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Basic equations of LEP for a silver strip nanolaser

In this paper we present basic equations of the lasing eigenvalue problem (LEP) for a silver strip nanolaser. Such laser is built as a silver nanostrip symmetrically embedded into an active cylinder (shell). Keeping in mind that at the threshold the lasing-mode frequency is real-valued (not attenuated emission), the LEP is formulated in terms of finding pairs of real numbers, where the first one corresponds to the emission wavelength and the second one is the associated threshold value of material gain in the cylinder. Due to the inherent two-fold symmetry of the cross-sectional geometry of this laser, we split the studied problem into four different independent classes of symmetry with respect to the x and y-axes: x-even/y-even case (EE), x-even/y-odd case (EO), x-odd/y-odd case (OO) and x-odd/y-even case (OE). On imposing two-side generalized boundary conditions (GBC) at strip's median line and taking into account continuity of the tangential field components at the circle contour we obtain four independent singular or hyper-singular integral equations (IE). Use of the Nystrom-type discretization enables us to derive four independent characteristic equations for different classes of symmetry and find their roots numerically.