Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf
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2.1 Homogeneous and Isotropic Particles |
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and using the relation Q3 Q3 = I, we deduce that
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M4 M4 = I . |
Because M is upper-triangular with real diagonal elements, it follows that M4 is unit upper-triangular (all diagonal elements are equal 1). Then, (2.24) implies that M4 = I, and therefore
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S = −Q3 Q3 .
In terms of the transition matrix, the above equation read as
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T = −Q3 Re Q3
and this new sequence of truncated transition matrices might be expected to converge more rapidly than (2.23) because the unitarity condition is satisfied for each truncation index.
2.1.3 Symmetries of the Transition Matrix
Symmetry properties of the transition matrix can be derived for specific particle shapes. These symmetry relations can be used to test numerical codes as well as to simplify many equations of the T -matrix method and develop e cient numerical procedures. In fact, the computer time for the numerical evaluation of the surface integrals (which is the most time consuming part of the T -matrix calculation) can be substantially reduced. The surface integrals are usually computed in spherical coordinates and for a surface defined by
r = r (θ, ϕ) , |
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we have |
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n (r) dS (r) = σ (r) r2 sin θ dθ dϕ , |
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where |
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1 ∂r |
1 ∂r |
σ(r) = er − r ∂θ eθ − r sin θ ∂ϕ eϕ
is a vector parallel to the unit normal vector n. In the following analysis, we will investigate the symmetry properties of the transition matrix for axisymmetric particles and particles with azimuthal and mirror symmetries.
1. For an axisymmetric particle, the equation of the surface does not depend on ϕ, and therefore ∂r/∂ϕ = 0. The vector σ has only er - and eθ - components and is independent of ϕ. The integral over the azimuthal angle can be performed analytically and we see that the result is zero unless m = m1.
94 2 Null-Field Method
Therefore, the T matrix is diagonal with respect to the azimuthal indices m and m1, i.e.,
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Tmn,mij |
1n1 = Tmn,mnij |
1 δmm1 . |
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Direct calculation shows that |
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Tmn,mnij |
1 = (−1)i+j T−ijmn,−mn1 |
(2.25) |
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and the symmetry relation (1.112) takes the form |
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Tmn,mnij |
1 = T−jimn1,−mn = (−1)i+j Tmnji |
1,mn . |
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2. We consider a particle with a principal N -fold axis of rotation symmetry and assume that the axis of symmetry coincides with the z-axis of the particle coordinate system. In this case, r, ∂r/∂θ and ∂r/∂ϕ are periodic in ϕ with period 2π/N . Taking into account the definition of the Qpq matrices, we see that the surface integrals are of the form
2π
f (ϕ) ej(m1−m)ϕdϕ ,
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where f is a periodic function of ϕ, |
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f (ϕ) = f ϕ + k |
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As a consequence of the rotation symmetry around the z-axis, the T matrix is invariant with respect to discrete rotations of angles αk = 2πk/N , k = 1, 2, . . . , N − 1. A necessary and su cient condition for the equation
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dϕ = e |
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f (ϕ) e |
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to hold is |m1 − m| = lN , l = 0, 1, . . ., and we conclude that
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T ij |
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Further, |
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2π |
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m)ϕ |
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f (ϕ) e |
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dϕ = e |
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k=1 |
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= N 2π/N f (ϕ) ej(m1−m)ϕdϕ , |
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for |m1 − m| = lN , l = 0, 1, . . ., and we see that |
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T ij |
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1n1 , |m1 − m| = lN, l = 0, 1, . . . , |
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rest |
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where (Trotsym)mn,mij |
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by integrating ϕ over the interval [0, 2π/N ]. |
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2.1 Homogeneous and Isotropic Particles |
95 |
3. We consider a particle with mirror symmetry and assume that the xy-plane is the horizontal plane of reflection. The surface parameterization of a particle with mirror symmetry has the property
r (θ, ϕ) = r (π − θ, ϕ)
and therefore
∂θ∂r (θ, ϕ) = −∂θ∂r (π − θ, ϕ) , ∂ϕ∂r (θ, ϕ) = ∂ϕ∂r (π − θ, ϕ)
for 0 ≤ θ ≤ π/2 and 0 ≤ ϕ < 2π. Taking into account the symmetry relations of the normalized angular functions πn|m| and τn|m| (cf. (A.24) and (A.25)),
πnm (π − θ) = (−1)n−m πnm (θ) , τnm (π − θ) = (−1)n−m+1 τnm (θ) ,
we find that
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1 + (−1) |
n+n1+|m|+|m1|+i+j |
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(Tmirrsym)mn,m1n1 |
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where (Tmirrsym)ijmn,m1n1 are the elements of the transition matrix computed by integrating θ over the interval [0, π/2]. The above relation also shows that
Tmn,mij 1n1 = 0 ,
if n + n1 + |m| + |m1| + i + j is an odd number.
Exploitation of these symmetry relations leads to a reduction in CPU time by three orders of magnitude from that of a standard implementation with no geometry-specific adaptations. Additional properties of the transition matrix for particles with specific symmetries are discussed by Kahnert et al. [111] and by Havemann and Baran [98].
2.1.4 Practical Considerations
The integrals over the particle surface are usually computed by using appropriate quadrature formulas. For particles with piecewise smooth surfaces, the numerical stability and accuracy of the T -matrix calculations can be improved by using separate Gaussian quadratures on each smooth section [8, 170].
In practice, it is more convenient to interchange the order of summation in the surface field representations, i.e., the sum
Nrank n
(·)
n=1 m=−n
96 2 Null-Field Method
is replaced by
Mrank Nrank
(·) ,
m=−Mrank n=max(1,|m|)
where Nrank and Mrank are the maximum expansion order and the number of azimuthal modes, respectively. Consequently, the dimension of the transition matrix T is
dim (T ) = 2Nmax × 2Nmax ,
where
Nmax = Nrank + Mrank (2Nrank − Mrank + 1) ,
is the truncation multi-index appearing in (2.7). The interchange of summation orders is e ective for axisymmetric particles because, in this case, the scattering problem can be reduced to a sequence of subproblems for each azimuthal mode.
An important part of the numerical analysis is the convergence procedure that checks whether the size of the transition matrix and the number of quadrature points for surface integral calculations are su ciently large that the scattering characteristics are computed with the desired accuracy. The convergence tests presented in the literature are based on the analysis of the di erential scattering cross-section [8] or the extinction and scattering crosssections [164]. The procedure used by Barber and Hill [8] solves the scattering problem for consecutive values of Nrank and Mrank, and checks the convergence of the di erential scattering cross-section at a number of scattering angles. If the calculated results converge within a prescribed tolerance at 80% of the scattering angles, then convergence is achieved. The procedure developed by Mishchenko [164] is applicable to axisymmetric particles and finds reliable estimates of Nrank by checking the convergence of the quantities
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Ce = − |
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(2n + 1) Re T0n,0n + T0n,0n |
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2π |
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&T 11 &2 + |
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The null-field method is a general technique and is applicable for arbitrarily shaped particles. However, for nonaxisymmetric particles, a semi-convergent
2.1 Homogeneous and Isotropic Particles |
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behavior is usually attended: the relative variations of the di erential scattering cross-sections decrease with increasing the maximum expansion order, attain a relative constant level and afterwards increase. In this case, the main problem of the convergence analysis is the localization of the region of stability. The extinction and scattering cross-sections can also give information on the convergence process. Although the convergence of Cext and Cscat does not guarantee that the di erential scattering cross-section converges, the divergence of Cext and Cscat implies the divergence of the T -matrix calculation.
2.1.5 Surface Integral Equation Method
The null-field method leads to a nonsingular integral equation of the first kind. However, in the framework of the surface integral equation method, the transmission boundary-value problem can be reduced to a pair of singular integral equations of the second kind [97]. These equations are formulated in terms of two surface fields which are treated as independent unknowns. In order to elucidate the di erence between the null-field method and the surface integral equation method we follow the analysis of Martin and Ola [155] and review the basic boundary integral equations for the transmission boundaryvalue problem. We consider the vector potential Aa with density a
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S a(r )g(k, r, r ) dS(r ) , r R |
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− S |
Aa(r) = |
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and evaluate the tangential components of the curl and double curl of the vector potential on S. For continuous tangential density, we have [39]
lim n(r) |
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A |
(r |
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hn(r))] = |
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a(r) + ( |
a)(r) , r |
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S , |
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h 0+ |
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a |
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M |
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where M is the magnetic dipole operator,
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a)(r) = n(r) |
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M |
× × S |
a(r )g(k, r, r ) dS(r ) , r |
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For a su ciently smooth tangential density, we also have
lim |
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0+ n(r) × [ × × Aa(r ± hn(r))] = (Pa)(r) , r S , |
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where the principal value singular integral operator P, called the electric dipole operator is given by
( |
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a)(r) = n(r) |
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× × × |
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a(r )g(k, r, r )dS(r ) , r |
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Considering the null-field equations for the electric and magnetic fields in Di and Ds, and passing to the boundary along a normal direction we obtain