Circuits as matrices

The circuits of this series are linear: the complex amplitude of the light at each output is a weighted sum of the amplitudes at the inputs, with weights that do not depend on how much light enters. A circuit with two inputs and two outputs is therefore described completely by a $2 \times 2$ matrix of complex numbers, and a circuit with $N$ inputs and outputs by an $N \times N$ matrix. This article builds these matrices from the components of the previous articles and shows how connecting components multiplies them; the lab computes every circuit in this way.

A component as a matrix

Write the complex amplitudes at a component’s inputs as a vector $\mathbf{a}$ and those at its outputs as $\mathbf{b}$. For a linear component,

The matrix $T$ is the component’s transfer matrix. Its entry $T_{ij}$ is the amplitude that arrives at output $i$ for light of amplitude 1 at input $j$: rows belong to outputs, columns to inputs. Each entry is an amplitude, not a power. The fraction of the power that goes from input $j$ to output $i$ is $|T_{ij}|^2$, and the powers of different inputs cannot be added in this way when the inputs are coherent: the amplitudes add first, as in the article §Interference.

The components of the series have simple matrices. A waveguide of length $L$ multiplies the amplitude by $e^{i\beta L}$, a phase shifter by $e^{i\Delta\varphi}$; two waveguides side by side form a diagonal matrix. A directional coupler with power coupling $\kappa^2$ mixes two amplitudes (§Splitters and combiners):

Connecting components

When the outputs of one component feed the inputs of the next, the amplitudes leaving the first are the amplitudes entering the second. A chain of components $T_1$, then $T_2$, then $T_3$ therefore has the matrix

a cascade, with the first component on the right, since it acts on the input vector first. The order matters: matrix products do not commute. A phase shifter placed before a coupler changes the phase relation of the light it mixes; the same phase shifter placed after the coupler only delays one of its outputs.

The MZI of the previous article, §The Mach-Zehnder interferometer, is the cascade of a coupler, two arms and a coupler, $S = C D C$. Figure 1 shows the product with numbers. Choosing an entry of $S$ shows how it arises.

Each entry of a product is a sum over the paths through the circuit: $S_{ij} = \sum_k C_{ik} D_{kk} C_{kj}$, one term for each arm $k$ the light can take from input $j$ to output $i$. The matrix product carries out, for every pair of input and output at once, the adding of phasors that the article §Interference introduced. At $\Delta\varphi = 0$, the two terms of $S_{00}$ are $1/2$ and $-1/2$: the two paths to the bar output cancel.

Lossless means unitary

A lossless circuit keeps the total power: for every input vector, $\sum_i |b_i|^2 = \sum_j |a_j|^2$. Written with the conjugate transpose $T^\dagger$, the total output power is $\mathbf{a}^\dagger T^\dagger T \mathbf{a}$, so the condition holds for all inputs exactly when

A matrix with this property is unitary. Its columns are vectors of length 1, since each input’s light leaves somewhere, and they are orthogonal to each other. The coupler of Eq. (2) is unitary for every $\kappa$, and so are the phase matrices; products of unitary matrices are unitary, so every circuit built from lossless parts is. A circuit that loses light satisfies the weaker condition that $T^\dagger T$ is at most $I$: no input vector gains power [1]Chrostowski & Hochberg 2015, Eq. (9.3), p. 324.

The scattering matrix

Real circuits are not always a simple chain. Light can travel in both directions in a waveguide, it can return to where it came from in a loop such as a ring resonator, and a component’s ports are not fixed as inputs or outputs. The general description is the scattering matrix (S-matrix). It lists every port of a component and relates the amplitudes of the waves entering all ports to those of the waves leaving them. For a directional coupler it is a $4 \times 4$ matrix in which the $2 \times 2$ transfer matrix appears twice, once for each direction.

For passive components made of ordinary materials, the S-matrix is symmetric, $S = S^\mathsf{T}$: light from port $j$ reaches port $i$ with the same amplitude as light from $i$ reaches $j$. This property is reciprocity; only components with magnetic materials or magnetic fields, such as optical isolators, break it [1]Chrostowski & Hochberg 2015, §2.2.4, p. 38.

The lab builds a circuit from the S-matrices of its components and the list of their connections. From these it computes the S-matrix of the whole circuit, including the light that circulates in loops, at one wavelength at a time. Every power shown in the lab is $|S_{ij}|^2$ of that matrix, times the power of the laser.

Matrices from meshes of MZIs

A single MZI with a phase shifter at one input sets both the splitting ratio and a phase: it can produce any $2 \times 2$ unitary matrix up to phases at its outputs. Meshes of such elements go further. Any unitary matrix of size $N \times N$ can be built from beam splitters and phase shifters [2]Reck et al. 1994, abstract, and a rectangular mesh does so with $N(N - 1)/2$ MZIs, arranged so that light passes through about half as many of them as in earlier designs [3]Clements et al. 2016, Fig. 1, p. 2; pp. 3–4. Programmable photonic circuits use such meshes [4]Bogaerts et al. 2020, Fig. 1, p. 208, and such a mesh has carried out the matrix multiplications of a small neural network [5]Shen et al. 2017, Fig. 1. The lab’s Clements mesh examples are meshes of this kind.

Approximations used on this page: APX-001lossless components by defaultWaveguides, couplers and phase shifters lose no light unless a loss is set. lossless components by default, APX-004wavelength-independent coupling ratioA coupler splits the light in the same ratio at every wavelength. wavelength-independent coupling ratio.