1.  laser linewidth | | 1. linewidth
The linewidth (or line width) of a laser, typically a single-frequency laser, is the width (typically the full width at half maximum) of its optical spectrum. In other words, it is the width of the power spectral density of the emitted electric field. It 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 phase undergoes unbounded drifts, as is the case for free-running oscillators. (Phase fluctuations which are restricted to a small interval of phase values lead to a zero linewidth and some noise sidebands.) Drifts of the cavity length (e.g. related to 1/f noise) 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.
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 spectroscopy (e.g. LIDAR), in coherent optical fiber communications, and for test & measurement.
2. Quantum Noise and Technical Noise
For simple cases, the fundamental limit for the laser linewidth arising from quantum noise has been calculated by Schawlow and Townes even before the first laser was experimentally demonstrated. According to the Schawlow-Townes formula
delta(v_l)=pi*h*v*(delta(v_c))^2/(P_out)
the linewidth (full width at half maximum) is proportional to the square of the cavity bandwidth divided by the output power (assuming that there are no parasitic cavity losses). The article on the Schawlow-Townes linewidth contains a somewhat 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 thus certain compromises in laser design for narrow linewidth. For example, a long laser cavity 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 linewidths of stable free-running solid state lasers (e.g. for a measurement time of one second) are a few kHz, while the linewidths of semiconductor lasers are often in the MHz range. Much smaller linewidths, sometimes even below 1 Hz, can be reached by stabilization of lasers e.g. using ultrastable reference cavities. Small linewidths are important e.g. for spectroscopic measurements or for application in fiber-optic sensors.
3.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 cavity are oscillating), one can use traditional techniques of optical spectrum analysis, e.g. based on diffraction gratings.
-Another technique is to convert frequency fluctuations to intensity fluctuations, using a frequency discriminator, which can e.g. be an unbalanced interferometer or a high-finesse reference cavity.
-For single-frequency lasers, one often uses the self-heterodyne technique, which involves recording a beat note between the laser output and a frequency-shifted and delayed version of it.
-For sub-kHz linewidths, the ordinary self-heterodyne technique usually becomes impractical, 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. This method is conceptually most 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, one may utilize frequency combs.
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, rather long fibers are often used; however, long fibers tend to introduce additional phase noise due to temperature fluctuations and acoustical 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.
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 state) (¡ú lifetime broadening) and is called natural linewidth. Significant linewidth broadening can be caused by movement of the atoms or ions (¡ú 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.
References
[1] A. L. Schawlow and C. H. Townes, "Infrared and optical masers", Phys. Rev. 112 (6), 1940 (1958) (contains the famous Schawlow-Townes formula)
[2] C. H. Henry, "Theory of the linewidth of semiconductor lasers" (¡ú introduction of the linewidth enhancement factor), IEEE J. Quantum. Electron. 18 (2), 259 (1982)
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