The plane-wave approximation

In many simulations the shape of the light beams or, in general, the geometric properties of a beam transverse to the optical axis are not of interest. In that case one can discard this information and restrict the model to the field on the optical axis. This is equivalent to a model where all light fields are plane waves traveling along one optical axis. This is the default mode of Finesse and is called the plane-wave approximation in the following.

In the plane-wave approximation all light fields are described in one dimension. All beams and optical components are assumed to be centered on the optical axis and of infinite size. Using plane waves, it is very simple to compute interferometer signals depending on the phase and frequency of the light, and of the longitudinal degrees of freedom. Furthermore, it can be easily extended to include other degrees of freedom, such as polarisation or transverse beam shapes.

This section introduces the plane-wave approximation as used in Finesse by default.

Description of light fields

A laser beam is usually described by the electric component of its electromagnetic field:

\[\vec{E}(t,\vec{x})\,=\,\vec{E}_0 \cos\left(\w\T - \vec{k} \cdot \vec{x}\right). \]

In the following calculations, only the scalar expression for a fixed point in space is used. The calculations can be simplified by using the full complex expression instead of the cosine:

\[E(t) = E_0 \exp\left(i(\omega t + \varphi)\right) = a \exp(i \omega t), \]

where \(a = E_0 \exp(i\varphi)\). The real field at that point in space can then be calculated as:

\[\vec{E}(t) = \text{Re}\left\{E(t)\right\} \cdot \vec{e}_{\rm pol} \]

with \(\vec{e}_{\rm pol}\) as the unit vector in the direction of polarisation. Note that Finesse currently does not model polarization effects.

Each light field is then described by the complex amplitude \(a\) and the angular frequency \(\w\). Instead of \(\w\), also the frequency \(f=\w/2\pi\) or the wavelength \(\lambda=2\pi c/\w\) can be used to specify the light field. It is convenient to define one default (laser) frequency \(f_0\), which in Finesse is done using the lambda command. The frequency of other light fields is then defined by an offset \(\Delta f\) to that default frequency (see e.g. the f parameter of the laser object). In the following, some functions and coefficients are defined using \(f_0\), \(\w_0\), or \(\lambda_0\) referring to the previously defined default frequency. The setting of the default frequency is arbitrary, it merely defines a reference for frequency offsets and does not influence the results.

The electric component of electromagnetic radiation is given in Volt per meter. From this, we can calculate the intensity \(I\) in \(\mathrm{Watt}/\mathrm{m}^2\) as the time-average of the Poynting vector \(S\)

\[I = \left| \bar{S} \right| = \frac{\epsilon_0 c}{2} ~E E^*, \]

with \(\epsilon_0\) the electric permeability of vacuum and \(c\) the speed of light.

In the plane-wave approximation we use the intensity of the field on axis as a stand-in or reference for the total power of the beam (remember, we are not modelling real plane waves, instead the plane-waves approximation just computes the field amplitudes on the optical axis as a model for a real beam). For convenience, unless noted otherwise, we re-define the electric field in units of \(\sqrt{\mathrm{Watt}}\). By definition, the total power of a beam \(P\) in Watt is then simply calculated as

\[P=E E^*. \]