Spatial correlation can be enabled in INTERCONNECT Monte Carlo analysis. Typically, circuit elements are considered as lumped elements for determining element-to-element distances and accordingly INTERCONNECT calculates element-to-element spatial correlation coefficients. For more information, please see Monte Carlo analysis with spatial correlation page.

Waveguide is one, and arguably the most important one, of the fundamental building blocks for photonic integrated circuits. Its nature of long physical path in layout makes it a distributed element. Therefore, the waveguide element needs a special approach for calculating its spatial correlation coefficients. Besides, when waveguide is a distributed element, its statistical parameters, such as \(\Delta w \) and \(\Delta h \) of the waveguide geometry, are non-uniform along waveguide path, which also require a special dealing.

This page describes how to enable waveguide as a distributed element for modeling spatial correlation effects in INTERCONNECT Monte Carlo analysis.

## Add properties to waveguide compact model

User needs to add the following properties to a waveguide compact model:

Name | Kind | Type | Description |
---|---|---|---|

distributed_model | NonQuantity | Logical | A Boolean option for enabling/disabling distributed waveguide effects. |

points | NonQuantity | String |
Parameter value is waveguide physical layout path, and the syntax is:
\([[x_1, y_1],[x_2, y_2],…,[x_n,y_n]]\)
where \([x_i, y_i]\) is the coordinates for a waveguide vertices, and the unit is micrometer. |

The parameter value of ‘points’ is typically given by a circuit netlist which is automatically generated through layout-driven simulation flow (such as Mentor integration and KLayout integration). Below is an example for a waveguide instance netlist:

WG1 net1 net2 waveguide_model wg_length=100 um, points=[[100,1000],[100,-1000],[-100, -1000]]

If ‘distributed_model’ is set to ‘false’, the element is treated as a regular lump element for modeling spatial correlation effects in INTERCONNECT Monte Carlo analysis. If ‘distributed_model’ is set to ‘true’ and the ‘points’ value exists, the element is treated as a distributed element for modeling spatial correlation effects. If 'distributed_model' is set to 'true' but 'points' value is either empty or has incorrect syntax, the element has no spatial correlation with other elements.

## Correlation coefficient for distributed waveguide

INTERCONNECT splits distributed waveguides into smaller segments, and each segment length equals 10% of the spatial correlation length. Then, it calculates spatial correlation coefficient between each waveguide segment and a target lump element, and finds the average value as the spatial correlation coefficient between the distributed waveguide and the target lump element:

$$C(P, Q)=\frac{1}{m} \sum_{j=1}^{m} C\left(P, Q_{j}\right)$$

where \(P \) is a point element; \(Q \) is a continuous element, and m is the number of segments in \(Q \). \(C(P,Q)\) is the correlation coefficient between \(P \) and \(Q \); See this reference paper for more information.

Similarly, the spatial correlation coefficient between two distributed waveguides can be calculated by:

$$C(P, Q)=\frac{1}{n m} \sum_{k=1}^{n} \sum_{j=1}^{m} C\left(P_{k}, Q_{j}\right)$$

where \(P \) and \(Q \) are both distributed element; \(n \) is the number of segments in \(P \), and \(m \) is the number of segments in \(Q \).

## Random values for the statistical parameters of distributed waveguides

Inside a distributed waveguide, INTERCONNECT applies spatial correlations to all the waveguide segments based on their coordinates at the segment centers and generates correlated random values to segments’ statistical parameters. For each statistical parameter, such as \(\Delta h \), the initial random value of the distributed waveguide is the average of random values for waveguide segments. Theoretically, this approach makes sense because perturbation of waveguide optical propagation constant, \(\beta\) , is almost linear to waveguide geometry variation, and the impact of \(\beta\) perturbation get accumulated when light propagating through all the waveguide segments:

$$E_{\text {out}}=E_{\text {in}} \sum_{i=1}^{N} e^{-j \beta_{i} \frac{L}{N}}=E_{\text {in}} e^{-j L \frac{\sum_{i=1}^{N} \beta_{i}}{N}}=E_{\text {in}} e^{-i \sum_{i=1}^{N} \beta_{\text {avg}} L}$$

where \(E_{in}\) is the electrical field amplitude at the waveguide input; \(E_{out}\) is the electrical field amplitude at the waveguide output; \(N\) is the total number of waveguide segment; \(L\) is waveguide length; \(\beta_i\) is the optical propagation constant for waveguide segment \(i\) . \(\beta_{avg}\) is the average optical propagation constant for all the waveguide segments.

Next, the initial random values of statistical parameters will be further processed by INTERCONNECT according to the correlations between the distributed waveguide and other circuit elements.

## Example

Below is an example that shows the \(\Delta h \) distribution along the path of a distributed waveguide under different spatial correlation lengths, where \(\Delta h \) is a statistical parameter of the distributed waveguide. The distributed waveguide is 200 µm. The average \(\Delta h \) for spatial correlation lengths of 20 µm and 200 µm are 0.203 nm and 0.657 nm, respectively.

Note that if waveguide length is much longer than spatial correlation length, e.g., when correlation length is 20 um in the above example, the average \(\Delta h \) approximates the true mean of the statistical parameter (i.e., 0 in this example), which is true because waveguide experiences both extremes of thickness perturbations and the impact is likely to be cancelled out.

If waveguide length is shorter than spatial correlation length, a waveguide approximates a lump element and the \(\Delta h \) for waveguide segments are uniform along the waveguide path. In this case, statistical distribution of the average \(\Delta h \) approximates the distribution of statistical parameter*.*