Modes of planar gradient-index hyperbolic secant waveguide

We obtained analytical expressions for TE and TM modes of the planar hyperbolic secant (HS) waveguide. These modes are partial case of the more general modes propagating in the Epstein's waveguides. Expression for the Talbot period (or focal distance) has been obtained for TE-modes in the planar HS-waveguide and we show that for TM-modes there is no analogous periodicity and focusing in HS-waveguide. We obtained expressions for full width at half-maximum of intensity and normalized intensity for the fundamental TE-mode in the HS-waveguide and waveguide parameters have been found for which the with of the fundamental mode is minimal. In silicon HS-waveguide this minimal mode has width at half-maximum 0.11 of wavelength. By using Full-WAVE simulation tool we show modal propagation of light fields with analytically obtained amplitudes in bounded HS-waveguides and we proved periodic reconstruction of non-modal light fields with the Talbot period. We also show by simulation that HS-lens, obtained by truncation of the HS-waveguide, focuses TE-polarized plane wave into subwavelength focal spot with half-maximum intensity 0.132 of wavelength.