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Using 3-dimensional numerical simulations, we investigated the characteristics and performance of essential photonic components based on the recently proposed quasi-coplanar waveguide (QCPW) including bends, couplers, and splitters. The results confirmed the QCPW’s potential in high density photonic integration. We also explored the application of various RF-originated concepts and techniques to QCPW-based photonic components to enhance and enrich their functionalities.
©2008 Optical Society of America
1. Introduction
Recently, significant research effort for high-density optical integration has been directed toward the utilization of the surface plasmon-polaritons (SPPs) due to their capability of subwavelength wave confinement. An efficient waveguiding structure is the most essential element for integration and a number of SPP guiding schemes based on various geometries and materials have been reported. They include gaps [1–3], V-grooves [4,5], deep-trenches [6,7], slots in metal films on dielectric substrates [8–10], metal [11,12] (or dielectric [13]) stripes on dielectric [11,12] (or metal [13]) substrates, and metal/dielectric heterostructures [14,15]. Important design criteria include the degree of confinement, the propagation loss, and the ease of fabrication. Support of telecom bandwidth is also considered valuable due to the availability of optical components developed for the wavelength regime. In particular, the V-groove waveguiding structure and photonic components based on it demonstrated successful subwavelength-scale SPP confinement and operations in telecom bandwidth in experiments [5]. Etching deep grooves into metal films, however, may be challenging in fabrication.
In our previous work, we proposed a new SPP waveguiding scheme which mimics the V-groove with three thin metal stripes [16]. As shown in Fig. 1(a), the center metal stripe was staggered to improve SPP confinement. Since monolithic microwave integrated circuit (MMIC) waveguides comprising three metal stripes on a flat surface are widely referred as coplanar waveguides [17], we called the proposed structure quasi-coplanar plasmonic waveguide (QCPW). In 2D finite element method (FEM) simulations, the QCPW structure exhibits a good level of SPP mode confinement over a wide range of spectrum including telecom bandwidth [16].
Fig. 1. (a) A schematic view of the quasi-coplanar plasmonic waveguide. Three metallic stripes are configured to mimic a V-groove structure. Laterally, however, the structure is partially open. (b) The electric field profile of the QCPW fundamental mode computed by 2D FEM calculation. The field amplitude is plotted in log scale. The arrows represent E fields. The operation wavelength is 1550 nm.
For practical applications in 3D settings, however, the partially open geometry of the QCPW structure may adversely affect the performance of photonic components based on it. Of particular concern are waveguide bending loss and mode coupling between waveguides. The question is: Is the QCPW’s confinement strong enough to avoid the loss of SPP power through the opening in the lateral direction? Based on the results of 3D FEM simulations, we answer the question affirmatively. We will also show that the performance of some QCPW-based photonic components can be further improved with the help of techniques and concepts borrowed from microwave and RF technologies. As examples, we will enhance the transmission through QCPW bends using air-bridges and numerically implement compact Mach-Zehnder interferometers based on half-modes of asymmetrized QCPWs.
2. 3D QCPW and Simulation setup
Figure 1(a) schematically shows the QCPW structure. A virtual groove is formed within an area enclosed by three metallic stripes. Two of the stripes are placed on the top and a dielectric spacer layer vertically separates them from the base stripe embedded in it. From the structural point of view, the QCPW is a vertical assembly of slot and nanowire waveguides. Figure 1(b) shows the electric field profile of the QCPW fundamental mode obtained by 2D FEM. Detailed mode characteristics can be found in Ref. [16].
For our 3D simulations, we set the structural parameters mostly identical to those of the 2D simulations in Ref. [16] which were optimized for the λo=1.55 µm: ts=200 nm, wg=300 nm, wts=800 nm, wbs=200 nm, nsup=1, nsub=1.495, and nspc=1.50. Only the top stripe width wts is reduced from 850 nm to 800 nm. The impact of wts on the propagation characteristics is minimal [16]. The substrate thickness, not restricted in 2D analysis, was set to 225 nm in the 3D setup to enclose mode field down to -30 dB in intensity. A rectangular port which measures 200×800×500 nm in x, y, and z, respectively, was excited by an x-polarized electric field to feed a z-propagating wave into the computational domain. Scattering boundary condition was adopted to cancel reflections. Tetrahedral elements were used with selective refinements along metallic stripes. We chose Au as our metal and modeled it based on [18]. The refractive indices of the spacer and the substrate were set to 1.5 and 1.495, respectively. We kept the index contrast very low to make sure that it does not play a role in waveguiding. We used a commercial, fully vectorial FEM package [19]. In geometrical modeling, the top two vertices of all metallic stripes were rounded to 10 nm radius of curvature to better approximate the real structures. All simulations are performed at λo=1.55 µm.
3. Straight QCPW and Directional Coupler
As a validation of the 3D model, we simulated a straight QCPW. The insets in Fig. 2 show a 3D view of a SPP wave (Ex component) propagating along the QCPW section (taken 50 nm above the base stripe) and the cross-sectional mode pattern which retains all salient features of the 2D result in Fig. 1(b). As expected, most modal energy is localized among the three stripes. The wavelength of the guided SPP wave measured along z-direction is approximately 890 nm which agrees with λo/neff, where neff is the effective index of the mode computed in 2D mode analysis. The attenuation characteristic of QCPW was investigated by measuring the mode field intensity within a 500 nm×700 nm area on x-y plane. As shown in Fig. 2, the total attenuation contains two decays components: one associated with the formation of the mode and the other with the attenuation of the mode itself. Even beyond the formation of mode (at z ~3 µm), the residual excitation not included in the mode keeps propagating and decaying and must be considered in the analysis. The curve-fitting results show that the intensity attenuation beyond the mode formation is exponential with approximately 13.2 µm propagation distance (Lprop) which is in reasonable agreement with the 2D simulation prediction of 12 µm.
Fig. 2. The change of power along 15 µm propagation. The dashed lines denote exponential decay. The inset shows the simulated amplitude of Ex. Also shown is the mode field distribution |E| at z=11.6 µm which retains the basic features of its 2D counterpart in Fig. 1(b). The fitting parameters are: C1=0.89, C2=0.064, Po=0.04.
The first passive photonic component we simulated was the parallel waveguide tap coupler. Since strong confinement hinders mode coupling, the tapping efficiency also provides information on the density of QCPW-based photonic integrated circuits. The tap coupler geometry is shown in the inset of Fig. 3(a). The edge-to-edge distance between the two base stripes (wETE) was used as a measure of separation. The coupling distance (Lc) is defined as the overlap length which results in a total transfer of energy from one channel to the other. For directional couplers with wETE>500 nm which could not achieve e-1 power transfer within the 15 µm of simulated propagation, we estimated their Lc by fitting the power decay curve of the main channel with exponential decay functions. Ohmic loss was excluded from the estimation. QCPWs separated by wETE>500 nm require overlap distance comparable to Lprop for effective tapping, indicating the possibility of very dense QCPW integration. As a tap coupler, however, the parallel QCPW coupler functions poorly, necessitating a different geometry for the purpose. As shown in Fig. 3(c) and (d), the good isolation can be attributed to the rapid attenuation of outreaching E field from the main channel.
Fig. 3. (a) The computed coupling distance (Lc) of a parallel QCPW tap coupler. (b) The top view E field pattern of a QCPW tap coupler with 500 nm wETE. The starting point of the tapping channel is placed 3 µm away from the source to avoid spurious coupling. (c) and (d) shows the changes in cross-sectional E field patterns of two QCPW tap couplers with different values of wETE.
4. QCPW Bends and Air-Bridging
The waveguide bend is another important passive photonic component. We numerically investigated QCPW bends as a function of R, the radius of curvature defined in the inset of Fig. 4. We attached straight QCPW sections before and after each curved section for proper QCPW mode formation and monitoring of the bend-induced mode distortion effect, respectively. The transmission T, defined as the ratio of power in output (averaged over 2 µm post-curve) to input (averaged over 500 nm pre-curve) is plotted in Fig. 4(a) as a function of R. The transmission considering only the bending loss is estimated using Lprop and is superimposed. The two curves indicate that the bending loss is dominant up to R~3 µm but becomes negligible by R~5 µm, beyond which we expect a gradual decrease in T due only to the Ohmic loss. For R>3 µm, the T is comparable to other similarly-sized, non-trench type SPP waveguides [20–23]. When λo<R<3 µm, the loss is higher than other plasmon waveguides. For R≤λo, T is maintained at 40%. Figures 4(b) and (c) show the changes in mode field patterns along the propagation through two curved QCPW sections with different radii. When R=1 µm, the electric field pattern shows a significantly higher level of distortion than that of R=1.65 µm case. The distortion in field pattern stems from the phase mismatch between the mode’s outer-track and inner-track components which travel over different path lengths. Accumulation of phase mismatch leads to the excitation of an odd mode which will not be supported in the straight post-curve section, causing degradation in the transmission T. The change in mode power along the propagations in pre-curve, curve, and post-curve sections, plotted in Fig. 4(d), further corroborates the prediction. As expected, the mode in a bend with smaller R decays faster not only within the curved section but also within the straight post-curve section due to the higher level mode mismatch.
Fig. 4. (a) The transmission T of QCPW bends as a function of the radius of curvature R defined in the inset. (b) and (c) show the amplitude of Ex along the QCPW bends with R=1 µm and R=1.65 µm, respectively. The Ex patterns show that the smaller the radius of curvature R, the severer the mode pattern distortion within the curve section. (d) The change in mode power due to propagations in pre-curve, curve, and post-curve sections. As expected, the bend with smaller R causes steeper power loss both within the curve and the post-curve sections.
The degradation of transmission T due to the phase mismatch between the inner- and outertrack electric field components and subsequent odd mode excitation has also been identified as the main drawback of MMIC coplanar waveguides [17,24]. The “air-bridge” which physically connects the left and right stripes over or under the center stripe and forces equal electric potential between them has been utilized to suppress the odd mode excitation. In this work, we explored an analogous scheme to enhance the transmission of QCPW bends with R≤λo.
As shown in Fig. 5(a) and (b), air-bridges were implemented in QCPWs by connecting the two top stripes. The extent of the air-bridge was restricted to the area directly over the QCPW bend. The insets show the electric field pattern along the propagation through the bends. Under the air-bridge, the distortion of the field pattern is significantly reduced. The change of mode power along the propagation plotted in Fig. 6(a) corroborates the improvement in T due to incorporation of the air-bridge. Clearly the loss is reduced not only in the curved section but also in the post-curve section thanks to the increased mode matching.
There are two possible explanations for the observed enhancement in T: (i) As suggested in the MMIC theory (Section 9.8 of [17], for example), the air-bridge equalizes the electric potential on both top stripes and suppresses the emergence of the second, odd mode which degrades T. (ii) As proposed by Manolatou et al, incorporating low-Q resonant cavities into the bends of high-index dielectric waveguides can improve the transmission through resonant coupling [25,26]. The air-bridged QCPW can also be considered as a partially open, low-Q resonant cavity and the transmission improvement can be attributed to the resonant coupling. While it is too early to reach a conclusion, the authors are in favor of the former interpretation based on the lack of frequency dependence in T improvement over 400 nm wide spectral range as shown in Fig. 6(b). The bending efficiency enhancement in V-groove waveguides by incorporation of a pillar defect [27] may also be traced to the same phase mismatch mitigation effect.
Fig. 5. A QCPW bend and the electric field pattern before (a) and after (b) incorporation of an “air-bridge.” The radius of curvature R is 1 µm. The reduction in mode distortion in the air-bridged QCPW bend is clearly shown.
Fig. 6. (a) The change of mode power along the propagation through QCPW bends with or without an air-bridge. (b) The frequency dependence of air-bridge induced T enhancement.
5. Asymmetrized QCPW and Half-Modes
From the numerical simulations of SPP propagation along curved QCPW sections, we identified a new set of modes which exhibit theoretically interesting and practically useful properties. When the originally symmetric QCPW fundamental mode entered a section with widened base stripe width (the gap between the two top stripes is widened as well) or with one of the top stripes removed, it will gradually lose half of its mode field and begin to assume a field pattern which can be best described as the QCPW half-mode. The half-modes can also be obtained directly by solving for modes in a QCPW with one of the top stripes removed and can be obtained from 2D mode analysis as shown in Fig. 7(a) and (b). Interestingly, a similar mode was predicted in MMIC coplanar waveguide structure which was the original inspiration for the QCPW [17].
It is interesting to point out that the half-mode can recover its original, symmetric mode pattern, which can be referred as the full-mode, in two different ways. As shown in Fig. 7(c), the half-mode recovers its full-mode when the symmetric top stripe configuration is restored. In addition, when two similar but complementary half-modes are brought together as shown in Fig. 8, they can form a full-mode by combining with each other. Despite its high loss, the QCPW half-modes can play an important role in short distance applications.
Fig. 7. The log-scale |E| patterns of (a) the left half-mode and (b) right half-mode computed by 2D simulations. (c) The full-mode recovery capability of a half-mode is demonstrated using a QCPW section with a partially open top stripe. Due to the disappearance of the right top stripe, the mode becomes a left half-mode at a-a’. Once the top stripe is recovered, a full-mode is recovered along the propagation. At b-b’, the recovery is completed. No noticeable loss in excess of the Ohmic loss was caused by the recovery process.
Fig. 8. Another example of the half-mode’s capability of full-mode recovery is demonstrated using a QCPW with curved top and base stripes. The widening of the gap between the top stripes split a full-mode (a-a’) into two half-modes. Once the gap narrows back (b-b’), a full-mode is formed again. At b-b’, a 3.2 dB penalty was measured beyond the Ohmic loss. It was mostly due to the half-mode propagation rather than the recovery itself.
6. QCPW Y-Branches and Interferometers
The QCPW’s ability to handle the left and right portions of its mode as separate half-modes and to recover a full mode from one half-mode plays an important role in reducing the size of Y-splitter/combiner, another essential component for photonic integrated circuits. The inset of Fig. 9(a) shows the structure of a QCPW-based Y-splitter schematically. The splitter branches are shaped to a raised-cosine curve W/4·(1-cos(πz/Ls)) [22] with the splitting distance Ls varying from 0.5 to 4 µm. Based on the cross-talk analysis the previous section, the two branches were separated by wETE=1.7 µm (w=1.9 µm). For each Ls, we calculated the splitting efficiency, the ratio of the sum of mode power in the two split branches to that of the input branch and plotted them in Fig. 9(a). Unlike previously reported plasmonic splitters which exhibit higher efficiency with increasing Ls until the Ohmic loss kicks in, the splitting efficiency of the QCPW Y-splitter becomes maximizes at relatively short Ls~1 µm and declines afterward. The 83.7% splitting efficiency at such a short Ls can be attributed to the very efficient separation of the original mode into two half-modes, which is clearly shown in Fig. 9(b). The degradation in splitting efficiency for longer Ls (Ls>2 µm) is strongly related to La, the distance between the splitter starting point and the point at which the tip of the new top stripe appears. Without the top stripe, the half-modes cannot recover the symmetric full-mode and suffer from radiation loss. As shown in Fig. 9(c), the longer the La, the longer the length needed to recover the full-mode. By bridging two Y-splitters with a pair of QCPW sections, we realized a 3 µm long Mach-Zehnder interferometer shown in Fig. 9(d). Figure 9(e) shows the mode power measured along the red dotted line in Fig. 9(d) and indicates that the ultracompact Mach-Zehnder interferometer exhibits 40.4% overall transmission.
Fig. 9. (a) The splitting efficiency of QCPW-based Y-splitter as a function of splitting distance Ls. (b) and (c) show the Ex field patterns of Y-branches with Ls=1 µm and 4 µm, respectively. It is clear that a longer La impedes the subsequent recovery of a full-mode in each of the branches and results in a lower splitting efficiency. (d) A 3 µm long Mach-Zehnder interferometer made by bridging two Y-splitters with one straight section pair. (e) The mode power measured along the dotted line indicates 40.4 % overall transmission.
7. Conclusion
In this paper, we showed that the QCPW, a recently proposed SPP waveguiding structure which mimics a V-groove with three staggered metal stripes, can be successfully utilized for implementing various photonic components. Of initial concern was the QCPW’s partially open geometry which may allow radiation into dielectric spacers or mode coupling in the presence of another QCPW. The simulation results showed, however, that the confinement is strong enough to allow <2 µm waveguide-to-waveguide separation. The simulation results also revealed that a couple of concepts originally developed in the context of MMIC, the original inspiration for QCPW, can be utilized to improve the performance of the QCPW-based photonic components. We could enhance the transmission through QCPW bends by incorporating an air-bridge, a well-known technique in MMIC. The utilization of half-modes in asymmetrized QCPWs is another example. A QCPW full-mode can be split into two half-modes and each half-mode can recover the full-mode pattern by growing or combining with the other half-mode. These characteristics of the QCPW half-mode played important roles in implementing splitters and interferometers. All these features will make the QCPW structure a viable element for the plasmonic integration.
Acknowledgment
This work was supported by Battelle Fund.
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