Contrasting eigenvalue and singular-value spectra for lasing and antilasing in aPT-symmetric periodic structure

It has been proposed and demonstrated that lasing and coherent perfect absorption (CPA or ``antilasing'') coexist in parity-time ($\mathcal{PT}$) symmetric photonic systems. In this work we show that the spectral signature of such a CPA laser displayed by the singular value spectrum of the scattering matrix ($S$) can be orders of magnitude wider than that displayed by the eigenvalue spectrum of $S$. Since the former reflects how strongly light can be absorbed or amplified and the latter announces the spontaneous symmetry breaking of $S$, these contrasting spectral signatures indicate that near perfect absorption and extremely strong amplification can be achieved even in the $\mathcal{PT}$-symmetric phase of $S$, which is known for and defined by its flux-conserving eigenstates. We also show that these contrasting spectral signatures are accompanied by strikingly different sensitivities to disorder and imperfection, suggesting that the eigenvalue spectrum is potentially suitable for sensing and the singular value spectrum for robust switching. A differential light amplifier may also be devised based on these two spectra.