Definition of polaritonic fluctuations in natural hyperbolic media

The discovery of photonic hyperbolic dispersion surfaces in certain van der Waals bonded solids, such as hexagonal boron nitride and bismuth selenide (a topological insulator), offers intriguing possibilities for creating strongly modified light-matter interactions. However, open problems exist in quantifying electromagnetic field fluctuations in these media, complicating typical approaches for modeling photonic characteristics. Here, we address this issue by linking the identifying traits of hyperbolic response to a coupling between longitudinal and transverse fields that cannot occur in isotropic media. This description allows us to formulate a gauge theoretic description of the influence of hyperbolic response on electromagnetic fluctuations without explicitly imposing a characteristic size (model of nonlocality)—leading to formally bounded expressions so long as material absorption is included. We then apply this framework to two exemplary areas: the optical sum rule for modified spontaneous emission enhancement in a general uniaxial medium and thermal electromagnetic field fluctuations in hexagonal boron nitride and bismuth selenide. We find that while the sum rule is satisfied, it does not constrain the enhancement of light-matter interactions in either case. We also show that both hexagonal boron nitride and bismuth selenide possess broad spectral regions where the magnitude of electromagnetic field fluctuations are over 120 times larger, and over 800 times larger along specific angular directions, than they are in vacuum.

Figure 1

Wave conditions and fluctuation density. In isotropic media, the magnitude of fluctuations in the electromagnetic field are proportionally related to the magnitude of the wave condition $k=\sqrt{\epsilon \left(k,\omega \right)}\omega /c$. By inference, the $p$-polarized wave condition for uniaxial media suggests that hyperbolic response should be accompanied by strongly enhanced electromagnetic field fluctuations. The central goal of this article is to quantify this statement.

Figure 2

Sum rule for transverse spontaneous emission enhancement in hyperbolic media. Panel (a) displays the absolute relative permittivity values resulting from Eq. (48). The thin dashed lines and schematic dispersion surfaces highlight spectral regions of hyperbolic response where one of either $\text{Re}\left\{{\epsilon }_{{}_P}\left(\omega \right)\right\}$ or $\text{Re}\left\{{\epsilon }_{{}_A}\left(\omega \right)\right\}$ is negative. Panel (b) shows the resulting transverse spontaneous emission enhancement offset by vacuum, the integrand of Eq. (44). Panel (c) plots the integrated enhancement as function of the upper wave number considered. These result confirm that the enhancement sum rule is strictly obeyed inside hyperbolic media [89]. Accounting only for purely electromagnetic (transverse) contributions, emission enhancement in spectral regions of hyperbolic response is unremarkable. Panel (d) further highlights this fact by comparing the orientationally averaged enhancement from (b), black line, with the enhancement found by averaging two isotropic media with $\epsilon \left(k,\omega \right)={\epsilon }_{{}_P}\left(\omega \right)$ and $\epsilon \left(k,\omega \right)={\epsilon }_{{}_A}\left(\omega \right)$ weighted by factors of $2/3$ and $1/3$, respectively. The graphs are found to be nearly identical, even though the two situations correspond to very different electromagnetic environments.

Figure 3

Polaritonic spontaneous emission enhancement surpassing the sum rule in hyperbolic media. The figure displays the orientationally averaged spontaneous emission enhancement of the longitudinal part of the AP contribution, considering a uniaxial media with permittivity model Eq. (48). Contrasting with Fig. 2, this longitudinal enhancement dominates in spectral regions of hyperbolic response.

Figure 4

Scaling with material absorption of fluctuation densities in hyperbolic media. The figure depicts the power scaling of the transverse O-type Eq. (47), longitudinal C-type Eq. (42), and mixed AP-type (longitudinal part only) Eq. (49) contributions to the FD as a function of material absorption $\text{Im}\left\{{\epsilon }_{{}_A{,}_P}\left(\omega \right)\right\}$. For the C-type contribution, only the angular integrals in Eq. (36) have been computed as the $k$ integral diverges in the limit of local polarization response. The plot is computed by considering ${\epsilon }_{{}_A}=-1+i\left(\text{x-axis}\text{value}\right)$, ${\epsilon }_{{}_P}=1+i\left(\text{x-axis}\text{value}\right)$. The knee transitioning from a scaling of $\propto -1$ to a scaling of $\propto 0$ is set by the minimum magnitude of the real permittivity components $|\text{Re}\left\{{\epsilon }_{{}_A{,}_P}\left(\omega \right)\right\}|$. (This behavior is also seen the isotropic case [84].) The $x^{-3/2}$ scaling exhibited by the AP contribution is found to be stronger than either of the two pure solution types.

Figure 5

Relative thermal energy and fluctuation densities in natural hyperbolic media. The figure shows the contribution that AP-type modes, solid lines, and O-type modes, dashed lines, make to the total electric and magnetic thermal energy densities (total lines) inside hexagonal boron nitride (c) and bismuth selenide (d). (For comparison the energy densities are normalized by half the thermal energy density of vacuum.) The absolute value of the real and imaginary parts of relative permittivity components of these two materials, based on data from Refs. [51, 53], are plotted in panels (a) and (b). Each sharp peak and dip in (b) signals a sign flip of the real part as marked. The imaginary part of each component remains positive throughout. The green electric lines (bold polariton, dashed ordinary) double as the respective FDs. Both media show broad spectral regions where this quantity is over 120 times larger than it is in vacuum.

Figure 6

Angular polaritonic fluctuation density in natural hyperbolic media. The figure depicts the polaritonic FD inside hexagonal boron nitride at 1500 ${\text{cm}}^{-1}$, and bismuth selenide at 110 ${\text{cm}}^{-1}$, as a function of polar angle on a logarithmic scale. The inset shows this same quantity as a polar plot on a linear scale. Although the integrated FDs of these two cases are almost equal, Fig. 6, as hexagonal boron nitride more closely approaches the resonance condition $|{\epsilon }_{{}_U}\left(\theta ,\omega \right)|=0$, but possess less polarization anisotropy, the angular distribution of its FD is much more radical. Both materials show angular regions where the relative polar FD is over 800 times larger than vacuum.

Figure 7

Near-field electromagnetic energy density of natural hyperbolic media. The figure plots the sum of the near-field electric and magnetic energy densities above half-spaces of hexagonal boron nitride (a) and bismuth selenide (b) for increasingly small observation distances. (For comparison with the FD results this energy is normalized by half the energy density of vacuum.) The inset in panel (a) shows a schematic representation. Recalling that surface mode contributions are not included in Eq. (50), the relative spectral characteristics of the near-field energy density are nevertheless seen to be in good agreement with FDs plotted in Fig. 5 for small observation distances.