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 standard 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 often convenient to define one default (laser) frequency \(f_0\), which can be done in Finesse using the lambda command. The frequency of other light fields can then be 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 a 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 practice, we are never dealing with true plane waves that have a constant intensity over an infinite space, but with beams that are limited in size by e.g. the aperture of a laser or a mirror. The total power is then obtained by integrating the (position dependent) intensity over the finite area of the beam. We normally ignore these details and, unless noted otherwise, 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

\[P=E E^*. \]