Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006

200 3 Simulation Results

are discretized in an analogous manner. The di erential equations are formulated in terms of the magnetic grid voltages hα,ijk along the dual edges Lα,ijk , the dielectric fluxes dα,ijk and the conductive currents jα,ijk through the dual facets Sα,ijk , α = x, y, z, and the electric charge qijk in the dual

Collecting the equations of all element surfaces of the complex pair {G, G} and introducing the integral voltage-vectors e and h, the flux state-vectors b and d, the conductive current-vector j, and the electric charge-vector q, we derive a set of discrete matrix equations, the so-called Maxwell grid equations

The transformation into frequency domain for the Maxwell grid equations with e(t) = Re{ee−jωt}, yields

Ce = jωb , Ch = −jωd + j .

The discrete curl-matrices C and C, and the discrete divergence matrices S and S are defined on the grids G and G, respectively, and depend only on the grid topology. The integral voltageand flux state-variables allocated on the two di erent volume element complexes are related to each other by the discrete material matrix relations

d = M εe + p , j = M κe , h = M ν b − m ,

where M ε is the permittivity matrix, M κ is the matrix of conductivities, M ν is the matrix of reluctivities, and p and m are the electric and magnetic polarization vectors, respectively. The discrete grid topology matrices have the same e ect as the vector operators curl and div. For instance, the discrete analog of the equation · × = 0 (div curl = 0) is SC = 0 and SC =

0. Transposition of these equations together with the relation between the

T

discrete curl-matrices C = C , yields the discrete equations CST = 0 and

T

CS = 0, both corresponding to the equation × = 0 (curl grad = 0). Basic algebraic properties of the Maxwell grid equations also allow to prove conservation properties with respect to energy and charges.

The finite integration technique su ers somewhat from a deficiency in being able to model very complicated cavities including curved boundaries with high precision, but the usage of the perfect boundary approximation eliminate this deficiency [123].

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Z z

a

β

O

Y

α

b

Fig. 3.3. Geometry of a prolate spheroid

Localized Sources

In our first example, we consider prolate spheroids in random and fixed orientation, and compare the results obtained with the TAXSYM code to the solutions computed with the codes developed by Mishchenko [167–169]. The orientation of the axisymmetric particle with respect to the global coordinate system is specified by the Euler angles of rotation αp and βp, and the incident field is a linearly polarized plane wave propagating along the Z-axis (Fig. 3.3). The rotational semi-axis (along the axis of symmetry) is ksa = 10, the horizontal semi-axis is ksb = 5, and the relative refractive index of the spheroid is mr = 1.5. The maximum expansion order and the number of integration points are Nrank = 17 and Nint = 100, respectively. In Figs. 3.4 and 3.5 we plot some elements of the scattering matrix for a randomly oriented spheroid. The agreement between the curves is acceptable. For a fixed orientation of the prolate spheroid, we list in Tables 3.1 and 3.2 the phase matrix elements Z11 and Z44, and Z21 and Z42, respectively. The Euler angles of rotation are αp = βp = 45◦, and the matrix elements are computed in the azimuthal planes ϕ = 45◦ and ϕ = 225◦ at three zenith angles: 30◦, 90◦, and 150◦. The relative error is around 10% for the lowest matrix element and remains below 1% for other elements.

In the next example, we show results computed for a perfectly conducting spheroid of size parameter ksa = 10, aspect ratio a/b = 2, and Euler angles of rotation αp = βp = 45◦. The perfectly conducting spheroid is simulated from the dielectric spheroid by using a very high value of the relative refractive index (mr = 1.e+30), and the version of the code devoted to the analysis of perfectly conducting particles is taken as reference. For this application, the

Fig. 3.4. Scattering matrix elements F11 and F22 of a dielectric prolate spheroid

Scattering Angle (deg)

Fig. 3.5. Scattering matrix elements F21 and F43 of a dielectric prolate spheroid

maximum expansion order is Nrank = 18, while the number of integration points is Nint = 200. The normalized di erential scattering cross-sections presented in Fig. 3.6 are similar for both methods.

To verify the accuracy of the code for isotropic, chiral particles, we consider a spherical particle of size parameter ksa = 10. The refractive index of the particle is mr = 1.5 and the chirality parameter is βki = 0.1, where ki = mrks. Calculations are performed for Nrank = 18 and Nint = 200. Figure 3.7 compares the normalized di erential scattering cross-sections computed with the

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Table 3.1. Phase matrix elements 11 and 44 computed with (a) the TAXSYM routine and (b) the code of Mishchenko

Table 3.2. Phase matrix elements 21 and 42 computed with (a) the TAXSYM routine and (b) the code of Mishchenko

Fig. 3.6. Normalized di erential scattering cross-sections of a perfectly conducting prolate spheroid

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Fig. 3.8. Variation of the normalized di erential scattering cross-sections with the axial position of a prolate spheroid illuminated by a Gaussian beam

Fig. 3.9. Variation of the normalized di erential scattering cross-sections with the o -axis coordinate of a prolate spheroid illuminated by a Gaussian beam

Distributed Sources

While localized sources are used for not extremely aspherical particles, distributed sources are suitable for analyzing particles with extreme geometries, i.e., particles whose shape di ers significantly from a sphere. Extremely deformed particles are encountered in various scientific disciplines as for instance

Fig. 3.10. Variation of the normalized di erential scattering cross-sections with the beam waist radius

Fig. 3.11. Particles with extreme geometries: prolate spheroid, fibre, oblate cylinder and Cassini particle

astrophysics, atmospheric science and optical particle sizing. For example, light scattering by finite fibres is needed in optical characterization of asbestos or other mineral fibres, while flat particles are encountered as aluminium or mica flakes in coatings.

Figure 3.11 summarizes the particle shapes considered in our exemplary simulation results. The spheroid is a relatively simple shape but convergence problems occur for large size parameters and high aspect ratios. The finite fibre is a more extreme shape because the flank is even and without convexities. This shape is modeled by a rounded prolate cylinder, i.e., by a cylinder with two half-spheres at the ends. In polar coordinates, a rounded prolate cylinder as shown in Fig. 3.12 is described by

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z

2b

z1

z2

x

O

2a

zN

Fig. 3.12. Geometry of a prolate cylinder and the distribution of the discrete sources on the axis of symmetry

where θ0 = arctan[b/(a − b)]. The rounded oblate cylinder is constructed quite similar, i.e., the flank is rounded and top and bottom are flat. An oblate cylinder as shown in Fig. 3.13 is described in polar coordinates by

where θ0 = arctan[(b − a)/a]. Cassini particles are a real challenge for light scattering simulations because the generatrix contains concavities on its top and bottom. The Cassini ovals can be described in polar coordinates by the equation

and the shape depends on the ratio b/a. If a < b the curve is an oval loop, for a = b the result is a lemniscate, and for a > b the curve consists of two separate loops. If a is chosen slightly smaller than b we obtain a concave, bone-like shape, and this concavity becomes deeper as a approaches to b.

For the prolate particles considered in our simulations, the sources are distributed on the axis of symmetry as in Fig. 3.12, while for the oblate particles, the sources are distributed in the complex plane as in Fig. 3.13. The wavelength of the incident radiation is λ = 0.6328 µm, the relative refractive index