# Experimental demonstration of coupled nano-Fabry-Perot groove resonators

Jules Lackner, Baptiste Fix, Nathalie Bardou, Christophe Dupuis, Patrick Bouchon

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Jules Lackner, Baptiste Fix, Nathalie Bardou, Christophe Dupuis, Patrick Bouchon. Experimental demonstration of coupled nano-Fabry-Perot groove resonators. Optics Letters, 2025, 50 (13), pp.4298-4301. 10.1364/OL.561647. hal-05191677

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# Experimental demonstration of coupled nano Fabry-Perot grooves resonators

JULES LACKNER<fnref n="1" />,<fnref n="2" />, BAPTISTE FIX<fnref n="1" />,*, NATHALIE BARDOU<fnref n="2" />, CHRISTOPHE DUPUIS<fnref n="2" />, AND PATRICK BOUCHON<fnref n="1" />

Compiled May 13, 2025

Fabry-Perot (FP) resonances are ubiquitous in plasmonic resonators, however their quality factor is mostly driven by the losses of the material and has typical value of 10 for noble metals. The coupling of two FP nanocavities (cFP) was theoretically shown by Fix et al., Opt. Lett. 42, 5062 (2017) to enable the control of the quality factor thanks to a geometrical parameter. Here, we experimentally demonstrate the cFP resonance with a quality factor of 30 in a structure made of metallic grooves with a trapezoidal-shaped profile that is simultaneously easier to fabricate and more complex to model. An analytical one-mode model is derived that undoubtedly attributes the resonance to the coupling of interferences between the two nanocavities. Finally, we also experimentally demonstrate the combination of two cFP either in orthogonal polarizations or for distinct wavelengths. © 2025 Optica Publishing Group

http://dx.doi.org/10.1364/ao.XX.XXXXXX

Fabry-Perot resonances, dating back to the original discovery by Charles Fabry and Alfred Perot at the dusk of the nineteenth century[1], are among the most common phenomenon in nanophotonic resonators [2–4]. They are in particular present in all the variant of metal-insulator-metal (MIM) resonators [5, 6] as well as in metallic groove resonators [7, 8] that have been used for various applications ranging from phase control [9], thermal emission [10] or chemical sensing [11]. In all these cases, an optical cavity is defined between two metallic mirrors that sustain resonances which main properties are the resonance

wavelength, the quality factor and the coupling efficiency [12]. The resonance wavelength is determined by both the cavity length and its effective optical index that also depends on the plasmons propagating on the sidewalls coupled into a gap-plasmon[13]. For its part, the resonance quality factor is directly related to the lifetime of the photons in the cavity, which is limited by both the losses at the mirrors interfaces of the cavity, as well as the propagation losses and experimental demonstrations of MIM resonance have typically led to quality factors between 5 and 15.

In order to reach higher values, the propagation losses should be decreased, i.e., the coupling between plasmons should be reduced by increasing the distance between the two sidewalls. Such a modification simultaneously increases the radiative losses and decreases the nonradiative losses, leading to the over-coupling of the resonator with the free space. The addition of another groove acting as an overcoupled resonator in the same period was shown to bring back the structure to critical coupling, thanks to a three-waves interference coined as coupled Fabry-Perot resonance (cFP) [14]. This leads to higher quality factors that can be set by the geometrical parameters at least up to a value of 100. However, this groove architecture is a technological challenge since an array of two grooves per period with different depth for each of the two grooves has to be fabricated.

In this letter, we demonstrate experimentally the coupled nano Fabry Perot (cFP) in grooves resonators with a straightforward technological fabrication. The gap plasmon present in each metallic groove has an effective index that depends on the width. Thus, two grooves of same depth exhibit different optical propagation, as long as their width is different, and this allows the three waves interference to occur. Due to the simple technological

<footnote n="1">¹ DOTA, ONERA, Université Paris-Saclay, F-91123 Palaiseau, France</footnote>

<footnote n="2">² Center for Nanosciences and Nanotechnology (C2N) - CNRS, Université Paris-Saclay, 10 Boulevard Thomas Gobert, 91120 Palaiseau, France</footnote>

<footnote>* baptiste.fix@onera.fr</footnote>
process, the grooves have inclined sidewalls leading to
a trapezoidal profile, and we adapt a simple analytical
model based on a one mode per groove that pretty well
described the resonance mechanism. We also demon-
strate a 2D version of the resonator independent to the
polarization. Eventually, we experimentally show that
two cFP resonances can be combined in a 3-grooves struc-
ture.

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** a) Scheme of the periodic coupled nano Fabry Perot made of two trapezoidal grooves (inclination angle $\varphi = 15^\circ$). The incoming light is TM-polarized and impinging through an objective (numerical aperture of 0.15). b) Scanning electron microscope (SEM) image of the fabricated coupled nano Fabry Perot sample made of gold (dimensions $w_1 = 885$ nm, $w_2 = 370$ nm, $h = 380$ nm, $P = 2650$ nm). c) Absorption spectra of the cFP measured (continuous blue line) and simulated (dashed blue line). For the sake of comparison, the simulated absorption spectrum of a single groove ($d = 2650$ nm, $h = 440$ nm, $w = 180$ nm) is also shown (dashed orange line).</figcaption></figure>

The scheme of the coupled nano Fabry Perot (cFP) is
depicted in Fig. 1a. Light is impinging on the structure
with a transverse magnetic (TM) polarization. It is made
of two metallic grooves of the same depth per period.
This geometry allows a very simple 2-steps fabrication
process but that produces inclined sidewalls leading to a
trapezoidal-shape profile of the grooves. First, electron
beam lithography was performed on a gold coated (200
nm thick) silicon wafer (Spin coated PMMA resist was
insoled by the Vistec LEICA at 100 kV). Then, after de-
velopment in MIBK:IPA, a lift-off of a 370 nm metallic
multilayer is done with two layers of gold and two tita-
nium adhesion layers (Au 270/Ti 20/Au 70/Ti 20 nm). A
tilted view of the SEM image is shown in Fig. 1b on which
the angle of inclination of the sidewalls is measured as
$\varphi = 15^\circ$. The measured absorption spectrum of the cFP
is plotted in Fig. 1c, obtained as $A = 1 - R$, where R is
the measured specular reflection focalized through an
objective (NA=0.15). It shows an absorption resonance
peak at 3.29 µm with a quality factor of 30. The spectrum
has also been simulated with the B-spline modal method
(BMM) [15] combined with the resolution of the modal
equation of the waveguide in the groove [16, 17] and ac-
counting for the 8.6° of aperture angle of the experimental
setup. The gold model used here is a Drude model in

agreement with experimental data in the infrared range: $\epsilon_{Au}(\lambda) = 1 - 1/(\lambda_p/\lambda(\lambda_p/\lambda + i\gamma))$ with $\lambda_p = 159$ nm and $\gamma = 0.0077$ [18]. The agreement between the experiment and the exact electromagnetic simulations is fair. We attribute the reduction of the quality factor to the surface roughness and to the unique incidence that are not taken into account. A comparison is also done with a periodic single nanocavity per period, optimized to have a similar resonance wavelength and amplitude ($d = 2650$ nm, $h = 440$ nm, $w = 180$ nm). The simulated spectrum (orange curve) has a broader resonance with a quality factor of 7.

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** a) Illustration of the scalar coefficients used in the analytical model. All of the reflectivity and transmission coefficients are scalar elements from S-matrices, and $P_1$ (resp. $P_2$) is the mode propagator element in the first (resp. second) groove. b) Comparison of the reflectivity spectrum computed from the analytical model (dashed line with crosses) with the exact calculation (full blue line). c) Amplitude and (b) phase difference ($\Delta\Phi_{1-2} = \Phi(G_1 t_{AB}^{(1)}) - \Phi(G_2 t_{AB}^{(2)} + r_{AA})$) of the three terms involved in the analytical reflectivity 1.</figcaption></figure>

To further understand the mechanism at stake in the
cFP resonance, we derived an analytical resolution of
the reflectivity response based on the hypothesis of a
single propagating mode per groove with inclined side-
walls [14, 19, 20]. The following discussion is done on
an optimized structure ($w_1 = 262$ nm, $w_2 = 812$ nm, $h =$
$350$ nm, $P = 2.65$ µm) that is near critical coupling with
the free space (i.e., $R = 0$). The scalar parameters used
are illustrated in Fig. 2a and represent the transmission
and reflection Fresnel coefficients between the incoming
wave of layer A (air) and the two propagating modes
of the patterned layer B (one per groove), the reflection coefficient ($\rho$) at the bottom of each groove, the reflection coefficients $r_{BB}^{(12)}$ and $r_{BB}^{(21)}$ between the two electromagnetic modes and the propagation in each groove. In practice, the Fresnel coefficients are obtained from the exact modal computation. To compute the propagations $P_1$ and $P_2$, the trapezoidal profile is approximated by a staircase profile, and in each sub-layer the propagation is computed by solving the waveguide modal equation [16, 17]. Eventually, $P_1$ and $P_2$ are computed as the product of the propagations in all the sublayers.

The reflectivity of the cFP structure can be written as the sum of three terms: the direct reflection at the first interface ($r_{AA}$), and the electromagnetic field that escape from each groove after one or more roundtrip in either one or both of them ($G_1 t_{AB}^{(1)}$, $G_2 t_{AB}^{(2)}$).

$$R = |r_{AA} + G_1 t_{AB}^{(1)} + G_2 t_{AB}^{(2)}|^2 \quad (1)$$

At steady state, the field at the top of each groove writes as the superposition of the interferences in the twin grooves:

$$\left\{ \begin{aligned} G_1 &= \frac{\rho P_1^2 [t_{BA}^{(1)} + \rho P_2^2 (r_{BB}^{(12)} t_{BA}^{(2)} - r_{BB}^{(2)} t_{BA}^{(1)})]}{(1 - \rho r_{BB}^{(1)} P_1^2) (1 - \rho r_{BB}^{(2)} P_2^2) - P_1^2 P_2^2 (r_{BB}^{(12)} \rho)^2} \\ G_2 &= \frac{\rho P_2^2 [t_{BA}^{(2)} + \rho P_1^2 (r_{BB}^{(12)} t_{BA}^{(1)} - r_{BB}^{(1)} t_{BA}^{(2)})]}{(1 - \rho r_{BB}^{(1)} P_1^2) (1 - \rho r_{BB}^{(2)} P_2^2) - P_1^2 P_2^2 (r_{BB}^{(12)} \rho)^2} \end{aligned} \right. \quad (2)$$

As illustrated in Fig. 2b, this simple analytical resolution is in perfect agreement with the complete modal simulation and can thus provide deeper insights into the physical behavior of the cFP resonance.

Indeed, a further study of the spectral amplitude and phase of the three terms of Eq. 1 is presented in Fig. 2c and Fig. 2d. At the resonance wavelength, the light in the two grooves are in phase opposition and of similar amplitudes (with the constant direct reflection added to one of the two terms). Thus, the resonance is arising from a three plane waves destructive interference which trap the light within the structure.

Since the one dimensional structure only interact with the TM polarized light, it is possible to extend this study to a 2D structure in order to have cFP resonances for both polarizations.

Polarization independance of the cFP resonance can be obtained by simply extending the 1D patterning into two orthogonal directions). The scheme of this 2D structure and an SEM view of the sample are shown in Figs 3a-b. It consists of identical bi-grooves in the two horizontal directions with parameters $P = 1.8$ µm, $h_{Au} = 371$ nm, $w_1 = 620$ nm, $w_2 = 337$ nm and $\phi = 15^\circ$. As shown in Fig. 3c, the reflectivity spectrum under focused light (NA=0.15) is identical for both polarization and exhibits a quality factor of about 28. This is in very good agreement

<figure><img src="image_4.png" /><figcaption>Fig. 3. a) SEM view and b) scheme of the 2D coupled nano Fabry Perot grooves geometry in gold. c) Measured reflection spectra for both polarizations along the structure axis in blue and green and the diagonal polarization in orange (continuous lines) compared to the simulation spectrum (dashed lines).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. a) Tilted SEM image of the fabricated three grooves coupled nano Fabry Perot sample made of gold (dimensions $w_1 = 140$ nm, $w_2 = 296$ nm, $w_3 = 665$ nm, $h = 340$ nm, $P = 2650$ nm). b) Scheme of the periodic coupled nano Fabry Perot made of three trapezoidal grooves (inclination angle $\phi = 15^\circ$). c) Measured angular dispersion of the TM reflectivity of the three-grooves structure. For comparison, the simulated resonant wavelengths are added (crosses).</figcaption></figure>

with the simulated spectrum (also shown on Fig. 3c) that has been computed with Comsol Multiphysics.

Interestingly, since the cFP resonance results from the coupling of two over-coupled groove resonators, it is possible to add additional grooves to engineer additional resonances. So as to maintain a subwavelength period, instead of considering two pairs of grooves, we investigate the case where one common groove is acting in two distinct cFP resonances.

The scheme of the structure and an SEM image are shown in Fig. 4a-b. It has been fabricated with the same process as previously described and the inclination angle remains at $\phi = 15^\circ$. The measured angular reflectivity diagram is plotted in Fig. 4b. It has been measured on a homemade goniometer coupled to a FTIR spectrometer (vertex 70 Bruker). The incident light is collimated with a divergence angle of $1^\circ$. The angular dispersion of the two resonances wavelengths simulated with the BMM (crosses) is in perfect agreement with the experimental
<figure><img src="image_6.png" /><figcaption>**Fig. 5.** a) Simulated reflectivity spectrum of a critically-coupled three-grooves structure (dimensions $w_1 = 140$ nm, $w_2 = 296$ nm, $w_3 = 665$ nm, $h = 340$ nm, $P = 2650$ nm, $\phi = 0$) under normal incidence TM polarized light. c) Amplitude and d) phase difference of the three terms involved in the analytical reflectivity while ignoring the thinner (resp. wider) groove parameters in the equations.</figcaption></figure>

result. Due to technological variations, the experimental reflectivity at the resonances remains high.

To confirm the cFP nature of the two resonances, the previous analytical study is applied to a three-grooves structure without inclined sidewalls ($\varphi = 0^\circ$) which is close to critical coupling as shown on Fig. 5a. In fact, the model can be applied to each grooves pair using Eqs. 2-1. It can be observed in Fig. 5b and Fig. 5c that the behavior of each cFP resonance is very similar to the one described in the two-grooves structure. Indeed, the first resonance at 2.81 µm can be described by the coupling of the two thinnest grooves while the resonance at 3.11 µm is described by the coupling of the two wider grooves. This confirms that the intermediate groove is acting in both resonances. Rigorously, for each resonance, the inactive groove has a slight impact on the electromagnetic environment leading to a slight spectral shift of the resonance between the analytical and full computation: 2.85 µm instead of 2.81 µm and 3.11 µm instead of 3.14 µm.

To conclude, we have experimentally demonstrated a coupled nano Fabry-Perot resonance in the mid-infrared range. It leads to an improvement of the quality factor from 7 to 30. Even if the two grooves have the same depth, the propagation length is different due to the effective index caused by the coupling of the plasmons on

the metallic inclined sidewalls. It permits the three waves interference to appear, while greatly simplifying the fabrication procedure. We experimentally demonstrate the straightforward adaptation of the structure to a 2D geometry with an unpolarized response at normal incidence. Finally, we also show that two coupled nano Fabry-Perot resonances can be combined in the same subwavelength period, one of the three grooves participating to both coupling. This new implementation of the coupled nano Fabry-Perot resonance paves the way to new applications for metallic nano-grooves based structures. Indeed, nano-groove Fabry-Perot resonators can find usage for either their higher Q factor in filtering for example or the field concentration in small volumes which can enhance SEIRA or infrared absorption [21].

**Funding.** This work was supported by an EDOM scholarship and ANR ASTRID HEMERE. This work was done within the C2N micro nanotechnologies platforms and partly supported by the RENATECH network and the General Council of Essonne.

**Disclosures.** The authors declare no conflicts of interest.

**Data availability.** Data underlying the results presented in this paper can be obtained from the authors upon reasonable request.

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