Optical properties of cylindrical nanowires

Chapter 2.Small dielectric cylindersthe starting point in this paper, see Figure 2.2. For six fixed values of λ 0 , sofor six different values of ɛ, the extinction efficiencies in Case I and Case IIup to third order in x at normal incidence are plotted as a function of x.QIext1.41.210.80.60.4608 nm508 nm422 nm395 nm376 nm354 nma:QIext,Θ ⩵ 00.200 0.01 0.02 0.03 0.04 0.05xQIIext0.0150.010.005608 nm508 nm422 nm395 nm376 nm354 nmb:QIIext,Θ ⩵ 000 0.01 0.02 0.03 0.04 0.05xFigure 2.2: Extinction efficiencies Q ext I and Q ext II at normal incidenceas a function of the size parameter x = kR. In every plot k, and so ɛ,is fixed. The corresponding values for the wavelength are showed in theleft corner. The efficiencies are expanded up to third order in x.In the illustrated domain the linear terms (2.19) and (2.20) in the extinctionefficiencies dominate and the total removal of EM energy from theincident beam increases with the size parameter.The slope of this relation depends on the wavelength and in a quite differentway for the two cases. It is explained by looking closely to the dependenceon the complex dielectric function:• In Case I the slope increases to a maximum around λ 0 = 400 nmafter which it decreases for increasing wavelength. This is caused bythe factor Im{m 2 − 1} = ɛ ′′ in (2.19), see Figure 2.1.• In Case II the slope decreases in the whole domain for increasing λ 0 .The increase of the denominator of Im{m2 −1} ɛ∝ ′′in (2.20)|m 2 +1| 2 (ɛ ′ ) 2 +(ɛ ′′ ) 2dominates the increase of the numerator to ∼ 400 nm. Afterwardsɛ ′′ falls off so fast that the slope remains decreasing for increasingwavelength.It even seems from the figure that the third order terms can be neglectedin the given domain. Nevertheless, a closer look gives the opposite: forincreasing x the third order terms start to give significant corrections. Thisis illustrated in Figure 2.3: here the correction by the third order terms withrespect to (2.19) and (2.20) are shown in percentages.The figure reveals that also the magnitude of the deviation depends onthe wavelength: in both cases the influence of the third order correctionbecomes larger for increasing wavelength. This is explained with the same32