\\
  \Sigma_{21} & \Sigma_{22}
\end{array}
\right]
\left[
\begin{array}{c}
  E_1 \\
  E_2
\end{array}
\right]$$ with the following scattering matrix: $$\label{dip2}
\Sigma =
\left[
\begin{array}{cc}
  r & jt \\
  jt & r
\end{array}
\right]
-
\frac{(r+jt)/\tau}{j(\omega-\omega_0)+1/\tau}
\left[
\begin{array}{cc}
  1 & 1 \\
  1 & 1
\end{array}
\right]$$ The scattering matrix is composed of two contributions. The first matrix describes the direct coupling between input and output modes, given by direct reflection or transmission by the membrane (coefficients $r$ and $t$, respectively). The second matrix, on the other hand, describes the coupling through the resonator mode. The reflectivity of the structure described by Eqs. (\[dip1\]) and (\[dip2\]) is: $$R (\omega) = \frac{r^2 (\omega -\omega_0)^2 +(t/\tau)^2 + 2 r t (\omega-\omega_0)/\tau}{(\omega-\omega_0)^2+(1/\tau)^2}$$ In the case of $r=0$ (or $t=0$), we just see the cavity resonance, with a Lorentzian lineshape. In all other cases, the interference between the resonance-assisted path and the path directly reflected by the membrane gives rise to a characteristic asymmetric Fano lineshape. In our case the membrane is $133$ nm thick, which is approximately half the central wavelength of the resonator. A $\lambda/2$-membrane is completely transmissive and the cavity resonance should have a Lorentzian lineshape.

![Model for asymmetric Fano lineshapes, using a 