# Saturation, Allowed Transitions and Quantum Interference in Laser Cooling of Solids
## 1 Introduction  
Optical refrigeration was first achieved using laser-induced anti-Stokes fluorescent (ASF) emission in Yb\({}^{3+}\)-ZBLANP glass in 1995 [1]. Since then, advances in experimental methods and the purification of materials have led to the optical cooling of numerous crystals [2], the reduction of attainable temperatures to the cryogenic range [3], the first optical cryocooler [4], and self-cooling or radiation-balanced lasers (RBLs) [5; 6; 7]. However, unintended impurities incorporated during sample preparation are still a key limitation to applications of ASF cooling [8]. Saturation of the absorption and reduction in the associated heating caused by background impurities can overcome this limitation under certain conditions. Hence, it is important to understand and characterize such nonlinear effects. In this study, the enhancement of laser cooling due to differential absorption saturation is analyzed and observed by two experimental methods. We also note that the cooling rate is limited by the excited state emission rate of coolant ions as the excited state lifetime determines the number of absorption/emission cycles each ion can undergo per unit time during ASF cooling. Electric-dipole-allowed (ED) transitions have very short upper state lifetimes; so, they can be cycled more rapidly for faster cooling. However, allowed transitions suffer from configuration relaxation and often mediate excited state absorption, both of which contribute to the internal heating of the sample. Consequently, while laser cooling on such transitions has been investigated previously [9], net cooling by this approach has not been reported until now. Finally, quantum interference has not been investigated for possible applications in laser cooling. In this paper, Fano resonance and induced transparency are analyzed for their potential to mediate self-cooled lasing without inversion in a 3-level system with two allowed transitions, modeled after trivalent Cerium.  
Several advances are discussed here that enable cooling to lower minimum temperatures or more rapid cooling than has been possible to date with optical refrigeration based on anti-Stokes fluorescence (ASF). The first topic to be analyzed is optical saturation, relevant to attaining lower minimum temperatures by ASF cooling. As the laser cooling of solids is the only known method for refrigeration in space below the thermo-electric limit, any improvements in its capabilities will be important for future technology. Increased cooling efficiency through saturation is also demonstrated experimentally in Yb:LiYF\({}_{3}\) near room temperature. Optical refrigeration of Ti:Al\({}_{2}\)O\({}_{3}\) on the allowed \({}^{2}\)E-\({}^{2}\)T\({}_{2}\) transition is realized for the first time in a crystal with an exceptionally high figure of merit. Thisresult demonstrates that cooling on electric-dipole transitions is possible in a bulk solid and should enable more rapid cooling than can be achieved with ASF cooling on forbidden transitions of rare-earth ions. This advance should facilitate the development of imaging arrays with an improved signal-to-noise performance at cryogenic temperatures for sensing applications in outer space. In addition, it should encourage the development of new methods of cooling solids, such as Raman cooling. Analysis of the potential role of quantum interference in self-cooled lasers is also presented. Specific conditions are identified to enable self-cooled lasing without inversion in a Ce:LiCaAlF\({}_{6}\) laser.