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Abstract
Uniaxial hyperbolic materials enable excitation of phonon polaritons with utrahigh wavevectors that have been shown to be promising for many optical and thermal radiative applications and thus have attracted much attention recently. However, the characteristics of surface and volume phonon polaritons excited with uniaxial hyperbolic materials that exhibit in-plane anisotropy or in-plane isotropy have not been discussed thoroughly and some issues have so far remained elusive. In this paper, we conducted a comprehensive investigation on surface and volume phonon polaritons in a bulk or a thin slab of hexagonal boron nitride (hBN). We clarified the excitation, characteristics and topology of surface and volume phonon polaritons in such a uniaxial hyperbolic material. In particular, we showed that hyperbolic surface phonon polaritons (HSPhPs) can exist in the Type I hyperbolic band of hBN with confined wavevectors when the optic axis (OA) is parallel to the surface. For a thin hBN slab, we revealed a split of HSPhPs and a smooth transition between HSPhPs and HVPhPs in the Type II hyperbolic band. Furthermore, we also identified non-Dyakonov surface phonon polaritons excited without evanescent ordinary waves. These findings may extend the understanding of phonon polaritons in hyperbolic materials and offer new theoretical guidance for the design of infrared optical devices with hyperbolic materials.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Uniaxial hyperbolic materials possess some distinctive electromagnetic properties and have received extensive investigations in the past decade [1–3]. Usually, the permittivity tensor of a uniaxial material is described by two varibles $\varepsilon _{\parallel }$ and $\varepsilon _{\perp }$, where the former represents the principal component along the optic axis (OA) while the latter is the principal component perpendicular to the OA. For a uniaxial hyperbolic material, its hyperbolic bands refer to frequency bands where $\varepsilon _{\parallel }$ and $\varepsilon _{\perp }$ have opposite signs. In particular, a frequency band with $\varepsilon _{\parallel }<0$ and $\varepsilon _{\perp }$>0 is known as Type I hyperbolic band, while that with $\varepsilon _{\parallel }>0$ and $\varepsilon _{\perp }<0$ is called Type II hyperbolic band [4–8]. Uniaxial hyperbolic materials exhibit hyperbolic dispersion in both Type I and Type II hyperbolic bands. Therefore, ultrahigh propagating wavevectors as well as surface wave excitation can be realized with such materials [8–11]. Such peculiar electromagnetic properties have been reported extensively for artificial hyperbolic materials fabricated with periodic multilayers or nanowires [12–18]. However, the unit cells of hyperbolic metamaterials are usually tens to hundreds of nanometers in size. The hyperbolicity of metamaterials will break down when electromagnetic wavelength is comparable to the dimension of the unit cell. Unlike artificial hyperbolic materials, the structural periods of natural hyperbolic materials are those of crystal lattice constants, which are usually on the atomic scale. As a result, hyperbolic responses of natural hyperbolic materials can occur in much higher frequency bands than those of artificial ones [1,19–22]. Recently, van der Waals (vdW) crystals such as hexagonal boron nitride (hBN) [11,23,24] and $\alpha$-MoO$_3$ [25] in which atomic layers are bonded together by weak vdW forces [11,19,26]) have been explored extensively for spectral tailoring of thermal emission and enhancing near-field thermal radiative heat transfer [23,26] because such materials exhibit hyperbolic response in the mid-infrared region.
Hyperbolic phonon polaritons (HPhPs), which are treated as quasiparticles induced by coupling of electromagnetic fields with optical phonons, can be categorized as hyperbolic surface phonon polaritons (HSPhPs) and hyperbolic volume phonons polaritons (HVPhPs), depending on the polaritons being confined at the interfaces or in the volume of hyperbolic materials [8,18]. Potential applications based on HPhPs can be found in sensing [27,28], super-resolution imaging [5,21], super-Planckian heat transfer [29–31], wavefront control [32,33], to name a few.
HSPhPs are a kind of surface polaritons which can be launched when the hyperbolic materials exhibit in-plane anisotropy (i.e., the OA parallel to the interface) [8]. The study of surface polaritons on uniaxial crystal can be traced back to 1988 when Dyakonov reported his new finding on surface waves excited at isotropic-uniaxial interfaces [34]. Later, such surface waves were also termed Dyakonov surface waves (DSWs) or surface polaritons. Dyakonov pointed out that such surface polaritons, driven by in-plane anisotropy, do not require either of $\varepsilon _{\parallel }$ and $\varepsilon _{\perp }$ to be negative. However, excitation of such surface polaritons involves conversion of polarization as well as existence of both ordinary and extraordinary waves [34,35]. The first experimental observation of DSWs was reported by Takayama [36] in 2009. Later, Zapata-Rodríguez [37] and Cojocaru [38] found that HSPhPs, a special type of DSWs, can be excited in Type II hyperbolic band. However, the possibility of exciting HSPhPs in Type I hyperbolic band still remains elusive. Furthermore, although both HSPhPs and HVPhPs in Type I and Type II hyperbolic bands have been investigated extensively, their dispersion topology has not been well clarified.
In this paper, we numerically and theoretically investigated surface and volume phonon polaritons in bulk and thin slab of hBN. By carefully analyzing the dispersion relation of electromagnetic waves propagating in uniaxial hyperbolic materials, we discussed in detail these modes in Type I and Type II hyperbolic bands when the OA is parallel and perpendicular to the material surface. Especially, the dispersion curves of surface and volume phonon polaritons were identified and highlighted clearly by comparing the analytical solution to the dispersion relation with the intensity contour of the imaginary part of the Fresnel reflection coefficient Im($r_{ pp}$) [39], which was computed with the enhanced $4\times 4$ transfer matrix method [35,40]. Furthermore, we found that there exist surface phonon polaritons excited only with decaying extraordinary wave or ordinary wave (unlike DSWs) when the OA is parallel to the surface. When the OA is perpendicular to the surface, existence of HSPhPs in Type II hyperbolic band was also identified.
2. Theory
Figure 1 is the schematic of a uniaxial hBN slab. If its thickness $d$ tends to infinity, the slab becomes a bulk. When the OA is parallel to the surface, shown in Fig. 1(a), the permittivity tensor can be expressed as diag($\varepsilon _\parallel,\varepsilon _\perp,\varepsilon _\perp$). In Fig. 1(b), the OA is perpendicular to the surface and the permittivity tensor is written as diag($\varepsilon _\perp,\varepsilon _\perp,\varepsilon _\parallel$). The components $\varepsilon _\parallel$ and $\varepsilon _\perp$ can be obtained from the following equation [39,40]
Fig. 1. Schematic of a uniaxial hBN slab: (a) OA is parallel to the interface and (b) OA is perpendicular to the interface. $d$ is the thickness of the slab.
Fig. 2. Real parts of the permittivity components $\varepsilon _{\perp }$ and $\varepsilon _{\parallel }$ of hBN varying with angular frequency. Either of the Type I hyperbolic band (1.47$\sim$1.56$\times 10^{14}$ rad/s) and Type II hyperbolic band (2.58$\sim$3.03 $\times 10^{14}$ rad/s) is illustrated between two dashed lines. The three light blue regions, 1.47$\sim$1.56$\times 10^{14}$ rad/s, 2.58$\sim$2.97$\times 10^{14}$ rad/s and 3.03$\sim$3.14$\times 10^{14}$ rad/s, correspond to the frequency bands where surface phonon polaritons can be excited when the OA is parallel to the surface when the adjacent medium is air.
The dispersion relation for electromagnetic waves propagating in uniaxial crystals can be written as [41]
The ordinary wave dispersion relation has the same expression in both cases as
where $k_{oz}$ is the wavevector compnent (normalized by $k_0$) of ordinary waves along the $z$-axis. In Type I and Type II hyperbolic bands, the isofrequency surfaces described by Eq. (3) and Eq. (4) are hyperboloids, and outside the two bands are ellipsoids in the momentum space. As for Eq. (5), the isofrequency surface is always a sphere, if solution exists. When the imaginary parts of $\varepsilon _\parallel$ and $\varepsilon _\perp$ are neglected, Eq. (3) reduces to a hyperbolic equation in the $k_x$-$k_y$ plane ($k_{ez}$=0) with two asymptotes given byHVPhPs describe the volume modes of extraordinary waves that propagate following a hyperbolic dispersion. Therefore, projected regions where HVPhPs can exist on the $k_x-k_y$ plane are confined between the two asymptotes [30], viz.
For bulk uniaxial materials, the dispersion relation for surface modes at an isotropic-uniaxial interface can be written when the OA is parallel to the interface as [34]
When the OA is perpendicular to the interface, the surface mode dispersion relation can be expressed as [42]
where the expression of ${q_{1}}$ remains the same as above but ${q_{e}}$ should be calculated as ${q_{e}} = \sqrt {\frac {{{\varepsilon _ \bot }}}{{{\varepsilon _\parallel }}}{\beta ^{2}} - {\varepsilon _ \bot }}$. Here, $\beta = \sqrt {k_x^{2}+k_y^{2}}$ is the normalized wavevector parallel to the interface. From Eq. (9), the dispersion relation can be equivalently expressed with $\beta$ as [43]For surface polaritons, all $q_{1}$, $q_{e}$ and $q_{o}$ are positive [34]. Therefore, a necessary condition for surface polaritons excited in Type II hyperbolic band of a uniaxial hyperbolic material with its OA perpendicular to the interface is $\sqrt {\varepsilon _1} < \beta < \sqrt {\varepsilon _\parallel }$. However, no surface polaritons can be excited in Type I hyperbolic band since the above necessary condition cannot be satisfied.
For uniaxial crystal slabs, under the condition of $\beta \gg 1$, the analytical approximate modal dispersion relation is given as [1,42]
Note that this relation holds for the OA either parallel or perpendicular to the interface. When the OA is along $x$-axis, $\varepsilon _z =\varepsilon _y =\varepsilon _\perp$, $\varepsilon _x=\varepsilon _\parallel$ , while $\varepsilon _z =\varepsilon _\parallel$, $\varepsilon _x=\varepsilon _y=\varepsilon _\perp$ when OA is along $z$-axis.
3. Surface and volume phonon polaritons: OA parallel to the $x$-axis
3.1 Bulk hBN
3.1.1 SPhPs in hBN for positive $\varepsilon _{\parallel }$ and $\varepsilon _{\perp }$
As mentioned above, SPhPs can be excited at an air-hBN interface in the frequency band where $\varepsilon _{\parallel } > \varepsilon _1 > \varepsilon _{\perp }>0$, i.e., in the right light blue region marked in Fig. 2. Here, we investigated in detail the dispersion characteristic of SPhPs in this frequency band by focusing on a specific angular frequency at 3.07$\times 10^{14}$ rad/s ($\varepsilon _{\perp }=0.4+0.05i$, $\varepsilon _{\parallel }=2.83+0.004i$). In addition to obtaining the dispersion curves of SPhPs by solving Eq. (8), another method of illustrating the dispersion relation of SPhPs is to investigate the imaginary part of the $p$-polarized Fresnel reflection coefficient $r_{pp}$, since this quantity is proportional to the density of states of SPhPs, which shows very large values for excited SPhPs and has been widely adopted in literature [26]. Shown in Fig. 3(a) is the intensity contour of Im($r_{pp}$) varying with the wavevector components $k_{x}$ and $k_{y}$. Several bright regions can be seen in the figure, and the four bright segments marked in green color highlight excitation of SPhPs. In fact, the bright color pattern in Fig. 3(a) is very similar to the structure shown in Fig. 3(b), which was drawn with the dispersion equations of electromagnetic waves propagating in air and in hBN assuming $k_{1z}$, $k_{ez}$ and $k_{oz}$ equal to zero, together with the four dispersion curve segments of SPhPs obtained from Eq. (8). It can be seen that the SPhP dispersion curves appear only around the four intersection points of and intercepted by the outer circle and the ellipse. Therefore, SPhPs can only be excited in a certain range of angles relative to OA [34], as shown clearly in Fig. 3(a).
Fig. 3. (a) Intensity contour of the imaginary part of the p-polarized Fresnel reflection coefficient Im($r_{pp}$) at 3.07$\times 10^{14}$ rad/s. (b) Theoretical dispersion curves in the $k_x-k_y$ plane at the same angular frequency. $R_1$, $R_2$ and $R_3$ correspond to the bright regions in (a)
The regions $R_1$, $R_2$ and $R_3$ marked in Fig. 3(b) correspond to different propagating characteristics of electromagnetic waves in air and hBN. In $R_1$ region, $q_{1}^{2}<0$, $q_{e}^{2}<0$ and $q_{o}^{2}>0$, which means the wave in air and the extraordinary wave in hBN are propagating waves while the ordinary wave in hBN is evanescent wave. In the special case when a $p$-polarized wave is incident with $k_{x}=0$, the transmitted wave is totally an ordinary wave which is evanescent and no extraordinary wave exists in this case. This explains the high value of Im($r_{pp}$) in $R_1$ around $k_{x}$ close to zero. Similarly, in $R_2$ region, both ordinary wave and extraordinary wave are evanescent, but the wave in air is propagating. Therefore, total reflection occurs, resulting in the bright color shown in Fig. 3(a). In $R_3$ region, only extraordinary wave can propagate in hBN, the wave in air and the ordinary wave in hBN are evanescent. But the extraordinary wave component is very small for incidence of a $p$-polarized wave when $k_{x}$ is close to zero, leading to the low intensity of Im($r_{pp}$) in this region.
3.1.2 HVPhPs and HSPhPs in Type I and Type II hyperbolic bands
Figure 4 is the schematic of isofrequency surfaces in the Type I and Type II hyperbolic bands. As the OA is along the $x$-axis, the red isofrequency surfaces shown in Fig. 4(a), which were drawn with Eq. (3), exhibit the shape of hyperboloid extending to infinity, indicating that an extraordinary wave can propagate with unbounded wavevector in the hyperbolic band as long as the wavevector falls inside the hyperboloids. The hyperbolicity of the dispersion assures the extraordinary waves propagating in the volume of the material with unbounded wavevectors. Such volume modes are termed HVPhPs. The green sphere in Fig. 4(a) depicts the isofrequency surface drawn from Eq. (5) for ordinary wave, which illustrates that the ordinary wave can only propagate with its wavevector confined in the sphere. Furthermore, the sphere has two tangent points with the hyperboloids on the $k_{x}$-axis, indicating that the extraordinary wave and the ordinary wave can propagate with the same wavevector only when the propagation is along the $x$-axis. The red isofrequency surface shown in Fig. 4(b) also illustrates the dispersion of extraordinary wave propagating in the volume of the material, which also exhibits the shape of hyperboloid, but is different from that shown in Fig. 4(a). In this case, Eq. (5) has no real solutions and thus ordinary waves propagating in the volume of the material are prohibited. Note that the blue regions in Fig. 4 correspond to projection of the hyperboloids on the $k_x-k_y$ plane, whose borders are a pair of hyperbolas whose foci are inside the blue region in Fig. 4(a) but are outside the blue region in Fig. 4(b). Here, we choose two angular frequencies, $1.5\times 10^{14}$ rad/s ($\varepsilon _{\perp }=7.67+0.01i$, $\varepsilon _{\parallel }=-6.14+1.12i$) in Type I hyperbolic band and $2.9\times 10^{14}$ rad/s ($\varepsilon _{\perp }=-2.19+0.11i$, $\varepsilon _{\parallel }=2.82+0.005i$) in Type II hyperbolic band, to elaborate the HVPhPs and HSPhPs.
Fig. 4. Schematic of isofrequency surfaces showing the dispersion of electromagnetic waves propagating in hBN in (a) Type I and (b) Type II hyperbolic bands. The red hyperboloids are calculated with Eq. (3) and the green sphere comes from Eq. (5). The intersecting curves of the hyperboloids with the $k_x-k_y$ plane form a pair of hyperbolas. The blue region is the projection of the hyperboloids on the $k_x-k_y$ plane.
Figure 5(a) is the intensity contour of Im($r_{pp}$) varying with the wavevector components $k_{x}$ and $k_{y}$ at 1.5 $\times 10^{14}$ rad/s. The bright color regions look similar to those of the isofrequency surfaces projected on the $k_x-k_y$. In fact. there are several details need to be clarified. First of all, the bright color regions illustrating excitation of HVPhPs are bounded by the blue dashed curves, which are a pair of hyperbolas as discussed above. The two white dashed lines denote the asymptotes of these hyperbolas and were calculated with Eq. (6). However, even brighter color can be found around the borders of the regions for HVPhPs. For clearer observation, the region in the rectangle is locally enlarged and is illustrated in Fig. 5(b). It can be seen that the bright strip is actually located outside the region for HVPhPs. Therefore, the bright strip does not result from HVPhPs. We attribute it to the excitation of HSPhPs, because we solved Eq. (8) for the dispersion curves of HSPhPs in the Type I hyperbolic band and the solution is plotted in Figs. 5(a) and (b) as the green solid curves, which can be seen to agree excellently with the bright strip. It should be noted that, unlike HVPhPs, the wavevectors of HSPhPs are bounded in the segments determined by intercept of the green solid curves by the blue dashed curves. In fact, it has been pointed out that HSPhPs of very large wavevectors cannot be excited in Type I hyperbolic band [31]. Here, we should stress that, although the word ’hyperbolic’ is used to describe the the surface phonon polaritons in hyperbolic band, strictly speaking, the dispersion curves are not hyperbolic. Furthermore, although HSPhPs can be excited in the whole Type I hyperbolic band, their dispersion curves are broken at $k_y=0$. For better analysis, Eq. (8) can be rewritten as [34]
Fig. 5. Intensity contour of the imaginary part of the p-polarized Fresnel reflection coefficient Im($r_{pp}$) varying with $k_{x}$ and $k_{y}$: (a) the plot is at 1.5 $\times 10^{14}$ rad/s; (b) local enlargement of the rectangular region marked in (a); (c) the plot is at 2.9 $\times 10^{14}$ rad/s; (d) local enlargement of the region around the origin in (c). The white dashed lines denote the asymptotes calculated with Eq. (6). The solid green curves and the dashed black curve represent analytical solutions to Eq. (8). The blue dashed curves are analytical solutions to Eq. (3) with $k_{ez}=0$.
Assuming $k_y = 0$, Eq. (13) has no real solutions, which means HSPhPs cannot be excited in the OA direction in Type I hyperbolic band. In fact, it has been pointed out that when the surface normal, the OA and the wavevector lie in the same plane, no surface phonon polaritons (DSWs) can be excited [34].
As for the bright region of $\beta <1$, which is confined in a certain angle around the $k_{x}$-axis, this high value of Im($r_{pp}$) can be interpreted as high directional reflection caused by in-plane anisotropy of hBN under $p$-polarized incidence. Because within $\beta <1$, both the wave in air and the ordinary wave in hBN are propagating, while the extraordinary wave in hBN is evanescent. If a $p$-polarized wave is incident with $k_{y}=0$, the transmitted wave contains only extraordinary wave given that the OA is along the $x$-axis, which results in the high value of Im($r_{pp}$) confined in the certain angle around the $k_{x}$-axis. In the annular region $1<\beta <\sqrt {\varepsilon _\perp }$, both the wave in air and the extraordinary wave in hBN are evanescent, only the ordinary wave in hBN is propagating. As such, the value of Im($r_{pp}$) is relatively high except around the $k_{x}$-axis where the ordinary wave is negligible. Interestingly, a very bright circular strip is seen for $\beta$ close to 1. After careful analysis, we found that the mechanism for this bright strip is due to excitation of surface phonon polaritons at the air-hBN interface. Because this bright strip matches excellently with the solution to Eq. (8), shown in Fig. 5(a) as the black dashed circle. However, such surface phonon polaritons are not the same type investigated by Dyakonov [34] since the ordinary wave in this case is not an evanescent wave.
Figure 5(c) shows the intensity contour of Im($r_{pp}$) varying with the wavevector components $k_{x}$ and $k_{y}$ at 2.9$\times 10^{14}$ rad/s. Four bright strips can be clearly seen. We attributed these bright strips to excitation of HSPhPs, because they agree very well with the dispersion curves of HSPhPs obtained from Eq. (8), represented by the green solid curves. It should be pointed out that the regions for HVPhPs are marked in Fig. 5(c), but no bright color can be seen in the corresponding regions. This is because the intensity of Im($r_{pp}$) due to HVPhPs does not exceed unity while that due to HSPhPs can be up to 50. Therefore, the large difference in the scale of Im($r_{pp}$) for HVPhPs and HSPhPs makes the regions for HVPhPs appearing quite dark. Figure 5(d) is the local enlargement of Fig. 5(c) around the origin. One can find that the dispersion curves of HSPhPs are cut off by the HVPhPs dispersion curves. On the other hand, two bright short bars appear around $k_x = 0$ in the HVPhPs regions which, however, are not caused by HVPhPs, but by excitation of SPhPs associated with the ordinary wave. These excited SPhPs can couple with the HSPhPs.
3.2 Thin hBN slab
A hBN slab of finite thickness becomes a waveguide. Hence, HVPhPs and HSPhPs in this case can be termed the waveguide modes, whose dispersion relation can be described by Eq. (11) under the condition of $\beta \gg 1$. In this work, the slab was assumed to be suspended in air. For a thin slab of thickness 500 nm, the intensity contour of Im($r_{pp}$) varying with the wavevector components $k_{x}$ and $k_{y}$ at 1.5 $\times 10^{14}$ rad/s is shown in Fig. 6(a). Discretized bright color strips appear in the regions to the left and right of the origin. These bright color strips correspond to the waveguide modes of different orders, because they agree well with the dispersion curves of the waveguide modes solved from Eq. (11), denoted by the green solid curves and the blue dashed curves in Fig. 6(a). In order to distinguish the modes associated with HSPhPs and HVPhPs, we amplify the square region marked in Fig. 6(a), as shown in Fig. 6(b). The green solid curves in Fig. 6(b) correspond to the solution of Eq. (11) with $l=0$, which can be found to deviate slightly from the bright strips at small $\beta$. This is because Eq. (11) is the approximated dispersion relation under $\beta \gg 1$. The blue solid curves drawn in Fig. 6(b) are a pair of hyperbolas, which represent the solution to Eq. (3) with $k_{ez}=0$ and highlight the regions where HVPhPs can exist. Clearly, the green solid curves are partly located outside the regions for HVPhPs, indicating that the bright color strips of this part are not caused by HVPhPs, but by HSPhPs. Interestingly, the green solid curves cross over the hyperbolas and extend into the regions for HVPhPs smoothly. Therefore, the 0th-ordered dispersion curves are associated with both HSPhPs and HVPhPs, depending on the values of $k_{x}$ and $k_{y}$. In order to verify the transition of modes, we choose three points A(2.18$k_0$, −0.66$k_0$), B(3.46$k_0$, −1.85$k_0$) and C(4.67$k_0$, −3.07$k_0$) on the bright strip shown in Fig. 6(b), where B is on the hyperbola while A and C are located outside and inside the region for HVPhPs, respectively. The normalized amplitude distribution of the field component E$_\textrm {x}$ across the slab is shown in Fig. 6(c) for these three points. It can be seen that for point A, the E$_\textrm {x}$ decays on both sides from the interface, indicating that the waveguide mode is dominated by HSPhPs. For point B, the amplitude of E$_\textrm {x}$ is flat across the slab, manifesting zero value of $k_{ez}$ in this case. On the other hand, the amplitude distribution of E$_\textrm {x}$ exhibits a convex shape for point C. The convex shape shows that the waveguide mode in point C is a TM0-like mode, which is dominated by HVPhPs. From point A to point C, the waveguide mode changes from HSPhPs to HVPhPs smoothly. The higher-ordered waveguide modes are located in the regions for HVPhPs. Therefore, they are dominated by HVPhPs. The solutions to Eq. (11) for $l=1,2$ are portrayed in Fig. 6(a) as the blue dashed curves, which can be found to agree very well with the bright color strips. Note that the two asymptotes divide the $k_x-k_y$ plane into four regions, but HVPhPs and HSPhPs exist only in the two regions to the left and right of the origin for angular frequencies in Type I hyperbolic band. No polaritons exist in the other two regions above and below the origin.
Fig. 6. Intensity contour of the imaginary part of the p-polarized Fresnel reflection coefficient Im($r_{pp}$) varying with $k_{x}$ and $k_{y}$: (a) the plot is at 1.5 $\times 10^{14}$ rad/s for a 500 nm thick hBN slab; (b) local enlargement of the square region marked in (a); (c) normalized amplitude distribution of electromagnetic field E$_\textrm {x}$ at points A(2.18$k_0$, −0.66$k_0$), B(3.46$k_0$, −1.85$k_0$) and C(4.67$k_0$, −3.07$k_0$) shown in (b); (d) the plot is at 2.9$\times 10^{14}$ rad/s for a 50 nm thick hBN slab. The numbers in (a) and (d) represent the orders of waveguide modes. The green solid curves and the blue dashed curves denote solutions to Eq. (11) with $l=0$ and $l>0$, respectively. The bule solid curves in (b) are the solutions to Eq. (3) with $k_{ez}=0$. The magenta curves in (d) are the solutions to Eq. (8). The white dashed lines represent the asymptotes from Eq. (6).
Figure 6(d) compares the intensity contour of Im($r_{pp}$) with the dispersion curves obtained from Eq. (11) at $2.9\times 10^{14}$ rad/s. In the calculation, the slab thickness was set at 50 nm. It can be observed that all the bright strips agree very well with the dispersion curves. Here, the 0th-ordered dispersion curves are located in the regions above and below the origin. Therefore, these 0th-ordered dispersion curves represent the dispersion curves of HSPhPs, which are similar to the dispersion curves of HSPhPs for a bulk hBN, illustrated as the magenta solid curves. However, the 0th-ordered dispersion curve can be seen to split into two branches When $k_{x}$ and $k_{y}$ are not large, due to coupling of HSPhPs excited on the two surface of the slab. More interestingly, one branch of the 0th-ordered dispersion curve merges with the 1st-ordered dispersion curve of HVPhPs smoothly on the asymptote line. In order to get a better understanding, we chose four points D(53.33$k_0$,79.4$k_0$), E(66$k_0$,79.4$k_0$), F(52$k_0$,49.15$k_0$) and G(44$k_0$,16.4$k_0$), and calculated the distribution of the electric field component E$_\textrm {z}$ across the slab. Points D and E are on the two splitting branches of the dispersion curve for HSPhPs, while points F and G are on the 1st-ordered dispersion curve for HVPhPs. The distributions of E$_\textrm {z}$ associated with the four points are displayed in Figs. 7(a)-(d), respectively. We can find in Fig. 7(a) that the distribution of E$_\textrm {z}$ across the slab exhibits an antisymmetric pattern, while that in Fig. 7(b) shows a symmetric pattern. Therefore, it is clear that the branch with point A corresponds to the antisymmetric branch while that with point B is the symmetric branch for the dispersion curves of HSPhPs. Figures 7(c) and (d) show that the the electric field distribution in the slab indeed appears in the shape of a standing wave of the 1st order. This confirms our analysis that point F and point G are located on the dispersion curve of HVPhPs.
Fig. 7. Distributions of the electromagnetic field component E$_\textrm {z}$ in a hBN slab of thickness equal to 50 nm at: (a) point D(53.33$k_0$,79.4$k_0$), (b) point E (66$k_0$,79.4$k_0$), (c) point F(52$k_0$,49.15$k_0$) and (d) point G(44$k_0$,16.4$k_0$) shown in Fig. 6(d), respectively.
4. HVPhPs and HSPhPs in Type I and Type II hyperbolic bands: OA parallel to the $z$-axis
For a comprehensive understanding of HVPhPs and HSPhPs in hyperbolic materials, we also considered the case when the OA is along the $z$-axis, i.e., the OA is perpendicular to the surface of hBN as shown in Fig. 1(b). Figure 8 is the schematic of isofrequency surfaces showing the dispersion of electromagnetic waves propagating in hBN in Type I and Type II hyperbolic bands, respectively. In this case, projection of the isofrerquency surfaces onto the $k_x-k_y$ plane is a circular region, meaning that the electromagnetic response of hBN depends only on the magnitude of $\beta$, but not on its orientation.
Fig. 8. Schematic of isofrequency surfaces showing the dispersion of electromagnetic waves propagating in hBN in (a) Type I and (b) Type II hyperbolic bands. The red hyperboloids are calculated with Eq. (4) and the green sphere comes from Eq. (5).
Figure 9(a) and 9(b) are the intensity contour of Im($r_{pp}$) varying with $k_{x}$ and $k_{y}$ for bulk hBN at $1.5\times 10^{14}$ rad/s and $2.9\times 10^{14}$ rad/s, respectively. From Fig. 9(a), the bright areas are distributed in the region of $\beta >1$, showing the contribution of HVPhPs. In Fig. 9(b), we can see a bright circle which is consistent with solution to Eq. (10) (green dashed curve). The radius of the bright circle, $\beta$, satisfies the condition $\sqrt {\varepsilon _1}<\beta <\sqrt {\varepsilon _\parallel }$, indicating that $q_{e}>0$, $q_{o}>0$, and $q_{1}>0$. Therefore, the bright circle is caused by excitation of HSPhPs which, however, are also not the same type of surafce phonon polaritons discussed by Dyakonov because they are not induced by in-plane anisotropy. Furthermore, such HSPhPs can only be excited in Type II hyperbolic band, but not in Type I hyperbolic band.
Fig. 9. Intensity contour of the imaginary part of the p-polarized Fresnel reflection coefficient Im($r_{pp}$) varying with $k_{x}$ and $k_{y}$: (a) Bulk hBN at $1.5\times 10^{14}$ rad/s. (b) Bulk hBN at $2.9\times 10^{14}$ rad/s, where the green dashed circle denotes solutions to Eq. (10). (c) 50-nm-thick hBN slab at $1.5\times 10^{14}$ rad/s. (d) 50-nm-thick hBN slab hBN at $2.9\times 10^{14}$ rad/s. For hBN slab, analytical dispersion relations solved from Eq. (11) are shown as the dashed curves with the numbers denoting the orders of waveguide modes.
Figures 9(c) and 9(d) show the intensity contour of Im($r_{pp}$) varying with $k_{x}$ and $k_{y}$ for a hBN slab of 50 nm thick at $1.5\times 10^{14}$ rad/s and $2.9\times 10^{14}$ rad/s, respectively. Unlike Fig. 6, the dispersion curves of the waveguide modes are in the form of a series of concentric circles. The dashed curves denote the dispersion curves obtained from Eq. (11), which are seen to be in excellent agreement with the bright color strips of Im($r_{pp}$). The numbers marked on the figures denote the corresponding order of waveguide modes. We can find that the 0th-ordered waveguide mode can be excited in Type II hyperbolic band, but cannot exist in Type I hyperbolic band.
5. Conclusion
In this paper, taking hBN as an example, we conducted a comprehensive study on surface and volume phonon polaritons excited with a bulk or a thin slab of uniaxial hyperbolic material, and discussed the influence of optic axis orientation on these polaritons. The excited phonon polaritons, their characteristics and dispersion topology were clarified. Existence of HSPhPs in Type I hyperbolic band was confirmed and it was shown that HSPhPs cannot be excited in the direction of the OA when the OA is parallel to the interface. In the $k_{x}-k_{y}$ plane, HSPhPs and HVPhPs are bounded on the same sides of the asymptotes in Type I hyperbolic band, while in Type II hyperbolic band, they are distributed on different sides of the asymptotes. For a thin hBN slab, it was found that dispersion curves of HSPhPs can split into two branches and one branch merges smoothly with the 1st-ordered dispersion curve of HVPhPs in Type II hyperbolic band. Furthermore, non-Dyakonov surface phonon polaritons were identified which can be excited without evanescent ordinary waves. Finally, our results reveal that HSPhPs can be launched in Type II hyperbolic band but not in Type I hyperbolic band when the OA is perpendicular to the interface. These findings may give new insights about surface and volume phonon polaritons excited with hyperbolic materials and offer new guidance for design of novel infrared optical devices with hyperbolic materials.
Funding
National Natural Science Foundation of China (51576004).
Disclosures
The authors declare that they have no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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