Optical spectral singularities as zero-width resonance frequencies of a Fabry-Perot resonator

Spectral singularities are spectral points of non-Hermitian Hamiltonians with the real energies, that annihilate the completeness of the eigenfunctions. The complex $\mathcal{P}\mathcal{T}$-symmetric potentials with real spectrum have such Hamiltonians. It was identified that the spectral singularities of complex potentials such as an optical gain medium correspond to zero-width resonance modes of the reflected and transmitted beams from the medium. We show that a Fabry-Perot resonator with a gain medium under inclined incident beam exhibits spectral singularities that are measurable by changing the incident angle of the beam to the resonator. The results are realized by theory and simulations.

Figure 1

Sketch of a Fabry-Perot resonator consisting of a plan parallel slab gain medium with the width $L$ and the complex refractive index $n^{\prime }$. It is illuminated with a plan electromagnetic wave with amplitude $A_{-}$ under incident angle $\theta$. $B_{-}$ and $A_{+}$ are the reflected and transmitted amplitudes from the resonator.

Figure 2

Logarithmic plots of the reflection (dashed curve) and transmission (full curve) coefficients of the Fabry-Perot resonator as functions of wavelength, $\lambda \left(\text{nm}\right)$, for $n_0=3.4$, $L=300\mu \text{m}$, ${\lambda }_0=1500\text{nm}$, and $\frac{\gamma }{{\omega }_0}=0.02$.

Figure 3

Logarithmic plots of the transmission coefficient of the Fabry-Perot resonator as the function of wavelength, $\lambda \left(\text{nm}\right)$, for $\theta =0^{\circ }$ to $\theta =6^{\circ }$ with the step $0.5^{\circ }$ that have been sketched from the right to the left.