- Email: [email protected]
Optical scattering investigation of a nonabsorbent silicon nanoparticles array
Optik - International Journal for Light and Electron Optics 201 (2020) 163477
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Optical scattering investigation of a nonabsorbent silicon nanoparticles array
T
Fereshteh Rahimi Moghadam Faculty of Science, Lorestan University, Khorramabad, Iran
A R T IC LE I N F O
ABS TRA CT
Keywords: Plasmonic effects Scattering cross-section Spherical nanoparticle Silicon Finite element
In this article, plasmonic effects of spherical nanoparticles were studied. Since the dielectric nanoparticles are nonabsorbent, so scattering cross section and extinction cross-section are the same. Optical scattering of spherical nanoparticles of silicon in the visible range of the electromagnetic spectrum using Maxwell's equations and finite element method was investigated. At the start, the suitable radius for single nanoparticle (due to exposure peaks in the visible range of the electromagnetic spectrum) was selected by comparing four different radiuses. In the following the distance between the pairs of particles for the study of behavior was examined and, ultimately, based on optimized parameters for radius and the distance between the nanoparticles, linear arrays, and the square was designed. Numerical calculations have been done for a plane wave with linear polarization indicates that the peak of the scattering cross section due of resonance between the incident field and surface plasmon nanoparticles. By studying the spectrum of scattering cross section of the single spherical nanoparticle, the nanoparticle pairs, a linear array and a square of spherical nanoparticles were observed that the scattering cross section depends on the particle size, the distance between particles, the polarization and the type of lattice.
1. Introduction Nanoparticles consist of tens or hundreds of atoms or molecules of varying structures and shapes (amorphous, crystalline, spherical, needle-shaped, etc.). Often, the nanoparticles that are used are in the form of dry powder or liquid form. Of course, the combined nanoparticles, which are in the form of an organic or aqueous solution, are also considered. The optical properties of the nanoparticles in the visible and infrared regions are adjustable by their shape and size. Also, based on quantum limitations, the interaction of electromagnetic waves with nanoparticles can be placed in the desired wavelength. Due to the interconnection of the optical properties of metallic nanostructures with the cumulative fluctuations of conduction electrons, the interaction of light with surface electric charges is considered. The cumulative oscillations of conducting electrons depend on the geometric properties and the surrounding environment around the nanostructure. Nanoparticles are placed between atoms and solid-state bulk and exhibit very different properties than their solid state. In response to size variations, nanoparticles show two different types of behavior. Intrinsic effects, which are related to changes in the surface-to-volume ratio of matter, and the non-intrinsic effects that the nanoparticle shows in response to external fields and forces. In fact, the non-intrinsic effects are dependent on particle size, but independent of the intrinsic effects. Nanoparticles have a large number of surface atoms compared to atoms within their volume. This itself increases the significance of surface effects compared to volumetric effects. Usually, nanoparticles with a diameter greater than 10 nm have a dielectric constant equal to the mass value of the material and independent of size. Therefore, it is observed that the dependence of
E-mail address: [email protected]. https://doi.org/10.1016/j.ijleo.2019.163477 Received 21 June 2019; Accepted 24 September 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 201 (2020) 163477
F. Rahimi Moghadam
the optical spectrum of large nanoparticles is as significant as an external effect, which is controlled only by the particle size relative to the electromagnetic beam. Due to the interaction of light with nanoparticles, its surface electrons are stimulated and oscillate. This fluctuation causes the light scattering from the surface. In order to investigate the optical properties of plasmonic nanoparticles, Cross-sectional spectra of dispersion, absorption, and extinction [1–3] and nonlinear optical properties are used [4]. Recently, the properties of plasmonic nanoparticles have been studied for their many applications in optical devices [5–8], nonlinear optics [9–11], optical data storage [12,13], and advanced surface spectrometry[14], catalysts [15], biological labels and sensors [16] and it is essential for the treatment of cancer [17]. Silicon is a promising material for photonic dielectrics due to the highest refractive index in the visible and near infrared region and low optical loss [18]. Silicon is one of the most widely used materials for the formation of polymers and has shown absorbing properties at high and low temperatures. In 2011, Gomez and his colleagues [19] examined the electric and magnetic bipolar response of silicon dielectric spheres with a radius of 200 nm at near-infrared frequencies. Their research showed that, in the spectrum dispersed from silicon multipolar due to the overlapping between the multipolar peaks, no more polarized sentences are observed. Here, the influence of various parameters such as structural parameters and radiation parameters on the dispersion of arrays of multipolar silicon nanoparticles has been investigated. Besides, smaller spheres have been used to observe the peaks of higher multipolar in the visible range of the electromagnetic spectrum. 2. The Mie theory To study the absorption and scattering of light by small particles, the Maxwell equations are applied by applying boundary conditions. The spherical particle scattering was first reviewed by Mie in 1908 [20]. The Mie theory is based on the analytical solution of the Maxwell equations for the dispersion of spherical nanoparticles by a wave with linear polarization. For simplicity, Mie assumed that the spherical particles have an equal size to ignore the interactions between different dispersing particles. Also, it assumed that the total dispersed energy is equal to the sum of scattered energies of each particle. One of the reasons why this theory has been considered so far is that the only precise and straightforward way is to solve the Maxwell equations for spherical particles. For the radiation of the flat waves, the field is considered as follows
E = E0z ei (kx − ωEt ) B = B0y ei (kx − ωt )
(1)
In this theory, the scattered field by the sphere is expressed regarding multipolar terms with the Mie coefficients an and bn which are called electric and magnetic coefficients, respectively.
1 (1 − e−2iαn ) = isinαn e−iαn 2 1 bn = (1 − e−2iβn ) = isinβn e−iβn 2
an =
(2)
That
tan αn =
m2jn (y )[xjn (x )]ˊ − jn (x )[yjn (y )]ˊ m2jn (y )[xyn (x )]ˊ − yn (x )[yjn (y )]ˊ
tan βn =
jn (y )[xjn (x )]ˊ − jn (x )[yjn (y )]ˊ jn (y )[xyn (x )]ˊ − yn (x )[yjn (y )]ˊ
(3)
nsi nair
= 3.5 the ratio of the refractive index of the silicon sphere to the surrounding environment, x = ka is parameter Here m = particle size and y = mx. jn (x ) and yn (x ) respectively represent the functions of spherical Bessel and Newman the ' symbol represents the derivative of these functions. For a non-absorbing environment, the relative refractive index and the phase shift are real. The optical properties of the particle, i.e., the cross-section of absorption and scattering, are strongly related to the radiation light intensity (due to the absorption and scattering of the elastic). In practice, absorption and dispersion accumulate together and lead to a cross-section of extinction [19]. The dispersion and extinction cross-section is described as follows ∞
σS = σext =
2π ∑ (2n + 1){sin2αn + sin2βn } = k2 n=1
∞
∑ {σE,n +
σM , n }
n=1
(4)
In this paper, we seek to investigate the higher polarization in a spectrum dispersed from linear and square arrays of spherical silicon nanoparticles in the visible region of the electromagnetic spectrum. In the present paper, depending on the behavior of silicon metal in the visible region (due to the small electron gap), the conduction electron is discussed. 3. Numerical computational results To study the propagation of electromagnetic waves by the nanoparticle and its surroundings, the finite element computing method (FEM) has been used. The model consists of two parts of the spherical nanoparticle and a matrix layer (PML) with a thickness of the incidence wavelength, which absorbs the reflection of the outer boundaries. In other words, the calculations are performed for a non- absorbing dielectric sphere with a radius a, a refractive index n and an electrical permeability, ε = n2, which is placed in a 2
Optik - International Journal for Light and Electron Optics 201 (2020) 163477
F. Rahimi Moghadam
Fig. 1. Cross section scattering spectrum for four radii of spherical nanoparticle.
uniform environment (air). It should be noted that in order to ensure the accuracy of the calculations, the first proposed dielectric sphere was simulated in [19] and, given the consistency of the results obtained from the method described below, this method will be used for the structures described in this paper. In this paper, first, the dependence of the dispersion cross-section to nano particle size is studied. In Fig. 1, the dependence of the dispersion cross section as a function of wavelength is compared for a nanoparticle with different radii. The peaks seen in this figure are due to aggravation on the surface of these nanoparticles and the heterogeneous distribution of free charges. As the nanoparticle size increases, the number of conduction electrons increases and as a result the influence of the electric field applied to the nanoparticle decreases. In other words, the electric field only penetrates the surface of these nanoparticles, producing superficial and inhomogeneous effects in the distribution of free charges (conduction electrons) of the nanoparticle. In each figure of the first peak, related to the magnetic octupoles, the peak is related to the quadrilateral electric, and the third peak is the quadrilateral magnetic [19]. According to the shape, it is seen that the position and magnitude of the peaks associated with electric and magnetic multipolar depend on the radius of the nanoparticle so that it increases with the increase in the radius and the peaks move towards longer wavelengths. In the resonance of nanoparticle plasmon's, due to the coupling of electric fields applied to particle plasmons at certain wavelengths, the plasmon peaks in these wavelengths occur. According to Fig. 1, and the placement scattering cross-sectional spectra of the nanoparticle with a radius of 80 nm in the visible region, the 80-nanometer radius continues to be used for nanoparticles. In Figs. 2 and 3, the dispersion of two spherical nanoparticles with an optimal radius (80 nm), spaced along the x-axis and spaced apart d, for two different polarization directions (z and x, respectively) and at different distances is shown. According to Fig. 2, it is observed that for a wave with linear polarization along the z-axis, the magnitude of the distance between the nanoparticles (d) of the magnetic octupoles magnitude increases in the cross-sectional scattering area and the magnitude of the quadrupole magnetic peak due to the strong overlap between the distribution different polygons are almost eliminated. While there is no regular change in the size of the quadrupole electric peak. From the comparison of Figs. 1 and 2 It can be seen that the presence of the second nanoparticle is effective on the magnitude of the eight magnetic poles peak of the magnetic and quadrupole electrons, because the second nanoparticle has increased the number of conduction electrons, however, it does not change the peak location because the wavelength resonance is dependent on the type of material. In Fig. 3, the cross-section of the scattering for the
Fig. 2. Cross section scattering spectrum regarding wavelength for two spherical nanoparticle silicon with a radius of 80 nm at different distances (the field is polarized in z). 3
Optik - International Journal for Light and Electron Optics 201 (2020) 163477
F. Rahimi Moghadam
Fig. 3. Cross section scattering spectrum regarding wavelength for two spherical nanoparticle silicon with a radius of 80 nm at different distances (the field is polarized in x).
case where the wave is linearly polarized along x and released in the z-direction, at different intervals, for two nanoparticles are plotted as a function of wavelength. It is observed that with the increase in the intermediate distance the peak corresponding to the eight magnetic poles is the larger magnitude and the magnitude of the quadrilateral magnetic peak decreases (the opposite changes), while the mid-peak (quadrupole electric) is almost eliminated. In this case, it was also observed that changing the distance to the peak size is effective, but does not change the wavelength of the resonance. According to the results, the distance of 80 nm is chosen as the optimum distance. Fig. 4 shows the comparison between the spectra of the scattering cross section of a single nanoparticle and the pair of nanoparticles with different polarization directions. It is observed that the direction of polarization is effective in the size of the cross-sectional of the dispersion. When the field is propagated along the z-axis, the wave interacts with both nano-particles simultaneously, and in this case (propagation along z), the light hits the two nano-particles with the same intensity while in the second case (propagation along x) light after interaction with the first nanoparticle with less intensity, and it interacts with the second nanoparticle. Considering the similarity of the cross-sectional spectra of the nanoparticle pair for the polarization x with the spectrum of a single nanoparticle, the z-direction is considered as the direction of field propagation. In order to reach the larger peaks, linear arrays consisting of 3 and 4 spherical nanoparticles with a radius of 80 nm were considered for the X polarization direction. As shown in Fig. 5, the cross-sectional dispersion spectrum of linear arrays is similar to that of a single nanoparticle spectrum. Besides, with the increase in the number of nanoparticles in the array, the cross-sectional area in which the light interacts with it increases, increasing the number of surface electrons and, consequently, an increase in the magnitude of the eight magnetic poles and quadrilateral magnetic magnitude. In Fig. 6, the cross-sectional dispersion area of a square array is comprised of nine spherical silicon nanoparticles. As expected, the dispersion of a square array is far greater than the linear array. Moreover, it was observed that there is a complete similarity between the spectrums of a square array with a single nanoparticle spectrum. In other words, in this spectrum, there are three peaks of eight magnetic poles, quadrilateral, and four-polar magnetic magnitudes. While in linear arrays the middle of the peaks is almost eliminated. A square array is arranged in xy. Given the similarity of this spectrum with the spectrum of a single nanoparticle, the square array behaves just like a larger nanoparticle. Therefore, in order to achieve a larger dispersion cross-section, a square array with appropriate parameters can be selected.
Fig. 4. Comparison of cross section scattering spectrum from single nanoparticles and nanoparticle pair with 80 nm radius in two polarization align. 4
Optik - International Journal for Light and Electron Optics 201 (2020) 163477
F. Rahimi Moghadam
Fig. 5. Comparison of scattering cross-sectional spectra from a linear array consisting of 3 and 4 spherical nanoparticles with a radius of 80 nm.
Fig. 6. Dispersion cross-sectional area of a square array consisting of 9 spherical nanoparticles with a radius of 80 nm.
4. Conclusion In this paper, due to the importance of alternative structures than single dielectric cores, after selecting the appropriate radius, first pair of nanoparticles in the following, linear and square arrays of silicon nanoparticles have been investigated. Numerical calculations show that the peak of the dispersion cross section is due to the resonance between the incident field and the surface plasmons of the nanoparticles. It is well observed that the light scattered by spherical nanoparticles, based on the eight-pole magnetic, quadrupole, and a quadrilateral magnetic field is described; in the visible region, factors such as the size of the nanoparticle, the intermediate distance, the number of nanoparticles, and the direction of the polarization of the field are dependent. In other words, it can be said that the dispersion cross section is maximized due to the resonance of a plasmon in a given wavelength; besides, the dispersion cross-section is strongly sensitive to nanoparticles. The type of multi-polarities is determined according to the theory of Mie. The results play an important role not only in achieving the field distribution in metamaterials and optical antennas but also in regulating the transmission of light through a dielectric environment, such as photonic glasses [21,22]. References [1] S. Link, M.A. El-Sayed, Size and temperature dependence of the plasmon absorption of colloidal gold nanoparticles, J. Phys. Chem. B 103 (21) (1999) 4212–4217. [2] C. Voisin, N. Del Fatti, D. Christofilos, F. Vallee, Ultrafast electron dynamics and optical nonlinearities in metal nanoparticles, J. Phys. Chem. B 105 (12) (2001) 2264–2280. [3] G.V. Hartland, Measurements of the material properties of metal nanoparticles by time resolved spectroscopy, Phys. Chem. Chem. Phys. 6 (23) (2004) 5263–5274. [4] B. Lamprecht, J.R. Krenn, A. Leitner, F.R. Aussenegg, Resonant and off-resonant light driven plasmons in metal nanoparticles studied by femtosecond-resolution third-harmonic generation, Phys. Rev. Lett. 83 (21) (1999) 4421–4424. [5] M. Salerno, J.R. Krenn, B. Lamprecht, G. Schider, H. Ditlbacher, N. Felidj, A. Leitner, F.R. Aussenegg, Plasmon polaritons in metal nanostructures: the optoelectronic route to nanotechnology, Opto-Electron. Rev. 10 (3) (2002) 217–224. [6] J.R. Krenn, Nanoparticle waveguides watching energy transfer, Nat. Mater. 2 (4) (2003) 210–211. [7] S.A. Maier, P.G. Kik, H.A. Atwater, S. Meltzer, E. Harel, B.E. Koel, A.A.G. Requicha, Local detection of electromagnetic energy transport below the diffraction
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Optik - International Journal for Light and Electron Optics 201 (2020) 163477
F. Rahimi Moghadam
limit in metal nanoparticle plasmon waveguides, Nat. Mater. 2 (4) (2003) 229–232. [8] W.L. Barnes, A. Dereux, T.W. Ebbesen, Surface plasmon subwavelength optics, Nature 424 (6950) (2003) 824–830. [9] Y.H. Liu, A.N. Unterreiner, Q. Chang, N.F. Scherer, Ultrafast dephasing of single nanoparticles studied by two-pulse second-order interferometry, J. Phys. Chem. B 105 (11) (2001) 2135–2142. [10] D. Yelin, D. Oron, S. Thiberge, E. Moses, Y. Silberberg, Multiphoton plasmon resonance ev microscopy, Opt. Express 11 (12) (2003) 1385–1391. [11] J. Nappa, G. Revillod, I. Russier-Antoine, E. Benichou, C. Jonin, P.F. Brevet, Electric dipole origin of the second harmonic generation of small metallic particles, Phys. Rev. B 71 (16) (2005) pp. 165407:1-165407:4. [12] H. Ditlbacher, J.R. Krenn, B. Lamprecht, A. Leitner, F.R. Aussenegg, Spectrally codedvz optical data storage by metal nanoparticles, Opt. Lett. 25 (8) (2000) 563–565. [13] J.W.M. Chon, C. Bullen, P. Zijlstra, M. Gu, Spectral encoding on gold nanorods doped in a silica solgel matrix and its application to high-density optical data storage, Adv. Funct. Mater. 17 (6) (2007) 875–880. [14] K. Kneipp, H. Kneipp, I. Itzkan, R.R. Dasari, M.S. Feld, Surfaceenhanced Raman scattering and biophysics, J. Phys. Condens. Matter 14 (18) (2002) 597–624. [15] H.G. Boyen, G. Kastle, F. Weigl, B. Koslowski, C. Dietrich, P. Ziemann, J.P. Spatz, S. Riethmuller, C. Hartmann, M. Moller, G. Schmid, M.G. Garnier, P. Oelhafen, Oxidationresistant gold 55 clusters, Science 297 (5586) (2002) 1533–1536. [16] G. Raschke, S. Kowarik, T. Franzl, C. Sonnichsen, T.A. Klar, J. Feld ¨mann, A. Nichtl, K. Kurzinger, Biomolecular recognition basedon single gold nanoparticle light scattering, Nano Lett. 3 (7) (2003) 935–938. [17] P.K. Jain, I.H. El-Sayed, M.A. El-Sayed, Au nanoparticles target cancer, Nano Today 2 (1) (2007) 18–29. [18] D.E. Aspnes, A.A. Studna, Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV, Phys. Rev. B 27 (2) (1983) 985–1009. [19] A. Garc ıa-Etxarri, R. Gomez-Medina, L.S. Froufe-Perez, C. Lopez, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, J.J. Saenz, Strong magnetic response of submicron Silicon particles in the infrared, Opt. Express 19 (6) (2011) 4815–4826. [20] G. Mie, Beitrage zur Optik truber Medien, speziell kolloidaler metallosungen, Ann. Physik 330 (3) (1908) 377–442. [21] P.D. García, R. Sapienza, Á. Blanco, C. López, Photonic glass: a novel random material for light, Adv. Mater. 19 (18) (2007) 2597–2602. [22] M. Reufer, L. Fernando Rojas-Ochoa, S. Eiden, J.J. Sáenz, F. Scheffold, Transport of light in amorphous photonic materials, Appl. Phys. Lett. 91 (17) (2007) pp 171904:1-171904:3.
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Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Optical scattering investigation of a nonabsorbent silicon nanoparticles array
T
Fereshteh Rahimi Moghadam Faculty of Science, Lorestan University, Khorramabad, Iran
A R T IC LE I N F O
ABS TRA CT
Keywords: Plasmonic effects Scattering cross-section Spherical nanoparticle Silicon Finite element
In this article, plasmonic effects of spherical nanoparticles were studied. Since the dielectric nanoparticles are nonabsorbent, so scattering cross section and extinction cross-section are the same. Optical scattering of spherical nanoparticles of silicon in the visible range of the electromagnetic spectrum using Maxwell's equations and finite element method was investigated. At the start, the suitable radius for single nanoparticle (due to exposure peaks in the visible range of the electromagnetic spectrum) was selected by comparing four different radiuses. In the following the distance between the pairs of particles for the study of behavior was examined and, ultimately, based on optimized parameters for radius and the distance between the nanoparticles, linear arrays, and the square was designed. Numerical calculations have been done for a plane wave with linear polarization indicates that the peak of the scattering cross section due of resonance between the incident field and surface plasmon nanoparticles. By studying the spectrum of scattering cross section of the single spherical nanoparticle, the nanoparticle pairs, a linear array and a square of spherical nanoparticles were observed that the scattering cross section depends on the particle size, the distance between particles, the polarization and the type of lattice.
1. Introduction Nanoparticles consist of tens or hundreds of atoms or molecules of varying structures and shapes (amorphous, crystalline, spherical, needle-shaped, etc.). Often, the nanoparticles that are used are in the form of dry powder or liquid form. Of course, the combined nanoparticles, which are in the form of an organic or aqueous solution, are also considered. The optical properties of the nanoparticles in the visible and infrared regions are adjustable by their shape and size. Also, based on quantum limitations, the interaction of electromagnetic waves with nanoparticles can be placed in the desired wavelength. Due to the interconnection of the optical properties of metallic nanostructures with the cumulative fluctuations of conduction electrons, the interaction of light with surface electric charges is considered. The cumulative oscillations of conducting electrons depend on the geometric properties and the surrounding environment around the nanostructure. Nanoparticles are placed between atoms and solid-state bulk and exhibit very different properties than their solid state. In response to size variations, nanoparticles show two different types of behavior. Intrinsic effects, which are related to changes in the surface-to-volume ratio of matter, and the non-intrinsic effects that the nanoparticle shows in response to external fields and forces. In fact, the non-intrinsic effects are dependent on particle size, but independent of the intrinsic effects. Nanoparticles have a large number of surface atoms compared to atoms within their volume. This itself increases the significance of surface effects compared to volumetric effects. Usually, nanoparticles with a diameter greater than 10 nm have a dielectric constant equal to the mass value of the material and independent of size. Therefore, it is observed that the dependence of
E-mail address: [email protected]. https://doi.org/10.1016/j.ijleo.2019.163477 Received 21 June 2019; Accepted 24 September 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 201 (2020) 163477
F. Rahimi Moghadam
the optical spectrum of large nanoparticles is as significant as an external effect, which is controlled only by the particle size relative to the electromagnetic beam. Due to the interaction of light with nanoparticles, its surface electrons are stimulated and oscillate. This fluctuation causes the light scattering from the surface. In order to investigate the optical properties of plasmonic nanoparticles, Cross-sectional spectra of dispersion, absorption, and extinction [1–3] and nonlinear optical properties are used [4]. Recently, the properties of plasmonic nanoparticles have been studied for their many applications in optical devices [5–8], nonlinear optics [9–11], optical data storage [12,13], and advanced surface spectrometry[14], catalysts [15], biological labels and sensors [16] and it is essential for the treatment of cancer [17]. Silicon is a promising material for photonic dielectrics due to the highest refractive index in the visible and near infrared region and low optical loss [18]. Silicon is one of the most widely used materials for the formation of polymers and has shown absorbing properties at high and low temperatures. In 2011, Gomez and his colleagues [19] examined the electric and magnetic bipolar response of silicon dielectric spheres with a radius of 200 nm at near-infrared frequencies. Their research showed that, in the spectrum dispersed from silicon multipolar due to the overlapping between the multipolar peaks, no more polarized sentences are observed. Here, the influence of various parameters such as structural parameters and radiation parameters on the dispersion of arrays of multipolar silicon nanoparticles has been investigated. Besides, smaller spheres have been used to observe the peaks of higher multipolar in the visible range of the electromagnetic spectrum. 2. The Mie theory To study the absorption and scattering of light by small particles, the Maxwell equations are applied by applying boundary conditions. The spherical particle scattering was first reviewed by Mie in 1908 [20]. The Mie theory is based on the analytical solution of the Maxwell equations for the dispersion of spherical nanoparticles by a wave with linear polarization. For simplicity, Mie assumed that the spherical particles have an equal size to ignore the interactions between different dispersing particles. Also, it assumed that the total dispersed energy is equal to the sum of scattered energies of each particle. One of the reasons why this theory has been considered so far is that the only precise and straightforward way is to solve the Maxwell equations for spherical particles. For the radiation of the flat waves, the field is considered as follows
E = E0z ei (kx − ωEt ) B = B0y ei (kx − ωt )
(1)
In this theory, the scattered field by the sphere is expressed regarding multipolar terms with the Mie coefficients an and bn which are called electric and magnetic coefficients, respectively.
1 (1 − e−2iαn ) = isinαn e−iαn 2 1 bn = (1 − e−2iβn ) = isinβn e−iβn 2
an =
(2)
That
tan αn =
m2jn (y )[xjn (x )]ˊ − jn (x )[yjn (y )]ˊ m2jn (y )[xyn (x )]ˊ − yn (x )[yjn (y )]ˊ
tan βn =
jn (y )[xjn (x )]ˊ − jn (x )[yjn (y )]ˊ jn (y )[xyn (x )]ˊ − yn (x )[yjn (y )]ˊ
(3)
nsi nair
= 3.5 the ratio of the refractive index of the silicon sphere to the surrounding environment, x = ka is parameter Here m = particle size and y = mx. jn (x ) and yn (x ) respectively represent the functions of spherical Bessel and Newman the ' symbol represents the derivative of these functions. For a non-absorbing environment, the relative refractive index and the phase shift are real. The optical properties of the particle, i.e., the cross-section of absorption and scattering, are strongly related to the radiation light intensity (due to the absorption and scattering of the elastic). In practice, absorption and dispersion accumulate together and lead to a cross-section of extinction [19]. The dispersion and extinction cross-section is described as follows ∞
σS = σext =
2π ∑ (2n + 1){sin2αn + sin2βn } = k2 n=1
∞
∑ {σE,n +
σM , n }
n=1
(4)
In this paper, we seek to investigate the higher polarization in a spectrum dispersed from linear and square arrays of spherical silicon nanoparticles in the visible region of the electromagnetic spectrum. In the present paper, depending on the behavior of silicon metal in the visible region (due to the small electron gap), the conduction electron is discussed. 3. Numerical computational results To study the propagation of electromagnetic waves by the nanoparticle and its surroundings, the finite element computing method (FEM) has been used. The model consists of two parts of the spherical nanoparticle and a matrix layer (PML) with a thickness of the incidence wavelength, which absorbs the reflection of the outer boundaries. In other words, the calculations are performed for a non- absorbing dielectric sphere with a radius a, a refractive index n and an electrical permeability, ε = n2, which is placed in a 2
Optik - International Journal for Light and Electron Optics 201 (2020) 163477
F. Rahimi Moghadam
Fig. 1. Cross section scattering spectrum for four radii of spherical nanoparticle.
uniform environment (air). It should be noted that in order to ensure the accuracy of the calculations, the first proposed dielectric sphere was simulated in [19] and, given the consistency of the results obtained from the method described below, this method will be used for the structures described in this paper. In this paper, first, the dependence of the dispersion cross-section to nano particle size is studied. In Fig. 1, the dependence of the dispersion cross section as a function of wavelength is compared for a nanoparticle with different radii. The peaks seen in this figure are due to aggravation on the surface of these nanoparticles and the heterogeneous distribution of free charges. As the nanoparticle size increases, the number of conduction electrons increases and as a result the influence of the electric field applied to the nanoparticle decreases. In other words, the electric field only penetrates the surface of these nanoparticles, producing superficial and inhomogeneous effects in the distribution of free charges (conduction electrons) of the nanoparticle. In each figure of the first peak, related to the magnetic octupoles, the peak is related to the quadrilateral electric, and the third peak is the quadrilateral magnetic [19]. According to the shape, it is seen that the position and magnitude of the peaks associated with electric and magnetic multipolar depend on the radius of the nanoparticle so that it increases with the increase in the radius and the peaks move towards longer wavelengths. In the resonance of nanoparticle plasmon's, due to the coupling of electric fields applied to particle plasmons at certain wavelengths, the plasmon peaks in these wavelengths occur. According to Fig. 1, and the placement scattering cross-sectional spectra of the nanoparticle with a radius of 80 nm in the visible region, the 80-nanometer radius continues to be used for nanoparticles. In Figs. 2 and 3, the dispersion of two spherical nanoparticles with an optimal radius (80 nm), spaced along the x-axis and spaced apart d, for two different polarization directions (z and x, respectively) and at different distances is shown. According to Fig. 2, it is observed that for a wave with linear polarization along the z-axis, the magnitude of the distance between the nanoparticles (d) of the magnetic octupoles magnitude increases in the cross-sectional scattering area and the magnitude of the quadrupole magnetic peak due to the strong overlap between the distribution different polygons are almost eliminated. While there is no regular change in the size of the quadrupole electric peak. From the comparison of Figs. 1 and 2 It can be seen that the presence of the second nanoparticle is effective on the magnitude of the eight magnetic poles peak of the magnetic and quadrupole electrons, because the second nanoparticle has increased the number of conduction electrons, however, it does not change the peak location because the wavelength resonance is dependent on the type of material. In Fig. 3, the cross-section of the scattering for the
Fig. 2. Cross section scattering spectrum regarding wavelength for two spherical nanoparticle silicon with a radius of 80 nm at different distances (the field is polarized in z). 3
Optik - International Journal for Light and Electron Optics 201 (2020) 163477
F. Rahimi Moghadam
Fig. 3. Cross section scattering spectrum regarding wavelength for two spherical nanoparticle silicon with a radius of 80 nm at different distances (the field is polarized in x).
case where the wave is linearly polarized along x and released in the z-direction, at different intervals, for two nanoparticles are plotted as a function of wavelength. It is observed that with the increase in the intermediate distance the peak corresponding to the eight magnetic poles is the larger magnitude and the magnitude of the quadrilateral magnetic peak decreases (the opposite changes), while the mid-peak (quadrupole electric) is almost eliminated. In this case, it was also observed that changing the distance to the peak size is effective, but does not change the wavelength of the resonance. According to the results, the distance of 80 nm is chosen as the optimum distance. Fig. 4 shows the comparison between the spectra of the scattering cross section of a single nanoparticle and the pair of nanoparticles with different polarization directions. It is observed that the direction of polarization is effective in the size of the cross-sectional of the dispersion. When the field is propagated along the z-axis, the wave interacts with both nano-particles simultaneously, and in this case (propagation along z), the light hits the two nano-particles with the same intensity while in the second case (propagation along x) light after interaction with the first nanoparticle with less intensity, and it interacts with the second nanoparticle. Considering the similarity of the cross-sectional spectra of the nanoparticle pair for the polarization x with the spectrum of a single nanoparticle, the z-direction is considered as the direction of field propagation. In order to reach the larger peaks, linear arrays consisting of 3 and 4 spherical nanoparticles with a radius of 80 nm were considered for the X polarization direction. As shown in Fig. 5, the cross-sectional dispersion spectrum of linear arrays is similar to that of a single nanoparticle spectrum. Besides, with the increase in the number of nanoparticles in the array, the cross-sectional area in which the light interacts with it increases, increasing the number of surface electrons and, consequently, an increase in the magnitude of the eight magnetic poles and quadrilateral magnetic magnitude. In Fig. 6, the cross-sectional dispersion area of a square array is comprised of nine spherical silicon nanoparticles. As expected, the dispersion of a square array is far greater than the linear array. Moreover, it was observed that there is a complete similarity between the spectrums of a square array with a single nanoparticle spectrum. In other words, in this spectrum, there are three peaks of eight magnetic poles, quadrilateral, and four-polar magnetic magnitudes. While in linear arrays the middle of the peaks is almost eliminated. A square array is arranged in xy. Given the similarity of this spectrum with the spectrum of a single nanoparticle, the square array behaves just like a larger nanoparticle. Therefore, in order to achieve a larger dispersion cross-section, a square array with appropriate parameters can be selected.
Fig. 4. Comparison of cross section scattering spectrum from single nanoparticles and nanoparticle pair with 80 nm radius in two polarization align. 4
Optik - International Journal for Light and Electron Optics 201 (2020) 163477
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Fig. 5. Comparison of scattering cross-sectional spectra from a linear array consisting of 3 and 4 spherical nanoparticles with a radius of 80 nm.
Fig. 6. Dispersion cross-sectional area of a square array consisting of 9 spherical nanoparticles with a radius of 80 nm.
4. Conclusion In this paper, due to the importance of alternative structures than single dielectric cores, after selecting the appropriate radius, first pair of nanoparticles in the following, linear and square arrays of silicon nanoparticles have been investigated. Numerical calculations show that the peak of the dispersion cross section is due to the resonance between the incident field and the surface plasmons of the nanoparticles. It is well observed that the light scattered by spherical nanoparticles, based on the eight-pole magnetic, quadrupole, and a quadrilateral magnetic field is described; in the visible region, factors such as the size of the nanoparticle, the intermediate distance, the number of nanoparticles, and the direction of the polarization of the field are dependent. In other words, it can be said that the dispersion cross section is maximized due to the resonance of a plasmon in a given wavelength; besides, the dispersion cross-section is strongly sensitive to nanoparticles. The type of multi-polarities is determined according to the theory of Mie. The results play an important role not only in achieving the field distribution in metamaterials and optical antennas but also in regulating the transmission of light through a dielectric environment, such as photonic glasses [21,22]. References [1] S. Link, M.A. El-Sayed, Size and temperature dependence of the plasmon absorption of colloidal gold nanoparticles, J. Phys. Chem. B 103 (21) (1999) 4212–4217. [2] C. Voisin, N. Del Fatti, D. Christofilos, F. Vallee, Ultrafast electron dynamics and optical nonlinearities in metal nanoparticles, J. Phys. Chem. B 105 (12) (2001) 2264–2280. [3] G.V. Hartland, Measurements of the material properties of metal nanoparticles by time resolved spectroscopy, Phys. Chem. Chem. Phys. 6 (23) (2004) 5263–5274. [4] B. Lamprecht, J.R. Krenn, A. Leitner, F.R. Aussenegg, Resonant and off-resonant light driven plasmons in metal nanoparticles studied by femtosecond-resolution third-harmonic generation, Phys. Rev. Lett. 83 (21) (1999) 4421–4424. [5] M. Salerno, J.R. Krenn, B. Lamprecht, G. Schider, H. Ditlbacher, N. Felidj, A. Leitner, F.R. Aussenegg, Plasmon polaritons in metal nanostructures: the optoelectronic route to nanotechnology, Opto-Electron. Rev. 10 (3) (2002) 217–224. [6] J.R. Krenn, Nanoparticle waveguides watching energy transfer, Nat. Mater. 2 (4) (2003) 210–211. [7] S.A. Maier, P.G. Kik, H.A. Atwater, S. Meltzer, E. Harel, B.E. Koel, A.A.G. Requicha, Local detection of electromagnetic energy transport below the diffraction
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