The authors report here a significant progress in side-pumped solar laser slope efficiency by pumping a 3.0 mm diameter, 30 mm length Nd:YAG single-crystal laser rod through a heliostat-parabolic mirror solar energy collection and concentration system with 0.9 m(2) effective collection area. A large fused silica aspheric lens allowed an efficient focusing of the concentrated solar power from the focal zone of the parabolic mirror into the laser rod mounted within a novel two-dimensional semi-cylindrical pump cavity. 15.3 W continuous-wave 1064 nm solar laser power was measured, resulting in 5.40% slope efficiency and 2.43% solar-to-laser conversion efficency, being 2.08 and 2.30 times, respectively, more than the previous records by side-pumping configuration. The 5.40% slope efficiency is also 1.08 times more than the previous record with Nd:YAG single-crystal rod by end-side-pumping approach. By adopting an asymmetric laser resonator, this novel side-pumping scheme also enabled an efficient production of high-quality solar laser beams in either TEM11 (4.0 W), TEM01* (doughnut-shaped, 2.9 W), TEM10 (3.2 W) or TEM00 (2.8 W) mode profiles.