Circular quantum wire symmetrically loaded with a graphene strip as the plasmonic micro/nano laser: threshold...

We study, apparently for the first time, the threshold conditions for the time-harmonic natural modes of the micro-to-nanosize plasmonic laser shaped as a circular quantum wire with a flat graphene strip, placed symmetrically inside it, in the H-polarization case. We suppose that the quantum wire is made of a nonmagnetic gain material, characterized with the aid of the "active" imaginary part of the complex refractive index. The emergence of lasers integrating plasmonic effects marks a significant trend in contemporary photonics. Here, the graphene offers a promising alternative to the noble metals as it exhibits the capacity to sustain plasmon-polariton natural surface waves across the infrared and terahertz (THz) spectra. The used innovative approach is the lasing eigenvalue problem (LEP), which is classical electromagnetic field boundary-value problem, adapted to the presence of active region. It is tailored to deliver both the mode-specific emission frequency, which is purely real at the threshold, and the value of the gain index of the active region, necessary to make the frequency real-valued. The conductivity of graphene is characterized using the quantum Kubo formalism. We reduce the LEP for the considered nanolaser to a hyper-singular integral equation for the current on the strip and discretize it by the Nystrom-type method. This method is meshless and computationally economic. After discretization, a matrix equation is obtained. The sought for mode-specific pairs {the frequency and the threshold gain index} correspond to the zeros of the matrix determinant. It should be noted that the convergence to exact LEP eigenvalues is guaranteed mathematically if the discretization order is taken progressively larger. Two families of modes are identified and studied: the modes of the quantum wire, perturbed by the presence of the graphene strip and the plasmon modes of the strip. The frequencies of all plasmon modes and the lowest mode of the quantum wire are found to be well-tuned by changing the chemical potential of graphene. Engineering analytic formulas for the plasmon-mode frequencies and thresholds are derived. We believe that the presented results can be used in the creation of single-mode tunable micro and nanolasers.