The structure of an experimentally fabricated MIMIM double cavity consisting of silver (Ag) as metal and Al_{2}O_{3} as insulator (dielectric) is sketched in Figure 1A. Here, the dielectric cavity layers have nearly similar thickness of 100 and 115 nm, and the three metal layers of Ag were chosen with a thickness of 20 nm. Such a double cavity manifests two distinct peaks in absorbance and transmittance, and two corresponding minima in reflectance that occur at 470 and 610 nm, and which can be associated to the high- and low-energy cavity modes (Figure 1B). The real (*ε*′) and imaginary (*ε*″) parts of the ellipsometrically measured dielectric permittivity are depicted Figure 1C and D, respectively. The curve of *ε*′ crosses the zero at four wavelengths, two of which (*ω*_{HE-ENZ} and *ω*_{LE-ENZ}) are characterized by a small imaginary part, which renders them high-quality ENZ modes. Such ENZ resonances correspond to Ferrell-Berreman modes that occur naturally in thin Ag films (at 327 nm) and that can be designed within certain spectral bands in layered metamaterials [5], [38]. Interestingly, the wavelengths of these ENZ modes coincide with the cavity features in the optical spectra in Figure 1B, as highlighted by the vertical dashed lines, which indicates that the cavity modes of the MIMIM correspond to ENZ resonances. For an analytical modeling of the effective dielectric permittivity of the MIMIM structure, we extended the harmonic oscillator model that we developed for a single MIM system in Ref. [5]. Then, the double cavity system can be described by adding two damped oscillator terms, one for each MIM, to the Drude expression of the dielectric permittivity of Ag:

$$\begin{array}{c}{\epsilon}_{\text{eff},\text{MIMIM}\mathrm{(}\text{Ag}\mathrm{)}}={\epsilon}_{\infty}-\frac{{\omega}_{\text{p}}^{2}}{\mathrm{(}{\omega}^{2}+i\gamma \omega \mathrm{)}}-\frac{{\alpha}_{1}{\omega}_{\text{MIM}}^{2}}{\mathrm{(}{\omega}^{2}-{\omega}_{0,\text{MIM}1}^{2}+i{\gamma}_{M\text{IM}1}\omega \mathrm{)}}\\ -\frac{{\alpha}_{2}{\omega}_{\text{MIM}}^{2}}{\mathrm{(}{\omega}^{2}-{\omega}_{0,\text{MIM}2}^{2}+i{\gamma}_{\text{MIM}2}\omega \mathrm{)}};\end{array}$$(1)

Figure 1: Optical properties of the MIMIM structure.

(A) Architecture of the MIMIM structure with illumination from the top. (B) Ellipsometrically measured p-polarized transmittance (blue), reflectance (green), and absorbance (black), detected at *θ*=40°, showing two absorbance maxima at the two low-loss ENZ wavelengths. (C and D) Theoretically modeled (empty circles) and ellipsometrically measured (solid line) real (C) and imaginary (D) effective dielectric permittivity of an MIMIM cavity. Low-loss ENZ wavelengths are highlighted with blue (high-energy) and red (low-energy) dashed circles.

Here, *γ*_{Ag}=0.021 eV and *ω*_{p}=9.1 eV are the Drude parameters of Ag. *ε*_{∞} is taken as a fitting parameter, and we obtain *ε*_{∞}=6.8, which is slightly larger than the value of Ag due to the residual polarizability of the system. The parameters *ω*_{0,MIM1}=2.452 eV, *ω*_{0,MIM2}=2.7382 eV, *γ*_{MIM1}=0.075 eV, and *γ*_{MIM2}=0.07 eV are obtained from the experimental ellipsometry spectra. Moreover, it is convenient to fix the parameter ${\omega}_{\text{MIM}}^{2}$ (at 3.53 eV) and to express the difference between the two oscillators in the numerator by a coefficient *α*_{i} (where *i* is the *i*th resonance). In this case, *α*_{1}=0.35 and *α*_{2}=0.3. The fitting with this approach is depicted by the open circles in Figure 1C and D and shows very good agreement with the experimental data. We note that Eq. (1) allows to describe the MIMIM system as one homogenized layer with an effective dielectric permittivity.

The modeling of the norm of the electric field with finite-element methods (COMSOL) at the resonance wavelengths shows that the cavity modes are strongly confined in the dielectric layers, as demonstrated in Figure S1 in the Supporting Information (SI).

The MIMIM double cavity can be seen as two MIM cavities stacked on top of each other that are connected by the central metal layer. This configuration resembles two coupled oscillators that lead to a mode splitting when the individual resonance frequencies are similar or only slightly detuned. In the case of MIM and MIMIM cavities, the resonance frequencies are mainly determined by the thickness of the dielectric layers; therefore, we can control the detuning in MIMIM cavities by varying the thickness of one dielectric layer while keeping the other one fixed.

Figure 2A shows the absorbance maxima of the high-energy (blue stars) and low-energy (red stars) resonances for five samples, where the bottom dielectric layer thickness varied from 60 to 160 nm, whereas that of the top layer was constant at 112 nm. We clearly observe the anticrossing behavior that is expected for two coupled modes, corroborated by scattering matrix method (SMM) simulations (dashed lines) [27], [42], [43]. A careful inspection of the experimental and simulated frequencies in the anticrossing region reveals that the shift of the high-energy (HE-ENZ) mode from the unperturbed frequency (solid gray line) is larger than that of the low-energy (LE-ENZ) mode. Such behavior deviates from the classical coupled oscillator model where a symmetric mode splitting occurs. The corresponding spectra are shown in Figure 2B, where we notice that the resonance associated to the top layer manifests a more pronounced absorbance peak outside the strong anticrossing region. The coupling strength of the ENZ modes is determined by the thickness of the central metal layer, as evident in Figure 2C. Here, symmetric MIMIM cavities were fabricated with fixed dielectric and outer metal layer thicknesses, whereas the thickness of the central Ag layer varied from 20 to 100 nm. Clearly, the mode splitting decreases with increasing central layer thickness. The anticrossing behavior of MIMIM structures with different mode splitting is shown in Figure S2 in the SI. Another important factor for photonic cavities is the quality of the resonances in terms of linewidth and quality (*Q*) factor, where the latter is evaluated as the ratio of the full-width at half-maximum over central resonance frequency. In the black spectrum in Figure 2C, we obtain a *Q*-factor of 35 at 540 nm (2.3 eV) with a linewidth of 80 meV. However, such a performance can be noticeably improved by acting on the thickness of the external metal layers. The SMM calculations in Figure 3 demonstrate that the quality of the resonances is mainly determined by the thickness of the outer metal layers, whereas the mode splitting, as determined by the central layer thickness, remains roughly constant. ENZ resonances with linewidth as low as 25 meV are achievable, together with very high *Q*-factors of about 100 (Figure 4B) that correspond to plasmon relaxation times of the order of 200 fs.

Figure 2: Mode splitting in a MIMIM double cavity.

(A) Mode anticrossing in a MIMIM system with 30 nm Ag layers and 112 nm Al_{2}O_{3} as top dielectric layer, whereas the thickness of the bottom dielectric layer is varied. Experimentally measured data is shown by stars and the simulated dispersion via SMM is shown by dashed lines. The gray markers (experimental) and lines (SMM simulations) show the case of noninteracting cavities. (B) Absorbance spectra measured in p-polarization and obtained as (1-transmittance-reflectance) for the five MIMIM structures with different dielectric bottom layer thickness displayed in (A). (C) Ellipsometrically measured p-polarized absorbance curves for different thickness of the central layer. For a MIMIM with 20 nm central Ag layer, we obtained a mode splitting of 447 meV that corresponds to about six times the linewidth of the antisymmetric mode (see also Figure S2 in the SI for MIMIM systems with other metal layer thicknesses).

Figure 3: Dependence of the cavity resonances on the thickness of the external Ag layers.

SMM simulations of the (A) linewidth, (B) quality factor, and (C) mode splitting of a MIMIM system with 20 nm thickness of the central metal layer as a function of the thickness of the external metal layers.

Figure 4: Quantum mechanical model of the MIMIM cavity.

(A) Structure of a MIMIM with thick external metal layers as used for SMM calculations. (B) Double quantum well that is the quantum mechanical analogue of the MIMIM structure. The *y*-scale for the optical potential is –(*k*/*k*_{0})^{2}, and the origin of the *x*-axis is chosen in the middle of the central barrier. The barrier height of the potential for the photons induced by the metal layers is given by the square of the imaginary part of the refractive index ${\kappa}_{\text{m}}^{2},$ whereas the potential in the dielectric is at $-{n}_{\text{d}}^{2}.$ (C and D) Resonance frequencies calculated with the quantum approach (markers) and numerical SMM simulations (dashed lines) for different thickness of the dielectric layers (C) and the external Ag layers. (E) Experimental (large markers), quantum calculated (small markers), and SMM simulated resonance wavelengths of MIMIM structures with different central metal layer thicknesses. The mode splitting can be tuned from uncoupled resonators, through weak coupling, into the strong coupling regime. The largest experimental mode splitting is 477 meV for a central metal layer thickness of 18 nm, which is among the highest values reported in the literature [12], [13], [44], [45], [46], [47], [48], [49], [50], [51], [52].

The MIMIM structure is therefore a highly versatile photonic cavity, where the frequency of the resonance modes, their quality factor, and the mode splitting can be tailored to a very large degree throughout the visible and near-infrared spectral range. In particular, the spectral range of the resonances can be extended by an adequate choice of the dielectric material, as demonstrated in Figure S3 where the resonances for MIMIM cavities with TiO_{2} and SiO_{2} are shown. Interestingly, for TiO_{2} also the asymmetric resonances manifest hybridization and mode splitting. Therefore, MIMIM double cavities are appealing when the spectral overlap with dyes, quantum emitters, and other photonic systems is sought.

Motivated by the discrepancies of the optical properties of the MIMIM double cavity with the classical analytical model, we outline its analogy to a double quantum well in quantum mechanics that allows to resolve this problem. Furthermore, this treatment leads to analytical expressions for the cavity resonances that describe the MIMIM with high accuracy and provides a more intuitive physical insight to such a complex photonic system. The geometry for the SMM calculation for MIMIM with thick external metal layers is shown in Figure 4A. The corresponding double quantum well with infinitely thick external barriers is depicted in Figure 4B together with the symmetric and antisymmetric eigenmodes that such a system sustains. In Ref. [5], we demonstrated that the square of the imaginary part of the dielectric permittivity of the metal, *κ*, can be seen as the optical equivalent of the potential that defines the barrier height. The semiclassical treatment for the MIM cavity that we developed on this basis can be straightforwardly extended to the MIMIM structure, thus allowing to find the dispersion relations that define the resonant modes:

$$\text{tanh}\left({k}_{0}{\kappa}_{\text{m}}\frac{{t}_{\text{cm}}}{2}\right)=\frac{{\kappa}_{\text{m}}}{{n}_{\text{d}}}\text{tan}\left[-{k}_{0}{n}_{\text{d}}\left(\frac{1}{{k}_{0}{\kappa}_{\text{m}}}+{t}_{\text{d}}\right)\right]$$(2)

$$-\text{coth}\left({k}_{0}{\kappa}_{\text{m}}\frac{{t}_{\text{cm}}}{2}\right)=\frac{{\kappa}_{\text{m}}}{{n}_{\text{d}}}\text{tan}\left[-{k}_{0}{n}_{\text{d}}\left(\frac{1}{{k}_{0}{\kappa}_{\text{m}}}+{t}_{\text{d}}\right)\right]$$(3)

The details on the derivation of these relations are reported in the SI. Figure 4C shows the resonance frequencies obtained by solving Eqs. (2) and (3) for different dielectric layer thicknesses, in the case of external metal layers with 100 nm thickness. The comparison to the numerical electromagnetic (SMM) simulations shows excellent agreement. As we consider illumination of the structures from the top, i.e. through the metal layers, MIMIM structures with thinner Ag layers are of large practical interest. In this case, the tunneling of the photon through the barrier has to be taken into account by an additional phase factor of –exp(–2*k*_{0}*κ*_{m}t_{m}). Then, the dispersion relation for a “leaky” MIMIM structure is

$$\begin{array}{c}\text{tanh}\left({k}_{0}{\kappa}_{\text{m}}\frac{{t}_{\text{cm}}}{2}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{{\kappa}_{m}}{{n}_{\text{d}}}\text{t}a\text{n}(-{k}_{0}{n}_{\text{d}}\left(\frac{1}{{k}_{0}{\kappa}_{\text{m}}}+{t}_{\text{d}}\right)\\ \begin{array}{c}\\ \end{array}-\text{\hspace{0.17em}}\text{e}x\text{p(}-2{k}_{0}{\kappa}_{\text{m}}{t}_{\text{m}}\text{)})\end{array}$$(4)

$$\begin{array}{c}-\text{coth}\mathrm{(}{k}_{0}{\kappa}_{\text{m}}\frac{{t}_{\text{cm}}}{2}\mathrm{)}\text{\hspace{0.17em}}=\frac{{\kappa}_{\text{m}}}{{n}_{\text{d}}}\text{tan}\mathrm{(}-{k}_{0}{n}_{\text{d}}\mathrm{(}\frac{1}{{k}_{\text{0}}{\kappa}_{\text{m}}}+{t}_{\text{d}}\mathrm{)}\\ \begin{array}{c}\\ \end{array}-\text{\hspace{0.17em}}\text{exp(}-2{k}_{0}{\kappa}_{\text{m}}{t}_{\text{m}}\text{)}\mathrm{)}\end{array}$$(5)

The resulting resonance frequencies for the symmetric and asymmetric modes are plotted in Figure 4D as a function of the thickness of the external Ag layers. Here, the thickness of the dielectric layers was fixed at 140 nm, and the central metal layer was 20 nm thick. We clearly observe a decrease in resonance frequency when the thickness of the external layers is smaller than 40 nm, which identifies the leaky regime, where the additional phase correction in Eqs. (4) and (5) is necessary. Again, a very good agreement with the numerical (SMM) modeling, as shown by dashed lines, is obtained. The electric and magnetic field profiles of the resonant modes in such a leaky MIMIM structure, as calculated by finite-element method simulations (COMSOL Multiphysics), are reported in Figure S2 in the SI. From the experimental data in Figure 2C, we already saw that the coupling strength of the modes is determined by the thickness of the central metal layer. The double quantum well model of the MIMIM illustrates this coupling as resonant tunneling of the photons between the two wells, and Figure 4E shows that also the asymmetric mode splitting is well described by the quantum mechanical treatment (as depicted as small symbols). The experimental and theoretical data in Figure 4E demonstrate that the coupling in the MIMIM cavity can be tuned through the entire range, from uncoupled, to weakly coupled and strongly coupled by acting on the thickness of the central layer. The strong coupling regime is reached if $\frac{4{g}^{2}}{{\gamma}_{{\text{MIM}}_{1}}\cdot {\gamma}_{{\text{MIM}}_{2}}}>1,$ where *g* is the coupling constant $\mathrm{(}g=\frac{\hslash \Omega}{2}\mathrm{)},$ with *ℏ*Ω being the mode splitting in meV, and ${\gamma}_{{\text{MIM}}_{1}}$ and ${\gamma}_{{\text{MIM}}_{2}}$ are the linewidths of the uncoupled modes. We note that, for an infinitely thin central metal layer, the resonances approach the symmetric and asymmetric modes in a MIM system and that, for a very thick central metal layer, the bottom cavity is shielded by the top one, which leads to the optical response of a MIM superabsorber [5], [53]. Figure 4 demonstrates that the semiclassical approach, treating the MIMIM as a double quantum well for photons, allows for accurate analytical calculation of the resonance energies and provides physical insight in the mode coupling and mode confinement as related to photon tunneling through the metal barriers.

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