Cavity eigenmodes

A cavity eigenmode is defined as the optical field whose spatial properties are such that the field after one round-trip through the cavity will be exactly the same as the injected field [12]. In the case of resonators with spherical mirrors, the eigenmode will be a Gaussian mode, defined by the Gaussian beam parameter qcav. For a generic cavity (an arbitrary number of spherical mirrors or lenses) a round-trip ABCD matrix \(M_{\mathrm{rt}}\) can be defined and used to compute the cavity’s eigenmode.

The change in the \(q\) parameter after one round-trip through a cavity is given by:

\[\frac{A q_{1} + B}{C q_{1} + D} = q_{2} = q_{1} \]

where \(A\), \(B\), \(C\) and \(D\) are the elements of a matrix \(M_{\mathrm{rt}}\). If \(q_1 = q_2\) then the spatial profile of the beam is recreated after each round-trip and we have identified the cavity eigenmode. We can compute the parameter \(q_{\mathrm{cav}} \equiv q_1 = q_2\) by solving:

\[C q_{\mathrm{cav}}^2+(D-A)q_{\mathrm{cav}} - B = 0, \]

An example of this is shown in Fig. 10 where the round trip matrix is given at the top of the figure. From this, we can compute the \(A\), \(B\), \(C\) and \(D\) coefficients for the round-trip matrix to solve the eigenmode equation above. This quadratic equation generally has two solutions, one being the complex conjugate of the other.

Fig. 10 Cavity round trip ABCD matrix for a Fabry-Perot cavity, with corresponding wavefront curvature.

When the polynomial above has a suitable solution the optical resonator is said to be “stable”. The stability requirement can be formulated using the Gaussian beam parameter: a cavity is stable only when the cavity’s eigenmode, \(q_{\mathrm{cav}}\), has a real waist size. The value for the beam waist is a real number whenever \(q_{\mathrm{cav}}\) has a positive non-zero imaginary part, as this defines the Rayleigh range of the beam and therefore the beam waist, \(\Im{(q_{\mathrm{cav}})} = \pi w_0^2/\lambda\). A complex \(q_{\mathrm{cav}}\) is ensured if the determinant of the cavity eigenmode equation is negative.

This requirement can formulated in a compact way by defining the parameter \(m\) as:

\[m \equiv \frac{A+D}{2}, \]

where \(A\) and \(D\) are the coefficients of the round-trip matrix \(M_{\mathrm{rt}}\). The stability criterion then simply becomes:

\[m^2 < 1. \]

The stability of simple cavities are often described using g-factors. These factors are simply rescaled versions of the more generic \(m\) value:

\[g \equiv \frac{m+1}{2} = \frac{A+D+2}{4}, \]

Note

Finesse uses g-factors for stability attributes in the finesse.components.cavity.Cavity class. Individual g-factors for both the tangential and sagittal planes are provided by this class.

For the cavity to be stable the g-factor must fulfil the requirement:

\[0 \leq g \leq 1 \]

The closer \(g\) is to 0 or 1, the smaller the tolerances are for any change in the geometry before the cavity becomes unstable.

For a simple two-mirror cavity, such as the one given in Fig. 10, its g-factor is

\[\begin{array}{l} g_1 = 1 - \frac{L}{R_{c,1}}, \\ g_2 = 1 - \frac{L}{R_{c,2}}, \\ g = g_1 g_2. \end{array} \]

Where \(g_{1,2}\) are the individual g-factors of the cavity mirrors and \(g\) is the g-factor of the entire cavity.