1. Introduction

Metamaterials provide us with a unique platform to manipulate electromagnetic (EM) wave within a subwavelength dimension. The optical response of a metamaterial arises from the localized EM resonance in its subwavelength constituent element. Sharp dispersion and strong field enhancement can be achieved, which are of critical importance for various applications, open up tremendous opportunities for the realization of a wide range of metal-based optical devices.

Recently, attention has been paid to the functions and applications of photonic angular momentum states (PAMSs) in metamaterials [1–14]. In cylindrical or coaxial subwavelength elements, PAMSs refer to these resonant EM modes with a helical phase distribution of exp(−imϕ), where ϕ denotes the azimuthal angle. Each PAMS is characterized by the integer m, and the sign of m (plus or minus) determines the handedness of the helical EM field. Note that in beam optics m is frequently termed the topological charge, which represents the orbital-angular-momentum degree of freedom of light [12, 13]. Tribelsky et al. [1] discussed light scattering by a small particle when close to the plasmon resonance frequencies. In particular, they showed that the resonance extinction cross section increases with an increase in m, and the characteristics of the Poynting vector field is very sensitive to fine changes in the incident light frequency. Peng et al. [2] demonstrated that by utilizing displacement currents in simple resonators and by additionally exciting modes with proper m, left-handed properties can be observed in an array of high dielectric cylindrical resonators. Cao et al. [3] showed that PAMSs can gently confine light within high-refractive-index semiconductor subwavelength nanostructures, and are ideally suited to enhance and spectrally engineer light absorption at length scales comparable to the wavelengthes of electrons (∼1 nm) and photons (∼ 1 μm). Ruan et al. [4] showed that an arbitrarily large total cross section, i.e. superscattering, can be achieved provided that one maximizes contributions from a sufficiently large number of channels with different m. Du et al. [5] demonstrated that an optical beam transmits in a negative direction when passing through a single array of high-refractive-index dielectric nanorods provided that the m = ±1 modes properly interfere. Furthermore, it is shown that a single-layer array of high-electric-permittivity rods is capable of reflecting more than 97% of the energy of an optical wave with an arbitrary incident angle [6], benefiting from the highly independent manipulability of PAMSs with opposite signs of m.

Transmission properties of EM waves along the axial direction z of cylindrical and coaxial subwavelength resonators have also been discussed. Extraordinary optical transmission via the excitation of PAMSs for normal and oblique incidences have been studied both theoretically and experimentally [7–10]. Novel phenomena, including the realization of a wide-angle negative-index metamaterial [11], have been demonstrated. Furthermore, on account of the darkness nature of m ≠ ±1 PAMSs for all polarizations of normal incidence, Guo et al. [12] proposed a slow-light scheme by indirectly exciting dark PAMSs. A group index n_g greater than 500 is achieved [12]. Similar configuration has then been utilized to manipulate the polarization of transmitted optical wave [13] via selective excitation of bright PAMSs (m = ±1), in which the orbit-spin conversion of the optical angular momentum determines the ellipticity.

As the optical eigenstates of coaxial or cylindrical resonators, PAMSs possess an unique characteristic, that the +m and −m PAMSs are degenerated in their resonant frequencies. Although sometimes this degeneracy helps to realize attractive applications, for example, in superscattering [4], it is still desirable to break this degeneracy so as to take advantage of the individual unique properties of each PAMS. A nice example is the circular polarizer proposed by Guo et al. [13], which is based on the fact that the effective dipoles of the m = +1 and m = −1 PAMSs radiate left- and right-handed circular polarizations, respectively.

To lift this degeneracy, we can resort to the resonant condition of PAMSs, as φ₁ +φ₂ + kh = 2lπ, where φ₁ (φ₂) is the reflection phase from the front (rear) aperture, h is the length of the resonator, and l is an integer representing the longitudinal order of the resonance. By introducing additional elements into the front or rear apertures we can break the rotational symmetry of the whole structure. This geometric method modifies the values of φ_1,2 for different PAMSs by different degrees [12, 13], thus lifts the degeneracy in the resonant frequencies of the ±m PAMSs. The excitation coefficient, or extinction cross section σ_m, also gets modified. Although such a geometric method has been successfully applied in applications such as slow light [12] and polarization manipulation [13], the performance of the metamaterials could not be further tuned once the structural pattern is fabricated.

Another way in lifting the degeneracy of PAMSs is to use an external excitation to modify the optical properties of the media forming the metamaterials. Feasibility of this method is of a greater importance than that of the geometric one, because we can actively tune and modify the performance of the metamaterial whenever necessary. Recently, Wang et al. [14] showed that under the presence of a static magnetic field B, the eigenfrequencies of PAMSs in a gyromagnetic cylinder experience a splitting that is proportional to m. Such a splitting is similar to the Zeeman splitting of electronic states in atoms, and can lead to some unusual decoupling properties [14]. However, further applications of this classic analogue of Zeeman effect have not been seen in any literatures.

In this article, we prove the feasibility in utilizing the classic analogue of Zeeman effect to manipulate the performance of slow light associated with dark PAMSs in magneto-optical (MO) metamaterials. We show that the positions of the induced transmission dips and slow-light windows can be tuned by changing the magnitude of the applied magnetic field B. The classic analogue of Zeeman effect contributes to the split in the transmission dips, although the structure possesses a reflection symmetry. Two unique features are demonstrated. First, the lifted degeneracy from the classic analogue of Zeeman effect enables us to achieve a high-contrast excitation of PAMSs with opposite signs of m. The difference in the weights of the +m and −m PAMSs can reach > 70%, much greater than that obtained by using the geometric method (< 0.1%) [12]. Secondly, associated with the classic analogue of Zeeman effect the distributions of field magnitude and transverse energy flux for each pair of transmission dips are greatly different, which can be explained by considering the nonreciprocal propagating effect in MO media [14–16]. The performance of slow-light effect are also discussed, including the variation of phase delay near the induced transmission dips, the dependence of resonant frequencies with the magnitude of B, the dispersion of group index n_g, its abnormal feature and the associated strong absorption. Because the topological charges m of PAMSs form a high-dimensional Hilbert space, our investigation might push forwardly the advances of optical-angular-momentum science and engineering in nano-scaled structures.

2. Simulation and discussion

2.1. Structural design

Geometry of the metamaterial structure under investigation is shown in the inset of Fig. 1. Each unit cell, with a width a of 1.70 mm, contains a coaxial element and a rectangular element. Thickness h of the perfect-electric-conductor plate is 0.20 mm. Dimensions of the rectangular element are l =0.90 mm and w =0.20 mm, and the inner and outer radii of the coaxial element are r₁ = 0.60 mm and r₂ =0.65 mm, respectively. Distance between the two elements is 0.05 mm. The structure possesses a reflection symmetry with respect to the central x axis. Electric field of the normally incident EM wave parallels to the short side w of the rectangular element, i.e. it is x-linearly polarized.

 

Fig. 1 Transmission (T) and absorption (A) spectra of the metamaterial when ω_B = 0 (black curve), 0.19ω_p (blue curve) and 0.20ω_p (red curve), respectively. Inset shows a schematic view of the metamaterial design.

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As a special kind of coupled slot antennas, this structure is an excellent candidate for manipulating EM field at the subwavelength scale, especially in the terahertz region [17–22]. PAMSs, with a phase dependence of exp(−imϕ) in the cylindrical coordinate frames of (r, ϕ, z), are supported in the coaxial element due to its annular symmetry [12, 13], where r = 0 represents the center and ϕ denotes the azimuthal angle with respect to the x axis. Under normal incidence, the bright modes denoted by m = ±1 can be directly excited [13], while the other PAMSs with m ≠ ±1 are subradiant and can be only indirectly excited by the near-field coupling between the coaxial and rectangular elements [12]. The resonance frequency of each PAMS can be estimated by considering the cutoff frequency of coaxial waveguides [7–10], which is a function of r₁, r₂, m and the refractive index inside the coaxial resonator. We can eliminate the influence from the bright modes by properly designing the structure so that only the frequencies of subradiant PAMSs within our interesting, e.g. m = ±2 and m = ±3, overlap with that of the resonant mode inside the rectangular element [12]. The fundamental PAMS, which is in fact a cylindrically symmetric TEM mode without cutoff, is also dark because m = 0 [8].

Here, in order to lift the degeneracy of PAMSs via the classic analogue of Zeeman effect [14], we propose to insert a MO medium inside the coaxial aperture and apply a static magnetic field B along z direction. The permittivity tensor of the MO medium would become anisotropic when B is nonzero [15, 16, 23, 24], as

Note that the parameters appeared in Eqs. (2) to (4) can be manipulated in purpose to meet the values we desire. For example, proper semiconductors can be utilized to get the desired ε_∞. The bulk plasma frequency ω_p, which reads (ne^*2/m^*ε₀)^1/2 in the free electron model, can occupy the frequency regime from infrared to GHz since the carrier concentration n can be adjusted by properly doping. Value of ω_p can be pushed to low frequency regime (e.g. GHz) by using a low carrier concentration n, so that a moderate value of B can appreciably change the values of the tensor elements [15]. Realization of the proposed scheme in infrared and even visible light regime is also possible. However, besides the requirement of fabricating metamaterials with a subwavelength geometric size according to the cutoff condition [7–10], a high doping density n and a strong field B are generally required in order to achieve the required high values of ω_p and ω_B, which inevitably lead to a strong absorption. Novel material with a small m^*, e.g. graphene [25, 26], can be utilized to break this bottleneck.

2.2. Transmission spectra

Consider the situation with ε_∞ =15.68, ω_p = 2π ×464.45 GHz, and Γ =0.0001, which can be obtained by properly doping InSb [23, 24] with a doping density n of 3.747 × 10¹³ cm⁻³. From m^* = 0.014m_e we can see a magnetic field B of 0.05 Tesla can render ω_B = 0.20ω_p. We simulate the transmission properties of the structure by full-field three-dimensional finite-element optical simulations (COMSOL Multiphysics 4.3b). Figure 1 shows the transmission spectra when ω_B = 0, 0.19ω_p and 0.20ω_p, respectively. When no magnetic field is applied, a broad transmission peak at 152.95 GHz can be observed. We check the situation without the coaxial element, and find that the transmission spectrum hardly changes. It implies that within this transmission window only the bright EM resonance inside the rectangular element is excited [12, 13].

When B increases, the values of the tensor elements get changed, and the resonant frequencies of PAMSs can approach that of the rectangular element. Some pairs of sharp transmission dips are produced in the background transmission window when ω_B is greater than 0.18ω_p. For example, when ω_B = 0.19ω_p, two pairs of transmission dips can be observed from 148 GHz to 160 GHz. One pair is around 150 GHz, and the other pair is around 156 GHz. These sharp transmission dips are closely related to the darkness nature of m ≠ ±1 PAMSs that are subradiant for all polarizations of normal incidence [12]. By checking the spectra of m at these dips [12, 13], as we will show below, the pair at 150 GHz is associated with m = ±2 PAMSs, while the other pair at 156 GHz is for m = ±3. When ω_B further increases with B, the transmission dips blueshift mainly due to the changed value of ε [14]. Consequently, we can tune the optical properties of the system by means of adjusting the applied magnetic field B.

Because the elements in the permittivity tensor of the MO medium are complex, absorption also exists. We evaluate the absorption coefficient A from the transmission coefficient T and reflection coefficient R by A = 1 − T − R. As shown in Fig. 1, A is greatly enhanced around the induced transmission dips. Below we will show that this high absorption is associated with the high field enhancement and high-Q factor of the dark PAMS resonance.

2.3. Distributions of field and energy flux

The most interesting feature in the transmission spectra shown in Fig. 1 is the emergence of split transmission dips. It is well known that if the medium inside the coaxial element is isotropic, e.g. air, only a single transmission dip can be observed due to the excitation of ±m PAMSs with the same amplitudes [12]. On the other hand, the classic analogue of Zeeman split effect on the resonant frequencies of ±m PAMSs takes place if an external static magnetic field is applied [14]. Consequently, each pair of transmission dips shown in Fig. 1 should be associated with the lifted degeneracy between the ±m PAMSs via the classic analogue of Zeeman effect.

To prove the above logic, we analyze the distributions of field amplitude |E| and transverse energy flux S_⊥ inside the structure at these induced transmission dips and the adjacent peaks. Simulation results are shown in Fig. 2 for the two transmission dips with m = ±2, and in Fig. 3 for the adjacent three transmission peaks, respectively, when ω_B = 0.20ω_p. Note that our numerical simulation shows that the excited PAMSs inside the coaxial element are transverse electric, and the dominant field components are E_r and H_z. The azimuthal component E_φ, albeit greater than that of incidence, is much weaker than E_r.

 

Fig. 2 Distributions of field amplitude |E| and transverse energy flux S_⊥ (by arrows) in a quarter of the coaxial element, at the transmission dips of (a) 152.618 GHz and (b) 152.945 GHz, respectively, when ω_B = 0.20ω_p. Insets show the distribution of field amplitude |E| in the whole unit cell.

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Fig. 3 Distributions of field amplitude |E| when ω_B = 0.20ω_p at the transmission peaks of (a) 152.475 GHz, (b) 152.89 GHz, and (c) 153.49 GHz, respectively. Plots (d) to (f) are the corresponding distribution of transverse energy flux S_⊥.

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From Fig. 2 we can see that at the two transmission dips, strong field exists inside the coaxial element, much stronger than that in the rectangular element (see the inset). It implies that the optical energy is switched from the bright waveguide resonance inside the rectangular element to PAMSs inside the coaxial element. The directions of transverse energy flux S_⊥ of the two transmission dips are remarkably different. The energy flux S_⊥ of the low-frequency dip is anticlockwise, while that of the high-frequency one is clockwise.

The field inside the coaxial element can be expressed as a combination of PAMSs with different m values. To see the components of the excited PAMSs, we analyze the distribution of electric field E_r(ϕ) inside the coaxial element by $E_r(\varphi )={\sum }_{-\infty }^{+\infty }{A}_m\text{exp}(-jm\varphi )$ [12, 13]. This analysis proves the excitation of dark PAMSs at each transmission dip. For the transmission dip at 152.618 GHz, the weights of m = −2 and m = +2 PAMSs are 85.61% and 14.24%, respectively. For the transmission dip at 152.945 GHz, the corresponding weights are 15.12% and 84.80%, respectively. The m-spectra of the high-frequency transmission dips around 158 GHz are also investigated. For the lower frequency dip at 158.34 GHz, weight of the m = −3 can even approach 99.929%; while for the higher frequency one at 158.685 GHz, weight of the m = +3 reads 99.925%. Evidently, the degeneracy of the ±m PAMSs is broken.

From above analysis and from the distributions of field amplitude and transverse energy flux shown in Fig. 2, we can observe some unique features in each pair of MO-induced transmission dips. First, the weight difference between the two dark PAMSs is very high, in general, greater than 70%. In [12] Guo et al. have utilized the geometric method, i.e. break the reflection symmetry of the input and output surfaces by shifting the rectangular element, to lift the degeneracy. However, only the degeneracy of reflection phases φ_1,2 in the two PAMSs are distinctly lifted, while the magnitudes of excitation coefficients are almost identical. The difference in the weights of ±m PAMSs is very small, e.g. < 0.1% [12]. Here we can see the anisotropy from the MO effect greatly breaks the degeneracy in the excitation coefficients. This feature might possess great potential since each PAMS can be utilized separately for its own merits.

Second, associated with the lifted degeneracy of ±m PAMSs, the field amplitude |E| and transverse energy flux S_⊥ for each pair of transmission dips show asymmetric distributions. For example, at 152.618 GHz where the m = −2 PAMS dominates, the field amplitude and transverse energy flux distribute mostly near the outer radius r = r₂ of the coaxial element [see Fig. 2(a)]. On the other hand, at 152.945 GHz where the m = +2 PAMS dominates, they prefer the inner radius at r = r₁ [see Fig. 2(b)].

This asymmetric feature on field amplitude and transverse energy flux could not be observed if the medium inside the coaxial element is isotropic [12]. It can be attributed to the anisotropy induced into the MO medium by the magnetic field B, and can be understood by referring to the nonreciprocal field and energy flux distributions in the resonant transmission through MO subwavelength structures [15] and the explicit connection of coaxial waveguide with the planar metal-insulator-metal geometry [27]. The off-diagonal element γ determines the phase difference between E_r and E_ϕ, leading to the transverse MO nonreciprocal phase shift that depends on the direction of transverse energy flux [16]. The field profile is also influenced by γ [15]. Because under a full-loop variation in ϕ a net phase advance of integer times of 2π should be achieved, the degeneracy of +m and −m PAMSs gets broken, leading to the different distributions of field amplitude |E| and transverse energy flux S_⊥ [15] in each pair of transmission dips.

We also check the dependencies of transmission spectra, distributions of field amplitude |E| and transverse energy flux S_⊥ versus the direction of B. Reversing the direction of B is equivalent to changing the signs of ω_B and γ. Under this operation, we find that the transmission spectra and distribution of field amplitude do not change. However, the transverse energy flux changes its rotational direction. In other words, the topological charge m changes its sign. These results can be understood readily. According to [14], the dispersion of a PAMS with the classic analogue of Zeeman effect is determined by the factor mγ. In order to get the same dispersion, the topological charge m should reverse its sign if γ changes to −γ. Also from the numerical simulation we find that the phase difference between H_z and E_r flips by π. Consequently, in consistent with the changed sign of m, the dominant component S_ϕ in the transverse energy flux, defined by ${\mathit{S}}_{\varphi }\propto -{\mathit{E}}_r\times {\mathit{H}}_z^{*}$, also reverses its direction.

Figure 3 shows the distributions of field amplitude |E| and transverse energy flux S_⊥ at the three adjacent transmission peaks. In sharp contrast with those shown in Fig. 2 for transmission dips, here the EM field is mostly confined inside the rectangular element. By noticing the difference between the colormaps of Figs. 2 and 3, we can see the field enhancement inside the rectangular element at the transmission peaks is much weaker than that inside the coaxial element at the transmission dips, which is understandable because the bright resonance of the rectangular element has a low-Q factor with a high leaking rate.

From Fig. 3 we can see at the three transmission peaks the transverse energy flux S_⊥ are also different. The directions of energy flux at the inner radius r₁ and outer radius r₂ are opposite with each other. We also analyze the m spectra of these transmission peaks. For the 1st peak at 152.475 GHz, the weights of m = −2 and m = +2 PAMSs are 74.81% and 22.23%, respectively. For the 2nd peak at 152.89 GHz, the weights of m = −2 and m = +2 PAMSs are 58.51% and 39.01%, respectively. The other m components are dominated by m = ±1, which is around 2.5%. The most interesting case is for the 3rd peak at 153.49 GHz, where the m = −1 and m = +1 PAMSs are excited with more significant weights. The corresponding weights of m = −2 −1, +1 and +2 PAMSs are 22.79%, 12.81%, 12.98% and 42.84%, respectively. However, the field inside the coaxial element for this peak is much weaker than those of the other two transmission peaks, see Fig. 3.

Resorting to the field distributions at transmission dips and peaks shown in Figs. 2 and 3, we can then discuss how the high absorption A shown in Fig. 1 is achieved. From Fig. 1 we can see high absorption is present only near the induced transmission dips. From above analysis we can see these dips are obviously associated with the excitation of dark PAMSs with m ≠ ±1. As stated in [12], because the darkness nature of PAMSs is closely related to the existence of m ≠ ±1 topological charges, their resonances possess a high-Q factor. Consequently, the width of the induced transmission dips are very narrow, and the localized field intensity is greatly enhanced. Because for the structure under investigation the absorption of EM field is due to the presence of imaginary components in ε and γ of the MO medium, A is then very strong near these induced transmission dips, where the light energy is stored inside the MO coaxial element by a relatively long time.

2.4. Phase delay and group index

The most-promising application of the sharp dispersion around the induced transmission dips is slow light [12, 28]. Before ending this article we would like to briefly analyze the performance of slow light, especially the dispersion of phase delay δ and group index n_g versus the magnitude of B.

By investigating the transmission phase delay δ through the structure, we obtain the group index n_g versus frequency [12]. As shown in Fig. 4, due to the resonance of the EM wave inside the rectangular and coaxial elements and their mutual interaction, the phase delay δ and group index n_g varies abruptly around the transmission dips, where a high group index n_g can be achieved [12]. It is interesting to emphasize that with the introduction of imaginary parts in ε and γ, the evaluated group index n_g can be negative. This phenomenon is absence in [12] where only non-absorption media are utilized. To understand this phenomenon, for comparison we also plot in Fig. 4(c) the associated absorption coefficient A. We can see when the group index n_g is negative, the absorption A is also very large. The negative group index thus indicates the existence of abnormal dispersion, with a high reflection coefficient and a stronger absorption [28]. The mechanism of this induced abnormal dispersion is due to the excitation of dark PAMSs inside the MO coaxial element, and possesses a geometric-resonance nature.

 

Fig. 4 (a) Phase delay δ, (b) group index n_g and (c) absorption coefficient A versus frequency when ω_B = 0.19ω_p (red lines) and 0.20ω_p (blue lines), respectively.

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Because the slow-light applications are realizable only when the transmission is relatively high, we must pay attention to the frequency regimes near the transmission peaks. Referring to the results shown in Fig. 4, we can see the group index n_g can reach a great value. When ω_B = 0.19ω_p, at the frequency range from 149.987 GHz to 149.993 GHz, where the transmission coefficients T are higher than 0.7, group index n_g from 3360 to 4400 can be achieved. When ω_B = 0.20ω_p, at the three transmission peak frequencies of 152.475, 152.89 and 152.49 GHz, where the transmission coefficients T are greater than 0.97, 0.93 and 0.99, respectively, n_g equals to 488, 554 and 117, respectively. The variation of group index n_g versus ω_B proves that the slow-light effect is controllable by manipulating the magnitude of B.

3. Conclusion

In summary, we prove the feasibility in achieving tunable slow-light performance by applying an external magnetic field to manipulate the dark PAMSs in metamaterials. We show that with increased magnetic field, some pairs of sharp transmission dips emerge in the background transparency window of the complex metamaterial, and the frequencies of these resonances blueshift with increased magnetic field. Each pair of transmission dips are related to the dark PAMSs with opposite topological charges −m and +m, with a giant lifted degeneracy from the classic analogue of Zeeman effect. Novel nonreciprocal features shown in the distributions of field amplitude and transverse energy flux are discussed. The tunable feature of the dark PAMSs and the associated slow-light effect can be subsequently utilized in various applications.