The complex refractive index (m ¼ nðλÞ þ ikðλÞ) canprovide important information required for the con-trol of processes involving industrial and biologicalsuspensions. Also, for the estimation of particle sizeand size distribution using light scattering methods,accurate values of the optical constants (complex re-fractive index) are required [1]. It is highly desirableto obtain the optical constants without submittingthe samples to high dilutions. A method that utilizesthe radiative transfer theory to model multiple scat-tering in combination with theories to describe singleparticle characteristics have been developed for ex-tracting the complex refractive index of sphericalparticles in suspensions [2–4]. For most biologicalsuspensions, the particles are not spherical and atheory to account for the particle shape has tobe used.
We present a general method for the estimation ofoptical constants in the UV-Vis-NIR region of non-spherical particles in a suspension at concentrationssuch that multiple scattering is significant. Theoptical constants are obtained by an inversion tech-nique using the adding-doubling method to solve theradiative transfer equation [5] in combination withthe single scattering theories for modeling scatteringby nonspherical particles. The method is thenapplied to obtain the optical constants of Bacillussubtilis spores in the wavelength region 400–1200nm from measurements of total diffuse reflec-tance and transmittance using an integrating spheresetup. Two methods for describing scattering by sin-gle scattering are considered: the T-matrix method[6,7] and the less accurate but computationally fasterRayleigh–Gans–Debye (RGD) approximation [8,9].
2. Method
The radiative transfer equation (RTE) is solvedusing the adding-doubling method to obtain simu-lated values of total diffuse reflectance (Rtds) and
transmittance (Ttds). The inputs required for the cal-culations are the mean diameter and standard devia-tion (for polydisperse samples) of particles, particleshape, cuvette pathlength (ℓ), and the optical con-stants, nwðλÞ and kwðλÞ, of water. In addition, initialguess values n0ðλÞ and k0ðλÞ of the optical constantsof the particle have to be provided. These inputs areused to compute the single particle parameters—theabsorption cross section (σabs), the scattering crosssection (σsca), and the anisotropy factor g. The albedoα ¼ σsca=ðσabs þ σscaÞ and the optical depth τ ¼ ðσabs þσscaÞℓ are then computed and, along with g, form theinputs to the adding-doubling routine. The outputsfrom this routine (Rtds and Ttds) are then comparedwith the measured values (Rtdm and Ttdm):
X¼ absðRtdm − RtdsÞ þ absðTtdm − TtdsÞ: ð1Þ
The guess values are then updated and the iterationscarried out until convergence (Σ ≤ 1:0e − 7) isachieved. This iteration was carried out using thefunction “fmincon” of the MATLAB Optimizationtoolbox. Previously, this inversion scheme was usedto extract the optical constants of spherical particles[3,4]. The single particle parameters in those studieswere computed using the exact Mie theory for sphe-rical particles [3] and approximations to speed up thecalculations [4]. In this study, the inversion scheme isextended to apply to nonspherical particles by imple-menting the T-matrix method and the RGD approx-imation to compute the single particle parametersfor ellipsoids.
A. Particle Scattering Parameters Using the T-MatrixMethod
In the T-matrix method [6,7], the incident andthe scattering fields are written in terms of vectorspherical functions Mmn and Nmn:
EincðrÞ ¼X∞n¼1
Xnm¼−n
½amnRgMmnðκrÞ þ bmnRgNmnðκrÞ�;
ð2Þ
EscaðrÞ ¼X∞n¼1
Xnm¼−n
½pmnMmnðκrÞ þ qmnNmnðκrÞ�;
jr̂j > r0: ð3Þ
Making use of the linearity property of the Maxwellequations and its boundary conditions, the coeffi-cients of the scatterer field, p and q can be linearlyrelated to the incident field coefficients, a and b,by a transition matrix (T matrix):
�pq
�¼
�T11 T12
T21 T22
��ab
�: ð4Þ
The extinction and the scattering cross sectionaveraged over the uniform orientation distributionof a nonspherical particle are given by
σext ¼ −2πκ2 Re
Xnmax
n¼1
Xnm¼−n
½T11mnmn þ T12
mnmn�; ð5Þ
σsca ¼ 2πκ2
Xnmax
n¼1
Xnmax
n0¼1
Xmaxðn;n0Þ
m¼−maxðn;n0Þ
Xi;j¼1;2
jTijmnmn0 j2; ð6Þ
where nmax is the dimension of the matrix T. Theabsorption cross section is computed as
σabs ¼ σext − σsca: ð7Þ
The phase function in the T-matrix method is thefirst element (a1ðθÞ) of the Stokes scattering matrix.The asymmetry factor g ¼ hcos θi in the T-matrixmethod is computed by
g ¼ 12
Z þ1
−1a1ðθÞ cos θdθ: ð8Þ
Because of computational considerations, the phasefunction was calculated using the Henyey–Greenstein function. It has been shown [3], usingthe case of spherical particles, that the error intro-duced due to this approximation in the estimationof optical constants is small. The T-matrix methodwas implemented in MATLAB using the methoddescribed by Mischenko et al. [6,7,10].
B. Particle Scattering Parameters Using the Rayleigh–Gans–Debye Approximation
The RGD approximation applies when the followingconditions are satisfied:
j ̑n − 1j ≪ 1 and xj ̑n − 1j ≪ 1; ð9Þ
where ̑n is the relative refractive index of the particlewith respect to the medium and x is the size para-meter. Under these conditions, the differential scat-tering cross section Fun for unpolarized incidentwave is given by [8,9]:
Fun ¼ κ48π2 V
2pð1þ cos2θÞðmr − 1Þ2PðθÞ; ð10Þ
where mr is the complex refractive index of the par-ticle relative to that of the medium, κ ¼ 2π=λ0, λ0 isthe wavelength of incident radiation in free space,and Vp is the volume of the particle, which for a pro-late or oblate ellipsoid is given by
where a, c, and b are the dimensions of the particlealong the three axes. The spores are modeled as pro-late spheroids, for which a ¼ b < c. The form factorfor an ellipsoid of revolution is given by [8]
PðθÞ ¼�3
u3 ðsinu − u cosuÞ�
2; ð12Þ
u ¼ 2ka sinθ2
�cos2β þ b2
a2 sin2β�
1=2; ð13Þ
cos β ¼ − cos α sin θ2þ sin α cos θ
2cosφ; ð14Þ
where α is the angle of the incident radiation. Forperpendicular radiation α ¼ π=2, and for parallel in-cident radiation α ¼ 0. The form factor for randomlyoriented particles can be computed using [8,9]
PðθÞrandomly oriented ¼Zπ=2
0
PðθÞ sin βdβ: ð15Þ
The scattering and absorption cross sections aregiven by
σsca ¼Z2π
0
Zπ
0
Fðθ;a; ̑n; λÞ sin θdθdϕ; ð16Þ
σabsðλÞ ¼4πnðλÞkðλÞVp
λ : ð17Þ
The phase function in the RGD approximation isgiven by
pðcos θÞ ¼ 1
4π2a4
Vp2x4ð ̑n − 1Þ2ð1 − cos2θÞPðθÞ
σsca: ð18Þ
The anisotropy factor g ¼ hcos θi is given by
g ¼ hcos θi ¼Z2π
0
Zπ
0
pðcos θÞ cos θ sin θdθdϕ: ð19Þ
3. Experiments
Bacillus subtilis spores in water suspension wereobtained from the Institute for Cell and MolecularBiosciences, Faculty of Medical Sciences, NewcastleUniversity. The population of the spores in water sus-pension was 1:1 × 1010 cfu=ml (colony forming unitsper milliliter), where cfu is equivalent to the number
of spores. Figure 1 shows a scanning electron micro-graph (SEM) of the spores used in this study. In thisfigure, it can be seen that the spores have a rod shapethat can be simulated as a prolate ellipsoid [11]. Toobtain an estimate of the error in the extractedoptical constants, two more samples were preparedby diluting the original suspension in deionizedwater to obtain samples with concentrations of 5:5 ×109 cfu=ml and 2:75 × 109 cfu=ml. The sizes of thespores were estimated by measuring the lengthand the width of 56 spores from several SEM pic-tures. The mean of the longest axes is 2 � amean ¼1:39 μm with a standard deviation of sa ¼ 0:1 μm,and the mean of the width is 2 � bmean ¼ 0:71 μmwitha standard deviation of sb ¼ 0:06 μm. Given thenarrow distribution, the particles were consideredmonodisperse.
Figures 2(a) and 2(b) show measurements of totaldiffuse reflectance and transmittance of the spores insuspension obtained at the three concentrations. Thesamples were measured in a 2mm optical glass cuv-ette using a DRA-2500 accessory (single integratingsphere setup) attached to the Cary 5000i UV-VIS-NIR spectrophotometer over a wavelength rangeof 300–1200nm.
4. Results and Discussion
A. Optical Constants of Bacilis Subtilis Spores
The optical constants for Bacillus subtilis sporeswere estimated by applying the inverse methodusing both the T-matrix approach and the RGDapproximation for ellipsoidal particles to computethe albedo (Wo), optical depth (τ), and the anisotropyfactor (g). The estimated values of nðλÞ and kðλÞ areshown in Figs. 3(a) and 3(b). The values reportedwere estimated as the mean value over three differ-ent concentrations (1:1 × 1010, 5:5 × 109, and2:75 × 109 cfu=ml), and the error bars are plottedas two times the standard deviation computed usingthe values estimated at the three concentrations. In
Fig. 1. SEM of Bacillus subtilis spores in the suspension used inthis study.
these figures the optical constants published byTuminello et al. [12] for Bacillus subtilis spores inthe region 200 to 2500nm are also included for com-parison purposes. It is seen that the inversionscheme using the RGD approximation for single par-ticle characteristics leads to nðλÞ and kðλÞ valueswhich are similar to the ones obtained using the T-matrix method, with the differences between thembeing less than the experimental error.Comparing the values obtained in this study with
those of Tuminello et al. from Fig. 3(a), it is seen thatthe inversion scheme used here results in nðλÞ valuesthat are significantly higher. According to the cur-rent work, in the wavelength range 400–1200nm,nðλÞ varies from 1:63 − 1:55, whereas Tuminelloet al. obtained values around 1.52. The differencein the values estimated by the two methods isprobably due to the differences in the methodology.Tuminello et al. computed the imaginary part kðλÞby measuring collimated transmittance and assum-ing that the scattering is negligible as
kðλÞ ¼ λ lnðTwater=TÞ4πf vd
; ð20Þ
where Twater is the transmittance of pure water andT is the transmittance of the spores suspension, f v isthe concentration, and d is the path length of thesample. The nðλÞ values were then computed usingthese values of kðλÞ through the Kramers–Kronigrelations.
Another work that is found in the literature forestimating nðλÞ is that of Katz et al. [13]. They esti-mated the refractive index of Bacillus subtilis sporesusing the Gaussian ray approximation of anomalousdiffraction theory for two cases: assuming the sporesas solid spherical particles and as two concentricnonabsorbing spheres in the wavelength range from400–1000nm. They reported a refractive index valueof 1.55 for the first case, and 1.515 for the second.
From Fig. 3(b) it is seen that compared to the kðλÞvalues reported by Tuminello et al., the values ob-tained in this work are significantly smaller. Againthese differences could be attributed to the differentmethodologies used.
Fig. 2. (a) Total diffuse reflectance and (b) total diffuse transmit-tance for three different concentrations of Bacillus subtilis sporesin water suspensions.
Fig. 3. Optical constants (a) n and (b) k as a function of the wa-velength of Bacillus subtilis spores in water suspensionsestimated using the T-matrix method and RGD approximationalong with those reported by Tuminello et al. [12].
The close agreement of nðλÞ and kðλÞ obtainedthrough the RGD approximation compared tothose obtained using the Tmatrix is surprising, espe-cially since, for the wavelength range and particlesize considered, the validity conditions for theRGD approximation, viz. j̑n − 1j ≪ 1 andxj ̑n − 1j ≪ 1, are not met. For the wavelength regionconsidered in this work, we get 0:1831 ≤ j ̑n − 1j ≤0:2065 and 0:88 ≤ xj ̑n − 1j ≤ 3:08 from which it canbe seen that the second condition is not met by theBacillus subtilis suspension. There could be tworeasons for this agreement. One is that the errorin calculating the single particle characteristicsσabs, σsca, and g using the RGD approximation isnot large enough to cause significant deviations inthe extracted optical constants despite the suspen-sion not satisfying the validity conditions for theRGD approximation, or it may be due to a cancella-tion of errors. To investigate the reason, the values ofσabs, σsca, and g obtained using the two methods werecompared.
B. Comparison of Single Particle Parameters Computedby the T-Matrix and Rayleigh–Gans–Debye Approximation
Figure 4(a) shows the absorption cross section σabsfor the Bacillus subtilis spores, modeled as prolateellipsoids, using the T matrix and the RGD approx-imation for the range of size parameters encounteredin this study. It is observed that the values obtainedby the T-matrix approach are much higher than thevalues given by the RGD approximation. The percenterror in using the RGD approximation instead of theT-matrix method to compute the absorption crosssection is over 40%, with the error increasing asthe size parameter is increased [Fig. 4(b)]. In the caseof scattering cross section, it is seen from Fig. 5(a)that the RGD underestimates the magnitude ofσsca for most of the size parameter range relevantfor the current study. The error decreases with in-creasing size parameter [Fig. 5(b)] and at a size para-meter of about 13.5, there is a crossover point wherethe RGD approximation starts to overestimate thescattering cross section. For the anisotropy factor,
Fig. 4. (a) Absorption cross section computed using the T-matrixmethod and the RGD approximation for prolate ellipsoids in water.(b) Percent error in using the RGD approximation instead of the T-matrix method.
Fig. 5. (a) Scattering cross section computed using the T-matrixmethod and the RGD approximation for prolate ellipsoids.(b) Percent error in using the RGD approximation instead ofthe T-matrix method.
the RGD overestimates the values over the entiresize parameter range considered, though the errorsover the entire range is less than 5%.This analysis indicates that the errors in using the
RGD approximation in computing single particlecharacteristics, especially the scattering and absorp-tion cross sections, are high for the range of size para-meters encountered in this study. Both theseparameters are underestimated by the RGD approx-imation. The fact that, despite such large discrepan-cies, the estimated optical constants obtained usingthe RGD approximation does not significantly differfrom those obtained using the T-matrix methodsuggests that the agreement is probably due to can-cellation of errors. Another reason could be that theextracted optical constants are not very sensitive toerrors in the single particle parameters. However,this seems unlikely, especially for the estimation ofkðλÞ. Studies with spherical particles [3] have shownthat when the RGD approximation is used insteadof the exact Mie theory for computing particle
characteristics it resulted in very large errors (about20%) in kðλÞ, and the errors could be traced backto the errors in the computation of single particlecharacteristics.
5. Summary and Conclusions
An inversion method to estimate the optical con-stants of nonspherical particles from measurementsin the multiple scattering regime that uses the radia-tive transfer theory along with the T matrix or theRGD approximation for describing single particlescattering characteristics was implemented. Thismethod was applied to obtain the values of nðλÞand kðλÞ of Bacillus subtilis spores in the wavelengthregion of 400–1200nm. It was seen that the inversionscheme using the RGD approximation for single par-ticle characteristics leads to nðλÞ and kðλÞ values thatare similar to the ones obtained using the T-matrixmethod, with the differences between them beingless than the experimental error. This work leadsto values of nðλÞ that are higher than those foundin the literature [12,13]. The values of kðλÞ obtainedin this work are significantly smaller than has beenreported previously by [13].
This work was funded through Engineering andPhysical Science Research Council (EPSRC) grantsGR/S50441/01 and GR/S50458/01 and Marie CurieFP6 (INTROSPECT). The authors thank ColinHarwood at the Institute for Cell and MolecularBiosciences, Newcastle University for providingtheBacillus subtilis spores sample used in this study.
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