4.2: Mach-Zehnder Interferometers
- Page ID
- 113802
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A key component in integrated optical circuitry is the Mach-Zehnder interferometer (MZI). An MZI consists of two beam splitters that first split light so that it travels by two different paths, and is then recombined at the second beam splitter. The length of the two different paths changes the relative phase of the two light fields and, therefore, dictates the intensity balance between the two outports. A schematic for a generic MZI is shown in Figure \(\PageIndex{1}\)(a) and the integrated optical version is shown in Figure \(\PageIndex{1}\)(b).
Figure \(\PageIndex{1}\): (a) a classical free space MZI. (b) an integrated optical MZI.
In Figure \(\PageIndex{1}\)(a) it is clear that an MZI has two inports and two outports, while for Figure \(\PageIndex{1}\)(b) only one inport and one outport are visible. However, the missing ports still exist. When light is supposed to exit the second outport, it is scattered out of the integrated optical circuit, while the second inport is also scattered light coming into the integrated optical circuit. Since light scattered into the circuit is tiny, it can be neglected, while the purpose of the MZI is to control the light intensity at the guided outport, so we don't need to care about light scattered out of the outport.
The beam splitters are also different. In Figure \(\PageIndex{1}\)(a), the beam splitters are 50% reflective mirrors, meaning that they divide the power across the whole mode evenly between the two paths. In Figure \(\PageIndex{1}\)(b), the mode is split by Y connectors, which act a bit like a scalpel, sending the left half of the mode along one track, and the right half of the mode along a different track. Thus, to obtain 50/50 splitting/combining, a Y connector must be highly symmetrical and the optical mode must also be symmetric about the splitting point.
MZIs are generally used as modulators and coarse filters. They also form an integral part of the fine filters that are used in multiplexers. From the previous section, we know that the intensity at the outport of an MZI is given by
\[\begin{equation}I_{o} = \frac{1}{4}\left(E_1E_1^* + E_2E_2^* + E_1^*E_2 + E_1E_2^*\right)\label{eqn:mziOut}\end{equation}\]
where \(I_o\) is the output intensity and \(E_1, E_2\) are the complex amplitudes of the light fields that travel by the two different paths to the combiner. At the Y combiner, the two fields are given by
\[\begin{equation} E_i = E_0\exp\left(\frac{2\pi}{\lambda_0}n_il_i\right)\label{eqn:fields}\end{equation}\]
where the subscript \(i\) denotes the path, \(\lambda_0\) is the wavelength in free space. In this formulation, the phase difference between the two paths is given by the difference in the factor \(n_il_i\). Following the same mathematical arguments as the previous section, we get the same formula for the output, but with an emphasis on the physical form and changes of the MZI
\[\begin{equation}I_o = \frac{I_i}{2}\left(1 + \cos\left(\frac{2\pi}{\lambda_0}(n_1l_1-n_2l_2)\right)\right)\label{eqn:MZIoutput}\end{equation}\]
An MZI with thermally controlled optical path length differences is often used as a highly stable but tunable filter, while an electro-optically controlled optical path length difference is more suitable for modulation, with GHz modulation frequencies are common.