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Abstract
We propose an original method of suppressing thermally induced beam distortions in composite active elements of disk geometry. The main idea of the method is to use a heatsink of special geometry to provide heat removal from the active element only in the pump area. In this way, the heat flows and temperature gradients in the thick active element, and, as a consequence, thermally induced distortions are reduced. The wavefront distortions, temperature, and gain in the composite element mounted in a profiled heatsink have been simulated numerically and measured in experiment. A 2.5-fold reduction of thermally induced lensing compared to a flat heatsink was demonstrated in the experiment at the same small signal gain.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The available concept of solid-state lasers based on active elements (AE) with thin-disk geometry allowed attaining record radiation power in the cw mode [1–3] thanks to effective heat removal from the active medium. To be more specific, 10 kW output from a single disk at M2 = 110 was reported in [1]; 7 kW at M2 = 7 in [2]. Combining several disks in one laser setup resulted in over 10 kW output at M2 = 1.76 [3]. Large AE aperture and pump spot size (12 mm) [4] makes it possible to operate with high peak powers [5,6]. However, the potential of the disk geometry for generating powerful pulses has not been fully utilized, as the efficiency of energy storage is limited by the effect of amplified spontaneous emission (ASE) [7,8]. The use of a beveled disk reduces roundtrip gain for the photons emitted at the angle of total internal reflection [9], thus mitigating ASE. Another technique for ASE suppression is adding a layer of an undoped medium, i.e. creating composite Yb:YAG/YAG disk active elements [10,11], but this gives rise to stronger thermally induced beam distortions due to a much larger optical thickness of the undoped layer compared to a thin disk. This, in turn, restricts the average radiation power of disk laser amplifiers rather than pulse energy. To reduce thermally induced distortions in composite disk elements, it was proposed to use composite AEs made of dissimilar materials (Yb:YAG/Sapphire) [12]. However, the magnitude of thermally induced lensing is still much higher than in disk active elements [13].
In this work, we present results of the studies aimed at further suppression of thermal lensing and concomitant thermally induced beam distortions in composite disk AEs by controlling heat flows in them by shaping the surface of a heatsink. We calculated the temperature distribution, deformations, and the corresponding wavefront distortions for different heatsink profiles. The obtained theoretical results were used in experiments on reducing thermal lensing in a composite disc active element.
2. Theoretical modeling
At a high average power, an inhomogeneous temperature distribution arises in the active elements, resulting in mechanical stresses, deformations, and distortions of laser beam. Thermally induced phase distortions in an optical element of disk geometry are made up by three components: (1) deformations of optical surfaces, including mirror ones; (2) changes in the refractive index due to its temperature dependence ($dn/dT \ne 0$); and (3) a photoelastic effect associated with the influence of mechanical stresses on the dielectric permeability tensor. All the three components are related to the inhomogeneous temperature distribution in the optical medium. These inhomogeneities on the optical path of the radiation may be minimized by controlling heat removal from the active element. For this we propose to use a heatsink with a definite surface profile [Fig. 1(c), 1(d)] made of silicon carbide.
Fig. 1. a – Conventional disk element on a flat heatsink; b,c,d – composite active elements on heatsinks of different profiles: flat (b), type 1 (c), type 2 (d). Green arrows – rays of luminescence, orange arrows – heat flow, heat sources are shown by the red color. The photographs of the manufactured heatsinks are presented in Fig. 4.
The simple disk active element [Fig. 1(a)] is subject to the ASE due to the effect of total internal reflection, since a beam propagating at the corresponding angle passes a considerable distance in the active medium, is amplified and carries away substantial energy. The presence of an undoped layer in the composite active element [Fig. 1(b)] ensures absence of the effect of total internal reflection, thereby reducing the ASE effect, but leads to pronounced thermally induced distortions, as the flat heatsink removes the heat through the entire rear surface, creating temperature gradients in the thick optical medium. With the use of profiled heatsinks of type 1 [Fig. 1(c)] and type 2 [Fig. 1(d)], heat is removed only in the area of heat sources; consequently, there are no transverse heat flows in the thick undoped layer. The main sources of heat are concentrated in the region of the pump spot. However, there are also minor sources distributed throughout the doped layer and the AE periphery. The heat sources in the doped layer are caused by nonradiative decay of the Yb3+ excited states [14], created under the illumination from the pumped region [15]. The side surface of the active element is opaque, therefore certain fraction of incident radiation is converted to heat. The magnitude of the said heat sources is not known precisely, and they have little effect on the temperature and wavefront distortions in the case depicted in Fig. 1(a), 1(b), but they may lead to overheating of the periphery shown in Fig. 1(c). For this reason, we consider two types of profiled heatsinks. The temperature in the studied geometries was numerically simulated on the basis of the stationary equation of thermal conductivity:
Hooke's law in the isotropic medium is written as
Deformation tensor field ${\mathrm{\varepsilon }_{\textrm{ik}}}$
Boundary conditions:
- 1. Free boundary: $\mathop \sum \limits_{k = 1}^{k = 3} {\sigma _{ik}}{n_k} = 0$ at i ɛ {1,2,3} and $\vec{r} \in {\Gamma _1}$,
which means zero stress applied through the boundary;
- 2. Fixed boundary (symmetric boundary condition): $\overrightarrow {{u_1}} = \overrightarrow {{u_2}} $ at $\vec{r} \in {\Gamma _2}$
which means that the displacement is a continuous function at this boundary;
- 3. Boundary with free tangential displacement: $({\overrightarrow {{u_1}} \cdot\vec{n}} )= ({\overrightarrow {{u_2}} \cdot\vec{n}} )$, which means that the normal component of the displacement is continuous, which means only the normal component of the force is applied through the boundary;
Here, $\nu $ is Poisson ratio, $\vec{u}$ is the displacement vector field, $\overrightarrow {{u_1}} ,\; \; \overrightarrow {{u_2}} $ are the displacement vector fields from two different sides of the interface $\Gamma $ and ${u_k}$ is the k-th component of the vector field $\vec{u}$, ${x_i}$ is the i-th coordinate in the 3D space, ${\mathrm{\varepsilon }_{ik}}$ is the deformation tensor field, indices i, k, l, m can take values from the set {1,2,3}, which correspond to 3 basis vectors, $\mathrm{\alpha }$ is the linear coefficient of thermal expansion, T is the temperature scalar field obtained from Eq. (1), ${\sigma _{ik}}$ is the stress tensor field, E is Young modulus, ${\delta _{ik}}$ is Kronecker delta, and $\vec{n}$ is the normal to the boundary. Equation (2) was solved taking into consideration the temperature distributions obtained by solving Eq. (1), both of them were solved utilizing finite element method. The geometry of the problem used for solving Eqs. (1) and (2) is shown in Fig. 1(b), 1(c), and 1(d). The boundary ${\Gamma _1}$ is the external boundary on Fig. 1, and ${\Gamma _2}$, ${\Gamma _3}$ is the interface between the active element and the heatsink. The heatsink was made of silicon carbide, the doped layer of Yb:YAG, and the undoped layer of sapphire. The problem was regarded to be axisymmetric; the heat sources were only in the doped layer and were uniformly arranged along the optical axis; the radial dependence was adopted from the experiment [Fig. 2(a)]. The fraction of pump power released in the form of heat was equal to the quantum defect and amounted to 8.7%. When solving Eq. (1), the lower boundary of the heatsink was assumed to have the heat transfer coefficient h = 3.3 W/cm2K. The given heat transfer coefficient was obtained in earlier experiments [16]. The ambient temperature was 20°С (cooling water). All other external boundaries were thermally isolated, which corresponds to h = 0. The internal boundary between the optical element and the heatsink had the heat transfer coefficient h = 8.7 W/ cm2K, which was estimated via an independent experiment with a simple thin-disk active element. Thermal conductivity of Yb:YAG was temperature-dependent, according to data from [17]. The properties of all other materials were constant and presented in the Table 1:
Fig. 2. (a) Pump spot profile and heatsink profile obtained in experiment; (b) calculated thermally induced lensing and temperature of the AE active zone versus pump spot diameter.
Table 1. Material properties used in the model
Equation (2) was solved assuming that all external boundaries of the region were free, which corresponds to the boundary condition ${\Gamma _1}$. The interface between the doped and the undoped layers had a continuous field of displacements, i.e. the boundary condition ${\Gamma _2}$ was valid. Free relative tangential displacements were allowed at the boundary between the active element and the heatsink, only the normal component of the displacement field was continuous, which corresponds to the boundary condition ${\Gamma _3}$. The arguments favoring the use of the boundary condition ${\Gamma _3}$ are following. The active element and the heatsink are bonded together with acrylate-based polymer glue, solidified under UV illumination. Organic polymers are known to have low rigidity compared to sapphire, YAG and silicon carbide. In particular, Young modulus of the glue is 4-5 orders of magnitude smaller. Although the chosen model with ${\Gamma _3}$ boundary condition fits the experiment with large relative error, the use of the boundary condition ${\Gamma _2}$ yields a completely inconsistent simulation result. To be more specific, the latter model shows negative thermally-induced lensing in a case where the experiment shows positive one, and no reasonable variation of the model parameters can attain consistency. The heatsink and pump profiles corresponding to the experiment are shown in Fig. 2(a). In the section “Measurement results”, the diameter of this pump profile is 5.2 mm (Figs. 8, 7). The calculations were made for different pump spot profiles obtained from the profile depicted in Fig. 2(a) by linear scaling along the radial coordinate. The scaling factor is laid off along the x-axis in Fig. 2(b), “1” corresponds to the pump profile in Fig. 2(a). The density of the absorbed pump power was constant in all calculations. This implies that total pump power grows along the abscissa on Fig. 2(b). The calculation of distortions for different pump spot sizes makes it possible to find out at what value a zero lensing is reached. The results of the calculations are presented in Fig. 2(b). In this paper we define thermal lensing optical power through second-order polynomial approximation of the wavefront distortion after 1 V-pass (1 reflection) through the active element. The area of approximation is round, 4.5 mm in diameter for all cases. Note that optical power defined in this way equals inverse focal length of a collimated beam after 1 reflection from the disk.
The results of wavefront distortion calculations [Fig. 2(b)] indicate that, by profiling the heatsink, it is possible to achieve a zero value of the thermally induced lens, if the pump diameter is approximately equal to the diameter of the central plateau of the heatsink. One can see in the Fig. 3(a,b) how scaling up the pump diameter affects temperature gradients. Significant suppression of the radial gradient is clearly visible between Fig. 3(a) and Fig. 3(b). Note that the effect is, actually, achieved by reducing the cooling efficiency of the active element as a whole, which leads to an increase in the maximum temperature in the composite disk element [see the temperature graph in Fig. 2(b)]. The plot on Fig. 2(b) is derived for one specific pump power density; however, calculations in a wide range of pump densities and constant pump profile scaling factor of 1.05 demonstrate only minor change (<0.01 1/m) in the thermal lensing optical power. This means that athermal design is possible for this specific pump spot diameter. The presented model allows for qualitative estimate of the behavior of the composite disk element under pumping; however, it is insufficiently accurate and we do not present a direct comparison of the simulated and experimental results.
Fig. 3. Simulated temperature distribution in the composite active element on the heatsink type 2 [Fig. 1(d)], for pump profile scaling factor of 0.8 (a) and 1.2 (b), which correspond to the extreme cases on Fig. 2(b)
3. Experimental setup
For practical implementation of the proposed idea, heatsinks with a special profile were made of silicon carbide [Fig. 4 (a), 4(b)]. The active elements were mounted on the heatsinks with UV-curable glue [18,19], providing a good thermal contact between them (heat transfer coefficient 5-10 W/cm2K, measured in a separate experiment) in a circle with a diameter of 4.5-5 mm, which approximately corresponds to the pump spot of the available laser heads [20] for disk AE. The heatsink with type 2 profile [Figs. 1(d), 4(b)] also ensured thermal contact along the AE periphery, in a 0.5-mm wide ring. The composite AE mounted on the heatsink was placed in the head of a disk laser. The laser head provided 12 V-shaped passes of pump radiation at a wavelength of 940 nm through the AE; and water cooling was applied through the back surface. The AE was located in the Michelson interferometer [21] used to detect wavefront distortions (optical path difference) by means of phase-shift interferometry [22] at a wavelength of 1064 nm. In addition, a probe beam at a wavelength of 1030 nm was directed to the AE to measure signal gain. The AE surface temperature was monitored using an IR camera (thermal imager). The scheme of the entire installation is shown in Fig. 5. It should be noted that pump alignment in this case is a more difficult task than in the case of a flat heatsink, as it is necessary to precisely align the pump spot and the spot of the heat contact.
Fig. 4. Profiled heatsinks aimed at limiting heat flows in the active element: (a) type 1 profile [Fig. 2(c)]; (b) type 2 profile [Fig. 2(d)]; (c) active element on heatsink with type 1 profile 1.
Fig. 5. Schematic of a setup for measuring temperature, phase distortions and signal gain in the active element of a disk laser.
Two composite active elements were studied in experiment. The first one, (Yb:YAG/YAG) mounted on the heatsink, is depicted in Fig. 4(c). It was shaped as a truncated cone with base diameters of 10 and 12 mm. Its doped layer was made of a 0.2-mm thick Yb:YAG disk with 10% doping level. The undoped layer was a YAG disk 2 mm thick. It was mounted on the heatsink with a profile of type 1 [Fig. 4(a)], which had a central plateau 5 mm in diameter. We measured temperature and wavefront distortions in this active element as a function of pump power.
The second composite active element (Yb:YAG/sapphire) had a cylindrical shape and a diameter of 9 mm. Its doped layer was made of Yb:YAG (0.4 mm thick and 5% doping level). The undoped layer was made of 1-mm thick sapphire. This active element was mounted on a heatsink with type 2 profile [Fig. 4(b)]. The heatsink had an annular groove with inner and outer diameters of 4.5 and 8 mm, respectively. We measured temperature, wavefront distortions and small signal gain in the second active element as a function of pump power for two different pump spot diameters. Finally, the second composite active element (Yb:YAG/sapphire) was mounted on a flat heatsink, and the corresponding temperature and wavefront distortions of the signal were measured.
4. Measurement results
First, we performed measurements on an Yb:YAG/YAG composite AE on a heatsink with type 1 profile [Fig. 4(a), 4(c)]. The obtained 2D-profile of wavefront distortions (optical path difference) at a pump power of several hundred watts is shown in Fig. 6(a), and the surface temperature in Fig. 6(d). A strong negative thermally induced lensing was observed in the experiment (Fig. 7); the periphery of the active element was heated stronger than the center, see Fig. 6(d).
Fig. 6. (a,b,c) Experimentally obtained optical path difference and (d,e,f) temperature fields on the surface in the composite AE on the heatsinks: for profiles of type 1 and pump diameter 4.8 mm (a,d); for profiles of type 2 and pump diameter 5.2 mm (b,e); for the flat heatsink and pump diameter 5.2 mm (c,f); (g) photograph of the active element in the holder. The artifacts on the OPD profiles (a-c) are present due to damaged coating.
Fig. 7. Thermal lensing power versus pump radiation power for different heatsinks and pump spot diameters.
Next, similar studies were carried out with a heatsink of the second type shown in Fig. 4(b). It was assumed that a narrow (0.5 mm) thermal contact strip at the AE periphery will reduce its overheating. A Yb:YAG/sapphire composite active element with a doped layer thickness of 0.39 mm and an undoped layer thickness of 1 mm was mounted on the heatsink; the experiments were made with two pump spot diameters: 4.8 and 5.2 mm. The wavefront distortions in the Yb:YAG/sapphire composite at a pump diameter of 5.2 mm are shown in Fig. 6(b) at the pump power of 440 W. The temperature distribution is illustrated in Fig. 6(e). The above results indicate that the heating of the AE periphery decreased, indeed, and the thermally induced lensing became positive. The experimental dependence of the temperature and the thermally induced lensing on the pump power, corresponding to pump spot diameters of 4.8 and 5.2 mm, are plotted in Figs. 8,7. The lensing power decreased with increasing spot diameter. The lensing power with a spot diameter of 5.2 mm and a pump power of 440 W was 0.086 1/m. For comparison, a similar experiment was carried out with the same composite AE mounted on a flat heatsink with a pump spot diameter of 5.2 mm. The results are shown in Figs. 7(c), 7(f), 8, 7. The thermally induced lensing proved to be stronger than in the case of the profiled heatsink. At a pump power of 440 W, the lensing power was 0.2 1/m. During the experiment, it was ascertained that the front surface of the composite AE on the heatsink of the second type [Fig. 6(e)] was heated more uniformly than on the flat one [Fig. 6(f)], whereas the maximum surface temperature was higher in the first case (Fig. 8). It is worthy of note that the temperature of the doped layer [23] rather than the surface temperature affects the signal amplification. The experimental data demonstrate a decrease in the thermally induced lensing in a composite AE from 0.2 1/m to 0.086 1/m due to the use of a profiled heatsink instead of a flat one, all other things being equal. However, these values are still higher than in a simple disk AE with a similar pump spot size [13], where the thermally induced lensing is in the range of 0.02–0.03 1/m.
Fig. 8. Experimental maximum surface temperature of composite active element as a function of pump power for different heatsinks and pump spot diameters.
It worth noting that the thermograms on Fig. 6(d) ,6(e) show asymmetric temperature field of the active elements despite their axial symmetry. This may be because the pump spot is slightly biased off-center. And as we already mentioned, the heat sources located on the periphery of the active element are caused by light absorption on the opaque surface, therefore the magnitude of those sources may be irregular and asymmetric. However, this is not an issue for the flat heatsink, because in this case the maximum temperature is in the pumped region, and peripheral heating is negligible.
Despite the increased surface temperature, the composite AE on the profiled heatsink demonstrated gain comparable to that in the composite on a flat heatsink at the same pump density (Fig. 9). Due to the increased temperature in continuous mode, there was a risk of destruction of the active element on the profiled heatsink. Therefore, Fig. 10 features no points with a pump density over 2 kW/cm2.
5. Conclusion
A new approach to suppressing thermally induced beam distortions in composite active elements of a disk laser was proposed and studied. The key idea is to control heat flows and temperature gradients in an active element by choosing a proper heatsink profile. To determine a needed profile, the temperature, deformations and wavefront distortions in the active element were calculated. It was shown analytically that thermal lensing may be completely suppressed by optimizing the profile of the heatsink surface. Experiments with composite active elements on heatsinks of different shape were carried out, the wavefront distortions, temperature and gain were measured. The efficiency of the proposed idea–2.5-fold thermal lensing reduction with a constant signal gain–was demonstrated. A concomitant negative effect is an increase in the average temperature of the active element, which limits the maximum pump power. However, such a pump level makes it possible to provide almost maximal gain (in terms of ASE limitation) and will not adversely affect multipass amplifiers based on composite disk elements. It is also worth noting that the presented results may be improved significantly with the use of a profiled heatsink made of polycrystalline diamond. This will allow reducing the negative effect of the concomitant heat increase due to a much higher heat conduction of the heatsink. Another challenging application of the proposed approach may be disk multipass amplifiers with pulsed pumping, in which disk thickness may be increased several-fold from 200-300 µm to a thickness close to that of composite disk elements.
Funding
Institute of Applied Physics, Russian Academy of Sciences (State research task Project No. 0030-2021-0029).
Acknowledgments
The research was supported by the State Research Task for the Institute of Applied Physics of the Russian Academy of Sciences (Project No. 0030-2021-0029).
Disclosures
The authors declare no conflicts of interest
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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