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 can be defined and used to compute the cavity’s eigenmode.
The change in the parameter after one round-trip through a cavity is given by:
where , , and are the elements of a matrix . If 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 by solving:
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 , , and 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, , has a real waist size. The value for the beam waist is a real number whenever has a positive non-zero imaginary part, as this defines the Rayleigh range of the beam and therefore the beam waist, . A complex is ensured if the determinant of the cavity eigenmode equation is negative.
This requirement can formulated in a compact way by defining the parameter as:
where and are the coefficients of the round-trip matrix . The stability criterion then simply becomes:
The stability of simple cavities are often described using g-factors. These factors are simply rescaled versions of the more generic value:
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:
The closer 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
Where are the individual g-factors of the cavity mirrors and is the g-factor of the entire cavity.