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High-precision laser transverse differential confocal radius measurement method

Shun Tang, Yanhong Li, Lirong Qiu, Yin Song, Weiqian Zhao, and Shuai Yang

Open Access Open Access

Abstract

To meet the current need for high-precision and environment-insensitive measurement of the radius of curvature (ROC), we proposed a transverse differential confocal radius measurement (TDCRM) method based on the optical system of the confocal ROC measurement. Using a D-shaped aperture and the virtual pinhole technology, two signals, analogous to the pre-focus and post-focus signals in the two-detector-based differential confocal radius measurement (DCRM), can be obtained from two segmentations of a single CCD image. The difference of these two signals can be used to precisely determine the cat’s-eye and confocal positions, thereby achieving the high-accuracy ROC measurement as DCRM with a relative repeatability of 3.4 ppm. Furthermore, compared to DCRM, no optical alignment is needed after replacing the objective lens, which significantly reduces the time cost of measurements. We believe this novel and high-precision ROC measurement method will widen its application to optical manufacturing and provide an exciting opportunity for mass production of the ROC measurement instrument.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Owing to their excellent manufacturing machinability, spherical optics are used as a core component in high-performance optical systems, such as laser fusion systems and UV lithography objective lenses [1]. The key parameters of spherical optics include the refractive index, central thickness, radius of curvature (ROC), and surface shape. With the improved performance requirements of optical systems, high-precision ROC measurements are in high demand. At present, improving the accuracy of ROC measurement is a fundamental problem in the field of optical manufacturing and measurement [2,3].

At present, the ROC measurement methods can be divided into two categories: contact and non-contact types. The contact ROC measurement methods include the ball diameter instrument [4], coordinate [5], and laser tracking methods [6]. These methods, with simple measuring principles, are flexible and easy to use; however, they have relatively low accuracy and slow speed. Furthermore, it is easy to damage the surface of the optics during these measurements. The non-contact ROC measurement methods can be further divided into interferometric and non-interference methods, and have the advantages of zero damage and high accuracy. Due to the diffraction limit, the measurement accuracy of the non-interference method is approximately 20 ppm, such as the knife edge [7], autocollimator, and microscope-scanning methods [8,9]. Compared to the non-interference method, the interferometric method has the characteristics of higher accuracy and a larger measuring range, and is currently the most widely used ROC measurement method [2,10]. The phase measurement interferometer can determine the cat’s-eye position and the confocal position of the measured spherical surface, allowing the measurement of its ROC across scales [11,12]. The wavelength-tuned interference ROC measurement method does not require any mechanical movement, avoiding the uncertainty error introduced by the motion of the guide rail, and its measurement accuracy can reach 10 ppm. However, because this method requires the measurements of interferences at multiple wavelengths, the system structure is often more complicated [13]. In addition, the ROC measurement methods based on the interference principle are sensitive to environmental disturbances, especially in high-precision detections [14,15].

To achieve environment-insensitive and high-precision ROC measurement, we proposed a differential confocal ROC measurement method in the early stage [16,17]. The method divides the detection optical path into two, in which the axial light intensity signals are detected at the pre-focus and post-focus positions with the same defocus distance. Thereafter, we performed differential subtraction processing on the two signals. The absolute zero in the differential confocal curve is used to replace the peak position with zero slope in the confocal response curve, which effectively suppresses the interference of the external environment, significantly enhances the ability to find the focus, and improves the measurement accuracy of the ROC to 5 ppm [16]. However, to ensure the high-precision measurement of the ROC, the deviation of the individual CCD’s position must be controlled within 10 µm, which requires a complicated assembly process. In addition, when the differential confocal ROC measurement system changes the objective lens, the pinhole position should be readjusted. The complexity and the high cost of the entire system hinder mass production and high-precision detection in optical processing industry.

To address these issues, here we proposed a transverse differential confocal radius measurement (TDCRM) method. The optical path of this method is the same as the confocal ROC measurement method except that it uses a D-shaped aperture to block half of the detection beam. Two circular regions, which are symmetric about the optical axis, on the focal plane are selected as the virtual pinholes for off-axis segmentation detection using virtual pinhole technology. In this way, the original two-way axial defocusing detection method is transformed into a one-way off-axis detection scheme. The TDCRM significantly simplifies the optical system, which prevents the error caused by the physical pinhole position, reduces the difficulty of the system installation and alignments, and provides the possibility for the mass production.

2. TDCRM principle

The principle of high-precision transverse differential confocal focusing is illustrated in Fig. 1.

 figure: Fig. 1.

Fig. 1. Transverse laser differential confocal focusing principle.

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After passing through the beam splitter (BS), the light emitted by the point light source S is collimated into parallel light by the collimating lens Lc. The parallel beam enters the standard converging lens Ls and then converges into a measuring beam. The measuring beam is reflected on the surface of the test sample. After passing through the converging lens Ls and collimating lens Lc, the reflected beam is split by the BS to form the detection beam. The detection beam is blocked by a D-shaped aperture before the detection system.

The symbols z0, z1, and z2 represent the position of the measured surface before, in, and after the focal plane, which correspond to the three magnified Airy Spots. When the axial position of the sample moves from z0 to z2, the Airy disk in the focal plane moves laterally along a certain direction, and the sample defocus change is reflected as the lateral deviation of the Airy disk in the focal plane. A microscopic objective lens M is used to magnify the Airy disk in the focal plane, and a clear image of the light spot with a magnification β is collected by the CCD. IA and IB are the corresponding light intensities from the two virtual pinholes A and B in the light spots.

According to the theory of partially coherent imaging, the complex amplitude response function of the focal plane is as follows.

$$\begin{aligned} U(\nu ,\varphi ,u) &= \frac{{{D^2}}}{4}\int_0^\pi {\int_0^\textrm{1} {{P_1}({{\rho_0},\theta } )} } \cdot {P_2}({{\rho_0},\theta } )\exp ({ju{\rho^2}} )\times \\ &\exp [{ - j\nu \rho \cos ({\theta - \varphi } )} ]\rho d\rho d\theta \end{aligned}$$
where $u = \pi {D^2}z/2\lambda f_s^2$ is the normalized axial coordinate, z is the axial displacement of the test sample, λ is the wavelength of the light source, $\nu = \pi Dr/\lambda {f_c}$ is the normalized radial coordinate, ${\rho _0} = D\rho /2$ is the normalized pupil radius, ${f_c}$ is the focal length of the collimating lens Lc, and ${f_s}r$ is the focal length of the converging lens Ls. $rD/{f_c}$ is the relative aperture of the collimating lens. $rD/{f_s}$ is the relative aperture of the Ls. ${P_1}({{\rho_0},\theta } )= 1r$ is the normalized pupil function of the illumination optical path. $r{P_2}({{\rho_0},\theta } )= 1,\theta \in ({0,\pi } )$ is the normalized pupil function of the detection optical path.

As shown in Fig. 1, two virtual pinholes A and B are set symmetrically about the optical axis on the detector plane, and are used to detect the off-axis light intensity of two specific segmentation areas of the light spot. The light intensities obtained from the two virtual pinholes are normalized and the transverse differential confocal curve obtained by the difference of these signals is used to achieve the focusing.

As shown in Fig. 2, when the radial offsets of the two selected virtual pinholes are the same, the two angular offsets then differ by π, and the obtained corresponding light intensity signals can be regarded as the axial response near the focus with the same defocus distance. When the sample is near the focal plane, the light intensity detected by the virtual pinholes A and B under normalized axial coordinate u are:

$$\left\{ \begin{array}{l} {I_A}(u )= {\left|{\frac{{{U_A}({\nu_M},\varphi ,u)}}{{{U_{A\textrm{ - }\max }}}}} \right|^2}\\ {I_B}(u )= {\left|{\frac{{{U_B}({\nu_M},\varphi \textrm{ + }\pi ,u)}}{{{U_{B - \max }}}}} \right|^2} \end{array} \right.$$

Substituting (1) into (2), the transverse differential light intensity response function ${I_{diff}}(u )$ is:

$$\begin{aligned} {I_{diff}}(u) &= {I_B}(u) - {I_A}(u)\\ &= \frac{\textrm{4}}{{{\pi ^\textrm{2}}}}\left\{ {{{\left|{\int_0^\pi {\int_0^\textrm{1} {\exp ({ju{\rho^2}} )} } \cdot \exp [{ - j{\nu_M}\rho \cos ({\theta - \varphi - \pi } )} ]\rho d\rho d\theta } \right|}^2} - } \right.\\ &\left. {{{\left|{\int_0^\pi {\int_0^\textrm{1} {\exp ({ju{\rho^2}} )} } \cdot \exp [{ - j{\nu_M}\rho \cos ({\theta - \varphi } )} ]\rho d\rho d\theta } \right|}^2}} \right\} \end{aligned}$$

 figure: Fig. 2.

Fig. 2. A schematic diagram of TDCRM.

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The absolute zero of the transverse differential confocal curve corresponds exactly to the cat’s-eye position and the confocal position of the measured spherical element. The distance R between the cat’s-eye position and the confocal position is the ROC of the measured lens, which is accurately measured by a laser interferometer. The principle of the transverse differential confocal ROC measurement method is shown in Fig. 2.

In the linear region of the transverse differential confocal curve, the coordinate of the absolute zero is calculated by the straight-line fitting method, which significantly improves the system’s axial response ability. In the TDCRM system, the position and size of the virtual pinhole can be set using software, which can prevent the shortcomings of traditional physical pinhole alignment difficulties, simultaneously simplify the optical path, and improve the assembly efficiency of the optical system.

3. Key parameter analysis

In the polar coordinate, the position of the optical axis is considered as the coordinate center, and the position of the virtual pinhole is determined by the two parameters of the angular offset $\varphi$ and the radial offset $\nu$.

3.1 Centroid calculation of the light spot

To reduce the environmental noise on the CCD and obtain a light spot image with a suitable signal-to-noise ratio, the images are processed before calculating the centroid coordinates. Because only the middle part of the image is exposed to light and exhibits a larger gray value, the maximum between-class variance method can be used to obtain the binarization of the global light image. This method mainly sets a threshold to segment the image into the target and background parts. The best edge contour of the light spot is obtained when the variance between the two parts is the largest. Finally, the centroid coordinates $({{i_k},{j_k}} )$ of each spot image are obtained using the center of gravity method and the formulas are as follows.

$$\left\{ \begin{array}{l} {i_k} = \frac{{\sum\nolimits_m {\sum\nolimits_n {g({i,j} )\cdot i} } }}{{\sum\nolimits_m {\sum\nolimits_n {g({i,j} )} } }}\\ {j_k} = \frac{{\sum\nolimits_m {\sum\nolimits_n {g({i,j} )\cdot j} } }}{{\sum\nolimits_m {\sum\nolimits_n {g({i,j} )} } }} \end{array} \right.$$

Here, m and n are the numbers of pixels in the horizontal and vertical directions of the light spot image, respectively.

3.2 Angle offset determination of the virtual pinhole

According to the optical path of the TDCRM, the centroid of the light spot moves in the vertical direction along the chord side of the D-shaped aperture. The aperture does not have a fixed placement angle in the actual installation and alignment process, resulting in a certain angle $\varphi $ between the movement trajectory of the light spot and the horizontal direction on the detection surface, as shown in Fig. 3. After obtaining the centroid coordinates of the light spots, the motion trajectory is fitted to determine the angle $\varphi $.

 figure: Fig. 3.

Fig. 3. Effect of D-shaped aperture rotation. (a) Integration area with an angular offset (b) Centroid movement track of light spot.

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To determine the angular offset in actual use, the centroid coordinate $({{x_i},{y_i}} )$ motion trajectory is linearly fitted using the least-squares method.

$$\left\{ \begin{array}{l} k = \frac{{\frac{1}{n}\sum\nolimits_i {{x_i}{y_i}} - \frac{1}{n}\sum\nolimits_i {{x_i}} \cdot \frac{1}{n}\sum\nolimits_i {{y_i}} }}{{\frac{1}{n}\sum\nolimits_i {{x_i}^2 - {{\left( {\frac{1}{n}\sum\nolimits_i {{x_i}^2} } \right)}^2}} }}\\ b = \frac{1}{n}\sum\nolimits_i {{y_i} - } k \cdot \frac{1}{n}\sum\nolimits_i {{x_i}} \end{array} \right.$$

According to Eq. (5), a linear trajectory with the function relationship $y = kx + b$ is obtained, from which the angular offset $\varphi $ can be obtained as Eq. (6).

$$\varphi = \arctan (k )$$

3.3 Radial offset determination of the virtual pinhole

In the actual measurement, the focus position on the CCD image plane is unknown. The centroid of the light spots with the largest gray value is used as the focal position of the test optical path with a coordinate $({{i_o},{j_o}} )$.

The radial offset of the virtual pinhole changes, the slope at absolute zero of the transverse differential confocal axial response curve changes continuously. Figure 4(a) shows the transverse differential confocal curve with different radial offsets ν. The position of the virtual pinhole is mainly determined based on the focusing sensitivity at absolute zero of the transverse differential confocal axial response curve. It is necessary to maximize the slope at the zero position of the transverse differential confocal axis to achieve high-precision axial focusing. The slope at the zero position is calculated using the following equation.

$$k(0,v) = \frac{{\partial {I_{diff}}(u )}}{{\partial u}}|{_{u = 0}} $$

Figure 4(b) shows the focusing sensitivity of the normalized transverse differential confocal curve at the zero position when the radial offset of the virtual pinhole continues to increase. The highest focusing sensitivity ${k_{max}} = 5.18$ is achieved at ${\nu _M}$=4.2.

 figure: Fig. 4.

Fig. 4. (a) Axial response curves with different radial offsets (b) the measurement sensitivity at different virtual pinhole offsets changes.

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In addition, the actual optimal radial offset of the virtual pinhole is related to the magnification of the objective lens and the pixel size of the CCD. The optimal off-axis amount of the virtual pinhole ${v_{pixel}}\; $ on the CCD detection surface can be calculated as follows.

$${v_{pixel}} = \frac{{4.2\lambda }}{\pi }\frac{{{f_C}}}{D}{ \times }\frac{\beta }{p}$$

Here β is the magnification of the objective lens, and p is the size of the CCD pixel.

The TDCRM mainly sets the three parameters of the center of symmetry and radial offset of the virtual pinhole, and the angle $\varphi $ through the software, to prevent the measurement error caused by the pinhole adjustment and can significantly improve the ROC measurement accuracy and efficiency.

3.4 Determination of the virtual pinhole size

As the measurement sensitivity of the TDCRM depends on the virtual pinhole size, we need to find the optimal virtual pinhole size before ROC measurement. The relationship between the detected light intensity and the virtual pinhole size is shown as follows:

$$\mathop {{I_{diff}}}\limits^{\_\_\_\_} ({r_p},u) = \frac{{2\pi \int_0^{{r_p}} {{I_{diff}}({v,u} )} vdv}}{{\pi r_p^2}}$$
where rp is the normalized virtual pinhole radius. Based on Eq. (7) and Eq. (9), we can obtain the relationship between measurement sensitivity k (rp) and the virtual pinhole size, which is shown in Fig. 5:

 figure: Fig. 5.

Fig. 5. The influence of virtual pinhole radius on measurement sensitivity

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As the virtual pinhole radius increases, the measurement sensitivity decreases. Therefore, the virtual pinhole size should be as small as possible. But it is necessary to ensure the signal intensity and the detection accuracy. The experiments indicate that a virtual pinhole radius of 3 pixels that corresponds to a physical pinhole radius of about 12.5 µm gives the best result.

3.5 Influence of D-shaped aperture deviation on measurement accuracy

As shown in Fig. 6, when the light intensity is exactly half of the intensity without the diaphragm i.e. the normalized offset of the D-shaped aperture position e=0, it is the exact position of the D-shaped aperture.

 figure: Fig. 6.

Fig. 6. Aperture with an offset e

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When the normalized offset e of the D-shaped aperture position exists, the image complex amplitude response function becomes:

$$\begin{aligned} {U_1}(r,\varphi ,z) &= \frac{{{D^2}}}{4}\int_{\arcsin e}^{\pi + \arcsin ({ - e} )} {\int_{\frac{e}{{\sin \theta }}}^1 {{P_1}\left( {\frac{D}{2}\rho ,\theta } \right) \cdot {P_2}\left( {\frac{D}{2}\rho ,\theta } \right)\exp (ju{\rho ^2})} } \\ &\quad\exp [{ - j\nu \rho \cos ({\theta - \varphi } )} ]\rho d\rho d\theta \end{aligned}$$
where ${P_2}\left( {\frac{D}{2}\rho ,\theta } \right) = \left\{ \begin{array}{l} {1 - \frac{e}{\sin \theta },\arcsin e \le \theta \le \pi + \arcsin ({ - e} )}\\ \;\;\;\;\; 0 \qquad ,\pi \le \theta \le 2\pi \end{array} \right.$

According to the Eq. (10), the light intensity response curve for different offsets of the D-shaped aperture is simulated as shown in Fig. 7.

 figure: Fig. 7.

Fig. 7. The light intensity response curve for different offsets of the aperture.

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As shown in Fig. 7, the response curve has the steepest slope for offset e=0, i.e. the measurement has the highest sensitivity. We also find that the zero-point position of the response curve does not change with the offset e of the D-type aperture. When the absolute value of the offset e is less than 0.2, the signal intensity does not change significantly. In other words, the error caused by the deviation of the D-shaped aperture can be ignored.

3.6 Numerical aperture change of the objective lens

According to Eq. (1) and the definition of the normalized axial coordinate u, the numerical aperture (NA) of the converging lens Ls only influences the amplitude of k(0,v), but not changing the optical ${\nu _M}$ corresponding to maximal k(0,v). The simulation results of the optimal offset under a converging lens with different NA values are shown in Fig. 8.

 figure: Fig. 8.

Fig. 8. The effect of N.A. on k$({0,{v_M}} )$

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The simulation shows that when the detection beam remains circularly symmetrical, the optimal radial offset ${\nu _M}$ does not depend on the NA of the converging lens Ls. Therefore, there is no need to reset the virtual pinhole aperture and radial offset when the system changes a different objective lens to measure the ROC of different lenses.

4. Experiment

To verify the correctness and feasibility of the proposed method, a TDCRM system was constructed, as shown in Fig. 9. Specifically, a 632.8 nm He-Ni laser was used as the light source in the optical path. A D-shaped aperture was placed in the detection optical path to achieve transverse differential confocal detection. A standard objective lens produced by Zygo (NA=0.33) with a focal length of 151.74 mm was selected for the measurement objective lens. The experiments were performed at Lab temperature (20.0 ± 0.5°C), pressure = (102540 ± 60) Pa and humidity (42 ± 4%). The ROC of the outermost mirror of another standard objective lens produced by Zygo was used as a proof-of-principle system to test the accuracy and reliability of the proposed method.

 figure: Fig. 9.

Fig. 9. The experimental setup of TDCRM.

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4.1 Single-point focusing experiment with TDCRM

A single-point focusing experiment was performed at the cat’s-eye position to verify the TDCRM’s focusing accuracy. A servo motor was used to drive the test lens on an air-bearing guide to the cat’s-eye position along the optical axis. Axial scanning was performed based on this position, and a laser interferometer was used to monitor the lens position.

The pixel coordinates of the light spot centroid with the highest gray value (294, 341) were set as the center point. According to Eq. (5), the curve of the movement track of the spot is fitted as follows.

$$x ={-} 0.044 \cdot y + 308.9916 \;\;({x,y \in ({0,500} )} )$$

According to Eq. (6), the angle offset of the virtual pinhole is approximately 87.48°. The detector used an AM1160 gigabit black and white camera from the JoinHope Company. The CCD size was 500×500 pixels, and the pixel size was 8.3×8.3µm. The objective lens magnification was 25x. The equation for pixels on the actual CCD with a normalized radial offset is:

$$r(pixel) = \frac{\lambda }{\pi }\frac{{{f_C}}}{D}v \times {25/8.3}$$

The radial offset of the virtual pinhole on the CCD image plane was calculated to be approximately 27 pixels. The center coordinates of the two selected off-axis virtual pinholes were (295, 314) and (293, 398), respectively. A circular area with a radius of three pixels was used as the detection area of the virtual pinhole. As shown in Fig. 10, the axial differential response light intensity was obtained by the difference between the two curves of the average gray value detected from the two circular areas. The linear functional relationship of the axial response light intensity curve at the (−0.002236 mm, 0.007298 mm) area is fitted as follows.

$$I(z) = 197.9138 \cdot z - 0.548369,\;\;z \in ({ - 0.002236,0.007298} )\textrm{mm}$$

The optical coordinate of the cat’s-eye position of the test lens was calculated to be 0.002771 mm using Eq. (11).

 figure: Fig. 10.

Fig. 10. TDCRM focusing curve at the cat’s-eye position.

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According to the experimental results, the measured transverse differential confocal curve is consistent with the theoretical simulation and can be used to precisely detect the cat’s-eye and confocal positions. The linear measurement range of TDCRM is 9.53µm, and it has the highest fixed focus sensitivity.

4.2 ROC measurement

Before the measurement, the optical axis of the test lens needs to be aligned with the measurement optical axis. To do so, we first place the test lens at the cat’s eye position and then record the light spot position on the CCD. Then move the test lens to the confocal position and adjust its posture so that the light spot on the CCD is located at the same position as previously. If the position error of the two light spots is within one pixel, we assume that the optical axis of the test lens and the optical axis are in alignment.

The interferometer was set to zero when the test lens was near the cat’s-eye position. The coordinates of the cat’s-eye point and confocal point measured by the laser interferometer were Z1 and Z2, respectively. The ROC of the measured surface was the distance between the two positions, R = Z1-Z2. Figure 11 shows the transverse differential confocal curve of the cat’s-eye position and confocal position of the measured surface.

 figure: Fig. 11.

Fig. 11. Light intensity curve of cat’s-eye and confocal position of the measured surface.

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The ROC of the test lens was measured 10 times. The measurement process does not require manual intervention after the cat’s eye and confocal positions are set and a single ROC measurement takes only about 1 minutes. The average ROC was found to be −298.110460 mm with a standard deviation of 0.001014 mm and a relative standard deviation of 3.4 ppm. The results of 10 repeated measurements are shown in Fig. 12.

 figure: Fig. 12.

Fig. 12. Results of 10 repeated ROC measurements.

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In order to verify the accuracy of the TDCRM results, the same lens is tested by traditional differential confocal equipment with the relative measurement accuracy of 5 ppm in our lab. The average result obtained from ten-time differential confocal ROC measurements is −298.111024 mm which has a difference of 0.000564 mm between the TDCRM results. Therefore, TDCRM has the same measurement accuracy as the traditional differential confocal ROC measurement method.

5. Uncertainty analysis

In the process of the ROC measurements using the TDCRM, the uncertainty is primarily caused by the length measurement error of the laser interferometer (${u_1})$, the different axes between the optical axis of the laser and the optical axis of the system (${u_2}$), the surface shape error of the test lens (${u_3})$, misalignment of the center of the D-type aperture and optical axis (${u_4})$, the repeated measurement (${u_5})$.

The system selects a RENISHAWXL-80 laser interferometer, which has a measurement uncertainty of up to ${\pm} 0.5\;\textrm{ppm}\; ({k = 2} )$, for position monitoring. Thus, the uncertainty ${u_1}$ with the measurement rang L is:

$${u_1} = 1 \times {10^{ - 6}} \times L$$

Assuming the angle γ between the measuring optical axis and the optical axis of the interferometer, with the assistance of the CCD, the adjustment accuracy of the γ angle can reach 4′. The introduced measurement uncertainty ${u_2}$ is:

$${u_2} = L({1 - \cos \gamma } )\approx 0.68 \times {10^{ - 6}} \times L$$

When measuring the cat’s-eye position, the laser converged on a small area on the surface of the test lens. If this area does not coincide with the best reference sphere, the introduced error can reach the wavelength level. The PV value of standard reference mirror-form can be used to compensate for the error. Here, PV=0.1λ and the uncertainty ${u_3}$ with compensation is:

$${u_3} = 0.1PV \approx 0.006\;\;\mathrm{\mu}\textrm{m}$$

The misalignment between the center of the D-shaped aperture and the optical axis also influences the selection of the virtual pinhole position, and the resulting uncertainty can be negligible (${u_4} \approx 0$). The errors caused by the environments can be included in repeated measurement.

$${u_\textrm{5}} = \sqrt {\frac{{\sum\nolimits_{i = 1}^{10} {{{({{x_i} - \bar{x}} )}^2}} }}{{10 - 1}}} \approx 1.014\;\;\mathrm{\mu }\textrm{m}$$

The final composite uncertainty is:

$${u_c} = \sqrt {u_1^2 + u_2^2 + u_3^2 + u_4^2 + u_5^2} \approx 1.014\;\;\mathrm{\mu }\textrm{m}$$

The relative uncertainty of the measurement result of the ROC is:

$$u = \frac{{{u_c}}}{R} \times 100\%\approx 3.4\;\;\textrm{ppm}$$

It is difficult to adjust the defocusing amount of the two detectors to be exactly the same during the installation and alignment process for the differential confocal instrument, which is the main source of the measurement uncertainty of the differential confocal method. This TDCRM uses only one detection optical path to achieve differential confocal radius measurement, significantly reducing the uncertainty introduced by the defocusing error.

6. Conclusion

We propose a laser TDCRM based on the virtual pinhole technology, which detects two segments of the focal spot on the CCD. In this way, the axial response intensity of the sample position before and after the focal point is obtained. The absolute zero of the transverse differential confocal curve accurately corresponds to the differential confocal cat’s-eye and confocal positions, respectively, enabling high-precision and high-sensitivity ROC measurements. A TDCRM instrument was constructed, and its performance was tested. The average value of the ROC measurement was −298.110460 mm, with a standard deviation of 0.001014 mm and a relative standard deviation of 3.4 ppm. The linear range of TDCRM can reach approximately 9.53µm. The position and size of the virtual pinhole in TDCRM can be set by software, avoiding the disadvantages of traditional physical pinhole adjustment, and realizing single-channel off-axis detection, which significantly reduces the alignment difficulty and cost of the system.

Funding

China National Funds for Distinguished Young Scientists (51825501); National Key Research and Development Program of China (2017YFA0701203); Special Fund on Scientific Instruments of the National Natural Science Foundation of China (61827826).

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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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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Figures (12)
Fig. 1.
Fig. 1. Transverse laser differential confocal focusing principle.

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Fig. 2.
Fig. 2. A schematic diagram of TDCRM.

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Fig. 3.
Fig. 3. Effect of D-shaped aperture rotation. (a) Integration area with an angular offset (b) Centroid movement track of light spot.

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Fig. 4.
Fig. 4. (a) Axial response curves with different radial offsets (b) the measurement sensitivity at different virtual pinhole offsets changes.

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Fig. 5.
Fig. 5. The influence of virtual pinhole radius on measurement sensitivity

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Fig. 6.
Fig. 6. Aperture with an offset e

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Fig. 7.
Fig. 7. The light intensity response curve for different offsets of the aperture.

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Fig. 8.
Fig. 8. The effect of N.A. on k$({0,{v_M}} )$

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Fig. 9.
Fig. 9. The experimental setup of TDCRM.

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Fig. 10.
Fig. 10. TDCRM focusing curve at the cat’s-eye position.

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Fig. 11.
Fig. 11. Light intensity curve of cat’s-eye and confocal position of the measured surface.

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Fig. 12.
Fig. 12. Results of 10 repeated ROC measurements.

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Equations (19)

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U ( ν , φ , u ) = D 2 4 ∫ 0 π ∫ 0 1 P 1 ( ρ 0 , θ ) ⋅ P 2 ( ρ 0 , θ ) exp ⁡ ( j u ρ 2 ) × exp ⁡ [ − j ν ρ cos ⁡ ( θ − φ ) ] ρ d ρ d θ
{ I A ( u ) = | U A ( ν M , φ , u ) U A  -  max | 2 I B ( u ) = | U B ( ν M , φ  +  π , u ) U B − max | 2
I d i f f ( u ) = I B ( u ) − I A ( u ) = 4 π 2 { | ∫ 0 π ∫ 0 1 exp ⁡ ( j u ρ 2 ) ⋅ exp ⁡ [ − j ν M ρ cos ⁡ ( θ − φ − π ) ] ρ d ρ d θ | 2 − | ∫ 0 π ∫ 0 1 exp ⁡ ( j u ρ 2 ) ⋅ exp ⁡ [ − j ν M ρ cos ⁡ ( θ − φ ) ] ρ d ρ d θ | 2 }
{ i k = ∑ m ∑ n g ( i , j ) ⋅ i ∑ m ∑ n g ( i , j ) j k = ∑ m ∑ n g ( i , j ) ⋅ j ∑ m ∑ n g ( i , j )
{ k = 1 n ∑ i x i y i − 1 n ∑ i x i ⋅ 1 n ∑ i y i 1 n ∑ i x i 2 − ( 1 n ∑ i x i 2 ) 2 b = 1 n ∑ i y i − k ⋅ 1 n ∑ i x i
φ = arctan ⁡ ( k )
k ( 0 , v ) = ∂ I d i f f ( u ) ∂ u | u = 0
v p i x e l = 4.2 λ π f C D × β p
I d i f f _ _ _ _ ⁡ ( r p , u ) = 2 π ∫ 0 r p I d i f f ( v , u ) v d v π r p 2
U 1 ( r , φ , z ) = D 2 4 ∫ arcsin ⁡ e π + arcsin ⁡ ( − e ) ∫ e sin ⁡ θ 1 P 1 ( D 2 ρ , θ ) ⋅ P 2 ( D 2 ρ , θ ) exp ⁡ ( j u ρ 2 ) exp ⁡ [ − j ν ρ cos ⁡ ( θ − φ ) ] ρ d ρ d θ
x = − 0.044 ⋅ y + 308.9916 ( x , y ∈ ( 0 , 500 ) )
r ( p i x e l ) = λ π f C D v × 25 / 8.3
I ( z ) = 197.9138 ⋅ z − 0.548369 , z ∈ ( − 0.002236 , 0.007298 ) mm
u 1 = 1 × 10 − 6 × L
u 2 = L ( 1 − cos ⁡ γ ) ≈ 0.68 × 10 − 6 × L
u 3 = 0.1 P V ≈ 0.006 μ m
u 5 = ∑ i = 1 10 ( x i − x ¯ ) 2 10 − 1 ≈ 1.014 μ m
u c = u 1 2 + u 2 2 + u 3 2 + u 4 2 + u 5 2 ≈ 1.014 μ m
u = u c R × 100 % ≈ 3.4 ppm
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