## Introduction

The S-Parameter Simulator (SPS) solver is a frequency domain simulator that calculates the overall frequency domain response of a circuit. This is done by solving a sparse matrix that represents the circuit as connected scattering matrices, each one of them representing the frequency response of an individual element. Sophisticated analysis and visualization tools allow for the analysis of circuits behavior, including amplitude, phase, group delay and dispersion.

## Solver physics

### Overview and Problem Formulation

This section will introduce the basic mathematical and physics formalism behind the SPS algorithm. The SPS algorithm is modeled on the same type of scattering analysis used in the high-frequency electrical domain for solving microwave circuits, thus enabling bidirectional signals to be accurately simulated [1]. However, it has been extended to allow for an arbitrary number of modes supported in the waveguide elements along with the possible coupling between those modes that can occur in any element.

In INTERCONNECT, a schematic diagram is used to describe the system to be simulated. The schematic diagram is comprised of a number of elements, each having a certain number of ports, as well as connections linking various ports on different elements. These connections can pass signals from one port to another in either or both directions. In the case of connections linking electrical ports, each signal can only contain a single mode that represents the Fourier transform of a real, scalar, time-varying quantity, which is typically a voltage or current. However, in the case of connections linking optical ports, the signals can be comprised of multiple modes, where each mode represents the Fourier transform of the time-varying complex envelope of a particular optical mode. Owing to this last case, it is thus possible for more than one mode to be passed along by a single connection between ports, since optical ports, themselves, may transmit more than one optical mode (different polarizations and/or different waveguiding orders).

Figure 1 shows an example of an INTERCONNECT schematic diagram that has been annotated with arrows depicting the multiple optical modes (two, in this example) at the bidirectional inputs and outputs of each element. Red indicates an outgoing (or reflected) mode, propagating out the element. Blue indicates an incoming (or incident) mode, propagating into the element. The grey lines between modes on connected elements, which are also annotations, highlight the equality between a particular outgoing mode (red arrow) on the port of one element and a particular incoming mode (blue arrow) on a connected port on another element. This equality is implied by the green connection between ports in the INTERCONNECT schematic, itself.

A complete solution to the problem is comprised of knowing the spectra of each of the modes traveling in both directions on each of the connections in the schematic diagram. Note, therefore, that there is one such spectrum for each grey line in Figure 1.

### Connection Matrix

The Connection Matrix is extracted from the schematic diagram and characterizes the relationships (equality or general non-equality) between the various modes flowing into and out of different elements based on the connections made between ports on the various elements in the schematic diagram. It is a \(P \times Q\) matrix, where \(P\) and \(Q\) are the total number of outgoing and incoming modes, respectively, on all the elements in the schematic diagram. It is essentially a matrix representation of the relationships depicted by the grey lines in Figure 1.

### Scattering Matrices (also called S-Parameters or S-Matrices)

Each element in INTERCONNECT can be viewed as an N-port network where each port receives an incoming mode, \(a_j\), and scatters or reflects an outgoing signal, \(b_j\), as shown in Figure 2. In the case of electrical signals, the \(a_j\) and \(b_j\) typically represent a voltage or current, but they can also sometimes be used to represent other real, scalar quantities of interest such as temperature. In the case of optical signals, the \(a_j\) and \(b_j\), represent the complex envelope of a particular optical mode.

The relationship between \(b_j\) and \(a_i\) at each frequency, \(f\), is knows as the Scattering Matrix, \(S(f)\), and is defined by Equation 1:

$$\begin{bmatrix}b_{1}(f) \\ b_{2}(f) \\ \vdots \\b_{N}(f) \end{bmatrix} = \begin{bmatrix} S_{11}(f) & S_{12}(f) & \cdots & S_{1M}(f) \\ S_{21}(f) & S_{22}(f) & \cdots & S_{2M}(f) \\ \vdots & \vdots & \ddots & \vdots\\ S_{N1}(f) & S_{N2}(f) & \cdots & S_{NM}(f) \end{bmatrix} \begin{bmatrix} a_{1}(f) \\ a_{2}(f) \\ \vdots \\ \vdots\\a_{M}(f) \end{bmatrix}$$

Equation 1

where, \(M\) and \(N\) are the number of incoming and outgoing signals, respectively.

Each INTERCONNECT element has its own embedded algorithm based on a mathematical model that mimics the behavior of the physical or logical component it represents. It is important to note that when the INTERCONNECT element has optical ports, the total number of incoming and reflected signals, \(N\), will not simply equal twice the total number of ports in the INTERCONNECT element, \(N_P\). Therefore, the Scattering Matrix for the element may be larger than \(2N_P \times 2N_P\). This is because each optical port may have more than one signal, owing to the fact that there may be multiple optical modes traveling through each of the optical ports.

### Solution Method

The SPS calculates the solution using a method called Scattering Analysis based on the combination of Scattering Matrices and a Connection Matrix. As previously discussed, the Scattering Matrix for a particular element characterizes the relationships between each incoming and outgoing mode for that particular element at a particular frequency. The SPS first performs a calculation in which these Scattering Matrices are evaluated for each of the elements in the schematic diagram. At each frequency, the individual Scattering Matrices are combined with the Connection Matrix to construct a set of linear equations for the spectral values of each of the modes traveling in both directions on each of the connections in the schematic diagram. As mentioned above, for the example shown in Figure 1, there is one such value to be solved for (at each frequency) per grey line in the figure. The set of linear equations is then solved iteratively at each frequency.

## Video resources

INTERCONNECT introductory webinar

## References

[1] D. M. Pozar, Microwave Engineering, Third edition. John Wiley & Sons (2004).