Double Corner Cube Method for Enhancing the Sensitivity of Straightness Measurement Tung-Shiuan Pan1, Ju-Yi Lee1*, Szu-Wei Wu1, Hung-Lin Hsieh2, Cheng-Chih Hsu3

1Department of Mechanical Engineering, National Central University, Taiwan (Tel: 886-3-4267307; E-mail: juyilee@ncu.edu.tw)

2 Department of Mechanical Engineering, National Taiwan University of Science and Technology, Taiwan 3Department of Photonics Engineering, Yuan Ze University, 135 Yuan-Tung Road, Chung-Li, 32003 Taiwan

Abstract:

A method of straightness measurement utilizing two corner cubes is proposed. The architectural design of the system leads to the magnification of the displacement of a light spot on the detector with the same offset of the determinand. This method enhances the sensitivity of the system and raises the precision to about 5 m.

Keywords: straightness, corner cube, quadrant detector

1. INTRODUCTION In straightness measurement we search for the degree

similarity of a segment to a straight line. This kind of measurement is one dimensional and can be expressed in many kinds of measurement terminologies, for example, flatness measurement which is straightness measurement with two dimensions, roundness, sphericity and so on. All of these are extensively applied in industrial manufacturing.

Long-range motion stages are widely used in many fields, such as precision engineering, metrology, nanotechnology, lithography applications, as well as advanced scientific applications [15]. Normally, these motion stages are expected to move in a straight and precise manner in each direction. However, in practice, fabrication errors in the motion system can result in motion error that appears as movement in an orthogonal direction from the proposed movement path. This kind of error is called straightness error [6] and will influence the real performance of the application system. To achieve the high-resolution positioning needed, methods for straightness or linear displacement measurement are becoming more important.

The two basic types of straightness measurement methods usually applied are optical interferometry and the geometric optics method. For example, in our previous work, we designed a heterodyne grating interferometer based on a quasi-common-optical-path (QCOP) [7] for two-degree-of-freedom straightness measurement. In spite of the fact that our heterodyne grating interferometer provides the high measurement sensitivity and resolution, its cost is quit high and the optical configuration is complex due to the optical modulator and demodulator used. In contrast, the optical configuration of the geometric optics method is much simpler, and is used in autocollimators, alignment telescopes and optical theodolites [8]. Numerous studies in the literature [9-10] have reported results related to such measurement systems. Straightness measurement can be carried out by detecting the lateral displacement of a laser beam with a four-quadrant detector and corner-cubes. Generally, the geometric optics method is low cost and has a sub-micrometer accuracy.

2. MEASUREMENT PRINCIPLES The purpose of this study is to enhance the

straightness measurement sensitivity of the traditional geometric optical system. As shown in Fig.1, corner cube #1 is placed on a moving linear stage for straightness measurement. The light beam from the laser propagates between corner cubes #1 and #2, to be finally received by the four-quadrant detector. The path of propagation of the light beam in this system is 1 2 3 4 5 6 7 8 (see Fig. 1).

Fig. 1 Double corner cube configuration for straightness measurement.

Given their characteristics, the distances from the

center axis of the corner cube to the incident light and reflected light are equal. As shown in Fig. 2, D is the distance between the incident laser beam 12 and the center axis of corner cube #2. Suppose the distance between the incident beam 12 and the center axis of corner cube #1 is a, and the distance between the reflected beam 34 and the incident beam 12 is 2a. We can obtain the distance between the original incident beam 12 and the final reflected beam 78 as follows: d=2(2a-D). When the corner cube #1 moves down x, the laser beam has lateral displacement (dashed line), and the length of a will change to a+x. The distance between the original incident beam and the final reflected beam is d=2(2(a+x)-D). Therefore, the displacement of the final reflected beam is x=d-d=4x. The beam displacement is four times the displacement of corner cube 1. It is twice as large as the result in reference [8].

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Fig. 2 Illustration of the corner cube measurement principle.

The intensity distribution of the light beam on the

four-quadrant detector can be written as a Gaussian function [8]

I = I0 exp(2(x2 + y2 ) w2 ), (1)

as shown in Fig. 3, where w is the beam waist, I0=2P/w2 and P is the total light power of the laser beam. The four-quadrant detector is made up of the quadrant detectors for A, B, C and D, and the light power distributed on these four detectors can be written as

q = I0 exp(2(x2 + y2 ) w2 )q dxdy, (2)

q=A, B, C, or D is the integral range of the quadrant. The relationship between the beam displacement and the currents is determined using [11]

fx =(A +D ) (B +C )(A +B + C +D )

, (3)

fy =(B +A ) (C +D )(A +B + C +D )

. (4)

Fig. 3 Intensity distribution of the light beam on the four-quadrant detector. As shown in Fig. 3, if the center (x0, y0) of the four-quadrant detector is not located at the beam center (0,0), the light powers on these four detectors are not symmetric, resulting in the change of fx and fy. Fig. 4 shows the relationship between the displacement and fx which is calculated by MATLAB, with a beam waist w=0.5 mm. Compared with Feng and Fans method, the proposed method has twice the measurement sensitivity. Future experiments will demonstrate the performance of this measurement system.

Fig. 4 Relationship between the displacement x and fx.

3. EXPERIMENTAL SETUP The experiments demonstrating the capability of

the architecture design described above are now discussed. As shown in Fig. 5, in the experiments with one corner cube, the laser beam propagating from the laser to corner cube #1 is 25.4 mm diameter. It is then reflected back through the beam splitter to the detector. The beam splitter, which can be replaced with a mirror, solves the problem of the detector being so large in size that it may block the propagation of the beam from the laser. This problem also can be solved by enlarging the size of corner cube #1.

The two corner cube experiment is shown in Fig 6. The design is similar to the framework of the one corner cube experiment but with one more corner cube #2 which has a diameter of 10mm being fixed near the laser. The beam is again reflected back to corner cube #1 to again enhance the beam spot displacement on the detector.

The entire apparatus, except for corner cube #1, is still fixed on the motorized stage so it can move in a direction orthogonal to the incident beam from the laser. With the motion of corner cube #1, there will be a corresponding displacement of the light spot received by the detector.

The long range motion experiment is designed to change the direction of the stage to parallel to the direction of the laser beam. The distance of the travel path is 300 mm.

In many studies of straightness measurement it is claimed that interference from the offset of the corner cubes angle can be ignored. To check on this inference related to angle offset, we design an experiment for rotational motion as shown in Fig 7. In the experimental design the moving stage is replaced, as shown in the architecture in Fig 6, with a rotational stage to let corner cube #1 rotate.

y

(0,0)

x (x0, y0)

B A

C D

-1 -0.8 -0.6 -0.4 -0.2

x displacement (mm) 0 0.2 0.4 0.6 0.8 1

-0.8-0.6-0.4-0.2

00.20.40.60.8

1

fx

Fen's

Feng's

Ours

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Fig. 5 Experiments for step motion and long-range motion with one corner cube.

Fig. 6 Experiments for step motion and long-range motion with two corner cubes.

Fig. 7 Experiment for rotational motion with two corner cubes.

4. DISCUSSION OF EXPERIMENTAL RESULTS

To show the utility of enhancing the displacement of the light spot on the detector by the additional corner cube, we respectively measured the results of the two systems.

The voltage signals from the detector with the step motion of corner cube #1 are shown in Fig. 8. The distance of one step is 10 m. We can see that the signals with two corner cubes are always larger than the signals with one corner cube. This can indicate that the displacement of the light spot on the detector is enlarged, so the signal the detector receives will be correspondingly enhanced.

The distance of the step is the same in the two systems, but with the enhancement of the voltage signal, the sensitivity will be enlarged. From the results in Fig 8 we see that the variation of the voltage signal per each

step in the one corner cube system is 0.04 V, meaning the sensitivity is about 4 V/mm. Compared to the results for the two corner cube system, the signal per step is 0.08V. The sensitivity is about 8 V/mm. We can also conjecture that the precision of the two corner cube system will be about 5 m, as can be seen from Fig 10.

Fig. 8 Results of the step motion experiment with the voltage signal received by the detector. The distance of one step is 10 m.

Fig 9 shows the voltage signal which represents the straightness error for corner cube #1 with long-range motion. It is obvious that the direction of the movement of the stage is not parallel to the direction of the laser beam propogating through corner cube #1. Though both of the two systems have straightness error, the amount of voltage from the two corner cube system is always larger. This again proves that the displacement of the beam received by the detector is enhanced.

Fig. 9 Voltage signal received by the detector with long-range motion.

We also tested the systems stability. Fig. 10 shows the signal vibration of the two systems. The range of vibration is larger with the two corner cube system than with the one corner cube system. This means that not only the displacement by motion but also the influence from the surroundings has been enlarged.

two corner cubes

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one corner cube

two corner cubes

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0

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volt signal(V)

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Fig. 10 Displacement of the light spot on the detector.

The straightness error for corner cube #1 with

rotational motion is shown in Fig. 11. The signals show vibration in a range similar to the results shown in Fig. 10 which has no motion. We learn from this that the angle of offset of corner cube #1 has almost no interference on the straightness measurement.

Fig. 11 Straightness error when corner cube #1 has rotation motion.

5. CONCLUSION

We propose a double corner cube straightness

measurement method that enlarges the sensitivity of the system and frees it from the influence of rotational motion.

Compared with the one corner system, the two corner system magnifies the displacement of the light beam to enhance the sensitivity of the system, but the measurement range is correspondingly reduced. The experimental results demonstrate the precision of our measurement system which can reach about 5 m.

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