Fabricated silicon photonic device performance often varies from their designed-for performance. This is mainly due to variations in the fabrication process in foundries, as well as SOI chip variations. As a photonics designer, it is important to understand how these variations will affect the expected output. Statistical simulations provide insight to these variations. In this example, we demonstrate INTERCONNECT’s capability to run simulations based on statistical models, including Monte Carlo (MC) and Corner analysis. This is demonstrated on a PAM4 transceiver circuit. For the MC analysis, the signal extinction ratio (ER) will be studied and for the Corner analysis, the difference of the eye diagrams between the corners will be studied. This example includes spatial correlations between the elements’ statistical parameters (see the Important Model Settings section).

## Overview

Understand the simulation workflow and key results

We are simulating a PAM4 transceiver using a CML that includes statistical models for the MMIs, the PN phase shifters and the photodetector. The figures of merit for this circuit are the PAM4 signal levels and the extinction ratio. The models include statistical variations and manufacturing corners. We perform Monte Carlo and Corner analyses to obtain an understanding of the impact of manufacturing variability on the figures of merit.

**Note:** It is not necessary to run the Monte Carlo analysis prior to the Corner analysis. They are independent calculations.

## Run and Results

Instructions for running the model and discussion of key results

Prior to running any statistical simulations, it is important to have the test circuit setup and functioning properly. It is also necessary to install the CML library:

- For this example, install the “PAM4_library.cml” into the design kits. See
**Install CML**for instructions. - Reload the file“PAM4_transmitter_statistical.icp” once the CML is installed.
- To verify the CML is installed properly, run the file and view the results from the eye diagram. The results produced here are for the nominal values.
- Before running the Monte Carlo analysis and Corner analysis, edit the Root Element and add the following results to the
**Result**tab by clicking on the**Add**button:- ER
- Level_1
- Level_2
- Level_3
- Level_4

Following is the nominal eye diagram:

and here is the results setup in the Root Element:

### Step 1: Monte Carlo Analysis

- In the
**Optimizations and Sweeps**tab, right click on**Monte Carlo analysis**and select**Edit**. - In the
**Libraries**tab, click**Add**. Then under**Library**, double click on**<select library...>**and select the "PAM4_library.lib" file contained within the CML. Under**Variant**, choose**statistical**. - Under the
**Correlations**tab, make sure the**Enable spatial correlations**setting is selected. - Click
**OK**to exit this window. At the top of the**Optimizations and Sweeps**windows hit RUN. This will take several minutes to complete. - The raw data can be access in the
**Results View**tab once the “Monte Carlo analysis” sweep object is selected. Go to the entry “Result/ histogram” and plot the PAM4 signal levels and the extinction ratio results.

The following plot displays the histogram of the mean values from each level of the eye diagram, as evaluated by the vector signal analyzer (VSA). We can see that each level approximates a gaussian distribution, with the tails slightly overlapping. We can define a yield standard for the signal levels and then using the histogram, we can calculate the yield rate for the statistical models. We can also estimate the probability density function (pdf) of the signal levels based on the histogram, and estimate the signal overlapping for each level to calculate the symbol error rate (SER) and dispersion eye closure penalty quaternary (TDECQ).

The extinction ratio is the ratio of power from one level to the next, calculated in dB as:

$$

\mathrm{ER}(\mathrm{dB})=10 \log _{10}\left(\mathrm{P}_{4} / \mathrm{P}_{0}\right)

$$

The following plot displays the extinction ratio between the signal level 4 and level 1:

Ideally we want the ER to be high to give us an open eye.

### Step 2: Corner Analysis

- In the
**Optimizations and Sweeps**tab right click on the "Corner sweep" and select**Edit**. - In the
**Libraries**tab, choose**Add**. Then under**Library**, double click on the entry and the .lib file contained in the used CML will be automatically selected. In the**Corners**tab, check the “corner_1”, “corner_2” and “nominal” corners. - Click
**OK**to exit this window. At the top of the**Optimizations and Sweeps**window hit RUN. - The raw data can be access in the
**Results View**tab once the Corner Analysis sweep object is selected. Use the script “PAM4_corner_eye.lsf” to plot the eye diagram for the 3 simulations.

The nominal result is the expected result for the design and the corner_1 and corner_2 corner results estimate the best and worst cases of the result given the model statistical variant.

## Important Model Settings

Description of important objects and settings used in this model

### CML

This example requires the PAM4_library Compact Model Library (CML) to be installed before running the example files. The CML file “PAM4_library.cml” is provided with this example. See ** Install CML ** for detailed instructions. The statistical variation of the models is defined in the LIB file “PAM4_library.lib” which is associated with the CML.

### Correlation

In Monte Carlo analysis, parameters can be correlated or uncorrelated. The correlation coefficients of the statistical parameters can be specified directly in the Monte Carlo analysis edit dialog window, under the **Correlations** tab.

It is also possible to specify spatial correlations between the elements’ statistical parameters in the LIB file by placing the parameters in correlation groups. The correlation coefficient between the parameters is then determined from the positions of the elements and the correlation length of the parameters’ correlation group. The correlation lengths can be seen in the **Value** column of the **Correlations** tab in the **Monte Carlo analysis edit dialog** window. In this example, the phase shifter parameters are spatially correlated, with a correlation length of 2 cm.

### Variant

The statistical variants for the statistical models are defined in the LIB.X file. There are three types of variant defined in the file, namely the "nominal" variant, the "corner" variant and the "statistical" variant. When the file is loaded to the MC or Corner analysis object, the variant properties defined in the LIB.X file will automatically be imported. The "nominal" option is for the nominal values; the "corner" option is for the corner values and these two options are for the Corner analysis. The "statistical" option is for the model statistical values and it is for the MC analysis.

### Results

Results of interest (PAM4 signal levels and the extinction ratio) are defined in the Root Element using the Analysis script.

## Updating the Model With Your Parameters

Instructions for updating the model based on your device parameters

This example is based on a PAM4 transceiver with the "PAM4_library.cml" CML, and users can apply the methodology described in this example to any circuit of interest, assuming it is based on a CML that contains statistical models.

## Additional Resources

Additional documentation, examples and training material