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Abstract
We investigated Goos-Hänchen (GH) and Imbert-Fedorov (IF) shifts on a uniaxial hyperbolic crystal, where a circularly-polarized beam was incident on the crystal from the free space. The GH- and IF-shifts were analytically obtained and numerically calculated for the hexagonal boron nitride. Our results demonstrate that the GH- and IF-shift spectra are complicated and completely different in and out the hyperbolic frequency-bands (the reststrahlen bands in the infrared region). At the critical or Brewster angle, concisely analytical expressions of GH-shift was found, which explicitly state the optical-loss dependence of GH-shift at these special angles. We found the GH-shifts are very large at the critical and Brewster angles. It is very necessary to know these effects since hyperbolic materials are usually applied in the nano- and micro-optics or technology fields.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The reflection and refraction of electromagnetic radiation at a plane dielectric interface are the most basic optical phenomena described in optical textbooks. Behind these phenomena, one can find two kinds of interesting effects from the reflective and refractive beams (the secondary beams), i.e. the spatial and angular shifts [1]. In disregard of the angular shifts, there are two types of spatial shifts with respect to the incident plane. The first type is the in-plane shift or the Goos-Hänchen (GH) shift [1–4], originally found for the total internal reflection. The GH shift was theoretically described in both the classical or quantum cases [1,5–8] and was experimentally observed [3,9–11]. This shift is tightly related to the natures of the interface, material and incident beam, originating from the dispersion of reflective or refractive coefficient and mathematically determined by the gradient of the coefficient phase with respect to the incident angle. It also exists in the case of partial reflection for some special interface or optical-loss material, for example the monolayer MoS2-coated surface and grapheme-coated surface [12–13], or a metal or metal-metamaterial [14–15]. The investigation of this shift was first extended to anisotropic transparent media [16–17], metals [14–15,18], photonic crystals [19–20] or metamaterials [1–2,11]. The second type is the out-plane shift, or the Imbert-Fedorov (IF) shift [21]. This shift exists upon a circularly- polarized or elliptically-polarized beam of incidence [1,21–25]. It is one of spin-Hall effects of electromagnetic radiation related to the geometric Berry phase [26], originating from the spin-orbit interaction in the beam. The relevant theory was experimentally verified twelve years ago [25]. For GH- and IF-shifts based on a dielectric interface, the optical loss can be generally ignored [4–7], but it should be considered at the interface with metals and dispersive materials as well as artificial structures including metal or dispersive components [13–15,18] since optical absorption and scattering from the internal imperfection (impurities and defects) in the materials are not easy to be eliminated [27] and the electric conductance leading to optical-loss in metals always exist [27–28].
In the past decade, the GH-shift was found on the surface of photonic crystals in a forbidden band [19–20] and on the surface of metal or metamaterial with negative permittivity [14–15,18]. It has been well- known that ionic crystals possess total- reflection bands or the so-called reststrahlen bands [29]. Our recent work investigated the GH-shift on these crystals [30]. Natural hyperbolic crystals possess at least a reststrahlen band in which the principal values of the individual permittivity are opposite in sign [31–35], so the dispersion equation of electromagnetic radiation is hyperbolic. As a representative example, the hexagonal boron nitride crystal (hBN) has two separate reststrahlen bands, and its longitudinal and transverse principal values are opposite in sign for frequencies in either reststrahlen band [32–33,35]. It is a strong uniaxial and dispersive crystal. We will focus on the GH and IF shifts on the surface of such a natural hyperbolic crystal. For a circularly-polarized paraxial incident-beam, each radiation in the beam is composed of one s-wave (the TE wave) and one p-wave (the TM wave), and the phase difference between them is π/2. Due to the strong anisotropy and hyperbolicity, there are three cases: (1) both the s- and p-waves are partially reflected from the crystal surface, (2) one is approximately totally reflected and the other is partially reflected and (3) both are approximately totally reflected due to optical loss. As demonstrated in our previous work [36], the reflective spectra are very complicated in amplitude and phase, so we think that the GH- and IF-shift spectra of reflection also will be complicated and impenetrable for a natural hyperbolic crystal. We will discuss these effects in frequency regions including a reststrahlen band.
2. Model and analytical calculations
The permittivity of a uniaxial hyperbolic crystal is a diagonal matrix with elements ${\varepsilon _t}$, ${\varepsilon _t}$ and ${\varepsilon _l}$ in the principal-axis frame. If the c-axis of the crystal lies in the incident plane and is at an angle θ relative to the z-axis (the direction normal to the crystal surface), as shown in Fig. 1(a), the permittivity is a non-diagonal matrix in the present coordinate system, i.e.
Fig. 1. Configuration of incidence-reflection and sketch of shifts wherein the c-axis of crystal lies in the x-z plane (the incident plane) and is at a angle θ relative to the z-axis, and the x-y plane is the surface : (a) the geometry of incidence-reflection and coordinate system, (b) GH-shifts and (c) IF-shift, where the light line indicated with kr is the geometric light line (G-Line).
Subsequently, we discuss several special angles that are always met in reflective-refractive optics, i.e. the critical angle and Brewster angle.
For a loss-less dielectric interface, the total inner reflection occurs for an optical wave incident on an optically thin medium from an optically thick one if the incident angle exceeds the critical angle. For our issues, the critical angle involves that the principal part of ko or ke becomes an imaginary quantity from a real quantity. Therefore, the critical angles can be obtained with the following equations
where the superscript R indicates the real part of a relevant quantity. At the critical angle $\beta _c^s$, the GH shift of the s-wave is reduced toFor an incident p-wave, there is another special angle, the Brewster angle (${\beta _b}$). At this angle, the p-wave is totally transmitted (i.e. no reflection) through the loss-less dielectric interface. However, in the present case it should be noted that the reflective ratio of the p-wave is very small but nonzero at the Brewster angle owing to the existence of optical loss. According to the above description, this angle is approximately satisfies ${\gamma ^R} = 1$ on the basis of Eq. (3). At the same time ${\gamma ^I}$ is a small quantity that is demonstrated later. We first find an explicit expression fulfilled by the Brewster angle to be
3. Numerical results and discussion
We use the hexagonal boron nitride crystal (hBN) to offer numerical results. It has a layered structure along its c-axis. In its any layer, alternate B and N atoms are connected with the covalent bond to form the hexagonal profile. The Van der Waals bond links adjacent layers. We assume that its dimension and thickness are much larger than wavelength and attenuation length of the transmitted wave so that the boundary effect can be ignored, and meanwhile its surface is an ideal and smooth plane. The hBN is an interesting van der Waals crystal and is a naturally hyperbolic material [32–33,35]. It possesses two separate reststrahlen frequency bands (RBs). It is a type-I HM in the RB-I with negative $\varepsilon _l^R$ and positive $\varepsilon _t^R$, but it is a type-II HM in the RB-II where $\varepsilon _l^R$ is positive and $\varepsilon _t^R$ is negative. Therefore, the hBN is representative. Its longitudinal and transversal principal-values of permittivity can be uniformly expressed with the function $\varepsilon = {\varepsilon _\infty }[1 + (f_{LO}^2 - f_{TO}^2)/(f_{TO}^2 - {f^2} - i\tau f)]$ [32,35–36,38,39] where $f_{LO}^{}$ is the frequency of longitudinal optical phonon, and ${f_{TO}}$ is that of transverse optical phonon. The damping constant τ is responsible for optical loss in theory since it leads to the imaginary part of the permittivity. In fact, the optical loss mainly originates from impurities and defects in the crystal that bring on optical absorption and scattering. The damping constant that better matches with the experimental results is situated in the range of $\textrm{1}\textrm{.0}c{m^{ - 1}} < \tau < 10.0c{m^{ - 1}}$[27,35–40]. It is fixed at $\tau \textrm{ = 5}\textrm{.0}\,\,c{m^{ - 1}}$ in our numerical calculation. For the longitudinal principal-value ${\varepsilon _l},$ the physical parameters are ${\varepsilon _\infty } = 4.95$, ${f_{LO}} = 825c{m^{ - 1}}$ and ${f_{TO}} = 760c{m^{ - 1}}$; for the transversal principal- value ${\varepsilon _t},$ the parameters are ${\varepsilon _\infty } = 4.52$, ${f_{LO}} = 1610c{m^{ - 1}}$ and ${f_{TO}} = 1360c{m^{ - 1}}$[27,32,36,38,39]. For convenience, we take ${f_t} = 760c{m^{ - 1}}$ as the reference frequency. The two separate RBs of the hBN are ${f_t} < f < 1.0855{f_t}$ (RB-I) and $1.7895{f_t} < f < 2.118\textrm{5}{f_t}$ (RB-II). The hBN exhibits either mainly metallic or mainly dielectrically- optical response in either RB, depending on the polarization and frequency of electromagnetic radiation. It is an elliptical material outside the RBs. It needs to be further pointed out that the hBN is a uniaxial anisotropic crystal whose anisotropic axis is just its c-axis. Compared with metals, its optical loss is much lower [27,38–40].
We are interested in the GH and IF shifts in frequency ranges including either RB. Subsequently, we are going to discuss numerical results. We first focus on the two high-symmetry geometries, i.e. geometry-I ($\theta \textrm{ = }{\textrm{0}^o}$) with the c-axis vertical to the surface and geometry-II ($\theta \textrm{ = 9}{\textrm{0}^o}$) with the c-axis parallel to the x-axis and the surface. Figure 2 illustrates the GH and IF shifts in a frequency range including the RB-I. It should be explained that the reflective coefficient of the s-wave depends on only ${\varepsilon _t}$ positive and slowly changeable in the range. In consequence, the imaginary part of this coefficient is a slowly-changing small quantity, so the GH shift of the reflective s-wave is an invisibly tiny value, induced by the optical loss in both the geometries. Due to $Re ({\varepsilon _t}) \gg 1$ in this frequency range, the critical angle is absent for the s-wave. In contrast to the s-wave, the reflective p-wave is closely related to not only ${\varepsilon _t}$ but also ${\varepsilon _l}$ that is dramatically changeable with frequency. In addition to these, the reflective p-wave changes with θ. Consequently, both the critical and Brewster angles exist for this wave, and the GH shift is significant and possesses its peak values in either geometry. Figure 2(a) shows that the maximum (negative) of the shift reaches about one-hundred times the vacuum wavelength in the geometry-I. Figure 2(c) illustrates that the maximum (positive) in the geometry-II is about ten times the vacuum wavelength. In order to know why the shift-peaks of the GH-shift exist, we illustrate the GH-shift at the critical and Brewster angles, as demonstrated by Fig. 3. Combining Fig. 2(a) with Figs. 3(a) and 3(b), we find that the shift-peaks emerge at the critical and Brewster angles. The sharper peaks correspond to the Brewster angle {see Fig. 3(b)}, but the blunter peaks are located at the critical angle {see Fig. 3(a)}. For example, the sharper peak on the black curve in Fig. 2(a) corresponds to the Brewster angle and the blunter one is situated at the critical angle. The peak-value of GH-shift in Fig. 2(c) is situated at the critical angle, where the Brewster angle is absent. For example, Fig. 2(c) shows that the peak-value $\Delta {x_p}/{\lambda _0} = \textrm{9}\textrm{.75}$ corresponds to $\beta \textrm{ = }{\beta _c} = {45^o}$.
Fig. 2. GH and IF shifts for various angles of incidence in a frequency range including the RB-I: (a) and (b) in the geometry-I; (c) and (d) in the geometry-II. The left two diagrams illustrate the GH shift and the right those show the IF shift. Here there almost is not the GH shift of the reflective s-wave.
Fig. 3. GH-shift corresponding to the critical angle and the Brewster angle in different geometries: the red curves show the critical or Brewster angle and the black curves represent the relevant shift.
The eigen modes of IF-shift are the left and right circularly-polarized waves, as described above, so the reflective IF shift is commonly contributed by the reflective s-wave and p-wave. Every curve of the IF shift carries a dip at the right boundary of the RB-I. The dips in the two geometries are different. The dip is small in the geometry-I, but that is large and even reaches the zero-point in the geometry-II.
Combining Eq. (6b) with Figs. 5(a) and (b), we think the dip is caused by the rapid decrease of the e-wave reflection. This rapid change makes the spin-orbit interaction abnormal to be responded in IF-shift. This phenomenon also is present in the frequency range corresponding to the RB-II.
In a frequency range including the RB-II, Fig. 4 illustrates the frequency-dependence of GH- and IF-shifts for various incident angles. Obviously, there is not only the GH-shift of the p-wave but also the significant shift of the s-wave.
In the geometry-I, the shift of the s-wave is negative (backward), but that of the p-wave is positive (forward). The shift of the s-wave nearly disappears at the position of the peak value of the p-wave shift, vice versa, as shown in Fig. 4(a). The IF shift exhibits two dips. The left dip is situated at the right boundary of the RB-II. The right one approximately corresponds to the Brewster angle, where the p-wave reflection is induced by the damping or the loss and is very weak, as shown by the dot lines in Fig. 5(c).
Fig. 4. GH and IF shifts for various angles of incidence in a frequency region including the RB-II. (a) and (b) in the geometry-I. (c) and (d) in the geometry-II.
Fig. 5. Reflective ratios of s-wave and p-wave: the upper two diagrams exhibits those in the region including RB-I and the bottom two diagrams offer those in the range including the RB-II. The reflective ratio of the s-wave is given only in the geometry-I since it is the same in the two geometries.
In the geometry-II, the GH shifts of the s-wave and the p-wave both are negative (backward). The s-wave GH shift possesses one peak value that corresponds to the critical angle. The GH shift of the p-wave exhibits two peaks. The tall and sharp peak is related to the Brewster angle and the short and blunt peak is located at the critical angle. The IF shift has a zero-point located at the right boundary of the RB-II. This shift does not exhibit clearly abnormal behaviors at the Brewster and critical angles.
Unlike the GH shift that results from only the phase gradient of the relevant reflective coefficient (or from the dispersion of the Fresnel reflective coefficient), the IF shift is determined by multiple factors. It originates from the spin-orbit interaction and is closely related the geometrical Berry phase. The IF shift should be abnormal at the Brewster angle since the reflection of the p-wave rapidly become especially weak and further the spin angular momentum also is very small at this angle, for example the sharper dip of the IF shift exhibited by Fig. 4(b) in the geometry-I and frequency range related to the RB-II. However, we do not find its relevant dip in other cases. It is because of the damping and different anisotropies of the hBN crystal in different frequency-ranges and geometries.
In practical application, another important physical quantity must be considered, which is the reflective ratio defined by ${R_{s,p}} = r_{s,p}^\ast {r_{s,p}}$. The GH shifts of the s-wave and p-wave fulfill different regulations, so we are going to separately illustrate numerical results for the s-wave and p-wave. It should be noted that the ratio of the s-wave is independent of geometry (θ), so the relevant curves are offered in the Geometry-I, as illustrated in Fig. 5.
We first discuss the reflective ratios in the upper two diagrams related to the RB-I. The reflective ratio of the s-wave is smooth and almost unchangeable with frequency, but obviously increases with incident angle. The reflective ratio of the p-wave in the geometry-I is higher in the vicinity of the right boundary of the RB-I. It is completely different in the geometry-II, or it rapidly changes with frequency in the same vicinity and is higher in the RB-I, as shown in Figs. 5(a) and (b). The bottom two diagrams related to the RB-II demonstrates the picture of physics opposite to that in the upper two diagrams. The reflective ratios of the two waves in the geometry-I both rapidly change with frequency in the vicinity of the right boundary of the RB-II, but that of the p-wave in the geometry-II is higher in the same vicinity, as shown in Figs. 5(c) and (d).
Finally, we examine the reflective ratios corresponding to the peak values of GH-shifts. Figures 5(a) and (b) correspond to Fig. 2. We take several examples to indicate relations between the reflective ratio and the peak values of GH-shift. In the geometry-I, Fig. 2(a) and Fig. 5(a) demonstrate that the largest peak value of the shift of p-wave (see the black solid curve) is about -108 at the Brewster angle $\beta \textrm{ = }{\beta _b}\textrm{ = 7}{\textrm{5}^o}$, corresponding to ${R_p} \approx \textrm{1}{\textrm{0}^{ - 4}}$; the secondary peak value is equal to -3.4 at the critical angle $\beta \textrm{ = }{\beta _c}\textrm{ = 7}{\textrm{5}^o}$, corresponding to ${R_p}\textrm{ = 0}\textrm{.39}$. In the geometry-II, the shift peak- value of the p-wave is about 10 at ${\beta _b}\textrm{ = 4}{\textrm{5}^o}$, and the reflective ratio also is merely ${R_p} \approx \textrm{1}{\textrm{0}^{ - 4}}$. Similar to the analytical conclusion, the GH shift can be very large at the Brewster angle, but the relevant reflective ratio is very small. In the frequency range including the RB-II, Figs. 5(c) and (d) correspond to Fig. 4. The shift at the Brewster angle also is companied by a very small reflective ratio, so we ignore it here and turn to focus on the GH shift at the critical angle in this range. In the geometry-I, Fig. 4(c) and Fig. 5(c) show the shift of the s-wave is equal to -2.4 at the critical angle $\beta \textrm{ = }{\beta _c} = {60^o}$, corresponding to Rs = 0.47. In the geometry-II, Fig. 4(c) and Fig. 5(d) indicate that the shifts of the s- and p-waves are -5.32 and -3.71 at the critical angle, respectively, companied by relevant reflective ratios Rs=0.51 and Rp=0.58. The critical angle is situated outside either RB in the geometry-I and II, but it can be located inside either RB and the GH shift of the p-wave can be larger at the critical angle, as illustrated in Figs. 3(c) and (d) in an ordinary geometry (θ=30°).
We should mention one work [41] that investigated the GH shift on the hyperbolic metal/dielectric metamaterial with spatial dispersion. This paper numerically calculated the GH shift of the reflective p-wave with or without spatial dispersion, but ignored that of the s-wave because nonlocal effect on the s-wave does not exist. The material model used by them corresponds to our geometry-I ($\theta \textrm{ = 0}$) and does not involve the critical angle in the used frequency rang. The high-frequency region corresponds to our RB-I, where the metamaterial is a type-I hyperbolic material and the low-frequency region to our RB-II, where it is of type-II. Similar to the relevant result in Ref.41, the GH shift obtained by us is positive at the Brewster angle in the geometry-I, indicated in Fig. 2(a). The GH shift is very large at the Brewster angle, but the reflective ratio is very small. In addition, the optical-loss in our crystal very low and working frequency is situated in infrared range in the present work.
The imaginary parts of the principal values of permittivity are directly proportional to τ. It means that the shift is enhanced at the critical angle, according to Eqs. (11) and (13), as τ is decreased. τ is determined by the quality of the hBN crystal, as described previously. If we take τ=2.0cm−1, as did in Refs. [27,33–34], the GH-shift at the critical angle is 1.6 times the present result. Moreover if we take τ=1.0 obtained recently [40], the shift is about 2.4 times the present result.
4. Summary
We have analytically and numerically calculated the GH and IF shifts on the surface of a hyperbolic crystal. The shift spectra is complicated, especially the GH-shift spectra. The GH shifts of the s-wave and p-wave contained in the reflective beam obey completely different equation, so their GH-shifts were separately calculated. In the frequency range including the RB-I, the GH shift of the s-wave is too small to be invisible, but that of p-wave possesses a much larger shift. In the frequency region including the RB-II, their shifts both are evident. It is more interesting that the GH-shifts are very large at the critical and Brewster angles. However, the reflective ratio is very small at the Brewster angle. The IF-shift spectrum is simpler and is abnormal at the Brewster angle in the Geometry-I and near the RB-II. The GH- and IF-shifts both are important in the nano-optics and infrared technology.
Funding
Natural Science Foundation of Heilongjiang Province (ZD2009103).
Disclosures
The authors declare no conflicts of interest.
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