Assessment of refractive index gradients by standard rainbow thermometry

$: Assessment of refractive index gradients by standard rainbow thermometry$
Assessment of refractive index gradients bystandard rainbow thermometry

Maria Rosaria Vetrano, Jeronimus Petrus Antonius Johannes van Beeck, andMichel Léon Riethmuller

Up to now the application of rainbow thermometry has been limited to particle systems possessing auniform refractive index. This is mostly due to the absence of an appropriate data inversion algorithmthat takes into account the presence of a refractive index gradient. In this paper, for the first time to ourknowledge, exploiting a generalization of the Airy theory, a data inversion algorithm for a single droplet,presenting a parabolic refractive index gradient, is proposed. This data inversion algorithm is used tocompute the diameter and the refractive index at the core and at the surface of a simulated burningdroplet. The results are compared to the analytical solutions showing a satisfactory agreement. © 2005Optical Society of America

OCIS codes: 120.5820, 290.5850, 120.6780.

1. Introduction

Rainbow thermometry is used to measure the meantemperature and the mean size of transparent parti-cles immersed in a transparent environment pos-sessing a lower refractive index. Standard rainbowthermometry (SRT) is used to measure the refractiveindex (hence temperature) and diameter of a singletransparent particle. The extension of the SRT tech-nique to multiple particles, the so-called global rain-bow thermometry (GRT), Ref. 1, provides the meanvalues and the dispersion factors of the refractiveindex and size distributions associated with an en-semble of particles. The refractive index and size de-terminations for both techniques are based on theanalysis of interference fringe patterns generated bythe interaction of monochromatic laser light, linearlypolarized, and the particles themselves. The first ex-ploitations of the rainbow technique were done at thebeginning of the 1990s by Roth et al.2 In the past 15years the rainbow techniques have been applied tospray combustion,3 to the detection of nonsphericaldroplets,4 and to the measurement of the thickness ofthin films in spherical or cylindrical geometry.5–7

The Lorenz–Mie theory, which describes in a rig-orous way the phenomenon of light scattering bysmall particles, has been extended to multilayeredspheres or cylinders8 and to nonspherical parti-cles.9,10

Another theoretical approach frequently used forthe rainbow is based on the Airy theory11 and calcu-lates the electromagnetic scattered wave in the ob-servation point by means of the Huygens–Fresnelprinciple. Recently the Airy theory for the rainbowhas been generalized to nonuniform particles,12 i.e.,particles presenting an internal profile of a refractiveindex.

The goal of this paper is to show new developmentsin the assessment of a refractive index gradient by theSRT technique. The reasons why previous results re-garding the temperature of burning droplet streams bythe rainbow technique were in disagreement with thetheoretical ones will be explained. Moreover, it will beshown that a data inversion algorithm can be used todetermine with good precision the diameter and therefractive index at the core of a droplet presenting aninternal profile of a refractive index. The refractiveindex at the particle surface is obtained with lessaccuracy.

2. Generalized Airy Theory

Consider a uniform transparent spherical particlepossessing a refractive index higher than its environ-ment and illuminated by a plane wave. Usinggeometrical optics the relation between the scatter-ing angle � and the incident one, �, can be written as

The authors are with the Von Karman Institute for FluidDynamics, Chaussée de Waterloo 72, B-1640, Rhode Saint Genèse,Belgium. M. R. Vetrano’s e-mail address is [email protected].

Received 20 January 2005; revised manuscript received 23 May2005; accepted 2 June 2005.

0003-6935/05/347275-07$15.00/0© 2005 Optical Society of America

1 December 2005 � Vol. 44, No. 34 � APPLIED OPTICS 7275

follows11:

�� 2� � 2�N � 1��, n�, (1)

where n is the relative refractive index of the particlewith respect to its environment, N is the number oflight–particle interaction (i.e., N � 3 for a primaryrainbow), and �� is the refractive angle of �, related toit by Snell’s law. If the particle presents a sphericallysymmetric refractive index gradient, Eq. (1) can begeneralized as follows12:

�� 2� � 2�N � 1��1, (2)

where �1 is the fourth part of the internal angleformed by the light ray in the particle.

The mathematical expression of �1 can be found byintegrating the differential equation of the light raysinside a sphere:

�1�� 1

r̃m ne cos �

r�n�r̃�2r̃2 � �ne cos ��2�r̃, (3)

where n�r̃� is the refractive index profile inside theparticle, ne is the refractive index of the particle en-vironment, r̃ � r�R is the nondimensional particleradius, and r̃m is the minimal distance of the light rayfrom the center of the particle (Fig. 1).

The light scattered by a single sphere using thegeneralized Airy theory (GAT) for nonuniform drop-lets can then be expressed by the following set ofequations12:

I � ��z��2 ��0

cos12�� rg�

� �16D2h�1 �2�1�3� � � �3��d��2

, (4)

h �23

�2�

��2��rs

1

sin2 �rg, (5)

�rg � ��rg�, (6)

where is the light wavelength and �rg �r̃� representsthe solution of the equation

��1

�� N � 1��1, (7)

�rg and �rg are the coordinates of the minimum ofEq. (2) and are, respectively, called the rainbow inci-dent angle and rainbow scattering angle. Equation (4)is the theoretical form of the experimental pattern ob-tained by plotting the distribution of gray levels acrossthe interference fringe pattern generated by the light–particle interaction. This pattern is generally calledthe standard rainbow pattern.

3. Parametric Study

The scattered light intensity of a nonuniform parti-cle, expressed by Eq. (4), depends on the scatteringangle �, on particle diameter D, its refractive index atsurface ns, its refractive index at core nc, and theshape of its internal profile of a refractive index. Inthis section the effect of these parameters on thestandard rainbow pattern are analyzed. In Fig. 2 the

Fig. 1. Light path inside a particle presenting an internal spher-ical symmetric profile of refractive index. Due to this symmetryfour equal branches constitute the light path (dashed curve, dottedcurve, dashed–dotted curve, solid curve). � and � are, respectively,the scattering angle and the incident angle. r̃ and � radial are thepolar coordinates of the spherical system of coordinates used.

Fig. 2. Effect of the particle diameter on the standard rainbowpattern. The particle is supposed to possess a parabolic refractiveindex profile. The refractive indexes at the surface and at thecenter of the particle are, respectively, ns and nc, and is the lightwavelength. The only point of the rainbow pattern independentfrom the particle diameter is the rainbow scattering angle �rg.

7276 APPLIED OPTICS � Vol. 44, No. 34 � 1 December 2005

influence of the particle diameter can be seen. Thestandard rainbow patterns reported there come froma perfectly spherical particle possessing an internalparabolic profile of refractive index. The diametervariation reveals a harmonic effect of the patternpreserving a constant value only for one point of thepattern. This point is the rainbow scattering anglethat, as expressed in Eqs. (6) and (7), depends only onthe refractive index profile.

The dependence of the rainbow pattern from the val-ues of the refractive indices at the core and at the surfaceof the particle is presented in Fig. 3. In this figure pat-terns are obtained for a particle of 100 �m diameterpossessing either a parabolic internal profile of a re-fractive index as n�r̃� � nc � �nc � ns� r̃2, or a uniformrefractive index. The presence of a refractive indexnonuniformity in the droplet is mostly characterizedby a shift of the standard rainbow pattern. Theshifted rainbow pattern is closer to that of a particle(with the same diameter) with uniform refractive in-dex equal to nc.

The dependence of the standard rainbow patternon the shape of the internal profile of a refractiveindex is also important because it strongly affects thevalue of the rainbow scattering angle. The paramet-ric study performed supposes an internal refractiveindex of the type

n�r̃ � � nc � �nc � ns�r̃ m, (8)

where m is the polynomial order. Some of the profilesare shown in Fig. 4. It is important to note that forlarge values of m � m � 12� the refractive indexprofile, obtained using Eq. (8) is similar to the one ofa droplet at the beginning of a burning process. Form � 2 the profile is parabolic and it is similar to theone of a droplet at the end of a burning process. Theother values of m correspond to intermediate states.

In Fig. 5 the rainbow scattering angle �rg is repre-sented as a function of the polynomial order m forfixed values of nc and ns. To calculate �rg, Eq. (3) is

numerically integrated, substituted in Eq. (2), andthen the scattering angle � is computed for a largenumber of impact angles. The minimum of the curveobtained in such a way corresponds to ��rg, �rg� and itis obtained with the least-squares-fit method. A sur-prising result is that if the polynomial order m isbetween 1 and 4 the rainbow scattering angle in-creases; then starting from m � 5 it decreases, reach-ing a constant value corresponding to nc for very highvalues of m � m � 100�. This behavior can be due tothe fact that starting from m � 5 the refractive indexprofile becomes flat in the core of the particle and theregion where the refractive index profile varies de-creases. As a consequence the geometric rainbow an-gle starts to decrease and it approaches to the oneobtained for n�r� � nc.

Moreover, the parabolic profile �m � 2� possesses a

Fig. 3. Effect of the particle internal profile of refractive index onthe standard rainbow pattern. Fig. 4. Example of polynomial refractive index profiles inside a

particle.

Fig. 5. Representation of the rainbow scattering angle �rg as afunction of the polynomial order m of the refractive index profileinside a particle. The dashed-dotted curve and the dotted curverepresent, respectively, the value of the rainbow scattering anglein the case of a uniform refractive index equal to nc and ns.


rainbow scattering angle that is very close to the onecorresponding to profiles with a polynomial orderranging from 9 to 13. It is interesting to note that thesame effect is present for the variable h presented inEq. (5).

4. Rainbow Temperature of Nonuniform Particles

Up to now the data inversion algorithms used in therainbow techniques only provide a single value ofrefractive index even in the presence of a gradient.These data inversion algorithms relate the rainbowscattering angle to the particle refractive index asfollows:

�rg � 2nrg2 � 13 �0.5

� 4 arccos� 1nrg

cosnrg2 � 13 �0.5�,

(9)

where nrg is called rainbow refractive index.If a refractive index profile exists, nrg is not its

value at the particle surface or at the particle core.Moreover, it has been shown that in some cases SRTmeasurements of burning droplets give a result thatis outside of the range of refractive indices deservedin the droplet.13 The reason for this problem is anerroneous hypothesis (uniform refractive index)made in the existing data inversion algorithms thatare used to analyze the standard rainbow pattern.Figure 6 shows the values of nrg obtained if a stan-dard rainbow pattern, issued from a particle with anonuniform refractive index, is inverted using a datainversion algorithm for uniform particles. The com-putation is restricted to refractive indices varying inthe range n � �1.340, 1.399�. The correlation lawthat expresses the relation between nrg, nc, and ns canbe written as follows:

nrg �nc, ns� � ns � �ns�2�nc � ns�, (10)

with � � 2.332 � 10�4. The value of � is obtained by

applying the least-squares-fit method to the surfaceof Fig. 6.

For any value of the ns in the range �1.34, 1.399�,the quantity ��ns

2 exceeds unity, hence nrg is largerthan the refractive index in the core of the particle. Ifthe refractive index at the particle’s core is also largerthan the one at its surface, such as in burning dropletprocesses, nrg is greater than both of them and thusoutside the range �nc, ns�. This limitation of theformer data inversion algorithms has to be avoided ifone aims to apply the rainbow technique to the studyof physical phenomena, such as combustion, evapo-ration, and droplet dissolution, where a significantrefractive gradient is present inside the particles.

5. Data Inversion Algorithm

The use of the standard rainbow thermometry tech-nique for phenomena involving refractive index gra-dients inside particles is possible if a new datainversion algorithm, which takes into account thedroplet nonuniformity, is introduced. The data inver-sion algorithm that will be proposed here is capable ofdetermining the particle diameter and the refractiveindex values at the core and at the surface of a par-ticle presenting an internal profile of refractive index.However, the hypothesis of a priori knowledge of therefractive index profile shape has to be made. In thispaper we will build the data inversion algorithm as-suming a parabolic profile of refractive index. Subse-quently the algorithm will be tested on the nonlinearinternal refractive index profiles of a burning dropletand error estimation will be carried out.

One of the first requirements for a data inversionalgorithm is to have no ambiguity. Figure 7 revealsthat the distance between the first five maxima of astandard rainbow pattern is constant while varyingthe refractive index gradient in the particle. Thismeans that the particle diameter can be deduced

Fig. 6. Three-dimensional representation of the rainbow refrac-tive index nrg as a function of the refractive indices nc and ns. Thissurface can be well approximated by Eq. (10). Fig. 7. Angular position of the first five maxima of a standard

rainbow pattern as functions of the variable nc. The standardrainbow pattern is generated by a particle of 100 �m diameterwith a constant surface refractive index ns.


from any fringe spacing without knowing the internalrefractive index profile.

The next step is to prove that, given the particlediameter, any couple �ns, nc� gives rise to only onestandard rainbow pattern and vice versa. We proceedwith a proof by contradiction. Assuming the existenceof a standard rainbow pattern generated by twodifferent couples of refractive indices �n1s

, n1c� and

�n2s, n2c

�, this will imply that for any value of thescattering angle � [Eq. (4)],

�� 1rg�h1�1�3 � �� 2rg�h2

�1�3 � k� ∀ k � �1, �,(11)

where �1rgand h1 are calculated using the first

couple of refractive indices, and �2rgand h2 are cal-

culated using the second one. If Eq. (11) has to bevalid for any value of the scattering angle �, then ithas to be valid also for � � k�. For such a value of thescattering angle Eq. (11) becomes

�1rg

�2rg

� h1

h2�1�3

, (12)

which is valid only if contemporaneously �1rg� �2rg

andh1 � h2. This necessarily implies that �n1s

, n1c� �

�n2s, n2c

�, contradicting the hypothesis.The data inversion algorithm we propose is based

on the least-squares-fit method using as a fitting pa-rameter the particle diameter ns and nc. The shape ofthe refractive index profile is assumed a priori to beparabolic. The choice of a parabolic profile is essen-tially due to the fact that the values of �rg and hobtained for this profile are very close to the onesobtained for other refractive index profiles, such asthe profile presented in Fig. 4 for a value of m equalto 12. Moreover, a parabolic refractive index profilecorresponds to the one of evaporating spherical drop-lets in quasi-steady conditions.14 The assumption of a

least-squares-fit model in a data inversion algorithmhas already been used by the authors for the GRTinversion of the data in a liquid–liquid suspension.15

6. Results

The data inversion algorithm proposed in Section 5 isused to compute the diameter and the refractive in-dex at the surface and in the core of an n-octanedroplet burning in a standard atmosphere. Some ofthe refractive index profiles of such a droplet duringthe burning process are shown in Fig. 8 (Refs. 16 and17), with the quantity � � t�t0 representing the drop-let heating time. The standard rainbow patterns, nu-merically generated using the GAT model on therefractive index profiles presented in Fig. 8, areshown in Fig. 9. As shown in Section 4, using theformer data inversion algorithm for standard rain-bow patterns on those patterns, based on Eq. (9), theresulting temperature erroneously shows that the

Fig. 8. Internal refractive index profiles of an n-octane dropletburning in a standard atmosphere. The quantity � represents thenondimensional time t�t0, where t0 is the heating time.

Fig. 9. Standard rainbow patterns obtained for the refractiveindex profiles of Fig. 8.

Fig. 10. Rainbow temperature evolution, obtained with theformer data inversion algorithm, as a function of the n-octanedroplet heating time.


droplet was cooling down during the initial burningstage (Fig. 10).

Applying the new data inversion algorithm to theprofiles of Fig. 9, the values of the particle diameterand refractive index in the core and at the surface areobtained. Figure 11 presents the comparison betweenthe values of the refractive index imposed by theprofiles of Fig. 8 and the corresponding values ob-tained using the former data inversion algo-rithm and the new one. The values corresponding tothe core of the droplet are in better agreement�discrepancy 2.7 � 10�3� than those at the surface�discrepancy 1.3 � 10�2�. Translated in tempera-ture for the n-octane droplet, the disagreement forthe core value is approximately 5.6 K � 1.5%� whilefor the surface this value rises to 27 K � 7%�. Thereason why the fitting procedure works better for

the core values rather than for the surface ones has tobe found in the stronger dependence of the rainbowscattering angle �rg on nc than on ns. The resultsconcerning the diameter evaluation, shown in Fig. 12,show that the discrepancy between the real value andthe one obtained by the data inversion algorithm is atmaximum 5%.

7. Application of the Technique and Conclusions

The GAT model and the data inversion algorithmthat have been proposed make the standard rainbowthermometry technique a powerful laser-based non-intrusive technique applicable to many scientific con-tests. In Section 6 the special case of a burningdroplet has been exploited to show a practical use ofthe technique obtaining satisfactory results. Otherpractical applications of the standard rainbow ther-mometry technique in nonuniform spherical systemscan be either in the description of the dissolution rateof multicomponent droplets suspended in partiallymiscible liquids or the measurement of the evolutionof evaporating films over spheres or cylinders. It isimportant to point out that for the first time, to ourknowledge, an experimental technique based on elas-tic scattering is used to measure the refractive indexgradient inside a particle, even if its shape is as-sumed to be known a priori, without the addition ofexperimental constraints, and maintaining the sim-plicity of the data inversion algorithm.

M. R. Vetrano gratefully acknowledges the financialsupport of this work by the Fonds pour la formation àla Recherche dans l’Industrie et dans l’Agriculture,Belgium.

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Fig. 11. Comparison of the theoretical refractive index values inthe core and at the surface of an n-octane droplet burning in astandard atmosphere and the values obtained with the old andwith the new data inversion algorithm.

Fig. 12. Comparison between the theoretical n-octane droplet di-ameter and the one obtained using the new data inversion algo-rithm.


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Assessment of refractive index gradients by standard rainbow thermometry

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