Analytical solution of the fundamental waveguide mode of one-dimensional transmission grating for TM polarization

TL;DR

This paper derives an analytical expression for the effective index of the fundamental TM mode in 1D transmission gratings, simplifying analysis compared to numerical methods and revealing key dependencies on physical parameters.

Contribution

It provides the first analytical solution for the effective index of the fundamental waveguide mode in 1D TM gratings, contrasting with previous numerical approaches.

Findings

Effective index squared is inversely proportional to slit width and ridge refractive index.

Effective index varies linearly with wavelength.

Dependence on grating period and incident angle is minimal.

Abstract

This article presents an analytical solution of the effective index of the fundamental waveguide mode of $1$D metallo-dielectric grating for Transverse Magnetic (TM) polarization. In contrast to the existing numerical solution involving transcendental equation, it is shown that the square of the effective index ($n_{Eff}$) of the fundamental waveguide mode of $1$D grating is inversely proportional to the slit width ($w$) and the refractive index ($n_m$) of the ridge material and varies linearly with the incident wavelength ($\lambda$). Further, it has also been demonstrated that the dependence of $n_{Eff}$ on the grating period ($P$) and the incidence angle ($\theta$) is minimal. Agreement between the results obtained using the solution presented in this article and published data is excellent.