## Modeling of beam width in passively Q-switched end-pumped lasers

Optics Express, Vol. 11, Issue 6, pp. 552-559 (2003)

http://dx.doi.org/10.1364/OE.11.000552

Acrobat PDF (256 KB)

### Abstract

An iterative model based on the *LambertW* function was developed for estimating the fundamental mode parameters of resonators with saturable gain guiding. The process of pulse buildup in passively *Q*-switched, end-pumped lasers was analyzed. The effective *ABCD* cavity matrix for consecutive round-trips was calculated, taking into account spatially variable saturated gain in an active medium and absorption bleaching in a saturable absorber. The twofold decrease in beam width, as compared with the fundamental mode of the bare cavity, was demonstrated. The application of such a model for resonators with other nonlinear elements is feasible.

© 2002 Optical Society of America

## 1. Introduction

*Q*-switched lasers. In Section 2 an iterative model is developed, with the gain saturation taken into account, and applied to the analysis of a laser operating in a free-running regime [16]. In Section 3, a similar procedure for cavities with variable gain and saturable losses is derived and applied for the analysis of passively

*Q*-switched lasers. Conclusions are drawn in Section 4.

## 2. Iterative model of a cavity with radially variable saturated gain

2. T. Y. Fan and R. L. Byer, “Diode laser-pumped solid state lasers,” IEEE J. Quantum Electron. **24**, 895–912 (1988). [CrossRef]

3. P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state lasers,” IEEE J. Quantum Electron. **27**, 2319–2326 (1991). [CrossRef]

*et al.*[4

4. R. Kapoor, P. K. Mukhopadhyay, and J. George, “A new approach to compute overlap efficiency in axially pumped solid state lasers,” Opt. Express **5**, 125–133 (1999), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-5-6-125. [CrossRef] [PubMed]

*ABCD*model of a cavity with gain guiding was first proposed by Salin and Squier [8

8. F. Salin and J. Squier, “Gain guiding in solid state lasers,” Opt. Lett. **17**, 2319–2326 (1992). [CrossRef]

*et al.*[12

12. O. Denchev, S. Kurtev, and P. Petrov, “Modes of unstable resonators with a saturable gain guide,” Appl. Opt. **40**, 921–929 (2001). [CrossRef]

*et al.*[13

13. E. J. Grace, G. H. New, and P. M. W. French, “Simple ABCD matrix treatment for transversely varying saturable gain,” Opt. Lett. **26**, 1776–1778 (2001). [CrossRef]

*g*is the small-signal logarithmic gain coefficient and

*I*

_{r}=

*I/I*

_{sat}is the relative intensity. The general integral of this type of first-order ordinary differential equation is known as the

*LambertW*function

*W*(see, e.g., Ref. [14

14. D. A. Barry, J. Y. Parlange, L. Li, H. Prommer, C. J. Cunningham, and F. Stagnitti, “Analytical approximations for real values of the LambertW-function,” Math Comput. Simulations **53**, 95–103 (2000). [CrossRef]

13. E. J. Grace, G. H. New, and P. M. W. French, “Simple ABCD matrix treatment for transversely varying saturable gain,” Opt. Lett. **26**, 1776–1778 (2001). [CrossRef]

*I*

_{r,1}after passage through an active medium of length

*l*as an explicit function of the incident intensity

*I*

_{r,0}and the small-signal logarithmic gain

*gl*as follows:

*ABCD*matrix

*M*

_{SG}as follows:

*w*

_{SG}is the effective radius of the gain diaphragm,

*w*

_{inp}is the radius of the incident Gaussian beam, and

*w*

_{out,sg}is the radius of the output approximated Gaussian beam. The value of

*w*

_{out,sg}can be determined according to Siegman’s definition [15] with the second moments of the output intensity profile:

*M*

_{BC}including the effective thermal lensing power of a gain medium to determine the incident Gaussian beam profile at the point of entrance. We take the small value of the incident intensity magnitude compared with the saturation one. In each step of the iterative procedure we calculate the output intensity profile after passage through the gain medium according to Eqs. (3) and (4) and determine the effective gain matrix

*M*

_{SG}; next, we pass the beam through the cavity, applying the product of both matrices

*M*

_{SG}

*M*

_{BC}. At each step we introduce the logarithmic passive losses of cavity δ

_{pas}, multiplying the intensity profile at the output mirror by the factor exp(-Γ

_{pas}). Because of the saturated gain profile and passive losses, the peak intensity converges with the number of round-trips to finite value, for which the procedure stops (see Figs. 2 and 3).

*ABCD*rule. As a result, we dealt with the “effective” Gaussian beam of a cavity whose parameters change in each round-trip, converging to a stationary value. The solution for a laser operating in the free-running regime depends both on parameters of bare cavity

*M*

_{BC}and on the magnitude of gain and its profile and existing linear losses.

## 3. Iterative model of a passively Q-switched laser

*g*=α

_{abs}<0 also describes the change of intensity in the saturable absorber of the small-signal absorption coefficient α

_{abs}. Thus a similar iterative procedure was developed (see Fig. 4) with the additional components describing the transformation of the beam in the saturable absorber. The passive

*Q*switch is placed at the opposite end of a cavity; thus the round-trip matrix

*M*

_{BC}was divided into two components,

*M*

_{BC1}forward from the gain medium to the absorber and

*M*

_{BC2}backward to the gain medium. In the procedure, for the free-running regime we were looking for a stationary solution, assuming that the shape of gain profile does not change (i.e., radially variable gain depletion was neglected). Now we intend to describe the process of pulse formation in a passively

*Q*-switched laser, and so both radially variable depletion of gain and bleaching of saturable absorbers should be taken into account.

*e*

^{2}definition of beam width in the model, calculating relative intensity in two points and assuming a Gaussian approximation. Let us note that saturation intensities of gain and absorber media can differ significantly. The condition for efficient passive

*Q*switching (see Refs. [1] and [17

17. J. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. **31**, 1890–1901 (1995). [CrossRef]

_{a}/σ

_{e}should be much greater than 1. Thus, to build the iterative procedure for modeling the passively

*Q*-switched laser, we should modify the function of the gain/absorption profile, adding the depletion/bleaching effect, and scale the relative intensity before and after the passive

*Q*switch by the ratio of saturation intensities of the gain medium and absorber.

*W*

_{SA}of the passive

*Q*-switch diaphragm rapidly decreases (see Fig. 5, dashed curve) causing narrowing of the pulse width during the pulse-formation process. Thus the effective width of the pulsed mode is narrower compared with the value of the bare cavity fundamental mode width 2

*W*

_{00}(see Fig. 5, dotted curve). The change of instantaneous beam width in the process of pulse formation depends on the parameters of the

*Q*switch (initial transmission) and parameters of the bare cavity. Examples of quasi–two-dimensional intensity maps of pulse formation for two passively

*Q*-switched lasers having the same resonators but different initial saturable losses are shown in Fig. 6.

*ABCD*matrix near 0) the changes in effective pulse beam width are the lowest and they depend mainly on initial losses of the

*Q*switch. In contrast, for a passively

*Q*-switched laser operating near the stability limit, the effective pulse width significantly varies with bare cavity parameters.

*Q*switch was placed near the gain medium. Such a scheme is typical for passively

*Q*-switched microchip lasers. The calculation results are shown in Fig. 7.

## 4. Conclusions

*Q*-switched lasers. The simple iterative procedure was proposed to calculate effective fundamental mode parameters of a cavity under gain guiding for given bare cavity

*ABCD*matrix and pumping parameters, including gain saturation and passive cavity losses. Application of such a method for resonators of passively

*Q*-switched lasers and cavities with other nonlinear elements such as OPO or Raman crystals is possible. For free-running and passively

*Q*-switched lasers, the change of beam width compared with fundamental mode width is in the range of 50%. The gain guiding effect should be taken into consideration in designing cavities destined for high-gain, high-power lasers, especially with decreased thermal load. The iterative model based on the

*LambertW*function can be applied for low-and medium-power passively

*Q*-switched microchips, lasers with intracavity conversion, and the like.

## Acknowledgments

## References

1. | A. E. Siegman, |

2. | T. Y. Fan and R. L. Byer, “Diode laser-pumped solid state lasers,” IEEE J. Quantum Electron. |

3. | P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state lasers,” IEEE J. Quantum Electron. |

4. | R. Kapoor, P. K. Mukhopadhyay, and J. George, “A new approach to compute overlap efficiency in axially pumped solid state lasers,” Opt. Express |

5. | J. J. Zayhowski. “Thermal guiding in microchips,” in |

6. | M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave-end-pumped solid-state lasers,” Appl. Phys. Lett. |

7. | X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of passively Q-switched lasers,” J. Opt. Soc. Am B |

8. | F. Salin and J. Squier, “Gain guiding in solid state lasers,” Opt. Lett. |

9. | V. Magni, G. Cerullo, and S. De Silvestri, “Closed form gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,” Opt. Commun. |

10. | A. J. Kemp, R. S. Conroy, G. J. Friel, and B. Sinclair, “Guiding effects in Nd:YVO |

11. | C. Serrat, M. P. van Exter, N. J. van Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. |

12. | O. Denchev, S. Kurtev, and P. Petrov, “Modes of unstable resonators with a saturable gain guide,” Appl. Opt. |

13. | E. J. Grace, G. H. New, and P. M. W. French, “Simple ABCD matrix treatment for transversely varying saturable gain,” Opt. Lett. |

14. | D. A. Barry, J. Y. Parlange, L. Li, H. Prommer, C. J. Cunningham, and F. Stagnitti, “Analytical approximations for real values of the LambertW-function,” Math Comput. Simulations |

15. | A. E. Siegman, “New developments in laser resonators,” in |

16. | J. K. Jabczyński, J. Kwiatkowski, and W. Zendzian, “Laser beam propagation in gain media,” Appl. Opt. (to be published). |

17. | J. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. |

**OCIS Codes**

(140.3410) Lasers and laser optics : Laser resonators

(140.3480) Lasers and laser optics : Lasers, diode-pumped

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

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 11, 2002

Revised Manuscript: March 13, 2003

Published: March 24, 2003

**Citation**

Jan Jabczynski, Jacek Kwiatkowski, and Waldemar Zendzian, "Modeling of beam width in passively Q-switched end-pumped lasers," Opt. Express **11**, 552-559 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-6-552

Sort: Journal | Reset

### References

- A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
- T. Y. Fan and R. L. Byer, ??Diode laser-pumped solid state lasers,?? IEEE J. Quantum Electron. 24, 895-912 (1988). [CrossRef]
- P. Laporta and M. Brussard, ??Design criteria for mode size optimization in diode-pumped solid-state lasers,?? IEEE J. Quantum Electron. 27, 2319-2326 (1991). [CrossRef]
- R. Kapoor, P. K. Mukhopadhyay, and J. George, ??A new approach to compute overlap efficiency in axially pumped solid state lasers,?? Opt. Express 5, 125-133 (1999), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-5-6-125">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-5-6-125</a>. [CrossRef] [PubMed]
- J. J. Zayhowski. ??Thermal guiding in microchips,?? in OSA Proceedings on Advanced Solid State Lasers, G. Dube and H. P. Jensen, eds. (Optical Society of America, Washington, D.C., 1990), pp. 9-13.
- M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, ??Thermal modeling of continuous-wave-endpumped solid-state lasers,?? Appl. Phys. Lett. 56, 1831-1833 (1990). [CrossRef]
- X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, ??Modeling of passively Q-switched lasers,?? J. Opt. Soc. Am B 17, 1166-1175 (2000). [CrossRef]
- F. Salin and J. Squier, ??Gain guiding in solid state lasers,?? Opt. Lett. 17, 2319-2326 (1992). [CrossRef]
- V. Magni, G. Cerullo, and S. De Silvestri, "Closed form gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,?? Opt. Commun. 101, 365-370 (1993). [CrossRef]
- A. J. Kemp, R. S. Conroy, G. J. Friel, and B. Sinclair, ??Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,?? IEEE J. Quantum Electron. 35, 675-681 (1999). [CrossRef]
- C. Serrat, M. P. van Exter, N. J. van Druten, and J. P. Woerdman, ??Transverse mode formation in microlasers by combined gain- and index-guiding,?? IEEE J. Quantum Electron. 35, 1341-1320 (1999). [CrossRef]
- O. Denchev, S. Kurtev, and P. Petrov, ??Modes of unstable resonators with a saturable gain guide,?? Appl. Opt. 40, 921-929 (2001). [CrossRef]
- E. J. Grace, G. H. New, and P. M. W. French, ??Simple ABCD matrix treatment for transversely varying saturable gain,?? Opt. Lett. 26, 1776-1778 (2001). [CrossRef]
- D. A. Barry, J. Y. Parlange, L. Li, H. Prommer, C. J. Cunningham, and F. Stagnitti, ??Analytical approximations for real values of the LambertW-function,?? Math Comput. Simulations 53, 95-103 (2000). [CrossRef]
- A. E. Siegman, ??New developments in laser resonators,?? in Optical Resonators, Proc. SPIE 1224, 4-14 (1990).
- J. K. Jabczynski, J. Kwiatkowski, and W. Zendzian, ??Laser beam propagation in gain media,?? Appl. Opt. (to be published).
- J. J. Degnan, ??Optimization of passively Q-switched lasers,?? IEEE J. Quantum Electron. 31, 1890-1901 (1995). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.