Micro v-groove has found wide applications in optical areas as one of the most important structures. However, its performance is significantly affected by its angular geometry accuracy. The diamond cutting has been commonly used as the fabrication method of micro v-groove, but it is still difficult to guarantee the cutting tool angle, which is limited by the measurement accuracy in the manufacture and mounting of the diamond tool. A cutting tool alignment method based on the on-machine measurement is proposed to improve the fabricated quality of the v-groove angle. An on-machine probe is employed to scan the v-groove geometrical deviation precisely. The system errors model, data processing algorithm and tool alignment methods are analyzed in details. Experimental results show that the measurement standard deviation within 0.01° can be achieved. Retro-reflection mirrors are fabricated and measured finally by the proposed method for verification.
© 2015 Optical Society of America
Micro v-groove has been widely used in optical fiber positioning , retro-reflection , grating , and other light guiding . The increasing demands arouse the high precision fabrications, in which the ultra-precision single point diamond cutting is the popular method, including fly-cutting , non-rotational cutting [6, 7 ] and raster milling . The performance of a v-groove optical component is significantly affected by the deviation value of its included angles. For instance, the irradiance distribution from a retro-reflector could be scattered by the deviation errors of the included angles . Figure 1 shows how the angle deviation of 0.06° causes a significant scatter of the irradiance distribution. The cutting tool should be aligned to achieve high accuracy in micro v-groove cutting, because its high accurate measurement is the most important guarantee.
The optical surfaces with array features are usually scanned in contact mode by a microprobe. The single-line-scanning, however, cannot define unambiguously the shape of micro-grooves unless the alignment accuracy of the probe is better than the resolution of the profiler on the whole. Another alternative yet viable method is to use multi-line-scanning method at different angles. The alignment error of the probe related to the measured surface would not affect the accuracy of the measurement as long as the relative angles could be precisely determined . Non-contact methods could also be considered to measure the v-groove angle. However, the AFM and SEM with high resolution have small measurement range, and the computer vision method based on CCD and image algorithms is too low in resolution. The white light interferometry (WLI) and laser scanning can process relative large measurement range and high enough accuracy. However, the secondary reflection is inevitable due to the acute angle of v-groove . In recent years, the on-machine measurement has attracted much attention especially when drum lathe has been used to fabricate the micro v-groove walls. The CCD-based machine vision method was also used in the on-machine measurement. It has been proved to have sub-micrometre accuracy and be suitable for the width measurement of micro v-groove . One novel interferometry method based on Wavelength Division Multiplexing (WDM) was also studied, which can measure the shape of v-groove walls in nanometric precision and has vertical range of about 100µm . However, there are still not the suitable on-machine method to obtain the micro v-groove angle.
This study aims at obtaining the integrated 3D profile to calculate the micro v-groove dihedral angle accurately, which is unaffected by the alignment error in the conventional single-line scanning method. The accurate 3D data were scanned by the on-machine probe to control the cutting tools expediently. Moreover, the on-machine mode can provide the proper scanning path to extend the measurement range in nanometric accuracy.
2. Tool alignment scheme
Figure 2 shows the motivation and scheme of the measurement and fabrication control of the v-groove angle, in which the fly-cutting method is used. It also presents an orthogonal section line of v-groove, which indicates the parameters of the v-groove. In theory, a v-groove shape structure has three angles, including the included angle α between the adjacent groove side faces, which is always an acute angle, and two obtuse angles β 1, β 2 between the planes and groove walls (called tilting angles). The common approach for calculating the included angle of the v-groove cross-section lines is prone to error, because it is difficult to align the axis of the probe to be exactly coplanar with vertices A, B and O. In view of this, a fitting plane of the 3D contour is adopted to calculate the v-groove angle. In addition, the on-machine measurement is also used for the cutting tool alignment conveniently.
The v-groove angle is the dihedral angle of the two adjacent groove faces, which is calculated by the normal vectors n 1, n 2 as the following formula,Fig. 3 . The probe is mounted on the machining system simultaneously in the measuring process. The probe has an air-bearing slide and a high-precision displacement sensor. The ruby tip moves on the surface along the designed path. The stylus of the probe moves up and down along the air-bearing slide in friction-free state due to the surface variation, whose position can be tracked by the displacement sensor, such as laser interferometer, Linear Variable Differential Transformer (LVDT), capacitance micrometer and many others. There are still some other on-machine probes can be considered to be employed in the proposed method [14–16 ].
In this paper, a Moore Nanotech’s on-machine measuring system is adopted. Its probe with LVDT sensor has been verified to have good linearity and a resolution of 20nm in the measurement range of ± 156μm. The small measurement range enables the probe to obtain small measurement uncertainty, and the reference step measurement has proved that the standard deviation varies from 2nm to 6nm for the step height from 30nm to 10μm. Therefore, the measurement range of 10μm is suggested for better accuracy.
3. Metrology method
The sag value of micro v-groove is generally larger than the above suggested measurement range. To solve this problem, the probe was proposed to move along the ideal design profile in scanning process, as shown in Fig. 4 . The dotted line represents the scanning path which is defined by the design profile of the v-groove, while the solid line stands for the actual profile of the fabricated v-groove. The probe moves along the scanning path to obtain one profile with the control of Y and Z slides, and then feeds along the X axis to acquire the three dimension surface data of the v-groove. In this process, the displacement sensor only captures the difference between the actual and ideal profiles in high precision. For instance, the deviation δy can be measured when the probe moves along the ideal profile in Fig. 4. Therefore, the coordinates (x, y, z) of the measured profile can be calculated based on the machine motion, sensor reading δy and nominal angle, expressed as Eq. (2).
As a result, only the deviation δy between the scanning path and the actual profile of the manufactured v-groove will be captured by the sensor, in which a large scale measurement range is transformed into a small one. Meanwhile, as the sensor working in a small distance range, the proposed method can reach a relatively high precision with a relative stable measuring force due to the small motion of the stylus.
Firstly a reasonable scanning path should be proposed to avoid wrong measured results caused by the large deviation of the actual profile from the nominal one, which may exceed the measurement range of the probe. In Fig. 4, CD is a part of the designed path, while CF is the actual scanning path. ∠CDF = θ = β 1-90. Generally, β 1≈β 2, so θ is about a half of the v-groove angle α and ∠DCF = θʹ is the deviation of actual profile from the designed path. ∠CFG≈∠CDF = θ, since θʹ is relative small. The following formula can be deduced according to the sine law.
For example, θ is about 35.264° and the measuring length CD is larger than 150μm for the micro v-groove used in a retro-reflector. When the suggested measurement range DF is 10μm, the calculated result of θˊ equals to 2.2°. In other words, the probe can be employed to measure the angle deviation up to 2.2°, which is much larger than the error of the ultra-precision machining. Therefore, the proposed scanning path is quite competent to measure the v-groove angle.
The probe radius and measurement speed have a great impact on the measuring force and accuracy. If the probe radius is too small, the probe would generate a large local pressure to produce the press-in areas and deform the measured surface. The pressure effect is different for different workpiece materials . In addition, the quick scanning speed would bring in the measurement instability due to the micro fluctuation of the measured surface. Low scanning speed could make the measurement much stable, but results in long measuring time and significant sensor drift. Experimental results prove that the probe radius of 0.2mm and the scanning speed of 1mm/min provide a nice measurement performance for the brass material by Moore Nanotech’s probe.
It is difficult to make probe exactly vertical, as shown in the right side of Fig. 4. However, because the tip of the probe is a sphere, the tilting angle γ of the probe only affects the contact point of the probe on the surface and it does not affect the measured coordinates. Therefore, the measuring error induced by the tilt is negligible.
4. Data analysis
The side planes of v-groove are fitted by the robust least square algorithm, which has the excellent stability and noise immunity. Each fitting plane is expressed by its unit normal vector n and shift distance d in Z axis. For the measured points p i, the plane parameters can be calculated by searching the minimum value of objective function,
Commonly, several v-grooves are measured simultaneously in the measuring process in order to get reliable data efficiently. Meanwhile, the plane data between the adjacent v-grooves are also captured in the same scanning step. Therefore, the measured data consist of the v-groove and plane data linked together, which need to be segmented for the angle calculation. Mean curvature is used in the data segmentation, because it can describe the bending degree of surface precisely. The large mean curvature means the discontinuity on the surface. The borderline between the planes and v-groove walls can be positioned conveniently by searching the local maximum value of mean curvature.
Figure 5 shows the analysis result of the measured data. The mean curvatures of the 3D data points are calculated to separate the data of the plane area and v-groove area. The data in the middle part of each segment are fitted as a plane. The angle formed by two adjacent faces of the v-groove is computed according to Eq. (1).
5. Experimental results
A retro-reflection mirror was fabricated experimentally to verify the proposed method. The retro-reflection mirror is a micro prism array, which has three mutually perpendicular reflective surfaces. It is commonly fabricated through three v-grooves in different 120° directions by diamond cutting. The v-groove included angle α is designed as 70.5288°, and both of the tilting angles β 1 and β 2 are 125.2644°. The sag value of the micro prism array is about 410μm.
A testing sample was fabricated for aligning tool to ensure the accuracy of v-groove angle before machining the retro-reflection mirror. The proposed measurement method was adopted in the tool alignment. Figure 6 displays the experimental system setup. Two diamond tools were mounted for the precision and rough fly-cutting considering the large sag value and cutting area.
5.1. Measurement accuracy verification
In the measurement accuracy verification, three groups of the v-grooves with different angles were studied. Each group consists of five v-grooves with the same angle because they were fabricated under the same conditions. The scanning path was designed according to the nominal angle. The results of v-groove angles are shown in Fig. 7 . The angle deviation from the design angle of each v-groove is less than 1°, so the probe can work in the required status. The standard deviation of less than 0.005° proves the proposed measurement method has a good repeatability. The calculated fitting errors of each v-groove planes have an average value of less than 50nm and standard deviation of less than 10nm, which proves that the measured data and fitting algorithm have good reliability.
To confirm the measuring accuracy of the proposed method, a comparison experiments were carried out. The above v-grooves are measured by the Taylor Hobson 1250GPI, which is a common meausrement instrument for micro-structure based on the interferometer displacement with sub-nanometric resolution. Therefore, its measurement results can be regarded as the reference values. To obtain the accurate results in the Taylor Hobson profiler measurment, the tip radius is 2μm and measuring speed is 0.1mm/min. The results were compared to the above results correspondingly, which are also shown in Fig. 7. The compared results show that the maximum deviation is 0.0091°, the average deviation is less than 0.006° and the standard deviation is less than 0.004°. Therefore, the results prove that the measurement standard deviation of the proposed method is less than 0.01°.
5.2. Retro-reflection mirror fabrication and evaluation
The proposed method was used to fabricate the final retro-reflection mirror in order to verify its effectiveness. The tools were rotated manually according to the measured angle values in order to make the angles approaching the design values. The manual vertical adjustment mount δ is generated by the gentle knocking at the end of the workpiece holder where is guided by the sliding screw, as shown in Fig. 8(a) . In the tool alignment model, D 1 is the space between the spindle axis center and the sliding screw and D 2 is the distance from the sliding screw to the locking screw. There is δ/D 2 = sinγ, in which γ equals to the tool tip rotation angle, which resulting the changing of tool rake angle. It would change the v-groove dihedral angle finally, as shown in Fig. 8(b). The distance from the axis of the locking screw to the bottom of the v-groove is H = D 3cosγ, where D 3 is the distance from the axis of the locking screw to the tool tip. The v-groove dihedral angle can be calculated as α = 2arctan(W/2H), in which W is the theoretical width of the tool at the locking screw position. When the manual vertical adjustment δ = 10μm, the change of the v-groove dihedral angle is about 0.001°~0.004° according to the different W, D 2, D 3 of the tool holder. In addition, the tilting angles β 1 and β 2 of the v-groove is adjusted synchronously by measuring the planeness of the pre-machined block. A precise dial indicator walks along a circle path on the surface and judge the tilt direction and then rotate the B axis of the machine to make the tilt direction agreed with the machining direction of the fly-cutting. So that the tilt angles of the v-groove can be reduced as much as possible. In the actual experiment, there were altogether six adjustment steps implemented, as shown in Table 1 . The final results show that the deviation of the v-groove included angle from the design value is about 0.006° and less than 0.013° overall including two tilting angles.
The convex brass block was fabricated by the fly-cutter on the 5-axis ultra-precision machine after tools alignment, as shown in Fig. 9(a) , which has a square contour with the edge length of 50mm. The next-step electroforming provided the final concave nickel workpiece, which realized the retro-reflection function. Figure 9(b) shows the good fabricated shape of the final concave nickel workpiece, which is the 3D profile measured by Zygo Nexview. The bottom right image of Fig. 9(b) also proves a relative uniform for each element in retro-reflection, where the white zone is the retro-reflection light. Meanwhile, the retro-reflection performance was evaluated by the laser casting mode, as shown in Fig. 9(c). The laser beam is projected on the retro-reflection mirror from a distance of about 5 meters. The retro-reflection light distribution is captured on the screen at the back of laser device, shown in Fig. 9(d). The result shows that the retro-reflection light has the good convergence performance with the hexagonal distribution, which is similar to Fig. 1(b) and conforms to the retro-reflection theory well . The retroreflective sheeting for traffic control is very important application of the fabricated retro-reflector, and the coefficient of retro-reflection is the most important parameter. Finally, the retro-reflection coefficient was measured by the dedicated measurement device. The results prove that the retro-reflection coefficient is mostly better than 3000, which is much better than the standard value of the top-class retro-reflector in ASTM international standard . The experimental results prove that the proposed method provides a good precision of tool alignment to guarantee the high angular accuracy manufacture method of micro v-grooves.
The study proposes a high angular accuracy fabrication of micro v-groove. The proposed method guarantees the good tool alignment based on the on-machine measurement. The measurement accuracy is verified by the compared results with the Taylor Hobson profilometer. And the final retro-reflection mirror experiment proves the effectiveness of the proposed method. The experimental results also prove that the probe with an air-bearing slide and a high-precision displacement sensor is competent to the on-machine measurement on the ultra-precision machine. And the scanning path along the ideal design profile reduces the measurement range to improve the measurement accuracy, which just utilizes the high-precision motion of the machine axis in the on-machine mode. The proposed method is expected to be used for the angle and shape measurement of many other kinds of micro-structures, especially in the promising drum turning.
The authors express their sincere thanks to Y. B. Lu and Y. X. Xiang for the dedicated efforts on the experiments. This work has been funded by the State Key Development Program of Basic Research of China (“973” Project, Grant No. 2011CB706700), the National Natural Science Foundation of China (Grant No. 51375337), Tianjin Research Program of Application Foundation and Advanced Technology (Grant No. 14JCQNJC05200) and the ‘111’ project by the State Administration of Foreign Experts Affairs of China (Grant No.B07014).
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