定义：
只具有量子噪声的单频激光器的线宽。

第一台激光器在实验上实现之前，Schawlow和Townes已经理论上计算了激光器线宽的基本限制因素。这就是著名的Schawlow-Townes方程：

光子能量为 h ν，Δνc 是谐振腔带宽（半高半宽，HWHM），Pout是输出功率。这里假设不存在任何寄生腔内损耗。注意这表示的是半高半宽。
后来Melvin Lax指出激光器工作（高于阈值泵浦功率）的线宽比Schawlow和Townes得到的要小两倍。如果考虑因子1/2，并且将线宽值转化为半高全宽（激光器线宽和腔线宽同时改变），得到修正后的线宽方程为：

另一个更常用的形式为:

其中Toc 代表输出耦合器的透射率，ltot是谐振腔总的损耗（可能比Toc 大），Trt是谐振腔往返时间，θ是考虑了三能级增益介质中自发辐射会增大的自发辐射因子。
对应相位噪声的双边功率谱密度为：

白频率噪声为：

通常认为Schawlow-Townes线宽对应的相位噪声是由于自发辐射进入激光器模式引起的。尽管这种解释很直观，但不是完全正确的。因为激光器增益和激光器谐振腔的线性损耗对腔内光场的量子噪声的影响是相同的。
也就是说，即使将激光增益区换成无噪声的放大过程，相位噪声也只会减小到Schawlow-Townes值的一半。
精心设计制作的固体激光器可以达到几kHz的窄线宽，但是仍然高于Schawlow-Townes极限，因为技术上的附加噪声使其很难达到该极限。
根据以上的方程，半导体激光器的线宽通常也很大，因为：
存在自发辐射因子，例如，由于半导体的重吸收，在2-3个量级之间
存在很强的振幅相位耦合效应，可以定量的由线宽增强因子α来表示，会将线宽提高至少一个数量级。
还存在瞬时频率的功率谱密度的1/f形式的附加噪声。这与半导体中的带电载流子的涨落有关。
外腔二极管激光器中，存在机械振动引起的附加噪声。

在锁模激光器中的应用
Schawlow–Townes公式甚至适用于有源锁模激光器，这时Pout为总的平均功率（而不是频率梳某一条线对应的功率）。后一定律可能比较难以理解，但是可以认为光谱中单条线的相位不是单独变化的，它们由调制器锁定在一起。（不然循环脉冲会散开）
它还可以进一步推广到无源锁模激光器中。这里根据Schawlow–Townes公式可以推测光谱中心附近的线宽，而光谱两翼的线宽则由于量子噪声引起的时间抖动而增大。无源锁模激光器中存在的非线性动力学过程会引起很强的附加噪声，在这种情况下Schawlow–Townes结果就不适用了。

Definition: linewidth of a single-frequency laser with quantum noise only

More general term: linewidth

Even before the first laser was experimentally demonstrated, Schawlow and Townes calculated the fundamental (quantum) limit for the linewidth of a laser [1]. This lead to the Schawlow–Townes equation, which became one of the most famous equations of laser physics:

with the photon energy hν, the resonator bandwidth Δν_c (half width at half-maximum, HWHM), and the output power P_out. It has been assumed that there are no parasitic cavity losses. Note that the result is interpreted as a half width at half-maximum.

Later, Melvin Lax showed that the linewidth in lasing operation (above the threshold pump power) must be two times smaller than derived by Schawlow and Townes [2]. If we take into account that factor 1 / 2 and also convert the linewidth values to full width at half maximum (FWHM) (for both the laser linewidth and the cavity linewidth), which is more common, we obtain the modified linewidth equation

A more general form of the equation [6, 8] is

where T_oc denotes the output coupler transmission, l_tot the total resonator losses (which may be larger than T_oc), T_rt the resonator round-trip time, and θ the spontaneous emission factor, which takes into account increased spontaneous emission in (quasi-)three-level gain media.

The corresponding two-sided power spectral density of the phase noise is

This corresponds to white frequency noise with

Essentially, the Schawlow–Townes linewidth is determined by the strength of fluctuations of the optical phase which are caused by quantum noise.

It is often claimed that the phase noise level corresponding to the Schawlow–Townes linewidth is a result of spontaneous emission into the laser mode. Although this picture is intuitive, it is not completely correct. Both the laser gain and the linear losses of the laser resonator contribute equal amounts of quantum noise to the intracavity light field. This means that even when replacing laser gain with some noiseless amplification process, the phase noise would only decrease to half of the Schawlow–Townes value [3].

Carefully constructed solid-state lasers can have very small linewidths in the region of a few kilohertz, which is still far above their Schawlow–Townes limit: technical excess noise makes it very difficult to reach that limit.

The linewidth of semiconductor lasers is also normally much larger than according to the original equation (without the α factor):

• There is also a spontaneous emission factor, e.g. of the order of 2 or 3, due to reabsorption in the semiconductor.
• More importantly, there are strong amplitude-to-phase coupling effects, which can be quantified with the linewidth enhancement factor α and can increase the linewidth by a factor 1 + α², which can contribute more than an order of magnitude.
• In addition, there can be excess noise with 1 / f-type shape of the power spectral density of the instantaneous frequency. This is related to fluctuations of the charge carriers in the semiconductor.
• Particularly for external-cavity diode lasers, there can also be significant excess noise due to mechanical vibrations.

Application to Mode-locked Lasers

Interestingly, the Schawlow–Townes formula can be applied even to actively mode-locked lasers [4], using the total average power (rather than the power in a particular line of the frequency comb) for P_out. The latter rule may be surprising, but it can be understood considering that the phases of the individual lines in the optical spectrum can not evolve independently; they are locked together by the modulator. (Otherwise, the circulating pulse would fall apart.)

A further generalization is possible for passively mode-locked lasers [5]. Here, the Schawlow–Townes formula provides an estimate for the linewidth near the center of the spectrum, whereas the linewidth in the spectral wings is somewhat increased due to quantum-noise-induced timing jitter. Note, however, that the nonlinear dynamics occurring in many passively mode-locked lasers can lead to strong excess noise, invalidating the Schawlow–Townes results for such cases.

Bibliography