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Abstract
A novel chromatic confocal measurement (CCM) method using a hybrid diffractive- refractive lens is presented. This hybrid diffractive- refractive lens is designed to optimize the linearity of chromatic dispersion and minimize the size of the optical system. The hybrid diffractive- refractive lens is fabricated by etching a diffractive surface onto a quartz aspheric lens through lithography, which combines the high numerical aperture (NA) of a refractive lens with the unique dispersion properties of the diffractive optical elements (DOE). The lens is incorporated as a dispersive objective lens in a CCM experimental system. The system has a measurement range of 514.8 µm, calibrated using a laser displacement interferometer. The experimental results show that the wavelength-to-axial position coding of the CCM system achieves high linearity (R2= 0.9999) in the working wavelength range (500-700 nm). The system has an axial resolution of 0.08 µm and a displacement measurement nonlinear error of less than 2.05 µm.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The CCM method is a non-contact measurement technique. This method is widely used in the field of displacement measurement, thickness detection, and three-dimensional shape scanning with the characteristics of integrability, wide measurement range, and high axial resolution. It offers higher accuracy than laser triangulation displacement sensors, compared with the laser interferometer, it is an absolute measurement method, and has better integration, larger measurement angle [1–3]. The CCM system generates an axial chromatic dispersion of a broadband spectrum through dispersive lenses, thereby coding the axial position to the wavelength [4]. The most common method for dispersive lenses involves the use of multiple refractive lenses [5–8]. These lenses are made of materials with different Abbe numbers. The use of such materials helps control the measurement range and reduce the nonlinear dispersion. However, the use of multiple refractive lenses increases the size and weight of the probe, as well as the complexity of the optical fabrication and alignment.
Some researchers added DOE to refractive lenses to improve the optical performance of dispersive lenses owing to their unique focusing and imaging capabilities [9–12]. Researches indicate that the chromatic dispersion of a DOE is at least one order of magnitude greater than that of an equivalent refractive lens [13]. Such a strong dispersion of the DOE provides more sensitive wavelength-to-axial position coding [14]. Consequently, incorporating the DOE into the CCM optical system can increase the sensitivity and measurement range [15–17]. In early ideas, DOE was frequently used as a dispersive element integrated within groups of refractive lenses [18–21]. However, a DOE with a large numerical aperture (NA) is difficult to fabricate, which limits its resolving power, making it unsuitable for use as an objective lens for CCM systems [13]. Accordingly, the current DOE-based CCM systems still require a refractive lens as the objective. Besides, only part of the axial chromatic dispersion of the DOE in its working spectrum is linear [16].
To address the limitations of the current methods, a novel method based on a hybrid diffractive-refractive lens for the dispersive lens of a CCM system was proposed. This method effectively integrates the optical characteristics of both the refractive lens and DOE. The hybrid diffractive-refractive lens has excellent dispersion linearity, high NA and high axial resolution, small size, and low weight.
The following sections describe the principle of the hybrid diffractive-refractive lens, along with the design and fabrication procedures. The validation experiments are presented, followed by an analysis of the experimental result.
2. Hybrid diffractive-refractive lens
2.1. Principle
The basic idea of a hybrid diffractive-refractive lens is to use the focal power of the refractive lens to increase the NA and combine it with the diffractive surface to produce a linear dispersion. The dispersion properties of a lens are characterized by a function of its focal power versus wavelength. The hybrid diffractive-refractive lens comprises a refractive lens and diffractive surface. For the refractive lens, the function can be expressed as [14]:
where λ is the incident wavelength; Φref, n, and fref(λ) are the power, refractive index, and focal length of the lens at wavelength λ; C0 is a constant for the lens, which is related to the surface curvature. It’s readily seen that the dispersive properties of a refractive lens depend on the design and material characteristics.When the thickness of lens is reduced to a scale similar to the wavelength of the incident light, conventional ray-tracing methods prove insufficient for characterizing its performance. Instead, light must be treated as waves. The zone radius coordinates of the diffractive surface can be determined using the Fresnel zone plate equation, which involves solving the Fresnel diffraction integral [22].
The design wavelength of a diffractive surface is denoted as λ0, f0 represents the main focal length corresponding to the first diffraction order and λ0/2 signifies the optical path difference between adjacent bands. r denotes the radius of the m-th diffraction order. Since m0λ0≪ f0, Eq. (2) can be approximated as:
Generalize f0 to a wide spectral band. For the first diffraction order, the focal power of a diffractive surface follows the relation:
where Φdif(λ) and fdif(λ) are the focal power and the focal length, respectively. Equation (4) shows that the focal power of a diffractive surface is dependent on the incident wavelength and is not affected by material characteristics [15]. Another dispersion characteristic of a diffractive surface is typically described by its Abbe number [14]. Where λd = 0.5876 µm, λF = 0.4861 µm, λC = 0.6563 µm. The negative sign of the Abbe number reflects the opposite dispersion order compared to a refractive lens, and its low amplitude indicates the large dispersion characteristic of a diffractive surface. For a hybrid diffractive-refractive lens, the focal power Φhyb of the hybrid diffractive-refractive lens can be expressed as:By adjusting the focal power distribution of the diffractive surface and refractive lenses, any Abbe number can be simulated, achieving any dispersion characteristics, including linear dispersion [14], which is highly suitable for CCM systems.
2.2. Design and fabrication
The hybrid diffractive-refractive lens was optimized for dispersion and focusing using ZEMAX (Fig. 1). The main wavelength of the lens was 587.6 nm, with a diameter of 15 mm and a focal length of 6.1 mm. The working wavelength range is 500-700 nm, and the NA range is 0.57-0.615.
The diffractive surface was designed as a Fresnel zone plate on the second surface of the lens, consisting of a series of radially arranged circular bands with increased in radius and decreased in width. This surface imparts a phase shift ϕ to the incident ray, which can be described as:
The central portion of the designed diffractive surface is illustrated in Fig. 2.
To fabricate the hybrid diffractive-refractive lens, an aspherical surface was machined onto a fused silica substrate (Corning HPFS 7980 with a refractive index of 1.4584 at λ=587.6 nm). The fabrication sequence involved two cycles of a pin-coating process, UV exposure, and reactive ion etching. This sequence was repeated to create a precise four-step diffractive structure on the planar surface of the aspheric lens. The final structure comprised 546 concentric annuli, each representing a diffractive zone.The etching width of the outmost step is 1.55 µm, and the etching depth of each step is 320 nm, a dimension determined by the central wavelength of 587.6 nm. A microscopic view of the central area of the diffractive surface is shown in Fig. 3.
Fig. 3. Design profile and actual machining profile of the diffractive surface. Theoretical profile function of the designed diffractive surface and its approximation using 4 levels. (b)Diffractive surface, measured by a microscope.
Energy efficiency is an important attribute of the DOE. A DOE with a segmented, continuous phase distribution achieved 100% diffraction efficiency at the corresponding order of the design wavelength. However, the diffraction efficiency of a DOE with a step structure varies with the number of phase levels N. The diffraction efficiency of the first diffraction order is given by Eq. (10), indicating that a DOE with four phase levels exhibits an efficiency of 81%. For wide spectral incidence light, the diffraction efficiency decreases relative to the central wavelength. However, in the range of 500-700 nm, the decrease in broadband diffraction efficiency is less than 10%, which is acceptable for detection [23].
3. Experiment
A CCM system based on a hybrid diffractive-refractive lens was set up, as illustrated in Fig. 4.
The CCM system has a fiber-coupled LED spanning a wavelength range of 500-700 nm as its light source, a spectrometer with a spectral resolution of 0.025 nm for spectral detection, and a hybrid diffractive-refractive lens as the confocal optical module (Fig. 5). A three-port Y-type fiber (a multimode fiber with a core diameter of 100 µm) connects these components. A plane mirror was used as the measured sample, which was placed within the measurement range. A laser displacement interferometer was used to calibrate the accurate displacement of the mirror.
3.1. Calibration of the CCM system
A system calibration experiment is necessary to establish the correlation between the wavelength and axial position. Using a motorized positioning stage, the plane mirror was displaced at intervals of 25 µm increments along the axial direction. Data were collected five times after each displacement to record the displacement values detected by the laser interferometer and the spectral data from the spectrometer. 22 positions of the plane mirror and their corresponding spectral data were recorded.
The next step is to extract the peak wavelength from the spectral data. The comparative analysis was performed, utilizing polynomial fitting, Gaussian fitting, and centroid methods. The results of analysis are presented in Fig. 6, highlighting the repeatability of peak wavelength extraction across the 22 spectral datasets using the different methods. Among the methods tested, Gaussian fitting demonstrated the highest repeatability, with a maximum error of 0.05 nm.
Fig. 5. Assembled hybrid diffractive-refractive lens:1- mounting unit,2- hybrid diffractive-refractive lens.
Fig. 6. Repeatability of peak wavelength extraction by polynomial fitting, Gaussian fitting, and centroid method.
Wavelength-axial position calibration data were plotted based on the acquired displacement and peak wavelength data. To ensure the availability of a complete calibration curve for accurate measurements, 1000 spline interpolation data points were generated between each pair of adjacent calibration data. A linear fit was performed on the calibration curve to obtain a linear function between the wavelength and the axial position. As shown in Fig. 7, the dispersion range of the hybrid diffractive-refractive lens within the 500-700 nm wavelength range is 514.8 µm. The calibration curve showed excellent linearity throughout the measurement range, with a linear correlation coefficient (R2) of 0.9999.
The nonlinear error of the CCM system is evaluated, and the residual of the linear fitting function is shown in Fig. 8. The maximum nonlinear residual of the system is less than 2.05 µm, the RMSE is 1.264 µm, and the relative residual error is below 0.39% over the entire measurement range. The results indicate that the hybrid refractive-diffractive lens has good dispersion linearity, which leads to good measurement accuracy and resolution consistency in the CCM system.
3.2. Axial resolution test experiment
The CCM system showed a highly linear relationship between the response wavelength and measured axial position. The theoretical axial resolution function is as follows [24].
Where Δl is the measurement range of the system, σλ is the spectral resolution of the spectrometer, σd is the axial resolution of the CCM system, and Δλ is the wavelength range. According to Eq. (9), the theoretical axial resolution of the CCM system was 0.064 µm. To verify the real axial resolution of the CCM system, the motorized positioning stage in the experimental system was replaced with a piezo stage.The minimum detectable displacement of the plane mirror determined the axial resolution of the CCM system. The measurement method uses a piezo stage to perform incremental step movements; if a significant change in the peak wavelength is detected, the displacement step interval is further reduced until the peak wavelength is not clearly distinguished. Step movements were respectively implemented at an interval of 0.08 µm, 0.07 µm, 0.06 µm, and 0.05 µm while recording 50 response-peak wavelengths at each position using the Gaussian method. The peak response wavelength was collected after multiple averaging processes to increase the precision of the measurements. The results, shown in Fig. 9, demonstrate that axial resolution of 0.08 µm was achieved.
4. Conclusions
A CCM method based on a hybrid diffractive-refractive lens is proposed. The focal power distribution and dispersion characteristics of hybrid diffractive and refractive lenses are derived and analyzed. The refractive substrate and diffractive surface are designed, and a hybrid diffractive-refractive lens is fabricated. The hybrid diffractive-refractive lens has the characteristics of a high NA, small size, easy integration, and low cost. The CCM measurement system was developed. The experimental results show that it is feasible to use the hybrid diffractive-refractive lens as the dispersion element in the CCM system. The wavelength-axial position relationship encoded by the lens is highly linear, and the CCM system has high axial resolution and small probe size. Therefore, it has wide potential for precision measurement applications in geometric quantities. Improving the optical resolution of the CCM system by upgrading the lithography process could be a future direction for developing this method.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. J. Luo, H. D. Zhang, S. M. Hou, et al., “Confocal hyperspectral microscopic imager for the detection and classification of individual microalgae,” Opt. Express 29(23), 37281–37301 (2021). [CrossRef]
2. R. J. Zhou, D. R. Shen, P. Huang, et al., “Chromatic confocal sensor-based sub-aperture scanning and stitching for the measurement of microstructured optical surfaces,” Opt. Express 29(21), 33512–33526 (2021). [CrossRef]
3. B. Dong, Y. Zhang, W. Zhang, et al., “Highly sensitive, wide dynamic range displacement sensor combining chromatic confocal system and phase-sensitive spectral optical coherence tomography,” Opt. Express 25(5), 5426–5430 (2017). [CrossRef]
4. G. Molesini, G. Pedrini, P. Poggi, et al., “Focus-wavelength encoded optical profilometer,” Opt. Commun. 49(4), 229–233 (1984). [CrossRef]
5. J. Bai, X. H. Li, X. H. Wang, et al., “Chromatic Confocal Displacement Sensor with Optimized Dispersion Probe and Modified Centroid Peak Extraction Algorithm,” Sensors 19(16), 3592 (2019). [CrossRef]
6. M. Browne, O. Akinyemi, and A. Boyde, “Confocal surface profiling utilizing chromatic aberration,” Scanning 14(3), 145–153 (1992). [CrossRef]
7. Z. Zhang and R. Lu, “Initial structure of dispersion objective for chromatic confocal sensor based on doublet lens,” Opt. Lasers Eng. 139(4), 106424 (2021). [CrossRef]
8. K. B. Shi, P. Li, S. Z. Yin, et al., “Chromatic confocal microscopy using supercontinuum light,” Opt. Express 12(10), 2096–2101 (2004). [CrossRef]
9. C. Li, K. Li, J. Liu, et al., “Design of a confocal dispersion objective lens based on the GRIN lens,” Opt. Express 30(24), 44290–44299 (2022). [CrossRef]
10. J. Wu, F. Qin, and X. P. Li, “Atomically thin Fresnel zone plate with broadband focusing property,” Opto-Electron Eng 49(4), 220011 (2022). [CrossRef]
11. R. Akan, T. Frisk, F. Lundberg, et al., “Metal-assisted chemical etching and electroless deposition for fabrication of hard X-ray Pd/Si zone plates,” Micromachines 11(3), 301 (2020). [CrossRef]
12. Q. Liu, J. Zhao, J. Guo, et al., “Sub-5 nm lithography with single GeV heavy ions using inorganic resist,” Nano Lett. 21(6), 2390–2396 (2021). [CrossRef]
13. M. Rayer and D. Mansfield, “Chromatic confocal microscopy using staircase diffractive surface,” Appl. Opt. 53(23), 5123–5130 (2014). [CrossRef]
14. T. Stone and N. George, “Hybrid diffractive-refractive lenses and achromats,” Appl. Opt. 27(14), 2960–2971 (1988). [CrossRef]
15. G. Berkovic, E. Shafir, M. A. Golub, et al., “Multi-wavelength fiber-optic confocal position sensor with diffractive optics for enhanced measurement range,” Proc. SPIE 6619, 66190U–1572 (2007). [CrossRef]
16. S. L. Dobson, P. C. Sun, and Y. Fainman, “Diffractive lenses for chromatic confocal imaging,” Appl. Opt. 36(20), 4744–4748 (1997). [CrossRef]
17. P. C. Lin, P. C. Sun, L. Zhu, et al., “Single-shot depth-section imaging through chromatic slit-scan confocal microscopy,” Appl. Opt. 37(28), 6764–6770 (1998). [CrossRef]
18. J. Garzón, D. Duque, A. Alean, et al., “Diffractive elements performance in chromatic confocal microscopy,” J. Phys.: Conf. Ser. 274(1), 012069 (2011). [CrossRef]
19. A. K. Ruprecht, C. Pruss, H. J. Tiziani, et al., “Confocal micro-optical distance sensor: principle and design,” Proc. SPIE 5856, 128–135 (2005). [CrossRef]
20. M. Hillenbrand, B. Mitschunas, C. Wenzel, et al., “Hybrid hyperchromats for chromatic confocal sensor systems,” Adv. Opt. Technol. 1(3), 187–194 (2012). [CrossRef]
21. T. Liu, J. Wang, Q. Liu, et al., “Chromatic confocal measurement method using a phase Fresnel zone plate,” Opt. Express 30(2), 2390–2401 (2022). [CrossRef]
22. B. D. Guenther, Modern optics (Oxford University, 2015).
23. H. P. Herzig, Micro-optics: elements, systems and applications (Taylor and Francis, 1997).
24. M. C. Hutley and R. F. Stevens, “The use of a zone-plate monochromator as a displacement transducer,” J. Phys. E: Sci. Instrum. 21(11), 1037–1044 (1988). [CrossRef]
