Broadband directional coupler¶

In the rapidly evolving field of silicon integrated photonics, the directional coupler (DC) stands out as a crucial building block for a multitude of applications, such as optical signal processing, sensing, and communication systems. As a passive device, a DC allows for the precise manipulation and distribution of light between two parallel waveguides within close proximity, enabling highly efficient and compact coupling with minimal loss. This elegant, yet simple structure capitalizes on the waveguiding properties of silicon, leveraging the evanescent coupling between the propagating modes to achieve precise control over the flow of optical power, thereby playing a pivotal role in shaping the future of photonic integrated circuits.

Conventional compact DCs often face limitations in terms of narrow bandwidth, while broadband designs tend to require a significantly larger footprint. This document explores a design for compact, broadband DCs, as proposed in Zeqin Lu, Han Yun, Yun Wang, Zhitian Chen, Fan Zhang, Nicolas A. F. Jaeger, and Lukas Chrostowski, "Broadband silicon photonic directional coupler using asymmetric-waveguide based phase control," Opt. Express 23, 3795-3808 (2015), DOI: 10.1364/OE.23.003795. The key innovation in this design, as compared to traditional DCs, lies in its incorporation of an asymmetric-waveguide-based phase control section. To demonstrate a concrete example, we will design a 2x2 DC for the TE mode, with a 50%/50% (-3 dB) splitting ratio, operating within the 1500 nm to 1600 nm range. Different polarizations and splitting ratios can be achieved in a similar design process.

Initially, we employ the transfer matrix method (TMM) to model the DC in a semi-analytical manner. TMM necessitates the calculation of effective indices for various waveguide configurations, for which we utilize Tidy3D's waveguide plugin, as it offers a convenient means of performing mode analysis. TMM provides a computationally efficient and accurate preliminary estimation of the ideal design parameters, which are then further optimized using rigorous 3D FDTD simulations.

Transfer Matrix Method (TMM) Analysis¶

The broadband DC primarily consists of a symmetric coupler, an asymmetric phase control section, and another symmetric coupler. The three sections are connected via linear tapers. The inputs and outputs are from 90 degree circular bends.

We can use TMM to analyze this system. Assuming the input and output fields on the waveguide are $E_1$, $E_2$, $E_3$, and $E_4$, each section of the DC can be represented by a matrix. That is, we can write down the following equation that connects the input and output fields:

$\begin{bmatrix}E_3\\E_4\end{bmatrix} = C \cdot P_t \cdot P \cdot P_t \cdot C \cdot \begin{bmatrix} E_1\\E_2 \end{bmatrix}$.

$C =\begin{bmatrix}t & -jk\\-jk & t\end{bmatrix} e^{-j\frac{\pi}{\lambda}(n_++n_-)L_1}$ represents the symmetric coupler. $t$ and $k$ are the transmission and coupling coefficients given by $t = cos(\frac{\pi\Delta n_{eff}}{\lambda}L_1) $ and $k = sin(\frac{\pi\Delta n_{eff}}{\lambda}L_1) $, where $\Delta n_{eff}=n_+-n_-$. $n_+$ and $n_-$ are the effective indices of the symmetric and anti-symmetric modes, which we will determine from mode analysis using the waveguide plugin.

$P = \begin{bmatrix}e^{-j\frac{2\pi n_1}{\lambda}L_2} & 0\\0 & e^{-j\frac{2\pi n_2}{\lambda}L_2}\end{bmatrix}$ represents the asymmetric phase control section. Here $n_1$ and $n_2$ are the effective indices of the lowest order modes, which correspond to modes confined in the top waveguide and bottom waveguide, respectively. We will determine $n_1$ and $n_2$ from mode analysis too.

Lastly, $P_t = \begin{bmatrix}e^{-j\theta_{t1}} & 0\\0 & e^{-j\theta_{t2}}\end{bmatrix}$ approximately represents the linear tapers. $\theta_{t1}$ and $\theta_{t2}$ are the phase shifts in the top and bottom tapers, which we will determine numerically from quick FDTD simulations.

In the reference, a small propagation loss of 2.7 dB/cm is also considered. Since this is a small factor, we choose to ignore it in the analysis without a significant loss of accuracy.

After we perform the TMM calculation, we can determine the power splitting ratio by $\eta_{cross} = \frac{|E_4|^2}{|E_3|^2+|E_4|^2}$ and $\eta_{through} = \frac{|E_3|^2}{|E_3|^2+|E_4|^2}$, assuming the input field is $\begin{bmatrix}E_1\\0\end{bmatrix}$.

We are interested in the wavelength range of 1500 nm to 1600 nm.

The silicon layer is 220 nm. The waveguide width and gap size in the symmetric coupler are 500 nm and 200 nm, respectively.

To define the media, we directly use the dispersive medium model of silicon and oxide from Tidy3D's material library for convenience.

Now we use the waveguide plugin to perform the mode analysis on the symmetric coupler. Alternatively, one can use the ModeSolver plugin to do the same analysis. The waveguide plugin just provides a fast and convenient way of setting up the mode analysis for dielectric waveguides.

The effective indices of different modes can be directly calculated. The corresponding mode profiles can be visualized. Here we focus on the two lowest order modes, which are the symmetric and anti-symmetric modes.

After obtaining $n_+$ and $n_-$, we will calculate $n_1$ and $n_2$ in a similar fashion for the phase control section, where the waveguide widths are 600 nm and 400 nm. The gap is 300 nm.

To perform the TMM analysis, we still need to obtain $\theta_{t1}$ and $\theta_{t2}$. This can be done by setting up FDTD simulations for the tapers. To get the phase shift, we place one ModeMonitor right before the taper and another ModeMonitor right after the taper. The phase difference calculated from the mode amplitudes is the phase shift we are looking for. Since the tapers are only 1 $\mu m$ in length, the FDTD simulations are very fast.

First, let's do the top taper that transitions from a 500 nm waveguide to a 600 nm waveguide.

Compute and plot $\theta_{t1}$. As expected, the phase shift is linear. Note that the phase is only significant mod 2$\pi$.

Similarly, we simulate the taper that transitions from a 500 nm waveguide to a 300 nm waveguide.

Compute and plot $\theta_{t2}$.

Now that we have calculated $n_+$, $n_-$, $n_1$, $n_2$, $\theta_{t1}$, and $\theta_{t2}$, we are ready to assemble the matrices and calculate $\eta_{cross}$. First, we focus on the wavelength of 1550 nm and explore $\eta_{cross}$ as a function of $L_1$ and $L_2$.

Plot $\eta_{cross}$ as a function of $L_1$ and $L_2$. Since we are looking for a specific power splitting ratio, we add contours to the plot. For a desirable splitting ratio, one only needs to pick design parameters on the specific contour.

In this case, we aim for a 50%/50% splitting ratio. Since we want the DC to work in a wide wavelength range, we also need to calculate $\eta_{cross}$ at other wavelengths to make sure the same splitting ratio is maintained. Here we skip explicitly showing this step to avoid making the notebook excessively long.

From this analysis, the optimal $L_1$ and $L_2$ are found to be 12.8 $\mu m$ and 4.4 $\mu m$. Now we fix $L_1$ and $L_2$ and calculate $\eta_{cross}$ as a function of wavelength.

Plot $\eta_{cross}$ as a function of wavelength. Here we can see if we define +- 1 dB as the bandwidth, the bandwidth is larger then 100 nm.