Considering light propagation at a specific wavelength, the governing equation inside the wave guide can be written as:
\(E=E_0e^{i\left(\omega t-\beta z\right)}\)
where \(\beta=\frac{2\pi n}{\lambda}\) is the propagation constant, in which n is the effective index of a given wavelength in the waveguide.
Splitter
To create an MZI structure, light wave needs to be split and delivered onto two arms of the inteferometer. Consider using two Y-branch components as splitter (combiner) in this MZI design, the desired function of the splitter is to split the input electric field \(E_i\) into \(E_1\) and \(E_2\) that propagates on arm1 and arm2, respectively, i.e., \(E_1=E_2=\frac{E_i}{\sqrt{2} }\)
similarly, the Y-branch on the other side combines \(E_1\) and \(E_2\) into \(E_o\), i.e. \(E_o=\frac{E_1+E_2}{\sqrt{2}}\).
Interferometer
The theoretical estimation of the MZI's frequency response can be derived as below. Consider an unbalanced MZI depicted in Fig.\ref{202795}, the length of two arms are \(L_1\) and \(L_2\), and their length difference \(\Delta L=L_2-L_1\). Consider lossless waveguide, where \(E_{1,2}=E_0e^{-i\beta L}\), the electrical field at output of integerometer can be written as: