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A fresh look at continuous-wave lasers: How they really work!

Markus Pollnau

Published Oct 1, 2013• 0 citations

Abstract

This lecture describes the operation principle of a continuous-wave (cw) laser. In order to keep the photon rate equation and its solution simple, usually the spontaneous-emission rate is neglected with the argument that it is so much smaller than the stimulated-emission rate. The direct consequence is that in a cw laser the gain would equal the losses. Yet, additional implications are that the light emitted by such a laser would be a pure sine wave with an infinite coherence length, its linewidth would become a delta function, its Q-factor would assume an infinite value, the coherent photon number would build up and coherence would manifest itself inside the resonator only when pumping above the laser threshold, and the threshold inversion would depend only on the total resonator losses. None of these implications holds true for any laser that mankind has ever created. Starting from vacuum fluctuations, we consider spontaneous emission directly in the photon rate equation, thereby a priori avoiding all these inconsistencies. It is then straight-forward to see that in a cw laser the gain is smaller than the losses, a cw laser does not only have a finite linewidth that can be derived in a very simple manner, but also a finite Q-factor, as well as two laser thresholds which are both different from the commonly assumed “threshold inversion‿. This work was performed in collaboration with Dr. Marc Eichhorn from the French-German Research Institute of St. Louis, France.

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This lecture describes the operation principle of a continuous-wave (cw) laser. In order to keep the photon rate equation and its solution simple, usually the spontaneous-emission rate is neglected with the argument that it is so much smaller than the stimulated-emission rate. The direct consequence is that in a cw laser the gain would equal the losses. Yet, additional implications are that the light emitted by such a laser would be a pure sine wave with an infinite coherence length, its linewidth would become a delta function, its Q-factor would assume an infinite value, the coherent photon number would build up and coherence would manifest itself inside the resonator only when pumping above the laser threshold, and the threshold inversion would depend only on the total resonator losses. None of these implications holds true for any laser that mankind has ever created. Starting from vacuum fluctuations, we consider spontaneous emission directly in the photon rate equation, thereby a priori avoiding all these inconsistencies. It is then straight-forward to see that in a cw laser the gain is smaller than the losses, a cw laser does not only have a finite linewidth that can be derived in a very simple manner, but also a finite Q-factor, as well as two laser thresholds which are both different from the commonly assumed “threshold inversion‿. This work was performed in collaboration with Dr. Marc Eichhorn from the French-German Research Institute of St. Louis, France.

We extend the existing theory of continuous-wave lasers by systematically considering spontaneous emission. In a simple rate-equation approach, the laser eigenvalue, defined as the ratio of coherent photons coupled out of the resonator divided by the number of photons coupled in via spontaneous emission, emerges as the fundamental parameter describing a continuous-wave laser. We derive a general equation for the laser eigenvalue and confirm the point at which coherence manifests itself in the resonator. The theory describes all types of lasers of three-level, four-level, or any intermediate nature.

We review the physics underlying the process of spontaneous emission, with a special focus on spontaneous emission into a resonator mode. We define the mode volume, verify the fundamental modal dimensions, present the spectral mode profile, the coherence time, the Q-factor, the Füchtbauer-Ladenburg equation, and the Purcell factor, and discuss their influence on different types of lasers. We obtain the relation between peak emission cross section, radiative lifetime, and emission linewidth. By interpreting spontaneous emission as stimulated emission driven by vacuum fluctuations, we derive the spontaneous-emission rate into a resonator mode and establish physical expressions for the fractions of spontaneous emission and total decay from the upper laser level into this mode. Furthermore, we discuss coupling of the atomic system with the coherent field inside a lasing resonator mode, resulting in the formation of a Mollow triplet, and demonstrate that it leads to a reduction of the spontaneous-emission rate by a factor of 2.

The processes of absorption and stimulated emission are investigated via (i) energy conservation in a Fabry-Perot resonator, (ii) the Lorentz-oscillator model, and (iii) the Kramers-Kronig relations. Vacuum fluctuations are added and the relation between stimulated and spontaneous emission is clarified. It is shown that previous models start from incorrect assumptions about the physical foundations of spontaneous emission. The non-orthogonality of longitudinal optical modes in a linear resonator is investigated and its consequences for the laser linewidth are discussed. Based on these results, a semi-classical approach to the laser linewidth below, around, and above the laser threshold which is consistent with resonator theory is presented. These results also shed new light on the usually considered linewidth-narrowing and -broadening factors.

A photon, unlike an electron, can stimulate an excited atom to emit a second photon into the same electromagnetic mode. This stimulated emission process is the foundation for light amplification and oscillation (i.e., self-generation). Initially, the term LASER referred to Light Amplification by Stimulated Emission of Radiation. Nowadays laser often means Light Oscillation by Stimulated Emission, which should literally be called "LOSER" instead of "LASER." To distinguish the above two devices, the former is called a laser amplifier, the latter a laser oscillator (Siegman 1986). Laser oscillation occurs when the photon generation rate exceeds the photon loss rate in a system. If gain saturation were absent, the photon number in a laser oscillator would diverge in time. In other words, the rate equation for the photon number would acquire an unstable solution above the oscillation threshold. In reality, gain saturation reduces the photon generation rate to the photon loss rate so that the number of photons in the oscillator remains at a finite value.

The standard quantum limit to the linewidth of a laser for which the gain medium can be adiabatically eliminated is ${l}_{0}=\ensuremath{\kappa}/2\mathrm{n\ifmmode \bar{}\else \={}\fi{}}.$ Here $\ensuremath{\kappa}$ is the intensity damping rate and $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ the mean photon number. This contains equal contributions from the loss and gain processes, so that simple arguments which attribute the linewidth wholly to phase noise from spontaneous gain are wrong. I show that an unstimulated gain process actually introduces no phase noise, so that the ultimate quantum limit to the linewidth comes from the loss alone and is equal to ${l}_{\mathrm{ult}}=\ensuremath{\kappa}/4\mathrm{n\ifmmode \bar{}\else \={}\fi{}}.$ I investigate a number of physical gain mechanisms which attempt to achieve gain without phase noise: a linear atom-field coupling with a finite interaction time, a nonlinear atom-field coupling, and adiabatic photon transfer using a counterintuitive pulse sequence. The first at best reaches the standard limit

{l}_{0},

the second reaches

\frac{3}{4}{l}_{0},

and the third reaches the ultimate limit of

{l}_{\mathrm{ult}}=\frac{1}{2}{l}_{0}.