Characterization of the scatteringproperties of a mirror by speckle-field statistics
Vincenzo Greco, Giuseppe Molesini, and Franco Quercioli
The statistics of partially developed speckle patterns produced by reflection under coherent illuminationcan be used to determine the scattering properties of a mirror. The theory of the approach is presented,showing that in proper conditions the standard deviation of the speckle-field distribution expresses thefinish grade of the reflecting surface. Interferometric measurements are described, and experiments onselected samples are reported.
The total reflectance of a mirror is generally analyzedin terms of a specular component plus a diffusecontribution. The diffuse contribution is due tolight scattering from a surface microstructure andlocal defects such as dust praticles, scratches, stain-ing, pits, and digs. The measurement of the diffuselight strength against the specular reflectance ac-counts for the actual condition of the reflectingsurface, including features of the optical polishingand degradation induced by age and use.
At some distance from the mirror the diffuse lightand the specular component are superimposed. Incoherent illumination, because of the statistics char-acter of the diffuse contribution being propagated,the result is a partially developed speckle field. Sucha field can be accurately mapped by means of phase-shift interferometry. 1' 2 Based on this standard tech-nique, the statistics properties of speckle fields havebeen verified experimentally. 3 4 As a special applica-tion, in this paper a proper version of the interferomet-ric approach is described, showing the statistics pa-rameters that are significant for the strength of thediffuse light against the specular reflectance. Re-sults of experiments on a series of selected samplesare also presented.
Roughness-measuring methods based on differentworking principles are thoroughly reviewed in the
The authors are with the Instituto Nazionale di Ottica, Largo E.Fermi 6, Firenze 50125, Italy.
Received 6 May 1992; revised manuscript received 29 September1993.
literature. 5 Noninterferometric techniques in par-ticular are available that are used in research ondiffuse light such as total integrated scattering6 andangle-resolved scattering. 7 Interferometric tech-niques at 10.6 .Lm, operating on the contrast of theinterference fringes,8 are also described. Still basedon diffuse light, the interferometric technique is hereused in the visible, and the scattering properties ofoptical surfaces are given in terms of speckle-fieldstatistics.
Theory
In discussing the speckle approach, we refer to theoptical geometry in Fig. 1. A plane wave is cast on aweakly scattering plane mirror through a beam split-ter. The reflected light is processed by an imagingsystem made of two lenses in a 4-f configuration.According to classical treatments,9 -1 2 the optical dis-turbance at the image plane is written in complexform:
a(x, y) = a(x, y)exp[ jO(x, y)], (1)
where a is the modulus, 0 is the phase, and j is theimaginary unit. Instead of using a and 0, it iscustomary to decompose a in its real and imaginaryparts r and i:
r = a cos 0, i = a sin 0.
In the present case a(x, y) is made of a strong meanphasor s resulting from the specular wave plus arandom contribution caused by the diffuse light.The random contribution qualifies for a speckle field;the nonzero mean phasor s makes it only partiallydeveloped. Without loss of generality such a meanphasor can be taken along the real axis of the complex
with n = 1, 2, 3, 4. Such light buckets are usuallymade available in digitized form as a standard outputof the interferometer. In the present application asimple algorithm is defined, producing the quantitiesU, v:
respectively, focal lengths); A, stop aperture; D, detector array.
plane. It is now assumed that the object correlationarea is much smaller than the resolution area of theimaging system, as is shown to be the case in theconditions of the experiments reported here. Thelocal field is thus made of a high number of statisti-cally independent contributions. By virtue of thecentral limit theorem, the expected field statistics isGaussian.'3 The joint probability density functionp(r, i) takes the general form
1 [(r- S)2 i2 ]p(r, i) = 2 7r exp- (3)[7oa 2
LT, 2 jui
where arr2 , U,2 are the variances of the r, i distribu-tions, respectively. The mean local intensity I = a2
is expressed as
I = s2[1 + + ()] (4)
Light-scattering losses result in departures fromunity of the quantity in square brackets. As ameasurement of the strength of the diffuse lightagainst the specular reflectance, r/S and ui/s aredetermined in experiments. Using an interferomet-ric approach, we map a(x, y) by means of a referencefield, b = b exp(j4), also injected in the optical path(Fig. 1). The intensity I(x, y) then becomes
I(x, y) = a2 + b2 + 2ab cos(0 - j). (5)
The measuring technique referred to here is phase-shift interferometry," 2 according to which the phase+ = +(t) of the reference field is linearly shifted withtime. Typically the detector is made of a charge-injection-device camera, so that the elements of thedetector array sample the interference pattern I(x, y)at regularly spaced locations. In a 21r phase shifteach element is read at four equal time intervals,producing the output signals
I(x,y)d4 = (a2+ b2)2
As long as b is uniform over the image plane and thedetector array is operated in the linear region of itsresponse computations can be carried out on the (u, v)data set in place of the (r, i) ensemble. The datamanipulation includes several steps 4:
(a) Converting the (u, v) data to amplitude andphase by computing (U2 + 2)'/2 and tan-'(v/u).
(b) Connecting the phase map, solving for ambigu-ities modulo 2 rr.
(c) Computing and removing the tilt on the phasemap.
(d) Applying a high-pass filter both to the amplitudeand to the phase maps to remove low-order aberra-tions and intensity spatial variations of the laserbeam.
(e) Normalizing the amplitude and subtracting thephase of a prestored reference to reduce the effect ofthe spatial noise caused by system optics.
(f) Computing (r, i) after the amplitude and thephase according to Eqs. (2).
(g) Reducing the (r, i) data about the point (1, 0) bycomputing the mean values r, i rotating the axes bythe angle tan-'(i/r), and dividing the amplitude ofeach data point by (r2 + i2)1/2
(h) Generating a two-dimensional histogram of (r, i)distribution. In the current case, 30 x 30 squarechannels were considered.
(i) Fitting the histogram to a two-dimensionalnoncircular Gaussian by standard chi-square meth-ods.'5 The procedure provides the best values for s,ur, and ri.
(j) Computing the correlation coefficient ri/(arcri) tocheck the speckle-field decorrelation.1 2 Valuessmaller than 0.1 (absolute value) were accepted here.
(k) Computing the end values (Yr/s and i/s.Normalization to s provides freedom from arbitraryunits used in light detection, signal digitization, anddata manipulation. Such end values compare withunity according to Eq. (4).
One performs the high-pass filtering operationmentioned in step (d) by defining at each point of thedata map a square mask centered about it. The datapoints inside the mask define a local mean planewave. The phase belonging to the central point isthen subtracted. A similar operation is carried outon the amplitude: one finds the linear two-dimen-sional component of the spatial distribution andnormalizes the amplitude of the central point to it.To make sure that the structure of the specklepattern is not significantly modified by this operation,
the side of the mask is chosen to be at least 1 order ofmagnitude wider than the correlation length of thespeckle field. This rule has been taken as a criterion,after one has carried out a number of attempts withdifferent masks and computed the end correlationlength of the speckle. It was found in particularthat, starting with masks whose side was 41 pixels,the correlation length remained reasonably stable at
4 pixels when the mask side was further increased.In this study the mask used for filtering had a61-pixel side. The correlation length was taken asthe half-width mean radius of the autocorrelationfunction, computed over 128 x 128 data points byFourier transform, square modulus, and inverse Fou-rier transform.
In the laboratory a standard interferometer foroptical testing applications has been used. Its de-sign includes apertures that stop spurious reflectionsand in fact prevent the high-spatial-frequency spec-trum from reaching the detector. This low-pass-filtering operation is represented by aperture A (diam-eter d) in Fig. 1. In the present application d affectsthe number N of correlation areas of the surface thatcontribute to the intensity observed at a given imagepoint. If it is assumed that the correlation functionof the surface heights is Gaussian with correlationlength r,, N is given as'3
/rrdr, -2N = I Cl = (rrr p ) 2 (8)
2XfA/8
where f, is the focal length of lens L, in Fig. 1, X is thewavelength, and Pc = d/(2Xfl) is the spatial-frequencycutoff of the interferometric processor. For the ac-tual instrument in use it is d = 1.5 mm, f, = 500 mm,X = 632.8 nm; correlation lengths as long as r, = 40,um then allow for N 2 10, reasonably accounting forthe appeal to the central limit theorem.
In terms of scattered-light throughput, because ofthe small value of d available, the configuration ofFig. 1 performs poorly. The measurements reportedhere refer instead to the modified optical setup in Fig.2, where a spherical mirror of radius g is tested andthe first half of the interferometric processor matchesthe curvature of the mirror with a system lens Lo offocal length fo. The spatial-frequency cutoff Pc isthus scaled times the magnification m = fo/g of lensLo, allowing for a more effective throughput of thescattered light. Besides, using thin-lens formulas,
0
R
Fig. 2. Interferometric arrangement for testing convex surfacesin reflection. Lo, system lens (focal length fo); S, sample mirror(radius g).
lens Lo images the sample surface at a front dis-tance 1:
fo(fo - g) (9)g
When the axial separation of lenses Lo and L isneglected, the setup in Fig. 2 is equivalent to that ofFig. 1 in the condition = f. In such a geometry,however, the resulting N is critically reduced m2
times.Experiments refer to fo = 355 mm and g = 8.25
mm, giving m = 40.6 and = 13268 mm > fl. As aresult the detector plane is considerably out of focus.Although this circumstance allows for a high numberof uncorrelated contributions to make up the localspeckle field, the noncircularity of the Gaussian statis-tics predicted in the literature about the image plane13
may not show up.
Experiments and Results
With the aim of testing the interferometric approachto surface scattering, a series of samples of differentfinishes has been studied. The sample surfaces be-long to biconvex lenses made of BK7 optical glass:One surface (radius g = 8.25 mm) is prepared andinspected, and the other is thoroughly polished.Samples are taken from an optical workshop, wheresuch lenses are produced in high quantity with quitea traceable polishing process. In particular a seriesof 12 samples has been studied with a process thattakes 3 min to go from finely ground surface (a 10-imgrit size) to an optical finish. Samples are obtainedby interruption of the pitch polishing process atdifferent times, as shown in detail in Table 1.Typical surfaces are shown in Fig. 3 as they appearwith a Nomarski interference contrast microscopewith a 16 x objective. A finely polished sample sur-face is also used to generate the prestored referencementioned in step (e) of the data-processing sequence.
The data acquisition is performed with a commer-cial digital phase-shift interferometer. The configu-ration of Fig. 2 is set up with a system lens Lo ofrelative aperture f/3.3, so that the sample area usedis 4.9 mm2. Typical interference patterns with tiltfringes are shown in Fig. 4.
Table 1. Statistics Data of the Sample Surfaces Inspected
Fig. 3. Nomarski micrographs of optical surfaces polished differ-ently. The area represented is 200 pLm x 100 plm: (a) sample S1,(b) sample S4, (c) sample S7.
The charge-injection-device detector array samplesthe interference pattern at 288 x 210 equally spacedlocations, providing a reasonable amount of datapoints to statistics. A zoom position is chosen sothat the correlation length of the speckle field at thedetector plane extends over a proper number ofphotodetectors. In the measurements reported heresuch a correlation length is of the order of 4 pixels, sothat a single speckle grain is sampled at 50 loca-tions, whereas the number of correlation areas in aframe is of the order of 1200.
The interferometer is used only to acquire four288 x 210 light data sets, 8 bits each, according to Eq.(6). The data sets are then transferred to an exter-nal workstation for processing. The (r, i) distribu-tion of the speckle field for a typical sample (S1) isshown in Fig. 5. Each dot represents the (r, i)components obtained from a single photodetector ofthe array after application of the data-processingsequence to step (g). The cloud is centered about thenormalization point (1, 0) and appears quite circular.The separate fitting of r and i histograms to Gauss-ians of equal standard deviation is presented in Fig. 6.
Sample S1 still corresponds to a rough surface. As
(b)
(C)
Fig. 4. Interference fringes of partially developed speckle fieldsproduced by reflection from surfaces that received a differentoptical polish: (a) sample S1, (b) sample S4, (c) sample S7.
1.0
0.5
i n n
-0.5 -
- .0 - _0.0
. . . .0.5 1 1.5 2.0
Fig. 5. Sampling of a typical speckle field (sample SI) in its realpart r and imaginary part i.
Fig. 6. Frequency histograms (dots) of the speckle field presentedin Fig. 5: (a) real part, (b) imaginary part. Continuous curves:the fit to Gaussians of the same standard deviation.
the polishing progresses, the data cloud shrinks andshould ideally collapse at the point (1, 0) for a perfectsurface. In practice, owing to various noise sources,even with sample S12 some spread is observed (Fig.7).
The results of the measurements are summarizedin Table 1. The uncertainty on the nominal valuesindicates the average departure from the mean over10 independent measurements. The standard devia-tions, UJr, oi, do not differ significantly and can beaccounted for altogether by their mean (Fig. 8). Thegeneral trend as a function of polishing time can berelated to a process that progressively goes to comple-
1.0
0.5
I 0.0
-0.5 -
..0 .5 1. 0 1. 5 2. 0-i.ng oftespcl fl b 812.
Sampling of the speckle field produced by S 12.
0.25
z0
i 0.20 -
0a 0.15 -
0
4 0.10I-v)
0.05
0.00 _20 40 60 80 1o 120 1 o 160 160 200
POLISHING TIME (sec.)
Fig. 8. Standard deviation (mean of rr/s, Jils) of the speckle fieldversus polishing time as shown in detail in Table 1.
tion and then, within the sensitivity of the measure-ments, reaches a stationary condition.
To clear up whether the tail of Fig. 8 indicates theconvergence of the polishing process to a well-polished surface or residual noise of the interferomet-ric technique, the finely polished sample used as areference has been repeatedly inspected, still takingthe prestored data into account. Values of 0.020 0.002 for the standard deviations were obtained, closeenough to the values of sample S12, so that the tailcan be attributed to both concurring causes. Fur-ther insight into the final polishing behaviors can beexpected from the availability of phase-shift interfer-ometers of the last generation for which 10-bit digiti-zation is used.
Conclusions
An interferometric technique has been presentedthat gives information on the scattering properties ofoptical surfaces. The optical setup is based on aphase-shift interferometer routinely used as an opti-cal testing facility. The measuring configurationrefers to a spherical sample surface, magnified by asystem lens in front of a 4-f processor to increase thespatial-frequency cutoff. Experiments are reportedin out-of-focus conditions of the magnified surface.
The finish grade of the surface is given in terms ofthe standard deviation of the speckle field observed.While the absolute value of the result depends on themagnification in use, the technique can be applied tomonitor the optical polishing process or to evaluatethe actual finish of a given surface compared with afinely polished sample of the same radius of curvature.The technique has been demonstrated on a series ofsamples prepared on purpose and proves effectiveuntil the scattered light reduces to the noise level ofthe interferometer.
The authors are grateful to Alberto Righini forvaluable discussions and comments throughout thecourse of this research. This research has beenpartially supported by the National Research Councilof Italy under the Progetto Finalizzato on Electroop-tical Technologies.
References1. J. H. Bruning, D. R. Herriott, J. E. Gallager, D. P. Rosenfeld,
A. D. White, and D. J. Brangaccio, "Digital wave-front measur-ing interferometer for testing optical surfaces and lenses,"Appl. Opt. 13, 2693-2703 (1974).
2. K. Creath, "Phase-measurement interferometry techniques,"in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam,1988), Vol. 26, pp. 349-393.
3. G. Molesini, M. V. Pires de Souza, F. Quercioli, and M. Trivi,"Digital phase-shifting interferometry applied to partiallydeveloped speckle fields," Opt. Commun. 75, 14-17 (1990).
4. G. Molesini, M. V. Pires de Souza, F. Quercioli, and M. Trivi,"Experimental statistics of partially developed speckle fields,"Opt. Lett. 15, 646-648 (1990).
5. J. M. Bennett and L. Mattsson, Introduction to SurfaceRoughness and Scattering (Optical Society of America, Wash-ington, D.C., 1989).
6. H. E. Bennett and J. 0. Porteus, "Relation between surfaceroughness and specular reflectance at normal incidence," J.Opt. Soc. Am. 51, 123-129 (1961).
7. J. M. Elson and J. M. Bennett, "Vector scattering theory,"Opt. Eng. 18, 116-124 (1979).
8. 0. Kwon, J. C. Wyant, and C. R. Hayslett, "Rough surfaceinterferometry at 10.6 jim," Appl. Opt. 19, 1862-1869 (1980).
9. J. C. Dainty, ed., Laser Speckle and Related Phenomena(Springer-Verlag, Berlin, 1975).
10. R. K. Erf, ed., Speckle Metrology (Academic, New York, 1978).11. M. Frangon, Laser Speckle and Applications in Optics (Aca-
demic, New York, 1979).12. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
13. J. W. Goodman, "Statistical properties of laser speckle pat-terns," in Laser Speckle and Related Phenomena, J. C. Dainty,ed. (Springer-Verlag, Berlin, 1975), pp. 9-75.
14. V. Greco, "Evaluation of optical surface finish by speckletechniques," M. Sc. thesis in physics (University of Florence,Florence, Italy, 1991).
15. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T.Vetterling, Numerical Recipes (Cambridge U. Press, Cam-bridge, 1986), Chap. 14, pp. 498-546.
