Optimization of white light interferometry on rough surfaces based on error analysis

Transcription

1 ELSEVER Optik 115, No. 8 (2004) nternational Journal for Light and Elertron Optics Optimization of white light interferometry on rough surfaces based on error analysis Zehra Sara~ 1 2, Reinhard Groß 2, Claus Richter 2, Bernhard Wiesner 2, Gerd Häusler 2 1 Department of Electronics Engineering, Gebze nstitute of Technology, lstanbul C. 101, Cay1rova, Kocaeli,Turkey 2 nstitute for Optics, nformation and Photonics, University of Erlangen, Staudtstraße 7/82, Erlangen, Germany The signal, reflected from the object surface with shape z(x,y) is given by u(x,y) = exp [2ikz(x,y)], (1) where x, y are the lateral coordinates, and 2kz is the phase cp of the reflected signal. At a smooth surface, the local phase variations are much smaller than unity, hence by eq. (1): u(x,y) "'1 + 2ikz(x,y). (2) 1. lntroduction White light interferometry (WL) is a meanwhile established principle to measure the shape of small objects with high precision [1]. WL at smooth (polished) surfaces is called "classical interferometry". White light interference microscopes have been available for about 70 years. n the last 10 years, this technology has been largely perfected, and commercial instruments are available with measuring errors below 1 nanometer. Since 1992 it is known [2] that it is possible to measure optically rough surfaces with WL as well. Rough surfaces in an interferometer do not produce any interference fringes. However, with proper illumination and observation conditions we see speckles even with white light illumination. By evaluation of the interference contrast within each speckle, we can measure the shape of rough surfaces. The signal generation mechanism for smooth objects and for rough objects is quite different, as well as the achievable measuring uncertainty [3]: Received 20 April 2004; accepted 23 June Correspondence to: G. Häusler Fax: haeusler@physik.uni-erlangen.de Because of the diffraction limited resolution, we do not see the "local" signal u(x, y), but a laterally averaged signal Usmooth = 1 + 2ik(z(x,y)). The consequence is, that with less lateral resolution the measured signal becomes smoother and smoother. The linear averaging makes the system quite robust and it turns out that with classical WL the measurement uncertainty is not given by the object under test, but by technological imits ( as far as we do not reach quantum imits ). Presently, sub-nanometer uncertainty can be achieved. This is quite different at rough surfaces. Here, the approximation of eq. (2) cannot be done. nstead, we get: Urough = (exp [2ikz(x,y)]). (3) This averaging ( over the diffraction spot) is a highly nonlinear and nonmonotonic function of z. The corresponding signal Urough depends in a "chaotic" way on the true shape z(x,y). As a consequence, it is not possible to measure the true shape. We have to cope with an intrinsic "noise" which turns out to be equal to the roughness of the surface [4]. t is remarkable that this "noise" does not depend on the size of the diffraction spot ( or the aperture of the observation). n other words, we can measure the roughness even if we cannot laterally resolve the microstructure of the surface. The consequence of these considerations is that we cannot build "rough surface interferometers" (we call this type "coherence radar" [2]) with an accuracy much better than the surface roughness. We can make /04/115/ $ 30.00/0

2 352 Zehra Sara~ et al., Optimization of white light interferometry on rough surfaces based on error analysis a virtue of necessity from this drawback: Since we cannot get longitudinal nanometer-accuracy, we may utilise low cost technology and "low cost algorithms" (i.e. fast and simple algorithms ). The aim of this paper is to investigate the influence of the major components of WL, which are the translation stage, the light source, and the evaluation algorithm, on the measuring uncertainty of the total system. n this paper, we will not use the theoretical approach [4], but supplement it by a "top-down" strategy and experimentally investigate the potential sources of errors. For this purpose we implement different experimental setups for WL, as well as different decoding algorithms". We determine the measuring uncertainty of the different schemes and try to identify the major sources of those uncertainties. The goal of the investigation is to find an optimal device and evaluation algorithm under the given technicallimitations. For these investigations we first use as a test object a smooth object, because the intrinsic errors caused by the nonlinear components of eq. (3) do not come into account. Hence, we can see directly which components display the best results. With the best "low cost setup" we will investigate in a second step the measuring uncertainty that can be achieved at rough surfaces. 2. Experimental setup The setup is shown in fig. 1. Light from the source is split into two separate beams. One beam is reflected by the test object, the other by a reference mirror. After recombination of the beams the resulting interference pattern is recorded by a CCD camera. The object is moved along the z-axis and the images are captured at different positions zn. White light interferometry uses a broadband light source with a small coherence length lc. nterference fringes are localized within a small range of optical path differences smaller than lc. Typical light sources used in white light interferometry are broadband LEDs with a Gaussian spec- Fig. 1. Reference Mirror bd&.. it r j1 '\ \~Light Source Schematic diagram of the white-light interferometer. tral distribution or incandescent lamps like halogen lamps. The resulting intensity in one camera pixel can be described as (z) = A(z - zo) cos (kz + <p), (4) where k = 2rc/ A. is the central wave number of the light source and A(z - zo) is the visibility envelope function with maximum at the object position z 0 The intensity (z) is often called "correlogram". For the determination of the height value z 0, we have to find the maximum of the envelope A(z - zo). Depending on the lateral resolution, different interferometric setups can be used. For large areas [5] up to 10 x 10 cm 2, a Michelsan setup is used. For high magnifications, Mirau or Linnik microscope setups are used. 3. Evaluation algorithms n recent years a variety of algorithms has been developed to evaluate the correlograms. We distinguish "real time methods" and "post processing methods". "Real time methods" deliver the depth map of the surface immediately after finishing the scan, whereas "post processing methods" need the whole dataset for the analysis and start with the evaluation just after the scan. An extensive list of evaluation algorithms was discussed by J. Schraud [6]. We will briefly sketch the algorithms that are suited to be compared in our study Real time methods "Contrast method" The "contrast method" utilizes the intensity difference C between two consecutive sample points captured at zn and zn+l. We allocate the measured height value Zm to the distance zm with the highest difference C = (zm+l) - (zm) The results are sensitive to the sampling distance, to the phase of the correlogram, and to noise. However, because of the intrinsic uncertainty mentioned in the introduction, this uncertainty may be neglected for very rough objects. "Siiding average method" The sliding average method is an improvement of the contrast method. A sliding average filter is smoothing the contrast values. The maximum of the smoothed contrast gives the sample no with the height value Zm The filtering reduces the noise. Like sensitivity to the contrast method, this algorithm is not accurate enough to measure with nanometer uncertainty, however, it can be easily implemented to work in real time. Our experience is that this algorithm works with sufficient accuracy for rough objects. "Maximum method" The maximum method only utilizes just the maximum of all measured intensity values. An improvement can

3 Zehra Saras: et al., Optimization of white light interferometry on rough surfaces based on error analysis 353 be achieved by finding the true maximum via an interpolation over the highest intensity value and its neighbouring points. t should be noted that the maximum method is prone to "outliers", since it may occasionally happen that there are two maxima of nearly equal intensity - and then the noise will determine which of these maxima is chosen by the algorithm. However, this error can be compensated by a proper unwrapping algorithm Post processing methods "Single Side Band Demodulation" The "Single Side Band Demodulation Method" is derived from the weil known AM - demodulation. t consists of subsequent band pass filtering, rectifying and low pass filtering. The result is the envelope of the correlogram. ts maximum is used to find Zm This method requires considerable computing time. However, the computing time can be reduced by storing only a certain number of samples in a range around the maximum of the envelope. This range can approximately be found by one of the real time methods. There are other post processing methods such as the evaluation of the correlogram phase or of the phase slope [7]. These will not be discussed here, because they are mainly used to achieve sub-nanometer accuracy. 4. Description of the performed experiments n order to investigate the influence of the different components mentioned above, we use a microscopic white-light interferometer in a Michelsan setup. The optical path difference z can be varied by a piezo. The whole setup was mounted on a mechanical translation stage. This allows the direct comparison of the results achieved by a mechanical and a piezo driven setup. We use two different light sources: A super luminescent diode (SLD, wavelength 837 nm, coherence length 15 f!m) and a halogen lamp (wavelength 733 nm, coherence length 3 f!m). To obtain sampled correlograms that satisfy the Nyquist theorem, at least 4 sample points in one oscillation period have tobe recorded. For our video camera with a frame rate of 50 Hz the maximum scanning speed is 4f,lm/s. We also use scanning speeds of 1 f!rnls and 2 f!m/s. A step gauge with calibrated steps from the PTB Braunschweig [8] is used as a "gold standard" test object. The object is measured 25 times with the same setup, i.e. the same scanning speed, translation stage and light source. The number of 25 datasets measured under the same conditions fulfils the MSA standard for the characterization of measuring processes [9]. For each measurement the shape of the object is evaluated by using the algorithms described above. The standard deviation of the measured step height is calculated from the series of measurements. By varying one of the mentioned parameters (scan speed, light source, translation stage) we achieve different standard deviations that are used to characterize the measuring uncertainty of each modification. The standard deviation is taken as a measure for the repeatability. 5. Measurements on smooth surfaces The primary goal of the measurements is to obtain an overview over the parameters that may have an influence on the measuring uncertainty. Figs. 2, 3, and 4 show the repeatability for different experimental set- 1},1 '! 0,~ 10 '0 t:: l'o 0,2 0, 1 (),0 Fig. 2. Piezo Halogen Lamp LEO Translation Stage Repeatability for a scanning speed of 1 '-'m/sec, different setups and different evaluation algorithms. The different evaluation algorithms are abbreviated as C = contrast method, S = sliding average, M = maximum method, and SSB = single side band evaluation. Fig. 3. 0,~, r------r , , Piezo Translation Stage Translation Sta.ge M $ $1,3 M SSB ~ S 1\,1 sse. C S M -$$& Halogen t.amp Le:D Repeatability for a scanning speed of 2 '-'m/sec, different setups and different evaluation algorithms. The different evaluation algorithms are abbreviated as C = contrast method, S = sliding average, M = maximum method, and SSB = single side band evaluation.

4 354 Zehra Sara~ et al., Optimization of white light interferometry on rough surfaces based on error analysis G.O Fig. 4. Tran$lation Stage c.s M ssa c s M sse c s M ssa c s M sse Halogen Lamp L.ED Repeatability for a scanning speed of 4 ~m/sec, different setups and different evaluation algorithms. The different evaluation algorithms are abbreviated as C = contrast method, S = sliding average, M = maximum method, and SSB = single side band evaluation. ups from a series of 25 measurements. To get a better overview, the name of the evaluation algorithms are abbreviated: C (Contrast), S (Sliding Average), M (Maximum method) and SSB (Single Side Band). Each one scanning speed and every bar diagram shows one of the evaluation methods described above. Several conclusions can be drawn from figs. 2, 3, and 4: A: ncandescent lamps should be used as light sources: ndependently from of the evaluation algorithm the best results are achieved by using a halogen lamp as light source. The reason is the narrow visibility envelope function with gives a clear maximum of the correlogram. Yet, the use of an LED is often justified because it is less sensitive to dispersion [10] and bacause its onger coherence length is more suitable for high speed scanning, e.g. sample distances greater than ).j4. B: A piezo should be used as translation stage: The use of a mechanical translation stage causes small position errors. They result from the interaction of the mechanical parts like cogwheels and therefore cannot be avoided. For high accuracy results a broadband light source should be used in combination with a piezo. However, for certain objects like deep holes requiring long distance translation, a mechanical translation stage has to be used. C: Post processing evaluation algorithms display higher stability: For all scanning speeds the single side band demodulation provides the best repeatability. Apart from the experiments with the halogen lamp, where the best results of the whole experiment were acquired, the single side band algorithm delivers a very good repeatability, too, using an LED in contrast to the other evaluation algorithms. Nevertheless, the results become worse by using a mechanical translation stage. Since this behaviour is independent from the light source, we may conclude that the algorithm is sensitive to position errors. D: The influence of the scanning speed is nearly negligible: Since all three are figures show equal behaviour, i.e. the good values are on the left side and the bad ones on the right, the scanning speed seems to have only a minor influence. With higher speed the repeatability becomes worse. By taking a closer look a few exceptions can be seen, but they can be explained by the characteristics of the evaluation algorithms. E: The behaviour of the real time evaluation algorithms strongly depends on the experimental setup: As can be easily seen, that the contrast method delivers the worst repeatability, especially with the LED. Since this algorithm looks for the highest intensity difference between two consecutive sample points, usually not the maximum but one of the zero-crossings of the oscillation is detected and the noise determines which one. The best combination of a real time algorithm and the experimental setup can be obtained from the three figures. For low scanning speeds the maximum method is a good choice, especially when a halogen lamp is used. For higher scanning speeds the use of the sliding average algorithm is advantageous. To summarize these results and to provide a basis for further measurements on rough surfaces we conclude: For high quality results a halogen lamp should be used in combination with either the maximum method or a post processing method. The post processing method is sensitive to position errors and should therefore not be used with a mechanical translation stage. The use of an LED in combination with the contrast method should be avoided. Hence we did not use this algorithm in our further measurements. 6. Measurements on rough surfaces According to our goal, further investigations are now done on rough surfaces. With the best 'low cost setups' derived from the results of the previous chapter, i.e. a piezo in combination with a halogen lamp, we examine the measuring uncertainty that can be achieved. We replace the step gauge by a rough diffusing ground glass with gold coating having a surface roughness of Ra= 0.74 f.tm. Additionally, we use an LED to compare the results with those obtained for the halogen lamp. Before we state the results, we shall briefly discuss some principal problems of WL on rough surfaces.

5 Zebra Sara~ et al., Optimization of white light interferometry on rough surfaces based on error analysis 355 () z/ljm Fig. 5. Measured correlogram intensity for 25 pixels in x-direction. The black line shows the height value calculated with the sliding average algorithm, the white line with the single side band algorithm. A denotes a bright speckle, B a dark speckle. Fig. 5 displays the measured correlogram intensity for a few pixels along the x-direction versus the z-position. We therefore get the correlograms for different speckles. n each correlogram, the phase of the speckle and of the correlogram is arbitrary. Between the speckles the phase varies strongly or even "jumps". Dark speckles cause low interference contrast. n fig. 5 the black and white lines display the profile z(x) from two evaluation algorithms. n fig. 6 we show the superposition of 10 measured profiles at the same line along x. The repeatability for some speckles is quite good, while other positions display a big uncertainty. Two exemplary pixels are abelied with A and B in the two figures. "A" derrotes a bright speckle with high contrast leading to a good repeatability as in fig. 6. "B" derrotes a dark speckle with low contrast displaying a big uncertainty in fig. 6. n the following we measure the standard deviation of a height value at a camera pixel from 25 measurements. The figs. 7 to 10 need some further explanation: The signal generation mechanism causes statistical errors as discussed in the introduction. Hence, we consider not only one pixel to determine the repeatability but display a statistics over all pixels; figs. 7 to 10 display the accumulated probability of the repeatability below a certain value. For example, in fig. 7, 90% of all pixels display a repeatability better than 0,74 ~-tm (vertical line) using the sliding average algorithm. The other 10% of the pixels display bigger uncertainties, because these are speckles with very low brightness. SSB f -:::-:: ::::: s l,. l ' ', : Halogen tamp, v = 1f.lm/s M -- Single Side Band S~idlng Average Maximum ~«-"'T~r-~...,...,...,,... "r... ~~~to 1,5 :Z.O 2.5 3J) Standard Deviation 1 pm Fig. 7. Accumulated probability for a pixel to display a repeatability than a certain value using ahalogen lamp and v = 1 ~mjs. 100 r:<t, $(')-. ~ i..,.0 e a ~~ ~,. _., : Halogen lamp, v = 2JJm/s SS8 M ' j ' ;'. (},(l tt:s 0 ':,0 1,$ :to 'l.s :M Standard Deviation pm -- Single Side Band, Sliding Average - - ~ - Maximum Fig. 6. Superposition of 10 measured profiles at the same line along the x-direction. A and B denote the same speckles as in fig. 5. Fig. 8. Accumulated probability for a pixel to display a repeatability than a certain value using ahalogen lamp and v = 2 ~mjs.

6 356 Zebra Saray et al., Optimization of white light interferometry on rough surfaces based on error analysis LEO, v = 1)Jmls SSB s M i H1 Halog~n Lamp tel) ;t' i.20 t ) f.~- Single Sld~ SaM,. " "' S!iding Av1;1r~,ge,,,.,, Max:imum Standard Deviation /pm Fig. 9. Accumulated probability for a pixel to display a repeatability than a certain value using an LED and v = 1!lm/ s. Fig % of all measured pixels display a repeatability better than the length of the bar. ~'&. 100 $() ~ ' :ä (\()! f t J E a.. 40 f ' j 20. l j!.f ), : ~.. LED, v :::: 2JJm/s sss fv1 -- Single Side Band ~ Sliding Average,,,.., Maximum Ui U5 2;0 c2j5 \Hl Standard Deviation /Jm Fig. 10. Accumulated probability for a pixel to display a repeatability than a certain value using an LED and v = 2 11m/ s. The verticalline at 0.74 ~-tm derrotes the roughness of the surface. n all four figures the maximum evaluation algorithm attracts attention. n cantrast to the measurements on smooth surfaces it gives only a bad repeatability and even the change of the light source or the scanning speed does not improve the results. This is due to the low cantrast within the modulation. For correlograms with several nearly equal maxima, noise will decide which one is taken. Another difference between the results on rough and smooth surfaces is that the sliding average method displays a performance nearly as good as the single side band demodulation algorithm. Since outliers have a strong influence on the shape of the curves in the region close to 100% probability, fig. 11 shows a bar chart of the standard deviations for a probability of 90%, i.e. 90% of all measured standard deviations are smaller or equal than the length of the bar. n cantrast to the maximum evaluation algorithm, sliding average and single side band algorithm display satisfying results. The repeatability achieved by the sliding average algorithm is nearly as good as the repeatability of the single side band algorithm. Since the sliding average algorithm is a real time method and delivers the result immediately after the scan, it seems to be the most suitable algorithm for measuring rough surfaces. n fact, the results with the single side band algorithm are slightly better, but this improvement is caused by an increase of computing time. On rough surfaces the influence of the light source is nearly negligible. This is different on smooth surfaces, where there is a strong influence of the light source. However, it is in good agreement with theoretical Observations [11 ]. As on smooth surfaces, the influence of the scanning speed is negligible. The higher value of the sliding average algorithm using a halogen lamp and a scanning speed of 2 ~-tm/ s can be explained by looking at fig. 8: n the region around the 90% probability the curve shows a flat slope causing a comparatively high standard deviation. To summarize these results we can say that for measurements on rough surfaces the choice of the light source does not matter as long as it fulfils the basic conditions for white light interferometry [2]. Best results can be achieved using post processing algorithms. However, they are only negligibly better than the results achieved with a suitable real time evaluation algorithm. The sliding average method has proven to be the best real time algorithm. 7. Conclusions An experimental analysis of the influence of essential parameters on the repeatability of white light interferometry was performed. The parameters under test were the light source, the translation stage and the evaluation

7 Zehra Sarac; et al., Optimization of white light interferometry on rough surfaces based on error analysis 357 algorithm. We used a "low cost setup" causing typical errors of a white light interferometer. As test objects we used a smooth and a rough object: The smooth surface to identify the best combination of the given parameters and the rough surface to investigate the measuring uncertainty with these parameters. A ist of the results is given below. t can be seen as an "instruction to use" for different measuring problems: Smooth surfaces Light Source: A halogen lamp should be used, particularly for real time evaluation algorithms. LEDs should only be used if only low accuracy is required. Translation Stage: For high accuracy measurements with post processing evaluation algorithms, the use of a piezo is essential since these algorithms are sensitive to position errors caused by a mechanical translation stage. Evaluation algorithms: n combination with a halogen lamp best results are achieved by using the maximum method or a post processing method. Rough surfaces Light source: The influence of the light source on rough surfaces is negligible. Evaluation algorithms: Post processing methods barely improve the repeatability compared to a suitable real time method, e.g. the sliding average method. References [1] Kino GS, Chim S: Mireau correlation microscope. Appl. Opt. 29 (1990) [2] Dresel T, Häusler G, Venzke H: Three-dimensional sensing of rough surfaces by coherence radar. Appl. Opt. 31 (1992), [3] Ettl P, Schmidt B, Schenk M, Laszlo, Häusler G: Roughness parameters and surface deformation measured by coherence radar. nternational Conference on Applied Optical Metrology (1998) Balatonfüred, Hungary [4] Ettl P: Über die Signalentstehung bei Weißlichtinterferometrie. Doctoral Thesis, Erlangen, Germany 2001 [5] Ammon G, Andretzky P, Bahn G, Ettl P, Habermeier HP, Harand B, Häusler G: Coherence radar - new modifications of white light interferometry for large object shape acquisition. Proceeding of the EOS Topical Meeting on Optoelectronics Distance Measurement and Applications (1997) [6] Schraud J: Optimierung und Vergleich der Datenaufnahme und -auswertung am optischen 3D-Sensor Kohärenzradar. Diplama Thesis, Erlangen, Germany 2000 [7] de Groot P, Colonna de Lega X, Kramer J, Thrzhitsky M: Determination of fringe order in white-light interference microscopy. Appl. Opt. 41 (2002) [8] Brand U, Hinzmann G, Schnädelbach H, Feist C, Stuht P, Krüger-Sehm R, Jäger V: Rückführbare Präzisions-Tiefen Einstellnormale für Messbereiche von 1 f.tm bis 1 mm. Tech. Mess. 66 (1999) [9] Dudschke W: Fertigungsmesstechnik Teubner, Wiesbaden 2002 [10] Pavlicek P, Soubusta J: Measurement of the influence of dispersion on white-light interferometry. Appl. Opt. 43 (2004) [11] Pavlicek P, Soubusta J: Theoretical measurement uncertainty of white light interferometry on rough surfaces. Appl. Opt. 42 (2003)