A side resonator cuts a notch into a lossless waveguide

A mirror-symmetric, lossless side resonator couples equally to the two directions of an otherwise transparent single-mode waveguide. In a rotating-wave normalization where \(|a|^2\) is stored energy and \(|s|^2\) is power, \[ \dot a=(i\omega_0-\gamma)a+\sqrt\gamma\,(s_{1+}+s_{2+}), \] \[ s_{1-}=s_{2+}-\sqrt\gamma\,a,\qquad s_{2-}=s_{1+}-\sqrt\gamma\,a. \] Here \(\gamma>0\) is the total amplitude decay rate into the two ports. Send a monochromatic wave \(s_{1+}=s_0e^{-i\omega t}\) from port 1 and set \(s_{2+}=0\). Derive the steady resonator amplitude and the complex reflection and transmission coefficients. Prove power conservation for every detuning and explain the destructive interference that makes the through amplitude vanish on resonance even though the device has no loss. Find the reflected-power full width at half maximum, the stored-energy residence ratio \(|a|^2/|s_0|^2\), and its on-resonance value. Identify what would cease to hold if an unmodeled intrinsic loss rate were inserted only in the resonator equation without the associated dissipative power channel. For \(\gamma=1.00\times10^6\,\mathrm{s^{-1}}\) and \(\Delta=\omega-\omega_0=0.750\times10^6\,\mathrm{s^{-1}}\), compute the complex \(r,t\), reflected and transmitted powers, relative scattering phase, and residence ratio.

Insert \(a(t)=a_0e^{-i\omega t}\) and keep \(\Delta=\omega-\omega_0\) signed.