## 2 Review of Cryogenic-Cooling Benefits
### Spectroscopic, Lasing, Linear and Nonlinear Optical Parameters
#### 2.7.2 Linear and Nonlinear Optical Parameters
For the calculations reported below in this paper, the appropriate index equation was converted to angular frequency, resulting in the following frequency dependent Sellmeier equation:  
\[E_{out}\left(t\right)=\left(\frac{1}{2\pi}\right)\cdot\int A_{in}\left(\omega \right)e^{i\omega t}e^{i\psi\left(\omega\right)}d\omega \tag{81}\]5  
Table 5 shows calculated values of the index of refraction, the wavelength the values were calculated at, and the first, second, third, and fourth order dispersion coefficients. In addition, references are provided to allow interested readers to know what form of the index equation was used, as well as the values of the constants. We have included crystals and optical materials often used in ultrafast lasers, including Ti:Al\({}_{2}\)O\({}_{3}\), Cr:LiSAF, Cr:LiCAF, and a number of Yb laser materials. Some entries in Table 5 have also served as a check on the calculational accuracy. For example, for BK-7 at 800 nm and fused silica (SiO\({}_{2}\)) at 800 nm, the calculated dispersion parameter values agree closely with previously published results [32]. For Ti:Al\({}_{2}\)O\({}_{3}\), the dispersion parameters were calculated for a Sellmeier equation used for sapphire grown using the heat exchanger method (HEM) [79], and are similar to previously published results [32], but a one-to-one comparison cannot be made because the data and Sellmeier equations used in [32] were not revealed. The variation of the index of refraction with wavelength can also be slightly different depending upon the crystal growth method, and the corresponding Sellmeier equations will be somewhat different as well, and it is likely that such differences explain the variation between our calculations and [32]. We have included in Table 5 not only legacy materials like LiSAF, LiCAF, YAG, YALO, and YLF, but also the newer sesquioxide laser materials Y\({}_{2}\)O\({}_{3}\), Sc\({}_{2}\)O\({}_{3}\), and Lu\({}_{2}\)O\({}_{3}\), and the tungstate materials KGW, KYW, and KYbW. One case deserves mentioning here: that of Cr:LiSAF. We found published values for the second (21.0 fs\({}^{2}\)/mm) and third-order (185.0 fs\({}^{3}\)/mm) dispersion coefficients for the e-axis [70] that do not agree well with our results, nor with those found elsewhere [80]. Values in [70] were calculated using the Sellmeier equations of [81]. We calculated \(\beta_{2}\) and \(\beta_{3}\) using both the Sellmeier equations of Peterson [81] and Payne [82]. While both Sellmeier equations differ substantially from the measured values, the data of Peterson [81] gave the worst agreement, so in Table 5 we show our calculated values (7.9 fs\({}^{2}\)/mm) and (15.7 fs\({}^{3}\)/mm) obtained using the Payne Sellmeier equation. It has been noted however [80], that even the Payne Sellmeier equations give results considerably different from the measured results. In addition, our results differ from the calculated results of [80], where the reported calculated values using the same Sellmeier equation are (9.7 fs\({}^{2}\)/mm) and (28.6 fs\({}^{3}\)/mm). While we have endeavored to calculate accurate values for the dispersion parameters reported in Table 5, and have checked and compared our results to other values found in the literature where available, readers should be aware that coding long and tedious Sellmeier, Cauchy, Laurent, and other equations and their derivatives up to the fourth order may lead to some inadvertent errors being present. We conclude this section of the paper by again pointing out that all of the Sellmeier equations and calculated values for the dispersion parameters are for room-temperature operation. It is now well-known that the value of the index of refraction at room temperature are diminished at lower (cryogenic) temperatures by the thermo-optic coefficient \(\mathrm{d}n/\mathrm{d}T\). Calculation shows that if the change in index of refraction for YAG for example by integrating the thermo-optic coefficient over the range 300 to 100 K, the index value at 300 K is affected in the third decimal place.