droplet profiles obtained from the intensity distribution of refraction patterns

$: Droplet profiles obtained from the intensity distribution of refraction patterns$
Droplet profiles obtained from theintensity distribution of refraction patterns

Luis P. Thomas, Roberto Gratton, Beatriz M. Marino, and Javier A. Diez

5840 APPLIED OPTICS @

A noninterferometric method for obtaining profiles of axially symmetric transparent liquid droplets isdescribed. The drops are illuminated along the symmetry axis by a uniform parallel beam whoseintensity distribution is recorded at the focal plane of a lens placed behind the drop. In some conditionsand within the geometrical optics approach, it is possible to reconstruct the profile of the drop from thisintensity distribution except for the length scale factor, which, if necessary, may be provided by anadditional simple measurement. Because of CCD cameras and digital image processing, this method isan interesting alternative technique for measuring drop profile shapes with considerable accuracy wheninterferometry is unwieldy. We also analyze the diffraction features of the intensity distribution toclarify the extent that they affect the approach that we used and to establish additional information thatthey may provide.

1. Introduction

Optical techniques are widely used in research ontransparent liquid droplets supported by plane sub-strates for measuring thickness profiles h1r2 and someother parameters that characterize the drops.1–8 Ingeneral, interferometricmethods are employed,1,2,6,9–12but they are often unwieldy because of the complexityof the optical systems and the need for a high degreeof mechanical stability. However, although interestis on determining the entire profile, interferometry isfully adequate for studying thin small drops 1h < 0.1mm2, but for larger drops observing the completefringe pattern requires special procedures.12 As aresult, some different optical techniques have beendeveloped and are frequently used.13,14 An interest-ing possibility is the determination of drop param-eters from the angular pattern, i.e., the patternformed on a far screen or on a screen placed at thefocal plane of a lens, by an incident parallel beam1usually an expanded laser beam2 impinging on thesubstrate from below and refracted by the drop.Angular patterns have been used15mainly formeasur-ing the maximum slope 1≠h@≠r2m, which is related tothe radius Rm of the circular pattern and to the

The authors are with the Instituto de Fisica Arroyo Seco,Facultad de Ciencias Exactas, Universidad Nacional del Centro dela Provincia de BuenosAires, Pinto 399, 7000 Tandil, Argentina.Received 31 October 1994; r1evised manuscript received 1 March

1995.0003-6935@95@255840-09$06.00@0.

r 1995 Optical Society of America.

Vol. 34, No. 25 @ 1 September 1995

refractive index h of the liquid. However, it has beenshown5 that these patterns may provide much moreinformation.This research is an attempt to clarify what may be

actually obtained from angular patterns in view of thefact that they are easily and currently recorded instudies on the spreading of drops. In particular, weanalyze a technique that allows the reconstruction ofdrop profiles from the intensity I1R2 of angular pat-terns 1an inverse problem2 by making use of digitalimage processing. Within the geometric optics ap-proach and in some conditions that prevent rays fromoverlapping, the solution of the above inverse problemis elementary and nonambiguous, except for a lengthscale factor. In this way, it is possible to obtain froma single image nondimensional thickness profiles formany cases of interest, such as spreading sessiledrops and static drops on completely and partially wetsubstrates,7 respectively. As we see below, the ap-proach does not involve a special averaging procedureto cancel the diffraction fringes that modulate thepattern. The method has been tested by comparingits results with profiles obtained by a standard inter-ferometric technique and proved to be fairly good,provided I1R2 is determined from an average over theentire azimuthal angle.The method is especially convenient only if the

shape of profiles 1or nondimensional profiles2 is neces-sary to characterize the flow, but not their lengthscale. For example, a relevant parameter is theso-called shape factor,1,2,7,8 G 5 V@horf 2, where V isthe drop volume, ho 5 h102 is the thickness at the

center, and rf is the radius of the drop front. ClearlyG is invariant under scale transformations of both hoand rf, so that it can be obtained directly from theangular pattern. However, if necessary, the actualsize of the drop can be obtained from a single addi-tional measurement, for example, the drop radius at agiven time or the drop volume.The angular pattern corresponding to typical drops

1rf $ 1 mm2 shows only few sharply defined fringes inthe peripheral region, whereas in the remaining largecentral region the fringes are so dense and poorlymodulated that they become virtually unobservablein normal conditions. Therefore the use of the geo-metric optics approach appears in a natural way.However, as we discuss in Section 3, we find that thisapproximation provides good results even if the fringesare well defined throughout the entire pattern with-out the need of a specific averaging process. In thissection we report on numerical experiments consist-ing of the calculation of I1R2 for a given h1r2 1analyticalor experimental2 with the scalar diffraction theory, toour knowledge never used before in this context.Next, we employ the geometric approach to the so-obtained angular patterns, thus recovering the origi-nal profile with good accuracy even in cases where theangular pattern is formed by few sharply definedfringes. Therefore the use of the geometric opticsapproach for solving the inverse problem is justified.An important result from this analysis is that therelative intensity and modulation of the peripheralfringes are sensitive to the shape of drops near thefront, but not to their spacing, which depends verylittle on h1r2.In Subsection 2.A we describe the method, and

then, by using the experimental arrangement inSubsection 2.B, we simultaneously obtain angularpatterns and interferograms of sessile drops to testthe reliability of the method. The results are pre-sented in Subsection 2.C. Section 3 is devoted to thecalculation of angular patterns produced by somedrop profiles by use of the scalar diffraction theory.Finally, Section 4 contains the conclusions.

2. Geometric Optics Approximation: A Method forDetermining the Drop Thickness Profile

A. Basis of the Method

Let us consider a transparent drop on a horizontalsubstrate illuminated by a vertical parallel monochro-matic light beam of uniform intensity, as shown inFig. 1. We shall deal with the intensity distributionof the pattern formed on screen S placed at the focalplane of lens L with focal distance f.For an actual drop formed by a liquid that does not

completely wet the substrate, i.e., the equilibriumcontact angle is between 0 and p@2, the equilibriumconfiguration exhibits a slope 0≠h@≠r 0 that increasesmonotonically from zero at r 5 0 to a maximum at thefront position rf.7 For a drop spreading on a wethorizontal surface, the absolute value of the profileslope increases1,2,5,7 monotonically until it remainspractically constant in a small peripheral region and

defines what is called the dynamic contact angle.7Therefore, even for the latter case, we can assumethat the absolute value of the slope increases mono-tonically from r 5 0 to r 5 rf. This fact assures that,except for a small central spot produced by rays thatdo not pass through the drop, the light intensityinside a given circle on the screen is due solely to thecontribution of all the rays from inside a correspond-ing circle on the substrate; i.e., there is no overlappingof rays. Note, however, that hanging drops7 or dropsin which the equilibrium contact angle is greater thanp@2 do not satisfy this condition, so the methoddescribed here cannot be applied in these cases.Let us suppose that the slope is small everywhere

so as to make negligible the intensity variations of thetransmitted wave from Fresnel’s law. We will notconsider the rays passing outside the drop, which areall focused in a single small spot. Because theaxisymmetric intensity distribution I1R2 on the screenmust satisfy energy conservation, we have

e0

r

I0r8dr8 5 x e0

R

I1R82R8dR8,

where r and R are radial coordinates on the substrateand the screen, respectively. Here x is a coefficientthat accounts for losses caused by reflection at theoptical interfaces and the absorption has been ne-glected. By assuming that the incident light inten-sity I0 is uniform, it follows that

r 5 32x

I0 e0

R

I1R82R8dR841@2

5 F1R2

and R may be correlated with r through the inversefunction of F, that is, R 5 F211r2. Because F1R2 isrelated to the integral of I1R2, F1R2 is a smootherfunction than I1R2. Note that, as we see below, thequantity directly obtained from the experimental

Fig. 1. Optical system and definition of the parameters.

1 September 1995 @ Vol. 34, No. 25 @ APPLIED OPTICS 5841

patterns simply by adding the light intensity of thepixels contained within the radiusR is proportional toF21R2.On the other hand, as Snell’s law is applied to the

liquid-free surface, we obtain

≠h

≠r<

R

1h 2 12 f5

F211r2

1h 2 12 f

for small angles.Because h1r2 is supposed to be a monotonic function

of r, by integrating the last equation, we mightcalculate the thickness profile provided that x isknown and the intensity on the screen is given interms of Io. This would require careful calibrations,so that in practice F21R2 is determined except for aconstant factor. As a result, the method gives onlythe nondimensional profile, i.e., h@ho as a function ofr@rf, but neither the actual length scale rf nor ho.Note that the thickness scale and the radius scale arerelated through a known factor because slopes areprovided in absolute units. If necessary, the scalemay be obtained from an additional independentmeasurement, for example, the drop radius at a giventime or the drop volume.As an example, let us consider a drop with a

parabolic thickness profile 1a shape factor G 5 p@22,because this is the case for very small drops 1v & e3,where e is the capillary length2 when the Laplacepressure is the dominant force.1,2,7 Optically thedrop is equivalent to a plano–convex lens. In thiscase and within the geometric optics approach, theangular pattern is a circle of uniform intensity whoseradius is given by Rm 5 1h 2 12 f 1dh@dr2m, where h isthe refraction index of the liquid and 1dh@dr2m is themaximum value of the derivative of the drop profile.Therefore F21R2 is a linear function and the thicknessprofile can be retrieved by an elementary analyticalintegration. Instead, for drops with rf . a, gravitybecomes important and the profile is modified7,8 by aflattening of the central region of the drop 1G increases2.This circumstance is visualized in the angular pat-tern by an increase in I1R2 toward the center.

B. Experimental Arrangement

In principle, the use of sources other than lasers ispossible in this method, but they should be almostmonochromatic to avoid chromatic effects. Besides,the size of the source image on the screen must besmaller than the size of a pixel of the system used forthe acquisition of the intensity distribution, a condi-tion that could be satisfied at the price of a strongreduction in the total intensity. Therefore it is conve-nient to use a laser as a source in spite of theappearance of speckles and fringes resulting fromdiffraction. The effect of the fringes does not invali-date the method and is analyzed in detail in Section 3.Speckles are overcome because F21R2 is the totalintensity over a given circle of radius R.We have carried out drop spreadings for comparing

thickness profiles as measured by a known interfero-

5842 APPLIED OPTICS @ Vol. 34, No. 25 @ 1 September 1995

metric technique with those resulting from the abovemethod. Interferometric measurements were per-formed with a Mach–Zehnder arrangement9 with a15-mW He–Ne laser as the light source 3see Fig. 21a24.In a natural way the arrangement also provides asecond light output 3see Fig. 21b24, which is used toform an angular pattern on a second screen, where itis recorded and digitized by a Sony CCD camera and aframe grabber 1512 3 512 pixels, Imaging TechnologyCorporation2. The reference beam and the portion ofthe beam that passes out of the drop converge at thefocus of lens 2 on screen 2 3Fig. 21b24. This straightfor-ward light is reduced because of a diaphragm placednear the focal point of the drop 1which in a firstapproximation works as a lens2 and by stopping thereference beam when the angular pattern was re-corded.A first point is the uniformity of the incident light

intensity Io throughout the drop. The intensity dis-tribution measured with our experimental arrange-ment 12 m between the laser and the beam expander2is virtually independent of the azimuth, and theintensity averaged over 2p rad shows small variationsfor rf # 0.7 cm, which may be safely neglected. Theintroduction of a known nonuniform distribution inthe theory is in principle possible, so that the unifor-mity of the incident beam is not a strictly necessary

Fig. 2. Experimental setup: 1a2 Mach–Zehnder interferometerwith a drop in one arm. Lens 1 forms the drop image on Screen 1using interferometer Output 1. 1b2 Angular pattern obtained byemploying interferometer Output 2. Note that the parallel lightis stopped by a black screen and a diaphragm.

condition for the present technique. However, werestrict ourselves to this simplest case here.

C. Practical Performance of the Method

Once the pattern is digitally recorded, we add thelight intensities Ii, j in arbitrary units correspondingto the pixels 1i, j2 contained in circles of radii R. Inthis way we obtain a magnitude that is proportionalto F21R2:

oRi, j,R

s1Ii, jDi, j ;1

s0I0 e0

R

s1I1R822pR8dR8

5 C2F21R2 5 C2r2.

Here Ri, j and Di, j are the radius and the area of thepixel i, j, respectively, s0 and s1 are the ratiosbetween the measured intensity and the absoluteintensity I, and C is a constant to be determined.The constant C can be calculated, for example, from

knowledge of the drop radius rf corresponding to agiven angular pattern. In Fig. 3 we plot C2F2 as afunction of R; R 5 Rm is shown by the long-dashedline. Within the geometric optics approximation anideal parabolic profile would result in a straight linefor F21R2, but this is not the case for actual dropprofiles. In the case corresponding to the figure,≠F21R2@≠R decreases when R increases, thus indicat-ing that the average intensity is less for high slopes.Moreover small changes appear in the derivative nearRm from the modulation related to the diffractionfringes. The value of C2F21R2 should be a constantfor R . Rm if the background intensity on screen 2were negligible. Even though some low backgroundintensity is unavoidable, the change in the increasingtendency of C2F21R2 is marked enough to determinewithout ambiguity the value 1maximum2 correspond-ing to rf.In Fig. 41a2 we show a nondimensional profile

obtained with the interferometric technique and withthe current method. The discrepancy between bothshape factors G 5 V@horf 2 is very low 10.4%2. Theconstant C 1i.e., the scale2 may be determined by

Fig. 3. Function C2F2 obtained by numerical integration of pixelintensities. Small changes in the curve derivative for R , Rm aredue to the presence of fringes.

measuring rf with a simple 1noninterferometric2complementary method, for example, by shadowgra-phy. In Fig. 41b2 we show the thickness profile ob-tained by introducing this parameter. The goodagreement near the front is lost at the center wherethe maximum thickness ho is 4% below the interfero-metric experimental data. The error in ho is due tothe profile being built up by integration of the slopestarting from the front radius, so that small differ-ences in the slope near the front can introduceconsiderable discrepancies in ho. Note that bothintegrations virtually cancel the fringe effects. Thecorresponding modulations near the average valuesnear the front 1see Fig. 32 slightly affect ≠h@≠r, butthey are not noticeable in the profile 1Fig. 52.The method loses accuracy when the local average

intensity Ia1R2 undergoes a strong variation betweenthe center and the periphery of the angular pattern.In this case much care must be taken to avoidsaturation of the CCD camera near the center, thusimplying a loss of accuracy in the determination ofF21R2. This situation arises for extended drops 1rf ismuch larger than the capillary distance2, whose pro-files 1from gravity effects2 are markedly flattened inthe central region with respect to a parabolic profile.In Fig. 6 we show three intensity distributions of theangular pattern corresponding to one of the extended

Fig. 4. 1a2 Nondimensional thickness profile obtained by thegeometric method 1curve2 compared with the interferometric profile1circles2. 1b2 Dimensional profile 1curve2 obtained by an additionalmeasurement of the front radius.


drops for different apertures of the CCD camera.In this case, as the average intensity varies by 1 orderof magnitude between R 5 0 and R 5 Rm, it is difficultto appreciate details both near the front and near theaxis with the same aperture. The correspondingprofiles as given by the current method are shown bythe filled symbols in Fig. 7. The best agreement withthe interferometric profile 1the open circles2 is achievedfor the minimum diaphragm aperture, i.e., minimumsaturation. The errors in the shape factors 3see Fig.71a24 are 8%, 7%, and 3.5%. The errors in h0 are muchhigher when the scale is introduced, as we can see inFig. 71b2.

3. Drop Diffraction Pattern

We consider the angular pattern as a diffractionpattern in the Fraunhofer approximation and calcu-late the intensity distribution I1R2 for a given profileh1r2 according to the scalar diffraction theory.Several authors have considered the diffraction pat-tern generated by small-size fluid configurations.Da Costa and Calatroni16 analyzed the far-field pat-

Fig. 5. Enlarged view of the profile of Fig. 41a2 and its derivativenear the front. The variations of ≠h@≠r associated with the outerfringes are visible. In this case a maximum modulation of 5% for≠h@≠r does not produce apparent effects on the profile h1r2.Dashed curve, enlarged view of the profile in Fig. 41a2; continuouscurve, its derivative near the front.

Fig. 6. Experimental angular pattern of an extended drop1ho 5 5.956 3 1023 cm, xf 5 0.7223 cm2 for three diaphragm aper-tures of the CCD camera.


tern caused by reflection from the surface of a holeformed on an initially horizontal surface of heavy oilheated by a vertical laser beam. They showed thatthis pattern can be regarded as an interference pat-tern produced by a wave coming from an annularvirtual source near the hole contour and a sphericalwave proceeding from the principal focal point of thehole. A straightforward adaptation of this model todrop patterns shows fringe positions in good agree-ment with experiments except for the outer fringes.On the other hand, Tanner3–5 studied oil drops on aflat vertical surface flowing down under gravity andestimated oil viscosity by means of thickness profilesgiven by interferometric measurements. He calcu-lated the optical path difference between equallydeviated rays arising from the positions on both sidesof the inflection point of h1r2. This calculationmethod can be used in our case for obtaining theseparation of the outer fringes. Both interferencemodels may be understood if we consider that thediffraction pattern of a hole in a screen can beassumed, in a first approximation, as formed by thesuperposition of the original wave and a wave comingfrom the hole periphery.17,18 This approach allowsfor the calculation of fringe positions but does notshow intensity distributions as the scalar diffraction

Fig. 7. 1a2 Nondimensional profiles calculated by the method1solid circles, triangles, and squares2 compared with the interfero-metric data 1open circles2. 1b2 Dimensional profiles for the samecases.

theory does. We see below that intensity distribu-tions provide important information about the drops.We use the well-known scalar diffraction theory18

for linearly polarized waves, so that both the electric-and the magnetic-field strengths can be representedby a complex scalar function U. Assuming a thinphase transformation, i.e., a transformation in whichray translation through the object is negligible, thefield distribution Ud immediately behind the drop isrelated to the constant incident field magnitude U0by18Ud 5 t0U0, with

t0 5 exp32ikh1x, y2

h 2 1 4 .In this equation k 5 2p@l, where l is the wavelengthof light and h1x, y2 is the thickness at the position 1x, y2on the substrate. By considering an axisymmetricdrop centered on the optical axis of the lens, we maywrite the field at screen S as the Fourier transform13

of t01r2:

U1R2 5U0

ikf e0`

t01r2P1r, D, f, d2J01krRf 2rdr,where P and D are the pupil function and the diam-eter of the lens, respectively; d is the distance be-tween the drop and the lens 1see Fig. 12; and Jo is thezero-order Bessel function. In this equation a phasefactor preceding the integral and depending on d hasbeen omitted, because it does not affect the lightintensity 1I , U22. The function to be integrated isstrongly oscillatory, and a suitablemethod for integra-tion has to be chosen.The Fourier transform for the case of an ideal drop

with a parabolic thickness profile, optically identicalto a plano–convex convergent lens, is useful for test-ing the reliability of the calculations 1see Fig. 82. In afirst approximation for rf : l each edge portion of thedrop produces a diffraction pattern like the oneproduced by a half-plane 1see, for example, Ref. 17,Chap. 10, p. 4102, in which I1Rm2 5 Ia1R 5 02@4,

Fig. 8. Angular pattern calculated by diffraction theory for a dropwith a parabolic thickness profile 1ho 5 5.76 3 1023 cm, rf 5 0.7356cm2.

where Ia1R2 is the local average intensity. In the lastcase, when R decreases from Rm the intensity reachesthe maximum value Im < 1.35Ia, and successivemaxima of less magnitude appear, whereas for R .Rm the intensity decreasesmonotonically. These fea-tures can be seen in Fig. 8, showing that the calcula-tion is quite accurate near Rm, but numerical oscilla-tions affect the precision of the results near thecenter. As expected a parabolic profile leads to aconstant average intensity. However, the principalmaximum is not exactly at R 5 Rm, but it is slightlydisplaced toward the center.In Fig. 9 we show the intensity distribution 1in

arbitrary units2 of the experimental angular patternof a spreading drop averaged over 2p rad 1circles2, thesame distribution averaged only in the first quadrant1dashed curve with dots2 and finally the distributioncalculated 1solid curve2 for the corresponding interfero-metric thickness profile 1as given in Fig. 42. Asexpected, limiting the averaging process over a singlequadrant leads to more pronounced maxima thanaveraging over the four quadrants. Then, even ifthe last process is more suitable for drop profilecalculations, the first one is preferable for comparingintensity distributions in the peripheral region withthe interest on the fringe configuration. Figure 9shows good quantitative agreement for positions andrelative amplitudes of the maxima, especially in thezone near the maximum radius Rm. We also observethat, according to the geometric approach, the aver-age intensity increases when R decreases, differingfrom the case of a parabolic profile. 1In the actualdrop under consideration both gravity and surfacetension are important.2 The relative intensity of theoutermost maximum with respect to the followingmaxima is greater than for the case of a parabolicprofile, and also the position of the outermost maxi-mum is closer to Rm than in the latter case. Thesefeatures are related to the presence of a relativelylarge peripheral region of almost constant slope andmust be taken into account when angular patterns

Fig. 9. Intensities of an experimental angular pattern averagedover four quadrants 1circles2 and over the first quadrant 1linesquare2 compared with the corresponding angular pattern 1continu-ous curve2 calculated from the interferometric profile of an actualdrop 1ho 5 5.76 3 1023 cm, rf 5 0.2664 cm2.


are used to determine apparent contact angles.15Finally, we point out that the separation between agiven maximum and the outermost fringe seems tofollow an N2@3 law,5 where N is the fringe numbercounted from the periphery, and that this dependenceis almost insensitive to the profile shape 1see Fig. 102.So the spacing of the fringes is of little utility for thedetermination of specific features of the drop periph-ery.The scalar diffraction theory allows for a numerical

test of the method for very thin drops in which theangular pattern shows few sharply defined fringes; inthis case the interferometric techniques produce largeerrors. For example, let us consider a parabolic dropwith ho 5 0.0005 cm. The angular pattern calcu-lated for a realistic choice of rf is shown in Fig. 11.Only three principal maxima and some secondarymaxima appear due to diffraction of the aperture,separated by a distance of the order of fl@rf. Now weemploy the method described in Section 2 to thispattern without any averaging process. Because of

Fig. 10. Separation between the Nth fringe and the outermostfringe 1N 5 02 corresponding to the cases analyzed in this research:circles, parabolic profile; squares, drop of Fig. 4; diamonds, ex-tended drop of Fig. 6. The lines correspond to theN2@3 law.

Fig. 11. Angular pattern of a thin drop 1ho 5 5 3 1024 cm, rf 5 0.1cm2 calculated by diffraction scalar theory. The three principalmaxima are due to the phase shift introduced by the drop, whereasthe secondary maxima are due to diffraction of the aperture 1Airy’spattern2.


the integration process, the calculated slope is onlyslightly modulated near the right average value asseen in Fig. 12, whereas the corresponding thicknessprofile does not show appreciable modulations andagrees well with the assumed profile. The shapefactor obtained with the method differs by only 0.75%with respect to the assumed one for the drop, adiscrepancy of the same order as that found for theactual drop in Fig. 4. From this example we candeduce that the method does not need a specialaveraging process to find shape factors with errors ofless than a fraction of 1%.Note that the intensity distribution in the angular

pattern is able by itself to put in evidence theexistence of an inflection point in the thicknessprofile, i.e., the presence of overlapping rays. Withthis aim we compare the intensity distribution forthree typical profiles with the same ho and 1dh@dr2m to

Fig. 13. Three similar profiles with slight differences near thefront for ho 5 6 3 1023 cm, 1dh@dr2i 5 0.06, and rf 5 0.1cm. Profile a is parabolic for 0 # r # rf, profile b is parabolicbetween the center and an inflection point at ri 5 0.95rf, whereit is connected with another parabola of opposite curvature anddh@dr 5 0 at r 5 rf; profile c is parabolic between r 5 0 and ri 5

0.9rf, where it is continued by a straight line.

Fig. 12. Nondimensional profile and its slope calculated by thegeometric method 1continuous curves2 compared with the theoreti-cal corresponding curves 1dashed curves2 for the drop of Fig.11. Note that the existing differences between the lines for theslopes virtually disappear in the profile comparison.

obtain equal numbers of fringes and equal patternsizes 1see Fig. 132. Profile a is parabolic for 0 # r #

rf; profile b starts with a parabola from the center toan inflection point at ri 5 0.95rf, where it is continuedby another parabola with opposite curvature leadingto dh@dr 5 0 at r 5 rf; and profile c is parabolic fromr 5 0 to ri 5 0.90rf 1the maximum slope2 where it iscontinued by a straight line 1a wedgelike peripheralregion2. In Fig. 141a2 we show the angular patternsfor cases a and b calculated by means of the scalardiffraction theory. The positions of the fringes arealmost the same, but a much stronger modulationoccurs for the profile with the inflection point result-ing from the interference between equally deviatedrays coming from both sides of that point. On theother hand, in the angular pattern corresponding toprofile c 3see Fig. 141b24 there is a shift of the fringepositions toward the periphery, but the fringe spacingis nearly the same as in the angular pattern corre-sponding to the parabolic profile. Note that in case cthe outermost maxima are stronger too. In conclu-sion the angular pattern corresponding to the casewith an inflection point 1i.e., overlapping ray2 showsclear distinctive features.

Fig. 14. Angular patterns calculated for the analytic profiles a, b,and c of Fig. 13 1the continuous, dashed, and dashed–dot curves,respectively2.

4. Conclusions

The angular pattern of spreading drops was studied.A method based on the geometric optics approachallowed the calculation of the drop profiles from theintensity distribution of the pattern, which werecompared with those determined by interferometry.Because the frame grabbers and CCD cameras are nottoo expensive nowadays, the method may be used asan alternative of interferometry to measure dropprofiles 1if the volume or the drop radius is given2.A study based on the scalar diffraction theory showedthat the geometric optics approach can be used with-out averaging processes on the intensity distribution,although the patterns may be modulated by sharplydefined fringes especially in the peripheral region.A numerical experiment showed that the methodshows a profile with good accuracy even for an ex-treme case in which the 1ideal2 drop is so thin that theangular pattern is formed by few fringes.The above-mentioned diffraction features can be

related to details of the drop profile near the front.We showed that the position of the outermost fringegenerally shifts a little with respect to the radius thatcorresponds to the maximum deviation angle accord-ing to the geometrical approximation. This fact mayintroduce some systematic errors when the contactangle is measured by means of diffraction patterns.15An interesting point is that the spacing of the fringesnear the front shows very little sensitivity to smallchanges in the drop profile, so that it cannot berelated directly to drop properties near the front.Instead the relative intensities of the maxima giveinformation about drop features even by simple visualinspection; for example, in drops presenting a periph-eral region with an almost constant slope 1a wedgelikeprofile2, the intensity of the outermost maximum isenhanced with respect to that of parabolic drops.The method could be generalized to determine the

thickness profile of surfaces in a rather wide variety offlows, for example, the thickness h1r2 along the axis ofa linear channel. The main limitations are that inthe region studied h1r2 should not contain inflectionpoints and that the slope must be small everywhere.A practical limit for a refraction index of ,1.5 is thatthe slope be less than 30°.

L. P. Thomas and R. Gratton were supported by theConsejo Nacional de Investigaciones Cientificas yTecnicas 1CONICET2 as members of its research staff.J. A. Diez is a fellow of CONICET. The research waspartially supported by grants from CONICET andComision de Investigaciones Cientıficas de la Provin-cia de BuenosAires.

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droplet profiles obtained from the intensity distribution of refraction patterns

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