© The Author(s), published by EDP Sciences, 2026

1 Introduction

Coupled Surface Plasmons (CSP) in metal-dielectric-metal (MDM) and dielectric-metal-dielectric (DMD) geometries started attracting theoretical interest since the late 20th century [1]. Initial experimental validations [2] opened the path for posterior applications of both arrangements. Specifically, referring to the case of MDM geometry that we will be addressing in this work, applications range from guiding systems [3] to spectral filtering [4], including sensors [5] or spectroscopy [6] as well.

Our recent studies [7] have focused on developing and validating an analytical model for transverse magnetic (TM)-polarized light transmission through an optical plasmonic microcavity (MC), involving an MDM structure. The MC is formed by two thin plane metallic mirrors (M) separated by a low-index dielectric gap (L) and surrounded by two semi-infinite higher-index dielectric media (H). An analytical expression for T – see equation (1) – was obtained by application of Fresnel coefficients at the interfaces, and the subsequent results were calculated by implementing Fresnel formalism for coherent optical scattering at an interface and validated by comparison with simulations using in-house developed software based on transfer-matrix-method [8]. Experimental validation was also carried out, using two identical symmetrically arranged prisms playing the role of the external semi-infinite high-index media, which couple and decouple light to the possible plasmonic resonances of the inner structure.$$T=\frac{{\left|t_{\mathrm{LMH}}\right|}^2 {\left|t_{\mathrm{HML}}\right|}^2}{4 {\left|r_{\mathrm{LMH}}\right|}^2 \left[ {\sinh }^2\left(k_{L,\perp }^{\mathrm{\text{'}\text{'}}}d-\ln \left|r_{\mathrm{LMH}}\right|\right) + {\sin }^2\left(k_{L,\perp }^{\text{'}}d+{\phi }_{\mathrm{LMH}}^{ }\right) \right]},$$(1)where r_LMH and t_LMH are reflection and transmission field amplitude coefficients for each of the two three-medium mirror structures; φ_LMH represents the phase of r_LMH; d denotes the intracavity thickness; and $k_{L,\perp }=k_{L,\perp }^{\text{'}}+i{k}_{L,\perp }^{\mathrm{\text{'}\text{'}}}$ is the component of the intracavity wavevector in the direction perpendicular to the interfaces. This equation can be seen as a generalization of the common formula for transmittance in Fabry-Pérot (FP) interferometers to all incidence angles, from which the different resonant modes in the system can be obtained. On the one hand, k_L,⊥ is real for incidence angles θ below the critical angle for incidence from H to L, θ_cr, for which volume resonances are allowed. That case corresponds to the well-known FP regime, in which harmonic waves can propagate between the two mirrors at photonic resonance conditions. On the other hand, for θ > θ_cr, k_L,⊥ becomes imaginary, and only CSP resonances are allowed. In both cases, the dependence on the separation between the mirrors, d, in equation (1) becomes straightforward, since it only appears inside the argument of either the sine or the hyperbolic sine, respectively. As a result, CSP resonances correspond to zeros of the hyperbolic sine, leading to two distinct solutions for each d at two different θ, as shown in Figure 1a. It is also remarkable that one of these two curves is the continuation of the first FP resonance, forming a hybrid branch. For thick enough cavities these two plasmonic resonances become degenerated, merging at the same angular position – named the coalescence angle θ_co – for d > d_co = max[(ln|r_LMH|)/k_L,⊥].

Fig. 1 (a) Angle of incidence at which the transmittance of each CSP is maximum as a function of cavity thickness. (b) Maximum transmittance at each CSP resonance peak as a function of cavity thickness, FTIR transmission at θco without Ag for comparison. (c) Modulus and (d) phase of rLMH (lighter) and rLM (darker) as functions of the incident angle. All results correspond to a setup using silver as metal (dM = 39 nm), BK7 as prism and air filling the gap between the mirrors, at λ = 800 nm.

In this work, we emphasize the high values of transmittance in the system even for over-wavelength-thick cavities. The presented theoretical and experimental studies of maximum transmittance as a function of d reveal that it remains not only noticeable but also large for d exceeding by far the predicted penetration depth of light in the intracavity L medium. Thus, transmittance of CSP resonances in this MC can be said to resemble some kind of enhanced light tunneling. Analysis of equation (1) reveals that r_LMH is the key parameter determining this phenomenon, especially in the plasmonic regime. A brief theoretical analysis on the behavior of that reflection coefficient is also presented. Its modulus becomes much larger than unity near the coalescence. This may seem surprising if one thinks of Fresnel coefficients as the square root of reflectance (R). However, there is no issue with energy conservation here because the waves involved are evanescent and $\left|r\right|=\sqrt{R}$ does not hold.

2 Material and methods

The experimental setup used to test our theoretical predictions is shown in Figure 2. A supercontinuum light source (SuperK COMPACT, NKT Photonics, 450–2400 nm) illuminates the MC, with power adjusted by a variable neutral optical density. A cube polarizer provides a transmitted TM beam, which is a requirement for the excitation of surface plasmons. Simultaneously, part of the original beam is deflected towards a photodiode, used to monitor the beam power. The MC consists of two identical rectangular BK7 glass coupling prisms, whose largest faces are coated with thin silver films, facing each other and creating a micron-scale air gap in between. That cavity thickness is determined by iteratively comparing with subsequent theoretical configurations, varying d with the desired precision. It is modified in ~20 nm steps using a piezoelectric actuator (Thorlabs PIAK25), which is connected to the prisms through custom 3D-printed mounts. These are mounted on a rotation platform (URS75BCC from Newport Optics), that allows us to control the angle of incidence (±6 mdeg accuracy). Finally, the transmitted signal is collected with a photodiode preceded by a spectral filter, enabling monochromatic mapping of T across a 2D θ − d space, using incident light as a reference.

Fig. 2 3D schematic drawing of the experimental setup. (1) Output of the supercontinuum light soruce, (2) variable optical density filter, (3) cube acting as polarizer and beam splitter, (4) power monitoring photodiode, (5) couple of prisms conforming the MC, (6) rotation platform, (7) narrrowband spectral filter, (8) photodiode detector for measuring transmittance.

The thin silver layers on the prisms, used as MC mirrors, were deposited by physical vapor deposition (PVD), evaporating the metal through heating. The prisms were placed inside the vacuum chamber of a Baltec BAE250 Coating System by Balzers, and the pressure was reduced to ~10⁻³ Pa. Afterwards, the silver was heated until it started to evaporate (~140 °C), and an approximately constant deposition rate was used to obtain silver films with a nominal thickness around 39 nm. Later, this value was found to be correct within an uncertainty range of ±5 nm with a Dektak3 Surface Profilometer, as an extra check.

3 Results and discussion

The analytical model developed for transmittance of the MC was experimentally validated. Angular positions of the transmittance maxima for the two CSP resonances as a function of d at a fixed wavelength λ = 800 nm are displayed in Figure 1a, while their peak values are presented in Figure 1b. Both graphs show the agreement between theoretical predictions and experimental measurements. Those results correspond to a BK7-Silver-Air MC, with metallic mirror thickness d_M = 39 nm. Peak values for transmittance remain high and almost constant until the coalescence thickness, which is well beyond the wavelength (in this case, d_co = 2.1 μm). Besides, the maximum observed transmittance decreases slowly for d > d_co, staying above 10% until d = 2.9 μm, exceeding by far the theoretical penetration depth of light into the air gap, calculated to be $\delta ={\left[k_{L,\perp }^{\mathrm{\text{'}\text{'}}}\left({\theta }_{\mathrm{co}}\right)\right]}^{-1}\approx 652 \mathrm{nm}$ in this situation, with θ_co ≈ 42.4°.

The resonance conditions derived from equation (1) are clearly highly dependent on r_LMH, whose modulus and phase appear in the arguments of the hyperbolic sine and sine, respectively (with the former determining CSP resonances and the latter defining FP resonances). This three-layer reflection coefficient can be expressed in terms of Fresnel coefficients r_ML, r_MH:$$r_{\mathrm{LMH}}=\frac{-r_{\mathrm{ML}} + r_{\mathrm{MH}} e^{i{\varphi }_M}}{1 - r_{\mathrm{ML}} r_{\mathrm{MH}} e^{i{\varphi }_M}}\hspace{1em}\mathrm{with}\hspace{1em}{ \varphi }_M={2 d}_M\cdot k_{M\perp }={2 d}_M\cdot k_0\sqrt{{\epsilon }_M-{\epsilon }_H{\sin }^2\theta },$$(2)where ε_i stands for the permittivity of medium i and k_M⊥ denotes the component of the wavevector inside the metal normal to the interfaces. The modulus and phase of r_LMH are plotted in Figure 2c and 2d as functions of the incident angle θ, together with those for Fresnel coefficient r_LM, corresponding to the limiting case where the delimiting metallic mirrors are semi-infinite (d_M  → ∞). The modulus |r_LMH| is seen to behave differently below and above θ_cr. When harmonic propagation inside the cavity is allowed, it stays slightly below 1. In turn, it grows far greater than 1 above θ_cr, reaching its maximum value around θ_co. Separation from θ_co is due to the role played by the rest of the argument of the hyperbolic sine in equation (1). The thicker the metallic mirrors, the higher the peak in |r_LMH|, converging to |r_LM| for large d_M. For θ > θ_cr, |r_LMH| and |r_LM| being greater than 1 means no problem in terms of energy conservation, since the amplitudes related by them correspond to the electric fields of evanescent waves [9]. Focusing on the phase, it grows gradually for θ < θ_cr, reaching π at θ = θ_cr. For larger angles, it starts decreasing slowly until θ approaches θ_co, when it suddenly drops to nearly zero, undergoing a slow decline again at the end. Finally, comparison with r_LM reveals that the drop is steeper for larger d_M. This anomalous behavior of r_LMH above θ_cr – large modulus and step-like phase – happens especially at the same angles θ ≈ θ_co at which outstanding transmittance occurs.

4 Conclusions

The experimentally validated analytical formula for T through an MDM structure predicts high and constant transmittance for cavity thicknesses far above the penetration depth of light. This outstanding transmittance is key for the use of CSP-based arrangement in devices for different applications. These include sensing thin media, for which measurements based on light transmission instead of reflection can simplify experimental setups; highly selective optical filtering, spectroscopy or light control at the nanoscale.

Funding

This research was funded through projects by the Spanish Ministry of Science, Innovation and Universities (PID2024-156552OA-I00), Universidade de Santiago de Compostela (USC 2024-PU031) and Xunta de Galicia (GRC ED431C 2024/06), respectively. Finally, AD thanks the Spanish Ministry of Science, Innovation and Universities for the financial support through FPU21/01302, as well as YA acknowledges Xunta de Galicia for the postdoctoral fellowship ED481D-2024-001.

Conflicts of interest

The authors have nothing to disclose. They certify that they have no financial conflicts of interest (e.g., consultancies, stock ownership, equity interest, patent/licensing arrangements, etc.) in connection with this article.

Data availability statement

Data associated with this article is available under request.

Author contribution statement

Conceptualization, R.F.; Methodology, Y.A.; Software, A.D., Y.A.; Validation, A.D.; Formal Analysis, R.F.; Investigation, A.D., Y.A., L.S., R.F.; Data Curation, Y.A., L.S.; Writing – Original Draft Preparation, A.D.; Writing – Review & Editing, R.F, Y.A.; Visualization, Y.A., A.D.; Supervision, R.F.; Project Administration & Funding Acquisition, Y.A., R.F.

References

All Figures

Fig. 1 (a) Angle of incidence at which the transmittance of each CSP is maximum as a function of cavity thickness. (b) Maximum transmittance at each CSP resonance peak as a function of cavity thickness, FTIR transmission at θco without Ag for comparison. (c) Modulus and (d) phase of rLMH (lighter) and rLM (darker) as functions of the incident angle. All results correspond to a setup using silver as metal (dM = 39 nm), BK7 as prism and air filling the gap between the mirrors, at λ = 800 nm.

In the text

	Fig. 2 3D schematic drawing of the experimental setup. (1) Output of the supercontinuum light soruce, (2) variable optical density filter, (3) cube acting as polarizer and beam splitter, (4) power monitoring photodiode, (5) couple of prisms conforming the MC, (6) rotation platform, (7) narrrowband spectral filter, (8) photodiode detector for measuring transmittance.
In the text