Design of a Silicon based Mach-Zehnder Interferometer and Two-stage optical filter

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.

In this section a brief theoretical explanation of the operation of a Mach-Zehnder interferometer is given in terms of their gain spectrum response.

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: (Chrostowski 2015)

\(I_{0}=\frac{I_{i}}{4}\left|\exp\left(-j\beta_{1}L_{1}-\frac{\alpha_{1}}{2}L_{1}\right)+\exp\left(-j\beta_{2}L_{2}-\frac{\alpha_{2}}{2}L_{2}\right)\right|^{2}\)

Assuming no propagation loss along the waveguides, one can simplify the output intensity as:

\(I_{0}=\frac{I_{i}}{2}\left\{ 1+\cos\left(\beta_{1}L_{1}-\beta_{2}L_{2}\right)\right\} \)

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:

\(I_{0}=\frac{I_{i}}{2}\left\{ 1+\cos\left(\Delta\beta L\right)\right\} \)

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:

\(I_{0}=\frac{I_{i}}{2}\left\{ 1+\cos\left(\beta\Delta L\right)\right\} \)

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:

\(FSR=\frac{\lambda^{2}}{\Delta Ln_{g}}\)

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: (Agrawal 2005)

\(\left|H\left(\omega\right)\right|^{2}=\overset{M}{\underset{m=1}{\prod}}\cos^{2}\left(\frac{\omega\tau_{m}}{2}\right)\)

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:

\(\left|H\left(\omega\right)\right|^{2}=\cos^{4}\left(\frac{\omega\tau_{m}}{2}\right) = \frac{1}{2}[1+\cos(\omega \tau_{m})]^2 \)

\(\left|H\left(\omega\right)\right|^{2}=\frac{1}{4}[\frac{3}{2} + 2 \cos(\omega \tau_{m}) + \frac{1}{2} \cos(2 \omega \tau_{m})] \)

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:

\(\left|H\left(\lambda\right)\right|^{2}=\frac{1}{4}\left[\frac{3}{2}+2\cos\left(\beta\Delta L\right)+\frac{1}{2}\cos\left(2\beta\Delta L\right)\right]\)

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:

\(\left|H\left(\omega\right)\right|^{2} = \cos ^2 (\frac{\omega \tau_1}{2}) \cos ^2 (\frac{\omega \tau_2}{2}) = \frac{1}{4}[1 + \cos(\omega \tau_{1})] [1 + \cos(\omega \tau_{2})] \)

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:

\(\left|H\left(\lambda\right)\right|^{2}= \frac{1}{4} \left[ \cos\left(\beta\Delta L\right)^2 \times \cos\left(2\beta\Delta L\right)^2 \right]\)

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