定义：
光学或微波谐振腔中的模式。

谐振腔模式是光学谐振腔中的模式。无论谐振腔是否是几何上稳定的，模式都存在，但是不稳定谐振腔中的模式性质相对比较复杂。下面只考虑稳定谐振腔中的模式。 

TEMnm模式，轴向和高阶模式 
最简单的情况是谐振腔只包含抛物线型反射镜和均匀光学介质，谐振腔模式（腔模）为厄米-高斯模式。其中最简单的是高斯模式，它的场分布由高斯方程得到（参阅高斯光束）。

光束半径和波前曲率半径的变化由谐振腔决定。例如，图1和图2给出了两个高斯共振模式，其中谐振腔仅包含平面反射镜，激光晶体和曲面反射镜。图2中曲面反射镜曲率半径更小，左端反射镜处的模式半径变小。 

图1：一个简单谐振腔和其高斯模式的电场分布。在左端的平面反射镜处的波前是平面，并且左端反射镜的光束半径需要使波前与右侧反射镜的曲率相符。 

图2：与图1相同，但是右侧反射镜的曲率半径更小。模场也会相应的变化。 
除了高斯模式，谐振腔中还存在高阶模式，具有更加复杂的强度分布。在束腰处，电场分布可写为两n阶和m（非负整数，分别对应于x和y方向）阶厄米多项式与两高斯方程的乘积。
（这里仍然假设谐振腔只包含抛物线型反射镜和光学均匀介质。）这些模式也称为TEMnm模式，词条高阶模式中给出了它们准确的数学形式。该模式的光强（图1）在水平方向具有n个节点，在数值方向有m个节点（图3）。 

图3： TEMnm 模式的强度分布。 
在光学谐振腔中，不仅振幅分布在光往返一周后需要保持形状不变，并且相位变化也需要是2π的整数倍。这只能在某些频率处能满足条件。
因此，模式可以由三个指标表征：横向模式指数n和m，轴向模式数目q。q加1代表往返相移增加2π。采用记号TEMnmq就包含了轴向模式数目。当n = m = 0时称为轴向模（或基模，高斯模式），其它的都称为高阶模式或者高阶横向模式。

由于存在古依相移，光频率不仅依赖于轴向模式数目，还与横向模式指数n和m有关（见图4）： 

其中Δν 是自由光谱范围（轴向模式间距），δν 是横向模式间距。后者可由下式计算： 

其中，φG是单圈的古依相移。古依相移的大小与谐振腔设计有关。 

图4：光学谐振腔的模式频率。自由光谱范围 Δν对应的是蓝线间的距离，δν是高阶模式的间距（这里只显示了四个低阶模式）。
由于存在色散和衍射效应，模式间距也与频率有关（很弱），一般不需要考虑。 

入射光的共振增强效应，在一定光学频率范围内是可能得到的。该范围的宽度称为谐振腔带宽，它由光功率损耗决定。 
当古依相移为某一具体值时，会产生模式频率简并。在激光器中，简并会使光束质量下降很多，由于轴向模式与高阶模式之间发生谐振腔耦合。采用合适的谐振腔设计，可以尽量避免频率简并，因此会提高激光光束质量。

频率简并也有其用途，例如，将法布里-珀罗干涉仪用作光谱分析仪时（光谱仪），只需准确调节反射镜，而不需要模式匹配。另外，简并腔可以极大的提高在谐振腔中往返路径长度，并不需要改变总体的谐振腔设计。 

激光器只会在某一个或多个模式频率处振荡。但是，由于增益与频率有关，因此会产生频率牵引效应（非共振振荡），并且模式频率也会受到增益介质中的热透镜效应影响。 

模式叠加 
在激光物理中，将谐振腔中的辐射看做一个或多个谐振腔模式的叠加非常方便。例如： 

• 激光器的单模工作表示只有一个谐振腔模式（几乎都是高斯模式）被激发，这时的辐射带宽比多谐振腔模式被激发的情况低很多。 
• 模式锁定的脉冲产生也可以由谐振腔模式解释；这时，同时激发了很多基本模式，它们具有固定的相位关系。在这种情况下，激光器辐射的是周期性的脉冲列。 
• 当激光器的光束质量差时，通常是由于激发了高阶横向腔模。 

一定要区分相干和非相干模式叠加，它们具有非常不同的性质。 
如果同时激发了激光器谐振腔的不同模式，通常存在模式竞争现象。 

激光器谐振腔模式的其它性质 
激光器谐振腔中的模式与空谐振腔中的模式差别很大，因为激光器中存在横向变化的增益和损耗。这不仅会导致其空间形状变形，还导致谐振腔模式之间不再相互正交。存在一组伴随模式，与真正的谐振腔模式之间存在双正交关系。

这种双正交关系会得到很多奇怪的结果。例如，激光器中的总功率不再是不同模式功率的简单相加。

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

More general terms: modes

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.

TEM_nm 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 2π.

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 TEM_nm 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).

Figure 3: Intensity distributions of TEM_nm modes.

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 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 (the round-trip phase shift divided by 2π). 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 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).

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

where φ_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 [3]. 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.

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.

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