### 3.7 Coupling matrices for beams with multiple frequency components

The coupling between electromagnetic fields at optical components introduced in Section 2 referred only
to the amplitude and phase of a simplified monochromatic field, ignoring all the other parameters of the
electric field of the beam given in Equation (1). However, this mathematical concept can be extended to
include other parameters provided that we can find a way to describe the total electric field as a sum of
components, each of which is characterised by a discrete value of the related parameters. In the case of the
frequency of the light field, this means we have to describe the field as a sum of monochromatic
components. In the previous sections we have shown how this could be done in the special case of an initial
monochromatic field that is subject to modulation: if the modulation index is small enough
we can limit the amount of frequency components that we need to consider. In many cases it
is actually sufficient to describe a modulation only by the interaction of the carrier at
(the unmodulated field) and two sidebands with a frequency offset of to the carrier.
A beam given by the sum of three such components can be described by a complex vector:
with , and . In the case of a phase modulator that applies a
modulation of small modulation index to an incoming light field , we can describe the coupling of
the frequency component as follows:
which can be written in matrix form:
And similarly, we can write the complete coupling matrix for the modulator component, for example, as