Enhanced displacements in reflected beams at hyperbolic metamaterials

Chenran Xu; Jingping Xu; Ge Song; Chengjie Zhu; Yaping Yang; Girish S. Agarwal

doi:10.1364/OE.24.021767

1. Introduction

A beam of light reflected off an interface experiences a longitudinal Goos-Hänchen (G-H) shift [1] as well as a transverse Imbert-Fedorov (I-F) shift [2,3]. Both the G-H and I-F shifts have been widely studied in different systems, such as the dielectric [4], metallic [5] and photonic crystal [6] slabs. Shifts at the surface of more complex systems like topological insulators are also studied now [7]. Even the control of the G-H shifts by using a pump beam has been considered [8]. In general, it is difficult to observe the G-H shift at a plane metallic interface in experiments since the shift has the same order as the wavelength of the incident light. However, Merano et al. [9] succeeded in observing the shifts at a bare metallic surface. Various schemes have been proposed to enhance the G-H shift in order to make a direct measurement. These include the metasurfaces [10], thin films [11], and left-handed metamaterials [12]. In recent years, the beam shifts also stimulated the demonstration of weak measurement of shifts [13–15], and led to applications in sensing and switching [16, 17].

As another interesting effect of the lateral displacement, the spin Hall effect of light (SHEL) has been studied both theoretically [18, 19] and experimentally [20–22]. When a linearly polarized beam is reflected, the beam will split into left- and right-circularly polarized components. The splitting results from an effective spin-orbit coupling [18]. The SHEL may help manipulate the photon spin effectively and thus encourage the development of new generations of all-optical devices.

In this paper we propose a scheme using a thin slab made of hyperbolic metamaterial (HMM) to enhance the lateral displacements. Recently, hyperbolic metamaterials (HMMs) have provoked much interest due to their hyperbolic dispersion relation [23]. As a result of the novel property, this metamaterial can support the propagation of high wavevectors. Thus, many interesting properties and applications appeared, including negative refraction [24], sub-diffraction imaging [25, 26], sub-wavelength modes [27, 28], and thermal emission engineering [29]. It is known that there are two most popular HMM structures: a multilayer structure made of alternating metallic and dielectric layers [32], and a nanowire structure fabricated by embedding metallic nanowires in a dielectric medium [33]. In this paper, we focus on multilayer HMM since it’s easier to fabricate and realize.

For the light displacement on HMM, the recent literature [34, 35] mainly discussed the impact of the volume fraction and nonlocal effect on the G-H shifts of reflected light at the surface of infinite multilayer HMM. In this paper, not only have we studied the G-H shift of a reflected Gaussian beam at the surface of a thin HMM slab made of Ag and TiO₂, but we have also calculated the I-F shift and the SHEL in the same scheme. The G-H shift, I-F shift and SHEL are calculated, analyzed and compared between slabs made of HMM, Ag and TiO₂.

2. Effective medium theory of HMM

The structure of multilayer HMM is shown in Fig. 1(a). In general, an effective medium theory [36, 37] can be used to describe the optical properties of multilayer or nanowire media. In HMM, the diagonal elements of the permittivity tensor satisfy the hyperbolic relation. Here, the HMM is made of alternating layers of silver and TiO₂ used in the experiment [32]. According to the effective medium theory, the effective permittivities of the multilayer structure are given by [38],

ϵ_{⊥} = f ϵ_{Ag} + (1 - f) ϵ_{{TiO}_{2}},

ϵ_{∥} = \frac{ϵ_{Ag} ϵ_{{TiO}_{2}}}{f ϵ_{{TiO}_{2}} + (1 - f) ϵ_{Ag}},

where f is the filling fraction of silver and the permittivity tensor has the form of diag(ϵ_⊥, ϵ_⊥, ϵ_∥) with respect to the principal axis. In our case, z-axis is the optical axis of the material. Within the Drude model, the permittivity of the silver is

ϵ_{Ag} = ϵ_{\infty} - ω_{P}^{2} / [ω (ω + i τ^{- 1})]

, with the parameters ϵ_∞ = 3.7, the bulk plasmon frequency ω_P = 1.4 × 10¹⁶rad/s and the relaxation time τ = 0.45 × 10⁻¹⁴s. The permittivity of TiO₂ can be well described by the formula

ϵ_{{TiO}_{2}} = 5.913 + 0.2441 / (λ^{2} - 0.0803)

, where λ is the wavelength in micrometer [39].

Fig. 1 (a) Illustration of a multilayer HMM; (b)-(d) describe the effective permittivities ϵ_⊥ and ϵ_∥ for a Ag/TiO₂ multilayer HMM with a silver filling fraction of f = 0.35. (b) plots the real part of ϵ_∥ and ϵ_⊥; (c) and (d) plot the imaginary part of ϵ_∥ and ϵ_⊥, respectively. The red and blue lines correspond to ϵ_∥ and ϵ_⊥, respectively.

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Figures. 1(b)–1(d) show the real and imaginary parts of the effective permittivities for Ag/TiO₂ multilayer HMM. In Fig. 1(b), as the wavelength increases, the medium changes from type I HMM to elliptical medium at 395 nm wavelength. And then at 551 nm wavelength, it changes from elliptical medium to type II HMM. As shown in Fig. 1(c) and 1(d), the imaginary part of ϵ_∥ is extremely large at the transition point 395nm, while the imaginary part of ϵ_⊥ remains small. This is because of the denominator in Eq. (2). When the real part of the denominator becomes zero, then the Im ϵ_∥ becomes large.

3. Goos-Hänchen shift on HMM

As shown in Fig. 2(a), a light beam is incident on a thin slab made of HMM with an incident angle θ, and is reflected at the surface of HMM. The reflected beam experiences a short longitudinal shift in the incident plane δ_GH, known as the Goos-Hänchen (G-H) shift. In order to calculate the shift of the beam, we first resolve the incident beam into its plane wave components [10],

E_{I} (r, ω) = \iint d^{2} \vec{κ} e^{i (k_{x} x + k_{y} y) + i w (k_{x}, k_{y}) z} E_{I} (\vec{κ}, ω) .

where

\vec{κ} = k_{x} \hat{x} + k_{y} \hat{y}, w (k_{x}, k_{y}) = \sqrt{k_{0}^{2} - k_{x}^{2} - k_{y}^{2}}

and k₀ = 2π/λ. Here, we assume that the center of the field distribution locates at x = 0 in the plane of the interface and the beam propagates with a central wave-vector k_x₀. The field distribution can be described by the following expression,

E_{I} (k_{x}, k_{y}) = E_{0} \sqrt{\frac{2 σ_{x} σ_{y}}{π}} \exp [\frac{- σ_{x}^{2} {(k_{x} - k_{x 0})}^{2}}{4}] \exp [\frac{- σ_{y}^{2} k_{y}^{2}}{4}] .

σ_y is the waist radius of the incident beam which is chosen as 160 µm in our model. As a result of the oblique incidence, σ_x = σ_y/cos θ. Then, the amplitude of the reflected field can be expressed as,

E_{R} (r, ω) = \iint d^{2} \vec{κ} e^{i (k_{x} x + k_{y} y) - i w (k_{x}, k_{y}) z} E_{R} (\vec{κ}, ω) .

Fig. 2 (a) Geometry of the beam reflection at the surface of a slab made of HMM. The reflected beam will undergo a longitudinal shift δ_GH and transverse shift δ_IF. (b), (c) and (d) show the G-H shift δ_GH at the surface of a slab made of Ag/TiO₂ multilayer HMM, Ag and TiO₂, respectively. The slabs have the same thickness of 120 nm.

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Consequently, the expression of the G-H shift is given by [4, 10, 40],

δ_{GH} = \frac{i \iint d^{2} \vec{κ} E_{R}^{*} (k_{x}, k_{y}) \cdot \frac{\partial}{\partial k_{x}} E_{R} (k_{x}, k_{y})}{\iint d^{2} \vec{κ} E_{R}^{*} (k_{x}, k_{y}) \cdot E_{R} (k_{x}, k_{y})} \approx - \frac{λ}{2 π} \frac{\partial φ}{\partial θ} .

φ is the phase of the reflection coefficient. It is clear to see that the shift reaches its peak when the phase of the reflected field experiences a sharp variation over the incident angle. It should be noted that the incident angle for achieving a maximum shift is close to the Brewster angle. In all the calculations we use the rigorous formula (the middle one in Eq. (6)). We will show one case in Fig. 3(a) which exhibits a major difference between exact and approximate results.

Fig. 3 (a) plots the G-H shifts calculated using the exact (solid) and approximate (dashed) results at the surface of HMM near the Brewster angle for 345 nm. (b) plots the values of |r_p|, and (c) plots the relative phases φ.

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In traditional cases, the G-H shift for s− polarized beam can be ignored since it’s very small. Thus, we only focus on the case of the p− polarized beam. This is due to the fact that we are not considering here the angular G-H shift. As is pointed out by Merano et al. [41] and Dennis et al. [15], if the angular G-H shift is considered also, the s− polarization alone can be appreciable. Based on Eqs. (3)–(6), Fig. 2(b) shows the G-H shifts on the Ag/TiO₂ multilayer HMM for p− polarized beam with different incident wavelengths. The thickness d of HMM is chosen as 120 nm. Figure. 2(b) shows that as the wavelength of the incident field increases, the G-H shift changes from positive to negative. The maximum positive value of the shift is 38.4 µm at 69.104° for λ = 330 nm, while the maximum negative value is −38.6 µm at 67.022° for λ = 345 nm. As λ continues to increase, the maximum value and critical angle decrease gradually. Comparing with the G-H shift of traditional materials, the HMM is a very good candidate to enhance G-H shifts.

To compare the G-H shifts at HMM and its two constituents: metal and dielectric, we also calculate the shifts on Ag and TiO₂ slabs of the same thickness. As shown in Fig. 2(c) and 2(d), with λ changing from 345 nm to 600 nm, the G-H shifts on the Ag slab are smaller than 1 µm, while on the TiO₂ slab, the shifts vary from −1 µm to 3 µm. It is clear that the shifts on HMM are significantly enhanced for specific wavelengths ranging from 330 nm to 395 nm.

Now we relate the well-defined peaks in the G-H shift for HMM to the existence of the Brewster modes [42]. For an anisotropic slab, the reflection coefficient r for p−polarized light is given by [43],

r_{p} = \frac{(k_{e z} - ϵ_{⊥} k_{0 z}) (k_{e z} + ϵ_{⊥} k_{0 z}) - (k_{e z} - ϵ_{⊥} k_{0 z}) (k_{e z} + ϵ_{⊥} k_{0 z}) e^{2 i k_{e z} d}}{(k_{e z} + ϵ_{⊥} k_{0 z}) (k_{e z} + ϵ_{⊥} k_{0 z}) - (k_{e z} - ϵ_{⊥} k_{0 z}) (k_{e z} - ϵ_{⊥} k_{0 z}) e^{2 i k_{e z} d}},

where

k_{e z} = \sqrt{ϵ_{⊥} (k_{0}^{2} - κ^{2} / ϵ_{∥})}, k_{0 z} = \sqrt{k_{0}^{2} - κ^{2}}

. By setting the numerator of r_p to be zero, we can easily get the complex Brewster mode and the corresponding complex Brewster angle θ_B. The real value of the Brewster angle corresponds to the position of the peak in the shift. Moreover, it’s found that the absolute imaginary value of the complex Brewster angle is equal to the half width at half maximum (HWHM) of the shift. This result indicates that the width of the shift is related to the dissipation of the medium, which gives rise to the imaginary part of the angle. For example, the complex Brewster angle at the surface of HMM for λ = 345 nm is 67.022° + 0.052°i. In Fig. 3(a), we show the calculated G-H shifts near this Brewster angle using the exact and approximate results. It shows that the HWHM is 0.104° when the approximate result is used, exactly equal to the imaginary part of the angle. When the exact result is used, the HWHM is 0.130°, a little larger due to the Gaussian distribution of the wavevector. However, for the lossless dielectric slab considered here, the imaginary value vanishes, leading to the disappearance of the peak in the G-H shift, as shown in Fig. 2(d).

It is well known that the large G-H shift originates from the phase change of the reflection coefficient. The absolute value and phase of r_p with λ = 345 nm are plotted in Fig. 3(b) and 3(c). For the HMM slab,|r_p| is close (but not equal) to zero at θ_B = 67.022°, accompanied by a steep phase change (see black curves). This corresponds to the position of the peak in Fig. 2(b), indicating the relation between the shift and phase change. For the lossless dielectric slab, the value of r_p is exactly zero and the phase has an abrupt jump (see red curves). In this case, the phase change of an infinite slope without reflection has no physical meaning [4]. For the metal slab, no Brewster mode is found due to high reflection (see green curves). This corresponds to small displacements at the metal slab. It can be seen that the combination of multilayer metal and dielectric slabs modifies the effective permittivities, which leads to large shifts due to the change of the reflection coefficients and to the existence of the Brewster modes. This property can not be found in the metal and dielectric slabs.

Also, it can be seen that the phase transition point from type I HMM to elliptical medium is 395 nm. This wavelength is not equal to transition point of the G-H shifts from positive to negative (i.e., 335 nm). This indicates that the sign of the shift has no direct relevance to the phase of the medium. In order to explain this point, we need to consider the reflection coefficient r_p which determines the shift. Equation. (7) shows that both ϵ_⊥ and ϵ_∥ affect the reflection coefficient, indicating that the phase of the medium may determine the shift of the reflection beam. However, as mentioned above, Fig. 1(b) and 1(c) show that ϵ_∥ always has a large real or imaginary part. In this case, κ²/ϵ_∥ is very small and thus can be ignored, leading to $k_{e z} \approx \sqrt{ϵ_{⊥}} k_{0}$ . That’s why the shift doesn’t show the similar property of a sharp transition as ϵ_∥ exhibits. In addition, the reason why the shift and critical angle of the shift change smoothly as λ increases, is that the ϵ_⊥ changes smoothly.

4. Imbert-Fedorov shift on HMM

The geometry of the I-F shift is also shown in Fig. 2(a). Contrary to the longitudinal G-H shift, a circularly (or elliptically) polarized incident beam experiences a transverse shift perpendicular to the incident plane on reflection. This is known as the Imbert-Fedorov (I-F) shift. In our calculation, we consider the initial state to be right circularly polarized, $| + 〉 = (\hat{H} + i \hat{V}) / \sqrt{2}$ . The horizontally and vertically polarized incident beams Ĥ and $\hat{V}$ can be expressed in terms of unit polarization vectors ŝ and $\hat{p}$ [20],

\hat{H} = \frac{\hat{p} - \frac{w k_{y}}{k_{x} k} \hat{s}}{\sqrt{2} \sqrt{1 + {(\frac{w k_{y}}{k_{x} k})}^{2}}},

\hat{V} = \frac{\hat{s} + \frac{w k_{y}}{k_{x} k} \hat{p}}{\sqrt{2} \sqrt{1 + {(\frac{w k_{y}}{k_{x} k})}^{2}}} .

The unit ŝ and $\hat{p}$ polarization vectors are defined by $\hat{s} = \hat{z} \times \vec{κ} / κ$ and $\hat{p} = \hat{s} \times \vec{k} / k$ . Similar to the G-H shift, the I-F shift can be expressed as [8, 40],

δ_{IF} = \frac{i \iint d^{2} \vec{κ} E_{R}^{*} (k_{x}, k_{y}) \cdot \frac{\partial}{\partial k_{y}} E_{R} (k_{x}, k_{y})}{\iint d^{2} \vec{κ} E_{R}^{*} (k_{x}, k_{y}) \cdot E_{R} (k_{x}, k_{y})} .

Assuming the slow variation of the reflection coefficients, we get the well-known expression for the I-F shift,

δ_{IF} = - \frac{\cot θ}{k} [1 + 2 \cos (φ_{s} - φ_{p}) \frac{| r_{p} | | r_{s} |}{{| r_{p} |}^{2} + {| r_{s} |}^{2}}],

where φ_s, φ_p are the phases of the reflection coefficients r_s and r_p, respectively.

Figures. 4(a)–(c) depict I-F shifts on the HMM, Ag and TiO₂ slabs. The negative value indicates that the shift is along the −y direction at the interface. As λ increases, the displacement increases and reaches a peak, and then decreases to zero as the incident angle increases to 90 degrees. it can be seen that as λ increases from 350 nm to 600 nm, the I-F shifts on Ag and TiO₂ slabs are all under 40 nm. For type I HMM (λ less than 395 nm) and type II HMM (λ larger than 551 nm), the I-F shifts are around 40 nm. The similar results of the shifts on metal, dielectric and HMM slabs implies that the HMM doesn’t affect the transverse shifts much. However, for λ = 500 nm which is in the phase of elliptical medium of the multilayer slab, the I-F shift has a large value of 135 nm. This is enhanced by around 3 times larger than on the metal or dielectric slabs at the same λ. Though this enhancement has no relation to the property of hyperbolic dispersion, this still offers us a way of enhancing the transverse I-F shift by using the multilayer dielectric/metal slab.

Fig. 4 (a), (b) and (c) show the I-F shift δ_IF at the surface of a slab made of Ag/TiO₂ multilayer HMM, Ag and TiO₂, respectively. The slabs have the same thickness of 120 nm. The black, red, green and blue lines denote the wavelengths of 350 nm, 400 nm, 500 nm and 600 nm, respectively.

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5. Spin Hall effect of light on HMM

The geometry of the SHEL is shown in Fig. 5(a). As described in the introduction, when reflected at the interface, a linearly polarized incident beam splits into two spin components. The two components acquire opposite displacements perpendicular to the incident plane. In the spin basis, the horizontal and vertical states |H〉 and |V〉 can be expressed as $| H 〉 = \frac{| + 〉 + | - 〉}{\sqrt{2}}$ and $| V 〉 = \frac{| + 〉 - | - 〉}{\sqrt{2} i}$ . Upon reflection, the two incident states |H〉 and |V〉 evolve as [20, 44],

| H 〉 \to \frac{r_{p} + \frac{i w k_{y}}{k_{x} k} r_{s}}{1 - \frac{i w k_{y}}{k_{x} k}} \frac{| + 〉}{\sqrt{2}} + \frac{r_{p} - \frac{i w k_{y}}{k_{x} k} r_{s}}{1 + \frac{i w k_{y}}{k_{x} k}} \frac{| - 〉}{\sqrt{2}}

| V 〉 \to \frac{- i r_{s} + \frac{w k_{y}}{k_{x} k} r_{p}}{1 - \frac{i w k_{y}}{k_{x} k}} \frac{| + 〉}{\sqrt{2}} + \frac{i r_{s} - \frac{w k_{y}}{k_{x} k} r_{p}}{1 + \frac{i w k_{y}}{k_{x} k}} \frac{| - 〉}{\sqrt{2}}

Fig. 5 (a) Scheme of the SHEL on a HMM slab. (b) and (c) show the $δ_{| + 〉}^{H}$ at the slab of HMM and Ag, respectively; (d), (e) and (f) show the $δ_{| + 〉}^{V}$ at the slab of HMM, Ag and TiO₂, respectively. The black, red, green and blue curves denote the wavelengths of 350 nm, 400 nm, 500 nm and 600nm, respectively.

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When $\frac{| r_{s} |}{| r_{p} |} \frac{w k_{y}}{k_{x} k} ≪ 1$ , which is satisfied as long as the incident angle is not too close to the Brewster angle, Eqs. (12) and (13) have the approximate form [44, 45],

| H 〉 \to \frac{r_{p}}{\sqrt{2}} [\exp (- i k_{y} δ_{| + 〉}^{H}) | + 〉 + \exp (- i k_{y} δ_{| - 〉}^{H}) | - 〉],

| V 〉 \to \frac{- i r_{s}}{\sqrt{2}} [\exp (- i k_{y} δ_{| + 〉}^{V}) | + 〉 - \exp (- i k_{y} δ_{| - 〉}^{V}) | - 〉] .

The opposite displacements of the two spin components depend on the input polarization state, resulting from the polarization-dependent Fresnel reflections at the interface. For |H〉 and |V〉 input polarizations, the displacements for different spins $δ_{| \pm 〉}^{H}$ and $δ_{| \pm 〉}^{V}$ are given by the approximate expression [45],

δ_{| \pm 〉}^{H} = \mp \frac{λ}{2 π} [1 + \frac{| r_{s} |}{| r_{p} |} \cos (φ_{s} - φ_{p})] \cot θ,

δ_{| \pm 〉}^{V} = \mp \frac{λ}{2 π} [1 + \frac{| r_{p} |}{| r_{s} |} \cos (φ_{p} - φ_{s})] \cot θ .

Figures. 5(b) and 5(c) describe the displacements $δ_{| + 〉}^{H}$ for the case of horizontal polarization on the HMM and Ag slabs, respectively. Figure. 5(b) shows that the shift rises rapidly as the incident angle approaches to the Brewster angle. As the incident angle surpasses the Brewster angle, the shift becomes opposite and decreases rapidly. It should be noted that in the phase of type I HMM (λ less than 395 nm), the displacements is larger than 2 µm when the incident angle is close to the Brewster angle. Compared with the shifts on the metal, the shifts on the HMM slab are remarkably enhanced for λ less than 400 nm. This attributes to the change of the effective permittivities due to the combination of multilayer metal and dielectric slabs. In the case of TiO₂ slab, Eq. (14) is not satisfied because of the small reflection coefficient for p polarization.

Finally, we calculated the displacements of the |+〉 spin component $δ_{| + 〉}^{V}$ in the case of vertical polarization. In Figs. 5(d)–5(f) we plot $δ_{| + 〉}^{V}$ on the HMM, Ag and TiO₂ slabs, respectively. When the incident wavelengths change from 350 nm to 600 nm, the shifts on Ag and TiO₂ slabs are smaller than 60 nm. For type I HMM and type II HMM, the SHEL shifts are around 40 nm. This is similar to the case of I-F effect, which also implies that the HMM also doesn’t have a big impact on the displacements for the initial vertical polarization. However, for a wavelength 500 nm which is in the elliptical phase of the multilayer slab, the displacement is about 120 nm, which is enhanced by almost 2 (4) times than the dielectric (metal) slab at the same wavelength. Similar to the case of I-F shift, this also offers us a method of enhancing the SHEL for the initial vertical polarization by using the multilayer dielectric/metal slab.

6. Conclusion

To summarize, we first studied the Goos-Hänchen shift of a reflected Gaussian beam at the surface of a thin slab made of Ag/TiO₂ multilayer HMM. The effective medium theory was used to describe the anisotropic permittivities of HMM. We presented a rigorous calculation of the shifts without any approximations. The G-H shift on HMM slab can be enhanced to a value of about 40 µm, which is much larger than the G-H shifts at either metallic or dielectric surfaces. The enhanced shifts are explained in terms of the characteristics of the Brewster modes in HMM. We demonstrated how the calculation of G-H shifts using approximate formula can give quite erroneous result when the Brewster modes are excited. We also presented the first calculations of the Imbert-Fedorov (I-F) effect as well as the spin Hall effect of light (SHEL) at the surface of a thin HMM slab. We have found that for specific wavelength the I-F shift at HMM can be enhanced by almost 3 times than the result for metal or dielectric slabs. For the SHEL, compared with the shifts on the metal, the shifts on the HMM slab are found to be enhanced remarkably for wavelength less than 400 nm. Thus a thin HMM slab can be used as a candidate to enhance the lateral displacements.

Funding

Joint Fund of the National Natural Science Foundation of China (Grant No. U1330203); National Natural Science Foundation of China (Grants No. 11274242, No. 11474221, No. 11574229, No. 11104075 and No. 11504272); 973 program (Grant No. 2013CB632701); Shanghai Science and Technology Committee (Grant No. 15XD1503700, No. 15YF1412400); International Exchange Program for Graduate Students of Tongji University; Shanghai Education Commission Foundation.

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Enhanced displacements in reflected beams at hyperbolic metamaterials

Abstract

1. Introduction

2. Effective medium theory of HMM

3. Goos-Hänchen shift on HMM

4. Imbert-Fedorov shift on HMM

5. Spin Hall effect of light on HMM

6. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (17)

Optics Express