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We numerically analyzed the power-coupling characteristics between a high-index-contrast dielectric slot waveguide and a metal-insulator-metal (MIM) plasmonic slot waveguide as functions of structural parameters. Couplings due mainly to the transfer of evanescent components in two waveguides generated high transmission efficiencies of 62% when the slot widths of the two waveguides were the same and 73% when the waveguides were optimized by slightly different widths. The maximum transmission efficiency in the slot-to-slot coupling was about 10% higher than that in the coupling between a normal slab waveguide and an MIM waveguide. Large alignment tolerance of the slot-to-slot coupling was also proved. Moreover, a small gap inserted into the interface between two waveguides effectively enhances the transmission efficiency, as in the case of couplings between a normal slab waveguide and an MIM waveguide. In addition, couplings with very wideband transmissions over a wavelength region of a few hundred nanometers were validated.
© 2015 Optical Society of America
1. Introduction
Nanoscale optical waveguides, which are able to deliver highly confined optical modes, can be used as the basic elements of photonic circuits and have attracted considerable attention in recent years. High-index-contrast dielectric slot waveguides and plasmonic slot waveguides with metal-insulator-metal (MIM) structures are two representative, nanosized, optical waveguides [1–10]; they are based on different principles but can confine light in nanometer-wide slots with the same polarization and similar mode profiles. The dielectric slot waveguide is a kind of silicon-on-insulator (SOI) nanowire. Since it was put forward by Almeida in 2004 [1], many related studies of basic analyses [2–4] and feature-based devices [11–15] have been proposed. The structure of the dielectric slot waveguide comprises a thin, low-index dielectric slot embedded between two high-index dielectric slabs. Because of the large discontinuity in the quasi-TM mode at the high-index-contrast interface, the electric field is more effectively confined in the low-index slot than in the high-index slabs. Therefore, in such waveguide geometry, light fields can propagate with constant nanosized mode profiles [1–4]. MIM waveguides based on oscillation and propagation of surface-plasmon polariton (SPP) optical waves at metal-insulator interfaces can also tightly confine lights within the nanosized slot region [5–10]. Because the slot is sufficiently narrow, the gap distance between two metal-insulator interfaces is less than the characteristic decay length of an evanescent plasmonic wave. The plasmonic waves around two interfaces interact and overlap, which results in a highly confined symmetrical mode.
Comparing the propagation characteristic of dielectric slot waveguides and MIM waveguides, the former (which consists of dielectric materials) has less propagation loss and is a true eigenmode [1]. The latter has such large loss due to the absorption of metal that its propagation length is limited to just tens of micrometers at the optical communication wavelength (~1550 nm) [16, 17]. On the other hand, MIM waveguides can confine almost all the light in the slot and provide high propagation efficiency through the sharp bends of the waveguides [18, 19]. However, dielectric slot waveguides typically confine about 30%, at most, of the power within a slot region [1, 2] and cannot avoid large bending loss [20]. Since dielectric slot waveguides and MIM waveguides have their own unique advantages and performances, the overall performance of a photonic circuit that incorporates both will be improved. In short, a very small or sharp bending part utilizes an MIM waveguide, and a transmission line with a relatively long span employs a dielectric slot waveguide. To achieve effective integration, highly efficient coupling between dielectric slot waveguides and MIM waveguides is of great importance.
It is noteworthy that numerous papers have demonstrated highly efficient coupling between high-index slab waveguides with cores of hundreds of nanometers and MIM waveguides. An efficiency of 68% was achieved by directly connecting the two types of waveguides [21], and that efficiency has been improved upon by the use of various types of couplers [22–24]. Although high-index slab waveguides and dielectric slot waveguides have similar structures, remarkable light-power confinement within a small slot core is achieved in dielectric slot waveguides, but not in the other. In an MIM waveguide, it is interesting that light coupling occurs between evanescent light components propagating in slot cores with similar scales. As mentioned above, coupling characteristics have mainly been analyzed for couplings between normal, high-index-contrast slab waveguides and MIM waveguides; however, a few slot-to-slot couplings between dielectric slot waveguides and MIM waveguides have been reported [25]. The authors of [25] proposed a fabrication method and experimentally verified the feasibility of slot-to-slot coupling between their waveguides; however, to our knowledge, a comprehensive analysis has not been reported until now.
In this paper, we evaluate, in detail, the transmission characteristics of the slot-to-slot connection between dielectric slot waveguides and MIM waveguides. We first investigate transmission efficiency with respect to variable waveguide parameters (such as slot width) in simple waveguide connections. We discuss the optimum combinations of those parameters that provide the highest transmission efficiency. Then, we present the estimation of the alignment tolerance of the slot-to-slot coupling, which contributes to practical fabrication. We also report on the transmission enhancement effect resulting from inserting a compact gap into the interface between two waveguides. Finally, we discuss the transmission efficiency’s dependence on wavelength over a broadband wavelength range. The remainder of this paper is organized as follows. In Section 2, the simulation method is described. The results and analysis are presented in Section 3, and conclusions are summarized in Section 4.
2. Simulation method
Coupling between dielectric slot waveguides and MIM waveguides was simulated by a two-dimensional finite-difference time domain (FDTD) method at a 1550 nm wavelength. The uniform grid resolution was 1 nm, both in the transmission and transverse directions. This size was small enough to accurately capture field changes at the high-index-contrast interface while also ensuring an acceptable processing time. The refractive index of the background and slot was 1, which corresponds to air. The employed high-index dielectric material and metal material were both made from silicon and silver, whose dielectric constants are 12.376 and −103.715 + i8.231, respectively. These dielectric constants were achieved from Material Editor Library of simulation software [31]. Because of the non-zero imaginary parts of the refractive index, material loss was taken into account in our simulation. The coupling efficiency used in this investigation was defined as the ratio of the power transmitted into the MIM waveguides to launch power. The power flux in the MIM waveguides was measured by a monitor immediately after the interface between these two waveguides. The launch light into the dielectric slot waveguides was used as the fundamental mode for the waveguide profile. Propagation loss in the dielectric slot waveguide, which has an extremely short length, was ignored in the simulation.
3. Results and analysis
First, the coupling properties between the dielectric slot waveguides and MIM waveguides were evaluated when the slots in each waveguide had the same width, as shown in Fig. 1(a). Here, Wslot and Wslab indicate the widths of the slot core and side core. The widths of the silver layer in MIM waveguides are much larger than Wslab. The transmission efficiency, which is equivalent to the coupling efficiency in our model, was measured as a function of various values of Wslab and Wslot. Figure 1(b) shows the transmission efficiency’s dependence on Wslab when Wslot = 20, 50 and 80 nm. We can see that transmission efficiency is optimized at 62% when Wslab reaches about 150 nm for all Wslot values. In [1], 30%, at most, of the total power could be confined in the slot regions of dielectric slot waveguides. In our simulation, a higher confinement (up to 40%) resulted from the higher index contrast, which was itself caused by the air slot having a lower refractive index than that found in [1]. However, the transmission efficiency is much higher than the confinement ratio of 40%, which indicates that light delivered into MIM waveguides not only originates from the slot of the dielectric slot waveguide but also from the slabs on both sides of the slot. This characteristic accounts for the effect that extraordinary optical transmission (EOT) observed for couplings between a normal slab waveguide and an MIM waveguide [21] and is verified by the behavior of the time-averaged Poynting vector near the interface, as shown in the inset of Fig. 1(a). When the light transmit through a metal slit with a subwavelength width, light from an area larger than the slit width will be delivered into the slit, and an extraordinary optical transmission can be observed. This effect is found and researched for a long time [26, 27], and applied by severalfields. For convenience, in the following text, we call this effect as “funnel effect”. In the inset of Fig. 1(a), the power flow in the slot region of the MIM waveguide is apparently enhanced by the power flow from the pair of slabs in the dielectric slot waveguide. Furthermore, because the width of the slab is the main determining factor in the total mode size of the dielectric slot waveguide and because a relatively small mode size is propitious to coupling light into MIM waveguides, Wslab which directly determines the mode size has a large effect on transmission efficiency. We see in Fig. 1(b) that the modal energy spreads out of the high-index region into its surroundings when Wslab < 140 nm; when Wslab > 160 nm, the modal energy spreads over the cladding, and the mode size increases with Wslab, which results in a decrease in the coupling efficiency. In Fig. 1(c), we show the transmission efficiency’s dependence on Wslot when Wslab = 150 nm, 180 nm, and 200 nm. As Wslot increases from 0 nm to 10 nm, the transmission efficiency rapidly increases and becomes almost constant for Wslab = 150 nm and 180 nm. In contrast, the transmission efficiency peaks around Wslot = 10 nm when Wslab = 200 nm. This result shows that a high coupling efficiency of about 62% can be retained over a wide range of Wslot and that an appropriately designed Wslab is tolerant to Wslot. The differences in transmission for different Wslab values are mainly caused by the mode size in the dielectric slot waveguides, as explained above. The transmission and reflection as functions of Wslot at Wslab = 150 nm are shown in Fig. 1(d). The curve calculated for an MIM consisting of lossless silver is also plotted in Fig. 1(d); we can see that the most loss is caused by reflection at the silicon-silver interface because there is little difference between the cases with and without material loss of silver. In Fig. 2, we show a typical contour map of an electric field propagating from the left-side dielectric slot to the right-side MIM waveguide for Wslab = 150 nm and Wslot = 50 nm. Very high light confinement can be clearly observed within the two slot areas.
Fig. 1 (a) Schematic model of joint between dielectric slot waveguide and MIM waveguide with same slot width. (b) Transmission efficiency as function of Wslab when Wslot = 20 nm, 50 nm, and 80 nm. Behavior of time-averaged Poynting vector near interface is shown in inset. (c) Transmission efficiency as function of Wslot when Wslab = 150 nm, 180 nm, and 200 nm. (d) Transmission efficiency and reflection coefficient as function of Wslot for Wslab = 150 nm. Transmission efficiency calculated for MIM with lossless silver is also plotted for reference.
Fig. 2 Field distribution around joint between dielectric slot waveguide and MIM waveguide for Wslot = 50 nm and Wslab = 150 nm.
Next, we discuss the transmission efficiency for general slot-to-slot couplings when the slot widths Wsd of a dielectric slot waveguide and Wsp of an MIM plasmonic waveguide are different, as shown in Fig. 3(a). Figure 3(b) shows the transmission efficiency’s dependence on Wsd for Wsp = 20 nm, 50 nm, and 80 nm. The dielectric slot waveguide for Wsd = 0 corresponds to a conventional high-index slab waveguide. One can see that the transmission takes maximum values of 62%, 72%, and 73% for Wsp = 20 nm, 50 nm, and 80 nm at Wsd = 5 nm, 10 nm, and 12 nm, respectively. The Wsd values that provide the maximum transmissions are shorter than every Wsp. Furthermore, the maximum transmissions are approximately 10% higher than those when Wsd = 0, which corresponds to the coupling between a normal dielectric slab and an MIM waveguide. Here, the transmission for Wsd = 0 is in good agreement with the data from [21]. Compared with those data, our results suggest that a thin slot inserted into a dielectric waveguide can effectively enhance the transmission efficiency between the dielectric waveguide and MIM waveguide. However, as Wsd continues to increase, the transmission efficiency tends to decrease constantly, becoming even lower than that forWsd = 0. This behavior is caused by the optimization of the power fluxes from the dielectric slot waveguide into the MIM waveguide (which flow both from a thin slot (confining very large light power) and from the side cores (mainly inducing a funnel effect)) at adequate combinations of Wsd and Wsp. A separate change of Wsd will lead transmission efficiency alter with mode size of dielectric slot waveguides. In order to investigate this point and to determine the relationship between mode size and Wsd, we considered the effective width of the fundamental eigenmode as a mode-size evaluation criterion. Effective width can be understood as effective area with a reduced dimension. A smaller effective area indicates a smaller mode size. According to the definition of effective area in dielectric slot waveguides [28, 29], the effective width is expressed as
where F(x) is the field profile of the fundamental mode of the dielectric slot waveguide [30] (F(x) = Ex(x), for TM polarization), and HW corresponds to the half width of dielectric slot waveguides (HW = Wsd/2 + Wslab). Figure 3(c) shows Weff as a function of Wsd in the case of Wslab = 150 nm. The field profiles at Wsd = 10 nm, 50 nm, and 100 nm are also shown as typical examples. One can see that the minimum effective width is obtained around Wsd = 10 nm, which is consistent with the previous results, in which the maximum transmission is observed at Wsd = 10 nm, as shown in Fig. 3(b). Although mode size is not the only factor determining the coupling efficiency, the highest coupling is observed when the input dielectric waveguide has the smallest effective width of about 0.31 μm under the simulation conditions. It is noted that the smallest effective width was not achieved in the normal slab waveguide but only in the slot waveguide structure. In Fig. 3(d), we show the transmission efficiency’s dependence on the slot width Wsp of the MIM waveguide when Wsd = 0 nm, 10 nm, and 50 nm. As mentioned above, the input slot waveguide with a 10 nm slot width provides a higher transmission than does a normal slab waveguide without a slot. The optimal Wsp that provides the maximum transmission shifts towards larger values as Wsd increases, and a high transmission of more than 60% is obtained for Wsd = 50 nm across the range of Wsp (which is approximately 150 nm). This flatness will facilitate the practical production of a MIM waveguide.Fig. 3 (a) Schematic model of waveguide connection with slots of different widths. (b) Transmission efficiency as function of Wsd when Wsp = 20 nm, 50 nm, and 80 nm. (c) Effective width as function of Wsd for Wslab = 150 nm. Insets show profiles of propagation mode in dielectric slot waveguides with slot widths of 10 nm, 50 nm, and 100 nm. (d) Transmission efficiency as function of Wsp when Wsd = 0 nm, 10 nm, and 50 nm.
Figures 4(a) and 4(b) illustrate the behavior of the time-averaged Poynting vector (indicated by arrows) and the field distribution at the interface region between dielectric slot waveguides with Wsd = 10 nm or 100 nm and MIM waveguides with Wsp = 50 nm. Althoughthe funnel effect is observable in both cases, a higher power distribution can be seen around the joint region of the two slot cores in Fig. 4(a).
Fig. 4 Behavior of time-averaged Poynting vector (arrows), and distribution for (a) Wsd = 10 nm and (b) Wsd = 100 nm when Wsp = 50 nm.
Due to the practical difficulty in accurately aligning nanosized waveguides, a minimum alignment tolerance of the coupling between dielectric slot waveguides and MIM waveguides is required. To evaluate the influences of misalignments, we built a coupling model with displacement D (distance between the center positions of two cores), as shown in Fig. 5(a). Here, the slot width Wsp of MIM waveguides is a constant 50 nm, while the slot width Wsd of dielectric slot waveguides takes three values: 0 nm, 10 nm, and 50 nm. Figure 5(b) shows the transmission efficiency as a function of D. First, in the waveguide where Wsd = 0, which is equivalent to normal slab waveguides, we find that the transmission monotonically decreases with D and falls to one-half of its original value at about D = 50 nm. In contrast, in the case of dielectric slot waveguides with Wsd = 10 nm and 50 nm, the transmission efficiency maintains approximately 72% and 68% when the critical D = 30 nm and 50 nm, respectively, and quickly decays beyond the critical D. Thus, the critical D indicates that the two slot cores of the input and output waveguides are entirely misaligned; as long as D < Wsd/2 + Wsp/2 (corresponding to the alignment tolerance range), high transmission can be expected in slot-to-slot coupling. Figures 5(c) and 5(d) show the electric field distributions at D = 20 nm and 40 nm in the case of Wsd = 10 nm and 50 nm, respectively. We can see that light fields fully transmit through the reduced contact region between the two slot cores.
Fig. 5 (a) Schematic model of waveguide connection with position misalignment D between dielectric slot waveguide and MIM waveguide. (b) Transmission efficiency as function of D when Wsd = 50 nm, 10 nm, and 0 nm for Wsp = 50 nm. Field distributions for (c) Wsd = 10 nm and D = 20 nm, (d) Wsd = 50 nm and D = 40 nm, when Wsp = 50 nm, respectively.
Next, we consider the light coupling through the air gap at the tail end of a dielectric slot waveguide, with a structure similar to that reported in [22]. In Fig. 6(a), we show a waveguide bond model with a gap of length Lg and width Wg. The transmission efficiency is presented as a function of Lg and Wg in Fig. 6(b). Here, the slot widths Wsd and Wsp of the two waveguides are 50 nm. It is evident from Fig. 6(b) that the air gap in dielectric slot waveguides effectively improves the transmission efficiency in slot-to-slot waveguide coupling. When Wg = 250 nm, the coupling efficiency rises to above 73% for 15 nm < Lg < 55 nm. In particular, the gap’soptimal dimensions appear at Lg = 35 nm and Wg = 250 nm, for which the transmission efficiency is enhanced from 62% in the waveguide without the gap to 75% in the waveguide with the gap. Figure 6(c) shows a contour map of the field distribution when Lg = 35 nm and Wg = 250 nm. We can see the light fields slightly spreading around the gap region and highly confined within both slot cores. To investigate the effective transmission though this gap, the behaviors of the time-averaged Poynting vectors near the gap region delineated by white dotted lines in Fig. 6(c) are shown in Figs. 6(d) and 6(e). By comparing arrows, we can see that more power is distributed over a greater width of about 220 nm with a gap headed to the slot of the MIM waveguide than is distributed over a shorter width of about 160 nm without a gap. This difference indicates an enhanced funnel effect and explains the improvement in transmission efficiency that is caused by a gap. As is already known for couplings between normal dielectric waveguides and MIM waveguides, besides the air gap [22] inserted at the interface between two waveguides, taper couplers [23, 24] and multi-section taper couplers [21] also have an extraordinary effect on the coupling efficiency. All these couplers may be applied in slot-to-slot coupling. Moreover, other devices—such as optical splitters or mode field converters—may be created by dielectric slot waveguides and MIM waveguides. These devices will be examined in future work.
Fig. 6 (a) Schematic model of waveguide connection through air gap with width Wg and length Lg. (b) Transmission efficiency as function of Wg and Lg. (c) Field distribution around joint with air gap of Wg = 250 nm and Lg = 35 nm. (d) Behavior of time-averaged Poynting vector in local region delineated by white dotted lines in (c). (e) Behavior of time-averaged Poynting vector without air gap.
All of the investigations discussed above were conducted at a wavelength of λ = 1550 nm. Finally, we present the transmission efficiency’s dependence on wavelength in slot-to-slot couplings. Figure 7 depicts the transmission efficiency with respect to wavelength for three typical configurations, as shown in Fig. 1(a) (Wslab = 150 nm, Wslot = 50 nm), Fig. 2(a) (Wslab = 150 nm, Wsd = 10 nm, Wsp = 50 nm), and Fig. 6(a) (Lg = 35 nm, Wg = 250 nm). Since a waveguide with a fully long length is assumed in our model, none of the curves show any resonance. In Fig. 7(a), we can see a very high transmission over a broad wavelength range. In particular, the application of the air-gap coupler effectively broadened the spectrum over a range of about 1 μm. As the wavelength increases, changes in transmission efficiency are mainly caused by the variation of two factors. The first factor is the effective width of the dielectric slot waveguides. A greater wavelength directly leads to an increase in the effective width, as shown in Fig. 7(b). The second factor is the funnel effect at the entrance of the MIM waveguide. It is difficult to accurately calculate the intension of funnel effect. We demonstrated an approximated changing trend of it by simulation in a wavelength range of 1200 nm to 2200 nm. To realize this, we removed the cladding in two sides, and launched plane wave through the silicon-air-silicon region into the MIM waveguide. Duo to the waveguides structure was removed, light cannot be confined before transmitting into MIM waveguides, and the funnel effect became the only factor influencing the transmission efficiency. The transmission efficiency was measured and draw as the changing trend of funnel effect, as shown in Fig. 7(b). As the wavelength increases, the increased effectivemode width impairs the transmission efficiency between the dielectric slot waveguide and the MIM waveguide; meanwhile, the enhanced funnel effect favors coupling. These opposing effects indicate the presence of optimal wavelength regions. Although the transmission efficiency tends to decline for an air gap with a large width Wg (where the effective width is expanded for slot width increase and slab width decrease), the funnel effect due to an air gap can fully surpass the decline as long as the length Lg of the air gap is sufficiently small. In various wavelengths, the influence on coupling caused by the air gap is related to many parameters and requires more detailed study, which will be done in future work.
Fig. 7 (a) Comparison of transmission efficiency as function of wavelengths obtained from different structures (Figs. 1(a) (Wslab = 150 nm, Wslot = 50 nm), 2(a) (Wslab = 150 nm, Wsd = 10 nm, Wsp = 50 nm), and Fig. 6(a) (Wslab = 150 nm, Wslot = 50 nm, Lg = 35 nm, Wg = 250 nm)). (b) Effective width and intension of funnel effect for various wavelengths. (c) Schematic model of waveguide connection used to estimate funnel effect. Plane wave launches from the left.
4. Conclusion
We evaluated the slot-to-slot coupling between dielectric slot waveguides and MIM waveguides. We demonstrated that, when these two types of waveguides are directly connected, slabs of dielectric slot waveguides (Wslab) that are about 150 nm wide provide optimum coupling, under which a maximum coupling efficiency of 62% can be obtained for slots with the same width of 20–80 nm. We also proved that a relatively high coupling efficiency can be obtained when the slots of dielectric waveguides are smaller than those of MIM waveguides. For dielectric slot waveguides with 150-nm-wide slabs, the maximum transmission efficiency reached 72% and 73% at slot widths Wsd of 10 nm and 12 nm and Wsp of 50 nm and 80 nm, respectively. The slot width for optimum transmission is in good agreement with the minimum mode size (~10 nm) of the dielectric slot waveguide. In addition, we showed that slot-to-slot coupling has excellent alignment tolerance and that high coupling efficiency is maintained as long as the displacement is held within half the width of each slot of the two waveguides. Besides direct connection, we reported that an air gap in the interface between two waveguides can effectively improve the coupling efficiency by enhancing the funnel effect. The coupling efficiency with 50-nm-wide slots in the two waveguides increased from 62% to 75% by inserting an air gap that was 250 nm wide and 35 nm long (Wg = 250 nm, Lg = 35 nm). Finally, we showed that all of the couplings listed above are broadband. Our investigation determined the optimal coupling and proved the superior coupling between dielectric slot waveguides and MIM waveguides. These results provide the theoretical basis and motivation for integration of dielectrics slot waveguides and MIM plasmonic waveguides.
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