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Abstract
A chromatic confocal measurement method based on a phase Fresnel zone plate (FZP) is described. Strong dispersion of FZP results in significant axial focal shift. The axial dispersion curve is close to linear within a certain wavelength range determined by the quantitative calculation using the vectorial angular spectrum theory. A 11.27 mm diameter phase FZP with a primary focal length of 50 mm was processed using standard photolithography technology and used as the dispersive objective in a homemade chromatic confocal measurement system. The calibrated axial measurement range exceeds 16 mm, the axial resolution reaches 0.8 µm, and the measurement accuracy of displacement is better than 0.4%. This chromatic confocal sensor has been practically used in the measurement of step height, glass thickness, and 3D surface profile. The proposed method has the obvious characteristics of simplicity, greatly reduced cost and superior performance. It is believed that this sensing method has broad application prospects in glass, coating, machinery, electronics, optics and other industries.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Chromatic confocal measurement (CCM) method is an important branch of confocal microscopy. It is almost suitable for the measurement of any reflectance samples and can especially realize single-point real-time measurements [1]. From this point of view, CCM outperforms other high-precision 3D measurement methods, e.g. confocal microscopy [2] and white light interferometry [3]. Usually, CCM requires high-cost, complicated objectives composed of multiple refraction lenses and with linear dispersion characteristics [4–6]. In recent years, wide attention has been paid on super-focusing and nano-imaging of a classic binary diffractive optical element, i.e., Fresnel zone plates (FZP) [7–9] and a variety of applications have been found in atomic optics, confocal imaging, X-ray nanoscopy, and holographic microscopy [10–15]. Special phenomena for FZP include pronounced chromatic dispersion [16] and single focus for high-NA (numerical aperture) FZPs [17]. The chromatic dispersion of FZP has been studied in depth in Refs. [8,18]. In this paper, a CCM measurement method based on a phase Fresnel zone plate is reported. In principle, this technology avoids the use of complex and expensive objectives. A CCM prototype based on a single FZP has been developed and tested in detail. In comparison, the technology has made great progress from the earlier idea [19]. So far, focusing a white light beam directly by a FZP in a chromatic confocal system has not been reported. A vision chromatic confocal sensor based on a commercial geometrical phase lens has been reported recently in Ref. [20]. This is different from the current method. On one hand, the core components are significantly different. The dispersive focusing component, a kind of metalens, in Ref. [20] is polarization sensitive and its resonant units are basically at the sub-wavelength scale. FZP, independent of polarization, is much simpler, less costly, and has a wide range of options. On the other hand, the theoretical basis in this research is quantitative and more accurate as the vectorial angular spectrum theory is used to select the most suitable linear dispersion region. The exact focal spot distribution at each shifted focal plane as well as the upper and lower limits of the illumination wavelength can be accurately determined. Further, there is also a difference in the visual realization. In this research, a FZP acts as the chromatic dispersion lens as well as a wide-field imaging objective in contrast to the polarization method in Ref. [20]. Experimental results show that the current method has the advantages of simple structure, good performance and extremely low cost. So, it can be widely used in precision measurement fields such as machinery, electronics, optics, glass materials, etc.
2. Principle
2.1 Fresnel zone plate
Fresnel zone plate is a binary diffractive optical element, with its radius coordinates of each annulus being defined by ${r_n} = \sqrt {n\lambda f + {n^2}{\lambda ^2}/4} ,\textrm{ }n = 0,1,2,\ldots ,N$. f is the main focal length, N is the total annulus number and λ is the light wavelength in the medium where a FZP is immersed. λ=λ0/η with λ0 being the illumination light wavelength and η the refractive index of the immersion medium. The effective numerical aperture (NAeff) of a FZP is calculated by NAeff=ηsinα, similar to the case for an objective lens. α is the maximum focusing semi-angle, satisfying tanα=rN/f. rN is the FZP radius and the diameter is therefore $D = 2{r_N} = 2{{f \cdot \textrm{NA}} / {{{({\eta ^2} - \textrm{N}{\textrm{A}^2})}^{1/2}}}}$. For a binary phase-type FZP, the transmission function t(r) = 1, when r2m<r ≤ r2m+1 and t(r)=−1 when r2m+1 < r ≤ r2m+2, m = 0, 1,…, N/2-1 (N is supposed to be an even number).
Under the scalar diffraction theory, the transverse size (full width at half maximum, FWHM) of the main focus can be evaluated as [21]
Meanwhile, it is well known that the transverse spatial resolution of a FZP in vacuum, according to scalar diffraction theory and Rayleigh’s criterion, has been simply expressed by [22]
where, $\Delta r = {r_N} - {r_{N - 1}}$ is the radial width of the outmost annulus. The transverse spatial resolution is merely determined by one structural parameter, $\Delta r$. It should be noted that Eqs. (1) and (2), hold only for low-NA FZPs where the influence of polarization can be ignored. When the effective numerical aperture becomes large (e.g. NAeff > 0.4), the secondary foci constructed by multi-order diffraction beams disappear [16,17], and only one main focus remains. The underlying reason has been clearly revealed recently [18]. Super-resolution focusing is easily obtained for a high-NA FZP compared with a finely-corrected objective lens with the same numerical aperture due to the difference of apodization factor [23]. However, restricted by current micromachining technology, only high-NA micro-FZPs or low-NA macro-FZPs can be fabricated within an acceptable processing cost in the visible light band.2.2 Negative linear dispersion
Usually for a FZP, the illumination wavelength equals or is close to the design wavelength. However, when the illumination wavelength deviates from the design wavelength, significant chromatic dispersion occurs for FZPs [8]. The axial dispersion curve is negative and close to linear within a certain wavelength range
Three low-NA phase FZPs (FZP1-3) are studied for the purpose of dispersion focusing. The electric field intensity distribution behind FZPs are accurately calculated by vectorial angular spectrum theory (VAS) [24]. FZP parameters are summarized in Table 1. λc denotes the center wavelength, and fc is the main focal length. λmin and λmax are the minimum and maximum illumination wavelengths, fmin and fmax are the corresponding focal lengths. R = rN is the FZP radius. Here, NAeff is the numerical aperture evaluated at the focal plane for λc.
Table 1. Parameters for FZP1-3
According to the VAS theory, dispersion focal shift curves for FZP1-3 are plotted in Fig. 1 and the transverse spot sizes (FWHM) at each focal plane are shown in Fig. 2. A linearly polarized light (x-polarized) has been used. As the numerical aperture is low, the spot sizes along the x and y directions (FWHMx and FWHMy) are very close. The accuracy of the calculations by VAS has already been validated in previous publications [17,18,25].
Although the polarized light focusing characteristic of FZPs has bee calculated, the actual imaging and measurement process is rarely affected by polarization especially under low-NA optical systems. The reason is that the longitudinal polarization component Ez will not be able to propagate through low-NA optical systems [25]. Three-dimensional light field distributions at any plane behind a FZP can be known from the vectorial angular spectrum theory.
It can be clearly seen from Fig. 1 that significant, negative focal shift occurs and the dispersion curve is close to linear. The axial dispersion range is evaluated (VAS) as 16.31 mm, 6.46 mm, and 0.67 mm for FZP1-3, respectively. This phenomenon can be particularly used in chromatic confocal microscopy. The traditional refraction objective with complex structure is replaced by one single FZP.
When the illumination wavelength deviates from the design wavelength, the focal length of a FZP can be qualitatively evaluated, under the framework of scalar diffraction theory, from [19,20]
where, $\Delta \lambda$ and $\Delta f$ denote the changes in the illumination wavelength and the focal length, respectively. For FZP1-3, the approximate dispersion range is 15.80 mm, 6.16 mm, and 0.63 mm, respectively, according to Eq. (3). So, more accurate results have been rigorously obtained by VAS. More importantly, previous publications cannot give the accurate prediction for the maximum and minimum illumination wavelengths which can be practically used in chromatic confocal systems. By detailed analysis of the lateral and axial distribution of the focal spot at each plane along the axial direction, upper and lower limits of the illumination wavelength have been determined as shown Fig. 1 and Fig. 2.3. Experiment
3.1 FZP objective
FZP1-3 are fabricated using silicon nitride (Si3N4, refractive index 2.008 at λ = 633 nm) on a glass substrate (BF33, 0.5 mm thickness). In order to increase the light transmittance of the substrate, anti-reflection coating has been realized by alternately coating four layers of titanium pentoxide and silicon dioxide. The measured reflectivity is less than 0.5% in the wavelength range of 500-700 nm. The FZPs are then prepared by a standard lithography processing. The etching depth of Si3N4 is 314 nm calculated according to the center wavelength of 633 nm. The phase difference between adjacent zones is therefore 180 degrees in order to maximize the focal intensity [26]. FZPs are then finely adjusted and mounted, as shown in Fig. 3 (FZP1).
3.2 Experimental prototype
Based on the dispersion phenomenon of a phase FZP (Fig. 4(a)), an experimental prototype of CCM has been built, as shown in Fig. 4(b). It is composed of the main chromatic confocal measurement module and an auxiliary wide-field imaging module. The latter is realized using critical illumination (Halogen light through a fiber bundle) and the infinity optical imaging unit.
Fig. 4. Schematic diagram of dispersion focusing and experimental prototype of CCM. (a) Schematic diagram of the negative dispersion phenomenon of FZP; (b) experimental prototype: 1-XY translation stage with servo motor; 2-Sample; 3-FZP objective; 4-Y-type fiber; 5-tube lens and CCD; 6-splitting optical path; 7-wide-field illumination lens.
The CCM sensing system is simply integrated using a three-port Y-type fiber (multi-mode fiber with 50 µm core diameter). This fiber also acts as a confocal pinhole filter, which is essential for a practical confocal microscope. The first port is connected to the illumination white-light LED source, the second port is connected to a high-speed spectrometer, and the outgoing beam from the third port is collimated by an achromatic doublet (75 mm focal length) and then focused by the FZP objective, FZP1 (Fig. 3). The reflected/scattered light from the sample is again collected by the same FZP objective, and then refocused to the third port of the Y-type fiber. A high-speed spectrometer is used to collect light signals.
3.3 System calibration
The basic work for a CCM sensing system is to carry out the system calibration experiment and an accurate calibration curve is to be determined for the actual measurement afterwards. The experimental setup for calibration is shown in Fig. 5. A laser interferometer (XL-80, Renishaw) is used in the experiment to feedback the actual displacements in real time. The calibration process is to accurately determine the corresponding relationship between the peak wavelength and the current displacement. A flat mirror (sample) was axially displaced at equal intervals of 200 µm. 90 locations were collected resulting in a 17.8 mm calibration range in total. Five repeated measurements were taken at each location. The light signal collected by the spectrometer typically presents single-peak intensity response characteristics. The averaged value is used as the final displacement. Four peak extraction algorithms have been particularly compared in Fig. 6, and the Gaussian fitting method has the best peak extraction accuracy among the four algorithms. A calibration curve can be obtained reasonably by the fitting or interpolation method using 90 discrete uniform sampling measurement points. And the final calibration curve for measurement is plotted in Fig. 7 using the Gaussian fitting method for each position. Further, 1000 data points are generated between every two adjacent position points ensuring sufficient interpolation data. The calibration curve (Fig. 7) becomes the actual measurement curve.
Fig. 5. Experimental platform for calibration: 1-Laser interferometer; 2-Beamsplitter and corner prism; 3-Corner prism; 4-Translation stage; 5-Reflection mirror; 6-FZP objective; 7-White light source; 8-Spectrometer.
Although the calibration curve (Fig. 7) is non-linear, high-precision measurement can still be achieved through the above accurate calibration experiment. The axial measurement range exceeds 16 mm (at most 17.8 mm). Meanwhile, a calibration curve with good linearity can also be extracted from Fig. 7. The linear calibration range for measurement is greatly reduced to be less than 3 mm, as shown in Fig. 8. The main difference between linear and nonlinear sensors is that the sensitivity of the measurement is not the same. The sensitivity is variable for a nonlinear calibration curve. Therefore, there is an optimal measurement interval for Fig. 7. It is also difficult to achieve high-speed measurement because the height calculation from a nonlinear fitting curve needs more time.
3.4 Axial resolution test
The axial resolution of the experimental prototype is to be determined using a reflection mirror as the sample (Fig. 5). Five reciprocating step movements were implemented using a manual stage at a step interval of 0.8 µm, 1 µm, 1.5 µm, and 2 µm, respectively. At each mirror position, the spectral response curve was repeatedly acquired for 50 times. The relative displacements are calculated according to Fig. 7 and the peak extraction method is based on Gaussian fitting. It can be clearly seen from Fig. 9 that the axial displacement or height resolution reached 0.8 µm.
3.5 Displacement measurement accuracy
The experiment setup is the same with Fig. 5. The flat mirror is stepped at 1 mm intervals. At each position, the spectral response curve was repeatedly acquired for 50 times. The measured displacements were calculated using Fig. 7 and the results are shown in the second column in Table 2. The peak wavelengths were calculated using the Gaussian fitting method. The true displacement positions were given according to the readings of the laser interferometer (the third column in Table 2). The actual displacement range is 14 mm in this experiment. The results are compared in Table 2 and Fig. 10. The relative error is less than 0.4% within the whole range of 14 mm (mostly less than 0.1%).
Table 2. Experimental data of axial displacements
4. Measurement results
4.1 Step height
The step height of a 0.1 mm feeler gauge (rough surface) is measured. The sample is made of two feeler gauges tightly attached to each other by a magnet. The rectangular area inside the red box in Fig. 11(a) was scanned by a two-dimensional servo motor stage with a 10 µm step. The scanned area is 2 mm × 8 mm. 3D surface profile of the feeler gauge is shown in Fig. 11(b). The one-dimensional cross-sectional data corresponding to the red dashed line is shown in Fig. 11(c). The average height difference is evaluated as 0.106 mm.
Fig. 11. Measurement result of a feeler gauge: (a) photo of the sample; (b) 3D surface profile; (c) 1D cross section.
4.2 Thickness
A set of quartz glass with thickness from 1 mm to 10 mm is measured. The measured thicknesses have been compensated with the refractive index (1.456). Each quartz glass is measured by 10 times. The measured results are plotted in Fig. 12 with the standard deviations. Compared with the readings of the thickness gauge (nominal thickness in Fig. 12), the maximum relative error is only 0.5% (the inset within Fig. 12, the standard deviation is 0.0397 mm).
4.3 MEMS
Two MEMS (micro-electro-mechanical system) samples are measured. The first MEMS sample is an accelerometer sensor. The scanned area is 2 mm × 1.56 mm with 20 µm step. The height of the structure is approximately 320 µm. The second MEMS sample is a pressure sensor. The scanned area is 1.5 mm × 1.2 mm with 15 µm step. The height of the structure is approximately 280 µm. It is noted that the abnormal points in Fig. 13 have been filtered.
Fig. 13. Measurement results of MEMS samples: (a) accelerometer MEMS; (b) 3D point cloud data; (c) 3D reconstruction result; (d) pressure MEMS; (e) 3D point cloud data; (f) 3D reconstruction result.
The effectiveness of the new measurement system has been demonstrated by the above three sets of measurement data.
5. Conclusion
A chromatic confocal measurement method has been reported based on a phase FZP. The dispersion focusing properties of FZPs are accurately calculated using the vectorial angular spectrum theory. Phase FZPs are fabricated using the standard lithography technology so as to ensure the realization of low-cost batch processing. The principle prototype has been built and experimentally tested in detail for the system calibration, axial resolution, and displacement measurement accuracy. Actual measurements of step height, glass thickness, and 3D topography fully demonstrates that the proposed method has the characteristics of advanced principle, simple system structure, large measurement range and high precision. Therefore, it is a very promising measurement method in the fields of optics, mechanics and electronics, etc. This method can be applied to other electromagnetic wavebands, such as the use of near infrared wavebands to directly measure the thickness of opaque silicon wafers. Multi-level, high-NA phase FZPs and metasurfaces (or metalenses) can be used to effectively improve the diffraction efficiency and the transverse spatial resolution.
Funding
Key Research and Development Program of Shaanxi Province (2020ZDLGY04-02, 2021ZDLGY12-06); Program for Science and Technology Innovation Team of Shaanxi Province (2019TD-011).
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.
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