## 2 Review of Cryogenic-Cooling Benefits
### Spectroscopic, Lasing, Linear and Nonlinear Optical Parameters
#### 2.7.2 Linear and Nonlinear Optical Parameters
A much larger nonlinear index for many commonly mode-locked materials like YVO4, KGW, and KYW makes Kerr lens mode locking easier to achieve, but will provide challenges to suppress nonlinear effects in scaled up ultrafast systems consisting of multiple amplifiers. Minimizing B-integral contributions in ultrafast systems utilizing some of these newer materials will likely force designers to consider thin lasing elements such as thin-disk lasers for some applications, however, the thinner the lasing element the more troublesome the suppression of parasitics and amplified-spontaneous-emission, so undoubtedly tradeoffs will need to be made. Also, thin lasing elements have low gain as well, and require multi-passing of the element for good extraction, thus increasing the B-integral and negating any advantage. The system B-integral, \(B_{\text{sys}}\), obtained by adding up the nonlinear phase contributions from all the laser and optical elements encountered by a beam, is a useful measure of the susceptibility of a system to small-scale-self-focusing damage. In general, \(B_{\text{sys}}\) values of \(<\)1.5 radians are very safe, whereas values greater than about 2.5 are very concerning. While these observations are only generally true, minimizing the possibility of system damage due to nonlinear effects is very dependent on system cleanliness and the avoidance of diffraction effects. In very short-pulse ultrafast CPA systems such as Ti:Al2O3, the generation of pulsewidths \(<\)20-30 fs requires that the accumulation of second, third, and fourth order dispersion phase in the amplifier portion of the system be compensated, a feat normally accomplished by the use of compressors that can be adjusted to cancel out second, third, and fourth order phase. In femtosecond laser systems the accumulation of nonlinear phase due the B-integral is also an important consideration, and Equation (77) is re-written  
\[B\left(r,z,t\right)=\Delta\varphi_{NL}\left(r,z,t\right)=2\pi\cdot\int\left( \frac{1}{\lambda}\right)\gamma_{N}I\left(r,z,t\right)dz \tag{80}\]  
reflecting the fact that temporal nonlinear phase Fourier maps into spectral phase. Because of the resulting nonlinear group delay, reducing the B-integral value to \(<\)1 has advantages in that the compensation of small amounts of nonlinear group delay in the compressor can be achieved. Dispersion Effects in Ultrafast Lasers  
The understanding of dispersion is a very important aspect of ultrafast laser technology, particularly for pulsewidths \(<\)about 10 ps, and into the fs and as regimes, and the successful management of dispersion is prerequisite to the success of ultrafast laser systems. Here, we review the description of dispersion and apply the formalism to the calculation of the first, second, third, and \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Crystal** & **Crystal** & **E Field** & **Wavelength** & **\(n_{0}\)** & **\(n_{2}\) (esu)** & **\(\gamma_{N}\) (cm\({}^{2}\)/W)** & **Reference** \\
**Type** & **Crystal** & **Orientation** & **(nm)** & **\(n_{0}\)** & **10\({}^{-13}\)** & **10\({}^{-16}\)** & **Reference** \\ \hline SC & YVO4 & Unoriented & & & & & & \\  & Measurement & & & & & & \\ SC & GdVO4 & Unoriented & & & & & \\  & Measurement & & & & & & \\ SC & KGW & E \(\mid\) \(\mid\) Nm & 800–1600 & 1.99 & 9.50 & 20.00 & [77] \\ SC & KGW & E \(\mid\) \(\mid\) Np & 800–1600 & 2.03 & 7.27 & 15.00 & [77] \\ SC & KYW & E \(\mid\) \(\mid\) Nm & 1080 & 2.01 & 4.80 & 10.00 & [75] \\ Gas & Air & I & 400 & 1 & 0.00128 & 0.00536 & [78] \\ Gas & Air & I & 800 & 1 & 0.00072 & 0.00301 & [78] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._fourth-order dispersion for a number of important laser materials. We first consider the variation of the linear index of refraction with wavelength or frequency.