Photon coupling-induced spectrum envelope modulation in the coupled resonators from Vernier effect to harmonic Vernier effect

Abstract

The Vernier effect and harmonic Vernier effect have attracted ever-increasing interest due to their freely tailored spectrum envelope in tunable laser, modulator, and precision sensing. Most explorations have mainly focused on configuring two isolated optical resonators, namely the reference and tunable resonator. However, this configuration requires a stable reference resonator to guarantee robust readout, posing a significant challenge in applications. Here, we discover the coupled-resonators configuration enabling a reference-free envelope modulation to address this problem. Specifically, all parameters of one resonator theoretically span a hypersurface. When the resonator couples to another one, photon coupling merit an escaped solution from the hypersurface, resulting in an envelope modulation independent of reference. We have first experimentally verified this mechanism in a coupled air resonator and polydimethylsiloxane resonator by inserting a semi-transparent 2-mercaptobenzimidazole-modified silver nanowire network. In addition, this novel mechanism provides a new degree of freedom in the reciprocal space, suggesting alternative multiplexing to combine more envelope modulations simultaneously. This study facilitates the fundamental research in envelope multiplexing. More importantly, the combination of silver nanowire network and flexible microcavity experimentally progress the spectral envelope modulation in optoelectronic integration inside resonators.

Keywords: Vernier effect and harmonic Vernier effect; envelope multiplexing modulation; photon coupling; reference-free envelope modulation; silver nanowire.

Figure 1:

(a) Physical model of two coupled resonators. We mathematically treat two coupled resonators as two BSs and one mirror. (b) Structure of the coupled resonators composing a length-tunable air resonator and a length-fixed PDMS resonator. According to Fresnel’s formula, the calculated $r_1=\frac{n_{{\text{SiO}}_2}-n_{\text{Air}}}{n_{{\text{SiO}}_2}+n_{\text{Air}}}\approx 0.18$ and $r_2=\frac{n_{\text{PDMS}}-n_{\text{Air}}}{n_{\text{PDMS}}+n_{\text{Air}}}\approx 0.17$. (c) Hypersurface of Fabry–Pérot interferometer model, that is, non-modulated spectra. The black curve indicates that reflection varies with the parameters in resonator 2 when the length of resonator 1 is zero. The red curve indicates that the reflection escapes the hypersurface due to the photon coupling. (d) Envelope modulation depends on resonator 1 due to the photon coupling.

Figure 2:

Typical reflections with envelope modulation in (a) experiments and (b) theory. From top to bottom, the reflections in (a) are the air resonator length 2.70, 65.20, 71.45, 77.70, 133.95, 141.45, and 151.45 μm. The reflections in (b) are the length of air resonator 2.70, 63.10, 69.58, 75.20, 131.00, 139.10, and 151.45 μm. The dashed curves guide the eye toward the modulation. Fourier transform shows the mode components in the reciprocal space of (c) wavelength and (d) frequency. The colored and black curves are the theoretical and experimental results.

Figure 3:

Mode evolution in the reciprocal space (a) experiments and (b) theory. The intensities of the three modes in (c) experiments and (d) theory. pp stands for the positive pulse, and 2pp equals 1.25 μm.

Figure 4:

(a) Schematic illustration of the coupled resonators containing a AgNW-based transparent heater (left) and the scanning electron microscope (SEM) image of the MBI SAM-modified AgNW network on the PDMS film resonator (right). The cross-linked AgNW indicates a high-quality network. (b) Schematic of the AgNW modified with the MBI SAM via Ag–S and Ag–N bonds. (c) XPS S2P spectra of the raw AgNWs and MBI-modified AgNWs. A source meter applied voltage (d) and responded current (e) to the AgNW-based transparent heater. (f) The calculated power (left) and the corresponding temperature (right). (g) The mode components are observed from reciprocal space depending on the temperature. (h) The calculated OPL of modes I, II, and III. (i) Zoom in the OPL to compare the OPL1 and OPL3-OPL2.

Figure 5:

Experimental envelope multiplexing modulation of two sets of coupled resonators with (a) a fixed air resonator length of ∼13.2 μm in the thin PDMS resonator group, and we changed air resonator length in the thick PDMS resonator group. (b) A typical spectral component in the reciprocal space corresponds to the dashed white line in (a), where the air resonator length changes 12.5 μm. (c) The blue curve corresponds to experimental reflections from two coupled resonators. The curves with red and orange are the measured reflections individually, and the black curves are the demodulated signals via extracting the mode components inside the window of (b). (d)–(f) maintain a similar meaning to (a)–(c). In (d), the fixed air resonator length was ∼8.3 μm in the thick PDMS resonator group, and we changed air resonator length in the thin PDMS resonator group. (e) A typical spectral component in the reciprocal space corresponds to the dashed white line in (d), where the air resonator length changes 7.5 μm.