A Monte Carlo Ray Tracing Study of Polarized Light Propagation in Liquid Foams

ARTICLE IN PRESS

Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 277–287 www.elsevier.com/locate/jqsrt

A Monte Carlo ray tracing study of polarized light propagation in liquid foams

J.N. Swamya,Ã, Czarena Crofchecka, M. Pinar Mengu¨c- b,ÃÃ aDepartment of Biosystems and Agricultural Engineering, 128 C E Barnhart Building, University of Kentucky, Lexington, KY 40546, USA bDepartment of Mechanical Engineering, 269 Ralph G. Anderson Building, University of Kentucky, Lexington, KY 40506, USA

Received 10 July 2006; accepted 28 July 2006

Abstract

A Monte Carlo ray tracing scheme is used to investigate the propagation of an incident collimated beam of polarized light in liquid foams. Cellular structures like foam are expected to change the polarization characteristics due to multiple scattering events, where such changes can be used to monitor foam dynamics. A statistical model utilizing some of the recent developments in foam physics is coupled with a vector Monte Carlo scheme to compute the depolarization ratios via Stokes–Mueller formalism. For the simulations, the incident Stokes vector corresponding to horizontal linear polarization and right circular polarization are considered. It is observed that bubble size and the polydispersity parameter have a signiﬁcant effect on the depolarization ratios. This is partially owing to the number of total internal reﬂection events in the Plateau borders. The results are discussed in terms of applicability of polarized light as a diagnostic tool for monitoring foams. r 2006 Elsevier Ltd. All rights reserved.

Keywords: Polarized light scattering; Liquid foams; Bubble size; Polydispersity; Foam characterization; Foam diagnostics; Cellular structures

1. Introduction

Liquid foams are random packing of bubbles in a small amount of immiscible liquid [1] and can be found in a wide variety of applications. Many of the processed foods like chocolate bars, bread, cakes, beer, whipped cream, etc. are essentially foams [1]. In contrast, specially designed detergent foams are being developed to decontaminate areas exposed to biological weapons. Owing to their low density and unique rheological properties, foams have a wide range of applications. It is widely accepted that the quality of foam products and the efﬁciency of processes involving foams are largely dependent on foam structure. Hence, there is a well- motivated need to develop techniques to monitor foam properties (density and consistency) during production and stabilization.

ÃCorresponding author. Tel.: +1 859 257 3000; fax: +1 859 257 5671. ÃÃAlso to be corresponded. Tel.: +1 859 257 6336; fax: +1 859 257 3304. E-mail addresses: [email protected] (J.N. Swamy), [email protected] (M.P. Mengu¨c-).

0022-4073/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2006.07.022 ARTICLE IN PRESS 278 J.N. Swamy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 277–287

Nomenclature

A plateau border area E number of edges I stokes intensity L length of edges/borders P degree of polarization PM phase matrix p polydispersity parameter Q tendency for horizontal polarization R radius S scattering matrix or minimized surface area TIR total number of total internal reﬂections in borders TSE total number of scattering events U tendency for +451 linear polarization

Greek symbols

Y scattering angle F gas or liquid fraction d turning angle of Plateau borders m moment of a distribution s standard deviation f vertical opening angle

Subscripts

b bubble f foam g gas l liquid p Plateau border t total

Foams are continuously evolving systems far from equilibrium, which require measurement using non- intrusive diagnostic techniques. Being cellular structures, foams exhibit a highly multiple scattering behavior resulting in a familiar white appearance. This limits the utility of conventional measurement techniques like photography, video imaging, etc. Optical tomography has been used with some success to study the topology of three dimensional foams [2]. However, such techniques require complex image reconstruction algorithms, which are computationally expensive, limiting its utility as a research tool. Several studies have been conducted in the past to investigate multiple scattering of light in foams using a diffusion approximation [3–7]. Using such techniques, Durian et al. [3] studied the transient behavior of foam due to coarsening and drainage. In this method, foams are modeled as air bubbles separated by liquid ﬁlms and a model for photon transport based on random walk is applied. The resulting photon transport mean free path is correlated with the foam microstructure. Thus, the validity of correlations is largely dependent on the mechanism underlying the random walk. However, it is suggested that implementing the rules from geometric optics for transmittance and reﬂectance of light results in a persistent random walk [6]. The premise of the current approach is that cellular structures like foam are likely to alter the polarization of the incident radiation due to successive scattering events. Thus, monitoring polarization changes in addition to the attenuated intensity can lead to additional information about foams. If the properties of the foam layer ARTICLE IN PRESS J.N. Swamy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 277–287 279 can be related to the changes in polarization of the incident light, an intelligent diagnostic scheme can be developed to monitor foam properties. The changes in polarization due to multiple scattering events, assuming foam to be an emulsion of spherical bubbles, has been investigated earlier using a geometric optics approximation [8,9]. However, in these studies only a few or non-contact bubbles were considered and hence did not include the scattering effects due to the Plateau borders and vertices. In the current work, a similar approach is utilized with a more realistic physical foam model, where the effects of the Plateau borders are included.

2. Theoretical background and problem formulation

2.1. Overview

The main goal of this study is to provide a theoretical understanding of the interaction of light with foam using a geometric optics model coupled with a statistical model for foam structure using a Monte Carlo scheme. Recent developments in foam physics and availability of tools like surface evolver have led to theoretical calculations of the geometric properties of three dimensional polydisperse foams [10,11]. This has lead to significant improvements in the overall understanding of foam structure. A temporal approach is used for ray tracing considering the computational expense for a spatial approach. In a temporal approach, the foam microstructure is generated as the photon travels through it, based on relevant statistics available from theoretical calculations. A bubble size distribution (or cell size distribution; the terms bubbles and cells are used interchangeably here) is assumed to start with; based on this the statistics for shape and structure are computed to provide for the geometrical constraints required for ray tracing simulations. The bubbles/cells are modeled as polyhedra with F faces with finite film curvature and E edges constrained by the rules of Euler and Plateau. Further, to simplify the ray tracing calculations, it is assumed that a given film has a constant mean curvature (i.e., a spherical cap) following the isotropic plateau polyhedra theory [10]. A polarized laser beam is assumed to impinge normally on a cylindrical system (containing the foam), at the circumferential surface and the scattered signal is assumed to be collected on the same plane. The medium temperature is assumed low, such that there is no radiative emission. A geometric optics approach is used instead of a more detailed physical optics analysis, given that the wavelength of light is much shorter than the bubble sizes considered (Rb 40.5 mm). The effects of dependent scattering, diffraction, and interference phenomena are neglected. The typical thickness of Plateau borders in foams varies from 0.1 to 2 mm depending on their liquid content [1]. Hence, with exception of extremely dry foams (Fgo0.02), the clearance between bubbles is at least one-half of the wavelength making the neglect of dependent scattering effects reasonable [8]. Stokes–Mueller formalism is used to track the changes in intensity and polarization via Vector Monte Carlo method presented by Vaillon et al. [12]. The elements of the scattering matrix for individual reflection/ transmission events follow Fresnel equations.

2.2. Physical model for foam

Very dry foam is made up of cells that are considered to be polyhedral in shape; while very wet foams are made up of spherical bubbles. For most practical applications, foams are neither completely dry, nor completely wet. Considering the differences between the two extremes, the overall foam structure is difficult to model. However, recent developments have led to interesting geometric correlations which relate the structural parameters to the number of faces and volume [10]. Some of the numerical experiments conducted by Kraynik et al. [11] have revealed very useful statistics of random polydisperse foam, utilized here. All the topological and geometric parameters are expressed in terms of the polydispersity, p, based on the Sauter mean diameter. Fig. 1 illustrates the simplified microstructure of foam, used in this work, where three bubbles (B1–B3) come together to form a Plateau border. Fig. 1(a) illustrates the three dimensional view in which four of these Plateau borders meet in a tetrahedral network (which of course requires the presence of another bubble B4 below the visible plane). Fig. 1(b) shows the cross-section of one of these Plateau borders. The bubbles are modelled as polyhedra of F faces with weakly curved films (radius of curvatures R1–R3) which touch each other. The shaded region in Fig. 1(c) is shown to emphasize this aspect. The equilibrium of ARTICLE IN PRESS 280 J.N. Swamy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 277–287

(a) (c)

(b)

Fig. 1. (a) The Plateau borders or the liquid network inside foams, (b) cross-sectional view of the Plateau border, (c) illustration of partial contact between faces. surface tensions requires that this region of contact must have the same surface normal on either side. Thus, the direct photon flights from one bubble to another are fairly simplified. The scattering events inside the Plateau borders are known to involve several total reflection events which lead to photon channelling [13,14]. To follow the scattering effects in the plateau border, the geometrical constraints required to determine dependencies between subsequent events of reflection and transmission need to be computed. The effect of vertices where these borders meet is unaccounted for in this study. To start with an average bubble volume /VbS corresponding to equivalent volume sphere of radius /RbS is assumed. The polydispersity parameter is known to be well representative of the topological disorder [2,11]. Hence, instead of assuming a distribution and then computing p, the available empirical relationship given by Eq. (1) is used to compute the standard deviation in bubble size based on an assumed value of p: s Rb ¼ 0:95p1=2. (1) hiRb The mean and standard deviation values facilitate modelling the bubble size as a probability distribution function. A log-normal distribution of bubble sizes is assumed for the purpose of simulations. The mean 2 number of faces /FS and the normalized variance (m2//FS ) are obtained from data in [11] based on the value of p, and modelled as a normal distribution. The number of edges in a cell with F faces follows from Euler’s theorem and Plateau’s law and is given as 3F-6. This allows for temporal generation of foam structure using random numbers as the photon ensemble travels. In terms of geometrical constraints, it is desirable to obtain the film curvatures and the average length of the plateau borders, for the ray tracing simulation. For this purpose, the bubbles are approximated as an Isotropic Plateau polyhedra, which allows the usage of ARTICLE IN PRESS J.N. Swamy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 277–287 281

Eqs. (2)–(4). Details can be found in [10]: 1=3 1=2 1=2 LtV b ¼ 4:1728F þ 1:2313 þ 2:0455F , (2)

S L þ pffiffiffit ¼ 2pF 6ðF 2Þd, (3) R2 3R ÀÁ 2 1=3 S ¼ b 36pV b . (4) The value of the constant b is given as 1.170.008 [10,11]. d is the turning angle of the Plateau borders in radians and is given by the Plateau’s laws.

2.3. System geometry

The conﬁguration of the system containing the foam (Fig. 2) is assumed to be cylindrical for the purpose of Monte Carlo simulations [9,12]. The laser beam is assumed to impinge normally on a cylindrical system (containing the foam) at the circumferential surface and the scattered signal is assumed to be collected from the same plane. The detection plane is subdivided into pie sections with cone angles deﬁned by DY and Df as shown in Fig. 2. For the purpose of simulation, only the photons exiting through this cone angle contribute to the detected signal, which is assigned to the direction cosine corresponding to the central axis of the pie section. The radius of the cylindrical system can be varied to study the effect of medium thickness.

2.4. Stokes– Mueller formalism

The Stokes vector can be used to fully describe the intensity and polarization state of light and can be expressed using the parallel and perpendicular components of the electric field [15]. The utility of Stokes vector to the current application lies in its phase independence. This allows for addition of Stokes parameters of thousands of photon ensembles exiting a given pie section to compute statistical means. Further, the total degree of polarization can be calculated as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q2 þ U 2 þ V 2 P ¼ . (5) I

Incident laser beam Reference Frame (632.8 nm) z

x ∆Θ ∆ϕ Scattered light

Fig. 2. Schematic of the system conﬁguration. ARTICLE IN PRESS 282 J.N. Swamy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 277–287

For a ﬁxed frame of reference, the incident and scattered Stokes vectors corresponding to a single scattering event are related by a 4 4 phase matrix PM(Y) which is dependent on the scattering angle Y [9]: ÀÁT ÀÁT IQUV ¼ PMðYÞ IQUV . (6) sca inc The phase matrix can be computed based on the scattering matrix S(Y) and two rotational matrices L(ps2) and L(s1) and is given as

PMðYÞ¼Lðs2ÞSðYÞLðp s1Þ. (7)

The angles s1 and s2 are related to the angle of incidence and angle of scattering defined with respect to fixed reference frame [16]. For a randomly oriented symmetric particle, the scattering matrix is defined by six independent elements (S11, S12, S22, S33, S34,andS44) and is given by Eq. (8). The foam structure can be considered as a network of randomly oriented Plateau borders and vertices which connect them. The Plateau borders are assumed to have a plane of symmetry. Hence, such an approach can be applied by considering an S(Y) of the form 2 3 S11ðYÞ S12ðYÞ 00 6 7 6 7 6 S12ðYÞ S11ðYÞ 007 SðYÞ¼6 7. (8) 4 00S33ðYÞ S34ðYÞ 5

00S34ðYÞ S33ðYÞ The elements of the scattering matrix can be computed for individual reflection and transmission events using the Fresnel equations [9]. Eqs. (6)–(8) can now be used to compute the resultant Stokes vector (in the fixed frame of reference) after every event of reflection or transmission as the photon ensemble travels through the foam. This enables tracking of the intensity and polarization history of the ensembles and computation of resultant Stokes vector (statistical mean with respect to incident) for ensembles exiting along a given direction cosine.

3. Monte Carlo simulations

Forward Monte Carlo simulations are carried out based on the physical models chosen in Section 2. The incident laser beam is considered to be divided into several photon ensembles, whose intensities are sampled from a Gaussian distribution. Each ensemble is launched, one at a time, at the point of incidence of the laser beam. The scattering direction and resultant Stokes parameters for each scattering event is dictated by the foam geometry (primarily by the Plateau Borders) and is determined by a ray tracing scheme. As an initial step, three bubble/cell sizes are sampled from the distribution along with the number of faces, and the corresponding geometric descriptors are computed. There are two possible cases: (1) the photon is incident at the Plateau border–bubble interface and (2) the photon is incident at the bubble–bubble interface. The probability of case (1) happening is directly proportional to the liquid fraction of the foam.

3.1. Case 1: incidence at Plateau border– bubble interface

When the polarized light is incident at the Plateau border–bubble interface, there is a possibility that the light will be reflected or transmitted [9], with each event having equal probability, except for the case of incidence angles larger than the critical angle resulting in total internal reflection. Consider Fig. 3, the three bubbles B1–B3 come together to form the Plateau border defined by V12V13V14 with a photon incident at point P1 with an angle of incidence y1. For either reflection or transmission, computing the resulting direction is defined by the Snell’s laws.

3.1.1. Transmission In the case of transmission, the photon would subsequently hit the point P3 with an incidence angle y3, which needs to be determined. Consider the point of intersection N13 of the two surface normals. If the angle of intersection w is known, y3 is given as (pwy2), y2 being the transmission angle. The value of w can be ARTICLE IN PRESS J.N. Swamy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 277–287 283

CR3

Fig. 3. Schematic of the Plateau border used to compute subsequent event parameters. written as p-angle (V12V13V23) if the Plateau border is considered to be a triangle, which is a reasonable assumption given that surfaces are very weakly curved for relatively dry foam. Since the Plateau border considered is shared by three bubbles of unequal sizes, the vertex angles cannot be determined directly. For this purpose, let us assume the angle subtended by the V13V12 at the center of CR1 is o (note that R1bV13V12). Then, using the argument of very weak curvature, V13V12EoR1. If it is assumed that the angle subtended by the arcs in the neighboring bubbles, at their respective centers, is o, then it follows that the triangle laws can be used to compute the vertex angles based on R1–R3 independent of the value of o. Thus, the effect of the radius of curvature is still captured in the subsequent incident angle (y3) despite the triangle assumption. The distance traveled by the photons between subsequent events is dependent on the point of incidence of the photon relative to the vertices. The value of o can be related to the Plateau border areas by the following equation: o2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ¼ ðR1 þ R2 þ R3ÞðR1 þ R2 R3ÞðR1 þ R3 R2ÞðR2 þ R3 R1Þ. (9) p 4 The average Plateau border area and liquid fraction of foam are directly related by the total edge length per unit volume of given foam, given by the following empirical equation [12]:

2=3 1=3 Lf hiV b ¼ Sf hiV b þ 0:063, (10)

ApLf ¼ fl. (11)

For a foam with an assumed liquid fraction Fl, o can be calculated. A random number oran is generated from a uniform distribution (o/2, o/2) to determine the point of incidence relative to the central axis of the arc. This enables computation of the distance P1P3 following triangle laws. Obviously, all subsequent events occurring within a given Plateau border are constrained by the triangle laws. Following each event, the corresponding scattering matrix is computed using the Fresnel equations and the Stokes vector is computed. The photon is then moved based on the computed travel distance.

3.1.2. Reﬂection In this case, the direction follows Snell’s law. Calculation of the distance of interaction or in other words the photon travel distance requires signiﬁcant computational effort. Hence, a look-up table is created as a function of the angle of incidence and number of faces for various bubble sizes. The travel distance is then sampled based on geometry of the bubbles under consideration. The subsequent interface of incidence is based on the liquid fraction and the angle of incidence is chosen randomly from the interval (0,p/2). Details of the scheme can be found in [17]. ARTICLE IN PRESS 284 J.N. Swamy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 277–287

3.2. Case 2: incidence at bubble– bubble interface

In this case, transmission is simple to handle as the equilibrium of films demands the touching bubble surfaces to share a common normal. A transmitted photon is assumed to travel undisturbed into a neighboring bubble. For the case of reflection, the direction is determined by Snell’s law and the distance of interaction follows the same procedure as in Section 3.1.2. The exit of the ensemble is decided by the system geometry previously defined. A total of 2 107 photon ensembles are launched and tracked to obtain the resultant Stokes vector (statistical mean) for a given scattering angle. Particular attention is paid to the number of total internal reflections each photon ensemble undergoes inside the Plateau border and an average percentage is computed and referred to as

7 2X10 TIR TIREð%Þ¼ k 100. (12) TSE k¼1 k

4. Results and discussion

The wavelength of the laser beam considered for the simulations is 632 nm. The refractive index of water is taken to be 1.33 and that of air to be 1. The incident Stokes vectors considered for the simulations correspond to horizontal linearly polarized light and right circularly polarized light. The medium thickness used for simulations is r ¼ 15 mm. A total of 32 exit direction cosines are considered at equal intervals with a vertical opening angle of Df ¼ 0:04 radians. A total of four mean bubble diameters (/2RbS ¼ 0.5, 0.75, 1.0, and 1.5 mm) were considered with a polydispersity parameter p ¼ 0:05 to study the depolarization effects due to bubble size. To study the effect of the width of bubble size distribution, five polydispersity values (0, 0.025, 0.05, 0.075, and 0.1) were considered for a mean bubble size of 0.75 mm. For each given condition, simulations were repeated five times. The results converged within 5–10% standard deviation indicating the requirement to include more photon ensembles. However the results obtained still provide information about the effect of foam characteristics on the Stokes parameters. The effect of bubble size on the normalized Stokes intensity is shown in Fig. 4. The profiles are not quite distinguishable for different bubble sizes, for both reflection and transmission angles. The backscattered intensity are an order of magnitude higher than forward and side angles (side angles not shown in Fig. 4) indicating higher reflectance values. Fig. 5 shows the degree of depolarization of a 100% polarized incident

10-1 10-1 1.5 mm 1.0 mm ) 0 0.75 mm 10-2 0.5 mm 10-2

10-3 10-3

10-4 10-4

Normalized Stokes Intensity (I/I Reflection Transmission

10-5 10-5 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 0.5 0.6 0.7 0.8 0.9 1.0 Direction cosine

Fig. 4. Effect of bubble size (with fixed p ¼ 0:05) on normalized Stokes intensity at reflection and transmission angles for linearly polarized incident beam (error bars, about the size of the symbols, show the standard error based on five simulations per bubble size). ARTICLE IN PRESS J.N. Swamy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 277–287 285 beam as a function of the scattering angle (each point is within o7% error, error bars are excluded for sake of clarity). The effect of bubble size on the linearly polarized light can be seen in Fig. 5(a). In foam with a larger mean bubble size, the scattered light seems to retain its polarization for the forward and backward scattering angles. For a given container size and geometry, as the mean bubble size decreases, the total number of bubbles increases and hence the number of scattering events. This contributes to higher depolarization ratios for a smaller bubble size. Similar effects of bubble size are seen for circularly polarized light at the backward scattering angles (as shown in Fig. 5(b)). These effects are more evident in Figs. 6(a) and (b). It should be noted that there is an inversion of results for strongly forward angles. Much of the side scatter data consists of overlapping results with little or no sensitivity to bubble size. Figs. 5(c) and (d) show the depolarization effects due to polydispersity. The linear depolarization ratios indicate that as the polydispersity increases the amount of polarization retained decreases. The monodisperse case of p ¼ 0, has the highest degree of polarization for most angles compared to polydisperse systems, for both the cases of linear and circular polarization. Such effects are clearly seen for the backscattering angles in Figs. 6(c) and (d). This is expected as the topological disorder increases with increasing polydispersity value leading to higher depolarization effect. Fig. 6 shows the depolarization effects at backscattering angles and their sensitivity to the foam parameters. It is observed that the degree of polarization, for the case of horizontally polarized light, has a dynamic range with respect to bubble size at scattering angles of 1051 and 1351. In the case of incident right circular polarized light, the degree of polarization values show higher sensitivity to the bubble size for all the backscattering angles considered. Results corresponding to 1351 exhibit a large dynamic range over the bubble sizes considered. The sensitivity of depolarization ratios to polydispersity at backscattering angles, for both cases of horizontal and right circular polarized light, is shown in Figs. 6(c) and (d). It can be seen that the amount of linear polarization retained for a given polydispersity value decreases with increasing value of scattering angle. These trends are inverted for the case of circular polarization, where the polarization retention increases with

(a) Horizontal 1.5 mm (c) Horizontal p=0 p=0.025 0.5 1.0 mm 0.5 0.75 mm p=0.05 0.5 mm p=0.075 0.4 0.4 p=0.1

0.3 0.3

0.2 0.2

0.1 0.1 Degree of Polarization Degree of polarization 0.0 0.0

1.5 mm 1.0 (b) Right Circular 1.0 (d) Right Circular p=0 1.0 mm p=0.025 0.75 mm p=0.05 0.8 0.8 0.5 mm p=0.075 p=0.1 0.6 0.6

0.4 0.4

0.2 0.2 Degree of polarization Degree of Polarization

0.0 0.0

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 Direction cosine Direction cosine

Fig. 5. Effect of cell/bubble size (with p ¼ 0:05) on depolarization of (a) horizontally polarized and (b) circularly polarized light. Effect of polydispersity (with ﬁxed /2RbS ¼ 0.75) on depolarization of (c) horizontally polarized and (d) circularly polarized light. ARTICLE IN PRESS 286 J.N. Swamy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 277–287

(a) Horizontal 105 (c) Horizontal 105 120 120 135 135 0.2 150 0.2 150 165 165

0.1 0.1 Degree of Polarization Degree of Polarization

0.0 0.0

0.7 (b) Right Circular 0.7 (d) Right Circular

0.6 0.6

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2 Degree of Polarization Degree of Polarization 0.1 0.1

0.0 0.0 0.50 0.75 1.00 1.25 1.50 0.000 0.025 0.050 0.075 0.100 Average Cell/Bubble Size (mm) Polydispersity

Fig. 6. Sensitivity of depolarization ratios to (a,b) average cell/bubble size (at constant p ¼ 0:05) and (c,d) polydispersity (at constant /

2RbS ¼ 0.75) at backscattering angles; (a) and (c) correspond to depolarization of horizontally polarized light; (b) and (d) correspond to right circularly polarized light; error bars are based on standard error over ﬁve simulations for each cell/bubble size and polydispersity value.

45 Bubble size Avg. %TIRE (mm) 0.50 45.2±3.6 40 0.75 44.5±3.5

% TIRE 1.00 36.2±3.4 35 1.5 36.6±2.4

25 0.50 0.75 1.00 1.25 1.50 Mean cell/bubble size (mm)

Fig. 7. Percentage of events being total internal reflection in Plateau borders for each mean cell/bubble size (/2RbS) considered (fixed p ¼ 0:05). Error bars are based on standard error over all ensembles for a given simulation (total of 5 simulations per cell/bubble size are considered). ARTICLE IN PRESS J.N. Swamy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 277–287 287 the scattering angle. The degree of polarization decreases with increasing polydispersity and has a dynamic range, at all angles, over the polydispersity values considered. Additionally, the statistics for the number of total internal reflections are shown in Fig. 7. It is observed that 30–50% of the scattering events are a result of total internal reflection in the Plateau borders, depending on the bubble size and polydispersity. To a certain extent, these results explain the higher depolarization ratios for foam with smaller bubbles. Similar phenomena have been observed by others, whereby the photons channel through the Plateau borders due to total reflections [13]. This implicates that some of the photons do not make it to the cone angles, resulting in much smaller Stokes intensity values.

5. Conclusions

The initial results from simulations are encouraging and suggest the need for a more detailed study with a broad range of bubble size distributions. The sensitivity of depolarization ratios to both bubble size and the polydispersity parameter suggest that the polarized light could be used to characterize foams, if the backscattering angles are considered for measurement. The scattering angle of 1351 seems particularly useful for measurement of both bubble size and the polydispersity of foams. Currently, efforts are directed towards increasing the number of photon ensembles incorporated in simulations to be able to compute the effective Mueller matrices accurately following the approach of Vaillon et al. [18]. Further, experimental studies are underway to investigate the utility of polarization data for foam characterization.

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