1. Introduction

With the rapid development of the micro manufacturing technology, micro parts or microstructure arrays with high aspect ratios have been widely used in various fields. For example, the crucial dimensions of the micro-groove arrays in aerospace micro-propulsion engines are tens of microns with an aspect ratio of about 40:1. For a binary optics lens, there are many annular binary microstructures with features of more than hundred microns. The diameters of fuel nozzles in high performance engines are less than 150 μm, with an aspect ratio of more than 6:1. In a spatial light modulator, the crucial dimensions of the micro lens arrays ranges from tens microns to hundred microns. The measuring requirements of dimension and form of the above micro parts are becoming urgent. However, the measurability of these micro parts is limited by the contradiction between the tiny feature, high aspect ratio and limited space. For existing scanning probe microscopy and conventional tactile probes are inappropriate to realize the measurement because of its small measurable aspect ratio or the oversize probe. So, it is significant to find effective measurement methods for these micro parts with a resolution up to nanometer level.

During the last 20 years, different methods have been developed to solve this issue. For example, the diffraction measuring method [1], the laser scanning method, and tactile methods with various probes [2], wherein, methods based on optical fiber probe became one of the mainstream for its excellent flexibility, lower prices, and easy making. For instance, H. Schwenke et al. used in 2001 an optical fiber method based on detecting the feeler element optically [3], and measured a gear with 0.5mm in diameter. Kao proposed in 2007 a fiber probe with a more slender fiber stylus based on the same principle [4], and micro holes about 160 μm in diameter were measured at a depth of 0.9 mm. Currently, this fiber probe has been commercialized by Werth [5]. However, due to the shadowing effect, which means that the radiation beam from the probing sphere is obstructed or reflected by the sidewall, the photodetector cannot obtain enough light energy for imaging and the aspect ratio of this fiber probe is limited. Tan et al. proposed in 2011 a fiber probe based on spherical coupling for the measurement of micro-holes and achieved a radial resolution of 50 nm [6]. Cui et al. modified in 2014 this probe though fabricating axion lens on the effluent end of the probe [7], and this probe finally achieves a resolution of 30 nm. But the resolution is difficult to further improve because of the excessive loss of emergent light in coupling. Depiereux proposed in 2007 a fiber probe based on white-light interferometer for the measurement of plane structures [8]. Tilo Pfeifer et al. also proposed in 2011 a fiber probe based on white-light interferometer with grinded fiber tip [9], and a borehole with a diameter of 125 μm was measured. But these two probes only have measuring capability in one dimension. Muralikrishnan reported in 2004 a fiber probe by detecting the deflection of the fiber which works as a focusing cylindrical lens for measuring a micro-hole that was 0.5 mm in depth and 0.129 mm in diameter [10], and this fiber probe was extended to achieve a 3-dimensional measuring capability via buckling [11], however, its low optical magnification reduces the displacement sensitivity in sensing the deflection of the fiber probe. Dr. Wang proposed in 2011 a fiber deflection probing method that uses fiber as a collimating cylindrical lens [12], and it achieved a resolution of 3 nm, but it is lack of capability in z-axis and has coupling between the x-axis and y-axis. So, all the methods above have their drawbacks. In order to address this issue, a 3-dimensional fiber probe based on micro focal-length collimation (MFL-collimation) is proposed in this paper. Combining this fiber probe with the triggering method mentioned in [13], precision measurement of the micro parts with high aspect ratios can be achieved.

2. Principle

2.1 Measurement principle

As shown in Fig. 1, the 3-dimensional fiber probe based on MFL-collimation consists of a fiber probe and two orthogonal MFL-collimation optical paths. The fiber probe is composed by a fiber stylus which works as a micro focal-length cylindrical lens (MFLC-lens) and a probe tip which works as an element for touching the workpiece. The arrangements of both MFL-collimation optical paths are the same, a laser beam from the laser diode (LD) is converged by the reflective objective lens (ROL) which features a long working distance into an annular point light source, and meanwhile, the annular point light source locates at the focus of the fiber stylus which works as a MFLC-lens. The light from the annular point light source is collimated by the MFLC-lens, and then the collimated light forms a fringe image on the detecting screen. When the probe tip is moved to touch the workpiece, the fiber stylus will deflect or buckle which means the defocus of the MFLC-lens at the observation point (OP), and then the centroid position of the fringe image will change accordingly. Therefore, the displacement of the probe tip is transformed into the centroid position shift of the fringe image.

 

Fig. 1 Schematic diagram of the 3-dimensional fiber probe based on MFL-collimation.

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2.2 Analysis of the light transmission of the MFL-collimation optical path

When the MFLC-lens defocus along x and y direction, the light transmission of the MFL-collimation is analyzed using the geometrical optics method as shown in Fig. 2. Wherein, $\Delta x$ is the longitudinal defocus displacement of the MFLC-lens, $\Delta y$ is the transversal defocus displacement of the MFLC-lens, f is the focal-length of the MFLC-lens, L is the imaging distance, $\theta$ is the emergence angle of point light source, ${\beta }_1$ is the incidence angle, ${\beta }_2$ is the refraction angle, r is the radius of the MFLC-lens, n is the refractive index of the MFLC-lens.

 

Fig. 2 Light transmission of the MFL-collimation optical path.

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In $△P\text{'}{A}_{1}O$, $P\text{'}O=f+\Delta x-\Delta y/\tan \theta$, according to the sine law and refraction law, the incidence angle ${\beta }_1$ and the refraction angle ${\beta }_2$ can be written as

The incidence point $A_1\left(x_1,y_1\right)$ of the point light source to the MFLC-lens is given by

The emergence point $A_2\left(x_2,y_2\right)$ of the point light source to the MFLC-lens is given by

Finally, the intersection point $A_3\left(x_3,y_3\right)$ of the point light source to the detecting screen can be written as

The range of the emergence angle of point light source is $\theta \in \left[-\phi ,\phi \right]$, by Eqs. (1)-(5), the distribution of lights collimated by the MFLC-lens can be obtained. When L = 125 mm, r = 62.5 μm, $\theta \in \left[-0.5,0.5\right]$, the distribution of the collimated light is shown in Fig. 3, and the collimated light is formed into a fringe image. It can be seen that, when the MFLC-lens has a transversal defocus displacement along y direction, the centroid position of fringe image shifts accordingly; when the fiber has a longitudinal defocus displacement along x direction, the distribution width of the fringe image change without any shift of the centroid position. So, it can be concluded that the centroid position of the fringe image is sensitive to the transversal defocus displacement of the fiber stylus, but insensitive to the longitudinal defocus displacement of the fiber stylus. Based on this conclusion, the centroid position of the fringe image is used as the output response signal of the MFL-collimation optical path to the defocus of the MFLC-lens.

 

Fig. 3 Normalized intensity of the fringe image of the MFL-collimation optical path (a) situation of transversal defocus displacement along y direction (b) situation of longitudinal defocus displacement along the positive direction of x-axis (c) situation of longitudinal defocus displacement along the negative direction of x-axis.

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2.3 Analysis of the response property of the MFL-collimation optical path

In order to analysis the response property of the MFL-collimation optical path accurately, an accurate fringe image of the MFL-collimation optical path must be got, so a simulation on the propagation of light based on raytracing instead of geometrical optics method is run. The fringe image of the MFL-collimation optical path which is an interference pattern with diffraction effect is shown in Fig. 4. Here, considering the symmetry and simplicity of the fringe image, we choose the image located in the central-horizontal position (the yellow dash line) as the measured image, and the image is captured by a linear CCD camera.

 

Fig. 4 Fringe image of the MFL-collimation optical path.

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The fringe image captured by the linear CCD camera is shown in Fig. 5. It can be seen that, when the MFLC-lens has a transversal defocus displacement along y direction, the position of the fringe image shifts accordingly, and it is clearly that the fringe image captured by the photosensitive area of the CCD changes with the shift of the fringe image. Obviously, it is inappropriate to represent the centroid position of the fringe image of the MFL-collimation optical path by the centroid position of the fringe image captured by the CCD as a result of the moving-in and moving-out non-zero-order fringe image on the photosensitive area of the CCD.

 

Fig. 5 Fringe image captured by the linear CCD camera.

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As shown in Fig. 5, the zero-order fringe image is characterized in the maximum intensity, so it is extracted from the fringe image captured by the CCD to locate the centroid position of the fringe image. Figure 6 shows the response curve of the centroid position of the zero-order fringe image to the defocus displacement of MFLC-lens along x and y direction. It can be seen in Fig. 6(a) that, the transversal response curves of the MFLC-lens with different longitudinal defocus displacement along x direction are consistent with each other, and the centroid position of the zero-order fringe image is linear to the transversal defocus displacement along y direction with a sensitivity of 137 pixel/μm. It can be seen in Fig. 6(b) that, all the longitudinal response curves of the MFLC-lens with different transversal defocus displacement along y direction are horizontal lines. Combining Fig. 6(a) and 6(b), we can conclude that the centroid position shift of the zero-order fringe image is sensitive to the transversal defocus of the MFLC-lens along y direction, but insensitive to the longitudinal defocus displacement of the MFLC-lens along x direction.

 

Fig. 6 Response curve of the centroid position to the two-dimensional defocus of the MFLC-lens along x and y direction (a) transversal response curves with different longitudinal defocus displacement (b) longitudinal response curves with different transversal defocus displacement.

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Finally, we take use of the response property of the transversal defocus displacement of the MFLC-lens, and build a transformation model of the MFL-collimation optical path: the transversal defocus displacement of the MFLC-lens is transformed into the centroid position shift of the zero-order fringe image with an equation expressed as

3. Analysis of the operating mode

3.1 Deflection mode

When the probe tip subjects to a radial displacement, the fiber probe works in the deflection mode. As shown in Fig. 7, the radial displacement of the probe tip along x direction causes a radial deflection of the fiber stylus, and the radial deflection of the fiber stylus at the OP-X is detected by the MFL-collimation optical path along y direction. The radial deflection $\Delta x_o$ of the fiber stylus at the OP-X is expressed with a deflection formula as

 

Fig. 7 (a) Deflection mode of the fiber probe (b) simplification of the deflection mode.

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Finally, the transfer function of the radial displacement $\Delta x_t$ to the radial deflection $\Delta x_o$ can be expressed as

As discussed in section 2.3, the MFL-collimation optical path is only sensitive to the transversal defocus displacement of the MFLC-lens. It can therefore be concluded that the two orthogonal MFL-collimation optical paths are decoupled in two-dimensional defocus displacement along x and y direction: The MFL-collimation optical path along y direction is sensitive to the radial displacement of the probe tip along x direction, and the MFL-collimation optical path along x direction is sensitive to the radial displacement of the probe tip along y direction. As a consequence, the radial displacements of the probe tip are transformed into the centroid position shifts of the zero-order fringe images of the two orthogonal MFL-collimation optical paths. From Eqs. (6)-(8), the transfer function can be calculated as

Due to the decoupling capability of the two orthogonal MFL-collimation optical paths of the fiber probe, a radial displacement along x-y plane can be decomposed into components along x direction and y direction which can be measured through the MFL-collimation optical path along the corresponding axis.

3.2 Buckling mode

When the probe tip subjects to a compressive force $F_a$ along the axis of the fiber stylus, the fiber stylus will be compressed or buckled, and it is decided by the compressive force $F_a$. Here, we expect a buckling of the fiber stylus. As shown in Fig. 8, when the probe tip is subjected to an axial displacement $\Delta z_t$, the fiber probe will operate in the buckling mode. The buckling of the fiber stylus will cause defocus displacement of the fiber stylus (acts as a MFLC-lens), and the defocus displacement of the fiber stylus at the OP-X and OP-Y will be detected by the MFL-collimation optical path along the corresponding axis.

 

Fig. 8 (a) Buckling mode of the fiber probe, (b) simplification of the buckling mode.

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As describing in [11], the fiber stylus can be simplified as a slender rod with two fixed ends, and the elastic buckling mode of the slender rod is shown in Fig. 9. The slender rod subjects to a compressive force $F_a$ acting at both ends and is bent into a curve with a shape given by $w(v)$, the differential equation of bending based on small deflection bending theory is expressed as

 

Fig. 9 Elastic buckling mode of the slender rod with two fixed ends.

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For the situation of two fixed ends which means no slipping and no rotation, the standard solution of Eq. (10) is expressed as

Meanwhile, the arc length of the buckling slender rod can be expressed as

Therefore,

The slender rod will not buckle until the compressive force $F_a$ reaches the critical value $F_{cr}$. When $F_a<F_{cr}$, the slender rod will be compressed as a function expressed as $\Delta l=F_{a}l/\left(EA\right)$, where A is the cross sectional area of the slender rod. The maximum compressive length of the slender rod is $\Delta l_{cr}=F_{cr}l/\left(EA\right)=4{\pi }^{2}I/\left(lA\right)$. Hence, the arc length of the bend slender rod can be modified as $s=s_0-\Delta l_{cr}$, where $s_0$ is the uncompressed arc length. And then, the Eq. (13) can be expressed as

Combining Eqs. (11) and (14), the transfer function of the defocus of the fiber stylus at the OP and the axial displacement of the probe tip can be expressed as

Taking Eq. (15) into Eq. (6), we will get the transfer function of the axial displacement $\Delta z_t$ of the probe tip and the centroid position shift $P_c$ of the zero-order fringe images of the MFL-collimation optical path:

From Eq. (16), it can be concluded that the axial displacement of the probe tip cannot be measured until it reaches the critical value, the compression region is the dead zone of our fiber probe, but it can be measured by the method proposed in [14]. We have no idea of the direction that the fiber stylus buckles from Eq. (16). In reality, the buckling direction of the fiber stylus is decided by the assembling of the fiber probe, and the defocus of the fiber stylus at the OP along x-y plane can be decomposed into components along x and y direction which can be measured through the MFL-collimation optical path along the corresponding axis. We will focus on the measurement of the compression of the fiber probe in the further work.

In consideration of the dead zone and the nonlinearity of the axial response curve of the fiber probe, we can take advantage of the critical point of the fiber probe to provide a trigger signal of the contacting of the probe tip and the micro part. When the probe tip subjects to an axial displacement larger than the maximum compressive length of the fiber probe, both the MFL-collimation optical paths can output the trigger signal of contact.

3.3 Probing force

For the measurement of micro parts, the probing force is a critical parameter of the probe. For our fiber probe, the contact force is less than 10 μN when the probe tip subjects to a radial displacement of 10 μm; the maximum compressive length of the fiber stylus $\Delta z_{cr}$ is ~1 μm, and the critical value of compressive force $F_{cr}$ is ~26 mN when the probe tip subjects to an axial displacement of 1 μm. A low probing force ranging from tens of millinewtons to a few micronewtons can be achieved.

4. Experiment

4.1 Fabrication of the fiber probe

The structure diagram and photo of the fiber probe is shown in Fig. 10. The single mode fiber can be used to make fiber probe by melting probe tips of different diameters at the ends of the fiber stylus which is etched by the hydrofluoric acid. The diameter of the single mode fiber is d_s = 125 μm, the diameter d_E of the etched fiber stylus is about 30 μm - 120 μm, and the length of the fiber stylus being etched is ~2mm. The diameter of the probe tip is in a range of 50 μm-200 μm, and the working length of the fiber probe is about 2 mm-5 mm. Thus the aspect ratio of the fiber probe is about 10:1-100:1 which can be used to achieve high measurable aspect ratio. Here, the fiber probe is fabricated with a fiber stylus length of 15 mm, and a probe tip diameter of 131.52 μm.

 

Fig. 10 Structure diagram and photo of the fiber probe.

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4.2 Experimental setup

The experimental setup of the 3D fiber probe is shown in Fig. 11. The working distance of the ROLs is 24.5 mm. The distances of the OP-X and OP-Y to the probe tip are ~6 mm, and the focal length of the fiber stylus is ~93 μm. The resolution of the linear CCD cameras is 2048 × 1 pixel, and the size of one pixel is 10 × 10 μm. The imaging distance of the fiber stylus to the detecting screen of linear CCD camera is ~125 mm. Experiments are made to evaluate the performance of the 3D fiber probe. A glass cover is used to reduce the influence of air. All the devices are deposited on an active self-leveling isolation system. A plane mirror is moved forward along the X, Y and Z direction to touch the probe tip.

 

Fig. 11 Experimental setup of the 3D fiber probe.

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4.3 Experiment on performance of the fiber probe

The response curves of the fiber probe to the radial displacements are shown in Fig. 12. It can be seen that the output of CCDX has no response to the radial displacement of the probe tip along y direction, and is linear to the radial displacement of the probe tip along x direction with a sensitivity of 86 pixel/μm; Similarly, the output of CCDY has no response to the radial displacement of the probe tip along x direction, and is linear to the radial displacement of the probe tip along y direction with a sensitivity of 73 pixel/μm. It is clear that the fiber probe is decoupled in x and y probing. As shown in Fig. 13, the radial resolutions of the fiber probe along x and y direction can reach 5 nm.

 

Fig. 12 Response curves of the fiber probe to the radial displacements (a) along X direction and (b) along Y direction.

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Fig. 13 Radial resolutions of fiber probe (a) along X direction and (b) along Y direction.

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The response curves of the fiber probe to the axial displacement along Z direction is shown in Fig. 14. It can be seen that both the output of CCDX and CCDY have the similar response. The response curve can be classified into three sections. Section I is the section that the probe tip approaching and being compressed without buckling by the workpiece, the outputs of CCDX and CCDY have no response to these axial displacements. In the transitory section II, the outputs of CCDX and CCDY change suddenly with a large gradient which conflicts with the theoretical analysis of the critical state of the compressed slender rod. Assumption is made that there is a position vibration of the fiber probe causing a sudden change of the response curve when the probe tip touches the workpiece, further research will be down in the future on this aspect. Section III is the buckling section. The fiber probe operates in the buckling mode, and the outputs of the CCDX and CCDY have obvious variation. The sensitivity of section III is not a constant as a result of the nonlinear buckling model, but the sensitivity of the initial approximate linear zone of CCDX and CCDY is about 230 pixel/μm and 170 pixel/μm respectively. Figure 15 shows the experimental results of the axial resolution and an achievable axial resolution can reach 3 nm in the initial approximate linear zone.

 

Fig. 14 Response curves of the fiber probe to the axial displacement along Z direction: Output of CCDY (top) and output of CCDX (bottom).

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Fig. 15 Axial resolution of the fiber probe along Z direction: Output of CCDY (left) and output of CCDX (right).

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4.4 Measurement on micro-holes

Experiments on the performance of the fiber probe are conducted with three micro-holes on a ceramic circular disk, as shown in Fig. 16 (a). The nominal diameters of the micro-holes to be measured are 0.3, 0.5 and 0.7 mm, respectively. The measured diameters of three micro-holes at a depth of 500 μm are 300.33, 500.52 and 683.00 μm, and the standard deviations of the experiments are 0.024, 0.032 and 0.050 μm, respectively. It can be seen that precision measurements of micro-holes can be made with the fiber probe.

 

Fig. 16 (a) Measurement of three micro-holes on a ceramic circular disc (b) Measurement results of three micro-holes.

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5. Conclusions

A 3-dimensional fiber probe based on orthogonal micro focal-length collimation is proposed and fabricated for the measurement of micro parts with high aspect ratios. The model of the light transmission and response property of the MFL-collimation optical path are built, and the deflection and buckling of the fiber probe are analyzed. Experimental results indicate that the fiber probe has a 3-dimensional measuring capability, and a radial and axial resolution of 5nm and 3nm respectively. Three micro-holes with nominal diameters of 0.3, 0.5 and 0.7 mm on a ceramic disc are measured at a depth of 500 μm; the measurement results are 300.33, 500.52 and 683.00 μm with standard deviations of 0.024, 0.032 and 0.050 μm, respectively. The probe also has advantages in low cost, high measurable aspect ratio, and low probing forces. Currently, the main obstacle of the probe is the mutation around the critical state of the axial buckling of the fiber probe. Further studies will focus on the measurement of the axial compression of the fiber probe to perfect the measuring capability of the fiber probe in z direction.