# Engineering of Ti:Sapphire Lasers for Dermatology and Aesthetic Medicine
## 2 Experimental Results and Discussion
### The Influence of Pump Fluence Distribution on Ti:Sapphire Laser Operation
The thermal component of RIC was determined as \(\sim\)1/3 of the total value. Our laser operates on \(\pi\)-polarization at 785 nm. According to [26], the electronic component of RIC for this polarization is reduced to zero at \(\sim\)780 nm, so only the thermal component should be taken into consideration in our case. Supposing low dependence of the thermal component on wavelength and light polarization, in this estimation, we used the value calculated from [25]: RIC (thermal) = \(2\times 10^{-24}\) cm\({}^{3}\times\) N. The measured gain coefficient, K\({}_{0}\), in our Ti:Sapphire crystals was 3 cm\({}^{-1}\) at W\({}_{\text{PUMP}}\) = 1.5 J/cm\({}^{2}\). Using well-known relation: K\({}_{0}\) = \(\sigma\times\) N, where \(\sigma\) = 4.1\(\times\) 10\({}^{-19}\) cm\({}^{2}\)[31] is Ti:Sapphire emission cross section, for K\({}_{0}\) = 3 cm\({}^{-1}\), we obtained N = 7.3 \(\times\) 10\({}^{18}\) cm\({}^{-3}\), and RIC = 1.46 \(\times\) 10\({}^{-5}\). Then, we calculated [\(\delta\Psi\)]\({}_{\text{max}}\) = (2\(\pi\)/\(\lambda\)) \(\times\) RIC \(\times\) t = 0.35 rad at \(\lambda\) = 785 nm. Here, t = 3 mm is crystal thickness. It is possible to estimate the thermal component of the phase shift even more simply, calculating the average heating of the Ti:Sapphire crystal by single pump pulse. Supposing that the heating is caused only by Stokes losses, we obtained the energy dissipated into the heat, W\({}_{\text{HEAT}}\) \(\sim\) 0.32 W\({}_{\text{PUMP}}\). Then, supposing 100% absorption of pump energy in the crystal and using data for pure sapphire crystals from Crystran Ltd. (www.crystran.co.uk, accessed on 14 March 2016): density 4 g/cm\({}^{3}\), specific heat capacity 0.763 J \(\times\) g\({}^{-1}\times\) K\({}^{-1}\), dn/dT = 13 \(\times\) 10\({}^{-6}\), for 3 mm thick crystal at W\({}_{\text{PUMP}}\) = 1.5 J/cm\({}^{2}\), we obtained [\(\delta\Psi\)]\({}_{\text{AVE}}\) = 0.16 rad. The magnitude [\(\delta\Psi\)]\({}_{\text{MAX}}\) should be 1.3-1.5 times larger: [\(\delta\Psi\)]\({}_{\text{MAX}}\) \(\sim\) 0.2-0.24 rad, depending on fluence modulation depth. To estimate the scattered part of laser radiation in the case of nonuniformities, created by MLA, we supposed the sinusoidal character of phase modulation. Then, we calculated total diffraction losses into both \(\pm\)1 st orders of phase grating with a modulation depth [\(\delta\Psi\)/2]\({}_{\text{max}}\) = 0.175 rad, using the relation [32]: \(\eta\) = [J\({}_{1}\) (\(\delta\Psi\))]\({}^{2}\), where \(\eta\) is the efficiency of diffraction into the 1 st or -1 st order, and J\({}_{1}\) is the first-order Bessel function. We obtained 2 \(\eta\) = 1.6% per single pass. It is notable that we believe the structure of w(x, y) is unchangeable within the Ti:Sapphire crystal. In practice, one can observe slow structure variationswhen the distance from the condenser lens to the crystal changes. We can neglect these variations (at least for MLA and DD with parameters that produce a pattern with spatial frequencies \(\sim\)25 mm\({}^{-1}\)) in calculations, because the Ti:Sapphire crystal thickness is much lower than the condenser lens focal length of 60 mm. For pseudo-random W\({}_{\text{PUMP}}\) distribution, like in the case of DD, we can calculate the relative value of scattered radiation, supposing that \(\delta\)V(x, y) is linearly proportional to w(x, y), and calculate the standard deviation StD (w) directly from experimental beam profiles, using beam profiler software. For our DD pump pattern, we obtained StD(w) \(\sim\)0.29. Then, we calculated the relative value of scattered radiation: \(\{\text{StD}\,[\delta\text{V}(x,y)]\}^{2}=\{\lvert\delta\text{V}(x,y)\rvert_{ \text{max}}\times\text{StD}\,(W)\rvert^{2}=2.3\%\). Expressing this value in standard form, we obtained an integral scattering coefficient \(\alpha=0.077\) cm\({}^{-1}\), related to a gain coefficient K\({}_{0}=3\) cm\({}^{-1}\).