Reflection and diffraction at the end of a cylindrical dielectric nanowire: Exact analytical solution

The rigorous theory of reflection and diffraction at the end of a semi-infinite dielectric circular cylinder is developed. An exact solution of this problem is found by the use of fictitious electric and magnetic current sheets located at the end of the cylinder. The solution has the form of the Fourier integral along the integration path in the complex plane of propagation constants. The theory assumes an arbitrary ratio between the cylinder radius and the wavelength and hence it can be used for the description of the nanowire optical properties. The case when the incident wave is a transverse magnetic (TM)/transverse electric (TE) waveguide mode is analyzed in detail. It is found that such a mode is completely converted into TE/TM-polarized field on reflection from the nanowire facet. It is shown that the corresponding angular diffraction pattern forms a cone which is specific for a given waveguide mode. The obtained results can be used for the determination of the threshold gain and quality factor as well as Fabry-Perot modes of a nanolaser.