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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 11 — Jun. 2, 2014
  • pp: 14022–14030
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Tunable bulk polaritons of graphene-based hyperbolic metamaterials

Liwei Zhang, Zhengren Zhang, Chaoyang Kang, Bei Cheng, Liang Chen, Xuefeng Yang, Jian Wang, Weibing Li, and Baoji Wang  »View Author Affiliations


Optics Express, Vol. 22, Issue 11, pp. 14022-14030 (2014)
http://dx.doi.org/10.1364/OE.22.014022


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Abstract

The tunable hyperbolic metamaterial (HMM) based on the graphene-dielectric layered structure at THz frequency is presented, and the surface and bulk polaritons of the graphene-based HMM are theoretically studied. It is found that the dispersions of the polaritons can be tuned by varying the Fermi energy of graphene sheets, the graphene-dielectric layers and the layer number of graphene sheets. In addition, the highly confined bulk polariton mode can be excited and is manifested in an attenuated total reflection configuration as a sharp drop in the reflectance. Such properties can be used in tunable optical reflection modulation with the assistance of bulk polaritons.

© 2014 Optical Society of America

1. Introduction

Graphene is a two-dimension substance composed of a single layer carbon atom arranged in a honeycomb lattice and it is actually a gapless semiconductor [1

1. A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009). [CrossRef]

]. It has attracted intensive scientific interest owing to its incredible physical properties, such as optical transparency, flexibility and extraordinary electrical properties [2

2. K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, “A roadmap for graphene,” Nature 490(7419), 192–200 (2012). [CrossRef] [PubMed]

4

4. Z. Zhang, H. Li, Z. Gong, Y. Fan, T. Zhang, and H. Chen, “Extend the omnidirectional electronic gap of Thue-Morse aperiodic gapped graphene superlattices,” Appl. Phys. Lett. 101(25), 252104 (2012). [CrossRef]

]. With the recent developments in the fabrication of graphene with large lateral dimensions [5

5. V. Chabot, D. Higgins, A. P. Yu, X. C. Xiao, Z. W. Chen, and J. J. Zhang, “A review of graphene and graphene oxide sponge: material synthesis and applications to energy and the environment,” Energy Environ. Sci. 7(5), 1564–1596 (2014).

,6

6. A. Reina, X. T. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus, and J. Kong, “Large area, few-layer graphene films on arbitrary substrates by chemical vapor deposition,” Nano Lett. 9(1), 30–35 (2009). [CrossRef] [PubMed]

], there have been numerous graphene applications at optical, infrared and terahertz frequencies as tunable waveguiding interconnects, polarizer, optical modulator, cloaking and strong light–matter interaction platform [7

7. A. Vakil and N. Engheta, “Transformation Optics Using Graphene,” Science 332(6035), 1291–1294 (2011). [CrossRef] [PubMed]

11

11. F. Liu and E. Cubukcu, “Tunable omnidirectional strong light-matter interactions mediated by graphene surface plasmons,” Phys. Rev. B 88(11), 115439 (2013). [CrossRef]

]. Because the imaginary part of graphene conductivity can attain positive values in a certain frequency, a graphene layer can effectively behave as a “metal” layer capable of supporting a p-polarization surface plasmon polariton [3

3. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012). [CrossRef]

,7

7. A. Vakil and N. Engheta, “Transformation Optics Using Graphene,” Science 332(6035), 1291–1294 (2011). [CrossRef] [PubMed]

]. Although graphene sheet shows excellent metal-like characteristics under electric/magnetic biasing or chemical doping, it is generally believed that graphene does not conduct well enough to replace metals, since it is too “thin” to sustain an intense resonance [12

12. P. Tassin, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “A comparison of graphene, superconductors and metals as conductors for metamaterials and plasmonics,” Nat. Photonics 6(4), 259–264 (2012). [CrossRef]

]. While a hyperbolic metamaterial (HMM) can be realized by employing a multilayer graphene-dielectric composite [13

13. K. V. Sreekanth, A. De Luca, and G. Strangi, “Negative refraction in graphene-based hyperbolic metamaterials,” Appl. Phys. Lett. 103(2), 023107 (2013). [CrossRef]

17

17. B. Zhu, G. Ren, S. Zheng, Z. Lin, and S. Jian, “Nanoscale dielectric-graphene-dielectric tunable infrared waveguide with ultrahigh refractive indices,” Opt. Express 21(14), 17089–17096 (2013). [CrossRef] [PubMed]

], where the graphene sheets are substituted to mimic essential metallic behavior. HMM is a uniaxial anisotropic medium, the diagonal elements of the permittivity tensor have different signs (e.g., εxx=εyy<0, εzz>0), which causes the dispersion relation to become hyperbolic rather than elliptical [18

18. C. J. Zapata-Rodríguez, J. J. Miret, S. Vuković, and M. R. Belić, “Engineered surface waves in hyperbolic metamaterials,” Opt. Express 21(16), 19113–19127 (2013). [CrossRef] [PubMed]

]. It can be realized at optical frequencies using metal-dielectric multilayers, where the unit cells are much smaller than the operation wavelength [19

19. O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic metamaterials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012). [CrossRef]

]. But the performance of metals is hampered by the difficulty in tuning their permittivity functions and the existence of optical losses. Unlike the metals, the electronic and optical properties of graphene can be tuned through electrical or chemical modification of the charge carrier density [20

20. A. A. Avetisyan, B. Partoens, and F. M. Peeters, “Electric-field control of the band gap and Fermi energy in graphene multilayers by top and back gates,” Phys. Rev. B 80(19), 195401 (2009). [CrossRef]

,21

21. J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Plasmon-Induced Doping of Graphene,” Nature 487, 77–81 (2012).

]. And the conductivity of graphene can be readily tuned not only at THz frequencies but also at optical frequencies. The loss of graphene can be very low in the frequency region of interest, it seems to be a good candidate for designing the tunable HMM operates in both THz, infrared and optical frequency ranges.

The recent study about graphene-based HMMs composed of stacked graphene sheets separated by thin dielectric layers have demonstrated that such HMMs can be used in negative refraction [13

13. K. V. Sreekanth, A. De Luca, and G. Strangi, “Negative refraction in graphene-based hyperbolic metamaterials,” Appl. Phys. Lett. 103(2), 023107 (2013). [CrossRef]

], tunable broadband hyperlens [16

16. T. Zhang, L. Chen, and X. Li, “Graphene-based tunable broadband hyperlens for far-field subdiffraction imaging at mid-infrared frequencies,” Opt. Express 21(18), 20888–20899 (2013). [CrossRef] [PubMed]

], tunable infrared plasmonic devices [17

17. B. Zhu, G. Ren, S. Zheng, Z. Lin, and S. Jian, “Nanoscale dielectric-graphene-dielectric tunable infrared waveguide with ultrahigh refractive indices,” Opt. Express 21(14), 17089–17096 (2013). [CrossRef] [PubMed]

], and so on. In addition, such HMM structure can also support surface (Bloch) waves, which have been theoretically studied in Ref [18

18. C. J. Zapata-Rodríguez, J. J. Miret, S. Vuković, and M. R. Belić, “Engineered surface waves in hyperbolic metamaterials,” Opt. Express 21(16), 19113–19127 (2013). [CrossRef] [PubMed]

,22

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

]. In this paper, we investigate the bulk polaritons of the HMM based on the graphene-dielectric layered structure. It is found that such graphene structures support controllable bulk polaritons, which can be excited by attenuated total reflection (ATR). Furthermore, the excitation of the bulk polaritons can be used in tunable optical reflection modulation.

2. Model and theory

A graphene monolayer is electrically characterized by its surface conductivityσ(ω,EF), which is highly dependent on the working frequency and Fermi energy. Within the random-phase approximation and without external magnetic field, the graphene is isotropic and the surface conductivity σcan be written as a sum of the intraband σintraand the interband term σinter [23

23. G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008). [CrossRef]

],
σintra=ie2KBTπ2(ω+i/τ)(EFKBT+2ln(1+eEFKBT)),σinter=ie2π2ln|2EF(ω+i/τ)2EF+(ω+i/τ)|
(1)
whereωis the frequency of the incident electromagnetic wave, e andare the electron charge and reduced Planck’s constants, respectively. EF andτare the Fermi energy (or chemical potential) and electron-phonon relaxation time, respectively. KBis the Boltzmann constant, and T is temperature. The Fermi energy EF can be straightforwardly obtained from the carrier density in a graphene sheet which is tunable by an applied gate voltage and doping [20

20. A. A. Avetisyan, B. Partoens, and F. M. Peeters, “Electric-field control of the band gap and Fermi energy in graphene multilayers by top and back gates,” Phys. Rev. B 80(19), 195401 (2009). [CrossRef]

,21

21. J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Plasmon-Induced Doping of Graphene,” Nature 487, 77–81 (2012).

,24

24. Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012). [PubMed]

,25

25. B. D. Guo, L. Fang, B. H. Zhang, and J. R. Gong, “Graphene Doping: A Review,” Insciences J. 1(2), 80–89 (2011). [CrossRef]

].

Consider the periodic stack of graphene-dielectric layers as depicted in Fig. 1(a)
Fig. 1 (a) Schematic of graphene-based HMM consisting of alternating graphene sheets and dielectric layers, the layers are infinite in x-y plane. (b) Illustration of the graphene-based HMM (medium B) with thickness of d sandwiched by media A and C.
, where graphene sheets are separated by dielectric layers with thicknesstdand relative permittivityεd. It is assumed that the thickness of the dielectric layers are deep subwavelength but thick enough to avoid the interaction between graphene layers (e.g., interlayer transitions). Hence the graphene’s effective permittivityεgis written as εg=1+iσ/(ωε0tg) [13

13. K. V. Sreekanth, A. De Luca, and G. Strangi, “Negative refraction in graphene-based hyperbolic metamaterials,” Appl. Phys. Lett. 103(2), 023107 (2013). [CrossRef]

,22

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

], where ε0is the permittivity in vacuum,tgis the effective graphene thickness. In the calculations,tgis taken to be 0.5nm similar to that used in [13

13. K. V. Sreekanth, A. De Luca, and G. Strangi, “Negative refraction in graphene-based hyperbolic metamaterials,” Appl. Phys. Lett. 103(2), 023107 (2013). [CrossRef]

,26

26. W. R. Zhu, I. D. Rukhlenko, and M. Premaratne, “Graphene metamaterial for optical reflection modulation,” Appl. Phys. Lett. 102(24), 241914 (2013). [CrossRef]

]. For simplicity, we use a constant value of relaxation time τ = 1 ps, which is similar to that in experiment [27

27. S. Winnerl, M. Orlita, P. Plochocka, P. Kossacki, M. Potemski, T. Winzer, E. Malic, A. Knorr, M. Sprinkle, C. Berger, W. A. de Heer, H. Schneider, and M. Helm, “Carrier Relaxation in Epitaxial Graphene Photoexcited Near the Dirac Point,” Phys. Rev. Lett. 107(23), 237401 (2011). [CrossRef] [PubMed]

]. Under the subwavelength limit, the graphene-dielectric composite (GDC) can be modeled as a homogeneous uniaxial anisotropic medium. And the effective medium theory (EMT) is valid to study the wave propagation in the GDC, the effective relative permittivity tensorεeffis a diagonal matrix in Cartesian coordinates,εxx=εyy=εp,εzz=εt,which are approximated as follows [13

13. K. V. Sreekanth, A. De Luca, and G. Strangi, “Negative refraction in graphene-based hyperbolic metamaterials,” Appl. Phys. Lett. 103(2), 023107 (2013). [CrossRef]

,19

19. O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic metamaterials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012). [CrossRef]

]:

εp=(tgεg+tdεd)/(tg+td),εt=(tg+td)εgεd/(tgεd+tdεg)
(2)

Since the thickness of a graphene sheet is negligible compared with that of the dielectric layer, the dielectric tensor in the perpendicular direction can be considered equal to the permittivity of dielectric layer (εtεd). However, the dielectric tensor in the parallel direction (xy-plane) varies with frequency. As an example, in Fig. 2(a)
Fig. 2 (a) Effective permittivity of the graphene-based HMM. (b) The EFCs of a graphene-based HMM at 8THz, 9THz and 10THz respectively, the solid (dotted and dashed) line is calculated from EMT, the hollow circles are the results based on Bloch’s theorem.
we plot the real and imaginary parts of effective permittivities εp andεt with frequency. Here, we assume that the Fermi energy of graphene EF = 0.2 eV, and T = 300 K. Silica is selected as the dielectric layer with relative permittivityεd = 2.2 and thickness td = 50nm, where the absorption loss of dielectric has been neglected. According to Fig. 2(a), the hyperbolic dispersion can be obtained for certain frequencies of interest in which the real part of εpandεt are negative and positive respectively. Figure 2(b) is the calculated equi-frequency contours (EFCs) of a graphene-based HMM at 8THz, 9THz and 10THz respectively. The solid (short dotted and the short dashed) line is calculated from the EMT, and the hollow circles represent the results based on Bloch’s theorem, which agree well with each other and indicate the validity of EMT. It is evident that the graphene-dielectric composite possesses hyperbolic dispersion relation, hence the name of effective graphene-based HMM. Note that under the same kx, kz decreases with the frequency, kz is not zero between −1.48k0 and 1.48 k0 (k0 is the wavenumber in free space) because of the loss in the graphene sheets which decreases with frequency as shown in Fig. 2(a).

In the lossless case, hyperbolic dispersion only occurs when εp<0, and the HMM allows for propagation of extraordinary waves with a large transverse wavenumber with kx>εtk0. So the graphene-based HMM will support highly confined bulk plasmon modes in addition to the long-and short-range surface plasmon modes for p polarization like that in metal-dielectric based HMMs [28

28. I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402 (2007). [CrossRef]

]. Now, we present the derivation of the polariton dispersion relations for a graphene-based HMM slab with thickness of d sandwiched by two semi-infinite media (i.e., ABC), as shown in Fig. 1(b). The effective anisotropic permittivity tensor of HMM are given by the approximate model of Eq. (2), where the slab B has the same thickness as the layered metamaterial as shown in Fig. 1(a). P-polarized wave is characterized by the existence of a Hy component of the magnetic field together with Ex and Ez components of the electric field. Here, z is the direction normal to the HMM, and x, y are the directions parallel to the slab. According to Maxwell equations, the expressions of electric and magnetic fields are given as follows:
{HAy(z)=A0eiKAzzEAx(z)=KAzεAε0ωA0eiKAzz(z<0)
(3)
{HBy(z)=B0eiKBzz+C0eiKBzzEBx(z)=KBzε0εpω(B0eiKBzzC0eiKBzz)(0<z<d)
(4)
{HCy(z)=D0eiKCzzECx(z)=KCzεCε0ωD0eiKCzz(z>d)
(5)
Where εA(εC) is the permittivity of the medium A(C),kBz2=ω2εp/c2εp/εtkx2, kA/CZ2+kx2=ω2εA/C/c2, d=N(tg+td), N is the periods of the graphene-dielectric composite, kx and kjz are the transverse and longitudinal components of the wave vector kj, respectively. The boundary conditions require the tangential components of the electric and magnetic fields to be continuous at z = 0 andz=d, yielding a set of homogeneous linear equations for coefficients A0, B0, C0 and D0. The determinant of this set must be zero for nontrivial solutions to exist. After some manipulations, the following equation, which is the surface polariton dispersion relation for p polarization, is obtained [29

29. K. Park, B. J. Lee, C. J. Fu, and Z. M. Zhang, “Study of the surface and bulk polaritons with an egative index metamateria,” J. Opt. Soc. Am. B 22(5), 1016–1023 (2005). [CrossRef]

,30

30. H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, and T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100(24), 243110 (2012). [CrossRef]

]:
(εCkBzεpkCz)(εCkBz+εpkCz)(εAkBzεpkAz)(εAkBz+εpkAz)=ei2kBzd
(6)
A bulk polariton occurs when both kAzand kCz are purely imaginary, but kBz is real. The dispersion relation can be obtained if 2kBz is substituted fori2kBzin Eq. (6) such that
(εCkBzεpkCz)(εCkBz+εpkCz)(εAkBzεpkAz)(εAkBz+εpkAz)=e2kBzd
(7)
Notice that Eq. (7) is the same as the dispersion equation of the waveguide, whose solutions exist only when all the phase shifts that the guided wave undergoes are summed up to be a multiple of2π. In the case of a bulk polariton, the electromagnetic fields inside the slab are oscillatory as stationary waves.

3. Numerical results and discussions

From Eqs. (6) and (7), the effective permittivity εpis one important parameter which determines the existence of surface and bulk polaritons for p polarization wave in HMM structure as shown in Fig. 1(b). Fortunately, the condition for εp<0 can be controlled by the Fermi energy of the graphene sheet, the filling ratio of the graphene sheet and dielectric layer, and the number of graphene sheets Ns in one unit, that’s the graphene-based HMMs are tunable. Figure 3
Fig. 3 The dependences of dispersion relations of p-polarized surface and bulk polaritons on (a) the Fermi energy EF, (b) the number of graphene sheets (i.e., Ns). and (c) the period number of the graphene-based HMM (i.e., N), where Ns = 1, N = 6, td = 50nm in (a), N = 6, td = 50nm, EF = 0.2eV in (b), and Ns = 1, td = 50nm, EF = 0.2eV in (c). Curves (I), (II) and (III) correspond to kz = 0 in air, HMM and prism, respectively, curve (IV) in (a) denotes kx=npk0sin(θ) with incident angle θ=68.550 and np=4.
depicts a regime map in a kxf space, illustrating the regions where surface or bulk polaritons may exist. Here, HMM layer (i.e., medium B) is sandwiched by air, for the HMM layer, it is realized by the graphene-dielectric multilayer and characteristic with the effective permittivity tensor, that’s εp andεt. Note that no surface polaritons can be excited when εp>0, where the graphene-dielectric composite behaves like a pure dielectric with an elliptical EFC. Two dotted curves (I) and (II) separate three different regions. Curves (I) and (II) correspond tokz=0 for air and HMM, respectively. In the left region of curve (I), kA/Cz2>0, and in the right region of curve (II), kBz2>0, no surface modes exist. In regions between curves (I) and (II), evanescent waves emerge in both air and HMM, hence, surface polaritons can exist at dual boundaries of HMM based on the positive εA/C and negativeεp. While at the right region of curve (II), bulk polaritons can also exist since kA/Cz2<0 and kBz2>0, here we focus on the bulk polaritons in this paper.

Except for EF, the bulk polariton properties are also dependent on the fill factions of dielectric layer and graphene sheet. If change the thickness of dielectric layer in one unit, εp will vary with td remarkedly as shown in Ref.13

13. K. V. Sreekanth, A. De Luca, and G. Strangi, “Negative refraction in graphene-based hyperbolic metamaterials,” Appl. Phys. Lett. 103(2), 023107 (2013). [CrossRef]

. Here we keep the thickness of td = 50nm, and increase the thickness of graphene by controlling the number of layers of graphene sheets. We can estimate the conductivity following the approximation σ'=Nsσ for a few layer graphene, where Ns is the number of layers (Ns <6) in one period [32

32. G. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008). [CrossRef]

], tg'=Nstg. Figure 3 (b) shows the dependences of dispersion relations of p-polarized surface polaritons and bulk polaritons on the number of graphene sheets (i.e., Ns), where N = 6, td = 50nm and EF = 0.2eV respectively. As observed, the existence region obviously increases to high frequency band with the number of the sheet Ns. Furthermore, the bulk polariton dispersion relations are affected by the thickness of the HMM (i.e., the period number, N). The polariton dispersion relations are shown in Fig. 3(c) with different N, N = 5, N = 6, N = 7 and EF = 0.2eV respectively. It is seen that the polaritons curve regions decrease to lower frequency with N. On the whole, the polariton dispersions exhibit an obvious tunability.

Since HMM allows for propagation of extraordinary waves with a large transverse wavenumber kx>εtk0 with εp< 0, the bulk polaritons do not interact directly with an incoming electromagnetic plane wave because of the momentum mismatch. In order to study the bulk polaritons in the graphene-based HMM, ATR technique (Otto configuration) can be employed [33

33. C. H. Gan, “Analysis of surface plasmon excitation at terahertz frequencies with highly doped graphene sheets via attenuated total reflection,” Appl. Phys. Lett. 101(11), 111609 (2012). [CrossRef]

] as shown in the inset of Fig. 4. HMM is placed at a distance of h from the semi-cylindrical germanium prism withnp=4. The prism is used to access the wide range of effective indices and to match the wave vector of light in free space to the wave vector of the bulk polaritons. When the incidence angle θis greater than the critical angleθc=arcsinεAμA/εprismμprism, kAz becomes purely imaginary and an evanescent wave emerges in the gap (i.e., medium A shown in Fig. 1(b)). If the gap width h approaches infinity, the evanescent wave will decay exponentially away from the prism and total internal reflection will occur. For a finite h, however the incident waves can be coupled to the surface polaritons or bulk polaritons, resulting in a reduction in the reflectance, especially near the resonance frequencies. Transfer matrix method has been used to calculate the angular reflectance spectra for different frequency with EF = 0.2eV. The calculated two-dimensional map of reflectance of the Otto configuration is shown in Fig. 4. The parameters used in the calculation are εA = εC = 1, h = 1um, the HMM is designed to consist of six pairs of silica/graphene sheet. As seen from the figure, the locations of minimum reflected intensity directly indicate the excitation of the bulk plasmon modes through the ATR method. When the frequency increases, the excitation of the bulk polariton needs larger incident angle, where the bulk polariton possesses nearly the same kz. The results are accordant with those in Fig. 2 (a). It can be seen from the map that the designed structure supports a bulk plasmon mode at incident angle of 68.55°for 10 THz. While the frequency of reflectance dip has somewhat deviation compared with the across between curve (IV) and the dispersion relation (dash-dotted line) shown in Fig. 3(a), because the electromagnetic fields in the medium A and the graphene-based HMM layer have been perturbed by the prism. Despite the deviations, the dispersion relation can guide the resonance frequency of bulk plasmon mode well.

Because the bulk plasmons can be effectively controlled by EF of the graphene, we investigate the behavior of bulk plasmon modes in the HMM at different EF under a certain incident angle and excitation frequency. Figure 5(a)
Fig. 5 (a) Reflectance of the Otto configuration with HMM vs incident angles at different Fermi energy EF. (b) Reflectance vs EF, the frequency of the incident wave is 10THz (11THz), θ=68.550(51.86°) for case 1 (2).
shows the reflectance spectrum of the Otto configuration as a function of incident angles, the excitation frequency is 10THz. When EF increases or decrease from 0.2 eV, the amplitude in the reflectance spectrum at the same incident angle increases significantly and the dip deviates 68.55°. The reflectance amplitude at 68.55°vs the Fermi energy are plotted as shown the Fig. 5 (b) (case 1). This tuning property of the considered HMM structure can be used in optical reflection modulation [26

26. W. R. Zhu, I. D. Rukhlenko, and M. Premaratne, “Graphene metamaterial for optical reflection modulation,” Appl. Phys. Lett. 102(24), 241914 (2013). [CrossRef]

,34

34. X. L. Shi, S. L. Zheng, H. Chi, X. F. Jin, and X. M. Zhang, “All-optical modulator with longrange surface plasmon resonance,” Opt. Laser Technol. 49, 316–319 (2013). [CrossRef]

]. As observed, the reflectance can be modulated from 0.02 to 1 when EF deviates ±0.1eVaround 0.2eV, near 100% modulation depth, which is higher than 64% reported in [35

35. B. Sensale-Rodriguez, R. Yan, S. Rafique, M. Zhu, W. Li, X. Liang, D. Gundlach, V. Protasenko, M. M. Kelly, D. Jena, L. Liu, and H. G. Xing, “Extraordinary Control of Terahertz Beam Reflectance in Graphene Electro-absorption Modulators,” Nano Lett. 12(9), 4518–4522 (2012). [CrossRef] [PubMed]

]. Due to the low propagation loss, the insertion loss for an ATR modulator is much lower than that in a waveguide system [36

36. D. G. Cooke and P. U. Jepsen, “Optical modulation of terahertz pulses in a parallel plate waveguide,” Opt. Express 16(19), 15123–15129 (2008). [CrossRef] [PubMed]

].

In the Otto configuration, the medium A and the separation h are employed as an additional degree of freedom to tune the bulk plasmon dispersion and it’s excitation. In practice, the optical reflection modulator can be realized by the following procedures: the MF2 dielectric layer with εA = 1.9, h = 1.5um is first coated on a glass substrate, and then the high quality multilayer is realized by stacking of many graphene/dielectric layers on the MF2 substrate. Figure 6(a)
Fig. 6 (a)The two-dimensional map of reflectance of the Otto configuration at different frequency and incident angles, (b) at different frequency and Fermi energy, where the dielectric A is MF2 withεA = 1.9, h = 1.5um.
is the corresponding two-dimensional map of reflectance for p-polarization wave, where EF = 0.3eV, N = 7. It is seen that by changing the parameters such as the thickness h of medium A, EF and N, the optimal reflectance dip can be obtained at different frequencies. We further study the excitations of the bulk polaritons with different Fermi energy. As shown in Fig. 6(b), the reflectance spectra became narrower when the excitation frequency is increased from 4 THz to 20 THz, where the incident angle is 51.86°. It is also possible to observe the reflectance minima for higher frequencies when the EF increases, which is in accordance with the theoretical analyses of the bulk polaritons as shown in Fig. 3(a). This is due to the increase in negative group index of such HMM at higher Fermi energy [13

13. K. V. Sreekanth, A. De Luca, and G. Strangi, “Negative refraction in graphene-based hyperbolic metamaterials,” Appl. Phys. Lett. 103(2), 023107 (2013). [CrossRef]

]. Furthermore, the reflectance amplitudes of 11THz at 51.86° vs the EF are also plotted as shown in Fig. 5 (b) (case 2). This tuning property of the considered HMM structure can be realized at different frequency, which indicates that large modulation bandwidth can be obtained by using the ATR modulator, such property is important in tunable reflectance modulation.

4. Conclusion

In conclusion, we have theoretically studied the electromagnetic properties of HMM that comprises graphene/dielectric multilayers based on the permittivity homogenization model. The p-polarized polariton dispersion relations for surface and bulk polaritons in a HMM layer surrounded by different media are obtained. We have quantitatively shown that the capability of tuning the polaritons properties via Fermi energy, the thickness of HMM, the layer number of graphene sheets. The calculated reflectance spectra in an ATR configuration involving an HMM shows that the bulk polaritons can be efficiently excited at the desired frequency by choosing appropriate parameters. The modifications of the optical reflectance properties in the Otto configuration with HMM may have practical applications in tunable optical reflection modulator.

Acknowledgment

This research was supported by NSFC (Nos.10904032, 11104063 and 11204068), by the Foundations of Henan Educational Committee (Nos. 14A140011 and 2012GGJS-060), by HPU Program for Distinguished Young Scholars (No.J2013-09) and by Science Foundation of Chongqing Jiaotong University (No. 2013kjc030).

References and links

1.

A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009). [CrossRef]

2.

K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, “A roadmap for graphene,” Nature 490(7419), 192–200 (2012). [CrossRef] [PubMed]

3.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012). [CrossRef]

4.

Z. Zhang, H. Li, Z. Gong, Y. Fan, T. Zhang, and H. Chen, “Extend the omnidirectional electronic gap of Thue-Morse aperiodic gapped graphene superlattices,” Appl. Phys. Lett. 101(25), 252104 (2012). [CrossRef]

5.

V. Chabot, D. Higgins, A. P. Yu, X. C. Xiao, Z. W. Chen, and J. J. Zhang, “A review of graphene and graphene oxide sponge: material synthesis and applications to energy and the environment,” Energy Environ. Sci. 7(5), 1564–1596 (2014).

6.

A. Reina, X. T. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus, and J. Kong, “Large area, few-layer graphene films on arbitrary substrates by chemical vapor deposition,” Nano Lett. 9(1), 30–35 (2009). [CrossRef] [PubMed]

7.

A. Vakil and N. Engheta, “Transformation Optics Using Graphene,” Science 332(6035), 1291–1294 (2011). [CrossRef] [PubMed]

8.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011). [CrossRef]

9.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011). [CrossRef] [PubMed]

10.

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011). [CrossRef] [PubMed]

11.

F. Liu and E. Cubukcu, “Tunable omnidirectional strong light-matter interactions mediated by graphene surface plasmons,” Phys. Rev. B 88(11), 115439 (2013). [CrossRef]

12.

P. Tassin, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “A comparison of graphene, superconductors and metals as conductors for metamaterials and plasmonics,” Nat. Photonics 6(4), 259–264 (2012). [CrossRef]

13.

K. V. Sreekanth, A. De Luca, and G. Strangi, “Negative refraction in graphene-based hyperbolic metamaterials,” Appl. Phys. Lett. 103(2), 023107 (2013). [CrossRef]

14.

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]

15.

M. A. K. 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]

16.

T. Zhang, L. Chen, and X. Li, “Graphene-based tunable broadband hyperlens for far-field subdiffraction imaging at mid-infrared frequencies,” Opt. Express 21(18), 20888–20899 (2013). [CrossRef] [PubMed]

17.

B. Zhu, G. Ren, S. Zheng, Z. Lin, and S. Jian, “Nanoscale dielectric-graphene-dielectric tunable infrared waveguide with ultrahigh refractive indices,” Opt. Express 21(14), 17089–17096 (2013). [CrossRef] [PubMed]

18.

C. J. Zapata-Rodríguez, J. J. Miret, S. Vuković, and M. R. Belić, “Engineered surface waves in hyperbolic metamaterials,” Opt. Express 21(16), 19113–19127 (2013). [CrossRef] [PubMed]

19.

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic metamaterials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012). [CrossRef]

20.

A. A. Avetisyan, B. Partoens, and F. M. Peeters, “Electric-field control of the band gap and Fermi energy in graphene multilayers by top and back gates,” Phys. Rev. B 80(19), 195401 (2009). [CrossRef]

21.

J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Plasmon-Induced Doping of Graphene,” Nature 487, 77–81 (2012).

22.

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

23.

G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008). [CrossRef]

24.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012). [PubMed]

25.

B. D. Guo, L. Fang, B. H. Zhang, and J. R. Gong, “Graphene Doping: A Review,” Insciences J. 1(2), 80–89 (2011). [CrossRef]

26.

W. R. Zhu, I. D. Rukhlenko, and M. Premaratne, “Graphene metamaterial for optical reflection modulation,” Appl. Phys. Lett. 102(24), 241914 (2013). [CrossRef]

27.

S. Winnerl, M. Orlita, P. Plochocka, P. Kossacki, M. Potemski, T. Winzer, E. Malic, A. Knorr, M. Sprinkle, C. Berger, W. A. de Heer, H. Schneider, and M. Helm, “Carrier Relaxation in Epitaxial Graphene Photoexcited Near the Dirac Point,” Phys. Rev. Lett. 107(23), 237401 (2011). [CrossRef] [PubMed]

28.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402 (2007). [CrossRef]

29.

K. Park, B. J. Lee, C. J. Fu, and Z. M. Zhang, “Study of the surface and bulk polaritons with an egative index metamateria,” J. Opt. Soc. Am. B 22(5), 1016–1023 (2005). [CrossRef]

30.

H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, and T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100(24), 243110 (2012). [CrossRef]

31.

C. F. Chen, C. H. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, and F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471(7340), 617–620 (2011). [CrossRef] [PubMed]

32.

G. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008). [CrossRef]

33.

C. H. Gan, “Analysis of surface plasmon excitation at terahertz frequencies with highly doped graphene sheets via attenuated total reflection,” Appl. Phys. Lett. 101(11), 111609 (2012). [CrossRef]

34.

X. L. Shi, S. L. Zheng, H. Chi, X. F. Jin, and X. M. Zhang, “All-optical modulator with longrange surface plasmon resonance,” Opt. Laser Technol. 49, 316–319 (2013). [CrossRef]

35.

B. Sensale-Rodriguez, R. Yan, S. Rafique, M. Zhu, W. Li, X. Liang, D. Gundlach, V. Protasenko, M. M. Kelly, D. Jena, L. Liu, and H. G. Xing, “Extraordinary Control of Terahertz Beam Reflectance in Graphene Electro-absorption Modulators,” Nano Lett. 12(9), 4518–4522 (2012). [CrossRef] [PubMed]

36.

D. G. Cooke and P. U. Jepsen, “Optical modulation of terahertz pulses in a parallel plate waveguide,” Opt. Express 16(19), 15123–15129 (2008). [CrossRef] [PubMed]

OCIS Codes
(230.4170) Optical devices : Multilayers
(160.3918) Materials : Metamaterials
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Metamaterials

History
Original Manuscript: April 28, 2014
Revised Manuscript: May 25, 2014
Manuscript Accepted: May 27, 2014
Published: May 30, 2014

Citation
Liwei Zhang, Zhengren Zhang, Chaoyang Kang, Bei Cheng, Liang Chen, Xuefeng Yang, Jian Wang, Weibing Li, and Baoji Wang, "Tunable bulk polaritons of graphene-based hyperbolic metamaterials," Opt. Express 22, 14022-14030 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-14022


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References

  1. A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009). [CrossRef]
  2. K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, K. Kim, “A roadmap for graphene,” Nature 490(7419), 192–200 (2012). [CrossRef] [PubMed]
  3. A. N. Grigorenko, M. Polini, K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012). [CrossRef]
  4. Z. Zhang, H. Li, Z. Gong, Y. Fan, T. Zhang, H. Chen, “Extend the omnidirectional electronic gap of Thue-Morse aperiodic gapped graphene superlattices,” Appl. Phys. Lett. 101(25), 252104 (2012). [CrossRef]
  5. V. Chabot, D. Higgins, A. P. Yu, X. C. Xiao, Z. W. Chen, J. J. Zhang, “A review of graphene and graphene oxide sponge: material synthesis and applications to energy and the environment,” Energy Environ. Sci. 7(5), 1564–1596 (2014).
  6. A. Reina, X. T. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus, J. Kong, “Large area, few-layer graphene films on arbitrary substrates by chemical vapor deposition,” Nano Lett. 9(1), 30–35 (2009). [CrossRef] [PubMed]
  7. A. Vakil, N. Engheta, “Transformation Optics Using Graphene,” Science 332(6035), 1291–1294 (2011). [CrossRef] [PubMed]
  8. Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics 5(7), 411–415 (2011). [CrossRef]
  9. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011). [CrossRef] [PubMed]
  10. P. Y. Chen, A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5(7), 5855–5863 (2011). [CrossRef] [PubMed]
  11. F. Liu, E. Cubukcu, “Tunable omnidirectional strong light-matter interactions mediated by graphene surface plasmons,” Phys. Rev. B 88(11), 115439 (2013). [CrossRef]
  12. P. Tassin, T. Koschny, M. Kafesaki, C. M. Soukoulis, “A comparison of graphene, superconductors and metals as conductors for metamaterials and plasmonics,” Nat. Photonics 6(4), 259–264 (2012). [CrossRef]
  13. K. V. Sreekanth, A. De Luca, G. Strangi, “Negative refraction in graphene-based hyperbolic metamaterials,” Appl. Phys. Lett. 103(2), 023107 (2013). [CrossRef]
  14. I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87(7), 075416 (2013). [CrossRef]
  15. M. A. K. Othman, C. Guclu, F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express 21(6), 7614–7632 (2013). [CrossRef] [PubMed]
  16. T. Zhang, L. Chen, X. Li, “Graphene-based tunable broadband hyperlens for far-field subdiffraction imaging at mid-infrared frequencies,” Opt. Express 21(18), 20888–20899 (2013). [CrossRef] [PubMed]
  17. B. Zhu, G. Ren, S. Zheng, Z. Lin, S. Jian, “Nanoscale dielectric-graphene-dielectric tunable infrared waveguide with ultrahigh refractive indices,” Opt. Express 21(14), 17089–17096 (2013). [CrossRef] [PubMed]
  18. C. J. Zapata-Rodríguez, J. J. Miret, S. Vuković, M. R. Belić, “Engineered surface waves in hyperbolic metamaterials,” Opt. Express 21(16), 19113–19127 (2013). [CrossRef] [PubMed]
  19. O. Kidwai, S. V. Zhukovsky, J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic metamaterials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012). [CrossRef]
  20. A. A. Avetisyan, B. Partoens, F. M. Peeters, “Electric-field control of the band gap and Fermi energy in graphene multilayers by top and back gates,” Phys. Rev. B 80(19), 195401 (2009). [CrossRef]
  21. J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, F. H. L. Koppens, “Plasmon-Induced Doping of Graphene,” Nature 487, 77–81 (2012).
  22. Y. J. Xiang, J. Guo, X. Y. Dai, S. C. Wen, D. Y. Tang, “Engineered surface Bloch waves in graphene-based hyperbolic metamaterials,” Opt. Express 22(3), 3054–3062 (2014). [CrossRef] [PubMed]
  23. G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008). [CrossRef]
  24. Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012). [PubMed]
  25. B. D. Guo, L. Fang, B. H. Zhang, J. R. Gong, “Graphene Doping: A Review,” Insciences J. 1(2), 80–89 (2011). [CrossRef]
  26. W. R. Zhu, I. D. Rukhlenko, M. Premaratne, “Graphene metamaterial for optical reflection modulation,” Appl. Phys. Lett. 102(24), 241914 (2013). [CrossRef]
  27. S. Winnerl, M. Orlita, P. Plochocka, P. Kossacki, M. Potemski, T. Winzer, E. Malic, A. Knorr, M. Sprinkle, C. Berger, W. A. de Heer, H. Schneider, M. Helm, “Carrier Relaxation in Epitaxial Graphene Photoexcited Near the Dirac Point,” Phys. Rev. Lett. 107(23), 237401 (2011). [CrossRef] [PubMed]
  28. I. Avrutsky, I. Salakhutdinov, J. Elser, V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402 (2007). [CrossRef]
  29. K. Park, B. J. Lee, C. J. Fu, Z. M. Zhang, “Study of the surface and bulk polaritons with an egative index metamateria,” J. Opt. Soc. Am. B 22(5), 1016–1023 (2005). [CrossRef]
  30. H. J. Xu, W. B. Lu, W. Zhu, Z. G. Dong, T. J. Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100(24), 243110 (2012). [CrossRef]
  31. C. F. Chen, C. H. Park, B. W. Boudouris, J. Horng, B. Geng, C. Girit, A. Zettl, M. F. Crommie, R. A. Segalman, S. G. Louie, F. Wang, “Controlling inelastic light scattering quantum pathways in graphene,” Nature 471(7340), 617–620 (2011). [CrossRef] [PubMed]
  32. G. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008). [CrossRef]
  33. C. H. Gan, “Analysis of surface plasmon excitation at terahertz frequencies with highly doped graphene sheets via attenuated total reflection,” Appl. Phys. Lett. 101(11), 111609 (2012). [CrossRef]
  34. X. L. Shi, S. L. Zheng, H. Chi, X. F. Jin, X. M. Zhang, “All-optical modulator with longrange surface plasmon resonance,” Opt. Laser Technol. 49, 316–319 (2013). [CrossRef]
  35. B. Sensale-Rodriguez, R. Yan, S. Rafique, M. Zhu, W. Li, X. Liang, D. Gundlach, V. Protasenko, M. M. Kelly, D. Jena, L. Liu, H. G. Xing, “Extraordinary Control of Terahertz Beam Reflectance in Graphene Electro-absorption Modulators,” Nano Lett. 12(9), 4518–4522 (2012). [CrossRef] [PubMed]
  36. D. G. Cooke, P. U. Jepsen, “Optical modulation of terahertz pulses in a parallel plate waveguide,” Opt. Express 16(19), 15123–15129 (2008). [CrossRef] [PubMed]

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