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\begin{document}
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\title{Design of a Silicon based Mach-Zehnder Interferometer and Two-stage optical filter}
gvargas008
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\section{Introduction}
% relevant application
This design proposal is about the study of an optical device known as the Mach-Zehnder interferometer (MZI) via simulation and to verify its operation experimentally. The device will be fabricated using Silicon over Insulator (SOI) technology.
These types of interferometers are commonly employed in telecommunications applications such as switches, modulators and filters. Also, in sensing applications, MZI's are commonly employed to allow one of the arms of a interferometer to interact with an analyte via a waveguide's evanescent field.
In this draft design, two devices were chosen. The first is a simple MZI with different path length differences $\Delta L$. An explanation of a MZI operation is thus presented.
The second device is a basic filter based on two cascaded interferometers. Also, the basic theory of a lossless two stage MZI filter is presented.
Here in this report simulations in terms of optical component and circuit device are shown, which later on will be compared with experimental results after device fabrication.
\section{Theory}
In this section a brief theoretical explanation of the operation of a Mach-Zehnder interferometer is given in terms of their gain spectrum response.
\subsection{Mach-Zehnder Interferometer basic operation}
Mach-Zehnder interferometers can be built with bulk optic elements, optical fibers with fiber couplers and using integrated optics. In this course we are going to build MZI interferometers using SOI integrated technology.
An MZI interferometer is formed by two couplers (or as in this course with two Y-branch) and two waveguide arms of different lengths, in order to have an optical phase difference. The intensity of the propagating wave at the output of an interferometer can be expressed as: \cite{chrostowski2015silicon}
\begin{center}
$I_{0}=\frac{I_{i}}{4}\left|\exp\left(-j\beta_{1}L_{1}-\frac{\alpha_{1}}{2}L_{1}\right)+\exp\left(-j\beta_{2}L_{2}-\frac{\alpha_{2}}{2}L_{2}\right)\right|^{2}$
\end{center}
Assuming no propagation loss along the waveguides, one can simplify the output intensity as:
\begin{center}
$I_{0}=\frac{I_{i}}{2}\left\{ 1+\cos\left(\beta_{1}L_{1}-\beta_{2}L_{2}\right)\right\} $
\end{center}
There are two ways to control the phase, either to modify the effective index of one of the arms of the interferometer, which gives to a propagation constant difference ($\Delta \beta$). The equation then becomes:
\begin{center}
$I_{0}=\frac{I_{i}}{2}\left\{ 1+\cos\left(\Delta\beta L\right)\right\} $
\end{center}
Another way to control the optical phase is by changing the path length difference of the two arms of the interferometer. In this course, this is the preferred method, because we are dealing only with passive silicon photonic designs. The path difference is denoted by $\Delta L$ and it should be no less than $21 \mu m$. The intensity at the output of the MZI is:
\begin{center}
$I_{0}=\frac{I_{i}}{2}\left\{ 1+\cos\left(\beta\Delta L\right)\right\} $
\end{center}
Finally, we can approximate the spacing between adjacent peaks of the interferometer intensity response. This is called the free spectral range or FSR. An expression is:
\begin{center}
$FSR=\frac{\lambda^{2}}{\Delta Ln_{g}}$
\end{center}
\subsection{Optical filter using two Mach-Zehnder interferometers}
%From Agarwal pp. 293
By putting multiple interferometers in cascade arrangement, a filter can be obtained. If we have M MZI devices connected in a chain, then a transfer function of the overall filter can be extracted following the sum of $2^M$ paths. An expression for the transfer function in frequency domain is: \cite{Agrawal_2005}
\begin{center}
$\left|H\left(\omega\right)\right|^{2}=\overset{M}{\underset{m=1}{\prod}}\cos^{2}\left(\frac{\omega\tau_{m}}{2}\right)$
\end{center}
For a two-stage MZI filter the transfer function in angular frequency can be derived according to the previous formula, and depending of the optical delays $\tau_{m}$ at the m-th stage, we can have two scenarios. The first results by assuming the delays at each stage to be equal, i.e. $\tau_1=\tau_2$, or in other words to have similar $\Delta L$. Accordingly, we can derive a transfer function as follows:
\begin{center}
$\left|H\left(\omega\right)\right|^{2}=\cos^{4}\left(\frac{\omega\tau_{m}}{2}\right) = \frac{1}{2}[1+\cos(\omega \tau_{m})]^2 $
$\left|H\left(\omega\right)\right|^{2}=\frac{1}{4}[\frac{3}{2} + 2 \cos(\omega \tau_{m}) + \frac{1}{2} \cos(2 \omega \tau_{m})] $
\end{center}
The transfer function can be rewritten as a function of wavelength, by noticing two important definitions: the propagation constant $\beta = \frac{n_{eff}\omega}{c}$ and the relative delay is related to the path difference of the MZI as $\tau = \frac{n_{eff}\Delta L}{c}$. Therefore $\omega \tau = \frac{\omega n_{eff}}{c}\Delta L = \beta \Delta L$. Therefore the Transfer function of the filter with two MZI with equal path difference is:
\begin{center}
$\left|H\left(\lambda\right)\right|^{2}=\frac{1}{4}\left[\frac{3}{2}+2\cos\left(\beta\Delta L\right)+\frac{1}{2}\cos\left(2\beta\Delta L\right)\right]$
\end{center}
The second scenario of the two-stage MZI filter is to have different path length differences at each stage. From the initial equation, this means that delays are different, therefore the transfer function becomes:
\begin{center}
$\left|H\left(\omega\right)\right|^{2} = \cos ^2 (\frac{\omega \tau_1}{2}) \cos ^2 (\frac{\omega \tau_2}{2}) = \frac{1}{4}[1 + \cos(\omega \tau_{1})] [1 + \cos(\omega \tau_{2})] $
\end{center}
Applying the previous variable conversion, the transfer function of two cascaded MZI with two unequal path length difference i.e. $\Delta L$ and $2\Delta L$ become as:
\begin{center}
$\left|H\left(\lambda\right)\right|^{2}= \frac{1}{4} \left[ \cos\left(\beta\Delta L\right)^2 \times \cos\left(2\beta\Delta L\right)^2 \right]$
\end{center}
\section{Modelling and Simulation}
% This should have:
% Compact equation for the waveguide
% Transfer function of our devices
% Simulation results
% neff / ng vs lambda
% Table with parameter variation (i.e. how FSR is affected by Delta L)
% Spectrum
% Waveguide and circuit geometry (layout)
In this design proposal, a strip waveguide structure was chosen with the following dimensions:
\begin{enumerate}
\item Width = 500 nm and Height = 220 nm
\end{enumerate}
This basic structure will be used to build the arms of the interferometer, the main MZI variable to explore is $\Delta L$ which corresponds to the difference between the two waveguide arms of the interferometer.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/XY-waveguide/XY-waveguide}
\caption{{Waveguide geometry in Lumerical's MODE GUI%
}}
\end{center}
\end{figure}
There will be 5 different Mach-Zhender Interferometer designs, and two cascaded MZI designs.
I decided to test the fiber couplers (TE/TM polarization) and Y-branch.
Table \ref{tab:mzi} shows the 7 different variations of the MZI design according to polarization and path length difference. also, the FSR at 1550 nm with a group index of 4.2 is estimated for each path length difference.\selectlanguage{english}
\begin{table}
\begin{tabular}{ c c c c c c } Device ID & Polarization & $L_1$ [$\mu m$] & $L_2$ [$\mu m$] & $\Delta L$ [$\mu m$] & $FSR$ [$nm$] \\ MZI1 & TE & 131.5 & 482.5 & 351 & 1.6 \\ MZI2 & TE & 131.5 & 162.5 & 31 & 18.4 \\ MZI3 & TE & 124.75 & 844.75 & 720 & 0.8 \\ MZI4 & TM & 131.5 & 162.5 & 31 & 18.4 \\ MZI5 & TM & 131.5 & 482.5 & 351 & 1.6 \\ MZI6 & TE & 70 & 190 & 120 & 4.7 \\ MZI7 & TE & 70 & 310 & 240 & 2.4 \\
\end{tabular}
\caption{{\label{tab:mzi} MZI design parameters.}}
\end{table}
A MZI device is comprised of two Y-branches, and waveguide segments either straight or with bends at certain radius. Fiber grating couplers at specified polarizations (e.g. TE r TM) are going to be used also in this design. Couplers allow to input light into the devices and captured light using optical fibers.
In this section, two types of simulations are presented. First a component simulation using Lumerical's MODE to obtain the mode profile, effective and group index for straight and bent waveguides are presented. From those simulations mathematical models for waveguide dispersion are obtained. The second type of simulation is related to the MZI device. For that, we use Lumerical's Interconnect. The MZI devices will incorporate circuit models for the straight and bend waveguides, and the provided S-parameter models for the Y-branch and the fiber grating couplers.
\subsection{Straight waveguide simulation results}
I used Lumerical's MODE software to simulate the straight waveguide and obtain a model of the effective index at quasi-TE mode. Although it may be necessary also to have the dispersion model of a waveguide bent at certain radius, I did not use it so i can simplify the analysis.
In figure \ref{fig:profile_TE} we have the mode profile of a 500 nm x 220 nm waveguide at quasi-TE mode at a wavelength of 1550 nm. Confinement of the mode is well in the center, with some of the field outside the waveguide which contributes with the propagation losses.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/wg-profile-TE/wg-profile-TE}
\caption{{Mode profile of straight waveguide (500 nm x 220 nm) quasi-TE mode, (a) Ex field, (b) Intensity profile.\label{fig:profile_TE}%
}}
\end{center}
\end{figure}
The effective index of this waveguide for the quasi-TE mode is depicted in figure \ref{fig:neff_TE} also it is shown the group index which contributes to the FSR later on in the MZI device response.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/neff-ng-TE/neff-ng-TE}
\caption{{Straight waveguide quasi-TE mode (a) Effective index $n_{eff}$ (b) Group index $n_g$. \label{fig:neff_TE}%
}}
\end{center}
\end{figure}
In this design, I also wanted to try to test some MZI's in quasi-TM mode. A profile of the mode is shown in figure \ref{fig:profile_TM}, as it can be seen some part of Ey field is outside the waveguide.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/wg-profile-TM/wg-profile-TM}
\caption{{Mode profile of straight waveguide (500 nm x 220 nm) quasi-TM mode, (a) Ey field, (b) Intensity profile.\label{fig:profile_TM}%
}}
\end{center}
\end{figure}
In figure \ref{fig:neff_TM} we have the effective index and group index response of the waveguide chosen. The value of effective index in less than the quasi-TM mode which demonstrates that the field is near the periphery of the core-cladding interface.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/neff-ng-TM/neff-ng-TM}
\caption{{Straight waveguide quasi-TM mode (a) Effective index $n_{eff}$ (b) Group index $n_g$. \label{fig:neff_TM}%
}}
\end{center}
\end{figure}
After the effective index information was obtained using simulation, the next step is to get a model based on a second order polynomial fit. As it was mentioned in the course, we can use MATLAB or Lumerical scripting tool to obtain the fit parameters. For the quasi-TE mode the effective index approximates to:
\begin{center}
$n_{eff} = 2.44763 - 1.12907(\lambda - 1.55) - 0.0354547(\lambda - 1.55)^2$
\end{center}
And for the quasi-TM mode an approximation of the effective index is:
\begin{center}
$n_{eff} = 1.77031 - 1.24432(\lambda - 1.55) + 1.92466(\lambda - 1.55)^2$
\end{center}
In the following sections, the responses of some MZI designs will be obtained.
\subsection{Mach-Zehnder interferometer circuit simulation}
Here, I applied Lumerical's Interconnect circuit simulation tool to get the responses of the different MZI designs mentioned before.
For the MZI simulations, I use the Fiber Grating coupler S-parameters given in this course as well as the simulated S-parameters for the Y-branch at quasi-TE mode. For the arms of the interferometers, I inserted Lumerical's MODE simulated data for the straight waveguide explained in the previous subsection. As I mentioned before, it may be needed to apply bend waveguide to be more accurate in these simulations. The reason is that in the layout I designed for this course, there are no 90 degrees sharp bends but nearly circular ones at certain radius (i.e. 5 $\mu m$).
MZI designs (MZI1 to MZI3) demonstrates three different path lengths. Figure \ref{fig:mzi1-3} illustrates the gain responses of those interferometers. Those results show the effect of the path length difference, where shorter differences gives few peak responses or large FSR. The FSR is reduced as the path length difference increase.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/GAIN-MZI1-3/GAIN-MZI1-3}
\caption{{Simulated gain response of three MZI designs: (a) $\Delta L = 351 \mu m$, (b) $\Delta L = 31 \mu m$ and (c) $\Delta L = 720 \mu m$.\label{fig:mzi1-3}%
}}
\end{center}
\end{figure}
In order to design an optical filter, I included two additional MZI designs, with path length differences of $\Delta L = 120 \mu m$ and $\Delta L = 240 \mu m$, named MZI6 and MZI7 respectively. Figure \ref{fig:mzi6-7} shows the simulated gain responses of each MZI design considering fiber coupler, Y-branch models from the course and 3dB/cm propagation loss for the waveguides.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/GAIN-MZI6-7/GAIN-MZI6-7}
\caption{{Simulated gain response of three MZI designs: (a) $\Delta L = 120 \mu m$, (b) $\Delta L = 240 \mu m$.\label{fig:mzi6-7}%
}}
\end{center}
\end{figure}
\subsection{Two-stage MZI optical filter circuit simulation}
Filter based MZI Gain Responses (MZ8, MZ9)
Here a demonstration of a filter response is shown by cascading two interferometers and considering no loss. Figure \ref{fig:mzi8-9}(a) shows the response when two interferometers with similar path length difference are chain together. The response is a bit similar to MZI6 interferometer. However, figure \ref{fig:mzi8-9}(b) shows an interesting response where clearly the transmission peaks are a bit narrower. If at each stage of interferometers the path difference follows a geometric progression i.e. $\Delta L$, $2\Delta L$, $4\Delta L$, ... then low passband loss occurs and clearly defined peaks are obtained.\cite{madsen1999optical} Another interesting application for these filters is as a channel selector or de-multiplexer. By cascading multiples of these devices, one can narrowly select closely spaced optical channels. \cite{Agrawal_2005}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/GAIN-MZI8-9/GAIN-MZI8-9}
\caption{{Gain response of two-stage MZI filter (a) with two equal $\Delta L$ MZI and (b) with two-MZI with different path length differences $\Delta L$ and $2\Delta L$. Here $\Delta L = 120 \mu m$. \label{fig:mzi8-9}%
}}
\end{center}
\end{figure}
At this part of the design, a basic explanation of how the MZI devices inserted in the layout were given using simulations. After fabrication, further experimental testing will be performed and results will be compared to these simulations.
\selectlanguage{english}
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