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Resonator Modes

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Definition: modes of an optical or microwave resonator

German: Resonatormoden

Category: general optics

How to cite the article; suggest additional literature

Resonator modes are the modes of an optical resonator (cavity), i.e. field distributions which reproduce themselves (apart from a possible loss of power) after one round trip. They can exist whether or not the resonator is geometrically stable, but the mode properties of unstable resonators are fairly complicated. In the following, only modes of stable resonators are considered.

TEMnm Modes, Axial and Higher-order Modes

In the simplest case of a resonator containing only parabolic mirrors and optically homogeneous media, the resonator modes (cavity modes) are Hermite–Gaussian modes. The simplest of those are the Gaussian modes, where the field distribution is defined by a Gaussian function (→ Gaussian beams). The evolution of the beam radius and the radius of curvature of the wavefronts is determined by the details of the resonator. As an example, Figures 1 and 2 show the Gaussian resonator modes for two versions of a simple resonator with a plane mirror, a laser crystal, and a curved end mirror. For a more strongly curved end mirror (Figure 2), the mode radius on the left mirror becomes smaller.

Gaussian resonator mode

Figure 1: A simple resonators and the electric field distribution of its Gaussian mode. The wavefronts must be plane on the flat left end mirror, and the beam radius on the left mirror is so that the wavefronts also match the curvature of the right mirror.

Gaussian resonator mode

Figure 2: Same as in Figure 1, but with a stronger curvature of the right mirror. The mode field adjusts accordingly.

In addition to the Gaussian modes, a resonator also has higher-order modes with more complicated intensity distributions. At a beam waist, the electric field distribution can be written as a product of two Hermite polynomials with orders n and m (non-negative integers, corresponding to x and y directions) and two Gaussian functions. (We still assume a simple resonator with only parabolic mirrors and optically homogeneous media.) These modes are also called TEMnm modes; the article on higher-order modes describes the exact mathematical form. The optical intensity distribution of such a mode (Figure 1) has n nodes in the horizontal direction and m nodes in the vertical direction (Figure 3).

intensity distributions of TEM modes

Figure 3: Intensity distributions of TEMnm modes.

For an optical resonance, the amplitude distribution not only has to maintain its shape after one round trip, but also to experience a phase shift which is an integer multiple of 2π. This is possible only for certain optical frequencies. Therefore, the modes are characterized by a set of three indices: the transverse mode indices n and m, plus an axial mode number q. Increasing q by one corresponds to an increase in the round-trip phase shift by 2π. A notation such as TEMnmq includes the axial mode number in cases where it is important. Modes with n = m = 0 are called axial modes (or fundamental modes, Gaussian modes), whereas all other are called higher-order modes or higher-order transverse modes. Note that due to the Gouy phase shift the optical frequencies depend not only on the axial mode number, but also on the transverse mode indices n and m (see Figure 4):

cavity mode frequencies

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

transverse mode spacing

where φG is the Gouy phase shift per round trip. The magnitude of that Gouy phase depends on the resonator design.

mode frequencies

Figure 4: Mode frequencies of an optical resonator. The free spectral range Δν corresponds to the distance of blue lines, and δν is the spacing of higher-order modes (of which only four with low orders are shown).

Due to chromatic dispersion and diffraction effects, the mode spacings actually have a (weak) frequency dependence, which, however, is often not of interest.

Resonant enhancement, e.g. of an incident light wave, hitting a partially transmissive mirror of a resonator from outside, is possible within a range of optical frequencies. The width of that range is called the resonator bandwidth, and this quantity is determined by the rate of optical power losses.

For certain values of the Gouy phase shift, mode frequency degeneracies can occur. In a laser, these can lead to a strong deterioration of beam quality by resonant coupling of the axial modes to higher-order modes. With proper resonator design, it is possible to avoid at least the particularly sensitive frequency degeneracies and thus to improve laser beam quality [3]. Such degeneracies also can have useful properties; e.g. when a Fabry–Pérot interferometer is used as an optical spectrum analyzer (spectrometer), precise adjustment of the mirrors (e.g. in a confocal configuration) allows the use without mode matching. Also, degenerate cavities can be used for Herriot-type multipass cells, which can be used e.g. for strongly increasing the round-trip path length in a laser resonator without changing the overall resonator design.

Laser oscillation usually occurs with one or several frequencies which correspond fairly precisely to certain mode frequencies. However, frequency-dependent gain can cause some frequency pulling (slightly nonresonant oscillation), and the mode frequencies themselves can be influenced e.g. by thermal lensing in the gain medium.

Superposition of Modes

In laser physics, it is usually convenient to describe the radiation within a resonator as a superposition of light in one or several resonator modes. Some examples:

Note that it is important to distinguish coherent and incoherent mode superpositions, which can have very different properties.

If different modes of a laser resonator are simultaneously excited, there is usually the phenomenon of mode competition.

Subtle Properties of the Modes of Laser Resonators

Modes of laser resonator can differ significantly from those of an empty resonator, because they are subject to transversely varying gain and loss. This not only results in some deformation of the spatial shape; it is also that the resonator modes are no longer mutually orthogonal. Instead, there is a set of adjoint modes, related to the actual resonator modes by some biorthogonality relations. This biorthogonal (non-normal) nature has a number of peculiar implications. For example, the total power circulating in the laser is no longer simply the sum of the powers propagating in the different modes. There are also effects on the laser noise.

Bibliography

[1]A. G. Fox and T. Li. “Resonant modes in a maser interferometer”, Bell Syst. Tech. J. 40, 453 (1961)
[2]A. G. Fox and T. Li, “Computation of optical resonator modes by the method of resonance excitation”, IEEE J. Quantum Electron. 4 (7), 460 (1968)
[3]R. Paschotta, “Beam quality deterioration of lasers caused by intracavity beam distortions”, Opt. Express 14 (13), 6069 (2006)
[4]A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986)

(Suggest additional literature!)

See also: cavities, modes, mode competition, free spectral range, finesse, single-frequency operation, mode hopping, mode locking, beam quality, resonator design
and other articles in the category general optics

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