Generally people use Plano-Convex Lens for focusing a High Power Q-Switched Laser Pulses onto the target. What are the advantages of Plano-Convex over Bi-Convex Lenses in this context?
Its due to the spherical abberation - its generally lower then in case of other usual lens shapes (with the same focal length).
Basic overview of this topic can be seen here (bottom figure on page 1.27): https://www.cvimellesgriot.com/Products/Documents/TechnicalGuide/Fundamental-Optics.pdf
As Jan Novotny mentioned above, it is related with sperical abberation. I use high power lasers(Nd:YAG, Nd:Glass and Ti:Sapphire) for laser ablation and depostion experiments. We face the curved surface of lens (planar convex) opposite to the direction of laser beam. The ultimate aim is focus the laser beam to a small area so that we get maximum intensity at focus. We measured spot size using different lenses, but we got minimum spot size for planar convex lens.( ie less abberation.). Minimum spot size is crutial only at hight intesity regime applications like ablation. but in nlo experiments like z scan, pump probe, four wave mixing etc people are still using bi convex lenses.
There is another kind of spherical singlet called "best form" lens. A best form lens has both surface convex but one side has shorter radius of curvature than the other. Among the three mentioned spherical singlets, best form lens has the minimum spherical aberration when it used to focus a collimated beam.
I just want to add a short comment about bi-convex lenses.
As Anoop K K said, to reduce spherical aberrations when focusing collimated beams, the best choice is to face the curved surface of the lens towards the laser.
Bi-convex lenses are generally designed to provide less aberrations when working with non-collimated beams e.g. when you want to work with 2f-2f configuration (imaging set-up). In that case, the beam is not collimated on both sides of your lens, and bi-convex lenses are the best choice.
in the case of a bi-Convex lens, the spot size of the focused beam is greater than the plano-convex lens. In addition, manufacturing a plano-convex lens is easier than bi-convex lenses. The polishing process and inspection process of a plane surface are easier than that of a spherical surface in terms of PV and scratch-dig parameters.