June 1, 2004 / Vol. 29, No. 11 / OPTICS LETTERS 1239

Analytical techniques for addressing forward and inverseproblems of light scattering by irregularly shaped particles

Xu Li, Zhigang Chen, Jianmin Gong, Allen Taflove, and Vadim Backman

Department of Biomedical Engineering and Department of Electrical and Computer Engineering,Northwestern University, Evanston, Illinois 60208

Received December 24, 2003

Understanding light scattering by nonspherical particles is crucial in modeling the transport of light in realisticstructures such as biological tissues. We report the application of novel analytical approaches based onmodified Wentzel–Kramers–Brillouin and equiphase-sphere methods that facilitate accurate characterizationof light scattering by a wide range of irregularly shaped dielectric particles. We also demonstrate that theseapproaches have the potential to address the inverse-scattering problem by means of a spectral analysis ofthe total scattering cross section of arbitrarily shaped particles. © 2004 Optical Society of America

OCIS codes: 290.5850, 290.3200, 170.6510.

Most natural particles have nonspherical geometries,and therefore light scattering by irregularly shapedparticles is of significant research interest in a varietyof disciplines such as astronomy, meteorology, remotesensing, and biomedicine. The development andapplication of accurate, yet easy to use, inversion-enabling approximations to model light scattering bynonspherical particles is of critical importance. Inthis Letter we focus our discussion on two such approxi-mate methods, namely, a modif ied Wentzel–Kramers–Brillouin (WKB) approximation1 and the recentlyproposed equiphase sphere (EPS) approximation.2

The mathematical simplicity of these methods makesthem especially appealing for application to inverse-scattering problems.

Our analysis is focused on the spectral propertiesof total scattering cross section (TSCS), as the TSCSspectrum is of particular importance in applicationssuch as spectroscopic tissue diagnosis.3 – 6 Previously,the feasibility of applying the modified WKB approxi-mation and the EPS approximation to calculate TSCSspectra of inhomogeneous and spheroidal particles wasdemonstrated.2,7 In this Letter we apply these twomethods to predict the orientation-dependent TSCSspectra of particles of arbitrary irregular shapes. Wecompare the validity and accuracy of these methodsand the rigorous numerical solutions to Maxwell’sequations, using the f inite-difference time-domain8

(FDTD) method for a wide range of irregular particlesof sizes in the resonance range. We also demonstratethe potential of using these methods to probe the sizesand geometric characteristics of arbitrarily shapedparticles from their light-scattering properties.

In both the modif ied WKB and the EPS approxima-tions the TSCS is given by the sum of two terms, sur-face effect term ss

�s� and volume scattering term ss�v�.

When the high-frequency ripple structure in the TSCSspectrum is neglected (the ripple structure is typicallyaveraged out in realistic experimental measurements),surface term ss

�s� can be approximated as7

ss�s� � 2Sm�kd�2�22�3, (1)

where Sm is the maximum transverse cross sectionarea, d is the mean diameter, and k � 2p�l is the

0146-9592/04/111239-03$15.00/0 ©

wave number. In the modif ied WKB approximation,volume term ss

�v� is given by

ss�v� � 2 Re

√√√ ZZS

�1 2 exp�ik�n 2 1�L�r���d2r

!!!(2)

for a homogeneous particle. Here, S is the projectionarea of the particle in the plane transverse to the in-cident light, r is a vector in plane S, and L�r� is thepath length of the light ray traveling inside the particlethrough position r. L�r� is equivalent to the longitu-dinal extent of the particle corresponding to position rif one neglects the refraction effect in determining thedirection of light propagation.

The EPS approximation simplif ies volume termss

�v� by using an explicit expression2:

ss�v� � 2S�1 2 2n sin r�r 1 4n sin2�r�2��r2� , (3)

where r � kd�n 2 1� and d is equal to the diameterof the equiphase sphere of the particle. Equation (3)is similar to the approximation given by van de Hulstfor a homogeneous spherical particle with a low re-fractive index and predicts an oscillatory interferencestructure in the TSCS spectra with a frequency propor-tional to the particle’s size. Previously, Chen et al.2

demonstrated that TSCSs of spheroids with certainbounds on major-to-minor axis ratios can be approxi-mated by those of their equiphase sphere counterpartswith diameters d equal to the spheroids’ longitudinalextent. In this Letter we extend the concept of theequiphase sphere to arbitrary particles by choosingd as the longitudinal extent of an ellipsoid that opti-mally f its the geometry of the particle [with minimumroot-mean-square (rms) error].

We use Gaussian random spheres as a test modelwith which to evaluate the validity and accuracy ofboth approximate methods. Following the formula-tion given by Muinonen et al.,9 the shape of a Gaussiansphere is determined by radial relative standard devia-tion D and by radius-to-angular-distance correlationangle g. As shown in Figs. 1(a)–1(c), a highervalue of D results in an increased magnitude ofdeformation from spheres, whereas, as illustrated

2004 Optical Society of America

1240 OPTICS LETTERS / Vol. 29, No. 11 / June 1, 2004

Fig. 1. Representative Gaussian sphere geometries.(a)–(c) Gaussian spheres with increasing D (g is f ixed at70±). (d)–(f ) Gaussian spheres with decreasing g (D isfixed at 0.1).

in Figs. 1(d)–1(f ), reducing g leads to increasedhigh-degree terms of the surface perturbation andthus results in increased numbers of valleys and hillson the particle surface. Thus one can represent awide range of natural and artif icial shapes by varyingD and g in the Gaussian sphere model.

We used FDTD simulated data as benchmarks tocharacterize accurately the light-scattering propertiesof various Gaussian spheres. FDTD simulations wereconducted for Gaussian spheres with D ranging from0.1 to 0.9 and g ranging from 10± to 90±. The par-ticles were assigned a mean diameter of the order of3.5 mm and a refractive index of 1.5. Following thesame procedure as described previously,3,10 we calcu-lated the frequency-dependent TSCS from the FDTDsimulations.

Figure 2 compares TSCS spectra calculated byFDTD simulation by a modified WKB method andby the EPS approximation for four Gaussian sphereswith shapes that deviate progressively from that ofa sphere. In Figs. 2(a)–2(c) the EPS approximationgives a slightly more accurate estimation of the TSCSthan does the modif ied WKB method. The improvedaccuracy is due to the fact that the EPS approximationimplicitly takes into account the refraction effectfor shapes that deviate moderately from spheres orellipsoids. For the severely deformed particle shownin Fig. 2(d) the oscillatory periodicity of the TSCSspectrum obviously departs from its EPS counterpart.However, the modified WKB approximation still givesa fairly good estimation of the TSCS spectrum.

We used rms error to parameterize the accuracyof the analytical approximations compared with theaccurate FDTD data. To investigate the inf luence ofparticle irregularity on the approximation accuracy weplotted the rms errors for both approximations as func-tions of the particle’s shape parameters D [Fig. 3(a)]and g [Fig. 3(b)]. As the particles deviate fromspherical shapes, the accuracies of both approxima-tions degrade. However, it is evident from Fig. 3 thatboth methods give TSCS spectra with rms errors of,5% for a wide range of irregularly shaped particles.

Because of their accuracy demonstrated above, alongwith their mathematical simplicity, these two approxi-

mate methods have great potential for being appliedin inverse-scattering problems for irregularly shapedparticles. For the range of shapes at which the EPSapproximation is valid, the diameter of the correspond-ing equiphase sphere d (an estimation of the longitudi-nal dimension of the particle) can be derived from theoscillation period of the TSCS given by Eq. (3), and, byfurther f itting the TSCS curve, one can readily findtotal transverse area S.

We can note from Fig. 3 that the modif ied WKB ap-proximation has a greater range of validity than theEPS approximation. However, because this methodinvolves numerical integration the inversion procedureis not so straightforward as that provided by the EPSapproximation. Here we propose a novel approach toprobing the size and shape of a particle from its TSCSspectrum by using the modif ied WKB approximation,as introduced in the following discussion.

For a homogeneous particle the modif ied WKBapproximation given by approximation (1) and Eq. (2)can be written as

ss � 2Sm�kd�2�22�3 1 2S

2 2ZZ

Scos�k�n 2 1�L�r��d2r . (4)

Fig. 2. Comparison of TSCSs calculated from FDTD simu-lations, the modified WKB approximation, and the EPSapproximation for Gaussian spheres with different shapes.The particles are illuminated with vertically directed in-cident light: (a) D � 0.1, g � 50±; (b) D � 0.1, g � 30±;(c) D � 0.1, g � 20±; (d) D � 0.7, g � 70±.

Fig. 3. TSCS spectrum rms error of the modified WKBapproximation and the EPS approximation. FDTD simu-lation results are used as the benchmark data. rms errors(%) as functions of (a) g (D is fixed at 0.1) and (b) D (g isfixed at 70±).

June 1, 2004 / Vol. 29, No. 11 / OPTICS LETTERS 1241

Fig. 4. WKB-reconstructed area distribution of the longi-tudinal extent, ai�Li�, for Gaussian spheres with variousvalues of radial relative standard deviations and correla-tion angles: (a) D � 0.1, g � 25±; (b) D � 0.4, g � 70±;(c) D � 0.2, g � 30±; (d) D � 0.7, g � 70±.

By dividing the possible range of longitudinal extentL�r� into discrete values Li we can replace the in-tegration over total transverse area S in Eq. (4) byP

i ai cos�k�n 2 1�Li�, where coefficient ai is the por-tion of transverse area that corresponds to Li. Thus,for a given wavelength lj , Eq. (4) can be rewritten as

ss�lj � �Xi2ai cos�2p�n 2 1�Li�lj �

1 2Sm�pd�22�3�lj �2�3 1 2S . (5)

If n is known, for a number of preselected values ofLi and wavelengths lj a set of linear equations can beformed from Eq. (5) to solve area coeff icients ai andtotal transverse area S simultaneously.

Because linear equation (5) constitutes an ill-posedproblem, one must employ additional a priori infor-mation to obtain a stable solution. We added twoimportant constraints, i.e., a nonnegativity constraint(ai $ 0, S . 0, Sm . 0, and d . 0) and an area-consistency constraint (

Pi ai � S 6 d, where d is a

small value chosen empirically) to form a constrainedlinear least-squares problem that can be solved withan active-set method.10

Figure 4 plots the area distribution of longitudinalextent reconstructed from TSCS spectra calculated byFDTD simulations for four Gaussian spheres. For allfour cases the inversion results are consistent withthe true distribution calculated from the original ge-ometry. From these graphs of area coeff icients, thevolume and the total transverse area of the particles

can easily be derived. Furthermore, the longitudinal-extent distribution profiles are especially useful indica-tors with which to infer the shape characteristics, suchas eccentricity and surface deformation, of particles.For the inversion results presented in this Letter, dwas chosen to be S�20. We noted that the robustnessof this approach is not sensitive to the choice of d.

In summary, in this Letter we have introduced novelanalytical approaches, based on the modified WKBapproximation and the EPS approximation, to thecharacterization of light scattering by a wide range ofirregularly shaped dielectric particles. The discus-sion in this Letter was focused on light scattering byisolated, nonabsorbing single particles of arbitraryshape and orientation. Improved knowledge of theoptical scattering properties of such particles formsthe foundation for understanding of light scatteringby ensembles of random particles. In turn, thisfacilitates general advances in the optics of randommedia, including remote sensing, light propagationin the atmosphere, random-media lasing, and opticaltissue diagnosis.

This study was supported in part by NationalScience Foundation grants BES-0238903 and ACI-0219925. We thank the authors of the originalGaussian-sphere generation code, K. Muinonen andT. Nousiainen, for making the code publicly available.X. Li’s e-mail address is xuli@northwestern.edu.

References

1. J. D. Klett and R. A. Sutherland, Appl. Opt. 31, 373(1992).

2. Z. Chen, A. Taf love, and V. Backman, J. Opt. Soc.Am. A 20, 88 (2004).

3. L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R.Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima,T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, andM. S. Feld, Phys. Rev. Lett. 80, 627 (1998).

4. A. Wax, C. Yang, and J. A. Izatt, Opt. Lett. 28, 1230(2003).

5. Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg,A. K. Kromine, K. Chen, and V. Backman, IEEE J. Sel.Top. Quantum Electron. 9, 243 (2003).

6. J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra,T. Aida, and J. P. Freyer, J. Biomed. Opt. 7, 378 (2002).

7. Z. Chen, A. Taf love, and V. Backman, Opt. Lett. 28,765 (2003).

8. A. Taf love and S. Hagness, Computational Electrody-namics: the Finite-Difference Time-Domain Method(Artech House, Boston, Mass., 2000).

9. K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, andJ. I. Peltoniemi, J. Quant. Spectrosc. Radiat. Transfer55, 577 (1996).

10. P. E. Gill, W. Murray, and M. H. Wright, PracticalOptimization (Academic, London, 1981).