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Nd:glass amplifiers are used in most of the existed petawatt laser facilities. A typical repetition rate of such lasers is 1 shot per 30 minutes or less. Limitations are thermally induced distortions of radiation and tensile stresses in Nd:glass. An increase of the repetition rate is an urgent problem. We have investigated thermally induced depolarization and thermal lens effects in Nd:glass rods up to 10 cm in diameter at a pump pulse repetition period of 3 minutes. It is shown that the rods have a safety factor of at least 5 before thermal stress induced damage would occur, and despite of their size phase and polarization distortions could be compensated.
©2011 Optical Society of America
1. Introduction
Nd-ion doped glass laser amplifiers are used in the majority of the existing petawatt laser facilities. All available petawatt lasers and those under development may be classified into three types: lasers with Nd:glass amplifying medium [1–5], with sapphire (corundum-titanium) amplifying medium [6–9], and with parametric amplification in KDP and DKDP crystals [10–15]. In the second and third cases, after transformation into the second harmonic, Nd:glass laser radiation is used either for pumping a sapphire crystal or parametric chirped pulse amplifiers, respectively.
The main merit of glass as an inversion medium is a possibility to produce large-aperture elements with high stored energy that can operate at moderate laser intensity well below the optical damage threshold. Damage of the active element above the threshold of admissible thermally induced elastic stresses is a basic, but normally not nearly reached restriction of pulse repetition rate. However, the small heat conductivity of Nd:glass restricts pulse repetition rate significantly. For example, the characteristic cooling time τ of a cylindrical phosphate-glass active element of 10 cm in diameter, corresponding to infinite heat exchange with the environment, is about 30 min [τ = R2/(5.76D), where R is the radius of the active element and D is the thermal diffusivity]. This explains why none of the available petawatt facilities can ensure more than several shots per hour. In practice, lasers operate with shorter pulse repetition rates. If the time interval between pump pulses is set to be shorter than the characteristic time of active element cooling, then gradual accumulation of heat inside the active medium gives rise to pronounced distortions in radiation polarization and phase caused by the photoelastic effect. Thermally induced birefringence in glass was studied in detail theoretically and in experiments in several publications [16–25]. However, this problem was left almost unnoticed in publications concerning facilities with pulse energies of 100 J and higher. For example, less than 3% of polarization losses in the entire facility were reported in the paper [26] devoted to the laser complex “Omega”. This situation is typical of single-shot systems in which heat is not accumulated in active elements from shot to shot, and heating by a single pump pulse in the region of radiation is quite uniform and does not result in significant elastic stress.
Increasing pulse repetition rate in powerful lasers is an important problem, where thermally induced radiation distortions start to play a fundamental role. The present paper reports results of theoretical and experimental study of thermally induced distortions of radiation in large-aperture Nd:glass active elements at high pump pulse repetition rate and their influence on the quality of amplified radiation. These elements are used as amplifiers in the 0.56 PW laser complex PEARL [10]. Our research demonstrates that it is possible to create a petawatt laser based on Nd:glass with a record small time interval between pump pulses of not more than three minutes.
2. Theory
Primary attention in our experiments was focused on the investigation of thermally induced depolarization of radiation. This phenomenon is encountered in an anisotropic medium where there exist eigenmodes with different indices of refraction. In an isotropic medium, such as glass, birefringence induced by the photoelastic effect may occur. In the case of interest to us, the matter concerns elastic stresses arising in Nd:glass rod active elements under inhomogeneous heating by pump pulses. This heating causes the change in the dielectric constant tensor [17]:
where σkl is the stress tensor, πijkl is the elastooptic tensor, n is the unperturbed refractive index, and ΔT is the difference between the rod temperature and the temperature of the surrounding medium.In the rod geometry there are only three nonzero stress tensor components [17]:
where α is the linear thermal expansion coefficient, ν is the Poisson’s coefficient, E is the modulus of elasticity, R is the rod radius, z, r, and ϕ are the cylindrical coordinates.Due to the symmetry of the stress tensor and the isotropic property of glass, there exist two eigen waves in the rod polarized in the radial and tangential directions. The thermally induced phase incursions of these waves may be written in the form:
where Δψ0 = const, L is the rod length, λ is the radiation wave length, C1 = – (n3/2)π1111 and C2 = – (n3/2)π1122 are photoelastic constants.The birefringence leads to depolarization, i.e. in the case of a linearly polarized wave incident on the active element, the output wave contains an orthogonally polarized component. The ratio of the intensity of this component to the total radiation intensity at the output of the active medium determines the depolarization factor:
where δ = Δψϕ – Δψr. In practice, it is convenient to consider the aperture-integrated depolarization factor γ – that is the ratio of the energy of the beam polarized orthogonally to the incident radiation to the total beam energy at the output of the active medium.Assuming that the rod is instantly heated by a pump pulse and that the boundary condition of the convective heat exchange with the environment at r = R is κdT/dr = − αcΔT (κ is the thermal conductivity coefficient and αc is the convective heat exchange coefficient), we can solve the equation for the thermal conductivity in the rod in the form of the series:
where is the decay time of the n-th term of the series; сT and ρ are the specific heat and density of the rod, respectively; and qn are the solutions of the equation qJ1(rR)/J0(qR) = αc/κ (J0, 1 are the zero- and first-order Bessel functions of the first kind). The coefficients Cn are determined by the initial temperature distribution in the rod and are proportional to the heat absorbed in the rod.Equation (3)-(4) connect the temperature and the depolarization distributions. Thus, having measured the depolarization Γ(r,ϕ), we can determine the temperature distribution ΔT(r) and the thermally induced phase for the waves of both polarizations [25]. From the experimental viewpoint, it is much easier to measure depolarization than temperature or phase. Equation (5) simulates the temperature diffusion during rod cooling. Equation (3)–(5) allow us to simulate the dynamics of thermally induced effects, i.e. the depolarization and the thermal lens.
3. Measurement of thermally induced distortions in large-aperture Nd:glass amplifiers
Nd phosphate glass rod laser amplifiers of large diameter were studied in experiments. Their key specifications are listed in Table 1 . Eight pulsed gas-discharge lamps in laser heads with mirror reflectors were used for pumping. The pump energy of up to 36.5 kJ was derived from two 200 μF energy storage capacitors charge up to 13.5 kV [27]. The active elements were cooled by running water, with the input temperature stabilized to an accuracy of ± 0.1 К. The measured transverse distributions of the gross small signal gain, G0, are depicted in Fig. 1 . Its average value <G0> over the cross-section and energy stored in population inversion Wstored are listed in Table 1. A principal optical scheme that was used to measure a small signal gain is shown in Fig. 2 . An Nd-YLF master oscillator produced 1 mJ of energy with a pulse duration of about 20 ns. The energy density distributions of the cold and amplified beams after the laser head were measured by a CCD camera. We used a photo diode to monitor the beam energy at the input of the rod.
Table 1. Characteristics of the studied active elements
Fig. 1 Distribution of small signal gain (G 0) over the aperture of active elements having diameter 6 cm (a), 8.5 cm (b), and 10 cm (c). The white curves designate cross-sections.
We investigated thermally induced depolarization in large-aperture Nd:glass active elements in the operating mode when the time interval between pump pulses was much less than the characteristic time of active element cooling. In other words, the rods did not have sufficient time to fully cool down by the next shot. As a result, heating was accumulated until a steady-state mode was established, when the thermal energy released in the rod during the shot was equal to the quantity of heat removed from the active element between successive pump pulses. The pulse repetition period was 3 minutes. The thermally induced depolarization of radiation in the rods was measured by means of linearly polarized probe pulses with a repetition rate of 0.25 Hz. To do so the active elements were placed between crossed polarizers, and laser radiation intensity was measured at the input and output of this scheme.
Dynamics of the integral depolarization γ is shown in Fig. 3 . The vertical lines of the grid correspond to the pump pulse moments (one in three minutes). The asterisks on the horizontal axes designate the last shots in the runs. The plots demonstrate both, establishment of a steady-state mode and the depolarization relaxation after the final pump pulses of a sequence of pulses. Transverse distribution of the depolarization Γ is shown in Fig. 4 (Multimedia).
Fig. 3 Dynamics of the integral depolarization γ in an active element having a diameter of 6 cm (a), 8.5 cm (b), and 10 cm (c) in a sequence of pump pulses with a repetition rate of 1 pulse per 3 min. The asterisks designate the last pump pulse in the run.
Fig. 4 (Multimedia online) The depolarization distribution Γ(r,ϕ) in an active element having a diameter of (a) 6 cm (Media 1), (b) 8.5 cm (Media 2), and (c) 10 cm (Media 3); t is the time passed from the instant of the corresponding pump pulse (hours:minutes:seconds)
The increase of the integral depolarization after the final pump pulse in the runs (Fig. 3) may be attributed to the rod geometry. Consider an element having a diameter of 10 cm. On completion of the pump pulse sequence, the elastic stresses in the rod are such that the depolarization is close to zero in a rather large area near the active element surface [Fig. 4(c) (Media 3)]. As the rod is cooling down, the stresses reduce according to the Eq. (2). It leads to the decrease of the thermally induced phases Δψϕ and Δψr (Eq. (3). The difference δ between them reduces as well. As a consequence, the number of depolarization fluctuations in the radial direction decreases [this statement follows from the Eq. (4)], i.e., the region occupied by the depolarized radiation broadens, expanding up to the surface of the active element, which is accompanied by an increase of the integral polarization loss γ [Fig. 3(c)].
In our earlier work [25] we proposed a technique for calculating temperature distribution in an isotropic active medium by measuring depolarization Γ (see section 2). Using this technique we calculated the changes in the rod temperature distribution ΔT introduced by a single pump pulse (the temperature of the cooling liquid was taken as zero). An example of such calculations for a rod 6 cm in diameter is given in Fig. 5(a) .
Fig. 5 The temperature distribution after a single pump pulse (a, solid line), a part of the temperature proportional to the population inversion (a, dash line), and the integral depolarization dynamics (b) in an active element 6 cm in diameter in a sequence of pump pulses with a repetition rate of 1 pulse per 3 min.
Note that the distribution ΔT(r) [Fig. 5(a), solid line] is not proportional to the inversion distribution in the active element [Fig. 5(a), dash line]. This is explained by the fact that, in addition to the temperature distribution component proportional to the inversion, i.e., stipulated by the quantum defect, there exists a heating component localized at the rod periphery that is related to absorption of the parasitic part of the lamp spectrum. The heat concentrated near the surface of the active element influences the depolarization dynamics during the first seconds after pump pulses. The effect is strongest during the first few pulses and is insignificant in the steady state, because this heat is rapidly removed from the active element and does not accumulate from pulse to pulse due to high efficiency of heat exchange with the cooling water.
4. Discussion
Knowing the distribution of heat release in an active element produced by a single pump pulse one can calculate the temperature dynamics in the rod. The results of such computations, namely, radial temperature distributions in active elements in a steady state mode are presented in Fig. 6 . As the heat release at the rod periphery is higher than in the center [Fig. 5(a)], temperature gradients after the shot in the steady state are less than before the shot (Fig. 6). This explains the depolarization decrease after the shot [Fig. 5(b) and the insert in it], in spite of the additional heat release. Then, despite rod cooling, the depolarization increases up to a steady state value.
Fig. 6 Temperature distribution in active elements having diameter 6 cm (a), 8.5 cm (b), and 10 cm (c) in a steady state mode. Blue curve – the temperature immediately before the pump pulse; red curve – the temperature immediately after the pump pulse.
With known temperature distribution one can assess the maximum value of elastic stress in experiments. From the viewpoint of thermomechanical damage, most vulnerable is the surface of the active element as it has a relatively large number of defects (microcracks and so on) [17]. For a cylindrical active element, the elastic stress on the surface is given [17] by the formula σ = 21/2αE(<T> – T)/(1 – ν), where α is the coefficient of linear heat expansion, E is the modulus of elasticity, ν is the Poisson’s coefficient, and T and <T> are, respectively, the surface temperature and the average temperature in the rod. The maximum admissible value of elastic stress σlim is determined by heat resistance of glass ΔT = (<T> – T)lim. For the KGSS-1621 phosphate glass used in our study, ΔT = 40 K (according to [28], heat resistance of the GLS-22 glass that has a similar composition is 38 K). The magnitude of maximum elastic stress will then be estimated to be 45 MPa.
Table 2 represents the calculated values of tensile stress σ in the rods and the focal lengths for the radially and tangentially polarized waves in a regime when the pump pulse repetition period was 3 minutes. It is clear from the Table 2 that the rods have a safety factor of at least 5 before damage. The regime with a repetition rate higher than 1 pulse per 3 minutes is possible but it may be risky. Operation at 15-20% of the damage threshold seems to be quite safe. Our experience shows that the active elements described in our paper can function in such a regime for a long time and none of them has been damaged.
Table 2. Investigated characteristics of the active elements
On the basis of the calculated temperature distributions in the active elements, components of the thermally induced stress tensor were found (Eq. (2), which enabled assessing phase distortions of radiation in the amplifiers. The thermally induced phase incursions Δψr and Δψϕ are plotted in Fig. 7 for radially and tangentially polarized waves. When the rod temperature depends only on the longitudinal and radial coordinates, these waves propagate in a cylindrical active element without changes in polarization. Arbitrarily polarized radiation may be decomposed into these eigenmodes, and the quantity δ = Δψr – Δψϕ determines its depolarization at the output of the active element (see section 2), thus allowing depolarization dynamics in the active element to be calculated [Fig. 5(b)]. Comparison of the theoretical and experimental values of depolarization verifies correctness of the calculation of the temperature distribution in the active element after a single pump pulse and of the temperature dynamics and steady-state solution. Examples of such tests are given in Fig. 4 and Fig. 5(b). Clearly, the depolarization distributions over the beam aperture obtained in an experiment and in calculations are in a good agreement (Fig. 4), and the solid theoretical curve for the integral depolarization in Fig. 5(b) coincides well with the experimental points. The error up to 20% is normal for the experiments devoted to the measurement of the radiation intensity. Calculations of the temperature and thermally induced effects were tested in the other active elements (8.5 and 10 cm in diameter) analogously. The values of thermal diffusivity were used in calculations are given in Table 2. Other optical, elastic and photoelastic characteristics of KGSS-1621 glass were chosen equal to the corresponding values for GLS-22 glass which are described in [28].
Fig. 7 Phase distortions of radiation in active elements having diameter 6 cm (a), 8.5 cm (b), and 10 cm (c) in a steady state mode (immediately after the pump pulse).
Except for the small periphery region, the functions Δψr,ϕ(r) in Fig. 7 are well approximated by a parabola: Δψr,ϕ(r) ≈– πr2/(λFr, ϕ). Under this approximation we calculated focal distances of the thermal lens, Fr, ϕ, that are given in Table 2. One can see from the table that the parabolic part of the thermal lens is rather small and may be readily compensated. Astigmatism (Fr ≠ Fϕ) is readily compensated by means of a 90° polarization rotator located between two identical active elements. Higher-order aberrations are strongly peripheral and their influence greatly depends on laser beam diameter and shape.
Note that the phase distributions plotted in Fig. 7 were obtained assuming that by the instant of laser pulse passage the heat from the pump had been fully released in active medium. In practice, the arrival time of the amplified laser pulse must coincide with the instant of maximum population inversion, which is prior to the pump pulse termination. Besides, there exists quantum-defect-generated heat that is released after the radiation transition when the electrons from the lower laser level relax to the ground state. The real temperature distribution at the time of laser pulse arrival is between the limiting distributions depicted in Fig. 6. Thus, the calculated aberration of the thermal lens (Fig. 7) is the upper estimate of real aberrations of laser radiation. The above considerations have almost no impact on calculation of focal distances Fr and Fϕ.
5. Conclusion
The thermally induced effects in large-aperture (6, 8.5 and 10 cm in diameter) Nd:glass rod laser amplifiers have been investigated. It was found in experiments that the pump pulse repetition period may be much lower than the effective time of active element cooling (30 minutes for a rod having a diameter of 10 cm). In our experiments the tolerable repetition rate was found to be 1 pulse per 3 minutes. The path-length difference on the surface and along the axis of the active element was measured to be of order laser wavelength (1 µm). Strong polarization distortions of radiation should be compensated, whereas appearance of a thermal lens may be neglected. The obtained data demonstrate that it is possible to create an Nd:glass laser with a subkilojoule energy of nanosecond pulse and a repetition rate of 1 shot per 3 minutes.
Acknowledgments
This work was supported by the program of the Presidium of the Russian Academy of Sciences “Extreme Light Fields: Sources and Applications”.
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