The roughness of machined surfaces is an importantcriterion for characterizing surface microstructure~e.g., ISO 4287y1!.The applications of engineering workpieces or tech-
nical products require roughness values that liewithin a given range of tolerance. The root-mean-square ~rms! roughness, as an example of a rough-ness parameter, corresponds to the standarddeviation of the microgeometric surface heights.Generally, surface-roughness parameters describethe quality of the corresponding workpiece, for exam-ple, with respect to its proper technical function, itsbehavior under mechanical loads, and its optical ap-pearance.In industrial applications roughness parameters
are usually determined by profilometers, either bymechanical stylus instruments or by optical autofo-cusing profilometers.1 These methods, however,cannot be applied to real in-process quality testing.Generally the surface under investigation must be
The authors are with the Institut fur Mess-, Regelungs- undSystemtechnik, University of Bremen, FB 4, Postfach 330440,28334 Bremen, Germany.Received 18 April 1996; revised manuscript received 12 August
2188 APPLIED OPTICS y Vol. 36, No. 10 y 1 April 1997
sampled while it is at a standstill, mostly in a sepa-rate laboratory that is free from vibrations. There-fore roughness measurements by profilometrictechniques are often carried out by spot checks only;even for postmortem analysis a 100% inspection can-not be achieved.An industrial in-process control of surface-
roughness parameters is of great interest because itwould lead to a more efficient use of production ma-chines and energy resources. Much work has al-ready been done over the past few decades to developmethods for such in-process roughness measure-ments.1 The most promising methods, however, arebased on optical phenomena, especially on the scat-tering behavior of light.2 In principle, these ap-proaches allow us to determine integral roughnessparameters, such as the rms value, directly, withoutreproducing the exact microgeometric shape of a sur-face profile.However, no measuring device based on light-
scattering principles could be established for indus-trial applications, as several fundamental problemsremained unsolved.These problems can be summarized as follows:
First, there is a lack of correlation of the roughnessparameters measured optically with the parametersgiven by industrial standards. This problem of op-tical light-scattering methods is caused by the influ-ence of vertical surface height fluctuations on theangular distribution of intensities of scattered light
and by parameters that describe horizontal surfacecharacteristics such as the autocorrelation function ofa rough surface.3,4 Second, the measuring range ofmany light-scattering methods is restricted to rmsroughness values of less than 1 mm.5In this paper, a new optical method is introduced
for surface-roughness determinations of rms valueslying within the 0.05–5-mm range. This method isbased on polychromatic light scattering and solvesthe problem mentioned above.If a rough surface is illuminated with coherent or
partially coherent light, the spatial distribution ofscattered light shows a grainy appearance that isknown as the speckle effect.6 Assuming that thesurface height fluctuations ~rms roughness! are nottoo large and that the illuminating light is dichro-matic or polychromatic within a narrow spectralrange, the speckle elongation phenomenon can beobserved in the far-field region as a fibrous radialstructure of the speckle pattern. This is illustratedin Fig. 1 by a photograph of a dichromatic specklepattern. The illumination beam ~diameter 0.65mm!was produced by an argon laser emitting dichromaticlight of 488- and 514-nm wavelengths and was inci-dent under an angle of 45° onto a ground metallicsurface of 0.5-mm rms roughness. The scatteredlight was recorded after propagation over a distanceof 30 cm. The center of the speckle structure corre-sponds to the direction of specular reflection, i.e., tothe direction of the optical axis.The radial structure is caused by the angular dis-
persion of polychromatic light. Because of the lackof correlation of the speckle patterns for differentlight wavelengths with increasing rms roughness ofthe scattering surface, the elongation phenomenonwill vanish more and more. For very rough surfacesthis decorrelation effect leads to nearly independentspeckle patterns for the different wavelengths so thatthe angular dispersion can no longer be observed andthe polychromatic spatial speckle intensity distribu-tion shows no radial structure.
Fig. 1. Photograph of a dichromatic speckle pattern ~wavelengthcombination 488 and 514 nm! produced by the scattering of light ofan argon laser from a groundmetallic surface of 0.5-mm rms rough-ness.
The roughness dependence of polychromatic speckleelongation was first demonstrated by Parry7 morethan two decades ago: Polychromatic speckle pat-terns are produced by the scattering of light from adiffuser by use of a multiline argon laser for illumina-tion. Parry presents photographs of such speckle pat-terns obtained with three transmitting diffusers ofdifferent roughness. For the smoothest diffuser~1-mm rms roughness! a distinct radial structure is tobe found in the speckle pattern, whereas for the rough-est diffuser ~10-mm rms roughness! no radial specklestructure appears. To explain this result, Parrytreats the effect theoretically by the spatial cross cor-relation of the intensities of two different light wave-lengths.Measurement arrangements to determine the
roughness of engineering surfaces were developed onthis basis. Commonly separated speckles of two dif-ferent wavelengths are detected on the optical axisand used to estimate the roughness by correlationtechniques.8–10 Because of the angular dispersion astrong roughness independent decorrelation of thetwo speckle intensities appears in practical applica-tions if the optical setup is not in perfect adjustment.For example, if a rough reflecting surface is slightlytilting over by an angular deviation from the adjustedposition of less than 1°, this decorrelation effect oc-curs and makes a reliable roughness measurementimpossible.10An approach to evaluating elongated speckle pat-
terns directly ~i.e., without wavelength separation! isgiven by Stansberg11: Polychromatic speckle pat-terns produced by the yellow line in the spectrum ofa high-pressure mercury lamp are recorded on a pho-tographic film and analyzed by an optical Fourier-transform system with a double aperture in the filmplane in order to obtain a roughness parameter in theFourier spectrum.However, all these measurement techniques are
hardly applicable to an in-process surface inspection.Nevertheless the principles of speckle correlation andspeckle elongation allow a fast and robust surfaceroughness determination by evaluation of polychro-matic speckle patterns. This is outlined below. Ina detailed description12 the relevant phenomenawereanalyzed by theoretical investigations and computa-tions of scattered light distributions by use of simu-lated surfaces with irregular height fluctuations ofgiven rms values and were proven experimentallythereafter.First, in Section 2, theoretical results that describe
the speckle elongation phenomenon in an analyticalway are presented and discussed. Based on theseresults, a compact experimental setup that consists ofonly a few optical standard components and that usesthe CCD technique for speckle pattern detection ispresented. The experimental analysis of the specklepatterns by digital image processing methods is de-scribed, and experimental results are given. Themeasurement results show both a rather good corre-lation of the roughness parameters determined opti-
1 April 1997 y Vol. 36, No. 10 y APPLIED OPTICS 2189
cally with the standardized rms roughness and a highreliability even in cases of maladjustment.
2. Theory
The theoretical procedure presented here is based onthe Kirchhoff approximation ~KA! or physical opticsapproximation, which is restricted to rather largeradii of curvature of the scattering surfaces. Thismeans that the correlation length of a surface profilemust be much greater than the standard deviation ofthe surface heights. Furthermore, shadowing andmultiple-scattering effects are neglected by the KA.The validity of the KA has been studied extensively.13At this point, we assume that for machined surfacesthe KA will lead to results that are accurate enoughto give a realistic description of the phenomena ob-served experimentally.
A. Scattering Equations
Applying the KA to rough surface scattering leads tothe expression of the normalized electric field um inthe Fourier plane of a lens system,14–17:
um~j, h! 52RF~Qe!~cos Qs 1 cos Qe!
8LxLy cos Qe
3 *2`
1`
*2`
1`
w~x, y!exp$2i@vxmx 1 vymy
1 vzmh~x, y!#%dxdy, (1)
with the Fresnel reflection coefficient RF~Qe!, the ap-erture function w~x, y! for a laser beam in the TEM00mode, where
w~x, y! 5 ~4yp!exp@2~x2yLx2 1 y2yLy
2!#, (2)
and
vxm 5 kmSsin Qs 2 sin Qe 1j
fcos QsD , (3)
vym 5 km~hyf ! , (4)
vzm 5 km~cos Qs 1 cos Qe!. (5)
Qe indicates the angle of incidence, Qs indicates theangle of observation ~Fig. 2!, km is the wave numberof the light wavelength lm, Lx and Ly are the radii of
Fig. 2. Scattering geometry with angle of incidence Qe, scatteringangle Qs, and the corresponding wave vectors ke and ks.
2190 APPLIED OPTICS y Vol. 36, No. 10 y 1 April 1997
the illuminated elliptic surface area in the x and they directions, respectively, f is the focal length of theFourier-transform lens ~Fig. 3!, and ~j, h! are thecoordinates in the lens Fourier plane.16The surface heights are described by the surface
function h~x, y!. For simplicity, the theoretical anal-ysis is restricted to surfaces that are rough in onedimension only, which yields h~x, y! 5 h~x!. Thescattered far field is observed within the plane ofincidence and the lens is positioned under the angleof specular reflection, so that Qs 5 Qe and h 5 0.Carrying out the y integration of Eq. ~1! leads to
um~j! 52RF~Qe!
ÎpLx*
2`
1`
exp@2~x2yLx2!#exp$2i@vxmx
1 vzmh~x!#%dx. (6)
B. Spectral Speckle Correlation
The speckle correlation coefficient r12 can be deter-mined from measured data with the following basicrelation of the speckle correlation technique:
r12 5^~I1 2 ^I1&!~I2 2 ^I2&!&
@^~I1 2 ^I1&!2&^~I2 2 ^I2&!
2y2
5^I1I2& 2 ^I1&^I2&
~^I12& 2 ^I1&
2!1y2 ~^I22& 2 ^I2&
2!1y2 , (7)
where the intensity Ij 5 ujuj*, j [ $1, 2%, for eachwavelength lj of dichromatic light used for illumina-tion and uj* is the complex conjugate of uj. Thestatistical expectation value of I is expressed by ^I&.For evaluating experimental data, the law of large
numbers18 is applied to replace the expectation val-ues in Eq. ~7! with arithmetic averages of a sufficientnumber of independent intensity values I1 and I2,which can be determined experimentally. Thesevalues are obtained when the rough surface is movedin the x direction and the two intensities for severalilluminated surface sections are recorded.The relation between the rms roughness and the
measured intensity values results from theoreticalcalculations based on statistical optics. First, thecorrelation length Dx 5 Lk of a surface profile isdefined as the displacement Dx. The normalized au-
Fig. 3. Geometric arrangement with the Fourier-transformproperties of a lens.
tocorrelation function rh~Dx! of the profile decreasesfrom its maximum value of 1 to the value 1ye.If the illuminated surface section contains a large
number of surface correlation lengths Lk and thestandard deviation sh of the surface heights is notconsiderably smaller than a quarter of the wave-length l, the central limit theorem of statistics can beapplied.19 This results in circular complex Gaussianelectric fields um.20 From Ref. 21 this gives the re-lationships
^I1I2& 5 u^u1u2*&u2 1 ^u1u1*&^u2u2*&,
^Im2& 5 2^Im&2, m [ $1, 2%. (8)
By use of Eqs. ~8!, it follows from Eq. ~7! that
r12 5u^u1u2*&u2
^u1u1*&^u2u2*&. (9)
In Eq. ~9! the expectation values ^umun*& with m,n [ $1, 2% can be expressed by use of Eq. ~6!:
^umun*& 5uRFu2
pLx2 * *
2`
1`
exp@2~xm2 1 xn
2!yLx2#
3 exp@2i~vxmxm 2 vxnxn!#
3 ^exp@2i~vzmhm 2 vznhn!#&dxmdxn. (10)
If we assume that the surface heights are homoge-neous and normally distributed with zero mean, anexpression for the characteristic function derived byPapoulis22 can be used:
^exp@2i~vzmhm 2 vznhn!#& 5 expH212
sh2@vzm
2
2 2vzmvznrh~xm 2 xn! 1 vzn2#J
5 expF212
sh2~vzm 2 vzn!
2Gexp$2sh2vzmvzn
@1 2 rh~xm 2 xn!#% , (11)
where the normalized autocorrelation function rh~xm2 xn! 5 rh~Dx! of the surface profile h~x!. For fur-ther evaluation of Eqs. ~10! and ~11!, a specific math-ematical shape of the autocorrelation function rh~Dx!must be introduced. In agreement with Beckmannand Spizzichino23 and Ogilvy,24 one can assume that
rh~Dx! < exp@2~DxyLk!2# < 1 2 SDx
LkD2 for Dx ,, Lk.
(12)
It must be added, however, that other models of theautocorrelation function do not change the final re-sults remarkably if all the other assumptions out-lined above are valid.25,26 For example, theexpression rh~Dx! ' exp~2uDxyLku!, which is alsowidespread in this context, yields the same result forthe speckle correlation coefficient as the relationshipgiven by approximation ~12! does.
Assuming that sh $ ly4 leads to sh2vzmvzn .. 1.
In this case and according to Eq. ~11!, the integrandin Eq. ~10! contributes only if the value of rh~Dx! liesclose to 1. This is fulfilled for DxyLk ,, 1 only. Nofurther contributions for DxyLk $ 1 exist if thesecond-order series expansion of rh~Dx! according toapproximation ~12! is used. Inserting approxima-tion ~12! and Eq. ~11! into Eq. ~10! results in
^umun*& 5
uRFu2expF212
sh2~vzm 2 vzn!
2GpLx
2 * *2`
1`
3 expS2xm
2 1 xn2
Lx2 Dexp@2i~vxmxm 2 vxnxn!#
3 expF2sh
2vzmvzn~xm 2 xn!2
Lk2 Gdxmdxn
5
uRFu2 expF212
sh2~vzm 2 vzn!
2
~1 1 2sh2vzmvznLx
2yLk2!1y2
3 expF2~vxm 2 vxn!
2Lx2
8 G3 expH2
~vxm 1 vxn!2Lx
2
8@1 1 2vxmvxn~shLxyLk!2#J . (13)
Because 2sh2vzmvznL
2yLk2 .. 1, the square root in
Eq. ~13! can be approximated by
~1 1 2sh2vzmvznLx
2yLk2!1y2 <
shLx
Lk~2vzmvzn!
1y2. (14)
According to Eq. ~9! this gives the final result forthe speckle correlation coefficient:
rmn < exp@2sh2~vzm 2 vzn!
2#expF2~vxm 2 vxn!
2Lx2
4 G ,m, n [ $1, 2%. (15)
For perfect adjustment of the measurement setup ~Qs5 Qe, j 5 0! the argument of the second exponentialfunction becomes zero such that
sh 5~2ln r12!
1y2
vz1 2 vz2. (16)
From approximation ~15! in conjunction with theother relations given above, the practical problemthat arises when the spectral speckle correlation isapplied experimentally can be understood.8–10 Ac-cording to Eq. ~3!, value j Þ 0 orQs Þ Qe results in vxmÞ vxn. In this case the second term in approxima-tion ~15! leads to large decorrelations that are causedby the angular dispersion instead of the surfaceroughness. This phenomenon is demonstrated inFig. 4, which shows the results of simulations basedon the KA.27 Figure 4 shows the intensity versusthe scattering angle for surfaces with two different
1 April 1997 y Vol. 36, No. 10 y APPLIED OPTICS 2191
Fig. 4. Simulated far-field speckle intensitycurves for several wavelengths of laser light as-suming 1-mm beam diameter, perpendicular inci-dence, and ~a! 0.1-mm rms roughness or ~b!1.25-mm rms roughness.
rms roughness values, Rq 5 0.1 mm @Fig. 4~a!# and Rq
5 1.25 mm @Fig. 4~b!#. The intensity curves for sev-eral wavelengths of an argon-ion laser, calculated forperpendicular incidence ~Qe 5 0! are plotted.12 InFig. 4~a!, all intensity curves have the same shapewithin the region of specular reflection, i.e., a scat-tering angle close to 0°. With increasing distance tothe 0° direction ~i.e., the optical axis!, angular disper-sion arises. Thereby the intensity maxima of thehigher wavelengths are shifted more and more com-pared with the corresponding maxima of the smallerwavelengths. If the surface is tilted by an angle of0.1° within the plane of incidence, for example, thedifference uQe 2 Qsu becomes 0.2°. At this value theintensity of the 476.5-nm wavelength shows a maxi-mum, whereas the corresponding maximum of the514-nmwavelength is shifted to higher angles. Thisproblem of adjustment can be overcome by use ofFourier-transform lenses with wavelength-adaptedfocal lengths.27,28 Another approach that uses an-gular dispersion as a measuring effect can also beseen by comparison of Figs. 4~a! and 4~b!: Notwith-standing the dispersion, the intensity curves of thedifferent wavelengths for the smoother surface @Fig.4~a!# behave similarly to each other so that the an-gular dispersion leads to a broadening of the totalintensity curve ~solid curve! for larger distances fromthe optical axis. The total intensity curve corre-sponds to the speckle structure; the broadening isknown as speckle elongation. A much rougher sur-face causes intensity decorrelations @Fig. 4~b!#, and nobroadening of the total intensity structures can beobserved.
C. Theory of Speckle Elongation
It is well known that spatial autocorrelation func-tions of intensity distributions describe speckle di-mensions.29 Hence, as the basis of an analytical
2192 APPLIED OPTICS y Vol. 36, No. 10 y 1 April 1997
description of the speckle elongation, the spatial au-tocorrelation function rDI of polychromatic far-fieldspeckle intensities must be evaluated.30 This auto-correlation function is defined analogously to thespeckle correlation coefficient according to Eq. ~7! byuse of DI 5 I 2 ^I&:
rDI~j1, j2! 5^DI~j1!DI~j2!&
@^DI2~j1!&^DI2~j2!&#
1y2 . (17)
The main difference between Eq. ~17! and Eq. ~7! isthat DI~j1! in Eq. ~17! expresses the deviation of thetotal speckle intensity of the polychromatic light fromthe affiliated expectation value for the location vectorj1 5 ~j1, h1! in the Fraunhofer plane, whereas Eq. ~7!refers to the intensities of monochromatic light of twodifferent wavelengths. Equation ~9!, which was ob-tained from Eq. ~7!, can be generalized as follows:
rmn 5 rmn~j1, j2; km, kn! 5u^um~j1; km!un*~j2; kn!&u2
^uum~j1; km!u2&^uun~j2; kn!u2&,
(18)
where km and kn give the different wave numbers andj1, j2 express different location vectors j in the Fou-rier plane of the lens ~see Fig. 3!.A polychromatic light source that is used in a dif-
fraction experiment behaves as a superposition ofincoherent monochromatic sources. As a result, thetotal intensity I~j! in the Fraunhofer plane is given bythe sum of the absolute squares of themonochromaticfield amplitudes uj~kj! at the location vector j:
I~j! 5 (j51
Nk
I~j, kj! 5 (j51
Nk
uuj~j, kj!u2 (19)
with the number Nk of the discrete wavelengths that
form the polychromatic light. If the spectrum of thelight source is continuous, the summation in Eq. ~19!must be replaced by an integration that considers thespectral profile of the light source.31 Analogous tothe derivation of Eq. ~9! or Eq. ~18! from Eq. ~7!, fromEqs. ~17! and ~19! the relationship
depend on other surface parameters, such as the cor-relation length Lk or the concrete shape of the auto-correlation function rh~x! of the surface profile.
D. Theoretical Results
In this subsection the results according to Eq. ~22!
rDI~j1, j2! 5(m51
Nk
(n51
Nk
SmSnrmn~j1, j2; km, kn!
F(m51
Nk
(n51
Nk
SmSnrmn~j1, j1; km, kn!G1y2F(m51
Nk
(n51
Nk
SmSnrmn~j2, j2; km, kn!G1y2 (20)
for the spatial autocorrelation function of the speckleintensity fluctuations can be derived, assuming thatthe field amplitudes uj~j, kj! are circularly complexnormally distributed.30In Eq. ~20! Sm and Sn give the normalized mean
intensity for the affiliated wavelength:
Sm 5 ^I~j, km!&y^I~j!&, Sn 5 ^I~j, kn!&y^I~j!&,
where I~j! is as defined in Eq. ~19!. When rDI~j1, j2!is evaluated according to Eq. ~20!, the speckle elon-gation can be calculated. For that, a fixed value j1 5j1a must be chosen, and j2 is varied, beginning withthe vector j2 5 j1a, until rDI~j1a, j2! at j2 5 j2abecomes 1ye, for example. Then the difference uj2a 2j1au gives the speckle elongation, depending on thelocation vector j1a.For a physical explanation of the speckle elonga-
tion phenomenon, the correlation coefficient rmn, ac-cording to approximation ~15!, can be inserted intoEq. ~20!, assuming a perfect adjustment of the opticalsetup ~Qs 5 Qe!:
rmn~j1, j2; km, kn! 5 exp@2sh24 cos2 Qe~km 2 kn!
2#
3 expF2Lx
2 cos2 Qe
4f 2~kmj1 2 knj2!
2G .(21)
This yields the final expression for the spatial auto-correlation function:
are investigated in detail. These investigationsare related to the optical setup of Fig. 3 in which thefar-field intensities of the scattered light are de-tected in the Fourier plane of a lens. Figure 5presents the normalized speckle elongation alongthe j axis for different surface roughness values,with perpendicular incidence and a focal length f of200 mm. The normalized speckle elongation isgiven by the multiple of the speckle radius jsp,which is defined by the 1ye2 width of the maximumintensity at j 5 0. Accordingly, the value of thespatial autocorrelation function rDI~j1 5 j1a 5 0, j2!is e21 at j2 5 jsp. Therefore for j1a . 0 the valuej2a was determined such that rDI~j1a, j2a! 5 e21
holds. The ordinate was subdivided by the result-ing ratio j2ayjsp. For Fig. 5 the values Sm and Snwere chosen such that the polychromatic light con-tains equal power for all wavelengths: Sm 5 Sn 51yNk, m, n [ $1, . . . , Nk%.For Fig. 5~a! the wavelength combination l1 5 488
nm, l2 5 514 nm, which is emitted by a multilineargon-ion laser, was assumed. Up to a distance j1aof 2.7 mm from the optical axis ~direction of specularreflection! an increasing speckle elongation can beobserved. For larger distances, all curves fall rap-idly down to the value of 1. This decrease is causedby the shift of two corresponding maxima for the twowavelengths that is due to the angular dispersion.If the distance from the optical axis is large enoughsuch that this shift is greater than the speckle diam-eter, an intensity minimum arises between the two
rDI~j1, j2! 5(m51
Nk
(n51
Nk
SmSn exp@2sh24 cos2 Qe~km 2 kn!
2#expF2Lx
2 cos2 Qe
4f 2~kmj1 2 knj2!
2G)j51
2 H(m51
Nk
(n51
Nk
SmSn exp@2sh24 cos2 Qe~km 2 kn!
2#expF2Lx
2 cos2 Qe
4f 2jj2~km 2 kn!
2GJ1y2 . (22)
In this context it should be remarked that rDI~j1, j2!according to Eq. ~22! as well as the correlation coef-ficient r12 @Eq. ~15!# depends on the standard devia-tion sh of the surface heights only. It does not
maxima @see Fig. 4~a! at larger values of the scatter-ing angle#. In Fig. 5~b! comparable results for poly-chromatic light of a wavelength combination of fourdifferent wavelengths are plotted. In this case the
1 April 1997 y Vol. 36, No. 10 y APPLIED OPTICS 2193
speckle elongation is more distinct and occurs at evenlarger distances j1.When dichromatic light is used, the total measur-
ing range increases with a decreasing difference be-tween the two wavelengths, whereas the resolution ofthe roughness measurement decreases.
3. Experimental Investigations
In Fig. 6 a scheme of an optical setup for generating,detecting, and analyzing polychromatic speckle pat-terns is illustrated.A laser light source emits a polychromatic beam.
A multiline argon-ion laser can be used as a lightsource. Alternatively, experimental investigationswere performed by use of two laser diodes emittingtwo different wavelengths ~e.g., 650 and 670 nm! andby the coupling of the two monochromatic beams by apolarizing beam splitter in order to produce a dichro-matic beam ~Fig. 7!. Advantages of the lattermethod are cost savings and compactness of the re-sulting measuring setup.The polychromatic laser light is scattered by the
rough surface. Because of the perpendicular inci-dence, a beam splitter is needed to lead the scatteredlight onto the CCD array, which is positioned in thefocal plane of a Fourier-transform lens ~Fig. 6!. Themaximum intensity of the scattered light has to beadapted to the dynamic range of the CCD array by
Fig. 5. Theoretical results of the normalized speckle elongationfor different wavelength combinations, depending on the j coordi-nate in the focal plane of a Fourier-transform lens.
2194 APPLIED OPTICS y Vol. 36, No. 10 y 1 April 1997
means of a filter. The distance d between the scat-tering surface and the lens has to be large enough toensure the validity of the Fresnel far-field approxi-mation.32,33 The mathematical relationship that de-scribes the transformation from the angulardistribution to the spatial distribution in the Fourierplane can be derived by analysis of Eq. ~3!: sin Qs3j cos Qsyf.For perpendicular incidence ~cos Qs 5 1! and
200-mm focal length the angular range of 61° willbe transformed to a range of ;7 mm in the Fourierplane of the lens. In addition, for resolving thespeckles by the CCD array, the mean speckle diam-eter, which depends reciprocally on the beam diam-eter, must be large compared with the pixel width.Generally this condition will be fulfilled if the beamdiameter is ;1 mm and the speckle pattern is re-
Fig. 6. Optical setup for generating, detecting, and evaluatingpolychromatic speckle patterns.
Fig. 7. Optical setup for coupling two beams of laser diodes emit-ting different light wavelengths ~e.g., 650 and 670 nm!.
Fig. 8. ~a! Dichromatic far-field speckle pattern ~nega-tive image! of a groundsurface of 0.5-mm rms rough-ness ~beam diameter 1.5 mm,lens focal length 150 mm,wavelength combination 650and 670 nm!, ~b! autocorrela-tion widths belonging to theimage segments indicated in~a!.
corded with a conventional CCD technique.Therefore the compact optical setup according toFig. 6 is quite satisfactory.The CCD data are received by a framegrabber and
stored at 8-bit gray levels. Further evaluation isperformed by a microcomputer ~Pentium!.
Fig. 9. Experimental results representing the roughness depen-dence of themean autocorrelation width T# when an argon-ion laseris used for polychromatic surface illumination.
A. Speckle Pattern Evaluation
The feasibility of the described method was provedby the detection of polychromatic speckle patternsfor different samples of the model, Rugotest 104,which represents ground surfaces of the roughnessclasses N1 ~0.0125 mm , Ra # 0.025 mm! to N8 ~1.6mm , Ra # 3.2 mm!. Because of the grinding pro-cess, the surface samples show an anisotropic tex-ture.34 This property is also expressed by theanisotropic structure of the speckle patterns.11 Anexample of a speckle pattern that shows the effect ofspeckle elongation is given in Fig. 8~a! as the neg-ative image of the speckle pattern of sample N5 ~0.2mm , Ra # 0.4 mm!, taken by a CCD array. Threeregions can be distinguished within the speckle pat-tern: In the center of the illuminated area thespeckles are distinct and nearly round. To theright and the left of this region one can observespeckles that are elongated in the horizontal direc-tion as mentioned above. Near the edges of thepattern the speckles show a fine structure, which iscaused by the spatial separation of two relatedspeckles of the two wavelengths.
Table 1. Results of Rough Surface Classification ~Sample: Rugotest 104!
Note: 11, in agreement with classification according to manufacturer; 1, sorting in correct transitional roughness class ~N5–N6 orN7–N8!.
1 April 1997 y Vol. 36, No. 10 y APPLIED OPTICS 2195
Performing the evaluation of the CCD record firstrequires the determination of the area of interestbetween the horizontal lines in Fig. 8~a! by a digitalimage processing algorithm, either with a gray-levelthreshold for the average gray level of a pixel line orby a Gaussian least-squares approximation of theaveraged gray level of each pixel line.This evaluation area is subdivided into 13 seg-
ments, for example, where an overlap of 50% betweenneighboring segments has been chosen. For eachsegment a mean autocorrelation function of the gray-level fluctuations is calculated, either as the averageof one-dimensional functions for each segment line orin two dimensions. This can be performed by theapplication of fast Fourier-transform algorithms ordirectly, as only a few values near the autocorrelationmaximum are needed for further computations.The results presented in this section were obtainedby the calculation of three values of the autocorrela-tion function, the maximum value, the values for Dj5 1 pixel, and Dj 5 2 pixels.Normalization gives amaximum value of 1 for each
autocorrelation function. We determine the widthsTj, j [ $1, . . . , 13%, of these autocorrelation functionsby approximating them by parabolas and by calcu-lating the zero crossings of the parabolic approxima-tions. These are given for each segment of Fig. 8~a!in a bar chart in Fig. 8~b!. It can be seen that thebars reflect the statistical properties of the specklestructures.Subsection 3.B demonstrates that characteristic
roughness parameters can be obtained from themean height of the bars as well as from their stan-dard deviation, as for rougher surfaces the elongationeffect vanishes and all bars tend to have the sameheight.
B. Experimental Results
Figure 9 shows results of the mean autocorrelationwidth T# obtained by the averaging of the autocorre-lation widthsTj of all segments. A 50-mWargon-ionlaser with 650-mmbeam diameter was used to realizeperpendicular incidence. The focal length of theFourier lens was 200 mm. This leads, in contrast toFig. 8~a!, to a wider region of elongation, while theregion of speckle separation nearly vanishes. A 2y3-in. ~1.69-cm! CCD chip consisting of 768 pixels 3 576pixels with a size of 11 mm3 11 mmwas used. It canbe seen that the given results clearly indicate the rmsroughness. The error bars that represent the stan-dard deviation of five independent measurements arerather small. The 2-mm roughness value lies out-side the measuring range of the wavelength combi-nation of 488 and 514 nm, as given by the argon laser@Fig. 5~a!#.Table 1 gives additional results that were obtained
with two laser diodes emitting at the wavelengths650 and 670 nm. The beams were coupled accordingto Fig. 7. The resulting beam diameter was ;1.5mm. An angle of incidence Qe of 45° and a focallength of 150 mm were chosen. The speckle pattern
2196 APPLIED OPTICS y Vol. 36, No. 10 y 1 April 1997
of Fig. 8~a! was taken from this series of measure-ments.The evaluation was expanded by use of the mean
autocorrelation widths and the standard deviation ofthe autocorrelation widths @Fig. 8~b!# in the j and theh directions in the Fourier plane. Speckle patternscorresponding to the roughness classes N3–N8 wereevaluated by this procedure. The results weresorted into characteristic intervals so that all sam-ples could be classified correctly with respect to theirroughness. This classification is documented in Ta-ble 1 for five independent surface areas of each sam-ple.
4. Conclusions
A roughness measuring system based on the speckleelongation phenomenon has been introduced and in-vestigated; it offers in-process capabilities within awide range of industrial applications. With digitalimage processing methods, a characteristic rough-ness parameter can be obtained with high reliabilityby statistical analysis of only one speckle pattern.The measuring range reaches from 0.05 to 5 mm forthe rms roughness if appropriate wavelength combi-nations are used for illumination. The correlation ofthe measured results with the roughness parameterRa is good. An extension of this method might alsogive information about the surface texture. For ex-ample, ground surfaces can be distinguished fromsamples produced by shot peening.The time required for data acquisition can be kept
extremely short, of the order of 1024 s with a conven-tionally shuttered CCD array, and even less withhigh-speed CCD technique. Therefore the measur-ing system is quite insensitive to surface vibrations.The system permits rather high lateral velocities ofthe rough surface relative to the optical system, sothat rapidly moving surfaces can be investigated.Maladjustments, such as tilts of the surface up to 1°,can be tolerated. Furthermore, the distance be-tween rough surface and optical system is noncriticaland can be chosen quite large ~more than 10 cm!.The size of the illuminated surface area is approxi-mately 1 mm 3 1 mm and can easily be kept clean bythe application of pressurized air.Finally, a versatile low-cost measuring device can
be developed by use of the above-mentioned methodsfor polychromatic speckle pattern generation andevaluation.
References1. T. V. Vorburger and E. C. Teague, “Optical techniques for
on-line measurement of surface topography,” Precis. Eng. 3,61–83 ~1981!.
2. J. M. Bennett and L. Mattsson, Introduction to Surface Rough-ness and Scattering ~Optical Society of America, Washington,D.C., 1989!, pp. 24–31.
3. R. Brodmann and G. Thurn, “Roughness measurement ofground, turned and shot-peened surfaces by the light scatter-ing method,” Wear 109, 1–13 ~1986!.
4. U. Persson, “Roughness measurement by means of the speckletechnique in the visible and infrared regions,” Opt. Eng. 32,3327–3332 ~1993!.
5. L. Cuthbert and V. M. Huynh, “Statistical analysis of Fouriertransform patterns for surface texture assessment,” Meas. Sci.Technol. 3, 740–745 ~1992!.
6. J. C. Dainty, “Introduction,” in Laser Speckle and Related Phe-nomena, J. C. Dainty, ed., Vol. 9 of Springer Series in Topics inApplied Physics ~Springer-Verlag, Berlin, 1975!, Chap. 1, pp.1–7.
7. G. Parry, “Some effects of surface roughness on the appearenceof speckle in polychromatic light,” Opt. Commun. 12, 75–78~1974!.
8. G. Bitz, Verfahren zur Bestimmung von Rauheitskenngrossendurch Specklekorrelation ~Fortschritt-Berichte VDI, VDI-Verlag, Dusseldorf, 1982!, Series 8, Number 47, Chap. 4, 5, pp.25–59.
10. P. Lehmann, Untersuchungen zur Lichtstreuung an technis-chen Oberflachen im Hinblick auf eine prozessgekoppelte laser-optische Rauheitsmessung ~Fortschritt-Berichte VDI, VDI-Verlag, Dusseldorf, 1995!, Series 8, Number 463, Chap. 7, pp.103–108.
11. C. T. Stansberg, “Surface roughness measurements by meansof polychromatic speckle patterns,” Appl. Opt. 18, 4051–4060~1979!.
12. Ref. 10, Chap. 8, pp. 130–148.13. J. A. Ogilvy, Theory of Wave Scattering from Random Rough
Surfaces ~Hilger, Bristol, 1991!, Chap. 4, pp. 100–117.14. Ref. 13, Chap. 4, pp. 74–85.15. P. Beckmann and A. Spizzichino, The Scattering of Electro-
magnetic Waves from Rough Surfaces ~Pergamon, Oxford1963!, Chap. 3, pp. 15–33.
16. J. W. Goodman, Introduction to Fourier Optics ~McGraw-Hill,New York, 1968!, Chap. 5, pp. 77–89.
17. Ref. 10, Chap. 2, pp. 8–31.18. A. Papoulis, Probability, Random Variables and Stochastic
Processes ~McGraw-Hill, New York, 1965!, Chap. 8, pp. 263–264.
19. Ref. 18, Chap. 8, pp. 266–267.20. J. W. Goodman, “Statistical properties of laser speckle,” in
Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol.9 of Springer Series in Topics in Applied Physics ~Springer-Verlag, Berlin, 1975!, Chap. 2, pp. 20–21.
21. J. W. Goodman, Statistical Optics ~Wiley, New York, 1985!,Chap. 2, pp. 40–44.
22. Ref. 18, Chap. 8, p. 254.23. Ref. 15, Chap. 5, p. 81.24. Ref. 13, Chap. 2, pp. 12–17.25. Ref. 8, Chap. 4, pp. 48–50.26. Ref. 10, Chap. 6, pp. 96–97.27. Ref. 10, Chap. 7, pp. 108–123.28. J. Peters, P. Lehmann, and A. Schone, “Specklekorrelation mit
einem dichromatischen Fouriertransformationssystem,” inLaser in der Technik, W. Waidelich, ed. ~Springer-Verlag, Ber-lin, 1994!, pp. 184–189.
29. Q. B. Li and F. P. Chiang, “Three-dimensional dimension oflaser speckle,” Appl. Optics 31, 6287–6291 ~1992!.
30. Y. Tomita, K. Nakagawa, and T. Asakura, “Fibrous radialstructure of speckle patterns in polychromatic light,” Appl.Opt. 19, 3211–3218 ~1980!.
31. H.M. Pedersen, “Second order statistics of light diffracted fromGaussian, rough surfaces with applications to the roughnessdependence of speckles,” Opt. Acta 22, 523–535 ~1975!.
32. Ref. 16, Chap. 4, pp. 59–61.33. Ref. 10, Chap. 2, pp. 28–29.34. J. Peklenik, “Neue statistische Verfahren zur topographischen
Erfassung von Oberflachen. 2. Teil,” WT Z. Ind. Fertigung 59,633–637 ~1969!.
1 April 1997 y Vol. 36, No. 10 y APPLIED OPTICS 2197
