Figure 1 exhibits the configuration of an asymmetric planar waveguide that consists of dissipative MTMs film having width *w* placed on a metal substrate and bounded from above by a Ferrite film with width *s* covered by air.

The dissipative MTM film has both permittivity *ε*
_{
M
} and permeability *μ*
_{
M
} depending on the frequency (*ω*) as follows:

{\epsilon}_{M}=1-\frac{{\omega}_{p}^{2}}{{\omega}^{2}+i\gamma \omega}

(1)

{\mu}_{M}=1-\frac{F{\omega}^{2}}{{\omega}^{2}-{\omega}_{r}^{2}+i\gamma \omega}

(2)

where *ω*
_{
r
} is the resonance frequency, *ω*
_{
p
} is the plasma frequency and *γ* is the damping factor. The values of the parameters *ω*
_{
r
}, *ω*
_{
p
} and *F* are chosen to fit the experimental data (Shebly2001) : *ω*
_{
p
} = 26.6π GHz, and *F* = 0.37. In the ferrite slab, a static magnetic field is applied in the *z* direction, resulting in a uniform intensity *H*
_{0} within the ferrite. The ferrite slab has positive permittivity *ε*
_{
f
} and permeability *μ*
_{
f
} defined as (Bestler1959, Courtois et al.1970, El-Khozondar et al.2010, Mansour et al.2009, Al-Sahhar et al.2013)

\Vert {\mu}_{f}\Vert =\left[\begin{array}{ccc}{\mu}_{11}& i{\mu}_{12}& 0\\ -i{\mu}_{12}& {\mu}_{11}& 0\\ 0& 0& {\mu}_{1}\end{array}\right],

(3)

where{\mu}_{11}=1+\frac{{\omega}_{M}{\omega}_{H}}{{\omega}_{H}^{2}-{\omega}^{2}},{\mu}_{12}=\frac{\omega {\omega}_{M}}{{\omega}_{H}^{2}-{\omega}^{2}}, *μ*
_{
1
} = 1, *ω* is surface wave frequency, *ω*
_{
H
} = *σH*
_{0} is the Larmor frequency, *ω*
_{
M
} = *4π σ M*
_{
0
} is the magnetic frequency, *σ* is the electromagnetic oscillation frequency, *4πM*
_{
0
} is the ferrite saturation magnetization, and *H*
_{0} is the applied magnetic field (Bespyatykh et al.2001).

We assumed transverse electric fields (TE) propagating in the y direction such that *E* = (0, 0, *E*
_{
z
}) e^{jωt} and *H* = (*H*
_{
x
}, *H*
_{
y
}, 0) e^{jωt}. The field equations are obtained by applying the TE fields into Maxwell’s equations as follows:

\frac{{\partial}^{2}{E}_{\mathit{\text{zi}}}}{\partial {x}^{2}}+\frac{{\partial}^{2}{E}_{\mathit{\text{zi}}}}{\partial {y}^{2}}+{q}_{i}^{2}{E}_{\mathit{\text{zi}}}=0,

(4)

Where the variation with *z*-direction is assumed to be zero, *i* indicates *f* for Ferrite layer, *M* for MTMs layer, and *l* for the linear cladding layer (air), *q*
_{
l
} = *k*
_{0},{q}_{f}={k}_{0}\sqrt{{\epsilon}_{f}{\mu}_{v}}, *q*
_{
M
} = *k*
_{0}
*ε*
_{
M
}, *k*
_{0} = *ω*/*c* and{\mu}_{v}=\frac{{\mu}_{11}^{2}-{\mu}_{12}^{2}}{{\mu}_{11}}, which is called the Voigt permeability. The dispersion equation (5) is derived by applying boundary conditions to the solutions of equation (4).

\begin{array}{l}\left({k}_{\mathit{\text{xf}}}+\frac{\nu}{\mu}{k}_{y}-{\mu}_{v}{k}_{\mathit{\text{xl}}}\right)\\ \left(\left(\frac{\nu}{\mu}{k}_{y}-{k}_{\mathit{\text{xf}}}\right)\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}tanh\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\left({k}_{\mathit{\text{xM}}}w\right)+{\mu}_{v}{k}_{\mathit{\text{xM}}}\right){e}^{-\left(2{k}_{\mathit{\text{fx}}}s\right)}\\ +\left({k}_{\mathit{\text{xf}}}+\frac{\nu}{\mu}{k}_{y}+{\mu}_{v}{k}_{\mathit{\text{xl}}}\right)\\ \left(\left(\frac{\nu}{\mu}{k}_{y}+{k}_{\mathit{\text{xf}}}\right)\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}tanh\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\left({k}_{\mathit{\text{xM}}}w\right)+{\mu}_{v}{k}_{\mathit{\text{xM}}}\right)=0,\end{array}

(5)

where{k}_{y}^{2}-{k}_{\mathit{\text{xi}}}^{2}={q}_{i}^{2} and *k*
_{
xi
} and *k*
_{
y
} are the components of the wave vector directed along the coordinate. The dispersion relation equation (5) relates the transverse wave numbers for each media. It is an implicit equation that gives the surface wave dispersion relation.

In the calculations, two limits are considered: *w* → ∞ which simplifies the structure to air-ferrite -MTM and *w* = 0 which reduces the structure to air-ferrite-metal.

At the limit *w* → ∞, equation (5) becomes,

\begin{array}{l}\left({k}_{\mathit{\text{xf}}}+\frac{\nu}{\mu}{k}_{y}-{\mu}_{v}{k}_{\mathit{\text{xl}}}\right)\left(\left(\frac{\nu}{\mu}{k}_{y}-{k}_{\mathit{\text{xf}}}\right)+{\mu}_{v}{k}_{\mathit{\text{xM}}}\right){e}^{-\left(2{k}_{\mathit{\text{fx}}}s\right)}\\ +\left({k}_{\mathit{\text{xf}}}+\frac{\nu}{\mu}{k}_{y}+{\mu}_{v}{k}_{\mathit{\text{xl}}}\right)\left(\left(\frac{\nu}{\mu}{k}_{y}+{k}_{\mathit{\text{xf}}}\right)+{\mu}_{v}{k}_{\mathit{\text{xM}}}\right)=0.\end{array}

(6)

While at the limit *w* = 0, equation (5) reduces to the following form,

{k}_{\mathit{\text{xf}}}\phantom{\rule{1pt}{0ex}}coth\phantom{\rule{1pt}{0ex}}\left({k}_{\mathit{\text{xf}}}s\right)-\frac{\nu}{\mu}{k}_{y}+{\mu}_{v}{k}_{\mathit{\text{xl}}}=0.

(7)

Equation (5) to equation (7) can only be solved numerically. The solutions of these equations give the MSSW at the different limits. The limiting frequencies for MSSW free ferrite film are expressed as follows (Damon and Eshbach1961):

{\omega}_{s}=\sqrt{{\omega}_{H}^{2}+{\omega}_{H}{\omega}_{M}}

(8)

{\omega}_{\mathit{\text{fin}}}={\omega}_{H}+{\omega}_{M}/2

(9)

where *ω*
_{
s
} is the starting frequency and *ω*
_{
fin
} is the final frequency.