# Laser linewidth

**Laser linewidth** is the spectral linewidth of a laser beam.

Two of the most distinctive characteristics of laser emission are spatial coherence and spectral coherence. While spatial coherence is related to the beam divergence of the laser, spectral coherence is evaluated by measuring the laser linewidth of the radiation. Although the concept of laser linewidth can have varied theoretical descriptions here this article provides a simple experimental description. One of the first methods used to measure the coherence of a laser was interferometry.^{[1]} An alternative approach is the use of spectrometry.^{[2]}

## Continuous lasers

Laser linewidth in a typical single-transverse-mode He-Ne laser (at a wavelength of 632.8 nm), in the absence of intracavity line narrowing optics, can be of the order of 1 GHz. On the other hand, the laser linewidth from stabilized low-power continuous-wave lasers can be very narrow and reach down to less than 1 kHz.^{[3]} Often this type of linewidth is limited by fundamental quantum processes.^{[4]} This limit is known as the Schawlow–Townes linewidth^{[5]} which can be lower than Hz for some kind of CW lasers. Nevertheless observed linewidths are larger due to a technical noise (from noise in current, vibrations etc.).

## Pulsed lasers

Laser linewidth from high-power high-gain pulsed lasers, in the absence of intracavity line narrowing optics, can be quite broad and in the case of powerful broadband dye lasers it can range from a few nm wide^{[6]} to as broad as 10 nm.^{[2]}

Laser linewidth from high-power high-gain pulsed laser oscillators, comprising line narrowing optics, is a function of the geometrical and dispersive features of the laser cavity.^{[7]} To a first approximation the laser linewidth, in an optimized cavity, is directly proportional to the beam divergence of the emission multiplied by the inverse of the *overall intracavity dispersion*.^{[7]} That is,

This is known as the *cavity linewidth equation* where is the beam divergence and the term in parenthesis (elevated to –1) is the overall intracavity dispersion. This equation was originally derived from classical optics.^{[8]} However, in 1992 Duarte derived this equation from quantum interferometric principles,^{[9]} thus linking a quantum expression with the overall intracavity angular dispersion.

An optimized multiple-prism grating laser oscillator can deliver pulse emission in the kW regime at single-longitudinal-mode linewidths of ≈ 350 MHz (equivalent to ≈ 0.0004 nm at a laser wavelength of 590 nm).^{[10]} Since the pulse duration from these oscillators is about 3 ns,^{[10]} the laser linewidth performance is near the limit allowed by the Heisenberg uncertainty principle.

## Linewidth equivalence

In the frequency domain the laser linewidth is denoted as and is given in units of GHz, MHz, or kHz. In the spectral domain the laser linewidth is denoted as and is often given in units of nm. In the field of spectroscopy the use of the reciprocal cm, or cm^{-1}, is widespread.

In more detail, the laser linewidth in frequency units can be written as^{[7]}

where *c* is the speed of light (in units of velocity) and is the coherence length (a length) so that the linewidth is in units of frequency. The equivalent definition in the wavelength domain is^{[7]}

Thus the quantity is the common factor between the equivalent linewidths in Hz and meter units. With these definitions it can be shown, for example, that ≈ 350 MHz is equivalent to ≈ 0.0004 nm at a laser wavelength of 590 nm.^{[10]}

## See also

- Beam divergence
- Multiple-prism dispersion theory
- Multiple-prism grating laser oscillator
- N-slit interferometric equation
- Oscillator linewidth
- Solid state dye lasers

## References

- ↑ O. S. Heavens,
*Optical Masers*(Wiley, New York, 1963). - ↑
^{2.0}^{2.1}F. P. Schäfer*et al.*, Organic dye solution laser,*Appl. Phys. Lett.***9**, 306-309 (1966). - ↑ L. W. Hollberg, CW dye lasers, in
*Dye Laser Principles*, F. J. Duarte and L. W. Hillman (eds.) (Academic, New York, 1990) Chapter 5. - ↑ H. Haken,
*Light and Matter*(Springer, Berlin, 1970). - ↑ http://www.rp-photonics.com/schawlow_townes_linewidth.html - Schawlow–Townes Linewidth
- ↑ M. L. Spaeth and D. P. Bortfeld, Stimulated emission from polymethine dyes,
*Appl. Phys. Lett.***9**, 179-181 (1966). - ↑
^{7.0}^{7.1}^{7.2}^{7.3}F. J. Duarte,*Tunable Laser Optics*(Elsevier Academic, New York, 2003). - ↑ J. K. Robertson,
*Introduction to Optics: Geometrical and Physical*(Van Nostrand, New York, 1955). - ↑ F. J. Duarte, Cavity dispersion equation: a note on its origin,
*Appl. Opt.***31**, 6979-6982 (1992). - ↑
^{10.0}^{10.1}^{10.2}F. J. Duarte, Multiple-prism grating solid-state dye laser oscillator: optimized architecture,*Appl. Opt.***38**, 6347-6349 (1999).