### 2.3 Coupling matrices

Computations that involve sets of linear equations as shown in Section 2.2 can often be
done or written efficiently with matrices. Two methods of applying matrices to coupling field
amplitudes are demonstrated below, using again the example of a two mirror cavity. First of all, we
can rewrite the coupling equations in matrix form. The mirror coupling as given in Figure 3
becomes

and the amplitude coupling at a ‘space’, as given in Figure 4, can be written as

In these examples the matrix simply transforms the ‘known’ impinging amplitudes into the ‘unknown’
outgoing amplitudes.
#### Coupling matrices for numerical computations

An obvious application of the matrices introduced above
would be to construct a large matrix for an extended optical system appropriate for computerisation. A
very flexible method is to setup one equation for each field amplitude. The set of linear equations for a
mirror would expand to
where the input vector
has non-zero values for the impinging fields and is the ‘solution’ vector, i.e., after solving the
system of equations the amplitudes of the impinging as well as those of the outgoing fields are stored in that
vector.
As an example we apply this method to the two mirror cavity. The system matrix for the optical setup
shown in Figure 5 becomes

This is a sparse matrix. Sparse matrices are an important subclass of linear algebra problems and many
efficient numerical algorithms for solving sparse matrices are freely available (see, for example, [13]). The
advantage of this method of constructing a single matrix for an entire optical system is the direct access to
all field amplitudes. It also stores each coupling coefficient in one or more dedicated matrix elements, so
that numerical values for each parameter can be read out or changed after the matrix has been constructed
and, for example, stored in computer memory. The obvious disadvantage is that the size of the matrix
quickly grows with the number of optical elements (and with the degrees of freedom of the system, see, for
example, Section 7).

#### Coupling matrices for a compact system descriptions

The following method is probably most useful for
analytic computations, or for optimisation aspects of a numerical computation. The idea behind the scheme,
which is used for computing the characteristics of dielectric coatings [28, 40] and has been demonstrated for
analysing gravitational wave detectors [43], is to rearrange equations as in Figure 6 and Figure 7 such that
the overall matrix describing a series of components can be obtained by multiplication of the component
matrices. In order to achieve this, the coupling equations have to be re-ordered so that the input vector
consists of two field amplitudes at one side of the component. For the mirror, this gives a coupling matrix of
In the special case of the lossless mirror this matrix simplifies as we have . The space
component would be described by the following matrix:
With these matrices we can very easily compute a matrix for the cavity with two lossless mirrors as
with and . The system of equation describing a cavity shown in
Equation (4) can now be written more compactly as
This allows direct computation of the amplitude of the transmitted field resulting in
which is the same as Equation (8).
The advantage of this matrix method is that it allows compact storage of any series of mirrors and
propagations, and potentially other optical elements, in a single 2 × 2 matrix. The disadvantage inherent
in this scheme is the lack of information about the field amplitudes inside the group of optical
elements.