# Transverse oscillation in laser cavities

## Abstract

Recent experiments on lasers with large Fresnel numbers have demonstrated a close correspondence between the transverse degrees of freedom (transverse modes) and the two-dimensional harmonic oscillator. Mode patterns that correspond to classical trajectories have been observed. These experiments suggest that such systems may serve as convenient platforms for studying quantum-classical correspondence. But why the paraxial wave equation for the laser leads to two-dimensional harmonic oscillation for the transverse degrees of freedom is yet to be clarified. We use two methods to show that such a correspondence is not a mathematical coincidence: the focusing mirror that stabilizes the laser gives rise to the transverse oscillation, and the reduced equation that governs the transverse degrees of freedom is the Schroedinger equation for the harmonic oscillator. For the fundamental Gaussian mode, the theory remains valid in the weakly nonlinear limit.

- Authors:

- Department of Mathematics, Tunghai University, Taichung 407, Taiwan (China)
- (China)

- Publication Date:

- OSTI Identifier:
- 20786984

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 73; Journal Issue: 3; Other Information: DOI: 10.1103/PhysRevA.73.033804; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; DEGREES OF FREEDOM; HARMONIC OSCILLATORS; LASER CAVITIES; LASER MIRRORS; LASER RADIATION; NONLINEAR PROBLEMS; OPTICS; OSCILLATIONS; RESONATORS; SCHROEDINGER EQUATION; STABILITY; TWO-DIMENSIONAL CALCULATIONS

### Citation Formats

```
Tsaur Ginyih, Wang Jyhpyng, and Institute of Atomic and Molecular Sciences, Academia Sinica, P. O. Box 23-166, Taipei 106, Taiwan.
```*Transverse oscillation in laser cavities*. United States: N. p., 2006.
Web. doi:10.1103/PHYSREVA.73.0.

```
Tsaur Ginyih, Wang Jyhpyng, & Institute of Atomic and Molecular Sciences, Academia Sinica, P. O. Box 23-166, Taipei 106, Taiwan.
```*Transverse oscillation in laser cavities*. United States. doi:10.1103/PHYSREVA.73.0.

```
Tsaur Ginyih, Wang Jyhpyng, and Institute of Atomic and Molecular Sciences, Academia Sinica, P. O. Box 23-166, Taipei 106, Taiwan. Wed .
"Transverse oscillation in laser cavities". United States.
doi:10.1103/PHYSREVA.73.0.
```

```
@article{osti_20786984,
```

title = {Transverse oscillation in laser cavities},

author = {Tsaur Ginyih and Wang Jyhpyng and Institute of Atomic and Molecular Sciences, Academia Sinica, P. O. Box 23-166, Taipei 106, Taiwan},

abstractNote = {Recent experiments on lasers with large Fresnel numbers have demonstrated a close correspondence between the transverse degrees of freedom (transverse modes) and the two-dimensional harmonic oscillator. Mode patterns that correspond to classical trajectories have been observed. These experiments suggest that such systems may serve as convenient platforms for studying quantum-classical correspondence. But why the paraxial wave equation for the laser leads to two-dimensional harmonic oscillation for the transverse degrees of freedom is yet to be clarified. We use two methods to show that such a correspondence is not a mathematical coincidence: the focusing mirror that stabilizes the laser gives rise to the transverse oscillation, and the reduced equation that governs the transverse degrees of freedom is the Schroedinger equation for the harmonic oscillator. For the fundamental Gaussian mode, the theory remains valid in the weakly nonlinear limit.},

doi = {10.1103/PHYSREVA.73.0},

journal = {Physical Review. A},

number = 3,

volume = 73,

place = {United States},

year = {Wed Mar 15 00:00:00 EST 2006},

month = {Wed Mar 15 00:00:00 EST 2006}

}