Optical spin-sensitive Zitterbewegung in bianisotropic metamaterials

Rui-Peng Guo; Qing-Hua Guo; Li-Ting Wu; Jing Chen; Dianyuan Fan

doi:10.1364/OE.24.013788

1. Introduction

In recent years, there have been rising interest in simulating coherent quantum effects of atoms, molecules and condensed matter by using optics via the quantum-optical analogies [1–3]. The advantages of these studies are that the Coulomb interactions and many-body effects in electrons can be avoided [2], and that many quantum parameters and constants can be equivalently simulated by designing proper photonic structures, to reach proper values that make the effects experimentally observable. Among all the quantum-optical analogies studied so far [2–25], Zitterbewegung (ZB) initially represented the rapid trembling motion of free electron obeying the Dirac’s relativistic wave equation [26, 27]. A direct experimental observation of electronic ZB is not possible because of the extremely high oscillation frequency (∼ 10²¹ Hz) and ultra-small amplitude (a Compton wavelength ∼ 10⁻¹² m). These obstacles can be overcome by using the quantum-optical analogies realized in various photonic systems, e.g. two-dimensional photonic crystals [28], photonic lattices [29,30], binary waveguide arrays [27, 31–34], negative-zero-positive index metamaterials [35], and anisotropic media [36]. The quantum constants c, m and h̄ in the electronic Dirac’s equation can be analogically imitated in optical ZB, for example, by utilizing the propagation constant mismatch and the coupling between two adjacent waveguides of the array [27]. The oscillation frequency of trembling is artificially reduced, while the effective Compton wavelength is sufficiently increased, enabling a feasibly experimental observation of ZB [27,31]. Similar merits also exist in photonic crystals [28] and metamaterials [35] because it is the characteristics of the Dirac-like dispersion (e.g. the splitting in frequency) that determines the trembling effect.

All the literatures in studying optical ZB to date are on the trembling motion of field intensity around the propagation direction [27–36], to the best of our knowledge. As a kind of vector fields, optical waves also possess some other inherent characteristics such as the optical spin angular momentum (polarization) and optical orbital angular momentum (transversal helical phase) [37–39], similar to electrons [40]. Spin-sensitive ZB for electron wave packets exhibiting a spin-orbit interaction (SOI) has been demonstrated in an electron waveguide [41], and in various condensed-matter systems such as spintronic semiconductors, graphene, and superconductors [42–45]. Based on the fact that light also supports SOI [38, 39], it is then of great interest to investigate the feasibility in realizing a spin-sensitive optical ZB.

Recently Guo et al. showed that a bulk metamaterial with a suitably designed bianisotropy exhibits line degeneracy in its equifrequency surfaces (EFSs) that consist of two ellipsoids of opposite helicity states intersecting with each other [46]. Here we would like to show that such a kind of bianisotropic metamaterials can be utilized to realize optical spin-sensitive ZB, which has not been noticed and explicitly discussed before. We present a theoretical analysis on how to manage the dispersions and eigenstates of optical modes in bianisotropic metamaterials, for example, to open a gap in the otherwise degeneracy Dirac dispersions. A close analogue of the developed analytical results with these of Dirac equation is also proposed. Numerical simulations prove that for a circularly polarized incidence, the refracted optical beam shows a strong ZB effect when propagating along a direction determined by the optical spin of incidence. Furthermore, we emphasize that when the optical field is linearly polarized, although ZB on field intensity does not exist, the optical spin possesses an interesting spatial split and trembling phenomenon. Importance of this investigation and potential applications are discussed.

2. Theory and simulation

2.1. Theory

Without loss of generality, let us consider a bianisotropic metamaterial with a coupling between the electric and magnetic responses [46]. Such a kind of bianisotropic metamaterials can be readily designed within the framework developed by Efrati and Irvine [47], e.g. by utilizing coupled orthogonally orientated unit elements with electrical and magnetic responses. A schematic example is shown in Fig. 1, which consists of two U shaped elements and can be easily fabricated and manipulated in experiments. The bianisotropic response can be simply understood, for example, by considering the fields and current distribution for a x-polarized electric field incidence, which induces a magnetic dipole m along the y direction [see Fig. 1]. The constitutive equations of this bianisotropic metamaterial are given by

\vec{D} = \overset{=}{ε} \vec{E} + \overset{=}{ξ} \vec{H},

\vec{B} = \overset{=}{μ} \vec{H} - \overset{=}{ξ} \vec{E},

where the uniaxial tensors read

\overset{=}{ε} = ε_{0} ε (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & η_{e}^{2} \end{matrix}), \overset{=}{μ} = μ_{0} μ (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & η_{m}^{2} \end{matrix}), \overset{=}{ξ} = \frac{γ}{c} (\begin{matrix} 0 & i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) .

Here ε, μ and γ characterize the interaction of the bianisotropic metamaterial with optical wave. Parameters η_e and η_m represent the degree of broken proportionality in tensors ε̿ and ξ̿, which might produce a gap in the EFS relation of (k_x, k_y). All the parameters in Eq. (3) are real, which can be realized when operating away from resonance.

Fig. 1 Schematic of the unit cell of a bianisotropic metamaterial design.

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The dispersion (k_x, k_y) of an optical eigenstate at a given ω can be found from the Maxwell’s equations. Consider an optical wave propagating in the x − y plane, its electric field can be expressed in general to E⃗ = ψ^T exp(ik_xx + ik_yy − iωt), where ψ^T =(E_x, E_y, E_z)^T. By using the constitutive equations of Eqs. (1) and (2), we can find that the EFS dispersion is given by a matrix operation in the form of Mψ = 0

[\begin{matrix} k_{0}^{2} η_{m}^{2} (ε μ - γ^{2}) - k_{y}^{2} & k_{x} k_{y} & - i γ η_{m}^{2} k_{0} k_{x} \\ k_{x} k_{y} & k_{0}^{2} η_{m}^{2} (ε μ - γ^{2}) - k_{x}^{2} & i γ η_{m}^{2} k_{0} k_{y} \\ i γ k_{0} k_{x} & - i γ k_{0} k_{y} & k_{0}^{2} η_{e}^{2} ε μ - k_{y}^{2} - k_{x}^{2} \end{matrix}] [\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \end{matrix}] = 0,

where k₀ = ω/c = 2π/λ₀.

The considered bianisotropic metamaterial has a C₄ symmetry in the x − y plane, and we are interested in the possible degeneracy points at the four corners of k_x = 0 or k_y = 0. Without loss of generality, let us pay attention to the region of positive k_x near k_y = 0.

When η_e = η_m = η, the metamaterial has a high symmetry in its electric and magnetic responses, and a line degeneracy exists in the dispersion relation (k_x, k_y). The variation of k_x around k_y = 0, to the 1st order of k_y, is given by

k_{x}^{\pm} = β_{0} \pm ν k_{y},

where

β_{0} = k_{0} η \sqrt{ε μ - γ^{2}}, ν = \frac{γ}{\sqrt{ε μ}} .

The dispersion curves are two linear ones with opposite slopes of ±ν that interact at the degeneracy point of (k_x = β₀, k_y = 0), forming a Dirac-like cone as shown in Fig. 2(a).

Fig. 2 Dispersion relations of (k_x, k_y) near k_y = 0 and the associated Stokes parameter $S_{3}^{(y z)}$ for (a, c) Δ = 0 and (b, d) Δ ≠ 0. Units of k_x and k_y are k₀. Here ε = 2.8, μ = 1.19, γ = 0.559, and η_m = 0.917. In (a, c) η_e = η_m, and in (b, d) η_e = 0.884. Thin lines in (a) and (b) are these given by Eq. (5), and Eqs. (8) to (9), respectively. Note that in (c) $S_{3}^{(y z)}$ jumps to zero when k_y=0 because here the eigenstates are linearly polarized.

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When η_e = η_m, the degenerated Dirac-like dispersion at k_y = 0 is broken, and a gap is open, see Fig. 2(b). Assume that

η_{e} = η - Δ, η_{m} = η + Δ,

the dispersions near k_y = 0 should be expanded to the 2nd order of k_y since the 1st order is zero,

k_{x}^{+} = β_{0} + σ + \frac{α^{+}}{2} k_{y}^{2},

k_{x}^{-} = β_{0} - σ + \frac{α^{-}}{2} k_{y}^{2},

where

σ = k_{0} Δ \sqrt{ε μ - γ^{2}},

α^{+} = \frac{1}{β_{0}} (ν^{2} \frac{η + Δ}{Δ} - \frac{η}{η + Δ}),

α^{-} = \frac{1}{β_{0}} (ν^{2} \frac{η - Δ}{Δ} + \frac{η}{η - Δ}) .

To verify above approximation, we numerically calculate the dispersion of (k_x, k_y) from Eq. (4), and fit them by using Eq. (5) if Δ = 0, and Eqs. (8) and (9) if Δ ≠ 0, respectively. Parameters used in our analysis are ε = 2.8, μ = 1.19, γ = 0.559, and η_m = 0.917. Figure 2(a) shows the situation when η_e = η_m, i.e. Δ = 0. With parameters β₀ = 1.664k₀ and v = 0.306, the linear Dirac-like dispersions of Eq. (5) match well with the numerical calculated ones. Figure 1(b) shows the results when η_e = 0.884 and Δ = 0.009. Now a gap with a width of 2σ is open at the otherwise degeneracy point. The calculated dispersion curves can be satisfactorily fitted by using Eqs. (8) and (9) with β₀ = 1.564k₀, $α^{+} = 2.691 k_{0}^{- 1}$ , and $α^{-} = 3.851 k_{0}^{- 1}$ .

With the opening of a gap in the otherwise degenerated Dirac-like dispersion, it is then a necessary to check the associated spin of the eigenstates ψ. At the zone center of k_y = 0 it is easily to find that the eigenstates given by Eq. (8) is linearly polarized with an non-vanishing E_y component only. As for the other solution given by Eq. (9), it is a hybridized mode with non-vanishing field components E_x and E_z that satisfy E_x = i(εμ − γ²)^−1/2γη_eE_z. Since we are interested in these modes propagating closely along the x direction, we can neglect the longitudinal field component E_x, and only consider the spin in the transverse y − z plane, which could be characterized by the Stokes parameter

S_{3}^{(y z)} = i (E_{y} E_{z}^{*} - E_{y}^{*} E_{z}) .

This parameter is zero at the zone center of k_y = 0, see Figs. 2(c) and 2(d).

When k_y is not zero, the solutions are very complex, and we should numerically calculate the eigenstates ψ and find the associated $S_{3}^{(y z)}$ . The results are shown in Figs. 2(c) for Δ = 0, and 2(d) for Δ ≠ 0, respectively. Sign of $S_{3}^{(y z)}$ can be also easily understood from Eq. (4). For example, because E_x is not within our interest here, we should pay attention to the second line of M in Eq. (8), where M(2,1) = k_xk_y is a small number near k_y = 0. Since Δ > 0 and γ > 0, when k_y > 0 the tensor element $M (2, 3) = i γ η_{m}^{2} k_{0} k_{y}$ is positive imaginary. By considering the fact that $M (2, 2) = k_{0}^{2} η_{m}^{2} (ε μ - γ^{2}) - k_{x}^{2} < 0$ for the upper branch $k_{x}^{+}$ , we can find that the associated eigenstate ψ = (E_y, E_z)^T is left-handed elliptically polarized with $S_{3}^{(y z)} < 0$ . Within the same branch the eigenstates at k_y < 0 are right-handed elliptically polarized. As for the lower branch $k_{x}^{-}$ given by Eq. (9), the opposite conclusion can be made. Above analysis is in agreement with the numerical calculated $S_{3}^{(y z)}$ shown in Figs. 2(c) and 2(d). It is interesting to emphasize that the variations of k_x and $S_{3}^{(y z)}$ versus k_y possess similar tendency with the frequency and broadening of quantum states in a non-Hermitian system around an exceptional point [12, 48, 49], that with the continuous varying of a proper parameter (Δ here), the branches of frequency (k_x here) change from a crossing to an anti-crossing, while the branches of broadening ( $S_{3}^{(y z)}$ here) behave in the opposite way. The only exception is that at the zone center k_y = 0 the values of $S_{3}^{(y z)}$ are always zero.

Up to now we have arrived the necessary conditions in achieving the spin-sensitive optical ZB, as an anti-crossing dispersion [27, 31] with varied eigenstate spins. Here we would like to argue that there exists a close analogue of the above analysis with these of Dirac equation. Consider a 1D Dirac equation in the x − y plane

i \frac{\partial ψ}{\partial x} = - i κ α \frac{\partial ψ}{\partial y} + σ β ψ,

where ψ =(E_y, E_z)^T, and α and β are the σ_x and σ_z Pauli matrices, respectively. Assuming that ψ possesses a dynamic phase of exp(ik_xx + ik_yy), this Dirac’s equation reads explicitly as

[\begin{matrix} k_{x} + σ & - i κ k_{y} \\ i κ k_{y} & k_{x} - σ \end{matrix}] [\begin{matrix} E_{y} \\ E_{z} \end{matrix}] = 0 .

The dispersion is given by

k_{x}^{\pm} = \pm \sqrt{σ^{2} + κ^{2} k_{y}^{2}} \approx \pm (σ + \frac{1}{2} \frac{κ^{2}}{σ} k_{y}^{2}) .

Besides the offset β₀, this dispersion is similar to Eqs. (8) and (9) if α^± = κ²/σ.

As for the eigenstates ψ_±, they generally read

ψ_{+} = A (\begin{matrix} i κ k_{y} \\ 2 σ \end{matrix}), ψ_{-} = A (\begin{matrix} 2 σ \\ i κ k_{y} \end{matrix}),

where the normalization constant A equals

{(κ^{2} k_{y}^{2} + 4 σ^{2})}^{- 0.5}

. The states of spin

S_{3}^{(y z)}

for ψ_± have the same characters with those from Eq. (4), that at a given k_y the eigenstates ψ_± have opposite spins, and change their spins when crossing k_y = 0. At k_y = 0 they are all linearly polarized with

S_{3}^{(y z)}

.

2.2. Zitterbewegung

From the analysis and calculation presented above we have arrived two necessary conditions in realizing the spin-sensitive optical ZB. The first condition is that a gap with a width 2σ is open in the (k_x, k_y) reciprocal space. In agreement with the theory of photonic ZB in waveguide arrays [27,31], and noting that here the EFS dispersion is discussed in term of wavevector other than the angular frequency ω, the spin-sensitive ZB effect should be associated with a spatial period d of

d = \frac{2 π}{2 σ} = \frac{λ_{0}}{2 Δ \sqrt{ε μ - γ^{2}}} .

The other necessary condition is that the dispersion is spin-sensitive, i.e. the spin

S_{3}^{(y z)}

is not a constant in each dispersion curve. The spin of incidence determines the distribution of excited eigenstates, which, in turn, finds the propagating direction of the transmitted field. For example, considering the case displayed in Fig. 2(b), if the incident field is left-handed circularly/elliptically polarized with

S_{3}^{(y z)} < 0

, only the eigenstates in the k_y > 0 region of the upper dispersion

k_{x}^{+}

and the k_y < 0 region of the lower one

k_{x}^{-}

can be efficiently excited, the gradients of which both point to the −k_y direction. The ZB effect for an incident left-handed circular/elliptical polarization is thus associated with a trembling of field intensity around a path inclined to −y direction. For the opposite polarization of incidence, the path is inclined to the +y direction.

We simulate the spin-sensitive optical ZB in bianisotropic metamaterials by using a full-field three dimensional finite element optical simulation (COMSOL Multiphysics 4.3b). An optical beam is normal incident along the normal x-direction of the vacuum-metamaterial interface. Frequency of the incident beam is assumed to be 6.8 GHz, i.e. λ₀ = 4.412 cm. To gain a clear understanding about the spin-sensitive optical ZB, we assume that μ = 1.19, γ = 0.559, η_m = 0.917, and $ε_{z} = ε η_{e}^{2} = 2.186$ throughout this article. We only change the value of ε, which, in turn, modifies η_e by $η_{e}^{2} = ε_{z} / ε$ , to see how the degree of bianisotropy influences the characteristics of optical ZB.

With proper values of ε and beam width w₀, the interesting phenomenon of optical analogue of spin-sensitive ZB can be observed. Figure 3 shows the results for circularly polarized incidences with parameters ε = 3.2 and η_e = 0.827 at a beam width of w₀ = 10cm. A trembling motion of the trajectory of beam center can be obtained for both polarized incidences. Propagating directions of the refracted beams are spin dependent. The distributions of field intensity and the trajectory of beam centers at these two opposite polarized incidences are symmetric with each other with respect to the y = 0 axis. Equation (18) predicts that the oscillation period is d = 5.931λ₀, i.e. 26.22 cm, which is in good agreement with the period of ∼ 25.8 cm obtained in the COMSOL simulation.

Fig. 3 Distributions of (a, c) field intensity and (b, d) the trajectory of beam center for (a, b) left-handed and (c, d) right-handed circularly polarized incidences, respectively. Here ε = 3.2, μ = 1.19, γ = 0.559, η_m = 0.917, η_e = 0.826, and w₀ = 10 cm.

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The trembling motion of the spin-sensitive optical ZB is sensitive to the variation of gap width 2σ. Figure 4 shows the results when ε = 2.8 and 3.0, respectively. The value of η_e equals 0.884 when ε = 2.8, and 0.854 when ε = 3.0, respectively. We can see that with the same spin of incidence, the refracted beams incline to the same direction. Besides the σ -sensitive refraction, our interest is focused on the period d of trembling. When ε = 2.8, Eq. (18) predicts a period of d = 17.377λ₀, while when ε = 3.0, d equals 8.784λ₀. Both values are in good agreement with our simulations. The reduced period d (or the increased spatial oscillation frequency) when εincreases is obviously associated with an increased gap width 2σ, which increases from 0.058k₀ when ε = 2.8 to 0.169k₀ when ε = 3.2.

Fig. 4 Distributions of (a, c) field intensity and (b, d) the trajectory of beam center for right-handed circularly polarized incidence with (a, b) ε = 2.8, and (c, d) ε = 3.0, respectively. Here ε = 1.19, γ = 0.559, and η_m = 0.917, and $η_{e}^{2} ε = 2.186$ . When ε = 2.8 the gap width 2σ equals 0.058k₀, and when ε = 3.0, 2σ = 0.114k₀.

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Although we do not pay much attention to the amplitude R of ZB here, R can be briefly expressed by using the analogue between the Dirac equation and the Maxwell’s equation discussed in the former subsection, which requires α^± = κ²/σ. By expressing κ with the averaged value of ᾱ = (α⁺ + α⁻)/2, and substitute it into R = κ/2σ [27, 31], we can get

R = \frac{1}{2} \sqrt{\frac{\bar{α}}{σ}} = \frac{1}{2} \sqrt{\frac{α^{+} + α^{-}}{2 σ}}

in term of α⁺ and α⁻ that characterize the quadratic dispersion. When ε increases from 2.8 to 3.2, as discussed in Figs. 2 and 3, the value of R decreases from 0.849λ₀ (3.746 cm) to 0.275λ₀ (1.213 cm). It sets an upper limit on the beam width w₀, which might conceal the trembling of field intensity and make an experimental observation difficult.

Besides the trembling of field intensity, the spin state $S_{3}^{(y z)}$ of the refracted beam is also investigated. Figure 5 shows the result corresponding to Fig. 3(b). Interestingly, along the intensity pathway of the trembling beam, $S_{3}^{(y z)}$ also oscillates and changes its sign periodically. Period of the oscillation is identical to d, the spatial trembling period on field intensity. It implies that the physical mechanism of this spatial oscillation of $S_{3}^{(y z)}$ is also closely related to the interference among the excited eigenstates of the split dispersion curves, similar to the ZB effect on field intensity. Here we provide a brief analysis on the distributions and spins of the excited eigenstates. Propagation directions of the excited eigenstates are normal to the corresponding EFSs, and from Eq. (8) and (9) we can see the angles of refraction θ^± are given by $arctan θ^{+} = \partial k_{x}^{+} / \partial k_{y} = α^{+} k_{y}$ and $arctan θ^{-} = \partial k_{x}^{-} / \partial k_{y} = - α^{-} k_{y}$ , respectively. Figure 3(b) shows that the refracted beam propagates approximately at arctanθ^± = 1/30, so the excited eigenstates in the two branches are around k_y = 0.08k₀ in the upper branch and k_y = −0.02k₀ in the lower branch, respectively. The corresponding $S_{3}^{(y z)}$ are −0.28 and −0.06, respectively. It is clear that the spins of the two sets of eigenstates are not identical. Because of the different phase velocities (∼ k_x) between these two groups of eigenstates, an interference effect takes place that produces the periodically spatial oscillation of $S_{3}^{(y z)}$ .

Fig. 5 Distribution of the Stoke’s parameter $S_{3}^{(y z)}$ for the situation displayed in Fig. 3(a).

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Above we have considered the situation of circularly polarized incidence, and have showed that the refracted beam possesses a spin-sensitive optical ZB effect. We also check the situations of linearly polarized incidences. Results for y- and z-directional linearly polarized incidences are shown in Fig. 6. We can see although all the parameters are identical to these of Fig. 3, no obvious ZB effect can be observed in the field intensity. The only noticeable feature on the field intensity is that the diffractions on the two orthogonal linear polarizations have different strengthes. This phenomenon can be explained because the excited eigenstates at these polarizations belong to different dispersion branches shown in Fig. 3(b). At the zone center of k_y = 0 the y polarized eigenstate is at the upper branch $y_{x}^{+}$ . The gradient of this branch is toward the central k_y = 0 axis, i.e. its dispersion is a convergent one. This convergent dispersion could compensate the diffraction effect. On the other hand, the z polarized mode is mainly located at the lower branch of $y_{x}^{-}$ , which, unfortunately, has a divergent dispersion. Consequently, the z-polarized linear incidence suffers a stronger diffraction than that of the y-polarized incidence, as shown in Figs. 6(a) and 6(c).

Fig. 6 Distributions of (a, c) field intensity and (b, d) Stoke’s parameter $S_{3}^{(y z)}$ when the incident wave is (a, b) y- and (c, d) z-linearly polarized. All the parameters are the same as these of Figs. 3(b) and Fig. 5.

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Interestingly, albeit the field intensity does not spatially tremble, the states of spin spatially splits and trembles. As shown in Figs. 6(b) and 6(d), the spin of the refracted beam is antisymmetrically distributed with respect to the central y = 0 axis. The fields with negative spin (left-handed helicity) and positive spin (right-handed helicity) behave opposite ZB effects. Field at the central y = 0 axis is linearly polarized, with $S_{3}^{(y z)} = 0$ . Such a phenomenon can be easily understood from the discussion provided in the former subsection, that although the dispersion is symmetric with respect to k_y = 0, the spins of eigenstates with opposite k_y values in the same branch are opposite with each other. Since a linear polarization can be expressed as the summation of two opposite circular polarizations, the linearly polarized incidence would excite two groups of eigenstates with opposite spins. Each group of eigenstates interferences with itself, rendering an optical ZB effect. The distributions of field intensity of these two groups are symmetric with respect to the central y = 0 axis, but the spins $S_{3}^{(y z)}$ are anti-symmetrically distributed. As for the observed overall distributions of field intensity and spin, they can be obtained by considering the interference between these two groups. Consequently, the spin states behave the unique anti-symmetrically spatially split and ZB phenomenon shown in Fig. 6.

3. Conclusion

Before ending this article, we would like to further discuss the importance of this work and the potential applications. The importance of quantum-optic analogues is that it enables people to prove the validity of quantum theory, and open a gateway for the advance of nano-optics. ZB, as one of quantum-optic analogues investigated so far, represents an interesting quantum interference effect between electrons satisfying the relativistic Dirac equation. Our investigation present here shows how the spin degree of freedom in photons can be utilized to influence and even control the trembling of optical beam in realizing spin-sensitive optical ZB. This work is different from former investigations on optical ZB effect and possesses some advantages. Unlike that in a zero-refractive index medium [35], the Dirac-like dispersion studied here is realized in the 2D wavevector space deduced from EFCs of a bianisotropic metamaterial. Also, since it is based on metamaterials made of subwavelength units, it avoids the introduction of the edge of Brillouin zone [27,31,33]. Consequently, the optical ZB can be observed under a normal incidence. From this aspect, our investigation further proves that metamaterials possess much unique functionality over ordinary media and optical structures. All these advantages, to our belief, can push forwardly the advances of various optical spin-controllable functionalities, not only for scientific curiosity but also for realistic applications. For example, the spatial trembling of spin surrounding the main pathway of field intensity can be unitized to design compact spin filters. Since the trembling of spin (as well as the intensity) is sensitive to the polarization of incidence, the performance of the spin filters can be easily manipulated via modulating the incident beam. Albeit loss would degrade the performance of metamaterials, we believe that this problem could be further mitigated here because a d/2-thick metamaterial slab is sufficient for a satisfactory realizing of the proposed compact spin filter.

In conclusion, here we prove the feasibility in obtaining an optical spin-sensitive optical ZB by utilizing designed metamaterials with proper bianisotropy. Under normal incidence the optical beam refracts to a direction determined by the optical spin of incidence, with noticeable trembling ZB feature. We emphasize that linear polarized incidence also possesses unique ZB effect, not on the field intensity, but on the optical spin. A close analogue of the developed analytical results with these of Dirac equation is proposed. Numerical simulation proves the existence of ZB. With the versatile of metamaterials, our investigation could be realized in different frequency region, especially from GHz to THz.

Acknowledgments

The authors thank Prof. Shuang Zhang at University of Birmingham for simulating discussions. This work was supported by the National Natural Science Foundation of China (NSFC) under grants 11574162, 61490713, and 61505111.

References and links

1. D. Dragoman and M. Dragoman, Quantum-Classical Analogies (Springer, 2004). [CrossRef]

2. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photonics Rev. 3, 243–261 (2009). [CrossRef]

3. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003). [CrossRef] [PubMed]

4. R. Khomeriki and S. Ruffo, “Nonadiabatic Landau-Zener tunneling in waveguide arrays with a step in the refractive index,” Phys. Rev. Lett. 94, 113904 (2005). [CrossRef] [PubMed]

5. H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006). [CrossRef] [PubMed]

6. S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103, 123601 (2009). [CrossRef] [PubMed]

7. N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008). [CrossRef] [PubMed]

8. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008). [CrossRef] [PubMed]

9. P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low-loss metamaterials based on classical electromagnetically induced transparency,” Phys. Rev. Lett. 102, 053901 (2009). [CrossRef] [PubMed]

10. B. M. Rodriguez-Lara and H. M. Moya-Cessa, “Optical simulation of Majorana physics,” Phys. Rev. A 89, 015803 (2014). [CrossRef]

11. S. Longhi, “Optical realization of relativistic non-Hermitian quantum mechanics,” Phys. Rev. Lett. 105, 013903 (2010). [CrossRef] [PubMed]

12. H. Cao and J. Wiersig, “Dielectric microcavities: model systems for wave chaos and non-Hermitian physics,” Rev. Mod. Phys. 87, 61–111 (2015). [CrossRef]

13. C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010). [CrossRef]

14. L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013). [CrossRef]

15. A. Regensburger, C. Bersch, M. Mohammad-Ali, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012). [CrossRef] [PubMed]

16. Y. D. Chong, L. Ge, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 106, 093902 (2011). [CrossRef]

17. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011). [CrossRef] [PubMed]

18. R. Fleury, D. L. Sounas, and A. Alu, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113, 023903 (2014). [CrossRef] [PubMed]

19. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009). [CrossRef] [PubMed]

20. M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014). [CrossRef] [PubMed]

21. F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008). [CrossRef] [PubMed]

22. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009). [CrossRef] [PubMed]

23. A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nature Mater. 12, 233–239 (2012). [CrossRef]

24. L. Lu, J. D. Joannopoulos, and M. Soljacic, “Topological photonics,” Nat. Photonics 8, 821–829 (2014). [CrossRef]

25. W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Beri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114, 037402 (2015). [CrossRef] [PubMed]

26. W. Greiner, Relativistic Quantum Mechanics (Springer-Verlag, 1990). [CrossRef]

27. F. Dreisow, M. Heinrich, R. Keil, A. Tunnermann, S. Nolte, S. Longhi, and A. Szameit, “Classical simulation of relativistic Zitterbewegung in photonic lattices,” Phys. Rev. Lett. 105, 143902 (2010). [CrossRef]

28. X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100, 113903 (2008). [CrossRef] [PubMed]

29. Q. Liang, Y. Yan, and J. Dong, “Zitterbewegung in the honeycomb photonic lattice,” Opt. Lett. 36, 2513–2515 (2011). [CrossRef] [PubMed]

30. H. Deng, F. Ye, B. A. Malomed, X. Chen, and N. C. Panoiu, “Optically and electrically tunable Dirac points and Zitterbewegung in graphene-based photonic superlattices,” Phys. Rev. B 91, 201402(R) (2015). [CrossRef]

31. S. Longhi, “Photonic analog of Zitterbewegung in binary waveguide arrays,” Opt. Lett. 35, 235–237 (2010). [CrossRef] [PubMed]

32. S. H. Nam, A. J. Taylor, and A. Efimov, “Diabolical point and conical-like diffraction in periodic plasmonic nanostructures,” Opt. Express 18, 10120–10126 (2010). [CrossRef] [PubMed]

33. S. Ding and G. P. Wang, “Plasmonic analogs of Zitterbewegung in nanoscale metal waveguide arrays,” J. Opt. Soc. Am. B 31, 603–606 (2014). [CrossRef]

34. Y. Fan, B. Wang, H. Huang, K. Wang, H. Long, and P. Lu, “Plasmonic Zitterbewegung in binary graphene sheet arrays,” Opt. Lett. 40, 2945–2948 (2015). [CrossRef] [PubMed]

35. L. G. Wang, Z. G. Wang, and S. Y. Zhu, “Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial,” EPL 86, 47008 (2009). [CrossRef]

36. K. Y. Bliokh, D. Y. Frolov, and Y. A. Kravtsov, “Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium,” Phys. Rev. A 75, 053821 (2007). [CrossRef]

37. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Advances in Optics and Photonics 3, 161–204 (2011). [CrossRef]

38. F. Cardano and L. Marrucci, “Spin-orbit photonics,” Nature Photonics 9, 776–778 (2015). [CrossRef]

39. K. Y. Bliokh, F. J. Rodriguez-Fortuno, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nature Photonics 9, 796–808 (2015). [CrossRef]

40. P. Brusheim and H. Q. Xu, “Spin Hall effect and Zitterbewegung in an electron waveguide,” Phys. Rev. B 74, 205307 (2006). [CrossRef]

41. C. C. Leary and K. H. Smith, “Unified dynamics of electrons and photons via Zitterbewegung and spin-orbit interaction,” Phys. Rev. A 89, 023831 (2014). [CrossRef]

42. J. Schliemann, D. Loss, and R. M. Westervelt, “Zitterbewegung of electronic wave packets in III–V zinc-blende semiconductor quantum wells,” Phys. Rev. Lett. 94, 206801 (2005) [CrossRef]

43. M. Lee and C. Bruder, “Charge- and spin-density modulations in semiconductor quantum wires,” Phys. Rev. B 72, 045353 (2005). [CrossRef]

44. Z. F. Jiang, R. D. Li, S-C. Zhang, and W. M. Liu, “Semiclassical time evolution of the holes from Luttinger Hamiltonian,” Phys. Rev. B 72, 045201 (2005). [CrossRef]

45. J. Cserti and G. David, “Unified description of Zitterbewegung for spintronic, graphene, and superconducting systems,” Phys. Rev. B 74, 172305 (2006). [CrossRef]

46. Q. Guo, W. Gao, J. Chen, Y. Liu, and S. Zhang, “Line degeneracy and strong spin-orbit coupling of light with bulk bianisotropic metamaterials,” Phys. Rev. Lett. 115, 067402 (2015). [CrossRef] [PubMed]

47. E. Efrati and W. T. M. Irvine, “Orientation-dependent handedness and chiral design,” Phys. Rev. X 4, 011003 (2014).

48. M. Philipp, P. von Brentano, G. Pascovici, and A. Richter, “Frequency and width crossing of two interacting resonances in a microwave cavity,” Phys. Rev. E 62, 1922–1926 (2000). [CrossRef]

49. H. X. Cui, X. W. Cao, M. Kang, T. F. Li, M. Yang, T. J. Guo, Q. H. Guo, and J. Chen, “Exceptional points in extraordinary optical transmission through dual subwavelength metallic gratings,” Opt. Express 21, 13368–13379 (2013). [CrossRef] [PubMed]

Optical spin-sensitive Zitterbewegung in bianisotropic metamaterials

Abstract

1. Introduction

2. Theory and simulation

2.1. Theory

2.2. Zitterbewegung

3. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (19)

Optics Express