An imbalanced MZI consists of a single input split into two branches of unequal length (ΔL) then recombined to a single output. In the Y-branch splitter the incident electric field, \(E_i\ \), and corresponding intensity, \(I_i\ \), are equally split into the two branches with electric field outputs of \(E_1 = E_2 = \frac{E_i}{\sqrt{2}}\) and intensity \(I_1=I_2=\frac{I_i}{2}\). For the Y-branch combiner the same equations apply and with two inputs the output field is a vector summation divided by \(\sqrt{2}\). For plane wave light propagation, \(E=E_oe^{i(\omega t-\beta z)}\), and the propagation constant of light, \(\beta=\frac{2\pi n}{\lambda}\), where n is the index of refraction. The light split into each arm observe different path lengths, \(L_1\) and \(L_2\), and therefore the expressions for electric fields at the end of each waveguide, in the lossless case, are: \[E_{o1}=E_1e^{-i\beta_1L_1}=\frac{E_i}{\sqrt{2}}e^{-i\beta_1L_1}\]
\[E_{o2}=E_2e^{-i\beta_2L_2}=\frac{E_i}{\sqrt{2}}e^{-i\beta_2L_2}\]
The combined light is therefore:\[E_o=\frac{1}{\sqrt{2}}(E_{o1}+E_{o2})=\frac{E_i}{2}(e^{-i\beta_1L_1}+e^{-i\beta_2L_2})\]
Therefore the intensity is: \[I_o=\frac{I_i}{4}\left|e^{-i\beta_1L_1}+e^{-i\beta_2L_2}\right|^{2}=\frac{I_i}{4}\left[2\cos\left(\frac{\beta_1L_1-\beta_2L_2}{2}\right)\right]^{2}\]\[=I_i\cos^{2}\left(\frac{\beta_1L_1-\beta_2L_2}{2}\right)=\frac{I_i}{2}\left[1+\cos\left(\beta_1L_1-\beta_2L_2\right)\right]\]
For an imbalanced interferometer the waveguides are the same and hence \(\beta_1=\beta_2\) however the lengths are different, therefore \(\Delta L = L_1-L_2\)
And the transfer function of the full MZI is: \[\frac{I_o}{I_i}=\frac{1}{2}\left[1+\cos\left(\beta \Delta L\right)\right]\]