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# Resonator Modes

Definition: modes (self-reproducing field configurations) of an optical or microwave resonator

More general term: modes

German: Resonatormoden

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Resonator modes are the modes of an optical resonator (cavity), i.e. electromagnetic field distributions which reproduce themselves (apart from a possible loss of optical power) after a full resonator round trip. Such modes 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.

Resonator modes are very important e.g. in the context of laser resonators, Fabry–Pérot interferometers and resonant mode cleaners.

## 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. In any case, one can see that a beam starting e.g. on the left side with flat wavefronts (fitting to the flat mirror there) will somewhat expand on its path to the right side, where the wavefronts again match the shape of the other mirror, and it is easy to see that the field configuration will reproduce itself, including the shape of the wavefronts, after a full round trip. In addition to what can be seen in the drawing, there is the condition that the round-trip phase shift is an integer multiple of <$2\pi$>. 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. 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 Hermite–Gaussian 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).

## Mode Frequencies

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 <$2\pi$>. 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 <$q$> (the round-trip phase shift divided by <$2\pi$>). A notation such as TEM_{nmq} 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). For now ignoring chromatic dispersion, we have: Figure 4: Mode frequencies of an optical resonator. The free spectral range <$\Delta \nu$> corresponds to the distance of blue lines, and <$\delta \nu$> is the spacing of higher-order modes (of which only four with low orders are shown). $${\nu _{nmq}} = {\nu _0} + q\;\Delta \nu + (n + m)\;\delta \nu$$

where <$\Delta \nu$> is the free spectral range (axial mode spacing) and <$\delta \nu$> the transverse mode spacing. The latter can be calculated as

$$\delta \nu = - \Delta \nu \cdot \frac{{{\varphi _{\rm{G}}}}}{{2\pi }}$$

where <$\varphi_\textrm{G}$> is the Gouy phase shift per round trip. The magnitude of that Gouy phase depends on the resonator design.

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 . Such degeneracies also can have useful properties; e.g. when a Fabry–Pérot interferometer is used as an optical spectrum analyzer, 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.

## Mode Frequencies in Lasers

Laser oscillation in continuous-wave operation 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.

Single-frequency operation of a laser means that only a single resonator mode (nearly always a Gaussian one) is excited; this leads to a much lower emission bandwidth than in cases where multiple resonator modes are excited.

If different modes of a laser resonator are simultaneously excited in continuous-wave operation, there is usually the phenomenon of mode competition.

When a laser has a poor beam quality, this is usually (although not always) the result of the excitation of higher-order transverse cavity modes. When the output light is sent to a fast photodiode, one can detect beat notes involving higher-order modes, i.e., with frequencies which substantially deviate from integer multiples of the round-trip frequency.

In mode-locked operation, the optical spectrum is a frequency comb, consisting of exactly equidistant spectral lines (ignoring possible laser noise), where the line spacing is the inverse pulse repetition rate. Due to the not exactly equidistant mode frequencies, there is some amount of mismatch between emission frequencies and mode frequencies; the larger that mismatch is, the stronger needs to be the effect of the mode-locking device, for example a saturable absorber. Based on this insight, it is easy to understand why in cases with substantial chromatic dispersion it is hard to achieve mode locking with a broad emission bandwidth and a correspondingly short pulse duration.

## Bandwidth of Modes

The modes of a passive resonator have a certain frequency bandwidth as a result of the damping of the intracavity field by power losses. If the optical power after one resonator round-trip is <$\rho$> times the original power (i.e., the fractional losses per round trip are <$1 - \rho$>) and the round-trip time is <$T_\textrm{rt}$>, the full width at half maximum bandwidth if the resonances is

$$\Delta \nu = \frac{2 \arcsin(\frac{1 - \sqrt{\rho}}{2 \rho^{1/4}})}{\pi \: T_\textrm{rt}}$$

## 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

  A. G. Fox and T. Li. “Resonant modes in a maser interferometer”, Bell Syst. Tech. J. 40, 453 (1961); https://doi.org/10.1002/j.1538-7305.1961.tb01625.x  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); https://doi.org/10.1109/JQE.1968.1075368  R. Paschotta, “Beam quality deterioration of lasers caused by intracavity beam distortions”, Opt. Express 14 (13), 6069 (2006); https://doi.org/10.1364/OE.14.006069  A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986)

## Questions and Comments from Users

2021-01-03

I use an OPO pumped with a femtosecond Ti:sapphire laser. The beam profile I obtain is TEM21, even though it is supposed to yield a Gaussian mode. How to fix this?

In some respects, such an OPO is similar to a laser. It may be that you misaligned the resonator such that the TEM21 mode is a better overlap with the pump beam than the fundamental resonator mode. So you probably need to optimize the alignment. There might also be a problem with the exact resonator length, which might be slightly different for higher-order modes. It is a special challenge in the initial alignment of a synchronously pumped OPO to get the transverse alignment and the resonator length correct at the same time.

2021-07-08

Are transverse modes lesser in number than longitudinal modes in a laser resonator? Why is it usually said that for a given transverse mode there can be a number of longitudinal modes, but not vice versa?

There is no well-defined number of transverse or longitudinal modes, so we cannot compare those numbers. Longitudinal refers to different optical frequencies while referring to a specific spatial shape, usually the Gaussian one for the fundamental modes. Transverse refers to different spatial shapes, as shown in Figure 3. According to that figure, you can say that for a given longitudinal mode, there are many transverse modes with similar optical frequency. However, you can also say that for any of those spatial shapes, there are many versions with different optical frequencies.

2023-09-13

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