## Abstract

An advanced Phase Measuring Deflectometry(PMD) is proposed to measure the three dimensional (3D) shape of the aspheric mirror. In the measurement process, a liquid crystal display(LCD)screen displaying sinusoidal fringe patterns and a camera observing the fringe patterns reflected via the tested mirror, are moved along the tested mirror optical axis, respectively. At each movement position, the camera records the fringe patterns of the screen located at two different positions. Using these fringe patterns, every camera pixels can find a corresponding point on the tested mirror and gets its coordinate and slope. By integrating, the 3D shape of the tested mirror can be reconstructed. Compared with the traditional PMD, this method doesn’t need complex calibration and can measure the absolute height of the aspheric mirror which has large range of surface geometries unambiguously. Furthermore, this method also has strong anti-noise ability. Computer simulations and preliminary experiment validate the feasibility of this method.

©2008 Optical Society of America

## 1. Introduction

The aspheric mirror can correct the aberrations and improve the imaging quality of the optical system. Moreover, using such a mirror can reduce the number of optical elements in an optical system and further reduce the weight and the size of the system. Because of its fine characteristics, the aspheric mirror is widely used in the modern optical design. In order to improve the accuracy of the aspheric mirror fabrication, there are many methods used to measure the aspheric mirror. Among these methods, the most popular one is the interferometry which can measure the surface with very high accuracy. However, the interferometry commonly requires complicated and expensive assistant optics, and its measurement range is limited. In order to measure aspheric mirror conveniently and effectively, we further evolve the well known approaches of ‘Phase Measuring Deflectomety’ (PMD) [1] to measure such surface.

The basic principle of the PMD is to display sinusoidal fringe patterns on a screen which is located far from the tested surface, and to observe the fringe patterns reflected via the tested surface. Any slope variation of the surface leads to distortions of the pattern, so the PMD can measure the slope of the surface with high accuracy [1, 2]. However, majority of the PMD approaches reported so far surfer from various disadvantages. Some of those methods only determine the slope, so they need additional assumption to reconstruct the height of the surface [3]. This causes the ambiguity of the measurement. Moreover, those methods often need complex calibration which requires high accuracy [4, 5].

In this study, we present a novel method based on the fringe reflection to measure the aspheric mirror. In the measurement process, the screen and the camera are moved along the tested mirror optical axis, respectively. In this paper, we propose the concept of a “dummy paraboloid”. We combine the ‘dummy paraboloid’ with the high precision phase measuring profilomety (PMP) [6] to reconstruct the 3D shape of the tested surface. Using this method, for each camera pixel, we can find its corresponding point on the tested surface, and get both the slope and coordinate of that point. By integrating, the absolute height of the tested surface can be reconstructed. Compared with the traditional PMD, this method can measure the absolute height of the aspheric mirror unambiguously and doesn’t need complex calibration. Computer simulations and preliminary experiment validate the feasibility of this method.

## 2. Principle

The measurement setup of our method is shown in Fig. 1. It mainly consists of a LCD screen, a beam splitter (BS), the tested mirror, and a camera. The LCD screen displaying circular sinusoidal fringe pattern is fitted perpendicular to the tested mirror optical axis. And it intersects with the optical axis at the circle center of the fringe pattern. Due to the use of the beam splitter, the camera can be located on the tested mirror optical axis, and observe the fringe patterns reflected via the tested mirror. In the measurement process, the screen and the camera are moved along the mirror optical axis, respectively. At first, the screen is kept immobile, and the camera is moved along the optical axis step by step. Then we stop moving the camera, and move the screen along the optical axis for a distance. After moving the screen, the camera is moved back to the point at the beginning step by step. At every movement point, the camera can get the fringe patterns from the screen located at two different positions. This process is described by the broken lines in the Fig. 1. If a pinhole projection model is assumed for the camera, a simplified, two dimensional case of the principle of our method is illustrated in Fig. 2.

In the Fig. 2, a Cartesian coordinate system is established. The origin *o* of it is located at the center of the tested mirror, and the y axis is the optical axis of the tested mirror. *l*
_{0}, *l*
_{1} and *p*
_{0} …… *p*
_{n} represent the coordinates of the screen and the camera pinhole positions in the movement, respectively. When the camera is located at every point of *p*
_{0} …… *p*
_{n}, one pixel of the camera records a point of the fringe pattern from the screen which is located at *l*
_{0} and *l*
_{1}. And the phases of these points are ϕ^{0}
_{0} …… ϕ^{0}
_{n} and ϕ^{1}
_{0} …… ϕ^{1}
_{n}. For this camera pixel, we can find a special position on the optical axis. When the camera is located at this position, the camera pixel gets the same phase from the screen located at two different positions. It means that the reflecting line is parallel with the optical axis. Therefore, the coordinate of the reflecting point on the tested surface can be gotten. Moreover, we can find a dummy paraboloid that is tangent to the tested surface at the reflecting point. Using this dummy paraboloid, we can get the slope of the reflecting point on the tested surface. This reflecting point is the corresponding point of the camera pixel on the tested surface.

The movement distance of the camera is

When the camera is located at each movement position, the divergence of the phases got from the screen located at two different places, can be expressed as:

By function fitting, we can get a function about Δ**h** and Δ**ϕ**. It can be represented by:

Using the *f*(Δ**h**), we can find a value Δ*h*
_{k} that satisfy the identity: Δ**ϕ** = 0. The corresponding position on the optical axis can be described as:

By function fitting, the function about **p**([o*p*
_{0}……o*p*
_{n}]) and **ϕ**
^{0} ([*ϕ*
^{0}
_{0}……*ϕ*
^{0}
_{n}]) can be gotten. It is expressed as

Bring o*p*
_{k} into the Eq. (5), we get value ϕ^{0}
_{k}.

When the camera is located at *p*
_{k}, one pixel of the camera observes the fringe pattern via the point A of the tested mirror. And the phases that the pixel gets from the screen located at two different positions are equal to ϕ^{0}
_{k}. As shown in the Fig. 2, the x coordinates of the point A can be calculate as

T is the length of one period of the fringe patterns.

Supposing that there is a dummy paraboloid tangent to the tested mirror at the point A, we can get its focus (R) as flowing

*y*
^{′} is the distance between the dummy paraboloid vertex and the coordinates origin. Since the *y*
^{′}≪*p*
_{k}
*o*, the focus can be approximate to

With the dummy paraboloid and the coordinates of point A, the slope of the tested mirror at point A can be calculated as

Using this method, for every pixel of the camera, we can find a corresponding point on the tested mirror and calculate its coordinates *x*
_{n} and slope
$\frac{\partial {y}_{n}}{\partial {x}_{n}}$. By integrating, the absolute height of the tested mirror can be reconstructed. The integrating process can be expressed as

## 3. Simulations and experiment

The feasibility of this method has been validated by computer simulations and experiments. In the simulation, the tested mirror is a paraboloid whose focus is 500mm and the aperture is about 400mm, as shown in Fig. 3. In the measurement, the camera is located at the position where is on the optical axis of the mirror and 510mm from the optical center. Then it is moved along the optical axis to the position 490mm from the optical center. In the movement, the camera is stopped per 1mm to record the fringe patterns. And there is 200mm between the two positions of the LCD along the optical axis. While 1% random noise is added into the recorded fringe patterns, we show the error percentage in Fig. 4. The error percentage shows the relationship between the error of the recovered height and the maximun height of the tested mirror. From this figure, it can be seen that the maximum error percentage is below 10^{-2}. Therefore, the tested surface is reconstructed with high accuracy.

In order to test the anti-noise ability of our method, 5%random noise is added into the recorded fringe patterns. Using our method, the absolute height of the tested mirror also can be reconstructed. And the error error percentage is shown in Fig. 5. From Fig. 5, we can see that the maximum error percentage is about 4 × 10^{-2}. It means that our method has strong anti-noise ability.

In this method, we use the dummy paraboloid to get the slope of the test surface. If the test surface is not paraboloid, the approximation of the Eq. (8) will cause error. In order to test the influence of the approximation, we reconstruct another surface which is not paraboloid. The tested surface is a spherical surface whose radius is 1000mm and the aperture is about 340mm as shown in Fig. 6. With 1% random noise in the fringe patterns, the error percentage of the recovered surface is shown in Fig. 7. It can be seen from Fig. 7 that the approximation doesn’t cause evident influence in the result. Therefore, our method is not only suitable for paraboloid surface, but also suitable for other kind of surface.

An experimental verification has been carried out. The tested surface is a concave mirror whose aperture is about 40mm. Some of the recorded fringe patterns are shown in Fig. 8. When recording the Fig. 8(a) and the Fig. 8(b), the camera is located at the same position where is on the optical axis and 1034mm away from the optical center, and the LCD screen is at two different positions. Figure 8(c) and Fig. 8(d) are the same situation. The camera is at the same position 1012mm from the optical center, and the two positions of the screen are different.

Using our method, the tested mirror is recovered and the height map of the result is shown in Fig. 9. In order to verify the result of our method, we reconstruct the same mirror by three-coordinate machine and the result is shown in Fig. 10. The good agreement in the height data of the two results can be seen from these two figures. Furthermore, the differences of the two results on the x axis and y axis are shown in Fig. 11 and Fig. 12, respectively. It can be seen that the differences are below 3.5um. So our method reconstructs the 3D shape of the tested mirror, successfully.

## 4. Conclusion and discussion

A method based on the fringe reflection is proposed to measure the aspheric mirror. It can measure the absolute height of the large range aspheric mirror with high accuracy. And this method doesn’t need complicated assistant optics and complex calibration in the measurement. Furthermore, this method has strong anti-noise ability. Since this method can measure aspheric mirror conveniently and effectively, it will be a promising method in the aspheric mirror measurement.

In the preliminary experiment, we just prove that essentially the 3D shape of the aspheric mirror can be recovered by this advanced phase measuring deflectometry. And we neglect the system calibration problem. This neglection takes error into the result. While our method and the coordinate machine reconstruct the same tested mirror, the differences between the two results are below 4 um. It means that though we don’t calibrate the system, our method reach the accuracy of the coordinate machine. Moreover, the coordinate machine measures the object point to point, but our method is a full-field measurement. In our method, most of the error comes from the error of the phase got from the fringe patterns. Since the PMP has high accuracy in getting the phase, our method can measure the aspheric mirror with high accuracy after system calibration. The system calibration includes the CCD calibration and the LCD calibration. Furthermore, the movement of the CCD and the LCD also must be accurately controlled. The further study about the system calibration and error analysis will be discussed in other papers.

## Acknowledgment

The authors wish to acknowledge the support by the National Natural Science Foundation of China (No. 60838002).

## References and links

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