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Abstract
A measurement method based on a confocal probe on the second harmonic generation that can measure linear and angular displacements in the focusing point is proposed. In the proposed method, a pinhole or an optical fiber placed in front of the detector in conventional confocal probes is replaced by a nonlinear optical crystal, which is utilized as a medium generating second harmonic wave whose light intensity changes by the linear and angular displacements of a target under measurement. The feasibility of the proposed method is verified by theoretical calculations and experiments with the newly designed optical setup. Experimental results have demonstrated that the developed confocal probe has a resolution of 20 nm and 5 arc-seconds for measurement of linear and angular displacements, respectively.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The confocal principle [1,2] has widely been used in a probe for measurement of a displacement and/or the step height of an object [3–8]. The most important feature of the confocal probe is the optical sectioning effect [9,10]. This is the effect of a small aperture that can transmit only signals in the focal plane and exclude signals outside the focal plane, like a pinhole, resulting in higher measurement resolution in the optical axis direction. This effect can be applied to highly reflective objects as well as translucent objects, enabling observation of the three-dimensional shape inside the object to be measured [9–12]. In addition, confocal probes have the advantage that they are optical probes utilizing converging light, allowing high-resolution measurements with in-plane resolution comparable to the diffraction limit [13,14].
However, the measurement principle limits the allowable tilt angle of the target surface in the confocal probes [15–17]. Fizeau interferometers and angle sensors have thus been employed to measure the surface profiles of inclined objects [5,18–21]. Fizeau interferometers reconstruct a surface profile from the interference fringes produced by the interference between the light rays reflected from the measurement object and a reference plane. The advantage of this technique is that it can measure a two-dimensional surface profile in a short time [18]. However, it requires a reference plane having a size comparable to that of the object to be measured, and the measurement accuracy depends on the quality of the reference plane [19]. The physical limitations of the reference plane made it difficult to measure large surfaces exceeding several hundred millimeters, such as silicon wafers and X-ray focusing mirrors. On the other hand, surface profilometry using angle sensors has been attracting attention as a method to directly measure surface topography without using a reference plane. Some national metrology institutes in various countries have been developing measuring instruments based on this method [22]. In the method, the surface profile of an object is obtained by scanning a reflector and measuring the surface local slope at each position; by integrating the obtained local slope information, the surface form can be reconstructed [23].
In general, angle sensors used in these measurement methods are based on the autocollimation method [20]. Angle sensors based on the autocollimation method are often referred to as autocollimators, which are important measuring instruments widely used for non-contact measurement of small angles [20,21]. Autocollimators employing a laser as the light source have shown potential for dynamic measurement of the angular error motion of precision stages, precision surface profile measurement, and ultra-sensitive tilt angle measurement, utilizing the good directivity and a high degree of collimation of the laser beam. However, due to the limitation of the millimetric or sub-millimetric laser beam diameter, the in-plane resolution of an autocollimator is limited [5]. It is thus difficult to measure local areas in a size of less than a few tens of micrometers. For more detailed shape evaluation, it is necessary to develop angle sensors that have better in-plane resolution and can measure local slopes in a small area.
Meanwhile, the angle sensors employing a converging second harmonic wave (SH wave) [24,25] generated by the phenomenon of second harmonic generation (SHG) is a promising one for the purpose. In SHG, when a ray of light with frequency ν1 enters the nonlinear optical medium, a light ν2 = 2ν1 with twice the frequency is generated [26]. The lights with frequencies ν1 and ν2 are referred to as the fundamental wave and the SH wave, respectively. In the angle sensors with the SH wave [24,25], the phase matching effect, which is one of the factors determining the magnitude of SH wave intensity generated in the SHG [24–26], is utilized. However, the angle sensors employing SH wave require a nonlinear optical crystal to be attached to the target objects; this means that the measurement target is limited to a surface to which a nonlinear optical crystal can be attached.
In this paper, a new confocal probe with SHG is proposed to realize the measurement of a local slope with a high in-plane resolution. The proposed confocal probe can measure the linear displacement along the optical axis and the angular displacement of a target under measurement with the same optical probe. It should be noted that the newly proposed confocal probe in this paper is completely different from the conventional ones with SHG [27–31]; the newly proposed SHG confocal probe generates an SH wave on the detector, whereas the conventional SHG confocal probes generate SH wave on the measurement object. At first, numerical simulation analysis based on the proposed principle is performed. A prototype optical system is then designed and constructed. Finally, the angular measurement range and measurement resolution are investigated in experiments with the developed prototype optical system to verify the feasibility of the proposed method.
2. Principle
The configuration of the newly proposed confocal probe based on SHG is almost the same as those of the conventional confocal probes [1,9,10], except for the detector. Figure 1 shows the comparison between a typical conventional confocal probe (Fig. 1(a)) and the newly proposed one based on SHG (Fig. 1(b)). A main feature of the conventional confocal probe is the use of two pinholes; one is placed just after the light source while the other is placed just before the detector. After passing through the pinhole, the beam from the light source becomes a point light source and is focused on the object by a lens. The reflected light from the target object is then captured by the detector after passing through the pinhole placed in front of the detector. It should be noted that the reflected light from the object surface is focused on the pinhole in front of the detector when the surface of the object to be measured is in the focal plane of the lens. The amount of light passing through the pinhole becomes maximum when the object surface is at the focal plane of the objective lens. From the intensity of the laser beam captured by the detector, the conventional confocal probe can measure the linear position of the target along the Z-axis. On the other hand, the newly proposed confocal probe in Fig. 1(b) can measure not only linear position but also angular displacement. As can be seen in the figure, the pinhole in front of the detector in the conventional confocal probe is replaced with a nonlinear crystal. The light focused on the nonlinear crystal generates an SH wave with a half wavelength of the fundamental wave [26]. By observing the light intensity of the SH wave, the linear position of the target along the Z-axis can be measured in the newly proposed confocal probe.
Fig. 1. Comparison between the configuration setup of the conventional confocal probe and that of our proposed confocal probe.
Figure 2 shows the measurement principle of the newly proposed confocal probe. The proposed confocal probe has two measurement modes. The first one shown in Fig. 2(a) is referred to as the linear displacement mode in this paper, where the z-displacement of the target object from the focal plane is measured by using the change of the intensity of the SH wave. The focusing position z’ of the reflected light in the nonlinear crystal will be changed geometrically depending on the Z-directional displacement of the target. Now we define the distance from the objective lens (with a focal length of f) to the target object as a0, and the distance from the objective lens to the center position in the nonlinear crystal as b0 under the condition where the target object has no z-displacement as shown in Fig. 1. In this condition, the shift of the focused position z’ can be estimated by the following equation based on the lens Eq. (1)/f = 1/a0 + 1/b0:
where b is the distance from the objective lens to the focus position in the nonlinear crystal when the target object has the Z-directional displacement z. The intensity of the SH wave generated in the nonlinear crystal changes according to this change in z’ as shown in Fig. 2(a). From this intensity change, it is possible to measure the Z-directional displacement of the object.Another measurement mode in the proposed confocal probe is referred to as the angular displacement mode in this paper (Fig. 2(b)). In this mode, the object angle θTilt, which is the angular displacement of the measurement target from the initial condition where the laser beam is projected onto the target surface at a right angle, can be determined from the change of the intensity of the SH wave. Corresponding to the object angle θTilt, the beam incident angle θTilt’ to the nonlinear crystal changes geometrically. Under the condition, the beam incident angle θTilt’ satisfies the following equation:
The theoretical equations are then derived to explain the change in the intensity of the SH wave corresponding to the Z-directional displacement and the angular displacement of the target. Now we consider the beam condition where the fundamental wave in the incident laser beam of diameter D is made incident to the nonlinear crystal and is focused into the crystal as shown in Fig. 3(a). Figure 3(b) shows the detail of the angle of incidence of the laser beam with respect to the nonlinear crystal. It should be noted that the lens in Fig. 3(a) corresponds to the objective lens in the newly proposed confocal probe in Fig. 2. L is the crystal length, and L’ is the distance from the crystal edge to the beam focusing position. In addition, θin and θr are the angle of incidence and the angle of refraction of the laser beam in the nonlinear crystal, respectively.
Fig. 3. The beam conditions in the nonlinear crystal; (a) a schematic diagram of a beam focused into a crystal, (b) a schematic diagram when the beam is obliquely incident on the crystal.
In this condition, the power of the second harmonic wave (extraordinary ray) P2 generated in the nonlinear optical crystal is proportional to that of the fundamental wave (the ordinary ray) P1, and is expressed by the following equation [24,25,32]:
3. Theoretical calculations
3.1 Optical system parameters
Figure 4 shows a schematic diagram of the prototype optical system that realizes the proposed principle. In this paper, a femtosecond laser with a pulse repetition rate νrep of 100 MHz, and a wavelength bandwidth from 1480 nm to 1640 nm from a homemade laser source is employed as a measurement laser beam. It should be noted that the use of pulsed lasers with short pulse widths as light sources leads to the construction of inexpensive measurement systems [33]. The femtosecond laser is guided through a single-mode fiber to the optical system. The laser beam emitted from the fiber end passes through a polarizing beam splitter (PBS) and a quarter-wave plate (QWP), and is focused onto the object surface by using an objective lens 1 (QL1) with a focal length f of 17.13 mm. Here, the object angle is set to θTilt=θX = 0 degree when the optical axis is perpendicular to the surface of the object to be measured, and the object is defined to be at the origin when the surface of the object coincides with the focal plane of QL1. The distance from the end surface of the fiber to QL1 is defined as b0, and the distance from QL1 to the object is defined as a0. In this study, for the feasibility verification of the proposed method as the first step, the optical system was set to the smallest feasible size of a0 + b0 = l = 105 mm. From the lens formula 1/f = 1/a0 + 1/b0, a0 and b0 are calculated to be approximately 21.55 mm and 83.45 mm, respectively. The reflected light from the object is again incident to the objective lens, and then reflected by the PBS and focused onto the nonlinear optical crystal. It should be noted that S-bend (the ordinary ray) light is focused on the nonlinear optical crystal because of PBS and QWP. A BBO (β-BaB2O4 (CASTECH Inc.)) crystal, which has been confirmed to generate an SH wave in previous studies [24,25], is employed as the nonlinear optical crystal. The thickness L of the BBO crystal is 2 mm with an angle θ0 of 19.8° ± 0.25°. The refractive indices n0 and ne of the BBO crystal are taken from the previous studies [24,25]. The SH wave generated in the BBO crystal is focused onto the surface of a photodiode (PD) using an objective lens 2 (QL2), the current I generated on the PD is converted to voltage V by using a trans-impedance amplifier, and the voltage output is measured by an oscilloscope. Here, we introduced the BBO crystal installation angle θBBO to obtain the maximum second harmonic wave intensity P2 expressed in Eq. (3) when θTilt = 0 degree and z = 0 µm. Note that the angle θin incident on the BBO crystal due to the presence of the BBO installation angle θBBO can be expressed by the following equation:
The values of the parameters employed in the theoretical calculations are summarized in Table 1. Here, a numerical simulation result showed that the maximum doubled wave intensity is obtained when the BBO installation angle θBBO is set to -0.375 degree.
Table 1. Specification of the prototype optical system
3.2 Numerical calculations for measurement of linear and angular displacements of the target objects
Based on the optical setup described in Section 3.1, numerical calculations are conducted to verify the feasibility of the newly proposed measurement method for linear and angular displacement measurement. Figure 5 shows the calculation result about the SH wave power of Eq. (3) in a range of -200 µm to +200 µm with a step of 2 µm, and in a range from -4° to +4° with a step of 0.125°. Figure 5(a) is h(σ, ξ, µ)/ξ in Eq. (3) illustrated by a 3D plot and Fig. 5(b) is a contour plot of Fig. 5(a). The results in Fig. 5 indicate that it is necessary to consider the crosstalk between the Z-directional displacement and the angular displacement of the object.
Fig. 5. A calculation result about the SH wave power of Eq. (3); (a): 3D plot, (b): a contour plot of Fig. 5(a).
A more detailed analysis of the linear displacement mode is then carried out. Figure 6(a) shows the cross-sectional view of the plot in Fig. 5 at θX = 0 degree. The sensitivity is calculated by linear approximation, defining the Z-directional measurement range as the range of 20% to 80% intensity for z > 0 and z < 0. The sensitivities in z > 0 and z < 0 are evaluated to be -0.03115 a.u./µm (k + z) and 0.02995 a.u./µm (k-z), respectively. Here, the root-mean-square fitting error (RMSE) Δz was considered as the minimum uncertainty of the linear displacement measurement [34]. From the results shown in Fig. 6(a), RMSE was estimated to be 368 nm in z > 0 and 400 nm in z < 0. Figure 6(b) shows the cross-sectional view of the plot in Fig. 5 at z = 0 µm for angular displacement mode. The sensitivities are also calculated in the same as the linear displacement mode. The sensitivities are evaluated to be -0.7278 a.u./arc-second (k+θ) at θTilt >0 and 0.5313 a.u./arc-second (k-θ) at θTilt < 0, respectively. RMSE (Δθ) was also calculated for estimating the minimum measurement uncertainty of the angular displacement in the same manner as that of the linear displacement measurement. From the results shown in Fig. 6(b), RMSE was estimated to be 35.7 arc-seconds in θX > 0 and 116 arc-seconds in θX < 0; these results indicate that it is possible to obtain the linear and angular displacements of the object by obtaining the constitutive curve experimentally.
Fig. 6. Cross-section views of Fig. 5; (a): a cross-section view for the linear displacement mode, (b) a cross-section view for the angular displacement mode.
4. Experimental verification
Figure 7 shows a schematic diagram of the prototype optical system. Figure 7(a) shows a three-dimensional image of the optical system and Fig. 7(b) shows a photograph of the optical system. A single mode fiber (SMF: SMF-28e + (Corning Inc.)) was employed to guide the femtosecond laser (FL) to the developed confocal probe. PBS (PBS104 (Thorlabs Japan Inc.)) was a broadband polarizing beam splitter cube. QWP (AQWP05M-1600 (Thorlabs Japan Inc.)) was a mounted achromatic quarter wave plate. A flat mirror (ME1-P01 (Thorlabs Japan Inc.)) was employed as the measurement object for the evaluation of the basic characteristics of the developed confocal probe. Objective lens (#43-903 (Edmund Optics Japan Co.)) was 10X DIN achromatic commercial grade objective. The plane mirror is fixed on a stepping motor stage (KX1250C-R5 (SURUGA SEIKI Co., Ltd.)) for giving Z-directional displacement to the mirror, and the stage was mounted on a manual rotation stage (RM07A-C3 (Kohzu Precision Co., Ltd.)) to match θTilt and θX. PD (SM05PD2A (Thorlabs Japan Inc.)) was connected to a trans-impedance circuit by SMA cable, and then, the voltage output was measured by an oscilloscope (OS: RA2300 (NIPPON AVIONICS Co., Ltd.)).
Fig. 7. A schematic diagram of the prototype optical system; (a): a three-dimensional CAD image, (b): a photograph.
At first, the verification of the linear displacement mode was conducted in experiments. The z-scanning range of the target was set to -200 µm to +200 µm following the simulation results, and the target mirror was made to travel along the Z-axis in a step of 2 µm. At each Z-position, the PD detected the power of the SH wave for 1 second at 200 ms intervals. Figure 8(a) shows the results of plotting the average value of the measured voltage at each Z-position. The z-displacement measurement range was defined as the range of 20% to 80% intensity at z > 0 and z < 0, respectively, and the sensitivity k was calculated by linear approximation. Here, the voltage was normalized by the maximum voltage. The sensitivity was evaluated to be -0.01533 a.u./µm (k + z) at z > 0 and 0.01441 a.u./µm (k-z) at z < 0, respectively. It was found that the sensitivity was approximately 50% smaller than that obtained in the theoretical calculations. This was believed to be due to axial chromatic aberration of the focusing lens [24]. The root-mean-square fitting error (RMSE) Δz corresponds to the uncertainty of the linear displacement measurement [34]. From the results shown in Fig. 8(a), the RMSE was evaluated to be 2.43 µm in z > 0 and 1.86 µm in z < 0. Measurement uncertainty should be further reduced and is a subject in future work. Noise measurements were then performed to evaluate the resolution. The results of voltage measurements taken every 1 ms for 1 second at z = -24 µm and +26 µm are summarized in Fig. 8(b). The noise level shown in Fig. 8(b) was assumed as the one-step displacement measurement result [35]. The axial displacement resolution can thus be obtained by converting the noise level 2σ into displacement using the sensitivity coefficient k. From the results, 2σ values at z=+26 µm and z = -24 µm were evaluated to be 2σ+z = 1.282 mV and 2σ-z = 1.226 mV, respectively, and the corresponding resolutions were calculated to be 19.4 nm (2σ+z/k + z) and 17.4 nm (2σ-z/k-z), respectively. The obtained axial resolution values in our newly proposed method were found to be comparable to those in conventional scanning confocal probes [13]. The above experimental results realized the feasibility of the proposed principle to measure the linear displacement of the target by utilizing the SHG.
Fig. 8. A result of the plot about the measured voltage; (a): the average value of at each z displacement, (b): voltage outputs for 1 second at z = + 26 µm and -24 µm.
The verification of the angular displacement mode was then conducted in experiments. The angular scanning range of θX (=θTilt) was set to -4° to +4° regarding the simulation results, and the scanning experiment was performed in a step of 450 arc-seconds. At each θX position, the PD was used to detect the light intensity of the SH wave for 5 seconds at 200 ms intervals. Figure 9(a) shows the results of plotting the average value of the measured voltage at each θX displacement. The angular measurement range was defined as the range of 20% to 80% intensity in each angular range of θX > 0 and θX < 0, and the sensitivity k was calculated by linear approximation. The sensitivities were evaluated to be -0.2664 a.u./degree at θX > 0 (k+θ) and 0.1960 a.u./degree at θX < 0 (k-θ). It was found that the sensitivity was approximately 36% smaller than that obtained in the theoretical calculations. This was believed to be due to the axial chromatic aberration of the focusing lens [24]. The RMSE Δθ was also calculated for estimating the uncertainty of the angular displacement measurement, in the same manner as the linear displacement measurement experiment. The RMSE is estimated to be 268 arc-seconds in θX > 0 and 308 arc-seconds in θX < 0 from Fig. 9(a). The measurement uncertainty should be further reduced, and will be considered in future work. Noise measurements were also conducted to evaluate the resolution. Figure 9(b) shows the results of voltage measurements taken every 1 ms for 1 second at θX = + 2 and -2 degrees. From the results, 2σ values at θX = + 2 degrees and θX = -2 degrees were evaluated to be 2σ+θ=1.371 mV and 2σ-θ=1.358 mV, respectively, and the corresponding resolutions were calculated to be 5.02 arc-seconds (2σ+θ/k+θ) and 6.50 arc-seconds (2σ-θ/k-θ), respectively. The above experimental results realized the feasibility of the proposed principle to measure the angular displacement of the target by utilizing the SHG. It should be noted that it is important for improving the measurement resolution to adjust the measurement range of the A/D converters or to use A/D converters with more bits. These considerations for the enhancement of the measurement resolution will be conducted in future work.
Fig. 9. A result of the plot about the measured voltage; (a): mean voltage output at each angular displacement, (b): voltage outputs for 1 second at θX = + 2 degrees and -2 degrees.
5. Conclusion
A new confocal probe that can measure linear and angular displacements in a focal area with the same optical probe by applying the principle of second harmonic generation to confocal optics have proposed. By replacing spatial filters, such as pinholes installed in front of the detector in conventional confocal probes, with BBO crystal working as a second harmonic generation media, linear and angular displacements of the object to be measured can be detected as second harmonic intensity changes. To verify the newly proposed measurement method, several theoretical equations have been derived, and then the feasibility of the proposed principle has been demonstrated through theoretical calculations. Furthermore, a prototype optical system has been constructed, and experiments were carried out to verify the feasibility of the proposed confocal probe. Resolutions of the linear displacement measurement and the angular displacement measurement have been evaluated to be approximately 20 nm and approximately 5 arc-seconds, respectively.
It should be noted that this paper has focused on the proposal of the new measurement principle for the confocal probe with a femtosecond laser source. Following these objectives, the feasibility of the proposed principle has been demonstrated through numerical simulations and basic characterization experiments. Future work includes the evaluation of the crosstalk between linear and angular displacement measurements, surface profile evaluation and displacement measurement of some calibrated samples and/or an object having arbitrary surface form, and the measurement uncertainty analysis of the proposed method. Considering the trade-off relationship between measurement range and resolution, a dual detection unit [6,7,13], which also makes scan-less displacement measurement possible, will be employed in the prototype optical system in future work to realize a high measurement range and high-resolution SHG confocal optical system.
Funding
Japan Society for the Promotion of Science (20H00211, 21J20652).
Acknowledge
The authors thank Dr. Hajime Inaba of AIST for his advice and support in developing the optical frequency comb in this research.
Disclosures
The authors declare no conflicts of interest.
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. M. Marvin, “Microscopy Apparatus,” U.S. Patent 3013467 (1962).
2. T. Wilson, “Imaging properties and applications of scanning optical microscopes,” Appl. Phys. 22(2), 119–128 (1980). [CrossRef]
3. T. Wilson, “Imaging of phase edges in conventional and scanning optical microscopes,” Appl. Opt. 20(18), 3238–3244 (1981). [CrossRef]
4. D. K. Hamilton and T. Wilson, “Surface profile measurement using the confocal microscope,” J. Appl. Phys. (Melville, NY, U. S.) 53(7), 5320–5322 (1982). [CrossRef]
5. W. Gao, H. Haitjema, F. Z. Fang, R. K. Leach, C. F. Cheung, E. Savio, and J. M. Linares, “On-machine and in-process surface metrology for precision manufacturing,” CIRP Ann. 68(2), 843–866 (2019). [CrossRef]
6. X. Chen, T. Nakamura, Y. Shimizu, C. Chen, Y.-L. Chen, H. Matsukuma, and W. Gao, “A chromatic confocal probe with a mode-locked femtosecond laser source,” Opt. Laser Technol. 103, 359–366 (2018). [CrossRef]
7. R. Sato, C. Chen, H. Matsukuma, Y. Shimizu, and W. Gao, “A new signal processing method for a differential chromatic confocal probe with a mode-locked femtosecond laser,” Meas. Sci. Technol. 31(9), 094004 (2020). [CrossRef]
8. J. Bai, Y. Wang, X. Wang, Q. Zhou, K. Ni, and X. Li, “Three-Probe Error Separation with Chromatic Confocal Sensors for Roundness Measurement,” Nanomanuf. Metrol. 4(4), 247–255 (2021). [CrossRef]
9. T. Wilson and C.J.R. Sheppard, “Theory and Practice of Scanning Optical Microscopy,” Academic Press, London, UK, (1984).
10. M. Gu, “Principles of Three Dimensional Imaging in Confocal Microscopes,” World Scientific Publishing Co. Pte. Ltd.: Hackensack, NJ, USA, (1996).
11. J. Garzón, T. Gharbi, and J. Meneses, “Real time determination of the optical thickness and topography of tissues by chromatic confocal microscopy,” J. Opt. A: Pure Appl. Opt. 10(10), 104028 (2008). [CrossRef]
12. M. Mikulewitsch, M. M. Auerswald, A. v. Freyberg, and A. Fischer, “Geometry Measurement of Submerged Metallic Micro-Parts Using Confocal Fluorescence Microscopy,” Nanomanuf. Metrol. 1(3), 171–179 (2018). [CrossRef]
13. C.-S. Kim and H. Yoo, “Three-dimensional confocal reflectance microscopy for surface metrology,” Meas. Sci. Technol. 32(10), 102002 (2021). [CrossRef]
14. Y. Shimizu, L.-C. Chen, D.W. Kim, X. Chen, X. Li, and H. Matsukuma, “An insight into optical metrology in manufacturing,” Meas. Sci. Technol. 32, 042003 (2020). [CrossRef]
15. J. F. Aguilar and E. R. Méndez, “On the Limitations of the Confocal Scanning Optical Microscope as a Profilometer,” J. Mod. Opt. 42(9), 1785–1794 (1995). [CrossRef]
16. M. Rahlves, B. Roth, and E. Reithmeier, “Systematic errors on curved microstructures caused by aberrations in confocal surface metrology,” Opt. Express 23(8), 9640–9648 (2015). [CrossRef]
17. R. Sato, Y. Shimizu, H. Matsukuma, and W. Gao, “Influence of Surface Tilt Angle on a Chromatic Confocal Probe with a Femtosecond Laser,” Appl. Sci. 12(9), 4736 (2022). [CrossRef]
18. M. P. Singh, M. Singh, and K. Khare, “Single shot interferogram analysis for optical metrology,” Appl. Opt. 53(29), 6713–6718 (2014). [CrossRef]
19. X. Chen, Y. Shimizu, X. Xiong, Y.-L. Chen, and W. Gao, “Self-calibration of Fizeau interferometer and planar scale gratings in Littrow setup,” Opt. Express 25(18), 21567–21582 (2017). [CrossRef]
20. W. T. Estler and Y. H. Queen, “Angle Metrology of Dispersion Prisms,” CIRP Ann. 49(1), 415–418 (2000). [CrossRef]
21. Y.-L. Chen, Y. Shimizu, Y. Kudo, S. Ito, and W. Gao, “Mode-locked laser autocollimator with an expanded measurement range,” Opt. Express 24(14), 15554–15569 (2016). [CrossRef]
22. F. Siewert, J. Buchheim, S. Boutet, G. J. Williams, P. A. Montanez, J. Krzywinski, and R. Signorato, “Ultra-precise characterization of LCLS hard X-ray focusing mirrors by high resolution slope measuring deflectometry,” Opt. Express 20(4), 4525–4536 (2012). [CrossRef]
23. R. D. Geckeler, “Optimal use of pentaprisms in highly accurate deflectometric scanning,” Meas. Sci. Technol. 18(1), 115–125 (2007). [CrossRef]
24. H. Matsukuma, S. Madokoro, W. D. Astuti, Y. Shimizu, and W. Gao, “A New Optical Angle Measurement Method Based on Second Harmonic Generation with a Mode-Locked Femtosecond Laser,” Nanomanuf. Metrol. 2(4), 187–198 (2019). [CrossRef]
25. W. D. Astuti, H. Matsukuma, M. Nakao, K. Li, Y. Shimizu, and W. Gao, “An Optical Frequency Domain Angle Measurement Method Based on Second Harmonic Generation,” Sensors 21(2), 670 (2021). [CrossRef]
26. R.W. Boyd, “Nonlinear Optics, Third edition,” Academic Press, The Netherlands (2008).
27. J. N. Gannaway and C. J. R. Sheppard, “Second-harmonic imaging in the scanning optical microscope,” Opt. Quantum Electron 10(5), 435–439 (1978). [CrossRef]
28. I. Veilleux, J. A. Spencer, D. P. Biss, D. Cote, and C. P. Lin, “In Vivo Cell Tracking With Video Rate Multimodality Laser Scanning Microscopy,” IEEE J. Select. Topics Quantum Electron. 14(1), 10–18 (2008). [CrossRef]
29. A. H. Reshak, “Second harmonic generation from thick leaves using the two-photon laser scanning microscope,” Micron 40(4), 455–462 (2009). [CrossRef]
30. C. Yang, K. Shi, H. Li, Q. Xu, V. Gopalan, and Z. Liu, “Chromatic second harmonic imaging,” Opt. Express 18(23), 23837–23843 (2010). [CrossRef]
31. P. J. Campagnola and C.-Y. Dong, “Second harmonic generation microscopy: principles and applications to disease diagnosis,” Laser & Photon. Rev. 5(1), 13–26 (2011). [CrossRef]
32. G. D. Boyd, “Parametric Interaction of Focused Gaussian Light Beams,” J. Appl. Phys. (Melville, NY, U. S.) 39(8), 3597–3639 (1968). [CrossRef]
33. R. L. Harzic, D. Breitling, M. Weikert, S. Sommer, C. Föhl, S. Valette, C. Donnet, E. Audouard, and F. Dausinger, “Pulse width and energy influence on laser micromachining of metals in a range of 100fs to 5ps,” Appl. Surf. Sci. 249(1-4), 322–331 (2005). [CrossRef]
34. C.-H. Lee and J. Wang, “Noninterferometric differential confocal microscopy with 2-nm depth resolution,” Opt. Commun. 135(4-6), 233–237 (1997). [CrossRef]
35. T. Liu, J. Wang, Q. Liu, J. Hu, Z. Wang, C. Wan, and S. Yang, “Chromatic confocal measurement method using a phase Fresnel zone plate,” Opt. Express 30(2), 2390 (2022). [CrossRef]
