Random laser properties of Nd crystal powders

This work presents an overview of near infrared random lasing emitters based on a variety of neodymium (Nd)-doped crystal powders with different Nd concentrations and different grain sizes. The pump-configuration used allows for an absolute measurement of both pumping and emitted energies. The results provide an absolute measure of the random laser efficiency and prove a relation of direct proportionality between the absorbance of the material and the laser slope efficiency. Likewise, they suggest a relationship close to an inverse proportionality between the absorbance and the threshold energy per unit area. The temporal behavior of the random laser emission shows noteworthy differences between local and spatially integrated registers. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (140.3530) Lasers, neodymium; (290.0290) Scattering. References and links 1. C. Gouedard, D. Husson, C. Sauteret, F. Auzel, and A. 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Introduction
Highly scattering materials have attracted attention as laser sources due to potential applications such as high definition speckle-free phosphors, holographic laser displays, optical or chemical sensors for medical diagnosis, nanoscale lithography, miniature spectroscopy, etc [1][2][3].
Conventional lasers are built on accurate reflecting cavities (resonators) so they have a well defined direction, very high coherence and mode stability, etc. In contrast, Random Laser (RL) sources are cavityless. Multiple light scattering drives the feedback of light (spatially distributed), and for that reason random lasing is emitted in all directions and with very low temporal and spatial coherences, pretty good for example, for high definition imaging [1][2]. As in conventional lasers, in RLs also, when pumping with energies below a given threshold energy, the obtained emission has relatively low intensity, broad spectrum and long decay time, but when pumping above a threshold value, the emission pulse becomes suddenly some orders of magnitude more intense, shorter in time, and narrower in spectrum.
A very wide range of materials have been tested as randomly emitting laser sources, like crystal powders, colloidal dye solutions, polymers, biological samples and even human tissues [4][5][6]. That so widespread kinds of emitting materials and scattering regimes, give very different behaviors in emission spectra, energy efficiencies, pulse time-ranges, spatial profiles etc, and furthermore, the parameters used by different authors to describe their results also tend to be different, and sometimes not completely described, which hinders understanding among different authors, as stated in [7].
Many efforts have been made to develop a general theoretical treatment for RL, for their spectral behavior [7][8][9], temporal dynamics [10][11][12][13], threshold and output-energies etc., but many fundamental questions remain unsolved, and some basic features of RLs are still unfixed.
The current results found in the literature about the investigations on random lasers based on Nd 3+ -doped micro-nano crystal powders display both quantitative and interpretative dispersion wide enough, so as to deserve a careful revision of their main features. In particular, of the absolute stimulated emission energy (and therefore slope efficiency), laser threshold energy and their dependences with the pump spot size and pump wavelength, and of the time dynamics of emission pulses and spatial distribution of pump and emission beams.
In this review we would like to clarify some of the basic properties of random lasing in rare earth-activated materials. For this purpose we will use a variety Nd 3+ doped crystal powders (in the 4 F 3/2 → 4 I 11/2 emission band of Nd 3+ ions around 1.064 μm) with different Nd 3+ concentrations and different grain sizes, with transport lengths in the 10-100 micron range. The investigated properties are the emission spectrum, the threshold energy and the absolute emitted energy and therefore, the slope efficiency, and their dependence on pump beam area, pump wavelength, Nd 3+ concentration and grain size. The dependence of threshold energy and laser slope with absorbance is well established.
Likewise, the spatial profiles of pump and emission areas and the temporal behavior of the RL emission have been investigated. Imaging experiments give an accurate measure of these areas and show a different behavior between high and low absorbent materials. In the last case a strong expansion of the laser emission area can be observed. Moreover, the temporal behavior of the detected RL emission shows significant differences between local and spatially integrated registers. Intensity fluctuations with characteristic times less than 100 ps are observed when detection is spatially resolved which could be attributed to coherent wave interferences.

Measurement system
In an amplifying random medium, where light is multiply scattered and absorbed, the relevant length scales that matter are the transport, inelastic, absorption and gain lengths. The transport length gives the mean length of diffusion, the inelastic length is the inverse to the absorption coefficient of the bulk material corrected by the filling factor, and the absorption length which depends on the transport and inelastic lengths, is the length of penetration of radiation into the sample. The gain length describes the amplification by stimulated emission. These magnitudes are defined in section 5. It is worth noticing that in the stoichiometric materials, transport, inelastic and absorption lengths are usually of the same order, whereas in the low-doped samples, the transport length is much shorter than the other two.
To estimate their relative grain sizes, related to the transport lengths, we took microscope photographs of surfaces of all the powdered samples. On the other hand, the diffuse reflectance spectrum of all the samples has been measured by means of a CARY 5-Varian spectrometer with an integrating sphere. As we shall see, these measurements allow to find relationships between the absorbance and RL slope and threshold.
The precise format of each measuring set-up is based on the specific objectives of the measurement. In Fig. 1, we show the four different ones used.
In our case, RL material consists of a compressed powder of an inorganic Nd 3+ doped chemical compound, with no addition of any specific diffusers, such as spheres, nanorods or others. Thus, the random nature of the material cannot be altered by any intervention other than grinding and compressing the powder itself. The filling factor is in all cases in the range of 30-50% and the transport length is in the 10-100 micron range [14]. The material is pumped by a pulsed tunable Ti: sapphire laser (BMI, TSA-802 model), 10 ns time width, with the aim of pumping 4 F 5/2 or eventually 4 F 3/2 bands (around 800 nm and 870 nm wavelengths respectively) and get laser emission around 1060 nm wavelength. The pumped area is controlled by means of a movable focusing lens. Maximal pumping pulse energy is about 35 mJ and its control is carried out by a variable attenuator by reflection (Lotis, VA-R model), to avoid changes in the pump beam size when changing the pumping energy. Lateral pumping (around 20°) has been chosen to prevent damage in the collecting optics because of the high intensity of the pumping beam. The relative intensity of the pump beam is obtained by using a silicon detector (Newport, 818-BB-21 model), which measures the radiation diffused by the folder mirror used to address the beam. The relative energy is given by the area under the curve of intensity registered in the oscilloscope (Tektronix, TDS 7104 model, 10 Gs/s). For absolute pumping energy measurements, the device has been previously calibrated for different pump wavelengths. For this purpose, the area under the intensity curve in the oscilloscope was compared with the one obtained by using a calibrated optical head detector (Ophir, PE25 model) placed at the powder sample position, defocusing the movable lens to avoid detector damage.
The RL emits its radiation in all directions, and therefore, it is difficult to collect a large proportion of the emitted energy. When the goal is simply to collect as much energy as possible, the scheme of Fig. 1(a) is suitable. The collecting lens receives part of the emitted energy (or reflected pumping) and projects it onto the head of the optical fiber. A low-pass filter removes the pump radiation. Sensibility and stability of the system depend on the collecting lens and optical fiber characteristics as well as on the distances between components. This is a suitable format to get the spectrum, by deriving the fiber to a spectrometer (Jobin Ybon, Instruments SA Inc., TRIAX-190 model), to work below threshold, to get a not too accurate temporal shape of the intensity, or even to measure the lifetime of spontaneous emission. In these cases we have used a fiber optic of 0.5 mm in diameter, coupled to a detector (Thorlabs, SIR-5 model) directly connected to the oscilloscope. The total temporal widening introduced by the system, both for the signal and the pumping is about 200-300 ps.
The 1a format is not appropriate when comparing emitted energies from slightly different positions or sizes of emission surface, or if we change the pumping wavelength, which could induce slight changes in the pumping beam direction and/or pump beam size, because the coupling between the RL emission concentrated by the collecting lens and the head of the optical fiber may considerably change. The problem is solved by using the simple setup shown in Fig. 1(b), where the collecting lens has been just suppressed. Though much signal is lost (usually this is not a great problem) the measurement is completely stable to all the above mentioned changes, and thus, different emission energy measurements are comparable among them. Of course, this is accomplished if the distance from the sample to the optical fiber is much longer than the size of the emitting surface, always fulfilled in our experiments.
Given the spatial incoherence of RL emission, its angular distribution is assumed to be Lambertian, (or cosine law) as has been reported before [15], in good agreement with the experimental results. This assumption also allows to calibrate the absolute energy emitted by the RL, because the energy per unit of area, M, measured at distance r and at angle θ with respect to the normal direction is: where E em is the total energy emitted by the RL. Therefore, if a calibrated optical head of sensible area S is placed at distance r (being the size of S much smaller than r) and angle θ, the relation between the total emitted energy E em and the energy measured by the optical head E mea is: A simultaneous measurement of the RL emission with the calibrated optical head (Ophir, PE9 model) and the collected one by the optical fiber and SIR5 detector connected to the oscilloscope, allows to achieve a calibration between the absolute emitted energy and the area under the curve of RL emission intensity registered in the oscilloscope. This method has been tested by using different active materials [16], because all of them must give the same calibration slope, and we have used it in several RL materials [16][17][18][19][20].
The format of Fig. 1(c) is used for local intensity measurements. In RLs there are differences between temporal intensity shapes depending on the detection mode, for example, if it is only collected from a very small area of the emitting surface or integrating all of it [21]. In this scheme, the image of the emitting surface is projected on the pin-hole, so that only the radiation emitted by a small area of the emitting surface (down to 10 micron) is captured by the fiber. In our experiments, it was possible to remove the pin-hole by directly using a single mode optical fiber (10 micron core). For a better temporal resolution (though less sensibility in energy) we use a "faster" system: a detector (Tektronix, TCA 292 D model) directly connected to an oscilloscope (Tektronix, DPO 72504 D model, 100 Gs/s). The total temporal widening introduced by the system is about 30 ps, getting a finer observation of the temporal shape of the intensity emitted by the RL, and therefore the detection of differences between different modes of measurement, not otherwise shown.
Finally, in the setup shown in Fig. 1(d), the image of the emitting surface is projected on two CCD cameras (Newport, LBP-3 USB model), directly obtaining its image (integrated in time). By using a beam-splitter and appropriate low-pass and high pass filters, the image of the reflected pumping and the RL emission can be obtained directly and simultaneously in both beam profilers. This scheme has been used to compare in each single pulse, sizes and spatial shapes of reflected pump beam and RL emission for different materials and working conditions, such as pumping beam focusing or energy, and also to get the accurate size of the pump beam itself, essential to know the pump energy per unit area because pump laser beams are divergent and therefore, the geometric estimation of the beam size can be erroneous.

Emission spectra
The emission spectra were obtained by using the setup of Fig. 1(a), and pumping at the wavelength of minimal diffuse reflectance (maximal absorbance).
In all cases, while the pumping energy is below a threshold value, the emission spectrum corresponds to the spontaneous emission: it has relatively low intensity and is relatively broad (about ten nm). But when the energy of the pump pulse exceeds the threshold value, the emission spectrum dramatically narrows, more than our spectral resolution (0.3 nm), and amazingly increases its intensity, becoming a single line located just at the peak of the spontaneous emission spectrum, as can be seen in Fig. 2  This feature is described in literature [22][23][24], and it is independent of the pump beam area, pumping wavelength, Nd 3+ concentration, and grain size. Possible changes in emission wavelength with concentration and/or pump-wavelength would be due to the properties of the compound itself and not to diffusion.
On the contrary to wide emission band systems, such as dyes, the absence of spectral structure is a characteristic of this kind of Nd based RLs.

Output energy and slope efficiency
The output energy of an RL as a function of pumping energy can be described by the same approximated equation of a conventional laser, that is: where E thr is the threshold energy and "m" is the slope efficiency.
We have performed slope efficiency measurements (output energy vs. input energy) using the variant of Fig. 1(b), with different pumping areas for many samples, and we found that the slope efficiency, m, is essentially independent of the pump beam area, for pump focusing diameters higher than 0.5 mm. This independence is no longer met in areas that are too small (less than 0.5 mm in diameter in low doped samples and 0.2 mm in stoichiometric samples). Moreover, depending on the material, the reproducibility of the results may be not good, the powder surface may be affected by pumping, small subsidence appears, and the slope and even the directionality of the emission are affected.
In Fig. 3 three input-output pictures at three different pump beam sizes, for oxysulfide La 2 O 2 S:Nd (9%) are shown. As can be seen in the figure, slope efficiencies of the three linear fits are very close to each other, even when the pump areas are quite different. In fact, the slope efficiency value inferred from the three input-output energy lines is 16% ( ± 1%). Taking into account that the absorbance η of that sample was about 23% at the pumping wavelength (λ pump = 819 nm), and that the energy ratio between absorbed and emitted photons is about 76% (λ emis = 1076 nm), we proposed the slope efficiency, m, of the RL emission to be given by [16]: if the branching ratio of excited state to the metastable level is close to unity. We have repeated slope efficiency measurements keeping the same experimental conditions, for all our crystal powder samples pumping around 800 nm ( 4 I 9/2 → 4 F 5/2 ) and also at 870 nm ( 4 I 9/2 → 4 F 3/2 ) for the borate powder (these two cases fulfill the previous condition). All experimentally obtained slope efficiencies show good agreement with the m value estimated by using Eq. (3) as can be observed in Table 1.
This result means that essentially every absorbed pump-photon over the threshold energy value is converted into a stimulated emission photon, that is, any loss channel of the upper laser level operates with a much longer time constant than the photon residence time of the RL emission. Or in other words, the probability of stimulated emission is much higher than the one corresponding to any other loss-channel of the upper level. To extend the validity of Eq. (3) to different pump wavelengths, and therefore explore if the slope efficiency remains proportional to the sample absorbance, slope efficiency measurements have been performed as a function of the pumping wavelength for several samples. As an example, Fig. 4 shows for comparison the predicted slope efficiency by Eq.
(3) for different pump wavelengths for the phosphate powder (NdPO 4 ) (black line) and the experimental slope efficiencies (blue dots). As can be seen, the predicted values very well fit the experimental ones.
This figure shows the proportionality between the slope efficiency m and the sample absorbance in the entire absorption band, predicted by Eq. (3), which is essentially true for all samples excited to 4 F 5/2 or 4 F 3/2 levels, always with the aforementioned exceptions of using too small pump beam sizes.
A special mention deserves the slope efficiency comparison when exciting at the 4 F 5/2 or 4 F 3/2 levels. As an example, Fig. 5 shows the energy input-output curves of NdAl 3 (BO 3 ) 4 obtained by pumping at 808 an 876 nm. As shown in Table 1, Eq. (3) is well satisfied, in contrast with what was observed by other authors [25] who found an increase of laser slope efficiency by changing λ exc from 810 to 884 nm. However, neither the sample grain sizes nor the experimental setups were the same. We did not use any collecting lens [ Fig. 1(b)] that could disturb coupling with the optical fiber.
The efficiencies cited in literature are in the 0.2-25% range [12,[26][27]. However, our results show efficiencies up to 42% [17], for high absorbent samples, usually stoichiometric ones. Nevertheless, it must be kept in mind that when exciting to other levels different from 4 F 5/2 or 4 F 3/2 (for example pumping at 532 nm), the effect of the branching ratio may be important, and the laser efficiency can drop significantly.   (3), account taken that absorbance at these two wavelengths are around 50% and 30% respectively. These measurements have been carried out with a pump beam area of 0.57 mm 2 .

Pump threshold energy
The results shown in Fig. 3 suggest that the threshold energy in the Nd 3+ doped oxysulfide sample is proportional to the pumped area. In order to study this effect in further detail, the threshold energy was registered as a function of the pumping beam area confirming the linear dependence between both magnitudes. Therefore, the threshold energy per unit area remains essentially constant, (we have tested it up to 3 mm diameters) while the pumping beam area is not too small. As an example, Fig. 6 shows the threshold energy per unit area of the stoichiometric borate and Nd doped YAG.
For small beam sizes, this linear behavior shows similar limits as those quoted at the beginning of Section 4. Table 1 shows threshold energies per unit area for several samples and pumping wavelengths.
In low doped materials, however, the level of compliance is not so strict, as can be seen in Fig. 6, where a slight drift towards higher threshold energy per unit area is observed in Nd 3+ doped YAG when decreasing the pump beam size. In Sections 6 and 7 we will go through these different behaviors between materials of high and low absorption. On the other hand, in order to define the relationship between RL threshold and the powder absorbance, we have performed measurements of the threshold energy in several samples as a function of pump wavelength around the absorbance peak. In Fig. 7 we show the measured threshold energies relative to its minimum value, at two different pump diameters and for two different powders, a) Nd:YAG, 1% doped and b) Phosphate (stoichiometric). First, it is observed that both pump diameters give almost identical curves when normalized, as expected from the results of Fig. 6 (the ratio between threshold energies must equal the ratio between pump areas).
Measuring the threshold energy as a function of pump wavelength in the same sample, we change the absorption coefficient and absorbance values, but not significantly the transport length, and so we can thus check whether the threshold spectrum matches the theoretical predictions.
The involved magnitudes are the transport length l t , the inelastic length l i , which is the inverse of the absorption coefficient, is the absorption length. The threshold condition is l g = l res , where l res is the mean residence length of the stimulated emission photons. Substituting the corresponding parameters, the threshold condition can be written as: According to the one-dimensional theory [28-29], the residence length is proportional to the inelastic length l i · · · ( ) This expression (4a) is tested in Fig. 7(a), (Nd:YAG), by dividing the square root of the absorption cross section by the absorbance of the Nd:YAG powder (red line), but the result is clearly different from the threshold spectrum experimentally measured.
We tried that expression also for stoichiometric powders, but it is difficult to obtain an absorption spectrum in this kind of crystals; fortunately, the ratio between transport and inelastic lengths can also be obtained from the diffuse reflectance spectrum by means of the Kubelka-Munk theory of reflectance [30], which gives the transport length/inelastic length ratio: When this expression is substituted to Eq. (4), the threshold condition applicable even for stoichiometric crystal powders is obtained: This expression (4b) is tested in Fig. 7(b) for the stoichiometric phosphate NdPO 4 (red line), but the result is again very different from the threshold spectrum experimentally measured. Figures 7(a) and 7(b) show that the experimental threshold fits quite well the inverse of the absorbance, especially in the stoichiometric material. We tried to fit experimental results, assuming that the residence length is proportional to the absorption length l abs and from (4): This last expression (6) was tested both in Fig. 7(a) and 7(b) (Nd:YAG and stoichiometric phosphate), just taking the inverse of the absorbance spectrum (black line), and the result fits quite well the threshold-energy spectrum experimentally measured in both cases.
In our opinion, the discrepancy with theory [expressions 4(a) and 4(b)] is due to the fact that the spatial inhomogeneity of the population inversion, and therefore of the amplification, is not taken into account (it is greater the closer to the exit surface). Expressions (4a) and (4b) are obtained from the passive diffusion theory. If the spatial inhomogeneity of the amplification is accounted for shorter paths are favored and the effective mean residence length must be shorter.

Effect of concentration and grain size
Crystals powders with a given grain size distribution and different dopant concentrations have different absorption properties (inelastic length), but the same diffusion characteristics (transport length). Therefore, changes in absorbance are related only to the inelastic length, and consequently lower concentrations must give lower absorbances, and therefore lower slope-efficiencies and higher thresholds and vice versa. Different Nd 3+ concentrations were synthesized in the borate Nd x Y 1-x Al 3 (BO 3 ) 4 [31] and in the Nd 3+ doped oxysulfide La 2 O 2 S [16] in order to study the effect of concentration on slope and threshold showing similar results. As an example, Table 2 displays a summary of those results for oxysulfide powder. The laser slope efficiency is well given by expression (3) and the threshold energy is in this case, also much better adjusted to the inverse of absorbance (expression (6)) than to expressions (4a and 4b). On the other hand, the values of l i /l t agree well with the inelastic length (about 1 mm at lowest concentration [32]) and the transport length (about 10 micron) of the samples. When regarding changes in the grain size, one must bear in mind that in principle this only affects the transport length and not the inelastic length. It is known that smaller grain sizes reflect the pump radiation more efficiently [33]. A shorter transport length causes more collisions to occur before the photon is absorbed, thereby increasing the probability of reemergence, and thus the absorbance decreases.
We had observed that when a stoichiometric powder is milled again, the threshold energy rises and slope efficiency falls. In these kind of samples, the rules given by expression (3) for slope efficiency and (6) for the threshold energy work well. However, this feature is not so clearly observed in low absorption samples. For example, more milled samples of 1% Nd:YAG, show lower absorbance as expected, but the threshold does not increase in the same proportion. This fact can be explained account taken that the absorption length (proportional to t i l l ⋅ ) is too long, being comparable to the size of the pump beam. In fact, in low doped samples, as Nd:YAG (1% doped), where the inelastic length in the absorption peak is about 1 mm, we have observed that these expressions do not work well if the pump beam size is small enough after milling the powder again. In contrast, this effect is not observed in stoichiometric samples in our usual working range due to their short absorption length, and we would have to use smaller pumping sizes to appreciate this behavior. In the following section we will consider this question again, which is related to the behavior of the threshold shown in Fig. 6 for low absorption samples.
To conclude this section, a brief comment on volume filling factor follows. In the early days of our research in this field, we tried different filling factors in the same powders, and the threshold energy and slope efficiency did not undergo major changes, that is, the volume filling factor affected them very little. In our opinion, the explanation comes from the fact that to change the volume filling factor in powders affects essentially to the mean concentration, and therefore, the transport and inelastic lengths in the same direction. As a consequence, the l i /l t ratio, which accounts for the threshold energy and slope efficiency, does not change significantly.

Emission imaging
The setup shown in Fig. 1(d) has been used to measure the size of the pump beams and thereby estimate the pumped energy per unit of area. We have measured, for each position of the focusing lens and for each powder sample, simultaneously the sizes of RL emission beam and reflected pump beam, which obviously includes its own scattering. Previously, we have measured in the same conditions the size of the reflected pump beam in a steel surface, which has very low scattering.
Using this set up, sizes of many samples under different focusing or energy conditions were measured. Note that in all the explored samples, we have not observed significant changes in the RL emission area when changing the excitation energy. The images and data to be presented in this section are obtained at a pumping energy about twice the threshold.
As shown in Fig. 8, stoichiometric materials (highly absorbent) show a different behavior if compared to low absorption ones. As has already been said, the main difference between the two kinds of materials is the inelastic length (few tens of microns for the stoichiometric materials and in the order of one mm for 1% doped Nd:YAG). The transport length is quite similar in both samples.
In the stoichiometric material, sizes of the pump beam (on steel), reflected pump beam and RL emission zone are similar, account taken of the experimental error. A moderate expansion of the reflected pumping respect to the incident beam is observed, and the RL emission is even "compressed" respect to the reflected pumping, surely due to the effect of a higher gain in central zones. For the smallest incident beams, the results show higher dispersions.
In contrast, in the sample of low absorption (Nd:YAG), an expansion of the RL emission is observed in all measured sizes if compared to the pump beam. The extent of the reflected pump beam is also larger than the original pump beam size.  The effect of changes in grain size (transport length) and/or pump wavelength (inelastic length) are displayed in Fig. 9 for the Nd:YAG powder.
As can be seen in this figure (the size of the pump beam is the same in all cases), the expansion of the emission area is larger in samples with larger grain size (longer transport length) and/or at pump wavelengths of less absorption (longer inelastic length), so that the rules obtained in Section 5 for threshold energy could not work well. The threshold energy per unit of area tends to be higher than the expected by Eq. (6), due to the effect of dispersion of stimulated emission photons towards zones of low gain.

Temporal measurements
Another fundamental aspect in RL research is the temporal behavior of the emission. Several authors have observed relaxation oscillations in RL emission and measured their period, and developed theories about the generation of relaxation oscillations in RL [12,13,26,28,[34][35][36][37].
In order to verify these results, we have employed the experimental setup shown in Fig. 1  (a or b) to simultaneously record the intensity as a function of time of the pumping and laser emission.
There is a time-delay between the signals coming from both detectors used (the one collecting the pump pulse and the other one which measures the emission pulse) depending on the lengths of the respective cables or optic fibers. The delay was calibrated by filtering the emission and observing the pump pulse in both detectors at the same time. The error induced in this calibration because of modal dispersion for multimode fibers, and chromatic dispersion in any case is estimated in the order of 10 ps. The pump pulses are 10 ns long and the temporal widening of the system used in variants of Fig. 1(a) and 1(b) is about 200-300 ps, both for pumping and lasing, and much shorter when we compare the temporal evolution between both. Consequently, the error is perfectly acceptable.
Vol. 26, No. 9 | 30 Apr 2018 | OPTICS EXPRESS 11800 Fig. 9. Sizes of RL emission zones for Nd:YAG (1% doped). Pumping energy is approximately twice the threshold. The diameter of the pump beam is 1.2 mm (steel surface) in the three cases. a) Pumping at maximal absorption wavelength (inelastic length 1 mm, transport length 25 micron). Emission area diameter 1.9 mm. b) Same pump wavelength than a) and larger grain size and (inelastic length 1 mm, transport length 50 micron). Emission area diameter 2.4 mm. c) Same grain size than a) and less absorbing wavelength (inelastic length 5 mm, transport length 25 micron). Emission area diameter 3.1 mm. Figure 10(a) shows typical pumping single pulse and RL for a stoichiometric or highly doped material. As can be seen, the pumping intensity has a clear mode beating and gives an excellent reference for comparing both signals. It is worth noticing the importance of registering the pump pulse temporal shape because otherwise we could be led to misinterpret these oscillations as corresponding to emission [26,28,36,37].
This figure shows that there is a delay time of about 2 ns before reaching the threshold (pulse build up time, as in conventional lasers) in which no emission is produced, after which the time profile of the laser emission and that of the pumping intensity are essentially synchronous. By measuring the RL pulse and spatially integrating all the entire emitting surface or just a significant portion of it, the observed oscillations come from pumping.
If a low doped material is used as RL emitter, a similar result is obtained, but a small delay of the RL emission peaks respect to the pumping ones can be detected, about a few hundreds of ps [18].
These results imply that the time response of the RL is in the range of one hundred of ps as much. The short response time of emitted photons in the "RL cavity", together with results of slope efficiencies in Section 4 suggest that any loss channel of the upper laser level operates with a much longer time constant than the photon residence time of the RL emission.
In order to observe if there is any change in the temporal behavior of the RL emission when it is locally observed, i.e., emission from a small area of the emitting surface, the setup of Fig. 1(c) (30 ps of temporal widening) is used with a pin-hole 10 micron in diameter and a lateral magnification of 1. In such a case, we register the laser emission obtained by imaging a zone of a similar size to the transport length itself [21]. Figure 10(b) shows an example of the RL pulse obtained by imaging a local zone in the NdVO 4 stoichiometric sample.
Concerning the delay between the laser emission and the pumping intensity, the behavior remains unchanged, but faster fluctuations, undetected before, now appear. The distribution of peaks is not repetitive at all from shot to shot. They do not have a defined period, but its characteristic time is in the range of 50-100 ps. These properties remain constant against any changes in pumping and doping level, but a reliable signal analysis is difficult, mainly due to modulation introduced by the pumping. We also measured in a reverse mode, i.e. instead of picking up the emission signal from a very small area, collecting it from the entire emitting surface, but in a very small detection area. For this purpose, we used the variant of Fig. 1(b), but with a single mode fiber (10 micron core diameter) and "fast" measurement device (30 ps of temporal resolution). The transverse coherence distance on the emitted wave front is given by: where λ is the wavelength, L is the distance between the emission and the detection points and φ is the diameter of the emitter surface [21]. In our case, λ ∼1μm, φ ∼1mm and L ∼100 mm, therefore, d ∼100μm, which is clearly larger than the core size of the single mode fiber. Measured in this way, no differences were observed in the temporal profile of the RL emission respect to the local measurement of Fig. 10(b). However, if a 0.5 core multimode fiber is used, the rapid fluctuations completely disappear.
In our opinion, these results suggest that the observed fluctuations are due to the interference of multiple waves, external overlapping in the last case and internal overlapping (within the RL active material) in the case shown in the Fig. 10b. In addition, it should be noted that the lifetime of oscillations is in the 50-100 ps time range, which is in the order of the temporal coherence measured in these lasers [38,39]. Whether the emission area or the reception area become larger, multiple intensities overlap and fluctuations disappear. This behavior is also observed in RL of organic dyes [40].

Summary and conclusions
We have performed spectral, spatial, energy, and temporal measurements, with an optimized experimental setup for each case.
The laser emission spectrum is always a single line, centered at the maximum of the emission spectra, and narrower than our experimental resolution (0.3 nm).
As regards to energy, our measurements prove that the laser slope is directly proportional to the absorbance of the material, being the constant of proportionality the quotient between pump and emission wavelengths (Figs. 4, 5 and Tables 1 and 2). This result says that almost all pumping photons absorbed above the threshold are re-emitted as stimulated emission, which means that other de-excitation channels from upper laser level are negligible (or almost) against the stimulated emission.
These measurements also prove that the pumping threshold energy is essentially proportional to the pumping area (large areas side in the Fig. 6). Likewise, they show that the pumping threshold energy per unit area is inversely proportional to the absorbance (Fig. 7 and Table 2). This result indicates that in the RL dynamics, the changes in pumping volume and hence population inversion per unit volume (because of the sample used and/or pumping wavelength), are compensated by unavoidable changes in the active residence length of the photons, which leads to a residence length being much closer to be proportional to the absorption length than to the inelastic length (see expressions (4) and (6)). In all cases, neither the slope nor the threshold energy per unit area depend on the size (area) of the pumping beam.
These rules are very well accomplished by any stoichiometric or high absorption materials in any pumping condition (beam size or wavelength). The only caveat is that the diameter of the pumping beam must be much longer than the absorption length. Consequently, if a high absorption powder is ground again, as its absorbance decreases, its benefits (slope and threshold) will worsen, always with the diameter of the pumping beam being much longer than the absorption length.
In the case of low absorption materials, absorption length and pumping beam diameter can be comparable, and therefore the emitted photons would be more likely to be present in peripherally weakly pumped zones. The observed result is a significant expansion of the RL emission area respect to the incident beam size, as is appointed in Section 7. The threshold energy per unit area is not as invariant as in the stoichiometric powders (Section 5).
Regarding the temporal behavior of the RL emission measured by spatially integrating the emitting surface, we have shown that RL emission intensity, once the threshold is reached, follows quite faithfully the pumping pulse [ Fig. 10(a)] with an undetectable delay in the case of stoichiometric powders and of the order of 100 ps in low doped samples, in any case very small for nanosecond pumping regime. The only oscillations observed come from fluctuations of the pumping itself. These response times of emitted photons, together with results of slope efficiencies, imply that every transfer-rate time of any loss channel of the upper laser level must be much longer than those 100 ps.
Something very different is observed when the measurement is local. Selecting a very small area of the emitting surface (about 10 micron, in the order of the transport length), fluctuations with characteristic times less than 100 ps are observed [ Fig. 10(b)]. Interestingly, the same fluctuations are observed if one collects all the spatially integrated emission in a sufficiently small area. These fluctuations are observed in both procedures both in stoichiometric materials and in low doped powders, with the same characteristic time. In our opinion, thus suggests that they are produced by interference of multiple waves, internal (within the sample) or external. It seems that these times are more related to temporal coherence (similar for stoichiometric materials and Nd (1%):YAG powders) than to length of residence (quite different).