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Comparing Monte Carlo simulation and pseudospectral time-domain

numerical solutions of Maxwell’s equations of light scattering

by a macroscopic random medium

Snow H. Tsenga兲and Boyeh Huang

Graduate Institute of Photonics and Optoelectronics and Department of Electrical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevett Road, Taipei 106, Taiwan

共Received 5 June 2007; accepted 10 July 2007; published online 1 August 2007兲

The Monte Carlo simulation of light scattering by a cluster of dielectric spheres is compared with numerical solutions of Maxwell’s equations via the pseudospectral time-domain technique. By calculating the total scattering cross-section 共TSCS兲 spectrum, respectively, the spectral light scattering characteristics are determined. Since the Monte Carlo simulation falls short to accurately account for coherent interference effects, it is shown that the Monte Carlo simulation yields TSCS spectra that significantly deviate from the numerical solutions of Maxwell’s equations. Therefore, it is necessary to resort to Maxwell’s equations in order to accurately determine the light scattering characteristics of a macroscopic geometry. © 2007 American Institute of Physics.

关DOI:10.1063/1.2767777兴

The problem of light scattering through macroscopic random media is commonly found in nature, including non-periodic structures such as clouds, biological tissues, etc. However, this problem has not been rigorously studied; due to the extreme complexity involved, heuristic approxima-tions based on the radiative transfer theory1 are commonly employed. Among such heuristic methods, the Monte Carlo technique2–6is widely used for the problem of light scatter-ing through macroscopic random media, particularly in the area of tissue optics.

However, all heuristic methods are fundamentally lim-ited by the imposed hypotheses. For example, in the study of light scattering by random media, the Monte Carlo technique assumes independent scattering events between stochastic,

pointlike scatterers; such assumptions involve modifications

of the physical nature of the original problem, and fall short to accurately account for all the physics of the problem, in-cluding polarization and coherent interference. As a result, the validity and applicability of such heuristic methods re-main to be determined.7,8In order to accurately account for the coherence effects and near-field interactions, a rigorous research method based on Maxwell’s equations is preferred. In this letter, we simulate the light scattering character-istics of closely packed dielectric spheres by employing two different approaches: the Monte Carlo technique and the nu-merical solutions of Maxwell’s equations via the pseu-dospectral time-domain 共PSTD兲 technique,9 respectively. The Monte Carlo technique is a widely used heuristic ap-proach based on the radiative transfer theory; by assuming light undergoes a sequence of independent scattering events, light scattering through random media is treated statistically.10,11On the other hand, the PSTD technique is a numerical method where the light scattering problem is simulated by solving Maxwell’s equations numerically. In this letter, we report the application of both simulation tech-niques, where the total scattering cross-section共TSCS兲 spec-tra are calculated and compared.

Light scattering by a cluster of dielectric spheres is simulated using the Monte Carlo technique. The Monte Carlo technique has been widely used in simulating light scattering through random media.2,6,12,13For a specific wave-length, a total of 1 000 000 photons are injected into a cluster 共cluster diameter d=50␮m兲 of N dielectric spheres; each sphere has a diameter d = 6␮m, with a refractive index n = 1.2. Each photon scattering angle is determined with a ran-dom number generator based on the Mie scattering phase function of a single dielectric sphere. The TSCS spectrum from 30 to 300 THz is calculated.

Alternatively, we calculate the light scattering character-istics of a cluster of dielectric spheres by employing the PSTD technique. The PSTD simulation is a grid-based tech-nique capable of solving Maxwell’s equations numerically. By assigning the spatial distribution of the refractive index, the PSTD technique can accurately determine the light scat-tering characteristics of macroscopic geometries, including irregular geometries. In this letter, we report the employing of the PSTD technique on a parallel computer to model

full-vector, three-dimensional共3D兲 scattering of light by a cluster

of dielectric spheres in free space. A standard anisotropic perfectly matched layer absorbing boundary condition14 is implemented to absorb outgoing waves to simulate light scattering in free space. A near-to-far-field transformation15 is employed, allowing scattered light for a broadband of wavelengths at all angles to be obtained in a single simula-tion. Based on Maxwell’s equations, the PSTD simulation is essentially an idealized optical experiment in a noiseless en-vironment, where the coherent effects can be accurately determined.

By employing the PSTD technique, light scattering by a cluster of closely packed dielectric spheres in free space is simulated, yielding the TSCS spectrum from 0.5 to 300 THz 共␭0= 600␮m – 1␮m兲 with a resolution of 0.5 THz. With a grid resolution of 0.33␮m, a PSTD simulation of light scat-tering by a 共60␮m兲3 _{cluster typically takes} _{⬃12 h with a} parallel computer cluster of 20 2.4 GHz Pentium 4 Xeon processors.

a兲_{Electronic mail: snow@cc.ee.ntu.edu.tw}

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A validation of the PSTD simulation of light scattering is shown in Fig. 1. Light scattering by seven randomly posi-tioned, dielectric spheres is calculated using PSTD simula-tions and compared with the multisphere expansion.16 The TSCS spectrum is determined. As shown in Fig. 1, the PSTD-computed TSCS spectrum shows good agreement with the analytical expansions of Maxwell’s equations.

The Monte Carlo technique and the PSTD technique are employed to simulate light scattering by a cluster of dielec-tric spheres, respectively. Firstly, as shown in Fig. 2, the TSCS spectra of a cluster of N dielectric spheres are obtained using the Monte Carlo technique. Five cases are shown: N = 25, 50, 75, 125, and 192. Notice that for sparse distribution of dielectric spheres共e.g., N=25兲, light impinging the cluster mostly encounters a single scattering event, and therefore the TSCS spectrum resembles the TSCS of a single dielectric sphere, showing that the Monte Carlo simulation yields re-sults consistent with Maxwell’s equations. However, as the cluster becomes closely packed with less space between ad-jacent spheres, the Monte Carlo assumption of “light under-goes independent scattering events” no longer holds and the

TSCS spectra obtained by the Monte Carlo simulation be-come less valid.

Secondly, as shown in Fig.3, the TSCS spectra are cal-culated using the PSTD technique. By numerically solving Maxwell’s equations, the TSCS spectrum of an arbitrary ge-ometry is accurately determined. Notice that, due to the co-herence effect, the magnitude of the TSCS spectrum based on Maxwell’s equations is larger than the Monte Carlo re-sults. Furthermore, as the cluster geometry becomes packed with more dielectric spheres, the TSCS spectra gradually ex-hibit optical characteristics that are significantly different from the Monte Carlo simulation共as shown in Fig.2.兲

To determine the relationship of the TSCS spectrum of a cluster of dielectric spheres and the size of the constituent spheres, we compare light scattering characteristics of two clusters, each consists of monodisperse dielectric spheres. As shown in Figs.4共a兲and4共b兲, light scattering by a cluster of

d = 6␮m and d = 8␮m dielectric spheres is simulated using the PSTD technique, respectively. Each 共60␮m兲3 cluster consists of N randomly positioned, closely packed, homoge-neous, dielectric共n=1.2兲 spheres of diameter d. Notice that even for an optically think cluster of closely packed dielec-tric spheres, the TSCS spectrum exhibits significantly differ-ent characteristics directly related to the size of the constitu-ent spheres of each cluster.

Research findings show that the Monte Carlo simulation yields TSCS spectrum that significantly differs from the nu-merical solutions of Maxwell’s equations, owing to the heu-ristic assumptions involved. Based on the radiative transfer theory, the Monte Carlo simulation of light scattering is treated as an energy transport problem by omitting the wave nature of light. Furthermore, light scattering through macro-scopic random media共e.g., biological tissues兲 is heuristically approximated as a stochastic sum of sequences of

indepen-FIG. 1.共Color online兲 Validation of the 3D PSTD simulation. Light scatter-ing by a cluster of seven randomly positioned, 6␮m diameter, n = 1.2 di-electric spheres is simulated, with a grid resolution of dx = 0.33␮m. The PSTD-computed total scattering cross-section共TSCS兲 as a function of fre-quency is compared with the multisphere expansion, which is based upon numerical expansions of Maxwell’s equations.

FIG. 2.共Color online兲 TSCS spectra obtained from the Monte Carlo simu-lation. By employing the Monte Carlo technique, the TSCS spectra are obtained, each corresponding to a共diameter d=50␮m兲 cluster consisting of

N randomly positioned, n = 1.2,共diameter d=6␮m兲 dielectric spheres. Five cases are shown共from bottom to top兲: N=25, 50, 75, 125, and 192.

FIG. 3.共Color online兲 PSTD-computed TSCS spectra. Each TSCS spectra corresponds to a共overall diameter d=50␮m兲 cluster consisting of N ran-domly positioned, n = 1.2,共diameter d=6␮m兲 dielectric spheres. Five cases are shown共from bottom to top兲: N=25, 50, 75, 125, and 192. Notice that as more dielectric spheres are packed together, the TSCS spectrum gradually exhibits optical characteristics due to the overall geometry which signifi-cantly differs from the TSCS spectrum obtained in the Monte Carlo simulation.

051114-2 S. H. Tseng and B. Huang Appl. Phys. Lett. 91, 051114共2007兲

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dent scattering events. By omitting the wave nature of light, the complex light scattering problem is reduced to a simpler problem that is solvable but distorting the physics of the problem. As a result, such heuristic approximation in prin-ciple excludes the possibility of accurately determining the coherent effects of light using the Monte Carlo technique.

For sparse distribution of spheres 共e.g., N=25兲, the Monte Carlo simulation yields a TSCS spectrum that is simi-lar to the TSCS spectrum determined by the PSTD simula-tion based on Maxwell’s equasimula-tions, as shown in Fig.2. This resemblance is anticipated, since with only a few dielectric spheres spaced far apart in space, the Monte Carlo assump-tion of independent scattering events is satisfied. However, for dielectric spheres closely packed in space, the assumption of light undergoes independent scattering events breaks down. By comparing Figs.2and3, it is readily seen that the TSCS spectrum determined via the Monte Carlo simulation deviates significantly from the PSTD numerical solutions of Maxwell’s equations. As the number of spheres increases and the cluster becomes closely packed, the overall TSCS spec-trum is gradually dominated by the optical characteristics of the cluster as a whole and shows less of the characteristics due to individual spheres.

Specifically,共in Fig.3兲 as the N increases, a TSCS peak

gradually forms around 55 THz; this peak is due to the co-herent interference effects of the overall cluster geometry as a whole.17 For long wavelengths, the electromagnetic wave is insensitive to the microscopic structural details and reacts to the cluster geometry as a whole. This phenomenon is simi-lar to the two-dimensional case, as reported in Ref.17. In addition, notice that the amplitude of the Monte Carlo calcu-lated TSCS spectra 共Fig. 2兲 is significantly lesser than the

PSTD calculated TSCS spectra 共Fig. 3兲. This difference is

anticipated because the Monte Carlo simulation treats each “photon” independently and does not accurately account for the coherent wave interference effects of light. As a result, the Monte Carlo simulation of light scattering falls short to determine the optical coherent effects.

Lastly, as shown in Fig. 4, the TSCS spectra exhibit optical characteristics that are related to the specific micro-scopic information of the cluster geometry共e.g., size of the constituent spheres兲. Such optical characteristics may pro-vide essential information for innovative optical techniques. However, due to the complexity involved, a thorough analy-sis is required. Further analyanaly-sis is currently ongoing and will be reported in future publications.

In summary, we report the comparison of the Monte Carlo simulation and PSTD numerical solutions of Max-well’s equations for the problem of light scattering by a clus-ter of monodisperse, dielectric spheres. Due to the heuristic assumptions, the Monte Carlo simulation falls short to ac-count for the wave interference phenomenon. As a result, Monte Carlo simulation of light scattering yields optical characteristics that deviate from the numerical solutions of Maxwell’s equations. On the other hand, based on Maxwell’s equations, the PSTD technique is robust and applicable to light scattering problems of macroscopic arbitrary geometry. To accurately determine the coherent optical characteristics of macroscopic random medium, it is necessary to resort to Maxwell’s equations.

The authors thank the Taiwan National Science Council under Grant No. 95-2112-M-002-039 for the support on this research. In addition, the authors would like to extend special thanks to all the computing facilities provided by the Na-tional Taiwan University Computing Center.

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051114-3 S. H. Tseng and B. Huang Appl. Phys. Lett. 91, 051114共2007兲