## 1 Introduction and History of Cryogenic Lasers
### The Modern Era  
The genesis of our cryogenic laser program was the publication of the Schulz and Henion paper in 1991 [11]. Starting in 1995, we investigated whether or not the favorable laser and thermal effects exhibited by Ti:Al\({}_{2}\)O\({}_{3}\) might not be operative in other laser crystals as well, and in particular for the well-developed laser material Y\({}_{3}\)Al\({}_{5}\)O\({}_{12}\). After a thorough literature search, we found enough high quality previously published thermal conductivity and thermal expansion coefficient data to begin to at first examine thermal and stress induced changes in undoped YAG, with a view towards understanding the detailed ramifications of lowering the temperature from room temperature to 77 K. That work led to the award and completion of a Phase I Small Business Innovation Research (SBIR) contract in 1997, with the Final Report published in 1998 [19]. Work completed prior to and during that contract led to three publications [20; 21; 22] during 1997 and 1998 in which the average power scaling behavior and the reduction of thermally-induced aberrations were thoroughly explored and quantified for the first time using a unified theoretical framework. In [20], we were the first group to use a finite element thermal and stress analysis code with continuously variable materials parameters (thermal conductivity, thermal expansion coefficient, and the elastic parameters Young's modulus and Poisson's ratio, as a function of temperature), to simulate the expected performance of solid-state YAG amplifiers at 77 K. The continuously variable thermal and elastic parameters were generated by good fits to the data existing at the time. Applying the theory to Nd:YAG and Yb:YAG lasers, we showed that the previous results obtained for Ti:Al\({}_{2}\)O\({}_{3}\) lasers [11] could indeed be emulated using other oxide laser hosts such as YAG, and that very favorable, but non-linear, scaling to very high average powers could be obtained by operating YAG at 77 K. We also showed for the first time that when a continuously varying thermal conductivity is employed in the finite-element simulations, non-quadratic temperature profiles are obtained. In addition to exploring the scaling properties of cryogenically-cooled YAG lasers, in [20] we also explored how thermally-induced aberrations vary with temperature, and showed that without taking the thermo-optic effect into account, approximately a factor of 10 improvement could take place by operating at 77 K, due to the deduction of the thermal profile by an increased thermal conductivity, and by the near elimination of stress-optic effects. We also lamented the apparent lack of thermo-optic (\(\beta=\mathrm{d}n/\mathrm{d}T\)) data for YAG and other optical materials as temperature is lowered from 300 to 77 K, but pointed out that the expected trend of \(\beta\) towards zero as temperature is lowered would further increase the reduction in thermo-optic effects in cryogenic lasers. The basic principles of low thermo-optic effects in the host YAG were correctly predicted in this article, confirming our previous conjecture that YAG and other oxide laser host crystals would behave very much like the oxide crystal Ti:Al\({}_{2}\)O\({}_{3}\) demonstrated experimentally previously [11]. The publication of the first accurate YAG thermo-optic coefficients by Fan and Daneu in 1998 [23] enabled us to further extend our results to include \(\beta\), and in the following papers in 1998 [21; 22], we presented the first comprehensive theoretical description of the expected reduced thermal distortion at 77 K in YAG for rod amplifiers, and nonlinear thermal and stress effects in YAG slabs, as well as the scaling behavior. The results presented in [20; 21; 22] assumed an undoped host YAG thermal conductivity variation with temperature, and did not include a description of the reduction in thermal conductivity in certain materials as the lasing ion dopant density increases. This effect will be further explored later in this article.