1. Introduction

Hyperbolic metamaterials are artificial nanostructures that support radiative modes with extremely large in-plane (lateral) wavevectors [1–10]. Their anisotropic properties create a hyperbolic dispersion that deviates from the usual elliptical dispersion of uniaxial medium. This unique property leads to potential applications in optical imaging systems, waveguides, near-field filters, and thermal radiation sources, which can be engineered by modifying the structural dimensions [5–12]. The energy exchange in the near field between two semi-infinite anisotropic media separated by a vacuum gap has also been investigated [13–19]. Both surface modes and hyperbolic modes can enhance radiative transfer through tunneling of photons with large lateral wavevectors [18–22]. Significantly enhanced near-field radiative transfer at nanometer distances has been demonstrated by recent experiments conducted worldwide as reviewed in [22,23]. The potentially large penetration depth and lateral shift of photons associated with these structures can lead to both lateral and vertical characteristic dimensions being quite large. While the lateral shift of energy streamlines and the penetration depth have been investigated for near-field radiative transfer between isotropic materials [24–29], the characteristic lateral dimensions at which three-dimensional effects become important for hyperbolic modes in these structures are unknown.

The present study focuses on the pertinent characteristic dimensions of such structures with a nanometer vacuum spacing between two metallodielectric photonic crystals, by analyzing the energy streamlines using the method of Green’s functions and the fluctuation-dissipation theorem. The energy streamlines trace the direction of the Poynting vector inside the structure to depict the propagation of evanescent waves that cannot be obtained simply by ray optics (i.e., geometric optics). The characteristic dimensions are then determined based on the overall lateral displacement of the electromagnetic energy within a few penetration depths.

2. Near-field radiative transfer in uniaxial media

Typically, uniaxial metamaterials are constructed from alternating metallic and dielectric layers or from metallic nanorods arrays surrounded by a dielectric matrix [5–11]. The term metallic here refers to a material whose dielectric function has a negative real part in a certain frequency region. For a multilayered structure, if the period is small enough that the electric displacement does not vary significantly over several layers (e.g., the period is sub-wavelength), the structures can be treated as an effective homogeneous medium with anisotropic properties [2, 3, 6, 10, 30]. This criterion may not be sufficient for near-field radiative transfer if the presence of surface polaritons between the metallic layers becomes important [31]. In the near field, due to the large transverse wavevector, the requirement is that the period be much smaller than the vacuum gap distance, which is usually much smaller than the characteristic wavelength. Detailed discussions and quantitative criteria are given in a separate publication by the present authors [32].

Consider two of the structured metamaterials separated by a vacuum gap of distance d as shown in Fig. 1. The dielectric tensor in Cartesian coordinates (x,y,z) for a uniaxial medium with a vertical optical axis is given by [33, 34]

 

Fig. 1 Illustration of near-field radiative transfer between two hyperbolic metamaterials whose optical axis is normal to the surface. A vacuum gap d separate the two semi-infinite media. Cylindrical coordinates are used in which ρ specifies the component parallel to the plane. The wavevector k is decomposed into a parallel component β and a perpendicular component γ. The source (medium 1) is at 300 K and the receiver (medium 2) is at 0 K.

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For plane waves in a uniaxial anisotropic medium, the dispersion relations are [3, 9]

When the spacing (gap width) d between the two media is comparable to the characteristic wavelength of thermal radiation, near-field effects become important. Coupled evanescent waves can enhance the heat flux beyond the far-field limit. Thermal radiation in both the near and far fields as described by the fluctuation-dissipation theorem is due to random charge fluctuations (or currents) inside of a medium at non-zero temperatures [35]. The fluctuational currents in the source region at location $r^{\prime }$ generate an electric field at a location $r$. A time-varying magnetic field is also generated according to Maxwell’s equations. In the frequency domain, the fields due to a current source can be evaluated by integrations with the assistance of the Green functions $\overline{\overline{G}}$ and $\overline{\overline{\Gamma }}$ as follows [20, 35]:

The Green function for the magnetic field $\overline{\overline{\Gamma }}$ is related to that for the electric field $\overline{\overline{G}}$ by $\overline{\overline{\Gamma }}=\nabla \times \overline{\overline{G}}$ for a nonmagnetic medium. The Green function $\overline{\overline{G}}$ can be expressed by the Weyl transform with the Fourier component $\overline{\overline{g}}$ according to [13, 36]

The calculation of the spectral Poynting vector for isotropic media is described in [36, 40–42]. For a prescribed frequency and lateral wavevector, the Poynting vector in the vertical and radial direction can be obtained from

The energy streamlines are determined by tracing a line tangent to the Poynting vectors inside of the medium [24–27], viz.

The method described above can be readily extended to multilayered media with each medium being either isotropic or uniaxial. In this case, the Green function in each sub-layer must be determined and Eqs. (13) and (14) need to be integrated within each single layer and then summed up for all the source layers.

It should be noted that $〈S_z〉$ is independent of z in vacuum and decays exponentially in the receiver. In the emitter, it increases as z approaches the interface. The behavior of the z-component Poynting vector allows the definition of a near-field penetration depth [28, 29]. The heat flux between the two uniaxial media can be obtained by integrating $〈S_z〉$ in vacuum over the wavevector plane and over all frequencies, giving

3 Energy streamlines in hyperbolic metamaterials

Consider a periodic multilayered structure consisting of alternating metallic and dielectric layers. Here, the metallic layer is either doped Si (D-Si) or SiC, whose dielectric function becomes negative at certain frequencies due to free carriers or lattice vibrations (or phonons). Ge is chosen as the dielectric layers for both structures. Assuming that the layer thicknesses are sufficiently thin for the effective medium theory to be valid, the two components of the dielectric tensor in Eq. (1) can be expressed as follows [3, 11]

The real part of the dielectric function for the two types of multilayered structures is shown in Fig. 2. The filling ratio for D-Si is 0.5 and that for SiC is 0.3. In both cases, two hyperbolic bands are shown in the shaded regions: I for ${{\epsilon }^{\prime }}_t>0>{{\epsilon }^{\prime }}_z$ and II for ${{\epsilon }^{\prime }}_z>0>{{\epsilon }^{\prime }}_t$. Negative refraction or backwards bending of the Poynting vector can only exist with type I hyperbolic dispersion, but both hyperbolic modes can support propagating waves with large β that may enhance near-field radiative transfer [14, 18, 19]. The hyperbolic bands in the D-Si are due to free electron absorption, and thus the type II band appears fairly broad toward long wavelengths. The hyperbolic modes of SiC are much narrower, as they are caused by the infrared phonon mode associated with the Restrahlen band of SiC [21, 48]. The predicted dielectric functions are used for future calculations with fixed filling ratio for the two multilayered structures, D-Si/Ge and SiC/Ge.The power transmission factor for extraordinary waves between two identical semi-infinite metamaterials is shown in Figs. 3(a) and 3(b), which corresponds to a D-Si/Ge and SiC/Ge multilayered structure, respectively. The contour plots show ξ_p versus frequency and in-plane wavevector β, divided by k₀, for a vacuum gap distance of 10 nm. Figures 3(c) and 3(d) show the spectral heat flux corresponding to the two structures. As seen from Fig. 3(a), ξ_p is broad in both type I and type II hyperbolic bands, although the values in the type II region are not as high as those in the type I region. Furthermore, due to loss, the distinction between the regions is not so clear. When ${{\epsilon }^{\prime }}_z>0>{{\epsilon }^{\prime }}_t$, evanescent waves exist in the metamaterials when $\beta <{\beta }_{\text{cr}}=k_0\sqrt{{\epsilon }_z}$ as can be seen from Eq. (3). Thus, the transmission factor is rather small for the type II band below the critical wavevector ${\beta }_{\text{cr}}$ [18]. Interestingly, the spectral heat flux for the D-Si/Ge structure shows a broad continuous peak, which can be seen in Fig. 3(c), presumably due to loss. The type I and type II hyperbolic bands can also be seen for SiC/Ge structures as shown in Fig. 3(c), although the magnitude of ξ_p is much higher and extends to greater β /k₀ values. The spectral heat flux for D-Si/Ge multilayers is several orders higher than that for blackbodies in the far field shown in Fig. 3(c) as the dashed line. For the SiC/Ge metamaterial, the distinction between the hyperbolic bands turned out to be clearer and two narrowband peaks can be seen corresponding to type I and type II bands in the spectral heat flux curve shown in Fig. 3(d). The frequencies of the two peaks correspond well to the transverse and longitudinal phonon modes in SiC [21]. Compared with the total heat flux of 460 W/m² between blackbodies, the near-field heat flux at d = 10 nm is 640 and 1270 times higher with D-Si/Ge and SiC/Ge multilayered structures, respectively.

 

Fig. 2 Real part of the dielectric function predicted by EMT: (a) doped Si and Ge with f = 0.5; (b) SiC and Ge with f = 0.3. The multilayered structure is illustrated by the inset for each case. These filling ratios are used in the property calculations throughout this paper.

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Fig. 3 Power transmission factor and spectral heat flux with a gap spacing d = 10 nm. Contour plots of the transmission factor for p-polarization between two semi-infinite hyperbolic metamaterials made of (a) D-Si/Ge and (b) SiC/Ge multilayered structures; (c) Spectral heat flux between D-Si/Ge multilayered structures plotted together with that between blackbodies in the far field; (d) Spectral heat flux between SiC/Ge multilayered structures. The hyperbolic bands for the multilayered structures are highlighted. Note that the unit of S_ω is heat flux [W/m²] per frequency [rad/s] interval.

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To study the lateral displacement, it is noted that the energy streamlines depend on the particular frequency and given parallel wavevector. For the SiC/Ge structure, the two peak frequencies at $1.82\times {10}^{14}$ and $1.54\times {10}^{14}$ rad/s can be readily picked. A frequency in between at $1.59\times {10}^{14}$ rad/s is also chosen since it represents elliptical dispersion where the spectral heat flux is much lower. For the D-Si/Ge structure, five representative frequencies are selected in the region of interest from $0.5\times {10}^{14}$ to $3\times {10}^{14}$ rad/s (wavelength λ from 37.7 to 6.3 μm). Two of the chosen frequencies are $0.5\times {10}^{14}$ and $1.0\times {10}^{14}$ rad/s, which fall inside the type II hyperbolic region. The third at $1.8\times {10}^{14}$ rad/s is at the edge of the type II band with ${{\epsilon }^{\prime }}_t$ close to zero. The fourth frequency of choice at $2.7\times {10}^{14}$rad/s falls in the type I hyperbolic band. The fifth frequency location is at $2.9\times {10}^{14}$ rad/s, where ${{\epsilon }^{\prime }}_z$ is near zero. These frequencies were chosen to demonstrate different and unique energy propagation characteristics. The selection of suitable and representative parallel wavevector is illustrated with Fig. 4 for the D-Si/Ge and SiC/Ge structures, respectively. Figures 4(a) and 4(b) plot the z-component of the Poynting vector calculated from Eq. (13) at each selected frequency for the two structures, respectively, as functions of the parallel wavevector. The magnitude corresponds well to the transmission factor shown in Figs. 3(a) and 3(b) for the given frequency. In the type II hyperbolic band, the Poynting vector is small but increases quickly and reaches a peak at β = β_cr, where k_z = 0. For the elliptical mode in SiC/Ge structure at $1.59\times {10}^{14}$ rad/s, as shown in Fig. 4(b), the Poynting vector is relatively large and increases with β for small β values.

 

Fig. 4 Poynting vector and cumulative heat flux: (a) z-component of the Poynting vector versus lateral wavevector at select frequencies for (a) D-Si/Ge and (b) SiC/Ge multilayer metamaterials; Integration of the total heat flux over the wavevector space for (c) D-Si/Ge and (d) SiC/Ge. The horizontal line in (c) and (d) shows where 50% of the heat flux falls above and below the median wavevector for each frequency. Note that line styles in (c) and (d) correspond to the line styles and frequencies in (a) and (b), respectively. The unit of S_z is the heat flux [W/m²] per unit frequency [rad/s], wavevector [rad/m], and azimuthal angle [rad].

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However, <S_z> drops quickly at $\beta =k_0\sqrt{{{\epsilon }^{\prime }}_z}\approx 6k_0$, beyond which only evanescent modes exist in the metamaterial. The enhancement at $\beta >k_0$ is still due to photon tunneling because the waves are evanescent in vacuum. Figures 4(c) and 4(d) show the cumulative spectral heat flux for the selected frequencies for both structures. The 50% lines intercept with each curve at the median wavevector ${\beta }_{\text{median}}$, which can be used as a characteristic value to plot the energy streamlines for the determination of the lateral displacement. Note that for the elliptical mode, as shown in Fig. 4(d), $\beta *\approx {\beta }_{\text{median}}/k_0=4.4$ is much smaller than other modes since most of the energy is transferred at $\beta <6k_0$.

The penetration depth is an important characteristic of the vertical dimension for the medium to be considered opaque [21]. Furthermore, to evaluate the lateral shift, one needs to choose a certain depth in the medium away from the vacuum gap. For near-field radiation, the power penetration depth can be calculated according to $\delta =1/\left(2{{\gamma }^{″}}_{\text{e}}\right)$, where ${{\gamma }^{″}}_{\text{e}}$ is the imaginary part of the z-component wavevector for the extraordinary wave, since the contribution of the ordinary wave is negligibly small [27, 28]. The penetration depth at each of the chosen frequencies is shown in Figs. 5(a) and 5(b) for the two structures, respectively.

 

Fig. 5 Penetration depth $\delta (\omega ,\beta )$and spectral penetration depth ${\delta }_{\omega }(\omega )$ for d = 10 nm: (a,b) $\delta /d$ for selected frequencies as a function of β/k₀ for D-Si/Ge and SiC/Ge, respectively; (c,d) ${\delta }_{\omega }/d$ as a function of frequency for D-Si/Ge and SiC/Ge structures, respectively. Note that the hyperbolic bands are highlighted in (c,d).

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In the type I hyperbolic band, $\delta$ decreases with β. A sharp peak in some of the curves corresponds to the critical wavevector in the type II hyperbolic band, where ${{\gamma }^{″}}_{\text{e}}$ has a minimum. At 1.8 × 10¹⁴ rad/s, where ${{\epsilon }^{\prime }}_t$ is near zero for the D-Si/Ge structure, the penetration depth is nearly a constant at small $\beta$, but decreases at large values of $\beta$. At 1.59 × 10¹⁴ rad/s for the SiC/Ge structure, the penetration depth is very high for small β values due to the existence of propagating waves with low loss. For large values of β, the lateral distance that the electromagnetic energy might travel in the medium may be very large because the slab acts as a waveguide. However, due to loss and the relation between ${\gamma }_{\text{e}}\text{and}\beta$ through Eq. (3), a large lateral wavevector results in a smaller penetration depth. At β >> k₀, the penetration depth for an isotropic medium is given by $\delta =1/\left(2\beta \right)$ [27, 28]. For a uniaxial medium, it can be shown that the asymptote of the penetration depth is

The spectral penetration depth as a function of frequency is plotted in Figs. 5(c) and 5(d) for the two metamaterials at d = 10 nm. For D-Si/Ge as shown in Fig. 5(c), the penetration depth is about 10 times that of the vacuum gap spacing in the spectral range of interest, with a dip around 3.2 × 10¹⁴ rad/s. Loss affects the spectral penetration depth significantly and it is well known that doped Si has broadband absorption due to free carriers [47]. For SiC/Ge, as shown in Fig. 5(d), the penetration depth is very large except at the two hyperbolic bands where near-field heat transfer is greatly enhanced. At 1.82 × 10¹⁴ rad/s where ${{\epsilon }^{\prime }}_z$ is near zero for the SiC/Ge structure and the spectral heat flux reaches a peak, the penetration depth is very small (about 0.5d) in agreement with Eq. (20). As $\left|{{\epsilon }^{\prime }}_z\right|$ increases toward lower frequencies, δ_ω sharply increases, reaching 100d at 1.68 × 10¹⁴ rad/s, which is the edge of the hyperbolic type I band. In the type II hyperbolic band, the penetration depth is a few tens of d, and the variation is presumably due to loss. It should be noted that the relatively large penetration depth in hyperbolic metamaterials for near-field radiation has also been recently studied [49].

The energy streamlines are evaluated for the selected frequencies at the median wavevector β*, and shown in Fig. 6 for both structures.The lateral dimension ρ is shown relative to the vacuum gap spacing d, which is 10 nm in the calculation. The vertical dimension z is divided by d in the vacuum region. In the emitter and receiver, the vertical dimension is based on the non-dimensional parameter defined as

 

Fig. 6 Energy streamlines at select frequencies based on the median wavevector for (a,b) D-Si/Ge and (c,d) SiC/Ge multilayer structures at d = 10 nm. Here, $\beta *={\beta }_{\text{median}}/k_0$ that corresponds to 50% of the cumulative heat flux for each frequency as shown in Figs. 4(c) and 4(d).

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At a given frequency and lateral wavevector, the lateral displacement is defined as

Figure 6(a) shows the streamlines for three frequencies for D-Si/Ge structure in the emitter, while the inset shows the streamlines in the vacuum gap and the receiver. Very large lateral displacements are found inside the emitter especially at the two lowest frequencies in the hyperbolic type II bands, where the lateral displacement of the streamline at two penetration depths is $\Delta /d=$3730 at ω = 0.5 × 10¹⁴ rad/s and 1600 at ω = 1.0 × 10¹⁴ rad/s. At ω = 1.8 × 10¹⁴ rad/s in the source region, the lateral shift is near 200d. As shown in Fig. 6(b), the type I hyperbolic mode (2.7 × 10¹⁴ rad/s) undergoes a negative refraction in terms of the Poynting vectors and $\Delta \approx$23d inside the emitter. At ω = 2.9 × 10¹⁴ rad/s where ${{\epsilon }^{\prime }}_z=0$, Δ is relatively small (about 3.7d). Usually, the lateral shift is the largest inside the emitter, while the shift inside the vacuum gap and receiver could also be significant in some cases. Similar results are shown in Fig. 6(c) for SiC/Ge structure, where the lateral shift for the type II mode becomes extremely large, close to 4900d at ω = 1.54 × 10¹⁴ rad/s. For propagating waves at ω = 1.59 × 10¹⁴ rad/s, $\Delta /d=$ 433 in the emitter. Negative refraction can be seen from Fig. 6(d) for the type I mode at ω = 1.82 × 10¹⁴ rad/s with a moderate Δ around 95d. Considering the large lateral shift for hyperbolic type II modes, the lateral dimension of the multilayer structures must be sufficiently large in order for the near-field radiative transfer to be treated as one-dimensional using the multilayer Green’s functions.

Note that except ω = 1.59 × 10¹⁴ rad/s for the SiC/Ge structure, the selected frequencies all have a relatively high z-component of the Poynting vector, as shown in Figs. 3(c) and 3(d). Therefore, the large lateral displacements at these frequencies are relevant to the requirement of the lateral dimension of the structures in order for them to be considered one-dimensional. Furthermore, the penetration depth will scale with the vacuum gap distance as well as the lateral displacement because the transmission coefficient varies with βd in the near field.

A weighted average of the lateral displacement can be defined using the energy streamline method, considering the transmission coefficient, according to

The lateral displacements across the vacuum gap and inside the emitter are plotted in Fig. 7 as a function of frequency. The lateral displacement varies significantly at different frequencies. For D-Si/Ge, the lateral displacement in the emitter is negative for 2.3 × 10¹⁴ rad/s < ω < 2.8 × 10¹⁴ rad/s, although the magnitude is relatively small. This corresponds to negative refraction in the type I hyperbolic band. For the SiC/Ge structure at certain frequencies, the magnitude of the lateral displacement can be very large even in the type I hyperbolic region. For example, |Δ_ω|/d exceeds 4000 at 1.7 × 10¹⁴ rad/s; see Fig. 7(b).

 

Fig. 7 Average lateral displacements in vacuum and emitter for (a) D-Si/Ge structure and (b) SiC/Ge structure, with a gap spacing d = 10 nm. The highlighted regions correspond to the hyperbolic band (type I or type II).

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Inside of the emitter or receiver, the lateral displacement is related to the eccentricity of the hyperbolic dispersion, since the Poynting vector is perpendicular to the hyperbola. In the case of negligible loss, the eccentricity for hyperbolic dispersions are given below

In the type II region, if the eccentricity is near one, then $〈S_z〉>>〈S_{\rho }〉$ and the line tangent $m$ defined in Eq. (15) becomes smaller and the lateral displacement is also expected to be smaller. This agrees with the trend for the D-Si/Ge at frequencies up to 1.8 × 10¹⁴ rad/s, where ${{\epsilon }^{\prime }}_t\approx 0$. Although it is expected that the penetration depth will have a maximum at this frequency according to Eq. (20), this did not happen due to loss. The lateral displacement in SiC/Ge between 1.50 × 10¹⁴ rad/s and 1.55 × 10¹⁴ rad/s in the narrow type II hyperbolic band also agrees with that predicted by Eq. (26).

The trend of the lateral displacement for type I hyperbolic dispersion is more complicated. According to Eq. (25) and the shape of the hyperbola, see [3] for example, m is large when $e_{\text{c}}\to 1$ or $\left|{{\epsilon }^{\prime }}_z\right|/{{\epsilon }^{\prime }}_t<<1$. However, the penetration depth also becomes very small when ${{\epsilon }^{\prime }}_z\approx 0.$ It is also expected that between positive refraction and negative refraction, there will be a narrow region where the magnitude m is very small, so is the lateral displacement. This is clearly true for D-Si/Ge in the whole type I hyperbolic band, especially at the band edges. Loss also plays a significant role in reducing the lateral displacement. The maximum magnitude of lateral displacement is at 2.6 × 10¹⁴ rad/s with ${\Delta }_{\omega }/d=-50$. For SiC/Ge at 1.82 × 10¹⁴ rad/s when ${{\epsilon }^{\prime }}_z\approx 0$, the lateral shift is also close to zero. As the frequency decreases, $\left|{{\epsilon }^{\prime }}_z\right|/{{\epsilon }^{\prime }}_t$ increases quickly and it is expected that m will reduce since e_c increases. However, the magnitude of the lateral shift increases as can be seen from Fig. 7(b). This is due to the increased penetration depth with decreasing frequency. At ω = 1.69 × 10¹⁴ rad/s, $\left|{\Delta }_{\omega }\right|/d$ reaches a maximum of 4755. Further reducing the frequency will result in a very large e_c or very small m. Subsequently, Δ_ω crosses zero at ω $\approx$ 1.665 × 10¹⁴ rad/s.

4. Conclusions

Hyperbolic metamaterials exhibit unique characteristics in terms of the energy pathways and the penetration depth for near-field radiative transfer. Surprisingly large lateral displacement of the energy streamlines has been demonstrated in the emitter, from about –5000 to + 5000 times the vacuum gap distance, depending on the hyperbolic region. The mechanisms are explained according to the dispersion relations that are obtained using the effective medium theory for multilayered metallodielectric photonic crystals, involving doped-Si and SiC with another dielectric. The energy streamline method enables an estimate of the critical lateral dimension for the structure to be treated as one-dimensional.