Super-Resonant Intracavity Coherent Absorption

The capability of optical resonators to extend the effective radiation-matter interaction length originates from a multipass effect, hence is intrinsically limited by the resonator’s quality factor. Here, we show that this constraint can be overcome by combining the concepts of resonant interaction and coherent perfect absorption (CPA). We demonstrate and investigate super-resonant coherent absorption in a coupled Fabry-Perot (FP)/ring cavity structure. At the FP resonant wavelengths, the described phenomenon gives rise to split modes with a nearly-transparent peak and a peak whose transmission is exceptionally sensitive to the intracavity loss. For small losses, the effective interaction pathlength of these modes is proportional respectively to the ratio and the product of the individual finesse coefficients of the two resonators. The results presented extend the conventional definition of resonant absorption and point to a way of circumventing the technological limitations of ultrahigh-quality resonators in spectroscopy and optical sensing schemes.

Firenze, Italy *Corresponding author: pietro.malara@ino.it In this supplemental materials we report the detailed derivations of equations 3a and 3b of the main text, describing the effective interaction pathlength of the symmetric and antisymmetric modes of a superresonant coherent perfect absorber (RCPA). Additional experimental details and data are also provided.
In this section we work out the sensitivity of the symmetric and antisymmetric modes of the RCPA modes to an intra-FP absorption in terms of equivalent absorption pathlength. For simplicity, we consider an internal FP with identical, lossless mirrors of reflectivity R and transmissivity T=1-R, and a refractive index nfp<nring (the optical field gains a  phase factor only upon reflection towards the ring). In these conditions, the fields reflected and transmitted by the FP are: Where is the optical wavenumber, d is the per-pass field intracavity transmission of the FP and l its length. In the close vicinity of a resonance, where ~, the above expressions become can be used to calculate | + | and | − |. After some algebraic passages, we get: In the presence of an absorbing sample (with absorption coefficient ) homogeneously distributed over the internal length of the FP resonator, we can consider = − 2 . For small absorption ( → 0), − 2~1 − 2 . Substituting d in Eqs. (S2): By dividing numerator and denominator by 1-R we obtain: At this point, we introduce the assumption that the single-pass absorption in the FP is negligible compared to the FP outcoupling: 1−~0 . In this approximation, we consider | + | and | − |as functions of 1− and linearize them around 0, obtaining As a final remark, we recall that the finesse of a Fabry-Perot is given by ℱ = √ 1− , and that  We can write the RCPA transmission by substituting and in equation (1) of the main text: and are the coupling coefficients of the ring optical couplers (assumed identical). For separated peaks (no interference), the intensity of the symmetric and antisymmetric resonances, normalized to their peak transmission 0 (in absence of absorption) is: Where Γ = 2 and = 2 . We note that the above expressions are analogous to the normalized transmissions of two independent rings with per-pass field intracavity transmission coefficient = | ± | respectively. We now substitute in equations (S7) the | ± | calculated in equation (S6). For ease of notation, we use a generic expressions 1 + A ± . We will distinguish symmetric and antisymmetric transmission at the end of the calculation, by substituting A ± with the coefficients of of equation (S6).
Neglecting the quadratic terms in we rearrange in Again the second term on the right-end side is negligible for → 0.
Then, we get to the absorbance: Analogously to what done previously for the internal FP cavity, if the single-pass absorption is small also compared to the coupling losses of the ring resonator ( 1−Γ~0 ), we can linearize the absorbance (S11) obtaining At this point we can finally write the absorbance of the symmetric and antisymmetric modes by substituting the coefficients A ± with the coefficient of equation (S6): Eqs.(S14) (corresponding to Eq.(2) of the main text) state that the pathlength enhancement in the antisymmetric mode scales with the product of the ring and the FP enhancement factors, and is therefore much larger than the traditional resonant enhancement. For the symmetric mode instead, the pathlength enhancement scales as the ratio of the ring and FP enhancement factors, and can therefore be even smaller than 1, which means that the effective absorption of the symmetric mode is smaller than the single-pass absorption.
The situation is illustrated in fig. S2. In the top layer, the modal absorption of a RCFP and the equivalent FPs are plotted for different values of the FP intracavity loss. In the bottom layer, the symmetric and antisymmetric mode absorbance is plotted for a fixed ℱ and different values of ℱ , along with the absorbance of the uncoupled FP mode and the single-pass absorbance (dashed black lines). In these measurements, in an attempt to reduce the saturation of the antisymmetric absorption at the center of the spectrum, the finesse of the internal FP resonator was also reduced from the FFP~25 to FFP~6 by tuning the reflectivity of the cavity FBG-mirrors at the laser wavelength to R=0.6.
For 4 different loss levels, the coupled-resonator transmission and the open-loop Fabry-Perot transmission were acquired. The recorded spectra are shown in Fig. S3. By fitting these data as described in the main text, and normalizing to the same I0 level, Fig.3a of the main text was obtained. The best fit parameters are shown in table S1. In table S2 instead the best fit parameters for the absorbance curves calculated in the central and lateral region of the acquired spectra are reported.    Table S2: linear-fit slopes for the absorbance measurements plotted in Fig.3b,3c of the main text.