deletions | additions       diff --git a/section_Theory_subsection_Mach_Zehnder__.tex b/section_Theory_subsection_Mach_Zehnder__.tex  index 81a68e4..a2728fe 100644  --- a/section_Theory_subsection_Mach_Zehnder__.tex  +++ b/section_Theory_subsection_Mach_Zehnder__.tex 
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$\left|H\left(\omega\right)\right|^{2}=\overset{M}{\underset{m=1}{\prod}}\cos^{2}\left(\frac{\omega\tau_{m}}{2}\right)$  \end{center}  For a two-stage MZI filter the transfer function in angular frequency is can be  derived according to the previous formula, and depending of the optical delays $\tau_{m}$ at the m-th stage, we can have two scenarios. The first results by assuming the delays at each stage to be equal, i.e. $\tau_1=\tau_2$, or in other words to have similar $\Delta L$. Accordingly, we can derive a transfer function  as follows: \begin{center}  $\left|H\left(\omega\right)\right|^{2}=\cos^{4}\left(\frac{\omega\tau_{m}}{2}\right) = \frac{1}{2}[1+\cos(\omega \tau_{m})]^2 $  $\left|H\left(\omega\right)\right|^{2}=\frac{1}{4}[\frac{3}{2} + 2 \cos(\omega \tau_{m}) + \frac{1}{2} \cos(2 \omega \tau_{m})] $  \end{center}  The transfer function can be rewritten as a function of wavelength, by noticing two important definitions: the propagation constant $\beta = \frac{n_{eff}\omega}{c}$ and the relative delay is related to the path difference of the MZI as $\tau = \frac{n_{eff}\Delta L}{c}$. Therefore $\omega \tau = \frac{\omega n_{eff}}{c}\Delta L = \beta \Delta L$. Therefore the Transfer function of the filter with two MZI with equal path difference is:  \begin{center}  $\left|H\left(\lambda)\right|^{2}=\frac{1}{4}[\frac{3}{2} + 2 \cos(\beta \Delta L) + \frac{1}{2}\cos(\beta \Delta L)] $  \end{center