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A method for surface profile measurement using a chessboard-shaped 2-D Ronchi grating is proposed. The gradients in two orthogonal directions of the measured surface can be obtained simultaneously using a chessboard-shaped 2-D Ronchi grating, so that it is possible to reconstruct the surface profile by means of only one Ronchigram with high accuracy. The measuring principle and the design of the chessboard-shaped 2-D Ronchi grating are described. Measurements of both stationary and varying surfaces using an instrument constructed on the basis of the method were conducted with satisfactory results.
©2005 Optical Society of America
1. Introduction
The Ronchi test is one of the simplest and the most powerful optical testing methods. This method has been widely used in the measurement of flat surface. In traditional Ronchi grating measurement, a rectangular (Ronchi) grating is employed to obtain light beam deflections in two orthogonal directions of the point on the measured surface respectively, with which the gradients of the point can be calculated and the profile of the measured surface can be then reconstructed. Since one Ronchi grating is employed in the test, only the gradient of one direction can be obtained each time. To obtain the gradient in another direction, the Ronchi grating must be rotated by an angle of 90°, therefore this method is only suitable for the measurement of stationary surface [1–4]. In the recent researches of Ronchi test, the Ronchi rulings are computer generated and displayed on the LCD. But in this method, the information of horizontal and vertical directions cannot be acquired at the same time either [5–8].
In order to obtain the information of both orthogonal directions at the same time, a two channel Ronchi test [9] was proposed. This system had two separate arms, one with a vertical grating and the other with a horizontal grating. Vertical and horizontal gradients information was acquired by these two gratings, respectively. The two Ronchigrams were then used to reconstruct the measured surface profile.
However, since the Ronchi test measures the light beam deflections, which contain not only the information on the magnitude of the slope of the measured point, but also the information on the direction of the slope, theoretically it is therefore possible to reconstruct the profile of the measured surface by use of one Ronchigram, if it contains the information of gradient for both x and y directions, and thus the method may be also suitable for measurement of changing surfaces. To reconstruct the measured surface from only one Ronchigram, the crossed grating [10,11] and the square grid [12] have been used in Ronchi test, respectively. But because of the structure of crossed grating and square grid, these methods cannot get continuous information from measured surface. The information of measured surface in the common opaque area formed by two orthogonal Ronchi patterns will be lost and cannot be obtained for the reconstruction.
We present a novel surface measuring method based on only one chessboard-shaped 2-D Ronchi grating, which is formed based on the principle of an exclusive-or logic operation of two orthogonal 1-D gratings. This 2-D Ronchi grating can completely convey the geometry information of the measured surface, and can be applied in the measurement of both stationary and changing surfaces.
2. Measuring principle
A chessboard-shaped 2-D Ronchi grating is the key component for 2-D Ronchi test we proposed. Since the gradients for both x and y directions should be obtained simultaneously, the whole set-up must have the same performance and parameters in the two orthogonal directions. Thus the chessboard-shaped 2-D Ronchi grating is designed to have a rectangular grid configuration, as is shown in Fig. 1. The key parameter of the grating is the side length of the rectangular grid, which is dependent on the dimension of the measured surface, the required resolution, and the size of the measuring instrument. The chessboard-shaped grating is formed based on the principle of an exclusive-or logic operation of two orthogonal 1-D gratings. It is not formed, like the crossed-grating or square grid, simply by crossing two 1-D grating patterns. With our proposed grating, it is possible to obtain the distorted white-black-crossed squares, which can completely convey the geometry information of the measured surface. The characteristic of our method is that the white-black-crossed squares are continuous.
In the above figure, for example, Line AC (marked red) is composed of Line AB and Line BC. Both AB and BC are the intersections of one white square and one black square, respectively. Any distortions of these two lines can be clearly obtained and used to calculate the surface profile. So the method can provide continuous information of the measured surface, and it is of much higher accuracy of measurement.
Figure 2 shows the measuring principle of the Ronchi test. A Ronchi grating R is placed near the focal point F of a mirror, and Ronchi fringes can be observed on the receiving matt glass S.
Figure 3 shows the measuring principle of our proposed chessboard-shaped 2-D Ronchi test method. According to Fig. 3, when the measured surface is an ideal surface, following geometrical relation will hold:
In practice, the Ronchi grating is often manufactured to have a duty ratio of 50%, thus and . Therefore the duty ratio of the projection of the grating on the receiving array area of a CCD camera is still 50%. In x-direction, the signal obtained by a CCD camera is a 50%-duty-raio square wave signal I(x), which can be expanded into
As is shown in Fig. 3, for an inclination θ, the reflected light beam from the surface is incident at the point P on the focal plane through a collimating lens and is intersected with the CCD receiving plane at the point D″. When there is no inclination, i.e., if it is the ideal situation, the light beam should be intersected with the CCD at the point D′, so the difference between the projected fringes for the deflection and the ideal situation is . According to the geometry, there will be
For an inclination of the measured surface, we have
then
For the reflected light, there will be
and
Thus the projecting process turns the inclination of the measured surface, θ, into a projected Ronchi fringe variation, Δ. To evaluate the inclination θ, we use the synchronous demodulation technique.
The obtained grating projection intensity can be expressed as
where ψk(x) is the phase change of the spatial frequency due to the change in Ronchi fringe position. Through a low-pass filtering to remove the dc component, the resulting light intensity becomes
where
where d is the projection period of the grating.
Making a generalization of I 1(x) yields
We use a synchronizing signal, whose intensity is
Multiplying Eq. (10) with Eq. (11) yields the product
Through the low-pass filtering of the product to remove the second harmonic component, we obtain
Solving Eqs. (6), (9) and (13) yields
Thus we obtained the inclination in x-direction of the measured surface through a single line of grating projection signal. For the whole surface, Eq. (14) can be changed to
With the same procedures for the demodulation of the inclination in x-direction, a similar expression for the inclination in y-direction can be obtained:
Although Eq. (15) and Eq. (16) have the similar form, they should be treated in different ways for data processing. The Ronchigram for x-direction must be processed line by line, whereas the Ronchigram for y-direction must be processed row by row.
After the inclinations in two orthogonal directions are obtained, slopes of the individual points of the measured surface can be evaluated and then the whole of the relative heights of the points, i.e., the surface profile, can be finally obtained through digital integrations.
Figure 4 shows the optical set-up for the measurement of surface profile based on the proposed 2-D Ronchi grating.
The light beam from the laser, after going through a beam expander and a pinhole to remove noise, is partly reflected by a beam splitter onto a planar reflector, and is there further reflected to an aspheric collimating lens. The collimated light beam contains the information of the measured surface after it has been reflected by the surface. This reflected light beam returns through the same path as the foregoing one to the reflector and is partly transmitted through the beam splitter, projecting the chessboard-shaped 2-D Ronchi grating and thereby forming a modulated grating image on the sensing array area of the CCD camera.
3. Experiment results of surface measurement
We developed a measuring device based on the afore-mentioned chessboard-shaped 2-D Ronchi grating with the following specifications:
a) max. diameter of the measured surface: Φ100mm;
b) measuring resolution: 0.1µm;
c) measuring range (peak-valley height): 15µm.
According to the requirement of the measuring resolution, and with the consideration of the changing rate of the surface (<3Hz), the designed 2-D Ronchi grating has a grid length a=0.25mm. A He-Ne laser device with a wavelength of λ=670nm, a black-white CCD camera with a shooting rate of 25frames/second, and a frame grabber board (8 bit A/D, 45MHz) are employed for the measuring system.
Surface profile measurements were performed with the system. Figure 5(a) shows the reconstructed 3-D profile of a measured optical flat, and Fig. 5(b) is its contour map. We have measured the flatness of optical flat for 30 times in the same condition. The average flatness of the optical flat is 1.33µm, and the mean square root deviation is 0.122µm. The repeatability precision of the proposed system is high. Measurement of changing medium (silicone oil) was also done with this system. Figure 6 shows the measurement results, where (a), (b) and (c) are the three obtained contour maps with a sampling interval of 0.04s between two successive samples, showing the changing conditions of the silicone oil surface.
Fig. 5. Measurement result of optical flat: (a) reconstructed 3-D profile of measured optical flat (b) contour map.
4. Conclusions
A novel surface profile measuring method based on a chessboard-shaped 2-D Ronchi grating is proposed. The measuring principle of the method and the structural design of the chessboard-shaped 2-D Ronchi grating are described. Measurements for both stationary object and changing medium with the developed system on the basis of the proposed method were performed with satisfactory measuring results.
References and links
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