## Abstract

Using V:YAG as the saturable absorber, a diode-pumped passively Q-switched and mode-locked Nd:GdVO_{4} laser at 1.34µm is realized. Nearly 100% modulation depth of mode-locking has been achieved. The width of the mode-locked pulse is estimated to be less than 460ps with 125MHz repetition rate within an about 1µs-long Q-switched pulse envelope. A maximum output power of 220mW and Q-switched pulse energy of 10.5µJ is obtained. Using the hyperbolic secant function methods, a fluctuation rate equation model considering the Gaussian distribution of the intracavity photon density and the population inversion in the gain medium as well as the ground-state population intensity of the saturable absorber has been proposed to describe the mode-locking process of diode-pumped Nd:GdVO_{4}/V^{3+}:YAG laser. With the space-dependent rate equations solved numerically, the theoretical calculations reproduce the laser characteristics well.

©2008 Optical Society of America

## 1. Introduction

Diode-pumped solid-state laser sources near 1.34 µm have wide applications in the fields of spectroscopy, micro machining, optical fiber communication, remote sensing, information storage, etc. Moreover, based on the 1.34 µm laser system, highly efficient radiation in the red wavelength range can be easily achieved by frequency doubling, which plays important role for applications such as medical treatment and laser display. Therefore, both continuous wave and pulsed laser operation at a wavelength of 1.34um, has attracted much attention in the last few years. Until now, many laser materials including Nd:KGW[1], Nd:YAG[1], Nd:GdVO_{4}[2], Nd:LuVO_{4}[3], and Nd:YVO_{4}[4], have been successfully used for 1.34 µm radiation; for this wavelength range, Nd:GdVO_{4} promising laser crystal material for not only the wavelength range around 1.06um but also for 1.34 um radiation, due to the large absorption coefficient at 808nm (78 cm^{-1}), large stimulated emission cross-section of 1.8×10^{-19}cm^{2} at 1.34 µm and 7.6×10^{-19}cm^{2} at 1.06 µm as well as unexpectedly high thermal conductivity of 11.7 Wm^{-1}K^{-1} along the <110> direction [2].

As for the pulsed laser operation at 1.34µm, due to the lack of proper saturable absorbers for continuous-wave mode-locking, much attention has been focused on the Q-switching performance, in the last few years. Up to now, several saturable absorbers such as Co^{2+}-doped crystals [4, 5], semiconductor saturable absorber mirrors (SESAMs) [6], and V^{3+}-doped crystals [7–10] have been employed as passive Q-switches. The reported experimental results show, that pulse widths obtained by Co-doped crystals such as Co:LMA are usually large [5], while a SESAM introduces higher loss and results in a lower damage threshold, which limits its application [6]. Among the V^{3+}-doped crystals, V^{3+}:YAG crystals have attracted great interest due to their excellent physical and optical performance at 1.34µm, such as high damage threshold, short absorption recovery time of about 5 ns, high ground state absorption cross-section of 7.2×10^{-18}cm^{2}, low saturable energy density of 0.05J/cm^{2} and low residual absorption at 1.34µm [8], which makes them an effective passive Q-switch for 1.34µm radiation. Previous studies showed that there are different absorption peaks including 425nm, 800nm, 1140nm and 1320nm in the absorption spectrum of V^{3+}:YAG crystal, which are attributed to two possible co-ordinations of the V^{3+} ions: tetrahedral and octahedral. For the passive Q-switching operation at 1.34µm, the corresponding transition is ^{3}A_{2}→^{3}T_{2} (^{3}F) of the tetrahedrally co-ordinated V^{3+} ions and the energy level ^{3}T_{2} (^{3}F) has a finite lifetime of 22 ± 6 ns [8]. Further studies show that there is excited-state absorption (ESA) in V^{3+}:YAG crystal, as is found in Cr^{4+}:YAG crystal, so it is also considered to be able to realize mode-locking when the ESA is saturated by strong intracavity intensity although the relaxation time of the ESA in V^{3+}:YAG is not measured precisely. In 2001, Agnesi etc. realized Q-switched mode-locked operation by using V^{3+}:YAG as saturable absorber in a diode-pumped Nd:YVO_{4} laser[11], which showed that the V^{3+}:YAG crystal could be used as mode-locker. Since then, however, it is rarely reported on the mode-locking characteristics of V^{3+}:YAG. Furthermore, there has not been any report on the theoretical model describing either the mode-locking mechanism or the Q-switching dynamics of V^{3+}:YAG as far as we know.

In this paper, we report, to the best of our knowledge, on the first diode-pumped passively Q-switched and mode-locked Nd:GdVO_{4} laser at 1.34µm using V^{3+}:YAG as saturable absorber. Nearly 100% modulation depth of mode-locking has been achieved. The width of the mode-locked pulse is estimated to be less than 460ps with 125MHz repetition rate within an about 1µs-long Q-switched pulse envelope. A maximum output power of 220mW and Q-switched pulse energy of 10.5µJ at 1.34µm is obtained. Using the hyperbolic secant function methods, a fluctuation rate equation model considering the Gaussian distribution of the intracavity photon density and the population inversion in the gain medium as well as the ground-state population intensity of the saturable absorber has been proposed to describe the mode-locking mechanism of V^{3+}:YAG for the first time to our best knowledge. With the space-dependent rate equations solved numerically, the theoretical calculations reproduce the laser characteristics well.

## 2. Experimental setup and results

The experimental setup is schematically shown in Fig. 1. The pump source is a fiber-coupled laser-diode (LYPE Inc., China) working at 808 nm. The diameter of the fiber core is 400µm with a numerical aperture of 0.22. The pump beam is collimated and focused into the crystal with diameter of 400µm by a focusing optics system (1:1 imaging module, Coherent Inc., USA), which results in a coupling efficiency of 90%. A 0.52at.%-doped a-cut Nd:GdVO_{4} crystal with dimension of 3×3×5 mm^{3} is employed. The pumped face of the Nd:GdVO_{4} crystal is antireflection coated at 808nm and high reflection coated at 1.34µm, which functions as the reflection mirror of the resonator. The back face of the laser crystal is antireflection coated at both 808nm and 1.34µm. The Nd:GdVO_{4} crystal is wrapped with indium foil and held in a copper block cooled at 20°C by water. The folded concave mirrors M_{1} (R=500mm) and M_{2} (R=100mm) are coated with high reflection at 1.34µm. A flat mirror M_{3} with transmission of 3% at 1.34µm is employed as the output coupler. To suppress the oscillation of the ^{4}F_{3/2}→^{4}F_{11/2} transition, the output mirror also had sufficient transmission (>90%) at 1.06 µm. The distances between the pumped face of Nd:GdVO_{4} crystal and M_{1} as well as between M_{1} and M_{2} are 400mm and 730mm, respectively. The folded angles of mirror M_{1} and M_{2} should be as small as possible to decrease the influence of astigmatism between the sagittal and tangential directions of the resonator. Because the thermal lensing effect in the gain medium can lead to the variation of oscillating mode size and instability of the oscillation, the distance between the output mirror M_{3} and M_{2} is designed to range from 55mm to 45mm, corresponding to the variation of pump power between threshold and maximum, respectively.

A 0.5mm-thick V^{3+}:YAG crystal with initial signal transmission of 94% is employed, which is AR coated at 1.34µm on both faces. The V:YAG crystal is placed near the output coupler with 3% transition at 1.34µm. Considering the effect of anisotropy in V:YAG [12], V:YAG is placed in a mount, which can be rotated regarding the optical axis of the laser cavity and fixed where the maximum output power is obtained. The total cavity length is about 120cm. According to the restrict ABCD matrix theory and considering the thermal-lensing effect of the laser medium, the oscillating mode in the laser crystal can be remained as 160µm, matching with the pump beam well at all time and the mode radius on V^{3+}:YAG is about 60µm. The temporal profile of the pulse is recorded by 1GHz-photodiode (New Focus Inc.) with a rising time of 400ps and a 1GHZ bandwidth digital oscilloscope DPO (Tektronix Inc., USA). A Field Max_{||} laser power meter (COHERENT Inc., USA) is used to measure the average output power.

The continuous-wave characteristics of the diode-pumped Nd:GdVO_{4} laser at 1.34µm are studied first. The threshold pump power is 140mW, and up to 1.23W average output power in a nearly diffraction-limited beam is obtained at the maximum pump power of 6W, corresponding to an optical-optical conversion efficiency of over 20%. At the maximum output power, the instability is observed to be less than 1% over an hour. Figure 2 shows the output power versus the pump power, corresponding to a slope efficiency of nearly 21%. With the V^{3+}:YAG inserted into the cavity, the oscillation threshold increases to be 1.15W and the laser operates in the regime of passively Q-switched mode-locking. The highest output power of 220mW is obtained at the pump power of 5.3W. However, the output power begins saturating when the pump power exceeds 5.3W, which is attributed to the thermal effect on V^{3+}:YAG. For the same reason, the repetition rates of the Q-switched pulse train show the similar variation tendency. As is shown in Fig. 3, the repetition rate decreases abruptly when the pump power reaches 6W. However, the decrease of the repetition rate can benefit the increase of the pulse energy. Estimated from the average output powers and the repetition rates, the Q-switched pulse energy is plotted in Fig. 4. From Fig. 4, we can see that the Q-switched pulse energy reaches the highest of 10.5µJ at the pump power of 6W, where the output power begins saturating and the repetition rate decreases abruptly. Figure 5 shows a typical Q-switched pulse train with the repetition rate of 32kHz at the pump power of 4.5W, in which the pulse to pulse instability is found to be less than 1%.

In our experiments, nearly 100% modulation depth can be yielded as long as the pump power reaches the oscillation threshold. Kartner etc have proposed a criterion for the onset of Q-switched mode-locking, which can be written as following [13]:

where σ and σ_{gs} are the stimulated emission cross-section of the gain medium and ground state absorption cross-section of saturable absorber, A_{G} and A_{s} are the mode areas in the gain medium and saturable absorber, q_{0} is the small-signal loss of saturable absorber. By substituting the related data in Table I into Eq.(1), the left side is much larger than the right one, so the laser demonstrates nearly fully modulation as long as the pump power reaches the threshold, which is in consistent with the conclusion of Ref.[11].

Figure 6 shows a nearly 100% fully modulated mode-locked pulse train within a 1-µs long Q-switched envelop obtained at the pump power of 6W. The corresponding expanded oscilloscope traces of the mode-locked pulse train are demonstrated in Fig. 6, from which we find that the mode-locked pulses within the Q-switched envelope are separated by 8ns, matching exactly with the cavity roundtrip transmission time and corresponding to a repetition rate of 125MHz. As the autocorrelator at 1.34µm is not available, we can not measure accurately the mode-locked pulse width. However, by using the formula ${t}_{\mathrm{measure}}=\sqrt{{t}_{\mathrm{real}}^{2}+{t}_{\mathrm{probe}}^{2}+{t}_{\mathrm{oscilloscope}}^{2}}$ which describes the relationships among the measured rise time *t*
* _{measure}*, the real rise time

*t*

*of the pulse, the rise time*

_{real}*t*

*of the photodiode and*

_{probe}*t*

*of the oscilloscope employed [14], we can estimate the pulse width roughly. The rise time of oscilloscope*

_{oscilloscope}*t*

*is determined by*

_{oscilloscope}*t*

*×*

_{oscilloscope}*BW*=0.35~0.4, where

*BW*is the bandwidth of the oscilloscope. The employed oscilloscope in our experiment has a bandwidth of 1GHz, corresponding to a rise time of 350ps. The rise time of employed probe is 400ps. With a close observation of Fig. 7, the rise time of the mode-locked pulse is estimated to be about 650ps. So the real rise time of the mode-locked pulse is estimated to be about 370 ps by using the above formula. According to the definition of the rise time and considering the symmetric shape of the mode-locked pulse, we can assume the mode-locked pulse width is approximately 1.25 times of the real rise time. Then the duration of the mode-locked pulse is estimated to be less than 460ps.

## 3. Theoretical analysis

The fluctuation mechanism has been proposed to explain the generation of picosecond pulses in simultaneously Q-switched and mode-locked laser with saturable absorber [15], in which the picture of the ultrashort pulse formation is described as follows: In the linear stage of generation, the fluctuations of intensity arise due to the interference of a great number of modes having a random phase distribution so that the radiation consists of a chaotic collection of ultrashort peaks, and in the nonlinear stage when the absorber is bleached, the most intensive fluctuation peaks are compressed and amplified faster than the weaker ones. Then one or several ultrashort pulses are amplified during each round trip in the cavity until all the stored energy in the active medium is emitted. The shortening of the pulses during the nonlinear stage has a finite value raging from 10 to 20 times [15]. After subsequent round trips the preferred pulse will not be much compressed in time. To simulate such process of ultrashort pulse formation, the hyperbolic secant function methods were employed [16,17], in which the theoretical calculations were in good agreement with the experimental results, however, the theoretical analysis is under plane-wave approximation. Recently, we have applied a rate equation model considering the Gaussian distribution of the intracavity photon density to demonstrate the self-mode-locking process of a Cr^{4+}:Nd^{3+}:YAG/KTP laser, and the theoretical calculations reproduces the laser performance well [18].

In comparison with Cr^{4+}-ions, V^{3+}-ions has the similr energy level, which indicates the V^{3+}:YAG has the similar mode-locking mechanism. So based on the rate equation model describing the mode-locking process of Cr^{4+}:YAG, the rate equations describing diode-pumped Q-switched and mode-locked Nd:GdVO_{4} laser with V^{3+}:YAG saturable absorber can be written as follows [16, 18]:

$$-2{\sigma}_{\mathrm{gs}}{n}_{s1}(r,{t}_{k}){1}_{s}\frac{{\omega}_{1}^{2}}{{\omega}_{s}^{2}}\mathrm{exp}\left(-\frac{2{r}^{2}}{{\omega}_{s}^{2}}\right)$$

$$-2{\sigma}_{\mathrm{es}}\left[{n}_{s0}-{n}_{s1}(r,{t}_{k})\right]{l}_{s}\frac{{\omega}_{1}^{2}}{{\omega}_{s}^{2}}\mathrm{exp}\left(-\frac{2{r}^{2}}{{\omega}_{s}^{2}}\right)$$

$$-\left[L+\mathrm{ln}\left(\frac{l}{R}\right)\right]\mathrm{exp}\left(-\frac{2{r}^{2}}{{\omega}_{l}^{2}}\right)\left]2\pi \mathrm{rdr}\right\}$$

where equation (2) describes the relative amplitude of the mode-locked pulses at time t_{k}=kT_{R} after an additional roundtrip through the cavity and Φ_{k} is the relative amplitude of the mode-locked pulses at the kth round trip, equations (3) and (4) describes the time rate of variation of the inverse population density *n*(*r*, *t*) in the gain medium and the absorber population ground state density *n*
_{s}
_{1}(*r*, *t*), respectively, *n*
_{s}_{0} is the total density of the absorber, and related to the small initial transmission of the saturable absorber by T_{o}=exp(-σ_{gs}n_{so}l_{s}), *l*
* _{s}* is the length of the absorber, σ

_{gs}and σ

_{es}are the ground-state and excited-state absorption cross section of the absorber,

*R*

_{in}(

*r*)=

*P*

*exp(-2*

_{in}*r*

^{2}/

*ω*

^{2}

*) [1-exp(-*

_{p}*αl*)]/

*hυ*

_{p}*πω*

^{2}

_{p}*l*is the pump rate, where

*P*

_{in}is the pump power,

*hυ*

*is the single-photon energy of the pump light,*

_{p}*ω*

*is the radius of the pump beam,*

_{p}*α*is the absorption coefficient of the gain medium,

*l*is the length of the laser crystal,

*τ*is the stimulated-radiation lifetime of the gain medium,

*τ*

*is the excited-state lifetime of the saturable absorber,*

_{s}*R*is the reflectivity of the output mirror, and

*L*is the nonsaturable intracavity roundtrip dissipative optical loss. ${\varphi}_{\mathit{G}}(r,t)=({\omega}_{\mathit{l}}^{2}\u2044{\omega}_{\mathit{G}}^{2})\mathrm{exp}(-2{r}^{2}\u2044{\omega}_{\mathit{G}}^{2})\sum _{k=0}{\Phi}_{k}f\left(t-{t}_{k}\right)$, $f\left(t\right)=\frac{1}{2\sigma c{\tau}_{p}}\mathrm{sec}{h}^{2}\left(\frac{t}{{\tau}_{p}}\right)$ is the mode-locked pulse evolving from the noise and satisfies $\underset{-\infty}{\overset{\infty}{\int}}c\sigma f\left(t\right)\mathrm{dt}=1$ [15], in which

*c*is the speed of light and

*σ*is the stimulated emission cross section of the gain medium,

*τ*

*is related to the FWHM mode-locked pulse duration*

_{p}*τ*at fundamental wavelength by

*τ*=1.76

*τ*

*,*

_{p}*ω*

*is the radius of TEM*

_{G}_{00}mode at the position of gain medium,

*ω*

*is the average radius of TEM*

_{l}_{00}mode oscillating,

*ω*

*is the radius of TEM*

_{s}_{00}mode at the position of saturable absorber, which can be calculated by ABCD matrix theory, t

_{k}=kT

_{R}, T

_{R}=2[

*n*

_{1}

*l*+

*n*

_{2}

*l*

*+(*

_{s}*L*

*-*

_{p}*l*-

*l*

*)/*

_{s}*c*is the round trip time of the resonator,

*n*

_{1}and

*n*

_{2}are the refractive indices of the gain medium and saturable absorber, respectively, and

*L*

*is the physical length of the cavity.*

_{P}Using the parameters shown in Table 1 and given an initial value of Φ_{0}, Φ_{k} can be obtained by numerically solving equations (2)–(4). Considering the bidirectional propagation of the fundamental wave in the laser cavity, the average power of the fundamental wave is:

where *A*=(1/2)*πω*
^{2}
* _{l}* is the average beam area of fundamental wave.

By integrating Equation (5) over time from zero to infinity, the total output energy of the Q-switched pulse envelop can be obtained as [18]:

We have estimated the width of the mode-locked pulse as about 460ps, so *τ*
* _{p}* is about 260ps according to the relation of

*τ*=1.76

*τ*

*. Considering*

_{p}*τ*

*=260ps and using equation (5), we can obtain the calculated temporal Q-switched pulse shape. Figure 8 shows the calculated temporal shape of a mode-locked pulse train within a 1µs-long Q-switched envelope at the pump power of 6W. The modulation depth of the mode-locked pulse is 100%, which reproduces the experimental pulse shape well. According to equation (6), we can also obtain the theoretical dependence of the total energy of the Q-switched pulse envelop on the incident pump power, which is shown as solid line in Fig. 4. From Fig. 4, it can be seen that the theoretical results have the similar tendency of variation with the experimental results.*

_{p}## 4. Conclusion

In summary, a Q-switched and mode-locked Nd:GdVO_{4} laser at 1.34µm with V^{3+}:YAG as saturable absorber has been demonstrated in a z-type cavity for the first time as far as we know. Nearly 100% modulation depth of mode-locking has been achieved. The width of the mode-locked pulse is estimated to be less than 460ps with 125MHz repetition rate within an about 1µs-long Q-switched pulse envelope. A maximum output power of 220mW and Qswitched pulse energy of 10.5µJ is obtained. Using a hyperbolic secant function method, a fluctuation rate equation model considering the Gaussian distribution of the intracavity photon density and the population inversion in the gain medium as well as the ground-state population intensity of the saturable absorber has been proposed to describe the mode-locking characteristics of the diode-pumped Nd:GdVO_{4}/V^{3+}:YAG laser. With the space-dependent rate equations solved numerically, the theoretical calculations reproduce the laser characteristics well.

## Acknowledgments

This work was partially supported by the China Postdoctoral Science Foundation (No. 20080430191), the National Natural Science Foundation of China under Grant No. 10534020, 50721002 and 60578010 as well as the Natural Science Foundation of Shandong Province (No. Y2005G10).

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