## 2 Review of Cryogenic-Cooling Benefits
### Spectroscopic, Lasing, Linear and Nonlinear Optical Parameters
#### 2.7.2 Linear and Nonlinear Optical Parameters  
The use of cryogenic laser technology to enable the development of advanced laser systems that have both high peak and high average with excellent beam-quality is an approach that has now been proven. So far in this paper we have outlined the many reasons to seriously consider the cryogenic-cooling approach, and have reviewed many of the relevant materials parameters and their dependence on temperature. The path to high-average-power lasers is now well-mapped, however, with regard to ultrafast operation we have just begun to develop the technology because many of the relevant ultrafast parameters for low temperature operation are not known and the database of such properties is at present very limited. The design of ultrafast laser systems requires a detailed knowledge of both the nonlinear optical properties of laser crystals as well as other optical materials used in constructing such systems, and in addition demands a detailed knowledge of the dispersion properties as well. For ultrafast operation, a detailed knowledge of the linear and nonlinear indices of refraction are needed to accomplish competent designs. It is fortunate that for most of the laser materials discussed here, good Sellmeier equations are available in the literature, and are discussed later in this paper, but readers should appreciate that at the present time most reported measurements of the index of refraction and available Sellmeier equations are at or near room temperature. Sellmeier (or Cauchy or Laurent equations) are used to calculate the first, second, third, and fourth-order dispersion parameters needed to achieve dispersion compensation and balancing in ultrafast laser systems, and unfortunately relatively few measurements can be found at cryogenic temperatures. Similarly, the values of the nonlinear index of refraction \(n_{2}\) for various crystals and optical materials are well-known only at room temperature. For ultrafast laser systems with pulsewidths down to the picosecond regime, nonlinear effects such as self-phase-modulation and self-focusing are the principal concerns, and dispersion effects can be mostly ignored. Below about 5-10 ps, however both nonlinear and dispersion effects must be taken into account to enable good ultrafast designs. Here we first consider just nonlinear effects, and the limitations they place on ultrafast system performance. Nonlinear Effects in Ultrafast Lasers  
Other than laser-induced optical damage, a subject that we shall discuss in Section 2.7.3, the major performance-limiting factor affecting ultrafast performance in solid-state laser systems is the existence of a finite nonlinear index of refraction \(n_{2}\). Much of the early work on inertial-confinement fusion involved the use of flashlamp-pumped laser glasses doped with Nd, driven by picosecond duration Nd:YLF mode-locked oscillators with typical FWHM pulse durations in the range 50-200 ps. When scaling such systems to ever larger energy/pulse and peak powers, it was discovered that whole-beam and small-scale self-focusing, caused by the nonlinear index, were the principal limiting factors, and managing and reducing such effects was a major design goal. Good summaries of the understanding and design tools used to design picosecond high-peak-power laser systems may be found in [37; 64]; in addition a very good summary of the most important papers addressing the phenomena of self-focusing and the design and performance of such systems may be found in [65]. It should be pointed out that in those early systems dispersion management was not a significant design issue and CPA was just in its infancy. For a laser pule traveling through a nonlinear medium, a finite nonlinear index leads to the temporal phenomena of self-phase-modulation, resulting in a temporally dependent phase modulation. Spatial phase modulation can lead to whole-beam and small-scale self-focusing [37; 64].