Application of Phase Shift Interferometry in Small Angle and Straightness Measurement

1 Introduction

Straightness measurement plays a very important role in the installation and commissioning of processing equipment and testing equipment. The accuracy of its measured value will directly affect the manufacturing accuracy of the processing equipment and the measurement accuracy of the testing equipment. In some high-precision applications, straightness measurement data directly participates in the accuracy compensation of processing and measuring equipment and is the data source of the accuracy compensation process. Its measurement accuracy determines the accuracy of the instrument and equipment. Straightness can generally be directly measured by special instruments (such as HP dual-frequency laser interferometer combined with straightness measurement accessories Wollaston prism and dihedral reflector) [1], or it can be obtained by conversion using a small angle measurement device (such as 702 photoelectric autocollimator) [2]. Due to the wide application of small angle measurement devices, the latter has become a common method for straightness measurement.

The general principle of small angle measurement is shown in Figure 1:

The measurement equation can be expressed as:

Since the angle is very small, the above formula can be written as:

In this way, when L or l is determined, the angle α can be obtained by measuring h. At present, when using an autocollimator to measure small angle values, the measurement accuracy of h is affected by the accuracy of components such as the micrometer eyepiece, and the measurement accuracy is generally δα≥1″[2]. There is a double reference mirror method for measuring small angles using the principle of optical interference[3], the core of which is a Twyman-Green interferometer. When using this method for measurement, the sensitivity provided by the fringe interval is the key to accurately reading the mantissa of the interval value. Since the brightness and shape of the fringe are very inconsistent, it is impossible to obtain high-precision detection results[4]. At the same time, affected by the conditions such as the range of the double reference mirror angle value, the error of the double reference mirror small angle measurement δα≥0.5″[3].

When measuring straightness, it is necessary to measure small angles in two directions (pitch and yaw) to achieve the straightness measurement. When using the above two measurement methods, it is impossible to obtain small angle values ​​in two directions at the same time. However, the measurement method proposed in this paper, which uses phase shift interferometry technology combined with Zernike wavefront fitting technology, can not only obtain small angle values ​​in two orthogonal directions at the same time, but also greatly improve the measurement accuracy and measurement speed.

2 Principle of small-angle phase-shift laser interferometry

The principle of measuring small angles using phase-shift interferometry is shown in Figure 2:

As shown in Figure 2, the laser 1 emits a stable frequency laser, which is expanded by the beam expander 2 and spatially filtered by the pinhole 3 to obtain a high-quality spherical wave. After the collimator 7 and the reference mirror 9, one beam of light is reflected by the reference mirror standard surface a to obtain a parameter wavefront, and the other beam of transmitted light is reflected by the reflector 10 to form a detected wavefront. After the two beams of light are acted upon by the beam splitter 4 and the projection mirror 5, interference fringes are generated on the image plane of the CCD detector 6. The so-called phase shift interferometry technology is an interferometry technology that realizes phase modulation in the time domain by modulating the optical path difference through PZT [1]. In Figure 2, the reference mirror 9 realizes the modulation of the optical path difference between the reference wavefront and the detected wavefront under the action of PZT 8. After obtaining the initial phase of the detected wavefront, the surface error of the reflector 10 and the spatial angle between the reflector 10 and the reference surface a can be obtained. The spatial angle here is the small angle value we want to measure. The initial phase of the reflector 10 can be obtained through the following four-step algorithm [1].

After obtaining the initial phase of each point on the reflector, the optical path difference of the corresponding point relative to the reference surface can be obtained by the following formula:

OPD (x, y) contains the reflector's surface error information and the reflector's tilt information. The tilt here is the small angle information we are interested in. The tilt in the orthogonal x and y directions is the pitch angle and yaw angle we want to measure in straightness measurement. The tilt in the x and y directions can be calculated and solved by Zernike polynomial wavefront fitting technology.

3. Small-angle Zernike wavefront fitting solution

Since optical surfaces tend to be smooth and continuous, they can be expressed as a linear combination of a complete basis function system or a combination of linearly independent basis function systems. In optical testing, Zernike orthogonal polynomials are often used as basis polynomials to represent the surface wave aberration of the detected wave[5]:

Where: W(x,y) is the wave aberration, Qk is the Zernike coefficient, Q is the column vector composed of the coefficients Qk, U is the column vector composed of the terms Uk of the Zernike polynomial, and K is the number of terms in the polynomial.

The first 6 terms of the Zernike polynomial can be expressed as:

It can be seen that the first two terms of equation (9) are tilt terms, and their corresponding coefficients are the tangent values ​​of the measured small angles in the x and y directions. The coefficients of the Zernike polynomials can be solved by the least square method using the optical path difference corresponding to each pixel point on the CCD. However, since the Zernike polynomials are orthogonal in the continuous domain and the data we measured are discrete, it would be very inconvenient to directly use equation (8) for calculation. To solve this problem, people proposed the Gram-Schmidt orthogonalization algorithm to find a set of discrete orthogonal basis function systems V that are linear combinations of Zernike polynomials at the measured data points, and get

Based on formula (14), the coefficient vector B can be obtained by using the least squares method and the orthogonality of the V matrix, and then the coefficient vector Q can be obtained by transforming it through formula (15).

In fact, in the detection of small angles and straightness, we only need to obtain the coefficients of the tilt term in the Zernike polynomial. It should be noted that what we have solved above is the Zernike polynomial coefficients of the reflected wave surface, and it has a two-fold relationship with the surface error of the reflector, that is:

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4. Guide rail straightness measurement experiment

The relationship between straightness error and angle change is as follows:

Where L is the span of the board bridge, that is, the angle measurement sampling length. Based on the above principle, we used the Zygo GPI digital wavefront interferometer to test a ball linear guide with a stroke of 1 m. The measurement results are shown in Table 1.

Using the data in Table 1, the straightness error curve of the guide rail in both directions is shown in Figure 3, and the least squares method is used to fit it. Linear fitting is performed on the straightness measurement data in the X direction to obtain:

The original data is rotated according to the slope of the fitting line to obtain the straightness error data without the tilt value. According to formula (24), the straightness of the guide rail in the X direction is 18·014μm, and the straightness in the Y direction is 32·327μm.

5. Accuracy Analysis

Since the number of pixels of the CCD used in the phase-shift interferometer is 512×512, the resolution of the system is λ/512. When the apertures of the reflector and the reference mirror are the same, both 100 mm, the angle measurement resolution is λ/(512×100), where λ=0·632 8μm. In small-angle measurement, tanα is equal to the tilt coefficient of the latter measurement minus the tilt coefficient of the previous measurement, and the tilt coefficient represents the relative position relationship between the reflector and the reference mirror, and has nothing to do with the surface accuracy of the parameter mirror and the reflector itself. In this way, the small-angle measurement accuracy using phase-shift interferometry technology is only related to the repeatability of the interferometer. The repeatability of the interferometer used in this experiment is better than λ/100, which is converted to an angle measurement accuracy of λ/(100×100).

6 Conclusion

This paper discusses and studies the measurement of small angles using phase-shift laser interferometer combined with Zernike wavefront fitting technology. It is proved that when this method is used for small angle and straightness measurement, compared with general methods, it can greatly improve the measurement resolution and measurement accuracy, providing a new way for high-precision measurement of small angles and straightness.

References

[1] Jin Guofan, Li Jingzhen. Laser Metrology[M]. Beijing: Science Press, 1998, 683-684.

[2] Wang Zhijiang et al. Handbook of Optical Technology[M]. Beijing: Machinery Industry Press, 1994, 1118-1123.

[3] Zhu Hongxi et al. Research on a new type of high-precision laser interferometer for small angle measurement[J]. Acta Metrologica Sinica, 1996, 17(2): 89-91.

[4] Wada Shang et al. Detection of linearity using heterodyne moiré fringe method [J]. Precision Machinery. 1985, 51(6): 100-107.

[5] Malacara D. Optical Shop Testing[M]. New York: JohnWiley, 1978.