Abstract

The tunability of slow light in graphene-based hyperbolic metamaterial waveguide operating in SCLU telecom bands is investigated. For the first time it has been shown that proper design of a GHMM structure forming waveguide layer and the geometry of the waveguide itself allows stopped light to be obtained in an almost freely selected range of wavelengths within SCLU bands. In particular, the possibility of controlling light propagation in GHMM waveguides by external biasing has been presented. The change of external electric field enables the stop light of the selected wavelength as well as the control of a number of modes, which can be stopped, cut off or supported. Proposed GHMM waveguides could offer great opportunities in the field of integrated photonics that are compatible with CMOS technology, especially since such structures can be utilized as photonic memory cells, tunable optical buffers, delays, optical modulators etc.

© 2017 Optical Society of America

1. Introduction

Phenomena of slow and stopped light have, in the last decade, attracted a growing interest due to a wide range of potential applications, including ultra-compact optical modulators and switches, enhanced optical nonlinearities, optical buffering, memory devices etc [1–3]. Initially, slow light was observed in Bose-Einstein condensates [4], Ruby crystals [5], ultra-cold atomic gases [6] and in solid media [7]. More recently, the new mechanisms of achieving slow and stopped light in optical waveguides have been developed [8]. Especially, this effect has been observed in photonic crystal waveguides [8–11]. Another solutions considered for this purpose are plasmonic [12–15] and negative-refractive–index (NRI) metamaterial waveguides [16–22]. In particular, it has been demonstrated both experimentally and theoretically that slow light effect can be achieved in various waveguide configurations, including single-negative [21] as well as double negative materials [18]. Moreover, the general concept of ‘rainbow trapping’ in negative index material (NIM), allowing for slowing or stopping of light over extremely large bandwidths, has been proposed in Ref [18]. What is more, it has been shown, that gain-enhanced nanoplasmonic metamaterials have been investigated in means of improved active imaging, ultrafast nonlinearities as well as cavity-free lasing [23]. Alternatively, the implementation of hyperbolic metamaterial (HMM) in the waveguide structures allows for field enhancing [24], plasmon propagation [25] as well as slowing and stopping light [26–28]. It is also worth to note, that the technology of this kind of materials is available.

In general, hyperbolic metamaterial represents a special class of uniaxially anisotropic metamaterials, realized mostly by sub-wavelength metal-dielectric subsequent layers, which can be described by a diagonal permittivity tensor ε¯¯=diag[ε||,ε||,ε], where the principal components have the opposite signs, i.e., ε|| > 0, ε < 0 (Type I of dielectric character) or ε|| < 0, ε > 0 (Type II of metallic character) [29–31].

Because of strong anisotropy and hyperbolic dispersion characteristic of HMM, it is shown that a waveguide with core layer formed by Type II HMM provide anomalous modal dispersion leading to superluminal and stopped light [26–28]. This effect is a consequence of TM modal degeneration resulting in the presence of two modes with contrary energy flows. Moreover, there exists certain core layer width for given wavelength for which two propagation constants are identical and energy flows are mutually compensated leading to effective light trapping (critical point of dispersion characteristics).

New functionality of HMM structures can be gained by substituting metal layer by graphene, so called Graphene-based HMM (GHMM) [32–40]. Since the conductivity of single layer graphene is frequency and chemical potential dependent [41,42], the effective tensor components of graphene/dielectric metamaterial structure also reveal similar dependence. Thus, by controlling chemical potential of graphene it is possible to change optical properties of this kind of structure. It is worth noting that, in general 2D materials, such as graphene, provide controllable plasmon propagation over long distances [43,44].

Recently, a novel GHMM-based waveguide has been proposed and numerically analyzed in context of local field enhancement [45], controlled plasmon propagation [46] as well as controlled light slowing in THz region [47].

The study presented in [47] concerned the tunable slow-light waveguide controlled by voltage or temperature and operating in THz range. Especially, the phenomenon of controlling speed of light from slow to fast may find exciting applications in photonic switch, optical buffers and memory devices operating in telecom frequency range.

In this paper we provide detailed study of slow light inside a graphene-based hyperbolic metamaterial (GHMM) waveguide operating in SCLU telecom bands. The analyzed structure is a symmetric planar waveguide composed of Type II GHMM core and air cladding. In our approach, we assume that both of the permittivity tensor components (i.e., ε||, ε) depend on frequency and GHMM structure parameters (i.e., thickness of dielectric and graphene layers) as well as higher order modes propagation is analyzed, in contrast to [47] where the perpendicular permittivity of GHMM, ε||, was taken to be constant and study was confined to fundamental oscillatory/bulk mode propagation. Moreover, without losing the generality, the considered structure is assumed to be lossless since it does not affect the existence of slow light phenomenon [14].

Our numerical results revealed possibility of voltage-controlled stopped light of selected wavelength within the SCLU bands range for certain waveguide width. Another proposed approach corresponds to employing waveguide with varying width (i.e., tapered waveguide) which allows for switchable light stopping of all wavelengths within the range of SCLU bands.

2. Theoretical model

Figure 1(a) shows an air/GHMM/air planar waveguide with schematically depicted oscillatory mode profile considered in this work. The GHMM consists of subsequent ultrathin graphene-dielectric stacks that single basic cell is composed of a graphene sheet and a dielectric layer (SiO2, ɛd = 2.08). In considered structure we assume “z” axis as a propagation direction.

 figure: Fig. 1

Fig. 1 Scheme of (a) analyzed waveguide and (b) its effective representation.

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The multilayer metamaterial can be treated as a homogeneous effective medium identified by the permittivity tensor:

ε=[ε||000ε||000ε],
where ǁ (x, y-axes) and ⊥ (z-axis) refer to the directions parallel and perpendicular to the interfaces in multilayer realization, respectively (see Fig. 1).

As the multilayer structure is subwavelength-scaled (i.e., the thickness of the unit cell is much less than the wavelength of light) the effective medium theory (EMT) can be applied to describe the optical properties of GHMM (so called effective medium homogenization model). By applying EMT method the diagonal components of the relative permittivity tensor are given as follows [25]:

ε||=tgεg+tdεdtg+td,ε=εgεd(tg+td)tgεd+tdεg,
where tg and εg are the thickness and dielectric permittivity of graphene layer in the unit cell, td and ɛd are the thickness and dielectric permittivity of dielectric layer, respectively. In contrast to model presented in [47], our approach predicts the influence of structure parameters on tensor components (see Eq. (2), since thicknesses of constituent layers are comparable. It is worth noting that our considerations are limited to real permittivity of dielectric layers, due to the fact that, in general, losses of constituent materials does not prevent slowing light effect from occurring [14].

According to Ref [39]. the graphene’s effective permittivity can be written as:

εg=1jσ(ω,μc)ωεotg,
where ɛ0 is the vacuum permittivity and σ is the conductivity of a single-layer graphene. Frequency and chemical potential dependent conductivity of a monolayer graphene can be given by Kubo formula [48]:
σ(ω,μc)=j4πq2kBTh2(ωj2τ)[μckBT+2ln(eμ/kBT+1)]+j4πq2(ωj2τ)h20fD(ξ)fD(ξ)(ωj2τ)216(πξh)dξ,
where fD(ξ) is the Fermi-Dirac function: fD(ξ) = [exp(ξ-μC/kBT) + 1]−1, μc is the chemical potential, T is temperature, kB and h are Boltzmann and Planck’s constant, ω is the angular frequency of the incident electromagnetic wave, and τ is the phenomenological scattering rate, which we set equal 0.1 meV. In case of a few layer graphene its multilayer conductivity can be described by σml = σNg, where Ng is the number of graphene layers. Considered model can be exploited for Ng ≤ 6 [35].

A change of graphene’s chemical potential and simultaneously the optical properties of the GHMM can be achieved by gate voltage Vg. Relationship between gate voltage and chemical potential can be given by the following formula [35]:

|μc|=υFπ|a0(VgVdirac)|,

where ħ is Dirac constant, υF is the Fermi velocity of Dirac fermions in graphene (∼106 m/s), a0 = 9 × 1016 m−1V−1, Vdirac is offset bias which reflects graphene’s doping and/or its impurities.

Considering a symmetric planar GHMM waveguide described by effective parameters (i.e., ɛ||, ɛ) (see Fig. 1.) with an air cladding layer (ε1 = 1) and a graphene-based hyperbolic metamaterial as a core layer, the propagation constant β of the transverse-magnetic (TM) oscillatory modes can be obtained by solving eigenmode equations [26].

tan(γfW2)=γ1γfεε1      forevenmodes,
cot(γfW2)=γ1γfεε1      foroddmodes,
where γf=ko2εεε||β2, γ1=β2ko2ε1 and W is the width of the waveguide.

Using above equations, Eqs. (6) and (7), the dispersion characteristics of symmetric GHMM waveguide, revealing the possibility of slow-light effect, are obtained.

3. Results and discussion

We present the dispersion characteristics of the analyzed waveguide with GHMM core layer. The propagation constants of the waveguide modes for different core-layer widths are obtained by solving the eigenmode equations given by Eqs. (6) and (7). Since the slow-light effect is achievable in waveguides with Type II HMM core, the first step of our analysis covers obtaining this kind of GHMM within SCLU bands.

As we presented in [40] the Type II of GHMM structure (i.e., ε||<0, ε >0) is achievable by proper tailoring of the parameters of basic cell (i.e., thickness and permittivity of dielectric layer, number of graphene layers), as well as applying external electric field. This is shown in Figs. 2(a) and 2(b) where the permittivity tensor components, ε|| and ε, are plotted as a function of the gate voltage Vg for the boundary wavelengths of SCLU bands, i.e., λ = 1.46 µm and λ = 1.675 µm, which are illustrated by blue and red line, respectively. In our case, the basic cell of hyperbolic metamaterial consists of SiO2 with thickness td and permittivity εd = 2.08 as well as graphene monolayers, which number is denoted by Ng.

 figure: Fig. 2

Fig. 2 Permittivity tensor components for boundary wavelengths of SCLU bands, λ = 1.46 μm and λ = 1.675 μm, as a function of gate voltage Vg and for different structure parameters (a) td = 8 nm, Ng = 4 and (b) td = 6 nm, Ng = 6.

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In particularly, the Type II is achievable for the structure consisted of dielectric layer with td = 6 nm and six graphene monolayer (Ng = 6), at the gate voltage Vg = 0.76 V for the wavelength λ = 1.46 µm and at the gate voltage Vg = 0.44 V for the wavelength λ = 1.675 μm, see Figs. 2(a) and 2(b). When we change the geometry of the basic cell (td = 8 nm, Ng = 4), the gate voltage required to achieve Type II is higher and for λ = 1.46 µm takes value Vg = 1.57 V, while for λ = 1.675 µm is equal Vg = 0.89 V.

The next two figures, Figs. 3(a) and 3(b) show the dispersion characteristics of the waveguide with GHMM core layer for the fundamental mode TM0 and different parameters of the basic cell. First of all, we can notice that waveguide of given width layer supports two different propagation constants for each TM mode corresponding to: the forward mode, whose energy flow and wave vector are in the same directions (lower branch), and backward mode, whose energy flow and wave vector are in opposite directions (upper branch). Moreover, for a given core layer width, a critical point appears in which two propagation constants are identical and power flows are mutually compensated leading to effective light stopping (trapping), see Figs. 3(c) and 3(d). In these figures, normalized total power flow P, defined by P=(P1+P2)/(|P|1+|P2|), where P1=2d/2Szdx, P2=d/2d/2Szdx and Sz is the z-component of the Poynting vector, is shown.

 figure: Fig. 3

Fig. 3 Normalized propagation constant β/k0 of the fundamental mode TM0 as a function of normalized waveguide width W/λ for (a) different numbers of graphene layers Ng, and (b) various values of dielectric layer thickness td. Normalized power flow P as a function of normalized waveguide width W/λ for (c) different numbers of graphene layers Ng, and (d) various values of dielectric layer thickness td.

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The negative power flow, P < 0, is related to backward modes possessing propagation constants corresponding to the upper branch of dispersion characteristic, i.e., above the critical point, while P > 0 is connected with the lower branch, i.e., below critical point, see Figs. 3(a)-3(d), and P = 0 at the critical point. Furthermore, an increase in the number of graphene layers (for given thickness of the dielectric layer) leads to the shift of the critical point towards greater values of waveguide width, see Fig. 3(a). On the contrary, with the increasing thickness of the dielectric layer the light stopping appears for lower values of waveguide width, see Fig. 3(b).

From a technological point of view we should look for such solutions for which light stopping could occur at the possibly largest values of the waveguide width, i.e., possibly large Ng and simultaneously small td. On the other hand the dielectric layer should not be too thin to avoid the tunneling effect. Thus, for further discussion we will concentrate on the waveguide with GHMM core layer consisting of subsequent bilayers of 6 nm SiO2 layer (no tunneling effect [49]) and six graphene monolayers.

In Figs. 4(a) and 4(b) the dispersion curves for the wavelengths within the SCLU bands are presented, where blue and green curves correspond to the boundary wavelengths of this range, i.e., λ = 1.46 µm and λ = 1.675 µm, respectively. It is shown that the proper biasing, in this case Vg = 1 V in Fig. 4(a) and Vg = 2 V in Fig. 4(b), as well as fabrication of the tapered waveguide with varying width including the range from 131 nm to 46 nm ([Fig. 4(a)]) and from 198 nm to 127 nm ([Fig. 4(b)]), allow to stop light in the whole SCLU range.

 figure: Fig. 4

Fig. 4 Slow light for GHMM waveguide propagation constant β as a function of waveguide width W for selected wavelengths from SCLU bands and different biasing voltage (a) Vg = 1 V and (b) Vg = 2 V.

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Figure 5 shows the dispersion curves for TM0 mode of the considered waveguide structure at λ = 1.55 μm and different biasing. It is worth noting that by changing gate voltage we can select the width of the waveguide for which the light stopping (critical point) can be observed. It means that for given waveguide width, by proper biasing, we can stop (slow down), cut off or provide propagation of the mode. In particularly, for the structure parameters and biasing range presented in Fig. 5, this effect can be obtained for the waveguide width within the range from W = 155 nm to W = 75 nm. Moreover, by proper tailoring of basic cell geometry and layer parameters this phenomenon can be obtained for different desirable wavelengths.

 figure: Fig. 5

Fig. 5 Normalized propagation constant β/k0 of the fundamental mode TM0 as a function of normalized waveguide width W/λ for various values of gate voltage Vg.

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Furthermore, in the waveguide structure with defined width, it is also possible to control wavelength of stopped light by voltage biasing. This is illustrated in Fig. 6, where the dependence of the stopped light wavelength on the biasing voltage for the waveguide width W = 0.12 μm is shown. Particularly, in this structure sweeping voltage bias from 0.9 V to 1.82 V allows to stop light in the whole SCLU range. Moreover, by proper tuning of the biased voltage, it is possible to select wavelength for which the stopping light effect is desirable.

 figure: Fig. 6

Fig. 6 Wavelength of stopped light as a function of biasing voltage.

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Finally, the dispersion characteristics of the same waveguide structure, for the first four guided modes, the selected wavelength (λ = 1.55 μm), and different biasing voltages, are presented in Figs. 7(a)-7(d). As we can see, if the structure is manufactured as a tapered waveguide, with varying width including the range from 620 nm to 8 nm, the biasing provides control of the number of modes which can be stopped, i.e., m = 0 for Vg = 630 mV, m = 0-1 for Vg = 660 mV, m = 0-2 for Vg = 700 mV and finally m = 0-3 for Vg = 800 V. What is worth noting, the higher order mode is stopped for greater values of the waveguide width.

 figure: Fig. 7

Fig. 7 Normalized propagation constant β/k0 as a function of normalized waveguide width W/λ for different orders of TM mode and different biasing voltage (a) Vg = 0.62 V, (b) Vg = 0.66 V, (c) Vg = 0.71 V and (d) Vg = 0.8 V.

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4. Conclusions

In this paper we investigate slow-light phenomenon in the GHMM waveguide operating in SCLU telecom bands. The considered structure is a symmetric planar waveguide composed of a Type II GHMM core and air cladding. It has been shown that proper design of GHMM structure forming waveguide layer and geometry of the waveguide itself allows for shaping the waveguide dispersion characteristics by applying external biasing. In particular, for the given width of waveguide it is possible to control the propagation of fundamental mode, i.e., its stopping, supporting or cutting off, as well as to shift critical point (stopping light) for desired wavelength within SCLU bands by employing the proper biasing voltage. Moreover, the realization of a tapered waveguide enables to stop light in the whole SCLU wavelength region and select the number of guided modes which can be stopped. The features described above can lead to potential applications of the proposed waveguides in any application requiring control of light propagation, especially it can be utilize in tunable optical buffers and photonic memory cells as well as optical switches and modulators. It is worth noting, that ultrafast response of complete device, due to graphene relaxation time in order of femtoseconds, is practically limited by driving electrical circuits [50], but still achievable at level of tens of GHz [51]. Moreover, such structures reveals compatibility with CMOS technology, in means of materials (i.e., graphene, SiO2) and technological processes (i.e., PECVD [52], Reactive Magnetron Sputtering [53] or Reactive Ion Etching [54]). We believe that waveguide structures considered within this paper could bring new quality in means of integrated optical systems, all the more being compatible with CMOS technology.

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50. W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014). [CrossRef]   [PubMed]  

51. C. T. Phare, Y. H. D. Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015). [CrossRef]  

52. M. Kalisz, R. Mroczyński, and R. B. Beck, “Specific features of fluorination of silicon surface region by RIE in rf CF 4 plasma—novel method of improving electrical properties of thin PECVD silicon dioxide films,” in Proceedings of IEEE Conference on Ultimate Integration on Silicon (IEEE,2011), pp. 1–4.

53. R. Mroczyński and R. B. Beck, “Reliability issues of double gate dielectric stacks based on hafnium dioxide (HfO2) layers for non-volatile semiconductor memory (NVSM) applications,” Microelectron. Reliab. 52(1), 107–111 (2012). [CrossRef]  

54. P. Wiśniewski, R. Mroczyński, and B. Majkusiak, “Reactive Ion Etching (RIE) of silicon for the technology of nanoelectronic devices and structures,” in Proceedings of SPIE on Electron Technology Conference (ELTE,2016), pp. 101750F.

References

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  1. J. T. Mok and B. J. Eggleton, “Photonics: Expect more delays,” Nature 433(7028), 811–812 (2005).
    [Crossref] [PubMed]
  2. T. F. Krauss, “Why do we need slow light,” Nat. Photonics 2(8), 448–450 (2008).
    [Crossref]
  3. B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3(4), 206–210 (2009).
    [Crossref]
  4. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 Metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).
    [Crossref]
  5. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90(11), 113903 (2003).
    [Crossref] [PubMed]
  6. C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
    [Crossref] [PubMed]
  7. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88(2), 023602 (2001).
    [Crossref] [PubMed]
  8. K. Inoue, N. Kawai, Y. Sugimoto, M. Carlsson, N. Ikeda, and K. Asakawa, “Observation of small group velocity in two-dimensional AlGaAs-based photonic crystal slabs,” Phys. Rev. B 65(12), 121308 (2002).
    [Crossref]
  9. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438(7064), 65–69 (2005).
    [Crossref] [PubMed]
  10. T. Baba and D. Mori, “Slow light engineering in photonic crystals,” J. Phys. D Appl. Phys. 40(9), 2659–2665 (2007).
    [Crossref]
  11. T. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D Appl. Phys. 40(9), 2666–2670 (2007).
    [Crossref]
  12. A. Karalis, E. Lidorikis, M. Ibanescu, J. D. Joannopoulos, and M. Soljacić, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett. 95(6), 063901 (2005).
    [Crossref] [PubMed]
  13. M. Sandtke and L. Kuipers, “Slow guided surface plasmons at telecom frequencies,” Nat. Photonics 1(10), 573–576 (2007).
    [Crossref]
  14. K. L. Tsakmakidis, T. W. Pickering, J. M. Hamm, A. F. Page, and O. Hess, “Completely stopped and dispersionless light in plasmonic waveguides,” Phys. Rev. Lett. 112(16), 167401 (2014).
    [Crossref] [PubMed]
  15. K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73(8), 085104 (2006).
    [Crossref]
  16. K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, “Sigle mode operation in the slow-light using oscillatory waves in generalized left-handed heterostructures,” Phys. Rev. Lett. 89(20), 201103 (2006).
  17. I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E.  67(5), 057602 (2003).
    [Crossref] [PubMed]
  18. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
    [Crossref] [PubMed]
  19. G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “TE and TM guided modes in an air waveguide with negative-index-material cladding,” Phys. Rev. E 71(4), 046603 (2005).
    [Crossref] [PubMed]
  20. S. A. Taya and I. M. Qadoura, “Guided modes in slab waveguides with negative index cladding and substrate,” Optik (Stuttg.) 124(13), 1431–1436 (2013).
    [Crossref]
  21. W. T. Lu, Y. J. Huang, B. D. F. Casse, R. K. Banyal, and S. Sridhar, “Storing light in active optical waveguides with single-negative materials,” Appl. Phys. Lett. 96(21), 211112 (2010).
    [Crossref] [PubMed]
  22. T. Jiang, J. Zhao, and Y. Feng, “Stopping light by an air waveguide with anisotropic metamaterial cladding,” Opt. Express 17(1), 170–177 (2009).
    [Crossref] [PubMed]
  23. O. Hess, J. B. Pendry, S. A. Maier, R. F. Oulton, J. M. Hamm, and K. L. Tsakmakidis, “Active nanoplasmonic metamaterials,” Nat. Mater. 11(7), 573–584 (2012).
    [Crossref] [PubMed]
  24. Y. He, S. He, and X. Yang, “Optical field enhancement in nanoscale slot waveguides of hyperbolic metamaterials,” Opt. Lett. 37(14), 2907–2909 (2012).
    [Crossref] [PubMed]
  25. S. Ishii, M. Y. Shalaginov, V. E. Babicheva, A. Boltasseva, and A. V. Kildishev, “Plasmonic waveguides cladded by hyperbolic metamaterials,” Opt. Lett. 39(16), 4663–4666 (2014).
    [Crossref] [PubMed]
  26. H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” in CLEO: QELS_Fundamental Science. Optical Society of America, (2013), paper QTu2A–4.
  27. B. Li, Y. He, and S. He, “Investigation of light trapping effect in hyperbolic metamaterial slow-light waveguides,” App. Phys. Exp. 8(8), 082601 (2015).
    [Crossref]
  28. A. D. Neira, G. A. Wurtz, and A. V. Zayats, “Superluminal and stopped light due to mode coupling in confined hyperbolic metamaterial waveguides,” Sci. Rep. 5, 17678 (2015).
    [Crossref] [PubMed]
  29. Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, “Applications of hyperbolic metamaterial substrates,” Adv. Optoelectron. 2012, 452502 (2012).
    [Crossref]
  30. V. P. Drachev, V. A. Podolskiy, and A. V. Kildishev, “Hyperbolic metamaterials: new physics behind a classical problem,” Opt. Express 21(12), 15048–15064 (2013).
    [Crossref] [PubMed]
  31. L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2015).
    [Crossref]
  32. L. Zhang, Z. Zhang, C. Kang, B. Cheng, L. Chen, X. Yang, J. Wang, W. Li, and B. Wang, “Tunable bulk polaritons of graphene-based hyperbolic metamaterials,” Opt. Express 22(11), 14022–14030 (2014).
    [Crossref] [PubMed]
  33. Y. Xiang, J. Guo, X. Dai, S. Wen, and D. Tang, “Engineered surface Bloch waves in graphene-based hyperbolic metamaterials,” Opt. Express 22(3), 3054–3062 (2014).
    [Crossref] [PubMed]
  34. M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-dielectric composite metamaterials: evolution from elliptic to hyperbolic wavevector dispersion and the transverse epsilon-near-zero condition,” J. Nanophotonics 7(1), 073089 (2013).
    [Crossref]
  35. A. Al Sayem, A. Shahriar, M. R. C. Mahdy, and M. S. Rahman, “Control of reflection through epsilon near zero graphene based anisotropic metamaterial,” in Proceedings of IEEE Conference on Electrical and Computer Engineering (IEEE, 2014), pp. 812–815.
    [Crossref]
  36. A. M. DaSilva, Y.-C. Chang, T. Norris, and A. H. MacDonald, “Enhancement of photonic density of states in finite graphene multilayers,” Phys. Rev. B 88(19), 195411 (2013).
    [Crossref]
  37. A. Andryieuski and A. V. Lavrinenko, “Graphene metamaterials based tunable terahertz absorber: effective surface conductivity approach,” Opt. Express 21(7), 9144–9155 (2013).
    [Crossref] [PubMed]
  38. R. Ning, S. Liu, H. Zhang, B. Bian, and X. Kong, “A wide-angle broadband absorber in graphene-based hyperbolic metamaterials,” Eur. Phys. J. Appl. Phys. 68(2), 20401 (2014).
    [Crossref]
  39. R. Ning, S. Liu, H. Zhang, B. Bian, and X. Kong, “Tunable absorption in graphene-based hyperbolic metamaterials for mid-infrared range,” Physica B 457, 144–148 (2015).
    [Crossref]
  40. B. Janaszek, A. Tyszka-Zawadzka, and P. Szczepański, “Tunable graphene-based hyperbolic metamaterial operating in SCLU telecom bands,” Opt. Express 24(21), 24129–24136 (2016).
    [Crossref] [PubMed]
  41. I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87(7), 075416 (2013).
    [Crossref]
  42. M. A. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express 21(6), 7614–7632 (2013).
    [Crossref] [PubMed]
  43. S. Xiao, X. Zhu, B.-H. Li, and N. A. Mortensen, “Graphene-plasmon polaritons: From fundamental properties to potential applications,” Front. Phys. 11(2), 117801 (2016).
    [Crossref]
  44. G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
    [Crossref] [PubMed]
  45. B. Zhu, G. Ren, Y. Gao, Y. Yang, B. Wu, Y. Lian, and S. Jian, “Local field enhancement in infrared graphene-dielectric hyperbolic slot waveguides,” IEEE Photonics Technol. Lett. 27(3), 276–279 (2015).
    [Crossref]
  46. H. Xu, L. Wu, X. Dai, Y. Gao, and Y. Xiang, “Tunable infrared plasmonic waveguides using graphene based hyperbolic metamaterials,” Optik-Int. J. Light Electron. Opt. 127(20), 9640–9646 (2016).
    [Crossref]
  47. A. Al Sayem, M. R. C. Mahdy, D. N. Hasan, and M. A. Matin, “Tunable slow light with graphene based hyperbolic metamaterial,” in Proceedings of IEEE Conference on Electrical and Computer Engineering (IEEE, 2014) pp. 230–233.
    [Crossref]
  48. L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007).
    [Crossref]
  49. R. Mroczyński, N. Kwietniewski, M. Ćwil, P. Hoffmann, R. B. Beck, and A. Jakubowski, “Improvement of electro-physical properties of ultra-thin PECVD silicon oxynitride layers by high-temperature annealing,” Vacuum 82(10), 1013–1019 (2008).
    [Crossref]
  50. W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
    [Crossref] [PubMed]
  51. C. T. Phare, Y. H. D. Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015).
    [Crossref]
  52. M. Kalisz, R. Mroczyński, and R. B. Beck, “Specific features of fluorination of silicon surface region by RIE in rf CF 4 plasma—novel method of improving electrical properties of thin PECVD silicon dioxide films,” in Proceedings of IEEE Conference on Ultimate Integration on Silicon (IEEE,2011), pp. 1–4.
  53. R. Mroczyński and R. B. Beck, “Reliability issues of double gate dielectric stacks based on hafnium dioxide (HfO2) layers for non-volatile semiconductor memory (NVSM) applications,” Microelectron. Reliab. 52(1), 107–111 (2012).
    [Crossref]
  54. P. Wiśniewski, R. Mroczyński, and B. Majkusiak, “Reactive Ion Etching (RIE) of silicon for the technology of nanoelectronic devices and structures,” in Proceedings of SPIE on Electron Technology Conference (ELTE,2016), pp. 101750F.

2016 (3)

B. Janaszek, A. Tyszka-Zawadzka, and P. Szczepański, “Tunable graphene-based hyperbolic metamaterial operating in SCLU telecom bands,” Opt. Express 24(21), 24129–24136 (2016).
[Crossref] [PubMed]

S. Xiao, X. Zhu, B.-H. Li, and N. A. Mortensen, “Graphene-plasmon polaritons: From fundamental properties to potential applications,” Front. Phys. 11(2), 117801 (2016).
[Crossref]

H. Xu, L. Wu, X. Dai, Y. Gao, and Y. Xiang, “Tunable infrared plasmonic waveguides using graphene based hyperbolic metamaterials,” Optik-Int. J. Light Electron. Opt. 127(20), 9640–9646 (2016).
[Crossref]

2015 (6)

R. Ning, S. Liu, H. Zhang, B. Bian, and X. Kong, “Tunable absorption in graphene-based hyperbolic metamaterials for mid-infrared range,” Physica B 457, 144–148 (2015).
[Crossref]

B. Zhu, G. Ren, Y. Gao, Y. Yang, B. Wu, Y. Lian, and S. Jian, “Local field enhancement in infrared graphene-dielectric hyperbolic slot waveguides,” IEEE Photonics Technol. Lett. 27(3), 276–279 (2015).
[Crossref]

C. T. Phare, Y. H. D. Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015).
[Crossref]

L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2015).
[Crossref]

B. Li, Y. He, and S. He, “Investigation of light trapping effect in hyperbolic metamaterial slow-light waveguides,” App. Phys. Exp. 8(8), 082601 (2015).
[Crossref]

A. D. Neira, G. A. Wurtz, and A. V. Zayats, “Superluminal and stopped light due to mode coupling in confined hyperbolic metamaterial waveguides,” Sci. Rep. 5, 17678 (2015).
[Crossref] [PubMed]

2014 (6)

S. Ishii, M. Y. Shalaginov, V. E. Babicheva, A. Boltasseva, and A. V. Kildishev, “Plasmonic waveguides cladded by hyperbolic metamaterials,” Opt. Lett. 39(16), 4663–4666 (2014).
[Crossref] [PubMed]

L. Zhang, Z. Zhang, C. Kang, B. Cheng, L. Chen, X. Yang, J. Wang, W. Li, and B. Wang, “Tunable bulk polaritons of graphene-based hyperbolic metamaterials,” Opt. Express 22(11), 14022–14030 (2014).
[Crossref] [PubMed]

Y. Xiang, J. Guo, X. Dai, S. Wen, and D. Tang, “Engineered surface Bloch waves in graphene-based hyperbolic metamaterials,” Opt. Express 22(3), 3054–3062 (2014).
[Crossref] [PubMed]

R. Ning, S. Liu, H. Zhang, B. Bian, and X. Kong, “A wide-angle broadband absorber in graphene-based hyperbolic metamaterials,” Eur. Phys. J. Appl. Phys. 68(2), 20401 (2014).
[Crossref]

K. L. Tsakmakidis, T. W. Pickering, J. M. Hamm, A. F. Page, and O. Hess, “Completely stopped and dispersionless light in plasmonic waveguides,” Phys. Rev. Lett. 112(16), 167401 (2014).
[Crossref] [PubMed]

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14(2), 955–959 (2014).
[Crossref] [PubMed]

2013 (8)

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[Crossref] [PubMed]

V. P. Drachev, V. A. Podolskiy, and A. V. Kildishev, “Hyperbolic metamaterials: new physics behind a classical problem,” Opt. Express 21(12), 15048–15064 (2013).
[Crossref] [PubMed]

I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87(7), 075416 (2013).
[Crossref]

M. A. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express 21(6), 7614–7632 (2013).
[Crossref] [PubMed]

M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-dielectric composite metamaterials: evolution from elliptic to hyperbolic wavevector dispersion and the transverse epsilon-near-zero condition,” J. Nanophotonics 7(1), 073089 (2013).
[Crossref]

A. M. DaSilva, Y.-C. Chang, T. Norris, and A. H. MacDonald, “Enhancement of photonic density of states in finite graphene multilayers,” Phys. Rev. B 88(19), 195411 (2013).
[Crossref]

A. Andryieuski and A. V. Lavrinenko, “Graphene metamaterials based tunable terahertz absorber: effective surface conductivity approach,” Opt. Express 21(7), 9144–9155 (2013).
[Crossref] [PubMed]

S. A. Taya and I. M. Qadoura, “Guided modes in slab waveguides with negative index cladding and substrate,” Optik (Stuttg.) 124(13), 1431–1436 (2013).
[Crossref]

2012 (4)

O. Hess, J. B. Pendry, S. A. Maier, R. F. Oulton, J. M. Hamm, and K. L. Tsakmakidis, “Active nanoplasmonic metamaterials,” Nat. Mater. 11(7), 573–584 (2012).
[Crossref] [PubMed]

Y. He, S. He, and X. Yang, “Optical field enhancement in nanoscale slot waveguides of hyperbolic metamaterials,” Opt. Lett. 37(14), 2907–2909 (2012).
[Crossref] [PubMed]

Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, “Applications of hyperbolic metamaterial substrates,” Adv. Optoelectron. 2012, 452502 (2012).
[Crossref]

R. Mroczyński and R. B. Beck, “Reliability issues of double gate dielectric stacks based on hafnium dioxide (HfO2) layers for non-volatile semiconductor memory (NVSM) applications,” Microelectron. Reliab. 52(1), 107–111 (2012).
[Crossref]

2010 (1)

W. T. Lu, Y. J. Huang, B. D. F. Casse, R. K. Banyal, and S. Sridhar, “Storing light in active optical waveguides with single-negative materials,” Appl. Phys. Lett. 96(21), 211112 (2010).
[Crossref] [PubMed]

2009 (2)

T. Jiang, J. Zhao, and Y. Feng, “Stopping light by an air waveguide with anisotropic metamaterial cladding,” Opt. Express 17(1), 170–177 (2009).
[Crossref] [PubMed]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3(4), 206–210 (2009).
[Crossref]

2008 (2)

T. F. Krauss, “Why do we need slow light,” Nat. Photonics 2(8), 448–450 (2008).
[Crossref]

R. Mroczyński, N. Kwietniewski, M. Ćwil, P. Hoffmann, R. B. Beck, and A. Jakubowski, “Improvement of electro-physical properties of ultra-thin PECVD silicon oxynitride layers by high-temperature annealing,” Vacuum 82(10), 1013–1019 (2008).
[Crossref]

2007 (5)

L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007).
[Crossref]

T. Baba and D. Mori, “Slow light engineering in photonic crystals,” J. Phys. D Appl. Phys. 40(9), 2659–2665 (2007).
[Crossref]

T. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D Appl. Phys. 40(9), 2666–2670 (2007).
[Crossref]

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[Crossref] [PubMed]

M. Sandtke and L. Kuipers, “Slow guided surface plasmons at telecom frequencies,” Nat. Photonics 1(10), 573–576 (2007).
[Crossref]

2006 (2)

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73(8), 085104 (2006).
[Crossref]

K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, “Sigle mode operation in the slow-light using oscillatory waves in generalized left-handed heterostructures,” Phys. Rev. Lett. 89(20), 201103 (2006).

2005 (4)

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “TE and TM guided modes in an air waveguide with negative-index-material cladding,” Phys. Rev. E 71(4), 046603 (2005).
[Crossref] [PubMed]

A. Karalis, E. Lidorikis, M. Ibanescu, J. D. Joannopoulos, and M. Soljacić, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett. 95(6), 063901 (2005).
[Crossref] [PubMed]

J. T. Mok and B. J. Eggleton, “Photonics: Expect more delays,” Nature 433(7028), 811–812 (2005).
[Crossref] [PubMed]

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438(7064), 65–69 (2005).
[Crossref] [PubMed]

2003 (2)

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90(11), 113903 (2003).
[Crossref] [PubMed]

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E.  67(5), 057602 (2003).
[Crossref] [PubMed]

2002 (1)

K. Inoue, N. Kawai, Y. Sugimoto, M. Carlsson, N. Ikeda, and K. Asakawa, “Observation of small group velocity in two-dimensional AlGaAs-based photonic crystal slabs,” Phys. Rev. B 65(12), 121308 (2002).
[Crossref]

2001 (2)

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88(2), 023602 (2001).
[Crossref] [PubMed]

1999 (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 Metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).
[Crossref]

Al Sayem, A.

A. Al Sayem, A. Shahriar, M. R. C. Mahdy, and M. S. Rahman, “Control of reflection through epsilon near zero graphene based anisotropic metamaterial,” in Proceedings of IEEE Conference on Electrical and Computer Engineering (IEEE, 2014), pp. 812–815.
[Crossref]

A. Al Sayem, M. R. C. Mahdy, D. N. Hasan, and M. A. Matin, “Tunable slow light with graphene based hyperbolic metamaterial,” in Proceedings of IEEE Conference on Electrical and Computer Engineering (IEEE, 2014) pp. 230–233.
[Crossref]

Andryieuski, A.

Aryal, D. P.

K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, “Sigle mode operation in the slow-light using oscillatory waves in generalized left-handed heterostructures,” Phys. Rev. Lett. 89(20), 201103 (2006).

Asakawa, K.

K. Inoue, N. Kawai, Y. Sugimoto, M. Carlsson, N. Ikeda, and K. Asakawa, “Observation of small group velocity in two-dimensional AlGaAs-based photonic crystal slabs,” Phys. Rev. B 65(12), 121308 (2002).
[Crossref]

Baba, T.

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Nat. Mater. (1)

O. Hess, J. B. Pendry, S. A. Maier, R. F. Oulton, J. M. Hamm, and K. L. Tsakmakidis, “Active nanoplasmonic metamaterials,” Nat. Mater. 11(7), 573–584 (2012).
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Nat. Photonics (4)

M. Sandtke and L. Kuipers, “Slow guided surface plasmons at telecom frequencies,” Nat. Photonics 1(10), 573–576 (2007).
[Crossref]

T. F. Krauss, “Why do we need slow light,” Nat. Photonics 2(8), 448–450 (2008).
[Crossref]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3(4), 206–210 (2009).
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C. T. Phare, Y. H. D. Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015).
[Crossref]

Nature (5)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 Metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).
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C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
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J. T. Mok and B. J. Eggleton, “Photonics: Expect more delays,” Nature 433(7028), 811–812 (2005).
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Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438(7064), 65–69 (2005).
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K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
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Opt. Express (7)

Opt. Lett. (2)

Optik (Stuttg.) (1)

S. A. Taya and I. M. Qadoura, “Guided modes in slab waveguides with negative index cladding and substrate,” Optik (Stuttg.) 124(13), 1431–1436 (2013).
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Optik-Int. J. Light Electron. Opt. (1)

H. Xu, L. Wu, X. Dai, Y. Gao, and Y. Xiang, “Tunable infrared plasmonic waveguides using graphene based hyperbolic metamaterials,” Optik-Int. J. Light Electron. Opt. 127(20), 9640–9646 (2016).
[Crossref]

Phys. Rev. B (4)

A. M. DaSilva, Y.-C. Chang, T. Norris, and A. H. MacDonald, “Enhancement of photonic density of states in finite graphene multilayers,” Phys. Rev. B 88(19), 195411 (2013).
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I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87(7), 075416 (2013).
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K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,” Phys. Rev. B 73(8), 085104 (2006).
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K. Inoue, N. Kawai, Y. Sugimoto, M. Carlsson, N. Ikeda, and K. Asakawa, “Observation of small group velocity in two-dimensional AlGaAs-based photonic crystal slabs,” Phys. Rev. B 65(12), 121308 (2002).
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Phys. Rev. E (2)

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E.  67(5), 057602 (2003).
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G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “TE and TM guided modes in an air waveguide with negative-index-material cladding,” Phys. Rev. E 71(4), 046603 (2005).
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Phys. Rev. Lett. (5)

K. L. Tsakmakidis, A. Klaedtke, D. P. Aryal, C. Jamois, and O. Hess, “Sigle mode operation in the slow-light using oscillatory waves in generalized left-handed heterostructures,” Phys. Rev. Lett. 89(20), 201103 (2006).

K. L. Tsakmakidis, T. W. Pickering, J. M. Hamm, A. F. Page, and O. Hess, “Completely stopped and dispersionless light in plasmonic waveguides,” Phys. Rev. Lett. 112(16), 167401 (2014).
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A. Karalis, E. Lidorikis, M. Ibanescu, J. D. Joannopoulos, and M. Soljacić, “Surface-plasmon-assisted guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett. 95(6), 063901 (2005).
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A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88(2), 023602 (2001).
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M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90(11), 113903 (2003).
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Physica B (1)

R. Ning, S. Liu, H. Zhang, B. Bian, and X. Kong, “Tunable absorption in graphene-based hyperbolic metamaterials for mid-infrared range,” Physica B 457, 144–148 (2015).
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Prog. Quantum Electron. (1)

L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2015).
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Sci. Rep. (1)

A. D. Neira, G. A. Wurtz, and A. V. Zayats, “Superluminal and stopped light due to mode coupling in confined hyperbolic metamaterial waveguides,” Sci. Rep. 5, 17678 (2015).
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Vacuum (1)

R. Mroczyński, N. Kwietniewski, M. Ćwil, P. Hoffmann, R. B. Beck, and A. Jakubowski, “Improvement of electro-physical properties of ultra-thin PECVD silicon oxynitride layers by high-temperature annealing,” Vacuum 82(10), 1013–1019 (2008).
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Other (5)

M. Kalisz, R. Mroczyński, and R. B. Beck, “Specific features of fluorination of silicon surface region by RIE in rf CF 4 plasma—novel method of improving electrical properties of thin PECVD silicon dioxide films,” in Proceedings of IEEE Conference on Ultimate Integration on Silicon (IEEE,2011), pp. 1–4.

A. Al Sayem, M. R. C. Mahdy, D. N. Hasan, and M. A. Matin, “Tunable slow light with graphene based hyperbolic metamaterial,” in Proceedings of IEEE Conference on Electrical and Computer Engineering (IEEE, 2014) pp. 230–233.
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P. Wiśniewski, R. Mroczyński, and B. Majkusiak, “Reactive Ion Etching (RIE) of silicon for the technology of nanoelectronic devices and structures,” in Proceedings of SPIE on Electron Technology Conference (ELTE,2016), pp. 101750F.

A. Al Sayem, A. Shahriar, M. R. C. Mahdy, and M. S. Rahman, “Control of reflection through epsilon near zero graphene based anisotropic metamaterial,” in Proceedings of IEEE Conference on Electrical and Computer Engineering (IEEE, 2014), pp. 812–815.
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H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” in CLEO: QELS_Fundamental Science. Optical Society of America, (2013), paper QTu2A–4.

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Figures (7)

Fig. 1
Fig. 1 Scheme of (a) analyzed waveguide and (b) its effective representation.
Fig. 2
Fig. 2 Permittivity tensor components for boundary wavelengths of SCLU bands, λ = 1.46 μm and λ = 1.675 μm, as a function of gate voltage Vg and for different structure parameters (a) td = 8 nm, Ng = 4 and (b) td = 6 nm, Ng = 6.
Fig. 3
Fig. 3 Normalized propagation constant β/k0 of the fundamental mode TM0 as a function of normalized waveguide width W/λ for (a) different numbers of graphene layers Ng, and (b) various values of dielectric layer thickness td. Normalized power flow P as a function of normalized waveguide width W/λ for (c) different numbers of graphene layers Ng, and (d) various values of dielectric layer thickness td.
Fig. 4
Fig. 4 Slow light for GHMM waveguide propagation constant β as a function of waveguide width W for selected wavelengths from SCLU bands and different biasing voltage (a) Vg = 1 V and (b) Vg = 2 V.
Fig. 5
Fig. 5 Normalized propagation constant β/k0 of the fundamental mode TM0 as a function of normalized waveguide width W/λ for various values of gate voltage Vg.
Fig. 6
Fig. 6 Wavelength of stopped light as a function of biasing voltage.
Fig. 7
Fig. 7 Normalized propagation constant β/k0 as a function of normalized waveguide width W/λ for different orders of TM mode and different biasing voltage (a) Vg = 0.62 V, (b) Vg = 0.66 V, (c) Vg = 0.71 V and (d) Vg = 0.8 V.

Equations (7)

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ε=[ ε || 0 0 0 ε || 0 0 0 ε ],
ε || = t g ε g + t d ε d t g + t d , ε = ε g ε d ( t g + t d ) t g ε d + t d ε g ,
ε g =1j σ(ω, μ c ) ω ε o t g ,
σ(ω, μ c )= j4π q 2 k B T h 2 (ωj2τ) [ μ c k B T +2ln( e μ / k B T +1 ) ]+ j4π q 2 ( ωj2τ ) h 2 0 f D (ξ) f D (ξ) ( ωj2τ ) 2 16( πξ h ) dξ,
| μ c |= υ F π| a 0 ( V g V dirac ) | ,
tan( γ f W 2 )= γ 1 γ f ε ε 1       for even modes,
cot( γ f W 2 )= γ 1 γ f ε ε 1       for odd modes,

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