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

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.