Roughness parameters and surface deformation measured by "Coherence Radar" P. Ettl, B. Schmidt, M. Schenk, I. Laszlo, G. Häusler

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1 Roughness parameters and surface deformation measured by "Coherence Radar" P. Ettl, B. Schmidt, M. Schenk, I. Laszlo, G. Häusler University of Erlangen, Chair for Optics Staudtstr. 7/B2, Erlangen, Germany ABSTRACT The "coherence radar" was introduced as a method to measure the topology of optically rough surfaces. The basic principle is white light interferometry in individual speckles. We will discuss the potentials and limitations of the coherence radar to measure the microtopology, the roughness parameters, and the out of plane deformation of smooth and rough object surfaces. We have to distinguish objects with optically smooth (polished) surfaces and with optically rough surfaces. Measurements at polished surfaces with simple shapes (flats, spheres) are the domain of classical interferometry. We demonstrate new methods to evaluate white light interferograms and compare them to the standard Fourier evaluation. We achieve standard deviations of the measured signals of a few nanometers. We further demonstrate that we can determine the roughness parameters of a surface by the coherence radar. We use principally two approaches: with very high aperture the surface topology is laterally resolved. From the data we determine the roughness parameters according to standardized evaluation procedures, and compare them with mechanically acquired data. The second approach is by low aperture observation (unresolved topology). Here the coherence radar supplies a statistical distance signal from which we can determine the standard deviation of the surface height variations. We will further discuss a new method to measure the deformation of optically rough surfaces, based on the coherence radar. Unless than with standard speckle interferometry, the new method displays absolute deformation. For small out-of-plane deformation (correlated speckle), the potential sensitivity is in the nanometer regime. Large deformations (uncorrelated speckle) can be measured with an uncertainty equal to the surface roughness. Keywords: coherence radar, white light interferometry, surface topology, rough surfaces, roughness, deformation measurement 1. INTRODUCTION: ABOUT SMOOTH AND ROUGH SURFACE INTERFEROMETRY The "coherence radar" 1 was introduced as a method to measure the topology of optically rough surfaces. The coherence radar is based on white light interferometry, with a significant difference: Classical interferometry can only measure smooth surfaces. "Smooth" means for our purposes that within the resolution cell of the imaging optics, the phase variation of contributing elementary waves is less than ± 90 (see figure 8). Then, we have only constructive interference in any pixel. No fully developed speckle occur. It has to be noted that "smoothness" is related both to the height variation of the object as well as to the lateral resolution of the imaging object. Even a ground glass can be optically "smooth", if we observe with high aperture and a sufficiently narrow diffraction spot. "Sufficiently narrow" means, that within the resolved area the surface profile z o does not vary more than by ± λ/8 (corresponding to the maximum phase variation mentioned above). We call this mode of operation "resolved mode". For our purposes, a completely "rough" object displays phase variations between at least ± 180 and causes fully developed speckles 2 ("unresolved mode"). The coherence radar can measure rough objects as well. For this goal we have to satisfy a couple of conditions: first, we generate high contrast subjective speckle 2. This is even possible with white light illumination and extended International Conference on Applied Optical Metrology, Balatonfüred, Hungary, June 8-11,

2 sources, if the illumination aperture is smaller than the observation aperture, and the coherence length of the source is larger than the surface roughness. Furthermore, the photodetector must not be much larger than the speckle size. We must evaluate the interference within each speckle, separately. However, there is a great difference to classical interferometry: the interference phase within each speckle is arbitrary, for objects with a big roughness. In other words, the phase is then equally distributed in the interval [-π, +π]. There is no useful distance information in the phase of the speckle interferogram, and it is completely useless to evaluate the phase. We, instead, evaluate the modulation M(x, y) of the interferogram versus the distance z between one object point (x, y) and the corresponding reference. This function M(z ) is called "correlogram". The position z(x, y), where M has its maximum, gives the height difference between the object and the reference (which is assumed to be planar in this paper). For further considerations we assume that the arithmetic mean of z equals zero. We call the standard deviation of the surface "σ o ", and we call the standard deviation of the measured signal z, "R q " to remind of the German DIN standard We write the optically acquired value in italics to distinguish it from the DIN parameter R q, as it is given by measuring with a mechanical stylus. A further roughness parameter given by DIN is R a : This is the mean of the magnitude of z (zero mean). We write the optically acquired value in italics, as before. Again, there is a big difference between smooth and rough surfaces (see also table 1): For smooth surfaces, white light interferometry averages the distance over the object area within the diffraction spot. Hence, the standard deviation R q of the measured signal will be smaller with decreasing observation aperture. We called this scaling behavior: type III 3. For rough objects, the coherence radar does not supply the microtopology itself, because this is not resolved. What we see is rather a statistical representation of the microtopology: The measured distance signal R a has the same value as the surface microtopology σ o itself 7. object: smooth (z o variations < ± λ/8) object: rough (z o variations > ± λ/4) observation: resolved mode R q = σ o R q = σ o observation: unresolved mode R q << σ o (see ref. 3) R a = σ o Tab. 1: Measurement properties of the coherence radar with regard to the roughness parameters R There is a remarkable and extremely useful property of the coherence radar in the unresolved rough surface mode": Although we do not have access to the optically unresolved microtopology, we can nevertheless measure the surface roughness, independent from the resolution cell of the optics 4. In other words, the unresolved microtopology introduces a "noise" R a on the measured signal which is equal to the surface roughness σ ο. The measured R q is independent from the observation aperture (we called that type of behavior type II 3 ). This is completely different from measuring with a mechanical probe: there the radius of the probe acts as a low pass filter. Hence we average over part of the microtopology. After we explained some basic properties of the coherence radar, the following chapters can easily be followed: In section 2 we will first describe the measurement of smooth objects with white light interferometry. Measuring smooth objects with white light interferometry is based on Michelson, already 5. New is that one can evaluate the position of the correlogram maximum with an uncertainty in the nanometer regime 6. We will introduce two further evaluation methods that are faster and sometimes even more accurate. In section 3 we will discuss the measurement of "roughness standards" and the evaluation of standard roughness parameters. In section 4 we will present one more new application of the coherence radar: the absolute measurement of out-of-plane surface deformation. One remarkable result is that we can measure deformations in the nanometer regime as well as deformations of millimeters. International Conference on Applied Optical Metrology, Balatonfüred, Hungary, June 8-11,

3 2. COHERENCE RADAR IN THE RESOLVED SMOOTH-SURFACE-MODE In this section we will describe and compare evaluation methods for the coherence radar in the "resolved smoothsurface mode". For the microtopology acquisition, we use high apertures, and work with a microscope. We use objective attachments of Michelson type (magnification 5x) and Mirau type (magnifications 10x, 20x and 40x). With the highest magnification it is possible to measure a field of 212µm x 212µm. To achieve high longitudinal resolution, a position controlled piezo is used (accuracy 12nm). We developed two new evaluation methods: (a) the three-point Gaussian interpolation, (b) the enhanced threepoint Gaussian interpolation and compare it with (c) the Fourier method 6. (a) With three-point Gaussian interpolation we sample in steps of about 30nm and search for the maximum intensity in the correlogram. We store the sample point with the maximum intensity and its two neighbours, for each CCD-pixel. Eventually, we have for each pixel three intensity values and the position of the maximum measured intensity (see figure 1). By simple Gaussian interpolation we calculate from these points the maximum position of the correlogram. The measurement uncertainty for this method is 2.6nm, but we have to keep in mind that errors of λ/2 are possible because the maximum of the correlogram and not that of the envelope is determined, and, finding the correct maximum is prone to errors, in the presence of noise. (b) The enhanced three-point Gaussian interpolation takes care of this error. Here we search for the five highest intensity modulations. For each of them we store the three highest measured intensities samples. From these we calculate by Gaussian interpolation the maximum of each modulation (z-position and intensity). Finally, we take the five interpolated points to determine the maximum of the envelope (see figure 2). The measurement uncertainty of this method is 6.4nm. (c) Our last method is the Fourier method. This is a very common method [6]. Here we take 32 points (around the maximum) to describe the correlogram. These points are then Fourier-transformed. In the Fourier domain the negative and zero frequencies are cut off and the single-sideband signal is back-transformed (see figure 3). Finally we interpolate over the three highest points. By this method we reach a measurement uncertainty of 4.4nm. Fig. 1: three-point Gaussian interpolation method Fig. 2: enhanced three-point Gaussian interpolation method International Conference on Applied Optical Metrology, Balatonfüred, Hungary, June 8-11,

4 Fig. 3: Fourier method These new methods so far work all on software basis and are therefore slower than the hardware evaluation of the macroscopical setup. Table 2 lists the measurement time, time for evaluation after the measurement and the measurement uncertainty for the different methods. Fig. 4. The roughness standard SR 13 Fig. 5. The roughness standard SR 13 at magnification 5x at magnification 40x Figures 4 and 5 are given as an example. They show gray scale height maps (the brighter the higher) of the "roughness standard SR 13" from the PTB (Physikalisch Technische Bundesanstalt). The measurement in figure 4 was done with magnification 5x, the measurement in figure 5 with magnification 40x. The profiles below the intensity encoded maps display the microtopology of the measurement. The measurement uncertainty is a few nm. We would like to explain again, that due to the high lateral resolution, the coherence radar here works in the "smooth-surface-mode", although the object has a peak-to-peak roughness of about 300 nm. International Conference on Applied Optical Metrology, Balatonfüred, Hungary, June 8-11,

5 Time for measurement [s/µm] time for evaluation [s] standard deviation σ z [nm] λ/2-error three-point Gaussian interpolation 21 or 31,5 2 < 2,6 yes enhanced three-point Gaussian interpolation 133 or < 6,4 no Fourier method 31,7 or < 4,4 no Tab. 2. Comparison of three methods to evaluate the height map of the object from of the interference signal. 3. ROUGHNESS PARAMETERS FROM OPTICAL MEASUREMENTS? Up to date mainly mechanically probing sensors are used to measure industrial surfaces. Therefore roughness standards were designed for these kind of sensors. They were not developed for optical sensors and their needs. But optical sensors have advantages (higher speed, non-contact, high lateral resolution). Therefore we tested the applicability of the mechanical standards to determine roughness parameters from data acquired by the coherence radar. Several important surface parameters (R a, R q, etc.) are defined in DIN standards 4768 and For these standards, just profile lines are considered. The measured line includes several deviations from the ideal form: roughness, waviness and form errors (Fig. 6). Fig. 6. The measured profile is the addition of roughness, waviness and form. To separate roughness from the other deviations, standardized filters are defined. Usually a Gaussian low pass filter is used. Low pass filters separate long wavelengths like form, form error and waviness from the roughness. Thus we obtain the roughness profile by subtracting the filtered line from the measured line. Further we obtain the waviness profile by subtracting the form errors and the form itself from the filtered line by another filter. The difficulties are mainly in the choice of the cutoff frequency of the low pass filter which is proposed by DIN. From the roughness profile the roughness parameters are obtained (analogous the waviness parameters from the waviness profile, etc.). The most common surface parameters, which can now be determined, are listed below: R z R a R q W t AVF arithmetical mean roughness depth arithmetical mean deviation root mean square deviation wave depth height amplitude curve We used a special PTB test sample (different from the sample before) with defined R a for measurements with the coherence radar in the "resolved mode". Fig. 7 shows measurements at the same location, with different magnifications (5x, 20x and 40x from left to right). In spite of the different magnification during the data International Conference on Applied Optical Metrology, Balatonfüred, Hungary, June 8-11,

6 acquisition, the displayed pictures show the same area, and are zoomed to have the same scale on the paper. The profiles below are taken along the white lines shown in the height maps. The speckled area at the top left corner was used to locate the same area, but not for the data evaluation. The surface of the object consists of grooves which can be seen in figure 7 from top to bottom. Fig. 7. PTB test object measured at different magnifications (5x, 20x and 40x) As explained in section 1, smooth variations h of the object height z o are averaged over the resolution cell. If h stays below ± λ/8, the phasors of the elementary object scatterers will all add up in a positive way (see figure 8 left). But if h is well above ± λ/8, the phasors will add up in a random way and speckle are observed (see figure 8 right). Fig. 8. Phasors for low height variation in an observation cell (left) and high height variation (right) The test sample can be considered as a smooth object. The profiles show that, from higher to smaller magnification, small height details are averaged whereas the coarse object shape is still visible, e.g. the large groove at the right is clearly visible in all three profiles. We calculated R q and R a with our program analogous to the specifications given by DIN 4768 and compared them to the values specified by mechanical measurements. The results are listed in table 3. The stylus radius of the mechanical sensor is 4µm, the cutoff-wavelength of the filter is 0.8mm. International Conference on Applied Optical Metrology, Balatonfüred, Hungary, June 8-11,

7 Roughness parameter PTB values R [µm] Coherence Radar values R [µm] 5x 20x 40x R a resp. R a R q resp. R q Tab. 3. Comparison between data measured by the coherence radar and the PTB specifications ("resolved mode") R a measured by the coherence radar is always larger than the specification given by the PTB. As explained above this is probably caused by the low pass filtering by the mechanical probe, compared to the higher lateral resolution of the optical measurement. R q was not specified from the PTB for this object. For measurements in the "unresolved mode" we used another roughness normal (RUGOTEST No. 3). We measured four different surfaces (N6, N7, N8 and N9) with the coherence radar and with the mechanical sensor PERTHOMETER S8P 7.1. The stylus radius of the mechanical sensor is 10µm. The cutoff-wavelength of the filter is 2.5mm, both for the mechanical sensor and the coherence radar. Table 4 compares the results. Coherence radar Mechanical sensor R a [µm] R z [µm] R max [µm] R q [µm] R z [µm] R max [µm] N N N N Tab. 4. Comparison between data measured by the coherence radar and a mechanical sensor ("unresolved mode") As mentioned in section 1, our R a relates to σ ο, and thus to R q measured by the mechanical sensor. It can be seen that the measured R a and R q correspond to each other. Besides the sample N6, R z and R max are close to R z and R max, respectively. Deviations can be explained by the inherent low pass filtering of the mechanical stylus. The data of the probe N6 measured with the coherence radar are probably too large. This is caused by the used fast evaluation technique, which can result in errors of a few 100nm. In the future, samples with a quite small roughness will be measured with a proper evaluation technique. 4. ABSOLUTE MEASUREMENT OF SMALL AND LARGE SURFACE DEFORMATION As discussed above, the coherence radar - as a conventional white-light interferometer - can measure the shape of smooth surfaces with nanometer precision. Out-of-plane deformation can be acquired by just subtracting two measurements taken before and after deformation. So far this is similar to classical interferometry, however we have no ambiguity, because we use white light illumination. Now we will discuss out-of-plane deformation of rough objects, and compare its properties with standard methods such as holographic interferometry and speckle interferometry. Preliminary results have already been demonstrated 4, here we show some more experiments. With rough objects, the coherence radar supplies data with R a equal to the surface roughness R q (see section 1). This "uncertainty" can be several microns, dependent on the object. An example is given in figure 9 (left). Nevertheless, we can measure deformations in the nanometer regime, provided, the speckle pattern is not decorrelated between the measurements. This is demonstrated in figure 9 (right). Figure 9 (left) displays the measured surface profile of an object with a roughness of about 5µm. After a small deformation, the object is measured again. Both measurements are subtracted. The result is displayed in figure 9 (right). The deformation can be seen as a tilt. The noise of about 50nm is partly due to measuring errors and partly due to speckle decorrelation. The experiment demonstrates that deformations in the nanometer regime can be measured, in spite of the large "roughness noise" on the single measurement. International Conference on Applied Optical Metrology, Balatonfüred, Hungary, June 8-11,

8 Fig. 9. Deformation measurement with correlated speckles: single measurement (left) and profile acquired by subtraction of measurements before and after deformation (right) For large deformations we cannot expect the two speckle patterns to be correlated. However, we still can measure the deformation - with reduced accuracy. Since the single measurements before and after deformation each display a statistical uncertainty equal to the surface roughness R q, the noise on the difference image displays a noise of 2 * R q. Fig. 10. Deformation measurement with uncorrelated speckles: single measurement (left) and profile acquired by subtraction of measurements before and after deformation (right) It should be noted that in contrast to other speckle methods (speckle holography, ESPI, holographic interferometry) the new method delivers absolute deformation, without ambiguity, for small and for big deformations. 5. ACKNOWLEDGEMENTS The coherence radar project is supported by BMBF (13N6667). We thank Jörg tom Felde from the Chair "Qualitätsmanagement und Fertigungstechnik" at the University of Erlangen for measuring the surface roughness parameters in the "unresolved mode" with the mechanical sensor. 5. REFERENCES 1. T. Dresel, G. Häusler, and H. Venzke: Three-dimensional sensing of rough surfaces by coherence radar, Appl. Opt. 31, pp , J. C. Dainty: Laser Speckle and related phenomena, in Topics in Applied Physics, Vol. 9, Berlin, G. Häusler: About the scaling behavior of optical range sensors, Proc. of the 3 rd. Int. Workshop on Automatic Processing of Fringe Patterns, Bremen, Germany, G. Ammon, P. Andretzky, S. Blossey, G. Bohn, P. Ettl, H. P. Habermeier, B. Harand, G. Häusler, I. Laszlo, B. Schmidt: New modifications of the coherence radar, Proc. of the 3 rd Int. Workshop on Automatic Processing of Fringe Patterns, Bremen, Germany, A. A. Michelson: Trav. Mem. Bur. Int. Poids Mes. 11, pp. 1-42, S. S. C. Chim, G. S. Kino: Phase measurements using the Mirau correlation microscope, App. Opt. 30, pp , P. Ettl: Studien zur hochgenauen Objektvermessung mit dem Kohärenzradar, Diploma thesis, Chair for Optics, University of Erlangen, 1995 International Conference on Applied Optical Metrology, Balatonfüred, Hungary, June 8-11,