Previously [1], a method to extract the optical con-stants n��� and k��� from multiple-scattered light wasdeveloped and applied to suspensions of sphericalpolystyrene latex particles. This method consisted ofinverting total diffuse reflectance and transmittancemeasurements collected with an integrating spheresetup using the adding-doubling method (ADD) tosolve the radiative transfer equation (RTE) combinedwith the exact Mie theory to describe single particleabsorption and scattering characteristics.
The RTE for light propagation through a slab isgiven by [2],
dI�r, s�ds � ��tI�r, s� �
�s
4��4�
p�s, s�I�r, s�d�, (1)
where I�r, s� is the specific intensity at a point r withradiation along the direction s, �t �� �s � �a� is thebulk extinction coefficient, �s is the bulk scatteringcoefficient, �a is the bulk absorption coefficient, andp�s, s� is the phase function which is a measure of theangular distribution of the scattered light. The bulkscattering and absorption coefficients for a suspen-sion of monodisperse particles in water are given by
�a��� � �a,p��� � �a,w���, �a,p��� � �a���,
�a,w��� �4�kw����1 � c�
�, �s��� � �s���, (2)
where a, s, and �, respectively, are the absorptionand scattering cross sections and number density ofthe particles, � is the wavelength of the incidentbeam, kw is the imaginary part of the complex refrac-tive index of water, and c is the volume fraction ofparticles in the sample. For spherical particles, if thecomplex refractive index and particle size are known,
the absorption and scattering cross sections can becalculated exactly using Mie theory, and if the num-ber density is known, the bulk scattering and absorp-tion coefficients for the particles can be computed.Then the RTE can be solved using the ADD withappropriate boundary conditions [3]. In our case, thecomplex refractive index of the particles is unknownand can be found by iteratively solving Eq. (1) togenerate total diffuse reflectance �Rcal� and transmit-tance �Tcal� and comparing these values with experi-mental measurements (Rmeas and Tmeas) such that thefollowing function is minimized [1]:
� � abs�Rmeas � Rcal� � abs�Tmeas � Tcal�. (3)
At each iteration step, the adding-doubling algo-rithm to solve the RTE requires as input �a, �s, andthe phase function, which are computed using Mietheory. The inversion methodology is computation-ally intensive, and it can take a few hours ofiterations to invert one sample (over a range of wave-lengths) to a desired convergence. Thus approxima-tions which significantly speed up the calculationswithout introducing large inaccuracies will be desir-able. The steps in the iterations that involve the com-putation of �a, �s, and the phase function throughMie calculations take a significant portion of the com-putational time. The reduction of computational ef-fort for this part of inversion scheme is investigatedin this paper. To avoid the complications involved inaccounting for interparticle interactions, in thisstudy the concentrations are kept below the levelsindicated by the analysis carried out previously [1],which indicated that above a weight fraction of�0.025, the refractive index values show a concen-tration dependence when the current inversionscheme is used. Thus the focus is on systems wherethe concentration is sufficiently high for multiplescattering effects to be significant without being af-fected by interparticle interactions.
2. Approximations to Single Particle ScatteringCharacteristics
To speed up the calculations, two approaches weretried. One was to replace the Mie calculations with theRayleigh–Gans–Debye (RGD) [4] approximation tocalculate the absorption and scattering cross sectionsand the phase function. The second approach was tocalculate the absorption and scattering cross sectionsand the anisotropy parameter (g) exactly using Mietheory and use approximations to the phase function,which are written in terms of g. The Henyey–Greenstein (H-G) [5] and the modified Henyey–Greenstein (H-Gm) [6,7] approximations were used inthis study.
A. Rayleigh–Gans–Debye Approximation
In the RGD approximation, the absorption and scat-tering cross sections are given by [4,8]
a��� �4�n���k���Vp
�, (4)
sca � 2��n� �1�2�52 � 2x2 �
sin 4x4x �
7
16x2�1 � cos 4x�
� � 1
2x2 � 2� � log 4x � Ci4x���, (5)
where Vp is the volume of the particle, n� is the realpart of the relative refractive index, � is the Euler’sconstant, and Ci is the cosine integral Ci�x� ���x
� �cos u�u�du, u � 2a� sin� �2�, � � 2���0, x ��a is the size parameter, a is the radius of theparticle, � is the scattering angle, and �0 is the wave-length in vacuum.
For symmetric particles, p�s, s�� varies only alongthe scattering angle � and is given by
p�s, s�� � p�cos � �F� , a, n�, ��
sca. (6)
The anisotropy factor g � cos � is given by
g � cos � �1
4�2
� x4�n� �1�2�1 � cos2 �P� �cos d
scat.
(7)
The phase function gives the angular distributionof light for a single scattering event. To solve the RTEusing the ADD, the angular distribution, which takesmultiple scattering into account, is required. This isprovided by the redistribution phase function, whichdetermines the fraction of light scattered from anincident cone with angle i into a cone with angle j.This function is calculated by averaging the phasefunction over all possible azimuth angles for fixedangles i and j [3,9],
h�cos i, cos j� �1
2� �0
2�
p�cos i cos j
� �1 � cos i2�1 � cos j
2 cos ��d�.(8)
The solution of this equation is not straightforwardwhen a phase function such as the RGD phase func-tion is used. The redistribution phase function can becalculated using the �-M method [3,9–11]. In thismethod the true phase function, p(cos �) is approxi-mated by a phase function P*�cos � consisting ofa Dirac delta function and M – 1 Legendre polyno-mials,
where P*�cos � is equivalent to the phase functionp�cos �, �m
* are the expansion coefficients, and f is thefractional scattering into the forward peak. The ex-pansion coefficients are found by matching this equa-tion with the first 2M terms of the true phase functionexpanded in terms of Legendre polynomials,
p�cos � � �n�0
�
�2n � 1��nPn�cos �, (10)
where the coefficients �n are computed from the fol-lowing expression:
�n �12 �
0
�
p�cos �Pn sin d . (11)
The first coefficient, �0 � 1, because the phase func-tion is normalized to 1 [11]. The following relation-ship is then obtained
�m* �
�m � f1 � f , m � 0, . . . , 2M � 1, (12)
where f is set to �2M. The redistribution of the phasefunction using the �-M approximation can then bewritten as [9]:
h*�cos i, cos j� � 2f��cos i � cos j�
� �1 � f� �m�0
2M�1
�2m � 1��m*
� Pm�cos i�Pm�cos j�. (13)
B. Phase Function Approximations
The H-G phase function approximation is a simpleanalytical expression which has been used exten-sively to describe forward scattering and is given by[3,5,6]:
pHG�cos � �1 � g2
1 � g2 � 2g cos �3�2. (14)
For this function we get
f � �m � gM, (15)
and �m* is given by
�m* �
gn � gM
1 � gM . (16)
From the last equation it is seen that the calculationof �m
* is very simple for this case as integration of Eq.(11) to compute �m are not required.
The H-G approximation does not describe well theforward or backward scattering peak, and it does nothave the right limiting behavior viz., it does not re-duce to the Rayleigh scattering phase function inthe Rayleigh regime when cos → 0. Cornette andShanks [7] proposed a H-Gm approximation, whichreduces to the Rayleigh phase function when cos �tends to 0. The H-Gm function is given by
pH-Gm�cos � �32
1 � g2
2 � g2
1 � cos2
1 � g2 � 2g cos �3�2. (17)
In this case, however, the calculation of the expansioncoefficients �m
* has to be carried out numerically byintegrating Eq. (11). Similarly, for the RGD approx-imation, the expansion coefficients have to be ob-tained by numerical integration.
3. Materials and Measurements
Suspensions of polystyrene and poly(methyl methac-rylate) (PMMA) latex spheres were used in this
Fig. 1. Complex refractive index, m��� � n��� � ik��� of PMMAmicrospheres (a) real part, n���, and (b) imaginary part, k���.
study. Polystyrene microspheres suspension of 10%by weight of solids of narrow particle size distribu-tions with coefficient of variance (CV), 3% of meandiameter of 0.45 �m was purchased from Duke Sci-entific. This was diluted using de-ionized water to0.075% by weight of solids. PMMA microspheres sus-pension of 2.7% by weight of solids, 0.324 �m of meandiameter, and 0.015 �m of standard deviation waspurchased from Polysciences, Inc. A sample was pre-pared by dilution to 0.56% by weight of solids. Thenumber density of these suspensions were calculatedusing the density of de-ionized water at room tem-perature �25 °C�, �medium � 1 g�ml, and the cor-responding density of the particles: polystyreneparticles, �particles � 1.05 g�ml, and PMMA particles�particles � 1.19 g�ml. The samples were placed in aspecial optical glass cuvette of 2 mm path length.Total diffuse reflectance and transmittance weremeasured using an integrating sphere of diameter150 mm (DRA-2500, Varian Instruments) attachedto an ultraviolet-visible-near-infrared (UV-Vis-NIR)spectrophotometer (Cary 5000, Varian Instruments)in the wavelength region 450–1320 nm at 20 nm in-tervals. Three replicates of each (polystyrene andPMMA) sample were obtained by repeating the sam-ple preparation. For each of these replicates, totaldiffuse reflectance and transmittance were measuredthree times, thus obtaining nine measurements foreach sample. The values of n��� and k��� reported arethe mean over the nine measurements, and the errorbars indicate two times the standard deviation. Therelative complex refractive index, mr is computed asthe ratio of the complex refractive index of the parti-cles to the complex refractive index of water. Forthese calculations the optical constants for the water,nw��� and kw��� published by Segelstein [12] wereused.
4. Results
The optical constants for polystyrene particles ob-tained using the inverse ADD along with the exactMie theory has been previously published [1]. ForPMMA, in this wavelength range, to our knowledgeonly the real part of the refractive index obtainedusing refractometer measurements has been pub-lished before [13]. Figure 1 shows the real and imag-inary parts of the refractive index estimated usingthe inversion method. In Fig. 1(a), the n��� valuesestimated in this study are compared with the pub-lished values of Sultanova et al. [13]. It can be seenthat while there are slight differences, the agreementis good considering the different methodologies usedto extract the refractive index.
Figures 2(a) and 2(c) show the values of n��� andk��� of polystyrene particles estimated using the RGDapproximation and those obtained using the Mie the-ory. In Fig. 2(a), it is seen that the n��� values esti-mated using the RGD approximation is slightlythough significantly higher than the Mie values be-
Fig. 2. Comparison between the complex refractive index esti-mated using the RGD approximation and the Mie theory for poly-styrene. (a) Real part, n, (b) percent relative error, �n � |nMie
� nRGD|�nMie � 100, (c) imaginary part, k and (d) percent relativeerror, �k � |kMie � kRGD|�kMie � 100.
Fig. 3. Complex refractive index of PMMA computed using Mietheory and the RGD approximation (a) real part n���, and (b)imaginary part k���.
tween 450 and 900 nm, whereas between 900 and1320 nm the difference is mostly within experimentalerror. The error over the wavelength range consid-ered is less than 2% as can be seen from Fig. 2(b). InFig. 2(c), it is seen that the RGD estimates of k��� areappreciably lower than the Mie estimates over theentire wavelength region considered with errorsabove 20% over most of the wavelengths considered.Similar trends are seen with the PMMA particles asseen in Figs. 3(a) and 3(b).
Next we consider the effect of approximating thephase functions, while the scattering and absorptioncross sections and the anisotropy factor are computedusing the Mie theory. Figure 4(a) compares the n���values estimated using the H-G and the H-Gm ap-proximation with those obtained using the Mie phasefunction. Both the H-G and the H-Gm functions over-estimate the values of n��� with the H-G approxima-tion showing better agreement with the exact Mievalues. From Fig. 4(b), it is seen that the H-G approx-imation gives values with less than 1.2% deviationover the wavelength range considered while theH-Gm approximation leads to less than 3% error. Forthe estimation of k���, it can be seen from Figs. 4(c)and 4(d) that the H-G approximation again performsbetter. Both approximations give larger errors be-yond �800 nm. Over the wavelength range consid-ered the H-G function gives errors of less than 15%,whereas with the H-Gm function the errors could beas high as 25% at some wavelengths. Figure 5 showsthe performance of these approximations when usedto estimate the optical constants of PMMA. Again theH-G approximation performs better than the H-Gmfor estimating n���. For k���, the agreement with theexact Mie theory estimates are very good for bothapproximations.
Table 1 summarizes the results of using the RGDand the phase function approximations in terms of
the average error and computing time for calculatingthe optical constants for 50 wavelengths. It is seenthat using the exact Mie theory to calculate singleparticle characteristics a, s, and g along with theH-G approximation to compute the phase functiongives the lowest average error both in n��� and k���.In this case, the average error in n��� is 0.62% com-
Fig. 4. Comparison between the optical constants estimated us-ing the Mie phase function, the H-G and the H-Gm approximationfor polystyrene. (a) n���, (b) percent relative error �n � |nMie
Fig. 5. Comparison between the optical constants estimated us-ing the Mie phase function, the H-G and the H-Gm approximationfor PMMA. (a) n��� and (b) k���.
Table 1. Performance Comparison for the Estimation of the OpticalConstants Using Mie Theory, RGD Approximation, Mie Theory
Combined with H-G, and H-Gm Approximation for the Phase Function
pared to 0.7% for the RGD, and 2.18% when H-Gm isused. The average error in k��� is 4.4% compared to23% when the RGD is used, and 7.9% when the H-Gmis used. In addition, Mie theory with the H-G approx-imation gives the greatest improvement in comput-ing time, 73% faster than using the exact Miecalculations for the phase function compared to 10%reduction in computing time when the RGD is used,and 3% reduction when the H-Gm phase function isused.
5. Discussion
To get a better understanding of the errors arisingdue to the approximations, we have to take a closerlook at the optical properties to which these approx-imations are applied. We first look at the effect ofusing the RGD approximation to calculate a and s.In Figs. 6(a) and 6(c) the scattering and the absorp-tion cross section (normalized by �a2), computedusing Mie theory and RGD approximation are com-pared for the polystyrene sample used in this studyfor the wavelength range of 450–1320 nm. In terms ofsize parameters this corresponds to a range of 1.409–4.222. The conditions of validity of the RGD approx-imation is given by |n� � 1| �� 1 and a�|n� � 1|�� 1. For the polystyrene suspension with a range ofsize parameters considered here, the range of valuesfor these terms are 0.1871 � |n� � 1| � 0.221, and0.2647 � a�|n� � 1| � 0.9375. These values do notstrictly satisfy the RGD conditions. Despite this, itcan be seen from Fig. 5(a) that the RGD values for s
are in good agreement with Mie theory. The percentdeviation [Fig. 6(b)] of s computed using the RGDapproximation from that calculated using the Mietheory is on average �8%. Figure 6(c) shows the nor-malized absorption cross section as a function of thesize parameter. It is seen that values obtained using
the RGD exhibits a large deviation from the Mie the-ory values with deviations higher than 40% at thelower size parameters and dropping to �20% towardthe high end of the size parameter as can be seen inFig. 6(d), with the average error of �30%. Such higherrors in absorption cross section may explain thevery large deviation in the estimated k��� valueswhen the RGD approximation is used.
Next we analyze the phase functions used in thiswork. Figure 7 compares the different phase func-tions: the H-G, H-Gm, and RGD approximations withthe Mie phase function for different values of sizeparameter and g that fall in the range of particle size,refractive index, and wavelength considered in thiswork. The parameters used in this analysis are givenin Table 2. Since the phase function varies by ordersof magnitude with the scattering angle, the y axis inthe plots of Fig. 7 are in log scale. Overall, the RGDphase function seems to be most accurate among thethree phase function approximations. Specifically atthe lower scattering angles, where the magnitude ofthe phase function is the highest, the RGD phasefunction matches the Mie phase function very well.Thus, the reason for the large deviation in the esti-mation of k��� appears to be solely due to the largeerror in the computation of a and to a lesser extenton the error in s using the RGD as was indicatedearlier through the examination of Fig. 6. Based onthis result, one could argue the use of the Mie the-ory to compute the absorption and scattering cross
Fig. 6. (a) Comparison between normalized scattering cross sec-tion computed by Mie theory and the RGD approximation. (b)Percent relative error in s. (c) Comparison between the normal-ized absorption cross section computed by Mie theory and the RGDapproximation. (d) Percent relative error in a.
Fig. 7. Comparison between phase functions computed usingMie, RGD, H-G, H-Gm, for particles of radius, a � 0.225 �m. (a)x � 4.1432, g � 0.86, and � � 450 nm. (b) x � 3.1731, g � 0.8, and� � 589 nm. (c) x � 2.1238, g � 0.69, and � � 880 nm. (d)x � 1.5574, g � 0.44, and � � 1200 nm.
Table 2. Data Used to Compute the Different Phase Functions Shownin Fig. 7
section and use the RGD phase function. However,since the reduction in computation time by usingthe RGD phase function is minimal, it does not leadto any advantage over the use of the exact Miecalculations.
Considering the H-G and H-Gm approximations, atfirst glance, it appears the H-Gm overall is a betterapproximation to the Mie phase function [except forthe case shown in Fig. 7(d)]. However, when we lookclosely, at scattering angles close to zero, where themagnitude of the phase function is orders of magni-tude higher than at larger angles, the H-G function ismuch closer (note that the y-axis is in log scale) thanthe H-Gm function. For example in Fig. 7(a), the Miephase function is �20, the H-Gm is �100, and theH-G value is �10. The error in the phase function willaffect the redistribution function through �m
* Eq. (12)and thus the calculated values of diffuse reflectanceand transmittance. Figure 8 shows the �m
* valuesobtained using the Mie, H-G, and H-Gm phase func-tions corresponding to Fig. 7(a). It is seen that theH-G phase function gives a slightly better agree-ment with the Mie values especially for m � 1–3after which the divergence from Mie values arequite high up to approximately m � 15. To considerthe impact of the errors in �m
* on the calculatedreflectance and transmittance, we take the condi-tions given by the first row in Table 2, which was
used to generate the phase function plot in Fig. 7(a).For this case the computed values of diffuse reflec-tance and transmittance are given in Table 3. It isseen that the H-G leads to a much smaller relativeerror in the computed values of reflectance andtransmittance compared to the H-Gm. Since theoptical constants are obtained by matching the ex-perimental values of diffuse reflectance and trans-mittance, larger errors in these calculated valuestranslate into larger errors in the estimated valuesof the optical constants.
6. Conclusions
The extraction of optical constants from multiple-scattered light using approximations to Mie theoryfor computing single particle characteristics was in-vestigated by considering two suspensions of spheri-cal particles viz., polystyrene and PMMA. It wasfound that, for the wavelength range considered�450–1320 nm�, using the exact Mie theory for calcu-lating the absorption and scattering cross sections,and the anisotropy parameter along with the H-Gapproximation to calculate the phase function gavegreatest improvement in terms of reduction in com-puting time, as well as the lowest error compared tousing the exact Mie theory when the reflectance andtransmittance measurements are inverted to esti-mate the optical constants. Analysis of the H-G andH-Gm phase functions in the context of how the de-viations of these approximations from the Mie phasefunction affects the redistribution function, thus thecomputed reflectance and transmittance measure-ments were carried out. The analysis suggests thaterrors in these computed values and thus on the es-timated optical constants depend mainly on howclosely the approximations match the Mie phasefunction at small scattering angles, since the phasefunction has a very high magnitude in this regioncompared to the larger scattering angles.
This work was funded by the EPSRC throughgrants GR�S50441�01 and GR�S50458�01. The au-thors also thank the Centre for Process Analytics andControl Technology for its support.
References1. M. A. Velazco-Roa and S. N. Thennadil, “Estimation of complex
refractive index of polydisperse particulate systems frommultiple-scattered ultraviolet-visible-near-infrared measure-ments,” Appl. Opt. 46, 3730–3735 (2007).
2. A. Ishimaru, Wave Propagation and Scattering in RandomMedia, IEEE�OUP Series on Electromagnetic Theory (IEEE,1997), pp. xxv, 574.
3. S. A. Prahl, “The adding-doubling method,” in Optical ThermalResponse of Laser Irradiated Tissue, A. J. Welch and M. J. C.van Gemert, eds. (Plenum, 1995), pp. 101–129.
4. C. Bohren and D. Huffman, “Rayleigh–Gans theory,” in Ab-sorption and Scattering by Small Particles (Wiley-VCH, 2004),pp. 158–165.
5. I. G. Henyey and J. L. Greenstein, “Diffuse radiation in thegalaxy,” Astrophys. J. 85, 70–83 (1941).
6. D. Toublanc, “Henyey–Greenstein and Mie phase functions inMonte Carlo radiative transfer computations,” Appl. Opt. 35,3270–3274 (1996).
Fig. 8. Expansion coefficients, �n*, computed using the �-M
method to compute the redistribution function of the Mie, H-G, andH-Gm phase function.
Table 3. Comparison between Total Diffuse Reflectance, R, andTransmittance, T Computed by the ADD Method Using Mie, H-G, and
7. W. M. Cornette and J. G. Shanks, “Physically reasonable an-alytic expression for the single-scattering phase function,”Appl. Opt. 31, 3152–3160 (1992).
8. M. Kerker, “Rayleigh–Debye scattering,” in The Scattering ofLight and Other Electromagnetic Radiation, M. L. Ernest, ed.(Academic, 1970), pp. 414–486.
9. W. J. Wiscombe, “The delta-M method: rapid yet accurate ra-diative flux calculations for strongly asymmetric phase func-tions,” J. Atmos. Sci. 34, 1408–1422 (1977).
10. J. H. Joseph and W. J. Wiscombe, “The delta-Eddington ap-
proximation for radiative flux transfer,” J. Atmos. Sci. 33,2452–2459 (1976).
11. K. N. Liou, “Approximations for radiative transfer,” in AnIntroduction to Atmospheric Sciences (Academic, 2002), pp.310–313.
12. D. Segelstein, “The complex refractive index of water,” M. S.thesis (University of Missouri, 1981).
13. N. G. Sultanova, I. D. Nikolov, and C. D. Ivanov, “Measuringthe refractometric characteristics of optical plastics,” Opt.Quantum Electron. 35, 21–24 (2003).
