Abstract
Keywords
1 Introduction
For laser applications in which high peak power and average power are necessary, diode-pumped lasers are currently considered the most viable option. Either as sources of picosecond, kilowatt-level average powers, or in combination with broadband optical parametric amplifiers, enabling terawatt-level few-cycle pulses, this technology has been paving the way for a generation of compact, cost-effective advanced coherent light sources[
A number of pumping schemes have been applied to diode-pumped solid-state lasers, such as side-pumped slabs[
After passing through a pumped laser crystal, paraxial rays are bent toward the axis due to a combination of the structural deformation of the crystal surfaces and the radial gradient of the thermally dependent refractive index. This could lead to an unexpected beam focus along the laser chain with associated intensities above the optical damage threshold, potentially damaging optical components. In the case of optical cavities, the presence of a power-dependent, variable focal length element can be detrimental to the overall performance. There has been a significant volume of both theoretical and experimental work toward describing and characterizing the thermal lens effect. Some models have aimed at obtaining a general analytical expression for the focal length, making use of the average power and discarding the time evolution of the temperature[
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In this work, we perform an experimental study and provide a theoretical description of thermal lensing in a diode-pumped amplifier operating at a low repetition rate (1–10 Hz), high pump power (max 4 kW) regime. This range of parameters is extremely relevant to high-intensity pulsed laser systems using diode-pumped laser either as primary or as pump sources. In these particular conditions, the heat deposited during the pumping time window leads to a dynamically changing temperature gradient and consequently a time-varying thermal lens in the laser rod. We characterize the dependence of thermal lensing on power and repetition rate within this range and develop a theoretical model assisted by a numerical analysis, with the capability of characterizing the pumping behavior in two spatial dimensions and one temporal dimension. The model describes adequately the observed results for a pulsed diode-pumped Yb-doped crystal.
2 Theoretical model and simulations
A wide range of conditions have been considered for studying the thermal lens effect. An exact analytical solution for the heat distribution is difficult to find. A series of assumptions and approximations are needed to reach a result. Nevertheless, many groups have attempted to solve the problem considering some simplifications. Schmid et al.[
Another approach to the problem is by solving the heat equation numerically. In a general way, it allows us to reduce the number of assumptions and approximations and provide an accurate result. In this work, we consider an end-pumped cylindrical crystal, where the heat generated within the laser rod by unabsorbed pump light absorption is removed by a coolant flowing along its cylindrical surface. Spatially uniform absorption is assumed, which will lead to an approximately uniform radial deposition of heat. In this way, an axially symmetric, steady-state temperature profile will be established in the crystal after a given number of pump cycles. Let us start by considering the time-dependent temperature equation, which is determined by the following differential energy-conservation law:
Material | Doping (at.%) | |||||||
---|---|---|---|---|---|---|---|---|
Yb:YAG | 8.4 | 590 | 4.560 | 3 | 940 | 1030 | 3 | 0.14 |
Table 1. Thermo-optical properties of Yb:YAG.
This equation was solved numerically by many groups, using commercial or in-house computational codes[
Considering cylindrical coordinates and assuming azimuthal symmetry, Equation (
To represent the temporal pulse profile, we considered a periodically pulsed top-hat distribution[
The boundary conditions for such a geometry have been intensively studied in previous works[
The above equation can represent several scenarios, including an insulating boundary (
We performed numerical simulations in order to obtain the temporal evolution of the temperature for different conditions, with a step size of
2.1 Pulsed regime
With the goal of finding the steady-state temperature profile in a pulsed regime, a range of simulations were performed covering different parameters. We first considered the temperature evolution with time for several pump rates. Figure
The resulting periodic profile is given by the pump pulse arrivals and the evolution between them. On the other hand, at 10 Hz the time between two consecutive pulses is shorter than the thermal relaxation time, leading to a steady-state profile. This is achieved when an equilibrium is reached between the heat source term and the dissipation of the heat due to the built-up gradient. It was observed that the characteristic time of the steady-state formation is below 10 s, which was therefore taken as the upper limit for the simulations.
One should mention that while the influence of the heat removal was not addressed, previous works have shown a small influence on the temperature gradient, and thus, thermal lens power[
The time evolutions of the radial profiles for two regimes are presented in a contour plot form in Figure
2.2 Thermal lensing
The temperature profile discussed above can influence the laser gain medium and its performance within an amplifier in different ways, e.g., optical and mechanical deformations. In particular, those affecting the propagation of light through the introduction of wavefront distortions are (i) the direct change of the temperature-dependent refractive index and (ii) the deformation of the end-face curvature, effectively resulting in the formation of a transient lens caused by the relaxation of the thermally induced stress in the end regions.
In the case of a steady-state temperature, a constant thermal lens magnitude arises, and we are able to calculate its equivalent focal length. For this goal, we use a model presented in Ref. [
The term
The above integral was implemented numerically for the computational model. Figure
3 Experimental setup
In order to verify the validity of the previous numerical simulations, we assembled an experimental setup as shown in Figure
4 Results and analysis
Since the probe beam at the crystal plane exhibits a small (
In order to retrieve the phase profile from the acquired image, the commercial sensor control software is used. It relies on the definition of a custom mask, after which the wavefront is projected onto a set of orthogonal wavefronts, the decomposition being unique. In the case of a circular pupil, as chosen, the Zernike polynomials act as the basic functions. In particular, the
Taking this into account, we compared the experimental data obtained from the thermal lens measurements with the calculations described in the previous Section
The measurements are repeated for pump repetition rates of 2 Hz, 5 Hz and 10 Hz. The corresponding results are shown in Figures
5 Conclusions
We have presented a comparative study and benchmarking on thermal lensing in a longitudinally pumped Yb-doped crystal at high energies and moderate repetition rates. A numerical model allowed us to simulate the temperature distribution inside the crystal and to calculate the induced thermal lens over a range of pump powers, repetition rates and pump beam sizes. The thermal relaxation time was measured and a study regarding its influence on the temperature buildup was performed. The influence of the pumping parameters, namely, the pumping waist and the repetition rate, was equated and we found, in our case, that a steady state with small oscillations can be achieved at rates equal or higher than 5 Hz.
Experimental work showed the main concerns regarding the measurement of the thermal lens: pump beam homogeneity is desirable, good alignment and temporal synchronization between pump and probe beam and known crystal properties for comparative purposes between simulations and experiment. These measurements performed under lasing/nonlasing conditions can be used for future design and implementation of high-power diode-pumped lasers.
References
[1] H. Fattahi, H. G. Barros, M. Gorjan, T. Nubbemeyer, B. Alsaif, C. Y. Teisset, M. Schultze, S. Prinz, M. Haefner, M. Ueffing, A. Alismail, L. Vámos, A. Schwarz, O. Pronin, J. Brons, X. T. Geng, G. Arisholm, M. Ciappina, V. S. Yakovlev, D.-E. Kim, A. M. Azzeer, N. Karpowicz, D. Sutter, Z. Major, T. Metzger, F. Krausz. Optica, 1, 45(2014).
[2] R. Voelkel, K. J. Weible. Proc. SPIE, 7102, 71020J(2008).
[3] M. Traub, H.-D. Hoffmann, H.-D. Plum, K. Wieching, P. Loosen, R. Poprawe. Proc. SPIE, 6104, 61040Q(2006).
[4] M. Franko, C. D. Tran. Rev. Sci. Instrum., 67, 1(1996).
[5] T. S. Rutherford, W. M. Tulloch, S. Sinha, R. L. Byer. Opt. Lett., 26, 986(2001).
[6] Y. Wang, H. Kan. Opt. Commun., 226, 303(2003).
[7] A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, H. Opower. Appl. Phys. B, 58, 365(1994).
[8] F. Song, C. Zhang, X. Ding, J. Xu, G. Zhang, M. Leigh, N. Peyghambarian. Appl. Phys. Lett., 81, 2145(2002).
[9] S. Chénais, F. Balembois, F. Druon, G. Lucas-Leclin, P. Georges. IEEE J. Quantum Electron., 40, 1217(2004).
[10] M. P. MacDonald, T. Graf, J. E. Balmer, H. P. Weber. Opt. Commun., 178, 383(2000).
[11] S. Wang, H. J. Eichler, X. Wang, F. Kallmeyer, J. Ge, T. Riesbeck, J. Chen. Appl. Phys. B, 95, 721(2009).
[12] R. Lausten, P. Balling. J. Opt. Soc. Am. B, 20, 1479(2003).
[13] H. Nadgaran, M. Sabaeian. Pramana, 67, 1119(2006).
[14] S. Chénais, F. Druon, S. Forget, F. Balembois, P. Georges. Prog. Quantum Electron., 30, 89(2006).
[15] R. Weber, B. Neuenschwander, H. P. Weber. Opt. Mater., 11, 245(1999).
[16] M. Schmid, T. Graf, H. P. Weber. J. Opt. Soc. Am. B, 17, 1398(2000).
[17] B. A. Usievich, V. A. Sychugov, F. Pigeon, A. Tishchenko. IEEE J. Quantum Electron., 37, 1210(2001).
[18] Z. Li, X. Huai, Y. Tao, Z. Guo. Appl. Opt., 48, 598(2009).
[19] P. Shi, W. Chen, L. Li, A. Gan. Appl. Opt., 46, 6655(2007).
[20] T. Liu, Z. M. Yang, S. H. Xu. Opt. Express, 17, 12875(2009).
[21] M. Sabaeian, H. Nadgaran, L. Mousave. Appl. Opt., 47, 2317(2008).
[22] M. Sovizi, R. Massudi. Opt. Laser Technol., 39, 46(2007).
[23] W. A. Clarkson. J. Phys. D, 34, 2381(2001).
[24] P. A. Loiko, K. V. Yumashev, N. V. Kuleshov, A. A. Pavlyuk. Opt. Laser Technol., 44, 2232(2012).
[25] A. H. Farhadian, H. Saghafifar, M. Dehghanbaghi. J. Russ. Laser Res., 36, 350(2015).
[26] P. Albrodt, X. Délen, M. Besbes, F. Lesparre, P. Georges. J. Opt. Soc. Am. B, 35, 3004(2018).
[27] I. Tamer, S. Keppler, J. Körner, M. Hornung, M. Hellwing, F. Schorcht, J. Hein, M. C. Kaluza. High Power Laser Sci. Eng., 7, e42(2019).
[28] M. M. Rezaee, M. Sabaeian, A. Motazedian, F. S. Jalil-Abadi, A. Khaldi-Nasab. Appl. Opt., 54, 1241(2015).
[29] Y. F. Chen, T. S. Liao, C. F. Kao, T. M. Huang, K. H. Lin, S. C. Wang. IEEE J. Quantum Electron., 32, 2010(1996).
[30] S. ChénaisNouveaux matériaux laser dopés à l’ytterbium: performances en pompage par diode et étude des effets thermiques. (Université Paris-Sud, 2002)..
[31] M. L. Baesso, J. Shen, R. D. Snook. J. Appl. Phys., 75, 3732(1994).
[32] M. Born, E. Wolf. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light(2000).
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