Spectral coherence, Part I: Passiveresonator linewidth, fundamental laser linewidth, and SchawlowTownes approximation
Abstract
The degree of spectral coherence characterizes the spectral purity of light. It can be equivalently expressed in the time domain by the decay time τ or the quality factor Q of the lightemitting oscillator, the coherence time τ^{coh} or length ℓ^{coh} of emitted light or, via Fourier transformation to the frequency domain, the linewidth Δν of emitted light. We quantify these parameters for the reference situation of a passive FabryPérot resonator. We investigate its spectral line shapes, mode profiles, and Airy distributions and verify that the sum of all mode profiles generates the corresponding Airy distribution. The FabryPérot resonator is described, as an oscillator, by its Lorentzian linewidth and finesse and, as a scanning spectrometer, by its Airy linewidth and finesse. Furthermore, stimulated and spontaneous emission are analyzed semiclassically by employing Maxwell‧s equations and the law of energy conservation. Investigation of emission by atoms inside a FabryPérot resonator, the Lorentz oscillator model, the KramersKronig relations, the amplitudephase diagram, and the summation of quantized electric fields consistently suggests that stimulated and spontaneous emission of light occur with a phase 90° in lead of the incident field. These findings question the quantumoptical picture, which proposed, firstly, that stimulated emission occurred in phase, whereas spontaneous emission occurred at an arbitrary phase angle with respect to the incident field and, secondly, that the laser linewidth were due to amplitude and phase fluctuations induced by spontaneous emission. We emphasize that the first derivation of the SchawlowTownes laser linewidth was entirely semiclassical but included the four approximations that (i) it is a truly continuouswave (cw) laser, (ii) it is an ideal fourlevel laser, (iii) its resonator exhibits no intrinsic losses, and (iv) one photon is coupled spontaneously into the lasing mode per photondecay time τ_{c} of the resonator, independent of the pump rate. After discussing the inconsistencies of existing semiclassical and quantumoptical descriptions of the laser linewidth, we introduce the spectralcoherence factor, which quantifies spectral coherence in an active compared to its underlying passive mode, and derive semiclassically, based on the principle that the gain elongates the photondecay time and narrows the linewidth, the fundamental linewidth of a single lasing mode. This linewidth is valid for lasers with an arbitrary energylevel system, operating below, at, or above threshold and in a cw or a transient lasing regime, with the gain being smaller, equal, or larger compared to the losses. By applying approximations (i)(iv) we reproduce the original SchawlowTownes equation. It provides the hitherto missing link between the description of the laser as an amplifier of spontaneous emission and the SchawlowTownes equation. Spontaneous emission entails that in a cw lasing mode the gain is smaller than the losses. We verify that also in the quantumoptical approaches to the laser linewidth, based on the densityoperator master equation, the gain is smaller than the losses. We conclude this work by presenting the derivation of the laser linewidth in a nut shell.
 Publication:

Progress in Quantum Electronics
 Pub Date:
 August 2020
 DOI:
 10.1016/j.pquantelec.2020.100255
 Bibcode:
 2020PQE....7200255P
 Keywords:

 Spectral coherence;
 Optical resonance;
 FabryPérot resonator;
 Resonator linewidth;
 Finesse;
 Lasers;
 Laser theory;
 Laser resonators;
 Laser linewidth;
 SchawlowTownes equation