1. Introduction

Hyperbolic optical metamaterials, highly anisotropic nanostructured composites with opposite signs of the dielectric permittivity in two orthogonal directions, represent one of the most active areas of current research [1–19]. These systems show a wide range of new physical phenomena – from extreme light localization in subwavelength waveguides [2] and resonant cavities [3], to enhanced radiative energy transport [4, 5] and quantum-electrodynamics phenomena [6, 7], and offer many intriguing applications – from far-field imaging with the resolution beyond the diffraction limit [8–10] to new stealth technology [11].

Generally, this behavior relies on two important aspects of the hyperbolic media – the ability to support propagating waves with the wavenumbers unlimited by the frequency [2] (leading to a broadband “super-singularity” [12] in the phonic density of states), and all-angle negative refraction [2, 13–16]. In particular, high-wavenumber modes in hyperbolic media allow for light confinement beyond the diffraction limit and enhanced light-matter interactions in “indefinite” resonant cavities – three-dimensional subwavelength resonators formed by a hyperbolic meta-material (HMM) [3]. However, these are inherently high-order modes with exponentially small radiative coupling in a high-loss environment, making practical applications of such devices a challenging problem.

There is however another aspect of the hyperbolic media which so far received relatively little attention, but which could be just as important: when a given interface of the hyperbolic medium shows positive refraction, the transmitted wave accumulates negative phase, leading to a zeroth-order Fabry-Perot resonance at subwavelength thickness – similarly to resonators based on simultaneously negative permittivity and permeability [20]. As opposed to high-order hyperbolic modes studied earlier [3], such a resonator is refractively coupled to the surrounding medium, and therefore allows an arbitrary degree of radiative coupling (by e.g. forming thin metallic “mirrors” of the required thickness on its surfaces).

In the present work, we introduce the concept and present the theoretical description of the hyperbolic 0th-order resonator. Our approach can combine arbitrary subwavelength resonator dimensions with strong radiative coupling, leading to possible applications in nanolasers and deep subwavelength waveguides.

2. Zeroth-order trasmission resonance in a Fabry-Perot hyperbolic resonator

Consider a plane electromagnetic wave incident on a planar interface between an isotropic dieletric medium (e.g. air) and a hyperbolic medium with negative in-plane dielectric permittivity component ε_x < 0 and positive normal permittivity component ε_y > 0 (Fig. 1(a)). For a TM (p-polarized) incident plane wave (represented by the magnetic field B⃗ = B₀ẑe^{i(k_xx+k_yy−ωt)} with the wavevector k = k_xx̂ + k_yŷ and the angular frequency ω), the associated electric field E and the Poynting vector S can be expressed as

Figure 1(b) shows the directions of the phase velocity (corresponding to the wave vector k⃗) and the energy flow (given by the Poynting vector S⃗) with respect to the iso-frequency curve of the medium. As refraction at a planar surface preserves the tangential (in-plane) momentum k_x and the direction of the energy flux of the transmitted wave S⃗_t is away from the boundary, for ε_x < 0 and ε_y > 0 the refraction is formally positive[2, 13, 14, 21]. However, in this case the wave vector k⃗_t lies in the opposite direction, and a wave propagating in this hyperbolic medium (ε_x < 0, ε_y > 0) will accumulate a negative phase – with dramatic consequences for interference and transmission resonances in this system.

 

Fig. 1 (a) The phase velocity and Poynting vector in regular dielectric (y < 0) and in hyperbolic medium (y > 0) with negative in-plane permittivity component (ε_x < 0). The wave vector k⃗₀ and Poynting vector S⃗₀ in an isotropic dielectric medium are collinear, while in HMM k⃗_t and S⃗_t are in two different directions. (b) The iso-frequency curves for an isotropic dieletric medium (circle) and an HMM (open hyperbolic curve) with ε_x < 0, ε_y > 0. The corresponding wave vectors (k⃗₀, k⃗_t) and Poynting vectors (S⃗₀, S⃗_t) are also shown. Note that refraction from the dielectric into such hyperbolic medium is only possible if the radius of the circle (proportional to the corresponding dielectric permittivity) exceeds the smallest wavenumber supported by the hyperbolic medium.

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In a planar Fabry-Perot resonator formed by a “core” (of thickness D) bounded by two thin conducting layers (barriers) with negative permittivity ε_m (Fig. 2(a)), the transmission peak positions can be approximated by

 

Fig. 2 (a) The schematics of a Fabry-Perot resonator with a hyperbolic metamaterial core. Conducting layers (ε_m < 0) serve as reflective “mirrors” and the wave is incident from the dielectric at the angle θ. (b) Transmission coefficients for a lossless HMM core, with different conducting layer thicknesses: d = 200 nm (red curve), 400 nm (blue curve), 600 nm (green curve) and 800 nm (violet curve). The dielectric permittivities of the HMM core are ε_x = −2 and ε_y = 1, and the dielectric permittivity in the conducting barriers is ε_m = −10. The free-space incident wavelength is λ₀ = 10 μm and incident angle θ = 60°. The inset shows the resonance position as a function of conducting layer thickness, with the dashed blue line corresponding to Eq. (4). (c) Transmission coefficients for the resonator with a lossy HMM core, with ε_x = −2 + 0.002i, ε_y = 1 + 0.001i (solid green curve), ε_x = −2 + 0.02, ε_y = 1 + 0.01i (dotted blue curve), and ε_x = −2 + 0.2i, ε_y = 1 + 0.1i (dashed red curve), while the conducting layer thickness d in this example is 600 nm. The inset shows the variation of the peak transmission with the loss in the metal barries Im[ε_m], for the core permittivity ε_x = −2 + 0.2i, ε_y = 1 + 0.1i.

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However, if the resonator core is formed by a hyperbolic medium with negative in-plane permittivity, we find that k_n < 0, and the resonance condition Eq. (3) allows a zeroth-order (m = 0) resonance. When the metal barrier thickness d is on the order of or larger than the corresponding penetration depth, for this resonance we obtain

3. Zeroth-order resonance in a multilayer HMM core

In contrast to 3D optical metamaterials where unit cell size is on the order of the wavelength in the component materials [22–25], planar metamaterials take advantage of fabrication technologies such as molecular-beam epitaxy [26] and chemical-vapor deposition [27] leading to the unit size on the scale of less than one-tenth of the wavelength [16]. As a result, such planar hyperbolic metamaterials (especially in the mid-IR range [16]) generally closely approach the predictions of the effective medium theory (EMT) [28] based of the effective dielectric permittivity tensor in terms of its conductive (permittivity ε_c) and dielectric (permittivity ε_d) components. In particlar, the tangential (in-plane) permittivty (ε_n) and normal permittivty (ε_τ) for a planar HMM with 1:1 conducting and dielectric layer thickness ratio can be expressed by

However, even in this limit, the fact that the total thickness of the resonator is also subwavelength and may thus only contain ∼ 10 individual layers, can still put the predictions of the EMT in question. To address this issue, we also calculate the transmission coefficient of the system using the T -matrix method, taking into account the actual geometry of the metamaterial using layer cells – see Figs. 3(b) and 3(c). Our results clearly confirm the existence of 0th-order transmission resonance even for a deeply subwavelength resonator core dimension (D ≈ 0.02 λ₀) and a relatively large metal barrier thickness d = 50 nm. Furthermore, the agreement with our theoretical description remains not only qualitative but quantitative.

 

Fig. 3 (a) The schematic structure of a 1:1 alternating Au-Al₂O₃ multilayer HMM resonator core with the single layer thickness d₀ ≪ λ₀. (b) Transmission coefficients using the HMM core in (a) with different metal barrier thicknesses: d = 30 nm (solid red curve), 35 nm (blue dashed curve), 40 nm nm (green dashed curve) and 45 nm nm (violet dashed curve). The free-space incident wavelength is λ₀ = 4 μm and the incident angle θ = 60°. (c) Comparison of the calculated transmission coefficients using the EMT and T -matrix methods. The EMT result is shown by solid red curve and the T -matrix results are shown by dashed curves with different single layer thicknesses of HMM core: d₀ = 2 nm (dashed blue), d₀ = 5 nm (dashed green), d₀ = 8 nm (dashed black), and d₀ = 10 nm (dashed violet). In this example, the metal barrier thickness d is set as 30 nm.

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Note that the zeroth-order resonance described in the current work can only be observed in the HMM core with a negative in-plane permittivity ε_n. In the opposite case of ε_n > 0 and ε_τ < 0, the system does not support such zeroth-order resonance.

4. Conclusions

In conclusion, we have presented a new approach to subwavelength light confinement – based on a novel zeroth-order resonance due to negative phase velocity in hyperbolic metamaterials. Our approach is robust to losses, is compatible with planar semiconductor metamaterial technology, and when necessary allows strong radiative coupling.