Calculation of the resonant radius of a dielectric cylinder under illumination by a plane TE-wave

When a dielectric circular cylinder of specific radius is illuminated by a plane wave, a whispering gallery mode (WGM) is generated in the cylinder. This resonant radius of the cylinder can be derived numerically on the assumption of the maximal value of the coefficient in a Bessel series expansion of the E-field amplitude of the TE-wave. Equations composed of cylinder functions are obtained, allowing the cylinder's resonant radius to be calculated approximately. For instance, based on the equations, one can accurately determine eight initial significant digits of the cylinder resonant radius for a WGM with mode number $N = 26$ to be generated. In a cylinder with refractive index $n = 1.59$ and resonant radius $R = 3.469239\lambda$ ($\lambda$ is wavelength), a WGM with mode number $N = 30$ is excited. Such a WGM cylinder can generate an external focal spot of size $0.15\lambda$, with its maximal intensity being $1500$ times higher than that of incident light.