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) A typical phase-sensing application of a Mach-Zender interferometer. (b) A monolithic implementation of a balanced MZI using two directional couplers.

(a) A typical phase-sensing application of a Mach-Zender interferometer. (b) A monolithic implementation of a balanced MZI using two directional couplers.

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.

Free Spectral Range (FSR) of an imbalanced MZI

Free Spectral Range (FSR) of an imbalanced MZI

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.