  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.
