定义：
光束光谱的宽度或者吸收特性。

激光器的线宽，尤其是单频激光器的线宽，是指光谱的宽度（同常为半高全宽，FWHM）。更准确的说是，辐射电场功率谱密度的宽度，以频率，波数或者波长表示。 
激光器的线宽与时间相干性关系很紧密，由相干时间和相干长度来表征。如果相位经历无界偏移，那么相位噪声产生一个线宽，自由振荡器就是这种情形。（限制在很小间隔相位范围内的相位涨落产生0线宽和一些噪声边带。）谐振腔长度的偏移也会对线宽有贡献，并且使其依赖于测量时间。
这表明，仅仅由线宽或者即使如意光谱形状（线型），也不能给出激光光谱的全部信息。（尤其是激光器的主要噪声为低频相位噪声的情况。）为了得到全部的噪声性能指标需要更多的数据。 
均方根线宽定义为瞬时光频率的均方根值： 

噪声频率的积分范围是有限的范围。采用瞬时频率的功率谱密度 SΔν(f)更易计算该量。采用均方根线宽并不都是合适的衡量方法，当SΔν(f)极大增强或者噪声频率下降时（闪烁噪声）只能采用它衡量，但是白噪声则不是。
均方根线宽和光谱宽度的关系是不重要的，依赖于频率噪声谱的形状。 
窄线宽激光器（很高的单色性）应用于很多领域，例如，作为各种光纤传感器、光谱学、相干光纤通讯的光源，或者用于测试和测量。 

目录 

1. 量子噪声和技术噪声
2. 激光器线宽测量
3. 激光器线宽最小化
4. 窄线宽带来的问题
5. 线宽用于其它情况

量子噪声和技术噪声 
最简单的情况是只有自发辐射（量子噪声）产生相位噪声。这时瞬时频率的噪声为白噪声，也就是，功率谱密度是常数，辐射光谱为洛伦兹型。在第一台激光器实现之前，Schwlow和Townes就计算出了对应的线宽。根据修正后的Schalow-Townes方程（M. Lax做了修正） 

线宽（FWHM）正比于共振带宽与输出功率比值的平方（假设不存在寄生谐振腔误差）。在词条Schawlow-Townes线宽中有更常用的方程形式。 
实际中很难实现Schawlow-Townes极限，因为有许多技术噪声（例如，机械振动，温度涨落，泵浦功率涨落）很难抑制。所以在激光器设置时需要在得到窄线宽方面做出一些妥协。例如，激光谐振腔长得到的Schawlow-Townes线宽窄，但是很难得到稳定的单频工作，也更难得到物理上稳定的装置。 
稳定的自由单频固体激光器的典型测量线宽（例如，测量时间为1s）约为几个kHz，远大于Schawlow-Townes极限。增加的线宽是由于存在一些技术上的噪声，例如，谐振腔长的涨落，激光晶体的泵浦功率或者温度涨落。 
单色半导体激光器的线宽通常在MHz范围，远大于Schawlow-Townes极限主要来自于振幅相位耦合，由线宽增强因子来描述。载流子涨落也会产生附加噪声，就是1/f频率噪声。 
线宽更窄时，甚至低于1Hz，可以采用超稳定参考腔来得到激光器的稳定。在光谱学和光纤传感器应用中，需要线宽很窄。 

激光器线宽测量
可采用很多技术测量激光器线宽： 

1. 线宽比较大时（>10 GHz，当多个激光器谐振腔中存在多个模式振荡），采用传统的采用衍射光栅的光谱分析仪就可以测量。采用这种方法很难得到高的频率分辨率。 
2. 另一种方法是采用频率鉴别器将频率涨落转化为强度涨落，鉴别器可以是不平衡的干涉仪或者高精细度的参考腔。这种测量方法的分辨率也很有限。 
3. 单频激光器通常采用自外差方法，是记录激光器输出与经过频率偏移和延迟的自身之间拍音。 
4. 几百赫兹线宽时，传统的自外差技术是不实际的，因为这时需要很大的延迟长度，可以采用循环的光纤环路和内置光纤放大器来延伸。 
5. 可以通过记录两个独立激光器的拍音得到非常高的分辨率，这时参考激光器的噪声远小于测试激光器，或者二者性能指标类似。可以采用锁相环路或者根据数字记录进行计算得到瞬时频率差。这种方法非常简单稳定，但是需要另一个激光器（工作在测试激光器频率附近）。如果测量的线宽需要光谱范围很宽，采用频率梳非常便利。 

光学频率测量通常会在某处需要某一频率（或者时间）参考。对于窄线宽激光器，只需要一个参考光就可以给出足够准确的参考。自外差技术是从测试装置本身通过施加足够长的时间延迟来得到频率参考，理想状况下会避免初始光束与其自身延迟光之间的时间相干。
因此，通常采用长光纤，由于稳定涨落和声学影响，长光纤会引起附加相位噪声。 
当存在1/f频率噪声时，仅线宽值不能完全描述相位误差。更好的方法是测量相位或者瞬时频率涨落的傅里叶光谱，然后采用功率谱密度来表征；可以参考噪声性能指标。1/f噪声（或者其他低频噪声的噪声谱）会引起一些测量问题。 

激光器线宽最小化 
激光器线宽与激光器类型有直接关系。可以通过优化激光器设计和尽量抑制外部的噪声影响来最小化激光器线宽。第一步需要判断量子噪声和经典噪声那个占据主导地位，因为这会影响接下来的测量。 
当腔内功率高，低谐振腔损耗和长谐振腔往返时间时，激光器的量子噪声（主要是自发辐射噪声）影响很小。经典噪声可能由机械涨落引起，涨落可以通过采用一个紧凑的短激光器谐振腔被减弱，但是有时在更短谐振腔中其长度涨落会产生更强的效应。合理的机械设计可以减小激光谐振腔与外界扰动之间的耦合，还会最小化热漂移效应。增益介质中也存在热涨落，由于泵浦功率涨落而引起的。对于更好的噪声性能，需要采用其他有源稳定装置，不过最初最好采用实际的无源方法。 
单频固态体激光器和光纤激光器的线宽为几千赫兹，有时甚至低于1kHz。采用有源稳定方法，可以得到小于1kHz的线宽。激光二极管的线宽通常在 MHz范围，也可以被减小到几kHz，例如外腔二极管激光器，尤其是存在高精细度参考腔的光学反馈的二极管。 
可参阅词条窄线宽激光器。 

窄线宽带来的问题 
有些情况下不需要激光光源产生光线宽很窄： 

1. 相干长度长时相干效应（由于寄生反射很弱）会破坏光束形状。在激光投影显示器中，斑点效应会干扰画面质量。 
2. 光在有源或者无源光纤中传输时，窄线宽由于存在受激布里渊散射而产生一些问题。这是需要增大线宽，例如通过电流调制激光二极管或者光调制器使瞬时频率快速抖动。 

线宽用于其它情况 
线宽也用于描述光跃迁的宽度（例如激光器跃迁或者一些吸收特征）。静止的单个原子或者离子的跃迁中，线宽与上能态寿命有关（更准确的说，是上能态和下能态的寿命），被称为自然线宽。原子或者离子的运动（参阅多普勒展宽）或相互作用会使线宽展宽，例如气体中的压力展宽或者固体介质中的声子间相互作用。如果不同原子或者离子受到的影响不同，会引起非均匀展宽。 
跃迁线宽与Q因子有关，它是频率与线宽的比值。

Definition: width of the spectrum of a light beam or an absorption feature

Alternative term: emission bandwidth

More specific terms: emission linewidth, laser linewidth, Schawlow–Townes linewidth

Formula symbol: Δν, Δλ

Units: Hz, nm

The linewidth (or line width) of a laser, e.g. a single-frequency laser, is the width (typically the full width at half-maximum, FWHM) of its optical spectrum. More precisely, it is the width of the power spectral density of the emitted electric field in terms of frequency, wavenumber or wavelength.

Similarly, other spectral lines e.g. from gas discharge lamps have certain linewidths, which can depend on the operation conditions.

The linewidth of a light beam is strongly (but non-trivially) related to the temporal coherence, characterized by the coherence time or coherence length. A finite linewidth arises from phase noise if the optical phase undergoes unbounded drifts, as is the case for free-running laser oscillators, for example. (Phase fluctuations which are restricted to a small interval of phase values lead to a zero linewidth and some noise sidebands.) Drifts of the resonator length can further contribute to the linewidth and can make it dependent on the measurement time. This shows that the linewidth alone, or even the linewidth complemented with a spectral shape (line shape), does by far not provide full information on the spectral purity of laser light. (This is particularly the case for lasers with dominating low-frequency phase noise.) More data are required for full noise specifications.

A simple way to define the linewidth would be to use the root-mean-square (r.m.s.) value of the instantaneous optical frequency:

where usually some limited integration range for the noise frequencies is chosen. This quantity can be easily calculated from the power spectral density S_Δν(f) of the instantaneous frequency. Note, however, that the r.m.s. linewidth is not always a sensible measure; one should only use it in cases with strongly increasing S_Δν(f) for decreasing noise frequency (flicker noise), but not e.g. for white frequency noise. It is more common to define the linewidth as the width of the optical spectrum, but the relation between the r.m.s. linewidth and the width of the optical spectrum (or with the phase noise PSD) is not trivial and depends on the shape of the frequency noise spectrum.

Lasers with very narrow linewidth (high degree of monochromaticity) are required for various applications, e.g. as light sources for various kinds of fiber-optic sensors, for laser spectroscopy (e.g. LIDAR), in coherent optical fiber communications, and for test and measurement purposes. Note that the achieved linewidth can be many orders of magnitude below the linewidth of the used laser transition.

Quantum Noise and Technical Noise

The simplest situation is one where only spontaneous emission (quantum noise) introduces phase noise. In that case, the noise of the instantaneous frequency is white noise, i.e., its power spectral density is constant, and the emission spectrum is of Lorentzian shape. The corresponding linewidth was calculated by Schawlow and Townes [1] even before the first laser was experimentally demonstrated. According to the modified Schawlow–Townes equation (with a correction from M. Lax)

Hear the Phase Noise!

In order to get some feeling for phase noise, you can listen to the following short sound samples of a 440-Hz tone with different linewidths (white phase noise, no amplitude noise):

  0 Hz (no noise), 1 Hz, 5 Hz, 10 Hz

You can download these .wav files (each being ≈ 215 KB long) and play them with any sound player software.

For comparison, if you temporally stretch the output of a 1064-nm Nd:YAG laser with a 10-kHz linewidth to 440 Hz, the linewidth will be ≈ 16 nHz – a pretty pure tone indeed!

the linewidth (FWHM) is proportional to the square of the resonator bandwidth divided by the output power (assuming that there are no parasitic resonator losses). The article on the Schawlow–Townes linewidth contains a more practical form of the equation.

The Schawlow–Townes limit is usually difficult to reach in reality, as there are various technical noise sources (e.g. mechanical vibrations, temperature fluctuations, and pump power fluctuations) which are difficult to suppress. There are therefore certain compromises in laser design for narrow linewidth. For example, a long laser resonator leads to a small Schawlow–Townes linewidth, but makes it more difficult to achieve stable single-frequency operation without mode hops, and to get a mechanically stable setup.

Typical measured linewidths of stable free-running single-frequency solid-state lasers (e.g. for a measurement time of 1 s) are a few kilohertz, which is far above the Schawlow–Townes limit. Various sources of technical noise, e.g. fluctuations of the resonator length, the pump power or the temperature of the laser crystal, can be responsible for the increased linewidth.

The linewidths of monolithic semiconductor lasers are often in the megahertz range and are strongly increased above the Schawlow–Townes limit mainly by amplitude-phase coupling, as described with the linewidth enhancement factor. There can also be excess noise from charge carrier fluctuations with a 1 / f characteristic of the PSD of the frequency fluctuations. In that case, the measurement time influences the measured linewidth value.

Much smaller linewidths, sometimes even below 1 Hz, can be reached by stabilization of lasers, e.g. using ultrastable reference cavities (→ narrow-linewidth lasers, frequency-stabilized lasers). Small linewidths are important, e.g. for spectroscopic measurements and for application in fiber-optic sensors.

Measurement of Laser Linewidth

A laser linewidth can be measured with a variety of techniques:

• For large linewidths (e.g. > 10 GHz, as obtained when multiple modes of the laser resonator are oscillating), traditional techniques of optical spectrum analysis, e.g. based on diffraction gratings, are suitable. A high frequency resolution is difficult to obtain in this way.
• Another technique is to convert frequency fluctuations to intensity fluctuations, using an optical frequency discriminator, which can be, e.g., an unbalanced interferometer or a high-finesse reference cavity. Again, the measurement resolution is quite limited.
• For single-frequency lasers, the self-heterodyne technique is often used, which involves recording a beat note between the laser output and a frequency-shifted and delayed version of it.
• For sub-kilohertz linewidths, the ordinary self-heterodyne technique usually becomes impractical, as one would require a very large delay length, but it can be extended by using a recirculating fiber loop with an internal fiber amplifier.
• Very high resolution can also be obtained by recording a beat note between two independent lasers, where either the reference laser has significantly lower noise than the device under test, or both lasers have similar performance. One can retrieve the instantaneous difference frequency e.g. with PLL (phase-locked loop) following the beat signal, or numerically from digitized recordings. This method is conceptually very simple and reliable, but the requirement of a second laser (operating at a nearby optical frequency) can be inconvenient. If linewidth measurements are required in a wide spectral range, a frequency comb source can be very useful.

Note that an optical frequency measurement always needs some kind of frequency (or timing) reference somewhere in the setup. For lasers with narrow linewidth, only an optical reference can give a sufficiently accurate reference. The self-heterodyne technique is a way to derive the frequency reference from the device under test itself by applying a large enough time delay, ideally avoiding any temporal coherence between the original beam and the delayed version. Therefore, long fibers are often used; however, long fibers tend to introduce additional phase noise due to temperature fluctuations and acoustic influences.

Particularly in cases with 1 / f frequency noise, a linewidth value alone may not be regarded as completely characterizing the phase noise. It may then be better to measure the whole Fourier spectrum of the phase or instantaneous frequency fluctuations and characterize it with a power spectral density; see also the article on noise specifications. Note also that 1 / f frequency noise (or other noise spectra with strong low-frequency noise) can cause problems with some measurement techniques.

Minimization of Laser Linewidth

The linewidth of a laser depends strongly on the type of laser. It may be further minimized by optimizing the laser design and suppressing external noise influences as far as possible. The first step should be to determine whether quantum noise or classical noise is dominating, because the required measures can depend very much on this.

The influence of quantum noise (essentially spontaneous emission noise) is small for a laser with high intracavity power, low resonator losses, and a long resonator round-trip time. Classical noise may be introduced via mechanical fluctuations, which can often be kept weaker for a compact short laser resonator, but note that resonator length fluctuations of a certain magnitude have a stronger effect in a shorter resonator. Proper mechanical construction can minimize the coupling of the laser resonator to external vibrations and also minimize effects of thermal drift. There can also be thermal fluctuations in the gain medium, introduced e.g. by a fluctuating pump power. For superior noise performance, various schemes for active stabilization can be employed, but it is often advisable first to use all practical passive methods.

Single-frequency solid-state bulk and fiber lasers can achieve linewidths of a few kilohertz, or sometimes even below 1 kHz. With serious efforts at active stabilization, sub-hertz linewidths are sometimes achieved. The linewidth of a laser diode is typically in the megahertz region, but it can also be reduced to a few kilohertz, e.g. in external-cavity diode lasers, particularly with optical feedback from a high-finesse reference cavity.

See also the article on narrow-linewidth lasers.

Problems Resulting from a Narrow Linewidth

A narrow linewidth from a laser source is not always desirable:

• A large coherence length implies that interference effects (e.g. due to weak parasitic reflections) can easily spoil the beam profile. In laser projection displays, laser speckle effects can disturb the image quality.
• For transmission of light in passive or active optical fibers, a narrow linewidth can cause problems due to stimulated Brillouin scattering. It is then sometimes necessary to increase the optical linewidth, for example by fast dithering of the instantaneous frequency via current modulation of a laser diode or with an optical modulator.

Linewidth in Other Context

The term linewidth is also used for the width of optical transitions (e.g. a laser transition or some absorption feature). For transitions in single atoms or ions at rest, the linewidth is related to the upper-state lifetime (more precisely, the lifetime of both upper and lower states) (lifetime broadening) and is called natural linewidth. Significant linewidth broadening can be caused by movement of the atoms or ions (→ Doppler broadening) or by interactions, e.g. pressure broadening in gases or interactions with phonons in solid media. If different atoms or ions are subject to different influences, this leads to inhomogeneous broadening.

The linewidth of a transition is often related to a Q factor, which is the frequency divided by the linewidth.

Bibliography