What are optical resonant cavity modes?

An optical resonant cavity is a cavity in which light waves reflect back and forth to provide optical energy feedback. It is an essential component of a laser and is usually composed of two plane or concave spherical mirrors perpendicular to the axis of the active medium.

The modes of an optical resonant cavity, also known as the electromagnetic field distribution within the cavity, are able to replicate themselves after a complete round trip. Specifically, this means that the amplitude distribution of the electromagnetic field (including the phase of the light) must remain consistent after a round trip, except for possible optical power losses. These modes exist in geometrically stable resonators, while in unstable resonators, their mode characteristics are more complex. These modes play a crucial role in applications such as laser resonators, Fabry-Perot interferometers, and mode cleaners (Figure 2).

In the most basic case, a resonator consisting only of a parabolic mirror and a homogeneous medium follows the Hermite-Gaussian mode. Among them, the most basic mode is the Gaussian mode, whose field distribution is described by a Gaussian function (i.e., a Gaussian beam). The variation of the beam radius and the radius of curvature of the wavefront depends on the specific design of the resonator.

For example, Figures 3 and 4 show the Gaussian modes of a simple resonator consisting of two different configurations of a flat mirror, a laser crystal, and a curved end mirror. In the case of a larger end mirror curvature (as in Figure 4), the mode radius on the left mirror decreases. It can be seen that a beam starting on the left with a flat wavefront (fitting a flat mirror) expands as it propagates to the right to match the shape of the other mirror, achieving self-replication of the field configuration including the wavefront shape. In addition, the round-trip phase difference should be an integer multiple of 2π.

If the right mirror is curved in the opposite direction (i.e. convex), there are no Gaussian modes and we are now dealing with an unstable resonator. This resonator also has modes, but their shapes are more complex. Most lasers operate in a stable state and in resonators with Gaussian modes. For the simple resonator described above, the mode parameters can be calculated by relatively simple equations. However, for more complex resonator configurations, involving multiple curved mirrors and other optical elements (e.g. laser crystals with thermal lensing), numerical software is usually required to calculate the mode properties. In addition to Gaussian modes, stable resonators also have higher-order modes with more complex intensity distributions. At the waist of the beam, the electric field distribution can be expressed as the product of two Gaussian functions of order n and m (non-negative integers, corresponding to the x and y directions, respectively). (We still assume a simple resonator consisting only of a parabolic mirror and an optically homogeneous medium. These modes are also called TEMnm modes. The light intensity distribution of such a mode (see Figure 3) has n nodes in the horizontal direction and m nodes in the vertical direction, as shown in Figure 5). If the Gaussian beam radius of the fundamental mode is known, the mode distributions of all higher-order modes can be easily calculated.

For optical resonance, the amplitude distribution must not only retain its shape after one round trip, but also undergo a phase change that is an integer multiple of 2π. This only applies to certain optical frequencies. Therefore, these modes are characterized by three indices: the transverse mode indices n and m, and the axial mode number q (the round-trip phase change divided by 2π). In some important cases, we use a notation like TEM_{nmq} to include the axial mode number. When n equals 0 and m equals 0, these modes are called axial modes (or fundamental modes, Gaussian modes), while all other modes are called higher-order modes or higher-order transverse modes. Note that due to the Gouy phase shift, the optical frequency depends not only on the axial mode number, but also on the transverse mode indices n and m (see Figure 6).

Neglecting dispersion, we have:

where Δν is the free spectral range (axial mode spacing) and δν is the transverse mode spacing. The latter can be calculated as:

where φG is the Gouy phase shift per round trip. The magnitude of the Gouy phase depends on the resonator design. Due to dispersion and diffraction effects, the mode spacing actually has a (weak) frequency dependence, but this is usually not critical. In the optical frequency range, resonant enhancement is possible, for example, when an incident light wave hits a partially transmitting mirror of the resonator from the outside. The width of this range is called the resonator bandwidth and this quantity is determined by the optical power loss rate. For certain values of the Gouy phase shift, mode frequency degeneracy may occur. In lasers, these degeneracy can lead to a severe degradation of the beam quality through resonant coupling of the axial mode to higher-order modes. By appropriate resonator design, at least particularly sensitive frequency degeneracy can be avoided, thus improving the laser beam quality. Such degeneracy may also have useful properties; for example, when the Fabry-Perot interferometer is used as an optical spectrum analyzer, precise adjustment of the mirrors (e.g. in a confocal configuration) allows its use without mode matching. Furthermore, degenerate cavities can be used in Herriot-type multichannel units, e.g. for substantially increasing the round-trip path length in a laser resonator without changing the overall resonator design.

Laser oscillations in continuous-wave operation usually occur at one or more frequencies that correspond fairly precisely to certain mode frequencies. However, the frequency-dependent gain leads to some frequency pulling (slightly off-resonant oscillations), and the mode frequencies themselves can be affected by, for example, thermal lensing effects in the gain medium. Single-frequency operation of the laser means that only one resonator mode (almost always a Gaussian mode) is excited; this results in a much lower emission bandwidth than in the case of exciting multiple resonator modes.

If different modes of a laser resonator are excited simultaneously in continuous-wave operation, mode competition phenomena usually occur. When the beam quality of the laser is poor, this is usually (although not always) a result of the excitation of higher-order transverse cavity modes. When the output light is sent to a fast photodiode, beat notes involving higher-order modes can be detected, i.e. frequencies that deviate greatly from integer multiples of the round-trip frequency.

In mode-locked operation, the spectrum is a frequency comb consisting of perfectly equidistant spectral lines (neglecting possible laser noise), where the line spacing is the reverse pulse repetition rate. Since the mode frequencies are not perfectly equidistant, there is a certain mismatch between the emission frequency and the mode frequency; the larger the mismatch, the stronger the effect of a mode-locking device (such as a saturable absorber). Based on this insight, it is easy to understand why it is difficult to achieve mode locking with a wide emission bandwidth and correspondingly short pulse duration in the case of large dispersion. The modes of a passive resonator have a certain frequency bandwidth due to the damping of the intracavity magnetic field by power losses. If the optical power after a resonator round trip is ρ times the original power (i.e., the fractional loss per round trip is 1 - ρ), and the round trip time is T_rt, then the resonant bandwidth is:

The modes of a laser resonator can be very different from the modes of an empty resonator because they are affected by the laterally varying gain and losses. This not only leads to some deformation of the spatial shape; it is also because the resonator modes are no longer mutually orthogonal. Instead, there is a set of companion modes, related to the actual resonator modes by some biorthogonal relationship. This biorthogonal (non-normal) property has many interesting implications. For example, the total power circulating in the laser is no longer just the sum of the powers propagating in the different modes.