Record Purcell factors in ultracompact hybrid plasmonic ring resonators

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Science Advances  02 Aug 2019:
Vol. 5, no. 8, eaav1790
DOI: 10.1126/sciadv.aav1790


For integrated optical devices and traveling-wave resonators, realistic use of the superior wave-matter interaction offered by plasmonics is impeded by ohmic loss, which increases rapidly with mode volume reduction. In this work, we report composite hybrid plasmonic waveguides (CHPWs) that are not only capable of guiding subwavelength optical mode with long-range propagation but also unrestricted by stringent requirements in structural, material, or modal symmetry. In these asymmetric CHPWs, the versatility afforded by coupling dissimilar plasmonic modes provides improved fabrication tolerance and more degrees of device design optimization. Experimental realization of CHPWs demonstrates propagation loss and mode area of 0.03 dB/μm and 0.002 μm2, corresponding to the smallest combination among long-range plasmonic structures reported to date. CHPW ring resonators with 2.5-μm radius were realized with record Purcell factor compared with existing plasmonic and dielectric resonators of similar radii.


Optical microcavities are key components for photonic integrated circuits and important for many applications ranging from nonlinear optics, quantum optics, signal processing, and sensing (1, 2). For these applications, desired attributes such as exaltation of optical nonlinearity, laser threshold reduction, or enhancement of detection sensitivity are governed by the Purcell factor, which is proportional to Q/Veff, where Q is the cavity quality factor and Veff is the effective mode volume (3). To date, the optimization of Purcell factor, particularly within traveling-wave cavities that are compatible with planar integrated photonic circuits, has mainly focused on increasing Q using low-loss dielectric structures (46). Although Q > 108 can be obtained within these integrated optical microcavities, their radii typically span over hundreds of micrometers. As a result, the tenable Q/Veff factors for ultrahigh-Q dielectric resonators tend to be smaller than their dielectric counterparts. This is because they offer orders of magnitude smaller Qs, but can be implemented with micrometer-scale radii (7, 8).

An alternative approach to increase the Purcell factor is to reduce Veff. To this end, subwavelength plasmonic structures have emerged as potential platforms for implementing high Purcell factor cavities (915). In contrast to dielectric waveguides that guide light through total internal reflection (TIR), electromagnetic waves are guided in the form of surface plasmon polaritons (SPPs) in plasmonic waveguides, which are guided along metal-dielectric interfaces (1619). As SPPs are interface modes and not restricted by the diffraction limit, higher field confinement and density of states can be obtained, the combination of which allows smaller Veff as well as enhancements of linear and nonlinear optical processes. Unfortunately, the high energy density near the metal region inevitably leads to substantial ohmic loss, which becomes increasingly severe as mode volume decreases (1820). Consequently, the experimental Q for micrometer-scale, traveling-wave plasmonic resonators is only in the few hundreds. For the light-matter interaction afforded by plasmonic waveguides to be effectively used, the waveguide loss needs to be reduced without sacrificing modal confinement. If achieved, the combination of modest Q-factor and subwavelength mode volume can result in Purcell factors that are competitive against dielectric resonators.

To date, numerous strategies have been proposed to alleviate the loss-confinement trade-off. In plasmonic structures designed for traveling-wave applications, coupled modes are commonly used to obtain favorable trade-offs. For example, metal-insulator-metal (MIM) waveguide with an insulator width of tens of nanometers can provide the highest mode localization, but at the cost of extremely short propagation length (22, 23). While gain media can be incorporated for loss compensation, this approach has seen limited success due to the high current densities required (24). Another approach is to form hybrid plasmonic waveguide (HPW), where the coupling between the SPP and TIR modes leads to smaller field overlap at the metal region (25, 26). More recently, coupled HPWs have been proposed, which further reduces the field overlap via destructive interference (2729). Although the design approach requires geometrically or modally symmetrical structures, subwavelength supermode with propagation loss akin to that of loosely confined, long-range plasmonic structures can be obtained.

In this work, we report a composite HPW (CPHW), which supports a low-loss supermode that is formed due to the coupling of the SPP and HPW modes through a common metal layer. Contrary to previous coupled-mode plasmonic structures where symmetry conditions are enforced, our asymmetrical structure provides additional degrees of design freedom to simultaneously optimize multiple waveguide attributes. By using the HPW side of the waveguide stack for modal confinement while manipulating the dimension on the SPP side to lower the field flux within the lossy metal layer, reduced loss and confinement can be simultaneously achieved. CHPW microring resonators with record Purcell factor are demonstrated and outperform existing plasmonic and dielectric counterparts of similar radii.


Theoretical analysis of the CHPW structure

The proposed CHPW structure consists of Si-Al-SiO2-Si layers, as schematically shown in Fig. 1A. The top Si-Al interface supports a single-interface SPP mode, while the bottom Al-SiO2-Si stack supports an HPW mode. The two modes have prominently different field distributions, as the SPP mode decays exponentially away from the interface, whereas the HPW mode is primarily confined within the SiO2 spacer region. However, for metal thickness less than the skin depth, the perturbation between the evanescent fields leads to the formation of transverse magnetic supermodes (Fig. 1B). The characteristics of these supermodes are primarily dictated by the interference between the longitudinal field component of the SPP and HPW modes (Ez) (30). In-phase coupling increases the field interaction with the metal layer and corresponds to a short-range supermode (TMSR), while an out-of-phase coupling reduces the overlap and results in a long-range supermode (TMLR). Note that transverse electric supermodes can also be supported by the structure. However, they are guided through TIR and will be cut off for sufficiently narrow CHPW width. Thus, only the TM supermodes will be considered here.

Fig. 1 Plasmonic supermodes and modal properties of one-dimensional (1D) CHPW.

(A) The CHPW structure is formed by coupling a single-interface SPP mode to an HPW mode through a common metal layer with a thickness smaller than its skin depth. The multilayer stack consists of Si (ε1), SiO22), and Al (ε3) forming the HPW side, and Al (ε3) and Si (ε4) forming the SPP side. (B) The coupling gives rise to transverse magnetic short-range (TMSR) and long-range (TMLR) supermodes. TMSR mode is highly absorptive as a result of in-phase coupling, leading to increased field interaction with the metal, while out-of-phase coupling in TMLR reduces the overlap, allowing for long-range propagation. (C) Effective mode index and (D) propagation loss as the top-side Si layer thickness are varied, calculated at an operating wavelength of 1550 nm. The stand-alone SPP and HPW modes are plotted for comparison. (E) The normalized 1D field profile of CHPW within the Al metal layer is plotted for the optimal field cancellation at h = 147 nm and when it deviates at h = 135 and 165 nm. The y axis is centered at the middle of the 10-nm-thick metal layer, while the x axis shows the antisymmetric field profile centered at the zero crossing. (F) Optimizing the top-side Si layer thickness can lead to a substantial decrease in energy flux through the metal layer, while no notable changes are observed in the other waveguide layers. As a result, the modal loss can be decreased by two orders of magnitude while modal area is relatively invariant.

The coupling between the SPP and HPW modes can be controlled by tuning the vertical dimension of the CHPW, which, in turn, can drastically alter the attributes of the supermodes. For example, the dispersion and loss of a one-dimensional (1D) CHPW structure are shown in Fig. 1 (C and D) as functions of the top Si layer thickness (h). The results are calculated using Lumerical MODE Solutions at λ = 1550 nm. The properties of stand-alone SPP and HPW modes are also plotted for comparison. As h increases, it is observed that the SPP and HPW modes will become phase matched and strongly coupled at h = 108 nm, after which the two effectively decouple despite a metal thickness of only 10 nm. Correspondingly, an anticrossing behavior is observed in the dispersion curves of the TMSR and TMLR supermodes, which converge toward that of the SPP and HPW modes at larger h, respectively. However, because of the asymmetry of the CHPW structure, the losses of the TMSR and TMLR are not maximized and minimized at the phase-matching point. Instead, the TMSR loss shows small sensitivity to the variation in h, while the TMLR loss can be tuned by over two orders of magnitude within the same range.

The strong tunability in the loss of the TMLR supermode can be explained by examining the Ez distribution within the waveguide. From Fig. 1E, it is observed that Ez is rendered antisymmetrically distributed across the metal at h = 147 nm, with a zero-crossing point at the center of the metal layer. As h deviates from 147 nm, the propagation loss becomes substantially higher, even though the zero-crossing point can still remain within the metal layer. Hence, it can be deduced from the antisymmetric field distribution that the loss of the TMLR supermode reaches a minimum when there is an equal amount of positive and negative modal fields within the metal. In a 1D CHPW, the field symmetry can be quantified via the field fluxf=tIm{Ez}Pz dl(1)where Pz is the Poynting vector in the propagation direction and t is the thickness of the waveguide layer of interest. As depicted in Fig. 1F, f within the metal layer shows strong dependence on h and approaches zero at h = 147 nm. As such, the loss of the TMLR supermode is reduced to 0.019 dB/μm, which is 17.5 and 85 times smaller compared with that of stand-alone HPW and SPP, respectively. In previous designs of long-range, coupled plasmonic waveguides, stringent geometrical, material, and modal symmetries are enforced to maximize the destructive interference between the coupled modes (2729). However, as shown here, given that loss is only dictated by the field overlap in the metal region, a low-loss supermode can be engineered even within a highly asymmetrical structure by simply manipulating the structural parameters to reduce the net field flux.

From Fig. 1B, it can also be observed that the modal energy of the 1D TMLR supermode is asymmetrically distributed and primarily confined within the bottom HPW stack. Specifically, for the continuity of the displacement field to be satisfied, strong field confinement is established inside the 20-nm low-index SiO2 region (25, 26). As a result, f in the metal region is orders of magnitude lower compared with those of the other waveguide layers (Fig. 1F). By only changing the parameters of the top SPP stack to engineer the field antisymmetry, minimal disturbance to the modal confinement between samples can be ensured. As such, energy distribution of the TMLR supermode will remain nearly constant despite the substantial variation in the modal loss. This is in stark contrast to the low-loss supermode guided by traditional long-range plasmonic waveguides, where destructive interference between the coupled modes increases the energy density within the nonmetallic waveguide layers and results in an increased modal area (23).

When extending to 2D CHPW structures (Fig. 2A), the minimization of the total Ez field remains an effective strategy for loss reduction. The field distribution of the TMLR supermode supported by a 200-nm-wide CHPW is shown in Fig. 2B. Because of the finite waveguide width, the modal field varies in the lateral direction, and the location where complete field cancellation (zero-crossing) occurs is not flat across the entire width of the metal layer. Nonetheless, by extending Eq. 1 to a surface integral, it is found that f in the metal layer can be reduced to 0.003 V⋅m2/W at h = 185 nm, which corresponds to a minimal loss of 0.02 dB/μm that is 20 and 75 times smaller compared with the losses of stand-alone HPW and SPP modes, respectively (Fig. 2A). Moreover, the long-range propagation condition is shown to be robust, as the TMLR loss remains <0.05 dB/μm even if h deviates from the optimal thickness by 10%. Last, although optimization has been carried out at λ = 1550 nm, the propagation loss of the optimized CHPW can remain <0.05 dB/μm across a 200-nm optical bandwidth (fig. S1).

Fig. 2 Schematic and modal properties of 2D CHPW.

(A) A schematic of the CHPW core cross section discussed in this work. This CHPW is a 200-nm-wide four-layer stack that consists of 220-nm bottom high-index Si (ε1), 20-nm low-index SiO22), 10-nm Al (ε3) metal, and 185-nm top high-index α-Si (ε4) layers. The long-range mode in this structure is primarily confined within the low-index layer ε2, as shown in the overlaid modal E-field intensity profile. (B) The cross-sectional area around the ε3 metal layer is expanded to plot the longitudinal Ez field profile. By varying the ε4 thickness h, the longitudinal electric flux f (in V⋅m/W) can be minimized by engineering the Ez within the metal. (C) Calculated modal area and propagation loss of the long-range mode as function of the layer ε4 thickness. The thickness for optimal propagation loss (h = 185 nm) corresponds to the minimum flux within the metal layer. Using this method, the propagation losses of modes in CHPWs can be drastically reduced for a wide range of structures, without any restrictions on symmetry, while maintaining an extremely localized effective mode area of 0.002 μm2. (D) Insertion loss calculations for 200-nm-wide CHPW 90° bend extracted via finite-difference time-domain (FDTD) simulations. The insertion losses are optimized between bend radii of 1.5 and 3 μm, as bend losses become presiding below 1.5 μm, while absorption losses start becoming dominant above 3 μm. (E) FDTD-simulated transmission spectra for ring resonators in all-pass filter configuration constructed using 200-nm-wide CHPW and 2.5-μm bend radius. Critical coupling is achieved at a gap spacing of 285 nm between the bus and the ring.

The effective mode area of the 200-nm-wide CHPW is shown in Fig. 2C. Specifically, a phenomenological mode area definition related to the Purcell factor is used (21)A=1max{W(r)}AW(r) dAW(r)=12Re{d[ωε(r)]dω}E(r)2+12μ0H(r)2(2)where W(r) is the mode energy density. It is observed that A will expand in the regime where h < 150 nm due to field leaking into the air cladding. However, as expected from the 1D analysis, it remains relatively constantly close to the region where loss is minimized as the majority of the field is localized within the low-index SiO2 layer. Specifically, subwavelength A of 0.002 μm2 is achieved at h = 185 nm, which changes by only 5% even with 10% deviation in h.

With strongly reduced propagation loss, the subwavelength modal confinement offered by plasmonic structures can now facilitate the enhancement of the Purcell factor. Using Lumerical 2.5D finite-difference time-domain (FDTD) simulations, the insertion loss of the 200-nm CHPW bend is calculated and shown in Fig. 2D. The insertion loss is ~0.25 dB for bend radius between 1.5 and 3 μm. For bend radius below 1.5 μm, radiation losses become dominant. However, even at a radius equal to the waveguide width, the insertion loss is still under 3 dB, thus highlighting the capability of the CHPW platform for designing compact devices. The transmission spectra for all-pass filter implemented via the 200-nm-wide CHPW ring and bus are shown in Fig. 2E. Under the critical coupling condition when the gap width is 285 nm, the theoretical extinction ratio and quality factor are calculated to be 30 and 775 dB, respectively. With an effective mode volume of 0.032 μm3, the Q/Veff for the CHPW ring is calculated to be ~16,000.

Table S1 compares the attributes of Si waveguides and exiting plasmonic waveguides with those of the CHPWs. By manipulating the coupling of dissimilar plasmonic modes within the same waveguide, the CHPW structure can achieve mode confinements similar to those found in the MIM structure as well as propagation loss that is an order of magnitude smaller than that of HPWs. Thus, the loss-confinement trade-off is heavily alleviated. Note that the use of an HPW as part of the coupled-mode structure enables highly efficient, instantaneous CHPW mode excitation without the need for taper structures. As shown in fig. S2, nonresonant excitation of CHPW devices can be obtained using Si nanowires with coupling efficiency of 71% at λ = 1550 nm.

Experimental realization of the CHPW structure

For proof of concept, 200-nm-wide CHPW waveguides and rings have been fabricated (Fig. 3A). First, 20-nm SiO2 and 10-nm Al layers are deposited onto silicon-on-insulator (SOI) wafer via plasma-enhanced chemical vapor deposition (PECVD) and sputtering, respectively. Next, α-Si is sputtered and partially etched down such that different samples have different α-Si thicknesses, ranging between 0 and 250 nm. Last, the waveguides are patterned using multiple electron beam lithography steps, and the dielectric and Al layers are etched via reactive ion etching and wet processing, respectively. CHPWs with lengths between 10 and 400 μm have been fabricated for cutback measurement. Light is coupled in and out of the CHPW devices via 800-nm-wide Si nanowires (Fig. 3B).

Fig. 3 Cutback measurement of CHPWs.

(A) Scanning electron micrograph of a cleaved CHPW. (B) Scanning electron micrograph of the Si nanowire–CHPW end-butt coupler. (C) Propagation loss for CHPWs with different α-Si thickness. For comparison, the theoretical values obtained via FDTD modeling (gray dashed line) and the measured propagation losses of a silver-air-silver plasmonic slot waveguide and single-sided HPW that does not have a top α-Si layer are also plotted. (D) Coupling efficiency between a junction formed by an 800-nm-wide Si nanowire and a 200-nm-wide CHPW. The theoretical values obtained via FDTD modeling are also plotted (gray dashed line).

The measured CHPW propagation loss at λ = 1550 nm is plotted in Fig. 3C. As predicted by the field-matching analysis, loss can be substantially reduced with increasing h and a minimum of 0.03 dB/μm is measured at h = 190 nm. This is slightly higher than the theoretical value of 0.02 dB/μm, and the deviation can be attributed to additional loss mechanisms such as sidewall roughness, layer interface roughness, and absorption from the α-Si layer. The experimental propagation losses for the MIM and HPW waveguides are also shown in Fig. 3C, which are orders of magnitude higher than that of the CHPW. Thus, the benefit of a coupled mode–based design approach is demonstrated. To our knowledge, this is the first experimental demonstration of long-range coupled plasmonic waveguides. Moreover, the strong fabrication tolerance is also illustrated, as the CHPW loss only increases to 0.07 and 0.09 dB/μm for h = 155 and 250 nm, respectively. On the basis of the cutback method, CHPW excitation efficiencies have also been extracted (Fig. 3D). The coupling efficiency is only 56% at h = 155 nm due to sample-specific fabrication imperfections such as trapezoidal sidewall profile and misalignment between CHPWs and Si nanowires. Nonetheless, a coupling efficiency of ~70% is maintained over a large range of h values.

The fabricated all-pass CHPW ring resonator is shown in Fig. 4A. The free spectral range is determined to be ~44.8 nm due to the micrometer-scale radius, and the resonance can be tuned across the C-band by varying the radius (Fig. 4B). Once critical coupling is established at a gap width of 270 nm, a maximum extinction ratio of 29 dB is observed at λ = 1574 nm (Fig. 4C). In addition, the full width at half maximum is measured to be 4.8 nm, which corresponds to a Q-factor of 320. Although the experimental Q is lower than the theoretical prediction of 775 because of additional losses incurred due to absorption and roughness from the sputtered α-Si layer, it is the highest for hybrid plasmonic ring resonators reported to date. Given a calculated A of 0.002 μm2, the 2.5-μm-radius ring resonator has Veff of only 0.032 μm3. Therefore, our experimental device can achieve Q/Veff as high as 6507, the highest reported for traveling-wave plasmonic resonator to date. Last, the temperature dependence of the CHPW ring resonators is shown in Fig. 4D. Extinction ratio is maintained >20 dB for temperatures up to 75°C, and the transmission minimum exhibits a linear shift of 0.052 nm/°C (Fig. 4E).

Fig. 4 Experimental CHPW microring resonator results.

(A) Scanning electron micrograph of an all-pass CHPW resonator with 2.5-μm radius. (B) Measured transmission spectra of CHPW ring resonators with 270-nm gap width and varying radii. (C) Measured transmission spectra of CHPW ring resonators with 2.5-μm radius for varying gap widths. (D and E) Measured transmission spectra and resonant wavelengths of CHPW ring at different substrate temperatures (2.5-μm radius and 270-nm gap width). The spectra and resonance wavelengths have been fitted to Lorentzian and linear functions, respectively.


In conclusion, we report a CHPW architecture where dissimilar plasmonic modes are coupled together at a single metal interface to simultaneously reduce field overlap with lossy metal and confine power within a subwavelength area. The waveguide platform does not require any structural or modal symmetry, therefore allowing a much more relaxed fabrication tolerance. As such, it enables the realization of long-range plasmonic modes in any platform with no material or structural restrictions. Experimental realization of the proposed CHPW structures has shown record-low propagation and record-high Q/Veff. Table 1 compares the experimental attributes of the different traveling-wave optical microcavities. Specifically, it can be seen how CHPW rings with 2.5-μm radius outperform their plasmonic and dielectric counterparts of similar radii and can enable Q/Veff and extinction ratio that are an order of magnitude higher compared with ultrahigh-Q dielectric cavities. Thus, the adverse effect of the ohmic loss inherent to plasmonic waveguides has been effectively alleviated. As a proof of concept, we have only examined how the thickness of the top Si layer can influence the supermode attributes. It is expected that the optimization of other waveguide layers or an extension to whispering-gallery disk structure may lead to additional performance enhancement (8).

Table 1 Normalized Purcell factor (Q/Veff) comparison of CHPW ring resonators against literature.

The measured propagation loss, ring resonator extinction ratio, and normalized Purcell factor of the CHPW ring resonators are compared against representative, previously reported plasmonic and silicon ring resonators. In typical plasmonic waveguides, highly localized plasmonic modes are afflicted by low-quality factors as a result of losses, while subwavelength confinement is not attainable in diffraction-limited dielectric waveguides. In contrast, the CHPW can achieve record-low losses of 0.03 dB/μm through flux engineering while concurrently asserting a nanoscale modal confinement of only 0.002 μm2, leading to an effective mode volume of 0.032 μm3 for 2.5-μm ring radius. Through this combination, 6507 normalized Purcell factor can be achieved as the loss-confinement trade-off inherent in plasmonics is drastically alleviated using our structure.

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The experimental demonstration of CHPW microring resonators with high Q/Veff ratio and, therefore, high Purcell factor opens the door to many potential applications. First, the CHPW is suitable for nanolaser application because (i) as reported in 28, the combination of low modal loss and nanoscale modal area within coupled-plasmonic waveguides can enable lower threshold gain and enhance the rates of spontaneous and stimulated emissions. (ii) The use of a ring resonator instead of Fabry-Perot–like cavities can avoid undesired loss due to imperfect reflection at end facets (31, 32). (iii) An efficient method for collecting the output light from the resonator is already in place. Second, the CHPW may be useful for cavity quantum electrodynamics, particularly in a coupling regime where the Purcell factor is still a limiting factor (33). Third, the CHPW can be exploited for nonlinear plasmonics by depositing nonlinear materials within the spacer layer where the energy density is the highest (13). Last, it is important to highlight that although the analysis here has focused on minimizing the field overlap with the metal layer, the same coupled-mode design approach can instead be used to maximize the field overlap for efficient photodetection (34). Alternatively, the waveguide loss can be dynamically tuned after fabrication using bias or current for optical modulation (35). Hence, CHPWs will no doubt have an impact that extends from ultracompact interconnects to integrated optical circuitry that is programmable and reconfigurable (14, 36).


CHPW device simulation

The effective index and propagation loss of the CHPW mode were calculated via 2D finite element method simulation using the commercial Lumerical Mode Solutions software. Metallic boundary conditions were used to terminate the 2 μm by 2 μm computational domain. Grid sizes of 0.1, 1, and 2.5 nm were used to mesh the Al thin film, the SiO2 spacer layer, and the rest of the waveguide structure, respectively. The top Si layer was taken to be crystalline in the simulations. The refractive indices of the materials at λ = 1.55 μm are as follows: nSi = 3.4784, nSiO2 = 1.44, and nAl = 1.44 + 16i.

CHPW device fabrication

The CHPWs were fabricated using standard 220-nm SOITEC wafer. Deposition of 20-nm-thick SiO2 was carried out using Oxford Instruments PlasmaLab System100 PECVD at a plasma temperature of 400°C using SiH4 and N2O. Deposition of 10-nm-thick Al was done using AJA International ATC Orion 8 Sputter Deposition System at room temperature in Ar plasma. Last, deposition of α-Si:H was performed using the MVSystems Multi-Chamber PECVD at a substrate temperature of 180°C using SiH4 plasma. Lithography to define etching patterns was carried out in Vistec EBPG 5000+ Electron Beam Lithography System. As Si nanowires and CHPW were built on the same substrate, multiple alignment steps were required. Waveguide patterning was realized using Oxford Instruments PlasmaPro Estrelas 100 deep reactive ion etching for silicon and oxide layers, and aluminum etchant type A for the metal layer.

CHPW device characterization

To perform cutback measurement, light from a C-band continuous-wave laser was first amplified using an erbium-doped fiber amplifier (EDFA). Next, the output fiber from the EDFA was wrapped through a Thorlabs paddle fiber polarization controller to ensure TM-polarized input. A single-mode lensed fiber that has a 2.5-μm spot diameter was used to couple light into the input Si nanowires. On the output side, the transmitted light from the output Si nanowire was collected with a 20× objective lens with 0.4 numerical aperture. Noise from the substrate was eliminated using an iris, and the output polarization was confirmed with a polarization beam splitter. Last, a germanium photodetector was used for power measurement.

Thermal measurement was performed using a custom copper stage with thermoelectric coolers, where temperature was controlled using the Keithley 2510-AT Autotuning TEC source meter through electrical feedback from a 10k thermistor.


Supplementary material for this article is available at

Section S1. Wavelength dependence of 2D CHPW

Section S2. Silicon nanowire-CHPW coupler

Section S3. Comparisons for various short-range and long-range waveguides against the CHPW

Fig. S1. The wavelength dependence of the CHPW supermode attributes.

Fig. S2. Mode matching and broadband power transfer between silicon nanowires and CHPW.

Table S1. Comparison of physical cross-sectional dimensions, propagation loss, and mode area of various waveguide designs.

Reference (37)

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Acknowledgments: Funding: This project was funded by the Natural Sciences and Engineering Research Council (NSERC) of Canada. Author contributions: A.S.H., C.L., P.H.C., and Y.W.S. have designed the structures. Y.W.S. and P.H.C. have electromagnetically modeled the structure. Y.W.S. has fabricated the samples. Y.W.S. and C.L. have characterized the devices. A.S.H., C.L., P.H.C., and Y.W.S. have analyzed the measurements and wrote the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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