This video is taken from the INT 100 course on Lumerical University.

## Transcript

In frequency domain simulations, every element is represented by an individual S-matrix that

describes how ingoing optical modes are transmitted into outgoing optical modes at the element

ports.

In the following, "a" and "b" represent the complex amplitudes of ingoing and outgoing

modes respectively.

The power carried in a mode is the magnitude squared of the complex amplitude.

The subscripts 1 and 2 represent ports 1 and 2.

For a simple two-port unidirectional element, supporting a single mode at either port, the

S-matrix is a single complex value, S21.

The output signal, b2 is equal to the input signal a1 multiplied by the scattering parameter

S21.

For a bidirectional two-port element, supporting a single mode at either port, the scattering

parameters create an S-matrix as shown in the diagram on the right.

The equations shown here describe how the outgoing signals are calculated from the incoming

signals by using the S-matrix

For example, b1 is the sum of the reflected signal at port 1, which is proportional to a1, and

the signal transmitted from port 2, which is proportional to a2.

Elements can support multiple ports, and each port can support an arbitrary number of modes.

The size of the S-matrix depends on the number of ports and modes.

For example, here is the S-matrix for the same unidirectional element we saw before

but now with three ports.

And here is the S-matrix when the same element supports 2 modes at port 1, 3 modes at port

2 and 2 modes at port 3.

Multiple elements with different S-matrices are connected to form circuits.

In the example shown here, matrices A, B, C and D represent the scattering of four

circuit elements.

The S-Parameter simulator implements a sparse matrix analysis to calculate the complex transmission

of the full circuit.

This analysis requires that the scattering matrix of each element in the circuit be known.

For opto-electronic devices like electro-optic modulators, this poses a challenge as the

scattering matrix depends on the driving voltage.

For this reason, INTERCONNECT performs a preliminary step during which the steady-state scattering

matrix for each element is calculated and reported back to the ONA.