Optical surfaces with central convex-concave features, such as tower-shaped toroids with a central protrusion or depression and compound aspheres with inflection zones, are essential for compact light-field manipulation in laser beam shaping, energy redistribution, and aberration correction. However, their non-monotonic radial height profiles and abrupt geometric transitions at the center-to-periphery boundaries cause severe air-cutting redundancy in slow tool servo (STS) diamond turning. In conventional Archimedean spiral tool paths, the diamond tool is forced to traverse the entire radial range during every cutting layer, including extensive regions where the tool hovers above the workpiece surface without material engagement. These non-productive motions nevertheless demand fully synchronized C-, X-, and Z-axis movements, significantly degrading machining efficiency, with the redundancy escalating for larger apertures and stronger undulations.
To overcome this intrinsic limitation, the work aims to propose a feature-adaptive dynamic path truncation optimization method. Firstly, the surface was partitioned radially into effective cutting and air-cutting zones based on the characteristic height distribution around the central convex-concave boundary, constructing a layered tool path structure. A discrete-point search was then executed along the planned spiral under the instantaneous tool-workpiece contact criterion, precisely identifying the critical transition points where air-cutting begins and ends. Finally, the roughing and semi-finishing spiral paths were radially truncated: only the effective cutting segments were retained and reconnected via smooth, machine-dynamics-compliant transition arcs that maintained C-axis continuity, respected acceleration and jerk limits, and guaranteed interference-free clearance. Crucially, the final finishing pass remained an unmodified full-coverage Archimedean spiral, ensuring the machined surface faithfully replicated the design form.
Experimental validation was performed on a representative tower-shaped toric surface by a Nanoform X ultra-precision lathe with a round-nose single-crystal diamond tool. Spindle speed, feed rate, and depth of cut were held constant. Two complete machining sequences were executed: one with conventional full-spiral roughing and semi-finishing, the other with the proposed truncated strategy, both followed by an identical full-coverage finishing pass. Surface form was measured with a Taylor Hobson PGI Optics contact profilometer, and areal micro-topography was characterized with a Zygo NewView 8000 white light interferometer. Results show that the dynamic truncation strategy reduces total machining time by approximately 31%, directly due to the elimination of extensive air-cutting motions. The areal surface roughness Sa values are 12.008 nm for the conventional path and 11.464 nm for the truncated path, while average maximum peak-to-valley heights Sz are 0.216 µm and 0.139 µm, respectively. Form accuracy remains at the same sub-micrometer level without statistically significant difference. The slightly lower Sz obtained under the truncated path confirms that no additional vibration or surface damage is introduced, preserving surface integrity. This work provides a practical and deterministic path optimization strategy for high-efficiency STS machining of central convex-concave optical surfaces, readily extendable to broader complex free forms.
Key words
ultra-precision cutting /
slow tool servo turning /
optical free form surfaces /
central convex-concave features /
tool path optimization /
air-cut truncation /
machining efficiency
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
References
[1] ZHANG Y C, SUN H T, CHEN Y Q, et al.Flexible Intensity and Wavefront Shaping with Freeform Optics[J]. Optics Letters, 2025, 50(20): 6317-6320.
[2] JIMENEZ C, TOULOUSE A, HERKOMMER A.Design of Freeform Beam Shaping Micro-Optics for Single Photon Sources[J]. Optics Express, 2025, 33(11): 22805.
[3] SHI Y, WU R N, XING L, et al.Designing an Off-Axis Freeform Imaging System with a Wide Field of View Based on an Equal Optical Path Surface Extension Method[J]. Applied Optics, 2025, 64(10): 2424.
[4] JIANG X, SCOTT P, WHITEHOUSE D.Freeform Surface Characterisation-a Fresh Strategy[J]. CIRP Annals, 2007, 56(1): 553-556.
[5] 顾培夫, 丁毅. 实现均匀照明的自由曲面反射器[J]. 光学学报, 2007(3): 540-544.
GU P, DING Y.Freeform reflector for uniform illumination[J]. Acta China Sinica, 2007(3): 540-544.
[6] BRINKSMEIER E, GLÄBE R, SCHÖNEMANN L. Review on Diamond-Machining Processes for the Generation of Functional Surface Structures[J]. CIRP Journal of Manufacturing Science and Technology, 2012, 5(1): 1-7.
[7] 邱昕洋, 张彦钦, 王宏斌. 慢刀伺服加工环曲面的刀具路径规划方法[J]. 国防科技大学学报, 2020, 42(3): 121-127.
QIU X Y, ZHANG Y Q, WANG H B.Tool Path Planning Method for Machining Toric Surface Based on Slow Tool Servo Technology[J]. Journal of National University of Defense Technology, 2020, 42(3): 121-127.
[8] MUKAIDA M, YAN J W.Fabrication of Hexagonal Microlens Arrays on Single-Crystal Silicon Using the Tool-Servo Driven Segment Turning Method[J]. Micromachines, 2017, 8(11): 121-127.
[9] CHEN L J, LIANG Y Y, WANG K W, et al.Research on the Optical Field Distribution and Micro-Fabrication Model in Convex-Surface Laser Lithography[J]. Optics Communications, 2010, 283(12): 2631-2639.
[10] HAYASHI N, OGIMOTO J, MATSUI K, et al.A GaN- Based VCSEL with a Convex Structure for Optical Guiding[J]. Physica Status Solidi (a), 2018, 215(10): 1700648.
[11] FAN R D, WEI S L, JI H R, et al.Automated Design of Freeform Imaging Systems for Automotive Heads-up Display Applications[J]. Optics Express, 2023, 31(6): 10758.
[12] LUNA PALACIOS Y Y, HOANG T N A, KHANDANI S, et al. Nonlinear Optical Microscopy with an Obscuration- Free, Freeform Reflective Objective[J]. Biomedical Optics Express, 2025, 16(11): 4530.
[13] JIMENEZ C, HERKOMMER A.3D Printed Freeform Micro-Optics Meets Generated Jacobian Equations Based Inverse Designs[J]. Journal of the European Optical Society-Rapid Publications, 2026, 22(1): 3.
[14] 牛恒泰, 康敏, 何成奎, 等. 离散曲面慢刀伺服车削刀具路径规划[J]. 机械科学与技术, 2018, 37(5): 721-728.
NIU H T, KANG M, HE C K, et al.Cutting Path Plan of Discrete Surface for Slow Tool Servo Turning[J]. Mechanical Science and Technology for Aerospace Engineering, 2018, 37(5): 721-728.
[15] WANG X S, FU X Q, LI C L, et al.Tool Path Generation for Slow Tool Servo Turning of Complex Optical Surfaces[J]. The International Journal of Advanced Manufacturing Technology, 2015, 79(1): 437-448.
[16] NEO D W K, KUMAR A S, RAHMAN M. CAx-Technologies for Hybrid Fast Tool/Slow Slide Servo Diamond Turning of Freeform Surface[J]. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 2016, 230(8): 1465-1479.
[17] NAGAYAMA K, YAN J W.Deterministic Error Compensation for Slow Tool Servo-Driven Diamond Turning of Freeform Surface with Nanometric Form Accuracy[J]. Journal of Manufacturing Processes, 2021, 64: 45-57.
[18] 郭航言, 康敏, 周玮. 慢刀伺服车削刀具补偿算法优化[J]. 表面技术, 2022, 51(4): 308-316.
GUO H Y, KANG M, ZHOU W.Optimization of Tool Compensation Algorithm for Slow Tool Servo Turning[J]. Surface Technology, 2022, 51(4): 308-316.
[19] TANIKAWA S, YAN J W.Fabrication of Micro-Structured Surface with Controllable Randomness by Using FTS-Based Diamond Turning[J]. Precision Engineering, 2022, 73: 363-376.
[20] LIU C, XUE C X, ZHANG Q, et al.Optimization Method of Tool Path Generation Considering the Edge of Lenslets for a Microlens Array in FTS Diamond Turning[J]. Applied Optics, 2019, 58(24): 6713-6719.
[21] AO S J, GONG H, HUANG K T.A Smooth Tool Path Generation Method for Ultraprecision Turning Discontinuous Multi-Freeform-Surfaces with High Machining Efficiency[J]. Journal of Materials Processing Technology, 2021, 298: 117280.
[22] ZHANG X D, LI Z X, ZHANG G X.High Performance Ultra-Precision Turning of Large-Aspect-Ratio Rectangular Freeform Optics[J]. CIRP Annals, 2018, 67(1): 543-546.
[23] JI S J, YU H J, ZHAO J, et al.An Optimal Method of Tool Path Generation for Radial Sinusoidal Surface[J]. Advanced Materials Research, 2014, 1027: 20-23.
[24] GONG H, FANG F Z, HU X T.Accurate Spiral Tool Path Generation of Ultraprecision Three-Axis Turning for Non-Zero Rake Angle Using Symbolic Computation[J]. The International Journal of Advanced Manufacturing Technology, 2012, 58(9): 841-847.
Funding
National Natural Science Foundation of China (52305445, 22FAA01871, U24A20126) and Science and Technology Development Plan Project of Jilin Province (YDZJ202401338ZYTS)
PDF(17170 KB)
