Optical properties of cylindrical nanowires

Chapter 2.Small dielectric cylindersWith condition (2.1) the following expressions for the Bessel and firstHankel functions can be used:J 0 (z) ≃ 1 − z24 , J ′ 0(z) ≃ − z 2 ,J 1 (z) ≃ −J 0 ′ (z) , J 1(z) ′ ≃ 1 2 − 316 z2 ,H 0 (z) ≃ 1 + 2iπ γ + 2iπ log z 2 , H′ 0(z) ≃ 2i 1π z ,H 1 (z) ≃ −H 0 ′ (z) , H′ 1(z) ≃ 2i 1π z 2 , (2.3)where | z | ≪ 1, H denotes the first Hankel function and γ is Euler’s constant.The Hankel functions in 2.3 are expanded to zeroth order in z, because thisis sufficient for a second order approximation of the scattering coefficients.With these expansions it can be easily shown that the scattering coefficients(1.41) and (1.42) up to second order in the size parameter x are approximatedby:Case Ia 0I (x, θ) = 0,a 1I (x, θ) = πx24(m 2 − 1)(m 2 + 1) sin θ + O(x4 ),b 0I (x, θ) = − iπx24 (m2 − 1) cos 2 θ + O(x 4 ),b 1I (x, θ) = − iπx24Case IIa 0II (R, θ) = O(x 4 ),a 1II (R, θ) = − iπx2 (m 2 − 1)4 (m 2 + 1) + O(x4 ),b 0II (R, θ) = 0,b 1II (R, θ) = − πx24(m 2 − 1)(m 2 + 1) sin2 θ + O(x 4 ), (2.4)(m 2 − 1)(m 2 + 1) sin θ + O(x4 ), (2.5)This result is in agreement with the earlier work of Wait [9], [10] andalso gives the expressions derived by Van de Hulst [7] and Kerker [8] fornormal incidence (θ = 0) . The internal coefficients for the fields inside thewire, c nI , d nI , c nII and d nII are not explicitly shown here because they havea complex form. They follow directly from equations (1.41) and (1.42).In principle now it is possible to proceed further and use equations (2.4) and(2.5) for the approximation of the fields (equations (1.21)-(1.26)). However,it is really important to be careful, because an expansion of the fields tosecond order in the size parameter needs more and further expanded coefficientsthen showed above.24