Exact ray-tracing algorithm for simultaneous defocus and tilt detection on planar and curved- surfaces in laser machining

Abstract

In laser machining, precisely maintaining the focal point on the workpiece surface is essential for high-quality processing but becomes challenging when the surface is tilted or curved. This paper presents a method for laser focal position detection based on exact ray tracing through a complete multi-surface optical system, using the full vector form of Snell’s law without paraxial approximation. The optical system incorporates a dual-aperture mask that splits the laser beam into two collimated sub-beams; the reflected spot coordinates on an image sensor are used to simultaneously determine the defocus displacement δ and the surface tilt angles ζ and η. The ray-tracing formulation is general to rotationally symmetric conic surfaces of arbitrary eccentricity (spherical, parabolic, hyperbolic); it is demonstrated numerically here for a tilted flat plane and for a spherical workpiece (radius 50 mm). Simulations over δ∈[−2,+2] mm, ζ,η∈[−3∘,+3∘] give a maximum absolute error of 0.039 mm on the flat plane ( 0.025 mm on the sphere) when spot centroids are extracted by image processing, falling to 0.007 mm when direct spot coordinates are used, identifying centroid extraction as the dominant practical error source. Axial defocus retrieval was additionally validated experimentally over 81 positions spanning ±2 mm on a bench-top prototype, yielding a linear correspondence between applied and reconstructed displacement ( R2=0.9999) after a two-parameter calibration, with a residual standard deviation of ∼6.08 μm. The method eliminates the need for prior surface-profile scanning and provides a computationally efficient basis for adaptive focal control in industrial laser systems, pending calibrated experimental validation of the tilt and curved-surface responses.