## Kinetics of a solid-state laser with polarizable saturable absorber

Optics Express, Vol. 9, Issue 9, pp. 428-435 (2001)

http://dx.doi.org/10.1364/OE.9.000428

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### Abstract

In this work we present a numerical study of a three-level laser containing a polarizable saturable absorber inside the cavity. This model allows us to study the kinetics of solid-state lasers in a general form. The stability of Q-switching regime is analyzed by means of numerical solution of rate equations; and main results of our analysis let us suggest that under certain conditions, the laser may pass from unstable relaxation oscillations to a stable CW operation by changing the mutual orientation of the absorber and polarizer, or by choosing the pump level.

© Optical Society of America

^{4+}:YAG crystal in practical Q-switching schemes of neodymium lasers [1

1. K. Spariosu, W. Chen, R. Stultz, M. Birnbaum, and A.V. Shestakov, “Dual Q-switching and laser action at 1.05 and 1.44 µm in a Nd^{3+}:YAG - Cr^{4+}:YAG oscillator at 300 K”, Opt. Lett. **18**, 814 (1993). [CrossRef] [PubMed]

4. A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccino, G.C. Reali, and Z. Sun, “Diode-pumped neodymium lasers repetitively Q-switched by Cr^{4+}:YAG solid-state saturable absorbers”, IEEE J. Sel. Top. Quantum Electron. **3**, 45 (1997). [CrossRef]

7. A.V. Kir’yanov, V. Aboites, and I.V. Mel’nikov, “Enhancing type-II second harmonic generation by the use of a laser beam with a rotating azimuth of polarization”, Appl. Phys. Lett. **78**, 874 (2001). [CrossRef]

^{4+}:YAG crystal, has not been addressed so far.

1. K. Spariosu, W. Chen, R. Stultz, M. Birnbaum, and A.V. Shestakov, “Dual Q-switching and laser action at 1.05 and 1.44 µm in a Nd^{3+}:YAG - Cr^{4+}:YAG oscillator at 300 K”, Opt. Lett. **18**, 814 (1993). [CrossRef] [PubMed]

3. Y. Shimony, Z. Burstein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetively Q-switching of a CW Nd:YAG laser using Cr^{4+}:YAG saturable absorbers”, IEEE J. Quant. Electron. **QE-32**, 305 (1996). [CrossRef]

8. N.N. Ilichev, A.V. Kir’yanov, E.S. Gulyamova, and P.P. Pashinin, “Polarization of a neodymium laser with a passive switch based on a Cr^{4+}:YAG crystal”, Quantum Electron. **28**, 17 (1998). [CrossRef]

^{4+}:YAG crystal which [001]-axis is assumed to be parallel to the longitudinal axis of the laser. The other two axes of the absorber, [100] and [010], lie in the transverse (

*x*-

*y*) plane of the cavity and are oriented with respect to the

*x*-axis at the angles

*θ*[100] and

*θ*+

*π*/

*2*[010], correspondingly. In turn, the orientation of the

*x*and

*y*axes are chosen such that the losses which are due to the presence of the partial polarizer (PP) (5) inside the cavity has minimum along the

*x*-axis and maximum - along

*y*-axis. The PP has the form of a glass plate forming the angle

*β*with the axis orthogonal to the longitudinal axis of the cavity. Therefore the linear anisotropy of the cavity is due to the PP specific orientation (angle

*β*), whereas its nonlinear anisotropy is due to the latent anisotropy of the SA and so to be determined by the angle

*θ*. We further assume that the laser output has an elliptic polarization, and the ellipse azimuth is denoted as the angle

*φ.*

*N*, an upper laser level with population density

_{1}*N*, and a pump level with population density

_{2}*N*. In addition, the laser contains a two-level saturable absorber of density of ions

_{3}*n*=

_{0}*n*+

_{1}*n*, where

_{2}*n*and

_{1}*n*corresponds to the ground-state population of ion dipoles oriented along [100]- and [010]-axis, correspondingly. The total density of the lasing states is

_{2}*γ*is the spontaneous emission capture ratio for photons trapped inside the cavity,

*g*is the gain factor for the active medium,

_{a}*g*is the absorption factor for the polarizable absorber,

_{s}*τ*is the decay time of level

_{32}*N*to level

_{3}*N*,

_{2}*τ*is the spontaneous decay time of level

_{sp}*N*,

_{2}*τ*is the decay time of absorber state

_{s}*n*

_{1,2},

*τ*is the transit time for lasing atoms pumped from level

_{p}*N*to level

_{1}*N*,

_{3}*r*is the reflection coefficient of the output mirror,

*α*

_{x,y}are the losses caused by the PP along x- and y-axis of the cavity, correspondingly.

*φ*(

*t*):

*α*-

_{y}*α*) can be found by means of the Fresnel formulas for a tilted glass plate.

_{x}*P*that is flowing into the laser cavity. This power is normalized with respect to the threshold power

*P*so that only the relative power

_{th}*P*/

*P*is needed for practical calculations. The second term in (3) accounts for the decay process from level

_{th}*N*to level

_{3}*N*. No other decay path is is provided for the population

_{2}*N*. The remaining rate equations (4) and (5) should be self-explanatory.

_{3}*γ*in solid-state lasers is small and has practically no influence on the steady-state solution. For this reason, we set

*γ*=

*0*for the purpose of computing the steady-state. However, for the dynamical solutions this factor is vital to allow the photon population to build up from zero values.

*d*/

*dt*=

*0*. In addition, we defined a normalized pump power as

_{s}appears on its right-hand side. However, even though the dependence of the right-hand side of (8) on F

_{s}is relatively weak and the method of, e.g., successive approximations seems to be applied, this procedure is cumbersome so that we prefer to find the solutions by a numerical one.

_{10}g

_{s}/g

_{a}along the horizontal axis and log

_{10}θ-β/θ on the vertical axis. The curve is computed for a relative pump power of P/P

_{th}=3. It is interesting to note that there is an unstable region within which the laser will pulse, that is surrounded by stable regions. For very small values of the relative absorber concentration the laser is seen to be always stable. But for the larger values of this ratio it is readily seen a region of instability. For the density of the same value as the density of lasing three-level particles, atoms or ions, the devise does not raise. The diagram clearly shows that for larger angles of the relative absorber orientation, the laser is also stable. These can be explained by the fact that, in these cases, the absorber remains essentially in its ground state so that bleaching does not occur. For intermediate values of θ-β/θ, the absorber can be bleached by high photon concentration, but it is able to return to its ground state sufficiently quickly and to turn the laser off as it reaches the low point of its relaxation oscillations. For very small values of θ-β/θ it is again stable, because now the absorber remains bleached and does not have time to turn the laser off.

_{s}/g

_{a}=0.1 and let the pump power vary along the horizontal axis. Again, there is an unstable region surrounded by stable regions. But this diagram shows that under certain conditions, the laser can pass from pulsing to stable CW oscillations simply by raising the pump power and by mutual orientation of the absorber and PP.

^{-3}in the first case and 10

^{-3}in the second, correspondingly. It is worth noticing that if being taken at a longer time scale, the both trains look like a decaying sequence of pulses.

^{4+}:YAG saturable absorber. We have developed a set of rate equations in order to describe the evolution of such a laser, in which the orientation-dependent interaction between the photon flux, saturable absorber, and partial polarizer have been taken into account. It is shown that inserting a polarizing saturable absorber may cause a unstable operation of the laser. The stationary solution is found and stability of the laser is analyzed by means of numerical solution of rate equations. It is also demonstrated that the laser may be turned from unstable relaxation oscillations into a stable CW ones either by means of mutual orientation of the absorber and partial polarizer or choosing the pump level. The coexistence of different types of behavior in the same nonlinear laser system is a remarkable feature that deserves further study. Our model may find application in other solid-state laser systems Q-switched by an anisotropic Cr

^{4+}- doped crystal [10]. Finally, the new regimes of quasi-chaotic pulse train generation and period doubling may find also application in optical communications with chaotic signal encoding/decoding at a 100 MHz bit rate.

## Acknowledgements

## References

1. | K. Spariosu, W. Chen, R. Stultz, M. Birnbaum, and A.V. Shestakov, “Dual Q-switching and laser action at 1.05 and 1.44 µm in a Nd |

2. | H.J. Eichler, A. Haase, M.R. Kokta, and R. Menzel, “Cr |

3. | Y. Shimony, Z. Burstein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetively Q-switching of a CW Nd:YAG laser using Cr |

4. | A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccino, G.C. Reali, and Z. Sun, “Diode-pumped neodymium lasers repetitively Q-switched by Cr |

5. | I.V. Klimov, I.A. Scherbakov, and V.B. Tsvetkov, “Control of the Nd:YAG laser output by Cr-doped Q-switches”, Laser Physics |

6. | A.V. Kir’yanov, V. Aboites, and I.V. Mel’nikov, “Second-harmonic generation by Nd |

7. | A.V. Kir’yanov, V. Aboites, and I.V. Mel’nikov, “Enhancing type-II second harmonic generation by the use of a laser beam with a rotating azimuth of polarization”, Appl. Phys. Lett. |

8. | N.N. Ilichev, A.V. Kir’yanov, E.S. Gulyamova, and P.P. Pashinin, “Polarization of a neodymium laser with a passive switch based on a Cr |

9. | M. Brunel, O. Emile, M. Vallet, F. Brtenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr |

10. | see, e.g., K.V. Yumashev, N.N. Posnov, I.A. Denisov, V.P. Mikhailov, and R. Moncorge, “Nonlinear spectroscopy and passive Q-switching of Cr |

**OCIS Codes**

(140.3530) Lasers and laser optics : Lasers, neodymium

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

(190.3100) Nonlinear optics : Instabilities and chaos

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 28, 2001

Published: October 22, 2001

**Citation**

Darwin Mayorga-Cruz and Igor Melnikov, "Kinetics of a solid-state laser with polarizable saturable absorber," Opt. Express **9**, 428-435 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-9-428

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### References

- K. Spariosu, W. Chen, R. Stultz, M. Birnbaum, and A.V. Shestakov, "Dual Q-switching and laser action at 1.05 and 1.44 m in a Nd3+:YAG - Cr4+:YAG oscillator at 300 K," Opt. Lett. 18, 814 (1993). [CrossRef] [PubMed]
- H.J. Eichler, A. Haase, M.R. Kokta, and R. Menzel, "Cr4+:YAG as a passive Q-switch for a Nd:YAG oscillator with an average repetition rate of 2.7 KHz, TEM00 mode and 13 W output," Appl. Phys. B 58, 409 (1994). [CrossRef]
- Y. Shimony, Z. Burstein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, "Repetively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers," IEEE J. Quant. Electron. QE-32, 305 (1996). [CrossRef]
- A. Agnesi, S. Dell'Acqua, C. Morello, G. Piccino, G.C. Reali, and Z. Sun, "Diode-pumped neodymium lasers repetitively Q-switched by Cr4+:YAG solid-state saturable absorbers," IEEE J. Sel. Top. Quantum Electron. 3, 45 (1997). [CrossRef]
- I.V. Klimov, I.A. Scherbakov, and V.B. Tsvetkov, "Control of the Nd:YAG laser output by Cr-doped Qswitches," Laser Phys. 8, 232 (1998).
- A.V. Kir'yanov, V. Aboites, and I.V. Mel'nikov, "Second-harmonic generation by Nd3+:YAG/Cr4+:YAG laser pulses with changing state of polarization," J. Opt. Soc. Am. B 17, 1657 (2000). [CrossRef]
- A.V. Kir'yanov, V. Aboites, and I.V. Mel'nikov, "Enhancing type-II second harmonic generation by the use of a laser beam with a rotating azimuth of polarization," Appl. Phys. Lett. 78, 874 (2001). [CrossRef]
- N.N. Ilichev, A.V. Kir'yanov, E.S. Gulyamova, and P.P. Pashinin, "Polarization of a neodymium laser with a passive switch based on a Cr4+:YAG crystal," Quantum Electron. 28, 17 (1998). [CrossRef]
- M. Brunel, O. Emile, M. Vallet, F. Brtenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, "Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:YAG absorbers," Phys. Rev. A60, 4052 (1999).
- see, e.g., K.V. Yumashev, N.N. Posnov, I.A. Denisov, V.P. Mikhailov, and R. Moncorge, "Nonlinear spectroscopy and passive Q-switching of Cr4+:doped SrGd4(SiO4)3O and CaGd4(SiO4)3O crystals," J. Opt. Soc. Am. B15, 1707 (1998), and references therein.

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