1. Introduction

Phase Measuring Deflectometry(PMD) can measure the three dimensional(3D) shape of the aspheric mirror without any reference surface, so it becomes a favorable method for measuring such surfaces with large aperture and high deviation [1]. There are some kinds of PMD [2–5]. To measure the aspheric mirror, the methods must be able to measure both slope and coordinate of test points on the surface, thus the tested surface can be measured unambiguously and with high precise. In this kind of PMD, the traditional methods [3,4] measure the tested surface by finding the intersections of the camera rays and the reflection rays. Those methods can be used to measure the absolute height of specular free-form surfaces. And they don’t need to fit tested surface to special position, so the measurements can be done flexibility. With very high calibration precise, they can measure the tested surface with high precise. However, in practice, the calibration is complex, and its precise may be not high enough to measure the tested surface with high precise. Therefore, those methods may be not suitable to measure the aspheric mirror if high accuracy measurements are desired. A method uses “dummy paraboloid” to measure the aspheric surface [5]. That method can measure the tested surface without complex calibration. But for having an approximation in the measurement, that method isn’t suitable to measure the surface with high deviation to paraboloid. Furthermore, in that method, the camera must be moved along the optical axis of the tested mirror. In practice, it is difficult to realize this, and the beam splitter used to solve the occlusion problem brings aberration and limits the measurement size of the test surface.

In this study, we present a novel PMD to measure the aspheric mirror with higher precise. In the measurement, a screen displaying fringe patterns is moved along the optical axis of the tested mirror and a camera is located beside the optical axis to observe the fringe patterns reflected via the tested surface. This method measures the surface with the reflection rays and the “dummy paraboloid”. Compared with the traditional methods, we calibrate the position of the camera pinhole instead of the camera rays. Therefore, our method doesn’t need to calculate the normalized image projection vectors which may be influenced easily by the calibration errors such as focal distance, principal point, distortions and so on. And the errors of the calibration influence our method less than the traditional methods. Compared with the method [5] which also measure the surface with the dummy paraboloid, our method doesn’t have approximation in the measurement, so it can be used to measure the high deviation aspheric mirror. Furthermore, by locating the camera beside the optical axis, our method can be realized more easily, and doesn’t have the limits caused by the beam splitter. Therefore, by this novel PMD, the high deviation aspheric mirror can be measured with high precise and less limits in the calibration precise and measurement fitment. The computer simulations and the optical test show the feasibility of this method.

2. Principle

In this study, we present a novel method to measure the aspheric mirror. For one point on an aspheric mirror, there must be a paraboloid whose optical axis coincides with the tested surface’s tangent with the aspheric on the point. The paraboloid of the point is called as its “dummy paraboloid”. The key point of our method is to find out the unique paraboloid to measure the tested point.

For the unique “dummy paraboloid” can make a relationship between the tested point’s coordinate and slope, we can measure the tested point without calibrating the camera rays. The camera rays are easily influenced by calibration errors. Therefore, compared with the traditional method, our method can measure the tested surface with higher precise even if the calibration isn’t very high.

The measurement setup of our method is shown in Fig. 1 . A screen displaying fringe pattern is fitted perpendicular to the optical axis of the tested mirror and moved along the optical axis for a distance of d. A camera is located beside the optical axis and observers the fringe patterns reflected via the tested mirror. The principle of our method is illustrated in Fig. 2 . As shown in Fig. 2, a Cartesian coordinate system is established. The origin o is the intersection of the tested mirror optical axis and the screen, and the z axis is the optical axis. A pinhole model is assumed for the camera and the pinhole coordinate C is known by calibration. Using well known phase shift algorithms, we can get the coordinate of all points that the camera observes from the screens precisely. Therefore, for each pixel of the camera, the reflection line can be calculated by moving the screen for a distance of d in the z direction.

 

Fig. 1 the structure of measurement setup

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Fig. 2 the schematic of measurement principle

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For the latter formula derivations, we make a rule of variable definitions as following: the bold symbol is a vector and its common types with x, y, z, in the subscript represent it components in the x, y, z direction respectively. For example, A is a vector and$A_x$,$A_y$,$A_z$ are the x, y, z coordinate of A, respectively.

As shown in Fig. 2, a pixel of the camera observes the fringe pattern via the reflection point S. For this pixel, the camera pinhole coordinate C and the reflection line T are known. And the reflection line T can be expressed as

Using those known quantity T and C, the slope and the coordinate of point S can’t be calculated directly. Therefore, we add another qualification which is that there is a dummy paraboloid P tangent with the tested surface at the reflection point S. The dummy paraboloid P can be expressed as

Using the known information, we make a function f and minimize it to measure the slope and coordinate of the reflection point. In the function f, there are two variable w and $S_z\text{'}$. The w is assumed to satisfied with the expression

The $S_z\text{'}$ is the coordinate in the z direction of an assumed reflection point $\colorbox[rgb]{}{$S$}\text{'}$ which is assumed to equal to the reflection point S.

From $S_z\text{'}$ and the reflection line T, we can calculate the coordinate in the x, y direction$S_x\text{'}$,$S_y\text{'}$ of the assumed reflection point $\colorbox[rgb]{}{$S$}\text{'}$.

Putting the $\colorbox[rgb]{}{$S$}\text{'}$ into the Eq. (2) and combining with Eq. (3), we can calculate the $R\text{'}$ and $h_0\text{'}$ of the assumed dummy paraboloid as following:

According to the reflection law, an assumed reflection line$\colorbox[rgb]{}{$T$}\text{'}$can be calculated from the assumed reflection point coordinate $\colorbox[rgb]{}{$S$}\text{'}$, slope$\colorbox[rgb]{}{$n$}\text{'}$and the camera pinhole coordinate C. The normalized direction vector $\colorbox[rgb]{}{$K$}\text{'}$ of the line$\colorbox[rgb]{}{$T$}\text{'}$ can be expressed as Eq. (7):

The function f mentioned at the beginning describes the difference between the assumed reflection line $\colorbox[rgb]{}{$T$}\text{'}$ and the real one T. It can be expressed as

Using this method, for every pixel of the camera, we can calculate the coordinate and the slope of the reflection point on the tested mirror. By integrating [6], the absolute height of the tested mirror can be measured.

3. Simulations and experiment

The feasibility of this method has been validated by computer simulations and experiments. In the simulation, the tested surface can be expressed as $-(1-{10}^2)z^2+2000z=x^2+y^2$, and its aperture is about 400mm. In Fig. 3 , we show the tested mirror.

 

Fig. 3 simulated tested surface

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In the whole simulation process, random noise as 1% of the fringe patterns amplitude is added into the fringe patterns to reconstruct the surface. When errors of calibration are set to 0, the error of surface reconstructed by our method is shown in Fig. 4 and its root mean square (RMS) of is 7.1625e-005. In order to test the influence of the calibration errors on our method, we add about 0.05mm random to the camera position to measure the surface with our method. And the value of the added noise is [-0.0792 0.0546 0.0414]mm. The error of the reconstructed surface from the only one noised camera position is shown in Fig. 5 , and RMS of the reconstructed surface error is 7.4231e-004.

 

Fig. 4 error of our method without error in the calibration

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Fig. 5 error of our method with error in the calibration

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In order to compare our method with the traditional method [4], we use the traditional method to reconstruct the surface with the same data. In the measurement, we calculate the slopes of the tested points at first, and then reconstruct the surface by integrating. When there isn’t error in the calibration, the error of the result is shown in Fig. 6 . and the RMS of it is 4.0935e-005. When there are errors in the calibration, we set the errors as below: the error of the camera position is the same as that in our method and random errors as five percent of the camera CCD size are added into camera principal point and focal distance. Then we calculate the normalized image projection vectors and measure the surface by the traditional method. The error of the reconstruct surface is shown in Fig. 7 , and the RMS is 0.0081mm.

 

Fig. 6 error of the traditional method without error in the calibration

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Fig. 7 error of the traditional method with error in the calibration

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It can be seen from Fig. 4 to Fig. 7 that without calibration error, both of our method and the traditional method can reconstruct the surface with high precise. However, when there are errors in the calibration, our method works better than the traditional method. It is because that our method uses the camera pinhole instead of the camera rays to measure the surface. The camera rays are determined not only by the pinhole position but also the normalized image projection vectors which are influenced by other values in the calibration such as focal distance, principal point, distortions and so on. As shown in Fig. 7, the error of the traditional method is higher even if we only add small errors into the camera principal point and focal distance. Therefore, compared with the traditional methods, our method can measure the aspheric mirror with higher precise and less limit in the calibration precise.

An experimental verification has also been carried out. The tested mirror is a concave mirror whose aperture is about 40mm and radius of curvature is about 180mm. In the experiment, the pixel size of LCD displaying fringe patterns is 0.2647mm, and the periodic of the displayed fringe pattern is 32 pixels. The screen is located about 400mm from the tested mirror and moved for 175mm by a linear translation stage. The camera is located about 380mm from the tested mirror. And horizontal and vertical fringe patterns observed by the camera are shown in Fig. 8 .

 

Fig. 8 recorded fringe patterns

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In the measurement, we measure the tested mirror by three-coordinate machine whose accuracy is um as the standard value, and the result is shown in Fig. 9 . Then, we reconstruct the tested mirror by our method, and the differences between the two results are shown in Fig. 10 . The RMS of the differences is 0.0018mm. To compare our method with the traditional method [4], we use the traditional method to recover the mirror with the same test data. The difference between the result of the traditional method and that of three-coordinate machine is shown in Fig. 11 , and the RMS of that is 0.0386mm. It can be seen from Fig. 10 and Fig. 11 that the precise of our method reaches that of the three-coordinate machine. Furthermore, compared with the traditional method, with the same calibrate precise, the result of our method is better than that of the traditional method. It is because that our method measures the surface by the camera position instead of the camera rays in the traditional method. The camera rays are determined not only by the camera position but also the image projection vectors of the camera. The image projection vectors are influenced easily by the errors of calibration factors such as focal distance, principal point, distortions and so on. Therefore, with the same tested data, our method works better than the traditional one.

 

Fig. 9 reconstruction surface using three coordinate machine

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Fig. 10 difference between the results of our method and three coordinate machine

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Fig. 11 difference between the results of the traditional method and three coordinate machine

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In the experiment, we use the result of the three-coordinate machine as the standard value to validate our method. Although there are errors in the result of the three-coordinate machine due to the machine accuracy, its accuracy is high enough to validate the feasibility of our method and clarify that compared with the traditional method, with the same tested data, our method works better. The accuracy of our method will be further researched in the next study.

4. Conclusion and discussion

A novel method based on fringe reflection is proposed to measure the absolute height of the aspheric mirror. This method measures the aspheric surface with dummy paraboloid, while the camera is located beside the optical axis of the tested surface. It can be use to measure the aspheric mirror with high deviation and doesn’t have occlusion problem. Compared with the traditional method, this method calibrates the camera pinhole instead of the camera rays to measure the surface. As a result of that, the calibration errors influence this method less than the traditional method. This method can measure the high deviation aspheric surface with high precise even if the calibration precise isn’t very high.

If the calibration precise is high enough, the principle of our method can be used in another way which measure reflection point by camera ray and one point observed from the screen. In that way, the surface can be measured without moving the screen. In this paper, in order to measure the aspheric mirror with lower calibration precise, we choose to measure the surface with the reflection rays and the camera pinhole. Furthermore, we discuss how to test optical axis before measuring absolute height of tested mirror in another paper [7], so the fitment problem of our method has been solved.