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

Physical Review A

Generalized radiation-field quantization method and the Petermann excess-noise factor

Yuh‐Jen Cheng, Anthony E.. Siegman

Published Oct 8, 2003• 17 citations
Publisher

Abstract

We propose a generalized radiation-field quantization formalism, where quantization does not have to be referenced to a set of power-orthogonal eigenmodes as conventionally required. This formalism can be used to directly quantize the true system eigenmodes, which can be non-power-orthogonal due to the open nature of the system or the gain/loss medium involved in the system. We apply this generalized field quantization to the laser linewidth problem, in particular, lasers with non-power-orthogonal oscillation modes, and derive the excess-noise factor in a fully quantum-mechanical framework. We also show that, despite the excess-noise factor for oscillating modes, the total spatially averaged decay rate for the laser atoms remains unchanged.

DOI
10.1103/physreva.68.043808

Related papers

Theory of the fundamental laser linewidth

Philip Goldberg, Peter W. Milonni, Bala Sundaram

The theory of the laser linewidth is formulated to account for arbitrarily large output couplings and spatial hole burning. We show explicitly that the linewidth can be interpreted in terms of either spontaneous-emission noise or the amplification of vacuum field modes leaking into the cavity, depending on the ordering of operators in the correlation function determining the laser spectrum. This allows us to derive the Petermann K factor associated with ``excess spontaneous-emission noise'' in a physically transparent and mathematically simple way, without the need to introduce adjoint modes of the resonator. It also allows us to straightforwardly include spatial-hole-burning effects, which are found to increase the K factor and the linewidth in high-gain systems appreciably.

Physical Review A•Aug 1, 1991•93 citations

Semiclassical dynamics of excess quantum noise

Martin P. van Exter, N. J. van Druten, A. M. van der Lee +3 more

A semiclassical theoretical framework is presented to describe the essential features of the excess quantum noise that occurs in systems with nonorthogonal eigenmodes. Excess noise is shown to be always spectrally colored, instead of white, so that the Petermann excess noise factor is best written as $K(\ensuremath{\omega})$ instead of K. The consequences of this spectral coloring are analyzed for lasers, both below and above the lasing threshold.

Physical Review A•Mar 5, 2001•10 citations

Theory of excess noise in unstable resonator lasers

Claus Lamprecht, Helmut Ritsch

We theoretically investigate the quantum dynamics of an unstable resonator laser. Compared to a stable cavity laser of equal gain and loss it exhibits a K-fold enhanced linewidth. This excess noise factor K is a measure of the nonorthogonality of the resonator eigenmodes and amounts to an enlargement of the quantum vacuum fluctuations. Using a quantum treatment starting from first principles based on the nonorthogonal eigenmodes, we put previous theoretical predictions onto a more firm ground. While we find a position-dependent enhancement of the spontaneous emission rate into an empty mode of only $\sqrt{K},$ the constructive quantum interference of the spontaneous emission with a single oscillating mode lets the Petermann excess noise factor K reappear in the phase diffusion (laser linewidth). Hence locally enhanced spontaneous emission as well as noise enhanced by interference (amplified spontaneous emission) play an equal role in the origin of excess noise.

Physical Review A•Nov 18, 2002•4 citations

Field quantization, photons and non-Hermitean modes

S. A. Brown, Bryan J. Dalton

Field quantization in unstable optical systems is treated by expanding the vector potential in terms of non-Hermitean (Fox-Li) modes. We define non-Hermitean modes and their adjoints in both the cavity and external regions and make use of the important bi-orthogonality relationships that exist within each mode set. We employ a standard canonical quantization procedure involving the introduction of generalized coordinates and momenta for the electromagnetic (EM) field. Three-dimensional systems are treated, making use of the paraxial and monochromaticity approximations for the cavity non-Hermitean modes. We show that the quantum EM field is equivalent to a set of quantum harmonic oscillators (QHOs), associated with either the cavity or the external region non-Hermitean modes, and thus confirming the validity of the photon model in unstable optical systems. Unlike in the conventional (Hermitean mode) case, the annihilation and creation operators we define for each QHO are not Hermitean adjoints. It is shown that the quantum Hamiltonian for the EM field is the sum of non-commuting cavity and external region contributions, each of which can be expressed as a sum of independent QHO Hamiltonians for each non-Hermitean mode, except that the external field Hamiltonian also includes a coupling term responsible for external non-Hermitean mode photon exchange processes. The non-commutativity of certain cavity and external region annihilation and creation operators is associated with cavity energy gain and loss processes, and may be described in terms of surface integrals involving cavity and external region non-Hermitean mode functions on the cavity-external region boundary. Using the essential states approach and the rotating wave approximation, our results are applied to the spontaneous decay of a two-level atom inside an unstable cavity. We find that atomic transitions leading to cavity non-Hermitean mode photon absorption are associated with a different coupling constant to that for transitions leading to photon emission, a feature consequent on the use of non-Hermitean mode functions. We show that under certain conditions the spontaneous decay rate is enhanced by the Petermann factor.

arXiv:quant-ph/0107039•Journal of Modern Optics•Jun 1, 2002•7 citations

Excess noise in gain-guided amplifiers

Ivan Deutsch, J. C. Garrison, Ewan M. Wright

A second-quantized theory of the radiation field is used to study the origin of the excess noise observed in gain-guided amplifiers. We find that the reduction of the signal-to-noise ratio is a function of the length of the amplifier, and thus the enhancement of the noise is a propagation effect arising from longitudinally inhomogeneous gain of the noise rather than from an excess of local spontaneous emission. We confirm this conclusion by showing that the microscopic rate of spontaneous emission into a given non-power-orthogonal cavity mode is not enhanced by the Petermann factor. In addition, we illustrate the difficulties associated with photon statistics for this and other open systems by showing that no acceptable family of photon-number operators corresponds to a set of non-power-orthogonal cavity modes.

Journal of the Optical Society of America B•Jun 1, 1991•35 citations

A physical explanation of excess quantum noise due to non-orthogonal modes

A M van der Lee, Martin P. van Exter, N. J. van Druten +1 more

We introduce a physical model of excess quantum noise in a laser with non-orthogonal polarization modes. We discuss the cause of the polarization excess quantum noise in terms of an injected wave excitation factor. Within this context the excess noise is due to the fact that the lasing mode is not the polarization state that experiences the highest gain.

New Journal of Physics•Mar 23, 2001•5 citations
We propose a generalized radiation-field quantization formalism, where quantization does not have to be referenced to a set of power-orthogonal eigenmodes as conventionally required. This formalism can be used to directly quantize the true system eigenmodes, which can be non-power-orthogonal due to the open nature of the system or the gain/loss medium involved in the system. We apply this generalized field quantization to the laser linewidth problem, in particular, lasers with non-power-orthogonal oscillation modes, and derive the excess-noise factor in a fully quantum-mechanical framework. We also show that, despite the excess-noise factor for oscillating modes, the total spatially averaged decay rate for the laser atoms remains unchanged.
The theory of the laser linewidth is formulated to account for arbitrarily large output couplings and spatial hole burning. We show explicitly that the linewidth can be interpreted in terms of either spontaneous-emission noise or the amplification of vacuum field modes leaking into the cavity, depending on the ordering of operators in the correlation function determining the laser spectrum. This allows us to derive the Petermann K factor associated with ``excess spontaneous-emission noise'' in a physically transparent and mathematically simple way, without the need to introduce adjoint modes of the resonator. It also allows us to straightforwardly include spatial-hole-burning effects, which are found to increase the K factor and the linewidth in high-gain systems appreciably.
We theoretically investigate the quantum dynamics of an unstable resonator laser. Compared to a stable cavity laser of equal gain and loss it exhibits a K-fold enhanced linewidth. This excess noise factor K is a measure of the nonorthogonality of the resonator eigenmodes and amounts to an enlargement of the quantum vacuum fluctuations. Using a quantum treatment starting from first principles based on the nonorthogonal eigenmodes, we put previous theoretical predictions onto a more firm ground. While we find a position-dependent enhancement of the spontaneous emission rate into an empty mode of only K,\sqrt{K},K​, the constructive quantum interference of the spontaneous emission with a single oscillating mode lets the Petermann excess noise factor K reappear in the phase diffusion (laser linewidth). Hence locally enhanced spontaneous emission as well as noise enhanced by interference (amplified spontaneous emission) play an equal role in the origin of excess noise.
Field quantization in unstable optical systems is treated by expanding the vector potential in terms of non-Hermitean (Fox-Li) modes. We define non-Hermitean modes and their adjoints in both the cavity and external regions and make use of the important bi-orthogonality relationships that exist within each mode set. We employ a standard canonical quantization procedure involving the introduction of generalized coordinates and momenta for the electromagnetic (EM) field. Three-dimensional systems are treated, making use of the paraxial and monochromaticity approximations for the cavity non-Hermitean modes. We show that the quantum EM field is equivalent to a set of quantum harmonic oscillators (QHOs), associated with either the cavity or the external region non-Hermitean modes, and thus confirming the validity of the photon model in unstable optical systems. Unlike in the conventional (Hermitean mode) case, the annihilation and creation operators we define for each QHO are not Hermitean adjoints. It is shown that the quantum Hamiltonian for the EM field is the sum of non-commuting cavity and external region contributions, each of which can be expressed as a sum of independent QHO Hamiltonians for each non-Hermitean mode, except that the external field Hamiltonian also includes a coupling term responsible for external non-Hermitean mode photon exchange processes. The non-commutativity of certain cavity and external region annihilation and creation operators is associated with cavity energy gain and loss processes, and may be described in terms of surface integrals involving cavity and external region non-Hermitean mode functions on the cavity-external region boundary. Using the essential states approach and the rotating wave approximation, our results are applied to the spontaneous decay of a two-level atom inside an unstable cavity. We find that atomic transitions leading to cavity non-Hermitean mode photon absorption are associated with a different coupling constant to that for transitions leading to photon emission, a feature consequent on the use of non-Hermitean mode functions. We show that under certain conditions the spontaneous decay rate is enhanced by the Petermann factor.
A second-quantized theory of the radiation field is used to study the origin of the excess noise observed in gain-guided amplifiers. We find that the reduction of the signal-to-noise ratio is a function of the length of the amplifier, and thus the enhancement of the noise is a propagation effect arising from longitudinally inhomogeneous gain of the noise rather than from an excess of local spontaneous emission. We confirm this conclusion by showing that the microscopic rate of spontaneous emission into a given non-power-orthogonal cavity mode is not enhanced by the Petermann factor. In addition, we illustrate the difficulties associated with photon statistics for this and other open systems by showing that no acceptable family of photon-number operators corresponds to a set of non-power-orthogonal cavity modes.