Optimization of femtosecond laser spot coverage: analytical modeling of spatial frequencies, anisotropy, and saturation boundaries

Keratorefractive lenticule extraction (KLEx) relies on dense, spatially uniform femtosecond laser photodisruption to create intrastromal cleavage planes that permit controlled mechanical separation of corneal tissue. While scanning trajectories vary between discrete meander scans and continuous Archimedean spirals, the geometric limits governing coverage efficiency remain unified under local planarity. In this work, we present a mathematical framework describing the asymptotic fill fraction achieved by circular laser spots. By modeling the fundamental 3-bubble dynamics via Boolean triangle tessellation, we derive exact closed-form equations for aligned and staggered spot arrangements. We determine the dependence of the row-to-row phase shift (δ) in the global average fill fraction. We establish analytical boundaries for full coverage (saturation) based on circumradius limits and introduce an invertible master equation to directly calculate required pulse densities for any target fill fraction. This comprehensive model confirms that hexagonal packing is the unique asymptotic optimum and geometrically supports physiological advantages of high-anisotropy scanning strategies.