1. Introduction

The investigation of lasing action in a large variety of disordered material structures including highly scattering powders, films, colloidal dye solutions, human tissues, etc is a subject of increasing interest [1–3]. In this type of cavityless lasers, multiple light scattering replaces the standard optical cavity of a conventional laser, and the interplay between gain and scattering determines the lasing properties. A detailed discussion about the latest results and theories concerning the mechanisms responsible for random lasing and the precise nature of the random laser modes can be found in Refs [1,4–6].

Since 1986 when Markushev et al. demonstrated laser-like behaviour in a powder of Na₅La_1-xNd_x(MoO₄)₄ at liquid nitrogen temperature [7], similar random laser experiments have been conducted at room temperature in numerous Nd-doped pulverized materials, mixtures of these powders, and highly scattering Nd-doped ceramics. The history and the state of the art of these neodymium-activated random lasers are reviewed by Noginov in Ref [2]. Among these materials, the stoichiometric NdAl₃(BO₃)₄ powder was regarded as a promising room temperature solid-state random laser material [8] because it presents many desirable features, such as a low laser threshold, a high gain, high Nd³⁺ concentration, and excellent physical and chemical properties [9,10]. In a recent paper the authors have studied the random laser performance of ground powders of the yttrium borate family, Nd_xY_1-xAl₃(BO₃)₄ (x = 0.5-1). In particular, the dependence of their random laser threshold, slope efficiency, and emission kinetics on the Nd³⁺ concentration. Although the obtained results are qualitatively similar in all the explored powders, our findings showed a reduction of the onset of laser-like emission and an increase of the slope efficiency when increasing the Nd³⁺ content [11]. At a first glance this behaviour seems to be unexpected due to the luminescence quenching at high concentrations; however, account taken of the short build up time of the random laser pulses, the lifetime shortening of the excited state, as concentration increases, does not affect the random laser process. It is therefore clear that for a random laser material high effective gain within the build up time scale of the pulse construction is the important issue. Following this idea, the authors have looked for alternative crystal laser materials having higher stimulated emission cross-sections than borates and which could allow using low rare-earth concentration doping but keeping the effective gain still optimal for random laser operation. Among those possible candidates Nd-doped orthovanadate crystals have been proved to be efficient laser materials for diode-pumped solid state lasers due to their large absorption and emission cross sections, high chemical stability, and high damage threshold. Recently Nd³⁺:LuVO₄ has attracted much attention since it was first reported by C. Maunier et al. in 2002 [12]. It has the highest absorption and emission cross-sections among vanadates, 6.9x10⁻¹⁹ cm² at 808 nm and 14.6x10⁻¹⁹ cm² at 1.06 µm respectively. Moreover, the relevance of Nd³⁺:LuVO₄ crystal has been recently confirmed by the demonstration of passively Q-switched laser-diode pumped nanosecond self-Raman laser operating at cascade downconverted frequency [13].

The main drawback of lutetium vanadate crystal is that it is difficult to be grown because of its high melting temperature and the high ion radii differences between neodymium and lutetium [14]. The Nd³⁺ doped lutetium vanadate single crystals used for obtaining the ground powder utilized in the random laser experiments presented in this work were grown by spontaneous nucleation in a Pb₂V₂O₇ flux [15]. Reagent-grade PbO and V₂O₅ and 99.99% Lu₂O₃ in the molar ratio 64.7:33.8:1.5 were used as starting materials. The dopant was added as 99.99% Nd₂O₃, and the concentration of Nd³⁺ was nominally 3 mol% with respect to Lu³⁺. A small amount of reagent-grade Na₂B₄O₇ was added to produce relatively thick rods [15]. The batch was put in a 50 cm³ platinum crucible and heated up to 1300 °C inside a horizontal furnace. After a soaking time of 15 h, the temperature was lowered to 900 °C at a rate of 1.8 °C/h. Then the crucible was drawn out from the furnace and quickly inverted to separate the flux from the crystals grown at the bottom of the crucible. This procedure permits fast recovery of most crystals and prevents them from being damaged as a consequence of flux solidification. The flux was then dissolved by use of hot diluted HNO₃. The crystals were grown in the form of clear blue-gray rods elongated in the direction of the c crystallographic axis. LuVO₄ crystallizes in the zircon structure and belongs to the tetragonal D _{4 h} ¹⁹ space group, with lattice parameters a = 7.0254(1) and c = 6.2347(1) Å [16]. The crystal size was approximately 3 × 1 × 0.8 mm³. The Nd³⁺ concentration in the sample used for random laser experiments was 3 mol%. The crystals were ground to powder of around 3 µm average grain size. The polydispersity of the measured powders were evaluated from SEM (scanning electron microscope) photographs like the one shown in the inset of Fig. 1 . The particle size has been computed calculating the mean of the major and minor axis lengths of the grains. As an example Fig. 1 shows the histogram of the particle size. By fitting the histogram to a log-normal function we obtain the average particle size.

 

Fig. 1 Histogram of the Nd³⁺:LuVO₄ particle size. The inset shows the SEM photograph of the laser crystal.

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It is worth noticing that in spite of the low concentration of neodymium in this vanadate powder, we have obtained a random lasing efficiency at room temperature which is comparable to that found in highly concentrated borates.

The present investigation provides a complete picture both in the spectral and temporal domain of the random lasing characteristics of Nd-doped vanadate powder together with a simplified model which accounts for the most relevant features of the random laser.

2. Laser experiments

The random laser experiments were carried out at room temperature by pumping the Nd³⁺ doped LuVO₄ crystal powder with a pulsed Ti-sapphire laser (10 ns pulse duration). The excitation pulse was tuned at the absorption peak wavelength of the ⁴I_9/2→⁴F_5/2 transition of Nd³⁺ (808 nm). The beam was focused to 3 mm in diameter. We have investigated both emission dynamics and output spectra as a function of pump energy.

The pumping beam energy was controlled by a half-wave plate combined with a polarizer and measured with an energy meter. The emission from the front face of the samples was collected in a backscattering arrangement by means of two lenses and a 500 µm diameter optical fiber. The image of the pumped area at the entrance face of the fiber was 1.5 mm diameter, so that only radiation from the central zone of the pumped area was registered. A long-pass filter was used to remove light at the pump wavelength. The spectral measurements of the emitted light were carried out by a TRIAX diode array analyzer (Jobin Yvon-Spex) with a spectral resolution of 0.3 nm and the emission kinetics were recorded by using a fast photodiode (Thorlabs SIR-5) connected to a digital oscilloscope (temporal resolution of 400 ps).

Figure 2 shows the emission spectra of the ⁴F_3/2 →⁴I_11/2 transition of Nd³⁺ in the powder sample obtained at low (5 mJ) and high (24 mJ) pumping energies. At low pump energy the spectrum presents the typical spontaneous emission features with a main peak around 1065.6 nm and some additional structures spreading over 10 nm. When increasing the pump energy, there is a threshold value of 9 mJ above which the emission peak intensity suddenly increases whereas the emission spectrum collapses to a narrow single line at 1065.6 nm and 0.3 nm HWHM, our spectral resolution limit.

 

Fig. 2 Normalized emission spectra from Nd³⁺(3%) LuVO₄ powder crystal obtained at pumping energies below (5 mJ blue) and above (24 mJ red) the laser threshold.

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The emission kinetics parallels the spectral behavior; at low pumping energy the spontaneous emission is single exponential with a decay time of 60 µs whereas above threshold the time profile shortens up and gives a fast emission pulse with a duration of about 800 ps. As pumping increases well above the threshold energy, more pulses appear in the emission temporal profile showing the typical laser relaxation oscillation behavior. As can be seen in Fig. 3 , we recorded, at maximum pump energy, up to six oscillations. As is common in conventional solid state lasers, the build-up time (time delay between the output pulses and the pumping pulse) of the random laser pulse decreases gradually as pumping energy increases. As its duration is about 10 ns, this figure clearly shows that the relaxation oscillations remain until the pump pulse is over.

 

Fig. 3 Stimulated emission intensity pulses of Nd³⁺ (3%): LuVO₄ powder crystal as function of time at some different pump energies. The zero time marks the start of the pump pulse.

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We have compared the random laser performance of this material with the one obtained in a ground laser powder of NdAl₃(BO₃)₄ under the same focusing and measuring conditions. The pumping wavelength used for the borate was 802 nm, which corresponds to its maximal absorption wavelength. To avoid damage in the vanadate sample surface the maximum pump energy used was 25 mJ (about 300 mJ/cm²). The damage threshold in borate is at least one magnitude order higher.

Figure 4 shows the time integrated intensity of the output pulses from these powders as a function of the pumping energy. The laser threshold was obtained with pump pulses of about 9 mJ for both samples being the slope efficiencies similar but slightly higher for the vanadate sample. The origin of this similar random lasing behaviour relies on the different Nd³⁺ concentrations of both compounds. Although the emission cross section in Nd-doped lutetium vanadate is higher than in the aluminium borate compound, NdAl₃(BO₃)₄ is a stoichiometric compound with higher Nd³⁺ concentration (around 20 times) than the 3% Nd³⁺-doped LuVO₄ and therefore it is possible to obtain an “accidental” equalized gain for the random lasing emission. Moreover, although the focussing conditions were the same for both compounds, the pumped volume, and therefore gain, may be different due to the different neodymium concentrations and refractive indices of both compounds. These results therefore suggest that better random laser performances of the vanadate could be achieved with more concentrated powders as has already been demonstrated in Nd-doped borate crystal powders [11].

 

Fig. 4 Laser output energy as a function of the pumping energy for Nd³⁺(3%):LuVO₄ (red dots) and NdAl₃(BO₃)₄ (blue dots) powders.

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3. Rate-equation model results

The simplest theoretical models applied to random lasers [10,17,18] are those based on the rate-equations applied to conventional lasers for long, which assume both uniform averaged population inversion and photon density inside the pumped volume. Their application to the present case is adequately described by Eqs. (1-2) displayed below:

The input values for the numerical calculations performed for the 3% Nd³⁺ doped LuVO₄ powder are the following: v = 2x10⁸ m/s, σ_em=1.46 x10⁻¹⁹ cm², L has been calculated using a filling factor of 60%, a transport length of l_t = 6 µm, an absorption cross-section of 6.9x10⁻¹⁹ cm² and a Nd³⁺ concentration of 2.6x10²⁰ atoms per cm³ [19], K=17000 s⁻¹ and β = 0.18 [20]. The pump pulse was simulated by the I_in(t)=I₀ t exp(-α t) function with a time constant α=5x10⁸ s⁻¹. The absorption efficiency is difficult to estimate, but account made of the absorption and diffusion properties of the material, a value of 25% has been used, mainly due to reflection on the sample [21].

For the NdAl₃(BO₃)₄ powder, we have used the following parameter values σ_em=1x10⁻¹⁸ cm², Nd³⁺ concentration = 5x10²¹ atoms per cm³, absorption cross-section = 6x10⁻²⁰ cm², obtained by extrapolation from optical density measurements in samples with low Nd³⁺ concentration. The filling factor was 50%, K=77000 s⁻¹, β = 0.2, l_t = 9 µm [11], being the rest of parameters, the same as those used for the vanadate.

In order to achieve the experimental threshold pumping energies shown in Fig. 4 for both powdered materials, l was used as an adjusting parameter in Eqs. (1-2). The obtained values were l = 130 µm for the vanadate and l = 280 µm for the borate. By using these values and solving Eqs. (1-2) for different pumping energies, we have obtained the same 0.19 slope efficiency for both compounds, which is the absorption efficiency multiplied by the relation between stimulated emission and pumping photon energy.

Although the threshold and slope efficiencies of both samples are well simulated by this simple model, the temporal behavior of the emitted pulses is not so well described. Figure 5(a) shows the temporal shape of the laser output of the vanadate sample predicted by the model whereas Fig. 3 displays the experimental result in this powder at the same pumping energies. Note that the temporal oscillations are much faster in Fig. 5(a) than in Fig. 3. An analogous result was obtained in the borate powder.

 

Fig. 5 Theoretical predictions of the laser output of Nd³⁺(3%):LuVO₄ powder crystal as a function of time and for different pumping energies: (a) using l = 130 µm and (b) using l = 1950 µm and σ_em/15.

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To solve this discordance, which also occurs in some other powdered materials [10,17], the model should yield a slower temporal evolution with no changes in the input/output properties of the system (threshold energy and slope) which on the contrary are well reproduced. The direct way to do this is to introduce a minor effective stimulated emission cross section (σ_em) and in the same proportion a longer mean path (l) for photons in the material which would affect the temporal behaviour, but as N₀ would not change, threshold and slope would also remain invariant as required.

Figure 5(b) shows the predicted oscillations for l = 1950 µm and an effective stimulated emission cross-section of σ_em/15. After this improvement on the model, quite good predictions of the experimental delay time for lasing, elapsed time among oscillations, and number of oscillations per pulse are attained. On the other hand, strong spatial modal interactions, not considered by this model, are probably the source of the broadening and irregularities of the experimental oscillations, which obviously deteriorate the quality of the comparison.

These results suggest that the zone with high population inversion, given by the absorption length L, is smaller than the zone with a relevant probability of presence of stimulated emission photons. The amplification of the photon is less than expected (lower σ_em than nominal value), because a great part of the path it travels is out from the amplifying zone. As a consequence to have an input/output energy and a threshold behavior compatible with the experimental temporal performance of random laser, the length of this travel must be necessarily greater than firstly obtained, which reinforces the idea that the volume of the effective laser cavity is greater than the pumped volume [17].

4. Conclusions

We have obtained random laser action from Nd³⁺ doped LuVO₄ crystal powder for the first time. We have compared its random laser performance with the one measured, under similar experimental conditions, in the stoichiometric NdAl₃(BO₃)₄ crystal powder. Account taken of the similar performance obtained and the very different dopant concentration of both powders, the results open up the possibility of using these Nd-doped vanadates as efficient random laser sources.

We have performed a numerical calculation using rate-equations to obtain the mean free path of the photons generated in both materials before they leave the sample. Although the model predictions are good for the energy threshold and slopes, it cannot describe the temporal behavior of each pulse. In order to improve the predictions of the temporal behavior of the emitted pulses, it is necessary to use minor effective stimulated emission cross-sections σ_em as well as longer mean paths l of the photons in these materials by keeping the threshold population inversion N₀ constant. As it is suggested in Ref [17], the necessity of these changes must be interpreted on the basis of a larger effective laser cavity volume if compared with the pumped volume.