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 3View Image becomes
View Image

Figure 6:

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

View Image

Figure 7:
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
( 1 0 0 0 ) ( a ) ( a ) | | | 1 | | 1| | − it 1 0 − r | | a2 | = | 0 | = Msystem ⃗asol = ⃗ainput, (9 ) ( 0 0 1 0 ) ( a3 ) ( a3) − r 0 − it 1 a4 0
where the input vector1 ⃗a input has non-zero values for the impinging fields and ⃗a sol 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 5View Image becomes

( ) ( ) ( ) 1 0 0 0 0 0 0 a0 a0 | − it 1 0 − r 0 0 0 | | a | | 0 | || 1 1 || || 1|| || || | − r1 0 1 − it1 0 0 0−ikD | | a4′| | 0 | || 0 0 0 1 0 0 − e || || a3|| = || 0 || (10 ) || 0 − e−ikD 0 0 1 0 0 || || a′1|| || 0 || ( 0 0 0 0 − it2 1 0 ) ( a2) ( 0 ) 0 0 0 0 0 − r 1 a 0 2 3
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 [2840] and has been demonstrated for analysing gravitational wave detectors [43], is to rearrange equations as in Figure 6View Image and Figure 7View Image 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
( a ) i( − 1 r ) ( a ) 1 = - 2 2 2 . (11 ) a4 t − r r + t a3
In the special case of the lossless mirror this matrix simplifies as we have r2 + t2 = R + T = 1. The space component would be described by the following matrix:
( a ) ( exp(ikD ) 0 ) ( a ) 1 = 2 . (12 ) a4 0 exp (− ikD ) a3
With these matrices we can very easily compute a matrix for the cavity with two lossless mirrors as
M = M × M × M (13 ) cav mir(ror1 space mirror2 ) -− 1 e+ − r1r2e− − r2e+ + r1e− = t1t2 − r2e− + r1e+ e− − r1r2e+ , (14 )
with + e = exp(ikD ) and − e = exp (− ikD ). The system of equation describing a cavity shown in Equation (4View Equation) can now be written more compactly as
( ) ( ) ( ) a0 − 1 e+ − r1r2e− − r2e+ + r1e− a2 a = ---- − r e− + r e+ e− − r r e+ 0 . (15 ) 4 t1t2 2 1 1 2
This allows direct computation of the amplitude of the transmitted field resulting in
a = a --−-t1t2-exp(−-ikD-)--, (16 ) 2 01 − r1r2exp (− i2kD )
which is the same as Equation (8View Equation).

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.


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