1. Introduction

With the rapid development of modern fabrication and testing, the three-dimensional coordinate measuring technique has become a powerful tool for high-precision measurement, design and machining, such as the space location, size measurement and reverse engineering, etc. The existing coordinate measuring technique mainly includes coordinate measuring machine (CMM) and multilateration technique [1, 2]. However, the CMM needs costly guide rails and huge platform, and it is not feasible for in situ measurement; several laser trackers are required to realize the absolute coordinate measurement in multilateration, and the form error and material inhomogeneity of the retro-reflector target could introduce additional measurement uncertainty. A new point-diffraction interferometer (PDI) with two point-diffraction sources [3–7] has been proposed to measure the absolute three-dimensional coordinates, in which the single-mode fiber is used as the point-diffraction source. The new PDI avoids the measurement uncertainty introduced by the imperfect target in multilateration, and allows the target (made of two point-diffraction sources) to take free movement within a volumetric space over the numerical aperture (NA) of point-diffraction wave.

The PDI method [8–13] employs point-diffraction spherical wavefront as ideal measurement reference, and it can achieve the precision better than ${10}^{-3}\lambda$ (the wavelength $\lambda =632.8\text{nm}$) [14–16]. The PDI requires no precise and costly standard parts, provides a feasible way to overcome the accuracy limitation of reference optics in traditional interferometers (such as Fizeau interferometer), and has got wide application in high-precision spherical surface testing. The achievable precision of PDI is mainly determined by the fundamental process of point diffraction, rather than the accuracy of standard reference parts. Thus, a good reproducibility of measurement precision can be realized with PDI. The pinhole and optical fiber are generally used as point-diffraction sources to generate an ideal spherical wave. In the pinhole PDI [8, 9], the spherical wavefront over high NA can be obtained with small pinhole of submicron (or even smaller) diameter; however, the poor diffraction light power due to the low transmittance (<0.1‰) of pinhole could introduce significant noise in pinhole PDI, and it also places ultra-high requirement on the CCD detector and adjustment of measurement system. In the PDI with single-mode fiber [17, 18], high light transmittance (>10%) can be obtained, however, the NA of diffraction wavefront is limited by the fiber NA (<0.20) [19]. Thus, neither the pinhole nor single-mode fiber can be applied to get high diffraction light power and high-NA diffraction wavefront, which limits the measurement range of PDI.

The measurement range of the new PDI for three-dimensional coordinate measurement could be significantly limited by the low NA of fiber-diffraction wavefront, and the lateral measurement range is much smaller than that in longitudinal direction. In this paper, we propose a fiber PDI with submicron aperture, which is based on the single-mode fiber with a narrowed exit aperture, for the absolute three-dimensional coordinate measurement within large aperture angle range. Both the high diffraction light power and high-NA spherical wavefront is obtained with the proposed PDI, and it provides a feasible way to extend the measurement range of the system. With the precise phase demodulated from point-diffraction interference field, a double-iterative method based on Levenbery-Marquardt (L-M) algorithm is applied to determine the coordinates of target. Section 2 presents the principle and basic theory of the proposed submicron-aperture fiber PDI. The analysis based on finite difference time domain (FDTD) method is carried out to determine the structure of fiber aperture. Section 3 and 4 show the numerical simulation and experimental results to verify the feasibility of the proposed measurement system, respectively.

2. Principle and basic theory of submicron-aperture fiber PDI

2.1 System layout

The system layout of the submicron-aperture fiber PDI for absolute three-dimensional coordinate measurement is shown in Fig. 1.The linearly polarized beam from a 532nm frequency-stabilized laser passes through a half-wave plate (HWP1), and then is divided into two parts by a polarized beam splitter (PBS). The p-polarized beam is coupled into a single-mode fiber (SF1) after passing through the PBS and half-wave plate (HWP2). The s-polarized beam is reflected by the PBS, and then is transformed to be p-polarized after passing through a quarter-wave plate (QWP) with fast axis oriented at −45° horizontally twice, respectively, before and after reflecting at a mirror mounted on the PZT scanner; and then, it passes through the PBS and is coupled into another single-mode fiber (SF2). Both the exit ends of the fibers SF1 and SF2 are integrated into a target with certain lateral offset, with the output point-diffraction waves $W_1$ and $W_2$ interfering on the CCD detector. The precise phase distribution corresponding to the point-diffraction interference field can be obtained with the phase-shifting method by translating the PZT scanner, from which the three-dimensional coordinates of target can be measured with numerical iterative reconstruction algorithm.

 

Fig. 1 System layout of the submicron-aperture fiber PDI for absolute three-dimensional coordinate measurement.

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2.2 Application of submicron-aperture fiber

Due to the limitation of the core diameter of single-mode fiber, the NA of diffraction wavefront in fiber PDI is generally less than 0.20. To increase the NA of diffraction wavefront with optical fiber, a single-mode fiber with submicron aperture is applied as point-diffraction source. Figure 2(a) shows a scanning electron microscope (SEM) photo of the single-mode-fiber-based diffraction source with narrowed exit end applied in the PDI, which is formed with the same processing technology as manufacturing of fiber-based probes for the scanning near-field optical microscopy [20], with the exit aperture diameter about 0.5 μm. Both the exit ends of two single-mode fibers are integrated in a target with lateral offset $d=125\text{μm}$, as is shown in Fig. 2(b).

 

Fig. 2 Single-mode fiber with submicron aperture. (a) SEM photo of submicron-aperture fiber, (b) structure of target in the PDI for coordinate measurement.

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2.3 Reconstruction of absolute three-dimensional coordinates

According to the PDI system shown in Fig. 1, the absolute three-dimensional coordinates of target can be measured from the phase distribution in interference field corresponding to the optical path difference (OPD), as is shown in Fig. 3.

 

Fig. 3 Model for three-dimensional coordinate reconstruction.

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Denote the plane of CCD detector as the xy plane and the central position o, with the distances $r_1$ and $r_2$ between an arbitrary point $P(x,y,z)$ on the CCD detector and the exit apertures of two fibers on the target, as is shown in Fig. 3, and we have the phase difference $\phi (x,y,z)$,

There are 6 unknowns in Eq. (2), and it needs at least 6 equations corresponding to 6 pixels to obtain the space coordinates $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ of two fiber apertures. To improve the measurement precision and anti-interference ability of PDI system, more than 6 pixels could be adopted to reconstruct the coordinates under measurement, and Eq. (4) can be transformed to a quadratic functional as

Thus, the space coordinates of two fiber apertures can be determined from the global minimum ${\Phi }^{*}$ of the function $F$ (that is the least-square solution of Eq. (4)), with a single true solution set.

Either the Gauss-Newton (G-N) algorithm or Levenbery-Marquardt (L-M) algorithm can be applied to solve the problem of the highly-nonlinear least squares in Eq. (5). Compared to the G-N algorithm, the L-M method is more stable and can avoid the problem of iterative matrix being singular, and it is used to determine the three-dimensional coordinate because of its robustness. Due to the high nonlinearity of Eq. (5), it is easy to be run into the local optimal solution, and a double-iterative method based on L-M algorithm is proposed to overcome the problem. Figure 4 shows the flow diagram for the double-iterative method based on L-M algorithm for three-dimensional coordinate reconstruction, which has double iterations. In the first iteration of the L-M-based double-iterative method, certain pixels (pixel number $n_1>6$) are selected to find the local optimal solution ${\Phi }_1$ rapidly with L-M algorithm, in which the common value ${\Phi }_0=[1,1,1,1,1,1]$ is taken as an arbitrary initial iterative value; more pixels (pixel number $n_2>n_1$) are chosen to find out the global optimal solution ${\Phi }^{*}$ (those are the coordinates under measurement) according to the L-M method in the second iteration, with the local optimal solution ${\Phi }_1$ obtained in the first iteration as the initial iterative value.

 

Fig. 4 Procedure for double-iterative method based on L-M algorithm for three-coordinate reconstruction.

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3. Numerical simulation results

3.1 Analysis of diffraction wavefront

The fiber with submicron aperture is a key element in the proposed PDI system, with which the ideal diffraction spherical wave with both high NA and high light power can be obtained. The achievable sphericity and NA of diffraction wavefront, as well as the light transmittance, are mainly determined by the exit aperture size of the fiber, which could be estimated by numerical analysis based on the FDTD method [16, 21].

Figure 5 shows the simulation results about the full aperture angle of diffraction wave and light transmittance for various fiber apertures. According to Fig. 5, the light transmittance decreases with the decreasing of fiber aperture, with an obvious increase in the aperture angle of diffraction wave. The full aperture angle of diffraction wave and light transmittance corresponding to the 0.5 μm fiber aperture are about 160° and 67%, respectively, and it provides the necessary conic boundary to extend the lateral measurement range almost within a half space. Figure 6 exhibits the diffraction wavefront error within various NA ranges for different fiber exit apertures. From the analysis results in Fig. 6, the diffraction wavefront error grows with both the increase of NA range and fiber aperture. The wavefront error RMS corresponding to the fiber aperture of 0.5 μm diameter over 0.70 NA is better than $\lambda /1000$, and it can be taken as ideal spherical wavefront for the measurement of three-dimensional coordinates.

 

Fig. 5 Aperture angle of diffraction wave and light transmittance for various fiber apertures.

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Fig. 6 Diffraction wavefront error within various NA ranges for different fiber apertures.

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3.2 Analysis of coordinate reconstruction

Computer simulation is carried out to check the proposed method for three-dimensional coordinate measurement, in which the pixel number of CCD detector is set to 640 × 480 and the pixel size $4.65\text{μm}\times 4.65\text{μm}$. For convenience of study, the middle point of two fiber exit apertures is taken as the target under analysis. The initial space coordinates of two fiber exit apertures are set to $(0,15,200)$ and $(-0.75,15,200)$, and those corresponding to the target is $(-0.375,15,200)$, with all the dimensions in unit of millimeters. In the proposed double-iterative method based on L-M algorithm for coordinate reconstruction, 20 pixels uniformly distributed over the whole CCD pixel range are selected to get the local optimal solution in the first step; and then, 48 pixels with uniform distribution in the same pixel region as that in the first step are applied to obtain the global optimal solution, with the corresponding measurement error of target coordinates about (−0.04 nm, 0.15 nm, 0.98 nm).

To study the effect of the pixel region selected for coordinate reconstruction on the measurement results, the pixels over various CCD pixel areas are taken to reconstruct the target coordinates. Figure 7 shows the measurement error over various pixel regions proportional to the area 640 × 480 pixels, with both the same ratio in horizontal and vertical directions. According to Fig. 7, the measurement precision increases with the pixel area, and it reaches the order of nanometer when the pixel area is larger than 320 × 240 pixels, with the coordinate reconstruction error negligible corresponding to the pixel area 640 × 480 pixels.

 

Fig. 7 Coordinate measurement error in various CCD pixel areas.

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To analyze the feasibility of the proposed coordinate reconstruction method, the random noise with root mean square (RMS) value $\lambda /100$ ($\lambda =532\text{nm}$) is added to the captured interferograms, and Fig. 8 shows the measurement results within the volumetric space 300 mm × 300 mm × 300 mm. According to Fig. 8, the RMS values of measurement error in x (and y) and z directions are 0.15 μm and 0.24 μm, respectively, with the corresponding RMS values about 0.056nm and 0.086nm in the cases without random noise. Thus, the measurement precision of the proposed method for coordinate measurement can reach the order of nanometer, and better than sub-micrometer even with the existence of random noise, which confirms the feasibility of the proposed measurement method with L-M-based double-iterative algorithm.

 

Fig. 8 Three-coordinate measurement error with random noise of λ/100 RMS in numerical simulation. Errors in (a) x and y directions, (b) z direction corresponding to the initial target position (−0.375 mm, 15 mm, 200 mm).

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4. Experimental results and analysis

An experimental fiber PDI system with aperture size about 0.5 μm, in which the pixel number of CCD detector is 1920 × 1080 and the corresponding pixel size being $4.65\text{μm}\times 4.65\text{μm}$, has been set up for the measurement of three-dimensional coordinates. The full aperture angle of diffraction wavefront and light transmittance about 155° and 3% are realized with the submicron-aperture fiber. According to Section 3.1, the sphericity of diffraction wavefront over 0.70 NA is better than $\lambda /1000$ RMS, and it can be taken as an ideal spherical wavefront.

The target in the proposed fiber PDI system is installed on the probe of a high-precision CMM, the positioning accuracy of which is about 1.0 μm, and the three-dimensional coordinate measurement is carried out to verify the proposed method. For the target is light in weight (about 3.5 g), its effect on the measurements is negligible. Figure 9 shows the measurement results about the coordinate deviations in x and z directions, where the nominal values are the coordinates measured with CMM. According to Fig. 9, a good agreement can be observed between the CMM results and those from the proposed fiber PDI system, with the RMS values of coordinate measurement errors 0.47 μm and 0.68 μm in x and z directions, respectively. Several factors could introduce the differences of coordinate measurement results in the control experiment, including the fringe demodulation error due to the limitation of CCD resolution and environmental disturbance, as well as the measurement error of CMM. Due to the reduction in the light intensity and broadening of fringe spacing, the coordinate measurement error would also grow with the increase of measurement volumetric space.

 

Fig. 9 Three-coordinate measurement error in experimental validation. Errors in (a) x direction, and (b) z direction corresponding to the initial target position (−0.375 mm, 15 mm, 200 mm).

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The whole experimental system is placed on an active vibration isolation platform and shielded with heat-insulating box, by which the influence of the environmental disturbance can be minimized. The achievable measurement range of the proposed system is limited by several hardware factors in practical application, and the major ones are the detector size and light power [6], though it is of no theoretical limit in principle. Compared to the traditional fiber PDI, the proposed method provides a feasible way to obtain the ideal diffraction spherical wavefront with the aperture angle almost covering half space, by which the lateral measurement range can be greatly extended. The performance of the PDI system can be further improved by increasing the size of CCD detector, lateral offset of two fiber apertures and source power.

To evaluate the stability of the proposed measurement method, repeated measurements at the positions (60 mm, 0 mm, 100 mm) and (140 mm, 0 mm, 100 mm), as well as those of (0 mm, 0 mm, 150 mm) and (0 mm, 0 mm, 300 mm), are carried out for 20 times at the time interval of 5 min, with the corresponding measurement errors shown in Fig. 10.From Fig. 10, the RMS values of repeatability measurement errors at the positions (60 mm, 0 mm, 100 mm) and (140 mm, 0 mm, 100 mm) are 0.064 μm and 0.29 μm, and those corresponding to the positions (0 mm, 0 mm, 150 mm) and (0 mm, 0 mm, 300 mm) be 0.11 μm and 0.39 μm, respectively, demonstrating good measurement repeatability with the proposed submicron-aperture fiber PDI system.

 

Fig. 10 Repeated measurement error in experimental validation. Errors in (a) x direction and (b) z direction.

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5. Conclusion

In this paper, we put forward a fiber PDI system with submicron aperture for the measurement of absolute three-dimensional coordinates. The proposed PDI utilizes the fiber with submicron exit aperture to get spherical wave with both high NA and high light power, by which the measurable range of the measurement system, especially in lateral direction can be extended. A double-iterative method based on L-M algorithm is proposed to reconstruct the coordinates of the target. Both the numerical simulation and experimental results are given to verify the accuracy and feasibility of the proposed measurement method, and the system achieves both high measurement precision and repeatability. The proposed PDI system decreases the difficulty in the adjustment of the system, and also provides a feasible way to realize the high-precision measurement of absolute three-dimensional coordinates and extend the achievable measurement range of PDI method.