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With the development of technology, ultrafast pulse lasers are increasingly used in many fields, such as material processing, imaging, and medical treatments. The precision of these applications often depends on the ability to focus the laser beam into a tight spot with a minimal divergence in a certain range along the optical axis. Therefore, accurate measurement of depth of focus (DOF) is crucial for optimizing the performance of ultrafast laser systems and ensuring the reliability of the results obtained in various experiments and applications. Traditional methods of measuring the DOF mainly rely on directly capturing the beam size, which is impractical in high-intensity environments of ultrafast pulse laser systems due to potential damage to sensors and limitations in measurement accuracy. Furthermore, using autocorrelation or moving sensors to measure DOF in ultrafast pulse lasers introduces complex optical paths, leading to measurement errors and making them unreliable in precise focusing applications. To solve the problem of the limitations of current DOF measurement techniques for ultrafast pulse laser, in this work we propose a novel method based on Z-scan technique. According to nonlinear optical theory, it is found that the transmittance curves obtained from open-aperture (OA) Z-scan measurements of samples exhibiting two-photon absorption (TPA) all follow a Lorentzian distribution. By fitting this curve by Lorentzian distribution, the DOF of ultrafast pulse lasers and the full widths at half maximum (FWHM) of the OA Z-scan curves can be determined rapidly. The transmittance curves of solid and liquid samples with TPA across different types of lenses and microscope objectives within ultrafast optical systems are measured. The results show that the FWHM of the OA Z-scan curves and the theoretical DOF values are well consistent. This method effectively relates the size of the DOF to the beam waist radius derived from the distribution of the Lorentzian function in the OA Z-scan experimental curves, eliminating the influence of other parameters on the measurement results. In conclusion, a novel method of measuring DOF in ultrafast pulse laser systems by using the OA Z-scan technique is proposed. It provides a rapid, accurate and reliable way for determining the DOF in ultrafast laser focusing systems, thereby precisely controlling the ultrafast laser beam for a wide range of applications. 
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Keywords:
- Z-scan /
- two-photon absorption /
- depth of focus /
- Lorentzian distribution
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图 1 测量DOF的开孔Z扫描实验装置
Figure 1. Open-aperture Z-scan setup for measuring DOF.
图 2 在(a)低光强和(b)高光强条件下, 开孔Z扫描计算公式前m项近似仿真的透射率分布曲线以及洛伦兹函数拟合; 取m = 3时, 理论模拟的不同(c)样品厚度、(d)非线性吸收系数下的透射率分布曲线. 插图为相应的归一化透射率分布
Figure 2. Under conditions of (a) low laser intensity and (b) high laser intensity, the simulated transmittance distribution curves for the first m terms of the open-aperture Z-scan calculation formula, and the Lorentzian function fits; simulated transmittance distributions for varying (c) sample thicknesses and (d) nonlinear absorption coefficients when m = 3. The insets show the corresponding normalized transmittance distributions.
图 3 (a)对光斑直径D进行测量的示意图; (b)光束沿光轴的光强分布模拟及DOF计算值; (c) 1 mm ZnSe的开孔Z扫描曲线及FWHM拟合值
Figure 3. (a) Schematic of measuring the diameter D of the light spot; (b) simulation of intensity distribution along the optical axis and calculated DOF; (c) open-aperture Z-scan curve for 1 mm ZnSe and measured FWHM.
图 4 (a)相同入射光强下, 不同厚度ZnSe的开孔Z扫描结果; (b)归一化至[0, 1]的结果
Figure 4. (a) Open-aperture Z-scan results for ZnSe with varying thicknesses at a constant incident light intensity; (b) results normalized to the range [0, 1].
图 5 (a)相同厚度ZnSe, 不同入射光强下的开孔Z扫描结果; (b)归一化到[0, 1]的结果
Figure 5. (a) Open-aperture Z-scan results for ZnSe with varying incident peak intensities at a constant thickness; (b) the results normalized to the range [0, 1].
图 6 (a)光束沿光轴的光强分布模拟及DOF计算结果; (b) 1 mm 比色皿中的荧光素染料的开孔Z扫描曲线及FWHM拟合值
Figure 6. (a) Simulation of intensity distribution along the optical axis and calculated DOF; (b) open-aperture Z-scan curve for fluorescein in a 1 mm cuvette and measured FWHM.
图 7 开孔Z扫描结果以及对应的洛伦兹拟合曲线 (a) 20倍消色差显微物镜(焦距10 mm); (b)平凸透镜(焦距50 mm); (c)消色差透镜(焦距50 mm); (d)平凸透镜(焦距100 mm)
Figure 7. Open-aperture Z-scan results and their Lorentzian fit results: (a) 20× apochromatic microscope objective lens (f = 10 mm); (b) planoconvex lens (f = 50 mm); (c) achromatic lens (f = 50 mm); (d) plano-convex lens (f = 100 mm).
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