ISSN: 0256-307X 中国物理快报 Chinese Physics Letters Volume 28 Number 5 May 2011 A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/cpl http://cpl.iphy.ac.cn C HINESE P HYSICAL S OCIETY Institute of Physics PUBLISHING CHIN. PHYS. LETT. Vol. 28, No. 5 (2011) 057303 A Trade-off between Propagation Length and Light Confinement in Cylindrical Metal-Dielectric Waveguides * SUN Bao-Qing(孙宝清), GU Ying(古=)**, HU Xiao-Yong(胡小[), GONG Qi-Huang(龚á煌)** State Key Laboratory for Mesoscopic Physics, Department of Physics, Peking University, Beijing 100871 (Received 17 April 2010) We theoretically investigate the hybrid plasmonic modes of cylindrical nanocables with gold nanocore and two dielectric nanolayers (SiO2 and BN). By solving a complete set of Maxwell’s equations, the propagation constants and effective radii depending on geometrical parameters are numerically calculated. By declining atrade-off between propagation length and light confinement, high quality hybrid modes which can travel a long range of 120–200휆 with a subwavelength effective radius are obtained at the optical wavelength. These modesin one-dimensional cylindrical waveguides should have potential applications in nanoscale optical device designs. PACS: 73.20.Mf, 78.67.−n DOI: 10.1088/0256-307X/28/5/057303 Surface plasmon polaritons (SPPs) are light waves the propagating and evanescent modes and generally coupled to free electron oscillations in metal-dielectric the evanescent part dominated. Due to the Ohmic interfaces. These waves are evanescent near surfaces, loss, the propagation lengths of these nanoscale 1D which makes it possible to localize and guide light in metallic waveguides were decreased to a small range, a subwavelength scale.[1;2] To guide SPPs, researchers which greatly limits the applications in nanoscale opti- have explored many plasmonic waveguide structures cal devices. The aim of this work is to find hybrid SPP such as thin metal films,[3−5] metal stripes,[6−8] two di- modes with a long range propagation length and good mensional metal-dielectric multilayers,[9;10] subwave- light confinement in 1D metallic-dielectric waveguides. length metal grooves,[11−13] nanoparticle chains,[14] In this work, a hybrid plasmonic waveguide com- and cylindrical waveguides.[15−20] Although the op- posed of a gold nanocore and two dielectric nanolayers tical energy is well confined, the propagation length is proposed, as shown in Fig. 1. The inner dielectric will be greatly reduced because of the Ohmic loss in layer SiO2 has a low dielectric constant and the outer metals. In fact, there is a trade-off between light con- dielectric layer BN has a high dielectric constant. The finement and propagation length in all kinds ofSPP whole system can be regarded as a gold nanocore nest- waveguides. To solve this problem, a hybrid waveg- ing in a BN nanotube, so the SPP modes in a metallic uide combining the advantages of both SPP modes nanowire can couple to the waveguide modes in a nan- and dielectric waveguide modes is suggested. With otube, leading to some hybrid SPP modes. By adjust- the coupling of plasmonic and dielectric waveguides, ing thicknesses of the layers, the high quality hybrid hybrid modes can be improved to achieve a long prop- modes which can travel a long range of 120–200휆 and agating length while keeping moderate light confine- have a subwavelength effective radius can be achieved ment in a ‘capacitor-like’ structure.[21;22] at the optical wavelength. These modes in 1D cylin- SPPs in one-dimensional (1D) metallic cylindri- drical waveguides should have potential applications cal waveguides have been extensively studied. In the in nanoscale optical device designs. 1970s, the first experiment using light scattering to The solution of Maxwell’s equations in a cylindri- excite SPP modes in 50 µm metallic cylinders was cal structure is well documented in the textbook.[23] performed[15] and also explained by means of the vir- Electromagnetic fields of the eigenmodes in the cylin- tual radiative modes with a complex frequency.[16] drical coordinates (r; 휃; z) can be expressed as Then, a clear mode classification in two-layer cylin- drical waveguide structures was thoroughly investi- hikz j 0 휇j!n j i gated where only negative real dielectric constant was Er = fn (훾jr) − 2 gn(훾jr) Sn; 훾j 훾 rc considered.[17] The SPP modes in a lossy cylindrical j h i metallic nanotube with a dielectric core were used to nkz j 푖휇j! j 0 E휑 = − 2 fn(훾jr) − 2 gn (훾jr) Sn; model light propagation of a scanning near field op- 훾j r 훾j c tical microscope (SNOM).[18;19] Mode coupling and j Ez = fn(훾jr)Sn; transformation of SPPs in a lossy metallic cylindri- hn" ! ik i cal nanocable were also studied.[20] However, in the H = j f j (훾 r) + z gj 0(훾 r) S ; r 훾2rc n j 훾 n j n above-mentioned works (Refs. [18–20]), instead of a j j clear distinction between pure propagating modes and hi"j! j 0 nkz j i H휑 = fn (훾jr) − 2 gn(훾jr) Sn; evanescent modes, the SPP modes mediated between 훾jc 훾j r *Supported by the National Natural Science Foundation of China under Grant Nos 10874004 and 10821062 and the National Basic Research Program of China under Grant No 2007CB307001. **To whom correspondence should be addressed. Email: [email protected]; [email protected] © 2011 Chinese Physical Society and IOP Publishing Ltd 057303-1 CHIN. PHYS. LETT. Vol. 28, No. 5 (2011) 057303 j Hz = gn(훾jr)Sn; (1) good confinement of Reff = 100 nm (the calculations related to this part are not shown). To overcome this where i is the imaginary unit, n is an integer (n = 0, dilemma that the SPP mode in nanowire has either a 1, 2, ···), kz is the propagation constant along the z long range with bad confinement or good confinement direction (the cylinder axis) and here it is complex due with a short traveling length, we set the gold nanowire to the loss of metal, "j is the dielectric constant of the in a high-permittivity dielectric nanotube, forming a medium j and c is the speed of light in vacuum. As we metallic-dielectric nanocable as shown in Fig. 1. The are not dealing with magnetic materials, the magnetic waveguide modes of the nanotube will pull the elec- permeability 휇 is set as unity. Sn is the exponential tromagnetic energy of the SPP mode of the nanowire factor, into the low-permittivity middle layer, leading to some Sn = exp(푖푛휑 + ikzz − i!t); (2) hybrid SPP modes. Then, by carefully choosing the 훾j is the transverse wave number in the jth medium, geometric parameters, compared to the SPP modes in nanowires, higher quality hybrid modes with both 2 2 2 훾j = 휇j"j(!=c) − kz ; (3) long propagation lengths and small effective radii are achieved. and f j , gj are the superposition of Bessel functions. n n 2.8 2.7 ) 2.0 Fig. 1. The schematic diagram of a hybrid cylindrical Re( metal-dielectric waveguide. 1.5 (a) By applying the boundary conditions that the tan- 1.0 gential components of E and bmH must be continu- 0.20 ous at the interface, a homogeneous system of linear HE Nanowire in SiO 0.16 11 2 j j HE Nanowire in BN ) equations for the expansion coefficients of fn and gn is 12 HE 13 0.12 EH obtained. The existence of the nontrivial solutions is 11 guaranteed by letting the determinant of the system 0.08 Im( vanish. Then we obtain the transcendental equation 0.04 (b) det[M(kz)] = 0 for the complex propagation constant [20] 0.00 kz. Using a numerical method, we can solve the k 100 200 300 400 value of z for a specific mode. With the propaga- (nm) tion constant, all the information of this mode can be obtained, including the EM fields, propagation length Fig. 2. Dispersion relations of nanocable versus the thick- ness d1 of the SiO2 middle layer. Here the gold nanocore and Poynting vector. is in size r0 = 40 nm and the BN outer layer is in thickness Propagation length is defined as the length Lm = d2 = 40 nm. (2Im(k ))−1 z after which the intensity of SPPs de- creases to 1=e.[1] Light confinement in the radial direc- 1.0 HE mode EH mode 11 11 tion is described by the effective radius Reff , defined 6 0.5 as a hypothetic radius inside which 86.5% of the power 3 0.0 0 is confined, R Reff S dA -0.5 z -3 0 R 1 = 86:5%; (4) -6 SzdA -1.0 0 HE mode S HE mode 9 z 12 where is the Poynting vector in the axis direction 4 [25] 13 Sz = (Er * H휑 − E휑 * Hr). 3 In a gold nanowire embedded in a dielectric host, 6 2 3 Electromagnetic fields low-permittivity dielectric tends to guide a long-range 1 SPP mode but with a large effective radius, while 0 0 high-permittivity dielectric tends to channel a short -3 -1 SPP mode but with a small effective radius. As an 0 200 400 600 0 200 400 600 (nm) example, we consider the n = 1 mode in the gold (nm) nanowire of r0 = 40 nm. The gold dielectric permit- Fig. 3. Field distribution at r0 = 40 nm, d1 = 260 nm, tivity of "m = −11:75 + 1:25i at the wavelength of d2 = 40 nm. (a) HE11 mode, (b) HE12 mode, (c) EH11 휆 = 632:8 nm is taken from Ref. [24]. With infinite mode, (d) HE13 mode.