Two-layer Spheroid Research Articles

The Condition of Applicability for the Extended Boundary Conditions Method for Small Multilayer Particles

We have examined an analog to the extended boundary conditions method (EBCM) with the standard spherical basis, which is popular in light scattering theory, with respect to its applicability to the solution of an electrostatic problem that arises for multilayer scatterers the sizes of which are smaller compared to the wavelength of the incident radiation. It has been found that, in the case of two or more layers, to determine the polarizability and other optical characteristics of particles in the far-field zone, the parameters of the surfaces of layers should obey the condition max{σ1(j)} < min{σ2(j)}. In this case, appearing infinite systems of linear equations for expansion coefficients of unknown fields have a unique solution, which can be found by the reduction method. For nonspheroidal particles, this condition is related to the convergence radii of expansions of regular and irregular fields outside and inside of the particle, including its shells—R1(j) = σ1(j) and R2(j) = σ2(j). In other words, a spherical shell should exist in which expansions of all regular and irregular fields converge simultaneously. This condition is a natural generalization of the result for homogeneous particles, for which such a condition is imposed only on expansions of the “scattered” and internal fields—R1 < R2. For spheroidal multilayer particles, which should be singled out into a separate class, the EBCM applicability condition is written as max{σ1(1), σ1(2), …, σ1(J−1), σ1(J)} < min{σ2(1), σ2(2), …, σ2(J−1)} and parameters σ2(j) of the surfaces of shells are not related to corresponding convergence radii R2j of irregular fields. Numerical calculations for two-layer spheroids and pseudospheroids have confirmed completely theoretical inferences. Apart from the EBCM algorithm, an approximate formula has been proposed for the calculation of the polarizability of two-layer particles, in which the polarizability of a two-layer particle is interpreted as a linear combination of the polarizabilities of homogeneous particles that consist of the materials of the shell and core proportionally to their volumes. The range of applicability of this formula is wider than that for the EBCM, and the calculation error is smaller than 1%.

On the applicability of a spherical basis for spheroidal layered scatterers

The applicability of an analog of the extended boundary condition method, which is popular in light-scattering theory, is studied in combination with the standard spherical basis for the solution of an electrostatic problem appearing for spheroidal layered scatterers the sizes of which are small as compared to the incident radiation wavelength. In the case of two or more layers, polarizability and other optical characteristics of particles in the far zone are shown to be undeterminable if the condition under which the appearing systems of linear equations for expansion coefficients of unknown fields are Fredholm systems solvable by the reduction method is broken. For two-layer spheroids with confocal boundaries, this condition is transformed into a simple restriction on the ratio of particle semiaxes a/b< \(\sqrt 2 \) + 1. In the case of homogeneous particles, the solvability condition is that the radius of convergence of the internal-field expansion must exceed that of the expansion of an analog of the scattering field. Since homogeneous spheroids (ellipsoids) are unique particles inside which the electrostatic field is homogeneous, it is shown that the solution can be always found in this case. The obtained results make it possible to match in principle the results of theoretical and numerical determinations of the domain of applicability for the extended boundary condition method with a spherical basis for spheroidal scatterers.

Scattering of electromagnetic radiation by a two-layer nonconfocal spheroid that is small compared to the wavelength

We have considered the electrostatic problem for a two-layer nonconfocal spheroid. The approach is based on surface integral equations that are similar to equations in terms of the extended boundary condition method for wave problems. Electrostatic fields are related to scalar potentials, which are represented as expansions in terms of eigenfunctions of the Laplace equation in two spheroidal coordinate systems, while unknown expansion coefficients are determined from infinite systems of linear algebraic equations. The constructed rigorous solution to the problem coincides with the known solution in a particular case of a confocal two-layer spheroid. In addition, for the nonconfocal two-layer spheroid, we have constructed an explicit approximated solution assuming that the field in the particle core is constant. This solution coincides with the rigorous solution if the scatterer shells are confocal. The formula found for the polarizability of the two-layer nonconfocal spheroid has a very simple form compared to the previously proposed cumbersome algorithm (B. Posselt et al., Measur. Sci. Technol. 13, 256 (2002)) and is more efficient numerically.

Attenuation, scattering, and depolarization of light by gold nanorods with silver shells

Gold nanoparticles with silver nanoshells are obtained by synthesizing gold nanorods in a growing solution containing cetyltrimethylammonium bromide, subsequent separation in a concentration gradient of glycerol, and reduction of silver nitrate by ascorbic acid under alkaline conditions in the presence of polyvinylpyrrolidone. The formation of silver nanoshells was monitored by the shift of plasmon resonances of extinction and differential light scattering, by the appearance of characteristics peaks of silver in the energy dispersive X-ray (EDX) spectra of samples, by the data of transmission electron microscopy, and by visual changes in the color of colloids. The spectrum of the intensity ratio of the co- and cross-polarized compo- nents of light scattered by gold-silver nanorods is measured for the first time, and it is observed that the maximum is shifted by 80–100 nm compared to previously published spectra of gold nanorods (Khlebtsov et al., J. Phys. Chem. C 112, 12760 (2008)). The extinction and light scattering spectra are calculated by the method of separation of variables using the model of a confocal two-layer spheroid and these calculations are found to agree with spectral measurements. A method for determining the thickness of a silver nanolayer by the spectral shift of an extinction longitudinal resonance is described. The obtained data of optical spectroscopy and transmission electron microscopy and estimations of the mass of the deposited metal show that the aver-age thickness of the silver layer varies from 0.12 to 4 nm as the Ag/Au ratio changes from 2/80 to 90/80 μg/μg.

Scattering of a point generated field by kidney stones

In this work, we examine the acoustic scattering problem of spherical waves by a two-layer spheroid simulating the kidney-stone system. Both the theoretical as well as the numerical treatment are presented. The outcome of the analysis is the determination of the scattered field along with its multivariable dependence on the several physical and geometric parameters of the system. A comparison with the simpler case of spherical geometry as well as the plane wave excitation case is realised.

New recursive solution of the problem of scattering of electromagnetic radiation by multilayer spheroidal particles

A new recursive algorithm for the solution of the problem of scattering of light (of an arbitrarily polarized plane electromagnetic wave) by multilayer confocal spheroidal particles is constructed. This approach preserves the advantages of the two approaches proposed earlier by us for single-layer and two-layer spheroids (special choice of scalar potentials and utilization of the basis of wave spheroidal harmonics) and for homogeneous axially symmetric particles (formulation of the problem in terms of surface integral equations, calculation of the potentials inside the particle from the potentials of the incident radiation, and calculation of the potentials of the scattered radiation from the potentials inside the particle). In the case of multilayer particles, the potential inside each shell is a sum of two terms. The first has the properties of the incident radiation (no singularities inside the volume enclosed by the external boundary of the shell), whereas the second term has the properties of the scattered radiation (satisfies the radiation conditions at infinity). Therefore, as the calculation progresses from one layer to the next (from the core to the outer shell), the dimensionality of the reduced linear matrix equations for the unknown expansion coefficients of the scattered field potentials does not increase with respect to the case of a homogeneous spheroid. The algorithm is particularly simple and lucid (as far as possible for such a complex problem). In the case of spherical multilayer particles, the solution can be found explicitly.

Absorption and luminescence of dye-coated silver and gold particles

Formulas are derived for the extinction cross section of small randomly oriented, two-layer spheroids and for the luminescence of small spheres coated with a luminescent layer. Numerical examples are presented showing the dependence of the extinction of dye-coated particles upon the particle size, the optical properties of the core (silver, gold, dielectric), the coating thickness, and the spheroid eccentricity. The interaction of metal and dye extinction bands, the splitting of these bands, and the enhancement of the extinction depend sensitively on these various parameters. The enhancement of the luminescence of dye layers coating metallic spheres is shown to be related to comparable enhancements in the extinction. The present model differs from earlier studies of surface-enhanced Raman scattering (SERS) in that the electric dipoles which simulate the dye molecules are embedded in a medium having the dielectric properties of the dye rather than dangling into the external medium. The present study utilizes air as the external medium. The enhancement effects would be much greater in water.