Highly-resonant two-polarization transmission guided-mode resonance filter

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Highly-resonant two-polarization transmission guided-mode resonance filter Léopold Macé, Olivier Gauthier-Lafaye, Antoine Monmayrant, Stéphane Calvez, Henri Camon, Hervé Leplan

We theoretically demonstrate a mid-infrared polarization-independent guided-moderesonance transmission filter. We designed a structure based on a deeply-etched 2D grating above a thin slab of the same material respectively supporting transverse magnetic and transverse electric fundamental modes with identical effective index, which propagate along orthogonal directions. This device relates to multi-resonant guided-mode-resonance filters, and more particularly to the concept of zero-contrast gratings (ZCG), which can operate either as wideband reflectors [R. Magnusson (2015)]. However, contrary to the latter, this new generation of filters is not bound by stringent material requirements inherent to conventional ZCGs. In particular, ZCGs are demonstrated with high to low refractive index ratio below 2, using germanium as high-index material over a low-index zinc sulfide substrate. These filters exhibit a transmission peak with a full-width at halfmaximum of about 30 pm, and a maximum transmission close to 100 % lying in a 46-nm-wide rejection window. © 2018 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5051621

I. INTRODUCTION
Our goal is to design polarization-insensitive transmission filters in the mid-IR range, between 3 and 4 µm, whose central wavelength could be easily varied for adjacent filters in a matrix. Zerocontrast gratings (ZCGs) have proven to be interesting solutions for the design of such filters: tunability of the resonance, few layers, planar structuration approach. Their principle, as described in Ref. 1, is to excite two distinct guided modes through two different grating orders in a single high-index partiallyetched layer. These 2 orders differ in coupling strength, resulting in two interfering/overlapping Fano resonances with different widths. By placing the resonant peak of the narrowband resonance in the middle of the low transmission band of the broader one, an efficient transmission filter with a broad rejection band can be obtained. ZCG can operate either as wideband reflectors 2 or bandpass filters. 3 Besides, such narrowband 1D ZCGs can easily be implemented as 2D transmission filters with a simple parametric optimization. 4 Another concept to design transmission filters in the long-IR range (8 − 14 µm) 5 relies on a high-contrast-grating (HCG) structure 6 that provides the broad rejection band.
The narrowband transmission resonance is obtained by breaking the in-plane symmetry of the system: either by using an off-normal incident plane wave, 5 or by introducing an asymmetry in the grating profile. 7,8 The difference in the coupling efficiency between the resonances responsible for the broad rejection band and the high-Q one originates from a variation in the overlap between the guided-modes and the incident plane wave. Depending on the polarization of the incident light and a leopold.mace@laas.fr the orientation of the incident k vector, TE or TM resonances can appear. However, it is likely that polarization-insensitive devices based on these structures will prove quite challenging to design and fabricate. Therefore, we choose to develop a concept based on ZCG-like designs functioning under normal incidence rather than symmetry-related devices. All ZCGs applied for filtering applications in the litterature exploit transverse electric (TE) guided-modes, 3,4,9,10 and will therefore be referred to as "pure TE ZCGs". Pure TE ZCGs get the required narrow-and large-band Fano resonances by coupling two guided modes through two different orders of the same grating: e.g. by coupling the fundamental TE 0 mode through the second grating order, and the TE 1 mode through the first grating order. Pure TE ZCGs thus need to support 2 modes with effective indices in a ratio of 2: the high-index layer supporting the guided-modes must have an index at least twice that of the low-index substrate. 1 In a previous study, 11 we demonstrated 1D narrowband filters in the mid-IR based on pure TE ZCG. Besides, we presented a method to extend both angular acceptance and resonance width by more than a decade for 1D pure TE ZCG filters using a double-corrugation scheme for the grating. However, we didn't manage to extend this concept to a polarization-insensitive filter. Indeed, as we switch from a 1D to a 2D grating, many more grating orders as well as transverse magnetic (TM) modes need to be taken into account and prevented us from easily obtaining an efficient filter with a single transmission peak.
The above-described filters require a high-index contrast between the guiding layers and the embedding materials. From a practical point of vue, this condition is very limiting in the considered mid-IR range. Indeed, there are very few low-index substrates with low enough absorption in this spectral domain that can fulfill this condition, even with germanium (n ≈ 4.15) as high-index material. The most appropriate ones are fluorides, 12 which are expensive, fragile, and hygroscopic.
A slightly different approach has been recently demonstrated. 13 Instead of relying on 2 guided modes of a single waveguide coupled by 2 orders of a single grating to generate 1 large and 1 narrow Fano resonances, this method uses 2 single-mode waveguides and a 2D grating for polarization independence. The waveguide closest to the grating provides the large Fano resonance (strong coupling), while the further provides the narrow Fano resonance (weak coupling). Contrary to single-layer multimodal waveguides, the fundamental modes in the two isolated waveguides can have their effective index adjusted independently. When the 2D grating is a hole-type (resp.rod-type) photonic crystal, then, the device is shown to couple two TE (resp. TM) guided-modes. It shows that both kinds of modes can be used to design bandpass filters.
In this article, we present an original approach for the design of transmission filters that combines the flexibility of the two-waveguide approach with the simplicity of the partially-etched single-layer designs. We design a ZCG that supports TE and TM fundamental modes with identical effective index, propagating along orthogonal directions, which we will call "TE/TM ZCGs". We use a deeply-etched 2D grating above a thin slab of the same material, which is designed such that the TE mode is mostly guided in the slab, while the TM one lies within the ridges of the grating (see Fig. 1). Because these modes are confined within separate parts of the high-index structure, their effective index can be tuned almost independently. This method allows us to couple guided-modes with close effective index, thereby relieving the stringent condition on the refractive index of the materials while using a simple, partially-etched, single-layer structure. The resulting filters are intrinsically polarization-insensitive and benefit from the tunability of guided-mode resonance filters.

II. DESIGN
We use germanium (Ge) as waveguide material and calcium fluoride (CaF 2 ) as substrate. Indeed, these materials are highly transparent in midIR range, which makes them very appropriate for transmission filters. Moreover, we want to show that we can achieve performance equivalent to that of purely TE ZCGs that require such materials. 11 The structure is composed of a slab waveguide above which lies a square grating of the same material with ridges along the x− and y − axis (see Fig. 1). For symmetry reasons, polarization independence under normal incidence is immediate for such a device. 14,15 Therefore, we can consider only one polarization state of light, for example E = E y and H = H x . The starting idea is schematically described in Fig. 1(a): we want to design a filter using two resonances originating from the coupling of one TE and one TM mode propagating in orthogonal directions. The TE mode is mainly guided in the slab, while the TM one is confined within the deep grating ridge (see Fig. 1(b)). These modes possess similar yet independently tunable effective indices, and have widely different overlaps with the grating, thus resulting in a large Fano resonance (TE mode) and a narrow one (TM mode) in the same wavelength range.
In order to obtain a single transmission peak, we want to limit the number of guided modes and coupling orders. Ideally, we thus want to have one single TM mode and one single TE mode in the whole structure and couple them through the first grating order. However, it is not possible to match the indices of the TE 0 and TM 0 modes in a slab waveguide. Thus, we choose to use a high aspect-ratio grating.
By doing so, we keep the waveguide thin enough to support only the fundamental TE mode, while the high grating ridges should support a fundamental TM mode of close effective index. The indices of the modes can then be tuned almost independently, by changing slightly either the ridge height t r or the slab thickness t s . We thus have the same flexibility as with two separated waveguides. 13

A. Waveguide design
The starting point is a Ge slab waveguide above a CaF 2 substrate. In the following we use for the Ge refractive indices data extracted by ellipsometry on sputtered thin films, and on manufacturer's data for the CaF 2 substrate. 16 We choose a slab waveguide thickness t s = 0.22 µm. For this value, and at λ = 3.5 µm: n slab TE 0 =2.67. Let us note that the fundamental TM mode is also present, but with n slab TM 0 =1.42 2.67. Since its index is significantly lower than that of the TE mode, it won't result in a resonance close to that arising from the coupling of the fundamental TE mode and can safely be ignored. For the rest of the study we can consider that the slab only supports the fundamental TE mode hereafter called the slab mode or TE mode.

B. Grating design
The next step consists in adding the deep ridge grating that will support a TM mode, hereafter called the ridge mode or TM mode. The first condition is that the grating should couple to free space the TE slab mode through the first order. We use the phase matching condition given in Ref. 1 to determine the period Λ ensuring a resonant coupling of the TE slab mode around λ = 3.5 µm: This parameter is not crucial, since the period is a tunable parameter that will allow us to eventually tune the central wavelength of the filter. Besides, as the grating ridges height increases, the effective index of the slab TE mode increases. The grating ridge fillfactor is chosen close to f = 0.26, which is small enough to limit the coupling between TM modes in adjacent ridges, but also high enough to keep a reasonable aspect ratio in order to ensure the fabricability of the structure. Fig. 2(a) shows the dispersion of the slab TE (red curve) and ridge TM (green curve) modes in the structure as a function of the grating ridge height t r (see Fig. 1(b)), together with the H-field principal component for each mode namely H x for the TM mode ( Fig. 2(b)) and H z for the TE mode (Fig. 2(c)). The calculation is performed for a single grating ridge on an infinite slab waveguide using an optical eigenmode solver for dielectric waveguides. 17 The grating is considered as a one-dimensional structure, therefore both modes are given in the (Oxz) plane. The lateral and vertical extension of the simulation domain as well as the mesh resolution are sufficient to ensure convergence.
We can see in Fig. 2(a) that the dispersion curves of both modes cross for t r = 0.87 µm. In the vicinity of this point, the effective index of the TM mode ( Fig. 2(a), red curve) is largely affected by the ridge height t r , while that of the TE mode ( Fig. 2(a), green curve) barely varies. This is due to the fact that the TM mode is mainly confined in the ridge, while the TE mode is mainly confined in the slab, as shown on Figs. 2(b) and 2(c). As a consequence the ridge and slab modes can be tuned almost independently: the grating ridge height t r controls the TM ridge mode effective index, while the slab waveguide thickness mainly determines that of the TE slab mode.

C. Parametric study
The crossing point determined above (see Fig. 2(a)) is only a starting point for an optimization procedure conducted with the S4 code, 18 which uses RCWA and the S-matrix algorithm. The filter transmission map T (λ, t r ) is shown in Fig. 3. This RCWA simulation is performed using 121 harmonics. The region in the vicinity of t r = 0.99 µm and λ = 3.62 µm is of particular interest to us. Indeed, a single high-transmission resonance lies within a broad rejection band.
The transmission of the filter with a ridge height of t r = 0.99 µm (corresponding to the light blue line in Fig. 3) is plotted in Fig. 4. A higher number of harmonics (501) is used to ensure full convergence of the solution, in order to resolve the narrow spectral peak. Fig. 4(a) shows a hightransmission peak around λ = 3.624 µm, having a FWHM of 30 pm within a 46 nm-wide rejection band of T < 1%. In Fig. 4(b), we can see that the transmission peak is flanked by 2 high-rejection dips, which means that there are indeed two Fano resonances in the system. Our aim is to identify the particular couplings occurring here. For this, we calculate the electric and magnetic fields within one period of the (Oxy) plane of the structure in the middle of the slab and ridge heights.
At the resonant peak, the most intense component of the fields are H x (shown in Fig. 4(c)) and E z (not shown) within the middle of the ridge. This confirms that the guided mode responsible for this narrow resonance is a TM mode, confined in the ridge. Moreover, the inspection of the isovalues in Fig. 4(c), shows that this mode propagates in the ridge along the y direction.
On the contrary, within the broad rejection band, the most intense components of the fields are H z (not shown) and E y (shown in Fig. 4(d)) and they are confined in the middle of the slab. The rejection band is thus due to the coupling with a TE mode, that propagates along the x direction, as can be deduced from Fig. 4(d).
Our ZCG filter is thus relying on the coupling of one TE and one TM mode, propagating along orthogonal directions, as was depicted in Fig. 1(a). Furthermore, Figs. 4(c) and 4(d) show that the coupled modes have the same periodicity than the coupling grating, confirming that the resonances arise from first-order couplings. Fig. 4 demonstrates a new filter design which, for the first time, simultaneously uses a TE and a TM resonance to respectively provide the broad resonance band and high-transmission peak of a single-layer photonic structure. In this regard, it significantly differs from pure TE ZCGs that rely only on TE modes. The main advantage of our approach is that it relies on eigenmodes with similar indices (see Fig. 2) coupled by the same first grating order (see Figs. 4(c) and 4(d)). As a consequence and contrarily to pure TE ZCGs, it is not limited by the stringent condition that the partially-etched high-index layer should have an index larger than twice that of the substrate. 1 Indeed, the sole restriction in our approach is a higher index than that of the substrate, like any conventional waveguide.
To demonstrate this significant difference, we have designed another TE/TM ZCG with a Ge layer ontop of ZnS substrate that has a significantly higher refractive index than the CaF 2 substrate used previously. The change in design related to the change in substrate from CaF 2 (see dimensions listed in the caption of Fig. 4) to higher-index ZnS (caption of Fig. 5) is quite straightforward. The thickness, t s , of the slab waveguide layer should be decreased in order to compensate for the increase in the TE mode effective index. The TM mode, which is confined within the grating ridge is barely affected by the substrate, and therefore we only perform a slight optimization of the grating ridge. The lateral dimensions are thus identical to that of the previous design (see Fig. 4).

D. Conclusion
A new approach for the design of ZCG filters, TE/TM-ZCGs has been proposed and numerically demonstrated. Contrary to pure TE ZCGs reported so far, it relies on a 2D deep-ridge grating which simultaneously couples one fundamental TE and one fundamental TM modes which exhibit similar effective indexes and whose propagation occurs along orthogonal directions. It differs from pure TE ZCGs in two ways. First, contrary to pure TE ZCGs that can be designed as 1D structures, TE/TM-ZCGs are inherently 2D structures providing polarization independence under normal incidence. Second, TE/TM-ZCGs can be designed on a wider range of material systems as they do not require a high index material with a refractive index at least twice higher than that of the underlying substrate. This approach was successfully used to design several narrowband transmission filters in the midIR, with different substrate materials.