## Mode calculations in asymmetrically aberrated laser resonators using the Huygens–Fresnel kernel formulation |

Optics Express, Vol. 19, Issue 20, pp. 19702-19707 (2011)

http://dx.doi.org/10.1364/OE.19.019702

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### Abstract

A theoretical framework is presented for calculating three-dimensional resonator modes of both stable and unstable laser resonators. The resonant modes of an optical resonator are computed using a kernel formulation of the resonator round-trip Huygens–Fresnel diffraction integral. To substantiate the validity of this method, both stable and unstable resonator mode results are presented. The predicted lowest loss and higher order modes of a semi-confocal stable resonator are in agreement with the analytic formulation. Higher order modes are determined for an asymmetrically aberrated confocal unstable resonator, whose lowest loss unaberrated mode is consistent with published results. The three-dimensional kernel method provides a means to evaluate multi-mode configurations with two-dimensional aberrations that cannot be decomposed into one-dimensional representations.

© 2011 OSA

## 1. Introduction

1. A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE **53**(3), 277–287 (1965). [CrossRef]

2. A. E. Siegman and R. W. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. **3**(4), 156–163 (1967). [CrossRef]

3. A. E. Siegman, “Unstable optical resonators,” Appl. Opt. **13**(2), 353–367 (1974). [CrossRef] [PubMed]

4. A. E. Siegman and H. Y. Miller, “Unstable optical resonator loss calculations using the prony method,” Appl. Opt. **9**(12), 2729–2736 (1970). [CrossRef] [PubMed]

7. L. W. Chen and L. B. Felsen, “Coupled-mode theory of unstable resonators,” IEEE J. Quantum Electron. **9**(11), 1102–1113 (1973). [CrossRef]

3. A. E. Siegman, “Unstable optical resonators,” Appl. Opt. **13**(2), 353–367 (1974). [CrossRef] [PubMed]

4. A. E. Siegman and H. Y. Miller, “Unstable optical resonator loss calculations using the prony method,” Appl. Opt. **9**(12), 2729–2736 (1970). [CrossRef] [PubMed]

## 2. Three-dimensional Huygens-Fresnel kernel formulation

*γ*is the eigenvalue associated with the complex amplitude (eigenvector)

*U*,

*λ*is the wavelength of the circulating light,

*L*is the length of the resonator,

*a*and

*b*of the

*x*and

*y*dimension as

*K*is the resonator kernel broken into

*x*and

*y*components, with eigenmodes

*Z*is the phase aberration, and

*W*provides a means to study variable reflectivity mirrors [10

10. V. Magni, G. Valentini, and S. De Silvestri, “Recent developments in laser resonator design,” Opt. Quantum Electron. **23**(9), 1105–1134 (1991). [CrossRef]

*w*are unity over the mirror diameter representing a hard edge.

## 3. Aberrated optical resonator studies

*L*= 10 m, and

*λ*= 1 μm. Results of the three-dimensional kernel calculation are shown in Fig. 1 . The three-dimensional mode order is from left to right, top to bottom, or inferred from the number of nulls along each axis. The beam waist of the fundamental mode (

11. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. **5**(10), 1550–1567 (1966). [CrossRef] [PubMed]

12. W. F. Krupke and W. R. Sooy, “Properties of an unstable confocal resonator CO2 laser system,” IEEE J. Quantum Electron. **5**(12), 575–586 (1969). [CrossRef]

*m*and the equivalent Fresnel number

2. A. E. Siegman and R. W. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. **3**(4), 156–163 (1967). [CrossRef]

13. D. B. Rensch and A. N. Chester, “Iterative diffraction calculations of transverse mode distributions in confocal unstable laser resonators,” Appl. Opt. **12**(5), 997–1010 (1973). [CrossRef] [PubMed]

*m*= 1.42, and

13. D. B. Rensch and A. N. Chester, “Iterative diffraction calculations of transverse mode distributions in confocal unstable laser resonators,” Appl. Opt. **12**(5), 997–1010 (1973). [CrossRef] [PubMed]

*x*), focus (

^{−2}.

13. D. B. Rensch and A. N. Chester, “Iterative diffraction calculations of transverse mode distributions in confocal unstable laser resonators,” Appl. Opt. **12**(5), 997–1010 (1973). [CrossRef] [PubMed]

*m*= 1.42, with a vertical

## 4. Conclusion

## References and links

1. | A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE |

2. | A. E. Siegman and R. W. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron. |

3. | A. E. Siegman, “Unstable optical resonators,” Appl. Opt. |

4. | A. E. Siegman and H. Y. Miller, “Unstable optical resonator loss calculations using the prony method,” Appl. Opt. |

5. | R. L. Sanderson and W. Streifer, “Unstable laser resonator modes,” Appl. Opt. |

6. | P. Horwitz, “Asymptotic theory of unstable resonator modes,” J. Opt. Soc. Am. |

7. | L. W. Chen and L. B. Felsen, “Coupled-mode theory of unstable resonators,” IEEE J. Quantum Electron. |

8. | A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. |

9. | P. J. Davis and I. Polonsky, “Numerical Interpolation, Differentiation, and Integration,” in |

10. | V. Magni, G. Valentini, and S. De Silvestri, “Recent developments in laser resonator design,” Opt. Quantum Electron. |

11. | H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. |

12. | W. F. Krupke and W. R. Sooy, “Properties of an unstable confocal resonator CO2 laser system,” IEEE J. Quantum Electron. |

13. | D. B. Rensch and A. N. Chester, “Iterative diffraction calculations of transverse mode distributions in confocal unstable laser resonators,” Appl. Opt. |

**OCIS Codes**

(140.3410) Lasers and laser optics : Laser resonators

(140.4780) Lasers and laser optics : Optical resonators

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: June 27, 2011

Revised Manuscript: August 16, 2011

Manuscript Accepted: August 18, 2011

Published: September 23, 2011

**Citation**

F. X. Morrissey and H. P. Chou, "Mode calculations in asymmetrically aberrated laser resonators using the Huygens–Fresnel kernel formulation," Opt. Express **19**, 19702-19707 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19702

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### References

- A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE53(3), 277–287 (1965). [CrossRef]
- A. E. Siegman and R. W. Arrathoon, “Modes in unstable optical resonators and lens waveguides,” IEEE J. Quantum Electron.3(4), 156–163 (1967). [CrossRef]
- A. E. Siegman, “Unstable optical resonators,” Appl. Opt.13(2), 353–367 (1974). [CrossRef] [PubMed]
- A. E. Siegman and H. Y. Miller, “Unstable optical resonator loss calculations using the prony method,” Appl. Opt.9(12), 2729–2736 (1970). [CrossRef] [PubMed]
- R. L. Sanderson and W. Streifer, “Unstable laser resonator modes,” Appl. Opt.8(10), 2129–2136 (1969). [CrossRef] [PubMed]
- P. Horwitz, “Asymptotic theory of unstable resonator modes,” J. Opt. Soc. Am.63(12), 1528–1543 (1973). [CrossRef]
- L. W. Chen and L. B. Felsen, “Coupled-mode theory of unstable resonators,” IEEE J. Quantum Electron.9(11), 1102–1113 (1973). [CrossRef]
- A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J.40, 453–488 (1961).
- P. J. Davis and I. Polonsky, “Numerical Interpolation, Differentiation, and Integration,” in Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, M. Abramowitz and I. A. Stegun, eds. (Dover, 1972) pp. 875–924.
- V. Magni, G. Valentini, and S. De Silvestri, “Recent developments in laser resonator design,” Opt. Quantum Electron.23(9), 1105–1134 (1991). [CrossRef]
- H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt.5(10), 1550–1567 (1966). [CrossRef] [PubMed]
- W. F. Krupke and W. R. Sooy, “Properties of an unstable confocal resonator CO2 laser system,” IEEE J. Quantum Electron.5(12), 575–586 (1969). [CrossRef]
- D. B. Rensch and A. N. Chester, “Iterative diffraction calculations of transverse mode distributions in confocal unstable laser resonators,” Appl. Opt.12(5), 997–1010 (1973). [CrossRef] [PubMed]

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