Note: this notebook requires Tidy3D version 2.10.0rc1 or later.

The co-planar waveguide (CPW) is a transmission line commonly used in RF photonics due to its compact planar nature, ease of fabrication, and resistance to external EM interference. In this notebook series, we will examine the CPW in the context of a Mach-Zehnder modulator (MZM).

In part one (this notebook), we will start with a simple CPW on a dielectric substrate. The goal is to demonstrate the typical 2D and 3D workflow in Tidy3D's RF solver. We will also benchmark key results against other commercial RF simulation software.

In part two, we will simulate the CPW with the MZM optical waveguide. A 2D mode analysis will first be performed on the conventional layout, followed by a 3D simulation of segmented electrode design.

Building the simulation¶

Key parameters¶

We start by defining some key parameters.

The CPW dimensions are defined below.

Material and Geometry¶

The substrate is assumed to be lossless. The metal is assumed to have constant conductivity over the frequency range.

The CPW geometry is created below. The substrate is assumed to occupy the lower half-space of the simulation domain.

Grid and Boundary¶

The simulation domain is surrounded by Perfectly Matched Layers (PMLs) by default. We add quarter-wavelength padding on all sides of the simulation to ensure that the external boundaries do not encroach on the near-field.

The simulation grid is automatically generated based on the minimum wavelength in the respective dielectric medium. In RF problems, the field is especially enhanced near metallic structures. For more accurate results, we use the LayerRefinementSpec feature to automatically refine the grid near metallic edges and corners.

To learn more about the LayerRefinementSpec feature, please refer to the API documentation page or the tutorial page.

Monitors¶

We define a field monitor for visualization purposes below.

Wave Ports¶

Wave ports are used to excite the CPW. It is important to ensure that the width of the ports are slightly smaller than the overall width spanned by the CPW side ground traces. We also define voltage and current integrals used to calculate the port impedance.

Define Simulation and TerminalComponentModeler¶

The base Simulation object contains information about the overall simulation environment. The TerminalComponentModeler object is used in 3D simulations to conduct a port and frequency sweep in order to obtain the full S-parameter matrix.

Visualization¶

We can inspect the grid to ensure a suitable level of refinement. The plots below show the full CPW (left) and the CPW gap (right).

2D Analysis¶

In this section, we will perform a 2D mode analysis on the CPW and extract key transmission line parameters.

Run Mode Solver¶

The 2D mode analysis is performed with a ModeSolver object. We can easily create one from the wave port using the to_mode_solver() method.

We run the mode solver below.

Mode Profile¶

First, let us examine the mode profile. The left plot depicts $\text{Re}(E_x)$ while the right plot shows the field magnitude in dB. (The divide-by-zero warnings are due to the field magnitude being zero in the metallic regions.)

Effective Index, Loss, and Line Impedance¶

The key properties that determine a guided mode are effective index $n_\text{eff}$, attenuation $\alpha$, and characteristic impedance $Z_0$. We demonstrate how to obtain those values below.

The first two quantities, $n_\text{eff}$ and $\alpha$, are computed directly by the mode solver. Below, we extract these values from the solution data and also use them to compute the complex propagation constant $\gamma = \beta + j\alpha$.

The characteristic line impedance $Z_0$ is calculated using the ImpedanceCalculator utility class. Note the use of np.conjugate() to convert from the physics phase convention (used in Tidy3D) to the standard electrical engineering phase convention.

The voltage and current integration paths are visualized below.

For comparison purposes, we import benchmark data obtained from other commercial RF simulation software.

The three quantities are plotted below. The results compare very well with the benchmark data.

Distributed RLCG transmission line parameters¶

Using the complex propagation constant $\gamma$ and the line impedance $Z_0$, we can calculate the RLCG parameters using the following relationships: $$ R = \text{Re}(\gamma Z_0), \quad L = \text{Im}(\gamma Z_0)/\omega, \quad G = \text{Re}(\gamma/Z_0), \quad C = \text{Im}(\gamma/Z_0)/\omega $$ where $\omega = 2\pi f$.

The RLCG parameters are shown below. Note that since the dielectric is lossless, $G$ is actually zero and the plotted values are due to numerical noise.

3D Analysis¶

Run Simulation¶

Let us simulate a short section of the CPW to demonstrate the 3D workflow. First, the grid resolution can be reduced from the 2D case. This is because S-parameters generally converge much more quickly than modal parameters. Below, we re-define the grid specification for the 3D case.

We make an updated copy of the base Simulation and TerminalComponentModeler using the reduced grid specification.

The simulation is executed below.

S-parameters¶

The S-matrix can be calculated using the smatrix() method from the TerminalComponentModelerData instance tcm_data returned by web.run().

Below, we define some convenience functions to extract the individual $S_{ij}$ entries. The port_in and port_out coordinates are used to specify the port name/index. Note the use of np.conjugate() to convert the S-parameter from the physics phase convention (Tidy3D default) to the standard electrical engineering convention.

We import some benchmark data for comparison below.

The insertion loss magnitude and angle are compared below. They agree very well.

Since this is a straight transmission line with no longitudinal variation, the return loss should theoretically be zero. Below, we plot the return loss magnitude in dB. The RF solver is able to achieve a very low level of numerical reflection. The propagating mode is very well matched to the modal absorbing boundary present in each wave port.

Field profile¶

To access the field monitor data, we need to first load the simulation dataset from the TerminalComponentModelerData instance. The simulation data is stored in a dictionary indexed by keys in the format <port name>@<mode number>. For example, to obtain data for the first (and only) excited mode in wave port 1, we use WP1@0.

Below, we plot the field magnitude in dB just below the CPW metal plane. (The divide-by-zero error is due to zero field magnitude in some metallic regions.)

Conclusion¶

In this notebook, we used the basic CPW as an example to demonstrate the typical Tidy3D RF workflow for modeling transmission lines. We calculated and benchmarked key quantities such as effective index, attenuation, characteristic impedance, and S-parameters.

In the next notebook, we will model the CPW in the context of a Mach-Zehnder modulator.