Fig. 1. (a) Representation of the Tamm cavity/LC metamaterial coupled resonators structure and unit cell pattern of the LC metamaterial. (b) Reflection and transmission spectra of the uncoupled Tamm cavity and LC metamaterial, respectively, alongside their representations (inset).

Fig. 2. (a) Transmission spectra of the Tamm cavity-LC metamaterial coupled resonators for decreasing (bottom to top) LC resonance frequencies fLC (indicated by color circles; see legend box). The Tamm cavity resonance is fixed, about 0.96 THz (dotted vertical line). The curves are offset for clarity. (b) Resonance frequencies of the upper (red squares) and lower coupled mode (blue squares) as a function of the uncoupled LC resonance frequency fLC (diagonal dotted line). Horizontal dotted line, uncoupled Tamm cavity resonance frequency.

Fig. 3. (a) Resonance peak linewidth of the two coupled modes from Lorentzian fit on data from Fig. 2(a). Dashed lines are guide for the eyes. (b) Schematic picture of the Tamm cavity and LC metamaterial directly on top, including radiative channels s1 and s2. Bottom, corresponding interaction scheme between the Tamm mode A and the LC metamaterial mode B including the relevant coupling rates.

Fig. 4. Reflection spectra of a Tamm cavity resonant at approximately 0.95 THz coupled with an LC metamaterial resonant at 0.92 THz, including an additional mirror blocking the transmission. Blue, two-layer Tamm cavity with Q=25.2±1.6 and Q=32.6±1.2 for the lower and upper frequency coupled modes, respectively. Red, three-layer Tamm cavity with Q=35±6 and Q=37±5, respectively (0.2 offset for clarity). Solid black lines, Lorentzian fits. Quality factor errors are evaluated from fitting standard deviation.

Fig. 5. (a) Distribution of the electric field in the LC metamaterial plane over a single unit cell from FEM simulations for the upper frequency coupled mode at fLC=1.01  THz. The figure represents the electric field enhancement factor, i.e., the electric field norm for an input wave of amplitude 1. r0 lies at the center of this picture. (b) Mode volume of upper (red) and lower (blue) frequency coupled modes, normalized by the mode volume of the uncoupled LC metamaterial, as a function of the detuning, from FEM simulation. Dashed line, model from Eqs. (10) and (11). The mode volume values at zero detuning are 3.2×10−4λ3 for V+ and 2.0×10−4λ3 for V−.

Fig. 6. Schematic coupling scheme between resonators A and B, as well as input and output propagation channels s1 and s2.

Fig. 7. Theoretical reflection and transmission spectra of the coupled resonator system from Eqs. (A23) and (A24). We used fA=fB=1  THz, G=0.1  THz, ΓA=0.01  THz, ΓB=0.05  THz, Γrad1,A=0.008  THz, and Γrad2,B=0.045  THz.

Fig. 8. Schematic pictures of the resonators under study, including input illumination “1” for (a) the LC circuit metamaterial on an infinite silicon substrate and (b) the Tamm cavity coupled to the LC circuit metamaterial directly on top. Bottom, corresponding interaction scheme between the Tamm mode A and the LC circuit metamaterial mode B including the relevant coupling rates.

Fig. 9. Evolution of the radiative coupling rate to the substrate Γrad1,LC and of the resonator coupling constant G from FEM simulations. (a) Linear scale; (b) log scale.

Fig. 10. Comparison of the radiative coupling rate to the substrate of a CSRR metamaterial Γrad1,LC versus the product G2CTammnSub from FEM simulations, showing excellent empirical agreement.

Fig. 11. (a) Resonance peak of an uncoupled LC metamaterial on an infinite silicon substrate with transmission blocked for increasing fLC from FEM simulations. (b) Radiative coupling rate to the substrate of the LC metamaterial Γrad1,LC as a function of fLC, deduced from (a).

Fig. 12. Electric field enhancement along the optical axis for the higher (red) and lower (blue) resonant coupled modes, without (left) and with (right) the additional mirror. The cut axes intersect the metamaterial plane at the center of the metamaterial unit cell. The cavity is illuminated from the left with a plane wave of unity amplitude, the LC metamaterial is located at z=0, and the additional mirror is located at z=+75  μm.

Fig. 13. (a) Mode volume of the uncoupled metamaterial (black) and of the upper (red) and lower (blue) frequency coupled modes, computed from COMSOL simulation. (b) Same, normalized by the value of λ3 at the resonance frequency.

Table 1. Comparison of V and Q (theoretical and experimental) performances for the LC metamaterial alone, coupled with a 2 silicon layers Tamm cavity, and coupled with a 3 silicon layers Tamm cavity + additional mirror^a