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 linewidth of laser radiation.
Theory
History: First derivation of the laser linewidth
The first human-made coherent light source was a maser. 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. One year later the same authors derived theoretically the linewidth of their device by making the reasonable approximations that their ammonia maser
Notably, their derivation was entirely semi-classical, 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
denoted here by an asterisk and converted to the full-width-at-half-maximum (FWHM) linewidth . is the Boltzmann constant, is the temperature, is the output power, and and 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"), Schawlow and Townes transferred the maser linewidth to the optical regime by replacing the thermal energy by the photon energy , where is the Planck constant and is the frequency of laser light, thereby approximating that
iv. one photon is coupled into the lasing mode by spontaneous emission during the photon-decay time ,
resulting in the original Schawlow–Townes approximation of the laser linewidth:
Again, the transfer from the microwave to the optical regime was entirely semi-classical. Consequently, the original Schawlow–Townes equation is entirely based on semi-classical physics and is a four-fold approximation of a more general laser linewidth, which will be derived in the following.
Passive resonator mode: Photon-decay time
We assume a two-mirror Fabry–Pérot resonator of geometrical length , homogeneously filled with an active laser medium of refractive index . We define the reference situation, namely the passive resonator mode, for a resonator whose active medium is transparent, i.e., it does not introduce gain or absorption.
The round-trip time of light travelling in the resonator with speed , where is the speed of light in vacuum, and the free spectral range are given by
Light in the longitudinal resonator mode of interest oscillates at the qth resonance frequency
The exponential outcoupling decay time and the corresponding decay-rate constant are related to the intensity reflectances of the two resonator mirrors by
The exponential intrinsic loss time and the corresponding decay-rate constant are related to the intrinsic round-trip loss by
The exponential photon-decay time and the corresponding decay-rate constant of the passive resonator are then given by
All three exponential decay times average over the round-trip time In the following, we assume that , , , , and , hence also , , and do not vary significantly over the frequency range of interest.
Passive resonator mode: Lorentzian linewidth, Q-factor, coherence time and length
Besides the photon-decay time , the spectral-coherence properties of the passive resonator mode can be equivalently expressed by the following parameters. The FWHM Lorentzian linewidth of the passive resonator mode that appears in the Schawlow–Townes equation is derived from the exponential photon-decay time by Fourier transformation,
The Q-factor is defined as the energy stored in the resonator mode over the energy lost per oscillation cycle,
where is the number of photons in the mode. The coherence time and coherence length of light emitted from the mode are given by
Active resonator mode: Gain, photon-decay time, Lorentzian linewidth, Q-factor, coherence time and length
With the population densities and of upper and lower laser level, respectively, and the effective cross sections and of stimulated emission and absorption at the resonance frequency , respectively, the gain per unit length in the active laser medium at the resonance frequency is given by
A value of induces amplification, whereas induces absorption of light at the resonance frequency , resulting in an elongated or shortened photon-decay time of photons out of the active resonator mode, respectively,