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Mach-Zehnder interferometers can be built with bulk optic elements, optical fibers with fiber couplers and using integrated optics. In this course we are going to build MZI interferometers using SOI integrated technology.  An MZI interferometer is formed by a two couplers (or in this course with two Y-branch) and two waveguide arms of different lengths, in order to have an optical phase difference. The intensity of the propagating wave at the output of an interferometer can be expressed as:  \begin{center}  $I_{0}=\frac{I_{i}}{4}\left|\exp\left(-j\beta_{1}L_{1}-\frac{\alpha_{1}}{2}L_{1}\right)+\exp\left(-j\beta_{2}L_{2}-\frac{\alpha_{2}}{2}L_{2}\right)\right|^{2}$  \end{center}  Assuming no propation loss along the waveguides, one can simplify the output intensity as:  \begin{center}  $I_{0}=\frac{I_{i}}{2}\left\{ 1+\cos\left(\beta_{1}L_{1}-\beta_{2}L_{2}\right)\right\} $  \end{center}  There are two ways to control the phase, either to modify the effective index of one of the arms of the interferometer, which gives to a propagation constant difference ($\Delta \beta$). The equation then becomes:  \begin{center}  $I_{0}=\frac{I_{i}}{2}\left\{ 1+\cos\left(\Delta\beta L\right)\right\} $  \end{center}  Another way to control the optical phase is by changing the path length difference of the two arms of the interforemeter. This is the method used here in this course, because we deal only with passive silicon photonics designs. The path difference is denoted as $\Delta L$ and it should be no less than $21 \mu m$. The intensity at the output of the MZI is:  \begin{center}  $I_{0}=\frac{I_{i}}{2}\left\{ 1+\cos\left(\beta\Delta L\right)\right\} $  \end{center}  Finally, we can approximate the spacing between adjacent peaks of the interferometer intensity response. This is called the free spectral range or FSR. An expression is:  \begin{center}  $FSR=\frac{\lambda^{2}}{\Delta Ln_{g}}$ 
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