Light is modeled as a plane wave as describe by the following equation:
\(E=E_0e^{i\left(\omega t-\beta z\right)}\)
where
\(\beta=\frac{2\pi n}{\lambda}\)
At the splitter, half of the light is directed down each branch.  The light propagates down each branch, and the combined field is
\(E_o=\frac{1}{\sqrt{2}}\left(E_{o1}+E_{o2}\right)=\frac{E_i}{2}\left(e^{-i\beta_1L_1-\frac{\alpha_1}{2}L_1}\ +\ e^{-i\beta_2L_2-\frac{\alpha_2}{2}L_2}\right)\)
 Converting the E-field to intensity, and assuming loss-less propagation, this simplifies to:
 \(\frac{I_o}{I_i}=\frac{1}{2}\left[1+\cos\left(\beta\Delta L\right)\right]\)
FSR is defined as the space between adjacent peaks, or where the phase shift is a multiple of \(2\pi\).
Solving for FSR gives 
 \(FSR=\Delta\lambda=\frac{\lambda^2}{\Delta Ln_g}\ \)

Simulation

Table \ref{616410} shows the devices that will be fabricated.  FSR were calculated using Lumerical INTERCONNECT simulations.  In addition, a de-embedding device is included which includes just an input grating coupler, and a Y-Branch connection to two output grating couplers.  From this device we can learn the response of the grating couplers themselves.
The included figures show the simulated gain response of the MZI devices as calculated in Lumerical INTERCONNECT.