### Abstract

Starting from the Maxwell-Bloch equations for a three-dimensional (3D) ring-cavity laser, we analyze stability of the nonlasing state and demonstrate that, at the instability threshold, the wave vectors of the critical perturbations belong to a paraboloid in the 3D space. Then, we derive a system of nonlinear evolution equations above the threshold. The nonlinearity in these equations is cubic. For certain sets of four spatial modes whose vectors belong to the critical paraboloid, the cubic nonlinearity gives rise to a resonant coupling between them. This is a nontrivial example of a nonlinear dissipative system in which the cubic terms are resonant. The equations for the four coupled amplitudes have two different solutions that are simultaneously stable: the single-mode one and a solution in which all the amplitudes are equal, while a certain combination of the phases is π. The latter solution gives rise to a quasiperiodic pattern in the infinite 3D cavity. We also consider effects of the boundary conditions and demonstrate that if the cavity's cross section is a trapezium it may support the quasiperiodic four-mode state rather than suppressing it. Using the Lyapunov function, we find that for the ring-laser configuration the four-mode state is metastable. However, we demonstrate that for a Fabry-Pérot cavity, where diffusion washes out the standing-wave grating, this state is absolutely stable. We also consider a number of more complicated patterns. We demonstrate that adding a pair of resonant vectors, or any number of nonresonant ones, always produces an unstable solution. A set containing several resonant quartets without resonant coupling between them may be stable, but it is less energetically favorable than a single quartet.

Original language | English (US) |
---|---|

Pages (from-to) | 803-812 |

Number of pages | 10 |

Journal | Physical Review A |

Volume | 56 |

Issue number | 1 |

State | Published - 1997 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Atomic and Molecular Physics, and Optics

### Cite this

*Physical Review A*,

*56*(1), 803-812.

**Resonant quasiperiodic patterns in a three-dimensional lasing medium.** / Komarova, Natalia L.; Malomed, Boris A.; Moloney, Jerome V; Newell, Alan C.

Research output: Contribution to journal › Article

*Physical Review A*, vol. 56, no. 1, pp. 803-812.

}

TY - JOUR

T1 - Resonant quasiperiodic patterns in a three-dimensional lasing medium

AU - Komarova, Natalia L.

AU - Malomed, Boris A.

AU - Moloney, Jerome V

AU - Newell, Alan C

PY - 1997

Y1 - 1997

N2 - Starting from the Maxwell-Bloch equations for a three-dimensional (3D) ring-cavity laser, we analyze stability of the nonlasing state and demonstrate that, at the instability threshold, the wave vectors of the critical perturbations belong to a paraboloid in the 3D space. Then, we derive a system of nonlinear evolution equations above the threshold. The nonlinearity in these equations is cubic. For certain sets of four spatial modes whose vectors belong to the critical paraboloid, the cubic nonlinearity gives rise to a resonant coupling between them. This is a nontrivial example of a nonlinear dissipative system in which the cubic terms are resonant. The equations for the four coupled amplitudes have two different solutions that are simultaneously stable: the single-mode one and a solution in which all the amplitudes are equal, while a certain combination of the phases is π. The latter solution gives rise to a quasiperiodic pattern in the infinite 3D cavity. We also consider effects of the boundary conditions and demonstrate that if the cavity's cross section is a trapezium it may support the quasiperiodic four-mode state rather than suppressing it. Using the Lyapunov function, we find that for the ring-laser configuration the four-mode state is metastable. However, we demonstrate that for a Fabry-Pérot cavity, where diffusion washes out the standing-wave grating, this state is absolutely stable. We also consider a number of more complicated patterns. We demonstrate that adding a pair of resonant vectors, or any number of nonresonant ones, always produces an unstable solution. A set containing several resonant quartets without resonant coupling between them may be stable, but it is less energetically favorable than a single quartet.

AB - Starting from the Maxwell-Bloch equations for a three-dimensional (3D) ring-cavity laser, we analyze stability of the nonlasing state and demonstrate that, at the instability threshold, the wave vectors of the critical perturbations belong to a paraboloid in the 3D space. Then, we derive a system of nonlinear evolution equations above the threshold. The nonlinearity in these equations is cubic. For certain sets of four spatial modes whose vectors belong to the critical paraboloid, the cubic nonlinearity gives rise to a resonant coupling between them. This is a nontrivial example of a nonlinear dissipative system in which the cubic terms are resonant. The equations for the four coupled amplitudes have two different solutions that are simultaneously stable: the single-mode one and a solution in which all the amplitudes are equal, while a certain combination of the phases is π. The latter solution gives rise to a quasiperiodic pattern in the infinite 3D cavity. We also consider effects of the boundary conditions and demonstrate that if the cavity's cross section is a trapezium it may support the quasiperiodic four-mode state rather than suppressing it. Using the Lyapunov function, we find that for the ring-laser configuration the four-mode state is metastable. However, we demonstrate that for a Fabry-Pérot cavity, where diffusion washes out the standing-wave grating, this state is absolutely stable. We also consider a number of more complicated patterns. We demonstrate that adding a pair of resonant vectors, or any number of nonresonant ones, always produces an unstable solution. A set containing several resonant quartets without resonant coupling between them may be stable, but it is less energetically favorable than a single quartet.

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M3 - Article

VL - 56

SP - 803

EP - 812

JO - Physical Review A

JF - Physical Review A

SN - 2469-9926

IS - 1

ER -