Abstract

We propose an ultra-compact electro-optic microring modulator based on a hybrid plasmonic waveguide. In comparison to previously proposed structures, the present structure utilizes aluminum-doped zinc oxide (AZO), rather than noble metals, for plasmon excitation. AZO can be used to simultaneously tune both the real and imaginary parts of the dielectric constant by changing the carrier concentration. The modulation depth and insertion loss of the microring modulator are 18.70 and 2.76 dB. The proposed modulator has a high modulation speed because its capacitance is 0.22 fF. This device could be used in high-density integrated optical circuits.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Surface plasmon polaritons (SPPs) have attracted considerable research attention over the last decade owing to their ability to surpass the diffraction limit and restrict the transmission of light waves to the nanoscale domain [13]. Some studies demonstrate that SPPs are among the most promising candidates for subwavelength optical confinement [4,5]. Recently, many SPP waveguide structures have been investigated, both experimentally and theoretically. Examples include nanowires [6,7], nanoparticles [8], stripes [9,10], slot waveguides [11,12], and metal nanostrips on dielectric substrates [13,14]. Nevertheless, SPP waveguides inevitably suffer from a very high propagation loss because they critically depend on the metal type and a considerable part of the light energy is confined to the metal film [15]. Although SPP waveguides limit the use of metals or utilize metals with smaller imaginary components of electric permittivity to achieve longer propagation lengths, their effect is limited, reducing the ability to confine light waves. It is difficult to provide a sufficient trade-off between the propagation length and the extent of field confinement. Therefore, the practical applicability of SPP waveguides has been quite limited. To overcome this, Oulton et al. [16] investigated a hybrid optical waveguide that comprised a dielectric nanowire separated from a metal surface by a nanoscale dielectric gap. However, although this hybrid plasmonic waveguide had a larger propagation distance than SPP waveguides, traditional hybrid waveguides still have issues owing to their use of metals. In addition, the problems associated with loss have significantly limited the application of the simple theoretical designs of plasmonic waveguides to practical engineering devices [17,18].

Recently, some solutions to the problem concerning loss have been proposed, based on the utilization of less traditional materials, such as transparent conductive oxides (TCOs) [19,20], alkali metals, and alkali-based alloys [19,21], as well as intermetallic compounds [19]. These materials, particularly, TCOs, exhibit much smaller losses than noble metals, such as gold and silver [22]. Furthermore, the optical properties of TCOs can be tuned by changing the carrier concentration or doping, and they can be grown by employing standard fabrication procedures [23]. Feigenbaum et al. demonstrated the refractive index changes in the accumulation layer of a metal–oxide–TCO heterostructure, and experimentally showed that the carrier concentration at the oxide/TCO interface could increase from ${1} \times {{10}^{21}}/{{\rm cm}^3}$ to ${1} \times {{10}^{22}}/{{\rm cm}^3}$ under conditions of a few volts across a 100 nm thick oxide layer [24]. Further, Melikyan et al. [25] demonstrated that the free carrier density of the indium-tin-oxide layer (ITO, with a thickness of about 30 nm) changes from ${9.25} \times {{10}^{26}}$ to ${9.34} \times {{10}^{26}}\;{{\rm m}^{- 3}}$ when applying 10 V. TCOs form an important category of plasmonic materials [22,23], and have been successfully applied to plasmonic devices such as modulators and switches [23,26]. Furthermore, aluminum-doped zinc oxide (AZO) has demonstrated superior optical properties to other TCOs (e.g., ITO) [27]. Optical modulators are important components in optical devices, and several optical modulators based on TCOs have been developed, including electro-optics modulators and ultra-fast switches.

Modulators are widely used in the field of optical interconnection. Several high-performance modulators have been proposed and fabricated based on different principles, including the thermo-optic effect [28,29] and electro-optic effect [3032]. However, most modulators use electro-optic polymers or silicon as the modulation medium. These devices have certain disadvantages, as follows: a large footprint size, low modulation speed, and complex structure. In this study, to obtain a better trade-off between performance and device size, we proposed a modulator based on AZO as the modulation medium. In comparison to other modulation media, for AZO, both the real and imaginary parts of the dielectric constant can be simultaneously tuned by changing the carrier concentration, which can provide a higher modulation efficiency. The tuning of AZO’s permittivity can provide further improvements and increase the propagation length. The results of this study show that the designed modulator has a high modulation speed ($C \approx {0.22}\;{\rm fF}$) and an ultra-compact structure. Furthermore, the proposed modulator has a broad modulation bandwidth (above 100 GHz). The proposed structure is CMOS-compatible and simple to fabricate, with good applicability in high-density integrated optical circuits owing to its high performance.

2. THEORY AND GEOMETRY OF THE PROPOSED MICRORING MODULATOR

The three-dimensional (3D) structure of the proposed waveguides and microring is shown in Fig. 1(a), while the two-dimensional (2D) cross section is shown in Fig. 1(b). The structure consists of a low-index thin layer and an AZO layer, sandwiched by high-index silicon regions. To better understand the proposed configuration, the width of the waveguides and the microring is denoted as $W$. The outer radius of the microring is denoted by $R$. The heights of the silicon or doped silicon layer, the low-index layer, and the AZO layer are denoted as ${H_{\text{Si}}}$, ${H_{\rm SiO_2}}$, and ${H_{\text{AZO}}}$, respectively. Two identical single-mode bus waveguides are placed at the two sides of the microring, and the gap is set as ${L_g}$.

 figure: Fig. 1.

Fig. 1. (a) Fundamental 3D structure and (b) the geometrical parameters of the proposed hybrid slot waveguide.

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 figure: Fig. 2.

Fig. 2. Schematic of the behavior of a light wave in the modulator.

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 figure: Fig. 3.

Fig. 3. Real and imaginary parts of AZO permittivity, for two different carrier concentrations: ${N_1} = {4.19} \times {{10}^{20}}\;{{\rm cm}^{- 3}}$ and ${N_2} = {7.089} \times {{10}^{20}}\;{{\rm cm}^{- 3}}$.

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The process followed by the modulation system is as follows. The light is launched into the first bus waveguide through the input port. The light that meets coupling conditions will transfer into the microring, loop clockwise along the microring, and a portion of the light will be transferred into the second bus waveguide and output through the drop port, as shown in Fig. 2. The resonant conditions can be defined as follows [33]:

$$\frac{{2\pi {{\rm Re}} \left({{n_{{\rm eff}}}} \right)}}{{{\lambda _m}}}L = 2\pi m,$$
where $L$ is the path length around the microring and ${n_{\text{eff}}}$ is the complex index of the microring. $m$ is an integer, which denotes that the light coupled into the ring returns to the coupling region, having accumulated a phase that is an integer $m$ times ${2}\pi$. Therefore, the resonant wavelengths of the modulator are given by
$${\lambda _m} = \frac{{L{\rm Re} \left({{n_{{\rm eff}}}} \right)}}{m}.$$

Next, the properties of the proposed microring modulator were investigated using the finite element method (FEM) and the finite-difference time-domain (FDTD) methods. In this study, the refractive index of the low-index layer and high refractive index layer were set as ${n_{\rm SiO_2}} = {1.444}$, and ${n_{\text{Si}}} = {3.478}$ [34], respectively. The permittivity of AZO can be approximated using the Drude model [22]:

$$\varepsilon = {\varepsilon _\infty} - \omega _p^2/(\omega (\omega + j\gamma)),$$
where ${\varepsilon _\infty}$ is the high-frequency dielectric constant; $\gamma$ is the electron damping factor; $\omega$ is the angular frequency of the light wave; ${\omega _p}$ is the plasma frequency, given by ${\omega _p} = N{e^2}/{\varepsilon _0}m^*$, which depends on the carrier concentration $N$; and $m^*$ is the effective electron mass. The real and the imaginary parts of AZO’s permittivity are plotted in Fig. 3 versus the wavelength for two different carrier concentrations, ${N_1}$ and ${N_2}$, using the Drude model [22,26]. Based on the experimentally measured dielectric constant of AZO, ${\varepsilon _\infty} = {4.0}$, ${\omega _{p}} = {2.22} \times {{10}^{15}}\;{{\rm s}^{- 1}}$, and $\gamma = {1.3} \times {{10}^{14}}\;{{\rm s}^{- 1}}$ are extracted [22]. The carrier concentration of AZO is ${N_1} = {4.19} \times {{10}^{20}}\;{{\rm cm}^{- 3}}$. In this case, the effective electron mass $m^* = {0.27}{m_0}$. With a suitable voltage (1–5 V), the carrier concentration can increase to ${N_2} = {7.089} \times {{10}^{20}}\;{{\rm cm}^{- 3}}$ [24,26]. Note that the imaginary part of the AZO permittivity increases with increases in the wavelength for two different carrier concentrations, implying that the propagation loss increases with the wavelength increase. However, the real part of the AZO permittivity constant exhibits the opposite trend, as shown in Fig. 3.
 figure: Fig. 4.

Fig. 4. (a), (b) 2D normalized electric field distribution of the fundamental modes of the microring waveguide with a wavelength of 1310 nm at ${N_1}$ and ${N_2}$, respectively; (c), (d) corresponding field profiles in the $Z$ axis ($E$) for the microring waveguide at ${N_1}$ and ${N_2}$, respectively.

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3. PROPERTIES OF THE ELECTRO-OPTIC MICRORING MODULATOR

To obtain a better understanding of the microring waveguide performance, the mode characteristics of the microring waveguide were investigated using FEM embedded in COMSOL Multiphysics. The free triangular mesh and the maximum element size of 3 nm were used in this simulation. The 2D electric field distributions and the 1D normalized electric fields of the waveguide are plotted in Figs. 4(a)–4(d) when the carrier concentrations are ${N_1}$ and ${N_2}$, respectively. It is important to note that most of the optical energy is confined in the AZO layer and the ${{\rm SiO}_2}$ layer when the carrier concentration is ${N_1}$. The AZO layer exhibits metal-like properties, indicating that the waveguide is excited by the plasmonic phenomenon. Figure 4(c) provides a visual understanding of the optical energy distribution in different layers. In comparison to the carrier concentration ${N_1}$, almost all of the optical energy is confined in the AZO layer when the carrier concentration is ${N_2}$, as shown in Figs. 4(b) and 4(d). Further, the AZO permittivity changes from ${1.636} + {0.214i}$ to ${0.004} + {0.361i}$ when the carrier concentration changes from ${N_1}$ to ${N_2}$. Correspondingly, the effective index of the microring changes from 2.0327 to 1.8465. As mentioned previously, we can utilize this characteristic of AZO to develop an electro-optic modulator. The properties of the waveguide directly affect the transmission performance of the modulator. To ensure the best possible properties of the proposed modulator, the waveguide should be optimized. The waveguide structure is optimized according to our previous optimization theories [35]. The heights of the Si, AZO, and ${{\rm SiO}_2}$ layers were set as ${H_{\text{Si}}} = {270}\;{\rm nm}$, ${H_{\text{AZO}}} = {10}\;{\rm nm}$, and ${H_{\rm SiO_2}} = {70}\;{\rm nm}$, respectively. The waveguide width was $W = {200}\;{\rm nm}$, to ensure a single transverse magnetic SPP mode of propagation [36]. Figure 5 shows the propagation length and the effective refractive index of the microring versus the wavelength for two different carrier concentrations. The propagation length for the carrier concentration ${N_1}$ is much shorter than that for ${N_2}$, implying a larger loss in the ${N_1}$ scenario. In addition, for a suitable applied voltage, the carrier concentration in AZO can increase from ${N_1}$ to ${N_2}$ [24]. In other words, the effective index of the waveguide can be changed by manipulating the applied voltage, implying that the resonant frequency can be changed by manipulating the applied voltage.

 figure: Fig. 5.

Fig. 5. Wavelength dependence of the waveguide properties, for different carrier concentrations, ${N_1}$ and ${N_2}$. The solid lines correspond to the real part of ${n_{\text{eff}}}$. The dashed lines correspond to the propagation length.

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 figure: Fig. 6.

Fig. 6. Distribution of the electric field, $|E|$, (a) for the carrier concentration ${N_1}$, and (b) for the carrier concentration ${N_2}$, in the $Z$ axis cross section (${\rm Z} = {305}\;{\rm nm}$).

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 figure: Fig. 7.

Fig. 7. Drop port spectra, insertion losses, and extinction ratio of the proposed ring modulator for carrier concentrations ${N_1}$ and ${N_2}$.

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To obtain better insight into the features of the modulator, we analyzed the transmission performances of the microring modulator using 3D-FDTD simulation embedded in Lumerical FDTD solutions. The Berenger’s perfectly matched layer boundary condition was applied, and the mesh accuracy was set as 8 to ensure the accuracy and reliability of the simulations. Further, convergence analysis testing was also conducted. The values of ${L_g}$ and $R$ were 80 nm and 700 nm, respectively. For carrier concentration ${N_1}$, the effective index of the model was ${n_{\text{eff}}} = {2.0327} + {0.005i}$. In this scenario, the center resonant wavelength was 1310 nm. When the carrier concentration increased to ${N_2}$, the effective index became ${n_{\text{eff}}} = {1.8465} + {0.0149i}$. Based on $L = \lambda /[{4}\pi {\rm Im}({n_{\text{eff}}})]$, the propagation lengths are 20.86 µm and 7 µm for carrier concentrations of ${N_1}$ and ${N_2}$, respectively. Figures 6(a) and 6(b) show the electric field distributions in the XY plane of the cross section of the ${{\rm SiO}_2}$ layer, which is perpendicular to the $Z$ axis for the wavelength of 1310 nm and carrier concentrations of ${N_1}$ and ${N_2}$, respectively. It is clear that, for the carrier concentration ${N_1}$, the microring modulator system exhibits strong resonance, and a considerable amount of light energy is transferred into the second bus waveguide via the microring.

Nevertheless, the light wave propagates through a relatively short distance for $N = {N_2}$. This relatively short propagation is because a significant fraction of the light wave is confined to the AZO layer, which induces a strong propagation loss in this system. To gain a better intuitive understanding, we investigated the drop port spectra via the second bus waveguide for wavelengths in the 1270–1350 nm region, and for $N = {N_1}$ and $N = {N_2}$. Figure 7(a) shows the drop port spectra as a function of the wavelength for carrier concentrations ${N_1}$ and ${N_2}$. The reflection power for the carrier concentration ${N_2}$ is quite low, reaching only 0.72% at the wavelength of 1310 nm. In contrast, the reflection power for the carrier concentration ${N_1}$ is much larger than that for the carrier concentration ${N_2}$, reaching 52.97%. Therefore, we can define the two states ${N_1}$ and ${N_2}$ as the on-state and in-state, respectively. These two states can be utilized for light modulation by manipulating the AZO carrier concentration. The calculated modulation depth of the designed modulator is 18.7 dB. This high modulation depth is achieved because the resonant wavelength of the microring changes after a voltage is applied to the device. Further, the propagation loss of the AZO layer with carrier concentration ${N_2}$ significantly increases in comparison to that with carrier concentration ${N_1}$. Generally, considering the overall system link budget and the balance of the system metrics, 4–5 dB is sufficient for the modulation depth. Figure 7(b) shows the insertion loss and extinction ratio of ${N_1}$ and ${N_2}$ as a function of wavelength. It is important to note that the calculated extinction ratio and the insertion loss are 18.70 dB and 2.76 dB, respectively, in the on-state. The maximum modulation speed of an electro-optic (EO) modulator is limited by its resistance-capacitance (RC) time constant. The capacitance can be defined as follows [37]:

$$C = S\frac{{{\varepsilon _0}{\varepsilon _{{\rm Si}}}{\varepsilon _{\text{AZO}}}{\varepsilon _{{{{\rm SiO}}_2}}}}}{{{\varepsilon _{{\rm Si}}}{\varepsilon _{\text{AZO}}}{H_{{{{\rm SiO}}_2}}} + {\varepsilon _{{\rm Si}}}{\varepsilon _{{{{\rm SiO}}_2}}}{H_{{\rm AZO}}} + {\varepsilon _{{\rm AZO}}}{\varepsilon _{{{{\rm SiO}}_2}}}{H_{{\rm Si}}}}},$$
where the cross-sectional area of the microring, $S$, is the approximate area of the plate, and ${\varepsilon _0}$ is the vacuum permittivity. The capacitance of the proposed microring modulator was 0.22 fF. The bandwidth of the modulator is determined by the corresponding RC time constant, and can be calculated by [3840]
$$f = \frac{1}{{2\pi {\rm RC}}},$$
where $R$ is the resistance of the modulator. According to Eq. (5), the proposed modulator bandwidth of the proposed modulator is above 100 GHz, given by a resistance below 7200 Ω achievable by proper electronics [41].

4. CONCLUSION

In conclusion, an ultra-compact electro-optic microring modulator based on hybrid plasmonic waveguides was presented and analyzed. The proposed microring modulator differs from traditional hybrid slot waveguides in that it induces the plasmonic phenomenon using a TCO, which enables a smaller-scale structure (submicrometer). The modulation depth of the proposed microring modulator reached 18.70 dB, while its insertion losses reached 2.276 dB in the on-state. The proposed microring modulator is fully compatible with standard semiconductor processes and can be used in integrated photonic chips and high-density photonic circuits.

Funding

Study Abroad Program for Graduate Student of Guilin University of Electronic Technology (GDYX2019001); Innovation Project of Guangxi Graduate Education (YCBZ2015037); National Natural Science Foundation of China (51366003, 61461014, 61764002).

Acknowledgment

J. Xiao thanks the Redhawk Super Computing Clusters at the Miami University.

Disclosures

The authors declare no conflicts of interest for this manuscript.

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.

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References

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  1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
    [Crossref]
  2. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
    [Crossref]
  3. S. Kim, S. Bailey, M. Liu, and R. Yan, “Decoupling co-existing surface plasmon polariton (SPP) modes in a nanowire plasmonic waveguide for quantitative mode analysis,” Nano Res. 10, 2395–2404 (2017).
    [Crossref]
  4. Y. Zhang, S. Li, Q. Xu, C. Tian, J. Gu, Y. Li, Z. Tian, C. Ouyang, J. Han, and W. Zhang, “Terahertz surface plasmon polariton waveguiding with periodic metallic cylinders,” Opt. Express 25, 14397–14405 (2017).
    [Crossref]
  5. B. Sturlesi, M. Grajower, N. Mazurski, and U. Levy, “Integrated amorphous silicon-aluminum long-range surface plasmon polariton (lr-spp) waveguides,” APL Photonics 3, 036103 (2018).
    [Crossref]
  6. W. Wang, W. Zhou, T. Fu, F. Wu, N. Zhang, Q. Li, Z. Xu, and W. Liu, “Reduced propagation loss of surface plasmon polaritons on Ag nanowire-graphene hybrid,” Nano Energy 48, 197–201 (2018).
    [Crossref]
  7. M. Song, J. Dellinger, O. Demichel, M. Buret, G. C. Des Francs, D. Zhang, E. Dujardin, and A. Bouhelier, “Selective excitation of surface plasmon modes propagating in Ag nanowires,” Opt. Express 25, 9138–9149 (2017).
    [Crossref]
  8. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003).
    [Crossref]
  9. N. R. Fong, M. Menotti, E. Lisicka-Skrzek, H. Northfield, A. Olivieri, N. Tait, M. Liscidini, and P. Berini, “Bloch long-range surface plasmon polaritons on metal stripe waveguides on a multilayer substrate,” ACS Photonics 4, 593–599 (2017).
    [Crossref]
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  11. X. Jing, W. Qiqin, Y. Daoguo, Z. Ping, H. Ning, Z. Guoqi, T. Ren, and C. Xianping, “A CMOS-compatible hybrid plasmonic slot waveguide with enhanced field confinement,” IEEE Electron Device Lett. 37, 456–458 (2016).
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  16. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
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  19. A. Boltasseva and H. A. Atwater, “Low-loss plasmonic metamaterials,” Science 331, 290–291 (2011).
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  20. X. Liu, J.-H. Kang, H. Yuan, J. Park, Y. Cui, H. Y. Hwang, and M. L. Brongersma, “Tuning of plasmons in transparent conductive oxides by carrier accumulation,” ACS Photonics 5, 1493–1498 (2018).
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  21. P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4, 795–808 (2010).
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  22. M. Noginov, L. Gu, J. Livenere, G. Zhu, A. Pradhan, R. Mundle, M. Bahoura, Y. A. Barnakov, and V. Podolskiy, “Transparent conductive oxides: Plasmonic materials for telecom wavelengths,” Appl. Phys. Lett. 99, 021101 (2011).
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  23. G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25, 3264–3294 (2013).
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  24. E. Feigenbaum, K. Diest, and H. A. Atwater, “Unity-order index change in transparent conducting oxides at visible frequencies,” Nano Lett. 10, 2111–2116 (2010).
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  25. J. Leuthold, W. Freude, C. Koos, A. Melikyan, and N. Lindenmann, “Surface plasmon polariton absorption modulator,” Opt. Express 19, 8855–8869 (2011).
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  26. Z. Lu, W. Zhao, and K. Shi, “Ultracompact electroabsorption modulators based on tunable epsilon-near-zero-slot waveguides,” IEEE Photonics J. 4, 735–740 (2012).
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  33. Y. S. Yi, B. J. Koch, and T. L. Smith, “Nanoparticle sensing utilizing back scattering reflection mode of an integrated micro-resonator,” IEEE J. Sel. Top. Quantum 22, 1–11 (2016).
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  36. H. S. Chu, E. P. Li, P. Bai, and R. Hegde, “Optical performance of single-mode hybrid dielectric-loaded plasmonic waveguide-based components,” Appl. Phys. Lett. 96, 221103 (2010).
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  37. F. Lou, D. Dai, L. Thylen, and L. Wosinski, “Design and analysis of ultra-compact EO polymer modulators based on hybrid plasmonic microring resonators,” Opt. Express 21, 20041–20051 (2013).
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  38. J. Witzens, T. Baehr-Jones, and M. Hochberg, “Design of transmission line driven slot waveguide Mach-Zehnder interferometers and application to analog optical links,” Opt. Express 18, 16902–16928 (2010).
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  39. J. Leuthold, W. Freude, J. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon organic hybrid technology—a platform for practical nonlinear optics,” Proc. IEEE 97, 1304–1316 (2009).
    [Crossref]
  40. T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, and A. Jen, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92, 163303 (2008).
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  41. L. Alloatti, D. Korn, R. Palmer, D. Hillerkuss, J. Li, A. Barklund, R. Dinu, J. Wieland, M. Fournier, and J. Fedeli, “42.7 Gbit/s electro-optic modulator in silicon technology,” Opt. Express 19, 11841–11851 (2011).
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2020 (2)

Z. Shi, J. Gong, J. Zhang, P. Xu, N. Yao, W. Fang, P. Wang, X. Guo, and L. Tong, “Single-nanowire thermo-optic modulator based on a Varshni shift,” ACS Photonics 7, 2571–2577 (2020).
[Crossref]

L. Ye, K. Sui, and H. Feng, “High-efficiency couplers for graphene surface plasmon polaritons in the mid-infrared region,” Opt. Lett. 45, 264–267 (2020).
[Crossref]

2019 (3)

2018 (4)

B. Sturlesi, M. Grajower, N. Mazurski, and U. Levy, “Integrated amorphous silicon-aluminum long-range surface plasmon polariton (lr-spp) waveguides,” APL Photonics 3, 036103 (2018).
[Crossref]

W. Wang, W. Zhou, T. Fu, F. Wu, N. Zhang, Q. Li, Z. Xu, and W. Liu, “Reduced propagation loss of surface plasmon polaritons on Ag nanowire-graphene hybrid,” Nano Energy 48, 197–201 (2018).
[Crossref]

L. Ye, K. Sui, Y. Liu, M. Zhang, and Q. H. Liu, “Graphene-based hybrid plasmonic waveguide for highly efficient broadband mid-infrared propagation and modulation,” Opt. Express 26, 15935–15947 (2018).
[Crossref]

X. Liu, J.-H. Kang, H. Yuan, J. Park, Y. Cui, H. Y. Hwang, and M. L. Brongersma, “Tuning of plasmons in transparent conductive oxides by carrier accumulation,” ACS Photonics 5, 1493–1498 (2018).
[Crossref]

2017 (6)

F. Vetrone and F. Rosei, “A low-loss origami plasmonic waveguide,” Science 357, 452–453 (2017).
[Crossref]

M. Song, J. Dellinger, O. Demichel, M. Buret, G. C. Des Francs, D. Zhang, E. Dujardin, and A. Bouhelier, “Selective excitation of surface plasmon modes propagating in Ag nanowires,” Opt. Express 25, 9138–9149 (2017).
[Crossref]

S. Kim, S. Bailey, M. Liu, and R. Yan, “Decoupling co-existing surface plasmon polariton (SPP) modes in a nanowire plasmonic waveguide for quantitative mode analysis,” Nano Res. 10, 2395–2404 (2017).
[Crossref]

Y. Zhang, S. Li, Q. Xu, C. Tian, J. Gu, Y. Li, Z. Tian, C. Ouyang, J. Han, and W. Zhang, “Terahertz surface plasmon polariton waveguiding with periodic metallic cylinders,” Opt. Express 25, 14397–14405 (2017).
[Crossref]

N. R. Fong, M. Menotti, E. Lisicka-Skrzek, H. Northfield, A. Olivieri, N. Tait, M. Liscidini, and P. Berini, “Bloch long-range surface plasmon polaritons on metal stripe waveguides on a multilayer substrate,” ACS Photonics 4, 593–599 (2017).
[Crossref]

J. Xiao, Q. Wei, D. Yang, P. Zhang, N. He, G. Zhang, and X. Chen, “Hybrid plasmonics slot THz waveguide for subwavelength field confinement and crosstalk between two waveguides,” IEEE J. Sel. Top. Quantum 23, 4601505 (2017).
[Crossref]

2016 (2)

X. Jing, W. Qiqin, Y. Daoguo, Z. Ping, H. Ning, Z. Guoqi, T. Ren, and C. Xianping, “A CMOS-compatible hybrid plasmonic slot waveguide with enhanced field confinement,” IEEE Electron Device Lett. 37, 456–458 (2016).
[Crossref]

Y. S. Yi, B. J. Koch, and T. L. Smith, “Nanoparticle sensing utilizing back scattering reflection mode of an integrated micro-resonator,” IEEE J. Sel. Top. Quantum 22, 1–11 (2016).
[Crossref]

2014 (1)

Y. Bian and Q. Gong, “Highly confined guiding of low-loss plasmon waves in hybrid metal-dielectric slot waveguides,” Nanotechnology 25, 345201 (2014).
[Crossref]

2013 (2)

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25, 3264–3294 (2013).
[Crossref]

F. Lou, D. Dai, L. Thylen, and L. Wosinski, “Design and analysis of ultra-compact EO polymer modulators based on hybrid plasmonic microring resonators,” Opt. Express 21, 20041–20051 (2013).
[Crossref]

2012 (2)

Z. Lu, W. Zhao, and K. Shi, “Ultracompact electroabsorption modulators based on tunable epsilon-near-zero-slot waveguides,” IEEE Photonics J. 4, 735–740 (2012).
[Crossref]

L. Chen, X. Li, G. Wang, W. Li, S. Chen, L. Xiao, and D. Gao, “A silicon-based 3-D hybrid long-range plasmonic waveguide for nanophotonic integration,” J. Lightwave Technol. 30, 163–168 (2012).
[Crossref]

2011 (5)

G. V. Naik and A. Boltasseva, “A comparative study of semiconductor-based plasmonic metamaterials,” Metamaterials 5, 1–7 (2011).
[Crossref]

M. Noginov, L. Gu, J. Livenere, G. Zhu, A. Pradhan, R. Mundle, M. Bahoura, Y. A. Barnakov, and V. Podolskiy, “Transparent conductive oxides: Plasmonic materials for telecom wavelengths,” Appl. Phys. Lett. 99, 021101 (2011).
[Crossref]

J. Leuthold, W. Freude, C. Koos, A. Melikyan, and N. Lindenmann, “Surface plasmon polariton absorption modulator,” Opt. Express 19, 8855–8869 (2011).
[Crossref]

A. Boltasseva and H. A. Atwater, “Low-loss plasmonic metamaterials,” Science 331, 290–291 (2011).
[Crossref]

L. Alloatti, D. Korn, R. Palmer, D. Hillerkuss, J. Li, A. Barklund, R. Dinu, J. Wieland, M. Fournier, and J. Fedeli, “42.7 Gbit/s electro-optic modulator in silicon technology,” Opt. Express 19, 11841–11851 (2011).
[Crossref]

2010 (4)

H. S. Chu, E. P. Li, P. Bai, and R. Hegde, “Optical performance of single-mode hybrid dielectric-loaded plasmonic waveguide-based components,” Appl. Phys. Lett. 96, 221103 (2010).
[Crossref]

J. Witzens, T. Baehr-Jones, and M. Hochberg, “Design of transmission line driven slot waveguide Mach-Zehnder interferometers and application to analog optical links,” Opt. Express 18, 16902–16928 (2010).
[Crossref]

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4, 795–808 (2010).
[Crossref]

E. Feigenbaum, K. Diest, and H. A. Atwater, “Unity-order index change in transparent conducting oxides at visible frequencies,” Nano Lett. 10, 2111–2116 (2010).
[Crossref]

2009 (1)

J. Leuthold, W. Freude, J. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon organic hybrid technology—a platform for practical nonlinear optics,” Proc. IEEE 97, 1304–1316 (2009).
[Crossref]

2008 (3)

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, and A. Jen, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92, 163303 (2008).
[Crossref]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
[Crossref]

E. Verhagen, A. Polman, and L. Kuipers, “Nanofocusing in laterally tapered plasmonic waveguides,” Opt. Express 16, 45–57 (2008).
[Crossref]

2006 (2)

R. Zia, J. A. Schuller, and M. L. Brongersma, “Near-field characterization of guided polariton propagation and cutoff in surface plasmon waveguides,” Phys. Rev. B 74, 165415 (2006).
[Crossref]

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[Crossref]

2003 (2)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref]

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003).
[Crossref]

Alloatti, L.

Atwater, H. A.

A. Boltasseva and H. A. Atwater, “Low-loss plasmonic metamaterials,” Science 331, 290–291 (2011).
[Crossref]

E. Feigenbaum, K. Diest, and H. A. Atwater, “Unity-order index change in transparent conducting oxides at visible frequencies,” Nano Lett. 10, 2111–2116 (2010).
[Crossref]

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003).
[Crossref]

Baehr-Jones, T.

J. Witzens, T. Baehr-Jones, and M. Hochberg, “Design of transmission line driven slot waveguide Mach-Zehnder interferometers and application to analog optical links,” Opt. Express 18, 16902–16928 (2010).
[Crossref]

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, and A. Jen, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92, 163303 (2008).
[Crossref]

Baets, R.

J. Leuthold, W. Freude, J. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon organic hybrid technology—a platform for practical nonlinear optics,” Proc. IEEE 97, 1304–1316 (2009).
[Crossref]

Bahoura, M.

M. Noginov, L. Gu, J. Livenere, G. Zhu, A. Pradhan, R. Mundle, M. Bahoura, Y. A. Barnakov, and V. Podolskiy, “Transparent conductive oxides: Plasmonic materials for telecom wavelengths,” Appl. Phys. Lett. 99, 021101 (2011).
[Crossref]

Bai, P.

H. S. Chu, E. P. Li, P. Bai, and R. Hegde, “Optical performance of single-mode hybrid dielectric-loaded plasmonic waveguide-based components,” Appl. Phys. Lett. 96, 221103 (2010).
[Crossref]

Bailey, S.

S. Kim, S. Bailey, M. Liu, and R. Yan, “Decoupling co-existing surface plasmon polariton (SPP) modes in a nanowire plasmonic waveguide for quantitative mode analysis,” Nano Res. 10, 2395–2404 (2017).
[Crossref]

Barklund, A.

Barnakov, Y. A.

M. Noginov, L. Gu, J. Livenere, G. Zhu, A. Pradhan, R. Mundle, M. Bahoura, Y. A. Barnakov, and V. Podolskiy, “Transparent conductive oxides: Plasmonic materials for telecom wavelengths,” Appl. Phys. Lett. 99, 021101 (2011).
[Crossref]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref]

Berini, P.

N. R. Fong, M. Menotti, E. Lisicka-Skrzek, H. Northfield, A. Olivieri, N. Tait, M. Liscidini, and P. Berini, “Bloch long-range surface plasmon polaritons on metal stripe waveguides on a multilayer substrate,” ACS Photonics 4, 593–599 (2017).
[Crossref]

N. Fong, M. Menotti, E. Lisicka-Skrzek, H. Northfield, A. Olivieri, R. N. Tait, M. Liscidini, and P. Berini, “Bloch long-range surface plasmon polaritons in metallic stripe waveguides,” in CLEO: Science and Innovations (Optical Society of America, 2017), paper SM3N.3.

Biaggio, I.

J. Leuthold, W. Freude, J. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon organic hybrid technology—a platform for practical nonlinear optics,” Proc. IEEE 97, 1304–1316 (2009).
[Crossref]

Bian, Y.

Y. Bian and Q. Gong, “Highly confined guiding of low-loss plasmon waves in hybrid metal-dielectric slot waveguides,” Nanotechnology 25, 345201 (2014).
[Crossref]

Boltasseva, A.

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25, 3264–3294 (2013).
[Crossref]

G. V. Naik and A. Boltasseva, “A comparative study of semiconductor-based plasmonic metamaterials,” Metamaterials 5, 1–7 (2011).
[Crossref]

A. Boltasseva and H. A. Atwater, “Low-loss plasmonic metamaterials,” Science 331, 290–291 (2011).
[Crossref]

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4, 795–808 (2010).
[Crossref]

Bouhelier, A.

Brongersma, M. L.

X. Liu, J.-H. Kang, H. Yuan, J. Park, Y. Cui, H. Y. Hwang, and M. L. Brongersma, “Tuning of plasmons in transparent conductive oxides by carrier accumulation,” ACS Photonics 5, 1493–1498 (2018).
[Crossref]

R. Zia, J. A. Schuller, and M. L. Brongersma, “Near-field characterization of guided polariton propagation and cutoff in surface plasmon waveguides,” Phys. Rev. B 74, 165415 (2006).
[Crossref]

Brosi, J.

J. Leuthold, W. Freude, J. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon organic hybrid technology—a platform for practical nonlinear optics,” Proc. IEEE 97, 1304–1316 (2009).
[Crossref]

Buret, M.

Cada, M.

Chen, L.

Chen, S.

Chen, X.

J. Xiao, Q. Wei, D. Yang, P. Zhang, N. He, G. Zhang, and X. Chen, “Hybrid plasmonics slot THz waveguide for subwavelength field confinement and crosstalk between two waveguides,” IEEE J. Sel. Top. Quantum 23, 4601505 (2017).
[Crossref]

Chu, H. S.

H. S. Chu, E. P. Li, P. Bai, and R. Hegde, “Optical performance of single-mode hybrid dielectric-loaded plasmonic waveguide-based components,” Appl. Phys. Lett. 96, 221103 (2010).
[Crossref]

Chung, S.

Cui, Y.

X. Liu, J.-H. Kang, H. Yuan, J. Park, Y. Cui, H. Y. Hwang, and M. L. Brongersma, “Tuning of plasmons in transparent conductive oxides by carrier accumulation,” ACS Photonics 5, 1493–1498 (2018).
[Crossref]

Dai, D.

Dalton, L.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, and A. Jen, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92, 163303 (2008).
[Crossref]

Daoguo, Y.

X. Jing, W. Qiqin, Y. Daoguo, Z. Ping, H. Ning, Z. Guoqi, T. Ren, and C. Xianping, “A CMOS-compatible hybrid plasmonic slot waveguide with enhanced field confinement,” IEEE Electron Device Lett. 37, 456–458 (2016).
[Crossref]

Davies, J.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, and A. Jen, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92, 163303 (2008).
[Crossref]

Dellinger, J.

Demichel, O.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref]

Des Francs, G. C.

Diederich, F.

J. Leuthold, W. Freude, J. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon organic hybrid technology—a platform for practical nonlinear optics,” Proc. IEEE 97, 1304–1316 (2009).
[Crossref]

Diest, K.

E. Feigenbaum, K. Diest, and H. A. Atwater, “Unity-order index change in transparent conducting oxides at visible frequencies,” Nano Lett. 10, 2111–2116 (2010).
[Crossref]

Dinu, R.

Dujardin, E.

Dumon, P.

J. Leuthold, W. Freude, J. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon organic hybrid technology—a platform for practical nonlinear optics,” Proc. IEEE 97, 1304–1316 (2009).
[Crossref]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref]

Emani, N. K.

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. 4, 795–808 (2010).
[Crossref]

Fang, W.

Z. Shi, J. Gong, J. Zhang, P. Xu, N. Yao, W. Fang, P. Wang, X. Guo, and L. Tong, “Single-nanowire thermo-optic modulator based on a Varshni shift,” ACS Photonics 7, 2571–2577 (2020).
[Crossref]

Fedeli, J.

Feigenbaum, E.

E. Feigenbaum, K. Diest, and H. A. Atwater, “Unity-order index change in transparent conducting oxides at visible frequencies,” Nano Lett. 10, 2111–2116 (2010).
[Crossref]

Feng, H.

Fong, N.

N. Fong, M. Menotti, E. Lisicka-Skrzek, H. Northfield, A. Olivieri, R. N. Tait, M. Liscidini, and P. Berini, “Bloch long-range surface plasmon polaritons in metallic stripe waveguides,” in CLEO: Science and Innovations (Optical Society of America, 2017), paper SM3N.3.

Fong, N. R.

N. R. Fong, M. Menotti, E. Lisicka-Skrzek, H. Northfield, A. Olivieri, N. Tait, M. Liscidini, and P. Berini, “Bloch long-range surface plasmon polaritons on metal stripe waveguides on a multilayer substrate,” ACS Photonics 4, 593–599 (2017).
[Crossref]

Fournier, M.

Frank, B.

J. Leuthold, W. Freude, J. Brosi, R. Baets, P. Dumon, I. Biaggio, M. L. Scimeca, F. Diederich, B. Frank, and C. Koos, “Silicon organic hybrid technology—a platform for practical nonlinear optics,” Proc. IEEE 97, 1304–1316 (2009).
[Crossref]

Freude, W.

J. Leuthold, W. Freude, C. Koos, A. Melikyan, and N. Lindenmann, “Surface plasmon polariton absorption modulator,” Opt. Express 19, 8855–8869 (2011).
[Crossref]

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Li, Q.

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N. Fong, M. Menotti, E. Lisicka-Skrzek, H. Northfield, A. Olivieri, R. N. Tait, M. Liscidini, and P. Berini, “Bloch long-range surface plasmon polaritons in metallic stripe waveguides,” in CLEO: Science and Innovations (Optical Society of America, 2017), paper SM3N.3.

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N. Fong, M. Menotti, E. Lisicka-Skrzek, H. Northfield, A. Olivieri, R. N. Tait, M. Liscidini, and P. Berini, “Bloch long-range surface plasmon polaritons in metallic stripe waveguides,” in CLEO: Science and Innovations (Optical Society of America, 2017), paper SM3N.3.

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R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008).
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X. Jing, W. Qiqin, Y. Daoguo, Z. Ping, H. Ning, Z. Guoqi, T. Ren, and C. Xianping, “A CMOS-compatible hybrid plasmonic slot waveguide with enhanced field confinement,” IEEE Electron Device Lett. 37, 456–458 (2016).
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X. Jing, W. Qiqin, Y. Daoguo, Z. Ping, H. Ning, Z. Guoqi, T. Ren, and C. Xianping, “A CMOS-compatible hybrid plasmonic slot waveguide with enhanced field confinement,” IEEE Electron Device Lett. 37, 456–458 (2016).
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G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25, 3264–3294 (2013).
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Z. Lu, W. Zhao, and K. Shi, “Ultracompact electroabsorption modulators based on tunable epsilon-near-zero-slot waveguides,” IEEE Photonics J. 4, 735–740 (2012).
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Z. Shi, J. Gong, J. Zhang, P. Xu, N. Yao, W. Fang, P. Wang, X. Guo, and L. Tong, “Single-nanowire thermo-optic modulator based on a Varshni shift,” ACS Photonics 7, 2571–2577 (2020).
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Z. Shi, J. Gong, J. Zhang, P. Xu, N. Yao, W. Fang, P. Wang, X. Guo, and L. Tong, “Single-nanowire thermo-optic modulator based on a Varshni shift,” ACS Photonics 7, 2571–2577 (2020).
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W. Wang, W. Zhou, T. Fu, F. Wu, N. Zhang, Q. Li, Z. Xu, and W. Liu, “Reduced propagation loss of surface plasmon polaritons on Ag nanowire-graphene hybrid,” Nano Energy 48, 197–201 (2018).
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J. Xiao, Q. Wei, D. Yang, P. Zhang, N. He, G. Zhang, and X. Chen, “Hybrid plasmonics slot THz waveguide for subwavelength field confinement and crosstalk between two waveguides,” IEEE J. Sel. Top. Quantum 23, 4601505 (2017).
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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.

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Figures (7)

Fig. 1.
Fig. 1. (a) Fundamental 3D structure and (b) the geometrical parameters of the proposed hybrid slot waveguide.
Fig. 2.
Fig. 2. Schematic of the behavior of a light wave in the modulator.
Fig. 3.
Fig. 3. Real and imaginary parts of AZO permittivity, for two different carrier concentrations: ${N_1} = {4.19} \times {{10}^{20}}\;{{\rm cm}^{- 3}}$ and ${N_2} = {7.089} \times {{10}^{20}}\;{{\rm cm}^{- 3}}$.
Fig. 4.
Fig. 4. (a), (b) 2D normalized electric field distribution of the fundamental modes of the microring waveguide with a wavelength of 1310 nm at ${N_1}$ and ${N_2}$, respectively; (c), (d) corresponding field profiles in the $Z$ axis ($E$) for the microring waveguide at ${N_1}$ and ${N_2}$, respectively.
Fig. 5.
Fig. 5. Wavelength dependence of the waveguide properties, for different carrier concentrations, ${N_1}$ and ${N_2}$. The solid lines correspond to the real part of ${n_{\text{eff}}}$. The dashed lines correspond to the propagation length.
Fig. 6.
Fig. 6. Distribution of the electric field, $|E|$, (a) for the carrier concentration ${N_1}$, and (b) for the carrier concentration ${N_2}$, in the $Z$ axis cross section (${\rm Z} = {305}\;{\rm nm}$).
Fig. 7.
Fig. 7. Drop port spectra, insertion losses, and extinction ratio of the proposed ring modulator for carrier concentrations ${N_1}$ and ${N_2}$.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

2 π R e ( n e f f ) λ m L = 2 π m ,
λ m = L R e ( n e f f ) m .
ε = ε ω p 2 / ( ω ( ω + j γ ) ) ,
C = S ε 0 ε S i ε AZO ε S i O 2 ε S i ε AZO H S i O 2 + ε S i ε S i O 2 H A Z O + ε A Z O ε S i O 2 H S i ,
f = 1 2 π R C ,