## 2 Materials and Methods  
### Experiment  
Figure 1 shows a schematic diagram of the Ti:sapphire laser. We employed a simple x-fold laser cavity, which is commonly used for ultrashort pulse laser oscillators. The laser cavity comprises two concave mirrors \(M_{1}\) and \(M_{2}\) (radius of curvature (\(R\)) = 100 mm) with dichroic coating, a high reflectance plane end mirror \(M_{3}\) and a plane output coupler \(M_{4}\). These mirrors were purchased from Layertec and are shown in the catalog. We used a 5-mm-long, Brewster-cut Ti:sapphire crystal (GT ADVANCED TECHNOLOGIES [16]) with a Ti\({}^{3+}\)-ion doping concentration of 0.25 wt.% and a FOM of 200 (measured by the manufacturer). The Ti:sapphire crystal was mounted on a copper heat sink and cooled at a room temperature of approximately 20 degrees Celsius.  
An intracavity-frequency-doubled Nd:YVO\({}_{4}\) laser (Spectra-Physics Millennia Pro) was used as the SHG laser pump source for high-intensity pumping. Since the two curvature mirrors are dichroic coated, it is considered that the pump beam passes through the gain medium almost only once. The pump absorption efficiency of the Ti:sapphire crystal at the pump wavelength of 532 nm \(\eta_{a}\) was measured to be 90.5% for single-pass pumping. If the pump beam is passed twice through the gain medium, the pump absorption efficiency \(\eta_{a}\) can be increased up to 99.1%. The double-pass pumping can be achieved by reflecting and focusing the pump beam transmitted from the concave mirror \(M_{1}\) back into the gain medium or by using a highly reflective concave mirror \(M_{1}\) and end mirror \(M_{4}\) for the laser and pump beams. This increase in the pump absorption efficiency \(\eta_{a}\) is effective for further increasing the laser efficiencies.  
The pump beam was focused at the center of the crystal. The beam qualities \(M^{2}\) of the pump beam were measured to be 1.0 for the vertical and horizontal axes. The spot radii were measured to be 10.1 \(\upmu\)m and 18.3 \(\upmu\)m for the vertical and horizontal axes, respectively. The maximum pump power in the experiment was 5.0 W, and the corresponding maximum pump intensity was approximately 860 kW/cm\({}^{2}\).  
The reasons for this high pump intensity are as follows: First, increasing the laser efficiencies by high-intensity pumping is particularly useful for lasers with gain media such as Ti:sapphire crystals, which have high losses and a small product of emission cross-section and fluorescence lifetime. With high-intensity pumping, we have already achieved the highest optical-to-optical conversion and slope efficiencies in continuous-wave lasers [27; 28] and pulsed lasers in the nanosecond range [29; 30; 31] using Yb:YAG crystals at room temperature. These efficiencies obtained by high-intensity pumping are close to the quantum limit of the gain medium and are higher than the efficiencies of lasers using other methods that are considered advantageous for higher efficiency, lasers with the gain medium cooled to cryogenic temperatures, thin-disk lasers and fiber lasers.  
The maximum pump intensity of those Yb:YAG laser experiments was above 100 kW/cm\({}^{2}\). For Ti:sapphire crystals, the product of emission cross-section and fluorescence lifetime is one order of magnitude smaller than that for Yb:YAG crystals. Therefore, in order to obtain a gain of the same order of magnitude as that of the Yb:YAG crystal in the Ti:sapphire crystal, the pump intensity was set to a value one order of magnitude higher than that of the Yb:YAG crystals. In order to lower the pump threshold by high-intensity pumping, the mode volume of the pump beam in the crystal should be minimized. The appropriate pump spot radii in the crystal are 12.2 \(\upmu\)m for both the vertical and the horizontal axes, which deviate from the experimental values. Therefore, if the pump spot radii are adjusted to the optimum values, the pump threshold should be much lower than that in this experiment.