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Ultraconfined Propagating Exciton–Plasmon Polaritons Enabled by CavityFree Strong Coupling: Beating Plasmonic TradeOffs
Nanoscale Research Letters volume 17, Article number: 109 (2022)
Abstract
Hybrid coupling systems consisting of transition metal dichalcogenides (TMD) and plasmonic nanostructures have emerged as a promising platform to explore exciton–plasmon polaritons. However, the requisite cavity/resonator for strong coupling introduces extra complexities and challenges for waveguiding applications. Alternatively, plasmonic nanowaveguides can also be utilized to provide a nonresonant approach for strong coupling, while their utility is limited by the plasmonic confinementloss and confinementmomentum tradeoffs. Here, based on a cavityfree approach, we overcome these constraints by theoretically strong coupling of a monolayer TMD to a single metal nanowire, generating ultraconfined propagating exciton–plasmon polaritons (PEPPs) that beat the plasmonic tradeoffs. By leveraging strongcouplinginduced reformations in energy distribution and combining favorable properties of surface plasmon polaritons (SPPs) and excitons, the generated PEPPs feature ultradeep subwavelength confinement (down to 1nm level with mode areas ~ 10^{–4} of λ^{2}), long propagation length (up to ~ 60 µm), tunable dispersion with versatile mode characters (SPP and excitonlike mode characters), and small momentum mismatch to freespace photons. With the capability to overcome the tradeoffs of SPPs and the compatibility for waveguiding applications, our theoretical results suggest an attractive guidedwave platform to manipulate exciton–plasmon interactions at the ultradeep subwavelength scale, opening new horizons for waveguiding nanopolaritonic components and devices.
Introduction
As an intriguing regime of the light–matter interaction, strong coupling between excitons and photons with the formation of polaritons enables great possibilities to modify the properties of the coupled systems, offering numerous opportunities for both fundamental research and technological applications including Bose–Einstein condensation [1], lowthreshold lasing [2], ultrafast modulation and switching [3, 4], and alloptical logic operation [5]. Recently, owing to their remarkable excitonic properties such as large binding energies and strong oscillator strengths [6], monolayer transition metal dichalcogenides (TMDs) are emerging as promising candidate twodimensional (2D) materials to sustain the exciton resonance for reaching the strong coupling regime. By combining them to plasmonic nanostructures with ultratight optical confinement, the great size mismatch between the optical field and the ultrathin thickness of monolayer TMDs can be bridged, providing the unprecedented ability to explore the strong plasmon–exciton interaction at the deep subwavelength scale [7].
Generally, in the plasmonicTMD system, the key for achieving strong coupling is to ensure a sufficiently large coupling strength that overcomes the overall damping of the coupled system. And a common strategy is to utilize tightly confined cavity modes or localized surface plasmon resonances (LSPRs), which have been previously realized by introducing plasmonic cavities or resonators including metallic FP cavities [8], periodic structures [9,10,11], plasmonic dimers [12, 13], single nanoparticles [14,15,16,17,18], and nanogap resonators formed by nanoparticleovermirror configurations [19,20,21]. However, the requisite cavity/resonator may introduce extra complexities [22, 23] and challenges for flexible mode engineering [24], onchip integration [25], and remote exciton–polariton transportations [26] for waveguiding applications.
On the other hand, besides the cavity modes and LSPRs, propagating modes can also be utilized to provide a nonresonant approach for strong coupling [23, 27,28,29,30], but have received little attention in the plasmonic nanowaveguiding system. As one of the simplest onedimensional (1D) nanowaveguides to support propagating surface plasmon polaritons (SPPs), metal nanowires (MNWs) possess unique advantages including excellent compatibilities to onchip nanophotonics [31] and deep subwavelength confinement (e.g., ~ 10^{–2} ~ 10^{–3} of λ^{2}) [32,33,34,35] for promoting light–matter interactions, offering a potential guidedwave platform for strong coupling. However, the utility of MNWs is limited by the wellknown tradeoff between the energy confinement and the loss of SPPs [33, 35]. In addition to the confinementloss tradeoff, another fundamental hurdle is the tradeoff between confinement and momentum mismatch to photons [36], leading to challenges for efficient photonSPP conversions and consequently weakened compatibilities for integrated hybrid components and devices.
Here, based on a MNWTMD system, we theoretically propose a cavityfree strong coupling approach for generating ultraconfined propagating exciton–plasmon polaritons (PEPPs) that beat the plasmonic confinementloss and confinementmomentum tradeoffs. We show that the strong coupling between SPPs in a single MNW and excitons in a monolayer WS_{2} results in a backbending dispersion with the complex momentum and an anticrossing dispersion with the complex frequency, exhibiting large Rabi splitting energies with tunability. Due to the strongcouplinginduced reformation in the energy distribution, the generated PEPPs are much more confined than the original SPPs in MNWs, offering the possibility to realize a full width at half maximum of the energy distribution at the ultradeep subwavelength scale (~ 1 nm). Meanwhile, as a mixture of SPPs and excitons, PEPPs are highly versatile that can be manipulated to exhibit excitonlike character with extremely tight confinement (~ 10^{–4} of λ^{2}) or SPPlike character with high quality and long propagation distance (up to ~ 60 µm). More importantly, we also show that PEPPs represent another class of waveguiding polaritons with much more efficient confinementloss and confinementmomentum tradeoffs that outperforms the original SPPs, which may offer new opportunities for waveguiding polaritonic applications such as ultracompact integrated circuits and highperformance polaritonic devices.
Methods
The proposed MNWTMD structure consists of a single MNW waveguide with a monolayer TMD cladding, which is schematically plotted in Fig. 1a. The MNW is assumed to have a uniform diameter with a smooth surface. In such configuration, the tightly confined SPP with strong field enhancement around the MNWTMD interface facilitates the plasmon–exciton interaction. As a model system for theoretical investigation, a WS_{2}clad Ag MNW is selected, in which the permittivities of WS_{2} (ε_{WS2}) and Ag (ε_{Ag}) are described by a Lorentz oscillator model [25] and an effective Drude model [37], respectively (see supporting information for details). The thickness of the WS_{2} layer is assumed to be 1 nm [12]. For simplicity and facilitating strong coupling, we only focus on the coupling of excitons to the fundamental mode in the Ag MNW, since the fundamental mode is more confined than the other order ones [35, 38], and the singlemode operation is favorable and can be readily realized in many applications [32, 33, 35, 39].
For theoretical investigation of the strong coupling and the formed PEPP in the proposed coupling system, the wave equations are numerically solved in both complexfrequency (complexω) and complexmomentum (complexk) planes to provide a comprehensive analysis. As to the complexω solution, the real (Re(ω)) and imaginary (Im(ω)) parts represent the eigenfrequency and the temporal damping of the PEPP in the coupled system. And for the complexk solution, the real (Re(k)) and imaginary (Im(k)) parts correspond to the propagation constant and the spatial damping along the propagation direction. Besides solutions from the wave equation, the PEPP can also be decomposed into the exciton resonance in the WS_{2} and the uncoupled SPP mode from the hybridization point of view (Fig. 1b). The bare SPP mode without coupling to excitons (refer to the SPP in further text) is obtained by using the nonresonant background permittivity ε_{b} of the WS_{2} with its oscillator strength being zero (see numerical methods in supporting information for details).
Results and Discussion
CavityFree Strong Coupling Between Excitons and 1DSPPs
Figure 2A gives the complexk solution of the PEPP with the MNW diameter of 50 nm. As to its dispersion curve (ħω vs. Re(k), left panel), the hybridization of the exciton (black dashed line) and the SPP mode (orange dashed line) gives rise to the anomalous dispersion in the vicinity of the exciton resonance with a significant backbending feature, clearly indicating strong coupling [23, 27, 29]. Meanwhile, when ω is approaching the exciton resonance, Im(k) of the PEPP dramatically increases (Fig. 2a, right panel), resulting in a drastic reduction in its propagation length that will be discussed later in waveguiding properties. For comprehensive characterization, Fig. 2b presents the corresponding complex ω solutions of the PEPP. Instead of the continuous dispersion curve in the complexmomentum plane, the dispersion in terms of ħRe(ω) versus k (Fig. 2b, left panel) exhibits two asymptotic branches (upper branch: blue dots; lower branch: red dots) disconnected by a polaritonic gap around the exciton resonance, manifesting itself in an anticrossing behavior with the Rabi splitting energy (ħΩ_{R}) of ~ 85.7 meV at the zerodetuning (green double arrow). Compared to the SPP, the PEPP exhibits a “leftpulling” trend in the complexω trajectory (Fig. 2b, right panel) and becomes highly damped around the excitonic resonance which corresponds well to other propagating polaritons previously reported [23]. Note that the aforementioned backbending and anticrossing behaviors in dispersions are not inconsistent with each other [23, 29, 40], and they are actually the results obtained at a given frequency or momentum and can be both experimentally measured by momentum and frequencyresolved spectroscopy [40].
For further verification, we approximate our system to the coupledoscillator model (COM) [41]:
from which the eigenvalues ω_{±} of the PEPP can be analytically obtained through diagonalization of the Hamiltonian matrix, and the eigenvectors \((\alpha ,\beta )_\pm^T\) are determined for revealing contributions from SPPs (α_{±}^{2}) and excitons (β_{±}^{2}). Here, ω_{SPP} and γ_{SPP} are eigenfrequency and damping frequency of the SPP mode (i.e., ω_{SPP} = Re(ω), γ_{SPP} = 2Im(ω) extracted from the orange dashed line in Fig. 2b). ω_{ex} and γ_{ex} are exciton resonance frequency and damping frequency of the WS_{2} material. With the above parameters, the analytically obtained results using the COM (solid line in Fig. 2b) exhibit excellent agreements to numerical simulations for both of the dispersion curve and the complexω trajectory. Note that Im(ω) around the excitonic resonance from the simulation is slightly larger than the one analytically obtained, which may be attributed to the extra dissipation caused by the extremely tight confinement [42, 43] near the excitonic resonance that is not considered in the analytical model. To claim the strong coupling, the strict criterion ħΩ_{R} > ħ(γ_{SPP} + γ_{ex})/2 [41] is well fulfilled comparing the Rabi energy ħΩ_{R} of ~ 85.7 meV to the overall damping in the system ħ(γ_{SPP} + γ_{ex})/2 of ~ 18 meV. As to fractions of SPPs and excitons in PEPPs, they are equally contributed (α_{±}^{2} =β_{±}^{2} = 0.5) for both upper and lower branches at the zerodetuning (Fig. 2c) corresponding to ħω_{±} = 2.057 (~ 603 nm in wavelength λ) and 1.971 eV (λ ~ 629 nm), respectively. Within this range (1.971 eV < ħω < 2.057 eV), PEPPs are dominant by excitons in terms of β_{±}^{2} > 0.5 (excitonlike) and they are otherwise SPPlike outside the range.
Mode Characteristics and Waveguiding Properties of PEPPs
The above results evidently show that SPPs in MNWs can be strongly coupled to excitons in the WS_{2} monolayer, creating PEPPs that are hybrid mixtures of SPPs and excitons. Due to the hybrid nature, the fractions of SPPs and excitons in PEPPs can be manipulated by the wavelength λ (which is discussed in Fig. 2c), offering opportunities to alter and even reform the energy distribution of PEPPs. To gain a deeper insight, Fig. 3a, b gives crosssectional mode profiles and λdependent fractional energy distributions of PEPPs, in which the fractional energy inside the MNW (η_{m}) and the WS_{2} layer (η_{l}) is calculated via their corresponding energy density W(x, y) [44] integrations (using Additional file 1: Eq. S1). For reference, the corresponding fractional energy for the SPP is also provided (pale red and blue dashed lines, Fig. 3b). It can be seen that at wavelengths distant away from the excitonic resonance (e.g., λ = 560 and 680 nm, Fig. 3a(i), (v)), where the plasmon–exciton interaction is relatively weak, the PEPP shows a SPPlike mode character with a much larger η_{m} than η_{l} (e.g., η_{m} ~ 0.46 vs. η_{l} ~ 0.10 at λ = 560 nm). As wavelengths approach the excitonic resonance, η_{l} rapidly increases with more energy being pulled out from the MNW and mode profiles shifted to the WS_{2} layer (e.g., Fig. 3a(ii), (iv) for λ = 596 and 636 nm, η_{m} = η_{l} ~ 0.32). And at wavelengths of 603 and 629 nm, η_{l} increases to 0.5 (which also coincides very well with the calculated result from β_{±}^{2} = 0.5), indicating that the PEPP mode enters the excitonlike region (bluefilling area, Fig. 3b). Finally, around the excitonic resonance wavelength (λ ~ 616 nm), η_{l} reaches its maximum (η_{l} ~ 0.94), enabling an extremely tight confinement with most of the energy inside the WS_{2} layer (Fig. 3a(iii)). To better visualize such strongcouplinginduced reformation in the energy distribution, Fig. 3c gives the normalized energy density along the x direction W(x, 0). Compared to the SPP, the PEPP further squeezes the energy into the atomicthin 2D materials with the full width at half maximum (FWHM) of the energy distribution at the ultradeep subwavelength scale (~ 1 nm), offering a promising route to enhance the light–matter interaction that may have great potentials for nonlinear applications.
For further quantitative characterization of the confinement, Fig. 4a gives mode areas A_{m} (calculated using Additional file 1: Eq. S2) of the PEPP (red dotted line). As is shown, benefitted from the strongcouplinginduced reformation in the energy distribution, A_{m} of the PEPP is always much smaller than the SPP (reddashed line), making it possible to realize an extremely small value down to 0.000169 µm^{2} (~ 4 × 10^{–4} of λ^{2}, see right yaxis for the normalized mode area) that is only ~ 1/20 the size of the corresponding SPP. On the other hand, the propagation lengths L_{m} (calculated using Additional file 1: Eq. S3) are shown in Fig. 4b. The profound dip in the L_{m} curve with a drastic reduction from ~ 6 to 0.24 µm is due to the most energy distributed in the WS_{2} layer with higher absorption, which may have potential applications in alloptical switching and modulation. For other applications where the longrange propagation is desired, L_{m} of the PEPP can be manipulated by increasing the MNW diameter, while the strong coupling still holds valid (e.g., L_{m} = ~ 60 µm can be achieved for a 400nmdiameter MNW which will be discussed in the next section). Moreover, as a mixture, the PEPP inherits both properties of the SPP and the exciton, offering opportunities to achieve higher versatility and superior quality than the bare SPP. For demonstration, the calculated figure of merit (FOM = \({{L_m} \mathord{\left/ {\vphantom {{L_m} {(2\sqrt {{A_m}/\pi } }}} \right. \kern\nulldelimiterspace} {(2\sqrt {{A_m}/\pi } }})\) [45]) of both PEPP and SPP is shown in Fig. 4c. Instead of the monotonic behaviors of the SPP, the FOM curve of the PEPP is divided into two types of regions (indicated by the bluefilling and nonfilling areas) according to the mode characters (excitonlike and SPPlike). In the bluefilling zone where the exciton dominates, the PEPP is able to exert its full potential for confining energy at the ultradeep subwavelength scale (e.g., at the maximum confinement wavelength of ~ 616 nm, blue star), while in the SPPlike region, the PEPP can offer higher FOM than SPP with two local maximum values of ~ 127 and 131 (at the maximum FOM wavelengths: ~ 596 and 642 nm, green diamond and square) around transitions of mode characters.
Tunability in Rabi Splitting Energy with Tailored Dispersions
Besides the versatility in operation wavelengths, the PEPP and strong coupling behaviors can also be manipulated by the diameter (D) of the MNW (Fig. 5), exhibiting large tunability with tailored dispersions. As shown in Fig. 5a, by increasing D from 50 to 400 nm, the backbending feature in the anomalous region becomes less profound (Fig. 5a(i)) with a smaller Rabi splitting (Fig. 5a(ii)) in dispersions. The corresponding ħΩ_{R} varies from ~ 85.7 to ~ 34.2 meV (dotted line in Fig. 5b). To understand this decline trend, we derive an analytical expression of Ω_{R} in a general form, which is calculated via the coupling strength between the SPP and the exciton resonance as [46, 47]
where g is the zerodetuning coupling coefficient, µ is the transition dipole moment of the exciton, N is the numbers of the excitons, and E_{m} is the electric field amplitude of the SPP per photon. Since the WS_{2} is described by the Lorentz oscillator model, the overall transition dipole moments term \(\mu \sqrt N\) can be estimated as [23, 48]
where ρ, fω_{p}^{2}, ω_{ex} represent the oscillator density, oscillator strength, and resonance frequency of the WS_{2}. V is the volume of the WS_{2} layer that can be obtained from its geometric thickness t_{l} as \(V = \pi (D + {t_l}){t_l}L = {A_l}{L_m}\), where A_{l} denotes the crosssectional area of the WS_{2} layer. On the other hand, E_{m} can be approximately calculated through the mode volume V_{m} [49]
where A_{m} is the mode area of the SPP mode. At the zerodetuning where ω = ω_{ex}, by substituting Eqs. (3–4) into Eq. (2), we can obtain the Ω_{R} as
As shown by the pale green dotted line in Fig. 5b, ħΩ_{R} obtained using Eq. (5) agrees reasonably well with the simulated one (green squared line), further validating our result. The decline trend in ħΩ_{R} is due to the increasing A_{m} in a thicker MNW with a consequent weaker plasmon–exciton interaction. Despite of the reduced ħΩ_{R}, the strong coupling condition is still fulfilled for every diameter within the range we presented compared to the overall damping of the system (gray dashed line). On the other hand, although ħΩ_{R} shown here (e.g., ħΩ_{R}/ħω_{ex} = ~ 4.2% for D = 50 nm) cannot reach the ultrastrong coupling regime (ħΩ_{R}/ħω_{ex} > 20% [50]), it can be further enhanced by decreasing A_{m}. And potential strategies for reducing A_{m} may include reducing the diameter of the MNW [32] and utilizing nanofocusing structures (e.g., tapered plasmonic waveguides) [51, 52]. Along with the tailored ħΩ_{R} and dispersions, waveguiding properties of PEPPs can also be engineered with varied MNW diameters, exhibiting large tunability with A_{m} (~ 0.000169 µm^{2} to ~ 0.09 µm^{2}) and L_{m} (~ 0.24 µm to ~ 60 µm) ranging across two orders of magnitudes, and FOM up to 250 (see Additional file 1: Fig. S4–S9 for waveguiding properties of MNW with D from 75 to 400 nm). Note that even for the thickest MNW (D = 400 nm) we discussed here, the energy can still be tightly confined within the ultrathin WS_{2} layer at the 1nm level due to the strong coupling (see Additional file 1: Fig. S10).
Exceptional ConfinementLoss and ConfinementMomentum TradeOffs
In this section, we show that the PEPP provided by our strongly coupled MNWWS_{2} structure represents another kind of waveguiding polaritons that is superior than the original SPP in MNWs. To understand its merits, parametric plots allowing direct comparison between different polaritons [53] are provided in Fig. 6. Due to two types of mode characters for the PEPP, operations in the excitonlike region at the maximum confinement wavelength (e.g., blue star in Fig. 4c for D = 50 nm) and in the SPPlike region with the two local maximum FOMs (e.g., green square and diamond in Fig. 4c for D = 50 nm) are considered. Figure 6a gives parametric plots of normalized propagation length (L_{m}/λ) versus normalized mode area (A_{m}/λ^{2}), showing the confinementloss trajectory over the range of D from 50 to 400 nm. As is shown, polaritons of the same character type follows the same trajectory, allowing a fair comparison between the PEPP and the SPP that is independent of the geometric size. As indicated by the inset, the trajectories toward the upperleft area indicate the best tradeoff between confinement and loss. Result shows that although the excitonlike PEPP follows the similar trajectory line with the SPP, it can reach an ultradeep subwavelength region (< 0.005 λ^{2}) that is challenging for the SPP, while for SPPlike PEPPs, they exhibit distinct trajectories from the SPP, offering higher qualities with a more efficient confinementloss relation.
Besides the tradeoff between confinement and loss, another fundamental hurdle in plasmonics is the tradeoff between confinement and momentum [36]. For the SPP, the momentum ħRe(k) is always larger than the momentum of the freespace photon ħk_{0} (k_{0} is the wavevector in vacuum), resulting in momentum mismatch that needs to be compensated for optical excitation [33]. However, the tight confinement is usually achieved at the cost of a large momentum mismatch to the photon, which hinders the efficient SPP excitation and may further limits its application (e.g., ultrathin MNW) [35]. Figure 6b gives parametric plots of normalized momentum (Re(k)/k_{0}) vs. normalized mode area (A_{m}/λ), where the trajectory towards the bottomleft area represents the best performance in confinementmomentum relations. As is shown, PEPPs outperform the SPP and offer the capability to realize a much tighter confinement with a smaller momentum mismatch to the freespace photon (e.g., for the SPP, A_{m} of ~ 0.01 λ^{2} with a Re(k)/k_{0} of ~ 1.62, while for PEPPs, A_{m} of ~ 0.01 λ^{2} (SPPlike) and ~ 0.0015 λ^{2} (excitonlike) can be achieved at a Re(k)/k_{0} as small as ~ 1.17). The smaller momentum mismatch indicates the less momentum needs to be compensated, which may facilitate a more efficient polariton excitation [36] with an improved compatibility for integrated photonic/plasmonic structures. Such compatibility offers the opportunity to realize highperformance hybrid polaritonic components and devices (e.g., by integrating with lowloss photonic waveguides), where ultradeep subwavelength confinement and low propagation loss can be simultaneously achieved.
Considerations for Practical Applications
The fabrication of the proposed structure is experimentally possible and can be realized by various techniques for the integration of nanowaveguides and 2D materials [54,55,56]. For instance, a bare Ag MNW with a uniform diameter and smooth surface can be chemically synthesized by a solution [57] or vaporphase [58] method. The monolayer 2D material can be wrapped around the MNW via micromanipulation under an optical microscope [54, 55] or a capillaryforcedriven rollingup process [56]. By selectively wrapping the WS_{2} monolayer on one segment of the MNW, we can seamlessly integrate our proposed WS_{2}clad MNW to the bare MNW for efficient external coupling. For demonstration, 3D simulations are performed (see Additional file 1: Fig. S11 for configurations). Energy density distributions of a bare MNW without cladding (Fig. 7a), a WS_{2}clad MNW (Fig. 7b), and the integrated structure (Fig. 7c) are, respectively, provided. Since the energy is highly concentrated in the 1nm WS_{2} cladding, energy densities are normalized and plotted in a color bar with saturation [59] for better visualization and comparison. As is illustrated by the schematic plot in Fig. 7c, the left part of the integrated structure is the bare MNW without cladding, while only the right part is wrapped with the WS_{2} layer. In this case, the plasmon mode of the bare part (inset P1 in Fig. 7c) is firstly excited and then efficiently converted to the PEPP mode (inset P2 in Fig. 7c) at the right part. Note that the simulated mode profiles for the plasmon and PEPP modes (insets P1 and P2 in Fig. 7c) in the integrated structure also agree well with the one individually obtained in the bare MNW (inset P1 in Fig. 7a) and WS_{2}clad MNW (inset P1 in Fig. 7b). The advantage of this external coupling strategy is that the excitation techniques of the plasmon mode in the bare MNW are very mature and have been extensively investigated (e.g., prism coupling, lensfocusing coupling, direct nanowiretonanowire coupling, and emitter coupling) [33, 35]. Hence, we only need to focus on the coupling efficiency (η_{ext}) from the bare MNW part to the WS_{2}clad MNW part, which can be calculated as:
where k_{1} (k_{2}) and L_{1} (L_{2}) are the propagation constant and the length of the bare (WS_{2}clad) MNW part in Fig. 7c. S_{21} is the transmission S parameter obtained from the 3D simulation (see Sect. 4 in supporting information for more details about 3D simulation). The calculated η_{ext} is ~ 90%, indicating an efficient coupling from the bare MNW to our proposed WS_{2}clad MNW. It is worth mentioning that for polarizations of the excitation mode in MNWs, although the fundamental TM and secondorder HE modes do not have cutoffs, we only focus on the fundamental TM mode under the following considerations: (1) The singlemode operation is usually favorable [32, 33] and can be readily realized for practical applications (e.g., by aligning the polarization of the incident light to the long axis of the MNW to only excite the TM mode) [35, 39]; (2) more importantly, the HE mode has a dramatically increasing A_{m} with the decreasing MNW diameter, making it nonconfined with an almost infinitely large A_{m} at the small diameter we discussed here [35], which is difficult to be excited and not suitable for strong coupling applications.
Finally, for guiding practical applications, we investigate three typical situations including a substratesupported WS_{2}clad MNW, a multilayer WS_{2}clad MNW, and a WS_{2}clad MNW with an insulating layer between the metal and WS_{2}, which are shown, respectively, in Fig. 8a–c.
For the substratesupported case (Fig. 8a), we calculate the situation of a WS_{2}clad MNW on a silica substrate (n = 1.45). As is shown, the energy can be well concentrated within the WS_{2} layer (Fig. 8a(i)), and the strong coupling effect is still valid at the presence of the substrate, exhibiting a similar Rabi energy (ħΩ_{R} = ~ 86.9 meV) compared to the freestanding case (ħΩ_{R} = ~ 85.7 meV) (Fig. 8a(ii)). For the waveguiding properties, the substratesupported MNW features asymmetric SPP mode with improved waveguiding properties [32]. Since the PEPP consists of both SPP and exciton, the mode profile of the PEPP also becomes asymmetric with more energy distributed towards the substrate side (Fig. 8a(i)). Meanwhile, compared to the symmetric PEPP mode in the freestanding WS_{2}clad Ag nanowire (black dashed lines in Fig. 8a(iii–v)), the asymmetric PEPP mode has a tighter confinement (red line in Fig. 8a(iii)), a slightly shorter L_{m} (red line in Fig. 8a(iv)), and an overall enhancement in FOM (red line in Fig. 8a(v)), which may be mainly due to the improved properties of the asymmetric SPP [45, 60, 61]. As can be seen, at the wavelengths far away from the excitonic resonance, PEPPs are mostly composed of SPPs, resulting in a relatively large difference in the waveguiding properties between the freestanding and the substratesupported cases (e.g., Fig. 8a(iii), (iv)), while at the wavelengths close to the excitonic resonance, such difference becomes almost negligible since the excitons contribute mostly to the PEPPs.
For the multilayer case, a Ag MNW with a multilayer WS_{2} cladding is investigated and schematically illustrated in Fig. 8b(i). The thickness (t_{l} = 4 nm) and permittivity parameters (ε_{b} = 20.25, ħ^{2}fω_{p}^{2} = 0.8 eV^{2}, ħω_{ex} = 2 eV and ħγ_{ex} = 50 meV) of the multilayer WS_{2} are taken from Ref. [62]. Due to the larger overall transition dipole moments (\(\mu \sqrt N\) which is proportional to the thickness t_{l}, see Eq. 3), the Rabi energy (ħΩ_{R} = ~ 127.5 meV) of the strong coupling is greater than the monolayer case (Fig. 8b(ii)). On the other hand, compared to the monolayer case, the increase in the t_{l} is not favorable for the energy confinement to achieve a small A_{m} (Fig. 8b(iii)). Moreover, the exciton damping ħγ_{ex} of the multilayer WS_{2} (50 meV) is much larger than that of the monolayer WS_{2} (22 meV), leading to extra loss in the coupling system. As a result, compared to the monolayer WS_{2}clad Ag MNW (black dashed lines in Fig. 8b(iv, v), PEPP modes in the multilayer WS_{2}clad Ag MNW (red lines in Fig. 8b(iv, v)) exhibit much shorter L_{m} and much poorer FOM, which may limit their waveguiding applications.
In some applications, the direct contact of metal and TMD materials may induce weak electronic coupling that affects the exciton formation in the WS_{2} and the electron dynamics in the metal [63, 64]. To minimize the influence, one may use the structure of a core/shell MNW with a TMD layer. As is schematically illustrated in Fig. 8c(i), a Agcore/silicashell MNW [65, 66] with a monolayer WS_{2} cladding is used, where the silica shell serves as an insulating layer between the Ag and the WS_{2}. For such configuration, the strong coupling can also be achieved (Fig. 8c(ii)) with comparable waveguiding properties to the ones without the dielectric insulating layer (Fig. 8c(iii–v)).
Conclusion
In summary, we have theoretically demonstrated plasmon–exciton strong couplings in a single Ag MNW with a monolayer WS_{2} cladding, generating PEPPs with exceptional properties. As to strong coupling behaviors, solutions in both complexmomentum and complexfrequency planes have been investigated, revealing the backbending and anticrossing features with tunable Rabi splitting energies that can be controlled by varying the diameter of MNWs. We have also shown that results obtained from numerical simulations exhibit very good agreement to the ones obtained from the COM model and the analytical estimation. For the generated PEPPs, fractions, model profiles, energy density distributions, waveguiding properties including mode areas, propagation lengths, and FOMs have been investigated to provide a comprehensive characterization. We have shown that energy distributions can be reformed by the strong coupling, yielding a much tighter confinement of the PEPP than the original SPP. Meanwhile, due to the hybrid nature of polaritons, the PEPP is highly versatile possessing SPPlike and excitonlike characters that can be operated at different wavelengths. When operated at the excitonlike region, the PEPP can exert its full potential to reach the ultradeep subwavelength confinement, while in the SPPlike region, the PEPP exhibits excellent FOM with long propagation distance and tight confinement. Moreover, by comparing trajectories in the parametric plots, we have also demonstrated that PEPPs represent another kind of waveguiding polaritons with exceptional confinementloss and confinementmomentum tradeoffs that outperform the SPP in MNW. Such exceptional properties are favorable for integrations with lowloss photonic waveguides to form hybrid photonicpolaritonic structures, making it possible to bypass the barriers of nanoplasmonics with simultaneous realization of ultradeep subwavelength confinement and low propagation loss. Note that this strong coupling scheme can be extended to other configurations of different nanowaveguides and TMDs (e.g., a TMDclad pentagonal MNW and a MNW on a flat TMD, see Additional file 1: Figs. S13–S14), offering a simple and promising guidedwave platform to manipulate the plasmon–exciton interaction at the ultradeep subwavelength scale. The generated PEPPs with exceptional properties may open new opportunities for various integrated polaritonic components and devices such as onchip polaritonic circuits, polariton lasers, and alloptical switches.
Availability of Data and Materials
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 TMD:

Transition metal dichalcogenides
 PEPPs:

Propagating exciton–plasmon polaritons
 SPPs:

Surface plasmon polaritons
 1D:

One dimension
 2D:

Two dimensions
 Ag:

Silver
 WS_{2} :

Tungsten disulfide
 FWHM:

Full width at half maximum
 MNWs:

Metal nanowires
 COM:

Coupledoscillator model
 FOM:

Figure of merit
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Acknowledgements
The authors thank Prof. Pan Wang and Dr. Junsheng Zheng for helpful discussion.
Funding
Funding was provided by National Natural Science Foundation of China (Nos. 62005031 and 62005032), Fundamental Research Funds for the Central Universities (Nos. 2021CDJQY046 and 2022CDJXY018), and Innovation Support Plan for Returned Overseas Scholars (No. cx2021058).
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YW designed the work, acquired the data, and drafted the manuscript. AL, CZ, ZL, and XW supplied help for data analysis and manuscript revision. YW and XW supervised the investigation. All authors read and approved the final manuscript.
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Supporting information including numerical methods and equations for characterization, parameters for the coupledoscillator (COM) model, waveguiding properties of PEPP with varied MNW diameters, 3D simulations, and extending the strong coupling strategy to other structures.
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Wang, Y., Luo, A., Zhu, C. et al. Ultraconfined Propagating Exciton–Plasmon Polaritons Enabled by CavityFree Strong Coupling: Beating Plasmonic TradeOffs. Nanoscale Res Lett 17, 109 (2022). https://doi.org/10.1186/s11671022037487
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DOI: https://doi.org/10.1186/s11671022037487
Keywords
 Strong coupling
 Exciton–plasmon polaritons
 Waveguiding
 Transition metal dichalcogenides
 Metal nanowires