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. 9 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. 9 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. 9, 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.