## Abstract

Using both acoustic-optic (AO) Q-switcher and GaAs saturable absorber, a diode-pumped doubly Q-switched Nd:YVO_{4}/KTP green laser is realized for the first time to our knowledge. This laser can generate a symmetric and shorter pulse when compared with purely AO and passive Q-switching. A rate equation model is introduced to theoretically analyze the results obtained in the experiment, in which the spatial distributions of the intracavity photon density, the pump beam and the population-inversion density are taken into account. The numerical solutions of the rate equations are in good agreement with the experimental results.

©2005 Optical Society of America

## 1. Introduction

Diode-pumped solid-state Q-switched lasers have attracted a great deal of attention in recent years because of their high efficiency, simplicity, compactness, good frequency stability. All solid-state actively and passively Q-switched lasers have wide applications in the fields of remote sensing, information storage, coherent telecommunications, medicine, etc. Acoustic-optic (AO) modulator is often used as the active Q-switcher and GaAs saturable absorber is often used as the passive Q-switcher [1–3]. Although single Q-switched lasers can obtain short pulses and high peak powers, experimental results show that the pulse temporal profile of single Q-switched lasers is usually asymmetric, with a fast rising edge and a slow falling edge in a AO Q-switched laser, and with a slow rising edge and a fast falling edge in a GaAs Q-switched laser [1,2,4,5]. In some applications, the symmetric pulse is more useful. For example, when such pulse is used in high power lasers, there is no need to reshape it after amplification. If we use both the AO Q-switcher and GaAs saturable absorber in the same cavity, according to their pulse characteristics, it is possible to obtain symmetric and shorter pulses and this has been experimentally proved in Ref. [5] in which the laser is operating at 1.06 µm wavelength. Nevertheless, the pulse performance of a doubly Q-switched green laser with AO and GaAs has not yet been reported as far as we know. In addition, the theoretical investigations of this type of laser have not been carried out.

In this paper, using both AO Q-switcher and GaAs saturable absorber, we realize a diode-pumped doubly Q-switched Nd:YVO_{4}/KTP green laser for the first time, to our knowledge. This laser can generate a symmetric and shorter pulse when compared with purely AO and passive Q-switching. To understand the results obtained in the experiment, we introduce a rate equation model in which the spatial distributions of the intracavity photon density, the pump beam and the population-inversion density are taken into account. The numerical solutions of the rate equations are well consistent with the experimental results.

## 2. Experimental setup and results

The experimental setup is shown in Fig. 1. The pump source is a fiber-coupled laser-diode (made by Semiconductor Institute, Chinese Academic, maximum output power 5 W) which works at the maximum absorption wavelength (808 nm) of the Nd:YVO_{4} crystal. The output pump beam from the fiber bundle end, which is 800 µm in diameter, is focused into the laser crystal with a spot size of about 440 µm at the focal plane and far-field half-angle of 18° by a focusing optics. The mirror M_{1} with 150-mm curvature radius is high antireflection coated at 808 nm and high reflection coated at 1064 nm. The Nd:YVO_{4} crystal doped with 1.0 at. % Nd^{3+} ions is 4×4×5 mm^{3} and its absorption coefficient at 808 nm is 5.32 cm^{-1}. Its front surface is antireflection coated at 808 nm and its rear surface is high antireflection coated at 1064 nm. It is near M_{1}. The distance between the front surface of the AO crystal and M_{1} is 7 cm and the distance between GaAs saturable absorber and M_{1} is 11 cm. The mirror M_{2} with 100-mm curvature radius is also used as the output mirror of the generated green light and the distance between M_{1}and M_{2} is about 22 cm. The KTP crystal cut for type-II phase matching (made by Coretech Crystal Company, Shandong University, China) is 3×3×10 mm^{3} and both of its surfaces are antireflection coated at 1064 nm and 532 nm. The temperatures of the Nd:YVO_{4} crystal and the KTP crystal are controlled at 20 °C and 22 °C by means of a temperature controller, respectively. M_{3} is a plane mirror and its surface is high reflection coated at 1064 nm and 532 nm. The KTP crystal is near M_{3} and the distance between M_{2} and M_{3} is about 8 cm. The filter is used for separating 532-nm green laser from the remainder 1064-nm fundamental wave leaking out from the resonator. A TED620B digital oscilloscope (Tektronix Inc., USA) is used to measure the generated-green-laser pulse width and a LPE-1B power meter (Institute of Physics, Chinese Academy of Science) is used to measure the generated-green-laser power.

Single-pulse temporal profiles for the AO, doubly and passively Q-switched lasers with a pump power of 2.67 W are shown by the solid lines in Fig. 2. The pulse width of the doubly Q-switched laser is 30.7 ns at *f*_{p}
=40 kHz as shown in Fig. 2(b). It is noticed that the pulse profile is rather symmetric with about 15 ns in both the rise and fall edges. Under the same conditions, the pulse width of the AO Q-switched laser is 71.6 ns as shown in Fig. 2(a), and the pulse profile is asymmetric with a fast rise time of about 28 ns and a slow falling edge of about 44 ns as well as a long decaying tail. The pulse width of the passively Q-switched laser is 56.8 ns as shown in Fig. 2(c), and the pulse profile is also asymmetric with a slow rise time of about 32 ns and a fast falling edge of about 25 ns. From Fig. 2, we can see that the doubly Q-switched laser has a symmetric pulse temporal profile and shorter pulse width compared to the other two Q-switched lasers.

The dependence of pulse width on incident pump power with the AO, passive and the double Q-switching for *f*_{p}
=10 kHz is shown by the scattered marks in Fig. 3, from which we can see that the double Q-switching considerably shortened the laser pulses when compared with the other two methods of Q-switching.

The dependence of the pulse width on the repetition rate with a maximum incident pump power of 3.77 W has been also measured, as shown in Fig. 4, from which we can see that the pulse width always increases with the repetition rate in both the AO and the doubly Q-switched laser while the pulse width of the double Q-switching is always shorter than that of the AO Q-switching. Fig. 4 also indicates that the pulse width variation as a function of repetition rate from the double Q-switching is much smaller than that from the AO Q-switching.

From Figs. 2–4, we can see that for a diode-pumped doubly Q-switched Nd:YVO_{4}/KTP green laser with AO and GaAs, the pulse profile is rather symmetric and the pulse width is obviously compressed when compared with the other two methods of Q-switching.

Figure 5 shows the dependence of average output power on incident pump power with the three types of Q-switching for *f*_{p}
=10 kHz. It indicates that although the average output power from the double Q-switching is smaller than that from the AO Q-switching, it is much higher than that from the passive Q-switching.

## 3. Theoretical analysis

#### 3.1 The spatial distribution of the photon density

We consider a diode-pumped doubly Q-switched Nd:YVO_{4}/KTP green laser depicted in Fig. 1, in which Nd:YVO_{4} works as gain medium, AO modulator works as active Q-switcher, and GaAs works as passive Q-switcher, KTP works as frequency-doubling crystal. If the intracavity photon density is assumed to be a Gaussian spatial distribution during the entire formatting process of the diode-pumped doubly Q-switched laser pulse, the intracavity photon density *ϕ*(*r*, *t*) for the TEM_{00} mode can be expressed as

where *r* is the radial coordinate; *w*_{l}
is the average radius of the TEM_{00} mode, which is mainly determined by the geometry of the resonator; *ϕ*(0, *t*) is the photon density in the laser axis.

To precisely describe the operation of the doubly Q-switched laser, we also take into account the longitudinal distribution of the intracavity photon density along the cavity axis caused by the variation of the beam radius along the cavity. So the photon densities *ϕ*_{g}
(*r*, *t*), *ϕ*_{a}
(r, t), *ϕ*_{s}
(*r*, *t*), and *ϕ*_{k}
(*r*, *t*) at the positions of Nd:YVO_{4} gain medium, AO crystal, GaAs saturable absorber, and KTP crystal can be expressed as [6]

where *w*_{g}
, *w*_{a}
, *w*_{s}
, and *w*_{k}
are the radii of the TEM_{00} mode at the above-mentioned four positions, respectively.

In our experiment, however, the pump power is focused into the gain medium with a spot size of a few hundred microns, so the phase difference of it becomes uneven due to the un-uniformity of the temperature field distribution in the gain medium. The phase difference is distributed as a parabola profile and the gain medium has thermal lens effect, and the thermal focal length is [7]

where *w*_{p}
is the average radius of the pump light in the gain medium; *η*=1-exp(-*αl*) is the absorptivity of the gain medium, in which *α* is the absorption coefficient and *l* is the length of the gain medium; *P*
_{in} is the incident pump power; *n*
_{1} is the refractive index of the gain medium; *K*_{c}
is the thermal conductivity; d*n*/d*T* is the thermal dispersion coefficient; *α*_{T}
is the thermal expansion coefficient; ξ is the fractional thermal load; and for our a-cut Nd:YVO_{4} crystal:

*K*_{c}
=5.23×10^{-3} Wmm^{-1}K^{-1}, d*n*/d*T*=3×10^{-6} K^{-1}, *α*_{T}
=4.43×10^{-6} K^{-1}, ξ=0.24 [7–9].

On the basis of the paraxial approximation, the radius of the pump light in the gain medium *w*_{p}
(*z*) may be given as

where *z* is the longitudinal coordinate and the pumped end of the laser crystal is taken as *z*=0; *w*_{p}
_{0} is the radius at the pump beam waist and can be obtained by using the method given in Ref. [10]; *θ*_{p}
and *z*
_{0} are the far-field half-angle and the distance between the focal plane of the pump beam in the gain medium and the pumped end of the laser crystal. If the absorption factor of the gain medium in the laser axis exp(-*αz*) is considered, we can obtain the dependence of the average pump beam radius in the gain medium *w*_{p}
on *z*
_{0}, and the minimum *w*_{p}
=216.6 µm is at *z*
_{0}=1.0 mm, where the maximum generated-green-laser power can be obtained at a certain pump power.

For our experimental configuration shown in Fig. 1, using the well-known ABCD matrix method and considering the thermal lens effect of the gain medium, we have simulated the radii of the TEM_{00} mode at the mirror M_{1} and M_{3}, that is, *w*_{g}
and *w*_{k}
as functions of incident pump power with *w*_{p}
=216.6 µm, and the results are shown in Fig. 6. *w*_{a}
, *w*_{s}
and the average radius of the TEM_{00} mode *w*_{l}
as functions of incident pump power are also shown in Fig. 6.

## 3.2 Nonlinear loss due to harmonic conversion

For a Q-switched intracavity-frequency-doubling laser, the harmonic conversion is always considered as a nonlinear loss of the fundamental wave. Assuming that the fundamental wave (FW) is plane wave with Gaussian amplitude profile

where *ω* is the angle frequency of the fundamental wave.

If the thermal effect and walking-off effect of KTP are neglected, the amplitudes for o and e light for KTP type-II phase matching are

So the amplitude of second-harmonic wave (SHW) satisfies the following equation under the small-signal and near-field approximation [11]

where *d*
_{eff} is the effective nonlinear coefficient; *c* is the light velocity in vacuum; *n*_{e}
2^{ω} is the refractive index of SHW.

Integrating Eq. (7) along the *z* direction, yields

where *l*_{k}
is the length of KTP.

From the relationship between intensity and amplitude for light

and the fundamental power

we can obtain the SHW intensity at the exit of end face

where *ε*
_{0} is the dielectric permeability of vacuum; ${n}_{o}^{\omega}$
and ${n}_{e}^{\omega}$
are fundamental-wave refractive indices of o and e light, respectively.

The FW intensity is

$$=\frac{2}{\pi {\omega}_{k}^{2}}\mathrm{exp}\left(-\frac{2{r}^{2}}{{\omega}_{k}^{2}}\right)\left({P}_{0}^{o}+{P}_{0}^{e}\right).$$

We assume

where *P*(*ω*,0)=(1/2)*A*_{k}*ħωcϕ*_{k}
(0,*t*) is the incident fundamental power in the axis of KTP, in which *A*_{k}
is the area of fundamental wave at the position of KTP, *ħω* is the single photon energy of the fundamental wave, *ϕ*_{k}
(0, *t*) is the photon density in the laser axis at the position of KTP.

The nonlinear loss due to harmonic conversion can be obtained

$$={K}_{N}\hslash \omega c{l}_{k}^{2}\mathrm{exp}\left(-\frac{2{r}^{2}}{{\omega}_{k}^{2}}\right){\varphi}_{k}(0,t)$$

$$={\delta}_{k}{\varphi}_{k}(r,t),$$

where

The corresponding parameters for type-II phase-matching KTP crystal are presented in Table 1.

## 3.3 Rate equations and solutions

For laser-diode end-pumped lasers, the pump light can be approximated by a Gaussian profile. The normalized function that describes the spatial distribution of the pump power can be expressed as [12]

So if neglecting the spontaneous radiation during the pulse formation and considering the single-photon absorption (SPA) and two-photon absorption (TPA) of GaAs saturable absorber, we can obtain the rate equations of a diode-pumped doubly Q-switched Nd:YVO_{4}/KTP green laser with AO and GaAs [6,13]

$$-2{\sigma}^{0}\lceil {n}_{0}-{n}^{+}(r,t)\rceil {l}_{s}{\varphi}_{s}(r,t)-B{l}_{s}{\varphi}_{s}^{2}(r,t)$$

$$-{\delta}_{a}\left(t\right){\varphi}_{a}(r,t)-{\delta}_{k}{\varphi}_{k}^{2}(r,t)-L\varphi (r,t)\}2\pi rdr,$$

where *n*(*r*, *z*, *t*) is the spatial distribution of the population-inversion density; *n*
_{0} is the total population density of the EL2 defect level (including EL2^{0} and EL2^{+}) of GaAs saturable absorber; *n*
^{+}(*r*, *t*) is the population density of positively charged EL2^{+};σ and *l* are the stimulated-emission cross section and length of Nd:YVO_{4} gain medium, respectively; σ^{0} and σ^{+} are the absorption cross sections of EL2^{0} and EL2^{+}, respectively; *l*_{s}
is the length of the saturable absorber; *t*_{r}
is the round-trip time of light in the resonator {*t*_{r}
=[2*n*
_{1}
*l*+2*n*
_{2}
*l*_{a}
+2*n*
_{3}
*l*_{s}
+2*n*
_{4}
*l*_{k}
+2(*L*_{e}
-*l*-*l*_{a}
-*l*_{s}
-*l*_{k}
)]/*c*}, in which *n*
_{1}, *n*
_{2}, *n*
_{3}, and *n*
_{4} are the refractive indices of Nd:YVO_{4} gain medium, AO crystal, GaAs saturable absorber, and KTP crystal, respectively, *L*_{e}
is the cavity length, *l*_{a}
is the length of the AO crystal, *c* is the velocity of light in vacuum; *B*=6*βh*
*γc*(*w*_{g}
/*w*_{s}
)^{2} is the coupling coefficient of TPA in GaAs [3], where *β* is the absorption coefficient of two photons, *hγ* is the single photon energy of the fundamental wave; *L* is the intrinsic loss; τ is the stimulated-radiation lifetime of the gain medium; *W*_{p}
=*P*
_{in}
*η*/*hγ*_{p}
is the pump rate, where *hγ*_{p}
is the single-photon energy of the pump light; *δ*_{a}
(*t*) is the loss function of the AO Q-switcher, which is defined as [14]

where *δ*_{a}
is the intrinsic diffraction loss of the AO Q-switcher; *t*_{s}
is the turnoff time of the AO Q-switcher.

The initial conditions of Eqs. (19) and (20) can be written as

where *n*(0, 0, 0) is the initial population-inversion density in the laser axis; *n*
^{+} is the initial population density of positively charged EL2^{+} of GaAs saturable absorber. From Eq. (19), we can deduce *n*(0, 0, 0) accumulated during a modulation period of the AO modulator

where *f*_{p}
is the modulation frequency of the AO modulator; *w*_{p}
(0) is the pump beam radius at *z*=0.

If neglecting the terms concerning GaAs in the above-mentioned rate equations, we can obtain the rate equations describing a diode-pumped AO Q-switched Nd:YVO_{4}/KTP green laser

$$={\int}_{0}^{\infty}\frac{1}{{t}_{r}}\left\{2\sigma \left({\int}_{0}^{l}n(r,z,t)\text{d}z\right){\varphi}_{g}(r,t)-{\delta}_{a}\left(t\right){\varphi}_{a}(r,t)-{\delta}_{k}{\varphi}_{k}^{2}(r,t)-L\varphi (r,t)\right\}2\pi rdr,$$

If neglecting the term concerning the AO Q-switcher in Eq. (18), we can obtain the rate equations describing a diode-pumped passively Q-switched Nd:YVO_{4}/KTP green laser with GaAs saturable absorber.

According to the corresponding parameters shown in Table 2, by numerically solving the above-mentioned rate equations, we obtain the theoretical pulse profiles for the AO, doubly, and passively Q-switched lasers with a pump power of 2.67 W as shown by the dotted lines in Fig. 2(a), (b), and (c), respectively. The pulse width of the doubly Q-switched laser is 30.2 ns at 40 kHz. Under the same conditions, the pulse width of the AO Q-switched laser is 72.4 ns and the pulse width of the passively Q-switched laser is 56.2 ns. The theoretical results indicate that the doubly Q-switched laser has a symmetric pulse temporal profile and shorter pulse width compared to the other two Q-switched lasers. The dependence of pulse width on incident pump power with the three types of Q-switching for *f*_{p}
=10 kHz is shown in Fig. 3 by the solid lines.

From Figs. 2 and 3, we can see that the theoretical calculations are in good agreement with the experimental results.

## 4. Conclusions

We have successfully realized a diode-pumped doubly Q-switched Nd:YVO_{4}/KTP green laser using both AO Q-switcher and GaAs saturable absorber for the first time, to our knowledge. This laser can generate a symmetric and shorter pulse when compared with purely AO and passive Q-switching. A rate equation model is introduced to theoretically analyze the results obtained in the experiment, in which the spatial distributions of the intracavity photon density, the pump beam and the population-inversion density are taken into account. The numerical solutions of the rate equations agree with the experimental results well.

## Acknowledgments

This work is supported by the Science and Technology Development Program of Shandong Province.

## References and links

**1. **T. T. Kajava and A. L. Gaeta, “Q switching of a diode-pumped Nd:YAG laser with GaAs,” Opt. Lett. **21**, 1244–1246 (1996). [CrossRef] [PubMed]

**2. **T. T. Kajava and A. L. Gaeta, “Intra-cavity frequency-doubling of a Nd:YAG laser passively Q-switched with GaAs,” Opt. Commun. **137**, 93–97 (1997). [CrossRef]

**3. **L. Chen, S. Zhao, and H. Zhao, “Passively Q-switching of a laser-diode-pumped inrtracavity-frequency-doubling Nd:NYW/KTP laser with GaAs saturable absorber,” Opt. & Laser Technol. **35**, 563–567 (2003). [CrossRef]

**4. **S. Zhao, X. Zhang, J. Zheng, L. Chen, Z. Cheng, and H. Cheng, “Passively Q-switched self-frequency-doubling Nd^{3+}:GdCa_{4}O(BO_{3})_{3} laser with GaAs saturable absorber,” Opt. Eng. **41**, 559–560 (2002). [CrossRef]

**5. **Z. Li, Z. Xiong, N. Moore, G. C. Lim, W. L. Huang, and D. X. Huang, “Pulse width reduction in AO Q-switched diode-pumped Nd: YVO_{4} laser with GaAs coupler,” Opt. Commun. **237**, 411–416 (2004). [CrossRef]

**6. **G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. **234**, 321–328 (2004). [CrossRef]

**7. **J. Zheng, S. Zhao, Q. Wang, X. Zhang, and L. Chen, “Influence of thermal effect in gain-media on optimum design of LD-end pumped solid state laser,” Acta Photonica Sinica **30**, 724–729 (2001) (in Chinese).

**8. **C. Pfistner, R. Weber, H. P. Weber, S. Merazzi, and R. Gruber, “Thermal beam distortions in end-pumped Nd:YAG, Nd:GSGG, and Nd:YLF rods,” IEEE J. Quantum Electron. **30**, 1605–1615 (1994). [CrossRef]

**9. **J. Harrison and R. J. Martinsen, “Thermal modeling for mode-size estimation in microlasers with application to linear arrays in Nd:YAG and Tm, Ho:YLF,” IEEE J. Quantum Electron. **30**, 2628–2633 (1994). [CrossRef]

**10. **F. Song, C. Zhang, X. Ding, J. Xu, G. Zhang, M. Leigh, and N. Peyghambarian, “Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO_{4} lasers,” Appl. Phys. Lett. **81**, 2145–2147 (2002). [CrossRef]

**11. **J. Zheng, S. Zhao, Q. Wang, X. Zhang, and L. Chen, “Influence of thermal effect on KTP type-II phase-matching second-harmonic generation,” Opt. Commun. **199**, 207–214 (2001). [CrossRef]

**12. **J. Dong, J. Lu, A. Shirakawa, and K. Ueda, “Optimization of the laser performance in Nd^{3+}:YAG ceramic microship lasers,” Appl. Phys. B **80**, 39–43 (2005). [CrossRef]

**13. **G. Li, S. Zhao, K. Yang, and H. Zhao, “Diode-pumped passively Q-switched Nd:YVO_{4} laser with GaAs saturable absorber,” Chin. Opt. Lett. **2**, 462–465 (2004).

**14. **X. Zhang, J. Yang, R. Han, and J. Yao, “Acousto-optic-dye double Q-switched laser: theory and experiments,” Chin. J. Lasers **19**, 241–246 (1992) (in Chinese).