- Email: [email protected]
Multiple scattering of a focused laser beam by a cluster consisting of nonconcentric encapsulated particles
Journal of Quantitative Spectroscopy & Radiative Transfer 219 (2018) 255–261
Contents lists available at ScienceDirect
Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt
Multiple scattering of a focused laser beam by a cluster consisting of nonconcentric encapsulated particles Hany L.S. Ibrahim a,∗, Elsayed Esam M. Khaled b a b
Telecom Egypt Company, Qina, Egypt Electrical Engineering Department, Faculty of Engineering, Assiut University, Assiut, Egypt
a r t i c l e
i n f o
Article history: Received 24 April 2018 Revised 16 July 2018 Accepted 9 August 2018 Available online 22 August 2018 Keywords: Multiple scattering T-matrix Nonspherical coated particles Laser beam
a b s t r a c t The optical characteristics of a cluster consisting of zinc sulfide (ZnS) particles doped with a nonconcentric spherical copper (Cu) cores illuminated with an arbitrarily focused Gaussian beam are investigated. The presented aggregations of nonconcentric doped particles (i.e. core with offset origin) form linear chains or densely packed clusters. The laser beam is modeled using angular spectrum of plane waves method and then combined with the cluster T-matrix method which is modified to solve such difficult multiple scattering problem. This combination provides a powerful mathematical technique to obtain the phase (scattering) matrix of a cluster illuminated with any incident electromagnetic fields. The scattering matrix provides complete descriptions of the scattering characteristics in the far field zone. The computed results are shown for different beam waists with respect to the cluster. The scattering processes and its results help understanding many cluster characteristics and nonlinear processes. The presented numerical results show that the elements of the scattering matrix are sensitive to the focusing of the incident beam and characteristics of the cluster constituents. The illustrated results are important for researches aim to improve polymer properties and to study several branches of practical sciences and industries such as nanotechnology, pharmaceuticals, chemistry, and biology. This paper represents the first attempt to study the multiple scattering from a cluster of nonspherical particles with nonconcentric spherical cores illuminated by an arbitrarily focused laser beam. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction The radiative properties of a cluster consisting of micro or nano coated particles are relevant to a wide variety of applications in many areas such as pharmaceuticals, nanotechnology, chemistry, biology, health sciences, and astrophysics [1]. Moreover various particles characteristics, such as shape, refractive index, composition, and surface roughness have important influences to the light scattering by such particles. In some cases, improving the handling properties of a material as in a doped zinc sulfide (ZnS) particle with a copper (Cu) core can enhance the photoluminescence process [2,3]. zinc sulfide (ZnS) is a common pigment of phosphorescent materials that has applications in the optoelectronics industry [4]. Also the photoluminescence results from the nonlinear optical characteristics of the ZnS particles make this material a suitable candidate for applications in the low voltage display technology
∗
Corresponding author. E-mail addresses: [email protected] esamk54_20 0 [email protected] (E.E.M. Khaled). https://doi.org/10.1016/j.jqsrt.2018.08.009 0022-4073/© 2018 Elsevier Ltd. All rights reserved.
(H.L.S.
Ibrahim),
[5]. A cluster consisting of ZnS axisymmetric particles encapsulated with Cu core can enhance the photoluminescence from such materials. In many cases, cores of the particles in the clusters are nonconcentric with the particles. Therefore in such cases, understanding light interactions processes are important keys to characterize the properties of the constituents of the clusters and to know how the polarization and frequency of the laser beam behave during the scattering processes. The analysis of scattering of light, especially focused laser beam, with a cluster is not an easy task. Some researchers illustrated a numerical analysis of the scattering problem of a cluster illuminated with a plane wave [6–8]. Ibrahim et al. [9] divided such problem into two parts; the first is the incident beam modeling using the angular spectrum of plane wave method [10], and the second is the interaction of the incident beam with the cluster through the use of a modified T-matrix technique [7]. Their method provide certain advantages: (1) the elements of the TMatrix are independent on the incident fields but depend only on the scatterer properties, therefore these elements are computed once for each case of the scatterer and then used for different
256
H.L.S. Ibrahim, E.E.M. Khaled / Journal of Quantitative Spectroscopy & Radiative Transfer 219 (2018) 255–261
cases of illuminations such as direction, polarization, and shape of the incidence light, (2) the incident field can be decomposed into simpler basis functions (spectrum of plane waves) and then manipulated in a modified T- matrix system to compute the scattered fields and different optical characteristics. Moreover using plane wave spectrum technique to model the incident light enables researchers to represent any physically realizable beam which has no mathematical representation such as beams with ripples. In our previous research [8] a cluster consisting of dielectric nonconcentric encapsulation particles illuminated with a plane wave only is investigated. Also in our previous work [9] the scattering of a focused laser beam by a cluster consisting of homogeneous particles is studied. So the new case in this paper is the investigation of the scattering processes of a focused laser beam by a cluster consists of nonspherical particles encapsulated with shifted cores. To the best of our knowledge this is the first attempt in the literature to tackle such type of problem. So we modified the cluster T-matrix method to deal with both a focused laser beam scattered by a cluster of nonspherical coated particles with shifted cores. This illustrated technique is applied to a practical case that is a cluster consisting of nonspherical zinc sulfide (ZnS) particles doped with nonconcentric spherical copper (Cu) cores illuminated with an arbitrarily focused laser beam. To check the validity of our technique the calculated results of a cluster consisting of dielectric nonconcentric encapsulation particles illuminated with a laser beam of very wide waist are contrasted with cases of a cluster illuminated with plane wave [8], and with cases of homogeneous axisymmetric particles illuminated with an arbitrarily focused electromagnetic Gaussian beam [9]. No differences were noticed in all the cases.
2. Method and theoretical analysis Fig. 1 shows a cluster consisting of N (N = 5 in this work) identical oblate spheroidal particles of zinc sulfide (ZnS) material with an off-centered spherical copper (Cu) core. The particles are centered at local origins of a right-handed Cartesian coordinate systems (xi ,yi ,zi ), where i indicates the particle’s index. In the chain cluster all particles are oriented along the x-axis of a righthanded Cartesian coordinate system. The greater radii of the particles (which is bs ) are along the x-axis and each particle touches the following one. The center of the middle particle is at the origin as shown in Fig. 1(a). In the packed cluster the center of the first particle is located in the origin and its greater radius is along the x-axis. The second particle touches only the first particle and its center is located on the x-axis. Also its greater radius is along the x-axis. The third particle touches only the first particle. Its center is located on the y-axis and its greater radius is along the y-axis. The centers of the fourth and fifth particles are located at the zaxis along each side of the first particle. Their smaller radii are on the z-axis and their greater radii are oriented along the x-axis, each of them touches only the first particle as shown in Fig. 1(b). The global Cartesian coordinate system of the cluster is (x, y, z) as shown in the figure. The radii of the shell are as and bs , and that of the core is ac . The cores are shifted by an offset distance L along the z-direction. The cluster is arranged as a linear chain as in Fig. 1(a) or as a packed cluster as in Fig. 1(b). The surface of the core can be described, with respect to the local coordinate axis (xi ,yi ,zi ) as [11],
rc (θi ) = ac pi cos(θi ) + (1 − pi 2 sin (θi )) 2
1 2
(1)
where pi = Li /ac , and θ i is local theta angle of the spherical coordinate system (ri , θ i , φ i ). By defining the core/particle size ratio R = ac /as , it can be seen that the core remains inside the shell as
long as the condition (Li /as + R) < = 1 is fulfilled. All the detailed analysis for the oblates constructing the cluster is given in [11]. The cluster is illuminated with a focused Gaussian beam of a waist 2w0 and wavelength λ propagating along the z-direction. The vector spherical harmonics (VSH) expansion of the plane wave spectrum which is used to model the incident beam [10] is,
Einc (kr ) = H
m
Dmn atemn M1emn (kr ) + atomn M1omn (kr )
n
+ btemn N1emn (kr ) + btomn N1omn (kr )
(2)
where H, and Dmn are normalization factors depend on the incident beam. The at emn , at omn , bt emn , and bt omn are the incident Gaussian beam expansion coefficients, and M1 emn , M1 omn , N1 emn , and N1 omn are VSH of the first kind. m (in italic) is the azimuthal mode index and n is the mode number. The symbols e and o stand for even and odd respectively. r is a position vector, and the wave number is k = 2∗ π /λ. In the case of a unit amplitude incident plane wave, H is unity and the azimuthal mode index m = 1. The advantages of this modeling method are: (1) it can be used for a beam whether or not it has a mathematical formula and (2) it can be used to model for physically realizable beams with ripples [10]. All the analysis of modeling the beam can be found in [10]. In general calculations of the elements of the fields are infinite. In practical computer calculations they must be limited to a finite size by truncating the expansion series to a maximum number n = nmax . If number of the terms in the series is greater than nmax then the computations will lead to the truncation error or a convergence error. For scattering of a single particle ni, max = x + 4.05x1/3 + 2 where x is the size parameter of the particle. The method for convergence and plane wave truncation can be found in details in Ref [10]. Because of electromagnetic interactions between the particles, the scattered fields from each particle i are interdependent, therefore the total incident field illuminating the particle j can be represented as a superposition of; (1) external incident field from the incident beam, and (2) sum of all fields scattered from all other particles illuminating the particle j, that is inc Einc + j = E
N
Esca l
j = 1, . . . , N.
(3)
l=1 l= j
where N is the number of the particles in the cluster. These jth particle illuminating fields can be expanded in VSH as,
Einc j =
jt amn +
mn
jl amn M1 mn (kr j )
l= j
jt bmn
+
+
jl bmn
N
1
mn
( kr j ) ,
j = 1, . . . , N.
(4)
l= j
jt
jt
The expansion coefficients amn and bmn describe the expansion coefficients of the external incident laser beam illuminating partijt jt cle j and amn , bmn describe the expansion coefficients of the field scattered from particle l and illuminating particle j. Then the expansion coefficients of the total scattered fields from the particle j can be written in form of T-matrix notation as,
fj gj
=T
j
a jl a jt + b jt b jl
,
j = 1, . . . , N.
(5)
l= j
where f j , g j are the scattered field expansion coefficients of the particle j, and T j is the T-matrix of the particle j and is given in detail in [7]. Following the analysis given in [8] and [9], we finally
H.L.S. Ibrahim, E.E.M. Khaled / Journal of Quantitative Spectroscopy & Radiative Transfer 219 (2018) 255–261
257
z
L
ac
bs
as
(a)
x
y z
4 x
2w0 1
2
(a)
3 5
y
(b)
Fig. 1. A cluster consisting of five identical oblate spheroidal particles with off-centered spherical cores. The radius of the shell is as and of the core is ac . The core is shifted a distance L long the z-direction. The cluster is illuminated with a focused Gaussian beam of minimum spot size w0 μm and propagates in the z- direction. The particles in the cluster are arranged in a Cartesian coordinate system (x, y, z) to form, (a) a linear chain, and (b) a packed cluster.
describe Eq. (5) in matrix notations as,
j
f =Tj gj
jt
a b jt
A krl j
+ B krl j l= j
given as,
l B krl j
fl ,
j = 1, . . . , N.
g
A krl j
(6) where A(krlj ) and B(krlj ) are the translation coefficients which given by analytical expressions in [12]. The matrix T j , that is for nonconcentric layered particle with refractive indices mc (core) and ms (shell) can be expressed as [11],
T j = − B ∗ A−1 = − Bs + Bcs ∗ − Bc ∗ A−1 c
∗ As + Acs ∗ − Bc ∗ A−1 c
−1 (7)
where: •
•
•
–Bc ∗ Ac −1 = Tc , is the T-matrix calculated for any particle in the cluster with refractive index m = mc /ms for the inner layer (core), and the size parameter as R∗ ms ∗ x; x = 2π as /λ. In other words, it is the T-matrix for the core alone without the shell, where R = ac /as , and x is the size parameter of the outer surface. Bs and As , are calculated using the refractive index equal to ms for outer layer and the size parameter equal to x. Matrices Acs and Bcs are calculated in the same way as As and Bs except that the Bessel functions of the first kind with argument kr are replaced by Hankel functions with the same argument.
Eq. (7) provides the T-matrix of the particle j in the cluster. Then the T-matrix of the whole cluster, after some mathematical procedures which are given in details in [7] and [9], can be
N A kr j0
T = B kr j0 j,l=1
B kr j0
T jl A(kr0l ) B ( kr0l ) A kr j0
B ( kr0l ) , A ( kr0l )
(8)
In this equation the translation coefficients A (kr0l ) and B (kr0l ) are based on spherical Bessel functions. The convergence check for a cluster consisting of axisymmetric particles faces two challenging numerical computational problems. First for each individual particle i in the cluster consisting of N particles the convergence criteria must be tested using the formula, ni, max = NO,i = xi + 4.05xi 1/3 + 2, i = 1,…,N, where NO , i is the maximum number of terms commonly used to obtain convergence. This number is proportional to the size parameter xi = 2π ai /λ; λ is the wavelength and ai is the radius of each particle (For a spheroidal particle the radius of an equal-volume sphere of the particle must be calculated). For a spheroidal particle the size parameter is given by (x)spheroid = (x)sphere (a/b)2/3 where a is the radius along the xaxis and b is the radius along the y-axis for a centered spheroids at an origin O of Cartesian coordinate system x, y, z. It has been proved that convergence can be obtained if the order of the particle’s T-matrix is 2 × NO,i [7]. Note that the summation over n is calculated for each mode m and the summation over the azimuthal modes m ranges from 0 to NO,i -1. In the case of the incident plane wave, the azimuthal mode m = 1. For a cluster, the radius ac of the smallest sphere centered at the origin of the coordinate system that containing all the particles in the cluster is considered as the radius of the cluster. Therefore the equivalent size parameter of the cluster is x = 2π ac /λ. Hence the maximum order nmax of the cluster T-matrix is nmax = No = x + 4.05x 1/3 + 2. This maximum order is proportional to the size parameter of the smallest sphere that circumscribes the entire cluster. Detailed description
258
H.L.S. Ibrahim, E.E.M. Khaled / Journal of Quantitative Spectroscopy & Radiative Transfer 219 (2018) 255–261
of the conversion criteria and maximum number of n and m can be found in [7,13]. Consequently, the elements of the S-matrix or the phase matrix can be completely expressed as [13]
S11 = 12 |F1 |2 + |F2 |2 + |F3 |2 + |F4 |2
S12 = 12 |F2 |2 − |F3 |2 + |F4 |2 − |F1 |2
S22 = 12 |F2 |2 − |F3 |2 − |F4 |2 + |F1 |2
∗ S33 = Re F2 F1 + F3 F4∗
S34 = Im F2 F1∗ + F4 F3∗
∗ S44 = Re F1 F2 − F3 F4∗
(9)
The four elements F1 up to F4 of the amplitude scattering matrix F are related to the cluster T-matrix given in (8) and are given in detail in [13]. The orientation averaged scattering matrix elements can be analytically obtained from the T-matrix. Following the procedures developed by Mishchenko [13] for axisymmetric scatterers in which one forms the products of the scattering amplitude elements Eq. (9) from the cluster T-matrix and then integrate the products over all directions and polarizations of the incident field. 3. Numerical results The scattering of a light beam by particles is fully characterized by a 4 × 4 phase matrix [s] (often called the Mueller or scattering matrix) [13] which, in the most general case, has 16 different real nonvanishing elements. Number of the elements in the scattering matrix reduces to six nonzero elements for a collection of identical randomly oriented axisymmetric particles. The random orientation matrix elements S11 , S22 , S33 and S44 , and S12 and S34 and their transposes are nonzero only for clusters (with on-centered cores) possessing a plane of symmetry (not necessarily axisymmetric chains or packed clusters). The configurations of both clusters (with on-cenetered cores as stated before [8]) used in computation for the results in Figs. 2 and 3 all possess a plane of symmetry which is xy-plane, and because of this we computed the elements S23 , S24 , S13 , and S14 and their transposes for both clusters. We found that these elements are very small approximation zeros and therefore we do not include them. The calculated S11 measures total scattered intensity for unpolarized incident light which gives general information of the particle size, S12 measures degree of the polarized light. The degree of polarization can vary from zero (completely unpolarized) to one (completely polarized). S22 , S33 , S34 , S44 measures the degree of depolarized light, when the incident beam fully linearly or circularly polarized. S22 express transformation of linearly polarized incident light (±90°) to linearly polarized scattered light (±90°); deviation from S11 is an important indication of the presence of nonspherical particles. S33 , S44 expresses transformation of linearly polarized incident light (±45°) (which is circularly) to linearly polarized scattered light (±45°) (which is circularly). The deviation of S44 from S33 indicates of nonspherical symmetry. S34 express transformation of circularly polarized incident light to linearly polarized scattered light (±45°); element is strongly dependent on size and complex refractive index of the scatterers [7,13,14]. In this paper, a cluster consisting of five oblate spheroidal particles made from zinc sulfide (ZnS) form a linear chain, shown in Fig. 1(a), or a packed of a hexagonal lattice, shown in Fig. 1(b) is considered. Each particle has a refractive index ms = 2.2, size parameter xs = 2π as /λ = 4, and axial ratio as /bs = 0.8 [15]. The particles are doped individually by off-centered spherical copper cores of a size parameter xc = 2π ac /λ = 2, and refractive index mc = 2.43 [16]. The offset parameter for the core for all the particles is p = L/ac = 0.5, L is the distance from the core’s center in its local origin to the center of its shell. The cluster is illuminated with
a Gaussian beam of light of wavelength λ = 1.0 μm propagating in the z-direction with an intensity Eo = 1 v/m at its focal point. The beam waist is chosen as w0 = 125bs , w0 = 5bs and w0 = bs to represent three different cases of illumination for each cluster. The first choice w0 = 125bs represent a plane wave illumination since w0 > > bs , the second choice w0 = 5bs represent a moderate focused laser beam, and the last one w0 = bs represent tightly focused beam. The computed results of the orientation averaged scattering matrix elements are shown in Figs. 2 and 3 for the chain cluster and packed cluster respectively. The results demonstrate that in the case of a very wide beam waist w0 = 125 bs ≈ 100 μm, (which simulates a plane wave illumination) the differences of the intensity of S11 between both clusters cases is very small at any scattered angle (there are tiny different in the front scattering) since the clusters are illuminated by the same field strengths (plane wave), which the strength of illumination is the same on all particles in the cluster for the case of plane wave illumination whereas the strengths of illumination are different on different particles in the cluster for the focused beam illumination. So the shape of the cluster has very little effect in intensity of S11 in the final scattered signal. Note that this observation is not true for the focused beam illuminations. The results show that the magnitudes of oscillations of S11 decrease as the waist of the beam gets narrower as illustrated in both Figs. 2 and 3. The magnitudes of oscillations means that the intensity of S11 oscillates between different higher values and different lower values at different consequence scattering angles between the front and the back scattering. These higher and lower values decrease as the waist of the beam gets narrower. This means that the interference of the multiple scattering is less as the beam gets more focusing since the beam may misses some particles to interact with as it gets narrower. The ratio S22 / S11 , equals 1 for the focused beam while it gets different values for the wider beam waist illumination. When S22 /S11 equal to 1 that means the cluster do not depolarized the incident beam and when S22 /S11 deviates from 1 that means the cluster can produce significant depolarization. When S22 /S11 equal to zero that means the cluster can produce fully depolarization [14]. Despite of the S22 /S11 indicate of higher or lower degree of sphericity for plane wave illumination, our paper shows that this is not true if the incident beam is very focused as shown in Figs. 2 and 3. So we can say that for a very focused incident beam the S22 /S11 can be closer to one and that does not mean the degree of nonsphericity decreases but means that the cluster is not depolarizing the incident beam. The ratio S12 /S11 for both clusters should be always zero at θ = π as expected. But the new observation is that the deviations of -S12 (θ )/S11 (θ ) as a function of beam waist depends on how the particles are arranged in the cluster, i.e. the results for the chain cluster is different than that of the packed cluster. Comparing the ratio -S12 /S11 for both clusters we can notice that for a very focused laser beam illumination the ratio S12 /S11 near the backscattering direction (θ = 135ο to 165ο ) for the packed cluster is attenuated greater than those for chain cluster which indicates that the packed-cluster configuration offers a much higher opportunity for multiple scattering and then this result can be used for cluster configuration identification. It is noted in the cases of the linear chain or densely packed clusters illuminated with a focused laser beam, Figs. 2 and 3 that S33 and S44 are identical and this indicates of fully symmetric for clusters (note that the beam and both clusters are symmetric along xy-plane and different field strengths are illuminating different particles). Since the case of a cluster illuminated of a focused laser beam is fairly a new case. This is unlike the case of incident plane wave (presented by wide beam waist, wo = 100b, at which the same field strength illuminates all particles) where a small deviation of S44 from S33 is obtained which shows nonspherical symmetry, see reference [7,14] for a cluster illuminated with a plane wave.
H.L.S. Ibrahim, E.E.M. Khaled / Journal of Quantitative Spectroscopy & Radiative Transfer 219 (2018) 255–261
259
Fig. 2. Orientation-averaged scattering matrix elements for a linear chain of five identical oblate spheroidal Zinc sulfide (ZnS) particles. The refractive index of each particle is m = 2.2, size parameter x = 4 and the axial ratio ρ = a/b = 0.8. Each particle is doped by a spherical copper (Cu) of size parameter x = 2 and refractive index m = 2.43. The offset parameter for the core is p = L/ ac = 0.5. Results are shown for different a Gaussian beam of waist w0 = 100 μm (i.e. plane wave), w0 = 5bs and w0 = bs of wavelength λ = 1.0 μm propagating in the z-direction.
260
H.L.S. Ibrahim, E.E.M. Khaled / Journal of Quantitative Spectroscopy & Radiative Transfer 219 (2018) 255–261
Fig. 3. Orientation-averaged scattering matrix elements for a packed cluster of five identical oblate spheroidal Zinc sulfide (ZnS) particles. The refractive index of each particle is m = 2.2, size parameter x = 4 and the axial ratio ρ = a/b = 0.8. Each particle is doped by a spherical copper (Cu) of size parameter x = 2 and refractive index m = 2.43. The offset parameter for the core is p = L/ ac = 0.5. Results are shown for different a Gaussian beam of waist w0 = 100 μm (i.e. plane wave), w0 = 5bs and w0 = bs of wavelength λ = 1.0 μm propagating in the z-direction.
H.L.S. Ibrahim, E.E.M. Khaled / Journal of Quantitative Spectroscopy & Radiative Transfer 219 (2018) 255–261
4. Conclusion The random-orientation scattering properties of a plane wave as well as a focused Gaussian beam scattered by a cluster consisting of nonspherical encapsulated particles are presented. Two examples, as applications, are studied in this paper; a chain cluster consisting of five oblate particles made from zinc Sulfide (ZnS) material injected with cores of spherical copper (Cu). The other cluster is a packed cluster consisting of the same particles as of the chain cluster. The scattering matrix elements are computed for three different cases of illuminations; a plane wave, a fairly focused Gaussian, and a focused beam. It noted that the behavior of the nonzero scattering matrix elements depend on the focusing of the incident beam. This behavior can be used to identify the type of a cluster under investigation. References [1] Quirantes A. Light scattering properties of spheroidal coated particles in random orientation. J Quant Spectrosc Radiat Transf 1999;63(2–6):263–75. [2] Ethiraj AS, Hebalkar N, Kulkarni SK, Pasricha R, Urban J, Dem C, Schmitt M, Kiefer W, Weinhardt L, Joshi S, Fink R, Heske C, Kumpf C, Umbach E. Enhancement of photoluminescence in manganese doped ZnS nanoparticles due to a silica shell. J Chem Phys 2003;118:8945. [3] Khaled EEM, Ibrahim HL. Photoluminescence enhancement from an axisymmetric zinc sulfide particle illuminated with a focused laser beam. Frontiers in Optics 2010/Laser Science XXVI conference, OSA Technical Digest (CD). Optical Society of America; 2010.
261
[4] Dumbrava A, Badea C, Prodan G, Popovici I, Ciupina V. Zinc sulfide fine particles obtained at low temperature. Chalcogenide Lett 2009;6:437–43. [5] Vacassy R, Scholz SM, Dutta J, Hofmann H, Plummer CJG, Carrot G, Hilborn J, Akinc M. Nanostructured zinc sulfide phosphors. In: Materials Research Society Symposium Proceedings, 501. Materials Research Society; 1998. p. 369–74. MRS. [6] Wriedt T, Schuh R, Doicu A. Scattering by aggregated fibres using a multiple scattering T-matrix approach. Part Part Syst Charact 2008;25:74–83. [7] Mackowski DW, Mishchenko MI. Calculation of the T matrix and the scattering matrix for ensembles of spheres. J Opt Soc Am 1996;13(11) November. [8] Ibrahim HLS, Khaled EEM. Light scattering from a cluster consists of dielectric nonconcentric encapsulation particles. Int J Electron Commun Eng Technol 2014;5:82–94. [9] Ibrahim HLS, Wriedt T, Khaled EEM. Light scattering by a cluster consisting of homogeneous axisymmetric particles illuminated with an arbitrarily focused electromagnetic Gaussian beam. J Quant Spectrosc Radiat Transf 2016;173:83–91. [10] Khaled EEM, Hill SC, Barber PW, Chowdhury DQ. Scattered and internal intensity of a sphere illuminated with a gaussian. TEEE Trans Antennas Propag 1993;41(3) March. [11] Quirantes A. A T-matrix method and computer code for randomly oriented, axially symmetric coated scatterers. J Quant Spectrosc Radiat Transf, 2005;92:373–81. [12] Friedman B, Russek J. Addition theorems for spherical waves. Q Appl Math 1954;12:13–23. [13] Mishchenko MI, Travis LD, Lacis AA. Scattering, absorption, and emission of light by small particles. New York: NASA Goddard Institute for Space Studies; July 2005. [14] Mishchenko MI, Hovenier JW, Travis LD. light Scattering by nonspherical particles: theory, measurements, and applications. Elsevier; 1999. [15] Mirabella FM. Internal reflection spectroscopy: theory and applications, 15. CRC Press; 1992. [16] http://www.raeco.com/training/refractive- index- values.htm.
Contents lists available at ScienceDirect
Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt
Multiple scattering of a focused laser beam by a cluster consisting of nonconcentric encapsulated particles Hany L.S. Ibrahim a,∗, Elsayed Esam M. Khaled b a b
Telecom Egypt Company, Qina, Egypt Electrical Engineering Department, Faculty of Engineering, Assiut University, Assiut, Egypt
a r t i c l e
i n f o
Article history: Received 24 April 2018 Revised 16 July 2018 Accepted 9 August 2018 Available online 22 August 2018 Keywords: Multiple scattering T-matrix Nonspherical coated particles Laser beam
a b s t r a c t The optical characteristics of a cluster consisting of zinc sulfide (ZnS) particles doped with a nonconcentric spherical copper (Cu) cores illuminated with an arbitrarily focused Gaussian beam are investigated. The presented aggregations of nonconcentric doped particles (i.e. core with offset origin) form linear chains or densely packed clusters. The laser beam is modeled using angular spectrum of plane waves method and then combined with the cluster T-matrix method which is modified to solve such difficult multiple scattering problem. This combination provides a powerful mathematical technique to obtain the phase (scattering) matrix of a cluster illuminated with any incident electromagnetic fields. The scattering matrix provides complete descriptions of the scattering characteristics in the far field zone. The computed results are shown for different beam waists with respect to the cluster. The scattering processes and its results help understanding many cluster characteristics and nonlinear processes. The presented numerical results show that the elements of the scattering matrix are sensitive to the focusing of the incident beam and characteristics of the cluster constituents. The illustrated results are important for researches aim to improve polymer properties and to study several branches of practical sciences and industries such as nanotechnology, pharmaceuticals, chemistry, and biology. This paper represents the first attempt to study the multiple scattering from a cluster of nonspherical particles with nonconcentric spherical cores illuminated by an arbitrarily focused laser beam. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction The radiative properties of a cluster consisting of micro or nano coated particles are relevant to a wide variety of applications in many areas such as pharmaceuticals, nanotechnology, chemistry, biology, health sciences, and astrophysics [1]. Moreover various particles characteristics, such as shape, refractive index, composition, and surface roughness have important influences to the light scattering by such particles. In some cases, improving the handling properties of a material as in a doped zinc sulfide (ZnS) particle with a copper (Cu) core can enhance the photoluminescence process [2,3]. zinc sulfide (ZnS) is a common pigment of phosphorescent materials that has applications in the optoelectronics industry [4]. Also the photoluminescence results from the nonlinear optical characteristics of the ZnS particles make this material a suitable candidate for applications in the low voltage display technology
∗
Corresponding author. E-mail addresses: [email protected] esamk54_20 0 [email protected] (E.E.M. Khaled). https://doi.org/10.1016/j.jqsrt.2018.08.009 0022-4073/© 2018 Elsevier Ltd. All rights reserved.
(H.L.S.
Ibrahim),
[5]. A cluster consisting of ZnS axisymmetric particles encapsulated with Cu core can enhance the photoluminescence from such materials. In many cases, cores of the particles in the clusters are nonconcentric with the particles. Therefore in such cases, understanding light interactions processes are important keys to characterize the properties of the constituents of the clusters and to know how the polarization and frequency of the laser beam behave during the scattering processes. The analysis of scattering of light, especially focused laser beam, with a cluster is not an easy task. Some researchers illustrated a numerical analysis of the scattering problem of a cluster illuminated with a plane wave [6–8]. Ibrahim et al. [9] divided such problem into two parts; the first is the incident beam modeling using the angular spectrum of plane wave method [10], and the second is the interaction of the incident beam with the cluster through the use of a modified T-matrix technique [7]. Their method provide certain advantages: (1) the elements of the TMatrix are independent on the incident fields but depend only on the scatterer properties, therefore these elements are computed once for each case of the scatterer and then used for different
256
H.L.S. Ibrahim, E.E.M. Khaled / Journal of Quantitative Spectroscopy & Radiative Transfer 219 (2018) 255–261
cases of illuminations such as direction, polarization, and shape of the incidence light, (2) the incident field can be decomposed into simpler basis functions (spectrum of plane waves) and then manipulated in a modified T- matrix system to compute the scattered fields and different optical characteristics. Moreover using plane wave spectrum technique to model the incident light enables researchers to represent any physically realizable beam which has no mathematical representation such as beams with ripples. In our previous research [8] a cluster consisting of dielectric nonconcentric encapsulation particles illuminated with a plane wave only is investigated. Also in our previous work [9] the scattering of a focused laser beam by a cluster consisting of homogeneous particles is studied. So the new case in this paper is the investigation of the scattering processes of a focused laser beam by a cluster consists of nonspherical particles encapsulated with shifted cores. To the best of our knowledge this is the first attempt in the literature to tackle such type of problem. So we modified the cluster T-matrix method to deal with both a focused laser beam scattered by a cluster of nonspherical coated particles with shifted cores. This illustrated technique is applied to a practical case that is a cluster consisting of nonspherical zinc sulfide (ZnS) particles doped with nonconcentric spherical copper (Cu) cores illuminated with an arbitrarily focused laser beam. To check the validity of our technique the calculated results of a cluster consisting of dielectric nonconcentric encapsulation particles illuminated with a laser beam of very wide waist are contrasted with cases of a cluster illuminated with plane wave [8], and with cases of homogeneous axisymmetric particles illuminated with an arbitrarily focused electromagnetic Gaussian beam [9]. No differences were noticed in all the cases.
2. Method and theoretical analysis Fig. 1 shows a cluster consisting of N (N = 5 in this work) identical oblate spheroidal particles of zinc sulfide (ZnS) material with an off-centered spherical copper (Cu) core. The particles are centered at local origins of a right-handed Cartesian coordinate systems (xi ,yi ,zi ), where i indicates the particle’s index. In the chain cluster all particles are oriented along the x-axis of a righthanded Cartesian coordinate system. The greater radii of the particles (which is bs ) are along the x-axis and each particle touches the following one. The center of the middle particle is at the origin as shown in Fig. 1(a). In the packed cluster the center of the first particle is located in the origin and its greater radius is along the x-axis. The second particle touches only the first particle and its center is located on the x-axis. Also its greater radius is along the x-axis. The third particle touches only the first particle. Its center is located on the y-axis and its greater radius is along the y-axis. The centers of the fourth and fifth particles are located at the zaxis along each side of the first particle. Their smaller radii are on the z-axis and their greater radii are oriented along the x-axis, each of them touches only the first particle as shown in Fig. 1(b). The global Cartesian coordinate system of the cluster is (x, y, z) as shown in the figure. The radii of the shell are as and bs , and that of the core is ac . The cores are shifted by an offset distance L along the z-direction. The cluster is arranged as a linear chain as in Fig. 1(a) or as a packed cluster as in Fig. 1(b). The surface of the core can be described, with respect to the local coordinate axis (xi ,yi ,zi ) as [11],
rc (θi ) = ac pi cos(θi ) + (1 − pi 2 sin (θi )) 2
1 2
(1)
where pi = Li /ac , and θ i is local theta angle of the spherical coordinate system (ri , θ i , φ i ). By defining the core/particle size ratio R = ac /as , it can be seen that the core remains inside the shell as
long as the condition (Li /as + R) < = 1 is fulfilled. All the detailed analysis for the oblates constructing the cluster is given in [11]. The cluster is illuminated with a focused Gaussian beam of a waist 2w0 and wavelength λ propagating along the z-direction. The vector spherical harmonics (VSH) expansion of the plane wave spectrum which is used to model the incident beam [10] is,
Einc (kr ) = H
m
Dmn atemn M1emn (kr ) + atomn M1omn (kr )
n
+ btemn N1emn (kr ) + btomn N1omn (kr )
(2)
where H, and Dmn are normalization factors depend on the incident beam. The at emn , at omn , bt emn , and bt omn are the incident Gaussian beam expansion coefficients, and M1 emn , M1 omn , N1 emn , and N1 omn are VSH of the first kind. m (in italic) is the azimuthal mode index and n is the mode number. The symbols e and o stand for even and odd respectively. r is a position vector, and the wave number is k = 2∗ π /λ. In the case of a unit amplitude incident plane wave, H is unity and the azimuthal mode index m = 1. The advantages of this modeling method are: (1) it can be used for a beam whether or not it has a mathematical formula and (2) it can be used to model for physically realizable beams with ripples [10]. All the analysis of modeling the beam can be found in [10]. In general calculations of the elements of the fields are infinite. In practical computer calculations they must be limited to a finite size by truncating the expansion series to a maximum number n = nmax . If number of the terms in the series is greater than nmax then the computations will lead to the truncation error or a convergence error. For scattering of a single particle ni, max = x + 4.05x1/3 + 2 where x is the size parameter of the particle. The method for convergence and plane wave truncation can be found in details in Ref [10]. Because of electromagnetic interactions between the particles, the scattered fields from each particle i are interdependent, therefore the total incident field illuminating the particle j can be represented as a superposition of; (1) external incident field from the incident beam, and (2) sum of all fields scattered from all other particles illuminating the particle j, that is inc Einc + j = E
N
Esca l
j = 1, . . . , N.
(3)
l=1 l= j
where N is the number of the particles in the cluster. These jth particle illuminating fields can be expanded in VSH as,
Einc j =
jt amn +
mn
jl amn M1 mn (kr j )
l= j
jt bmn
+
+
jl bmn
N
1
mn
( kr j ) ,
j = 1, . . . , N.
(4)
l= j
jt
jt
The expansion coefficients amn and bmn describe the expansion coefficients of the external incident laser beam illuminating partijt jt cle j and amn , bmn describe the expansion coefficients of the field scattered from particle l and illuminating particle j. Then the expansion coefficients of the total scattered fields from the particle j can be written in form of T-matrix notation as,
fj gj
=T
j
a jl a jt + b jt b jl
,
j = 1, . . . , N.
(5)
l= j
where f j , g j are the scattered field expansion coefficients of the particle j, and T j is the T-matrix of the particle j and is given in detail in [7]. Following the analysis given in [8] and [9], we finally
H.L.S. Ibrahim, E.E.M. Khaled / Journal of Quantitative Spectroscopy & Radiative Transfer 219 (2018) 255–261
257
z
L
ac
bs
as
(a)
x
y z
4 x
2w0 1
2
(a)
3 5
y
(b)
Fig. 1. A cluster consisting of five identical oblate spheroidal particles with off-centered spherical cores. The radius of the shell is as and of the core is ac . The core is shifted a distance L long the z-direction. The cluster is illuminated with a focused Gaussian beam of minimum spot size w0 μm and propagates in the z- direction. The particles in the cluster are arranged in a Cartesian coordinate system (x, y, z) to form, (a) a linear chain, and (b) a packed cluster.
describe Eq. (5) in matrix notations as,
j
f =Tj gj
jt
a b jt
A krl j
+ B krl j l= j
given as,
l B krl j
fl ,
j = 1, . . . , N.
g
A krl j
(6) where A(krlj ) and B(krlj ) are the translation coefficients which given by analytical expressions in [12]. The matrix T j , that is for nonconcentric layered particle with refractive indices mc (core) and ms (shell) can be expressed as [11],
T j = − B ∗ A−1 = − Bs + Bcs ∗ − Bc ∗ A−1 c
∗ As + Acs ∗ − Bc ∗ A−1 c
−1 (7)
where: •
•
•
–Bc ∗ Ac −1 = Tc , is the T-matrix calculated for any particle in the cluster with refractive index m = mc /ms for the inner layer (core), and the size parameter as R∗ ms ∗ x; x = 2π as /λ. In other words, it is the T-matrix for the core alone without the shell, where R = ac /as , and x is the size parameter of the outer surface. Bs and As , are calculated using the refractive index equal to ms for outer layer and the size parameter equal to x. Matrices Acs and Bcs are calculated in the same way as As and Bs except that the Bessel functions of the first kind with argument kr are replaced by Hankel functions with the same argument.
Eq. (7) provides the T-matrix of the particle j in the cluster. Then the T-matrix of the whole cluster, after some mathematical procedures which are given in details in [7] and [9], can be
N A kr j0
T = B kr j0 j,l=1
B kr j0
T jl A(kr0l ) B ( kr0l ) A kr j0
B ( kr0l ) , A ( kr0l )
(8)
In this equation the translation coefficients A (kr0l ) and B (kr0l ) are based on spherical Bessel functions. The convergence check for a cluster consisting of axisymmetric particles faces two challenging numerical computational problems. First for each individual particle i in the cluster consisting of N particles the convergence criteria must be tested using the formula, ni, max = NO,i = xi + 4.05xi 1/3 + 2, i = 1,…,N, where NO , i is the maximum number of terms commonly used to obtain convergence. This number is proportional to the size parameter xi = 2π ai /λ; λ is the wavelength and ai is the radius of each particle (For a spheroidal particle the radius of an equal-volume sphere of the particle must be calculated). For a spheroidal particle the size parameter is given by (x)spheroid = (x)sphere (a/b)2/3 where a is the radius along the xaxis and b is the radius along the y-axis for a centered spheroids at an origin O of Cartesian coordinate system x, y, z. It has been proved that convergence can be obtained if the order of the particle’s T-matrix is 2 × NO,i [7]. Note that the summation over n is calculated for each mode m and the summation over the azimuthal modes m ranges from 0 to NO,i -1. In the case of the incident plane wave, the azimuthal mode m = 1. For a cluster, the radius ac of the smallest sphere centered at the origin of the coordinate system that containing all the particles in the cluster is considered as the radius of the cluster. Therefore the equivalent size parameter of the cluster is x = 2π ac /λ. Hence the maximum order nmax of the cluster T-matrix is nmax = No = x + 4.05x 1/3 + 2. This maximum order is proportional to the size parameter of the smallest sphere that circumscribes the entire cluster. Detailed description
258
H.L.S. Ibrahim, E.E.M. Khaled / Journal of Quantitative Spectroscopy & Radiative Transfer 219 (2018) 255–261
of the conversion criteria and maximum number of n and m can be found in [7,13]. Consequently, the elements of the S-matrix or the phase matrix can be completely expressed as [13]
S11 = 12 |F1 |2 + |F2 |2 + |F3 |2 + |F4 |2
S12 = 12 |F2 |2 − |F3 |2 + |F4 |2 − |F1 |2
S22 = 12 |F2 |2 − |F3 |2 − |F4 |2 + |F1 |2
∗ S33 = Re F2 F1 + F3 F4∗
S34 = Im F2 F1∗ + F4 F3∗
∗ S44 = Re F1 F2 − F3 F4∗
(9)
The four elements F1 up to F4 of the amplitude scattering matrix F are related to the cluster T-matrix given in (8) and are given in detail in [13]. The orientation averaged scattering matrix elements can be analytically obtained from the T-matrix. Following the procedures developed by Mishchenko [13] for axisymmetric scatterers in which one forms the products of the scattering amplitude elements Eq. (9) from the cluster T-matrix and then integrate the products over all directions and polarizations of the incident field. 3. Numerical results The scattering of a light beam by particles is fully characterized by a 4 × 4 phase matrix [s] (often called the Mueller or scattering matrix) [13] which, in the most general case, has 16 different real nonvanishing elements. Number of the elements in the scattering matrix reduces to six nonzero elements for a collection of identical randomly oriented axisymmetric particles. The random orientation matrix elements S11 , S22 , S33 and S44 , and S12 and S34 and their transposes are nonzero only for clusters (with on-centered cores) possessing a plane of symmetry (not necessarily axisymmetric chains or packed clusters). The configurations of both clusters (with on-cenetered cores as stated before [8]) used in computation for the results in Figs. 2 and 3 all possess a plane of symmetry which is xy-plane, and because of this we computed the elements S23 , S24 , S13 , and S14 and their transposes for both clusters. We found that these elements are very small approximation zeros and therefore we do not include them. The calculated S11 measures total scattered intensity for unpolarized incident light which gives general information of the particle size, S12 measures degree of the polarized light. The degree of polarization can vary from zero (completely unpolarized) to one (completely polarized). S22 , S33 , S34 , S44 measures the degree of depolarized light, when the incident beam fully linearly or circularly polarized. S22 express transformation of linearly polarized incident light (±90°) to linearly polarized scattered light (±90°); deviation from S11 is an important indication of the presence of nonspherical particles. S33 , S44 expresses transformation of linearly polarized incident light (±45°) (which is circularly) to linearly polarized scattered light (±45°) (which is circularly). The deviation of S44 from S33 indicates of nonspherical symmetry. S34 express transformation of circularly polarized incident light to linearly polarized scattered light (±45°); element is strongly dependent on size and complex refractive index of the scatterers [7,13,14]. In this paper, a cluster consisting of five oblate spheroidal particles made from zinc sulfide (ZnS) form a linear chain, shown in Fig. 1(a), or a packed of a hexagonal lattice, shown in Fig. 1(b) is considered. Each particle has a refractive index ms = 2.2, size parameter xs = 2π as /λ = 4, and axial ratio as /bs = 0.8 [15]. The particles are doped individually by off-centered spherical copper cores of a size parameter xc = 2π ac /λ = 2, and refractive index mc = 2.43 [16]. The offset parameter for the core for all the particles is p = L/ac = 0.5, L is the distance from the core’s center in its local origin to the center of its shell. The cluster is illuminated with
a Gaussian beam of light of wavelength λ = 1.0 μm propagating in the z-direction with an intensity Eo = 1 v/m at its focal point. The beam waist is chosen as w0 = 125bs , w0 = 5bs and w0 = bs to represent three different cases of illumination for each cluster. The first choice w0 = 125bs represent a plane wave illumination since w0 > > bs , the second choice w0 = 5bs represent a moderate focused laser beam, and the last one w0 = bs represent tightly focused beam. The computed results of the orientation averaged scattering matrix elements are shown in Figs. 2 and 3 for the chain cluster and packed cluster respectively. The results demonstrate that in the case of a very wide beam waist w0 = 125 bs ≈ 100 μm, (which simulates a plane wave illumination) the differences of the intensity of S11 between both clusters cases is very small at any scattered angle (there are tiny different in the front scattering) since the clusters are illuminated by the same field strengths (plane wave), which the strength of illumination is the same on all particles in the cluster for the case of plane wave illumination whereas the strengths of illumination are different on different particles in the cluster for the focused beam illumination. So the shape of the cluster has very little effect in intensity of S11 in the final scattered signal. Note that this observation is not true for the focused beam illuminations. The results show that the magnitudes of oscillations of S11 decrease as the waist of the beam gets narrower as illustrated in both Figs. 2 and 3. The magnitudes of oscillations means that the intensity of S11 oscillates between different higher values and different lower values at different consequence scattering angles between the front and the back scattering. These higher and lower values decrease as the waist of the beam gets narrower. This means that the interference of the multiple scattering is less as the beam gets more focusing since the beam may misses some particles to interact with as it gets narrower. The ratio S22 / S11 , equals 1 for the focused beam while it gets different values for the wider beam waist illumination. When S22 /S11 equal to 1 that means the cluster do not depolarized the incident beam and when S22 /S11 deviates from 1 that means the cluster can produce significant depolarization. When S22 /S11 equal to zero that means the cluster can produce fully depolarization [14]. Despite of the S22 /S11 indicate of higher or lower degree of sphericity for plane wave illumination, our paper shows that this is not true if the incident beam is very focused as shown in Figs. 2 and 3. So we can say that for a very focused incident beam the S22 /S11 can be closer to one and that does not mean the degree of nonsphericity decreases but means that the cluster is not depolarizing the incident beam. The ratio S12 /S11 for both clusters should be always zero at θ = π as expected. But the new observation is that the deviations of -S12 (θ )/S11 (θ ) as a function of beam waist depends on how the particles are arranged in the cluster, i.e. the results for the chain cluster is different than that of the packed cluster. Comparing the ratio -S12 /S11 for both clusters we can notice that for a very focused laser beam illumination the ratio S12 /S11 near the backscattering direction (θ = 135ο to 165ο ) for the packed cluster is attenuated greater than those for chain cluster which indicates that the packed-cluster configuration offers a much higher opportunity for multiple scattering and then this result can be used for cluster configuration identification. It is noted in the cases of the linear chain or densely packed clusters illuminated with a focused laser beam, Figs. 2 and 3 that S33 and S44 are identical and this indicates of fully symmetric for clusters (note that the beam and both clusters are symmetric along xy-plane and different field strengths are illuminating different particles). Since the case of a cluster illuminated of a focused laser beam is fairly a new case. This is unlike the case of incident plane wave (presented by wide beam waist, wo = 100b, at which the same field strength illuminates all particles) where a small deviation of S44 from S33 is obtained which shows nonspherical symmetry, see reference [7,14] for a cluster illuminated with a plane wave.
H.L.S. Ibrahim, E.E.M. Khaled / Journal of Quantitative Spectroscopy & Radiative Transfer 219 (2018) 255–261
259
Fig. 2. Orientation-averaged scattering matrix elements for a linear chain of five identical oblate spheroidal Zinc sulfide (ZnS) particles. The refractive index of each particle is m = 2.2, size parameter x = 4 and the axial ratio ρ = a/b = 0.8. Each particle is doped by a spherical copper (Cu) of size parameter x = 2 and refractive index m = 2.43. The offset parameter for the core is p = L/ ac = 0.5. Results are shown for different a Gaussian beam of waist w0 = 100 μm (i.e. plane wave), w0 = 5bs and w0 = bs of wavelength λ = 1.0 μm propagating in the z-direction.
260
H.L.S. Ibrahim, E.E.M. Khaled / Journal of Quantitative Spectroscopy & Radiative Transfer 219 (2018) 255–261
Fig. 3. Orientation-averaged scattering matrix elements for a packed cluster of five identical oblate spheroidal Zinc sulfide (ZnS) particles. The refractive index of each particle is m = 2.2, size parameter x = 4 and the axial ratio ρ = a/b = 0.8. Each particle is doped by a spherical copper (Cu) of size parameter x = 2 and refractive index m = 2.43. The offset parameter for the core is p = L/ ac = 0.5. Results are shown for different a Gaussian beam of waist w0 = 100 μm (i.e. plane wave), w0 = 5bs and w0 = bs of wavelength λ = 1.0 μm propagating in the z-direction.
H.L.S. Ibrahim, E.E.M. Khaled / Journal of Quantitative Spectroscopy & Radiative Transfer 219 (2018) 255–261
4. Conclusion The random-orientation scattering properties of a plane wave as well as a focused Gaussian beam scattered by a cluster consisting of nonspherical encapsulated particles are presented. Two examples, as applications, are studied in this paper; a chain cluster consisting of five oblate particles made from zinc Sulfide (ZnS) material injected with cores of spherical copper (Cu). The other cluster is a packed cluster consisting of the same particles as of the chain cluster. The scattering matrix elements are computed for three different cases of illuminations; a plane wave, a fairly focused Gaussian, and a focused beam. It noted that the behavior of the nonzero scattering matrix elements depend on the focusing of the incident beam. This behavior can be used to identify the type of a cluster under investigation. References [1] Quirantes A. Light scattering properties of spheroidal coated particles in random orientation. J Quant Spectrosc Radiat Transf 1999;63(2–6):263–75. [2] Ethiraj AS, Hebalkar N, Kulkarni SK, Pasricha R, Urban J, Dem C, Schmitt M, Kiefer W, Weinhardt L, Joshi S, Fink R, Heske C, Kumpf C, Umbach E. Enhancement of photoluminescence in manganese doped ZnS nanoparticles due to a silica shell. J Chem Phys 2003;118:8945. [3] Khaled EEM, Ibrahim HL. Photoluminescence enhancement from an axisymmetric zinc sulfide particle illuminated with a focused laser beam. Frontiers in Optics 2010/Laser Science XXVI conference, OSA Technical Digest (CD). Optical Society of America; 2010.
261
[4] Dumbrava A, Badea C, Prodan G, Popovici I, Ciupina V. Zinc sulfide fine particles obtained at low temperature. Chalcogenide Lett 2009;6:437–43. [5] Vacassy R, Scholz SM, Dutta J, Hofmann H, Plummer CJG, Carrot G, Hilborn J, Akinc M. Nanostructured zinc sulfide phosphors. In: Materials Research Society Symposium Proceedings, 501. Materials Research Society; 1998. p. 369–74. MRS. [6] Wriedt T, Schuh R, Doicu A. Scattering by aggregated fibres using a multiple scattering T-matrix approach. Part Part Syst Charact 2008;25:74–83. [7] Mackowski DW, Mishchenko MI. Calculation of the T matrix and the scattering matrix for ensembles of spheres. J Opt Soc Am 1996;13(11) November. [8] Ibrahim HLS, Khaled EEM. Light scattering from a cluster consists of dielectric nonconcentric encapsulation particles. Int J Electron Commun Eng Technol 2014;5:82–94. [9] Ibrahim HLS, Wriedt T, Khaled EEM. Light scattering by a cluster consisting of homogeneous axisymmetric particles illuminated with an arbitrarily focused electromagnetic Gaussian beam. J Quant Spectrosc Radiat Transf 2016;173:83–91. [10] Khaled EEM, Hill SC, Barber PW, Chowdhury DQ. Scattered and internal intensity of a sphere illuminated with a gaussian. TEEE Trans Antennas Propag 1993;41(3) March. [11] Quirantes A. A T-matrix method and computer code for randomly oriented, axially symmetric coated scatterers. J Quant Spectrosc Radiat Transf, 2005;92:373–81. [12] Friedman B, Russek J. Addition theorems for spherical waves. Q Appl Math 1954;12:13–23. [13] Mishchenko MI, Travis LD, Lacis AA. Scattering, absorption, and emission of light by small particles. New York: NASA Goddard Institute for Space Studies; July 2005. [14] Mishchenko MI, Hovenier JW, Travis LD. light Scattering by nonspherical particles: theory, measurements, and applications. Elsevier; 1999. [15] Mirabella FM. Internal reflection spectroscopy: theory and applications, 15. CRC Press; 1992. [16] http://www.raeco.com/training/refractive- index- values.htm.