The Mach-Zehnder Interferometer (MZI) provides the ability for light in an optical path to be split and travel in two parallel optical paths and then recombined for the output.  The theory is if the two parallel optical path's are of the same length then the output light intensity (\(I_{o}\)) is equal to the input light intensity (\(I_{i}\)) due to constructive interference.  If the MZI is "imbalanced" with a differing length between the two parallel optical path's this causes a phase shift which theoretically with a phase difference of  \(\pi\) would lead to complete destructive interference and no power on the output.  Thus the MZI branches are considered phase shifters that through altering parameters such as physical length.  Altering other properties in an MZI branch can change the index of refraction and thus output phase.  These include temperature changing the index of refraction via the theromo optical affect or PN  diode altering the depletion region above the branch.
For our MZI silicon photonics application we pass light into a y-branch, 2 wave guides, and then recombine it in another opposite facing y branch.  The branch intensities can be modeled for constructive interference or light in phase as 
\(I_1=I_i/2\) 
\(I_2=I_i/2\)
and thus \(I_{o} = I_{i}\)
However given \(I \propto \mid{E}\mid^2\) we know then  the complex electric field in each branch is
\(E_{1} = E_{i}/\sqrt{2}\)
\(E_{2} = E_{i}/\sqrt{2}\) 
and thus \(E_{o} = \frac{E_{1}+E_{2}}{\sqrt{2}}\)
We can see our outputs will be dependent on the driving variables for the plane wave mainly 
\(E = E_{o}*e^{i(\omega t-\beta z)}\)
where \(\beta = \frac{2\pi n}{\lambda}\) where n is the index of refraction.  We see that the output varies sinusoidally with respect to the wavelength of the light and the index of refraction as well as time and space.  The length of the branches \(L_{1}\) and \(L_{2}\) are substituted into the \(z\) position variable to find the \(E_{o}\) in the above equations.
\(E_{o1} = \frac{E_{i}}{\sqrt{2}}e^{-i\beta_1L_1-\frac{alpha_1}{2}L_1}\) and \(E_{o2} = \frac{E_{i}}{\sqrt{2}}e^{-i\beta_2L_2-\frac{alpha_2}{2}L_2}\)
Plugging in the above to    \(I \propto \mid{E}\mid^2\) gives us   
\(I_{o} = \mid \frac{E_{o1}+E_{o2}}{\sqrt{2}}\mid^2\) simplifies down to when we consider the lossless case given \(\alpha\) = 1 and given identical wave guides with the same index of refraction and physical properties in our case then \(\beta_1 = \beta_2\) then we can derive an equation for an imbalanced  interferometer
\(I_o = \frac{I_i}{2}[1+cos(\beta\Delta L)]\)
Here we see our output sinusoidally varies based on wavelength, index of refraction, and length.

Free Spectral Range (FSR)

When characterizing an interferometer design the Free Spectral Range (FSR) is noted which basically describes the period or distance between two max \(I_o\) over varying wavelengths of light.  This describes for what bandwidth's and frequencies the output will be passed verses attenuated.
Following derivations given  by  \cite{Chrostowski_2015}  we are given a term for group index (\(n_g\)) related to group velocity based off of effective index (\(n\))
\(n_g = n - \lambda\frac{dn}{d\lambda}\)
\(FSR = \frac{\lambda^2}{\Delta L n_g}\)

MZI Transfer Function

The transfer function allows us to calculate the output power in dB based on the input power.  This function includes the "Waveguide compact model" namely the taylor series expansion of 
\(n_{eff}(\lambda) = n_1+n_2(\lambda-\lambda_0)+n_3(\lambda-\lambda_0)^2\)
and 
\(\beta(\lambda) = \frac{2\pi n_eff (\lambda)}{\lambda} + i\frac{\alpha}{2}\)
\(\alpha \) is the propagation loss within the waveguide where for this design we assume 3 to 4 dB/cm
\(T_{MZI-dB}(\lambda) = 10log_{10}(\frac{1}{4}\mid1+e^{-i\beta(\lambda)\Delta L})\mid^2\)