1. Introduction

In the past decade, with the development of high-power InGaAs laser diodes as the pump sources, ytterbium (Yb³⁺)-doped materials and laser diode pumped Yb³⁺-doped lasers have attracted much attention [1–10]. Although Yb³⁺-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 (Nd³⁺)-doped materials. Yb³⁺-doped materials are believed to be much more likely to yield high efficiencies at high powers [1–10]. Another advantage of Yb³⁺-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 Yb³⁺-doped hosts, Yb³⁺:YAG laser has been intensively studied [1–8]. Compared with Yb³⁺:YAG, Yb³⁺-doped gadolinium gallium garnet (Yb³⁺:GGG) is easier to grow even in large-size single crystals. Its thermal conductivity, although lower than that of Yb³⁺:YAG at weak Yb³⁺ concentration, becomes higher for high Yb³⁺ doping levels. S. Chenais et al. reported the free-running characteristics of a laser diode pumped Yb³⁺:GGG laser and the comparison between a Yb³⁺:GGG laser and a Yb³⁺:YAG laser, and considered Yb³⁺:GGG as a challenger of Yb³⁺:YAG for high power applications [10]. The present paper focuses on the passive Q-switching operation of the Yb³⁺: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 Yb³⁺: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 Cr⁴⁺:YAG saturable absorber is placed between the Yb³⁺:GGG crystal and the output coupler, close to the Yb³⁺:GGG crystal. Two pieces of Cr⁴⁺: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.

 

Fig. 1. Scheme of the experimental setup.

LD—laser diode, Fi—coupling fiber, L—lens; M—rear mirror, Yb—Yb³⁺:GGG crystal, Cr—Cr⁴⁺:YAG saturable absorber, OC—output coupler, F—filter, BS—beam splitter, P?power meter, Mo—monochromater, PMT—photomultiplier, O—oscilloscope.

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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 Cr⁴⁺: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].

 

Fig. 2. The variation of the pulse width with the absorbed pump power.

▼▼▼ T ₀=96.0%, R=94.0%, Re=5 cm; ■■■ T ₀=94.9%, R=94.0%, Re=5 cm; ▲▲▲ T ₀=96.0%, R=943%, Re=8 cm; ●●● T ₀=94.9%, R=943%, Re=8 cm.

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Fig. 3. The variation of the pulse energy with the absorbed pump power.

▼▼▼ T ₀=96.0%, R=94.0%, Re=5 cm; ■■■ T ₀=94.9%, R=94.0%, Re=5 cm; ▲▲▲ T ₀=96.0%,R=94.3%, Re=8 cm; ●●● T ₀=94.9%, R=94.3%, Re=8 cm.

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Fig. 4. The variation of the pulse repetition rate with the absorbed pump power.

▼▼▼ T ₀=96.0%, R=94.0%, Re=5 cm; ■■■ T ₀=94.9%, R=94.0%, Re=5 cm; ▲▲▲ T ₀=96.0%,R=94.3%, Re=8 cm; ●●● T ₀=94.9%, R=94.3%, Re=8 cm.

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Fig. 5. The variation of the average output power with the absorbed pump power.

▼▼▼ T ₀=96.0%, R=94.0%, Re=5 cm; ■■■ T ₀=94.9%, R=94.0%, Re=5 cm; ▲▲▲ T ₀=96.0%, R=943%, Re=8 cm; ●●● T ₀=94.9%, R=943%, Re=8 cm.

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3. Theoretical analysis and discussion1. Introduction

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

Figure 6 shows the related energy levels of Yb³⁺. The laser transition occurs between one sublevel of ²F_5/2 manifold and one sublevel of ²F_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) dz 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.

 

Fig. 6. Related energy levels of Yb³⁺.

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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]:

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 Yb³⁺ -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 TEM₀₀ 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:

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:

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 2σ 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 Yb³⁺: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 Yb³⁺:GGG crystal is about 229 μm. By measuring the relation between the threshold pump power and the output coupler reflectivity, we can obtain 2σ n _a0 l + L ≈ 0.10. We take 2σ n _a0 l = 0.08 and L = 0.02. Other parameters for calculating the theoretical results include σ _gsa= 4.3×10^-18 cm², σ _esa= 8.2×10^-19 cm² [23], γσ= 2.0×10^-20 cm² [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 Yb³⁺: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 Yb³⁺: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 Yb³⁺: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 Yb³⁺:GGG crystal will still keep about 229 μm. Because the exact sublevel splitting in the ²F_7/2 manifold of the Yb³⁺: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 2σ n _a0 l when other parameters keep unchanged. It can be seen both the pulse energy and the pulsewidth decrease with increasing 2σ 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 2σ 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 2σ 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.

Table 1. Comparison between the experimental results and the theoretical results of the Cr⁴⁺:YAG passively Q-switched Yb³⁺:GGG laser

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Fig. 7. The variation of the pulse energy with the reabsorption.

(a) T ₀=94.9%, R=94.3%, Re=8 cm; (b) T ₀=96.0%, R=94.3%, Re=8 cm; (c) T ₀=94.9%, R=94.0%, Re=5 cm; (d) T ₀=96.0%, R=94.0%, Re=5 cm.

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Fig. 8. The variation of the pulse width with the reabsorption.

(a) T ₀=94.9%, R=94.3%, Re=8 cm; (b) T ₀=96.0%, R=94.3%, Re=8 cm; (c) T ₀=94.9%, R=94.0%, Re=5 cm; (d) T ₀=96.0%, R=94.0%, Re=5 cm.

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Fig. 9. The variation of the pulse energy with the dissipative loss.

(a) T ₀=94.9%, R=94.3%, Re=8 cm; (b) T ₀=96.0%, R=94.3%, Re=8 cm; (c) T ₀=94.9%, R=94.0%, Re=5 cm; (d) T ₀=96.0%, R=94.0%, Re=5 cm.

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Fig. 10. The variation of the pulse width with the dissipative loss.

(a) T ₀=94.9%, R=94.3%, Re=8 cm; (b) T ₀=96.0%, R=94.3%, Re=8 cm; (c) T ₀=94.9%, R=94.0%, Re=5 cm; (d) T ₀=96.0%, R=94.0%, Re=5 cm.

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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 2σ 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 Yb³⁺:GGG crystal will be lower. And hence the value of 2σ _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 2σ _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.

 

Fig. 11. The relation between output pulse energy and R for different given ln(1/R)+ln(1/:$T_0^2$)+L in the case of 2σ n _a0 l = 0.08, L = 0.02 and ω_p , = ω_L . From top to bottom, ln(1/R)+ln(1/:$T_0^2$)+L is 0.22, 0.20, 0.18, 0.16, 0.14, respectively.

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Table 2. Some possible optimal combinations of R and T0

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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.

 

Fig. 12. The variation of the pulse energy with σ _esa/σ _gsa.

(a) T ₀=94.9%, R=94.3%, Re=8 cm; (b) T ₀=96.0%, R=94.3%, Re=8 cm; (c) T ₀=94.9%, R=94.0%, Re=5 cm; (d) T ₀=96.0%, R=94.0%, Re=5 cm.

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4. Conclusion

We have demonstrated for the first time the passively Q-switched operation of the Yb³⁺:Gd₃Ga₅O₁₂ 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.