A supermodes given by the coupled-mode equations, [Eq. [(\[sc\_sim\])]{}]{} and [Eq. [(\[sc\_antisim\])]{}]{}, at the input where the two defects are equal read: $$p_{+}(0)=\frac{1}{\sqrt{2}}\left (\begin{array}{c}1\\ 1\end{array}\right),
\label{sc_simin}$$ $$p_{-}(0)=\frac{1}{\sqrt{2}}\left (\begin{array}{c}1\\ -1\end{array}\right).
\label{sc_antisimin}$$ Projecting $\psi_{in}$ into the S and A supermodes and taking the modulus square, it is possible to find the sound intensity that excites each supermode, $I_+$ and $I_-$: $$I_+=|p_{+}(0)^T\ \psi_{in}|^2=\frac{1}{2}(1+\cos{\varphi})
\label{sc_Ip}$$ $$I_-=|p_{-}(0)^T\ \psi_{in}|^2=\frac{1}{2}(1-\cos{\varphi}).
\label{sc_Im}$$ Since at the output of the system the S supermode corresponds only to sound intensity in the upper linear defect ($I_+=I_{U,out}$), and the A supermode to sound intensity in the lower linear defect ($I_-=I_{L,out}$), it is possible to measure the initial phase difference $\varphi$ between the two sources of the same power by measuring the intensities at the two linear defect outputs: $$\varphi=\pm\arccos\left(\frac{I_{U,out}-I_{L,out}}{I_{U,out}+I_{L,out}}\right).
\label{sc_def}$$ Thus, this structure of linear defects can be used as a phase difference analyzer.

![image](fig7.pdf){width="0.75\linewidth"}

In order to enhance the working efficiency of the phase difference analyzer, the radius of the cylinders surrounding the linear defects (the cylinders corresponding to the rows above the upper linear defect, between the linear defects and below the lower linear defect) have values smoothly changing from $r_{dS}=2.5\,\rm{mm}$ to $r_{dS}=2.25\,\rm{mm}$ in the first $10$ columns of the 