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Amplified spontaneous emission (ASE) in large size, high gain Yb3+:YAG slabs severely impacts the gain/energy storage capability. We will discuss numerical simulations and experimental results obtained on large size Yb3+:YAG slabs. The spatial distribution and temporal evolution is shown under different conditions.
©2009 Optical Society of America
1. Introduction
The unavoidable Spontaneous Emission (SE) in any inverted medium is the subject of intense research since the early days of the development of the laser [1, 2, 3]. Energy storage capability in any gain medium is limited by SE, which is especially important when the pump duration is in the order of the radiative life time of the upper laser state. It is of special interest for Diode Pumped Solid State Lasers (DPSSL) used in high-efficient short pulse laser systems [4] and high-average power laser systems [5, 6].
While SE is created throughout the pumped medium it gets amplified on its way – leading to the so-called Amplified Spontaneous Emission (ASE). ASE can considerably reduce the locally stored energy or, with closed paths and sufficient feedback, generate parasitic lasing. The design of the laser gain media is thus one of the major challenges for amplifiers with a high energy storage capability [7, 8].
The LULI (Laboratoire pour l’Utilisation des Lasers Intenses) is currently building a kW average power DPSSL system relying on Yb3+:YAG, called LUCIA [9]. Power amplification is achieved with amplifiers using mm-thick slabs. Beside thermal management (100J at 10Hz) and laser damage threshold issues (100J within a 10cm 2 extraction aperture), ASE is the 3rd bottleneck to overcome. This paper aims at reporting on the experimental and numerical study of the energy storage capability on the LUCIA amplifiers. Figure 1 gives a view of the diode array (left) used for pumping the 38×32mm 2 Yb3+:YAG gain medium (right).
After the description of the experimental setup in part 2, we present the numerical model in part 3, its restrictions in part 4 and temporarily and spatially resolved gain results in part 5. Finally part 6 gives a conclusion.
Fig. 1. Overview over the amplifiers of the LUCIA laser system. As described in [9], we see the partially filled laser diode array with the multiplexing prisms (a), the concentration mirrors (b) on the left picture. The image on the right shows the illuminated laser gain medium in its cooling head (c).
2. Geometry and Pumping
Figure 2 shows the used slab geometry. The pump source is a laser diode array, hosting 88 stacks at maximum and delivers 264kW in final configuration [9]. A partially filled laser diode array (Fig. 1) is used during the experiments discussed in part 5, i.e. 30 stacks, emitting 90J within 1ms. This energy is concentrated on a surface of 40×10mm 2. The maximum average intensity can be adjusted up to 18kW/cm 2. As the experimental setup is in transmission, the crystals are not cooled, requiring to work under single shot conditions. The laser diode emission is tuned to fit the absorption cross section. Such spectral adjustment of the laser emission relies on the use of a bias current [10], offering the possibility to individually adapt every stack spectrally to the desired absorption region of the laser gain material.
As depicted in Fig. 2 the emission of the pump laser enters from the front. The material is homogenously doped and pumped more or less uniformly over its surface, as it is also shown in Fig. 2 (right). Gain probing is performed under an angle of incidence of 24°, i.e. the assumed extraction direction of the final layout, using a fibre-coupled cw laser source at 1030nm. The temporal evolution of the gain is recorded at a central horizontal line, the Line Of Interest (LOI).
The probed gain media are 7 different, 40 × 20mm2 Yb3+:YAG slabs with doping concentrations ranging from 1at.% to 10at.% and thicknesses ranging from 1mm to 6mm. The small lateral faces are not polished and cut under an angle of 37° concentrating the ASE in the corners and damping retro–reflection. The large surfaces are AR coated (reflectivity below 0.5%), the small lateral faces are not polished resulting in auxiliary losses for the SE due to diffusion and Fresnel refraction.
Under the same pumping conditions these different doping concentrations will introduce small–signal gains g 0 differing by more than one magnitude. Regarding transverse ASE, the [small–signal gain × lateral length] product called g 0 L plays an important role, as in ref. [11] a threshold value of g 0 L equal to 4 is derived, below which ASE will not play an important role. Values of 5 or even larger will bring up a serious impact of ASE, as the amplification of the SE may exceed the losses introduced by the surface reflection. A rich and detailed discussion regarding different conditions for parasitic lasing can be found in Ref. [7].
Values for g 0 L under the experimental conditions (L = 40mm) using a 1D model neglecting ASE are shown in table 1. In the case of low doping (1at.%), values of g 0 L between 2 and 4 imply a small impact of ASE. In the case of high doping levels (5at.% and 10at.%), the calculated large values for g 0 L are never achieved, thus the gain will stay significantly below the values estimated in the simple 1D model neglecting ASE.
Table 1. Maximum g 0 L assuming a uniform pump distribution of 10, 14 and 18kW/cm2 and a transverse size of L = 40mm. ASE is neglected during pumping.
3. Numerical Model
During pumping and below threshold regarding parasitic lasing, three major influences will affect the population density of the upper laser state: the action of the pump, the unavoidable losses due to the radiative life time of the upper laser state and the depopulation due to ASE.
Fig. 2. Pumping and probing setup. The gain medium is pumped perpendicular to the front surface (lower left picture). Different positions are probed within the Line Of Interest (LOI). The experimental pump light distribution is shown in the upper left figure. A vertical and horizontal line out is shown on the right side in the case of 18kW/cm 2 average pump intensity.
The influence of ASE can be interpreted as a change of the local life time as a function of the geometry and the surrounding population density.
We introduce a factor MASE [11, 12, 13] quantifying the local depopulation due to ASE and write the well known differential equation of the excited ion density n as follows:
In the case of a quasi-three level system, we can write for MASE at the position r⃗0 in the gain medium:
whereby the local flux of the ASE at this point can be expressed as the following:
Hereby we used the radiative lifetime of the upper laser state τrad, the emission and absorption cross section σa,e, the total doping density ntot, the wavelength λ, the distance ρ between r⃗ and the observed point r⃗0, the line shape function g(λ) and the gain G r⃗→r⃗0.
We might further simplify the equations derived above. In the monochromatic case and assuming that the radiative life time of the excited state is constant throughout the gain medium, we can combine Eq. (2) and (3) to get:
If ASE gets such important that it compensates the pump in Eq. (1) (L.H.S.=0), we can derive an expression for a maximum population density nmax:
Fig. 3. Two examples for a calculated gain distribution derived using the Monte-Carlo code. On the left (a) the result for a 1at.% doped, 30mm thick crystal is shown. On the right (b) the 10 at.%, 3mm case is shown. In both cases a uniform pump intensity of 14 kW/cm2 on an 10×40 mm2 area was used. The gain maps were computed by integration along the z-axis defined in Fig. 2. The horizontal line outs are shown above.
Hereby we used the intensity of the pump Ip and the saturation intensity Isatp at the pump wavelength of the laser material. As a quick check if ASE gets crucial, one can calculate the excitation n without taking ASE into account, estimate Mase using Eq. (2) and check the resulting maximum nmax. If the result is near or even below to the n used to calculate MASE, ASE may not be neglected.
Scaling laser gain media relies strongly on the evolution of MASE. If one takes the gain G r⃗→r⃗0 to be exponential with the lateral size G r⃗→r⃗0 = exp(g 0 L), Eq. (4) implies that, in the case of scaling the transverse size, the factor g 0 L should be kept constant, otherwise MASE grows exponentially.
Using a Monte–Carlo approach we developed a model capable of estimating the stored energy density for a given geometry in three dimensions. The calculations rely on Eq. (2) resulting in MASE as a function of the position within the gain medium.
Interpreting eqn 4 results in two major directions. In the case of a small amplification the gain can be assumed to be linear in an Taylor-series expansion. The factor MASE will be consequently dominated by the SE directly originating in the vicinity of r⃗0. As the center of the pumped zone will see the largest number of traveling rays, we can expect the strongest depopulation. A slightly higher gain will be found at the outer pumped zones.
The more common case found in literature [3] shows a large factor g 0 L, i.e. considerably larger than 1. A photon created on one side will be amplified throughout its path through in the gain medium. This holds true for a similar photon in the other direction too. Unlike the previous case, the amplification will be exponential or even saturated. In this scenario we expect a much stronger depopulation on the outer regions as they are far apart from each other. The highest gain will be thus found in the center.
As an illustration we will now discuss two examples in the case of a Yb+3: YAG slab, pumped monochromatically at 940nm. All surfaces are taken to be perfectly absorbing for the SE. The doping was 1at.% and the thickness was 30mm in the low gain case and the doping 10at.% with a thickness of 3 mm in the high gain case. For both slabs the [doping concentration × thickness] (cT) is equal to 30at.%mm. The pump power was taken uniformly to be 14kW/cm 2. The pumped surface is 40 × 10mm 2. In this model we probe the small signal gain at 1030nm along a direction perpendicular to the pumped surface (along z-axis as depicted in Fig. 2).
Gain estimation is performed using the emission and absorption cross sections at 1030nm to be σe = 2.46 × 10-20 cm 2 and σa = 1.35 × 10-21 cm 2. We focus our interest on the central horizontal line out of the computed 2D gain maps, since this line is probed experimentally.
Figure 3 (a) shows the calculated gain distribution in the low gain case. The upper picture shows the horizontal gain line out in the center (y = 0.5cm). As already discussed before, the gain in the central area appears to be lower compared to the outside. We see a positive curvature. The Fig. 3(b) shows the high gain case. With a g 0 L higher than the preceding case, the strongest influence of the ASE can be seen in the vicinity of the edges of the 3mm thick slab (x < 1cm & x > 3cm). The gain distribution shows a significantly stronger gain in the center than in these outer regions, therefore a negative curvature in the central horizontal gain line out.
As both examples carry the same factor cT, we estimate the same gain without the action of ASE. Obviously in the large size, high gain case, a simple estimation cannot be done anymore. Finally let us notice that the 1at.% case reveals a significantly higher gain compared to the 10at.% example (max. values are 1.56 vs. 1.2).
4. Restrictions to the numerical model
Within the model, multiple reflections on the surfaces are neglected since we considered all surfaces to be perfectly absorbing. While this might be feasible for laser gain materials with Aspect Ratios (AR) of approximately 1 (cubes, thick slabs and spheres) [14], this becomes problematic for thin slabs, where multiple reflections due to Total Internal Reflections (TIR) occur. We define the AR of the gain medium to be the thickness divided by the lateral size. In the case of one reflection due to TIR one can easily derive:
where n 1 is the index of refraction of the laser gain material and n 2 the index of refraction of the surrounding material. The index of refraction of Yb3+:YAG at 1030nm is n 1 ≈ 1.83 [15] and in the case of surrounding air n 2 = 1 we get an ARL ≈ 0.65. Typical lateral sizes are in the order of several centimeters and thicknesses in the order of a few millimeters, thus ARs for slabs used as amplifiers are in the order of 0.1 at most.
In order to get a feeling about the importance of TIR, one can calculate in a simplified scenario the gain between one point of the pumped surface and the opposite side. As in the case of TIR, the rays don’t suffer from the reflection losses anymore, thus it will show the maximum gain. Figure 4 shows the calculated gain as a function of the penetration depth (1D, without ASE). The doping is 1at.%, the incident pump intensity 10kW/cm 2. We observe that the maximum small signal gain is 0.5cm -1 at the pump entrance face. For a gain medium with a transverse size equal to 4cm, we get g 0 L = 2, justifying the omission of the ASE (g 0 L < 4). In the case of crystals thicker than ≈ 1.6cm, reabsorption can be seen (see Fig. 4(a)).
For a ray traveling under TIR we calculate the gain G between the lateral faces as a function of the Aspect Ratio (AR). The result is shown in Fig. 4(b). As the trajectory (dashed zig-zag arrow) length stays the same, but the average gain changes, we will see a different amplification. The thinner the crystal, the higher will be the gain until the photons strike the opposite lateral surface (end of the trajectory) acting as a perfect absorber. For AR = 0.4 or larger, reabsorption is expected. Especially in the case of an AR of 1 (lateral size = thickness = 4cm) the influence of reabsorption is predominant, showing a low gain of ≈ 1.2.
In the limit of an infinitely thin gain medium the gain will converge at ≈ 39. As a result, gain media which are thin compared to the transverse extends will show a considerably stronger depopulation due to TIR when compared to thick crystals.
Fig. 4. On left (a) the result of a 1D calculation omitting ASE in the transmission case of a 4cm thick Yb3+:YAG crystal doped at 1at.% is shown. The pump has an intensity of 10kW/cm 2 and a duration of 1ms. The horizontal axis shows the position z in the crystal, while the vertical axis represents the small signal gain g 0. The graph on the right (b) shows the gain G for a ray traveling under TIR and the conditions of (a) as a function of the Aspect Ratio (AR).
Neglecting the reflected rays will effectively underestimate the ASE-Flux. Especially the central region will show a stronger influence of the ASE. The thinner the crystal, the stronger will be the underestimation, the more distinctive will be the difference between the estimated gain distribution using the model and the experimental situation to be described in the next section.
5. Experiment
Three intensities were used in the experiment: 10, 14 and 18kW/cm 2 respectively. The transverse gain probing is performed using an 80mW cw fibre-coupled laser at 1030nm. The angle of incidence is 24°, i.e. the extraction direction of the final amplifier scheme. Measuring the temporal evolution of the transmitted signal and knowing the transmission without pumping, the gain G can be deduced.
From this point we will concentrate on two experimental cases: a crystal doped at 1at.% with a thickness of 5mm and a crystal with a doping of 10at.% with a thickness of 3mm. These two examples correspond to a low transverse gain factor g 0 L (1at.%) and a large g 0 L (10at.%) on the one hand and a factor cT = 0.5 (1at.%) and a cT = 3 (10at.%) on the other hand.
Figure 5(a) and 5(b) show the measured gain distribution in the central horizontal line - the Line Of Interest (LOI). Figure 5(a) depicts the lateral gain distribution in the case of a doping level of 1at%, 5mm thickness after 1ms pumping. The average g 0 L would be 2, 3 and 4 for the intensities of 10, 14 and respectively 18kW/cm 2 (see table 1) without taking ASE into account. A positive curvature is revealed, resulting in a higher gain on the outer edges of the probed gain medium. For a small g 0 L, the local effects are dominant, leaving less population in the central region, as already discussed in part 3.
Examination of the gain distribution in the case of a higher factors of g 0 L can be seen in Fig. 5(b). While neglecting ASE, the values of g 0 L would be 17, 25 and 33 after 1ms respectively (see table 1), and a strong influence of ASE is expected. Such high values for the factor g 0 L will never be achieved, thus the small signal gain will be restricted to much smaller values. The resulting gain will be strongly reduced compared to the ideal (non–ASE) case. As well as the gain distribution will change. In this high gain case (large g 0 L) SE gets strongly amplified resulting in a stronger depopulation at the edges, resulting in a negative curvature shown in Fig. 5(b). The maximum gain dropped by ≈ 10% compared to the low gain case.
Fig. 5. Experimental gain distributions for a Yb3+:YAG crystal doped at 1at.% (a) and 10at.% pumped at 10 (blue circle), 14 (green upright box) and 18 (red box) kW/cm 2. The gain corresponds to the horizontal line out in the center of the pumped region.
Each point in Fig. 5 corresponds to a single measurement using a photodiode allowing a sub-μs temporal resolution. Figure 6 shows the temporal evolution in the high gain and the low gain case for the three pump intensities and for the three different localizations along the LOI.
In the case of a low doping concentration (Fig. 6(a)-6(c)) the central gain stays always below the gain on the outer points. Let us now focus on the temporal gain evolution in the case the 1at.%, 5mm thick crystal (three top graphs in Fig. 6). Especially interesting are the gain drops in graphs (a) and (c) at pumping intensities of 18kW/cm 2, whereas a saturation is observed for the central position (b). At about 0.6ms after the pump started, the gain reaches its maximum in each of the three cases. In the remaining 0.4ms, the impact of ASE will differ with respect of the position. We have to remind that the crystal has a cT = 0.5, therefore we will still see even a significant amplification close to the back surface of the crystal (see sketch in 2). As the photons under internal reflection will see an amplification which gets stronger and stronger with the ongoing pumping, the depopulation due to ASE will increase on the vicinity of the edges much stronger compared to the center. This leads to the a decreasing gain on the outer edges and a saturation in the center after 0.6ms. Pump intensities superior to 14kW/cm 2 show saturation.
On the other hand we can see the temporal gain evolution for the crystal doped at 10at.% in Fig. 6(d)–6(f). Due to a higher reabsorption almost one third of the pump will be invested to bleach out the reabsorption. As the intensity distribution of the pump drops almost exponentially with the penetration depth, the populated parts will suffer from ASE, while the backside is still absorbing. An intensity of 10kW/cm 2 doesn’t show saturation, but is again influenced by ASE. Stronger pump intensities show saturation near the end of the pump duration of 1ms.
Figure 7 shows for the 7 different examined crystals at pump intensities of 10kW/cm 2 (a), 14kW/cm 2 (b) and 18kW/cm 2 (c) the resulting average gain calculated along the LOI in transmission compared to the simulation. While in the small gain example (1at.% with a low g 0 L, i.e. below 4) the expected gain fits quite well with the experimental results, we observe for large values of g 0 L a strong deviation, especially in the case of a high doping.
In the case of 10kW/cm 2 measured values for a large cT are in relatively good agreement with the simulation. As the pump is weak compared to the examples shown in (b) and (c), the trend of the gain as a function of the factor cT stays close to the expectations. It is obvious in the cases of high g 0 L and a small cT, that the deviation is significant. The thicker the crystal, the closer we get to the model, the less important are the internal reflections. Even higher pump intensities will further increase the gain and the influence of ASE as well. This can be clearly seen comparing Fig. 7(a) with 7(b): as the gain increases with increasing pump intensity, the more the measurements deviate from the model. The impact can be seen in (c). While in (a) the evolution of the gain as a function of cT shows a curvature for the distribution of the measured points, this feature evolves into a linear behavior in (c). The 5at.% doped crystal with a high cT (cT = 3) shows at 18kW/cm 2 the same performance as a crystal of 1at.%. We have to keep in mind that the crystal is too thin compared to the optimal thickness. We can expect a far superior performance for a gain medium with a low doping.
Fig. 6. The upper row shows the temporal evolution during pumping in the case of a 1at.% doped, 5mm thick crystal for the most outer point on the left (a), the center (b) and the most outer right measured point (c) in the case of 10 (blue), 14 (green) and 18 (red) kW/cm 2. The lower row shows the same measurement in the case of a 10at.% doped, 3mm thick crystal. The horizontal axis shows the time after pump begin in ms, while the vertical axis shows the measured gain. The pump duration was 1ms.
6. Conclusion
We discussed the influence of ASE on the energy storage capacity in the case of high gain, large size, millimeter thick Yb3+:YAG slabs. In the examples under studies, a strong influence of ASE can be found which is explained by the model derived before.
Large lateral size slabs pumped at high intensities combined with a high doping do not show benefits in terms of gain. As gain media with a low doping are preferred, they suffer due to their thickness in terms of thermal management. Unfortunately the combination of a high gain, a small thickness and a large lateral size cannot be achieved. In the limiting case, the factor g 0 L should be kept small, i.e. below 4, ensuring a small influence of ASE.
Reducing internal reflections is another important fact to consider. Reflections can be damped using absorbing cladding, index-matching cladding [7] or in the case of a gain material showing reabsorption by the use of sufficient non-pumped area. In the process of continuing development of the LUCIA laser system, large size crystals (60mm diameter), allowing a large peripheral region, will be used in the LUCIA amplifier head. Advantages of such a pump light distribution in terms of ASE management have already been discussed in Ref. [16].
Fig. 7. A comparison between the average measured gain along the Line Of Interest (LOI) (points) and calculations (lines) in the case of 10 (a), 14 (b) and 18 (c) kW/cm 2. The solid line corresponds to the non-ASE calculations, the dotted lines to the simulations on a 40× 10mm 2 surface for a doping of 1at.% (black), 5at.% (red) and 10at.% (blue). Indicated by the fasciated areas are the upper/lower limit corresponding to the maximum/minimum value found in the line outs (see Fig.3). The horizontal axis shows the factor [doping × thickness] (cT) of the crystal. On the vertical axis the gain G in transmission is shown.
Another promising candidate reducing transverse ASE might be a nonlinear doping gradient in pump direction [17, 18]. Properly used doping gradients might reduce the thickness of the gain material and the influence of ASE.
Acknowledgments
This work was supported by the Institut franco-allemand Saint-Louis (ISL). The authors like to thank B. Vincent for his help during the preparation of the experimental part.
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