Optical system, laser spot position determination process, and calibration process for deterministically shaping substrates using laser pulses

A method for determining laser pulse positions for shaping an optical element, the method comprising:

• determining, based at least on a computational model and a target substrate deformation, a stress field for at least a portion of a substrate, wherein the stress field comprises at least three components of stress, wherein the stress field comprises a plurality of stress states for the at least a portion of the substrate;
• determining, based on the stress field, one or more calibration constants;
• determining, based on at least on the one or more calibration constants, a spot density over the at least a portion of the substrate;
• determining, based on the spot density, a laser spot position allocation that arranges a number of laser pulses into a minimum number of lines that achieves the spot density;
• and
• causing, based on the laser spot position allocation, output of a machine program that coordinates a rotation stage, a translation stage, and a laser firing for surface shaping of the at least a portion of the substrate.