Thermal lens effect model of Ti:sapphire for use in high-power laser amplifiers

We mathematically model the thermal lens effect of Ti:sapphire for use in a high-power laser pulse amplifier. The model enables more accurate prediction with new interpretations and offers simplified equations for the optical path difference and thermally induced focal length. Our model is validated through comparisons with measurements of existing high-power laser facilities. Further, we apply the model to a 2 PW, 10 Hz Ti:sapphire laser amplifier design.

thermal distortion in the medium compared to existing methods. 18) The thermal lens effect is induced by a change in the refractive index caused by the temperature of the medium, surface curvature deformation caused by thermal expansion of the medium, and the photoelastic (PE) effect. 18) This has already been studied extensively in laser systems using YAG host media; however, to the best of our knowledge, these phenomena have not been studied for the Ti:sapphire medium, and only the change in the refractive index caused by the temperature has been considered to cause the thermal lens effect to date. [18][19][20][21][22] The reason seems to be the insignificance of the thermal effect, which results from both the excellent thermal stability of Ti:sapphire and the complexity of interpreting the hexagonal structure. However, to achieve higher than PW pulse powers and high repetition rates, the thermal lens effect in Ti:sapphire should be not ignored but rather seriously considered. To develop applications of high-power lasers using Ti:sapphire, it is necessary to study the properties of Ti:sapphire and the thermal lens characteristics arising from its crystal structure. 14) The thermal lens expression in this study was derived in that context and was verified by comparing it with experimental results for known high-power laser amplifiers. [19][20][21] The design of a laser amplifier is dependent on its target specifications such as the target intensity, energy, and spatiotemporal shape of the generated pulses. To generate a high power of a few tens of petawatts for a laser pulse using CPA, Ti:sapphire in the shape of a circular disk is used as the amplification medium. 2,23) As a flat-top pump beam is used to obtain a spatially uniform energy distribution and a shorter pulse width, heat is uniformly generated throughout the gain medium. 24) The heat generated during the laser pulse amplification process is generally discharged through the rim of the gain medium. An index-matched refrigerant encloses the rim to prevent amplified spontaneous emission. The medium is cooled by both cooling by the phase-matched refrigerant surrounding the rim and conductive cooling by surrounding air in contact with both sides of the medium. [18][19][20][21] The orientation of the Ti:sapphire was assumed to be ½11 " 20, and the polarized light in the C-axis direction [Figs. 1(a) and 1(b)] with the highest amplification cross section was used. 8) The thermal lens induced in the medium includes the refractive index change depending on the temperature of the medium, the curvature of the surface due to thermal expansion, and the PE effect. 14,18) Many conventional studies considered only changes in the refractive index with temperature. [19][20][21] In contrast, we considered all three factors to provide a more accurate model. The optical path difference (OPD) and the focal length of each element were derived, and the focal length was combined with the thin lens assumption to derive the focal length of the entire thermal lens. In these assumptions, the relationship between the heat, P h , generated by the stored pump beam and the temperature, Tðx; yÞ, of the medium are given by x @ 2 @x 2 Tðx; yÞ þ y @ 2 @y 2 Tðx; yÞ ¼ À where Tðx; yÞ is the temperature, A is the area of the medium, L is the thickness of the medium, h is the thermal coefficient, and κ is the thermal conductivity, which is an anisotropic value. Another point to note is that if the repetition rate is increased for practical application of ultrahigh-power lasers, the amount of heat applied to the medium increases proportionally with the repetition rate. Equation (2) is derived for a thermally steadystate condition and a rim-cooling condition; many laser amplifier models include both conditions. We changed to the polar coordinate system for convenience in the following formula: where R is the radius of the medium, and Q is the heat per volume, P h =AL.
Note that the thermal conductivity of sapphire varies depending on its direction; it can be expected that the temperature of the sapphire medium has an elliptical slope at the center. However, it is difficult to confirm the anisotropy of the temperature in Fig. 2. The reason is that the eccentricity of the temperature distribution gradient is negligibly small, only 0.04. The OPD due to the change in refractive index (Δ T ) is proportional to the temperature distribution, whereas the OPDs due to thermal expansion (Δ ε ) and the PE effect (Δ PE ) of the medium are not exactly proportional to the temperature distribution, but are influenced by the crystalline structure and its orientation and the shape of the medium. 25,26) More detailed research and measurement of sapphire, which has a hexagonal structure, are needed as it is developed for highpower applications. 19) The OPDs were derived as shown in Eqs. (3)-(5) using known properties, and the physical properties used are summarized in Table I. 22,27) Áðr; SðÞ ¼ 1 8 ð3s 11 þ 3s 12 À 4s 13 Þ cosð2Þ À 1 4 ½5s 11 þ 5s 12 þ 4s 13 þ 2 ffiffiffi 3 p s 14 sinð2Þ ð5Þ   where α is the thermal expansion coefficient, E is the Young's modulus, ν is the Poisson's ratio of the medium, Ω is the thermal expansion resistance of the medium, and C PE is the PE coefficient. A new variable (Ω) was introduced to express the thermal expansion of the hexagonal crystal. A similar expression for other crystal classes has been studied; however, to the best of our knowledge, this is the first research on Ti:sapphire as a hexagonal crystal. The Ω values derived by calculating the z-axis strain, ε m = S mn σ n + αT, 25,26) and C PE are shown with the relative dielectric impermeability (B m ). The PE coefficient, C PE , can be written as C PE ¼ 16ð1ÀÞ EQ ÁB m and ΔB m = π mn σ n . 14,18,22) f À1 f À1 The focal length equations, Eqs. (7) and (8), have been validated for the measured focal lengths of Ti:sapphire laser amplifiers from other facilities. We refer to the papers on the three facilities. [19][20][21] Table II shows the coefficients used in Eqs. (7) and (8), and Table III summarizes the specifications, predicted focal lengths, and measured focal lengths obtained using the equations in each paper. The superscript in Table III indicates the orientation; if there is no superscript, the orientation is not indicated. The OPDs induced through the medium in each condition are shown in Fig. 3. The measured wavefronts for Cases 1 and 2 are given in the respective papers, and they agree with the expected results obtained using Eq. (3). The expected change in the focal length with the laser pumping power is shown in Fig. 3.
Case 1 19) (Fig. 3, top) was expected to have thermally induced vertical and horizontal focal lengths of 33.9 and 32.0 m, respectively, and the measured focal length was 20 m. Under the conditions used in that study, we predicted that the vertical and horizontal focal lengths of the thermal lens would be 24.2 and 18.1 m, respectively. Although there are some differences, we can confirm that the predictions obtained from the known experimental values and the proposed model are consistent. In case 2 20) (Fig. 3, center), the astigmatism aberration of the Ti:sapphire thermal lens was also considered. As a result, the measured thermal focal lengths in the vertical and horizontal directions, including the astigmatism, are 60.  (Fig. 3, bottom), the measured thermal focal length was more similar to the conventional theory than ours is. In that study, only the refractive index change caused by temperature was considered, and the expected and measured focal lengths were 138 and 135 m, respectively. The expected vertical and horizontal focal lengths obtained using the proposed method are 104.7 and 78.4 m, which are somewhat different from the measurement values. This difference can be attributed to the difference in the cooling condition of the medium and the doping concentration of titanium ions.
The thermally induced focal lengths required in compensation design or realistic applications can be calculated using only the pump beam power and length of the Ti:sapphire crystal. Therefore, this paper is useful for researchers and developers using a high-power laser amplifier. Even if a cryogenic cooling system or highly doped crystal is used, our model can be used by applying the properties under the appropriate conditions. Because the elastic compliances and piezo-optical tensor depend slightly on the temperature and doping concentration, we have applied the specifications of Ti:sapphire with a low doping concentration. 14,23) However, because there is no information on the elastic compliance and piezo-optical tensor for various conditions, research on the relationships between the properties and the temperature and ion doping density is urgently needed. 24) We present a new interpretation of the thermal lens by describing the medium using a new thermal lens expression.   7) and (8).
Because the proposed model contains all three causes of the thermal lens effect, it also allows us to evaluate the contribution of each cause. Analysis of astigmatism aberration is a typical example. The known reason for the astigmatism of Ti:sapphire is temperature anisotropy due to the anisotropy of its thermal conductivity. However, according to Fig. 3, the astigmatism is affected by the effect of thermal expansion on the OPD, and the unique OPD shape due to the PE effect can be obtained, as shown in Fig. 4. Equation (6) gives the PE coefficient as a function of θ. The signs of the coefficients are different for θ values of 0 and π=2, that is, in the horizontal and vertical directions. The model can be applied to the design and maintenance of a petawatt-class amplifier, and the beam focusing shape can be estimated by applying the results of this study. To construct a multipass amplifier for an output power of 2 PW at 10 Hz, a 25-mm-thick Ti:sapphire crystal can be irradiated by a pump beam of 100 J with a diameter of 60 mm. For this amplifier, the expected vertical and horizontal focal lengths are 34.1 and 32 m, respectively. The simulation results of 2 PW system are in Fig. 5. These predictions may be useful for designing the compensation.
The PE effect not only deforms the wavefront, but also causes the depolarization effect. Laser facilities aiming at petawatt or higher powers require a thermal distortion compensation design to prevent degradation of the spatiotemporal shape and pre-pulsegeneration, which are induced by the thermal lens effect and depolarization. 23,28) Therefore, we will continue to study the precise physical properties of Ti:sapphire, an accurately derived mathematical model of depolarization, and the spatiotemporally distorted shape of the focused beam at a more precise level. 24,29) In summary, reasonable approximation equations for predicting the thermal effect of a Ti:sapphire amplifier were introduced. The equations include the change in refractive index with temperature, the surface curvature due to thermal expansion, and the PE effect. This is the first approximation to consider all of these effects on the Ti:sapphire crystal, to the best of our knowledge. By using the measurement results from existing facilities, the equations were verified at a reasonable error level. For a 2 PW system, the vertical and horizontal focal lengths are expected to be 34.1 and 32.0 m, respectively, for a pump beam energy of 100 J at 10 Hz. Because we can predict the thermal lens effect more accurately, we can also design the thermal lens compensation more effectively. Research on the optical properties of Ti:sapphire and the changes in the properties depending on the temperature or the doping density of active ions is necessary to predict the spatiotemporal shape of the temporally compressed and spatially focused PW pulse more precisely.