## Abstract

A new method of measuring external cylindrical surfaces is proposed, making use of a ring beam transform unit. The unit is composed of three cone mirrors, with which a parallel beam can be transformed into annular and convergent beams. The method has advantages over classic methods: It avoids physical contact with the cylinder surface and is fast and stitching-free. In experiments using a dynamic interferometer to demonstrate the feasibility of the method, the RMS difference of the axis contour from results using a Luphoscan is 0.0282 µm, while the roundness difference of the circumferential contour is 0.0962 µm.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Cylindrical surfaces are widely used in mechanical and optical systems such
as aerostatic cylindrical bearings, error calibration of rotary stages,
and interferometry systems. In most such systems, the profile of the
cylindrical surface directly affects the performance of the system.
Traditional instruments to measure cylindrical surfaces, e.g., a
coordinate measurement machine, cylindricity measuring instrument, and
contour graph with V-block, make contact with the surface, possibly
resulting in surface damage in the measurement process [1,2]. Moreover, because these instruments scan the cylinder circle
by circle along its axis, rotary state errors may be introduced into the
results. With the development of computer-generated holograms (CGH) and
subaperture stitching technology, interferometry methods have been
proposed to measure cylindrical surfaces. Peng *et
al.* sought to achieve high-precision measurements, including the
introduction of Legendre–Fourier polynomials, to offset the form error by
adopting a cylindrical projection to establish one-to-one mapping between
the subaperture data and points on the cylinder surface [3–5]. However, stitching technology is
inefficient because dozens of subapertures are needed, owing to the
limited $F$-number of CGH. Accordingly, other methods
have been proposed to achieve a higher measurement efficiency without
sacrificing accuracy. Weckenmann *et al.*
proposed a grazing-incidence interferometer equipped with CGH to measure
cylindrical surfaces. This incorporated parallel data acquisition of the
cylindrical surfaces and achieved excellent radial resolution [6]. White light interferometers have been
used with conic mirrors to measure both continuous and stepped cylindrical
surfaces, achieving an uncertainty as small as 1 µm [7,8]. Also,
deflectometry in cylindrical coordinates has been proposed to measure the
cross section of cylindrical parts and determine deviations from roundness
[9].

In this paper, a modified Twyman–Green interferometry system is proposed to realize efficient measurement of the external cylindrical surface. The key element of the system is a ring beam transform unit (RBTU) composed mainly of three conic mirrors, which transform the parallel beam into annular and convergent beams, so that information on the cylindrical surface can be reflected back to form useful interferometric fringes. The system is realized with dynamic interferometry, and the results demonstrate the feasibility and validity of the proposed method.

## 2. METHOD

#### A. Measurement System

The setup for the proposed method is shown in Fig. 1. It consists of a laser source, beam expander, reference mirror, RBTU, detector, and other lenses. The beam expander is optional because the size of the beam determines the height of the cylinder accessible for a single measurement. The laser beam is split by the polarizing beam splitter (PBS) into measuring and reference beams. The measuring beam is then transformed into annular and convergent beams by means of three reflections from conic mirrors A, B, and C. Reflected by the sample cylinder, which is set to be coaxial with the measured system, the measuring beam returns to the PBS carrying surface information and forms interference fringes with the reference beam on the detector.

Using the interference fringes and phase-stepping interferometry, the surface information can be obtained. Let ($x,y$) be the coordinates of a point on an interference fringe; the height ($x, y$) is the surface deviation at ($x,y$), and ${{ I}_1} - {{ I}_4}$ are the intensities at ($x, y$) for phase shifts of 0, ${0.5}\pi$, $\pi$, and ${1.5}\pi$, respectively. ${{I}_1} - {{ I}_4}$ can be obtained by the four interference fringes captured by the camera, and the surface deviation can be described by

#### B. Surface Reconstruction

The surface deviation results appear on a circular plane since the purpose is to measure surface information on a cylindrical surface; hence, surface reconstruction is needed to establish the relationship between the surface deviation and the cylindrical surface. Moreover, the eccentricity and tilt of the cylindrical surface relative to the RBTU directly affect the surface deviation, which needs to be corrected in the reconstruction process in order to obtain a more accurate result. As shown in Fig. 2, the reconstruction can be divided into three steps: (1) transforming the measured results to the 3D cylindrical surface; (2) fitting the surface with a spatial cylinder to remove the tilt and eccentricity between the sample and the RBTU; and (3) computing the distance between the measured points and the axis of the fitted cylindrical surface to check on the cylindricity.

According to the geometric relationships, the radial direction of the results corresponds to the axial direction of the cylinder. Let O be the center of the surface deviation results. Then, (${x_0}, {y_0}$) are the pixel coordinates of a point P, $\theta$ is the angle between OP and OX, as shown in Fig. 2(a), $d$ is the pixel size, and ${{\rm P}_1}$ is the projection of point P on the cylinder. The location of ${{\rm P}_1}$ can be described by

Let $q$ designate the axial line of a cylinder. ${{\rm P}_0}$ (${x_1}, {y_1}, {z_1}$) is a point on $q$, and ($l, m, n$) specify the direction vector of $q$, with $r$ being the radius of the cylinder. According to the definition of a cylinder, the equation of a cylinder can be described by Eq. (3).

Combining the point cloud found by Eq. (2) and the equation of a cylinder, the parameters of the cylinder can be obtained by minimizing Eq. (4) according to the nonlinear least-square method [11]. Then, the surface information corrected for eccentricity and tilt can be obtained by computing the distance between each point and the axis of the cylinder:

## 3. EXPERIMENT AND DISCUSSION

#### A. Experimental Setup

To verify the practicality of the proposed method, experiments based on dynamic interferometry were conducted. As shown in Fig. 3(a), the system included a dynamic interferometer (PhaseCam 6000 made by 4D Technology Corporation), a ring beam transform unit (RBTU), and several multiaxis adjustment stages. The camera of PhaseCam 6000 has an effective image size of ${996} \times {984}$ pixels, and the pixel size is 9.03 µm. The effective radius of the laser beam was 8.5 mm, which means the maximum theoretical measured range was 4.25 mm. To minimize the influence of the geometric position between three conic mirrors, the RBTU is specially designed, as shown in Fig. 3(b). Conic mirror A is glued to an optical flat with antireflection film on both sides, and the position can be adjusted by four screws. The relative position between conic mirrors B and C is ensured by the conic surface concentric to them. The cylindrical surface sample is an aluminum alloy part manufactured by single-point diamond turning with a diameter of 30 mm, in which an arc with 2 µm width and 5 mm length is manufactured as the measure region, as shown in Figs. 3(c) and 3(d).

The adjustment in the measurement process contains two main steps. (1) The sample is removed from the RBTU and the relative position of the RBTU and the dynamic interferometer is adjusted to make the optic axis parallel to the cylinder axis as well as to keep the laser at the center of the RBTU. In this step, the RBTU reflects the rays to the dynamic interferometer, so that the interference fringes indicate the relative position of the RBTU and the dynamic interferometer. (2) The sample is put back into the RBTU and adjusted, with the RBTU and dynamic interferometer fixed, until reasonable interference fringes appear.

#### B. Results and Discussion

Figure 4 shows the interference fringes of an experiment. There is a fuzzy part in the center due to (1) the top of conic mirror A in Fig. 1 can never be manufactured to its theoretical shape; (2) the lateral resolution along the angle theta is radius-dependent, and the effective surface information decreases with the decreases of radius. MATLAB is adopted to achieve the surface deviation computation, unwrapping, and surface reconstruction.

Figure 5(a) shows the original surface deviation unwrapped by phase-stepping interferometry based on the interference fringes in Fig. 4. The axial resolution is the pixel size, while the lateral resolution is the quotient of the radius of the cylindrical sample and the radius in terms of the number of pixels on the camera sensor according to Fig. 2. In order to obtain sufficient sampling density and a complete cylindrical surface, an annular region between two concentric circles is established as the region of interest, in which the radius of the inner circle is 150 pixels and the radius of external circle is 450 pixels; hence, the axial measure range is $({450} - {150}) \times {9.03} = {2.709}\;{\rm mm}$. The reconstructed surface corrected for eccentricity and tilt of region of interest is shown in Fig. 5(b), in which the colored bar indicates the radius deviation of the measure point relative the estimated $r$ in Eq. (3), and the maximum surface deviation is 1.3 µm within the region of interest.

As shown in Fig. 6(a), a Luphoscan 420 manufactured by Taylor Hobson is adopted to measure the cylindrical surface sample in order to verify the measured results. Figure 6(b) shows the results measured in this way, in which the colored bar is the radius deviation obtained by Luphoscan and reveals a maximum surface deviation of 2.2 µm over a 5 mm length region.

The proposed interferometry system and Luphoscan provide relative measurements not tied to the actual radius of the cylindrical surface. The contour lines adopted to compare the two methods must therefore be adjusted to the same reference value, which does not affect the comparison of surface deviations. Figure 7(a) shows comparative results for the axial contour line, for which the contours are integrally shifted and rotated. The RMS difference of the two contours is 0.0288 µm, indicating that the contours from the two methods match well. Figure 7(b) shows comparative results for the circumferential contour line, in which the two contours are adjusted to the reference radius of 1.65 µm. The roundness determined by Luphoscan is 0.1833 µm; it is 0.2795 µm for the proposed interferometric method.

## 4. CONCLUSIONS

A new interferometry method to measure radial deviation of an external cylindrical surface is presented. It makes use of a ring beam transform unit composed of three conic mirrors. An experiment using a dynamic interferometer demonstrates the feasibility of the method. In the experiment, the axial contour differs from the Luphoscan results by 0.0282 µm, and the roundness difference of the circumferential contour is 0.0962 µm. The proposed method can obtain the 360° surface map in a single measurement without physical contact, which improves measurement efficiency and avoids surface damage. However, the surface deviation of three conic mirrors is directly reflected in the results, though it is much smaller than the deviation of sample, which may be studied further to remove system errors and improve measurement accuracy.

## Funding

Science Challenging Program, China Academy of Engineering Physics (JCKY2016212A506-0106).

## Acknowledgment

The authors thank Shanghai Engineering Research Center of AI & Robotics and Engineering Research Center of AI & Robotics, Ministry of Education, for their support.

## Disclosures

The authors declare no conflicts of interest.

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