Calculating the resonance 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 resonance 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 cylindrical functions are obtained, allowing the cylinder's resonance radius to be calculated approximately. For instance, based on the equations, one can determine eight initial significant digits of the cylinder resonance radius for a WGM with mode number 26 to be generated. In a cylinder with refractive index 1.59 and resonance radius 3.469239 λ (λ is wavelength), a WGM with mode number 30 is excited. Such a cylinder can generate an external focal spot of size 0.15 λ, with its maximal intensity being 1500 times larger than that of incident light.