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We propose a novel plasmonic waveguide structure, which is referred to as a circular hybrid plasmonic waveguide (HPW) and consists of a metal wire covered with low- and high-index dielectric layers. The circular HPW exhibits two distinctly different modes, namely, the strongly localized mode and the extremely low-loss mode. Our numerical calculation demonstrates that the strongly localized mode exhibits 10−4 order scale in normalized mode area and can be performed even in tens of nanometer sizes of waveguide geometry. In the extremely low-loss mode, the HPW exhibits ultra-long propagation distance of more than 103μm that can be achieved by forming the dipole-like hybrid mode and properly adjusting the radius of the metal wire. It is also shown that, even with this long-range propagation, the mode area of the dipole-like hybrid mode can be maintained at subwavelength scale. The simultaneous achievement of a small mode area and ultra-long propagation distance contributes to the ultra-high propagation distance to mode size ratio of the waveguide. The HPW results are very helpful for plasmonic device applications in the fields of low-threshold nanolasers, ultrafast modulators, and optical switches.
©2013 Optical Society of America
1. Introduction
The need for faster and smaller optical devices has established nanophotonics branch, which has been extensively investigated along with the study of plasmonics in recent years. Plasmonics is one of the most promising avenues for device miniaturization that guarantees a number of merits such as low power consumption [1,2], ultra-fast operational speed [3–5], and efficient device integration [6–9]. Many theoretical and experimental researches have demonstrated that, for the dielectric-based photonics, large portion of electromagnetic waves spread over the space and cannot be confined in a small region due to the diffraction limit of light. This limits its ability of reducing optical modal volume to cubic half wavelength scale. In contrast, surface plasmon polaritons (SPPs), the electromagnetic waves propagating on the interface between metal and dielectrics, have allowed the modal confinement below the diffraction limit [10,11]. This allows low power operation and integration on a chip in a variety of applications such as nanolasers [12–15], modulators [9,16–18], and splitters [19,20].
However, when it comes to SPPs-based devices, we should pay critical attention to the huge amounts of metallic loss come from metal. It is inevitable for plasmonic devices to experience high loss due to the high conductance of metals, which creates problems over a wide range of applications based on the plasmonic effect. This high loss also limits the operation of many upcoming applications to only low temperatures [13–15]. As the fundamental building blocks for plasmonic devices, several types of plasmonic waveguide structures such as metal-dielectric-metal (MDM) waveguide [6,9,21], metal patch waveguide [22,23], plasmonic channel waveguide [24], and wire-typed waveguide [25–27] have been suggested and investigated. For most of current plasmonic waveguide structures, the propagation distance does not exceed a few microns at telecommunication wavelength regime without compromising the sub-diffraction-limit modal confinement. This implies that, to achieve subwavelength modal confinement, the performance of the plasmonic devices is limited to working at low temperature or requiring high operation power. The trade-off between propagation distance and modal confinement has been challenge for plasmonic waveguide research. It is believed that a huge reduction of propagation loss will be a stimulus in plasmonics research and lead to a remarkable development of nanophotonics.
In this respect, we propose an extremely low-loss novel plasmonic waveguide structure, the circular hybrid plasmonic waveguide (HPW). It consists of a metal wire covered with low- and high-index dielectric layers, which is an improved structure of a bare metal wire waveguide with the hybrid mode [26]. It has the merits of ultra-long propagation distance and subwavelength modal confinement simultaneously. We numerically analyze the modal characteristics of the circular HPW by using the finite element method with COMSOL Multiphysics. Furthermore, to demonstrate the effectiveness, the waveguide properties of the circular HPW have also been compared with those of other representative waveguide structures, such as the metal wire waveguide and dielectric wire HPW.
2. Various plasmonic waveguide structures
The various plasmonic waveguide structures reported in the literature are shown in Fig. 1. The most fundamental plasmonic waveguide is the planar metal-dielectric (MD) structure, a single interface plasmonic waveguide in the form of one-dimensional planar structure, as shown in Fig. 1(a). This structure has various applications based on plasmonic effect, including small-sized optical waveguides and on-chip operation. However, in this structure, mode size reduction inevitably increases its propagation loss [7,10,11]. This flaw is rather overcome by using the planar HPW [24], which is realized by inserting a low-index dielectric layer between the metal and high-index dielectric medium, as depicted in Fig. 1(b). Due to the insertion of the low-index dielectric layer, the loss is reduced and the modal confinement can be significantly improved. Both of these planar structures provide only one-dimensional confinement. In order to extend the planar MD plasmonic structure into a two-dimensional waveguide structure, one can think of a metal wire structure [25] depicted in Fig. 1(c). This metal wire structure has fair propagation distance when surrounded by air, but the trade-off between the mode area and the propagation loss still needs to be relieved as the planar MD structure.
Fig. 1 Various plasmonic waveguide structures. (a) A single interface plasmonic waveguide. The most fundamental plasmonic waveguide. (b) A planar HPW. A low-index dielectric is inserted in between a metal and high-index dielectric medium. (c) A metal wire waveguide. The metal wire surrounded by air. (d) A dielectric wire HPW. Two-dimensional modal confinement is accomplished in this structure. (e) A circular HPW, The metal wire is covered with low- and high-index dielectric layers.
A dielectric wire HPW depicted in Fig. 1(d) is an extension of the planar HPW into a two-dimensional waveguide by replacing the semi-infinite dielectric layer with a dielectric wire [26,27], which exhibits the best plasmonic waveguide performance reported thus far. In this structure, the hybrid plasmonic mode property provides low-loss characteristics, thereby leading to a propagation distance of tens of microns. From our point of view, it is necessary to further increase the range of the feasible propagation distance for proper waveguiding. For this, an alternative approach of the extension of the planar HPW into a two-dimensional waveguide is proposed in this work, which is depicted in Fig. 1(e). The proposed structure is formed by rolling-up the planar HPW in a similar way to get the metal wire structure from the planar MD structure.
3. Definition of mode area and propagation distance
In our modal analysis, the mode area, Am, is defined as the ratio of the total power to the maximum power density such that [22,26]
where Pave(x,y) is the time averaged Poynting vector, given byThe mode area is normalized by Ao, (λ/2)2. The definition of the propagation distance, Lp, is ½Im[β−1], where β is the wavevector component in the propagation direction. In this work, a propagation distance to mode size ratio, Lp/, is used as a performance metric of a plasmonic waveguide, which is also used in Ref [26].In all calculation, the permittivities of dielectrics are assumed to be εH = 13.84 (InGaAs), εL = 2.1025 (SiO2), and εA = 1 (air). The permittivity of metal (Au) is chosen as εM = −125.03 + 4.23i at λ = 1.55 μm [28].
4. Modal analysis of a circular hybrid plasmonic waveguide
In the proposed circular hybrid plasmonic waveguide, it has been found that there are two types of guided mode: one is a strongly confined mode which shows circular symmetry (the first mode) and the other is a mode of dipole-like shaped profile which shows an ultra-long propagation distance (the second mode). The characteristics of those modes are discussed in this section.
4.1 Mode with extremely enhanced confinement
Figure 2(a) shows the electric field profile of the first mode in the circular HPW, in which most of the field is confined in the narrow low-index dielectric region, just like the planar HPW depicted in Fig. 1(b) and thus, the mode area is extremely small. One can also see that the electric field is along the radial direction and this mode can be interpreted as a mode formed by rolling-up the mode of the planar HPW. In Figs. 2(b) and 2(c), the mode area and the propagation distance as a function of thigh are plotted for various low-index layer thicknesses (tlow) with r = 10 nm. The normalized mode area reaches ~10−4 scale. However, due to the strong modal confinement near the metallic region, the propagation distance of this mode is just a few microns. As shown in Fig. 2(d), Lp/ is relatively small and becomes almost flat for thigh > 55 nm. The short propagation distance of this type of mode may limit the usage as a waveguide, but its extremely small mode area may be useful for ultra-small resonators.
Fig. 2 Modal characteristics of the 1st mode in the circular HPW. (a) Electric field profile at the smallest mode area, where r = 10 nm, tlow = 1 nm, and thigh = 55 nm. The arrows indicate the polarization directions. (b) Normalized mode area, (c) propagation distance, and (d) propagation distance to mode size ratio as a function of thigh.
4.2 Mode with ultra-long propagation distance
There exists another type of guided mode in the circular HPW and Figs. 3(a)–3(c) show the electric field profiles of the modes for slightly different geometric parameters which are marked in Fig. 3(d). One can see that the planar HPW-like confinement is formed only in one direction (horizontal) and in the other direction (vertical), the modal confinement is formed by the index-guiding effect, which results in the dipole-like field profiles.
Fig. 3 Modal characteristics of the 2nd mode in the circular HPW. Electric field profiles for (a) tlow = 5 nm, thigh = 130 nm, (b) tlow = 5 nm, thigh = 170 nm, (c) tlow = 1 nm, thigh = 170 nm (r = 10 nm in all cases). One-dimensional graphs show the field distributions along the central lines and the insets show the blow-up of the electric field distribution near the metal wire region. (d) Normalized mode area, (e) propagation distance, (f) propagation distance to mode size ratio and (g) the effective index, neff, as a function of thigh. The curves of the same color correspond to the same parameter values.
The mode area and the propagation distance of the 2nd mode are depicted in Figs. 3(d) and 3(e), respectively, as a function of thigh for various tlow with r = 10 nm. As shown in Fig. 3(d), the mode area of the 2nd mode is about two orders of magnitude larger than that of the 1st mode. However, the loss is reduced by more than three orders of magnitude, and the propagation distance approaches a millimeter scale, as depicted in Fig. 3(e). Thus, Lp/, the performance metric of the waveguide, shows a remarkably high value exceeding 104 as shown in Fig. 3(f), which is about three orders of magnitude higher than that of the 1st mode. It is worth noting that, although this dipole-like hybrid mode exhibits the larger mode area compared to that of the 1st mode, its value is still in the subwavelength range. In this aspect, it can achieve the low-loss and subwavelength modal confinement characteristics at the same time.
Figure 3(g) shows the effective index, neff, as a function of thigh. The effective indices of the 1st mode substantially increase with decreasing the low-index dielectric thickness from 5 nm to 1 nm, and it implies that the 1st mode is highly affected by metal. However, the behavior of the 2nd mode is somewhat different, and it is similar to that of the dielectric wire HPW [26]. Because it is less influenced by metal, the effective indices of the 2nd mode show much lower values than those of the 1st mode for the same thigh.
So far, the effect of the thicknesses of the dielectric layers on the characteristics of the 2nd mode with a fixed metal wire radius has been investigated. Now, we investigate the effect of the metal wire radius. The mode area and the propagation distance are calculated and plotted for various r with tlow = 1 nm. One can see that as the metal wire radius decreases, the loss can be further reduced while the mode area becomes larger. An important feature to note here is that while the loss decreases remarkably as the metal wire radius decreases, the mode area does not show significant change. As shown in Fig. 4(b), when the metal wire radius decreases from 20 nm to 2 nm, the propagation distance is increased by 1000 times, whereas the mode area is increased just by a factor of 4.
Fig. 4 Modal characteristics of the 2nd mode with different r values in the circular HPW and the fundamental mode in the dielectric wire waveguide. (a) Normalized mode area and (b) propagation distance as a function of thigh. Electric field profiles for (c) r = 0 nm, tlow = 0 nm, and thigh = 165 nm, (d) r = 2 nm, tlow = 1 nm, and thigh = 180 nm, and (e) r = 20 nm, tlow = 1 nm, and thigh = 170 nm. One-dimensional graphs show the field distributions along the central lines and the insets show the blow-up of the electric field distributions near the metal wire region. (f) Propagation distance to mode size ratio and (g) the effective index of the mode as a function of thigh.
In Fig. 4(a), the modal characteristic of the fundamental mode (HE11) of the dielectric wire waveguide, which corresponds to the case of r = 0 nm and tlow = 0 nm in the circular HPW, is also plotted for a reference. The mode area of the dielectric wire waveguide is significantly larger than that of the circular HPW. The electric field profiles of the dielectric wire waveguide and the 2nd mode of circular HPWs are plotted in Figs. 4(c)–4(e). One can see that the 2nd mode of the circular HPW originates from a combination of the HE11 mode of the dielectric wire and the plasmonic mode of a metal wire, and this brings about the dipole-like hybrid mode feature. The metal wire in the center reduces the mode area significantly at the cost of propagation loss. Since even for a very small r, the strong field confinement in the low-index layer and the dipole-like hybrid mode feature remain, the mode area and the propagation distance can be tuned with r. This provides us another degree of freedom in subwavelength waveguide design.
Figure 4(f) shows Lp/, which increases remarkably fast due to the reduced loss as r decreases. As shown in Fig. 4(g), the effective index of the 2nd mode is not affected much by the metal wire thickness, which implies that little portion of the field experiences the metal wire in the 2nd mode. The effective index of the dielectric wire waveguide mode is also plotted in Fig. 4(g), which shows the smallest value.
5. Comparison among various waveguide structures
Figure 5 shows the overall comparison among various plasmonic waveguide structures. We plot the mode area versus propagation distance for a fair comparison. A structure with ideal performance should exhibit small mode area and long propagation distance at the same time. We compare the circular HPW with the metal wire waveguide [25] and dielectric wire HPW [26] since their waveguide performances are known to be superior to other plasmonic waveguides reported in the literature.
Fig. 5 Normalized mode area versus propagation distance. Top three trajectories indicate the circular HPWs, and bottom two indicate the dielectric wire HPW and metal wire, respectively. Data has been obtained by varying tlow in the circular HPWs, h in the dielectric wire HPW, and r in the metal wire. The rest parameters at the given geometries are thigh = 180 nm for the green and red lines, thigh = 180 nm for the purple line, and d = 200 nm for the blue line.
For the circular HPWs, three geometrical parameter sets are considered; (r, thigh) = (2, 180), (5, 180), (10, 180) nm. The value of thigh is chosen as the value near the minima in Fig. 4(a). The mode area and the propagation distance are calculated with tlow varied. For the dielectric wire HPW, the dielectric wire diameter of 200 nm is also chosen as the value close to the mode area minimum. In this case, the mode area and the propagation distance are also calculated with tlow varied. In the case of the metal wire waveguide, the radius is varied since it is the only geometrical parameter. As shown in Fig. 5, the circular HPW shows superior performance to other types of waveguides. For the circular HPW, the smaller metal wire radius, the better performance is achieved.
6. Applications of the circular hybrid plasmonic waveguide
Due to the low-loss and small mode area characteristics, the circular HPW can be used to form a plasmonic nanolaser cavity with a relatively high quality factor to volume ratio (Q/V) as well as a lightwave circuit element. As for the lightwave circuit element, the circular HPW structure is not easy to fabricate since the outermost InGaAs layer is hard to be grown. So, a planarized version of the circular HPW needs to be devised for the lightwave circuit element application as depicted in Fig. 6(a). The structure is devised from the fact that the dipole-like mode of the circular HPW is confined via index-guiding in one-direction. The analysis of the waveguide shown in Fig. 6(a) will be reported in another paper. As for the plasmonic nanolaser cavity application, the structure depicted in Fig. 6(b) can be considered, where the circular HPW structure provides the horizontal confinement. This structure can be fabricated by etching a hole in the center of an InGaAs mesa and filling the hole with the low-index dielectric shell and the metal. The volume of the structure can be much smaller than (λ/2)3 due to the small mode area of the circular HPW. The high propagation distance to mode size ratio of the circular HPW will also bring about a relatively high Q/V value. The detailed analysis and the fabrication of the plasmonic nanolaser structure are in progress and will be reported soon.
Fig. 6 Schematics of (a) the lightwave circuit element and (b) the plasmonic nanolaser cavity as the applications of the circular HPW. As for the lightwave circuit element, the planarized structure is devised from the fact that the dipole-like mode of the circular HPW is confined via index-guiding in one-direction.
7. Conclusion
By applying the concept of the hybrid plasmonic mode to a metal wire and forming the dipole-like hybrid mode, we have demonstrated that the subwavelength optical modal confinement and ultra-long propagation distance can be simultaneously achieved. By numerically analyzing the modal characteristics of the circular HPW and comparing its performance with other representative plasmonic waveguides, we have verified that the circular HPW exhibits several orders of magnitude longer propagation distance than those of the other structures of the same mode area. The first mode generated in the circular HPW is a strongly localized mode in a thin low-index dielectric region, representing the deep subwavelength modal confinement, down to the 10−4 scale in the normalized mode area. This extremely small optical mode area may enable efficient device miniaturization. On the other hand, the second mode shows extremely low-loss characteristics, centimeter-scale propagation distances, with the subwavelength modal confinement. A further reduction of the propagation loss is achievable as the radius of the metal wire is decreased while still maintaining the subwavelength modal confinement, and this renders the propagation distance to mode size ratio of the waveguide distinctly high. Thus the 2nd mode seems to be helpful in the realization and application of the novel low-loss plasmonic devices.
Acknowledgment
This work was supported by National Research Foundation of Korea Grant (NRF-2011-0014265).
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