Light scattering by ensemble of nonabsorbingcorrelated two-layered particles: specific

feature for spectral dependenceof extinction coefficient

Vladimir V. Berdnik1 and Valery A. Loiko2,*1Técnicas de Investigación Hidrogeológica S.A., Sibirsky Tract 34, Kazan, Russia

2Institute of Physics, Nezalezhnasti Ave. 68, 220072 Minsk, Belarus

*Corresponding author: loiko@dragon.bas‐net.by

Received 6 April 2011; accepted 3 May 2011;posted 17 June 2011 (Doc. ID 145408); published 21 July 2011

Scattering of light by an ensemble of nonabsorbing spherical concentric two-layered particles is consid-ered. It has been shown that exponent of the power function describing the wavelength dependence of theextinction coefficient of the medium with subwavelength-sized particles can exceed considerably thevalue of 4, which takes place for the Rayleigh scattering. Spatial correlation of particles enhances this“anomalous” dependence on the wavelength. Bleaching and darkening effects can be implemented. In thefirst case transmittance increases, while in the second case transmittance decreases with increasedvolume concentration. These effects can be used to get a sharp spectral dependence of transmittance.Comparison with the data for spatially correlated homogeneous particles is carried out. © 2011 OpticalSociety of AmericaOCIS codes: 160.4760, 290.5825.

1. Introduction

The process of interaction of radiation with naturaland artificial systems of spatially correlated particlesis of great interest due to its practical importancefor optics, optoelectronics, photonics, astrophysics,medicine, and biology.

The specific features in scattering arise if there is aspatial arrangement of particles in a medium [1–5].In particular, the so-called “anomalous” scattering isobserved for small volume concentration of particles,when their size is less than the light wavelength andtheir relative refractive index is close to unity [6–10].Anomalous scattering consists in that the exponentof the power function, describing dependence of ex-tinction coefficient on the wavelength, has a highvalue. According to the experimental data it can

be equal to 8–9, while under the theory of light scat-tering in a layer with sparse homogeneous sphericalparticles this exponent does not exceed 4. The phasefunction at the anomalous scattering regime imple-mentation is stretched in the backward direction[8]. The described features in scattering are observedin the phase-separated glasses [6–11], where the spa-tial correlation of particles arises at the phaseseparation process [7].

A model to describe light scattering by an ensem-ble of homogeneous spatially correlated optically softspherical particles is given in [11]. The arrangementin particle location was described by using the corre-lation radius Rcor, which exceeds the particle radius.The separation between particles cannot be smal-ler than 2Rcor. The developed model explains experi-mentally observed features of the scattering insodium borosilicate glasses with a small concentra-tion of scattering centers [6–10]. Such effects canbe realized in different media.

0003-6935/11/224246-06$15.00/0© 2011 Optical Society of America

4246 APPLIED OPTICS / Vol. 50, No. 22 / 1 August 2011

This research is aimed at consideration of scatter-ing in a layer with nonabsorbing concentric two-layered spherical particles. Spectral dependence ofthe extinction coefficient on particle characteristicsand particle concentration in the layer is analyzed.For the two-layered particles there are parameterswhen the phase function is extended in the backwarddirection. If such a situation is implemented, the ex-ponent of the power function can be more than 4 eventhough the layer has small volume concentration ofparticles. The exponent increases with increasedvolume concentration. Results for layers with homo-geneous particles are displayed as well.

2. Basic Relations

Let us consider a system of spherical particles dis-tributed in homogeneous nonabsorbing medium withthickness z0 and refractive index nm. The particlesconsist of a spherical homogeneous core of radius Rcwith the refractive index npc, enveloped by a spheri-cally symmetric homogeneous shell of radius Rs withthe refractive index nps. We indicate the relative re-fractive indices of the core and shell as nc ¼ npc=nmand ns ¼ nps=nm, respectively. The layer is schemati-cally shown in Fig. 1. When the distances betweenparticles are large there is no correlation in the par-ticles location and we deal with a sparse randommedium. When the distances between particles arecomparable with their size we deal with the close-packed partially ordered medium. The degree of par-ticle ordering increases with decreased interparticledistances [1,3,5]. We suppose that the particles arehard and separation between them cannot be smallerthan 2Rs.

The starting point in the simulation of charac-teristics of a close-packed medium is the knowledgeof the unit volume parameters of the medium inthe limit of low particle concentrations, where thereare no collective effects in light scattering. To calcu-late characteristics of scattering for a separate two-layered particle we used the algorithms describedin [12].

To calculate the extinction coefficient and the scat-tering phase function at a high concentration of par-ticles, it is necessary to solve the problem of lightdiffraction by a system of many bodies. This problemhas no general rigorous solution yet. There are someresults for light scattering and absorption by the den-sely packed groups of spherical particles [13,14].They are obtained by the T-matrix method [15]. Weconsider soft particles and use the interference

approximation [11]. It takes into account single scat-tering by particles and the interference of scatteredwaves in the far zone. Under this approach, we canwrite the following equations for the differentialscattering coefficient, the scattering coefficient, andthe extinction coefficient, respectively:

σhðγÞ ¼ wσ0lplðγÞS3ðγ;wÞ; ð1Þ

σh ¼ wσ0lu; ð2Þ

εh ¼ wðε0l − σ0l þ σ0luÞ; ð3Þ

u ¼Z π

0plðγÞS3ðγ;wÞ sin γdγ: ð4Þ

In Eqs. (1)–(4), w ¼ Nv=V is the volume concentra-tion of particles of the radius Rs, ν ¼ 4

3 πR3s is the

volume of a spherical particle of the radius Rs, Nis the number of particles in the volume V, σhðγÞ isthe differential scattering coefficient of medium vol-ume concentration of particles w; γ is the scatteringangle; σ0l ¼ Σs=v; ε0l ¼ α0l þ σ0l ¼ Σe=v; α0 ¼ Σa=v;Σa, Σs, and Σe are the absorption, scattering, andextinction cross sections of an individual particle, re-spectively; and plðγÞ is the scattering phase functionof an isolated particle normalized according to thecondition

R π0 plðγÞ sin γdγ ¼ 1.

The structure factor S3ðγ;wÞ takes into accountthe interference of light in a system of correlatedparticles:

S3ðγ;wÞ ¼ 1þ 4πnZ

∞

0½gðr;wÞ − 1� sin zr

zrr2dr: ð5Þ

Here, n is the number of particles in a unit volume;gðx;wÞ is the radial (two-particle) distribution func-tion, characterizing the spatial arrangement of par-ticles in a layer; x ¼ r=2Rs, where r is the separationbetween two particles; and z ¼ 4x sin γ=2, where λm ¼λ=nm is the wavelength in the medium, λ is thewavelength of the incident light.

We assume that the spatial distribution of sphereswith the radius Rs can be described at the Percus–Yevick approximation of hard spheres [3,16,17].Under this approximation, the structure factor canbe calculated by the formula

S3ðγ;wÞ ¼�1 − 24n

Z1

0gðr;wÞ sin zr

zrr2dr

�−1: ð6Þ

Here, the radial distribution function is given byequation

gðr;wÞ ¼�−a − b r

2Rs− c

�r

2Rs

�3; r ≤ 2Rs

0; r > 2Rs

; ð7Þ

where

a ¼ ð1þ 2wÞ2ð1 −wÞ4 ; ð8Þ

Rs, nps

Rc, npc

nmz0

Fig. 1. Schematic presentation of side view of the layer.

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b ¼ −6wð1þ 0:5wÞ2ð1 −wÞ4 ; ð9Þ

c ¼ 0:5wð1þ 2wÞ2ð1 −wÞ4 : ð10Þ

Thus, under the considered approximation theoptical properties of the medium of closely packedparticles are determined by the phase function ofa separate particle plðγÞ and the structure factorS3ðγ;wÞ.

The optical thickness of the layer τ0, the singlescattering albedo Λ, and the scattering phase func-tion of an elementary volume phðγÞ are determinedas follows:

τ0 ¼ τ0lð1 −Λlð1 − uðRs;wÞÞÞ; ð11Þ

Λ ¼ ΛluðRs;wÞ1 −Λlð1 − uðRp;wÞÞ ; ð12Þ

phðγÞ ¼plðγÞS3ðγ;wÞuðRs;wÞ : ð13Þ

Here, τ0l is the optical thickness of the layer andΛl isthe single scattering albedo at the independentscattering regime implementation. Pay attentionthat at the interference approximation, the absorp-tion coefficient

αh ¼ εh − σh ¼ wα0l ð14Þis proportional to the volume concentration ofparticles w.

Below, we use the asymmetry parameter g to char-acterize numerically the degree of the phase functionelongation. It is defined as the cosine–weighted aver-age of the phase function:

g ¼Z π

0phðγÞ cos γ sin γdγ: ð15Þ

Here, phðγÞ is the phase function of the ensemble ofparticles. Values of g can range from −1 at the

180 deg complete backward scattering to þ1 atcomplete forward scattering.

3. Asymmetry Parameter and u Parameter

Phase function has a characteristic maximum atnonzero scattering angle in the case of large concen-tration of particles. The position of such a maximumis shifted to large angles with increased volumeconcentration w and to small angles with increasedparticle size [3,17]. The results of calculation forthe asymmetry parameter and the u parameterare shown below. At some particle concentrationsthe asymmetry parameter g for homogeneous parti-cles can be equal or less than zero. It can attain thevalue of about −0:6 [11,18]. For a medium containingtwo-layered particles, when the values of (n − 1) forthe nucleus and shell have different signs (here nis the refractive index of the core or the shell), theasymmetry parameter g can be negative even atsmall particle concentration. The dependence ofthe asymmetry parameter of the ensemble of parti-cles on the relative wavelength (λr ¼ λ=R or λ=Rs)and volume concentration for homogeneous and two-layered particles (R ¼ Rs) are displayed in Figs. 2(a)and 2(b), respectively.

Deviation of parameter u from unity characterizesthe degree of particles interaction. If u ¼ 1, we dealwith the sparse media. For spherical homogeneousparticles the value of u decreases monotonically withincreased particle concentration. The dependence ofparameter u on relative wavelength λr ¼ λ=R andconcentration of homogeneous spherical particleswith refractive index n ¼ 1:108 is presented inFig. 3(a). As seen from this figure, spectral depen-dence has maximum in the range of λr from 3 to 5,if volume concentration w > 0:2. It is shifted to smallwavelengths with increased particle.

For two-layered particles, when the differences ofthe refractive index from unity for the core (nc − 1)and for the shell (ns − 1) have different signs, the de-pendence of uðλ=RsÞ strongly differs from the onefor homogeneous particles. Figure 3(b) illustratesdependence of u on λr ¼ λ=Rs and concentration for a

12

3

4

50

0.1

0.2

0.3

0.4

0.5

-0.4-0.2

00.20.40.60.8

g

λr

w

(a)

12

3

4

50

0.1

0.2

0.3

0.4

0.5

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

g

λr

w

(b)

Fig. 2. Dependence of asymmetry parameter g on λr and w. (a) Layer of homogeneous spherical particles with radius R ¼ 0:3mkmand refractive index n ¼ 1:108, (b) layer of two-layered spherical particles with core radius Rc ¼ 0:18mkm and refractive index of corenc ¼ 1:108, shell radius Rs ¼ R ¼ 0:3mkm and refractive index of shell ns ¼ 0:96.

4248 APPLIED OPTICS / Vol. 50, No. 22 / 1 August 2011

system of two-layered particles. Maximum value of ugrows up with the increased particle concentration.Its magnitude is about 2.5 for volume concentrationw ¼ 0:6. At u > 1 the effective optical thickness of thelayer τ0 ¼ τolu is more than the optical thickness ofsparse layer τ0l. So the darkening effect takes place.At u < 1 the effective optical thickness of the layer isless than the optical thickness of the sparse layer andthe bleaching effect is implemented [3,19].

As one can see from Fig. 3, the value of u can de-crease or increase with increased volume concentra-tion. It depends on the behavior of phase functionand structure factor. The u value is determined bythe functions pðγÞ and S3ðγÞ. These functions forsome particles are presented in Fig. 4 (see also datashown in [11] for homogeneous particles). The u val-ue can bemore than unity if the intensity of scatteredlight in the backward directions is rather high. Sucha value can be implemented in layers containingparticles of the core-shell type.

Dependence of parameter u on refractive indices ofthe core and the shell is displayed in Fig. 5. One cansee that there is a near-linear dependence on the re-fractive index of the shell ns and the core nc at max-imal values of u. WhenRc is increased the proper linerotates clockwise.

4. Spectral Dependence of Extinction Coefficient andTransmittance

Using the above equations, consider spectral depen-dences of the extinction coefficient of the layer withnonabsorbing particles εhðλÞ ¼ wε0lðλÞuðRs;wÞ. Spec-tral dependence of extinction coefficient εhðλÞ can bepresented by a power function [20,21]:

εhðλÞ ¼ Cλ−pðλÞ: ð16ÞHere,C is the constant, p is the exponent. Sometimesit is referred to as the wave exponent so λ is used as avariable base. The exponent can be a constant or afunction of λ. For the white cloud the exponentp ¼ 1. For the blue sky the magnitude of p ¼ 4 andwe say that we deal with the Rayleigh scattering.From Eq. (16) it follows

pðλ;wÞ ¼ −d ln εhðλÞd ln λ : ð17Þ

The dependence of the wave exponent on wave-length λ and concentration w of two-layered particlesat two values of the core radius Rc is illustrated inFigs. 6(a) and 6(b). If the value of Rc is close to zero,dependence of pðλ;wÞ looks like for the homogeneousparticles with refractive index n ¼ ns and radiusR ¼ Rs. In such a case the peak with maximum atλ=R ≈ 4:5 takes place. Its magnitude increases withincreased volume concentration. As it follows fromthe numerical calculations at Rc > 0:25mkm, func-tion pðλ;wÞ is similar to the one for homogeneousparticles with radius R ¼ Rs and refractive indexn ¼ nc.

The dependence of the wave exponent on relativewavelength λr and core radius Rc is presented inFig. 7. The calculations show that the values of pcan considerably exceed the Rayleigh value (which is

12

3

4

50

0.1

0.2

0.3

0.4

0.5

0.6

0

0.5

1

1.5

2

2.5

λr

w

u

(b)

1

2

3

4

50

0.1

0.2

0.3

0.4

0.5

0.6

0

0.2

0.4

0.6

0.8

1

λr

w

bu

(a)

Fig. 3. Dependence of u on λr and w. (a) Homogeneous spherical particles with R ¼ 0:3mkm and relative refractive index n ¼ 1:108,(b) two-layered spherical particles with Rc ¼ 0:18mkm, nc ¼ 1:108, Rs ¼ 0:3mkm, ns ¼ 0:96.

0 20 40 60 80 100 120 140 160 γ(°)

-3

-2

-1

0

lgp,

lgS31

2

34

56 7

Fig. 4. Angular dependence of scattered light intensity: phasefunction for homogeneous particles with R ¼ 0:3mkm and n ¼1:108 (curve 1) and two-layered particles with Rc ¼ 0:18mkm,nc ¼ 1:108, ns ¼ 0:96, Rs ¼ 0:3mkm (curve 2). Structure factorS3 for layer with particles R ¼ 0:3mkm and volume concentrationw ¼ 0:1 (curve 3), 0.4 (curve 4), 0.5 (curve 5), 0.55 (curve 6), and 0.6(curve 7).

1 August 2011 / Vol. 50, No. 22 / APPLIED OPTICS 4249

equal to 4). Maximum magnitude is implemented atthe relative wavelengths λr ¼ λ=Rs > 3 when Rc iscomparable with Rs.

Characteristic features of coherent transmittance

T ¼ expð−τ0ðλÞÞ ð18Þat different volume concentration of particles forlayers with homogeneous and two-layered particlesare displayed in Figs. 8 and 9, respectively.

For layers with homogeneous subwavelength-sized particles there is the increasing of transmit-tance with increased volume concentration. Thelayer bleaching is implemented [3,19,22].

0.9

0.95

1

1.05

1.1

ns

0.98

1

1.02

1.04

nc

0.5

1

1.5

2

u

(a)

0.9

0.95

1

1.05

1.1

0.98

1

1.02

1.040.5

1

1.5

2

u

ns

nc

(b)

Fig. 5. Dependence of the u value on nc and ns for two-layered spherical particles with Rs ¼ 0:3mkm, w ¼ 0:6, Rc ¼ 0:18mkm (a), Rc ¼0:20mkm (b) at λ ¼ 0:9mkm.

23

45

6

70

0.1

0.2

0.3

0.4

0.5

0

2

4

6

8

λ/Rs

p

w

(a)

23

45

6

70

0.1

0.2

0.3

0.4

0.5

0

5

10

15

20

p

λ/Rs

w

(b)

Fig. 6. Dependence of p on λ=Rs and w for two-layered spherical particles with nc ¼ 1:2, Rs ¼ 0:3mkm, ns ¼ 0:9. (a) Rc ¼ 0:05mkm,(b) Rc ¼ 0:2mkm.

23

45

67

0.1

0.12

0.14

0.16

0.18

0.2

0

5

10

15

20

p

λ/Rs Rc(mkm)

Fig. 7. Dependence of parameter p on λ=Rs andRc for two-layeredparticles at nc ¼ 1:1084, ns ¼ 0:9636, Rs ¼ 0:3mkm, w ¼ 0:5.

0.6 0.8 1 1.2 1.4

0

0.2

0.4

0.6

0.8

T

λ(mkm)

1

2

3

Fig. 8. Dependence of transmittance T on the wavelength for alayer with homogenous particles at w ¼ 0:01 (curve 1), w ¼ 0:4(curve 2), and w ¼ 0:6 (curve 3). R ¼ 0:3mkm, n ¼ 1:1, nm ¼ 1,z0 ¼ 20mkm.

4250 APPLIED OPTICS / Vol. 50, No. 22 / 1 August 2011

For a medium with two-layered subwavelength-sized particles it is possible to get conditions forincreased and decreased transmittance by changingthe size and refractive indices of the core and theshell. As seen from Fig. 9, there is another range ofwavelengths where transmittance increases (bleach-ing effect) and the range of wavelengths where trans-mittance decreases (darkening effect) with increasedvolume concentration. These effects result in morepronounced spectral dependence of transmittance.

5. Conclusion

Light scattering by a layer with closely packed non-absorbing spherical (homogeneous and two-layered)particles has been investigated. To describe thetransmittance of the layer, the interference approx-imation was used. The spatial correlation of particleswas described by the Perkus–Yevick approximationof hard spheres. The possibility of the anomalousscattering regime (anomalous high value of theexponent of the power function, describing the de-pendence of the extinction coefficient on the wave-length) implementation in a sparse medium withtwo-layered particles is displayed. The exponent ofthe power function increases with increased volumeconcentration.

Simulations disclose that in the layer with two-layered subwavelength-sized particles the bleachinganddarkening effects canbe implemented. In the firstcase transmittance increases,while in the second casetransmittance decreases with increased volume con-centration. The developed model can be used for lightscattering description in liquated glasses, filled liquidcrystals, polymer-filled nematics, optical filters, andpaints.

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0.6 0.8 1 1.2 1.4 1.6

0

0.2

0.4

0.6

0.8

T

2

3

1

λ(mkm)

Fig. 9. Dependence of transmittance T on the wavelength fora layer with two-layered particles at w ¼ 0:01 (curve 1), w ¼ 0:4(curve 2), and w ¼ 0:6 (curve 3). Rc ¼ 0:2mkm, nc ¼ 1:2, Rs ¼0:3mkm, ns ¼ 0:9, nm ¼ 1:0, z0 ¼ 20mkm.

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