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Abstract
Plasmonic waveguides can offer a promising solution beyond the optical diffraction limit. However, the cost of shrinking mode sizes reflects in metallic ohmic losses that lead to a short propagation distance of light, hindering the practical applications of plasmonic waveguides. Herein, we tackled the practicality of a novel CMOS-compatible all-dielectric waveguide structure that exploits electromagnetic boundary conditions of both the continuous normal component of the electric displacement field and the tangential component of the electric field at a high-index-contrast interface, which allows the attainment of mode areas comparable with those of plasmonic waveguides and theoretical lossless. The proposed waveguide comprises two oppositely contacted nanoridges with semicircular tops embedded in a conventional slot waveguide. By stepping on the strong electric field in the low-index slot region of the slot waveguides, the nanoridges squeeze the mode areas further with a guiding mechanism identical to that of a surrounding slot waveguide. Through the design of the geometry parameters, the calculated mode area of the reported structure achieved an unprecedented order of 4.21 × 10−5 A0, where A0 is the diffraction-limited area. The mode area dependence on fabrication imperfections and spectral response showed the robustness and broadband operation. Moreover, on the basis of extremely tight mode confinements, the present waveguide even outperformed the hybrid plasmonic waveguides in lower crosstalk. The proposed idea makes the realization of practically feasible nanoscale photonic integrated circuits without any obstructions by the limited propagation distance of light for plasmonic waveguides, thereby expanding its applications in various nanophotonic and optoelectronics devices requiring strong light–matter interaction within nanoscale regions.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The on-chip miniaturization of the mode sizes of light is of crucial importance in many aspects, such as the enhancement of light–matter interaction, the manipulation of light at the deep-subwavelength scale, and the construction of nanophotonic circuits [1] to resolve the inherent limitation of transmission bandwidth due to the RC-delayed and energy dissipation in modern nanoscale electronic circuits [2,3]. Significant breakthroughs in the field of nanophotonics have mainly benefited from the rapid advancements of nanofabrication technology in the past few decades [4]. However, shrinking photonic devices are intrinsically limited by the Abbe diffraction limit; thus, new strategies to squeeze light beyond the diffraction limit are necessary. Among the promising candidates include coupling light with free electrons in noble metals in order to form hybrid waves of photons and electron oscillations called surface plasmon polaritons (SPPs) propagating at a dielectric-metal interface, at visible and near-infrared wavelengths [5,6]. In this principle, light can be well confined, thus, breaking the diffraction limit in a dielectric gap while bringing two dielectric-metal interfaces close enough. This is the so-called plasmonic gap waveguides with nanoscale-mode confinements [7,8]. Various geometries of plasmonic waveguides have also been reported to accomplish distinct degrees of light confinements [9–12]. However, although the plasmonic waveguides [7–12] can realize nanoscale mode sizes, the cost is often reflected in the metallic ohmic losses, which result in the high propagation losses of light of approximately the order of dB/µm, as compared with approximately dB/cm to dB/m with the utilization of dielectric waveguides with no metals.
Hybrid plasmonic waveguides (HPWs) [13–16] were proposed to optimally leverage the mode size and propagation loss of conventional plasmonic waveguides. HPWs couple the dielectric waveguide with the SPP modes. Subsequently, plasmon lasers [17,18] were fabricated according to the HPW structure to realize nanolasers emitting spot sizes below the diffraction limit. The success with HPWs brought about a variety of modified HPWs mainly under two categories: (1) extremely small mode areas of approximately 10−3 to 10−5 A0 with propagation length of a few tens to a few micrometers [19–30], respectively, and (2) longer propagation length of a few millimeters to hundreds of micrometers with corresponding mode areas of approximately 10−2 to 10−3 A0 [31–36], where A0 = λ2/4 (λ is the working wavelength in free space) is defined as the diffracted-limit mode area, as intensively reported in the recent decade. The two strategies imply that a trade-off between mode area and propagation loss occurs in plasmonic waveguide structures, leading to an obstacle to practically accomplish large-area photonic integrated circuits (PICs). In light of this, more effort needs to be continuously devoted to resolving the propagation losses of HPWs in the future. In addition to the waveguide devices operating at telecommunication wavelength of λ = 1550 nm, the designing concept can be adapted to various silicon photonic platforms and extended to λ = 1310 nm for datacom and other applications [37–39].
Another guiding mechanism to achieve subwavelength mode confinement without metals was by the slot waveguides [40,41] proposed from the pioneering work of Lipson's group. In a conventional slot waveguide (CSW), the stronger electric field in the low-index side is formed by fulfilling the continuity of the normal component of the electric displacement field at high-index-contrast interfaces. Placing two high-index-contrast interfaces close enough with the gap smaller than a few tens of nanometers will result in the stronger electric fields in the low-index side coupling with each other to squeeze most of the energy in the low-index slot region that is different from concentrating most of the energy in the high-index regions for conventional dielectric waveguide modes. However, the slot waveguides achieved the mode areas only of the order of 10−2 A0. A recent approach [42–44] that exploits a similar mechanism but is a more comprehensive solution considers both the normal component of the electric displacement field and the tangential component of the electric field, to design ultrasmall mode volumes of optical resonators in photonic crystal structures. Herein, we extend the application of this new approach [42–44] in the design of an all-dielectric waveguide to overcome the inherent diffraction limit, which enables the realization of mode areas comparable with that of HPWs, but with theoretical lossless. In this work, we report a new structure constituting two oppositely contacted nanoridges with semicircular tops, to significantly improve the mode areas an order of magnitude higher. The geometry parameters of the proposed waveguide structure are extensively analyzed, and the fabrication tolerances are carried out. To evaluate the broadband operation, we investigate the mode size dependence on the wavelength. Finally, we looked into the crosstalk to show the degree of integration of the present design for building nanoscale PICs, the results of which are also compared with those of the HPWs.
2. Design principle and mode characteristics of the proposed waveguide
Although light in conventional dielectric waveguides is confined in high-index media clad by lower-index ones, in slot waveguides it is mainly squeezed in a subwavelength-scale low-index slot region surrounded by high-index materials [40,41], thus exploiting the electromagnetic boundary condition of discontinuity of the normal component of the electric field at high-index-contrast interfaces. Nevertheless, the shrinkage of the mode size of slot waveguides remains limited as compared with that of HPWs [13–16,19–30] which, for the last decade, have been capable of shrinking light into nanoscale regions. Although HPWs have been rewardingly successful with squeezing light to the nanoscale, the price has as well been reflected in the inevitable huge ohmic loss induced by the utilization of metals. A new approach [42–44] based mainly on all-dielectric materials, exploits two electromagnetic boundary conditions to design photonic crystal cavities with extremely small-mode volumes and a high-quality factor. As introduced above, we extend the application of this recent approach [42–44] to design all-dielectric waveguides with highly confined mode areas. Such a design principle is schematically shown in Fig. 1.
Fig. 1. (a) A conventional slot waveguide with a low-index medium (skin color) of permittivity εl surrounded by high-index media (deep blue color) of permittivity εh. (b) Bridging the high-index media of (a) by the identical high-index one. (c) Disconnecting the bridge of (b) by a low-index medium. Here, Dh,⊥ (Eh,⊥) and Dl,⊥ (El,⊥) are the electric displacement fields (electric fields) in the high- and low-index sides of the interface P1, respectively, which are perpendicular to P1. Likewise, Eh,‖ (D*h,⊥) and El,‖ (D*l,⊥) are the electric (electric displacement) fields in high- and low-index sides parallel (perpendicular) to the interface V1 (P2).
In CSWs [40,41], let us assume that the normal component of the electric field of a guided mode is Eh,⊥(El,⊥) in the high-(low-)index side (deep blue (skin) color) with permittivity εh (εl) of the interface P1, as shown in Fig. 1(a). Note that the boundary condition dictates the continuity of the normal components Dh,⊥ and Dl,⊥ (in the high- and low-index sides of the interface P1, respectively) of the electric displacement field in the interface P1:
From the constitutive relation of a dielectric, we have D = εE; therefore, the normal component of the electric field in the low-index side of the interface P1 can be expressed by
Equation (2) reveals that for a CSW, the enhancement ratio εh/εl of the electric field in the low-index side compares with that in the high-index side at P1. What if the slot region is bridged by the high-index medium of εh, as shown in Fig. 1(b)? Following the dictation of the electromagnetic boundary condition at the interface V1, we know that the tangential component of the electric field is continuous as represented by
Figures 1(a)–(c) show that El,‖ is identical to El,⊥; thus, we can express the formulation
Finally, by splitting the bridging region of Fig. 1(b) to form a minor slot region in Fig. 1(c) and by exploiting the electromagnetic boundary condition at the interface P2, we can obtain the continuity relation
Moreover, the electric field in the minor slot region is given by
Equation (6) reveals that the normal component of the electric field in the minor slot region can be enhanced further to that in the major slot region by εh/εl, as shown in Fig. 1(a). Theoretically speaking, the bridging and splitting processes can be carried out in an infinite amount of time in order to optimally enhance the electric field. However, note that time is restricted by the modern fabrication resolution, although the structure seems to become a bowtie after infinite times of bridging and splitting, mainly due to asymptotic geometry speculations [42–44]. Below is a comparison of the mode confinements of several structures. Here, we adopted the normalized mode area, Am = Ae/A0, which is widely used in evaluating the degree of light confinement of plasmonic waveguides [13–16,19–36]. The effective mode area Ae defined by Eq. (7) depicts the ratio of the total mode energy Wm over the peak of the energy density W(r), whose formulation is given by Eq. (8):
Figure 2 displays a comparison of the mode confinements of several structures. The relative permittivities of the high-index Si, low-index porous SiO2, and conventional SiO2 substrate used were εSi = 12.097 [45], εp-SiO2 = 1.103 [46], and εSiO2 = 2.085 [45], respectively, at a telecommunication wavelength of λ = 1550 nm. The geometry parameters are g = 20 nm, hSi = 220 nm, wSi = 180 nm, wr = 10 nm, gm = 2 nm, t = 2 nm, r = wr/2, hs = 5 nm, and hr = 5 nm. In a standard silicon-on-insulator platform, Si with a thickness of 220 nm (the reason the value of hSi = 220 nm) was deposited on a SiO2 substrate with 3 µm of a Si wafer. The numerical results were analyzed under the commercial COMSOL Multiphysics software based on the rigorous finite element method.
Fig. 2. Waveguide structures of (a) a conventional slot waveguide (CSW) with gap g = 20 nm; (c) a CSW with a high-index bridge (CSW-HIB) having a width wr = 10 nm; (e) a CSBW-HIB split by a low-index slot (CSW-HIBLIS) of thickness gm = 2 nm; (g) a CSW with an embedding bowtie structure (CSW-B), where the connecting width is t = 2 nm; (i) a CSW with an embedding single nanoridge (CSW-SNR) whose curvature of the radius of the top and the height are r = wr/2 and hs = 15 nm, respectively; and (k) a CSW with embedding two oppositely contacted nanoridges (CSW-TNR, this work), each with a height hr = 5 nm. The energy density profiles and the values of Am of (a), (c), (e), (g), (i), and (k) are illustrated in (b), (d), (f), (h), (j), and (l), respectively. (m) The normalized energy densities Wn = W(r)/Wmax_(k), where Wmax_(k) denotes the peak of W(r) of the structure (k).
Figures 2(a), 2(c), 2(e), 2(g), 2(i), and 2(k) show a schematic representation of a CSW, a CSW with a high-index bridge (CSW-HIB), a CSW-HIB split by a low-index slot (CSW-HIBLIS), a CSW with an embedding bowtie (CSW-B), a CSW with an embedding single nanoridge (CSW-SNR), and a CSW with embedding two oppositely contacted nanoridges (CSW-TNR, this work), respectively, whereas their corresponding W(r) profiles along with the normalized mode areas are shown in Figs. 2(b), 2(d), 2(f), 2(h), 2(j), and 2(l), respectively. Displayed in Fig. 2(m) are the one-dimensional normalized energy densities Wn of all the structures, which are the W(r) normalized by the peak of W(r) of the structure Fig. 2(k), at the central line of the slot region in the x-direction. As can be observed clearly, W(r) can be squeezed progressively from Am = 2.98 × 10−2 of the CSW structure [see Fig. 2(b)] to Am = 1.69 × 10−3 of the CSW-HIBLIS [see Fig. 2(f)] by more than one order of magnitude of Am. On the basis of the theoretical values resulting from Eqs. (2) and (6), the ratio of the peak values of El,*⊥and El,⊥ would be approximately εh/εl = 11.94, whereas this ratio [see Fig. 2(m)] according to the present numerical calculation would be approximately 8.12. The moderate deviation of the ratio from the theoretically and numerically calculated values is dictated by the chosen finite thickness of the minor slot gm = 2 nm. For instance, the theoretical ratio of 11.94 can be reached if gm tends to zero, which is limited by the practical fabrication resolution. Thus, the aforementioned result validates the correctness of the present design principle. Previously [42–44], it was reported that the ultimate geometry after infinite cycles of bridging [see Fig. 2(c)] and splitting [see Fig. 2(e)] is a bowtie structure with optimal enhancement and mode confinement. We thus analyzed the CSW-B structure in Fig. 2(g) with t = 2 nm not t = 0 nm due to the experimental difficulty to precisely fabricate the condition of t = 0 nm. For the CSW-B structure, we obtained Am = 1.05 × 10−3 after meshing around the narrowest region with sufficient fine grids sized 0.05 nm. This Am was moderately reduced from that of the CSW-HIBLIS structure. To further improve the mode confinement, we propose the novel waveguide structures of CSW-SNR [see Fig. 2(i)] and CSW-TNR [see Fig. 2(k)] in this paper. In the CSW-SNR structure, the semicircular top of the nanoridge contacts the bottom of the upper Si strip waveguide, and Am = 3.85 × 10−4 was reached with r = 5 nm and hs = 15 nm. For the CSW-TNR structure, we obtained Am = 1.24 × 10−4 at r = 5 nm and hr = 5 nm, which is one order of magnitude smaller than that of the CSW-B structure. This reduction could be attributed to the mode energy being squeezed to the varying nanoscale gap between the two nanoridges. As a result, the CSW-TNR structure showed the best mode confinement among those of other waveguide structures examined in this paper. We could also observe that the peak of Wn for the proposed CSW-TNR structure is far beyond those [Wn < 0.05; see the inset of Fig. 2(m)] of other structures. Specifically, CSW-SNR and CSW-B achieved nearly the same peak Wn = 0.045, whereas CSW-HIB, CSW-HIBLIS, and CSW had Wn = 0.026, 0.012, and 0.0032, respectively. Unprecedentedly, the Wn of the CSW can be significantly enhanced by stronger than 300 times using our proposed CSW-TNR structure.
Let us now investigate the CSW-TNR structure based on the aforementioned comparison of the mode sizes. Figure 3(a) shows a 3D schematic diagram of the structure, which comprises two contacted nanoridges embedded in a porous SiO2 slot layer sandwiched by two Si strip waveguides, with the front view with zoomed-in view detailed in Fig. 3(b). Here, the superstrate and substrate are air and conventional SiO2, respectively. The slot layer was drawn using a translucent color for a clear view of the interior structure. Moreover, porous SiO2 was used because of its lowest refractive index to date [46], making the highest contrast of refractive index to Si.
Fig. 3. (a) A 3D schematic diagram of the proposed CSW-TNR waveguide, which comprises two nanoridges embedded in a slot layer with porous SiO2 [46] sandwiched by two Si strip waveguides, deposited on a conventional SiO2 substrate. (b) The front view of (a) along with the zoomed-in view.
Before analyzing the performances of the proposed structure, the fabrication steps of the proposed waveguide are schematically shown in Fig. 4 including the following steps: (1) deposition of a Si and a photoresist (PR) films on a conventional SiO2 substrate; (2) definition of the lower Si layer of the SW, followed by PR exposure, development, and etching; (3) coating a PR film and then applying e-beam lithography to form a rectangular groove; (4) depositing a Si layer, followed by lifting off the PR to form a Si nanoridge; (5) rounding the top of the Si nanoridge using e-beam lithography; (6) coating a PR film; application of mask, PR exposure, and development to lift out the PR films next to the Si nanoridge; (7) evaporation of a porous SiO2 film using the oblique deposition technique [47], lifting out the PR film, and then using chemical mechanical polishing (CMP) to obtain a flat porous SiO2 plane; (8) applying e-beam lithography to form a groove with semi-circular top by controlling the exposure time and scanning speed; (9) repetition of step (6) and (7), but depositing Si film to form the upper Si nanoridge and SW.
We first study the geometrical effects on mode characteristics of the present CSW-TNR structure. Figures 5(a) and 5(b) show the Am and the effective refractive index ne versus hr for different values of wr of the nanoridges at hSi = 220 nm and wSi = 180 nm. It can be observed that the unprecedented small mode size of Am = 4.21 × 10−5 can be achieved at the condition wr = 2 nm and hr = 5 nm. To our best knowledge, this was the smallest Am obtained thus far with regard to only using all-dielectric materials. Compared with the latest results of Am with approximately 10−4 to 10−5 using various modified HPWs with noble metals [26], transition metal dichalcogenide [25,36], or graphene [27–30], the high ohmic losses lead to short propagation lengths from a few tens of micrometers to a few micrometers, making them unusable for the fabrication of practical PICs. Furthermore, we could see that for the proposed CSW-TNR structure, Am moderately increased to 9.12 × 10−5 as hr was increased from 5 to 20 nm [see Fig. 5(a)]. When hr = 5 nm was fixed, Am increased to 2.12 × 10−4 for values of wr up to 20 nm. Therefore, Am only moderately varied as hr or wr were increased to as large as 20 nm, which validates the robustness of the proposed CSW-TNR structure against the geometrical dimensions of the nanoridges. Moreover, note that the slops of Am and ne versus hr were larger for a smaller wr. At wr = 20 nm, the values of Am were nearly constant, regardless of the value of hr. Interestingly, the variation of Am was roughly proportional to the variation of wr. In addition, we could observe that the value of ne for the smaller wr was larger than that of the larger wr for hr < 10 nm, whereas the opposite result could be obtained for hr > 10 nm. The effect will be addressed in the next section.
Fig. 5. (a) Normalized mode area Am and (b) effective refractive index ne versus the height of nanoridge hr at several wr values of the nanoridges.
Figures 6(a)–(c) show the W(r) at hr = 0, 10, and 20 nm, respectively, for wr = 2 nm. Here, the mode profiles for the three conditions were relatively similar, except for the amplitudes of W(r), mainly because of the weaker field enhancement in the major slot region as hr was increased. To visually elucidate the energy profiles, we show the relative energy densities, Wn = W(r)/Wmax_(hr = 0), where Wmax_(hr = 0) is the peak of W(r) for hr = 0 nm, along the red dashed line indicated as q and the yellow dashed line indicated as s [see Fig. 3(b)] in Figs. 7(a) and 7(b), respectively. In particular, the smallest Am was observed when hr = 5 nm [see Fig. 5(a)], as more mode energy spread out of the slot region when the slot gap g [see Fig. 2(a)] was smaller than the threshold at hr < 5 nm. Quantitatively speaking, the peak of Wn was reduced by half roughly for every 10-nm increase of hr. It was also unprecedentedly expected that the full widths at half maximum (FWHMs) were smaller than 0.5 nm (depending on the wr) and 0.01 nm (due to the tiny gap between the nanoridges) along the x- and y-directions for the three conditions.
Fig. 6. Energy densities W(r) for the conditions (a) hr = 0 nm, (b) hr = 10 nm, and (c) hr = 20 nm at wr = 2 nm.
Fig. 7. Relative energy density, Wn = W(r)/Wmax_(hr = 0), where Wmax_(hr = 0) is the peak of W(r) for hr = 0 nm, along the (a) red dashed line q and (b) yellow dashed line s.
In Figs. 8(a)–(c), we further illustrate the W(r) values considering wr = 2, 10, and 20 nm, respectively, at hr = 5 nm.
As compared with the results when hr was increased for a fixed wr, the smallest value of wr = 2 nm led to both the strongest amplitude and the tightest mode profiles, attaining the smallest Am. We could validate the results by observing the relative energy densities, Wn = W(r)/Wmax_(wr = 2), where Wmax_(wr = 2) is the peak of W(r) at wr = 2 nm, along the lines q and s in Figs. 9(a) and 9(b). Quantitatively, the peak of W(r) was reduced to 1/e roughly for every 10 nm increase in wr. The FWHMs along the x-direction were approximately 0.5, 1, and 2 nm for wr = 2, 10, and 20 nm, respectively, whereas those along the y-direction were extremely small, at 0.01 nm.
Fig. 9. Relative energy density, Wn = W(r)/Wmax_(wr = 2), where Wmax_(wr = 2) is the peak of W(r) for wr = 2 nm, along the (a) red dashed line q and (b) yellow dashed line s.
Figures 10(a) and 10(b) show the Am versus the width wSi curves for several heights hSi at wr = 2 nm and hr = 5 nm. Am could be seen to be slightly dependent on wSi, whereas ne increased moderately with wSi and hSi. For the thickness of hSi = 340 nm used in other typical silicon photonic platforms [37], Am and ne are larger than that of smaller hSi. To leverage the Am and the dimension, we chose the condition hSi = 220 nm and wSi = 180 nm in the subsequent calculations unless stated otherwise.
Fig. 10. (a) Normalized mode area Am and (b) effective refractive index ne versus the width wSi for several heights hSi of the external Si waveguides.
Generally, the dimensions of nanoridges and the alignment between them are the most critical and challenging issues dealt with experimentally. Thus, we sought to determine the dependence of Am on the fabrication tolerance of the nanoridges. As far as the aforementioned analyses [see Fig. 5(a)] are concerned, the variation of Am is moderate from Am = 4.21 × 10−5 to Am = 2.1 × 10−4 for wr = 2 nm to wr = 20 nm, respectively, given hr = 5 nm. Likewise, Am varies from Am = 4.21 × 10−5 for hr = 5 and Am = 9.12 × 10−5 for hr = 20 nm at wr = 2 nm. On the basis of our results, the variation of Am was moderate within the ranges of δhr = 15 nm and δwr = 18 nm, thus, validating the robustness of the Am of the present waveguide. Next, we examined the alignment between the nanoridges. Figure 11 shows the curves depicting the Am versus the relative deviations Δx/wr, where Δx is the deviation in the x-direction between the nanoridges. We observe that Am varied from 1.24 × 10−4 to 1.87 × 10−3 for Δx/wr = 0 to 1, respectively, at wr = 10 nm and hr = 5 nm [see Fig. 11(a)]. Even though the maximum deviation of Δx/wr = 1, the variation of the Am values were still kept in one order of magnitude. We could observe similar results in Am at hr = 20 nm [see Fig. 11(b)], which reveals the accepted fabrication tolerance of the proposed structure.
Fig. 11. Am dependence on the relative deviation Δx/wr at different wr for (a) hr = 5 nm and (b) hr = 20 nm.
Moreover, the operating bandwidth is an essential issue in the design of wideband photonic devices. Here, we proceeded in addressing the spectral response of Am. Figure 12 shows plots of Am versus the working wavelength of the present waveguide, where the results demonstrate that the Am was almost wavelength independent within the λ = 1400–1650 nm, which signifies the ability of broadband operation of the proposed structure.
Fig. 12. Curves of Am versus the working wavelength of the proposed waveguide for three conditions: wr = 2 nm and hr = 0 nm, wr = 2 nm and hr = 5 nm, and wr = 10 nm and hr = 5 nm.
3. Waveguide crosstalk
Fig. 13. (a) Schematic of a coupled waveguide system composed of two parallel present waveguides with a center-to-center distance d. Field profiles of Ey of the (b) symmetric and (c) asymmetric modes at d = 500 nm, hSi = 220 nm, wSi = 180 nm, wr = 2 nm, and hr = 5 nm. Coupling length Lc versus d for (d) wr = 2 nm, (e) hr = 5 nm, and (f) hr = 20 nm and at (g) wr = 2 nm and hr = 0 nm for different dimensions of external Si of (i) wSi = 180 nm and hr = 220 nm and (ii) wSi = 200 nm and hr = 260 nm, along with that of the latest reported HPW by Bian et al. [26].
Crosstalk is a vital index to consider when assessing the on-chip compactness for the construction of high-density PICs. Let us analyze the crosstalk of a coupled waveguide system in Fig. 13(a) having two parallel waveguides with a center-to-center distance of d. The electric field components Ey of the symmetric and asymmetric modes supported by the coupled waveguide system are shown in Figs. 13(b) and 13(c), respectively, for d = 500 nm, hSi = 220 nm, wSi = 180 nm, wr = 2 nm, and hr = 5 nm.
Theoretically, the coupling length Lc = λ/[2(ne − no)] of a coupled waveguide system [48], where ne and no are the effective refractive indices of the symmetric and asymmetric modes, is used to quantitatively evaluate the degree of crosstalk. Figure 13(d) illustrates the Lc versus d curves for several hr values at wr = 2 nm. Unexpectedly, Lc for hr = 0 nm with a larger Am [see Fig. 5(a)] was larger than that for hr = 5 nm with a smaller Am. By principle, a smaller Am suggests better mode confinement, and thus, the weaker coupling strength between waveguides. Another factor must exist to influence the mode coupling of adjacent waveguides. Note that ne = 1.9511 for hr = 0 nm is larger [see Fig. 5(b)] than ne = 1.7076 for hr = 5 nm, which implies that more power remains within the high-index Si nanoridges than for a smaller ne. Therefore, a higher ne compensates the looser effect of a larger Am for guided modes, thereby moderately increasing the mode tightness to reduce the mode coupling. Consequently, both the ne and Am should be considered simultaneously when evaluating the mode coupling strength. To verify this deduction further, we considered the conditions hr = 5 and 20 nm for different values of wr; the results are illustrated in Figs. 13(e) and 13(f), respectively. For hr = 5 nm, Am was larger whereas ne was smaller for an increase in wr (see Fig. 5), resulting in a shorter Lc [see Fig. 13(e)]. By contrast, Am and ne were larger with an increase of wr for hr = 20 nm; a larger ne balances a larger Am for a larger wr, making a longer Lc [see Fig. 13(f)]. To demonstrate the crosstalk prevention capability from the adjacent waveguides, we compared the Lc values of the present waveguide with the latest reported HPW [26] having the smallest Am, as shown in Fig. 13(g). As could be observed, the Lc value at wSi = 200 nm and hr = 260 nm was longer than that at wSi = 180 nm and hr = 220 nm for wr = 2 nm and hr = 0 nm, which indicates weaker coupling strength between the waveguides. Referring to Fig. 10, we could see that the Si strip with the larger dimension has the larger ne as well, although it had almost the same value of Am as that of the smaller dimension, which reaffirms our deduction that Lc is determined by both Am and ne. Even after a comparison with the HPW having the best-ever mode confinement, the proposed waveguide still achieved lower crosstalk, demonstrating its superior advantage over available HPWs [19–36] for building highly dense photonic devices.
4. Summary
The present study has tackled the feasibility of an all-dielectric waveguide structure, constituting two oppositely contacted nanoridges with semicircular tops embedded in a CSW, to realize an ultrasmall mode size far beyond the Abbe diffraction limit through the utilization of electromagnetic boundary conditions of the continuous normal component of the electric displacement field and the tangential component of the electric field at a high-index-contrast interface. Instead of using plasmonic waveguides that can break the optical diffraction limit to squeeze light into nanoscale mode sizes, the proposed waveguide structure achieves an unprecedented mode area of 4.21 × 10−5 A0 without metallic ohmic loss confronted in plasmonic waveguides. The high fabrication tolerance and wavelength independence on the mode area allow the proposed design to alleviate the experimental difficulty and feature the broadband operation. The lower crosstalk of the present waveguide as compared with those of plasmonic waveguides show the high-density integration of PICs. The proposed idea paves a novel approach to realizing practical nanophotonic components with theoretical lossless, thus offering an alternative over other plasmonic waveguides. Notably, the deep-subwavelength mode confinement of the proposed waveguide has significant impacts in comprehensive applications, such as the enhancement of light–matter interaction, construction of higher-efficiency light sources, improvement of the sensitivity of optical sensing, decrease in power consumption for optical modulators, and increase in the resolution of near-field optical probes.
Funding
Ministry of Science and Technology, Taiwan (109-2112-M-005-005).
Acknowledgments
The authors would like to thank Enago (www.enago.tw) for the English language review.
Disclosures
The author declares no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. R. Kirchain and L. Kimerling, “A roadmap for nanophotonics,” Nat. Photonics 1(6), 303–305 (2007). [CrossRef]
2. D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88(6), 728–749 (2000). [CrossRef]
3. J. A. Conway, S. Sahni, and T. Szkopek, “Plasmonic interconnects versus conventional interconnects: a comparison of latency, crosstalk and energy costs,” Opt. Express 15(8), 4474–4484 (2007). [CrossRef]
4. A. F. Koenderink, A. Alu, and A. Polman, “Nanophotonics: shrinking light-based technology,” Science 348(6234), 516–521 (2015). [CrossRef]
5. A. D. Boardman, Electromagnetic Surface Modes, (John Wiley & Sons, 1982).
6. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef]
7. E. Moreno, F. J. G. Vidal, S. G. Rodrigo, L. M. Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31(23), 3447–3449 (2006). [CrossRef]
8. M. Yan and M. Qiu, “Guided plasmon polariton at 2D metal corners,” J. Opt. Soc. Am. B 24(9), 2333–2342 (2007). [CrossRef]
9. J. Dintinger and O. J. F. Martin, “Channel and wedge plasmon modes of metallic V-grooves with finite metal thickness,” Opt. Express 17(4), 2364–2374 (2009). [CrossRef]
10. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73(3), 035407 (2006). [CrossRef]
11. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010). [CrossRef]
12. Z. Han and S. I. Bozhevolnyi, “Radiation guiding with surface plasmon polaritons,” Rep. Prog. Phys. 76(1), 016402 (2013). [CrossRef]
13. M. Z. Alam, J. Meier, J. S. Aitchison, and M. Mojahedi, “Super mode propagation in low index medium,” Conference on Lasers and Electro-Optics/Quantum Electronics (Optical Society of America), paper JThD112 (2007).
14. R. F. Oulton, V. J. Sorger, D. Genov, D. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008). [CrossRef]
15. V. J. Sorger, Z. Ye, R. F. Oulton, Y. Wang, G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2(1), 331 (2011). [CrossRef]
16. M. Z. Alam, J. S. Aitchison, and M. Mojahedi, “A marriage of convenience: hybridization of plasmonic and dielectric waveguide modes,” Laser Photonics Rev. 8(3), 394–408 (2014). [CrossRef]
17. R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009). [CrossRef]
18. Y. J. Lu, J. Kim, H. Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C. Y. Wang, M. Y. Lu, B. H. Li, X. G. Qiu, W. H. Chang, L. J. Chen, G. Shvets, C. K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012). [CrossRef]
19. D. Dai and S. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement,” Opt. Express 17(19), 16646–16653 (2009). [CrossRef]
20. Y. Bian, Z. Zheng, Y. Liu, J. Zhu, and T. Zhou, “Dielectric-loaded surface plasmon polariton waveguide with a holey ridge for propagation-loss reduction and subwavelength mode confinement,” Opt. Express 18(23), 23756–23762 (2010). [CrossRef]
21. S. Zhu, T. Y. Liow, G. Q. Lo, and D. L. Kwong, “Silicon-based horizontal nanoplasmonic slot waveguides for on-chip integration,” Opt. Express 19(9), 8888–8902 (2011). [CrossRef]
22. C. C. Huang, “Metal nanoridges in hollow Si-loaded plasmonic waveguides for optimal mode properties and ultra-compact photonic devices,” IEEE J. Sel. Top. Quantum Electron. 20(4), 85–93 (2014). [CrossRef]
23. Y. S. Bian and Q. H. Gong, “Deep-subwavelength light confinement and transport in hybrid dielectric-loaded metal wedges,” Laser Photonics Rev. 8(4), 549–561 (2014). [CrossRef]
24. Y. S. Bian and Q. H. Gong, “Metallic-nanowire-loaded silicon-on-insulator structures: a route to low-loss plasmon waveguiding on the nanoscale,” Nanoscale 7(10), 4415–4422 (2015). [CrossRef]
25. K. Zheng, J. Song, and J. Qu, “Hybrid low-permittivity slot-rib plasmonic waveguide based on monolayer two dimensional transition metal dichalcogenide with ultra-high energy confinement,” Opt. Express 26(12), 15819–15824 (2018). [CrossRef]
26. Y. Bian, Q. Ren, L. Kang, T. Yue, P. L. Werner, and D. H. Werner, “Deep-subwavelength light transmission in hybrid nanowire-loaded silicon nano-rib waveguides,” Photonics Res. 6(1), 37–45 (2018). [CrossRef]
27. D. Teng, K. Wang, Z. Li, and Y. Zhao, “Graphene-coated nanowire dimers for deep subwavelength waveguiding in mid-infrared range,” Opt. Express 27(9), 12458–12469 (2019). [CrossRef]
28. D. Teng, K. Wang, and Z. Li, “Graphene-coated nanowire waveguides and their applications,” Nanomaterials 10(2), 229 (2020). [CrossRef]
29. D. Teng, K. Wang, Q. Huan, W. Chena, and Z. Li, “High-performance light transmission based on graphene plasmonic waveguides,” J. Mater. Chem. C 8(20), 6832–6838 (2020). [CrossRef]
30. X. He, F. Liu, F. Lin, and W. Shi, “Tunable 3D Dirac-semimetals supported mid-IR hybrid plasmonic waveguides,” Opt. Lett. 46(3), 472–475 (2021). [CrossRef]
31. A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. 23(1), 413–422 (2005). [CrossRef]
32. Y. Bian, Z. Zheng, X. Zhao, J. Zhu, and T. Zhou, “Symmetric hybrid surface plasmon polariton waveguides for 3D photonic integration,” Opt. Express 17(23), 21320–21325 (2009). [CrossRef]
33. L. Chen, T. Zhang, X. Li, and W. Huang, “Novel hybrid plasmonic waveguide consisting of two identical dielectric nanowires symmetrically placed on each side of a thin metal film,” Opt. Express 20(18), 20535–20544 (2012). [CrossRef]
34. C. C. Huang, “Ultra-long-range symmetric plasmonic waveguide for high-density and compact photonic devices,” Opt. Express 21(24), 29544–29557 (2013). [CrossRef]
35. Y. Ma, G. Farrell, Y. Semenova, and Q. Wu, “Hybrid nanowedge plasmonic waveguide for low loss propagation with ultra-deep-subwavelength mode confinement,” Opt. Lett. 39(4), 973–976 (2014). [CrossRef]
36. K. Zheng, Y. Yuan, J. He, G. Gu, F. Zhang, Y. Chen, J. Song, and J. Qu, “Ultra-high light confinement and ultra-long propagation distance design for integratable optical chips based on plasmonic technology,” Nanoscale 11(10), 4601–4613 (2019). [CrossRef]
37. K. Giewont, K. Nummy, F. A. Anderson, J. Ayala, T. Barwicz, Y. Bian, K. K. Dezfulian, D. M. Gill, T. Houghton, S. Hu, B. Peng, M. Rakowski, S. Rauch, J. C. Rosenberg, A. Sahin, I. Stobert, and A. Stricker, “300-mm monolithic silicon photonics foundry technology,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–11 (2019). [CrossRef]
38. M. Rakowski, C. Meagher, K. Nummy, A. Aboketaf, J. Ayala, Y. Bian, B. Harris, K. Mclean, K. McStay, A. Sahin, L. Medina, B. Peng, Z. Sowinski, A. Stricker, T. Houghton, C. Hedges, K. Giewont, A. Jacob, T. Letavic, D. Riggs, A. Yu, and J. Pellerin, “45nm CMOS - silicon photonics monolithic technology (45CLO) for next-generation, low power and high speed optical interconnects,” in Opt. Fiber Commun. Con., 2020, paper T3H.3.
39. Y. Bian, J. Ayala, C. Meagher, B. Peng, M. Rakowski, A. Jacob, K. Nummy, A. Stricker, Z. Sowinski, A. Sahin, A. Aboketaf, S. Hu, I. Stobert, K. Mclean, L. Medina, K. Dezfulian, B. Harris, S. Krishnamurthy, T. Houghton, W. S. Lee, Ma. Sorbara, D. Riggs, T. Letavic, A. Yu, K. Giewont, and J. Pellerin, “Towards low-loss monolithic silicon and nitride photonic building blocks in state-of-the-art 300mm CMOS foundry,” in Frontiers in Optics, 2020, paper FW5D.2.
40. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). [CrossRef]
41. Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett. 29(14), 1626–1628 (2004). [CrossRef]
42. S. Hu and S. M. Weiss, “Design of photonic crystal cavities for extreme light concentration,” ACS Photonics 3(9), 1647–1653 (2016). [CrossRef]
43. H. Choi, M. Heuck, and D. Englund, “Self-similar nanocavity design with ultrasmall mode volume for single-photon nonlinearities,” Phys. Rev. Lett. 118(22), 223605 (2017). [CrossRef]
44. S. Hu, M. Khater, R. S. Montiel, E. Kratschmer, S. Engelmann, W. M. J. Green, and S. M. Weiss, “Experimental realization of deep-subwavelength confinement in dielectric optical resonators,” Sci. Adv. 4(8), eaat2355 (2018). [CrossRef]
45. M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. V. StrylandBass, Handbook of Optics, Third Edition Volume IV: Optical Properties of Materials, Nonlinear Optics, Quantum Optics, (McGraw-Hill, 2009).
46. J. Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of Fresnel reflection,” Nat. Photonics 1(2), 176–179 (2007). [CrossRef]
47. E. F. Schubert, J. K. Kim, and J. Q. Xi, “Low-refractive-index materials: A new class of optical thin-film materials,” Phys. Status Solidi B 244(8), 3002–3008 (2007). [CrossRef]
48. W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11(3), 963–983 (1994). [CrossRef]
