1. Introduction

With a linearly polarized light incident on the surface of an anisotropic crystal, one can generally find the birefringence inside and a linearly polarized reflection outside the crystal [1,2]. This reflection-refraction phenomenon is one of the most elementary and useful optical phenomena, which was well-known long time ago. In recent years, the presence of hyperbolic materials (HMs) brought about one new and attractive field [3–7]. These materials have been paid great attention because of their wide-ranging applications including nano-imaging [8–10], surface phonon polaritons [11–14], sensing [15,16], wave-guiding [17,18], and thermal conductivity and emission [19–21]. In hyperbolic metamaterials (HMMs), hyperbolic phase matching can be achieved with a wide range of material parameters [22], offering access to the use of nonlinear media for which phase matching cannot be achieved by other means.

The HMs can be classified into two types, i.e. type-I and type-II, based on the signs of principal values of their dielectric permittivity [23,24]. The permittivity in the type-I HMs possesses one negative and two positive principal values (${\epsilon }_1\text{<}0$,${\epsilon }_2$and ${\epsilon }_3>0$), and the type-II HMs correspond to the permittivity with two negative and one positive principal values (${\epsilon }_1>0$,${\epsilon }_2$and ${\epsilon }_3\text{<}0$). There are currently two groups of HMs, i.e. HMMs artificially synthesized with a positive-permittivity dielectric and a negative-permittivity material (such as a metal), and naturally hyperbolic materials (NHMs) [25–28]. A HMM can be selectively synthesized to be either type, but a NHM belongs to different types in its different reststrahlen bands [25,30] and its optical loss is very low. In addition, a HM optically acts like a metal in the direction corresponding to one negative principal value, but it exhibits the optical response like an ordinary dielectric in the direction related to a positive principal value. Therefore, we believe the reflective and refractive properties of this HM will be very unique, and tightly depend on the polarization of incident wave and the orientation of HM anisotropic axis (the optical axis) with respect to the surface. Although we found some works describing the optical-reflective properties of HMs [23,29] a little for different aims, we have not found a more comprehensive investigation for reflection-refraction features of HMs, especially for those of NHMs. Our investigation will manifest two characters. The first is to use an NHM that belongs to different types in different hyperbolic frequency bands (HBs), i.e. to the type-I in the lower HB, but to the type-II in the higher HB. The second is that the optical axis of this material is arbitrary in orientation with respect to the material surface and the incident plane.

2. Model and analytical calculations

We propose that a linearly polarized radiation is obliquely incident on the surface of a NHM from an ordinary dielectric (OD) with a relative dielectric constant ${\epsilon }_i$, as shown in Fig. 1 where the optical axis (the dashed line) lies in the$x^{\prime }-y^{\prime }$ plane and is at an angle $\theta$ with respect to the surface. In the principal-axis coordinate system, the relative permittivity of the NHM is a diagonal matrix with elements${\epsilon }_l$,${\epsilon }_t$ and ${\epsilon }_t$. ${\epsilon }_l$ and ${\epsilon }_t$ elements can be uniformly expressed with $\epsilon ={\epsilon }_{\infty }[1+({\omega }_{LO}^2-{\omega }_{TO}^2)/({\omega }_{TO}^2-{\omega }^2-i\omega \tau )]$, but their physical parameters in this formula are different. In the formula, ${\epsilon }_{\infty }$ represents the high-frequency permittivity and ${\omega }_{LO}$ and ${\omega }_{TO}$ are the frequencies of longitudinal and transverse optical phonons with damping constant $\tau$.

 

Fig. 1 Coordinate systems and configuration where the dashed line represents the optical axis that lies in the $x^{\prime }-y^{\prime }$ plane and is at the angle $\theta$ with respect to the surface, the angle between the incident plane (the $x-y$ plane) and the $x^{\prime }-y^{\prime }$ plane is $\alpha$, and the incident angle is indicated with $\beta$.

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In the $xyz$ coordinate system shown in Fig. 1, the permittivity is an off-diagonally matrix, which can be obtained from its diagonal matrix in the principal coordinate system by means of the two rotation transformations. We find its expression to be a symmetrical matrix,

For the s-polarized incidence (the s-incidence, or the TE incidence), the electric field of incident wave is along the z-axis and the magnetic field lies in the incident plane. Based on the z-component of electric field in either space, we first show two magnetic components above the interface as

For the p-polarized incidence (the p-incidence, or the TM incidence), the magnetic field of incident wave is in the z-direction and the electric field lies in the incident plane. According to the requirement that the tangential components of magnetic field are continuous at the interface, we obtain two equations,

For both the s- and p-incidences, we have found the components of the reflective field and the refractive fields, which are some complex quantities. Here we define the reflective coefficient $R_s=\left|E_r\right|/E_{iz}$ and the reflective phase angle ${\gamma }_n=a\tan [Im(E_{rn})/Re(E_{rn})]$for the s-incidence, and define the reflective coefficient $R_p=\left|H_r\right|/H_{iz}$ and the reflective phase angle ${\gamma }_n=a\tan [Im(H_{rn})/Re(H_{rn})]$ for the p-incidence, where $n=x,yorz$. For the two refractive branches in the NHM, we will directly calculate their Poynting vectors and electric-field profile near the interface to explore their characters. The Poynting vector is defined with $S_j=0.5\mathrm{Re}(E_j^{*}\times H_j)$ where$j=o$ for the o-wave and $j=e$ for the e-wave.

3. Numerical results and discussions

In subsequent numerical simulations, we use the free space to take the place of the ordinary dielectric above the NHM, or ${\epsilon }_i=1$. The hexagonal boron nitride (hBN) is chosen as the NHM mentioned above, whose physical parameters are well-known. For the longitudinal component of the permittivity (${\epsilon }_l$), ${\epsilon }_{\infty }\text{=}4.95$, ${\omega }_{LO}=825c{m}^{-1}$ and ${\omega }_{TO}=760c{m}^{-1}$. For the transversal component (${\epsilon }_t$), ${\epsilon }_{\infty }\text{=}4.52$, ${\omega }_{LO}=1610c{m}^{-1}$ and ${\omega }_{TO}=1360c{m}^{-1}$. The damping constant is taken as $\tau \text{=}2.0c{m}^{-1}$ [7,10,23,30] responsible for optical loss. The longitudinal and transverse optical-phonon frequencies are measured in wave-number unit, so $f=\omega /2\pi c$ (c denotes the light velocity in the vacuum) and set $f_t=760c{m}^{-1}$ for numerical calculations. The hBN has two HBs. One is $f_t<f<1.0855f_t$ (HB-I) corresponding to $\mathrm{Re}({\epsilon }_l)<0$, where the hBN is the type-I HM. The other is $1.\text{7895}{f}_t<f<2.1184f_t$ (HB-II) related to $\mathrm{Re}({\epsilon }_t)<0$, where the hBN is the type-II HM. Figure 2 illustrates the HBs (see the shadow regions) and the longitudinal and transversal components versus f, where the damping is ignored. It will be helpful for understanding reflective and refractive spectra of the hBN.

 

Fig. 2 The longitudinal and transverse components in the relative permittivity versus frequency for the hBN, where $f_t=760c{m}^{-1}$and $\tau \text{=}0$. The two hyperbolic frequency bands are indicated by the two shadow regions.

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Before discussing numerical results of reflection and refraction, we must note that $K_{oy}$ and $K_{ey}$ are two complex quantities containing a real part and an imaginary part. For convenience, we take the incident-field amplitude to be $E_{iz}=1$ for the s-incidence and $H_{iz}=1$ for the p-incidence, and always hold that the imaginary parts of $K_{oy}$ and $K_{ey}$ are positive. For the energy-flux densities (the Poynting vectors) of the two refractive waves, we apply that of the incident wave as a unit, i.e. $S_i={({\epsilon }_0/{\mu }_0)}^{1/2}|E_{iz}{|}^2=1$ for the s-incidence and $S_i={({\mu }_0/{\epsilon }_0)}^{1/2}|H_{iz}{|}^2=1$ for the p-incidence. It should be reminded that one main character of evanescent waves is that the y-component of the Poynting vector is very small and originates from the absorption and the wave fields rapidly attenuates along the surface normal.

Figures 3 illustrate the reflective and refractive waves for the s-incidence. We first look into the reflective spectra in Figs. 3(a)-3(b’). The field-components in the plots represent the reflective amplitudes and are measured in incident amplitude. We first find that a special frequency point (SP) divides either HB into two different ranges wherein the reflective wave exhibits different behaviors. The SP is found in numerical calculation to correspond to $\mathrm{Re}({\epsilon }_{yy})=0$, depending on only the orientation of the optical axis. We know from Eq. (2a) and Fig. 2 that the principal part of ${\epsilon }_{yy}$ is a negative real number on the left side of the SP, but it is a positive real number on the right side. For the phase angle of reflective wave, we take the phase angle of the incident wave as the reference, which is 0 at the geometrically incident point on the NHM surface. The x- and y-components of the reflective wave are identical in amplitude and phase for the incident angle $\beta =\pi /4$, but the z-component is different from them in not only amplitude but also phase angle. Therefore we conclude that the reflection is a double reflection and the two reflective branches (a TE wave and a TM wave) compose an elliptically polarized reflective wave. The reflective spectra in the HB-I are illustrated in Figs. 3(a) and 3(a’) and the reflective spectra in the HB-II are shown in Figs. 3(b) and 3(b’), where these reflective spectra evidently reflect the boundaries of the HBs and the SP position. They also demonstrate that the z-component of reflective electric-field is obviously different in not only amplitude but also phase angle from the other two components. In the HB-I, the phase difference is approximately equal to π on the left side of the SP, but it evidently increases from π when f shifts right from the SP and then reaches 2π as f goes out the HB-I. In the HB-II, the phase difference obviously increases with f, but it is equal to about 0 out the HB-II.

 

Fig. 3 The reflection and refraction with $\alpha \text{=}\beta \text{=}\theta \text{=}\pi /4$ for the s-incidence, where the vertical point line in the four top plots indicates the frequency position of the SP and shows the interface in the two bottom plots. $E_{xy}$ and ${\gamma }_{xy}$ represent not only $E_x$ and ${\gamma }_x$ but also $E_y$ and ${\gamma }_y$. (a) The reflection versus frequency near or in the HB-I. (b) The reflection versus frequency near or in the HB-II. (c) The refractive energy-flux densities corresponding to (a) and (d) the refractive flux densities corresponding to (b). The electric-field profile near the interface (e) for a fixed frequency in the HB-I and (f) for a fixed frequency in the HB-II.

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Figures 3(c)-3(d’) illustrate the Poynting vectors of the refractive waves for the s-incidence. Its main features are described as follows. In the HB-I, the o-wave flux density is very weak and lies in the incident plane, where $K_{oy}$ contains a very small imaginary part introduced by the absorption and a relatively large real part as shown in Fig. 2 and Eq. (5a), so the o-wave is normal refractive wave. The e-wave flux density is much stronger and possesses a large z-component so that it seriously deviates from the incident plane, especially near the SP or at the right boundary of the HB-I. The y-component of the e-wave flux density is very small on the right side of the SP, so the e-wave principally is an evanescent wave there. The e-wave practically is a normally refractive wave on the left side since the y-component of the Poynting vector is larger and the principal part of $K_{ey}$ is a positive real number. In the HB-II, Figs. 3(d) and 3(d’) exhibit very interesting features of energy-flux density. We first realize that the flux density seriously deviates from the incident plane for both the o-wave and e-wave. The o-wave always has a very small y-component of the Poynting vector and the principal part of $K_{oy}$ is imaginary, as demonstrated in Eq. (5a) and Fig. 2, so the o-wave is an evanescent wave and its flux density approximately lies in the interface plane. The e-wave is a normal refractive wave on the left side of the SP, where the principal part of $K_{ey}$ is a positive real quantity. However, it is an abnormal evanescent wave on the right side of the SP where the Poynting vector approximately lies in the interface plane, as illustrated Fig. 3(d’), but the real and imaginary parts of $K_{ey}$ are matched. For a normal evanescent wave, the real part of $K_y$ should be minor and the imaginary part should be major, so we call the e-wave an abnormal evanescent wave there. The e-wave also exhibits other abnormal behaviors in the vicinity of the SP. The first behavior is that its energy-flux density is abnormally high at the SP, can reach about 6 times that of the incident wave. The second behavior is that the Poynting vector dramatically reverses when frequency goes through the SP and is roughly opposite in direction on the two sides of the SP. This reverse at the SP is a kind of switching effect which can be used to separate a beam of different-frequency lights into two beams. Although the flux density of the e-wave near the SP is much larger than that of the incident wave, the energy conservation law is still obeyed since this law requires the y-component continuity of total flux density at the interface.

We also illustrate the electric-field profile for a fixed frequency near the SP. For the fixed frequency in the HB-I, Fig. 3(e) shows that the o-wave is a weaker refractive wave and the e-wave is a much stronger near the interface. The e-wave obviously attenuates with the penetration depth but the o-wave does not, so the o-wave is principal and the e-wave can be ignored far from the interface. It is consistent with Figs. 3(c) and (c’). For the fixed frequency in the HB-II, Fig. 3(f) demonstrates that the o-wave is a very weak evanescent wave near the SP, but the e-wave is much stronger and is an oscillatory and attenuative wave. The y-component of the e-wave is much larger than the others and even the incident electric field. Therefore, the e-wave is major in the hBN. This plot is consistent with Figs. 3(d) and (d’). The incident and reflective waves do not attenuate and not obviously oscillate, especially for the frequency in the HB-II. The reason is that the absorption was ignored and the wavelength is relatively longer in the OD. We also realize that the abnormity of the e-wave energy-flux density comes from the unusual behavior of the y-component of the e-wave electric field near the SP and the e-wave is approximately polarized in the surface normal at the SP. Different from the o-wave, the e-wave is highly condensed at the inner surface of the NHM.

Figures 4 illustrate the reflective and refractive waves for the p-incidence. Although the differences between Fig. 3 and Fig. 4 are not very obvious in character, the numerical differences are evident. For example, in the HB-I, the reflective coefficient is manifestly larger for the s-incidence than for the p-incidence, as exhibited in Figs. 3(a) and 4(a). The o-wave is much weaker in energy-flux density for the s-incidence than for p-incidence, as explored in Figs. 3(c) and 4(c), but the e-wave is opposite to the o-wave, as shown in Figs. 3(c’) and 4(c’). In addition, in the HB-II, the o-wave is much stronger in energy-flux density for the s-incidence than for the p-incidence, but the e-wave is opposite, as shown in Figs. 3(d) and 3(d’) and 4(d) and 4(d’). We should emphasize that the energy-flux density of the e-wave in the HB-II is even 11 times that of the incident wave, as shown in Fig. 4(d’). On these points, the refractive waves for the p-incidence are obviously different from the refractive waves for the s-incidence. In addition, we illustrate that the electric-field profile near the surface for a given frequency near the SP in either HB. The e-wave and o-wave are matched in field-strength in the HB-I, as illustrated in Fig. 4(e). The e-wave is localized at the surface, but the o-wave is not. However, the o-wave is very weak and the e-wave is much stronger in the HB-II and strongly localized at the surface, as illustrated in Fig. 4(f).

 

Fig. 4 The reflection and refraction with $\alpha \text{=}\beta \text{=}\theta \text{=}\pi /4$ for the p-incidence, where the vertical point line indicates the frequency position of the SP in the four top plots and shows the interface in the two bottom plots. $H_{xy}$ and ${\gamma }_{xy}$ represent not only $H_x$ and ${\gamma }_x$ but also $H_y$ and ${\gamma }_y$. (a) The reflection versus frequency near or in the HB-I. (b) For the reflection versus frequency near or in the HB-II. (c) The refractive energy-flux densities corresponding to (a), and (d) those corresponding to (b). The electric-field profile near the interface (e) for a fixed frequency in the HB-I and (f) for a fixed frequency in the HB-II.

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The SP is a special frequency point corresponding to $\mathrm{Re}({\epsilon }_{yy})=0$ and situated in either HB. It is evidently important for the e-wave in the hBN, at which the e-wave carries a large energy-flux density. Based on the mathematical analysis in the above section and the numerical results in this section, we further investigate the features of the e-wave in the vicinity of the SP. In order to guarantee the reasonableness of numerical results in physics, the imaginary part of $K_{ey}$ must be positive, as described below Eq. (5). We found in the previous numerical simulations that this condition requires choosing the negative sign in Eq. (5b) in a frequency range including the SP and choosing the positive sign in the rest frequency ranges. Thus, we obtain $K_{ey}\approx -2{\epsilon }_{xy}{k}_x/{\epsilon }_{yy}$from Eq. (5b) as ${\epsilon }_{yy}$ is a tiny complex quantity in the vicinity of the SP. It is manifest that K_ey is comparatively very large at the SP, so we find $E_{ex}\approx k_x{E}_{ey}/K_{ey}$ from Eq. (4a) or (4b) and $E_{ez}\approx {\epsilon }_{yz}{f}^2{E}_{ey}/K_{ey}^2$ from Eq. (4c). It proves that $E_{ey}$ is much larger than the other electric-field components. Therefore, the electric field of the e-wave is approximately perpendicular to the surface in the vicinity of the SP. These are also demonstrated by Figs. 3(e) and 3(f) or 4(e) and 4(f).

Now we turn to discuss the reflection versus the incident angle. We illustrate the results obtained for the p-incidence in Fig. 5. All the components of the reflective magnetic field are not generally ignorable, but the z-component is principal. The reflective coefficient abnormally decreases as the incident angle is increased, and its minimum is situated at a larger incident angle. The x- and y-components are identical for $\beta =\pi /4$, which has been demonstrated by Eq. (18c).

 

Fig. 5 The reflective amplitudes and coefficient versus the incident angle for $\alpha \text{=}\theta \text{=}\pi /4$ and for the p-incidence: (a) For the reflection with a fixed frequency in the HB-I and (b) for the reflection with a fixed frequency in the HB-II.

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Finally, the dependence of reflection on the orientation of incident plane is shown in Fig. 6 for the p-incidence. We first find the asymmetry of reflection since turning the orientation of the incident plane by π is equal to chancing the incident angle from β into –β. Figure 6 shows that the z-component of the reflective magnetic field is a periodical function of α, with the period of π. However, the x- and y-components are the same anti-symmetrical function of α, with respect to the point of $\alpha =\pi$. Therefore, the reflection is asymmetric or it is different for $-\beta$ and $\beta$. Figure 6(a) exhibits four special orientations of the incident plane ($\alpha =0$, 1.295, π and 4.988 rad) in which the x- and y-components disappear so that the reflective wave is a p-polarized wave similar to the incident wave while only one refractive wave is found in the NHM, refer to Eqs. (14b) and (14c). The double reflection and the double refraction become the single reflection and single refraction in these special orientations. However, we see such two special orientations from Fig. 6(b), i.e. $\alpha =0$ and π. For the s-incidence, the similar phenomena also can be found.

 

Fig. 6 The magnetic-field components of reflective wave versus the orientation of the incident plane for $\beta \text{=}\theta \text{=}\pi /4$ and the p-incidence, where the solid and dot curves represent the real and imaginary parts of the magnetic-field components, respectively, and the red indicates the x- and y-components but the blue shows the z-component: (a) The reflective field for a fixed frequency in the HB-I and (b) the reflective field for a fixed frequency in the HB-II. The horizontal dashed line is the zero-point line.

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4. Conclusion

We predicted the double reflection from the surface of naturally hyperbolic uniaxial materials. The two reflective branches, a TE wave and a TM wave that are different in amplitude and phase, can generally constitute an elliptically-polarized reflective wave. In some special orientations of the incident plane customarily used in theory and experiment, the reflective wave is a TE wave or a TM wave. The reflective wave is generally asymmetric, i.e. it is different for a positive angle and a negative incident angle ($\beta$ and $-\beta$). The reflective coefficient continuously decreases to its minimum and then increases to 1 as the incident angle is enlarged from 0 to π/2. If the incident wave is a linearly polarized beam, the TE and TM branches contained by the reflective wave can be separated according to the Goos-Hänchen effect [32–34] since the phase difference between the TE and TM branches are evident and changes with the incident angle and frequency as well as the orientation of the incident plane. There is a special point (SP) in either hyperbolic band (HB) and it divides either HB into two frequency ranges. The reflection or refraction manifests different behaviors in the two different ranges. This SP is determined by only the orientation of the anisotropic axis.

For the two refractive waves, the o-wave is a normal refractive wave in the HB-I for either incidence. Its energy-flux density lies in the incident plane. It is an evanescent wave in the HB-II and its energy-flux density obviously deviates from the incident plane to be approximately parallel to the interface. The e-wave possesses completely different characters on the two sides of the SP, i.e. it is a normally refractive wave on the left side of the SP and is an abnormal evanescent wave on the right side. The energy-flux density of the e-wave is every high near the SP, is more even than 10 times that of incident wave and it seriously deviates from the incident plane, especially in the HB-II. The last main character of the e-wave is that it is highly condensed at the inner surface for frequencies near the SP. It is very interesting that the energy-flux density of the e-wave is opposite in direction on the two sides of the SP in the HB-II so that the switching effect is very obvious at the SP. These new characters of the NHM should be applicable in optical and optoelectronic technologies.