- Email: [email protected]
Experimental characterization of a random metallic rough surface by spectrophotometric measurements in the visible range
15 June 2000
Optics Communications 180 Ž2000. 191–198 www.elsevier.comrlocateroptcom
Experimental characterization of a random metallic rough surface by spectrophotometric measurements in the visible range Mady Elias a
a,b,)
, Michel Menu
a,b
Laboratoire de Recherche des Musees ´ de France, CRRMF, LRMF, UMR 171-CNRS, 6, Rue des Pyramides, 75041 Paris, Cedex 01, France b UniÕersite´ d’EÕry Val d’Essonne, 91025 EÕry Cedex, France Received 15 October 1999; received in revised form 13 April 2000; accepted 13 April 2000
Abstract Back-reflected light from a random rough surface is studied with the purpose of characterising the surface statistical properties. A new goniospectrophotometer with optic fibres is presented and a random rough copper surface is analysed. The probability density of the surface normal and the hrl ratio Žr.m.s. roughnessrautocorrelation length. are so obtained. Optical results are compared with mechanical ones, performed with a profilometer. For the first time, Abel’s transform method is used to connect the two-dimensional statistical information deduced from optical measurements and the one-dimensional profiles. q 2000 Elsevier Science B.V. All rights reserved. PACS: 07.60.Dq; 07.60.Hv; 07.60.Rd; 42.25.Dd; 68.35.Bs; 78.40.Kc; 78.66.Bz; 81.05.Bx; 81.70.Fy Keywords: Rough surface; Spectrophotometry
1. Introduction Among the various devices applied to increase our understanding of artists’ techniques, the Laboratory of Le Louvre Museum ŽLRMF. has recently acquired a new kind of goniospectrophotometer, to measure colours of patrimonial items in the visible range. The measurements are non-destructive, without any contact, and implemented in situ, if needed. Generally, colour depends not only on the nature of the bulk Ždiffusion of the incident light in the volume of the material., but also on the state of its surface Žreflection., which is often difficult to measure directly with a non-destructive test. Here, we suggest ) Corresponding author. Tel.: q33-01-40-20-84-41; fax: q3301-47-03-32-46; e-mail: [email protected]
that goniospectrophotometric measurements allow information on the state of the surface to be determined. For the sake of simplicity, and so as not to mix volume and surface effects, we choose a metallic random surface. In a first experiment, a random, rough copper surface is illuminated by plane waves and the back-reflected light is analysed. In a second experiment, the surface of the same sample is analysed by a profilometer. The connection between these two kinds of measurements is the theory of reflected waves by random surfaces, which we apply here to back-reflection. The theory allows the prediction of the statistical properties normal to the surface, involving the twodimensional density probability of the surface derivatives from the back-reflected light. On the other
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 7 1 4 - 8
192
M. Elias, M. Menu r Optics Communications 180 (2000) 191–198
hand, the profilometer gives statistical information in only one dimension. To compare both results, an Abel transform is used to deduce the required statistical properties from unidirectional profiles. The validity of the surface property predictions using optical measurements can be tested by mechanical means. First, the new goniospectrophotometer is quickly described. The theory of reflected light by random metallic surfaces is reviewed and applied to back-reflection. Optical and mechanical measurements performed on a copper sample are presented. Finally, the Abel transform is used to link the two kinds of measurements and to validate the optical method. 2. Experimental device A detailed description of the spectrophotometer may be found in w1x. A photograph of the equipment is presented in Fig. 1. Incident light is supplied by a 100 W regulated halogen lamp with quartz optic. A set of quartz optical fibres Ž1. illuminates a part of the sample. Another set of quartz optical fibres Ž2., randomly mixed with the first set, collects the backscattered light. The two sets, containing more than 2 = 100 individual fibres, are bifurcated in a Y-shape to allow the use of the same optic. Optical conjugation Ž3. is performed for both types of light by two coated cemented achromatic doublets Ž f f 10 mm.. Black velvet is fixed on the interior surfaces of the optic holder to avoid intrusive reflections. Finally, the reflected light illuminates a 200 mm slit and the light is dispersed by a reflecting grating, 600 groovesrmm Ž4.. The visible spectrum of the re-
flected light is then sampled via a CCD array, with 1200 linear photodiodes, which allows 500 kHz data acquisitions. Eventually, a goniometer Ž5. allows to change and measures the back-reflection angle with an accuracy of 0.58. The maximum beam angular aperture on the studied surface is 3.58 and the illuminated surface is 0.3 cm2 . The distance between sample and device is between 1 and 1.5 cm when the light is focused. In these conditions, the incident intensity can vary from 100 to 1000 lumenrm2 . Absolute and relative luminances are measured with an accuracy of 5%. Wavelength accuracy is 3 nm. Only the visible range is explored because we couple the measurements presented here with colour measurements which need white light Žilluminant D65..
3. Theoretical model for back-reflection We present here a brief review of the reflection of light by a perfectly conducting random rough surface, and we apply it to the case of back-reflection. 3.1. Random surface characterisation Let a random surface S ŽFig. 2a. around the plane z s 0 be described by z s g Ž x, y . with mean value ² g : s 0 Žwhere ²: means statistical average.. The ™ normal N to the surface has polar angles u N , w N such that:
°E g s ytgu
~ EE xg
¢E y
Fig. 1. The goniospectrophotometer. 1, incident light; 2, reflected light; 3, optical conjugation; 4, reflecting grating; 5, goniometer.
N
cos w N
Ž 1. s ytgu N sin w N
The surface, fully characterised by its multidimensional probability function, is assumed to be homogeneous. In the following, only the surface statistics to second order are needed. The surface r.m.s. height is h s Ž² g 2 :.1r2 . The correlation function of the surface C Ž j ,h . s ² g Ž x q j , y q h . g Ž x, y .: can be expanded as C Ž j , h . f h 2 Ž 1 y Ž j 2 r2 l x2 . y Žh 2r2 l y2 .. in the vicinity of j s h s 0, where l x and l y are the correlation lengths in the x and y direc-
M. Elias, M. Menu r Optics Communications 180 (2000) 191–198
193
ever, in the visible range, this case is seldom encountered in practice and will not be developed here. The second approximation is physical optics w4–6x, also called Kirchhoff’s approximation, valid when the radius of curvature of the surface is much larger than the wavelength. The scale of the radius of curvature being l 2rh, physical optics is valid when l h < l 2 . This is not the converse of the SPM validity condition, and physical optics may be applied to smooth surfaces Ž h < l., as well as to rough surfaces Ž h 4 l.. In this approximation, the total magnetic ™ field ™on the surface is twice the incident field Hr Ž0. s 2 Hi Ž0. on its illuminated part and zero outside. 3.3. Physical optics applied to back-reflected field Fig. 2. Notations. Ža. random surface characterisation definition of ™ the local normal N to the surface; Žb. reflection of light by a random surface definition of the back-reflected direction ™ u.
Without any assumption, the reflected field is given by Stratton–Chu integral w7x:
™
tions. If the surface is isotropic, then l x s l y s l. We will assume that l is small compared to the size of the surface. Finally, the relationship between angle u N and the correlation length is: ² tg 2u N : s 2
h
ž / l
2
.
Ž 2.
3.2. Prediction methods for the reflected field The surface S is illuminated a monochromatic ™ ™ by™™ incident plane wave Hi s Ho eyi k r.u i , ™ coming from ™. is the direction ™ u i with polar angles u , w i . Hr Ž Ru reflected magnetic field at a distance R in the direction ™ u with polar angles u , w ŽFig. 2b.. In the ™i . Numerical methods, back-reflection case ™ u s yu Ž such as BEM Boundary Element Method. w2x, are available to compute the reflected field without any assumption. However, such numerical methods cannot be implemented easily to give the relationship between the reflected field and the surface statistical properties. Fortunately, two kinds of approximation allow us to put this relationship in a closed form. The first approximation is the Small Perturbation Method ŽSPM. w3x, valid when h < l; that is, when the surface is smooth at the wavelength scale. How-
™
™
™. s jS Ž ™r . n =™r Gds, Hr Ž Ru ™
HS
™ ™
Ž™. where jS Ž™ r.sNn S r is the surface current den™ H 5 Ry™ r5 eyi k™ sity and G s 4p 5 R y ™r 5 the Green’s function. ™™ yi k R In the far field Ž R 4 r ., G f e4p R e i k r.u and ™Žin™r .the ™ ™ Ž . approximation of physical optics, H r s 2 H s S i ™ ™™ 2 Ho eyi k r.u i . We assume here that the whole surface is illuminated by the incident field and we have neglected the shading effect. In other words, we consider only incident directions ŽFig. 2b. close to the normal to the surface ŽFig. 2a. such that u - ² u N2 : . The reflected field becomes:
(
™
™. s Hr Ž Ru
™
ik 2p R
eyi k R Ho
ik y
2p R
eyi k R
™
™i . i k ™™ r Ž uyu
™
™. ™ i k™™ r Ž uyu ds.
HS Ž N.u™. e
ds
HS ž H .u™/ Ne o
i
™i , so for plane In the back-reflection case, ™ u s yu ™ ™ ™ ™ waves, H0 .u s H0 .u i s 0, the second integral is equal to zero, and: ™
™. s Hr Ž Ru
ik 2p R
™
eyi k R Ho
™
HS Ž N.u™. e
2 i k ™™ r .u
ds.
Ž 3.
M. Elias, M. Menu r Optics Communications 180 (2000) 191–198
194
With the notations of Fig. 2b and if S o is the projection of S on the plane z s 0, Ž3. can be written:
™ ™. s Hr Ž Ru
ik 2p R
™ eyi k R H
o
Ks
™
™ N.u
HS
0
Nz
Ž 4.
The intensity is proportional to ™ reflected ™.< 2 :; that is, ²< Hr Ž Ru to < H0 < 2 . We notice that in the back-reflection case and in the physical optics approximation for a perfectly conductive surface, the mean reflected light intensity does not depend on its possible polarisation.
3.4. Back-reflected intensity for a Gaussian surface If no assumption is made on ™ the relative values of ™.< 2 :, using Ž4., h and l, the computation of ²< Hr Ž Ru is complex and involves the Fourier transform of the two point probability density function of the random surface. A model is required for this function, which is often taken to be Gaussian. Unfortunately, even with such a model, the final result does not relate in a simple manner the reflected light and the surface properties. Only the two limits h < l and h 4 l remain tractable. In the visible range, we are only interested in the rough surface case h 4 l. If the surface is Gaussian and isotropic, the result is:
™
™
S0 2
™
™ N.u
ik 2p
HS
0
Nz
= ey2 i k sin u Ž x cos wqy sin w . ey2 i k g Ž x , y.cos u dxdy
= ey2 i k sin u Ž x cos w qy sin w . ey2 i k g Ž x , y .cos u dxdy.
™. < 2 : s < Ho < 2 ²< Hr Ž Ru
Consider the integral:
l2 2 2
l2
4
R 2p h cos u
2
ey 2 h 2 t g u .
Ž 5.
3.5. Back-reflected intensity for any rough surface There is another way to deal with Ž4. which gets rid of the Gaussian hypothesis. It consists in considering the rough limit h 4 l before performing the statistical average, and in using the method of stationary phase w8–11x.
and the function: G Ž x , y . s sin u Ž x cos w q y sin w . q g Ž x , y . cos u .
™
For k `, stationary points are those defined by Ž E GrE x . s Ž E GrE y . s 0. It appears then that the stationary points are the specular points on the™surwhere u N s u and w N s w ; that is, where N and ™face u are co-linear. Around one of these points ™ ri , a rotation of the coordinate axis allows us to write the above function as a function of the Gaussian radius of curvature: G Ž X ,Y . s™™ u.ri q cos u
ž
X2 2 R 12i
Y2 q
2 R 22 i
/
,
so that we get after integration K s Si Ž ey2 i kr cos 2 u . R 1i R 2 , where the summation is extended to all specular points. Performing the statistical average under the assumption of uncorrelated stationary points, the result is ²
(
™ S0 < H 0 < 2 ™ 2 ™ ²< Hr Ž Ru . < : s R
2
PN Ž u , w . cos u
.
Ž 6.
It appears that the reflected intensity does not depend on the wave length.
M. Elias, M. Menu r Optics Communications 180 (2000) 191–198
195
It is interesting to apply Ž6. to a Gaussian random isotropic surface. In this case, the density probability of the surface derivatives is also Gaussian. With:
Eg
2
Eg
2
¦ž / ; ¦ž / ; Ex
s
Ey
l2 s 2p h
h2
s l
2
exp y 2
, we get PD l2
2p h 2
ž
ž
Eg Eg , Ex Ey
E g2 Ex
E g2 q
Ey
/
/
and: PN Ž u N , w N . s
l2 2p h 2 cos 3 u N
ey
l2 2 h2
tg 2u N
relation Ž5. is then found again. However, we insist on the generality of the relation Ž6., which is not restricted to a Gaussian surface. The relevant characteristic of the surface is not the probability density of the surface itself, but the probability density of the normals to the surface. In conclusion, it appears that a close relationship between the mean back-reflected field intensity and the surface statistics exists when the surface is rough at the wave length scale Ž h 4 l., and when its curvature is small Ž l 2rh 4 l..
4. Experimental results on a copper sample As a simple test for the theoretical predictions, we choose a metallic random surface. It consists of a copper disk of about 1 cm2 , first polished then randomly scored with abrasive paper 220 Žthe diameter of the grains is about 60 mm.. Optical measurements with our spectrophotometer have been performed first, then with a profilometer Žwhich scratches the surface of the sample., in order to test the results. 4.1. Optical measurements In the spectrophotometer measurements, the irra™ diance L s ²< Hr < 2 : is measured for angles u ranging from y408 to q408 and for three different wavelengths. The maximum L m a x appears at u s 08. Plotting L1Ž u . s LŽ u .rL m a x for these three wavelengths, Fig. 3 shows that they collapse on a single
Fig. 3. Copper sample: relative irradiance versus back-reflection angle u .
curve. So L1Ž u . does not depend on the wavelength, in agreement with the theory. To go further and compute the probability density function of the polar angles of the normal to the surface, relation Ž6. must be adapted. First, the intensity of the incident light is not calibrated. Also, for a fixed illuminating device, the illuminated surface S0 is a function of angle u and increases as 1rcos u . Finally, for an isotropic surface, PN Ž u , w . is a function only of u . We have therefore PN Ž u . A cos 2 u L1Ž u .. The proportionality constant is then found using the normalisation H0p r2 PN Ž u . du s 1. Here, this normalisation is straightforward, because L1 is zero outside the measured angle range Ž08; um a x .. PN Ž u . deduced this way from the spectrophotometer measurements: PN Ž u . s
cos 2 u L1 Ž u . umax
H0
Ž 7.
cos 2 x L1 Ž x . dx
is presented in Fig. 4. We are now able, for the first test, to compute the r.m.s. value of u N , and we obtain ur .m .s.s 0.16 rd s 9.088. For small values of hrl, Ž tgu . r .m .s.( ur .m .s. and we deduce, using Ž2., hrl ( u r . m . s . s 0.11.
'2 4.2. Mechanical measurements The surface statistics is classically measured with a profilometer. Five profiles were performed parallel
196
M. Elias, M. Menu r Optics Communications 180 (2000) 191–198
To find a relationship between these two functions, we introduce the two-dimensional probability function Pd Ž g x , g y . of the surface derivatives along x and y. Following Ž1., the relation between Pd and PN is: Pd
Fig. 4. Copper sample: density probability surface slopes PN versus back reflection angle u .
to the same direction and five other profiles in the perpendicular direction. Each profile is 4.8 mm long with 1664 equispaced sampled points ŽFig. 5.. All the ten profiles happen to be homogeneous with respect to their r.m.s. height and their r.m.s. derivative values. The random surface may therefore be considered to be homogeneous and isotropic. The measured parameters on the whole set of profiles are respectively h s 1.18 mm, and hrl s 0.12, from which l s 9.46 mm follows. Our hypothesis are verified. In the visible range Ž0.4 mm–0.8 mm., h 4 l, so the surface is therefore very rough. Moreover, the radius of curvature is of order l 2rh f 76 mm, so that relation Ž4., valid for physical optics in the rough limit, applies. Finally, we obtained the histogram of the profiles derivatives along its axis, say Ox, to get an approximation of the probability density function Pd x Ž gx . of g x s Ž E grE x ..
ž (g q g / dg dg s P Ž u 2 x
2 y
x
y
N
N
. du N
Ž 8.
with dg x dg y s Žsin u N rcos 3 u N . du N . It remains then to find a relationship between the two-dimensional probability function Pd and the one-dimensional Pd x . We have first Pd x Ž g x . s HPd Ž g x , g y . dg y , and because the surface is assumed isotropic, we get also Pd Ž g x , g y . s Pd Ž g x2 q g y2 .. It appears that the relation between the one- and the two-dimensional probability functions is an Abel transform, namely:
(
`
Pd x Ž g x . s
Hy` P ž (g d
2 2 x qgy
/ dg . y
Ž 9.
In conclusion, Ž8. and Ž9. allow the comparison of the optical and mechanical measurements. Two ways are available: either compute PN from Pd x and make the comparison on PN , or compute Pd x from PN and make the comparison on Pd x . The first way requires an inverse Abel transform, the second a direct one. Because it is known that the inverse Abel transform is very sensitive to any noise or inaccuracy in the function to be processed, we present here the second type of comparison.
4.3. Comparison between optical and mechanical measurements The first simple comparison between the two sets of data is the value of the ratio hrl. The agreement is fairly good: Ž hrl .o p t.s 0.11 and Ž hrl . m ech s 0.12. However the full test consists in comparing the probability density functions obtained from both optical and mechanical measurements. This is not an easy task, because we measured optically PN Ž u N ., and mechanically Pd x Ž g x ..
Fig. 5. Copper sample: one of the profiles of the rough random surface, 1664 equispaced sampled points on 4.8 mm total length.
M. Elias, M. Menu r Optics Communications 180 (2000) 191–198
To compute more easily the Abel transform, PN Ž u . is first fitted using an analytical function. At first sight in Fig. 4, a simple Gaussian function seems appropriate, but adding a polynomial of second degree greatly improves the fit at large angles. 2 We get therefore PN Ž u . s A 0 eyu r A 1 q A 2 q A 3 u 2 . Then Pd is computed according to Ž8. and finally the Abel transform Ž9. is performed. Fig. 6 shows the comparison of Pd x Ž g x . deduced from the optical measurement and from the histogram performed on the whole set of the profiles derivatives. In detailed analysis, the curve deduced from the optical measurements Ždotted line. is symmetrical, because of the isotropic hypothesis used for the Abel transform. The histogram Žplain line. deduced from the profiles derivatives, which is only an approximation of the true probability density function, obeys this symmetry only in first approximation. Moreover, the maximum of the histogram does not occur exactly at g x s 0. These deviations from the expected shape may be due to the total number of profilometer sampled points Ž10 = 1664., insufficient to compute very accurately the probability density function, or more probably to some deviation in the surface homogeneity and isotropy. Indeed, it is very difficult, scoring manually a surface, to meet exactly such properties. This is why we think that the comparison, though not perfect, is acceptable. In conclusion, and for metallic isotropic surfaces, our optical and non-destructive measurements allow
197
the measurement of the relevant surface parameter hrl and of the surface normal angle probability function PN in fairly good accordance with the classical but destructive method of profilometry.
5. Conclusion In this paper, we have presented a method to characterise geometrical properties of a reflecting surface through the analysis of visible back-reflected light as a function of the incident angle. In the visible range, most encountered surfaces are rough and we have shown that the relevant characteristics of the surface are the probability density of its normal and the hrl ratio Žr.m.s. roughnessrautocorrelation length.. Experimental results on copper samples have been deduced from optical non-destructive measurements and confirmed by standard but destructive mechanical tests. Compared to similar methods, the originality of present results consists in the ability of our new goniospectrometer to give back-reflected measurements at various angles, in using the Abel transform to obtain the two-dimensions probability density of the normal from one-dimension profiles, and finally in the experimental demonstration of the validity of the theoretical model. Knowing the surface state of a work of art is an important issue for curators. These results will be directly applied to study paintings done with gold, with the purpose of differentiating schools and techniques, or to closely monitor the surface of a work of art which may be exposed to harsh conditions: in a gallery, or during UV photography, for instance. In the future, we plan to extend the theory to the case of rough dielectric surfaces, where diffusion in the bulk will be considered.
Acknowledgements Fig. 6. Copper sample: probability density function Žnormalised X by the maximum value. along the x x axis versus the x-derivative g x of the surface function z s g Ž x, y .. Dotted line: deduced from the Abel transform of optical measurements; solid line: histogram of the derivatives computed from the profile measurements.
Nicole De Dave ŽCMPMF–Laboratory of the Ecole Superieure des Mines de Paris, Evry. realised ´ the profilometer measurements. We want to thank her for her kindness.
198
M. Elias, M. Menu r Optics Communications 180 (2000) 191–198
References w1x A. Chiron, M. Menu, An approach to the spectrocolorimetric characterisation of patrimonial items and works of art, manuscript submitted, 2000. w2x D. Maystre, Rigorous vector theories of diffraction gratings, in: E. Wolf ŽEd.., Progress in Optics XXI, North Holland, Amsterdam, 1984. w3x D. Maystre, O. Mata Mendez, A. Roger, A new electromagnetic theory for scattering from shallow rough surfaces. Opt. Acta 30 Ž1983. 1707–1723. w4x P. Beckmann, A. Spizzichino, The scattering of electromagnetic waves from rough surfaces, Pergamon, New York, 1963.
w5x P. Beckmann, Scattering of light by rough surfaces, in: E. Wolf ŽEd.., Progress in Optics VI, Pergamon, New York, 1963. w6x S.O. Rice, Reflexion of electromagnetic waves from slightly rough surfaces. The Theory of Electromagnetic Waves, Interscience Publishers, 1951. w7x J.A. Stratton, Electromagnetic Theory, Chap. 9, McGraw– Hill, 1941. w8x P. Croce, L. Prod’homme, J. Optics ŽParis. 11 Ž5. Ž1980. 95–104. w9x P. Croce, L. Prod’homme, J. Optics ŽParis. 15 Ž2. Ž1984. 95–104. w10x P. Croce, J. Optics ŽParis. 21 Ž6. Ž1990. 273–286. w11x P. Croce, J. Optics ŽParis. 23 Ž4. Ž1992. 167–174.
Optics Communications 180 Ž2000. 191–198 www.elsevier.comrlocateroptcom
Experimental characterization of a random metallic rough surface by spectrophotometric measurements in the visible range Mady Elias a
a,b,)
, Michel Menu
a,b
Laboratoire de Recherche des Musees ´ de France, CRRMF, LRMF, UMR 171-CNRS, 6, Rue des Pyramides, 75041 Paris, Cedex 01, France b UniÕersite´ d’EÕry Val d’Essonne, 91025 EÕry Cedex, France Received 15 October 1999; received in revised form 13 April 2000; accepted 13 April 2000
Abstract Back-reflected light from a random rough surface is studied with the purpose of characterising the surface statistical properties. A new goniospectrophotometer with optic fibres is presented and a random rough copper surface is analysed. The probability density of the surface normal and the hrl ratio Žr.m.s. roughnessrautocorrelation length. are so obtained. Optical results are compared with mechanical ones, performed with a profilometer. For the first time, Abel’s transform method is used to connect the two-dimensional statistical information deduced from optical measurements and the one-dimensional profiles. q 2000 Elsevier Science B.V. All rights reserved. PACS: 07.60.Dq; 07.60.Hv; 07.60.Rd; 42.25.Dd; 68.35.Bs; 78.40.Kc; 78.66.Bz; 81.05.Bx; 81.70.Fy Keywords: Rough surface; Spectrophotometry
1. Introduction Among the various devices applied to increase our understanding of artists’ techniques, the Laboratory of Le Louvre Museum ŽLRMF. has recently acquired a new kind of goniospectrophotometer, to measure colours of patrimonial items in the visible range. The measurements are non-destructive, without any contact, and implemented in situ, if needed. Generally, colour depends not only on the nature of the bulk Ždiffusion of the incident light in the volume of the material., but also on the state of its surface Žreflection., which is often difficult to measure directly with a non-destructive test. Here, we suggest ) Corresponding author. Tel.: q33-01-40-20-84-41; fax: q3301-47-03-32-46; e-mail: [email protected]
that goniospectrophotometric measurements allow information on the state of the surface to be determined. For the sake of simplicity, and so as not to mix volume and surface effects, we choose a metallic random surface. In a first experiment, a random, rough copper surface is illuminated by plane waves and the back-reflected light is analysed. In a second experiment, the surface of the same sample is analysed by a profilometer. The connection between these two kinds of measurements is the theory of reflected waves by random surfaces, which we apply here to back-reflection. The theory allows the prediction of the statistical properties normal to the surface, involving the twodimensional density probability of the surface derivatives from the back-reflected light. On the other
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 7 1 4 - 8
192
M. Elias, M. Menu r Optics Communications 180 (2000) 191–198
hand, the profilometer gives statistical information in only one dimension. To compare both results, an Abel transform is used to deduce the required statistical properties from unidirectional profiles. The validity of the surface property predictions using optical measurements can be tested by mechanical means. First, the new goniospectrophotometer is quickly described. The theory of reflected light by random metallic surfaces is reviewed and applied to back-reflection. Optical and mechanical measurements performed on a copper sample are presented. Finally, the Abel transform is used to link the two kinds of measurements and to validate the optical method. 2. Experimental device A detailed description of the spectrophotometer may be found in w1x. A photograph of the equipment is presented in Fig. 1. Incident light is supplied by a 100 W regulated halogen lamp with quartz optic. A set of quartz optical fibres Ž1. illuminates a part of the sample. Another set of quartz optical fibres Ž2., randomly mixed with the first set, collects the backscattered light. The two sets, containing more than 2 = 100 individual fibres, are bifurcated in a Y-shape to allow the use of the same optic. Optical conjugation Ž3. is performed for both types of light by two coated cemented achromatic doublets Ž f f 10 mm.. Black velvet is fixed on the interior surfaces of the optic holder to avoid intrusive reflections. Finally, the reflected light illuminates a 200 mm slit and the light is dispersed by a reflecting grating, 600 groovesrmm Ž4.. The visible spectrum of the re-
flected light is then sampled via a CCD array, with 1200 linear photodiodes, which allows 500 kHz data acquisitions. Eventually, a goniometer Ž5. allows to change and measures the back-reflection angle with an accuracy of 0.58. The maximum beam angular aperture on the studied surface is 3.58 and the illuminated surface is 0.3 cm2 . The distance between sample and device is between 1 and 1.5 cm when the light is focused. In these conditions, the incident intensity can vary from 100 to 1000 lumenrm2 . Absolute and relative luminances are measured with an accuracy of 5%. Wavelength accuracy is 3 nm. Only the visible range is explored because we couple the measurements presented here with colour measurements which need white light Žilluminant D65..
3. Theoretical model for back-reflection We present here a brief review of the reflection of light by a perfectly conducting random rough surface, and we apply it to the case of back-reflection. 3.1. Random surface characterisation Let a random surface S ŽFig. 2a. around the plane z s 0 be described by z s g Ž x, y . with mean value ² g : s 0 Žwhere ²: means statistical average.. The ™ normal N to the surface has polar angles u N , w N such that:
°E g s ytgu
~ EE xg
¢E y
Fig. 1. The goniospectrophotometer. 1, incident light; 2, reflected light; 3, optical conjugation; 4, reflecting grating; 5, goniometer.
N
cos w N
Ž 1. s ytgu N sin w N
The surface, fully characterised by its multidimensional probability function, is assumed to be homogeneous. In the following, only the surface statistics to second order are needed. The surface r.m.s. height is h s Ž² g 2 :.1r2 . The correlation function of the surface C Ž j ,h . s ² g Ž x q j , y q h . g Ž x, y .: can be expanded as C Ž j , h . f h 2 Ž 1 y Ž j 2 r2 l x2 . y Žh 2r2 l y2 .. in the vicinity of j s h s 0, where l x and l y are the correlation lengths in the x and y direc-
M. Elias, M. Menu r Optics Communications 180 (2000) 191–198
193
ever, in the visible range, this case is seldom encountered in practice and will not be developed here. The second approximation is physical optics w4–6x, also called Kirchhoff’s approximation, valid when the radius of curvature of the surface is much larger than the wavelength. The scale of the radius of curvature being l 2rh, physical optics is valid when l h < l 2 . This is not the converse of the SPM validity condition, and physical optics may be applied to smooth surfaces Ž h < l., as well as to rough surfaces Ž h 4 l.. In this approximation, the total magnetic ™ field ™on the surface is twice the incident field Hr Ž0. s 2 Hi Ž0. on its illuminated part and zero outside. 3.3. Physical optics applied to back-reflected field Fig. 2. Notations. Ža. random surface characterisation definition of ™ the local normal N to the surface; Žb. reflection of light by a random surface definition of the back-reflected direction ™ u.
Without any assumption, the reflected field is given by Stratton–Chu integral w7x:
™
tions. If the surface is isotropic, then l x s l y s l. We will assume that l is small compared to the size of the surface. Finally, the relationship between angle u N and the correlation length is: ² tg 2u N : s 2
h
ž / l
2
.
Ž 2.
3.2. Prediction methods for the reflected field The surface S is illuminated a monochromatic ™ ™ by™™ incident plane wave Hi s Ho eyi k r.u i , ™ coming from ™. is the direction ™ u i with polar angles u , w i . Hr Ž Ru reflected magnetic field at a distance R in the direction ™ u with polar angles u , w ŽFig. 2b.. In the ™i . Numerical methods, back-reflection case ™ u s yu Ž such as BEM Boundary Element Method. w2x, are available to compute the reflected field without any assumption. However, such numerical methods cannot be implemented easily to give the relationship between the reflected field and the surface statistical properties. Fortunately, two kinds of approximation allow us to put this relationship in a closed form. The first approximation is the Small Perturbation Method ŽSPM. w3x, valid when h < l; that is, when the surface is smooth at the wavelength scale. How-
™
™
™. s jS Ž ™r . n =™r Gds, Hr Ž Ru ™
HS
™ ™
Ž™. where jS Ž™ r.sNn S r is the surface current den™ H 5 Ry™ r5 eyi k™ sity and G s 4p 5 R y ™r 5 the Green’s function. ™™ yi k R In the far field Ž R 4 r ., G f e4p R e i k r.u and ™Žin™r .the ™ ™ Ž . approximation of physical optics, H r s 2 H s S i ™ ™™ 2 Ho eyi k r.u i . We assume here that the whole surface is illuminated by the incident field and we have neglected the shading effect. In other words, we consider only incident directions ŽFig. 2b. close to the normal to the surface ŽFig. 2a. such that u - ² u N2 : . The reflected field becomes:
(
™
™. s Hr Ž Ru
™
ik 2p R
eyi k R Ho
ik y
2p R
eyi k R
™
™i . i k ™™ r Ž uyu
™
™. ™ i k™™ r Ž uyu ds.
HS Ž N.u™. e
ds
HS ž H .u™/ Ne o
i
™i , so for plane In the back-reflection case, ™ u s yu ™ ™ ™ ™ waves, H0 .u s H0 .u i s 0, the second integral is equal to zero, and: ™
™. s Hr Ž Ru
ik 2p R
™
eyi k R Ho
™
HS Ž N.u™. e
2 i k ™™ r .u
ds.
Ž 3.
M. Elias, M. Menu r Optics Communications 180 (2000) 191–198
194
With the notations of Fig. 2b and if S o is the projection of S on the plane z s 0, Ž3. can be written:
™ ™. s Hr Ž Ru
ik 2p R
™ eyi k R H
o
Ks
™
™ N.u
HS
0
Nz
Ž 4.
The intensity is proportional to ™ reflected ™.< 2 :; that is, ²< Hr Ž Ru to < H0 < 2 . We notice that in the back-reflection case and in the physical optics approximation for a perfectly conductive surface, the mean reflected light intensity does not depend on its possible polarisation.
3.4. Back-reflected intensity for a Gaussian surface If no assumption is made on ™ the relative values of ™.< 2 :, using Ž4., h and l, the computation of ²< Hr Ž Ru is complex and involves the Fourier transform of the two point probability density function of the random surface. A model is required for this function, which is often taken to be Gaussian. Unfortunately, even with such a model, the final result does not relate in a simple manner the reflected light and the surface properties. Only the two limits h < l and h 4 l remain tractable. In the visible range, we are only interested in the rough surface case h 4 l. If the surface is Gaussian and isotropic, the result is:
™
™
S0 2
™
™ N.u
ik 2p
HS
0
Nz
= ey2 i k sin u Ž x cos wqy sin w . ey2 i k g Ž x , y.cos u dxdy
= ey2 i k sin u Ž x cos w qy sin w . ey2 i k g Ž x , y .cos u dxdy.
™. < 2 : s < Ho < 2 ²< Hr Ž Ru
Consider the integral:
l2 2 2
l2
4
R 2p h cos u
2
ey 2 h 2 t g u .
Ž 5.
3.5. Back-reflected intensity for any rough surface There is another way to deal with Ž4. which gets rid of the Gaussian hypothesis. It consists in considering the rough limit h 4 l before performing the statistical average, and in using the method of stationary phase w8–11x.
and the function: G Ž x , y . s sin u Ž x cos w q y sin w . q g Ž x , y . cos u .
™
For k `, stationary points are those defined by Ž E GrE x . s Ž E GrE y . s 0. It appears then that the stationary points are the specular points on the™surwhere u N s u and w N s w ; that is, where N and ™face u are co-linear. Around one of these points ™ ri , a rotation of the coordinate axis allows us to write the above function as a function of the Gaussian radius of curvature: G Ž X ,Y . s™™ u.ri q cos u
ž
X2 2 R 12i
Y2 q
2 R 22 i
/
,
so that we get after integration K s Si Ž ey2 i kr cos 2 u . R 1i R 2 , where the summation is extended to all specular points. Performing the statistical average under the assumption of uncorrelated stationary points, the result is ²
(
™ S0 < H 0 < 2 ™ 2 ™ ²< Hr Ž Ru . < : s R
2
PN Ž u , w . cos u
.
Ž 6.
It appears that the reflected intensity does not depend on the wave length.
M. Elias, M. Menu r Optics Communications 180 (2000) 191–198
195
It is interesting to apply Ž6. to a Gaussian random isotropic surface. In this case, the density probability of the surface derivatives is also Gaussian. With:
Eg
2
Eg
2
¦ž / ; ¦ž / ; Ex
s
Ey
l2 s 2p h
h2
s l
2
exp y 2
, we get PD l2
2p h 2
ž
ž
Eg Eg , Ex Ey
E g2 Ex
E g2 q
Ey
/
/
and: PN Ž u N , w N . s
l2 2p h 2 cos 3 u N
ey
l2 2 h2
tg 2u N
relation Ž5. is then found again. However, we insist on the generality of the relation Ž6., which is not restricted to a Gaussian surface. The relevant characteristic of the surface is not the probability density of the surface itself, but the probability density of the normals to the surface. In conclusion, it appears that a close relationship between the mean back-reflected field intensity and the surface statistics exists when the surface is rough at the wave length scale Ž h 4 l., and when its curvature is small Ž l 2rh 4 l..
4. Experimental results on a copper sample As a simple test for the theoretical predictions, we choose a metallic random surface. It consists of a copper disk of about 1 cm2 , first polished then randomly scored with abrasive paper 220 Žthe diameter of the grains is about 60 mm.. Optical measurements with our spectrophotometer have been performed first, then with a profilometer Žwhich scratches the surface of the sample., in order to test the results. 4.1. Optical measurements In the spectrophotometer measurements, the irra™ diance L s ²< Hr < 2 : is measured for angles u ranging from y408 to q408 and for three different wavelengths. The maximum L m a x appears at u s 08. Plotting L1Ž u . s LŽ u .rL m a x for these three wavelengths, Fig. 3 shows that they collapse on a single
Fig. 3. Copper sample: relative irradiance versus back-reflection angle u .
curve. So L1Ž u . does not depend on the wavelength, in agreement with the theory. To go further and compute the probability density function of the polar angles of the normal to the surface, relation Ž6. must be adapted. First, the intensity of the incident light is not calibrated. Also, for a fixed illuminating device, the illuminated surface S0 is a function of angle u and increases as 1rcos u . Finally, for an isotropic surface, PN Ž u , w . is a function only of u . We have therefore PN Ž u . A cos 2 u L1Ž u .. The proportionality constant is then found using the normalisation H0p r2 PN Ž u . du s 1. Here, this normalisation is straightforward, because L1 is zero outside the measured angle range Ž08; um a x .. PN Ž u . deduced this way from the spectrophotometer measurements: PN Ž u . s
cos 2 u L1 Ž u . umax
H0
Ž 7.
cos 2 x L1 Ž x . dx
is presented in Fig. 4. We are now able, for the first test, to compute the r.m.s. value of u N , and we obtain ur .m .s.s 0.16 rd s 9.088. For small values of hrl, Ž tgu . r .m .s.( ur .m .s. and we deduce, using Ž2., hrl ( u r . m . s . s 0.11.
'2 4.2. Mechanical measurements The surface statistics is classically measured with a profilometer. Five profiles were performed parallel
196
M. Elias, M. Menu r Optics Communications 180 (2000) 191–198
To find a relationship between these two functions, we introduce the two-dimensional probability function Pd Ž g x , g y . of the surface derivatives along x and y. Following Ž1., the relation between Pd and PN is: Pd
Fig. 4. Copper sample: density probability surface slopes PN versus back reflection angle u .
to the same direction and five other profiles in the perpendicular direction. Each profile is 4.8 mm long with 1664 equispaced sampled points ŽFig. 5.. All the ten profiles happen to be homogeneous with respect to their r.m.s. height and their r.m.s. derivative values. The random surface may therefore be considered to be homogeneous and isotropic. The measured parameters on the whole set of profiles are respectively h s 1.18 mm, and hrl s 0.12, from which l s 9.46 mm follows. Our hypothesis are verified. In the visible range Ž0.4 mm–0.8 mm., h 4 l, so the surface is therefore very rough. Moreover, the radius of curvature is of order l 2rh f 76 mm, so that relation Ž4., valid for physical optics in the rough limit, applies. Finally, we obtained the histogram of the profiles derivatives along its axis, say Ox, to get an approximation of the probability density function Pd x Ž gx . of g x s Ž E grE x ..
ž (g q g / dg dg s P Ž u 2 x
2 y
x
y
N
N
. du N
Ž 8.
with dg x dg y s Žsin u N rcos 3 u N . du N . It remains then to find a relationship between the two-dimensional probability function Pd and the one-dimensional Pd x . We have first Pd x Ž g x . s HPd Ž g x , g y . dg y , and because the surface is assumed isotropic, we get also Pd Ž g x , g y . s Pd Ž g x2 q g y2 .. It appears that the relation between the one- and the two-dimensional probability functions is an Abel transform, namely:
(
`
Pd x Ž g x . s
Hy` P ž (g d
2 2 x qgy
/ dg . y
Ž 9.
In conclusion, Ž8. and Ž9. allow the comparison of the optical and mechanical measurements. Two ways are available: either compute PN from Pd x and make the comparison on PN , or compute Pd x from PN and make the comparison on Pd x . The first way requires an inverse Abel transform, the second a direct one. Because it is known that the inverse Abel transform is very sensitive to any noise or inaccuracy in the function to be processed, we present here the second type of comparison.
4.3. Comparison between optical and mechanical measurements The first simple comparison between the two sets of data is the value of the ratio hrl. The agreement is fairly good: Ž hrl .o p t.s 0.11 and Ž hrl . m ech s 0.12. However the full test consists in comparing the probability density functions obtained from both optical and mechanical measurements. This is not an easy task, because we measured optically PN Ž u N ., and mechanically Pd x Ž g x ..
Fig. 5. Copper sample: one of the profiles of the rough random surface, 1664 equispaced sampled points on 4.8 mm total length.
M. Elias, M. Menu r Optics Communications 180 (2000) 191–198
To compute more easily the Abel transform, PN Ž u . is first fitted using an analytical function. At first sight in Fig. 4, a simple Gaussian function seems appropriate, but adding a polynomial of second degree greatly improves the fit at large angles. 2 We get therefore PN Ž u . s A 0 eyu r A 1 q A 2 q A 3 u 2 . Then Pd is computed according to Ž8. and finally the Abel transform Ž9. is performed. Fig. 6 shows the comparison of Pd x Ž g x . deduced from the optical measurement and from the histogram performed on the whole set of the profiles derivatives. In detailed analysis, the curve deduced from the optical measurements Ždotted line. is symmetrical, because of the isotropic hypothesis used for the Abel transform. The histogram Žplain line. deduced from the profiles derivatives, which is only an approximation of the true probability density function, obeys this symmetry only in first approximation. Moreover, the maximum of the histogram does not occur exactly at g x s 0. These deviations from the expected shape may be due to the total number of profilometer sampled points Ž10 = 1664., insufficient to compute very accurately the probability density function, or more probably to some deviation in the surface homogeneity and isotropy. Indeed, it is very difficult, scoring manually a surface, to meet exactly such properties. This is why we think that the comparison, though not perfect, is acceptable. In conclusion, and for metallic isotropic surfaces, our optical and non-destructive measurements allow
197
the measurement of the relevant surface parameter hrl and of the surface normal angle probability function PN in fairly good accordance with the classical but destructive method of profilometry.
5. Conclusion In this paper, we have presented a method to characterise geometrical properties of a reflecting surface through the analysis of visible back-reflected light as a function of the incident angle. In the visible range, most encountered surfaces are rough and we have shown that the relevant characteristics of the surface are the probability density of its normal and the hrl ratio Žr.m.s. roughnessrautocorrelation length.. Experimental results on copper samples have been deduced from optical non-destructive measurements and confirmed by standard but destructive mechanical tests. Compared to similar methods, the originality of present results consists in the ability of our new goniospectrometer to give back-reflected measurements at various angles, in using the Abel transform to obtain the two-dimensions probability density of the normal from one-dimension profiles, and finally in the experimental demonstration of the validity of the theoretical model. Knowing the surface state of a work of art is an important issue for curators. These results will be directly applied to study paintings done with gold, with the purpose of differentiating schools and techniques, or to closely monitor the surface of a work of art which may be exposed to harsh conditions: in a gallery, or during UV photography, for instance. In the future, we plan to extend the theory to the case of rough dielectric surfaces, where diffusion in the bulk will be considered.
Acknowledgements Fig. 6. Copper sample: probability density function Žnormalised X by the maximum value. along the x x axis versus the x-derivative g x of the surface function z s g Ž x, y .. Dotted line: deduced from the Abel transform of optical measurements; solid line: histogram of the derivatives computed from the profile measurements.
Nicole De Dave ŽCMPMF–Laboratory of the Ecole Superieure des Mines de Paris, Evry. realised ´ the profilometer measurements. We want to thank her for her kindness.
198
M. Elias, M. Menu r Optics Communications 180 (2000) 191–198
References w1x A. Chiron, M. Menu, An approach to the spectrocolorimetric characterisation of patrimonial items and works of art, manuscript submitted, 2000. w2x D. Maystre, Rigorous vector theories of diffraction gratings, in: E. Wolf ŽEd.., Progress in Optics XXI, North Holland, Amsterdam, 1984. w3x D. Maystre, O. Mata Mendez, A. Roger, A new electromagnetic theory for scattering from shallow rough surfaces. Opt. Acta 30 Ž1983. 1707–1723. w4x P. Beckmann, A. Spizzichino, The scattering of electromagnetic waves from rough surfaces, Pergamon, New York, 1963.
w5x P. Beckmann, Scattering of light by rough surfaces, in: E. Wolf ŽEd.., Progress in Optics VI, Pergamon, New York, 1963. w6x S.O. Rice, Reflexion of electromagnetic waves from slightly rough surfaces. The Theory of Electromagnetic Waves, Interscience Publishers, 1951. w7x J.A. Stratton, Electromagnetic Theory, Chap. 9, McGraw– Hill, 1941. w8x P. Croce, L. Prod’homme, J. Optics ŽParis. 11 Ž5. Ž1980. 95–104. w9x P. Croce, L. Prod’homme, J. Optics ŽParis. 15 Ž2. Ž1984. 95–104. w10x P. Croce, J. Optics ŽParis. 21 Ž6. Ž1990. 273–286. w11x P. Croce, J. Optics ŽParis. 23 Ž4. Ž1992. 167–174.