## Abstract

The measuring probe integrated with multiple fiber point-diffraction sources can be applied to measure both the three-dimensional coordinates and highly accurate point-diffraction wavefront. The probe determines the achievable measurement accuracy of fiber point-diffraction interferometer (PDI), in which the fiber exit end plane is required to be parallel with the detector plane. The probe misalignment due to fabrication error could introduce significant measurement error. A high-precision method is proposed to calibrate the probe misalignment in fiber PDI, including the central positioning based on phase difference and tilt adjustment based on Zernike polynomials fitting. Both numerical simulation and experiments have been carried out to demonstrate the feasibility and accuracy of the proposed probe misalignment calibration method. The proposed method provides a feasible way to address the processing uncertainty on measuring probe in fiber PDI, and enables high-precision geometry alignment and misalignment calibration in the interferometric testing systems with case of no imaging lens.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

With the development of optical fabrication and testing, the interferometry has been widely applied in high-precision size measurement [1], three-dimensional (3D) positioning [2], optical surface and system testing [3,4], etc. Traditional interferometers, such as Fizeau interferometer and Twyman-Green interferometer, can be applied to achieve the high-precision wavefront testing in order of nanometers. The novel point-diffraction interferometer (PDI) has also been proposed to improve the testing accuracy better than sub-nanometers [5–8], and it further extends the application of interferometry in ultrahigh-precision testing. Different from traditional interferometers, PDI employs an ideal diffraction spherical wavefront from either a pinhole or optical fiber as the reference standard, rather than the standard lens, and it can achieve both high measurement accuracy and repeatability.

The PDI technology has been applied to achieve surface metrology [9,10], high-precision measurement of 3D absolute displacement [2,11–13] and diffraction wavefront error [7], etc. In the fiber PDI system, two fibers are integrated in measuring probe, and the diffraction spherical wavefronts generated from fiber sources interfere with each other. According to the phase distribution in point-diffraction interference field, the 3D coordinates of measuring probe can be precisely retrieved. Besides, a high-precision shearing-interferometry-based method, in which 3D coordinate reconstruction of measuring probe is performed to calibrate the systematic geometrical aberration, has been proposed to measure the point-diffraction wavefront [7]. In the 3D coordinate reconstruction of measuring probe, either in absolute displacement measurement or shearing wavefront measurement for point-diffraction wavefront retrieval, the fiber exit end plane (that is probe end face) is required to be parallel with detector plane. Otherwise, the probe misalignment could introduce significant measurement error. Traditionally, the spatial orientation of measuring probe can be evaluated and adjusted according to the light spots distribution. However, it cannot achieve high-precision misalignment calibration, and even no longer feasible in the case that the fibers integrated in probe are not parallel due to the fabrication error. A method based on ray tracing and digital image processing [14,15] can be applied to calibrate the systematic error introduced by misalignment, however, the tilt misalignment in fiber PDI system cannot be removed due to the unknown probe status.

The tilt misalignment between the detector plane and probe end face could introduce an additional optical path difference and systematic error in the measurement result. In this paper, a high-precision method based on phase difference and Zernike polynomials fitting is proposed to remove and calibrate the probe misalignment in fiber PDI for the measurement of 3D absolute displacement and shearing wavefront (which can be applied to retrieve the point-diffraction wavefront). In the proposed method, a central positioning method based on the rigorous model for phase difference is proposed to translate the probe to central optical axis, and then probe tilt is adjusted according to Zernike polynomials fitting to remove tilt misalignment. Section 2 presents the principle of the PDI, analysis of probe misalignment and the proposed calibration method. In Section 3, numerical simulation and experimental results are given to demonstrate the feasibility of the proposed method. Some conclusions are drawn in Section 4.

## 2. Principle of fiber PDI and probe misalignment calibration

#### 2.1 System layout

Figure 1 shows the basic schematic diagram of fiber PDI system, which can be applied to measure both the 3D absolute displacement [11–13] and point-diffraction wavefront [7]. According to Fig. 1, multiple fiber-diffraction sources with certain lateral displacement are integrated in measuring probe, and the fiber exit ends are coplanar on the probe end face. The coherent laser beams are coupled into single-mode fibers, and point-diffraction waves generated from submicron-aperture exit ends of fibers interfere on CCD detector. According to the phase distribution in the interference field of point-diffraction waves, the 3D coordinates of probe with respect to CCD detector can be measured with iterative reconstruction algorithm. Besides, the measured 3D coordinates can also be applied to calibrate the geometrical aberration due to the lateral displacement of fiber exit ends [7] and achieve accurate shearing wavefront measurement, in which no imaging lens is utilized. To improve the reconstruction accuracy and accelerate retrieve speed, the end face of measuring probe is required to be parallel with detector plane, by which the number of iterative variables to be solved is reduced to achieve rapid measurement. Thus, a general and effective calibration method is required to remove and eliminate the probe misalignment.

#### 2.2 Probe misalignment calibration method

Traditionally, the measuring probe can be adjusted according to bright spot distribution, by which the tilt error of fiber exit ends can be calibrated. For the ideal probe configuration, the fibers are parallel integrated in probe, and the midpoint of spot centers would remain stable when translating CCD detector back and forth along optical axis direction, as is shown in Fig. 2(a). However, the achievable calibration precision based on traditional image-processing-based method is very limited, for it is quite sensitive to image noise. Besides, it is not easy to obtain the ideally parallel fibers in the probe due to fabrication error, and the tilt misalignment could introduce significant measurement error, as is shown in Fig. 2(b).

Figure 3(a) shows the fringe pattern and corresponding light spots captured by CCD detector with actual measuring probe, in which long exposure time is set to exhibit the spot centers. According to Young’s interference, the connecting line between two fiber exit ends (or light spots on detector plane) should be perpendicular to fringe direction when the fibers in probe are parallel, as is shown in Fig. 2(a). Due to the fiber tilt in measuring probe, the connecting line between captured light spots in actual case is not perpendicular to fringe direction, as is shown in Fig. 3(a). In this case, the probe adjustment according to traditional calibration based on spot distribution would lead to the final probe orientation being shown in Fig. 3(b). Though the midpoint of spot centers remains unchanged when translating the probe along optical axis direction, there could be end face tilt in the measuring probe, and a significant residual error would be introduced in 3D absolute displacement and shearing wavefront measurement. A general method, including the central positioning based on phase difference and tilt adjustment based on Zernike polynomials fitting, can be applied to achieve high-precision calibration of probe misalignment.

### 2.2.1 Central positioning based on phase difference

To achieve high-precision probe misalignment calibration, the central positioning based on phase difference is firstly carried out to translate measuring probe to the central position (optical axis) with respect to detector plane in lateral directions. Without loss of generality, we take the tilt misalignment about *y*-axis as the case to be analyzed. Figure 4(a) shows the schematic diagram of the proposed central positioning method based on phase difference. In Fig. 4(a), ${S_1}({x_1},{y_1},{z_1})$ and ${S_2}({x_2},{y_2},{z_2})$ refer to the fiber exit ends before the adjustment, ${S^{\prime}_1}({x^{\prime}_1},{y^{\prime}_1},{z^{\prime}_1})$ and ${S^{\prime}_2}({x^{\prime}_2},{y^{\prime}_2},{z^{\prime}_2})$ are those after 180° rotation of measuring probe with lateral translation; the distances between fiber exit ends (${S_1}$ and ${S_2}$) and an arbitrary point $P(x,y,z)$ on detector plane are ${R_1}$ and ${R_2}$, respectively, and those between fiber ends (${S^{\prime}_1}$ and ${S^{\prime}_2}$) and the point *P* are ${R^{\prime}_1}$ and ${R^{\prime}_2}$. According to Fig. 4(a), we have phase distributions $\varphi $ and $\varphi ^{\prime}$,

*α*is the tilt angle of probe end face. The distance $2d$ could influence the measurement range [11] and introduce various geometrical aberrations [7], respectively in the 3D coordinate measurement and shearing point-diffraction wavefront measurement with fiber PDI.

According to Eq. (1), we may define the phase difference $\Delta \varphi = \varphi - \varphi ^{\prime}$ between the phase distributions $\varphi (x,y,z)$ and $\varphi ^{\prime}(x,y,z)$, which could be applied to evaluate the symmetry of the probe position before and after 180° rotation. By translating the probe in *y* direction (similar in *x* direction), the central positioning of probe in *y* direction can be achieved when ${x_\textrm{c}} = {x^{\prime}_\textrm{c}}$, ${y_\textrm{c}} = { - }{y^{\prime}_c}$ and ${z_\textrm{c}} = {z^{\prime}_\textrm{c}}$, as is shown in Fig. 4(b), and we have

### 2.2.2 Tilt adjustment based on Zernike polynomials fitting

After locating the measuring probe at the central position with respect to detector plane in lateral directions, the tilt adjustment based on Zernike polynomials fitting can be carried out to remove probe misalignment. According to Fig. 4(b), we have the coordinates of two fiber exit ends, ${S_1}(0,d\cos \alpha ,{z_c} + d\sin \alpha )$ and ${S_2}(0,{ - }d\cos \alpha ,{z_c}{ - }d\sin \alpha )$, and the optical path difference $OPD(x,y,z)$ between ${S_1}$ and ${S_2}$ can be obtained in the cylindrical coordinate system ($x = r\cos \theta $, $y = r\sin \theta $) as

*r*and $\theta$ are the polar radius and angle on detector plane. Based on the Taylor expansion, the $OPD$ in Eq. (6) can be written as

*y-*tilt terms, respectively, ${a_0}$ and ${a_2}$ are the corresponding coefficients. From Eqs. (6) and (8), we have the coefficient ${a_2}$ for

*y-*tilt term,

*x*and

*y*directions reach their maximum.

Figure 5 shows the procedure for the proposed probe misalignment calibration method, including the central positioning based on phase difference and tilt adjustment based on Zernike polynomials fitting. The phase distribution $\varphi $ corresponding to the probe position $({x_c},{y_c},{z_c})$ can be obtained by phase-shifting method. The probe is 180° rotated about *z*-axis and then translated in lateral directions to acquire various phase distributions $\varphi ^{\prime}$, with which the phase differences $\Delta \Phi = (\Delta {\varphi _x},\Delta {\varphi _y})$ can be obtained to locate the probe position in *x* and *y* directions according to Section 2.2.1. The central position of probe with respect to detector plane can be located at the position $[{{({x_c}{ + }{{x^{\prime}_c}})} \mathord{\left/ {\vphantom {{({x_c}{ + }{{x^{\prime}_c}})} 2}} \right.} 2},\textrm{ }{{({y_c} + {{y^{\prime}_c}})} \mathord{\left/ {\vphantom {{({y_c} + {{y^{\prime}_c}})} 2}} \right.} 2},{{\textrm{ }({z_c} + {{z^{\prime}_c}})} \mathord{\left/ {\vphantom {{\textrm{ }({z_c} + {{z^{\prime}_c}})} 2}} \right.} 2}]$, $({x^{\prime}_\textrm{c}},{y^{\prime}_c},{z^{\prime}_c})$ is the position where both the phase differences $\Delta {\varphi _x}$ and $\Delta {\varphi _y}$ reach their minimum, as is depicted in Fig. 4(b). After the central positioning of measuring probe based on phase difference, the tilt adjustment is carried out to remove probe misalignment. For the probe located at central position, the wavefront ${W^{({\boldsymbol \alpha })}}$ corresponding to the probe tilt angle position ${\boldsymbol \alpha } = ({\alpha _x},{\alpha _y})$ can be measured and fitted with Zernike polynomials, with the corresponding tilt coefficients ${\textbf T} = (a_1^{({\boldsymbol \alpha })},a_2^{({\boldsymbol \alpha })})$ in *x* and *y* directions. By adjusting probe tilt angle, the null-tilt angle position ${{\boldsymbol \alpha }_0}$ can be obtained with both the tilt coefficients ${\textbf T} = (a_1^{({{\boldsymbol \alpha }_0})},a_2^{({{\boldsymbol \alpha }_0})})$ reaching their maximum. Based on the calibration process mentioned above, the probe misalignment can be well calibrated to make the probe end face be parallel to detector plane.

## 3. Numerical simulation and experimental results

To validate the feasibility and accuracy of the proposed method for probe misalignment calibration in fiber PDI system, both the numerical simulation and experiments have been carried out.

#### 3.1 Numerical simulation results

In the numerical simulation, a probe integrated with two fiber exit ends is modeled and the ray tracing method is applied to analyze the measurement results. The distance of two fiber exit ends is 125 µm, and the connecting line is perpendicular to *x*-axis and the tilt angle between the connecting line and *xy* plane is set to 3°. The numerical analysis of measurement errors for both the displacement and shearing wavefront under various probe tilt angles has been performed, in which the probe being placed at the position (0 mm, 0 mm, 100 mm), and Fig. 6 shows the corresponding results. Figure 6(a) is the displacement measurement errors corresponding to the 5 mm probe movement in *x* and *z* directions from the original position, respectively. Figure 6(b) presents the PV and RMS values of obtained shearing wavefront errors with various NAs, which is based on coordinate-reconstruction-based systematic geometrical aberration calibration. It can be seen from Fig. 6 that the measurement error grows significantly with the probe misalignment, both in the displacement and shearing wavefront measurement. For the 1° probe tilt angle, the maximum displacement measurement error can reach tens of microns, and the PV and RMS values of shearing measurement error with 0.10 NA are 103.3831 nm and 16.7280 nm, respectively. Thus, high-precision probe misalignment calibration is required to minimize the measurement error.

In the central positioning of probe based phase difference, Figs. 7(a) and 7(b) show the RMS values of the phase differences $(\Delta {\varphi _x},\Delta {\varphi _y})$ corresponding to various translation positions in *x* and *y* directions, respectively, where the probe is originally placed at the position (5 mm, 8 mm, 100 mm). According to Fig. 7, both the phase differences $\Delta {\varphi _x}$ and $\Delta {\varphi _y}$ reach their minimum when the 180° rotated probe is located at the symmetric position (−5 mm, −8 mm, 100 mm) in both *x*- and *y*-axes with respect to the original position.

Based on the central positioning of probe, the tilt misalignment calibration based on Zernike polynomials fitting can be carried out according to Subsection 2.2.2. Figure 8(a) shows the Zernike tilt coefficients corresponding to various probe tilt angles in *y* direction, in which the tilt coefficient reaches its maximum as the tilt angle is 0°. Figure 8(b) presents the effects of central positioning error on the tilt misalignment calibration result. It can be seen from Fig. 8(b) that the tilt deviation grows linearly with central positioning error, with the corresponding sensitivity being about 8.1029×10^{−4} degree/µm. The probe tilt calibration accuracy can reach 2.9′′ for the 1 µm lateral positioning accuracy.

#### 3.2 Experimental results

According to Fig. 1, an experimental fiber PDI system for 3D absolute displacement measurement has been set up to validate the feasibility of the proposed probe misalignment calibration method. The pixel size of CCD sensor is 5.5 µm (H) × 5.5 µm (V), and the pixel number is 1920 (H) × 1080 (V). The distance between two fiber exiting ends integrated in measuring probe is 125.00 µm. To enable the precise adjustment of probe tilt and make probe end face be parallel with detector plane, the probe is installed on a precise tilt adjuster with the accuracy of 6'’ and 3D linear stage with 1 µm positioning precision. The probe is placed at the original stage-coordinate position (6.0000 mm, 6.0000 mm, 2.0000 mm), corresponding to the actual distance about 100 mm away from the CCD camera.

### 3.2.1 Probe misalignment calibration

According to Subsection 2.2.1, the central positioning based on phase difference is firstly carried out to translate the probe to the central position with respect to the detector plane in lateral directions, and the corresponding interferograms are captured with CCD detector. By translating the probe in *x* and *y* directions, respectively, the corresponding measured phase differences are presented in Fig. 9, and the phase differences $\Delta {\varphi _x}$ and $\Delta {\varphi _y}$ reach their minimum at the positions *x* = −13.3926 mm and *y* = −7.0368 mm. Thus, the symmetric stage-coordinate position (−13.3926 mm, −7.0368 mm, 2.0000 mm) can be obtained, and the central position with respect to detector plane is located at (−3.6963 mm, −0.5184 mm, 2.0000 mm).

After translating measuring probe to the central position, the tilt adjustment based on Zernike polynomials fitting according to Subsection 2.2.2 is performed to calibrate the misalignment. Figure 10 shows the Zernike tilt coefficients of measured wavefront corresponding to various adjusting tilt angles of probe. With the measured discrete data set, high-order polynomial fitting is applied to smooth the measured data and obtain the optimal maximum point. According to the six-order polynomial fitting curves shown in Figs. 10(a) and 10(b), the Zernike tilt coefficients in *x* and *y* directions reach their maximum at the adjusting angles 2.1306° and 1.9758° about *x-* and *y-*axes, respectively. By adjusting the tilt angle position of probe about *x*-axis to 2.1306° and that about *y-*axis to 1.9758°, the probe tilt can be well removed.

### 3.2.2 Evaluation of probe misalignment calibration in PDI measurement

To analyze the effect of probe misalignment on the 3D coordinate measurement in fiber PDI [16] and the coordinate-reconstruction-based systematic geometrical aberration calibration in shearing wavefront measurement (especially in the case of high NA and large lateral displacement) [7], the comparison experiment before and after probe misalignment is performed.

In the 3D coordinate measurement with fiber PDI, an additional spatial tilt is introduced in measuring probe. The probe is moved from the original position (0 mm, 0 mm, 100 mm) by 50 mm distance in *x*, *y* and *z* directions at the step of 5 mm, respectively. Besides, the measurement with a high-precision coordinate measurement machine (CMM) (HEXAGON Leitz PMM-C, positioning accuracy 0.50 µm) is carried out for comparison, and the measured displacement is taken as the nominal value. Figures 11(a), 11(b) and 11(c) show the displacement measurement errors for the movement in *x*, *y* and *z* directions before and after probe misalignment calibration, respectively, and the corresponding RMS values are summarized in Table 1. According to Fig. 11 and Table 1, a significant systematic error with RMS value greater than 5 µm due to probe misalignment can be seen in the displacement measurement results. With the proposed misalignment calibration method, the systematic error can be well eliminated and the RMS value less than 1.5 µm is achieved, making an obvious improvement in measurement accuracy.

To further validate the 3D coordinate measurement accuracy on the basis of probe misalignment calibration, the coordinate-reconstruction-based systematic geometrical aberration calibration is performed in the shearing point-diffraction wavefront measurement with fiber PDI. Both the lateral and longitudinal displacements between two fiber exit ends may introduce significant geometrical aberrations in the measurement of shearing point-diffraction wavefront, and it places ultrahigh requirement on the systematic error calibration. The measuring probe is placed at the position about 25 mm away from the CCD camera, and the measured shearing wavefronts are shown in Fig. 12, whose PV and RMS values before probe misalignment calibration are 23.9595 nm and 3.8396 nm, respectively, while those after probe misalignment calibration are 8.0539 nm and 1.1908 nm. According to Fig. 12(a), an obvious systematic geometrical aberration due to 3D coordinate measurement error, which is introduced by probe misalignment, can be seen in the measured shearing wavefront. After calibrating the probe misalignment calibration, the RMS value of measured shearing wavefront decreases from 3.8396 nm to 1.1908 nm, as is shown in Fig. 12(b). Thus, the probe misalignment calibration enables the significant accuracy improvement for 3D coordinate measurement and the corresponding systematic geometrical aberration calibration.

Several factors may lead to the calibration error for the probe misalignment in fiber PDI, including the environmental disturbance, CCD noise and stage positioning error, etc. To minimize the effect of the environmental disturbance and CCD noise, the experimental setup is placed on an active vibration isolation table and shielded in a heat-insulating box, and the measurement data is averaged from multiple measurements; besides, high-order polynomial curve fitting is also applied to fit measured discrete data set. Due to the fact that the probe misalignment calibration is achieved on the basis of central positioning and tilt adjustment, both the positioning errors of 3D linear translating stage and tilt adjuster could introduce an additional error in the calibration result. By employing a multi-axis motion stage with higher positioning precision and resolution, the calibration accuracy can be expected by further improvement.

## 4. Conclusion

In this paper, we put forward a high-precision method to calibrate the tilt misalignment of measuring probe in fiber PDI, which can be applied to measure both the three-dimensional absolute displacement and highly accurate point-diffraction wavefront. The double-step calibration, including the central positioning of probe based on phase difference and tilt adjustment based on Zernike polynomials fitting, is performed to remove probe misalignment. The calibration can be carried out with six-axis motion stage, requiring no additional high-precision and costly measuring instruments. Both the numerical simulation and experiments have been performed to demonstrate the feasibility of the proposed calibration method, and a good measurement accuracy is achieved. The proposed method enables high-precision probe misalignment calibration and loosens the requirement on the processing of measuring probe in fiber PDI. It also provides a feasible way to align the system geometry and calibrate the geometrical aberration in the interferometric testing systems with case of no imaging lens, especially those with large wavefront displacement and high NA.

## Funding

China Postdoctoral Science Foundation (2017M621928); National Natural Science Foundation of China (51775528, 61805048); Guangxi Key Laboratory of Optoelectronic Information Processing (GD18205).

## Disclosures

The authors declare no conflicts of interest.

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