## Pulse train characteristics of a passively Q-switched microchip laser

Optics Express, Vol. 5, Issue 7, pp. 149-156 (1999)

http://dx.doi.org/10.1364/OE.5.000149

Acrobat PDF (205 KB)

### Abstract

A study of passively Q-switched microchip laser pulse trains yields approximate, yet reliable, formulae for the peak power, pulse energy, half-width, period, and the pulse shape in time. The pulse gain differential equation is made integrable by assuming that the laser absorption cross sections for the gain and saturable absorber are equal. We compare our predictions with an experiment which uses Nd:YAG as a gain medium and Cr:YAG as a saturable absorber. The agreement between theory and experiment for the period, pulse width, and the pulse energy is within 10%.

© Optical Society of America

## 1. Introduction

2. John J. Zayhowski, “Microchip Lasers.” Optical Materials **11**, 255–267 (1999). [CrossRef]

3. J. J. Zayhowski and P. L. Kelly, “Optimization of Q-switched lasers,” IEEE J. Quantum. Electron. **27**, 2220–2225 (1991). [CrossRef]

4. J. J. Degnan, “Theory of the optimally coupled Q-switched laser,” IEEE J. Quantum Electronics **25**, 214–220, (1989). [CrossRef]

5. X. Zhang, S Zhao, Q. Wang, Q. Zhang, L. Sun, and S. Zhang, “Optimization of Cr^{4+} doped saturable absorber Q-switched lasers,” IEEE J. Quantum Electronics **33**, 2286–2294, (1997). [CrossRef]

6. A. Agnesi, S Dell’Acqua, C. Morello, G Piccino, G. C. Reali, and Z. Sun, “Diode-pumped Neodymium laser repetitively Q-switched by Cr^{4+}:YAG solid-state saturable absorbers,” IEEE J. Selected Topics in Quantum Electronics **1**, 45–52, (1997). [CrossRef]

7. P. Peterson, A. Gavrielides, M.P. Sharma, and T. Erneux, “Dynamics of passively Q-switched microchip lasers,” IEEE J. Quant.Electr. **35**, 1–10, (1999). [CrossRef]

7. P. Peterson, A. Gavrielides, M.P. Sharma, and T. Erneux, “Dynamics of passively Q-switched microchip lasers,” IEEE J. Quant.Electr. **35**, 1–10, (1999). [CrossRef]

8. J. J. Zayhowski and C. Dill III, “Diode-pumped passively Q-switched picosecond microchip laser ’,” Opt. Lett. **19**, 1427 (1994). [CrossRef] [PubMed]

9. W. G. Wegnar and B. A. Lengel, “Evolution of the giant pulse in a laser,” J. Appl. Phys. **42**, 2040–2046 (1963). [CrossRef]

10. A. Szabo and R. A. Stein, “Theory of giant pulsing by a saturable absorber,” J. Appl. Phys. **36**, 1562–1566 (1965). [CrossRef]

11. L. E. Erickson and A Szabo “Effects of saturable absorber lifetime on the performance of giant-pulse lasers,” J. Appl. Phys. **37**, 4953–4961 (1966). [CrossRef]

12. L. E. Erickson and A Szabo “Behavior of saturable absorber giant-pulse lasers in the limit of large absorber cross section,” J. Appl. Phys. **38**, 2540–2542 (1967). [CrossRef]

13. J. J. Degnan,“Optimization of passively Q-switched lasers, ”IEEE J. Quantum Electron. **31**, 1890–1902 (1995). [CrossRef]

14. G. J. Spuhler, R. Paschotta, R. Fluck, B. Braun, M. Moser, G. Zhang, E. Gini, and U. Keller, “Experimentally confirmed design guidelines for passively Q-switched microchip lasers using semiconductor saturable absorbers,” J. Opt. Soc. **16**, 376–388 (1999). [CrossRef]

7. P. Peterson, A. Gavrielides, M.P. Sharma, and T. Erneux, “Dynamics of passively Q-switched microchip lasers,” IEEE J. Quant.Electr. **35**, 1–10, (1999). [CrossRef]

**35**, 1–10, (1999). [CrossRef]

**35**, 1–10, (1999). [CrossRef]

## 2. Theory

**35**, 1–10, (1999). [CrossRef]

*G*≡2

*L*

_{gγe}N_{2}.

*N*

_{2}is the upper lasing level concentration, and

*N*

_{0}is the ground state concentration with

*N*

_{0}+

*N*

_{2}=

*N*where

_{T}*N*is the doping concentration. The gain length is

_{T}*L*and the laser effective emission cross section is

_{g}*γ*. On the other hand, we treat the saturable absorber as a two-level system described by the dimensionless gain

_{e}*G*≡2

_{s}*L*(

_{s}*σ*

_{e}N_{2}-

*σ*

_{a}N_{1}) where

*N*

_{2}is the upper level concentration and

*N*

_{1}+

*N*

_{2}=

*N*

_{0}, where the total doping concentration of the saturable absorber is

*N*

_{0}. The saturable absorber length is

*L*and the laser flux is coupled through the effective emission cross section

_{s}*σ*and the effective absorption cross section

_{e}*σ*. Finally, the gain decay rate is

_{a}*A*=1/

_{g}*τ*and the saturable absorber decay rate is

_{g}*A*=1/

_{s}*τ*. The pump effective absorption cross section is

_{s}*γ′*.

_{a}*γ′*,

_{a}γ_{e}*R*(

*t*) is the laser flux, and

*P*is the cw pump flux. The saturable absorber is a two-level atom with the rate equation for the upper lasing level given by

*Ṅ*

_{2}=-

*Ṅ*

_{1}=

*R*(

_{γa}N_{1}-

_{γe}N_{2})-

*N*

_{2}=

*τ*. Thus, the differential equation for the dimensionless saturable absorber gain

_{g}*G*is

_{s}*σ*

_{+}=

*σ*+

_{e}*σ*. Accompanying eqs. (2,3) is the differential equation for the sum of the forward and reverse laser fluxes

_{a}*R*(

*t*). In the mean field approximation this sum obeys

*τ*=2

_{c}*L*/

_{c}*v*where

*v*=

*c*/

*n*, c is the speed of light in vacuum,

*n*is the index of refraction, and

*L*=

_{c}*L*+

_{g}*L*. The laser outcoupling reflectivity is

_{s}*r*and all other losses are included in

*L*. Using the above equations we form the following three dimensionless differential equations

*I*,

*D*, and

*D̄*are the dimensionless laser intensity, dimensionless gain, and the dimensionless saturable absorber inversion defined by

*I*>>1, the three above normalized differential equations can be reduced to a single differential equation for the gain[7

**35**, 1–10, (1999). [CrossRef]

*D̄*when the first equation in eq. (5) is divided by the second equation. Integrating this last division gives

*dD*=

*ds*equation becomes

*D*is the dimensionless gain inversion just before the pulse starts; we discuss this in more detail later.

_{b}*m*is set equal to unity then the pulse shape is very close to the actual pulse shape obtained when microchip constants are used. Note that for the microchip experiment the exponent

*m*=

*αγ̄*/

*γ*=

*σ*

_{+}/

*γ*=3.2. The choice of

_{a}*m*=1 allows us to solve eq. (10) analytically. The restriction

*m*=1 requires the laser absorption cross sections of the gain to equal that of the saturable abosrber. This constraint, however, gives good results since the pulse shape is not strongly dependent on

*m*, rather the shape depends on other parameters as we discuss near the end of the next section. Figures (1b,c) show the pulses for the

*m*=3.2 and the

*m*=1 solutions, respectively. The approximate

*m*=1 pulse (curve c) turns on slightly sooner and is slightly higher than the exact solution (curve b). This small shift is inconsequential since the period is roughly six orders-of-magnitude greater than the pulse width. Simulations of eqs. (5) further show that at the initiation of the pulse the saturable absorber is bleached, that is

*D̄*=-1, and that the gain is extracted, i. e.

_{b}*D*<<

_{a}*D*where

_{b}*D*is the gain after the pulse.

_{a}*m*=1 and proceed to derive various approximations to the exact pulse shape including the peak power and the FWHM. For

*m*=1 the normalized gain differential equation, eq. (10), can be written as

*η*=1 where ln

*η*≈2(

*η*-1)=(

*η*+1). Thus, continuing with the algebra, we obtain the implicit solution

*C*=

*AD*. The pulse shape embodied in eq. (12,13) is very close to the full simulations of eqs. (5) and in a moment we will make this comparison. But before doing so we derive equations for the peak power

_{b}β*I*and the pulse width (FWHM). At the pulse peak the intensity differential equation gives -1+

_{p}*AD*+

_{p}*ĀD̄*=0. Also, at the peak our numerical simulations show that the saturable absorber is transparent, i. e.

_{p}*D̄*≈0. Thus,

_{p}*D*≈1/

_{p}*A*. Inserting this back into eqs. (12,13) gives the peak intensity

*I*as

_{p}*η*at the halfwidth are obtained from

*η*, and in the pulse tail,

_{r}*η*. In the tail region

_{t}*η*is small and in the rise region

_{t}*η*is near one and thus eq. (15) yields the approximate equations

_{r}*η*and forming the FWHM Δ

_{r,t}*=Δ*

_{sFW}*-Δ*

_{st}*. Completing this task gives our final result*

_{sr}8. J. J. Zayhowski and C. Dill III, “Diode-pumped passively Q-switched picosecond microchip laser ’,” Opt. Lett. **19**, 1427 (1994). [CrossRef] [PubMed]

*η*<<1 eqs. (12,13) show that the intensity

*I*is given by

_{t}*s*-

*s**=

*O*(1);

*s** is the integration constant and is unimportant in this development. In the pulse rise region substitution of the expansion ln

*η*≈2(1-

*η*)/(1+

*η*) into eqs. (9,10), for

*m*=1, followed by integration gives

*s*-

*s**=

*O*(1/(

*AD*))<<1.

_{b}β*T*and initial inversion

*D*in PQS lasers; for a more detailed discussion see reference [5

_{b}5. X. Zhang, S Zhao, Q. Wang, Q. Zhang, L. Sun, and S. Zhang, “Optimization of Cr^{4+} doped saturable absorber Q-switched lasers,” IEEE J. Quantum Electronics **33**, 2286–2294, (1997). [CrossRef]

*I*is much less than one, consequently the gain equations in eq. (5) integrate into exponentials. Furthermore our simulations assure that after the pulse

*D*;

_{a}*D̄*<<1. Also, the fact that the period is much longer than the saturable absorber decay time implies that before the pulse

_{a}*D̄*-1, as stated before. These assumptions lead to the inversion equation

_{b}≈*D*=1-exp(-

*γs*). After this equation is inserted into the flux equation in eq. (5) and integrated the resultant exponential function for the intensity

*I*is small until

*s*reached

*S*where

*S*satisfies

*X*=

*. When eq. (20) is satisfied a rapid pulse growth is initiated. This defines the period*

_{γ}S*S*. As we will see the value of

*S*agrees with the experiment[8

8. J. J. Zayhowski and C. Dill III, “Diode-pumped passively Q-switched picosecond microchip laser ’,” Opt. Lett. **19**, 1427 (1994). [CrossRef] [PubMed]

*A*,

*Ā*, , and

*γ̄*using eqs. (6, 7, 8). Next, eq. (20) is solved for

*X*from which the period

*T*, and initial population inverison

*D*can be calculated through

_{b}*T*=

*X*/

_{τc}*α*and

_{L}*D*=1- exp(-

_{b}*X*), respectively. After

*D*is determined the peak intensity is obtained using eq. (14) and the half-width using eq. (17). Finally, the pulse energy

_{b}*ε*can be found through[7

**35**, 1–10, (1999). [CrossRef]

*t*is the outcoupling intensity transmission,

*t*+

*r*=1, and

*hv*is the laser photon energy. Of course, all this can be done as a function of the pump power. Finally, the pulse profile can be generated by solving one of the differential equations eq.(10) or eq.(11), or by plotting the implicit solution as a simple loop over

*η*, see eq. (12). Alternatively, one could also use the explicit solution given by eq. (19).

## 3. Example

**19**, 1427 (1994). [CrossRef] [PubMed]

*L*=.05cm butt coupled to a Cr

_{g}^{+4}:YAG saturable absorber of length :025cm. The outcoupling reflectivity is given as

*r*=94%. The Nd doping is at 1.6at.%. The absorption coefficient[8

**19**, 1427 (1994). [CrossRef] [PubMed]

^{-1}for a wavelength of 1.064

*µ*m. The experimental report is brief but lists a laser repetition rate of 6kHz, pulse width of 330ps, pulse energy of 11

*µ*J, peak pulse power of 27kW at a peak intensity of 180MW/cm

^{2}. This behavior is observed for a pump power of 1.2W with a threshold at .8W. The laser emits at a wavelength of

*λ*=1.06

_{l}*µ*m and has an estimated beam waist of 70

*µ*m. The pump with a wavelength of

*λ*=808nm is double passed since it reflects from the Nd-Cr interface.

_{p}**35**, 1–10, (1999). [CrossRef]

**35**, 1–10, (1999). [CrossRef]

**35**, 1–10, (1999). [CrossRef]

*W*,

*A*=10.48 and the threshold (

*A*≡1+

_{th}*Ā*)=4.96. Also, we found

*D̄*=-1, and

_{b}*D̄*=10

_{a}^{-6},

*D*=10

_{a}^{-4}, and

*D*=.759. Furthermore these simulation of eqs. (5) gave a period of 5.9kHz, a peak power of 29.8kW, the pulse width as 343psec, and a pulse energy of 11.9

_{b}*µ*J. These values are very close to the above mentioned experimental values. Figure 2 shows the simulated pulse shape, the solid curve, compared to the experimental results shown as the dashed curve. Note that since our simulation does not include noise, the intensity drops to zero much more rapidly than in the experiment[8

**19**, 1427 (1994). [CrossRef] [PubMed]

**35**, 1–10, (1999). [CrossRef]

*D*=.759. At this point we employ our approximate equations. First, inserting the above values for

_{b}*A*,

*Ā*,

*γ*, and

*γ̄*into eq. (20) gives a period of 5.64kHz, which is within 7% of the experiment. This value of the period leads to a value of

*D*equal to .7584. We use this value as the initial condition for solving the inversion differential equation, eqs. (9,10) for the two values of

_{b}*m*=

*αγ̄*/

*γ*=3.2 and

*m*=1 which yields curves (b) and (c), respectively. The final curve (d) is our implicit solution [see eqs. (12,13)]. These four simulated pulse shapes are very close to one another. The two rigorous solutions, curve (a) and curve (b), are almost identical. The

*m*=1 differential equation, eqs. (9,10), over estimates the peak and the width by about 7%. Our implicit solution, eqs. (12, 13), closely replicates the simulated solutions in curves (a) and (b). In fact, further simulations show that the pulse shape remains within 10% of the experimental values for 1<

*m*<4.

*D*, eq. (14) gives a peak intensity of

_{b}*I*=206MW/cm

_{p}^{2}, eq. (17) gives the half width of Δ

*sFW*=321psec, and finally eq. (21) gives the energy

*ε*=12.1

*µ*J. The peak intensity matches the experiment to within less that 15% while the others agree to less than 10%.

*C*-2)), obtained from eqs. (18,19), is a strong function of pump power. However, as we have mentioned the pulse decay depends only on the cavity decay time which is 216psec. Thus, as the pumping is changed the only pulse shape changes occurs in the early part of the pulse. Note that for large pumping eq. (17) shows that the minimum pulse width is governed by the cavity and is given by (

*τ*/

_{c}*α*) ln(2). This value, of course, is not a practical limit since it corresponds to a delta function like pulse growth.

_{L}*m*=

*σ*

_{+}/

*γ*. Specifically for.5<

*m*<5 the pulse energy agrees with the

*m*=3.2 solution to within about 15%. However, the pulse is strongly dependent on

*τ*,

_{g}*τ*, the pump small signal gain

_{s}*g*=2

_{g}*, and the laser small signal gain*

_{γ}N_{T}L_{g}*g*=2

_{s}*σ*in the saturable absorber. Specifically eq. (21) shows that

_{a}N_{0}L_{s}*ε*∝ (

*g*-

_{g}P_{γ}_{e}τ_{g}*g*/

_{s}τ_{s}*τ*)/

_{g}*α*.

_{L}14. G. J. Spuhler, R. Paschotta, R. Fluck, B. Braun, M. Moser, G. Zhang, E. Gini, and U. Keller, “Experimentally confirmed design guidelines for passively Q-switched microchip lasers using semiconductor saturable absorbers,” J. Opt. Soc. **16**, 376–388 (1999). [CrossRef]

_{4}or Nd:LSB as a gain medium and a SESAM (seminconductor saturable absorber mirror). When we compare their pulse differential equations with our normalized pulse form, eqs.(5), we find that:

*γ̄*=9.3×10

^{-2},

*γ*=3.7×10

^{-7},

*Ā*=.36,

*A*=4.08,

*α*=4.×10

^{-3}, and that

*m*is large with a value of

*m*=1000. After these values are inserted in the exact gain differental equation eq.(10), for

*D̄*=-1, and in the

_{b}*m*=1 differential equation eq. (11) we find that the two pulse shapes are virtually identical. The reason that the

*m*=1 solution works so well is because the [1-

*η*] term is multipled by 1/

^{m}*m*, or

*β*≈1. An absolute comparison with their experiment[14

14. G. J. Spuhler, R. Paschotta, R. Fluck, B. Braun, M. Moser, G. Zhang, E. Gini, and U. Keller, “Experimentally confirmed design guidelines for passively Q-switched microchip lasers using semiconductor saturable absorbers,” J. Opt. Soc. **16**, 376–388 (1999). [CrossRef]

**16**, 376–388 (1999). [CrossRef]

## 4. Summary

## References

1. | John J. Zayhowski, “Passively Q-switched Microchip Lasers and Applications,”The Review of Laser Engineering,” 26, 841–846 (1998). |

2. | John J. Zayhowski, “Microchip Lasers.” Optical Materials |

3. | J. J. Zayhowski and P. L. Kelly, “Optimization of Q-switched lasers,” IEEE J. Quantum. Electron. |

4. | J. J. Degnan, “Theory of the optimally coupled Q-switched laser,” IEEE J. Quantum Electronics |

5. | X. Zhang, S Zhao, Q. Wang, Q. Zhang, L. Sun, and S. Zhang, “Optimization of Cr |

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

7. | P. Peterson, A. Gavrielides, M.P. Sharma, and T. Erneux, “Dynamics of passively Q-switched microchip lasers,” IEEE J. Quant.Electr. |

8. | J. J. Zayhowski and C. Dill III, “Diode-pumped passively Q-switched picosecond microchip laser ’,” Opt. Lett. |

9. | W. G. Wegnar and B. A. Lengel, “Evolution of the giant pulse in a laser,” J. Appl. Phys. |

10. | A. Szabo and R. A. Stein, “Theory of giant pulsing by a saturable absorber,” J. Appl. Phys. |

11. | L. E. Erickson and A Szabo “Effects of saturable absorber lifetime on the performance of giant-pulse lasers,” J. Appl. Phys. |

12. | L. E. Erickson and A Szabo “Behavior of saturable absorber giant-pulse lasers in the limit of large absorber cross section,” J. Appl. Phys. |

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

14. | G. J. Spuhler, R. Paschotta, R. Fluck, B. Braun, M. Moser, G. Zhang, E. Gini, and U. Keller, “Experimentally confirmed design guidelines for passively Q-switched microchip lasers using semiconductor saturable absorbers,” J. Opt. Soc. |

15. | T. Erneux, P. Peterson, and A. Gavrielides, “The pulse shape of a passively Q-switched microchip laser,” accepted Europ. J. Physics (1999). |

16. | Peter W. Milonni and Joseph H. Eberly, |

17. | Walter Koechner, |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

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

(140.3580) Lasers and laser optics : Lasers, solid-state

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 11, 1999

Published: September 27, 1999

**Citation**

Phillip Peterson and Athanasios Gavrielides, "Pulse train characterisitcs of a passively Q-switched microchip laser," Opt. Express **5**, 149-156 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-5-7-149

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### References

- John J. Zayhowski, "Passively Q-switched Microchip Lasers and Applications,"The Review of Laser Engineering," 26, 841-846 (1998).
- John J. Zayhowski, "Microchip Lasers," Optical Materials 11, 255-267 (1999). [CrossRef]
- J. J. Zayhowski, P. L. Kelly, "Optimization of Q-switched lasers," IEEE J. Quantum. Electron. 27, 2220-2225 (1991). [CrossRef]
- J. J. Degnan,"Theory of the optimally coupled Q-switched laser," IEEE J. Quantum Electronics 25, 214-220, (1989). [CrossRef]
- X. Zhang, S Zhao, Q. Wang, Q. Zhang, L. Sun, and S. Zhang, " Optimization of Cr 4+ doped saturable absorber Q-switched lasers," IEEE J. Quantum Electronics 33, 2286-2294, (1997). [CrossRef]
- A. Agnesi, S Dell'Acqua, C. Morello, G Piccino, G. C. Reali, and Z. Sun, "Diode-pumped Neodymium laser repetitively Q-switched by Cr 4+ :YAG solid-state saturable absorbers," IEEE J. Selected Topics in Quantum Electronics 1, 45-52, (1997). [CrossRef]
- P. Peterson, A. Gavrielides, M.P. Sharma and T. Erneux, "Dynamics of passively Q-switched microchip lasers," IEEE J. Quant. Electr. 35, 1-10, (1999). [CrossRef]
- J. J. Zayhowski, C. Dill III, "Diode-pumped passively Q-switched picosecond microchip laser," Opt. Lett. 19, 1427 (1994). [CrossRef] [PubMed]
- W. G. Wegnar and B. A. Lengel, "Evolution of the giant pulse in a laser," J. Appl. Phys. 42, 2040-2046 (1963). [CrossRef]
- A. Szabo and R. A. Stein, "Theory of giant pulsing by a saturable absorber," J. Appl. Phys. 36, 1562-1566 (1965). [CrossRef]
- L. E. Erickson and A Szabo "Effects of saturable absorber lifetime on the performance of giant- pulse lasers," J. Appl. Phys. 37, 4953-4961 (1966). [CrossRef]
- L. E. Erickson and A Szabo "Behavior of saturable absorber giant-pulse lasers in the limit of large absorber cross section," J. Appl. Phys. 38, 2540-2542 (1967). [CrossRef]
- J. J. Degnan,"Optimization of passively Q-switched lasers, "IEEE J. Quantum Electron. 31, 1890-1902 (1995). [CrossRef]
- G. J. Spuhler, R. Paschotta, R. Fluck. B. Braun, M. Moser, G. Zhang, E. Gini, and U. Keller, "Experimentally confirmed design guidelines for passively Q-switched microchip lasers using semiconductor saturable absorbers," J. Opt. Soc. 16, 376-388 (1999). [CrossRef]
- T. Erneux, P. Peterson, A. Gavrielides, "The pulse shape of a passively Q-switched microchip laser," accepted Europ. J. Physics (1999).
- Peter W. Milonni, Joseph H. Eberly, Lasers (John Wiley and Sons, New York, 1998).
- Walter Koechner, Solid-State Laser Engineering (Springer, New York, 1992).

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