# edx Phot1x report (2017/10)

Abstract

We present a design for an integrated, imbalanced Mach-Zehnder Interferometer (MZI) and propose a parameter sweep study of the MZI Free Spectral Range (FSR) as a function of path length difference. Experimental results are compared with theory and simulation data.

# Introduction

In its most general form, the MZI consists of two beamsplitters with the two outputs of the first optically connected to the inputs of the second (fig. 1a). A symmetric or ‘balanced’ MZI has no wavelength dependence in its transfer function, but an imbalance in path length between the beamsplitters gives rise to a spectral transmission that is periodic (fig. 2). Our proposed experiment will determine the FSR of a range of different path length imbalances in order to accurately characterise the group index $$n_g$$ of the waveguides.

A generic MZI can be implemented on a monolithic chip using 3dB directional couplers (or Multi-Mode Interferometers - MMIs) as beamsplitters (fig. 1b). However since we are only interested in the FSR of the MZI, the proposed design uses Y-splitters in place of directional couplers for compactness.

# Theory

The intensity at each output of a Y-splitter is simply $$I_1=I_i/2$$, $$I_2=I_i/2$$ where $$I_i$$ is the input intensity. The electric fields are therefore $$E_1=E_i/\sqrt{2}$$, $$E_2=E_i/\sqrt{2}$$ and hence the output of the second Y-splitter is $$E_o=(E_1+E_2)/\sqrt{2}$$. By approximating the fields as plane waves $$E=E_Ae^{\omega t - \beta x}$$ with $$\beta = 2 pi n$$ the output of the second Y-splitter is $E_o=E_i/2 (e^{-i \beta L_1}+e^{-i\beta L_2})$ The output intensity is $I_o=I_i/4 |(e^{-i \beta L_1}+e^{-i\beta L_2})|^2 = I_i/2 (1+cos{\beta \Delta L})$ The latter of which is is the transfer function of the MZI as a whole. Note for $$\beta \Delta L=\pi$$, $$I_o$$=0 for any $$I_i$$. Energy is, however, conserved as $$I_i$$ will be coupled into either higher order waveguide modes or radiative modes.

The free spectral range is distance between adjacent peaks in the transfer function (for a given $$\Delta L$$) and is given by $FSR(\lambda)=\lambda^2/\Delta L n_g(\lambda)$ where $$n_g$$ is the waveguide group index.