Measurement method for the eccentricity of an off-axis asphere with a laser tracker

Ruigang Li

doi:10.3788/COL201513.S22206

Abstract

In this Letter, we present a method to measure the eccentricity of an off-axis asphere using a laser tracker during optical null testing. We first adjust the optical path of the null testing, and then probe some necessary reference surface on the compensator or the off-axis mirror’s body with a laser tracker. Next, using the collected data to process, construct, build the coordinate system, solve, and so on, we obtain the eccentricity directly through comparing the central point of the asphere and the tested optical axis. A measurement experiment is conducted with a circular off-axis aspherical mirror; the result shows that the measurement accuracy can reach 0.1859 mm, 6.2% of its tolerance belt.

Aspherical optics have more variables than spherical optics during optical system design; consequently, using aspherical elements in optical systems can greatly reduce system mass and size with fewer pieces. Therefore, aspherical elements are increasingly used in optical systems such as space cameras, ground-based telescopes, lithographic objectives, and so on^[1–3].

Manufacturing technology of aspheres has been made great progress, but it is still more complicated than the spherical counterpart. Moreover, accurate measurement of the asphere’s parameters is very important but troublesome during the entire manufacturing period, especially when manufacturing an off-axis asphere. Because the vertex of the off-axis asphere is virtual, measurement of its eccentricity becomes increasingly difficult, and methods in use for spherical elements (reflective mirrors or refractive lenses) will lose efficacy.

Before high-precision measuring equipment appeared, a ruler was often used to measure the distance from the reference axis of the asphere to the reference light spot of the interferometer; then the eccentricity of off-axis asphere was obtained. However, this method is susceptible to human variability, so there are always different results from different individuals.

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We present a method that measures the eccentricity of an off-axis asphere with a laser tracker. The measurement principle is described, and a measurement experiment is given.

One important piece of equipment used in the measurement is the laser tracker, which is a kind of portable coordinate measurement machine with high precision. It is a distance-measuring interferometer (DMI) with a self-tracking ability. A laser tracker uses a DMI to measure the distance, and two angular encoders to measure the rotation angles. A laser beam (He–Ne laser with emission wavelength of 632.8 nm) is directed to and reflected by a sphere-mounted retroreflector (SMR). When the SMR moves, a feedback sensor detects the motion, and the mechanical and electronic devices ensure that the machine continues to track.

Laser trackers have many merits such as a large measurement range, fast response speed, high precision, and so on; they have been widely employed in scientific research and industrial production^[4–6]. However, they are rarely applied in optical manufacturing. In this Letter, we use a laser tracker as a critical tool to measure the eccentricity of an off-axis asphere.

During manufacture of large-aperture aspherical elements [Eq. (1)], the widely used testing method of the asphere’s figure error in the polishing stage is interferential null testing with a compensator^[7] $Z = \frac{c S^{2}}{1 + [1 - (K + 1) c^{2} S^{2}]^{1 / 2}} + A_{1} S^{4} + A_{2} S^{6} + A_{3} S^{8} + A_{4} S^{10} + \dots,$ (1)where $S^{2} = x^{2} + y^{2}, c = 1 / r, r$ is the vertex radius of curvature, $K$ is the conic coefficient, $A_{1}, \dots, A_{n}$ are high-order coefficients, and $Z$ is the theoretical height of the aspherical surface. When testing, a spherical or planar wavefront is emitted from the He–Ne laser light source (wavelength is 632.8 nm) of the interferometer, transmitted through a compensator specially designed and transformed into an aspheric wavefront, and then cast on the asphere under test and reflected with the asphere’s figure error information. After going through the compensator again, finally it returns to the interferometer and interferes with the reference wavefront, and then we can acquire the figure error of the asphere through image processing.

Null testing of an asphere is divided into three types according to the compensator that is used: reflective type, refractive type, and diffractive type. They are suitable for different situations. For an off-axis asphere’s optical null testing, the compensator is still an on-axis asphere, but that off-axis aphere testing uses part of the wavefront. In this context, we choose the refractive compensator to perform our measurement experiment; Fig. 1 is a typical off-axis asphere’s testing layout. The refractive compensator is often mounted into a cylindrical barrel for convenience. The mounting cylinder usually has a reference surface with high precision, which can be very helpful for optical testing, and we also can apply them to our eccentricity measurement.

Figure 1.Schematic of null compensating interferometry.

A SMR is placed on a testing target. The laser tracker receives the light reflected by the SMR and acquires the center position $P_{0} (x_{0}, y_{0}, z_{0})$ ; a group of these points is fitted according to the selected surface equation. After collecting all necessary features, a coordinate system is constructed, which requires two steps: translation and rotation. The matrices of the coordinate transformation for a three-dimensional coordinate system are as described next.

The translation and rotation matrix is $T = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ T_{x} & T_{y} & T_{z} & 1 \end{matrix}] .$ (2)

The rotation matrix is $R = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos α & \sin α & 0 \\ 0 & - \sin α & \cos α & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \cdot [\begin{matrix} \cos β & 0 & - \sin β & 0 \\ 0 & 1 & 0 & 0 \\ \sin β & 0 & \cos β & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \cdot [\begin{matrix} \cos γ & \sin γ & 0 & 0 \\ - \sin γ & \cos γ & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}],$ (3)where $T_{x}, T_{y}$ , and $T_{z}$ are the translations in the $x$ , $y$ , and $z$ directions, respectively; $α, β$ , and $γ$ are the angles of rotations around the $x$ , $y$ , and $z$ directions, respectively. Consequently, in order to transforming a point $P (x, y, z)$ in the old coordinate system to a point $P^{'} (x^{'}, y^{'}, z^{'})$ in the new coordinate system, we should use $[\begin{matrix} x^{'} & y^{'} & z^{'} & 1 \end{matrix}] = [\begin{matrix} x & y & z & 1 \end{matrix}] \cdot T \cdot R .$ (4)When measuring, reference features on the compensator and the aspherical mirror are probed, and thus the data of the optical axis and the geometrical center of the off-axis asphere are acquired. Then through processing, the eccentricity of the off-axis asphere is obtained.

Figure 2 is the diagram of the measurement procedure. The first step is optical path setting; i.e., adjusting the optical path according to the optical design parameters until the interferometer, compensator, and asphere are in an initial arrangement. The precision may be on the millimeter scale. More accurate adjustment must be conducted to analyze the interferogram and light spot. By translating or rotating the adjustable stages, the ultimate aim is to adjust the low-order aberrations until they are at a minimum.

Figure 2.Diagram of the testing procedure.

The second step is to collect feature data; i.e., moving the SMR to touch the compensator’s cylinder and acquire the optical axis’s data from the cylinder’s axis, moving the SMR to touch the reference surfaces on the asphere’s block, then obtaining the geometrical center of the off-axis asphere through coordinate processes such as translation, intersection, and so on.

The third step is to build a coordinate system; i.e., using the three reference surfaces of a standard cube which aligns to the optical path to establish a coordinate system. Let the $Z$ axis coincide with the optical axis. The $X$ axis is perpendicular to the $Z$ axis and directs to the geometrical center point of the off-axis asphere. The $Y$ axis is perpendicular to the $Z$ axis and $X$ axis. The coordinate system conforms to a right-hand system.

Finally, obtaining the distance from the geometrical center point of the off-axis asphere to the optical axis, then subtracting the nominal off-axis distance, the eccentricity of the off-axis asphere is acquired.

Figure 3 is sketch of the eccentricity measurement layout.

Figure 3.Sketch of the eccentricity measurement layout.

As aforementioned, we know that measurement with a laser tracker is a contact-type method, which acquires data indirectly by touching the reference surface. Factors that impact the measuring accuracy are as follows: a.Eccentricity error $e_{1}$ and cylindricity error $e_{2}$ of the compensator.b.Deflection error $e_{3}$ caused by the cylinder surface of the compensator with its optical axis, which is related to the length $L$ of the optical path; the consequent linear discrepancy is $L * \tan (e_{3})$ .c.Flatness $e_{4}$ of the aspherical mirror’s reference plane.d.Precision of the laser tracker. The laser tracker’s brand and number is FARO ION, when uses interferometer measurement mode; the typical values of the distance precision $e_{5}$ and angle precision $e_{6}$ are $2 μm + L * 0.4 μm / m and 10 μm + L * 2.5 μm / m .$ (5)

We measured the eccentricity of an off-axis aspherical mirror, it has a circular aperture, and the nominal off-axis distance is 550 mm. There are five reference planes on the block: four side planes and a bottom plane, and the flatness of surfaces are excellent; all are less than 10 μm. Before measurement, using a Zeiss coordinate measuring machining (CMM), and a PRISMO navigator, we tested the distances from the geometrical center point to the three references (two side planes and the bottom plane); they are, respectively, 410.833, 410.819, and 72.040 mm. Thus, conversely, when we measure the reference surfaces with a laser tracker, we can obtain the geometrical center point coordinates by processes such as translation, intersection, and so on.

First of all, we should set the interferometer, compensator, and off-axis aspherical mirror into a right position relationship. When measuring the eccentricity of an off-axis asphere, the laser tracker should be put at a position where it can “see” every feature involved. After all are well-settled, we measure according to the aforementioned procedure; Fig. 4 is a photo acquired during measurement.

Figure 4.Photograph of an eccentricity measurement.

After collecting all necessary information, next we should set a coordinate system as aforementioned. The coordinates system’s origin is constructed at the vertex of the off-axis asphere.

We have made five groups of measurement experiments separately; their results are as follows: 550.450, 550.413, 550.462, 550.447, and 550.433 mm. The average of these results is 550.441 mm. The uncertainty of the measurement is analyzed (Table 1), where the total optical length $L$ is about 7.5 m. Finally, the synthetic accuracy is 0.1859 mm, the designed tolerance of the eccentricity of the asphere is $\pm 1.5 mm$ , and the measurement accuracy is approximately 6.2% of the tolerance belt. According to the criterion of “testing accuracy is less than 1/3 of tolerance belt,” the measurement method clearly satisfies the design requirement.

Item	Error	Remark
Cylindricity of Compensator	e1<0.0100 mm
Eccentricity of Compensator	e2<0.0100 mm
Deflection of Compensator	<5′′, so e3<L*tan(5′′)=0.1818 mm
Flatness of Asphere	e4<0.0100 mm
Laser Tracker	Distance accuracy: e5=2 μm+L0.4 μm/m=0.0050 mm; Angle accuracy: e6=10 μm+L2.5 μm/m=0.0288 mm	FARO (ION), interferometer), distance accuracy and angle accuracy are all typical values
Total	e=Σi=16ei2=0.1859 mm	Root-sum squared (RSS) of individual errors.

Table 1. Uncertainty Analysis

View all Tables

A measurement method of the eccentricity of an off-axis asphere with a high-precision, portable laser tracker is presented in this Letter. The method has the following advantages: simple principle, convenient operation, good repeatability, and high accuracy. A measurement experiment is conducted by analyzing a circular off-axis asphere; the results show that the synthetic accuracy can reach 0.1859 mm, which is 6.2% of the tolerance belt ( $\pm 1.5 mm$ ). From the experiment we know that the main factor that affects the final accuracy is the total length of the optical setup; however, in general, the length is less than 10 m, and consequently the accuracy is satisfactory. The aformentioned method is not only suitable for an off-axis asphere, but can also measure the eccentricity of a sphere and an on-axis asphere.

References

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