1. Introduction

Plasmonics and semiconductor photonic technologies have already revolutionized communications [1,2] and have potential capabilities in spectroscopy, chemical and biological sensing as well as subwavelength laser devices [3,4]. Manipulating light on scales much smaller than the wavelength and achieving high-density integration of optical devices remain critical challenges in micro- and nanotechnology. Surface plasmon polaritons (SPPs) are regarded as promising candidates for guiding and confining light at the subwavelength scale at optical and near-infrared (IR) wavelengths [5]. In order to minimize the high optical loss of SPP waveguides, a hybrid plasmonic waveguide, which consists of a dielectric nanowire separated from a metal surface by a nanoscale dielectric gap, has been investigated [6,7]. For further improvement of the mode confinement and propagation distance, long-range hybrid surface plasmon polariton waveguide structures have been presented [8–11]. However, due to the intrinsic Ohmic loss of noble metals, the advantages of SPPs are prohibitive at mid-infrared (mid-IR) wavelengths. An infrared counterpart to the SPP is the surface phonon polariton, which results from the coupling of lattice vibrations (or phonons) of a polar dielectric crystal to incident mid-IR radiation [12]. The mid-IR spectrum is rich with vibrational and rotational molecular resonances which, when coupled to electromagnetic fields, provide informative probes that reveal spectroscopic material characteristics [3]. Phonon polaritons could also provide potential applications for high-density IR data storage [13], infrared phononic nanoantennas [14] and IR imaging [15,16]. The need for low-loss, subwavelength confinement and propagating modes in mid-IR nanostructures has motivated research into phonon-polariton based waveguiding.

Hyperbolic phonon polaritons (HPhPs) [17] exhibit a hyperbolic dispersion whose permittivity tensor possesses both positive and negative principal components. Owing to their strong confinement and enhancement of electromagnetic fields beyond the diffraction limit scale, HPhPs have gained a lot of attention recently [18–21]. Of particular interest is hexagonal boron nitride (h-BN), a typical van der Waals (vdW) material that exhibits natural hyperbolic dispersion which compensates the shortcomings of artificial hyperbolic metamaterials, such as high plasmonic losses and complex nanofabrication [19,20]. Another fascinating property of h-BN is that single-atom thick flakes can be achieved through standard exfoliation techniques [21]. It has been experimentally demonstrated that h-BN nanotubes can support one-dimensional surface phonon polaritons with deep-subwavelength field confinement in the mid-IR [22]. Being a typical low loss and anisotropic material, h-BN exhibits two reststrahlen bands in the mid infrared spectrum: (1) at lower frequency (Type I reststrahlen band $\text{ω}=746-819 c{m}^{-1}$), the real part of out-of-plane permittivity is negative (${\epsilon }_{\Vert }<0$) and the in-plane permittivity is positive (${\epsilon }_{\perp }>0$); (2) at higher frequency (Type II reststrahlen band $\text{ω}=1370-1610 c{m}^{-1}$), the real part of out-of-plane permittivity is positive (${\epsilon }_{\Vert }>0$) and the in-plane permittivity is negative (${\epsilon }_{\perp }<0$) [23–27].

Recent research has studied insulator-hyperbolic boron nitride-insulator (IHI) structures in the Type II reststrahlen band [28]. The IHI structure has been shown to support long-range propagating polariton modes with propagation lengths higher than a Ag/MgF₂ hyperbolic metamaterial waveguide [28]. H-BN has been considered as an alternative to hyperbolic metamaterials due to its naturally occurring hyperbolic dispersion and low loss characteristics in the infrared spectral region [28]. In this paper, we propose a hybrid long-range hyperbolic phonon polariton (LRHPhP) waveguide consisting of two identical dielectric cylindrical wires placed symmetrically about a h-BN slab embedded in a low-permittivity dielectric medium in the Type II reststrahlen band. To the best of the authors’ knowledge, this paper shows for the first time the implementation of recently theoretically-reported [28] long-range HPhP modes in a mid-IR hybrid waveguide using a h-BN slab.

Three-dimensional (3-D) simulations based on finite-element-method (FEM) using Ansys HFSS eigenmode solver are performed to calculate the field distributions. To improve the accuracy of the eigenmode results, sufficiently large dimensions in both the horizontal and vertical directions of 9${\text{λ}}_0$ by 6${\text{λ}}_0$, respectively, are used. In addition, measured dispersive material properties of h-BN provided by the US Naval Research Laboratory are imported in the model [18]. In [29], Xu et al. claims to achieve a hybrid waveguide using h-BN resulting from the coupling between type II surface phonon-polaritons in h-BN and a single high index dielectric cylinder. With the use of volumetric LRHPhP in h-BN, the proposed design can achieve three orders of magnitude longer propagation length while maintaining a similar degree of modal confinement as compared with the work by Xu et al. in [29]. This approach allows for ultra-compact and sensitive mid-IR waveguides with applications in sensing, spectroscopy, and lab-on-chip systems [16]. The hybrid waveguide approach utilizes field enhancement [30], which results in the shrinking of the polariton wavelength, from the phonon polaritons to reduce modal area. This paper is organized as follows: the properties of short and long-range h-BN HPhPs are analyzed and the hybrid LRHPhP waveguide properties are characterized in terms of the guide geometry parameters in section 2. In section 3, the hybridization analysis through coupled-mode theory is discussed. The conclusions are presented in section 4.

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

Figure 1 illustrates the structures of the hybrid long-range phonon polariton waveguide. Two identical high refractive index dielectric cylinder wires are symmetrically placed on each side of a thin h-BN slab with a small gap distance h. The LRHPhP mode that exists in this structure is a transverse magnetic (TM) mode that propagates in the z direction. The cylindrical wires with diameter d are gallium arsenide (GaAs) which has a negligible material loss in the window around 6.6μm with refractive index n = 3.5 [6] and the h-BN slab has thickness t = 0.5µm with in-plane permittivity ${\epsilon }_{\perp }=-3.38+0.19i$ and out-of-plane permittivity ${\epsilon }_{\parallel }=2.77+0.0003i$ at mid-infrared wavelength 6.6μm. Using the HFSS eigenmode solver, a thin cross-sectional slice of the geometry under consideration is taken. Master and slave boundaries are placed on the cross-sectional faces to extend the waveguide to infinity. Resulting resonant modes with a given complex eigen frequency are determined for a given phase delay between the master and slave boundaries. The quality factor, Q, of the eigenmode is used to determine the propagation length. Similarly, the phase delay between the master and slave boundaries is used to determine the real part of the complex index. The h-BN based hybrid phononic waveguide embedded in uniform dielectric medium air proposed here can be achieved in suspended waveguide techniques analogous to those seen in [31]. For easily realizable fabrication, the air gap can also be substituted with a low-index and low-loss dielectric spacer similar to that reported in [37] by Oulton et al. The diameter of the dielectric nanotube d, the gap height h and h-BN film thickness t can be controlled with high accuracy [23,36,37] which indicates the fabrication tolerance requirements are feasible.

 

Fig. 1 Schematic illustration of the proposed hybrid long-range phonon polariton waveguide which includes two identical GaAs cylinder wires with diameter d symmetrically placed on each side of a thin h-BN slab are separated with a spacer height h. The h-BN slab has thickness of t and this waveguide is embedded in air at mid-infrared wavelength ${\text{λ}}_0=6.6\text{μm}$.

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Before conducting hybrid waveguide analysis, we characterized the hyperbolic phonon polaritons that exist in an h-BN thin slab. Both the hybrid long-range and hybrid short-range HPhP refractive indices and propagation lengths are shown in Fig. 2. The propagation length of the hybrid phononic mode as a function of the wire diameter d and the air spacer height h is given by ${\text{L}}_{\text{m}}={\left[2\text{Im}({\text{k}}_{\text{hyb}}\left(\text{d},\text{h}\right))\right]}^{-1}$, where ${\text{k}}_{\text{hyb}}(\text{d},\text{h})$ is the wavenumber of the hybrid mode [6]. The long- and short-range HPhPs are akin to the respective long- and short-range modes in thin metal slabs that support SPPs [32–35]. For this thin hyperbolic h-BN slab surrounded by uniform dielectric medium in the Type II reststrahlen band there exists a symmetric mode and an anti-symmetric mode. The symmetric mode is the so-called long-range hyperbolic phonon-polariton (LRHPhP) mode providing long propagation distance with weak mode confinement and the anti-symmetric mode represents the short-range hyperbolic phonon polariton (SRHPhP) mode which provides shorter propagation distance with stronger mode confinement [28]. The dielectric cylinder wire mode couples with both modes, compensating the weak modal confinement of the pure LRHPhP mode and reducing the pure SRHPhP mode’s modal area. Here the diameter of both cylinders are kept constant at d = 1µm. Figure 2(a) shows the effective index varying with the thickness of the h-BN slab. As the h-BN slab thickness t increases, tending toward bulk behavior, the short- and long-range phonon polariton modes start to converge. Different colors of the effective index curve represent different spacer heights h in each waveguide. The dashed curves with increasing effective index as t decreases represent the SRHPhPs and solid curves represent LRHPhPs in Fig. 2(a). Figure 2(b) shows the propagation lengths for this waveguide for h-BN slab thickness less than 3µm. The LRHPhP modes in each case of the waveguide consistently has several orders of magnitude longer propagation length than that of SRHPhP modes. Therefore, the symmetric mode is of interest in this study since it supports much longer propagation lengths.

 

Fig. 2 (a) The effective indices as a function of h-BN slab thickness t for hybrid and non-hybrid LRHPhP and SRHPhP waveguides. (b) The propagation distance for hybrid and non-hybrid LRHPhP and SRHPhP changing with t. The solid lines and broken lines represent long- and short-range HPhPs, respectively. The colored curves (red and green) represent hybrid waveguides for different spacer heights h. The black lines represent a h-BN thin slab embedded in low dielectric material exhibiting long and short range HPhPs.

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In the following study, we choose h-BN slab thickness t = 0.5µm and we vary the cylinder wire diameter d and the dielectric spacer height h between the cylinder and the h-BN slab to manipulate the propagation distance ${\text{L}}_{\text{m}}$ and modal area ${\text{A}}_{\text{m}}$ while maintaining the electromagnetic field distribution of a single hybrid mode at wavelength 6.6$\text{μm}.$ The modal area, ${\text{A}}_{\text{m}}$, is defined as the ratio of the total mode energy to the peak energy [6],

Figures 3(a) and 3(b) show the normalized mode area and propagation distance, respectively, as a function of d and for various values of h. For a large cylinder wire diameter d and spacer height h (d ˃ 1μm, h ≥ 0.25μm), the hybrid LRHPhP waveguide supports a low–loss cylinder-like mode in which the electromagnetic energy is confined into the two high-permittivity cylinder wires [Fig. 4(a)]. Conversely, a small cylinder diameter (d < 1µm) leads to a single LRHPhP-like mode with very weak localization on both sides of h-BN/air interface and sustaining loss comparable to that of uncoupled LRHPhPs. At a specific value of d = 1μm, both the uncoupled mode indexes are equal to each other. As a result, the mode area is at its smallest as can been seen in Fig. 3(a). In addition, at d = 1μm, the mode character possesses equal contribution of both dielectric-cylinder mode and LRHPhP mode characteristics [Fig. 4(b)]. When h = 0.25μm and d = 1μm, the propagation distance is 2.46mm $\approx$ 370${\text{λ}}_0$, while confining the energy within the spacers between the dielectric cylinders and h-BN slab to a modal area of approximately ${10}^{-1}{\text{λ}}_{\text{o}}^2$. If the spacer height h is reduced to nanometer scale, the electromagnetic field is strongly confined in the nanometer air gap between the cylinder and the h-BN interface [Figs. 4(c) and 4(d)]. The minimum modal area can be achieved at cylinder diameter d = 1μm for different spacer heights h but at the cost of the shortest propagation distance [Figs. 3(a) and 3(b)]. For the case of d = 1μm and h = 0.01μm, the field has the highest confinement in the nanometer gap region with a modal area of approximately ${10}^{-2}{\text{λ}}_0^2$ while still maintaining a relatively long propagation distance that is 7 times longer than the free-space wavelength. This explicitly shows the known trade-off between modal area and propagation distance.

 

Fig. 3 (a) Normalized modal area (${\text{A}}_{\text{m}}/{\text{A}}_0$) versus cylinder wire diameter d for different spacer height h (colored lines), compared with a pure cylinder mode (black line). (b) Hybrid propagation distance versus cylinder wire diameter d for different spacer height h (colored lines), compared with LRHPhP modes in Air-hBN-Air is denoted by the black dashed line.

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Fig. 4 The cross-sectional electromagnetic energy density distribution for (a) [d,h] = [2.5, 0.25] μm, (b) [d,h] = [1, 0.25] μm, (c) [d,h] = [1, 0.01] μm and (d) [d,h] = [1.5, 0.01] μm. The lower right-hand corner shows the magnetic field vectors for the hybrid LRHPhP waveguides.

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Normalized energy densities along x = 0 and y = $\left(\text{t}/2\right)+\text{h}$ at d = 1μm for different spacer heights, h, are plotted in Fig. 5. This confirms that the stored electromagnetic energy is confined in the low-permittivity dielectric medium (here air) between the cylinder wire and the h-BN slab since the continuity of the displacement field at the material interfaces gives a strong normal electric-field component in the gap [6]. As h increases the electromagnetic energy in the gap region decreases, since the hybrid waveguide behaves as an effective optical capacitance.

 

Fig. 5 Normalized energy density along x = 0 [dashed line in inset in (a)] for h = 0.01μm (a), 0.05µm (b), 0.1μm (c), 0.25µm (d) shows the confinement in the air spacer. The shaded grey and blue areas represent the two dielectric wires and h-BN slab regions, respectively. The energy density along y = ($\text{t}/2$) + h [dashed line inset in (e)] for h = 0.01μm (e), 0.05µm (f), 0.1μm (g), 0.25μm (h).

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3. Coupling characteristics between long-range phonon polaritons and dielectric waveguide

For the purpose of gaining a deeper understanding, we analyzed the dependence of the hybrid LRHPhP mode’s effective index, ${\text{n}}_{\text{hyb}}$ (d,h) on cylinder wire diameter and air spacer height. Figure 6 shows the variation in d of hybrid mode effective index for different air spacer heights h. In the limit of large d, the effective index approaches that of the pure dielectric cylinder wire mode, ${\text{n}}_{\text{wire}}(\text{d})$, whereas for small d it converges to the pure LRHPhP mode, ${\text{n}}_{\text{LRHPhP}}$, in agreement with the electromagnetic energy density analysis in section 2. At the same time, the hybrid LRHPhP mode’s effective index is always larger than the index of both the pure dielectric cylinder wire mode and the pure LRHPhP mode, indicating that the dielectric cylinder wire waveguide mode is coupling with LRHPhP mode. The mode’s effective index can be increased by increasing the diameter of the cylinder wire, d, for a fixed spacer height, h, or by reducing the spacer height for a fixed d. This is because, as d increases or h reduces, the dielectric cylinder wire mode more effectively couples with the LRHPhP mode.

 

Fig. 6 Effective index of the hybrid LRHPhP waveguide for a range of spacer height h versus cylinder wire diameters d, ${\text{n}}_{\text{hyb}}$(colored lines). For comparison, the effective indices of pure cylinder wire, ${\text{n}}_{\text{wire}}$ (black solid line), and pure LRHPhP, ${\text{n}}_{\text{LRHPhP}}$ (black dashed line) are plotted.

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In order to explain the hybrid LRHPhP mode characteristics, coupled-mode theory is applied in which the hybrid mode is described as a superposition of the cylinder wire waveguide mode (without h-BN slab) and the LRHPhP mode (without cylinder wires) [6]. The hybrid mode can be expressed as

 

Fig. 7 (a) The cylinder mode character ${\left|{\text{a}}_{+}(\text{d},\text{h})\right|}^2$ depends on wire diameter d from Eq. (4) for different spacer height h. (b) The dependence of coupling strength κ on cylinder wire diameter d and spacer height h from Eq. (5).

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According to the coupled-mode theory, for the hybrid LRHPhP waveguide, the coupling strength κ(d, h) between the dielectric cylinder mode and LRHPhP mode can be computed as [10]:

The dependence of coupling strength κ on cylinder wire diameter d and spacer height h are plotted in Fig. 7(b). It is shown that the coupling strength increases with decreasing spacer height h, which conveys the fact that the two modes couple more effectively as h reduces. Also the maximum coupling strength occurs at $\text{d}\approx 1.25\text{μm}$ which is slightly shifted from ${\text{d}}_c\approx 1\text{μm}$. From Eq. (5), if ${\text{n}}_{\text{hyb}}\left(\text{d},\text{h}\right)$ did not change with the cylinder diameter d for a fixed spacer height h, the dielectric cylinder mode and LRHPhP mode satisfy the phase-matched condition and ${\text{n}}_{\text{LRHPhP}}= {\text{n}}_{\text{wire}}$ gives the maximum values of the coupling strength at $\text{d}\approx 1\text{μm}$. In fact, both ${\text{n}}_{\text{hyb}}$ and ${\text{n}}_{\text{wire}}$ are changing with d which causes the shifting of the coupling strength. The coupling strength between the dielectric cylinder mode and the LRHPhP mode determines the optical energy concentration in the gap region. The field enhancement in the gap regions allows for efficient trapping of nanoscale particles.

4. Conclusion

We have proposed a hybrid long-range phononic waveguide consisting of two identical dielectric cylinder wires symmetrically placed on each side of a h-BN slab at mid-IR wavelength: 6.6μm. The hybrid short-range and long-range h-BN based hyperbolic phonon polariton modes are shown to exist, however the hybrid LRHPhP mode provides propagation lengths that are significantly longer when compared with the hybrid SRHPhP mode while achieving comparable modal areas. The strong coupling between the LRHPhP mode and the dielectric cylinder wire waveguide mode results in a hybrid mode that can propagate distances up to 370λ₀ and maintain sub-wavelength modal area within the spacers between the dielectric cylinder wires and h-BN slab of approximately 10⁻¹λ₀². The relationship between the parameters d and h for the physical geometry of the hybrid structure and the propagation length and mode area are characterized. A hybrid mode with a mode character at 50%, where the two uncoupled modes are index matched, results in a hybrid mode with the most confined hybrid mode, however, the hybrid mode suffers from a reduction in the propagation length. Hybrid modes with mode character equal to 50% are useful for applications where physical device footprint is the fundamental limiting requirement or when high field enhancement is needed for applications like sensing or particle trapping in nano-capacitive spacer region between the high index dielectric waveguide and h-BN. Given the numerous examples of molecular fingerprints in the mid- to long-wave IR spectrum, hybrid waveguide designs for enhancing and sensing such molecules have great potential for impact in lab-on-chip technologies. The region where the hybrid waveguide’s mode character is above 50% is used to achieve sub-diffractional long-haul energy transport for integrated polaritonic devices like interferometers in the mid-IR spectrum wherein mode area is traded off for longer propagation length stemming from the high index waveguide attributes.