## Abstract

The passively Q-switched laser characteristics of a quasi-three-level Yb^{3+}:Gd_{3}Ga_{5}O_{12} (Yb^{3+}:GGG) crystal with Cr^{4+}:YAG saturable absorbers are studied experimentally and theoretically. The pulse parameters under different experimental conditions are measured. Some characteristics different from those of a four-level system are found. In the theoretical aspect, taking into account the spatial distributions of the pump light and intracavity laser mode, the rate equations describing the single Q-switched pulse characteristics of a quasi-three-level system are obtained. The obtained theoretical results are in fair agreement with the experimental results. Some topics such as the influence of the pumping power, the selection of the pump beam size, the optimal combination of output coupler reflectivity and saturable absorber initial transmission, the influence of the excited absorption of the saturable absorber, are discussed.

©2005 Optical Society of America

## 1. Introduction

In the past decade, with the development of high-power InGaAs laser diodes as the pump sources, ytterbium (Yb^{3+})-doped materials and laser diode pumped Yb^{3+}-doped lasers have attracted much attention [1–10]. Although Yb^{3+}-doped materials belongs to quasi-three-level system for 1 μm laser operation, they have much lower quantum defect, hence much lower fractional heating and smaller thermal load than the commonly used neodymium (Nd^{3+})-doped materials. Yb^{3+}-doped materials are believed to be much more likely to yield high efficiencies at high powers [1–10]. Another advantage of Yb^{3+}-doped materials is their long upper-laser-level lifetime, this makes them be able to store much energy and be particularly suitable for Q-switching operation. Other advantages include broad absorption bandwidth, broad emission bandwidth, high doping level, and no excited state absorption. Among the solid state lasers based on Yb^{3+}-doped hosts, Yb^{3+}:YAG laser has been intensively studied [1–8]. Compared with Yb^{3+}:YAG, Yb^{3+}-doped gadolinium gallium garnet (Yb^{3+}:GGG) is easier to grow even in large-size single crystals. Its thermal conductivity, although lower than that of Yb^{3+}:YAG at weak Yb^{3+} concentration, becomes higher for high Yb^{3+} doping levels. S. Chenais *et al*. reported the free-running characteristics of a laser diode pumped Yb^{3+}:GGG laser and the comparison between a Yb^{3+}:GGG laser and a Yb^{3+}:YAG laser, and considered Yb^{3+}:GGG as a challenger of Yb^{3+}:YAG for high power applications [10]. The present paper focuses on the passive Q-switching operation of the Yb^{3+}:GGG laser. The pulse repetition rate, average output power, pulse energy and width for different experimental conditions are measured.

The standard tools for analyzing the performance of a Q-switched laser are the rate equations. The rate equations obtained under the plane-wave approximation can reflect the basic characteristics of a Q-switched laser [11–13] while the rate equations by taking into account the spatial distributions of laser and pumping modes give more accurate results [14, 15]. For calculating single Q-switched pulse parameters such as pulse energy, pulsewidth and peak power, the rate equations for a quasi-three-level system obtained under the plane-wave approximation have no substantial difference from those of a four-level system [11–13, 16, 17]. In order to analyze the influence of the reabsorption of a quasi-three-level system on the Q-switched pulse parameters, it is necessary to consider the spatial distributions of laser and pumping modes in the rate equations. In this paper, the intracavity photon population density is assumed to be a Gaussian distribution during the entire formatting process of the passively
Q-switched pulse, the upper-level population density at *t* = 0 is also assumed to be a Gaussian distribution. The lower-level population density and the ground state population density of the saturable absorber at *t* = 0 is assumed to be uniform. These space-dependent rate equations are solved numerically and the obtained theoretical results are in fair agreement with the experimental results. Some topics such as the influence of the pumping power, the selection of the pump beam size, the optimal combination of output coupler reflectivity and saturable absorber initial transmission, the influence of the excited absorption of the saturable absorber, are discussed.

## 2. Experiment and results

The experimental setup is schematically shown in Fig. 1. The cavity is a plano-concave one with a length of 44 mm. The rear mirror is a plane dichroic mirror coated for high reflection at resonating 1039 nm laser wavelength and high transmission at pumping 971 nm wavelength. Two concave output couplers are used in the experiment, one has a radius of curvature of 5 cm and a reflectance of 94.0%, the other has a radius of curvature of 8 cm and a reflectance of 94.3%. The Yb^{3+}:GGG laser crystal (5.0 at. % doped, 3.12 mm long, AR coated on both faces) is mounted in a copper heat sink and located at 5 mm from the rear mirror. The temperature of the copper heat sink is maintained at 12 °C by a flow of water. The fiber-coupled pumping beam (HLU15F200, from LIMO GmbH, Germany, the fiber core-diameter: 200 μm, the N. A. value: 0.22, the maximum power available: 15 W) is focused into the laser crystal by a group of focusing optics. The average diameter of the pumping beam in the laser crystal is 230 μm. The Cr^{4+}:YAG saturable absorber is placed between the Yb^{3+}:GGG crystal and the output coupler, close to the Yb^{3+}:GGG crystal. Two pieces of Cr^{4+}:YAG, whose initial transmissions are 94.9% and 96.0%, respectively, are used in the experiment. The average output power is measured by a power meter, a filter being used to block the remaining pumping light. A photomultiplier tube (Hammamatsu, R1767) placed after a monochromater (JOBIN YVON, HRS2) and an oscilloscope (LeCroy, 9410, 150 MHz) are used to detect and display the Q-switched pulses.

2Figures 2–5 show, respectively, the variations of the pulse width, the pulse energy, the pulse repetition rate and the average output power with the increasing absorbed pump power for different output coupler reflectivities *R*, different output coupler curvature radii *Re* and different Cr^{4+}:YAG initial transmissions *T*
_{0}. The absorbed pump power is obtained by using the incident pump power and the absorption rate which is obtained under non-lasing condition. It can be seen from Fig. 2 that, for given *R*, *Re* and *T*
_{0}, the pulse width nearly keeps unchanged with increasing pump power. That is to say, no obvious dependence of the pulse width on the pump power can be found. It can be seen from Fig. 3 that the pulse energy decreases slightly with increasing pump power, particularly in the case of *Re* = 5 cm. The pulse energy is much larger than that of a four-level system of the same saturable absorber and cavity parameters [18]. As can be seen from Figs. 4, 5, the pulse repetition rate and the average output power increases rapidly with increasing pump power when the pump power is relatively small. But when the pump power is relatively large, the increases of these two parameters become gradually slow. The slope efficiency is larger than that of a four-level system of the same saturable absorber and cavity parameters [18].

## 3. Theoretical analysis and discussion1. Introduction

For clarity, we first list the symbols of the parameters.

n
_{0}
| total Yb^{3+} ion population density |

n
_{a0}
| initial population density in the lower-laser level |

f_{a}
| Boltzmann occupation factor of the lower-laser level |

f_{b}
| Boltzmann occupation factor of the upper-laser level |

n_{a}
(r,t) | population density in the lower-laser level |

n_{b}
(r,t) | population density in the upper-laser level |

ϕ(r,t) | intracavity photon density in the position of the gain medium (the corresponding value in free space) |

l
| length of the gain medium |

σ
| stimulated emission cross section of the gain medium |

c
| light speed in vacuum |

t
| time |

n
_{s0}
| total population density of the saturable absorber |

n
_{s1}(r,t) | ground state population density of the saturable absorber |

σ
_{gsa}
| ground state absorption cross section of the saturable absorber |

σ
_{esa}
| excited state absorption cross section of the saturable absorber |

l_{s}
| length of the saturable absorber |

t_{r}
| roundtrip transit time of light in the resonator of optical length l′ |

R
| reflectivity of the output mirror |

L
| roundtrip dissipative optical loss |

S_{g}
| resonating laser beam cross-section area in the gain medium |

S_{s}
| resonating laser beam cross-section area in the saturable absorber |

T
_{0}
| small signal transmission of the saturable absorber |

γ
| inversion reduction factor |

ω_{p}
| average radius of the pump beam in the gain medium |

ω_{L}
| laser mode radius |

hv
| photon energy |

Figure 6 shows the related energy levels of Yb^{3+}. The laser transition occurs between one sublevel of ^{2}F_{5/2} manifold and one sublevel of ^{2}F_{7/2} manifold [1–4, 6, 7, 16, 17, 19]. Even if the quasi-three-level nature, the pumped population density makes up of a small part of the total population density. When there is no pumping and laser operation in the cavity, the population density in the lower-laser level *n*
_{a0} will be:

When there is Q-switched laser operation in the cavity, by neglecting the pumping and the spontaneous emission, the derivatives of *n*_{a}
(*r*,*t*) and *n*_{b}
(*r*,*t*) with respect to *t* can be written as [20–22]:

For LD-end-pumped lasers, since the pump light intensity in the gain medium is attenuated along the longitudinal axis *z*, *n*_{a}
(*r*, *t*) and *n*_{b}
(*r*, *t*) are also functions of *z*, *n*_{a}
(*r*, *t*) and *n*_{b}
(*r*,*t*) in the above equations represents the average value, i.e., *n*_{j}
(*r*, *t*) = *l*
^{-1} ${\int}_{0}^{1}$
*n*_{j}
(*r*, *z*, *t*) d*z*
*j* = *a*, *b*. For side-pumped lasers, it is reasonable to consider *n*_{a}
(*r*, *t*) and *n*_{b}
(*r*, *t*) to be independent of *z*. For most LD-pumped lasers, the length of the gain medium is small compared with the cavity length, the change of the laser mode radius in the gain medium can be neglected. Also the reflectivity of the output coupler is fairly high, so the intracavity photon density does not vary too much with z, *φ*(*r*,*t*) can be considered to be independent of *z* in the gain medium.

The derivatives of *ϕ*(*r*,*t*) and *n*
_{s1}(*r*, *t*) with respect to *t* are the same with those of a four-level system, they can be written as [15]:

$$-2{\sigma}_{\mathrm{gsa}}{n}_{s1}\left(r,t\right){l}_{s}-2{\sigma}_{\mathrm{esa}}\left[{n}_{s0}-{n}_{s1}\left(r,t\right){l}_{s}-\mathrm{ln}\left(\frac{1}{R}\right)-L\right\}2\mathit{\pi rdr},$$

The initial photon density *ϕ*(*r*,0) should be zero. When the rate equations are solved numerically, *ϕ*(*r*,0) can be set at a value much smaller than the peak value of the photon density *ϕ*_{m}
(*r*, *t*) [for example, *ϕ*(*r*, 0)=10^{-4}
*ϕ*_{m}
(*r*, *t*)] [14, 15]. The initial value of *n*
_{s1}(*r*,*t*) can be considered to be *n*
_{s0}. This is because the ground state recovery times of the saturable absorbers are much shorter than the time interval between two successive pulses. Before a Q-switched pulse is generated, the saturable absorber can completely recover from its bleached status caused by the preceding Q-switched pulse. Considering that the pumped population density makes up of a small part of the total population density even if the quasi-three-level nature of the Yb^{3+} -doped lasers, the initial population density at the lower laser level *n*_{a}
(*r*,0) can be assumed to be *n*
_{a0} and the initial population density at the upper laser level *n*_{b}
(*r*,0) can be assumed to be proportional to the pumping light distribution (a Gaussian distribution [14, 15]). Thus, the initial conditions of Eqs. (2)–(5) can be written as:

where *n*_{b}
(0,0) is the initial upper laser level population density in the laser axis. For side-pumped lasers, it is reasonable to consider *n*_{b}
(*r*,0) to be uniform, which corresponds to the case of a Gaussian distribution when *ω*_{p}
→∞. *ϕ*_{m}
(*r*,*t*) in formula (8) can be estimated by consulting the results obtained under the plane-wave approximation or by preliminarily solving the equations. Considering the TEM_{00} mode operation, the intracavity photon density (*r*, *t*) can be expressed as:

where *ϕ*(0,*t*) is the photon density in the laser axis, *ω*_{L}
is mainly determined by the geometry of the resonator.

By using Eqs. (2), (3), (6), (7), (10), we can obtain:

$$\times \mathrm{exp}\left[-\mathit{\gamma \sigma c}\phantom{\rule{.2em}{0ex}}\mathrm{exp}\left(-\frac{2{r}^{2}}{{\omega}_{L}^{2}}\right){\int}_{0}^{t}\varphi \left(0,t\right)dt\right],$$

where *γ* = *f*_{a}
+*f*_{b}
is the inversion reduction factor, which actually corresponds to the net reduction in the population inversion resulting from the stimulated emission of a single photon. By using Eqs. (5), (9), (10), we can obtain:

Substituting Eqs. (10)–(12) into Eq. (4) and regrouping the result, we can obtain:

$$\times \mathrm{exp}\left[-\mathit{\gamma \sigma c}\phantom{\rule{.2em}{0ex}}\mathrm{exp}(-\frac{2{r}^{2}}{{\omega}_{L}^{2}}){\int}_{0}^{t}\varphi \left(0,t\right)dt\right]\phantom{\rule{.2em}{0ex}}2rdr$$

$$-\frac{4\left({\sigma}_{\mathrm{gsa}}-{\sigma}_{\mathrm{esa}}\right){n}_{s0}{l}_{s}\varphi \left(0,t\right)}{{\omega}_{L}^{2}{t}_{r}}$$

$$\times {\int}_{0}^{\infty}\mathrm{exp}\left[-\frac{{S}_{g}}{{S}_{s}}{\sigma}_{\mathrm{gsa}}c\phantom{\rule{.2em}{0ex}}\mathrm{exp}\left(-\frac{2{r}^{2}}{{\omega}_{L}^{2}}\right){\int}_{0}^{t}\varphi \left(0,t\right)dt\right]\phantom{\rule{.2em}{0ex}}\mathrm{exp}\left(-\frac{2{r}^{2}}{{\omega}_{L}^{2}}\right)\phantom{\rule{.2em}{0ex}}2rdr$$

$$-\frac{\varphi \left(0,t\right)}{{t}_{r}}\left[\mathrm{ln}\left(\frac{1}{R}\right)+\left(\frac{{\sigma}_{\mathrm{esa}}}{{\sigma}_{\mathrm{gsa}}}\right)\mathrm{ln}\left(\frac{1}{{T}_{0}^{2}}\right)+L\right],$$

where *T*
_{0} is the small signal transmission of the saturable absorber, i.e.,

Equation (13) is the basic differential equation describing *ϕ*(0, *t*) as a function of *t*. Since laser action begins at the moment that the population inversion density crosses the initial threshold value in a passively Q-switched laser, by setting Eq. (13) equal zero and *t*= 0 and using Eq. (14), we obtain the initial population density in the laser axis for the upper laser level:

It can be seen that *n*_{b}
(0,0) includes two parts. One is proportional to 2*σ*
*n*
_{a0}
*l*, it comes from the reabsorption of the quasi-three level system. The other is proportional to ln(1/*R*)+ln(1/${T}_{0}^{2}$)+*L*, it comes from the various losses in the cavity and is the same with that of a four-level system.

By numerically solving Eq. (13), we can obtain the relation between *ϕ*(0,*t*) and *t*. From the shape of *ϕ*(0,*t*), we can obtain the pulse duration *W*. The pulse energy *E* can be written as:

When the curvature radius of the output coupler is 5 cm, the laser beam diameter at the center of the Yb^{3+}:GGG crystal is about 146 μm. When the curvature radius of the output coupler is 8 cm, the laser beam diameter at the center of the Yb^{3+}:GGG crystal is about 229 μm. By measuring the relation between the threshold pump power and the output coupler reflectivity, we can obtain 2*σ*
*n*
_{a0}
*l* + *L* ≈ 0.10. We take 2*σ*
*n*
_{a0}
*l* = 0.08 and *L* = 0.02. Other parameters for calculating the theoretical results include *σ*
_{gsa}= 4.3×10^{-18} cm^{2}, *σ*
_{esa}= 8.2×10^{-19} cm^{2} [23], *γσ*= 2.0×10^{-20} cm^{2} [10], *t*_{r}
=0.32 ns. Table 1 gives the comparison between the experimental results and the theoretical results. It can be seen that the theoretical results are in fair agreement with the experimental results except that the theoretical results of the pulse width are a little smaller that the experimental ones. One of the causes leading to this difference is that the pump light distribution is not exactly the Gaussian distribution as the theory assumed. Another cause is that the values of *σ*
_{gsa} and *σ*
_{esa} at the wavelength of 1064 nm are used in the calculation while the actual laser wavelength is 1039 nm. We believe that the values of *σ*
_{gsa} and *σ*
_{esa} at these two wavelengths are not far from each other.

The thermal effect in the gain medium caused by the pumping can affect the pulse characteristics of the Q-switched laser. The increase of the pumping power will enhance the temperature of the central part of the Yb^{3+}:GGG crystal, and hence the lower-level population density *n*
_{a0}. Therefore *n*
_{a0} is a function of the pumping power. The inhomogeneous temperature distribution results in laser beam focusing as a lens does and diffraction loss. The equivalent thermal lens can change the laser mode size. So the laser mode size and the total dissipative loss *L* are also functions of the pumping power. For this laser, when pump beam diameter is 230 μm and the absorbed pump power is less than 4 W, the focal length of the equivalent thermal lens in the Yb^{3+}:GGG crystal caused by the pumping will be larger than 10 cm [10]. Considering this effect, the resonating laser beam diameter at the center of the Yb^{3+}:GGG crystal will still keep about 146 μm when the output coupler with a curvature radius of 5 cm is used. When the curvature radius of the output coupler is 8 cm, the laser beam diameter at the center of the Yb^{3+}:GGG crystal will still keep about 229 μm. Because the exact sublevel splitting in the ^{2}F_{7/2} manifold of the Yb^{3+}:GGG crystal is not known, we cannot obtain the exact variation of *n*
_{a0} with the temperature. Nevertheless it is evident that *n*
_{a0} and *L* will increase with increasing pumping power. Figs. 7 and 8 give the dependences of the pulse energy and pulsewidth on 2*σ*
*n*
_{a0}
*l* when other parameters keep unchanged. It can be seen both the pulse energy and the pulsewidth decrease with increasing 2*σ*
*n*
_{a0}
*l*. Figs. 9 and 10 give the dependences of the pulse energy and pulsewidth on *L* when other parameters keep unchanged. It can be seen the pulse energy decreases with increasing *L* while the pulsewidth increases with it. The slight decrease of the pulse energy with increasing pumping power in the case of *Re* = 5 cm can be attributed to the increase of *n*
_{a0} and *L* with increasing pumping power. Maybe the pulse energy also decreases slightly with increasing pumping power in the case of *Re* = 8 cm, but because of the scatteration of the experimental dada, it is undistinguishable. Because the pulsewidth decrease with increasing 2*σ*
*n*
_{a0}
*l* and increases with increasing *L*, the pulse width nearly keeps unchanged when the pumping power increases. When the pumping power is relatively large, the relatively large 2*σ*
*n*
_{a0}
*l* and *L* result in a relatively large *n*_{b}
(0,0) [see Eq. (15)], the increase of the pulse repetition rate becomes gradually slow. With the addition of the slight decrease of the pulse energy with the pumping power, the increase of the average output power certainly becomes gradually slow when the pumping power is relatively large.

By using Eqs. (7) and (15), we can obtain the threshold pump power:

where *η* is a parameter related to the pump efficiency. When the pump power is smaller than 4 W and the average pump beam diameter is 230 μm, the value of 2*σ*
*n*
_{a0}
*l* is about 0.08 while the value of ln(1/*R*)+ln(1/${T}_{0}^{2}$)+*L* is 0.16 ~ 0.20. If *ω*_{p}
becomes lager for a given pump power, the temperature at the center of the Yb^{3+}:GGG crystal will be lower. And hence the value of 2*σ*
_{n0}
*l* will be lower. But it cannot counteract the increase of *P*_{pumpth}
caused by the increase of ${\omega}_{p}^{2}$+${\omega}_{L}^{2}$. *ω*_{p}
= 230 μm is the smallest pumping beam size we can get and the corresponding threshold pump power is also the lowest.

The performance of a passively Q-switched laser can be optimized by selecting proper *R* and *T*
_{0}. That is to say, for a given ln(1/*R*)+ln(1/${T}_{0}^{2}$)+*L*, there is an optimal combination of *R* and *T*
_{0} to make the laser have a maximum output pulse energy. Fig. 11 gives the relation between output pulse energy and *R* for different given ln(1/*R*)+ln(1/${T}_{0}^{2}$)+*L* in the case of 2*σ*
_{n0}
*l* = 0.08, *L* = 0.02 and *ω*_{p}
= *ω*_{L}
. It can be seen that the output pulse energy is not so sensitive to *R* when *R* is not far from the optimal reflectivity *R*_{opt}
. Table 2 gives some possible optimal combinations of *R* and *T*
_{0}. It can be seen that the *R* and *T*
_{0} used in this experiment are not far from the optimal combination.

The excited state absorption of the saturable absorber has strong influence on the pulse characteristics of a Q-switched laser. Fig. 12 gives the relation between the output pulse energy and *σ*
_{esa}/*s*
_{gsa}. It can be seen that the pulse energy decreases with increasing *σ*
_{esa}/*σ*
_{gsa} Saturable absorbers with small excited state absorption are desired.

## 4. Conclusion

We have demonstrated for the first time the passively Q-switched operation of the Yb^{3+}:Gd_{3}Ga_{5}O_{12} crystal. The pulse parameters under different experimental conditions are measured. Some characteristics different from those of a four-level system have been found. Taking into account the spatial distributions of the pump light and intracavity laser modes, the rate equations describing the single Q-switched pulse characteristics of a quasi-three-level system have been obtained. These space-dependent rate equations have been solved numerically and the obtained theoretical results are in fair agreement with the experimental results. Some topics such as the influence of the pumping power, the selection of the pump beam size, the optimal combination of output coupler reflectivity and saturable absorber initial transmission, the influence of the excited absorption of the saturable absorber, have been discussed.

## Acknowledgements

This research is supported by the National Natural Science Foundation of China (No. 60478017), the Science and Technology Development Program of Shandong Province (No. 03BS095), and the Project-sponsored by SRF for ROCS, SEM. This research has benefited from the support of the Association Franco-Chinoise pour la Recherche Scientifique et Technique (PRA MX02-05) and of the CNRS network Cristaux Massifs et Dispositifs pour l’Optique (CMDO).

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