Extreme light confinement and low loss in triangle hybrid plasmonic waveguide

Optics Communications 319 (2014) 141–146

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Extreme light confinement and low loss in triangle hybrid plasmonic waveguide Qijing Lu a, Chang-Ling Zou b, Daru Chen a, Pei Zhou a, Genzhu Wu a,n a b

Institute of Information Optics, Zhejiang Normal University, Jinhua 321004, China Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China

art ic l e i nf o

a b s t r a c t

Article history: Received 5 October 2013 Received in revised form 21 December 2013 Accepted 24 December 2013 Available online 9 January 2014

A low-loss triangle hybrid plasmonic waveguide to confine light at an ultra-deep subwavelength scale is proposed and numerically investigated. Compared to other hybrid slot plasmonic waveguides based on cylinder or square semiconductor nanowires, the novel hybrid plasmonic waveguide based on triangle semiconductor nanowire has not only stronger field confinement, but also lower propagation loss. Detailed study of this structure reveals that these advantages originate from the tip enhancement of the triangle semiconductor waveguide. This mechanism of the waveguide permits tolerance for structural imperfection in actual experiments, which is very feasible for experimental realization. The extreme confinement of light can lead to strong electric field around the tip of the triangle semiconductor waveguide, thus can greatly enhance the light-matter interaction. Various applications will benefit from this triangle hybrid plasmonic waveguide, such as the laser, waveguide (cavity) quantum electrodynamics and optomechanics. & 2014 Elsevier B.V. All rights reserved.

Keywords: Plasmon waveguide Photonic integrated circuits Surface plasmon polariton

1. Introduction Surface plasmon-polariton (SPP) [1–3] is attracting more and more attentions due to that it offers the opportunity to confine and guide light beyond the diffraction limit. Thus, SPP-based devices have been studied extensively [4–7] and are regarded as the suitable candidates for guiding light in nano-photonic integrated circuits (PICs) [8,9]. Among them, various plasmonic waveguide structures have been proposed, such as self-assembled metallic nanowire [10], metal–insulator–metal [11,12], V-groove channel [13–15], and metal wedge [16–19]. However, there is a trade-off relation between the propagation loss and field confinement for these plasmonic waveguides, i.e., a deep subwavelength confinement of light is usually accompanied with a very short propagation distance. Hybrid waveguide, such as dielectric-loaded SPP waveguide (DLSPPW) [20–24], possesses longer propagation distance, but with field less confinement. Recently, a new kind of hybrid waveguides with slot (gap) structures [25–35] (in this paper, this particular type of structure is denoted as hybrid plasmonic waveguide, HPW) which show relatively longer propagation distance and strong field confinement have been proposed to improve the trade-off relation between the light confinement and propagation distance. The HPW structure usually consists of

n

Corresponding author. E-mail address: [email protected] (G. Wu).

0030-4018/$ - see front matter & 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.12.072

a low-index dielectric nanoscale gap which separates the metal layer and high-index dielectric waveguide. The nanoscale gap shows the capacitor-like energy storage ability due to the discontinuity at the high-low index dielectric interface and the surface plasmon polaritons (SPPs) at the metal–dielectric interface. Various integrated photonic devices based on HPW have been proposed and demonstrated experimentally, including plasmonic nanolaser [36], highly efficient optical modulator [37], polarization beam splitter [38]. In this paper, we propose a new kind of HPW, which consists of a triangle semiconductor waveguide embedded in a low-index dielectric cladding above a silver substrate. Compared with the previously studied HPWs in Refs. [25,26], the new triangle HPW shows stronger field confinement and lower propagation loss. Our analysis revealed that the mechanisms of the extreme confinement of light are the additional lateral confinement and extreme field enhancement around the tip of the triangle semiconductor waveguide. Such triangle semiconductor waveguides can be well prepared in practical experiments [39,40], which indicates that the proposed structure is very potential for ultra-strong light-matter interactions in future. 2. Structure and basics The schematic geometry of proposed triangle HPW is shown in Fig. 1, where a triangle semiconductor waveguide (Si) is separated from the metal (Ag) substrate by a nanoscale low-index dielectric

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Fig. 1. Schematic illustration of the proposed HPW consisting of a triangle semiconductor waveguide on a silver substrate. The origin is defined at the vertex of the triangle semiconductor waveguide.

gap with height of t. The height and vertex angle of the triangular semiconductor wedge are denoted as h and α, respectively. The characteristics of the triangle HPW are investigated at the telecommunication wavelength λ ¼ 1550 nm, and the relative permittivities of the triangle semiconductor waveguide and silver substrate are εt ¼12.25 and εm ¼ 129 þ3.3i [25], respectively. The whole hybrid waveguide is immersed into low-index cladding material (SiO2), with the permittivity εc ¼ 2.25. The triangle HPW is uniform along z-axis, therefore the confined propagation eigenmode can be characterized by the propagation constant k ¼Nhybk0, where k0 ¼2π/λ is the wave number in vacuum. The effective mode index Nbyb and corresponding mode profile in the cross section can be solved numerically by finite element method, with the commercial available software (COMSOL Multiphysics). The eigenmode solver is applied with the scattering boundary condition and a convergence analysis is done to ensure that the numerical boundaries do not interfere the solutions [25]. Since the metal absorbs electromagnetic wave energy, the light decays when propagating along the waveguide. Therefore, Nbyb is a complex number. We can introduce the propagation distance of hybrid mode as Lm ¼ λ=½4π ImðN hyb Þ

ð1Þ

The ability of light confinement in this HPW can be characterized by the mode area: Am ¼

Wm 1 2 ∬ WðrÞd r; ¼ max fWðrÞg max fWðrÞg

ð2Þ

n  2 ðrÞ EðrÞ þ where Wm is the electromagnetic energy and WðrÞ ¼ d½ωε dω   o μ0 HðrÞ2 =2 is the energy density, with E(r), H(r), ε(r), ω, μ0 being the electric field, magnetic field, dielectric permittivity, angular frequency, and vacuum magnetic permeability, respectively. In our realistic model, the tip of the triangle semiconductor waveguide is rounded with radius of curvature r for two reasons: (i) the waveguides fabricated in practical experiment always have a round corner with nonzero radius. (ii) The sharp corner will give rise to field singularity in numerical simulation [41,42].

3. Properties From the field distribution of the hybrid mode in the proposed waveguide structure (Fig. 2(a)), we can see great field enhancement in the gap between the silver film and triangle semiconductor waveguide. Here, the waveguide height h ¼200 nm, the vertex angle α ¼1001, radius of curvature r ¼10 nm and t¼ 2 nm. For a comparison, we also calculated the hybrid modes in the cylinder and square HPWs, as shown in Fig. 2(b) and (c), with the same

Fig. 2. (a)–(c) The field distributions of hybrid modes at the cross section of the triangle, cylinder and square HPWs, respectively. (d) The normalized propagation distance Lm/λ versus the normalized mode area Am/A0 for the hybrid modes in triangle (red line), cylinder (blue line) and square (green line) HPWs when gap t varies from 2 to 40 nm. (e) and (f) Normalized energy densities along x ¼0, y¼ 0 (t ¼2 nm) of the hybrid modes in triangle, cylinder and square HPWs, respectively. The shaded gray, green and wine areas in (e) represent silver, low-index dielectric, and semiconductor regions, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

height of semiconductor waveguide, which have been studied previously in Refs. [25,26] (The corners of the square semiconductor waveguide are all rounded with a 10 nm curvature to avoid

Q. Lu et al. / Optics Communications 319 (2014) 141–146

field singularity). With the diffraction-limited mode area denoted as A0 ¼ λ2/4, the normalized mode area Am/A0 in triangle, cylinder and square HPWs are 0.00274, 0.00585 and 0.0122, respectively. Obviously, the triangle HPW shows the best confinement ability even though its physical area of the semiconductor material is the largest. At the same time, the propagation distances of these waveguides are 46.2, 29.3 and 21.0 μm, respectively. Therefore, the proposed hybrid waveguide shows great advantages of better confinement and lower propagation loss. For a more intuitive comparison, the figure of merit (FOM) by a parametric plot of normalized propagation distance Lm/λ versus normalized mode area Am/A0 of these HPWs is shown in Fig. 2(d). Curves are obtained by varying the gap between silver substrate and semiconductor waveguide t (2–40 nm). This parametric plot allows direct comparison of a variety of waveguides without obscuring the absolute values of field confinement and propagation distance. All HPWs display the trade-off relation between Lm and Am. However, the triangle HPW is superior, as it shows longest propagation distance for confinement of the same degree, or has best confinement for the same propagation loss. Further studies of normalized energy density W(r)A0/Wm along the y-direction and x-direction are shown in Fig. 2(e) and (f). The field intensity of the hybrid triangle mode is more than two times and three times larger than those of the hybrid cylinder and square modes, as a result of smaller mode area. The y-direction (x ¼0) profiles are similar: the maximum field intensity is in the gap and exponentially decays in silver and semiconductor waveguide. As for x-direction, the mode profiles in cylinder and square HPWs are obviously much wider due to that the contact area between the silver and semiconductor waveguide increases, as the direct comparison shown in Fig. 2(f) (y¼0). From Fig. 2(a)–(c), we find that the normalized mode area of the hybrid mode is directly related to interface area between metal and semiconductor dielectric. For the triangle HPW, the interface area between metal and semiconductor changes with wedge angle α. Thus, we studied the properties of the triangle HPW against α. Fig. 3(a) and (b) show the dependences of the normalized mode area (Am/A0), the mode effective index (Nhyb) and the propagation distance (Lm) of the triangle hybrid mode on α. When α increases from 201 to 1601, Nhyb increases and Lm decreases monotonously. However, the behavior of Am/A0 is much different, which decreases first before it increases, leading to an optimized angle α of 1001. When α approaches to 01, the properties of the hybrid mode are similar to those of the pure SPP mode at metal–SiO2 interface (black dashed line in Fig. 3(a)), because there is no lateral confinement. When α ¼1801, there is no lateral confinement either. When α approaches 01 or 1801, the mode area increases significantly. To obtain a deeper understanding of the behaviors of the Am/A0, Lm and Nhyb, we analyzed the normalized energy density W(r)A0/Wm (Fig. 3(c)) and energy confinement ratio (Fig. 3(d)) in the low-index cladding (ηcla), triangle semiconductor nanowire (ηtri) and metal (ηmet) in the case of t¼ 2 nm, respectively. The energy confinement ratio η is defined as the ratio of the energy confined in certain area to the total energy. The increase of Nhyb is reasonable when α increases because the low-index cladding material is replaced by the high-index semiconductor. As a result, ηtri increases with increasing α which is consistent with the behavior of Nhyb (shown in right panel of Fig. 1(a)). The loss of the HPW mainly depends on the absorption of the metal, so the increasing ηmet shown in Fig. 3 (d) finally leads to smaller propagation distance (shown in Fig. 2(b)). For the Am/A0, its behavior at larger α(41001) is in consistent with the previous analysis that energy is better confined laterally in small interaction area and vice versa. However, for smaller α(o1001), the decrease of Am/A0 is hard to interpret, since the interaction area doesn0 t change much. So, there must be some mechanisms

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Fig. 3. (a) Normalized mode areas (solid lines), effective mode index (dashed lines, the black dashed line denotes the effective mode index of the pure SPP mode at metal–SiO2 interface) and (b) propagation distance with α for different gap width t. (c) Normalized energy densities along x-direction (y¼ 0) with angle α(t¼ 2 nm). (d) Energy confinement ratio of the hybrid mode with different angle α(t¼2 nm) in silver, triangle semiconductor waveguide and cladding regions, respectively. Here, h¼ 200 nm, r¼ 10 nm. Inset in (b): field distributions of hybrid modes when α¼201 and 1601, respectively.

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Fig. 4. (a) and (b) Field distributions of pure triangle modes when h ¼200, 400 nm, respectively. (c) |E| distributions of the pure triangle, cylinder, and square modes along the y direction with the height of the high-index regions fixed to 200 nm. The origin is the same as denoted in Fig. 1. (d) Normalized mode area of pure triangle mode with different α. Inset in (d): enlarged view of the tip of the triangle waveguide. Here, r ¼10 nm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

different from that in the previously studied cylinder [25] and square [26] HPWs, which give rise to better confinement and less absorption loss.

4. Tip enhancement To interpret the outstanding properties of the triangle HPW, we turn to analyze the pure triangle semiconductor waveguide. Fig. 4 (a) and (b) show the mode profiles of two triangle waveguides with h ¼200 nm and 400 nm, respectively. For larger waveguide (Fig. 4(b)), the mode is well confined in the dielectric and maximum energy density is located in the dielectric core. In contrast, the profile in smaller waveguide shows distinct aspect that there is an obvious enhancement of light around the tip of semiconductor waveguide. Here, the tip of the wedge is rounded with radius of curvature r so that the tip field enhancement is not singular (the inset of Fig. 4(d)). This tip field enhancement has been noticed in Ref. [41], where the field shows a maximum at the sharp tip of waveguide. As shown in Fig. 4(c) (red line), the field decays very quickly with increasing distance to the tip. And there is higher contrast of the electric field distribution at the interface of the high-low index in the pure triangle waveguide than those of cylinder and square waveguides. Here, the height of the highindex regions is fixed to 200 nm. This explains the higher field intensity in the gap of the triangle HPW than those of the cylinder and square HPWs in Fig. 2(e). Fig. 4(d) shows the normalized mode area of the pure triangle waveguide against α, which decreases as α increases and approaches to 0.3 when α > 100°. Combining with the mechanism that larger interface area will induce increase of Am/A0 of the hybrid triangle mode, Am/A0 shows an optimal value against α (Fig. 3(a)). As the local field intensity is greatly enhanced around the tip where the field distribution is not sensitive to the angle φ (denoted in the inset of Fig. 4(d)), we can expect that the properties of triangle HPW are also not sensitive to the offset angle θ (denoted in the inset of Fig. 5(a)). As shown in the inset of Fig. 5(a), the electromagnetic energy is still highly confined in the gap between the metal substrate and triangle waveguide despite of an offset angle of 101. Dependences of propagation distance (solid lines) and normalized mode area (dashed lines) on θ for different

Fig. 5. Dependence of the propagation distance (solid lines) and normalized mode areas (dashed lines) on (a) offset angle θ (denoted in the inset of (a)) and (b) the curvature radius r of the tip of the semiconductor waveguide. Red lines: t¼ 2 nm; blue lines: t¼ 10 nm; dark cyan lines: t¼25 nm. Two red dashed lines in the inset of (a) denote the y-axis and midperpendicular line of the triangle semiconductor waveguide. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

gap height are shown in Fig. 5(a), where we can find that the changes of propagation distance and normalized mode area can be ignored as θ varies from 01 to 101, and are about 10% as θ varies

Q. Lu et al. / Optics Communications 319 (2014) 141–146

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from 101 to 301. As θ increases further, the outstanding properties of the triangle HPW will be obviously degraded. Here, h¼200 nm, r ¼10 nm and α ¼1001. In addition, the tip field enhancement is a universal phenomenon for sharp corner. Thus, the varying of tip radius of curvature r will not destroy the advantages of triangle HPW. As shown in Fig. 5 (b), the changes of propagation distance and normalized mode area are less than 10% within the acceptable range, as r varies from 5 nm to 30 nm. Here, h ¼200 nm, α ¼ 1001 and θ ¼ 01. In practice, it is difficult to control the included angle between y-axis and the midperpendicular line of triangle semiconductor waveguide and the curvature of the tip. The above results show that the triangle HPW is excellently tolerant for these fabrication errors, thus the proposed HPW is feasible for experiments. In experiment, it is more feasible to fabricate a structure reversed of the HPW shown in Fig. 1. A possible fabrication process may be as following. First, the triangle semiconductor nanowire can be prepared by chemical synthesis [39] or fabricated by etching [40]. Then the triangle semiconductor nanowire is placed on the quartz substrate with the wedge tip up. Finally, the silver is deposited after depositing low-index cladding on the triangle semiconductor nanowire.

5. Extreme light confinement We have demonstrated that the triangle HPW exhibits both stronger field confinement ability and longer propagation distance due to the extreme light enhancement in the tip of the triangle semiconductor waveguide. Similarly, the light can also be enhanced in the tip of the triangle metal wedge, which is typically compromised with large propagation loss [19]. In the following, we replace the flat metal substrate with a triangle metal wedge (Fig. 6(a)) forming the tip-to-tip hybrid plasmonic waveguide (TTHPW) to obtain stronger confinement of light. The triangle metal wedge can be fabricated by using focused ion beam (FIB) technique [16] or standard UV lithography [18]. Similar to triangle semiconductor waveguide, the height and angle of the triangle metal wedge are denoted as hm and β, respectively. The tip of the metal wedge is rounded with radius of curvature rm. For simplification, we assume that the metal wedge is non-truncated (i.e., hm-1) and fix rm ¼ 10 nm. In this section, the angle of the triangle semiconductor waveguide α is fixed to 1001 which is optimal for light confinement according to the preliminary calculations. Fig. 6 (b)–(d) show the field distributions of the HPW with β ¼1801, 1001, 201, respectively. Dependences of normalized mode area (Fig. 6(e)) on β for different t indicate that a sharper metal wedge is preferred to achieve stronger confinement of light. For β ¼ 201, t¼ 2 nm, mode area (0.00057A0) more than three orders smaller than the diffraction-limited area (A0) in free space is achievable (Fig. 6(d)), which is less than a tenth of that of the cylinder HPW (Fig. 2(b)) [25]. For Nhyb, it monotonically increases as β decreases from 1801 to 201. For Lm, it increases first and then decreases as β decreases from 1801 to 201, and has a maximum as β is about 1501. The change of Lm results from the variation of the energy confined in the metal region (dashed lines in Fig. 6(f)). As compared with structure in Ref. [19] at the same conditions (e.g., metal wedge with angle β ¼201, gap width t ¼2 nm), the mode area is almost the same (Table 1). However, the normalized propagation distance is 10.45 longer than 7.68 of that structure [19]. As compared with structure in Ref. [35], the smallest normalized mode area in that structure is about three times larger than TTHPW and the normalized propagation distance is only two times larger than TTHPW. Thus, the TTHPW is superior to those structures in Refs. [19,35].

Fig. 6. (a) Schematic illustration of the hybrid plasmonic waveguide consisting of a triangle semiconductor wedge (α ¼ 1001) placed directly above triangle metal wedge. (b) and (c) Field distributions of hybrid plasmonic waveguide with β ¼ 1801, 1001, 201, respectively. (e) Dependences of normalized mode area (left side) and effective mode index (right side) on β at different gap width conditions. (f) Dependences of propagation distance (left side) and energy confinement ratio ηmet in the metal region (right side) on β at different gap width conditions.

Table 1 Comparisons of different HPWs. Structure

Ref. [19]

Ref. [35]

TTHPW

Am/A0 Lm/1.55

0.000569 7.68

0.00142 22.78

0.000573 10.45

As the effective mode area can be squeezed three orders below A0 (Fig. 6(e)), the density of states is strongly modified near the tip of the metal wedge. Therefore, the light-matter interaction can be

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greatly enhanced. When putting atoms or quantum dots in the gap of TTHPW, the spontaneous emission rates can be modified, which is known as the Purcell effect. The maximum of the emission enhancement [33] is Fp ¼

3N hyb ðλ=nÞ2 4π Am

ð3Þ

For β ¼201, t ¼2 nm, the enhancement of the spontaneous emission rate is Fp ¼1713.2 for a emitter placed in the cladding and Fp ¼314.7 for a emitter placed in the semiconductor wedge. The corresponding collection efficiency which is denoted as the ratio of the emission coupled to the hybrid mode is Fp/(1þFp)E 1 [33]. Thus, benefiting from such an ultra-small mode area, the proposed HPW can be applied for ultra-highly efficient single photon source and ultra-strong light-matter interaction.

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6. Conclusions In conclusion, we proposed a triangle HPW for ultra-deep subwavelength light confinement with low loss. The properties of the triangle HPW are numerically investigated, showing excellent properties (e.g., stronger light confinement, lower propagation loss) due to field enhancement at the tip of the triangle semiconductor waveguide, as compared with the reported HPWs. The proposed structure also shows high tolerance for fabrication errors, which is beneficial for its implement in experiment. In addition, mode area as small as 0.00057A0 can be achieved by replacing the flat metal substrate with a triangle metal wedge, which is more than three orders smaller than the diffractionlimited area. This proposed structure shows great potential in various applications, such as the laser [36], waveguide (cavity) quantum electrodynamics [43] and optomechanics [44]. Acknowledgement We gratefully acknowledge Fang-Jie Shu and Xiao Xiong for discussions. This work is supported partially by the National Natural Science Foundation of China under project Nos. 61007029 and 11274277, Projects of Zhejiang Province (Nos. 2011C21038, 2010R50007, 2011C22051, and Y1100041), the Open Project of the State Key Laboratory of Functional Materials for Informatics, the Program for Science and Technology Innovative Research Team in Zhejiang Normal University. References [1] E. Ozbay, Science 311 (2006) 189. [2] W.L. Barnes, A. Dereux, T.W. Ebbesen, Nature 424 (2003) 824.

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