Light scattering by dielectric bodies in the Born approximation
LLight scattering by dielectric bodies in the Born approximation
A. Bereza,
1, 2
A. Nemykin,
1, 2
S. Perminov, L. Frumin,
1, 2 and D. Shapiro
1, 2 Institute of Automation and Electrometry, Russian Academy of Sciences,Siberian Branch, 1 Koptjug Ave. Novosibirsk 630090, Russia Novosibirsk State University, 2 Pirogov Str., Novosibirsk 630090,Russia Rzhanov Institute of Semiconductor Physics, Russian Academy of Sciences,Siberian Branch, 13 Lavrent’yev Ave., Novosibirsk 630090,Russia
Light scattering is one of the most important elementary processes in near-field optics.We build up the Born series for scattering by dielectric bodies with step boundaries.The Green function for a 2-dimensional homogeneous dielectric cylinder is obtained.As an example, the formulas are derived for scattered field of two parallel cylin-ders. The polar diagram is shown to agree with numerical calculation by the knownmethods of discrete dipoles and boundary elements.PACS numbers: 03.50.De,42.25.Fx 1 a r X i v : . [ phy s i c s . op ti c s ] A p r . INTRODUCTION In the past decades a substantial progress has been achieved in nano-optics . However, asignificant methodological deficiency still persists even for basic problems, like scattering bynano-sized bodies. Unlike ”macroscopic” optics, where transverse waves (for instance, planeor spherical) are very useful to study, say, diffraction and interference, at sub-wavelengthregion the treatment of these phenomena becomes much more complicated. The reasonis evanescent waves near a boundary of illuminated objects. Such wave usually can be ne-glected while studying optical processes with large scatterers, but nano-optics is not the case.Strong coupling via evanescent wave is the key feature, which most practical nanophotonicstasks focus on. They include light energy concentration within few-nanometer range ; high-efficiency broad-band solar cells ; light-induced forces at nano-scale ; surfaces-enhancedRaman spectroscopy ; the tomographic reconstruction of a nano-structure. Only few problems allow analytical solution in photonics. Along with the classical papersdevoted to one cylinder, the scattering from two circular cylinders and two perfectlyconducting spheres can be found in the quasi-static limit using bipolar coordinates; aperfectly conducting cylinder near a surface was considered using expansion in the series ofcylindrical waves. In any more complicated cases, numerical or semi-analytical methodsbecome the only capable to calculate electromagnetic fields in both near and far regions, forinstance, in a system of several cylinders or in their periodic chain.
Analytical approximations are very useful for understanding the scattering properties ofa structure, at least for testing the numerical methods. There is a universal method toderive the formulas based on the Born approximation. It consists in taking the incidentfield in place of the total field at each point inside the scattering potential. If the scattereris not sufficiently weak, the next approximations are exploited. There are several recentoptical researches devoted to high-order terms of the approximation. In optical diffusiontomography the high orders are necessary for solving the nonlinear inverse problem. Thesecond-order approximation is needed for numerical reconstruction of a shallow buried objectby the scattered amplitude. The resonant-state expansion approximation uses the second-order terms to find eigen frequencies in an optical fiber waveguide. However, the traditionalBorn series is not applicable in a system of dielectric bodies with step edges, as not satisfyingthe boundary conditions. 2he main goal of the present work is to construct modified Born approximation for aset of dielectric bodies. The integral relations are derived and the series for two dielectriccylinders is obtained. We manage to account for the first cylinder exactly by means ofthe special Green function for a cylindrical dielectric, that intrinsically include multiplescattering processes with this cylinder. Thus, another aim of our paper is to derive thatspecial Green function.The Born series is constructed in Sec. II. The scattering by two cylinders, consideredin Sec. III, illustrates the application of developed approach. The obtained formulas are inagreement with numerical calculation using surface integral equations and discrete dipoleapproximation. The Green function is derived in Appendix A: the expressions for thesource point inside and outside the dielectric are given for both cases of p - and s -wave.The boundary element method has already been discussed in previous works devoted tothe scattering by cylinders on a dielectric substrate. The formulas for two-dimensionaldiscrete dipole method are derived in Appendix B.
II. BORN SERIES
The Helmholtz equations for magnetic field inside and outside the dielectric (denoted bysubscripts in and out , correspondingly) are( (cid:52) + k ) H in ( r ) = 0 , ( (cid:52) + k ) H out ( r ) = 0 , (1)where (cid:52) is 2-dimensional Laplace operator with respect to x and y variables, wavenumbers k = ω/c and k = √ εω/c , c is the speed of light, ω is the frequency, ε is the dielectricpermittivity. The field H out in free space is slightly changed due to a weak perturbation,which is small enough (i.e. k a (cid:28)
1, where a is its size) and/or low-polarizable ( | ε − | (cid:28) H in can be quite different. The Green function obeys the equation( (cid:52) + k ) G = δ ( r − r (cid:48) ) . (2)Here k is the wavenumber in corresponding region: k = k or k = k .We use Eq. (1), (2) to derive the relations between field amplitudes at the boundary: H in ( r ) = (cid:90) D + (cid:104) H in ( r (cid:48) ) (cid:52) G p ( r , r (cid:48) ) − G p ( r , r (cid:48) ) (cid:52) H in ( r (cid:48) ) (cid:105) dS (cid:48) , H out ( r ) = (cid:90) D − (cid:104) H out ( r (cid:48) ) (cid:52) G ( r , r (cid:48) ) − G ( r , r (cid:48) ) (cid:52) H out ( r (cid:48) ) (cid:105) dS (cid:48) , (3)3 + γ Γ D − FIG. 1. The domains of integration D + and D − for Eq. (3). The boundary of D − consists of γ = ∂ D + (solid line) and external infinitely remote contour Γ (dashed line). The arrows indicateits external normals n to ∂ D − . were dS (cid:48) is the element of integration over dielectric, D + , or free space, D − , domains, Fig. 1.The Green function G ( r , r (cid:48) ) describes free space, G p ( r , r (cid:48) ) is similar function that correspondsto dielectric of permittivity ε . The Green function of free space is the solution of Eq. (2)with k = k and can be written as G ( r , r (cid:48) ) = 14 i H (1)0 ( k | r − r (cid:48) | ) = 14 i ∞ (cid:88) m = −∞ e im ( ϕ − ϕ (cid:48) ) × H (1) m ( kr ) J m ( kr (cid:48) ) , r > r (cid:48) ,H (1) m ( kr (cid:48) ) J m ( kr ) , r (cid:48) > r , (4)where J m , H (1) m are Bessel and Hankel functions of the order m . The boundary conditions are H in (cid:12)(cid:12)(cid:12) γ = H out (cid:12)(cid:12)(cid:12) γ , ε ∂ H in ∂r (cid:12)(cid:12)(cid:12) γ = ∂ H out ∂r (cid:12)(cid:12)(cid:12) γ , (5)where γ is the contour separating D − and D + domains. Applying Green’s theorem to Eq.(3) we can reduce the surface integral as H in ( r ) = (cid:90) γ n (cid:104) H out ( r (cid:48) ) ∇ G p ( r , r (cid:48) ) − εG p ( r , r (cid:48) ) ∇H out ( r (cid:48) ) (cid:105) dl (cid:48) , H out ( r ) = − (cid:90) γ n (cid:104) H in ( r (cid:48) ) ∇ G ( r , r (cid:48) ) − ε G ( r , r (cid:48) ) ∇H in ( r (cid:48) ) (cid:105) dl (cid:48) ++ (cid:90) Γ n (cid:104) H out ( r (cid:48) ) ∇ G ( r , r (cid:48) ) − G ( r , r (cid:48) ) ∇H out ( r (cid:48) ) (cid:105) dl (cid:48) , (6)where dl (cid:48) is the element of path, n is the unit vector along the external normal, Γ is someremote contour (Fig. 1). The integral over Γ in the last line can be calculated explicitly by4he known relation for the Wronskian determinant: − kρ i π (cid:90) − π (cid:104) H (1)0 ( kρ ) + i cos ϕH (1)1 ( kρ ) (cid:105) e ikρ cos ϕ dϕ == 2 πkρ (cid:104) J ( kρ ) H (1)0 ( kρ ) − J ( kρ ) H (1)1 ( kρ ) (cid:105) = 1 . Then this integral reproduces the field of a plane incident wave, H (0) out = H e i kr . Eqs. (6) aresimilar to boundary integral equations; the only difference is the absence of the factor 1/2 inthe terms outside the integral. These terms are given within the external or internal limit,in contrast to boundary equations, where they are determined directly at the contour. The successive approximation series can be built up for both external and internal fields: H out = H (0) out + H (1) out + . . . , H in = H (0) in + H (1) in + . . . (7)Then from (6) we get the recurrent relations: H ( j ) in ( r ) = (cid:90) γ n (cid:104) H ( j ) out ( r (cid:48) ) ∇ G p ( r , r (cid:48) ) − εG p ( r , r (cid:48) ) ∇H ( j ) out ( r (cid:48) ) (cid:105) dl (cid:48) , H ( j +1) out ( r ) = − (cid:90) γ n (cid:104) H ( j ) in ( r (cid:48) ) ∇ G ( r , r (cid:48) ) − ε G ( r , r (cid:48) ) ∇H ( j ) in ( r (cid:48) ) (cid:105) dl (cid:48) . (8)The approximation exactly takes into account the boundary conditions that is distinguishedfrom the Born approach in quantum mechanics. It is to emphasize, that the shape of thecontour γ can be arbitrary; the circular cylinder (considered in the next section) is, basically,just the simplest example. The dielectric region D − could be inconnected; in that case thecontour γ is a sum of all the boundaries of dielectric domains. III. SCATTERING BY TWO CYLINDERS
Let us now consider two cylinders, see Fig. 2. There are three domains with differentdielectric permittivity. The Helmholtz equation (2) is valid for k = ω/c , k = √ εω/c or k p = √ ε p ω/c , and the boundary conditions (5) at the contour γ ∪ γ p is: H in (cid:12)(cid:12)(cid:12) γ = H out (cid:12)(cid:12)(cid:12) γ , ε ∂ H in ∂r (cid:12)(cid:12)(cid:12) γ = ∂ H out ∂r (cid:12)(cid:12)(cid:12) γ , H p (cid:12)(cid:12)(cid:12) γ p = H out (cid:12)(cid:12)(cid:12) γ p , ε p ∂ H p ∂ ˜ r (cid:12)(cid:12)(cid:12) γ p = ∂ H out ∂ ˜ r (cid:12)(cid:12)(cid:12) γ p . (9)Here, we treat the second cylinder as the perturbation. Let us obtain a number of suc-cessive approximations for the whole complicated configuration, shown in Fig. 2. We exploitthe Green function for cylindric geometry given by Eq. (A11). Using this Green function5 H k xyε, a ˜ x ˜ y ε p , b Rr ˜ rr (cid:48) ˜ r (cid:48) α FIG. 2. The scheme of p -wave scattering by two parallel cylinders. The dielectric permittivity andthe radius are indicated in the first (left) and second (right) cylinders. External infinitely remotecontour Γ is not shown. makes it possible to account for the first cylinder exactly including the multiple scattering.The second cylinder is described approximately in terms of the Born series. To found thenumber of terms, that would be sufficient to get the field with given accuracy, we compare itwith a known well-studied numerical solutions such as discrete dipole approximation (DDA)and boundary element methods (BEM).The coupled boundary integral equations are analogous to Eqs. (6). While the pertur-bation remains weak, the expansion (7) yields H ( j ) p ( ˜r ) = (cid:90) γ p n (cid:104) H ( j ) out ( r (cid:48) ) ∇ G p ( ˜r , ˜r (cid:48) ) − ε p G p ( ˜r , ˜r (cid:48) ) ∇H ( j ) out ( r (cid:48) ) (cid:105) d ˜ l (cid:48) , H ( j +1) out ( r ) = − (cid:90) γ p n (cid:104) H ( j ) p ( ˜r (cid:48) ) ∇ G ( r , r (cid:48) ) − ε p G ( r , r (cid:48) ) ∇H ( j ) p ( ˜r (cid:48) ) (cid:105) d ˜ l (cid:48) , (10)where n = n γ p , ˜ r = r − R . The recurrence relations (10) are valid for arbitrary shape ofthe perturber with a sharp boundary, provided its layout is in the external region of themain cylinder. Further generalization for arbitrary shape of the first cylinder requires otherGreen function.The integral over boundary of perturber can be calculated. The final relation is a series6ith a shift due to the axes offset: H ( j ) p ( ˜r ) = ∞ (cid:88) m = −∞ e im ˜ ϕ J m ( k p ˜ r ) D m ( j ) p , H ( j ) out ( r ) = ∞ (cid:88) m = −∞ e imϕ H m ( k r ) D m ( j ) out + ∞ (cid:88) m = −∞ e im ˜ ϕ H m ( k ˜ r ) ˜ D m ( j ) out . (11)Coefficients D are given by relations: D m (0) p = πk p b i (cid:104) J m ( k b ) H (cid:48) m ( k p b ) − √ ε p J (cid:48) m ( k b ) H m ( k p b ) (cid:105) ×× (cid:104) i m e ik R cos α + ∞ (cid:88) n = −∞ i n e i ( n − m ) α C n H n − m ( k R ) (cid:105) ,D m ( j ) p = πk p b i (cid:16) ˜ D m ( j ) out (cid:104) H m ( k b ) H (cid:48) m ( k p b ) − √ ε p H (cid:48) m ( k b ) H m ( k p b ) (cid:105) ++ ∞ (cid:88) n = −∞ D n ( j ) out e i ( n − m ) α H n − m ( k R ) (cid:104) J m ( k b ) H (cid:48) m ( k p b ) − √ ε p J (cid:48) m ( k b ) H m ( k p b ) (cid:105)(cid:17) ; D m ( j +1) out = − πk b i C m ∞ (cid:88) n = −∞ D n ( j ) p e i ( n − m ) α H m − n ( k R ) (cid:104) J n ( k p b ) J (cid:48) n ( k b ) − √ ε p J (cid:48) n ( k p b ) J n ( k b ) (cid:105) , ˜ D m ( j +1) out = − πk b i D m ( j ) p (cid:104) J m ( k p b ) J (cid:48) m ( k b ) − √ ε p J (cid:48) m ( k p b ) J m ( k b ) (cid:105) , where coefficients C m are given by Eq. (A14).Fig. 3 shows the angular dependence of scattered field square | H sc | . As the figure demon-strates, the first approximation gives rather correct qualitative description of the diagramwith a deviation of 15%. The error of the second order is nearly 3%. Fig. 4 shows the com-parison of 3-rd Born approximation with numerical calculations by BEM and DDA. Thedeviation for 3-rd order appears to be about 1%. IV. CONCLUSIONS
The Green function for a dielectric cylinder is found in the cases of p - and s -wave withsource points inside and outside the cylinder. High-order Born approximation of two di-electrics with step boundaries are reduced to recurrence relations. This technique is analyt-ically applied to the scattering by a pair of cylinders. The first approximation demonstratesits qualitative agreement in shape with numerical results. The second and third approxima-tions are shown to agree quantitatively with calculation by boundary elements and discretedipoles. 7 x - - y FIG. 3. Polar diagram of scattering by a pair of equal dielectric cylinders at a = b = 0 . µ m, ε p = ε = 2 . R = 0 . µ m at incidence angle α = − π/
4, the wavelength λ = 1 . µ m, the distancebetween observation point and center of first cylinder is r = 2 λ : first Born approximation (dotted),second (dashed), and BEM (solid line). x y FIG. 4. Polar diagram of the scattered field at the same parameters as in Fig. 3: the 3-rd Bornapproximation (dotted), BEM (solid), DDA (dashed line). The inset is the magnified part of themain plot, indicated by a square.
Appendix A: Scalar Green function
Let us consider a cylinder, which axis is along z direction, as shown in Fig. 5. Weare looking for scalar Green function G ( r , r (cid:48) ) that is the solution to the inhomogeneous2-dimensional Helmholtz equation (2) with k = k in free space and k = k in the dielectric.8 y E H k ( x, y ) FIG. 5. The xy -plane cross section of an infinite cylinder in homogeneous space. A circle indicatesthe observation point r = ( x, y ). The polarization of p -wave is shown at the left. Rewrite delta function (2) in polar coordinates δ ( r − r (cid:48) ) = 1 r δ ( r − r (cid:48) ) δ ( ϕ − ϕ (cid:48) ) , (A1)where r, ϕ r (cid:48) , ϕ (cid:48) are the polar coordinates of source and observation points and decomposethe angular factor into the Fourier series: δ ( ϕ − ϕ (cid:48) ) = 12 π ∞ (cid:88) m = −∞ e im ( ϕ − ϕ (cid:48) ) . (A2)The coefficient 1 / π is found from the delta-function normalization (cid:82) π − π δ ( ϕ ) dϕ = 1 . Expanding the Green function in partial waves G ( r, ϕ ; r (cid:48) , ϕ (cid:48) ) = ∞ (cid:88) m = −∞ g m ( r, r (cid:48) ) e im ( ϕ − ϕ (cid:48) ) (A3)and substituting into (2) we get an ordinary equation for each m : d g m dr + 1 r dg m dr + (cid:18) k − m r (cid:19) g m = 12 πr δ ( r − r (cid:48) ) . (A4)At r > r (cid:48) or r < r (cid:48) , the corresponding solutions can be expressed through the combinationsof Bessel and Hankel functions: g m ( r, r (cid:48) ) = a m J m ( k r ) , r < a,A m J m ( kr ) + B m H (1) m ( kr ) , a < r < r (cid:48) ,d m H (1) m ( kr ) , r (cid:48) < r. (A5)9onditions (5) are the continuity of the magnetic field’s and its weighted normal derivativeat the interface between dielectric and free space, r = a :[ g m ] r = a = 0 , (cid:20) ε dg m dr (cid:21) r = a = 0 , (A6)where the square bracket denotes a jump of the corresponding value. The conditions arewritten for p -wave, where the magnetic field is parallel to the z -axis. Next pair of conditionsfollow from the continuity of Green function and the jump of its first derivative at r = r (cid:48) :[ g m ] r = r (cid:48) = 0 , (cid:20) dg m dr (cid:21) r = r (cid:48) = 12 πr (cid:48) . (A7)We omit Hankel function in the first line and Bessel function in the third line in Eq. (A5)on a basis of regularity at r → r → ∞ .Substituting (A5) into boundary conditions (A6), (A7) we get the set for the coefficients: a m J m ( k a ) = A m J m ( ka ) + B m H (1) m ( ka ) , √ ε a m J (cid:48) m ( k a ) = A m J (cid:48) m ( ka ) + B m H (1) (cid:48) m ( ka ) , (A8) A m J m ( kr (cid:48) ) + B m H (1) m ( kr (cid:48) ) − d m H (1) m ( kr (cid:48) ) = 0 ,A m J (cid:48) m ( kr (cid:48) ) + B m H (1) (cid:48) m ( kr (cid:48) ) − d m H (1) (cid:48) m ( kr (cid:48) ) = − πkr (cid:48) . (A9)Here the prime means the derivative of cylindric functions with respect to their arguments.From (A8) we get B m = α m A m , where α m = J m ( ka ) J (cid:48) m ( k a ) − √ εJ (cid:48) m ( ka ) J m ( k a ) √ εH (1) (cid:48) m ( ka ) J m ( k a ) − H (1) m ( ka ) J (cid:48) m ( k a ) . (A10)Then the determinant of the set (A9) for coefficients A m , d m is { J m + α m H (1) m , H (1) m } =2 i/πkr (cid:48) , where the curly bracket stands for the Wronskian determinant { f, g } = f g (cid:48) − f (cid:48) g at r = r (cid:48) . The final form of (A5) is g m ( r, r (cid:48) ) = i (cid:104) J m ( ka ) + α m H (1) m ( ka ) (cid:105) H (1) m ( kr (cid:48) ) J m ( k r ) J m ( k a ) , r < a, i (cid:104) J m ( kr ) + α m H (1) m ( kr ) (cid:105) H (1) m ( kr (cid:48) ) , a < r < r (cid:48) , i (cid:104) J m ( kr (cid:48) ) + α m H (1) m ( kr (cid:48) ) (cid:105) H (1) m ( kr ) , r (cid:48) < r. (A11)The Green function for s -wave can be obtained in a similar way, replacing the boundaryconditions by the continuity of function g m together with its first derivative at r = a insteadof Eq. (A6). Besides, the results would differ when the source is outside the dielectric.10et us summarize the formulas for partial Green function. At r (cid:48) < a they are: g m ( r, r (cid:48) ) = i J m ( k r ) (cid:0) C m J m ( k r (cid:48) ) + H m ( k r (cid:48) ) (cid:1) , < r < r (cid:48) , i J m ( k r (cid:48) ) (cid:0) C m J m ( k r ) + H m ( k r ) (cid:1) , r (cid:48) < r < a,β m H m ( k r ) J m ( k r (cid:48) ) , a < r ; (A12) C m = − H m ( k a ) H (cid:48) m ( k a ) − ε ν H (cid:48) m ( k a ) H m ( k a )∆ , β m = ε ν πk a ∆ ;At r (cid:48) > a the formulas are: g m ( r, r (cid:48) ) = β m J m ( k r ) H m ( k r (cid:48) ) , r < a, i H m ( k r (cid:48) ) (cid:0) J m ( k r ) + C m H m ( k r ) (cid:1) , a < r < r (cid:48) , i H m ( k r ) (cid:0) J m ( k r (cid:48) ) + C m H m ( k r (cid:48) ) (cid:1) , a < r (cid:48) < r ; (A13) C m = − J m ( k a ) J (cid:48) m ( k a ) − ε ν J (cid:48) m ( k a ) J m ( k a )∆ , β m = 12 πk a ∆ ; (A14)∆ = J m ( k a ) H (cid:48) m ( k a ) − ε ν J (cid:48) m ( k a ) H m ( k a ) . Here the upper index (1) of Hankel function is omitted. The formulas with ν = − / , / p or s wave, respectively. Expressions for p - or s -wave differ in the factor ε − / or ε / due to distinct boundary conditions. The Eq. (A11) reduces to the particularcase of (A13) with ν = − / . Appendix B: DDA
Below we briefly recall 2-dimensional DDA approach to obtain here the particular re-lationships we used in our calculations. Let us have some scattering body, with the volume V (which is per unit length along z direction in 2-dimensional case) and the permittivity ε (which is constant within the body), placed in vacuum. From the Helmholtz equation weobtain the integral equation for isotropic medium: E ( r ) = E inc ( r ) + (cid:90) V \ V d r (cid:48) (cid:104) (cid:98) G ( r , r (cid:48) ) χ ( r (cid:48) ) E ( r (cid:48) ) (cid:105) + (cid:90) V d r (cid:48) (cid:104) (cid:98) G ( r , r (cid:48) ) χ ( r (cid:48) ) E ( r (cid:48) ) (cid:105) , (B1)where V is a small volume around singularity point R = r − r (cid:48) → V \ V is the volumeof dielectric without the singular part, E inc ( r ) is the given field of incident wave, χ ( r ) ≡ ( ε − / π is the polarizability, the Green tensor (cid:98) G ( r , r (cid:48) ) is the solution to Maxwell equations:rot rot (cid:98) G − k (cid:98) G = 4 πk δ ( r − r (cid:48) ) . (B2)11he Green tensor obeying (B2) can be expressed in terms of scalar Green function g that satisfies Eq. (2) G αβ = 4 π (cid:0) k δ αβ + ∇ α ∇ β (cid:1) g, (B3)where α , β — are Cartesian indices. Then, it is well known that the Green tensor actuallydepends on the difference R . Finally, in 2-dimensional case we have G αβ ( R ) = iπkR (cid:20) A ( kR ) δ αβ − B ( kR ) R α R β R (cid:21) , (B4) A ( x ) = xH (1)0 ( x ) − H (1)1 ( x ) , B ( x ) = xH (1)0 ( x ) − H (1)1 ( x ) , where H (1)0 ( x ), H (1)1 ( x ) are Hankel functions of the first kind.In (B1) we implicitly isolate the term, that includes the singularity of the Green tensorat r = r (cid:48) , by means of a small volume V , for which the point r is internal. Then, we rewritethis term, introducing the following quantities: (cid:99) M ( V , r ) = (cid:90) V d r (cid:48) (cid:20) G αβ ( r − r (cid:48) ) − R α R β − δ αβ R R (cid:21) χ ( r (cid:48) ) E ( r (cid:48) ) (B5)and (cid:98) L ( V , r ) = − (cid:90) V d r (cid:48) R α R β − δ αβ R R χ ( r (cid:48) ) E ( r (cid:48) ) , R ≡ r − r (cid:48) . (B6)Note that (cid:99) M is free from the singularity, thus (cid:99) M → V →
0. The fraction underintegration is, basically, the static limit (at k →
0) of the Green tensor. Also, we need todiscretize the whole scattering volume V into the parts V j (in such a way that V coincideswith one of them). With the use of (B5) and (B6) the equation (B1) becomes: E ( r i ) = E inc ( r i ) + (cid:88) j (cid:54) = i (cid:90) V j d r (cid:48) (cid:104) (cid:98) G ( r i − r (cid:48) ) χ ( r (cid:48) ) E ( r (cid:48) ) (cid:105) + (cid:99) M ( V i , r i ) − (cid:98) L ( V i , r i ) , (B7)where r i denotes a point lying inside the volume V i .Up to this line, the equations are fully correct as being exact consequences of the initialwave equation. Now we make two approximations: the first is that E ( r (cid:48) ) and χ ( r (cid:48) ) areconstant within the volume V j ; the second approximation assumes that1 V j (cid:90) V j d r (cid:48) (cid:98) G ( r i , r (cid:48) ) = (cid:98) G ( r i , r j ) . (B8)The condition (B8) is intrinsically contained in all DDA formulations, which initially dealwith replacing the scatterer with a set of point dipoles. If the volumes V i are square cells12we should keep in mind that we are treating 2-dimensional case) then we can place thepoints r i to the center of the corresponding squares.Below, we will neglect (cid:99) M , as most of authors do, choosing by that the simpler (or ”weak”)DDA formulation. Integrating (B6) we transform (B7) into its final form d i (cid:98) α − i = E i,inc + (cid:88) j (cid:54) = i (cid:98) G ( r i − r j ) d j , (B9)where we denote, for simplicity, the dependence on r i (and r j ) by the corresponding sub-script; d i = V i χ i E i — the polarization of the volume V i (basically, its dipole moment, as wetook χ i and E i being constant within V i ); and (cid:98) α i is the polarizability tensor defined as (cid:98) α i = (cid:98) I V i χ i (1 + 2 πχ i ) − ≡ a ε − ε + 1 (cid:98) I . (B10)The last term is the known quasi-static dipole polarizability of a cylinder (2-dimensionaldipole) with the cross section, πa , equal to V i .Thus, the calculations consisted in finding the dipole moments d i by solving (B9) with(B10) and (B4). Upon them, all the quantities of interest can be obtained. In our case, wecalculate the scattered magnetic field. ACKNOWLEDGEMENTS
Authors are grateful to O. V. Belai for helpful discussions. This work is supported bythe Russian Foundation of Basic Research
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