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J. Eur. Opt. Society-Rapid Publ. 22, 5( 2026)
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 transmittance maxima for the two CSP resonances as a function of d at a fixed wavelength k = 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 = 2.1lm). Besides, the maximum observed transmittance decreases slowly for d > d co, staying above 10 % until d = 2.9lm, exceeding by far the theoretical penetration depth of light into the air gap, calculated to
be d ¼½k 00 L;? ð h coÞŠ �1 652 nm in this situation, with h 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 threelayer reflection coefficient can be expressed in terms of Fresnel coefficients r ML, r MH:
r LMH ¼ �r ML þ r MH e i / M with
1 � r ML r MH e i / M pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi / M ¼ 2 d M k M? ¼ 2 d M k 0 e M � e H sin 2 h; ð2Þ
where e 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 h, together with those for Fresnel coefficient r LM, corresponding to the limiting case where the delimiting metallic mirrors are semi-infinite( d M? 1). The modulus | r LMH | is seen to behave differently below and above h cr. When harmonic propagation inside the cavity is allowed, it stays slightly below 1. In turn, it grows far greater than 1 above h cr, reaching its maximum value around h co. Separation from h 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 h > h 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 h < h cr, reaching p at h = h cr. For larger angles, it starts decreasing slowly until h approaches h 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 h cr – large modulus and steplike phase – happens especially at the same angles h h 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.