Optical properties of cylindrical nanowires

Chapter 1.General solutionon the basis of cylindrical unit vectors ê ρ , ê φ , ê z . Consequently, with u n =A n ϕ n and v n = B n ϕ n for certain A n , B n and taking the sum over all n, thecomponents of E and H areE ρ =H ρ =∞∑n=−∞∞∑n=−∞n=−∞inρ v n −g ∂u nmk 0 ∂ρ , (1.21)inmρu n + g ∂v nk 0 ∂ρ , (1.22)normal to the cylinder surface and∞∑E φ = − ∂v n∂ρ − ingmk 0 ρ u n, (1.23)E z =H φ =H z =∞∑n=−∞∞∑n=−∞∞∑n=−∞tangential to the cylinder surface.1.3 Scattering problemi(m 2 k 2 0 − g2 )mk 0u n , (1.24)−m ∂u n∂ρ + ingk 0 ρ v n, (1.25)− i(m2 k 2 0 − g2 )k 0v n (1.26)With the above formal solution it is now possible to solve the general scatteringproblem of an arbitrary polarized plane electromagnetic wave incidentobliquely on a circular cylinder of infinite length.For oblique incidence the direction of propagation of the incident wave makesan angle θ with the normal to the z-axis, see Figure 1.1. Furthermorethe cylinder is assumed to be surrounded by vacuum and the refractive indexof the cylinder is equal to m. In case of a surrounding homogeneousmedium with refractive index m 1 the solutions are of the same form if mis considered as the refractive index of the cylinder relative to the medium:m = m 2m 1. With the above definitions the incident wave, depicted inFigure 1.1 is represented by the scalar wave functionwhere12ψ 0 = Ẽ0 e −i(k 0x cos θ+k 0 z sin θ+ωt)= Ẽ0 e −i(hz+ωt) ∞ ∑n=−∞(−i) n J n (lρ)e ınφ , (1.27)h ≡ k 0 sin θ, (1.28)√l ≡ k 0 cos θ = k0 2 − h2 . (1.29)