Hybrid surface phononic waveguide using hyperbolic boron nitride

Yuancheng Xu; Navaneeth Premkumar; Yuchen Yang; Brian A. Lail

doi:10.1364/OE.24.017183

Optics Express
Vol. 24,
Issue 15,
pp. 17183-17192
(2016)
•https://doi.org/10.1364/OE.24.017183

Hybrid surface phononic waveguide using hyperbolic boron nitride

Yuancheng Xu, Navaneeth Premkumar, Yuchen Yang, and Brian A. Lail

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Yuancheng Xu, Navaneeth Premkumar, Yuchen Yang, and Brian A. Lail, "Hybrid surface phononic waveguide using hyperbolic boron nitride," Opt. Express 24, 17183-17192 (2016)

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Table of Contents Category
- Integrated Optics
Optics & Photonics Topics
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The topics in this list come from the Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Optical confinement and manipulation (170.4520)
- Polaritons (240.5420)
- Infrared (260.3060)

About this Article
History
- Original Manuscript: April 23, 2016
- Revised Manuscript: June 10, 2016
- Manuscript Accepted: July 9, 2016
- Published: July 20, 2016

Abstract

Sub-diffraction limited waveguides have been studied as a means to manipulate light into nanoscale regions. Hybrid waveguides are popular candidates in optical regimes for subwavelength confinement and long range propagation. However, advances in the mid-IR are lacking due to high propagation losses and limited confinement. Here we present the first analysis of hybrid phononic waveguide using a hyperbolic material h-BN to generate surface phonon polaritons. The strong coupling between the photonic cylinder and phononic surface enhances the confined field up to 10⁻³ λ_o² (λ_o is free-space wavelength) and enables propagation distances up to 100 λ_o. Our work is fully compatible with integrated polaritonic devices in the mid-IR and provides a systematic approach to design hybrid phononic waveguides.

1. Introduction

The field of nanophotonics has sought to solve the problem of manipulating light on scales beyond the diffraction limit [1]. Surface plasmon polaritons (SPPs) are among the most studied methods for subwavelength confinement at optical and near-IR wavelengths [2–5]. In order to improve upon the limitations of SPP waveguides, hybrid plasmonic waveguides, which consist of a high-index-contrast dielectric waveguide placed over the metal surface, were the alternative [6–9]. The coupling between the dielectric waveguide and plasmonic modes can produce highly confined modes with large propagation distance. This however is restricted to optical wavelengths because noble metal SPPs generally don’t offer significant field confinement at mid-IR wavelengths. The need for low-loss optical materials capable of supporting sub-diffraction limited, localized/confined, and propagating modes in such nanostructures in the mid-IR ranges motivates the study of surface phonon polaritons (SPhPs) [10,11]. The stimulation of SPhP modes in polar dielectrics enables a plethora of potential sensing and waveguiding applications in long wavelength infrared (IR) ranges including chemical sensing and IR imaging [12, 13]. Surface phonon polaritons are TM guided waves produced as a result of IR phonons- quasi particles produced from lattice vibrations in polar dielectrics subjected to an external field, and optical photons. They produce similar properties to SPPs and can be treated with similar mathematics [14]. There has been interest in phonon polariton based applications [15], however little has been done to look at phonon polariton based waveguiding.

In the search for materials that support SPhP modes, it has been found that crystalline dielectrics and semiconductor materials are prime candidates, specifically SiC, SiO₂ and III-V and III-nitride semiconductors [11–20]. While all of these candidates exhibit low-loss characteristics, it has been shown that hexagonal boron nitride (h-BN), is a very attractive material due to its dispersion characteristics, low optical losses and all the more because of the ability to deposit one atom thick unit cells of h-BN, essentially producing two dimensional surfaces that support SPhPs [21–23]. Boron nitride is a naturally hyperbolic dielectric material that exhibits two infrared resonances which span the transverse (ω_TO) and longitudinal (ω_LO) phonon frequencies of the out-of-plane mode (ω_TO = 760 cm⁻¹, ω_LO = 825 cm⁻¹) and the in-plane mode (ω_TO = 1370 cm⁻¹, ω_LO = 1614 cm⁻¹) [14]. Considerable work has been done to classify these resonances into a distinct lower frequency (Type I) and upper frequency (Type II) Reststrahlen band. Hexagonal BN (h-BN) possesses a strong surface phonon resonance resulting from a negative permittivity in the upper Reststrahlen, or in-plane, band. The excitation and control of these phonons has been studied in [14, 21–26]. Due to the large spectral separation between the out-of-plane and in-plane phonon resonances they are decoupled and do not interfere with one another. In order to enhance absorption/transmission and field confinement of h-BN polaritons in the areas of sensing and detection, graphene-based h-BN surface plasmon phonon polaritons have been used [27–32]. The high-field confinement of the tunable graphene plasmons in mid-IR allows for strong coupling with the h-BN monolayers. However, to the knowledge of the authors, h-BN structures have not been integrated into hybrid designs for mid-IR waveguide applications.

In this paper we present a hybrid phononic waveguide design by coupling h-BN SPhPs with a dielectric waveguide, significantly improving the propagation distance with enhanced modal confinement. For the purpose of a hybrid waveguide, we operate around the in-plane resonance owing to the strong surface phonon response on h-BN slab [26]. At wavenumber 1400 cm⁻¹, in the in-plane Reststrahlen band, the absolute magnitude of the negative h-BN permittivity is far greater than that of the air. Hence little evanescent field resides in the h-BN region and most of the wave energy flows through the air, significantly reducing, but not entirely eliminating volumetric losses inside the layer. The hybrid SPhPs propagate over large distances (up to 100 λ_o) with modal confinement reaching ~10⁻³ λ_o². This approach yields significant improvements to both guided phononics and photonics and has applications in phonon based IR lasers, bio-sensing and interferometry [12, 15, 34]. The hybrid phonon waveguide geometry and tuning is shown in section 2. In section 3, the hybridization is evaluated from the coupled-mode theory point of view and hybrid waveguide behavior at different frequencies is discussed. Concluding remarks follow in section 4.

2. Geometry and modal properties for the proposed hybrid phononic waveguide

Before attempting hybridization, we carried out initial calculations to estimate performance of h-BN as a SPhP supporting candidate. Figure 1(a) depicts the in-plane dielectric function of h-BN near the phonon resonance where the real part of the permittivity assumes a negative value over a narrow frequency band. Figure 1(b) depicts the computed dispersion curves of SPhP at a flat h-BN/air interface in the in-plane resonance band. Figure 1(b, inset) shows the region of interest where the data markers represent the wavenumbers at 1400 cm⁻¹, 1426 cm⁻¹ and 1483 cm⁻¹(red marker for 1400 cm⁻¹, purple marker for 1426 cm⁻¹ and orange marker for 1483 cm⁻¹).

Fig. 1 (a) Real and imaginary parts of the dielectric function of flat h-BN near the phonon resonance. (b) Dispersion of the SPhP mode of the flat h-BN-air boundary.

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Figure 2 depicts the proposed hybrid phononic waveguide structure. A high-refractive-index dielectric cylinder waveguide with diameter d is placed above the h-BN slab with nanoscale air gap distance h. The cylinder waveguide is silicon with refractive index n = 3.42, and the h-BN slab with an in-plane permittivity ε_∥ = −28.12 + 2.91i and out-of-plane permittivity ε_⊥ = 2.73 + 0.0004i at mid-IR wavenumber 1400 cm⁻¹. The simulated SPhP mode is along the interface between air and a 1 μm thick h-BN slab on a dielectric substrate. The interface of h-BN/air supports surface waves in the form of SPhPs as designated by the strong field concentration above the slab. Note that here we only consider the fundamental mode, since higher order modes will only exist in ultrathin h-BN [23]. The h-BN based hybrid phononic waveguide with an air gap presented here can be implemented in suspended waveguide designs akin to those seen in [36].

Fig. 2 The hybrid mid-IR waveguide which includes a dielectric cylinder of diameter d placed at a gap height h above an h-BN slab on a dielectric substrate.

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In the following study, we vary the cylinder diameter, d, and the air gap, h, to control the propagation distance, L_m, and modal area, A_m, while maintaining a transverse magnetic field distribution. The strong mode-coupling in the air gap between the dielectric waveguide cylinder mode and the SPhP mode results in an extremely confined hybrid phononic mode. The entire structure lies on a lossless dielectric substrate of permittivity ε_d = 2.25. It should be noted that SPhP modes can be supported by this h-BN/substrate boundary however in this work we are considering modes in which this additional SPhP mode is suppressed such that the air/h-BN SPhP and the cylinder mode dominate the hybrid coupling. Finite-element analysis method (Ansys HFSS) is used to calculate the eigenmode of the coupled waveguide system, with dispersive material properties of h-BN provided by the Naval Research Laboratory. The material properties of silicon were measured by J.A. Woollam’s IR-Variable Angle Spectroscopic Ellipsometry. Note that the loss of Si is negligible compared to h-BN loss at the wavenumbers of interest and has therefore been neglected in the numerical simulations.

Figures 3(a) and 3(b) show the modal area A_m and the propagation length L_m of the hybrid phononic mode as a function of the waveguide diameter d and the air gap h. The propagation length L_m and the modal area A_m are calculated from [2],

L_{m} = {[2 I m (k_{h y b} (d, h))]}^{- 1}

A_{m} = \frac{W_{m}}{\max {W (\vec{r})}}

where

k_{h y b} (d, h)

is the wavevector of the hybrid mode and

W_{m}

is the integrated electromagnetic energy over the modal distribution.

W (\vec{r})

is the local energy density at the position

\vec{r}

given by,

W (\vec{r}) = \frac{1}{2} R e {\frac{d (ω ε (\vec{r}))}{d ω}} {| E (\vec{r}) |}^{2} + \frac{1}{2} μ_{0} {| H (\vec{r}) |}^{2}

Here E(

\vec{r}

) and H(

\vec{r}

) are the electric and magnetic fields,

ε (\vec{r})

is the electric permittivity and is the magnetic permeability. Mode confinement can be gauged by the normalized modal area defined as

A_{m} / A_{o}

where

A_{o} = λ_{o}^{2} / 4

and

λ_{o}

is the free-space wavelength.

Fig. 3 (a) Modal area as a function of cylinder diameter d for different gap height h. (b) Hybrid propagation distance as a function of cylinder diameter d for different gap height h.

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Figure 4 shows the electromagnetic energy density distributions of the hybrid phononic mode for different waveguide diameter d and air gap h. For large d and h (for example, [d,h] = [2.5, 1] μm), the hybrid phononic waveguide supports a low-loss hybrid mode dominated by cylinder-like mode with optical energy confined in the dielectric cylinder. Conversely, a small diameter d results in a hybrid mode dominated by SPhP-like mode with very weak localization on the h-BN/air interface, suffering loss comparable to that of uncoupled SPhPs. At other cylinder diameters (for example, [d, h] = [1.4, 0.25] μm and [1.4, 0.1] μm), mode coupling results in a hybrid mode that has both cylinder and SPhP features i.e., its electromagnetic energy is distributed in the cylinder waveguide, the h-BN/air interface and inside h-BN slab. For h in nanometer scale, for example [d, h] = [1.4, 0.01] μm, the cylinder mode is strongly coupled to the SPhP mode, most of the optical energy being concentrated inside the air gap with ultrasmall modal area. The air gap between the cylinder dielectric waveguide and the h-BN slab provides a region of low-index material that can be used to strongly confine and propagate light. This arises from the need to satisfy the continuity of the normal component of electric flux density, D, giving rise to a large amplitude of normal component of electric field in the low-index region of a high-index-contrast interface [37]. Also, the uncoupled SPhP and dielectric geometries amplify this effect by having dominant normal electric field components to the material interfaces.

Fig. 4 Electromagnetic energy density distribution for (a) [d, h] = [2.5, 1] μm, (b) [d, h] = [1.4, 0.25] μm, (c) [d, h] = [1.4, 0.1] μm and (d) [d, h] = [1.4, 0.01] μm.

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3. Coupling characteristics between surface phonon polariton and dielectric waveguide

In order to explain the coupling between SPhP and dielectric cylinder modes, we analyzed the dependence of the hybrid mode’s effective index, $n_{h y b} (d, h)$ on d and h. Figure 5 depicts the effective index of the hybrid mode as a function of d for different gap height h. As expected, the waveguide mode index approaches that of the dielectric cylinder mode and SPhP mode at the extremities of the plot. Meanwhile, the hybrid mode’s effective index is always larger than dielectric cylinder mode and SPhP mode, indicating that the SPhP mode is coupled with the dielectric cylinder waveguide, which induces a much higher effective index. The mode’s effective index can be increased by reducing the gap distance for a fixed d, or enhancing the diameter of the cylinder nanowire, d, for a fixed gap distance, h. This can be explained by the observation that the coupling efficiency is increased between the cylinder and h-BN surface when the d increases or h reduces.

Fig. 5 Effective index of the hybrid waveguide for different gap height h versus cylinder diameter d. The dashed line indicates the index of the SPhP mode $n_{S P h P}$ .

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In order to explain the coupling efficiency with coupled mode theory, we describe the hybrid eigenmode as the superposition of the cylinder waveguide mode (without h-BN slab) and the SPhP mode (without cylinder) [2].

ψ_{h y b} (d, h) = a (d, h) ψ_{c y l} (d) + b (d, h) ψ_{S P h P}

Here

a (d, h)

and

b (d, h)

are the mode amplitudes of the cylinder mode

ψ_{c y l}

and the SPhP mode

ψ_{S P h P}

, respectively, and with

b = {(1 - {| a |}^{2})}^{1 / 2}

. The coupled waveguide system can be described as [2],

n_{c y l} (d) a (d, h) + κ (d, h) b (d, h) = n_{h y b} (d, h) a (d, h)

κ (d, h) a (d, h) + n_{S P h P} b (d, h) = n_{h y b} (d, h) b (d, h)

where

n_{c y l} (d)

and

n_{S P h P}

are the effective index of the cylinder mode and SPhP mode, respectively, and

n_{h y b} (d, h)

is the effective index of the waveguide mode. According to the coupled-mode theory for the hybrid waveguide, the coupling strength

κ (d, h)

between these two modes can be described as

κ (d, h) = \sqrt{(n_{h y b} (d, h) - n_{S P h P}) (n_{h y b} (d, h) - n_{c y l} (d))}

Figure 6 plots the dependence of coupling strength $κ$ on d and h for the hybrid mode. The coupling strength is maximum when the index of the cylinder mode equals that of the SPhP mode, which can be seen analytically from Eq. (7). At a critical diameter d_c for each gap height h, the coupling strength hits its maximum value when the cylinder mode and SPhP mode propagate in phase and the effective optical capacitance of the hybrid system is maximum. We can see from Fig. 6 that the coupling strength increases with decreasing gap distance h, which can be correlated with the fact that the modes couple more effectively as the gap size is reduced. Note that in Fig. 3(a), an uncharacteristic bump (d = d_c) in modal area can be observed which could be attributed to maximum coupling strength between the dielectric cylinder and the h-BN slab modes, causing apparent volumetric loss in h-BN which manifests itself as well confined hyperbolic polaritons (HPs). A corresponding small rise in propagation length (Fig. 3(b)) is also noted at this point of maximum coupling.

Fig. 6 The dependence of coupling strength κ on cylinder diameter d and gap height h.

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As seen in the dispersion diagram of h-BN from Fig. 1(b), the SPhP mode holds a relatively high effective index and confines the fields tightly, which suggests that as the dispersion curve deviates from the light line, the SPhP mode begins to get more confined on the surface of h-BN slab. This effect can also factor into the hybrid waveguide, where stronger field confinement is achieved at larger wavenumbers. This is illustrated in Fig. 7, where normalized electromagnetic field density is plotted for a fixed gap and cylinder diameter, for three different wavenumbers. We clearly see that for the largest wavenumber 1483 cm⁻¹, the largest amount of energy is highly confined within the gap between the cylinder and the h-BN slab, while for the shortest wavenumber 1400 cm⁻¹, the least amount of energy lies in the gap.

Fig. 7 Normalized energy density along z axis at wavenumber (a) 1400 cm⁻¹ (b) 1426 cm⁻¹ and (c) 1483 cm⁻¹. (d) Normalized energy density at z = -d/2 for different wavenumbers.

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Moreover when the coupling strength is maximum at the three different wavenumbers, as mentioned before, strong field penetration can be observed inside the h-BN slab, giving rise to apparent HPs, which is a sub-diffraction-limited phenomenon observed in h-BN. These HPs follow angular propagation along the y-direction, seen in Fig. 8(a) (the yellow and green dashed lines are used to trace the HPs in the h-BN slab for the θ₁ case), where the angle depends on the ratio of the extraordinary axis (z-axis) and ordinary (x-y plane) components of the anisotropic dielectric function of h-BN, as given by [25],

Fig. 8 (a) Critical angle θ₁ of the hyperbolic polaritons propagating inside the h-BN slab at 1400 cm⁻¹ for [d, h] = [1.4, 0.1] μm. (b) Frequency-dependent directional angles of the hyperbolic polaritons, where θ₁ is at 1400 cm⁻¹, θ₂ at 1426 cm⁻¹ and θ₃ at 1483 cm⁻¹.

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θ = \frac{π}{2} - arc \tan (\frac{\sqrt{ε_{z} (ω)}}{i \sqrt{ε_{x y} (ω)}})

Figure 8(b) shows the dependence of HP propagation angle versus wavenumber, where the solid blue line is computed from Eq. (8). The data markers represent the propagation angles at different wavenumbers from the simulation results (θ₁ at 1400 cm⁻¹, θ₂ at 1426 cm⁻¹ and θ₃ at 1483 cm⁻¹), which are in good agreement with the corresponding computed propagation angles from Eq. (8).

4. Conclusion

We have designed a hybrid surface phonon polariton waveguide using a dielectric Si cylinder placed at a height above a hyperbolic boron nitride slab on a substrate. The resulting hybrid mode can propagate distances of up to 100 λ_o, with normalized modal area of ~10⁻³ λ_o², which opens the way for various optomechanical applications in mid-IR such as nanoscale tweezers to trap nanoparticles (chemical species that have resonances in this region) analogous to those seen in the optical regime [37] and near-field optical imaging, waveguiding and focusing applications [25, 33]. Moreover, planar h-BN has proven easy to fabricate and incorporate on a variety of substrates; the air gap can also be substituted with a low loss, low index dielectric spacer which makes for easily realizable fabrications. Our work fulfills the apparent need for phononic waveguiding in integrated polaritonics and next-generation devices operating in the mid-IR spectrum.

Acknowledgement

The authors would like to acknowledge Joshua D. Caldwell of the Naval Research Laboratory for his valuable discussions and for providing measured h-BN material data.

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31. V. W. Brar, M. S. Jang, M. Sherrott, S. Kim, J. J. Lopez, L. B. Kim, M. Choi, and H. Atwater, “Hybrid surface-phonon-plasmon polariton modes in Graphene/monolayer h-BN heterostructures,” Nano Lett. 14(7), 3876–3880 (2014). [CrossRef] [PubMed]

32. Y. Wu, L. Jiang, H. Xu, X. Dai, Y. Xiang, and D. Fan, “Hybrid nonlinear surface-phonon-plasmon-polaritons at the interface of nolinear medium and Graphene-covered hexagonal Boron Nitride crystal,” Opt. Express 24(3), 2109–2124 (2016). [CrossRef] [PubMed]

33. J. D. Caldwell, L. Lindsay, V. Giannini, I. Vurgaftman, T. L. Reinecke, S. A. Maier, and O. J. Glembocki, “Low-loss, infrared and terahertz surface phonon polaritons,” Nanophotonics 4, 44–68 (2015). [CrossRef]

34. H. K. Choi, Long-Wavelength Infrared Semiconductor Lasers (Academic, 2004).

35. U. N. Roy, R. H. Hawrami, Y. Cui, S. Morgan, A. Burger, K. C. Mandal, C. C. Noblitt, S. A. Speakman, K. Rademaker, and S. A. Payne, “Tb³⁺-doped KPb₂Br₅: low-energy phonon mid-infrared laser crystal,” Appl. Phys. Lett. 86(15), 151911 (2005). [CrossRef]

36. S. A. Holmstrom, T. H. Stievater, M. W. Pruessner, D. Park, W. S. Rabinovich, J. B. Khurgin, C. J. K. Richardson, S. Kanakaraju, L. C. Calhoun, and R. Ghodssi, “Guided-mode phonon-polaritons in suspended waveguides,” Phys. Rev. B 86(16), 165120 (2012). [CrossRef]

37. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). [CrossRef] [PubMed]

38. X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011). [CrossRef] [PubMed]

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References

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Article Order
Year
Author
Publication

R. Kirchain and L. Kimerling, “A roadmap for nanophotonics,” Nat. Photonics 1(6), 303–305 (2007).
[Crossref]
R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]
F. Bigourdan, J. P. Hugonin, F. Marquier, C. Sauvan, and J. J. Greffet, “Nanoantenna for electrical generation of surface plasmon polaritons,” Phys. Rev. Lett. 116(10), 106803 (2016).
[Crossref] [PubMed]
T. J. Echtermeyer, S. Milana, U. Sassi, A. Eiden, M. Wu, E. Lidorikis, and A. C. Ferrari, “Surface plasmon polariton Graphene photodetectors,” Nano Lett. 16(1), 8–20 (2016).
[Crossref] [PubMed]
Y. Bian and Q. Gong, “Tuning the hybridization of plasmonic and coupled dielectric nanowire modes for high-performance optical waveguiding at sub-diffraction-limited scale,” Sci. Rep. 4, 6617 (2014).
[Crossref] [PubMed]
Y. Fang and M. Sun, “Nanoplasmonic waveguides: towards applications in integrated nanophotonics circuits,” Light Sci. Appl. 4(6), e294 (2015).
[Crossref]
L. Chen, X. Li, G. Wang, W. Li, S. Chen, L. Xiao, and D. Gao, “A Silicon-based 3-D hybrid long-range plasmonic waveguide for nanophotonic integration,” J. of Light. Tech. 30(1), 163–168 (2012).
[Crossref]
L. Chen, T. Zhang, X. Li, and W. Huang, “Novel hybrid plasmonic waveguide consisting of two identical dielectric nanowires symmetrically placed on each side of a thin metal film,” Opt. Express 20(18), 20535–20544 (2012).
[Crossref] [PubMed]
J. Chen, Z. Li, S. Yue, and Q. Gong, “Hybrid long-range surface plasmon-polariton modes with tight field confinement guided by asymmetrical waveguides,” Opt. Express 17(26), 23603–23609 (2009).
[Crossref] [PubMed]
X. G. Xu, B. G. Ghamsari, J. H. Jiang, L. Gilburd, G. O. Andreev, C. Zhi, Y. Bando, D. Golberg, P. Berini, and G. C. Walker, “One-dimensional surface phonon polaritons in Boron Nitride nanotubes,” Nat. Commun. 5, 4782 (2014).
[Crossref] [PubMed]
K. Feng, W. Streyer, S. M. Islam, J. Verma, D. Jena, D. Wasserman, and A. J. Hoffman, “Localized surface phonon polariton resonances in polar Gallium Nitride,” Appl. Phys. Lett. 107(8), 081108 (2015).
[Crossref]
K. M. Pitman, A. M. Hofmeister, A. B. Corman, and A. K. Speck, “Optical properties and applications of Silicon Carbide for astrophysics,” Astron. Astrophys. 483, 1–12 (2011).
R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light matter interaction at the nanometre scale,” Nature 418(6894), 159–162 (2002).
[Crossref] [PubMed]
Z. Shi, H. A. Bechtel, S. Berweger, Y. Sun, B. Zeng, C. Jin, H. Chang, M. C. Martin, M. B. Raschke, and F. Wang, “Amplitude- and phase- resolved nanospectral imaging of phonon polaritons in hexagonal Boron Nitride,” ACS Photonics 2(7), 790–796 (2015).
[Crossref]
T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz polaritonics,” Annu. Rev. Mater. Res. 37(1), 317–350 (2007).
[Crossref]
J. D. Caldwell, O. J. Glembocki, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme sub-diffraction photon confinement via Silicon Carbide surface phonon polariton nanopillar resonators,” Nano Lett. 13, 3690–3697 (2013).
[Crossref] [PubMed]
T. E. Tiwald, J. A. Woolam, S. Zollner, J. Christiansen, R. B. Gregory, T. Wetteroth, S. R. Wilson, and A. R. Powell, “Carrier concentration and lattice absorption in bulk and epitaxial Silicon Carbide determined using infrared ellipsometry,” Phys. Rev. B 60(16), 11464–11474 (1999).
[Crossref]
M. Haraguchi, M. Fukui, and S. Muto, “Experimental observation of attenuated-total-reflection spectra of GaAs/AlAs superlattice,” Phys. Rev. B Condens. Matter 41(2), 1254–1257 (1990).
[Crossref] [PubMed]
W. J. Moore and R. T. Holm, “Infrared dielectric constant of GaAs,” J. Appl. Phys. 80(12), 6939–6942 (1996).
[Crossref]
S. Adachi, The reststrahlen region.Optical Properties of Crystalline and Amorphous Semiconductors: Materials and Fundamental Principles (Science + Business Media. 1999).
[Crossref]
R. Geick, C. H. Perry, and G. Rupprecht, “Normal modes in hexagonal Boron Nitride,” Phys. Rev. B 146, 543 (1966).
Y. Cai, L. M. Zhang, Q. Zeng, L. Cheng, and Y. Xu, “Infrared reflectance spectrum of BN calculated from first principles,” Solid State Commun. 141(5), 262–266 (2007).
[Crossref]
S. Dai, Z. Fei, Q. Ma, A. S. Rodin, M. Wagner, A. S. McLeod, M. K. Liu, W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M. Thiemens, G. Dominguez, A. H. Castro Neto, A. Zettl, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Tunable phonon polaritons in atomically thin van der Waals crystals of boron nitride,” Science 343(6175), 1125–1129 (2014).
[Crossref] [PubMed]
E. Yoxall, M. Schnell, A. Y. Nikitin, O. Txoperena, A. Woessner, M. B. Lundeberg, F. Casanova, L. E. Hueso, F. H. L. Koppens, and R. Hillenbrand, “Direct observation of ultraslow hyperbolic polariton propagation with negative phase velocity,” Nat. Photonics 9(10), 674–678 (2015).
[Crossref]
P. Li, M. Lewin, A. V. Kretinin, J. D. Caldwell, K. S. Novoselov, T. Taniguchi, K. Watanabe, F. Gaussmann, and T. Taubner, “Hyperbolic phonon-polaritons in Boron Nitride for near-field optical imaging and focusing,” Nat. Commun. 6, 7507 (2015).
[Crossref] [PubMed]
J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
[Crossref] [PubMed]
S. Dai, Q. Ma, M. K. Liu, T. Andersen, Z. Fei, M. D. Goldflam, M. Wagner, K. Watanabe, T. Taniguchi, M. Thiemens, F. Keilmann, G. C. A. A. Janssen, S.-E. Zhu, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Graphene on hexagonal Boron Nitride as a tunable hyperbolic metamaterial,” Nat. Nanotechnol. 10(8), 682–686 (2015).
[Crossref] [PubMed]
Y. Jia, H. Zhao, Q. Guo, X. Wang, H. Wang, and F. Xia, “Tunable plasmon-phonon polaritons in layered Graphene-hexagonal Boron Nitride heterostructures,” ACS Photonics 2(7), 907–912 (2015).
[Crossref]
I. D. Barcelos, A. R. Cadore, L. C. Campos, A. Malachias, K. Watanabe, T. Taniguchi, F. C. B. Maia, R. Freitas, and C. Deneke, “Graphene/h-BN plasmon-phonon coupling and plasmon delocalization observed by infrared nano-spectroscopy,” Nanoscale 7(27), 11620–11625 (2015).
[Crossref] [PubMed]
A. Woessner, M. B. Lundeberg, Y. Gao, A. Principi, P. Alonso-González, M. Carrega, K. Watanabe, T. Taniguchi, G. Vignale, M. Polini, J. Hone, R. Hillenbrand, and F. H. L. Koppens, “Highly confined low-loss plasmons in Graphene-Boron Nitride heterostructures,” Nat. Mater. 14(4), 421–425 (2014).
[Crossref] [PubMed]
V. W. Brar, M. S. Jang, M. Sherrott, S. Kim, J. J. Lopez, L. B. Kim, M. Choi, and H. Atwater, “Hybrid surface-phonon-plasmon polariton modes in Graphene/monolayer h-BN heterostructures,” Nano Lett. 14(7), 3876–3880 (2014).
[Crossref] [PubMed]
Y. Wu, L. Jiang, H. Xu, X. Dai, Y. Xiang, and D. Fan, “Hybrid nonlinear surface-phonon-plasmon-polaritons at the interface of nolinear medium and Graphene-covered hexagonal Boron Nitride crystal,” Opt. Express 24(3), 2109–2124 (2016).
[Crossref] [PubMed]
J. D. Caldwell, L. Lindsay, V. Giannini, I. Vurgaftman, T. L. Reinecke, S. A. Maier, and O. J. Glembocki, “Low-loss, infrared and terahertz surface phonon polaritons,” Nanophotonics 4, 44–68 (2015).
[Crossref]
H. K. Choi, Long-Wavelength Infrared Semiconductor Lasers (Academic, 2004).
U. N. Roy, R. H. Hawrami, Y. Cui, S. Morgan, A. Burger, K. C. Mandal, C. C. Noblitt, S. A. Speakman, K. Rademaker, and S. A. Payne, “Tb3+-doped KPb2Br5: low-energy phonon mid-infrared laser crystal,” Appl. Phys. Lett. 86(15), 151911 (2005).
[Crossref]
S. A. Holmstrom, T. H. Stievater, M. W. Pruessner, D. Park, W. S. Rabinovich, J. B. Khurgin, C. J. K. Richardson, S. Kanakaraju, L. C. Calhoun, and R. Ghodssi, “Guided-mode phonon-polaritons in suspended waveguides,” Phys. Rev. B 86(16), 165120 (2012).
[Crossref]
V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004).
[Crossref] [PubMed]
X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011).
[Crossref] [PubMed]

2016 (3)

F. Bigourdan, J. P. Hugonin, F. Marquier, C. Sauvan, and J. J. Greffet, “Nanoantenna for electrical generation of surface plasmon polaritons,” Phys. Rev. Lett. 116(10), 106803 (2016).
[Crossref] [PubMed]

T. J. Echtermeyer, S. Milana, U. Sassi, A. Eiden, M. Wu, E. Lidorikis, and A. C. Ferrari, “Surface plasmon polariton Graphene photodetectors,” Nano Lett. 16(1), 8–20 (2016).
[Crossref] [PubMed]

Y. Wu, L. Jiang, H. Xu, X. Dai, Y. Xiang, and D. Fan, “Hybrid nonlinear surface-phonon-plasmon-polaritons at the interface of nolinear medium and Graphene-covered hexagonal Boron Nitride crystal,” Opt. Express 24(3), 2109–2124 (2016).
[Crossref] [PubMed]

2015 (9)

J. D. Caldwell, L. Lindsay, V. Giannini, I. Vurgaftman, T. L. Reinecke, S. A. Maier, and O. J. Glembocki, “Low-loss, infrared and terahertz surface phonon polaritons,” Nanophotonics 4, 44–68 (2015).
[Crossref]

E. Yoxall, M. Schnell, A. Y. Nikitin, O. Txoperena, A. Woessner, M. B. Lundeberg, F. Casanova, L. E. Hueso, F. H. L. Koppens, and R. Hillenbrand, “Direct observation of ultraslow hyperbolic polariton propagation with negative phase velocity,” Nat. Photonics 9(10), 674–678 (2015).
[Crossref]

P. Li, M. Lewin, A. V. Kretinin, J. D. Caldwell, K. S. Novoselov, T. Taniguchi, K. Watanabe, F. Gaussmann, and T. Taubner, “Hyperbolic phonon-polaritons in Boron Nitride for near-field optical imaging and focusing,” Nat. Commun. 6, 7507 (2015).
[Crossref] [PubMed]

S. Dai, Q. Ma, M. K. Liu, T. Andersen, Z. Fei, M. D. Goldflam, M. Wagner, K. Watanabe, T. Taniguchi, M. Thiemens, F. Keilmann, G. C. A. A. Janssen, S.-E. Zhu, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Graphene on hexagonal Boron Nitride as a tunable hyperbolic metamaterial,” Nat. Nanotechnol. 10(8), 682–686 (2015).
[Crossref] [PubMed]

Y. Jia, H. Zhao, Q. Guo, X. Wang, H. Wang, and F. Xia, “Tunable plasmon-phonon polaritons in layered Graphene-hexagonal Boron Nitride heterostructures,” ACS Photonics 2(7), 907–912 (2015).
[Crossref]

I. D. Barcelos, A. R. Cadore, L. C. Campos, A. Malachias, K. Watanabe, T. Taniguchi, F. C. B. Maia, R. Freitas, and C. Deneke, “Graphene/h-BN plasmon-phonon coupling and plasmon delocalization observed by infrared nano-spectroscopy,” Nanoscale 7(27), 11620–11625 (2015).
[Crossref] [PubMed]

Y. Fang and M. Sun, “Nanoplasmonic waveguides: towards applications in integrated nanophotonics circuits,” Light Sci. Appl. 4(6), e294 (2015).
[Crossref]

K. Feng, W. Streyer, S. M. Islam, J. Verma, D. Jena, D. Wasserman, and A. J. Hoffman, “Localized surface phonon polariton resonances in polar Gallium Nitride,” Appl. Phys. Lett. 107(8), 081108 (2015).
[Crossref]

Z. Shi, H. A. Bechtel, S. Berweger, Y. Sun, B. Zeng, C. Jin, H. Chang, M. C. Martin, M. B. Raschke, and F. Wang, “Amplitude- and phase- resolved nanospectral imaging of phonon polaritons in hexagonal Boron Nitride,” ACS Photonics 2(7), 790–796 (2015).
[Crossref]

2014 (6)

Y. Bian and Q. Gong, “Tuning the hybridization of plasmonic and coupled dielectric nanowire modes for high-performance optical waveguiding at sub-diffraction-limited scale,” Sci. Rep. 4, 6617 (2014).
[Crossref] [PubMed]

A. Woessner, M. B. Lundeberg, Y. Gao, A. Principi, P. Alonso-González, M. Carrega, K. Watanabe, T. Taniguchi, G. Vignale, M. Polini, J. Hone, R. Hillenbrand, and F. H. L. Koppens, “Highly confined low-loss plasmons in Graphene-Boron Nitride heterostructures,” Nat. Mater. 14(4), 421–425 (2014).
[Crossref] [PubMed]

V. W. Brar, M. S. Jang, M. Sherrott, S. Kim, J. J. Lopez, L. B. Kim, M. Choi, and H. Atwater, “Hybrid surface-phonon-plasmon polariton modes in Graphene/monolayer h-BN heterostructures,” Nano Lett. 14(7), 3876–3880 (2014).
[Crossref] [PubMed]

J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
[Crossref] [PubMed]

X. G. Xu, B. G. Ghamsari, J. H. Jiang, L. Gilburd, G. O. Andreev, C. Zhi, Y. Bando, D. Golberg, P. Berini, and G. C. Walker, “One-dimensional surface phonon polaritons in Boron Nitride nanotubes,” Nat. Commun. 5, 4782 (2014).
[Crossref] [PubMed]

S. Dai, Z. Fei, Q. Ma, A. S. Rodin, M. Wagner, A. S. McLeod, M. K. Liu, W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M. Thiemens, G. Dominguez, A. H. Castro Neto, A. Zettl, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Tunable phonon polaritons in atomically thin van der Waals crystals of boron nitride,” Science 343(6175), 1125–1129 (2014).
[Crossref] [PubMed]

2013 (1)

J. D. Caldwell, O. J. Glembocki, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme sub-diffraction photon confinement via Silicon Carbide surface phonon polariton nanopillar resonators,” Nano Lett. 13, 3690–3697 (2013).
[Crossref] [PubMed]

2012 (3)

L. Chen, X. Li, G. Wang, W. Li, S. Chen, L. Xiao, and D. Gao, “A Silicon-based 3-D hybrid long-range plasmonic waveguide for nanophotonic integration,” J. of Light. Tech. 30(1), 163–168 (2012).
[Crossref]

L. Chen, T. Zhang, X. Li, and W. Huang, “Novel hybrid plasmonic waveguide consisting of two identical dielectric nanowires symmetrically placed on each side of a thin metal film,” Opt. Express 20(18), 20535–20544 (2012).
[Crossref] [PubMed]

S. A. Holmstrom, T. H. Stievater, M. W. Pruessner, D. Park, W. S. Rabinovich, J. B. Khurgin, C. J. K. Richardson, S. Kanakaraju, L. C. Calhoun, and R. Ghodssi, “Guided-mode phonon-polaritons in suspended waveguides,” Phys. Rev. B 86(16), 165120 (2012).
[Crossref]

2011 (2)

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011).
[Crossref] [PubMed]

K. M. Pitman, A. M. Hofmeister, A. B. Corman, and A. K. Speck, “Optical properties and applications of Silicon Carbide for astrophysics,” Astron. Astrophys. 483, 1–12 (2011).

2009 (1)

J. Chen, Z. Li, S. Yue, and Q. Gong, “Hybrid long-range surface plasmon-polariton modes with tight field confinement guided by asymmetrical waveguides,” Opt. Express 17(26), 23603–23609 (2009).
[Crossref] [PubMed]

2008 (1)

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

2007 (3)

R. Kirchain and L. Kimerling, “A roadmap for nanophotonics,” Nat. Photonics 1(6), 303–305 (2007).
[Crossref]

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz polaritonics,” Annu. Rev. Mater. Res. 37(1), 317–350 (2007).
[Crossref]

Y. Cai, L. M. Zhang, Q. Zeng, L. Cheng, and Y. Xu, “Infrared reflectance spectrum of BN calculated from first principles,” Solid State Commun. 141(5), 262–266 (2007).
[Crossref]

2005 (1)

U. N. Roy, R. H. Hawrami, Y. Cui, S. Morgan, A. Burger, K. C. Mandal, C. C. Noblitt, S. A. Speakman, K. Rademaker, and S. A. Payne, “Tb3+-doped KPb2Br5: low-energy phonon mid-infrared laser crystal,” Appl. Phys. Lett. 86(15), 151911 (2005).
[Crossref]

2004 (1)

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004).
[Crossref] [PubMed]

2002 (1)

R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light matter interaction at the nanometre scale,” Nature 418(6894), 159–162 (2002).
[Crossref] [PubMed]

1999 (1)

T. E. Tiwald, J. A. Woolam, S. Zollner, J. Christiansen, R. B. Gregory, T. Wetteroth, S. R. Wilson, and A. R. Powell, “Carrier concentration and lattice absorption in bulk and epitaxial Silicon Carbide determined using infrared ellipsometry,” Phys. Rev. B 60(16), 11464–11474 (1999).
[Crossref]

1996 (1)

W. J. Moore and R. T. Holm, “Infrared dielectric constant of GaAs,” J. Appl. Phys. 80(12), 6939–6942 (1996).
[Crossref]

1990 (1)

M. Haraguchi, M. Fukui, and S. Muto, “Experimental observation of attenuated-total-reflection spectra of GaAs/AlAs superlattice,” Phys. Rev. B Condens. Matter 41(2), 1254–1257 (1990).
[Crossref] [PubMed]

1966 (1)

R. Geick, C. H. Perry, and G. Rupprecht, “Normal modes in hexagonal Boron Nitride,” Phys. Rev. B 146, 543 (1966).

Almeida, V. R.

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004).
[Crossref] [PubMed]

Alonso-González, P.

Andersen, T.

Andreev, G. O.

Atwater, H.

Bando, Y.

Barcelos, I. D.

Barrios, C. A.

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004).
[Crossref] [PubMed]

Basov, D. N.

Bassim, N. D.

Bechtel, H. A.

Berini, P.

Berweger, S.

Bezares, F. J.

Bian, Y.

Bigourdan, F.

Brar, V. W.

Burger, A.

Cadore, A. R.

Cai, Y.

Y. Cai, L. M. Zhang, Q. Zeng, L. Cheng, and Y. Xu, “Infrared reflectance spectrum of BN calculated from first principles,” Solid State Commun. 141(5), 262–266 (2007).
[Crossref]

Caldwell, J. D.

Calhoun, L. C.

Campos, L. C.

Carrega, M.

Casanova, F.

Castro Neto, A. H.

Chang, H.

Chen, J.

Chen, L.

Chen, S.

Chen, Y.

Cheng, L.

Y. Cai, L. M. Zhang, Q. Zeng, L. Cheng, and Y. Xu, “Infrared reflectance spectrum of BN calculated from first principles,” Solid State Commun. 141(5), 262–266 (2007).
[Crossref]

Choi, M.

Christiansen, J.

Corman, A. B.

K. M. Pitman, A. M. Hofmeister, A. B. Corman, and A. K. Speck, “Optical properties and applications of Silicon Carbide for astrophysics,” Astron. Astrophys. 483, 1–12 (2011).

Cui, Y.

Dai, S.

Dai, X.

Deneke, C.

Dominguez, G.

Echtermeyer, T. J.

T. J. Echtermeyer, S. Milana, U. Sassi, A. Eiden, M. Wu, E. Lidorikis, and A. C. Ferrari, “Surface plasmon polariton Graphene photodetectors,” Nano Lett. 16(1), 8–20 (2016).
[Crossref] [PubMed]

Eiden, A.

T. J. Echtermeyer, S. Milana, U. Sassi, A. Eiden, M. Wu, E. Lidorikis, and A. C. Ferrari, “Surface plasmon polariton Graphene photodetectors,” Nano Lett. 16(1), 8–20 (2016).
[Crossref] [PubMed]

Ellis, C. T.

Fan, D.

Fang, Y.

Y. Fang and M. Sun, “Nanoplasmonic waveguides: towards applications in integrated nanophotonics circuits,” Light Sci. Appl. 4(6), e294 (2015).
[Crossref]

Fei, Z.

Feng, K.

Ferrari, A. C.

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ACS Photonics (2)

Annu. Rev. Mater. Res. (1)

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz polaritonics,” Annu. Rev. Mater. Res. 37(1), 317–350 (2007).
[Crossref]

Appl. Phys. Lett. (2)

Astron. Astrophys. (1)

K. M. Pitman, A. M. Hofmeister, A. B. Corman, and A. K. Speck, “Optical properties and applications of Silicon Carbide for astrophysics,” Astron. Astrophys. 483, 1–12 (2011).

J. Appl. Phys. (1)

W. J. Moore and R. T. Holm, “Infrared dielectric constant of GaAs,” J. Appl. Phys. 80(12), 6939–6942 (1996).
[Crossref]

J. of Light. Tech. (1)

Light Sci. Appl. (1)

Y. Fang and M. Sun, “Nanoplasmonic waveguides: towards applications in integrated nanophotonics circuits,” Light Sci. Appl. 4(6), e294 (2015).
[Crossref]

Nano Lett. (4)

T. J. Echtermeyer, S. Milana, U. Sassi, A. Eiden, M. Wu, E. Lidorikis, and A. C. Ferrari, “Surface plasmon polariton Graphene photodetectors,” Nano Lett. 16(1), 8–20 (2016).
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X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011).
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Nanophotonics (1)

Nanoscale (1)

Nat. Commun. (3)

Nat. Mater. (1)

Nat. Nanotechnol. (1)

Nat. Photonics (3)

R. Kirchain and L. Kimerling, “A roadmap for nanophotonics,” Nat. Photonics 1(6), 303–305 (2007).
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Nature (1)

R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light matter interaction at the nanometre scale,” Nature 418(6894), 159–162 (2002).
[Crossref] [PubMed]

Opt. Express (3)

Opt. Lett. (1)

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004).
[Crossref] [PubMed]

Phys. Rev. B (3)

R. Geick, C. H. Perry, and G. Rupprecht, “Normal modes in hexagonal Boron Nitride,” Phys. Rev. B 146, 543 (1966).

Phys. Rev. B Condens. Matter (1)

Phys. Rev. Lett. (1)

Sci. Rep. (1)

Science (1)

Solid State Commun. (1)

Y. Cai, L. M. Zhang, Q. Zeng, L. Cheng, and Y. Xu, “Infrared reflectance spectrum of BN calculated from first principles,” Solid State Commun. 141(5), 262–266 (2007).
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Other (2)

H. K. Choi, Long-Wavelength Infrared Semiconductor Lasers (Academic, 2004).

S. Adachi, The reststrahlen region.Optical Properties of Crystalline and Amorphous Semiconductors: Materials and Fundamental Principles (Science + Business Media. 1999).
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Fig. 1 (a) Real and imaginary parts of the dielectric function of flat h-BN near the phonon resonance. (b) Dispersion of the SPhP mode of the flat h-BN-air boundary.

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Fig. 2 The hybrid mid-IR waveguide which includes a dielectric cylinder of diameter d placed at a gap height h above an h-BN slab on a dielectric substrate.

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Fig. 3 (a) Modal area as a function of cylinder diameter d for different gap height h. (b) Hybrid propagation distance as a function of cylinder diameter d for different gap height h.

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Fig. 4 Electromagnetic energy density distribution for (a) [d, h] = [2.5, 1] μm, (b) [d, h] = [1.4, 0.25] μm, (c) [d, h] = [1.4, 0.1] μm and (d) [d, h] = [1.4, 0.01] μm.

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Fig. 5 Effective index of the hybrid waveguide for different gap height h versus cylinder diameter d. The dashed line indicates the index of the SPhP mode

n_{S P h P}

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Fig. 6 The dependence of coupling strength κ on cylinder diameter d and gap height h.

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Fig. 7 Normalized energy density along z axis at wavenumber (a) 1400 cm⁻¹ (b) 1426 cm⁻¹ and (c) 1483 cm⁻¹. (d) Normalized energy density at z = -d/2 for different wavenumbers.

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Equations (8)

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

L_{m} = {[2 I m (k_{h y b} (d, h))]}^{- 1}

A_{m} = \frac{W_{m}}{\max {W (\vec{r})}}

W (\vec{r}) = \frac{1}{2} R e {\frac{d (ω ε (\vec{r}))}{d ω}} {| E (\vec{r}) |}^{2} + \frac{1}{2} μ_{0} {| H (\vec{r}) |}^{2}

ψ_{h y b} (d, h) = a (d, h) ψ_{c y l} (d) + b (d, h) ψ_{S P h P}

n_{c y l} (d) a (d, h) + κ (d, h) b (d, h) = n_{h y b} (d, h) a (d, h)

κ (d, h) a (d, h) + n_{S P h P} b (d, h) = n_{h y b} (d, h) b (d, h)

κ (d, h) = \sqrt{(n_{h y b} (d, h) - n_{S P h P}) (n_{h y b} (d, h) - n_{c y l} (d))}

θ = \frac{π}{2} - arc \tan (\frac{\sqrt{ε_{z} (ω)}}{i \sqrt{ε_{x y} (ω)}})