Symmetric metal-dielectric guided-mode resonators (GMR) can operate as infrared band-pass filters, thanks to high-transmission resonant peaks and good rejection ratio. Starting from matrix formalism, we show that the behavior of the system can be described by a two-mode model. This model reduces to a scalar formula and the GMR is described as the combination of two independent Fabry-Perot resonators. The formalism has then been applied to the case of asymmetric GMR, in order to restore the properties of the symmetric system. This result allows designing GMR-on-substrate as efficient as free-standing systems, the same high transmission maximum value and high quality factor being conserved.
© 2015 Optical Society of America
Metal-based nanostructured thin films have received a growing interest since the first theoretical and experimental evidences of extraordinary transmission [1–3]. This phenomenon originates from the coupling of the incident light to surface plasma waves and has been applied to various fields of research like microfluidic, photovoltaic devices [4–7]. Metallic gratings with narrow slits have been shown to exhibit high transmission  and can be used as an alternative to thin films for infrared band-pass filters, especially in the case of multispectral imaging . More recently, metal-dielectric guided mode resonances (GMR) filters have been introduced . They permit to design unpolarized filters [11, 12] or high angular tolerance filters  with an improved rejection ratio away from resonance. Nonetheless, one of the main drawback of these devices is that they are based on membranes, which complicates their fabrication and decreases their mechanical robustness. One solution consists in using a substrate . However, in most of the cases, the introduction of a dissymetry between the top and the bottom interfaces of these filters is to the detriment of its performances. Likewise, for lossless Fabry-Perot filters, 100% transmission are only reached for symmetric filters.
In this article, we develop a two modes model that accurately accounts for the various resonance peaks of metal-dielectric GMR filters, highlighting the paramount influence of the 0th and ±1st orders in the resonance mechanism. This model is further simplified to give a scalar expression of the transmitted intensity, and the GMR can be described by two coupled resonators. By analogy with a Fabry-Perot resonator, a symmetrisation criterion is formulated thanks to the scalar expression for either one of the coupled states. We show in the case of a GMR on substrate filter, that it is possible to satisfy this criterion for the first resonator and to restore the same performances than the symmetric filter.
First, we introduce the matrix formalism to compute the transmission spectra of the metal-dielectric GMR filters, and show that it can be reduced to a two modes model. It can be further simplified to a two terms scalar expression for the transmission. In the Sec. 3, we use this expression to formulate a symmetrisation criterion and apply it to various cases of asymmetric GMR filters.
2. Matrix formalism
GMRs are based on the coupling of the incident light to a waveguide via diffracted orders created by a periodic grating [15, 16]. In particular ref.  shows that the GMR is due to the coupling of the ±1st orders diffracted by a metallic grating to the dielectric layer guided-modes. The metal-dielectric GMR structure we consider consists in two identical one-dimensional gold gratings with a silicon carbide (SiC) layer in-between, as depicted on Fig. 1(a). The choice of the geometrical parameters stems from the procedure that was used in refs.  and  to obtain a transmission peak at 4.65μm. With the following values (grating period d = 2 μm, SiC layer thickness td = 700 nm, metal thickness tm = 100 nm and slit width a = 250 nm), we expect a high internal reflectivity of the metallic interfaces for TM-polarized light (with the magnetic field Hy parallel to the slits) [17,18], and no diffracted order in the 3 to 6 μm spectral range. The thickness td is large enough to prevent any coupling of the evanescent waves from the two sides, including when the surface plasmon-polaritons are excited at wavelength close to dnd ≈ 5.7μm, where nd the refractive index of the waveguide (2.84 for SiC).
2.1. Resonance analysis
A first transmission spectrum is calculated with the Rigorous Maxwell Constitutive Approximation (RMCA) method  considering all the diffracted orders using N mesh points (N ≈ 100), a SiC permittivity fixed at εd = 2.842 and a dielectric function of gold given by a Drude model : εm(λ) = 1 − [(λp/λ + iγ)λp/λ]−1 with λp = 159 nm and γ = 0.0048 [20, 21]. The full-calculation spectrum on Fig. 1(b) exhibits two transmission peaks at λa = 4.65 μm and λb = 4.04 μm. The resonance ”b” has a very low intensity compared to the resonance ”a” with a transmission up to 80%.
A second transmission spectrum is calculated by considering a smaller number of diffracted orders m, therefore the size of the S-matrices depicted on Fig. 1(c) is reduced. By using the scattering matrix formalism, the transmission amplitude S matrix of the all structure is written
From now on, we consider the case of normal incidence, for which orders +1 and −1 are excited symmetrically. The truncated response considering only the orders 0, +1 and −1 corresponds to a 2 × 2 matrix (m = 2), and is plotted (dotted line) on Fig. 1(b). It shows a very good agreement with the full calculation. In this approximation, S21 and S32 are considered as column and row vectors with two components, and P, S232 and S212 are 2 × 2 matrices.
On Fig. 1(b), the very good agreement between the exact response and the two-mode model spectrum on the 3.5 – 6 μm wavelength range demonstrates that only the propagating orders play a significant role in the optical response, while evanescent orders have a low influence. In fact, since only the three orders 0, +1 and −1 are propagating, all the propagation factors for j ≥ 3 tend to 0. Accordingly, the discrepancy below 3.5 μm between the resonance peaks can be corrected by taking the ±2nd diffracted orders into account in our simplified model. Figures 1(d) and 1(e) represent the magnetic field distribution over the GMR structure at λa and λb. The overall magnetic field is compared to the magnetic field contributions , and of the 0th order, the ±1st order and the evanescent orders. We also plot the magnetic field contribution which corresponds to the interference between the 0th and ±1st orders ( is given by the cross-coupling term appearing when developing the sum ). At λb, the prevalence of the 0th order is due to the Fabry-Perot resonance and therefore explains the intensity distribution of : the x–invariant field is slightly perturbed by the d-periodic field. At λa, the ±1st diffracted orders dominate. This is consistent with the observation of a d/2-periodic field of the standing-wave resulting from the interference of the d-periodic orders +1 and −1. Nonetheless, for the resonance ”a” we observe that evanescent orders are quite intense. We can still neglect them because we do not consider their propagation in the SiC layer but take them into account inside the transmission and reflection matrix coefficients. This might yet explain the small difference we see between the two spectra on Fig. 1(a).
2.2. Scalar expression for the transmission
The loop matrix M = P1/2S212PS232P1/2 is introduced. It refers to all the multiple reflections onto the metallic gratings and the propagation inside the SiC layer of the 0th and ±1st diffracted orders. Diagonalizing M allows to write M = ΠDΠ−1 with (Π is a transfer matrix). Finally the Eq, (2) is equivalent toEq. (3), we obtain a scalar expression for the transmission amplitude Figure 2(a) shows that the transmission spectrum can be divided in two parts in agreement with the two-term scalar expression found for the transmission intensity Fig. 2(b)). Therefore the 0th and ±1st orders can be viewed as coupled resonators that interfere constructively in the GMR structure when arg(da) ≡ 0[2π] or arg(db) ≡ 0[2π].
3. Symmetrisation of guided-mode resonance filters
3.1. Symmetry criterion
We want to study the effect of symmetry on the resonance ”a”. Although the Fabry-Perot peak corresponds to a single-mode resonance associated to a wavevector kz, we note two analogies between the Fabry-Perot resonance and the GMR resonance. (i) Considering the expression Ta we note that it is analogous to the classical expression |t32t21eikztd/(1 − r212r232e2ikztd)|2 for the intensity transmitted through a Fabry-Perot cavity of thickness td as a function of the transmission and reflection Fresnel coefficients t32, t21, r212 and r232. (ii) Fabry-Perot resonances occur similarly when arg(r212r232e2ikztd) ≡ 0[2π]. For a Fabry-Perot cavity, this condition defines similarly a resonance round trip.
The eigenvalue da of the matrix M also turns to be the eigenvalue of , P1/2 and being seen as the propagation matrices related to two dielectric layers of thicknesses t and t2, given that td = t + t2. Finally, as a symmetric Fabry-Perot cavity is defined by r212eikztd = r232eikztd, we introduce the eigenvalues d212 and d232 of the matrices P1/2S212P1/2 and . A symmetry criterion is formulated, the necessary condition to obtain a symmetric filter being: d212 = d232.
3.2. Symmetrisation results and discussions
Figure 3(a) presents three different freestanding GMR structures A, B and C. The structure A is the one originally shown on Fig. 1(a): it is made of two identical periodic gold gratings with a SiC layer in-between. The structure B only differs from A because the slits of the lower grating are filled with SiC. The structure B is therefore not symmetric, and we observe that d212≠d232 (on Fig. 3(b), solid lines refer to d212 and dotted lines to d232). To restore the equality, three degrees of freedom are available: the thickness t2 of the dielectric waveguide lower part and the slit width a2 and metal thickness tm2 of the lower grating. In theory, three parameters are more than enough to make the complex d232 equal to d212 at the resonance wavelength. For the structure C, the equalization is even obtained on a large wavelength range. As a consequence, the slopes of the arguments of d212 and d232 also coincide, which allows to restore the same quality factor.
To derive an expression for the quality factor, we consider the denominator of the scalar expression Ta. We introduce the length α by setting down: da = |da|eiδkaα where δka = k0 − ka with k0 = 2π/λ and ka = 2π/λa. Consecutively around the resonance wavelength λa, . The quality factor Q being defined as Q = k0/2ka,1/2, the Lorentzian shape of Ta allows to write . Finally we obtain the following expression for the quality factor
These 8 terms have been evaluated numerically. It appears that, contrary to the case of a simple Fabry-Perot filter where the phase variations at reflection onto the mirrors are negligible, the terms and must be taken into account. Furthermore, we notice that the first and last terms of this sum and are negligible.
The eigenmodes complex frequencies ωa and ωb are solutions of the implicit equation22, 23]. The expression (6) for the quality factor Q is analogous to the typical quality factor of a Fabry-Perot resonator of thickness td filled with a dielectric of refractive index nd :
The assumption that the system is separated in two independent resonators (indexed as a and b) is confirmed by Fig. 4, which shows the exact value of det(I − M) and the approximated value obtained as the product of terms (1 − da) and (1 − db). The two curves are in good agreement over a wide spectral range around the resonance peaks λa and λb.
Figure 5 shows the result of the symmetrisation approach for a GMR filter on a sapphire substrate. We manage to make d212 equal to d232 at the resonance wavelength λa. This allows to obtain the same transmission maxima for the original freestanding structure A and the structure on substrate C′, but not the same quality factor (on Fig. 5 (c) the transmission peak of the structure C′ appears larger than the one related to the original structure A). The equalization is obtained by enlarging the slits and reducing the thickness of the lower grating. Under these conditions, the hypothesis of a highly reflecting interface and the approximation at first order may no longer be valid, especially as the index contrast between the dielectric waveguide and the sapphire substrate is low. Therefore, second order terms should be considered. Finally, this explains why the the quality factor is not restored in addition to the transmission maximum value at the original resonance wavelength λa, even though the symmetry criterion is fulfilled: in this case, the influence of second order terms would require in addition to suppress the small discrepancy we observe on Fig. 5(b) between the slopes of the coefficients d212 and d232 (both for the modules and arguments), responsible for the slightly lower quality factor we observed for the structure C′.
We have proposed a mathematical framework to study the symmetry of metal-dielectric GMR filters. The analysis of the GMR resonance mechanism shows that only the 0th and ±1st diffracted orders play a significant role. As a consequence, we can reduce the size of the scattering matrices involved in the RMCA method computation. This reduction allows to write a scalar expression for the intensity transmitted through the GMR structures. A symmetry criterion is then formulated by noticing strong analogies between this expression and the one for standard Fabry-Perot filters. Satisfying this symmetry criterion by adapting the geometrical parameters at the resonance wavelength allows to restore the same transmission maximum value and quality factor than the original symmetric freestanding structure, at the same original resonance wavelength.
This simplification could also be applied to design GMR filters with a large angular acceptance , GMR filters exhibiting perfect extinction  or for example resonators to be integrated in VCSELs [26, 27], providing the minimum number of electromagnetic modes describing the system is small.
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