.