## Dynamic multimode analysis of Q-switched solid state laser cavities

Optics Express, Vol. 17, Issue 20, pp. 17303-17316 (2009)

http://dx.doi.org/10.1364/OE.17.017303

Acrobat PDF (579 KB)

### Abstract

We derive a new model and simulation technique called “Dynamic Multimode Analysis (DMA)” to simulate the 3-dimensional dynamic behavior of a laser. A Gaussian mode analysis is used to obtain resonator eigenmodes taking into account thermal aberrations. These modes are coupled by a set of rate equations to describe the dynamic behavior of the individual modes for cw and Q-switched lasers. Our approach analyzes mode competition and provides a detailed description of the laser beam in terms of output power, beam quality factor M^{2}, and pulse shape. Comparison of experimental data with our simulation results provides new insight into the effect of mode competition and the operation of Q-switched lasers.

© 2009 OSA

## 1. Introduction

1. P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. **16**(4), 196–198 (1991). [CrossRef] [PubMed]

5. X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of diode-pumped actively Q-switched lasers,” IEEE J. Quantum Electron. **35**(12), 1912–1918 (1999). [CrossRef]

6. O. A. Louchev, Y. Urata, and S. Wada, “Numerical simulation and optimization of Q-switched 2 mum Tm,Ho:YLF laser,” Opt. Express **15**(7), 3940–3947 (2007). [CrossRef] [PubMed]

7. G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. **234**(1-6), 321–328 (2004). [CrossRef]

8. K. Altmann, C. Pflaum, and D. Seider, “Three-dimensional finite element computation of laser cavity eigenmodes [corrected],” Appl. Opt. **43**(9), 1892–1901 (2004). [CrossRef] [PubMed]

9. B. Heubeck, C. Pflaum, and G. Steinle, “New finite elements for large-scale simulation of optical Waves,” SIAM J. Sci. Comput. **31**(2), 1063–1081 (2009). [CrossRef]

10. B. Heubeck and C. Pflaum, “Numerical simulation of multiple modes in solid state lasers,” Proc. SPIE **6190**, 61900X (2006). [CrossRef]

11. LAS-CAD GmbH Munich, Germany, http://www.las-cad.com/.

## 2. Model

### 2.1 Definitions

*σ*,

### 2.2 Space-dependent rate equations

#### 2.2.1 Rate equations for general field distributions

12. R. Paschotta, “Beam quality deterioration of lasers caused by intracavity beam distortions,” Opt. Express **14**(13), 6069–6074 (2006). [CrossRef] [PubMed]

14. T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. **29**(6), 1457–1459 (1993). [CrossRef]

#### 2.2.2 Computation of resonator eigenmodes

8. K. Altmann, C. Pflaum, and D. Seider, “Three-dimensional finite element computation of laser cavity eigenmodes [corrected],” Appl. Opt. **43**(9), 1892–1901 (2004). [CrossRef] [PubMed]

9. B. Heubeck, C. Pflaum, and G. Steinle, “New finite elements for large-scale simulation of optical Waves,” SIAM J. Sci. Comput. **31**(2), 1063–1081 (2009). [CrossRef]

11. LAS-CAD GmbH Munich, Germany, http://www.las-cad.com/.

8. K. Altmann, C. Pflaum, and D. Seider, “Three-dimensional finite element computation of laser cavity eigenmodes [corrected],” Appl. Opt. **43**(9), 1892–1901 (2004). [CrossRef] [PubMed]

#### 2.2.3 Accounting for anti-reflection coatings in the computation of the energy density

### 2.3 Dynamic Multimode Analysis

## Assumption 1

10. B. Heubeck and C. Pflaum, “Numerical simulation of multiple modes in solid state lasers,” Proc. SPIE **6190**, 61900X (2006). [CrossRef]

12. R. Paschotta, “Beam quality deterioration of lasers caused by intracavity beam distortions,” Opt. Express **14**(13), 6069–6074 (2006). [CrossRef] [PubMed]

15. Q. Zhang, B. Ozygus, and H. Weber, “Degeneration effects in laser cavities,” Eur. Phys. J. Appl. Phys. **6**(3), 293–298 (1999). [CrossRef]

12. R. Paschotta, “Beam quality deterioration of lasers caused by intracavity beam distortions,” Opt. Express **14**(13), 6069–6074 (2006). [CrossRef] [PubMed]

16. S. Wang, X. Wang, T. Riesbeck, and H. J. Eichler, “Thermal lensing effects in pulsed end pumped Nd lasers at 940 nm,” Proc. SPIE **7194**, 71940J (2009). [CrossRef]

## Assumption 2

*M*modes where

*Φ*can be computed by Eq. (10).

### 2.4 Numerical discretization

### 2.5 Application to active Q-switching

### 2.6 Computation of output power and beam quality

*i*-th mode can be calculated using the photon number

19. A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. **29**(4), 1212–1217 (1993). [CrossRef]

19. A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. **29**(4), 1212–1217 (1993). [CrossRef]

## 3 Experimental setup

## 4. Results and discussion

22. J. J. Degnan, “Theory of the Optimally Coupled Q-Switched Laser,” IEEE J. Quantum Electron. **25**(2), 214–220 (1989). [CrossRef]

5. X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of diode-pumped actively Q-switched lasers,” IEEE J. Quantum Electron. **35**(12), 1912–1918 (1999). [CrossRef]

7. G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. **234**(1-6), 321–328 (2004). [CrossRef]

23. R. B. Kay and D. Poulios, “Q-Switched Rate Equations for Diode Side-Pumped Slab and Zigzag Slab Lasers Including Gaussian Beam Shapes,” IEEE J. Quantum Electron. **41**(10), 1278–1284 (2005). [CrossRef]

## 5. Conclusion

## Acknowledgments

## References and links

1. | P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. |

2. | A. E. Siegman, |

3. | O. Svelto, |

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

5. | X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of diode-pumped actively Q-switched lasers,” IEEE J. Quantum Electron. |

6. | O. A. Louchev, Y. Urata, and S. Wada, “Numerical simulation and optimization of Q-switched 2 mum Tm,Ho:YLF laser,” Opt. Express |

7. | G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. |

8. | K. Altmann, C. Pflaum, and D. Seider, “Three-dimensional finite element computation of laser cavity eigenmodes [corrected],” Appl. Opt. |

9. | B. Heubeck, C. Pflaum, and G. Steinle, “New finite elements for large-scale simulation of optical Waves,” SIAM J. Sci. Comput. |

10. | B. Heubeck and C. Pflaum, “Numerical simulation of multiple modes in solid state lasers,” Proc. SPIE |

11. | LAS-CAD GmbH Munich, Germany, http://www.las-cad.com/. |

12. | R. Paschotta, “Beam quality deterioration of lasers caused by intracavity beam distortions,” Opt. Express |

13. | W. Koechner, |

14. | T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. |

15. | Q. Zhang, B. Ozygus, and H. Weber, “Degeneration effects in laser cavities,” Eur. Phys. J. Appl. Phys. |

16. | S. Wang, X. Wang, T. Riesbeck, and H. J. Eichler, “Thermal lensing effects in pulsed end pumped Nd lasers at 940 nm,” Proc. SPIE |

17. | R. J. LeVeque, |

18. | E. Hairer, and G. Wanner, |

19. | A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. |

20. | International Organization for Standardization EN ISO 11146:2000 and EN ISO 11146:1999,” Laser and laser-related equipment. Test methods for laser beam parameters. Beam width, divergence angle and beam propagation factor,” (ISO, Geneva, 2000). |

21. | F. Träger, |

22. | J. J. Degnan, “Theory of the Optimally Coupled Q-Switched Laser,” IEEE J. Quantum Electron. |

23. | R. B. Kay and D. Poulios, “Q-Switched Rate Equations for Diode Side-Pumped Slab and Zigzag Slab Lasers Including Gaussian Beam Shapes,” IEEE J. Quantum Electron. |

**OCIS Codes**

(140.3410) Lasers and laser optics : Laser resonators

(140.3430) Lasers and laser optics : Laser theory

(140.3540) Lasers and laser optics : Lasers, Q-switched

(350.6830) Other areas of optics : Thermal lensing

(140.3295) Lasers and laser optics : Laser beam characterization

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: July 20, 2009

Revised Manuscript: August 27, 2009

Manuscript Accepted: September 6, 2009

Published: September 14, 2009

**Citation**

M. Wohlmuth, C. Pflaum, K. Altmann, M. Paster, and C. Hahn, "Dynamic multimode analysis of Q-switched solid state laser cavities," Opt. Express **17**, 17303-17316 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-20-17303

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### References

- P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16(4), 196–198 (1991). [CrossRef] [PubMed]
- A. E. Siegman, Lasers, (University Science Books, Mill Valley, 1986).
- O. Svelto, Principles of lasers (Springer, New York, 1998).
- A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–458 (1961).
- X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of diode-pumped actively Q-switched lasers,” IEEE J. Quantum Electron. 35(12), 1912–1918 (1999). [CrossRef]
- O. A. Louchev, Y. Urata, and S. Wada, “Numerical simulation and optimization of Q-switched 2 mum Tm,Ho:YLF laser,” Opt. Express 15(7), 3940–3947 (2007). [CrossRef] [PubMed]
- G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234(1-6), 321–328 (2004). [CrossRef]
- K. Altmann, C. Pflaum, and D. Seider, “Three-dimensional finite element computation of laser cavity eigenmodes [corrected],” Appl. Opt. 43(9), 1892–1901 (2004). [CrossRef] [PubMed]
- B. Heubeck, C. Pflaum, and G. Steinle, “New finite elements for large-scale simulation of optical Waves,” SIAM J. Sci. Comput. 31(2), 1063–1081 (2009). [CrossRef]
- B. Heubeck and C. Pflaum, “Numerical simulation of multiple modes in solid state lasers,” Proc. SPIE 6190, 61900X (2006). [CrossRef]
- LAS-CAD GmbH Munich, Germany, http://www.las-cad.com/ .
- R. Paschotta, “Beam quality deterioration of lasers caused by intracavity beam distortions,” Opt. Express 14(13), 6069–6074 (2006). [CrossRef] [PubMed]
- W. Koechner, Solid-state laser engineering, (Springer, Berlin, 2006).
- T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29(6), 1457–1459 (1993). [CrossRef]
- Q. Zhang, B. Ozygus, and H. Weber, “Degeneration effects in laser cavities,” Eur. Phys. J. Appl. Phys. 6(3), 293–298 (1999). [CrossRef]
- S. Wang, X. Wang, T. Riesbeck, and H. J. Eichler, “Thermal lensing effects in pulsed end pumped Nd lasers at 940 nm,” Proc. SPIE 7194, 71940J (2009). [CrossRef]
- R. J. LeVeque, Finite volume methods for hyperbolic problems (Cambridge Texts in Applied Mathematics, 2007).
- E. Hairer, and G. Wanner, Solving ordinary differential Eqs. (2): Stiff and differential-algebraic problems, (Springer, Berlin, 1996).
- A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29(4), 1212–1217 (1993). [CrossRef]
- International Organization for Standardization EN ISO 11146:2000 and EN ISO 11146:1999,” Laser and laser-related equipment. Test methods for laser beam parameters. Beam width, divergence angle and beam propagation factor,” (ISO, Geneva, 2000).
- F. Träger, Springer Handbook of Lasers and Optics (Springer, Berlin, 2007), Chap. 11.2.
- J. J. Degnan, “Theory of the Optimally Coupled Q-Switched Laser,” IEEE J. Quantum Electron. 25(2), 214–220 (1989). [CrossRef]
- R. B. Kay and D. Poulios, “Q-Switched Rate Equations for Diode Side-Pumped Slab and Zigzag Slab Lasers Including Gaussian Beam Shapes,” IEEE J. Quantum Electron. 41(10), 1278–1284 (2005). [CrossRef]

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