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Since the cylinder surface is closed and periodic in the azimuthal direction, traditional stitching interferometry cannot be used to yield the 360° form map. This paper describes a full cylinder stitching interferometry based on the first-order approximation of cylindrical coordinate transformation. First, it introduces cylindrical projection, which allows us to determine the overlap region of the cylinder without ambiguity. Second, the relationship between the variations of radial coordinates and the movement errors of the rotational stage is derived from the first-order approximation of cylindrical coordinate transformation. Based on this relation, a cylinder stitching model is built to connect all sub-apertures together. Finally, we experimentally validate the proposed method by measuring a precision metal shaft. The high resolution and repeatability shown in the experimental results demonstrate our approach to be an attractive and promising technique in the field of precision measurement.
© 2017 Optical Society of America
1. Introduction
Precision cylinders arise in many applications, such as air-bearing spindles, ring gauges and rod lenses. A small form error or the cylindricity of the whole surface is vital for high-performance instruments. For example, the setting ring gauge with 50mm diameter requires a maximal roundness deviation of 100nm in the Germany industry [1]. Thus, precision measurement is desperately needed for the fabrication process of these precision cylinders.
Generally, the figure error of cylinder can be roughly estimated with tactile measurement methods, such as a coordinate measuring machine (CMM) or cylindricity measuring instrument [2]. Typically the number of sample points used is very small, because of the time required to collect them. Besides, the sampling strategy (factors including the sampling method and rate), has a significant impact on the final evaluation result [3, 4]. Moreover, according to the ISO 12180 standard, the results obtained from the assessment of selected roundness profiles cannot be considered sufficient to accurately determine cylindricity deviations. Other methods, which can acquire the 360° form map of cylinder with high resolution and accuracy, are therefore needed. Due to high sensitivity, high accuracy and repeatability, interferometry becomes a key technology for precision surface metrology. For instance, with two diffracting elements (e.g., computer generated hologram (CGH) plate), grazing incidence interferometer is able to measure the form deviation of cylindrical parts [5,6]. However, a strongly anamorphotic distortion occurs in the recorded interferogram, resulting in a low-density sampling rate along the rotation axis [7]. These drawbacks can be circumvented using the normal-incidence method with single CGH. Nevertheless, limited by the f/number of the commercial available CGHs (usually no less than f/1), [8], the 360° form map of a cylinder cannot be imaged in a single measurement. One way to overcome this limitation is to acquire a set of partially overlapping sub-apertures, taking them at different locations over the whole surface, and then mathematically connecting them to yield a stitched result.
Stitching interferometry was originally introduced to avoid the expense of fabricating large reference optics for testing astronomical mirrors [9]. It has since been extended to full aperture maps of large-aperture flat [10], spherical [11, 12], and aspherical surfaces [13, 14]. Despite some efforts were made to measure high-numerical-aperture (NA) cylindrical lenses with stitching interferometry [15], significant challenges appear when stitching interferometry of full cylinder. Firstly, the cylinder surface is closed in the azimuthal direction, whereas the curved face of cylindrical lens is a section of cylinder. In the case of high-NA cylindrical optics, the stitching equations were built with the misalignment aberrations removal models, such as the two-dimensional (2D) Legendre polynomials (LPs) [16]. Actually, these misalignment aberrations removal models are valid only when the NA of the tested optics is less than 1. Therefore, existing stitching models are not suitable for reconstructing the 360° form map of cylinder. Secondly, the cylinder surface is periodic in the azimuthal direction. When determining the overlapping regions with existing methods, a couple of points located on the front or back of the cylinder may have the same projections. Consequently, projection ambiguity in determining the overlap regions might arise in stitching interferometry of full cylinder.
This paper proposes a cylinder stitching algorithm to obtain the 360° form map. Different from the cylindrical stitching interferometry [15], cylindrical projection is introduced to establish a one-to-one mapping between the sub-aperture data and the points on the cylinder surface. Thus, the projection ambiguity mentioned above can be avoided. In addition, the first-order approximation of cylindrical coordinate transformation is used to decompose the sub-aperture data into the inherent form of partial cylinder surface and additional misalignment parameters. Based on this relation, a cylinder stitching model is built to mathematically tie all sub-aperture together. Finally, a metal shaft was measured, thus demonstrating the feasibility and validity of the proposed method.
Here is the outline for this paper. In Section 2, we propose an interferometric system and measurement procedure to collect a series of sub-aperture data covering the entire surface of cylinder. In Section 3, coordinate mapping is constructed from the local frame to the global frame. Then cylindrical projection is introduced to determine the overlap regions without ambiguity. Furthermore, a cylinder stitching model is built based on the first-order approximation of cylindrical coordinate transformation. In Section 4, we apply the proposed stitching algorithm and system to measure a metal shaft. Experimental results demonstrate that the proposed method possesses advantages of high resolution and accuracy compared with the contacting methods. Finally, Section 5 concludes by summarizing the benefits of our proposed stitching method.
2. Measurement of full cylinder using interferometry
2.1. System configuration and measurement procedure
Figure 1 sketches the measurement system, which consists of a Fizeau type interferometer with a transmission flat (TF), CGH cylinder null, and multi-axis stage to handle cylinder to be measured. The plane wavefront emitted from the interferometer is converted by the CGH cylinder null, and then the outgoing cylindrical wavefront hits the tested cylindrical surface and reflects back into the interferometer. Analyzing the recorded interferograms allows us to acquire the form error of the cylindrical surface. Limited by the f/number of the CGH, the 360° form map of cylinder cannot be acquired with the above system through a single measurement. To address this problem, a rotating stage is used to orient different partial surfaces of the cylinder in front of the CGH. Thus, the entire surface of cylinder can be measured in parts.
The measurement procedure for a cylinder is outlined in Fig. 2. It begins with placing the tested cylinder on a multi-axis stage. Then, we center and level the tested cylinder with the multi-axis stage, so that we can acquire resolvable interferograms during rotation. Next, the arrangement of collection (also called the lattice) is defined based on the f/number and clear aperture (CA) of the CGH. The lattice defines the sub-aperture location at which the interferometer must capture an interferogram. To ensure that the stitching algorithm yields a reliable and accurate result, each pair of adjacent sub-apertures should be partially overlapped. Following the lattice, we can orient the designated surface in front of the interferometer. After locating and aligning the first null, we utilize the interferometer to collect phase data for all sub-apertures. Finally, these data are connected to yield the 360° form map based on the stitching algorithm.
Motion errors of the rotating stage inevitably cause the cylinder to slightly deviate from the null position. Thus, the sub-aperture data may be contaminated with misalignment aberrations. The accuracy of the stitching result cannot be guaranteed unless these misalignment aberrations are removed from the sub-aperture data. The authors of this paper recently proposed a mathematical model to describe the contribution of small misalignment errors when null testing a cylindrical surface, reader can refer to [17] for details.
2.2. Problem in stitching interferometry of full cylinder
With the interferometric system and measurement procedure mentioned above, we can only obtain the form map of partial cylinder surface. To combine the individual sub-aperture into a 360° form map, we should transform the sub-aperture data from the local frame to the global frame in advance. Then we need to build a stitching model based on the corresponding points among overlap regions. Generally, we identify the corresponding points with the global coordinates. In the case of cylindrical stitching interferometry, we define a mid-plane that is parallel to the back plane of the cylindrical lens, project the measurement points onto the nominal cylinder, and map the projections onto the mid-plane. In this plane, the envelope of projections determines the overlap region. However, a new problem arises in the case of a cylinder, in which the whole surface is periodic in the azimuthal direction. As a result, a couple of points located on the front and back of the cylinder may have the same mapping points on the mid-plane. This problem can be avoided by using the cylindrical projection. In the next section, we will demonstrate why it is feasible and how to implement it.
Another concern is the cylinder stitching model. As mentioned in Section 1, existing stitching models are valid only when the numerical aperture (NA) of the cylindrical lens is less than 1. In addition, the LPs are orthogonal over the rectangular pupil, and like the classical aberrations, are inherent separable in the Cartesian coordinates of the pupil point [16]. In this case, the cylindrical stitching model was essentially built on the Cartesian coordinate system. In contrast, as the cylinder surface is closed and periodic in the azimuthal direction, the cylindrical coordinate seems to be a logical choice for describing a point on a cylinder. Thus the cylindrical stitching model built based on the LPs is not suitable for analyzing and reconstructing the form of cylinder. To address this problem, the first-order approximation of cylindrical coordinate transformation is introduced in the following section.
3. Stitching interferometry of full cylinder based on the first-order approximation of cylindrical coordinate system
3.1. Mapping sub-aperture data to global coordinate system
Unlike existing stitching interferometry methods, the proposed method operates in cylindrical system, which is more suitable for describing the 360° form map than the Cartesian system. Since the readout from the commercial interferometer is given in phase value, all sub-aperture data should be transformed into cylindrical coordinate for stitching.
As shown in Fig. 3, the local coordinate frame is set at the center of the CGH cylinder null. The sub-aperture data is denoted by phase triplets W (u, v, φ), where φ represents the phase difference at pixel (u, v). According to the test geometry, the local coordinates are related as follows:
where s is the scale factor, which can be calibrated based on the CGH fiducials [18]; rbf denotes the back focal distance of the CGH cylinder null; and Δρ is the radial deviation of the tested surface.
Fig. 3 The geometric sketch of coordinate transformation: (a) footprint on the CGH, (b) test configuration.
Applying Eq. (1) allows us to obtain the coordinates of measurements in the local cylindrical frames attached to the sub-apertures. To determine the 360° form error of the cylinder, we build a global frame attached to the tested cylinder as shown in Fig. 4, where the x-axis coincides with the center axis of the cylinder and the z-axis coincides with the optical axis of the interferometer. Based on this geometry, we transform local sub-aperture coordinates into global ones by
where x0 and θ0 denote the nominal translation of a sub-aperture along the x-axis and its rotational angle around the x-axis, respectively.3.2. Overlap calculation with cylindrical projection
To build a series of sub-aperture data covering the cylinder surface, a multi-axis stage is used to orient the tested cylinder. In practice, it is difficult to perfectly control the tested object in three-dimensional (3D) space. Thus, there is some ambiguity when connecting the individual sub-apertures based on the nominal movement parameters. To address this problem, a common strategy is to overlap adjacent sub-apertures, shifting the problem to the calculation of this overlap.
In contrast with cylindrical stitching interferometry, cylindrical projection is introduced to determine the overlap region. First, all sub-aperture data are projected onto a nominal cylinder; then, the projections are mapped onto the longitudinal plane, rather than the mid-plane, as shown in Fig. 5. The longitude of a cylinder is restricted within (−π, π). Therefore, we can build a one-to-one mapping between the sub-aperture data and points on the cylinder.
Assume N sub-apertures are used to cover the 360° surface of the cylinder. Generally, individual sub-apertures, whose longitude is within (−π, π), can be mapped onto the longitudinal plane with a single area. In this case, the overlap areas between adjacent sub-apertures (e.g., k and l) can be determined according to the envelope of the projections, as shown in Fig. 5. However, as the cylinder surface is periodic in azimuthal direction, there exists sub-apertures (e.g., sub-aperture N) whose longitude is beyond the range of (−π, π). Consequently, an incorrect conclusion—that the projections of sub-apertures 1 and N do not overlap—is drawn. To avoid this problem, the projection of sub-aperture N is first trimmed into two separate areas, N_0 and N_1. Then, N_1 is merged into the range of (−π, π) by subtracting a constant 2π, which yields two separate areas. This calculation determines the correct overlap region between sub-apertures 1 and N.
3.3. Stitching algorithm for full cylinder
Generally, six degrees of freedom (DOF), consist three translations (Δx, Δy, Δz) and three rotations (Δα, Δβ, Δγ), are usually used to determine the position and orientation of an object in 3-D space. To acquire resolvable interferograms at each sub-aperture, we need to center and level the tested cylinder in advance. In other words, the movement errors of the cylinder axis can be restrained to a small level. Under this circumstance, we can simplify the coordinate transformation between the adjacent sub-apertures with first-order approximation. According to [19], the first-order approximation of coordinate transformation in cylindrical coordinate system can be expressed as
where the point (ρ1, Θ1, X1) is in the overlap region and belongs to sub-aperture 1, its corresponding point in sub-aperture 2 is (ρ2, Θ2, X2).From Eq. (3), it can be seen that Δx and Δα do not affect the value of the radial coordinate ρ. Besides, although Δx and Δα will change the values of Θ and X, they can be restrained to a negligible level by using a commercial precision turntable. Thus the problem shifts to the suppression of the rest four errors (Δy, Δz, Δβ, Δγ). As mentioned above, to get the resolvable interferograms at each sub-aperture, the four movement errors (Δy, Δz, Δβ, Δγ) should be restrained to a small level. To accomplish this goal, we rotate the tested cylinder in contact with a dial indicator. According to the runout of indicator, we carefully adjust the multi-axis stage until the runout of indicator is close to zero. Since the minimum graduation of indicator is 0.01mm, Δy and Δz can be restricted within (−0.1mm, 0.1mm); Δβ and Δγ are restricted within (−0.01°, 0.01°). Substituting these values into Eq. (3), it is not difficult to draw the conclusion that the variations of Θ and X caused by the four movement errors (Δy, Δz, Δβ, Δγ) are much small, corresponding to 0.1 pixels in the interferograms. As a result, it is reasonably to just consider the variation of ρ in the stitching interferometry.
Assume ρ = ρn + Δρsub, where ρn is the nominal radius of the tested cylinder, Δρsub is the sub-aperture data after removing the misalignment aberrations. Then according to Eq. (3), a relation of the partial cylinder surface Δρstitched and the transformation parameters (Δy, Δz, Δβ, Δγ) are defined as follow,
In theory, if two sub-apertures k and l are measured with an overlap area, the figure errors in these two overlapping areas originally correspond to each other. However, movement errors induced by the rotation stage lead to discrepancies in the overlap areas. To stitch the two sub-apertures together, the sum of the squared difference for all points in the overlap areas should be minimized simultaneously using the following:
where Δρsub,l (X, Θ) and Δρsub,k (X, Θ) are the radial deviation of sub-apertures l and k, respectively. For convenience and simplicity, (X, Θ) is omitted in the following text and equations.The following matrix equations were obtained by differentiating Eq. (5):
where Δρ = Δρsub,k − Δρsub,l, Δy = Δyk − Δyl, Δz = Δzk − Δzl, Δβ = Δβk − Δβl, and Δγ = Δγk − Δγl. With matrix inversion, we can readily obtain the four relative errors from Eq (6).If N(N > 2) sub-apertures are used to cover the 360° surface, 4 × (N − 1) parameters need to be resolved. Here, we define Mi = [Δzi, Δyi, Δγi, Δβi]T, i = 1,…, N − 1. fi = ∑ Mi (Δρsub,i − Δρsub,i+1), g = ∑ M1 (Δρsub,1 − Δρsub,N), and Co = [cosΘ, sinΘ, −XsinΘ, XcosΘ]T. Subaperture 1 is set as the datum. In order to yield the 360° stitching result, the sum of the squared differences in N overlap areas should be simultaneously minimized [20], as
Therefore, the final matrix equation can be expressed as
where , .From Eq. (8), the unknown coefficients Mi for each sub-aperture can be solved readily by matrix inversion. Then, the radius deviations in each sub-aperture are updated using the following equations:
With Eq. (9), the discrepancies in the overlap regions are minimized and the 360° form map of the tested cylinder can be obtained by averaging the data in each overlap area.
4. Experimental demonstration
Several experiments were carried out to verify the practicality of the proposed stitching algorithm. Figure 6(a) shows the experimental setup, which consists of a Fizeau interferometer (Zygo GPI/XP 4″), CGH cylinder null with a CA of 80mm × 80mm (Diffraction international, H80F3C), and multi-axis stage. The CGH cylinder null can diffract a plane wavefront emitted from an interferometer into a f/3 cylindrical wavefront. As the multi-axis stage can rotate the tested cylinder during measurement, we can orient the designated surface in front of the interferometric system. It can also null and align the cylindrical surface, e.g., tilt and translate the cylinder in the horizontal plane, so that the fringe number of the recorded interferogram is minimal.
Fig. 6 Experimental setup and measurement region: (a) Experimental setup, (b) measurement region (unit: mm).
The target to be measured is a metal shaft with height of 120 mm and diameter of 75 mm. As the f/number of the CGH is 3, the maximum field of view (FOV) is 18.9° in a single measurement. To get the 360° form error, we take a series of sub-apertures covering the whole surface of the cylinder and mathematically tie these together. In order to yield consistent and reliable stitching result, there are overlap regions between adjacent sub-apertures. Here, we set the rotational interval between the adjacent sub-apertures to 10°; thus, 36 sub-apertures are required to cover the 360° surface. Limited by the DOF of the multi-axis stage, we did not translate the cylinder along the x-axis direction. The measurement areas are demonstrated in Fig. 6(b). The stitching measurement is performed by collecting the partial surfaces of the cylinder in sequence until its entire surface is covered. At each sub-aperture view, we carefully adjusted the tested cylinder so as to acquire the null fringe pattern image, which means the high-order misalignment aberrations can be constrained to a negligible level [21]. The interferogram and phase map of each sub-aperture are given in Figs. 7(a) and 7(b), respectively. After removing the misalignment aberrations from each sub-aperture, all sub-aperture data were stitched together with the proposed method. The 360° form map of the tested cylinder is shown in Fig. 8(a), where the PV and RMS values are 0.764 μm and 0.157 μm, respectively.
Fig. 7 Sub-aperture Data: (a) recorded interfergrams, (b) phase map after removing the misalignment aberrations.
To evaluate the effectiveness of the proposed stitching method, we observed the variation of the mismatch map within the overlap regions, as shown in Fig. 8(b). Since it was yielded by calculating the radial deviations of corresponding points in the overlap region, it can be thought of as the residual noise following stitching process. From Fig. 8(b), we can see the PV and RMS values are 0.143 μm and 0.005μm, respectively. Comparing with the PV and RMS values of the stitching result, the residual errors in the overlap areas are small and uncorrelated. The small magnitude of the mismatch also confirms that the proposed stitching model is able to eliminate the discrepancies among the overlap regions. In addition, to investigate the robustness of the proposed method, we set different sub-apertures as datum. To do this, we change the order of the sub-apertures. Then we used the proposed stitching model to yield the 360° form map, as shown in Fig. 9. Compared to the stitching result given in Fig. 8(a), it can be seen that the three stitching results conform to each other except that the latter two rotated with certain degrees. Furthermore, another experiment was carried out to investigate the reproducibility of the proposed method, where the metal shaft was measured 9 times repeatedly. In each measurement process, the meal shaft was rearranged so as to change the initial position. Table 1 summarized the PV and RMS values of the stitching results, where standard deviation of the PV and RMS values are 0.017μm and 0.008μm, respectively. It can be seen in the table that the stitching results fluctuate with a small amplitude and have high repeatability.
Fig. 9 Stitching results when different sub-apertures were set as datum: (a) Sub-aperture 310 was set as datum, (b) Sub-aperture 260 was set as datum.
To further verify the accuracy of the proposed method, a verification test was carried out, in which the same area of the metal shaft was measured by a cylindricity measuring instrument (Taylor Hobson 585 with radial accuracy of ±0.01μm) at Shanghai Institute of Quality Inspection and Technical Research (SQI). In practice, cylindricity is a 3D form deviation that controls how much a real shaft deviates from a perfect cylinder. To acquire cylindricity with this method, the roundness deviation was first measured at five different cross-sections. Then, based on the five circular cross-sections, the cylindricity was evaluated as 0.81 μm. Compared with the results listed in Table 1, the difference between the results obtained by the two different methods is less than 50nm. In addition, to visually compare the two results, the stitched result given in Fig. 8(a) is converted into a 3D plot, as shown in Fig. 10(a), and the 3D plot obtained by the cylindricity measuring machine is also given in Fig. 10(b). The reader can refer to Visualization 1 and Visualization 2 for detailed information. From these figures, it seems that the results obtained with these two measuring principles have good conformance. Most importantly, we can obtain a high resolution 360° form map for the tested cylinder using the proposed method, whereas the readouts from the roundness measurement machine are single cylindricity value and several roundness traces.
Table 1. Reproducibility of the proposed method (Unit: μm).
σ: Standard deviation.
Fig. 10 Comparison between the roundness measurement instrument and the proposed method: (a) 3D view of the stitched results (see Visualization 1), (b) 3D cylindricity map obtained by cylindricity measuring instrument (see Visualization 2).
5. Conclusion
We have developed a stitching interferometry algorithm and measurement system to acquire the 360° form map of a precision cylinder. Experimental results show good repeatability and surface characteristics of the cylinder. Through a comparison, it is demonstrated that the proposed technique achieves a high accuracy of λ/10(1λ = 632.8nm), with the differences of its measurement results from that of a high-precision tactile instrument being not more than 50nm. In addition, the newly proposed technique has advantages over the tactile one in being noncontact and having much higher lateral and axial resolutions.
Funding
National Natural Science Foundation of China (NSFC) (No. 61605126); China Postdoctoral Science Foundation (2016M590807); Scientific and Technological Project of the Shenzhen government (JCYJ20150625100821634).
Acknowledgments
We are very grateful to Dr. Zixin Zhao from Xi’an Jiaotong University for his thought-provoking discussions. We also acknowledge Mr. Qiangsheng Xu from SQI for his kind help in measuring the cylinder with a cylindricity measuring instrument.
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