Optical properties of cylindrical nanowires

Chapter 1.General solutionwhereC n ≡ (m2 − 1) 2 n 2 tan 2 θj 2 R 2 , (1.43)D n ≡ cos θ J ′ n(jR)J n (jR) , (1.44)K n ≡ J n(lR)′ ′J n (lR) , L n ≡ H(1) n (lR)H n (1) (lR) , (1.45)′M n ≡ H(1) n (lR)J n (lR) , N n ≡ H(1) n (lR)J n (lR) , (1.46)′O n ≡ H(1) n (lR)J n(lR)′(1.47)and the functions l (1.29) and j (1.39) depend on θ. Despite of the differentform, equations (1.30)-(1.37) are the same as derived by Bohren [11]. Theygive the complete, formal solution for the scattering problem of a planeelectromagnetic wave incident obliquely on a circular cylinder of infinitelength. In principle the electromagnetic fields and the intensities can beobtained by calculating the full expansion of (1.30)-(1.33) for Case I ((1.34)-(1.37) for Case II) and subsequently use these expressions to calculate thefields (1.21-1.26). However, in practice it is impossible to get an exactanalytic solution and a numerical procedure is the only way to solve thefull problem.In the special case of a normal incident wave (θ = 0), the scatteringcoefficients (1.30)-(1.37) reduce to:Case I16a nI (R, 0) = 0,b nI (R, 0) =mJ n (k 0 R)J n(mk ′ 0 R) − J n(k ′ 0 R)J n (mk 0 R)mH n (1) (k 0 R)J n(mk ′ 0 R) − H (1) ′n (k 0 R)J n (mk 0 R) ,c nI (R, 0) = 0,d nI (R, 0) =Case IIH (1) ′n (k 0 R)J n (k 0 R) − H n (1) (k 0 R)J n(k ′ 0 R)mH (1) ′n (k 0 R)J n (mk 0 R) − m 2 H n (1) (k 0 R)J n(mk ′ 0 R) (1.48) ,a nII (R, 0) =J n (k 0 R)J n(mk ′ 0 R) − mJ n(k ′ 0 R)J n (mk 0 R)H n (1) (k 0 R)J n(mk ′ 0 R) − mH (1) ′n (k 0 R)J n (mk 0 R) ,b nII (R, 0) = 0,H (1) ′n (k 0 R)J n (k 0 R) − H n (1) (k 0 R)J ′c nII (R, 0) =n(k 0 R)m 2 H (1) ′n (k 0 R)J n (mk 0 R) − mH n (1) (k 0 R)J n(mk ′ 0 R) ,d nII (R, 0) = 0. (1.49)