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The spectrum of the integral operator of electromagnetic scattering

Jussi Rahola Center for Scienti c Computing P.O. Box 405 FIN-02101 ESPOO FINLAND Jussi.Rahola@csc. The scattering of electromagnetic radiation is governed by an integral equation for the electric eld. The problem can be cast as a surface or a volume integral equation. Here we are interested in the volume integral equation because the potential applications involve scattering from inhomogeneous and anisotropic materials. We consider only the simplest discretization of the problem: the scatterer is divided into cubic cells and the electric eld is assumed constant inside each cell. Two example scatterers are given in Figures 1 and 2. The computational cells are shown as small spheres. The computational problem is the solution of a system of linear equations where the coe cient matrix

Introduction

is full and complex symmetric. We have applied several iterative methods in this setting and QMR was chosen as the iterative method for the production code. The matrix-vector product can be computed with two special methods: the fast Fourier transform and the fast multipole method. Figure 1: An example of a sphere where a number of particles have been removed. Figure 2: A cluster of particles generated by the di usion-limited aggregation (DLA) process.

The scattering integral equation for the electric eld E(r) is Z E(r) = Einc (r)+k 3 (m(r )2 ?1)G(r; r ) E(r ) d3 r ;

0 0 0 0

Scattering Integral Equation

where Einc is the incoming eld, G is the dyadic Green's function and

V

rr G(r; r ) = 1 + 2 g (jr ? r j) k

0 0

ikr e g(r) = 4 kr : Here m is the complex refractive index and k is the wave number. The dyadic rr contains the second-order partial derivatives, so it is essentially the Hessian matrix. The integral equation is a strongly singular one. We de ne the integral operator as Z (KE)(r) = k3 (m(r )2 ? 1)G(r; r ) E(r ) d3 r :

0 0 0 0

Now the integral equation can be written as (1 ? K)E = Einc

V

The spectrum of the integral operator K consists of those complex numbers z for which the operator (z ? K) does not have an inverse that is bounded and de ned everywhere. 1 K) which The operator (z ?K) is the same as z (1? z corresponds to a scattering problem with the same particle but with refractive index m de ned by

0

Spectrum of the integral operator

m 2 ? 1 = (m2 ? 1)=z:

0

The problem is now to nd those refractive indices m for which the scattering problem does not have an inverse that is bounded and de ned everywhere. Then the points of the spectrum are given by

0

z = (m2 ? 1)=(m 2 ? 1):

0

If the scatterer is a homogeneous sphere, the scattering problem has an analytical solution, the so called Lorenz-Mie solution.

When we computed the eigenvalues of the coe cient matrix arising from the simple discretization of a homogeneous sphere, we found that most of the eigenvalues line on a line. Figure 3 gives the numerically computed eigenvalues together with some analytically computed points of the spectrum. The points of the spectrum align nicely with the computed eigenvalues which lie outside the line. We were unable to indentify the line with the analytical spectrum. However, the line corresponds to a scattering problem with a purely imaginary refractive index which physically corresponds to non-oscillating decaying radiation. Figure 4 gives the numerical eigenvalues arising from the discretization of the DLA particle.

Numerical eigenvalues vs. spectrum

0.2

0.1

0

?0.1

?0.2

?0.3

?0.4

?0.5 0.5

1

1.5

2

2.5

3

Figure 3: Numerically computed eigenvalues (dots) and analytically computed points of the spectrum (rings) for the sphere

0.04 0.02 0 -0.02 -0.04 Imag -0.06 -0.08 -0.1 -0.12 -0.14 -0.16 0.7

0.8

0.9

1 Real

1.1

1.2

1.3

Figure 4: Numerically computed eigenvalues for the DLA particle.

The convergence of iterative methods is determined by the distribution of eigenvalues of the coe cient matrix. Let us consider the xed point problem

Convergence of Iterative Methods

x = Kx + b: (1) For any Krylov-subspace method, the residual rk = b ? (1 ? K)xk can decrease with the rate krk k = ( (K))k kr 0 k; (2) where the optimal reduction factor (K) is given by 1 : (3) (K) = ( ) = j (1) j Here is the conformal map that maps the unbounded component of the complement of spectrum of K, (K), to the outside of the closed unit disk (see Figure 5). Here we assume that the spectrum is simply connected. Figure 6 shows the ideal convergence rate together with the convergence of QMR and GMRES for a spherical scatterer.

K

φ

α

d1

0

d2

1

Figure 5: The conformal map maps the outside of the line to the outside of the unit circle.

10

1

10

0

10

-1

10 Residual

-2

10

-3

10

-4

10

-5

Ideal 10

-6

QMR GMRES

10

-7

2

3

4

5

6

7 Iterations

8

9

10

11

12

Figure 6: The convergence of QMR and GMRES together with the ideal convergence rate for a spherical scatterer.

References

1] K. Lumme and J. Rahola. Light scattering by porous dust particles in the discrete-dipole approximation. Astrophys. J. 425, 653{667, 1994. 2] O. Nevanlinna. Convergence of Iterations for Linear Equations. Birkhauser, Basel, 1993. 3] J. Rahola. Solution of dense systems of linear equations in electromagnetic scattering calculations. Licenciate's thesis, Helsinki University of Technology, 1994. 4] J. Rahola. Diagonal forms of the translation operators in the fast multipole algorithm for scattering problems. Helsinki University of Technology, Institute of Mathematics, Research Reports A 349, 1995. 5] J. Rahola. Solution of dense systems of linear equations in the discrete-dipole approximation. SIAM J. Sci. Comp. 17, No. 1, 1996. In press.

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