speckle correlation for the analysis of random processes at rough surfaces

*Corresponding author. Fax: #49-441-798-32-01.E-mail address: [email protected] (K.D. Hinsch).

Optics and Lasers in Engineering 33 (2000) 87}105

Speckle correlation for the analysis of randomprocesses at rough surfaces

K.D. Hinsch*, T. Fricke-Begemann, G. GuK lker, K. Wol!Applied Optics, FB8, Carl-von-Ossietzky Universita( t, D-26111 Oldenburg, Germany

Received 11 February 2000; accepted 12 April 2000

Abstract

The importance of technological processes like corrosion, ablation or deposition causesinterest in the quantitative monitoring of changes at rough surfaces. Thus, there is a need fore!ective methods to measure the statistical parameters characterizing changes in the pro"le orthe material composition of such objects. The speckle "eld scattered from the surface is used asinformation carrier and its change is measured by correlation. This is realized by sophisticateddata acquisition and digital processing techniques. An important issue is the interpretation ofthe correlation output in terms of statistical parameters describing the surface change. Formany random surfaces a geometrical relation between surface pro"le and optical phase provessatisfactory. This allows to determine the standard deviation of the pro"le change. Fora veri"cation, speckle decorrelation in model surfaces of known deviation is measured. Thepaper introduces the speckle correlation concept, outlines some history and current setups anddescribes methods for data evaluation. The reliability of the quantitative interpretation of thespeckle decorrelation is demonstrated. The method is illustrated by studies of metal corrosionand material removal in the cleaning of historical objects by laser ablation. ( 2000 ElsevierScience Ltd. All rights reserved.

Keywords: Speckle; Correlation; Surface process; Digital speckle photography; Corrosion; Laser ablation

1. Introduction

There is a great interest in the quantitative monitoring of minute changes in thesurfaces of test specimen. Many natural processes like the growth of plants or thedehydration of organic tissue and many technical processes like the corrosion ofa metal, the coating of a workpiece or peening, annealing and ablation of a material

0143-8166/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 1 4 3 - 8 1 6 6 ( 0 0 ) 0 0 0 3 5 - X

are associated with changes either in the micropro"le, the structure or the constitutionat the surface of the object. The study and control of such processes and theirdependence on ambient conditions bene"ts from a non-contact method for quantitat-ive monitoring of the surface changes. Furthermore, many applications requirereal-time measurements. An example is the control of removed patina in the lasercleaning of historical objects like valuable paintings.

Modern microscopic techniques like atomic force microscopy (AFM) or scanningnear-"eld optical microscopy (SNOM) provide surface data of unequaled high resolu-tion. These methods, however, are di$cult to employ in practical routine investiga-tions, because the equipment is very expensive and requires delicate adjustment,a controlled environment and extremely short working distance. Furthermore, theseare scanning techniques, and a measure for a surface change can only be obtainedfrom a subtraction of two states which requires very high accuracy in the reproducibil-ity of the scanning procedure which is not always guaranteed. Other techniques likethe classical light scattering techniques that measure either the total scattered inten-sity or its angular distribution rely on a change in the overall properties of the surface,for example, its roughness. There is no signal when the statistics of the surface remainconstant while its individual pro"le changes.

Microscopic properties of the surface can be transferred onto an interrogating lightwave when the object is illuminated with laser light. The resulting speckle pattern isa coded carrier of surface information, just as a "ngerprint identi"es its owner. Thespeckle pattern can be investigated remotely, its production hardly intrudes onto theobject and the desired surface change can be inferred from a comparison of specklepatterns obtained at di!erent states of the object. The required task is to obtaina quantitative measure for the similarity of speckle patterns and to relate their changeto corresponding changes in the surface.

The present paper gives an introduction into means to correlate the speckled light"elds scattered from laser-illuminated objects, recalls the historical achievements inspeckle correlation, and describes the power of digital speckle photography formacroscopic displacement analysis and especially its utilization for the study ofprocesses changing the microscopic surface structure of the specimen. Decorrelatione!ects that are due to object deformations or displacements must be separated fromthose attributed to the surface. The important task to relate the observed decorrela-tion with surface data is addressed theoretically for random surface variations. Theresults are backed by measurements on objects with controlled surface changes.Finally, practical applications in the monitoring of steel corrosion and laser ablationof historical patina show the power of the method and illustrate the scope of data thatcan be obtained by statistical interpretation of the complete set of correlation datathat are recorded during a surface process.

2. Classical speckle correlation

Speckle "elds generated by the scattering of laser light from rough surfaces havebeen the subject of a vast variety of investigations. The coverage relies on a statistical

88 K.D. Hinsch et al. / Optics and Lasers in Engineering 33 (2000) 87}105

treatment of the superposition of elementary scattered waves as treated by classicalscalar di!raction theory [1]. Various properties of the speckle "eld can be predicted,intensity distribution or speckle size, for example, can be derived from di!erent orderstatistics. In objective speckles, the random elementary wave phases propagate freelyin space. In subjective speckles, an imaging system is introduced that places restric-tions on the interfering waves. In this case, speckle size, for example, is essentiallya!ected by the imaging aperture. Generally, the micropro"le of the scattering surfaceis considered to produce corresponding geometrically related phase shifts where thelight wavelength acts as a scaling factor. Since the treatment is for random variablesand ensemble averages are considered, the individual structure of the surface does notenter into the analysis as long as it obeys certain statistical requirements. Yet, anyspeci"c speckle pattern is individually linked to the speci"c scattering surface.

The speckle e!ect is often considered as annoying noise deteriorating the quality ofimages recorded under coherent illumination. In holography, for example, a recon-structed image su!ers from inevitable speckle degradation. Due to the multiplicativecharacter of coherent noise it can be reduced only by speckle averaging. There is,however, a positive aspect in speckle that is the basis for many metrological applica-tions: a speckle "eld is a coded "ngerprint of the microstructure in the generatingsurface and can be used to tag or recognize a surface [2,3]. This is a prerequisite in anyof the methods where light "elds from rough objects are compared to derive macro-scopic object deformation or displacement "elds like in speckle photography [4],holographic interferometry [5] or electronic speckle pattern interferometry (ESPI)[6]. Any of these methods relies on an almost constant speckle "eld and the outputfrom such measurements, often di!raction, interference or correlation fringes, isgreatly deteriorated when the surface microstructure changes between the objectstates compared.

The bene"cial utilization of speckle analysis for the study of changes in thegenerating surfaces requires a quantitative comparison of the corresponding speckle"elds. The similarity of two speckle intensity patterns I

1and I

2is di$cult to estimate

directly because of their random character. It is best expressed by their correlationcoe$cient

c12

"

SI1I2T!SI

1TSI

2T

[(SI21T!SI

1T2)(SI2

2T!SI

2T2)]1@2

. (1)

When we take into account the special statistical properties of speckle "elds andassume that both "elds have the same statistics, this formula can be simpli"ed to give

c12

"

SI1I2T

SIT2!1, (2)

where SIT is the mean intensity value in either speckle "eld.It was stated above that the quality of fringes in coherent metrological techniques

re#ects the similarity of the underlying speckle "elds. Thus, the visibility of Young'sfringes (in the evaluation of double exposure speckle photography) or of holographicinterference fringes can be used to evaluate speckle similarity. At times, when the

K.D. Hinsch et al. / Optics and Lasers in Engineering 33 (2000) 87}105 89

means of image capture and digital processing were not available, as they are today,these were the only methods for speckle correlation.

Double exposure speckle photography (DSP) was originally introduced to mapin-plane displacement "elds. Two speckled images of the object before and after loadare superimposed on a single photographic frame. Upon evaluation small interroga-tion areas are scanned by an unexpanded laser beam and orientation and spacing ofthe resulting Young's fringes are utilized to give the displacement vector. When it isprimarily the correlation coe$cient between two speckle patterns that is of interest,the setup remains the same. By a spatial shift of either object or camera betweenexposures carrier fringes are introduced purposely, the visibility of which yields thedesired correlation coe$cient. Note that this similarity concerns only the specklepatterns and still has to be turned into information about the corresponding surfacechanges. Several early investigations have been made yielding some data aboutmicroscopic surface variations that were di$cult to measure by other means [7].

A similar approach has been utilized in holographic interferometry. Here thevisibility of holographic fringes was evaluated while the object was exposed to somesurface changing action [8]. Typical results concern cavitation erosion on metalsurfaces or plastic surface deformation by contact pressure. A mathematical analysisof fringe formation resulted in a formula relating the statistical distribution of therandom phase change on the surface micropro"le to the visibility of the fringes [9]. Itis interesting to note that similar results were obtained for the Young's fringesevaluation of double exposure particle image records in #ow velocimetry (particleimage velocimetry, PIV) [10,11].

We must mention here that fringe study for evaluation of the correlation coe$cienthas been taken up again in the modern version of holographic interferometry, i.e.,video holography or electronic speckle pattern interferometry (ESPI) [12]. In ESPI,two image plane video holograms of the object are compared. With phase shifting,a series of records is usually evaluated for the unknown phase di!erence between twoexposures that can be turned into a displacement value depending on the geometry ofthe optical arrangement. The same type of evaluation can also be used to determinethe modulation of the fringes, i.e., their visibility. In this way, a map of the correlationcoe$cient, i.e., the amount of `surface changea can be obtained. Since all data arestored digitally, they are available for various processing lines. A typical example isshown in Fig. 1, where a stone sample has been measured at instants of time 3 d apart.The speckle correlation coe$cient is mapped by the intensity value of the image,bright areas corresponding to high, dark to low correlation. A region in the stone thathad been soaked with an aqueous algae solution clearly shows up by its lowcorrelation because the microbiological activity of the algae has changed the scatteredlight "eld.

An alternative to the indirect determination of the speckle correlation coe$cientfrom an analysis of fringe visibility in optical metrology is the direct correlation oflight "elds by optical analog means. This method relies on the action of a holographicmatched "lter that was originally introduced for optical information processing. It canbe nicely described by an approach based on systems theory [13,14] and was used forpattern recognition at a time when digital techniques were still short in power for such


Fig. 1. Correlation map showing 3-d activity of a population of algae (dark area) on a 5]5 cm2 stonespecimen. Brightness is proportional to the correlation coe$cient. Data from modulation analysis of ESPIrecords.

a task. A simple understanding of the performance can also be obtained by inter-changing the role of reference and object wave in holography. Usually, a hologram isilluminated with a replica of the reference wave to reconstruct the object wave.Similarly, when the object wave illuminates the hologram we can expect a reconstruc-ted reference wave. Its existence can be interpreted as the successful recognition of theobject wave. The analogy is not quite exact and the practical realization requires somee!orts to make the performance shift invariant. In the "nal version, there resultsa so-called 4f optical correlator. The rough surface is imaged by two identical opticalsystems separated by twice their focal length from the input focal plane to the outputfocal plane (hence 4f ) with unit magni"cation. In the plane of symmetry, a Fouriertransform hologram is recorded of the initial state of the object. The object light isthen used to illuminate this hologram while the object surface is changing. Output ofthe correlator at the direction of the reference wave is a sharp point-shaped peakwhose intensity is proportional to the correlation coe$cient of the two light waves.This output can be monitored in real time.

The use of the optical correlator for speckle correlation was much more straightfor-ward than its application for pattern recognition. Here, the light waves are theprimary signals to be compared and it does not need any complicated light modulator


Fig. 2. Condensation and evaporation on the surface of highly porous SiO2

(average pore size 12nm)during changes in ambient humidity. Optical correlation with a holographic matched "lter.

device to produce the proper correlator input. It has been used for a variety ofinvestigations on surface e!ects [7,15,16]. Problems, of course, arose from the need forchemical development of a hologram and its very precise replacement. Re"nedversions therefore used photothermoplastic material or photorefractive crystals forrecording material. Optical correlators have participated in the study of dynamicprocesses on rough surfaces until not very long ago. Their inherent disadvantages,however, are that only a predetermined region in the object can be input to thecorrelator and that only a single reference state can be stored which makes updatingof data impossible. Furthermore, the correlator output is subject to object misalign-ments that are interpreted erroneously as surface decorrelations. Object tilt, forexample, has been compensated by an adjustable glass plate in the optical path [17].With the advent of powerful video and digital image processing equipment newtechniques have evolved to correlate speckle "elds. While the next section will bededicated to these, we may dismiss the analog optical correlator with yet a "nal resultindicating the high sensitivity of this elegant method. We studied water evaporationand condensation in the nanopores of an SiO

2sample where changes in ambient

humidity of just a few percent cause measurable decorrelations in the scattered light"elds (Fig. 2). Most certainly, the interactions of the laser light with structural changeson these small scales do not fall in the regime for a treatment by scalar di!ractiontheory and therefore the theoretical interpretation will be di$cult. Yet, there isenough change in the light "elds induced by the rearrangement of liquid menisci in thecracks and pores of the surface to be measurable.

In the age of electronic digital signal processing, light in metrology has obtainedchanged relevance. In several of the techniques mentioned above light was not onlythe information provider and carrier, but also served an essential purpose in theprocessing of the signals towards a "nal output value. The cumbersome numerical


calculation of a cross-correlation function for two-dimensional picture arrays waseither avoided by looking at Young's di!raction fringes (i.e., performing an opticalFourier transformation) or by interaction of waves in an optical correlator. Presently,most of the computational work can be left to electronic processors. Light serves asthe sensor, providing a most e!ective collection of the relevant information.

3. Digital speckle correlation

In digital speckle photography (DSP), usually two focused speckled images of theobject surface are recorded with a CCD camera for digital cross-correlation. Itsestablished application is in the mapping of in-plane displacement "elds. The positionof the correlation peak for a sub-window interrogation area indicates the relativedisplacement between these regions. Many such areas are evaluated to accumulatea displacement map. Meanwhile, the technique has been developed into a powerfultool for practical applications [18]. It might be mentioned that a similar task ondi!erent data has been solved in particle image velocimetry, where pair-wise images ofa "eld of tracer particles in #uid #ows must be evaluated as to their relativedisplacement [11]. This is why originally this technique was termed specklevelocimetry. An important consideration at the outset of an experiment is the specklesize in relation to the pixel elements on the camera target. It has been found that thegenerally accepted Nyquist sampling theorem can be relaxed to some extent withoutsacri"ce in performance. The quality of the correlation peak has been improvedessentially by iterative repositioning of the sub-windows that are compared. Finally,the resolution was raised by subpixel interpolation algorithms. Meanwhile, suchequipment is available commercially. Extensions have been proposed to obtainthree-dimensional displacement information by combination of two measurements ina stereo setup or by evaluating ESPI data to render ordinary irradiance distributionsthat can also be used for DSP.

Images of the object are recorded at a given rate by a high-resolution CCD cameraof typically 1000]1000pixel and processed in a computer. The algorithms developedhandle the data, subdivide the "eld into interrogation areas, do the window padding,calculate the cross-correlation functions, identify the peak and locate it with subpixelroutines. Several possibilities to obtain the subpixel location have been investigated.The data can be used in a center-of-mass calculation or they are "tted to someanalytical peak shape function. Alternatively, the use of a Fourier series expansion ofthe correlation function has been investigated thoroughly [19]. A formula for theerror associated with the peak position shows that a larger window and a narrowerpeak improve the performance. Furthermore, high correlation between the specklepatterns improves the accuracy considerably. Meanwhile, for small deformationsposition resolution even better than 1/100pixel can be guaranteed which gives some10nm when unity magni"cation is used. Thus, with the aid of speci"c data processingthe accuracy has become comparable to interferometric methods. The present statusof metrological quality is demonstrated in Fig. 3 where the motion of a high-precisiontranslation stage has been investigated. In this device operating on the inchworm


Fig. 3. Digital speckle photography for displacement analysis. Control of the performance of a translationstage operating on the inchworm principle.

principle there are two piezoelectric gripping elements interacting with a piezoelectricexpander. Upon maximum extension (some 1.5lm), the gripping piezos change roleand the expander is reset to begin a new cycle. The displacement curve duringconstant motion operation clearly shows these cycles and demonstrates the accuracyachieved with the method (1}3 nm).

According to Eq. (1) DSP can also be utilized for correlation investigations byevaluating the peak height, which is the basis of digital speckle correlation. The errorin the correlation coe$cient depends mainly on the stability of the setup. The greatadvantage of the digital data is that they are available for arbitrary pairing incorrelation. We will see that this allows to extract useful information about the actionof surface-changing processes. In spite of the large amount of data, fora 512]512pixel window results can be presented with a 0.5Hz rate that allows theon-line observation of a variety of interesting processes. The typical result of sucha correlation analysis is shown in Fig. 4. Here, two di!erent steel specimens wereexposed to an acid atmosphere in a climatic chamber and viewed through a window.The almost constant correlation coe$cient for stainless steel indicates that there arehardly any changes in its micropro"le. Pronounced corrosion, however, can beobserved on the surface of the less-re"ned (ordinary) steel. While these results are stillcomparable to output data from the optical correlator we will later see quite novelevaluations that are only possible with the digital data.

In the interpretation of the speckle decorrelation an important issue is the treat-ment of decorrelating e!ects other than those at the surface of the test specimen sincethey may be wrongly interpreted as surface changes. The e!ect of object deformationon speckle correlation has been the content of many thorough investigations in regardto speckle metrology [18,20,21]. Such studies obtain the decorrelation for a givenoptical setup. In the imaging setup used for correlation studies the predominant e!ectis produced by rigid body displacement and tilt of the object that can move the speckle"eld across the entrance pupil. In comparison, rotation and strain in the object can be


Fig. 4. Correlation coe$cient versus time for two di!erent types of steel in a humid atmosphere.

disregarded. Here, the adaptive window placement already mentioned is also impor-tant because correlation drops directly with the relative overlap area of the windows.Generally, it cannot be excluded that the object undergoes any deformation duringa speckle correlation study. If a very precise correlation result is required, knowledgeof the object deformation and motion must be known to correct for these e!ects.Transverse displacements are automatically measured by the peak position. Here, theESPI correlation method has the advantage that the three-dimensional displacement"eld is also provided with the measurement giving data for any corrections. In manystudies, however, the object remains rather passive and relaxed accuracy consider-ations allow to disregard the geometric decorrelation.

Care must also be taken to avoid polarization-induced decorrelations. When theobject is partly depolarizing, a change in the state of polarization of the illuminatinglight will show up in a changed speckle "eld and thus suggest a surface change. Wehave investigated such e!ects for objects of di!erent depolarization [22], an exampleis shown in Fig. 5: Here, the specimens are two types of stones of di!erent degrees ofdepolarization o"0.48 and 0.69 (o is the ratio of cross- to copolarized intensity of thescattered light; thus, o varies between the value 1 for total and value 0 for nodepolarization). The initial reference speckle pattern is recorded with linearly polariz-ed light, for the subsequent exposure the polarization angle of the incident light waschanged by the given amount h. The experimental correlation coe$cients werecalculated in the digital correlation con"guration and compared with theory [23]. It isquite obvious that in highly depolarizing objects polarization should be kept asconstant as possible. In "ber optic setups, for example, polarization-preserving "bersshould be used.

4. Quantitative interpretation of speckle decorrelation in terms of surface processes

Usually, DSP reveals the amount of speckle pattern change that has been invokedby the changes occurring at the specimen inspected. An important issue is the


Fig. 5. Speckle correlation coe$cient versus change in polarization angle h of illuminating linearlypolarized light. Stone specimen of depolarization o"0.48 (upper curve) and o"0.69 (lower curve).

quantitative interpretation of the speckle correlation coe$cient in terms of changes atthe surface. Quite generally this is not an easy task since there are so many possibleways by which the changed specimen can in#uence the scattered light "eld. Incomplicated cases, light}material interaction must not be restricted to the surface, butscattering can also involve bulk material within a certain subsurface region. This willalso change the state of polarization of the scattered light. Furthermore, in addition toa change in the geometry of the interface (micropro"le) the re#ectivity of the objectmay change locally between observations. All these e!ects, of course, are dependenton wavelength.

In view of these complications, it is quite obvious that there will be no way fora direct inversion of the scattering problem. Thus, simpli"ed models must be used topredict the speckle "eld changes that are connected to surface variations. Naturally,the commonly accepted model for speckle formation by a superposition of elementarywaves from discrete scatterers [1] can be utilized for the prediction of specklecorrelations in case of simple modi"cations of the scattering surface [24]. We assumethat the speckle pattern is e!ected by random phase changes *u (that are completelydetermined by the change in micropro"le geometry and the directions of illuminationand observation) and random re#ectivity changes b. Provided the usually acceptedconditions for speckle formation as a scalar di!raction process hold, we arrive at thefollowing expression for the correlation coe$cient:

c12

"

DSb exp(i*u)TD2Sb2T

. (3)

This equation can be evaluated for di!erent statistical distributions of *u andb [25]. If we disregard any re#ectivity changes and set b,1, then c

12is the squared


Fig. 6. Correlation coe$cient of the light scattered from two sample surfaces versus normalized standarddeviation of their di!erences in height. Squares give a numerical simulation.

modulus of the characteristic function U(u) of the probability density function (pdf) ofthe phase variations for u"1.

Assuming a direct transfer of surface pro"le h onto the light phase we obtain thefollowing relation:

*u"

2nj

(1#cos h)*h, (4)

where normal viewing direction has been assumed and wavelength j and illuminationangle h enter the formula. Thus the earlier result introducing the characteristicfunction of the phase distribution can be formulated accordingly for the pdf of thesurface pro"le, a result that was already discovered in the interpretation of thedecrease in fringe visibility V in holographic interferometry [9]. Recall that thecharacteristic function is the Fourier transform of the pdf. It is interesting to note thatan analog relation was derived for the visibility in the Young's fringes pattern of a PIVrecord in a turbulent #ow, i.e., locally varying displacement between particle pairs[10,11]. In our case, the resulting curves, shown in Fig. 6, resemble the `powerspectraa of the probability density functions and illustrate the sensitivity of thecorrelation coe$cient to changes in the micropro"le. For standard deviations of 0.1jthere is already a drop in correlation to about 20% and even standard deviations of0.01j produce still a few percent decrease in correlation. Thus, rms changes of theorder of magnitude of 10 nm are well resolved. For small changes, the type of statisticscan even be disregarded. The "gure also shows numerical points from a Monte Carlosimulation for a normal distribution that fall well onto the theoretical prediction.

Fig. 6 is a kind of interpretation curve in the handling of speckle correlation results.It can be extended to include also changes in re#ectivity [24]. Since several assump-tions enter into its derivation, however, that are di$cult to verify, it would becomfortable to check the assumptions and to obtain additional con"dence fromcomparative measurements on surfaces that are controlled by other means.


Fig. 7. Speckle correlation during cyclic application of an electric "eld to a specimen of piezoelectric grainsembedded in a plastic. First curve relative to situation during positive "eld, second curve relative tonegative "eld.

The results of speckle correlation on changing surfaces could be checked in severalways. It would be perfect to have an alternative measuring method that could be runin parallel to the speckle measurement. Since the di!erences in microrelief that mustbe measured are only fractions of a wavelength the accuracy and especially thereproducibility of the parallel method must be high. Suitable techniques would beAFM or white-light interferometry. Any of these, however, did not provide thenecessary working distance to allow simultaneous parallel monitoring. Thus it wasnecessary to perform consecutive measurements. To make sure that the same changesin microrelief were measured we considered three possibilities:

I. Design of a device with a controlled microscopic surface deformation that can bereversed and repeated.

II. Generation of a set of surfaces with well-controlled di!erences that can besubjected to the di!erent measurements.

III. Repetition of a surface process that can be well controlled in a statisticalsense.

The "rst two solutions require special e!orts, the third needs just a well-de"nedmaterial and surface that is exposed to a well-controlled action.

Various e!ects were examined for their suitability to provide a specimen of type I.We show one successful example where the piezoelectric e!ect was employed tochange the surface pro"le by application of an electric voltage. Small grains ofelectrically polarized lead}zirconate}titanate ceramics were embedded in a 0.2mmthickness slab of a plastic resin. Since the polarization direction of the grains can beexpected to vary randomly, an external voltage a!ects the surface relief throughpiezoelectric strain. By reversing the voltage, for example, we expect a correspondingchange in micropro"le. A typical correlation plot of the surface during such an


experiment is shown in Fig. 7. Quite clearly, there is a fairly reproducible periodicmodulation of the correlation coe$cient. An ESPI deformation control measurementguaranteed that the e!ects were not due to any large-scale deformation of the sample.Thus, the modulation is attributed to the change in pro"le. While it was not possibleto operate this sample under AFM observation because only a small electric "eld wasallowed in the AFM environment a comparative study in white-light interferometrycon"rmed the results that were also in agreement with an estimate of the e!ects on thebasis of the known piezoelectric coe$cient. Thus, the experiment provides a goodcon"rmation of the interpretation curve. It is interesting to note the delay mechanismsin the curves that suggest creeping motions in the compound sample.

For specimens of type II several pro"led surfaces were produced by exposinga photoresist material to speckle "elds that were slightly changed by a controlleddisplacement of a ground glass plate in the illuminating laser beam [25]. Assuminga linear relation between light intensity and pro"le depth the induced change in thepro"le and thus its standard deviation can be predicted. Furthermore, the pro"leswere controlled by AFM measurements. There were two obstacles for an idealrealization of this concept. First of all, small irregularities in the photoresist plates,their processing and in the illumination caused deviations from the expected pro"les.This became obvious, when two control pro"les that were produced with the samesetting clearly showed speckle decorrelation, in spite of a very precise positioning ofthe objects that required a special mount. Furthermore, in some cases repeated AFMrecords of identical specimen showed already some di!erences in pro"le that areattributed to insu$cient accuracy in the scanning stage. Therefore, a direct quantitat-ive comparison was di$cult. However, these measurements, too, proved to be a satis-factory con"rmation of the applicability of the interpretation curve. Besides, thepower of the speckle correlation method was emphasized because none of thehigh-resolution microscopic techniques was able to compete in accuracy for themeasurement of average pro"le di!erences.

Another good test can be made by checking the geometric consequences of theassumption that microrelief is turned directly into phase variations, cf. Eq. (4). For thispurpose, correlation measurements were made with the same surface but underdi!erent illumination and for a series of wavelengths [25]. Two photoresist surfaceswere taken as specimen. It was found that the correlation coe$cient increased withgrowing angle and at larger wavelengths as predicted when inserting the equation intoa surface model based on some AFM data. At large angles and large wavelengths,however, experimentally observed correlation remained short of the theoreticalcurves. Partly this could be attributed to multiple scattering and depolarizatione!ects.

In summary, it can be stated that these and additional experiments have providedstrong con"rmation that the interpretation curve of Fig. 6 is a good basis for thequantitative interpretation of speckle correlation measurements in terms of pro"lechanges at the scattering surfaces. Further evidence is furnished by specimens of typeIII as will be seen in the following section on applications where technical processes ofinterest have been studied and the results compared with additional information onthe situation.


5. Applications

Speckle correlation has been employed for several practical cases to monitorsurface changes during an external action. Details of some of these processes areknown from other measurements that are not at all as sensitive as the opticalmonitoring, but provide data suited for a rough check of the basis of our correlationevaluation. The overall mass loss in a corrosion or an ablation process, for example,can be set in relation to the optical results. In this regard, such objects act as type IIIspecimen. In addition, the corrosion results are used to show that a statisticalevaluation of a complete matrix of correlation coe$cients can be utilized for novelevaluation schemes, something that could not be done with the limited data set ofearlier techniques.

5.1. Corrosion of iron

A sample of pure iron has been sandblasted to a surface roughness of approximately0.5lmrms, cleaned in an ultrasonic bath of acetone, rinsed with distilled water, anddried in a #ow of nitrogen gas. After taking an initial speckle pattern for a referenceframe, the sample was removed from the optical setup and covered with a 30%solution of acetic acid for exactly 1 min. After another rinsing with distilled water anddrying with nitrogen the sample was replaced into the correlation setup to record thescattered light of the changed surface. This procedure was performed 15 times, thusrecording 15 consecutive states of the corrosion process. To ensure that the measureddecorrelation is solely due to microstructure changes and shows no errors producedby rigid body translation or tilt, a special mount has been constructed, which allowsthe repositioning of the iron sample to better than 0.5lm and to a tilt of less than0.1 arcsec. This guarantees a relative accuracy of the measured correlation coe$cientto better than 0.01. The procedure described is preferred to corrosion in a climaticchamber, as the conditions of the corrosion could be controlled more exactly, leadingto results that can be reproduced more easily.

In Fig. 8, the correlation coe$cients for the whole process are shown. Each imageserves as a reference state for all consecutive images, thus giving rise to one of theplotted curves. For example, the labeled data point in the third curve represents thecorrelation coe$cient between images 3 and 7. Thus, each curve plots the decorrela-tion with reference to one of the recorded states. For this representation, the completematrix of correlation coe$cients between all possible pairs of images has beencalculated. Except for the "rst step, for which the freshly prepared surface seems to bemore susceptible to the corrosion (and which is therefore excluded from the followingaverages), the process proceeds very uniformly as the curves in Fig. 7 run almostparallel to one another. We should mention that we have, for comparison, donesimilar measurements observing a stationary surface for a while. We "nd that there isa constant small decorrelation of the order of 1% at any time which we attribute toever present small changes in the water layers covering any object. Random #uctu-ations in the environment in direct contact with the surface as well as hysteresis e!ectsin the interaction of water with a rough surface are probably responsible for this


Fig. 8. Speckle correlation history for corrosion of a pure-iron surface in acetic acid. Each state of theprocess is compared with all subsequent states.

dynamic equilibrium. This could also explain part of the decrease in the correlationstudy on the stainless steel specimen shown in Fig. 4. In any case, the stability in thecomplete setup is su$ciently good so that it cannot account for these e!ects.

If one assumes that the decorrelation is mainly due to changes in the microtopogra-phy of the corroding surface following Gaussian statistics, the standard deviationp*h of the height changes between all of the recorded states is calculated by use of theinterpretation curve for a normal distribution function. A possible change in re#ectiv-ity is ignored because it has been shown that in the present situation there is nosigni"cant e!ect on the calculated changes in microrelief [24]. The normal distribu-tion can be used here because it was shown that, "rst of all, for small opticaldecorrelations the choice of distribution does not a!ect the calculated surface changes.For larger decorrelations, we have an addition of many small-step decorrelations thatare only partly correlated (see below), so that the distribution of the total changes willagain approach a Gaussian distribution due to the central limit theorem of statistics.

The standard deviation p*h is almost equal for equal time separation *t between thestates compared, irrespective of the absolute time. Thus, average values can becalculated that are displayed versus *t in Fig. 9, labeled experimental data. Within4min, for example, the surface topography has changed by p*h"30nm, regardless ofthe starting time. There is an interesting interpretation of the results shown in Fig.9 which demonstrates how the standard deviations of single-step processes (that areall equal) must be added to obtain the standard deviation of a combined processconsisting of the consecutive execution of N single steps. Statistics tells us that theaddition law for the standard deviations depends on the correlation between theprocesses summed up. We must not confuse this correlation with the speckle correla-tion that has been the overall topic of this contribution. Correlation in the presentcontext means that there is more or less relation between two subsequent corrosionsteps. It could be, for example, that corrosion starts at certain nuclei on the surface


Fig. 9. Averaged standard deviation of the microrelief changes on a sample of pure iron of 0.5 lm surfaceroughness versus corrosion time. Also shown are the theoretical values if changes at di!erent times arecompletely correlated or uncorrelated.

and continues for a while around these locations. This means that certain regions inthe surface do not change for a while whereas others continue to degrade. Each step isdescribed by a random variable *h

*and these steps may be interrelated to some

extent. We have included in Fig. 9, the two limiting cases for the addition law, i.e.,completely independent (uncorrelated) steps for which the squares of the standarddeviations (the variances) add, and completely correlated steps where directly thestandard deviations add. Obviously, the present data are located somewhere inbetween.

There is no space here to go into more details regarding this type of evaluation, ithas been covered elsewhere [24]. Yet, a few more results should be communicated. Itwas found that correlation of corrosion vanishes after about 10 steps, in which thecorresponding standard deviation in surface change amounts to some 60nm and thatfor specimens of the same material but di!ering roughness distinct di!erences wereobserved that could be related to microscopic surface features like cracks, for example.Finally, it should be noted that the measurements yield only the second moment of theheight change in the micropro"le. A direct value for the corrosion rate, for example,cannot be obtained. This is obvious for the case of constantly receding surface withunchanged pro"le, for example, for which the method would yield unit specklecorrelation.

5.2. Laser cleaning of historical objects

In the conservation of historical objects surface treatments are very important tomaintain the original appearance and to protect the delicate object from deterioratingsurface e!ects. Recently, lasers have been introduced successfully for such cleaningtasks. Still it remains a great challenge to monitor the surface changes in order to stopthe process at the right time when it threatens to harm the historical basis. Often, basis


Fig. 10. Correlation coe$cient of speckle "elds scattered from an encrusted sandstone surface duringcleaning with Nd : YAG laser pulses.

material and dirt layer di!er in their optical properties such that the interaction withthe cleaning laser light changes when the light "rst reaches the basis. Monitoring byspeckle correlation could reveal such a change and signal the need to end the cleaning.

We have made some "rst investigations with historical patina on carrier materialslike stone, glass or iron. Fig. 10 shows the typical set of curves, comparable to those ofFig. 6, that were obtained when a so-called black crust was removed from a sandstoneby green Q-switched Nd : YAG pulses of some 400mJ/cm2 each. The correlationcoe$cient is plotted versus the number of pulses. It is obvious from the curves that theprocess gradually slowed down (take, for example, the drop in correlation for each"rst pulse of the series of curves). These curves clearly show that the e!ects producedby laser ablation are good candidates for an analysis by speckle correlation. Obvious-ly, all additional evaluation techniques mentioned earlier can be applied to these data.Further work is in progress to assess the suitability of the method for routinemonitoring in the restoration "eld.

6. Conclusions

The detailed analysis of speckle "eld similarity by digital correlation of videorepresentations of the speckle patterns is providing a powerful method to studysurface processes. Meanwhile, it is possible to interpret many data quantitatively interms of a change of the microrelief of the surface. A "eld of continuing interest will bee!orts to separate e!ects when di!erent processes are candidates for an explanation.Interaction with a surface layer in the bulk material or re#ectivity changes, forexample, require additional object information for identi"cation that could be ob-tained by simultaneous observation in di!erent wavelengths and under varying statesof polarization.


Acknowledgements

Correlation studies have been supported by grants from Deutsche Forschungs-gemeinschaft and BMBF under various contracts. We acknowledge the support bythe Institut fuK r Optik und Feinmechanik of the Fraunhofergesellschaft in Jena, in thecontrol measurements with AFM and white-light interferometry and the help of A.Fuhrig and F. Nehrling in performing the DSP measurements on the inchwormtranslation stage.

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speckle correlation for the analysis of random processes at rough surfaces

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