Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Coherence manifestation in a continuous-wave laser

Marc Eichhorn, Markus Pollnau

Published May 1, 2014• 0 citations
Publisher

Abstract

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.

DOI
10.1117/12.2052081

Related papers

Laser Eigenvalue, Coherence Time, Q-factor, and Linewidth

Markus Pollnau, Marc Eichhorn

We generalize the theory of continuous-wave lasers by considering spontaneous emission in the photon rate equation. We relate the laser coherence time, Q-factor, and linewidth to the passive-resonator parameters, thereby unifying resonator and laser theory.

•Jan 1, 2015•2 citations

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

Markus Pollnau

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.

•Oct 1, 2013•0 citations

Spectroscopic Foundations of Lasers: Spontaneous Emission Into a Resonator Mode

Marc Eichhorn, Markus Pollnau

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.

IEEE Journal of Selected Topics in Quantum Electronics•Oct 9, 2014•26 citations

Cone emission from laser-pumped two-level atoms. I. Quantum theory of resonant light propagation

Li You, Jan Mostowski, J. Cooper

This is the first of a series of papers on the theory of cone emission from a laser-pumped two-level atomic medium. Starting from microscopic quantum theory for both the active medium and the field, equations for the slowly varying macroscopic quantities are derived. In the steady-state limit, which experimentally corresponds to cw excitation, we find that the sources for the paraxial fields of interest have a long coherence length despite the random nature of spontaneous emission. These radiation sources enter into the equations for the fields in the same form as the source generated by four-wave mixing and therefore may be considered as a ``spontaneous four-wave mixing,'' since the photons emitted in the Rabi sidebands are correlated. This has far-reaching consequences on the physics of cone emission and also sheds some new light on the fluoresence from strongly driven atomic systems.

Physical Review A•Sep 1, 1992•20 citations

Quantum theory of two-photon correlated-spontaneous-emission lasers: Exact atom-field interaction Hamiltonian approach

Ning Lü, Shi-Yao Zhu

A quantum theory of two-photon correlated-spontaneous-emission lasers (CEL's) is developed, starting from the exact atom-field interaction Hamiltonian for cascade three-level atoms interacting with a single-mode radiation field. We consider the situation where the active atoms are prepared initially in a coherent superposition of three atomic levels and derive a master equation for the field-density operator by using a quantum theory for coherently pumped lasers. The master equation is transformed into a Fokker-Planck equation for the antinormal-ordering Q function. The drift coefficients of the Fokker-Planck equation enable us to study the steady-state operation of the two-photon CEL's analytically. We have studied both resonant two-photon CEL for which there is no threshold, and off-resonant two-photon CEL for which there exists a threshold. In both cases the initial atomic coherences provide phase locking, and squeezing in the phase quadrature of the field is found. The off-resonant two-photon CEL can build up from a vacuum when its linear gain is larger than the cavity loss (even without population inversion). Maximum squeezing is found in the no-population-inversion region with the laser intensities far below saturation in both cases, which are more than 90% for the resonant two-photon CEL and nearly 50% for the off-resonant one. Approximate steady-state Q functions are obtained for the resonant two-photon CEL and, in certain circumstances, for the off-resonant one.

Physical Review A•Nov 1, 1989•38 citations

Nonlinear theory of a two-photon correlated-spontaneous-emission laser: A coherently pumped two-level–two-photon laser

Ning Lü, Fang-Xiao Zhao, János A. Bergou

We develop a nonlinear theory of a two-photon correlated-spontaneous-emission laser (CEL) by using an effective interaction Hamiltonian for a two-level system coupled by a two-photon transition. Assuming that the active atoms are prepared initially in a coherent superposition of two atomic levels involved in the two-photon transition, we derive a master equation for the field-density operator by using our quantum theory for coherently pumped lasers. The steady-state properties of the two-photon CEL are studied by converting the field master equation into a Fokker-Planck equation for the antinormal-ordering Q representation of the field-density operator. Because of the injected atomic coherence, the drift and diffusion coefficients become phase sensitive. This leads to laser phase locking and an extra two-photon CEL gain. The laser field can build up from a vacuum in the no-population-inversion region, in contrast to an ordinary two-photon laser for which triggering is needed. We find an approximate steady-state solution of the Q representation for the laser field, which consists of two identical peaks of elliptical type. We calculate the phase variance and, for any given mean photon number, obtain the minimum variance in the phase quadrature as a function of the initial atomic variables. Squeezing of the quantum noise in the phase quadrature is found and it exhibits the following features: (1) it is possible only when the laser intensity is smaller than a certain value; (2) it becomes most significant for small mean photon number, which is achievable in the no-population-inversion region; and (3) a maximum of 50% squeezing can be asymptotically approached in the small laser intensity limit. As a by-product we also study the ordinary two-photon laser and find, e.g., photon-number variance and laser linewidth.

Physical Review A•May 1, 1989•69 citations
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.
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 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.
This is the first of a series of papers on the theory of cone emission from a laser-pumped two-level atomic medium. Starting from microscopic quantum theory for both the active medium and the field, equations for the slowly varying macroscopic quantities are derived. In the steady-state limit, which experimentally corresponds to cw excitation, we find that the sources for the paraxial fields of interest have a long coherence length despite the random nature of spontaneous emission. These radiation sources enter into the equations for the fields in the same form as the source generated by four-wave mixing and therefore may be considered as a ``spontaneous four-wave mixing,'' since the photons emitted in the Rabi sidebands are correlated. This has far-reaching consequences on the physics of cone emission and also sheds some new light on the fluoresence from strongly driven atomic systems.
A quantum theory of two-photon correlated-spontaneous-emission lasers (CEL's) is developed, starting from the exact atom-field interaction Hamiltonian for cascade three-level atoms interacting with a single-mode radiation field. We consider the situation where the active atoms are prepared initially in a coherent superposition of three atomic levels and derive a master equation for the field-density operator by using a quantum theory for coherently pumped lasers. The master equation is transformed into a Fokker-Planck equation for the antinormal-ordering Q function. The drift coefficients of the Fokker-Planck equation enable us to study the steady-state operation of the two-photon CEL's analytically. We have studied both resonant two-photon CEL for which there is no threshold, and off-resonant two-photon CEL for which there exists a threshold. In both cases the initial atomic coherences provide phase locking, and squeezing in the phase quadrature of the field is found. The off-resonant two-photon CEL can build up from a vacuum when its linear gain is larger than the cavity loss (even without population inversion). Maximum squeezing is found in the no-population-inversion region with the laser intensities far below saturation in both cases, which are more than 90% for the resonant two-photon CEL and nearly 50% for the off-resonant one. Approximate steady-state Q functions are obtained for the resonant two-photon CEL and, in certain circumstances, for the off-resonant one.
We develop a nonlinear theory of a two-photon correlated-spontaneous-emission laser (CEL) by using an effective interaction Hamiltonian for a two-level system coupled by a two-photon transition. Assuming that the active atoms are prepared initially in a coherent superposition of two atomic levels involved in the two-photon transition, we derive a master equation for the field-density operator by using our quantum theory for coherently pumped lasers. The steady-state properties of the two-photon CEL are studied by converting the field master equation into a Fokker-Planck equation for the antinormal-ordering Q representation of the field-density operator. Because of the injected atomic coherence, the drift and diffusion coefficients become phase sensitive. This leads to laser phase locking and an extra two-photon CEL gain. The laser field can build up from a vacuum in the no-population-inversion region, in contrast to an ordinary two-photon laser for which triggering is needed. We find an approximate steady-state solution of the Q representation for the laser field, which consists of two identical peaks of elliptical type. We calculate the phase variance and, for any given mean photon number, obtain the minimum variance in the phase quadrature as a function of the initial atomic variables. Squeezing of the quantum noise in the phase quadrature is found and it exhibits the following features: (1) it is possible only when the laser intensity is smaller than a certain value; (2) it becomes most significant for small mean photon number, which is achievable in the no-population-inversion region; and (3) a maximum of 50% squeezing can be asymptotically approached in the small laser intensity limit. As a by-product we also study the ordinary two-photon laser and find, e.g., photon-number variance and laser linewidth.