($k_{L}$) is the propagation constant of the individual upper (lower) waveguide, $\Omega_{UL}$ ($\Omega_{LU}$) is the coupling coefficient from the upper to the lower (from the lower to the upper) waveguide, and $C_{U}$ ($C_{L}$) represents the amplitude function of the field in the upper (lower) waveguide. For the case of sound, the total pressure field in a system of two coupled sonic waveguides can be expressed as $$p(y,z)=\sum_{i=U,L}C_i(z)p_i(y),$$ where $p_U$ ($p_L$) is the mode of only the upper (lower) waveguide. The velocity field can be described similarly.

It is possible to diagonalize [Eq. [(\[sc\_coupled\])]{}]{} in order to obtain the expressions for the pressure field corresponding to the S supermode, $p_+(z)$, and the A supermode, $p_-(z)$, as a function of the parameters of the individual linear defects: $$p_+(z)=\left (\begin{array}{c}\cos\theta/\sqrt{\frac{\Omega_{UL}}{\Omega_{LU}}\sin^2\theta+\cos^2\theta}\\\sqrt{\frac{\Omega_{UL}}{\Omega_{LU}}}\sin\theta/\sqrt{\frac{\Omega_{UL}}{\Omega_{LU}}\sin^2\theta+\cos^2\theta}\end{array}\right),
\label{sc_sim}$$ $$p_-(z)=\left (\begin{array}{c}\sin\theta/\sqrt{\sin^2\theta+\frac{\Omega_{UL}}{\Omega_{LU}}\cos^2\theta}\\-\sqrt{\frac{\Omega_{UL}}{\Omega_{LU}}}\cos\theta/\sqrt{\sin^2\theta+\frac{\Omega_{UL}}{\Omega_{LU}}\cos^2\theta}\end{array}\right),
\label{sc_antisim}$$ where the mixing angle $\theta$ is defined by $$\tan{2\theta}=\frac{\sqrt{\Omega_{UL}\Omega_{LU}}}{\Delta k},
\label{sc_mixing}$$ with $\Delta k=k_L-k_U$. The propagation constants of the $p_+(z)$ and $p_-(z)$ supermodes are $$k_{\pm}=\frac{k_U+k_L\mp\sqrt{\Omega_{UL}\Omega_{LU}+{\Delta k}^2}}{2}.
\label{sc_kmodes}$$

In order to modify, along the propagation direction, the $p_+$ and $p_-$ supermodes obtained from the coupled-mode equations, it is necessary to change the couplings and thus, the mixing angle $\theta$ (see Eqs. (\[sc\_sim\]) and (\[sc\_antisim\])), which depends on the ratio $\sqrt{\Omega_{UL}\Omega_{LU}}/{\Delta 