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Probability density function of the intensity scattered by Rayleigh-particle aggregates. Evolution with optical properties
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Journal of Quantitative Spectroscopy & Radiative Transfer 101 (2006) 383–393 www.elsevier.com/locate/jqsrt
Probability density function of the intensity scattered by Rayleigh-particle aggregates. Evolution with optical properties O. Merchiers, J.M. Saiz, F. Gonza´lez, F. Moreno Grupo de O´ptica, Departamento de Fı´sica Aplicada, Universidad de Cantabria, Avda de los Castros, 39005 Santander, Spain
Abstract We study the probability density function of the statistical fluctuations of the intensity scattered by an aggregate freely floating in space and constituted by Rayleigh particles under the dipole approximation. Its evolution as a function of the optical properties of the particles (polarizability) and their separation distance is analyzed. Aggregate geometries with two and three particles will be considered. The influence of the multiple scattering effect on the statistics of the scattered intensity is especially studied. r 2006 Elsevier Ltd. All rights reserved. Keywords: Statistics; Light scattering; Multiple scattering; Aggregates
1. Introduction During the last decades, research on the interaction between electromagnetic radiation and matter has experienced important advances from the study of the scattering of electromagnetic waves from microstructures. In this sense, a number of theoretical and experimental techniques for obtaining information about the scattering system have been developed [1]. Many of them are based on the measurement of the radiation as a function of the scattering angle and concentrate on a particular scattering angle. This is the case of backscattering, for its interest in practical applications, like LIDAR [2] and astrophysics [3]. In these applications the scattering systems involved, present important multiple scattering effects, like the enhanced backscattering [4,5] and the negative polarization branch [6], for which many authors have proposed theoretical models (usually complicated geometries and/or particle aggregates) and numerical methods to solve this electromagnetic problem. Statistical properties of the scattered radiation have also been proposed for obtaining information about different scattering systems. For example, the statistics of the fluctuations of the scattered intensity can give information about the density of the scatterers for the non-Gaussian regime (low number of scatterers) [7]. This is because in this regime, the probability density function (PDF) of the intensity, and then its factorial moments, depends on the number of scatterers. On the contrary, in the Gaussian regime the PDF of the intensity is exponential and the factorial moments are given by the well-known factorial law. More recently, Corresponding author.
E-mail addresses: [email protected] (O. Merchiers), [email protected] (F. Moreno). 0022-4073/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2006.02.066
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the case of particles with inclusions has also been analyzed through the statistical fluctuations of the scattered intensity [8]. In addition, the measurement of the statistical properties of polarization fluctuations has shown to be very useful for determining morphological properties of anisotropic particles [9–12]. In a previous paper [13], statistical techniques based on the measurement and analysis of statistical parameters like the second order factorial moment nð2Þ of the intensity fluctuations (both co- and crosspolarized) and the probability of measuring no cross-polarized intensity PðI cross ¼ 0Þ were presented for scattering systems constituted by particles aggregates, where multiple scattering is present. Bearing in mind this kind of systems, the objective of this work is double, firstly to complement the previous one and secondly to show a complete analysis of the single interval statistics of the scattered intensity fluctuations through its PDF, PðIÞ, in the non-Gaussian regime from which other statistical parameters like either nð2Þ or PðI cross ¼ 0Þ can be obtained. The effect of multiple scattering on PðIÞ will be especially accounted for. The paper is organized as follows: in Section 2 the scattering model and the scattering geometry is presented. In Section 3 both the statistical parameters to be analyzed and the numerical method for calculating the scattered intensity fluctuations will be introduced. Section 4 is devoted to show the most relevant results and Section 5 will summarize the main conclusions. 2. Scattering model and scattering geometry Our scattering model consists of two dipole-like particles separated by an interdistance d (Fig. 1). Each particle has a polarizability a, given by a ¼ 4pa3
1 , þ2
(1)
where a is the particle radius and is the particle dielectric constant (assuming that the surrounding medium is vacuum). For a detailed discussion of the validity of Eq. (1), see Ref. [14]. These two parameters (a and d) will be used here to control the degree of multiple scattering between the two components. This is because the scattered intensity by a Rayleigh particle (a5l) decreases with distance as 1=r2 and scales with polarizability as a2 . We call this system Rayleigh bisphere (RBS). The reasons for using such a simple model is to simplify the calculation procedure in order to get a clear interpretation of the results and therefore to obtain as much information as possible about the physics of an electromagnetic problem where multiple scattering plays an important role. Several authors have used the dipole approximation for dealing with complex scattering problems like particles (isotropic and anisotropic) on substrates [15,16]. Because of its basic interest, the two Z θS
Ed
θ
Y φ
X 2a<<λ d Unpolarized incident beam Fig. 1. Scattering geometry for the Rayleigh bisphere (RBS).
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dipole-like particle system has been studied before [17] and has also been successfully used for giving more insight in problems involving multiple scattering effects [18,19]. In our model, the RBS will be assumed to be randomly oriented in the laboratory space and illuminated by an incident monochromatic unpolarized beam. This is the case for most practical situations, although the main conclusions would be also valid for other incident polarizations. The beam propagates along the Z direction and the scattered radiation is measured at an angle yS with respect to the incident direction. The scattering plane (formed by the incident and scattering directions) coincides with the ZY plane of the laboratory coordinate system (see Fig. 1). The statistics of the scattered intensity will be analyzed for both components: perpendicular (?), and parallel (k) to the scattering plane. 3. Numerical method and statistical parameters In order to calculate the scattered field from a system like a RBS, we have resorted to the coupled dipole method (CDM) [20–22], in which multiple scattering between the two scattering components is implicitly included. According to this method, the local electric field (the same can be done for the magnetic field) on particle i including the presence of its neighbor j (i; j ¼ 1; 2) (i.e. including multiple scattering) can be obtained by solving the two coupled equations Ei ¼ E0i þ a½AEj þ BðEj nji Þnji .
(2)
E0i
is the incident electric field on the ith particle, a the particle polarizability, nji the unit vector In Eqs. (2), connecting particle i to particle j and A and B are the interaction factors given by 1 ik expðikdÞ 2 , A¼ k 2þ d d d 3 3ik expðikdÞ . ð3Þ B ¼ k2 þ 2 d d d In these equations, k ¼ 2p=l, where l is the incident wavelength. From the solution of the system of equations (2), the scattered field can be written as k2 expðikRÞ ^ ðI nS nS Þ½E1 expðiknS r1 Þ þ E2 expðiknS r2 Þ, (4) R where I^ is the unit matrix, R the distance from the sample to the detector (we assume that kRb1), nS is a unit vector in the scattering direction and ri ði ¼ 1; 2Þ is the position vector of particle i (in Eq. (4), means dyadic product). From Eq. (4), the scattered intensity can be calculated in a straightforward manner for every random orientation of the RBS. The calculation procedure of the PDF of the intensity is based on conventional numerical techniques for the simulation of random orientations of the RBS. For a given number of samples, N T , of the scattered intensities (perpendicular (?) and parallel ðkÞ to the scattering plane), its experimental estimator is given by ES ¼ a
^ ?;k Þ ¼ NðI PðI
?;k
; I ?;k þ DÞ 1 , NT D
(5)
where NðI ?;k ; I ?;k þ DÞ is the number of cases with scattered intensity with values comprised between I and I þ D and N T is the total number of cases, i.e., random orientations of the RBS. Special attention must be paid to the value chosen for D in order to make the ratio NðI ?;k ; I ?;k þ DÞ=N T a good approximation of PðIÞ. Because this is well known for the non-interacting case, we have used this numerical result as a reference. 4. Results In this section, we discuss the results obtained for the PDF of the intensity for the case of a RBS. We begin with its evolution for different values of the interparticle distance, d while the scattering angle is fixed ðyS ¼ 30 Þ and we will analyze separately the cases dol and d4l. A brief discussion of the behavior of the PDF as a function of the scattering angle will also be given for different interaction regimes and a fixed
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interparticle distance (d ¼ l). For every situation we will show a 3D graph of the PDF, where in the X -axis we plot the evolution with the particle polarizability a, which is assumed to be the same for both particles. In the Y -axis we plot I=hIi where hIi represents the mean scattered intensity. 4.1. Evolution of the PDF as a function of the interparticle distance The case dol is especially interesting in biosensing applications where, for instance, the formation of nanoparticle dimers can give information about the hybridization process of DNA [23]. As a reference for the reader, we will take d ¼ 0:2l. Fig. 2 shows the evolution of the scattered intensity with the particle polarizability for this value of d. A resonance peak due to the coupling of local fields in the particles (not to a localized plasmon resonance [24]) is observed around a=l3 ’ 2:5 103 [13]. Its presence can also be detected in the evolution of the PDF (Fig. 3). This resonance divides the evolution in three regions. The first one (low polarizabilities) corresponds to the weak multiple scattering region, the second to the resonance itself and the third corresponds to strong multiple scattering. Each of these regions will be studied in detail for the most relevant interparticle distances. In the weak multiple scattering region, both parallel and perpendicular components show very similar behaviors (see Fig. 4). When a=l3 105 the PDF shows a sharp peak close to I=hIi ¼ 1. This peak is originated by the single particle scattering. It is also interesting to note that the scattered intensities never become zero. Both phenomena are the result of an interparticle distance much smaller than the incident wavelength (d ¼ 0:2l). Due to this small value of d, the phase differences between both scatterers never succeed to cover the whole interval ½0; 2p. This is why in this situation the expected maximum value of 2 for the normalized intensity for two independent scatterers is never reached. When a increases, the peak broadens. This is because of the increasing of multiple scattering. The phase differences between the two scattered fields increase and therefore higher and lower intensity values can be reached. However, the single particle scattering peak remains, which indicates that the internal structure of the aggregate remains partly hidden to the incident field. In the resonance region (see Fig. 5 for a=l3 2 ½103 ; 101 ), multiple scattering is strong due to the resonant coupling of the oscillating dipoles and, consequently, the probability of obtaining zeros in the scattered intensity increases due to the broad distribution of the phase differences between the two scattered fields. In fact, the second order factorial moment of the intensity fluctuations is larger than 2 and therefore larger than the value of 1.5 for the RBS without multiple scattering. This means that there are extra fluctuations due to the
Fig. 2. Scattered intensity versus polarizability for a RBS with unpolarized incident light. d ¼ 0:2l, yS ¼ 30 and N T ¼ 104 samples (see text for details).
ARTICLE IN PRESS O. Merchiers et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 101 (2006) 383–393
Perpendicular component
0.025
0.025
0.02
0.02
0.015
0.015
P(I)dI
P(I)dI
Parallel component
387
0.01 0.005
0.01 0.005 0
0 0
2 I/〈I〉
0 -2
log(α/λ3)
4
-4
0
2 I/〈I〉
0 -2
log(α/λ3)
4
-4
Fig. 3. 3D plot of the evolution with a (dipole polarizability) of the PDF of the intensity scattered by a RBS for incident unpolarized incident light and interparticle distance d less than l. d ¼ 0:2l, yS ¼ 30 and N T ¼ 5 104 samples.
Parallel component
Perpendicular component
0.02
0.015
0.015
P(I)dI
P(I)dI
0.01 0.01
0.005 0.005 0 0
0 0
-3.5
1
I/〈I〉
-4
-4.5 log(α/λ3)
-5
2
-3.5
1
I/〈I〉
-4
-4.5 log(α/λ3)
-5
2
Fig. 4. Same as Fig. 3 but only the weak multiple scattering regime is plotted.
resonant coupling [13]. Furthermore, the single scattering peak also appears but shifted to values of the scattered intensities such that I=hIio1. In Fig. 6 we show the strong multiple scattering regime for d ¼ 0:2l. A behavior similar to that commented in the resonant region can be observed. In this case, the single scattering peak still appears but shifted to even smaller values of I=hIi. For d ¼ 0:2l, the presence of a peak in both the resonant and strong multiple scattering regimes could be interpreted as if the incident radiation saw just one particle instead of two and the scattered intensity was lower than that of the isolated single particle due to the multiple scattering effect. For other distances dol, the general behavior of both the PDFs of I ? and I k is quite similar to that described previously for the resonant and strong multiple scattering regimes for d ¼ 0:2l. Perhaps, it is important to point out here that the single particle scattering peak sometimes is hardly present or even
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Perpendicular component
0.025
0.025
0.02
0.02
0.015
0.015
P(I)dI
P(I)dI
Parallel component
0.01 0.005
0.01 0.005
0 0 -1.5 -2
4
-2.5
log(α/λ3)
-3
2 I/〈I〉
0 0 -1.5 -2
4
-2.5
log(α/λ3)
2 I/〈I〉
-3
Fig. 5. Same as Fig. 3 but only the resonance region is plotted.
Fig. 6. Same as Fig. 3 but only the strong multiple scattering regime is plotted.
disappears, the PDFs being continuously decreasing functions. The PDF for the weak multiple scattering regime evolves very much as expected. As d increases, the position of the single scattering peak tends to I=hIi ¼ 2, while the probability of obtaining lower scattered intensities increases until the limit of the noninteracting case (no multiple scattering) where the typical U-shaped function for the PDF is obtained [7]. In Fig. 7, we show the evolution of the PDF for d ¼ l. In the weak multiple scattering regime, the typical Ushaped function is found again. As the interaction level increases (a increases), in the resonance region, high scattered intensity values larger than 2hIi start to be observed. This also happens in the strong multiple scattering regime where, in addition, the probability of obtaining zero values also increases. This corresponds to very strong fluctuations in the scattered intensity. When passing by the resonance, the two-particle peak moves downwards to zero and becomes undistinguishable from the peak in I=hIi ¼ 0 when passing through the center of the resonance. When leaving the resonance, the two particle peak becomes visible again but appears merely as a discontinuity.
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Perpendicular component
0.12
0.12
0.1
0.1
0.08
0.08
P(I)dI
P(I)dI
Parallel component
389
0.06 0.04
0.06 0.04
0.02
0
0
0.02
0
0 5
0 -2 3
log(α/λ )
-4
5
0
I/〈I〉
-2
10
3
log(α/λ )
-4
I/〈I〉
10
Fig. 7. Same as Fig. 3 for d ¼ l.
α/λ3 = 10-5
α/λ3 = 0.025
0.02 0 100 θS (º)
0 4
P(I)dI
0.04
0
(a)
0.2
0.1
P(I)dI
P(I)dI
0.06
α/λ3 = 5
0.05
0
0
2 I/〈I〉
(b)
100 θS (º)
0 10
0.1
0
0
5 I/〈I〉
(c)
100 θS (º)
0 10
5 I/〈I〉
Fig. 8. Evolution of the PDF as a function of the scattering angle and the polarizability for unpolarized incident light. d ¼ l and N T ¼ 5 104 samples in all cases.
4.2. Evolution of the PDF as a function of the scattering angle In this section, we analyze the evolution of the scattered intensity PDF for different scattering angles and fixed values of the polarizability. Fig. 8 shows the kind of plots obtained in these calculations for several interaction levels. In the weak multiple scattering regime we see for the PDF the typical U-form, except for the near forward direction where, again, only the single particle scattering peak is visible. For this scattering angle, the phase differences are not uniformly distributed between 0 and 2p. The maximum values of I=hIi oscillate due to interference effects. In the resonance and in the strong multiple scattering regimes we find mainly decaying PDF’s with oscillating maxima. 4.3. Three-dipole aggregate In this section, we analyze an aggregate geometry of three Rayleigh particles in the dipole approximation. It is assumed that they all have the same polarizability. The aggregate geometry is shown in Fig. 9. For brevity in this work we only consider geometries for which the interparticle distances 1–2 and 2–3 are the same, so our
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d
β d´ Fig. 9. Geometry of the three dipole aggregate with variable bending angle b. d 0 ¼ 2d sinðb=2Þ.
Perpendicular component
0.18 0.16 0.14
P(I)dI
0.12 0.1 0.08 0.06 0.04 0
0.02 0
5 0
-1
-2
log(α/λ3)
10 -3
-4
-5
I/〈I〉
15
Fig. 10. Same as Fig. 3 for a three-dipole aggregate in the linear geometry ðb ¼ 180 Þ. d ¼ l, yS ¼ 30 , perpendicular component and N T ¼ 5 104 samples.
analysis will be limited to the evolution of the PDF with the bending angle, b. We will show three representative examples, the linear chain ðb ¼ 180 Þ, the rectangle chain ðb ¼ 90 Þ and the folded chain (small b). In Figs. 10 and 11, the PDFs for the linear chain are represented. Observing the perpendicular component (Fig. 10) in the weak multiple scattering regime (low polarizabilities), we see three peaks for I=hIi 0; 0:3312; 3, the last one corresponding to the three-particle scattering (the coherent sum of three identical particles produces a maximum value equal to 9 times the value of a single particle scattering, i.e. 3 times the mean value for the three particles). When a increases, the three-particle peak broadens and for high values of the polarizability, only a discontinuity is visible which in the strong multiple scattering regime vanishes into the zero peak. The second peak (I=hIi 0:3312) moves inward to become undistinguishable from the zero peak when passing through the resonance. For the case of the parallel component (Fig. 11), a very different behavior is found. For low polarizabilities, only the broadened peak I=hIi 0:3312 is observed which moves over to the zero peak in the resonance and does not reappear for higher values of a. For the rectangle chain case (b ¼ 90 , Fig. 12), the parallel case is completely similar to the parallel case for the linear chain.
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Parallel component
0.18 0.16 0.14
P(I)dI
0.12 0.1 0.08 0.06 0.04 0
0.02 5
0 0
-1
10 I / 〈 I 〉 -2
15
-3
-4
log(α/λ3)
-5
20
Fig. 11. Same as Fig. 10 but for the parallel component.
Perpendicular component
2
2
1.5
1.5
P(I / 〈 I 〉 )
P(I / 〈 I 〉 )
Parallel component
1
1
0.5
0.5 0
0 0
0 10
0
I/〈I〉
-2
log(α/λ3)
-4
20
10
0
I/〈I〉
-2
log(α/λ3)
-4
20
Fig. 12. Same as Fig. 10 for the rectangular geometry (b ¼ 90 ) for both perpendicular and parallel components.
The perpendicular case shows some differences with respect to b ¼ 180 . Namely, the presence of a discontinuity in I=hIi ¼ 1:0 indicates that the single particle scattering is taking importance due to the geometrical configuration of the aggregate. For higher values of the polarizability, the evolution is the same as for the parallel case. Finally, for the folded chain (b has been fixed to 20 ) the tendency as described in for b ¼ 90 is set forth in the perpendicular case (Fig. 13), which is translated in the fact that the single particle scattering peak becomes even more visible.
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Parallel component
Perpendicular component
3
4
2.5
P(I / 〈 I 〉 )
P(I / 〈 I 〉 )
3 2 1.5 1
2
1 0.5
0
0 0
10
-2
log(α/λ3)
-4
15
5 I/〈I〉
0 0 5
0 -2
log(α/λ3)
-4
I/〈I〉
10
Fig. 13. Same as Fig. 12 for the folded geometry ðb ¼ 20 Þ.
For the rest of the evolution of both parallel and perpendicular components, it can be said that the behaviors are completely similar. 5. Conclusions We first studied the behavior of the PDF of the statistical fluctuations of the intensity scattered by a Rayleigh Bisphere (RBS) freely floating in space as a function of its optical and geometrical properties. For several values of the interparticle distance (geometrical property), we studied the evolution of the PDF when varying the dipole polarizability of the particles (optical property) assuming they are identical. In this work all calculations were performed considering unpolarized incident electromagnetic radiation and the scattered intensities were obtained for both parallel and perpendicular components to the scattering plane. For interparticle distances less than the incident wavelength, a single particle scattering peak was found in the weak multiple scattering regime (low polarizabilities). A qualitative change occurs when the resonance region ða=l3 2:5 103 Þ is reached. In this region, the single particle scattering peak moves to higher values of I=hIi. For higher values of a=l3 (strong multiple scattering regime), the peak recovers its original position. This behavior is similar for all distances less than l. In the case of interparticle distances larger than l we obtain in the weak multiple scattering regime the typical U-shaped PDF for two independent particles. When the polarizability increases, the discontinuity in I=hIi ¼ 2 disappears and we obtain scattering intensities larger than 2hIi. To study to behavior of the PDF as function of the scattering angle, we considered a fixed interparticle distance of d ¼ l and a representative value of the polarizability for each regime (weak multiple scattering, resonance and strong multiple scattering). The main observation are the oscillations of the maximum scattered intensities due to interferential effects. For the RBS, no significant differences between parallel and perpendicular components were found. We also studied some configurations of an aggregate constituted by three Rayleigh particles. Three fixed geometries (linear, rectangular and folded chains) were considered and their corresponding intensity PDFs analyzed. The linear chain in the weak multiple scattering regime, produced the most similar results to those of three independent particles freely moving in space. Depending on the distance between the two outer particles, the three particle peak will be visible or not. This is due to interferential effects. In this regime, strong differences between parallel and perpendicular components can be appreciated. The main difference lays in the absence of the three-particle peak for the parallel component.
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For increasing polarizability, the PDFs of the three geometrical configurations evolve to mainly decaying functions. In this regime, all differences between parallel and perpendicular components vanish. Acknowledgments The authors wish to thank the Direccio´n General de Ensen˜anza Superior for its financial support (projects BFM2001-1289 and FIS2004-06785). Olivier Merchiers wishes to thank the University of Cantabria for his research grant. References [1] Mishchenko M, Hovenier JW, Travis LD, editors, Light scattering by nonspherical particles: theory, measurements, and applications. San Diego: Academic Press; 2000. [2] McGill MJ, Lidar-remote sensing. In: Encyclopedia of optical engineering. New York: Marcel Dekker; 2003. p. 1103–13. [3] Wurm G, Relke H, Dorschner J. Experimental study of light scattering by large dust aggregates consisting of micron-sized SiO2 . Astrophys J 2003;595:891–9. [4] Sheng P, editor. Scattering and localisation of classical waves in random media, vol. 8. Singapore: World Scientific; 1990. [5] Ishimaru A, Wave propagation and scattering in random media, vols. 1 and 2. San Diego: Academic Press; 1978. [6] Shkuratov Y, Ovcharenko A, Zubko E, Miloslavskaya O, Muinonen K, Piironen J, Nelson R, Smythe W, Rosenbush V, Helfenstein P. The opposition effect and negative polarization of structural analogs for planetary regoliths. Icarus 2002;159:396–416. [7] Pusey PN, Statistical properties of scattered radiation. In: Photon correlation spectroscopy and velocimetry, NATO ASI series B: physics, vol. 23. New York: Plenum Press; 1977. p. 45–141. [8] Videen G, Prabhu DR, Davies M, Gonza´lez F, Moreno F. Light scattering fluctuations of a soft spherical particle containing an inclusion. Appl Opt 2001;40:4054–7. [9] Pitter MC, Hopcraft KI, Jakeman E, Walker JG. Structure of polarization fluctuations and their relation to particle shape. JQSRT 1999;63:433–44. [10] Bates AP, Hopcraft KI, Jakeman E. Particle shape determination from polarization fluctuations of scattered radiation. J Opt Soc Am A 1997;14:1–4. [11] Bates AP, Hopcraft KI, Jakeman E. Non-gaussian fluctuations of stokes parameters in scattering by small particles. Waves Random Media 1998;8:235–53. [12] Jakeman E. Polarization characteristics of non-gaussian scattering by small particles. Waves Random Media 1995;5:427–42. [13] Merchiers O, Saiz JM, Gonza´lez F, Moreno F. A two particle model to study fluctuations of the scattered radiation, multiple scattering effects. J Opt Soc Am A 2005;22:497–503. [14] Draine BT. The discrete-dipole approximation and its application to interstellar graphite grains. Astrophys J 1988;333:848–72. [15] Jakeman E, Jordan DL, Lewis GD. Fluctuations in radiation scattered by small spheroids above an interface. Waves Random Media 2000;10:317–36. [16] Videen G, Wolfe WL, Bickel WS. Light scattering mueller matrix for a surface contaminated by a single particle in the Rayleigh limit. Opt Eng 1992;31:341–9. [17] Muinonen K, Electromagnetic scattering by two interacting dipoles. In: Proceedings of URSI electromagnetic theory symposium (International Union of Radio Science). Ghent, Belgium, 1989. p. 428–30. [18] Ismagilov FM, Kravtsov YA. Backscattering enhancement polarization effects on a system of two small randomly oriented scatterers. Waves Random Media 1993;3:17–24. [19] Markel VA. Scattering of light from two interacting dipoles. J Mod Opt 1991;39:853–61. [20] Singham SB, Bohren CF. Light scattering by an arbitrary particle: the scattering order formulation of the coupled dipole method. J Opt Soc Am A 1988;11:1867–72. [21] Purcell EM, Pennypacker CR. Scattering and absorption of light by nonspherical dielectric grains. Astrophys J 1973;186:705–14. [22] Draine BT, Flatau PJ. Discrete-dipole approximation for scattering calculations. J Opt Soc Am A 1994;11:1491–9. [23] Storhoff JJ, Mirkin CA. Programmed materials synthesis with DNA. Chem Rev 1999;99:1849–62. [24] Raether H. Surface plasmons. Berlin: Springer; 1988.
Journal of Quantitative Spectroscopy & Radiative Transfer 101 (2006) 383–393 www.elsevier.com/locate/jqsrt
Probability density function of the intensity scattered by Rayleigh-particle aggregates. Evolution with optical properties O. Merchiers, J.M. Saiz, F. Gonza´lez, F. Moreno Grupo de O´ptica, Departamento de Fı´sica Aplicada, Universidad de Cantabria, Avda de los Castros, 39005 Santander, Spain
Abstract We study the probability density function of the statistical fluctuations of the intensity scattered by an aggregate freely floating in space and constituted by Rayleigh particles under the dipole approximation. Its evolution as a function of the optical properties of the particles (polarizability) and their separation distance is analyzed. Aggregate geometries with two and three particles will be considered. The influence of the multiple scattering effect on the statistics of the scattered intensity is especially studied. r 2006 Elsevier Ltd. All rights reserved. Keywords: Statistics; Light scattering; Multiple scattering; Aggregates
1. Introduction During the last decades, research on the interaction between electromagnetic radiation and matter has experienced important advances from the study of the scattering of electromagnetic waves from microstructures. In this sense, a number of theoretical and experimental techniques for obtaining information about the scattering system have been developed [1]. Many of them are based on the measurement of the radiation as a function of the scattering angle and concentrate on a particular scattering angle. This is the case of backscattering, for its interest in practical applications, like LIDAR [2] and astrophysics [3]. In these applications the scattering systems involved, present important multiple scattering effects, like the enhanced backscattering [4,5] and the negative polarization branch [6], for which many authors have proposed theoretical models (usually complicated geometries and/or particle aggregates) and numerical methods to solve this electromagnetic problem. Statistical properties of the scattered radiation have also been proposed for obtaining information about different scattering systems. For example, the statistics of the fluctuations of the scattered intensity can give information about the density of the scatterers for the non-Gaussian regime (low number of scatterers) [7]. This is because in this regime, the probability density function (PDF) of the intensity, and then its factorial moments, depends on the number of scatterers. On the contrary, in the Gaussian regime the PDF of the intensity is exponential and the factorial moments are given by the well-known factorial law. More recently, Corresponding author.
E-mail addresses: [email protected] (O. Merchiers), [email protected] (F. Moreno). 0022-4073/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2006.02.066
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the case of particles with inclusions has also been analyzed through the statistical fluctuations of the scattered intensity [8]. In addition, the measurement of the statistical properties of polarization fluctuations has shown to be very useful for determining morphological properties of anisotropic particles [9–12]. In a previous paper [13], statistical techniques based on the measurement and analysis of statistical parameters like the second order factorial moment nð2Þ of the intensity fluctuations (both co- and crosspolarized) and the probability of measuring no cross-polarized intensity PðI cross ¼ 0Þ were presented for scattering systems constituted by particles aggregates, where multiple scattering is present. Bearing in mind this kind of systems, the objective of this work is double, firstly to complement the previous one and secondly to show a complete analysis of the single interval statistics of the scattered intensity fluctuations through its PDF, PðIÞ, in the non-Gaussian regime from which other statistical parameters like either nð2Þ or PðI cross ¼ 0Þ can be obtained. The effect of multiple scattering on PðIÞ will be especially accounted for. The paper is organized as follows: in Section 2 the scattering model and the scattering geometry is presented. In Section 3 both the statistical parameters to be analyzed and the numerical method for calculating the scattered intensity fluctuations will be introduced. Section 4 is devoted to show the most relevant results and Section 5 will summarize the main conclusions. 2. Scattering model and scattering geometry Our scattering model consists of two dipole-like particles separated by an interdistance d (Fig. 1). Each particle has a polarizability a, given by a ¼ 4pa3
1 , þ2
(1)
where a is the particle radius and is the particle dielectric constant (assuming that the surrounding medium is vacuum). For a detailed discussion of the validity of Eq. (1), see Ref. [14]. These two parameters (a and d) will be used here to control the degree of multiple scattering between the two components. This is because the scattered intensity by a Rayleigh particle (a5l) decreases with distance as 1=r2 and scales with polarizability as a2 . We call this system Rayleigh bisphere (RBS). The reasons for using such a simple model is to simplify the calculation procedure in order to get a clear interpretation of the results and therefore to obtain as much information as possible about the physics of an electromagnetic problem where multiple scattering plays an important role. Several authors have used the dipole approximation for dealing with complex scattering problems like particles (isotropic and anisotropic) on substrates [15,16]. Because of its basic interest, the two Z θS
Ed
θ
Y φ
X 2a<<λ d Unpolarized incident beam Fig. 1. Scattering geometry for the Rayleigh bisphere (RBS).
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dipole-like particle system has been studied before [17] and has also been successfully used for giving more insight in problems involving multiple scattering effects [18,19]. In our model, the RBS will be assumed to be randomly oriented in the laboratory space and illuminated by an incident monochromatic unpolarized beam. This is the case for most practical situations, although the main conclusions would be also valid for other incident polarizations. The beam propagates along the Z direction and the scattered radiation is measured at an angle yS with respect to the incident direction. The scattering plane (formed by the incident and scattering directions) coincides with the ZY plane of the laboratory coordinate system (see Fig. 1). The statistics of the scattered intensity will be analyzed for both components: perpendicular (?), and parallel (k) to the scattering plane. 3. Numerical method and statistical parameters In order to calculate the scattered field from a system like a RBS, we have resorted to the coupled dipole method (CDM) [20–22], in which multiple scattering between the two scattering components is implicitly included. According to this method, the local electric field (the same can be done for the magnetic field) on particle i including the presence of its neighbor j (i; j ¼ 1; 2) (i.e. including multiple scattering) can be obtained by solving the two coupled equations Ei ¼ E0i þ a½AEj þ BðEj nji Þnji .
(2)
E0i
is the incident electric field on the ith particle, a the particle polarizability, nji the unit vector In Eqs. (2), connecting particle i to particle j and A and B are the interaction factors given by 1 ik expðikdÞ 2 , A¼ k 2þ d d d 3 3ik expðikdÞ . ð3Þ B ¼ k2 þ 2 d d d In these equations, k ¼ 2p=l, where l is the incident wavelength. From the solution of the system of equations (2), the scattered field can be written as k2 expðikRÞ ^ ðI nS nS Þ½E1 expðiknS r1 Þ þ E2 expðiknS r2 Þ, (4) R where I^ is the unit matrix, R the distance from the sample to the detector (we assume that kRb1), nS is a unit vector in the scattering direction and ri ði ¼ 1; 2Þ is the position vector of particle i (in Eq. (4), means dyadic product). From Eq. (4), the scattered intensity can be calculated in a straightforward manner for every random orientation of the RBS. The calculation procedure of the PDF of the intensity is based on conventional numerical techniques for the simulation of random orientations of the RBS. For a given number of samples, N T , of the scattered intensities (perpendicular (?) and parallel ðkÞ to the scattering plane), its experimental estimator is given by ES ¼ a
^ ?;k Þ ¼ NðI PðI
?;k
; I ?;k þ DÞ 1 , NT D
(5)
where NðI ?;k ; I ?;k þ DÞ is the number of cases with scattered intensity with values comprised between I and I þ D and N T is the total number of cases, i.e., random orientations of the RBS. Special attention must be paid to the value chosen for D in order to make the ratio NðI ?;k ; I ?;k þ DÞ=N T a good approximation of PðIÞ. Because this is well known for the non-interacting case, we have used this numerical result as a reference. 4. Results In this section, we discuss the results obtained for the PDF of the intensity for the case of a RBS. We begin with its evolution for different values of the interparticle distance, d while the scattering angle is fixed ðyS ¼ 30 Þ and we will analyze separately the cases dol and d4l. A brief discussion of the behavior of the PDF as a function of the scattering angle will also be given for different interaction regimes and a fixed
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interparticle distance (d ¼ l). For every situation we will show a 3D graph of the PDF, where in the X -axis we plot the evolution with the particle polarizability a, which is assumed to be the same for both particles. In the Y -axis we plot I=hIi where hIi represents the mean scattered intensity. 4.1. Evolution of the PDF as a function of the interparticle distance The case dol is especially interesting in biosensing applications where, for instance, the formation of nanoparticle dimers can give information about the hybridization process of DNA [23]. As a reference for the reader, we will take d ¼ 0:2l. Fig. 2 shows the evolution of the scattered intensity with the particle polarizability for this value of d. A resonance peak due to the coupling of local fields in the particles (not to a localized plasmon resonance [24]) is observed around a=l3 ’ 2:5 103 [13]. Its presence can also be detected in the evolution of the PDF (Fig. 3). This resonance divides the evolution in three regions. The first one (low polarizabilities) corresponds to the weak multiple scattering region, the second to the resonance itself and the third corresponds to strong multiple scattering. Each of these regions will be studied in detail for the most relevant interparticle distances. In the weak multiple scattering region, both parallel and perpendicular components show very similar behaviors (see Fig. 4). When a=l3 105 the PDF shows a sharp peak close to I=hIi ¼ 1. This peak is originated by the single particle scattering. It is also interesting to note that the scattered intensities never become zero. Both phenomena are the result of an interparticle distance much smaller than the incident wavelength (d ¼ 0:2l). Due to this small value of d, the phase differences between both scatterers never succeed to cover the whole interval ½0; 2p. This is why in this situation the expected maximum value of 2 for the normalized intensity for two independent scatterers is never reached. When a increases, the peak broadens. This is because of the increasing of multiple scattering. The phase differences between the two scattered fields increase and therefore higher and lower intensity values can be reached. However, the single particle scattering peak remains, which indicates that the internal structure of the aggregate remains partly hidden to the incident field. In the resonance region (see Fig. 5 for a=l3 2 ½103 ; 101 ), multiple scattering is strong due to the resonant coupling of the oscillating dipoles and, consequently, the probability of obtaining zeros in the scattered intensity increases due to the broad distribution of the phase differences between the two scattered fields. In fact, the second order factorial moment of the intensity fluctuations is larger than 2 and therefore larger than the value of 1.5 for the RBS without multiple scattering. This means that there are extra fluctuations due to the
Fig. 2. Scattered intensity versus polarizability for a RBS with unpolarized incident light. d ¼ 0:2l, yS ¼ 30 and N T ¼ 104 samples (see text for details).
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Perpendicular component
0.025
0.025
0.02
0.02
0.015
0.015
P(I)dI
P(I)dI
Parallel component
387
0.01 0.005
0.01 0.005 0
0 0
2 I/〈I〉
0 -2
log(α/λ3)
4
-4
0
2 I/〈I〉
0 -2
log(α/λ3)
4
-4
Fig. 3. 3D plot of the evolution with a (dipole polarizability) of the PDF of the intensity scattered by a RBS for incident unpolarized incident light and interparticle distance d less than l. d ¼ 0:2l, yS ¼ 30 and N T ¼ 5 104 samples.
Parallel component
Perpendicular component
0.02
0.015
0.015
P(I)dI
P(I)dI
0.01 0.01
0.005 0.005 0 0
0 0
-3.5
1
I/〈I〉
-4
-4.5 log(α/λ3)
-5
2
-3.5
1
I/〈I〉
-4
-4.5 log(α/λ3)
-5
2
Fig. 4. Same as Fig. 3 but only the weak multiple scattering regime is plotted.
resonant coupling [13]. Furthermore, the single scattering peak also appears but shifted to values of the scattered intensities such that I=hIio1. In Fig. 6 we show the strong multiple scattering regime for d ¼ 0:2l. A behavior similar to that commented in the resonant region can be observed. In this case, the single scattering peak still appears but shifted to even smaller values of I=hIi. For d ¼ 0:2l, the presence of a peak in both the resonant and strong multiple scattering regimes could be interpreted as if the incident radiation saw just one particle instead of two and the scattered intensity was lower than that of the isolated single particle due to the multiple scattering effect. For other distances dol, the general behavior of both the PDFs of I ? and I k is quite similar to that described previously for the resonant and strong multiple scattering regimes for d ¼ 0:2l. Perhaps, it is important to point out here that the single particle scattering peak sometimes is hardly present or even
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Perpendicular component
0.025
0.025
0.02
0.02
0.015
0.015
P(I)dI
P(I)dI
Parallel component
0.01 0.005
0.01 0.005
0 0 -1.5 -2
4
-2.5
log(α/λ3)
-3
2 I/〈I〉
0 0 -1.5 -2
4
-2.5
log(α/λ3)
2 I/〈I〉
-3
Fig. 5. Same as Fig. 3 but only the resonance region is plotted.
Fig. 6. Same as Fig. 3 but only the strong multiple scattering regime is plotted.
disappears, the PDFs being continuously decreasing functions. The PDF for the weak multiple scattering regime evolves very much as expected. As d increases, the position of the single scattering peak tends to I=hIi ¼ 2, while the probability of obtaining lower scattered intensities increases until the limit of the noninteracting case (no multiple scattering) where the typical U-shaped function for the PDF is obtained [7]. In Fig. 7, we show the evolution of the PDF for d ¼ l. In the weak multiple scattering regime, the typical Ushaped function is found again. As the interaction level increases (a increases), in the resonance region, high scattered intensity values larger than 2hIi start to be observed. This also happens in the strong multiple scattering regime where, in addition, the probability of obtaining zero values also increases. This corresponds to very strong fluctuations in the scattered intensity. When passing by the resonance, the two-particle peak moves downwards to zero and becomes undistinguishable from the peak in I=hIi ¼ 0 when passing through the center of the resonance. When leaving the resonance, the two particle peak becomes visible again but appears merely as a discontinuity.
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Perpendicular component
0.12
0.12
0.1
0.1
0.08
0.08
P(I)dI
P(I)dI
Parallel component
389
0.06 0.04
0.06 0.04
0.02
0
0
0.02
0
0 5
0 -2 3
log(α/λ )
-4
5
0
I/〈I〉
-2
10
3
log(α/λ )
-4
I/〈I〉
10
Fig. 7. Same as Fig. 3 for d ¼ l.
α/λ3 = 10-5
α/λ3 = 0.025
0.02 0 100 θS (º)
0 4
P(I)dI
0.04
0
(a)
0.2
0.1
P(I)dI
P(I)dI
0.06
α/λ3 = 5
0.05
0
0
2 I/〈I〉
(b)
100 θS (º)
0 10
0.1
0
0
5 I/〈I〉
(c)
100 θS (º)
0 10
5 I/〈I〉
Fig. 8. Evolution of the PDF as a function of the scattering angle and the polarizability for unpolarized incident light. d ¼ l and N T ¼ 5 104 samples in all cases.
4.2. Evolution of the PDF as a function of the scattering angle In this section, we analyze the evolution of the scattered intensity PDF for different scattering angles and fixed values of the polarizability. Fig. 8 shows the kind of plots obtained in these calculations for several interaction levels. In the weak multiple scattering regime we see for the PDF the typical U-form, except for the near forward direction where, again, only the single particle scattering peak is visible. For this scattering angle, the phase differences are not uniformly distributed between 0 and 2p. The maximum values of I=hIi oscillate due to interference effects. In the resonance and in the strong multiple scattering regimes we find mainly decaying PDF’s with oscillating maxima. 4.3. Three-dipole aggregate In this section, we analyze an aggregate geometry of three Rayleigh particles in the dipole approximation. It is assumed that they all have the same polarizability. The aggregate geometry is shown in Fig. 9. For brevity in this work we only consider geometries for which the interparticle distances 1–2 and 2–3 are the same, so our
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d
β d´ Fig. 9. Geometry of the three dipole aggregate with variable bending angle b. d 0 ¼ 2d sinðb=2Þ.
Perpendicular component
0.18 0.16 0.14
P(I)dI
0.12 0.1 0.08 0.06 0.04 0
0.02 0
5 0
-1
-2
log(α/λ3)
10 -3
-4
-5
I/〈I〉
15
Fig. 10. Same as Fig. 3 for a three-dipole aggregate in the linear geometry ðb ¼ 180 Þ. d ¼ l, yS ¼ 30 , perpendicular component and N T ¼ 5 104 samples.
analysis will be limited to the evolution of the PDF with the bending angle, b. We will show three representative examples, the linear chain ðb ¼ 180 Þ, the rectangle chain ðb ¼ 90 Þ and the folded chain (small b). In Figs. 10 and 11, the PDFs for the linear chain are represented. Observing the perpendicular component (Fig. 10) in the weak multiple scattering regime (low polarizabilities), we see three peaks for I=hIi 0; 0:3312; 3, the last one corresponding to the three-particle scattering (the coherent sum of three identical particles produces a maximum value equal to 9 times the value of a single particle scattering, i.e. 3 times the mean value for the three particles). When a increases, the three-particle peak broadens and for high values of the polarizability, only a discontinuity is visible which in the strong multiple scattering regime vanishes into the zero peak. The second peak (I=hIi 0:3312) moves inward to become undistinguishable from the zero peak when passing through the resonance. For the case of the parallel component (Fig. 11), a very different behavior is found. For low polarizabilities, only the broadened peak I=hIi 0:3312 is observed which moves over to the zero peak in the resonance and does not reappear for higher values of a. For the rectangle chain case (b ¼ 90 , Fig. 12), the parallel case is completely similar to the parallel case for the linear chain.
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Parallel component
0.18 0.16 0.14
P(I)dI
0.12 0.1 0.08 0.06 0.04 0
0.02 5
0 0
-1
10 I / 〈 I 〉 -2
15
-3
-4
log(α/λ3)
-5
20
Fig. 11. Same as Fig. 10 but for the parallel component.
Perpendicular component
2
2
1.5
1.5
P(I / 〈 I 〉 )
P(I / 〈 I 〉 )
Parallel component
1
1
0.5
0.5 0
0 0
0 10
0
I/〈I〉
-2
log(α/λ3)
-4
20
10
0
I/〈I〉
-2
log(α/λ3)
-4
20
Fig. 12. Same as Fig. 10 for the rectangular geometry (b ¼ 90 ) for both perpendicular and parallel components.
The perpendicular case shows some differences with respect to b ¼ 180 . Namely, the presence of a discontinuity in I=hIi ¼ 1:0 indicates that the single particle scattering is taking importance due to the geometrical configuration of the aggregate. For higher values of the polarizability, the evolution is the same as for the parallel case. Finally, for the folded chain (b has been fixed to 20 ) the tendency as described in for b ¼ 90 is set forth in the perpendicular case (Fig. 13), which is translated in the fact that the single particle scattering peak becomes even more visible.
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Parallel component
Perpendicular component
3
4
2.5
P(I / 〈 I 〉 )
P(I / 〈 I 〉 )
3 2 1.5 1
2
1 0.5
0
0 0
10
-2
log(α/λ3)
-4
15
5 I/〈I〉
0 0 5
0 -2
log(α/λ3)
-4
I/〈I〉
10
Fig. 13. Same as Fig. 12 for the folded geometry ðb ¼ 20 Þ.
For the rest of the evolution of both parallel and perpendicular components, it can be said that the behaviors are completely similar. 5. Conclusions We first studied the behavior of the PDF of the statistical fluctuations of the intensity scattered by a Rayleigh Bisphere (RBS) freely floating in space as a function of its optical and geometrical properties. For several values of the interparticle distance (geometrical property), we studied the evolution of the PDF when varying the dipole polarizability of the particles (optical property) assuming they are identical. In this work all calculations were performed considering unpolarized incident electromagnetic radiation and the scattered intensities were obtained for both parallel and perpendicular components to the scattering plane. For interparticle distances less than the incident wavelength, a single particle scattering peak was found in the weak multiple scattering regime (low polarizabilities). A qualitative change occurs when the resonance region ða=l3 2:5 103 Þ is reached. In this region, the single particle scattering peak moves to higher values of I=hIi. For higher values of a=l3 (strong multiple scattering regime), the peak recovers its original position. This behavior is similar for all distances less than l. In the case of interparticle distances larger than l we obtain in the weak multiple scattering regime the typical U-shaped PDF for two independent particles. When the polarizability increases, the discontinuity in I=hIi ¼ 2 disappears and we obtain scattering intensities larger than 2hIi. To study to behavior of the PDF as function of the scattering angle, we considered a fixed interparticle distance of d ¼ l and a representative value of the polarizability for each regime (weak multiple scattering, resonance and strong multiple scattering). The main observation are the oscillations of the maximum scattered intensities due to interferential effects. For the RBS, no significant differences between parallel and perpendicular components were found. We also studied some configurations of an aggregate constituted by three Rayleigh particles. Three fixed geometries (linear, rectangular and folded chains) were considered and their corresponding intensity PDFs analyzed. The linear chain in the weak multiple scattering regime, produced the most similar results to those of three independent particles freely moving in space. Depending on the distance between the two outer particles, the three particle peak will be visible or not. This is due to interferential effects. In this regime, strong differences between parallel and perpendicular components can be appreciated. The main difference lays in the absence of the three-particle peak for the parallel component.
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For increasing polarizability, the PDFs of the three geometrical configurations evolve to mainly decaying functions. In this regime, all differences between parallel and perpendicular components vanish. Acknowledgments The authors wish to thank the Direccio´n General de Ensen˜anza Superior for its financial support (projects BFM2001-1289 and FIS2004-06785). Olivier Merchiers wishes to thank the University of Cantabria for his research grant. References [1] Mishchenko M, Hovenier JW, Travis LD, editors, Light scattering by nonspherical particles: theory, measurements, and applications. San Diego: Academic Press; 2000. [2] McGill MJ, Lidar-remote sensing. In: Encyclopedia of optical engineering. New York: Marcel Dekker; 2003. p. 1103–13. [3] Wurm G, Relke H, Dorschner J. Experimental study of light scattering by large dust aggregates consisting of micron-sized SiO2 . Astrophys J 2003;595:891–9. [4] Sheng P, editor. Scattering and localisation of classical waves in random media, vol. 8. Singapore: World Scientific; 1990. [5] Ishimaru A, Wave propagation and scattering in random media, vols. 1 and 2. San Diego: Academic Press; 1978. [6] Shkuratov Y, Ovcharenko A, Zubko E, Miloslavskaya O, Muinonen K, Piironen J, Nelson R, Smythe W, Rosenbush V, Helfenstein P. The opposition effect and negative polarization of structural analogs for planetary regoliths. Icarus 2002;159:396–416. [7] Pusey PN, Statistical properties of scattered radiation. In: Photon correlation spectroscopy and velocimetry, NATO ASI series B: physics, vol. 23. New York: Plenum Press; 1977. p. 45–141. [8] Videen G, Prabhu DR, Davies M, Gonza´lez F, Moreno F. Light scattering fluctuations of a soft spherical particle containing an inclusion. Appl Opt 2001;40:4054–7. [9] Pitter MC, Hopcraft KI, Jakeman E, Walker JG. Structure of polarization fluctuations and their relation to particle shape. JQSRT 1999;63:433–44. [10] Bates AP, Hopcraft KI, Jakeman E. Particle shape determination from polarization fluctuations of scattered radiation. J Opt Soc Am A 1997;14:1–4. [11] Bates AP, Hopcraft KI, Jakeman E. Non-gaussian fluctuations of stokes parameters in scattering by small particles. Waves Random Media 1998;8:235–53. [12] Jakeman E. Polarization characteristics of non-gaussian scattering by small particles. Waves Random Media 1995;5:427–42. [13] Merchiers O, Saiz JM, Gonza´lez F, Moreno F. A two particle model to study fluctuations of the scattered radiation, multiple scattering effects. J Opt Soc Am A 2005;22:497–503. [14] Draine BT. The discrete-dipole approximation and its application to interstellar graphite grains. Astrophys J 1988;333:848–72. [15] Jakeman E, Jordan DL, Lewis GD. Fluctuations in radiation scattered by small spheroids above an interface. Waves Random Media 2000;10:317–36. [16] Videen G, Wolfe WL, Bickel WS. Light scattering mueller matrix for a surface contaminated by a single particle in the Rayleigh limit. Opt Eng 1992;31:341–9. [17] Muinonen K, Electromagnetic scattering by two interacting dipoles. In: Proceedings of URSI electromagnetic theory symposium (International Union of Radio Science). Ghent, Belgium, 1989. p. 428–30. [18] Ismagilov FM, Kravtsov YA. Backscattering enhancement polarization effects on a system of two small randomly oriented scatterers. Waves Random Media 1993;3:17–24. [19] Markel VA. Scattering of light from two interacting dipoles. J Mod Opt 1991;39:853–61. [20] Singham SB, Bohren CF. Light scattering by an arbitrary particle: the scattering order formulation of the coupled dipole method. J Opt Soc Am A 1988;11:1867–72. [21] Purcell EM, Pennypacker CR. Scattering and absorption of light by nonspherical dielectric grains. Astrophys J 1973;186:705–14. [22] Draine BT, Flatau PJ. Discrete-dipole approximation for scattering calculations. J Opt Soc Am A 1994;11:1491–9. [23] Storhoff JJ, Mirkin CA. Programmed materials synthesis with DNA. Chem Rev 1999;99:1849–62. [24] Raether H. Surface plasmons. Berlin: Springer; 1988.