## 2 Review of Cryogenic-Cooling Benefits
### Spectroscopic, Lasing, Linear and Nonlinear Optical Parameters
#### 2.7.2 Linear and Nonlinear Optical Parameters
In general the threshold for the formation of small-scale-self-focusing is significantly lower than that for whole-beam self-focusing. Small-scale self-focusing is caused by either phase or amplitude perturbations of a laser beam, due to dust particles, index inhomogeneities or inclusions. Such perturbations lead to localized self-focusing, resulting ultimately in self-trapped filaments that leave permanent \"angel hair\" tracks in the laser media. Another source of self-focused filaments is diffraction from hard apertures in the laser system, principally from the edges of laser amplifiers. If uncontrolled, small-scale-self-focusing can lead to a large fraction of the system output beam being lost to large-angle scattering in a halo surrounding the main laser beam. While we will not review in detail the methods for suppressing small-scale-self-focusing in this article, we mention that the imaging of apodized laser beams, the use of vacuum spatial filters, and relay-imaging throughout the entire laser system have been found to be effective tools [37]. Today, the use of the chirped-pulse-amplification (CPA) technique, invented at The Laboratory For Laser Energetics at the University of Rochester by Strickland and Mourou [66], in which ultrafast pulses are spectrally stretched to a much longer pulsewidth for amplification, and then re-compressed at the system output, can be used to largely avoid nonlinear effects as well as take advantage of a higher damage threshold obtained as pulsewidth increases. Calculating in advance the susceptibility of an ultrafast system to nonlinear and damage effects is an important part of ultrafast laser system design, and here we discuss the standard method for doing so. To manage whole beam and small-scale self-focusing in high peak power laser ultrafast systems, a knowledge of the nonlinear index n\({}_{2}\) and the corresponding nonlinear coefficient \(\gamma_{\text{N}}\) is essential. A standard and often-used measure of the severity of nonlinear effects in laser systems is the B-integral [37; 64], formally defined as  
\[B\left(r,z,t\right)=\Delta\varphi_{NL}\left(r,z,t\right)=\frac{2\pi}{\lambda} \cdot\int\gamma_{N}I\left(r,z,t\right)dz \tag{77}\]  
where \(B\) is in general a function of the beam radius \(r\), propagation location \(z\), and time \(t\), \(I\)(\(r\),\(z\),\(t\)) is the radially, longitudinally, and temporally varying laser intensity, the wavelength \(\lambda\), and \(z\) is the propagation length parameter. The B-integral is the nonlinear phase accumulation \(\Delta\varphi_{NL}\left(r,z,t\right)\) due to the nonlinear index \(n_{2}\), and in laser systems must be calculated for all optical and gain elements encountered by the propagating beam. It is customary to calculate the maximum value of the B-integral where the intensity \(I\) is maximum (\(t=0\)). For spatially Gaussian beams, the B-integral value at the beam edge is typically very small, and only the on-axis value for \(r=0\) is normally specified. The same holds true for higher-order super-Gaussian beams where the intensity at the beam edge becomes insignificant. For any arbitrary spatial beam profile, is easy to use Equation (72) in a computer code and calculate the B-integral profile at any location in a laser amplifier or system. The nonlinear index can be calculated using the relationship [37; 64]  
\[n_{2}=\frac{12\pi^{2}}{n_{0}^{2}c}\cdot\chi^{(3)} \tag{78}\]Here, \(\chi^{(3)}\) is the third-order susceptibility, and both \(n_{2}\) and \(\chi^{(3)}\) are expressed in electrostatic units (esu). The relationship between \(n_{2}\) and \(\gamma_{N}\) is given by the simple formula [37; 67]  
\[\gamma_{N}=\left(\frac{40\pi}{c}\cdot 10^{-6}\right)\cdot\left(\frac{n_{2}}{n_{0} }\right)=\left(4.19\;10^{-3}\right)\cdot\left(\frac{n_{2}}{n_{0}}\right) \tag{79}\]  
and \(\gamma_{N}\) is expressed in (cm\({}^{2}\)/W). We note that in some contributions to the laser literature, \(n_{2}\) is often used to describe \(\gamma_{N}\), a confusing practice since they have distinctly different values and units.