The acronym MASER stands for "Microwave Amplification by Stimulated Emission of Radiation".
More precisely, it was the ammonia maser operating at 12.5 mm wavelength that was demonstrated by Gordon, Zeiger, and Townes in 1954.
[1] One year later the same authors derived[2] theoretically the linewidth of their device by making the reasonable approximations that their ammonia maser Notably, their derivation was entirely semi-classical,[2] describing the ammonia molecules as quantum emitters and assuming classical electromagnetic fields (but no quantized fields or quantum fluctuations), resulting in the half-width-at-half-maximum (HWHM) maser linewidth[2] denoted here by an asterisk and converted to the full-width-at-half-maximum (FWHM) linewidth
are the HWHM and FWHM linewidths of the underlying passive microwave resonator, respectively.
In 1958, two years before Maiman demonstrated the laser (initially called an "optical maser"),[3] Schawlow and Townes[4] transferred the maser linewidth to the optical regime by replacing the thermal energy
Consequently, the original Schawlow–Townes equation is entirely based on semi-classical physics[2][4] and is a four-fold approximation of a more general laser linewidth,[5] which will be derived in the following.
, homogeneously filled with an active laser medium of refractive index
, the spectral-coherence properties of the passive resonator mode can be equivalently expressed by the following parameters.
of the passive resonator mode that appears in the Schawlow–Townes equation is derived from the exponential photon-decay time
of upper and lower laser level, respectively, and the effective cross sections
, respectively, the gain per unit length in the active laser medium at the resonance frequency
of photons propagating inside the lasing resonator mode, the stimulated-emission and photon-decay rates are, respectively,[5] The spectral-coherence factor then becomes[5] The photon-decay time of the lasing resonator mode is[5] The fundamental laser linewidth is[5] This fundamental linewidth is valid for lasers with an arbitrary energy-level system, operating below, at, or above threshold, with the gain being smaller, equal, or larger compared to the losses, and in a cw or a transient lasing regime.
[5] It becomes clear from its derivation that the fundamental laser linewidth is due to the semi-classical effect that the gain elongates the photon-decay time.
is always a positive rate, because one atomic excitation is converted into one photon in the lasing mode.
[5] This fact has been known for decades and exploited to quantify the threshold behavior of semiconductor lasers.
[8][9][10][11] Even far above laser threshold the gain is still a tiny bit smaller than the losses.
It is exactly this small difference that induces the finite linewidth of a CW laser.
[5] It becomes clear from this derivation that fundamentally the laser is an amplifier of spontaneous emission, and the cw laser linewidth is due to the semi-classical effect that the gain is smaller than the losses.
[5] Also in the quantum-optical approaches to the laser linewidth,[12] based on the density-operator master equation, it can be verified that the gain is smaller than the losses.
[5] As mentioned above, it is clear from its historical derivation that the original Schawlow–Townes equation is a four-fold approximation of the fundamental laser linewidth.
These extended equations often trade under the same name, the "Schawlow–Townes linewidth", thereby creating a veritable confusion in the available literature on the laser linewidth, as it is often unclear which particular extension of the original Schawlow–Townes equation the respective authors refer to.
mentioned above, thereby making steps towards the fundamental laser linewidth derived above.
[21] A typical method to measure the laser linewidth is self-heterodyne interferometry.
Rare-earth-doped dielectric-based or semiconductor-based distributed-feedback lasers have typical linewidths on the order of 1 kHz.
[27] Observed linewidths are larger than the fundamental laser linewidth due to technical noise (temporal fluctuations of the optical pump power or pump current, mechanical vibrations, refractive-index and length changes due to temperature fluctuations, etc.).
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[28] to as broad as 10 nm.
[29] 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.
is the beam divergence and the term in parentheses (elevated to −1) is the overall intracavity dispersion.
An optimized multiple-prism grating laser oscillator can deliver pulse emission in the kW regime at single-longitudinal-mode linewidths of
[32] Since the pulse duration from these oscillators is about 3 ns,[32] the laser linewidth performance is near the limit allowed by the Heisenberg uncertainty principle.