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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 21 — Oct. 21, 2013
  • pp: 25035–25044
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Manipulation of dark photonic angular momentum states via magneto-optical effect for tunable slow-light performance

Mu Yang, Teng-Fei Li, Qi-Wen Sheng, Tian-Jing Guo, Qing-Hua Guo, Hai-Xu Cui, and Jing Chen  »View Author Affiliations


Optics Express, Vol. 21, Issue 21, pp. 25035-25044 (2013)
http://dx.doi.org/10.1364/OE.21.025035


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Abstract

We propose a novel scheme in realizing tunable slow-light performance by manipulating dark photonic angular momentum states (PAMSs) in metamaterials via the magneto-optical effect. We show that by applying a static magnetic field B, some pairs of sharp transmission dips can be observed in the background transparency window of a complex metamaterial design. Each pair of transmission dips are related to the excitation of dark PAMSs with opposite topological charges −m and +m, with a lifted degeneracy due to the classic analogue of Zeeman effect. Nonreciprocal characteristics can be observed in the distributions of field amplitude and transverse energy flux. The performance of slow light, including the group index ng, its abnormal feature, the associated strong absorption and the dependence with B are also discussed.

© 2013 OSA

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.

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

7. F. I. Baida and D. Van Labeke, “Light transmission by subwavelength annular aperture arrays in metallic films,” Opt. Commun. 209, 17–22 (2002). [CrossRef]

10

10. P. Banzer, J. Kindler, S. Quabis, U. Peschel, and G. Leuchs, “Extraordinary transmission through a single coaxial aperture in a thin metal film,” Opt. Express 18, 10896–10904 (2010). [CrossRef] [PubMed]

]. Novel phenomena, including the realization of a wide-angle negative-index metamaterial [11

11. S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mat. 9, 407–412 (2010). [CrossRef]

], have been demonstrated. Furthermore, on account of the darkness nature of m ≠ ±1 PAMSs for all polarizations of normal incidence, Guo et al. [12

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

] proposed a slow-light scheme by indirectly exciting dark PAMSs. A group index ng greater than 500 is achieved [12

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

]. Similar configuration has then been utilized to manipulate the polarization of transmitted optical wave [13

13. Q. H. Guo, M. Yang, T. F. Li, T. J. Guo, H. X. Cui, M. Kang, and J. Chen, “Circular polarizer via selective excitation of photonic angular momentum states in metamaterials,” Appl. Phys. Lett. 102, 211906 (2013). [CrossRef]

] 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

4. Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. 105, 013901 (2010). [CrossRef] [PubMed]

], 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

13. Q. H. Guo, M. Yang, T. F. Li, T. J. Guo, H. X. Cui, M. Kang, and J. Chen, “Circular polarizer via selective excitation of photonic angular momentum states in metamaterials,” Appl. Phys. Lett. 102, 211906 (2013). [CrossRef]

], 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 = 2, 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 φ1,2 for different PAMSs by different degrees [12

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

, 13

13. Q. H. Guo, M. Yang, T. F. Li, T. J. Guo, H. X. Cui, M. Kang, and J. Chen, “Circular polarizer via selective excitation of photonic angular momentum states in metamaterials,” Appl. Phys. Lett. 102, 211906 (2013). [CrossRef]

], thus lifts the degeneracy in the resonant frequencies of the ±m PAMSs. The excitation coefficient, or extinction cross section σm, also gets modified. Although such a geometric method has been successfully applied in applications such as slow light [12

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

] and polarization manipulation [13

13. Q. H. Guo, M. Yang, T. F. Li, T. J. Guo, H. X. Cui, M. Kang, and J. Chen, “Circular polarizer via selective excitation of photonic angular momentum states in metamaterials,” Appl. Phys. Lett. 102, 211906 (2013). [CrossRef]

], 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

14. J. Wang, K. H. Fung, H. Y. Dong, and N. X. Fang, “Zeeman splitting of photonic angular momentum states in a gyromagnetic cylinder,” Phys. Rev. B 84, 235122 (2011). [CrossRef]

] 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

14. J. Wang, K. H. Fung, H. Y. Dong, and N. X. Fang, “Zeeman splitting of photonic angular momentum states in a gyromagnetic cylinder,” Phys. Rev. B 84, 235122 (2011). [CrossRef]

]. 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

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

]. 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

14. J. Wang, K. H. Fung, H. Y. Dong, and N. X. Fang, “Zeeman splitting of photonic angular momentum states in a gyromagnetic cylinder,” Phys. Rev. B 84, 235122 (2011). [CrossRef]

16

16. J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett. 89, 251115 (2006). [CrossRef]

]. 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 ng, 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 ωB = 0 (black curve), 0.19ωp (blue curve) and 0.20ωp (red curve), respectively. Inset shows a schematic view of the metamaterial design.

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

17. Y. M. Bahk, J. W. Choi, J. Kyoung, H. R. Park, K. J. Ahn, and D. S. Kim, “Selective enhanced resonances of two asymmetric terahertz nano resonators,” Opt. Express 20, 25644–25653 (2012). [CrossRef] [PubMed]

22

22. J. Nishitani, T. Nagashima, and M. Hangyo, “Terahertz radiation from antiferromagnetic MnO excited by optical laser pulses,” Appl. Phys. Lett. 103, 081907 (2013). [CrossRef]

]. 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

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

, 13

13. Q. H. Guo, M. Yang, T. F. Li, T. J. Guo, H. X. Cui, M. Kang, and J. Chen, “Circular polarizer via selective excitation of photonic angular momentum states in metamaterials,” Appl. Phys. Lett. 102, 211906 (2013). [CrossRef]

], 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

13. Q. H. Guo, M. Yang, T. F. Li, T. J. Guo, H. X. Cui, M. Kang, and J. Chen, “Circular polarizer via selective excitation of photonic angular momentum states in metamaterials,” Appl. Phys. Lett. 102, 211906 (2013). [CrossRef]

], 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

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

]. The resonance frequency of each PAMS can be estimated by considering the cutoff frequency of coaxial waveguides [7

7. F. I. Baida and D. Van Labeke, “Light transmission by subwavelength annular aperture arrays in metallic films,” Opt. Commun. 209, 17–22 (2002). [CrossRef]

10

10. P. Banzer, J. Kindler, S. Quabis, U. Peschel, and G. Leuchs, “Extraordinary transmission through a single coaxial aperture in a thin metal film,” Opt. Express 18, 10896–10904 (2010). [CrossRef] [PubMed]

], 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

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

]. The fundamental PAMS, which is in fact a cylindrically symmetric TEM mode without cutoff, is also dark because m = 0 [8

8. F. I. Baida, D. Van Labeke, G. Granet, A. Moreau, and A. Belkhir, “Origin of the super-enhanced light transmission through a 2-D metallic annular aperture array: a study of photonic bands,” Appl. Phys. B 79, 1–8 (2004). [CrossRef]

].

Here, in order to lift the degeneracy of PAMSs via the classic analogue of Zeeman effect [14

14. J. Wang, K. H. Fung, H. Y. Dong, and N. X. Fang, “Zeeman splitting of photonic angular momentum states in a gyromagnetic cylinder,” Phys. Rev. B 84, 235122 (2011). [CrossRef]

], 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

15. T. F. Li, T. J. Guo, H. X. Cui, M. Yang, M. Kang, Q. H. Guo, and J. Chen, “Guided modes in magneto-optical waveguides and the role in resonant transmission,” Opt. Express 21, 9563–9572 (2013). [CrossRef] [PubMed]

, 16

16. J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett. 89, 251115 (2006). [CrossRef]

, 23

23. Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008). [CrossRef] [PubMed]

, 24

24. H. Yin and P. M. Hui, “Controlling enhanced transmission through semiconductor gratings with subwavelength slits by a magnetic field: Numerical and analytical results,” Appl. Phys. Lett. 95, 011115 (2009). [CrossRef]

], as
ε¯¯=(εjγ0jγε000ε||),
(1)
where
ε=εωp2ω2ωB2+jω2Γ,
(2)
γ=ωBωp2ω(ω2ωB2+jω2Γ),
(3)
ε||=εωp2ω2+jω2Γ.
(4)
Here ε is the permittivity at high frequency, ωp is the bulk plasma frequency, and Γ represents damping. The static magnetic field B determines the cyclotron frequency ωB = e*B/m*, where e* and m* are the effective charge and mass of the charge carrier, respectively. With Eq. (1), we can show that the dispersions of transverse-electric PAMSs obtained from Maxwell’s equations and boundary conditions are different for +m and −m [14

14. J. Wang, K. H. Fung, H. Y. Dong, and N. X. Fang, “Zeeman splitting of photonic angular momentum states in a gyromagnetic cylinder,” Phys. Rev. B 84, 235122 (2011). [CrossRef]

] when γ is nonzero.

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 ωp, 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 ωp 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

15. T. F. Li, T. J. Guo, H. X. Cui, M. Yang, M. Kang, Q. H. Guo, and J. Chen, “Guided modes in magneto-optical waveguides and the role in resonant transmission,” Opt. Express 21, 9563–9572 (2013). [CrossRef] [PubMed]

]. 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

7. F. I. Baida and D. Van Labeke, “Light transmission by subwavelength annular aperture arrays in metallic films,” Opt. Commun. 209, 17–22 (2002). [CrossRef]

10

10. P. Banzer, J. Kindler, S. Quabis, U. Peschel, and G. Leuchs, “Extraordinary transmission through a single coaxial aperture in a thin metal film,” Opt. Express 18, 10896–10904 (2010). [CrossRef] [PubMed]

], a high doping density n and a strong field B are generally required in order to achieve the required high values of ωp and ωB, which inevitably lead to a strong absorption. Novel material with a small m*, e.g. graphene [25

25. A. K. Geim, “Graphene: status and prospects,” Science 324, 1530–1534 (2009). [CrossRef] [PubMed]

, 26

26. A. Bostwick, T. Ohta, T. Seyller, K. Horn, and E. Rotenberg, “Quasiparticle dynamics in graphene,” Nat. Phy. 3, 36–40 (2007). [CrossRef]

], can be utilized to break this bottleneck.

2.2. Transmission spectra

Consider the situation with ε =15.68, ωp = 2π ×464.45 GHz, and Γ =0.0001, which can be obtained by properly doping InSb [23

23. Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008). [CrossRef] [PubMed]

, 24

24. H. Yin and P. M. Hui, “Controlling enhanced transmission through semiconductor gratings with subwavelength slits by a magnetic field: Numerical and analytical results,” Appl. Phys. Lett. 95, 011115 (2009). [CrossRef]

] with a doping density n of 3.747 × 1013 cm−3. From m* = 0.014me we can see a magnetic field B of 0.05 Tesla can render ωB = 0.20ωp. 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 ωB = 0, 0.19ωp and 0.20ωp, 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

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

, 13

13. Q. H. Guo, M. Yang, T. F. Li, T. J. Guo, H. X. Cui, M. Kang, and J. Chen, “Circular polarizer via selective excitation of photonic angular momentum states in metamaterials,” Appl. Phys. Lett. 102, 211906 (2013). [CrossRef]

].

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 − TR. 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

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

]. 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

14. J. Wang, K. H. Fung, H. Y. Dong, and N. X. Fang, “Zeeman splitting of photonic angular momentum states in a gyromagnetic cylinder,” Phys. Rev. B 84, 235122 (2011). [CrossRef]

]. 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 ωB = 0.20ωp. Note that our numerical simulation shows that the excited PAMSs inside the coaxial element are transverse electric, and the dominant field components are Er and Hz. The azimuthal component Eφ, albeit greater than that of incidence, is much weaker than Er.

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 ωB = 0.20ωp. Insets show the distribution of field amplitude |E| in the whole unit cell.
Fig. 3 Distributions of field amplitude |E| when ωB = 0.20ωp 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.

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 Er(ϕ) inside the coaxial element by Er(ϕ)=+Amexp(jmϕ) [12

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

, 13

13. Q. H. Guo, M. Yang, T. F. Li, T. J. Guo, H. X. Cui, M. Kang, and J. Chen, “Circular polarizer via selective excitation of photonic angular momentum states in metamaterials,” Appl. Phys. Lett. 102, 211906 (2013). [CrossRef]

]. 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

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

] 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 φ1,2 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

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

]. 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

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

]. 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

15. T. F. Li, T. J. Guo, H. X. Cui, M. Yang, M. Kang, Q. H. Guo, and J. Chen, “Guided modes in magneto-optical waveguides and the role in resonant transmission,” Opt. Express 21, 9563–9572 (2013). [CrossRef] [PubMed]

] and the explicit connection of coaxial waveguide with the planar metal-insulator-metal geometry [27

27. P. B. Catryssea and S. H. Fan, “Understanding the dispersion of coaxial plasmonic structures through a connection with the planar metal-insulator-metal geometry,” Appl. Phys. Lett. 94, 231111 (2009). [CrossRef]

]. The off-diagonal element γ determines the phase difference between Er and Eϕ, leading to the transverse MO nonreciprocal phase shift that depends on the direction of transverse energy flux [16

16. J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett. 89, 251115 (2006). [CrossRef]

]. The field profile is also influenced by γ [15

15. T. F. Li, T. J. Guo, H. X. Cui, M. Yang, M. Kang, Q. H. Guo, and J. Chen, “Guided modes in magneto-optical waveguides and the role in resonant transmission,” Opt. Express 21, 9563–9572 (2013). [CrossRef] [PubMed]

]. 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

15. T. F. Li, T. J. Guo, H. X. Cui, M. Yang, M. Kang, Q. H. Guo, and J. Chen, “Guided modes in magneto-optical waveguides and the role in resonant transmission,” Opt. Express 21, 9563–9572 (2013). [CrossRef] [PubMed]

] 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 ωB 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

14. J. Wang, K. H. Fung, H. Y. Dong, and N. X. Fang, “Zeeman splitting of photonic angular momentum states in a gyromagnetic cylinder,” Phys. Rev. B 84, 235122 (2011). [CrossRef]

], the dispersion of a PAMS with the classic analogue of Zeeman effect is determined by the factor . 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 Hz and Er flips by π. Consequently, in consistent with the changed sign of m, the dominant component Sϕ in the transverse energy flux, defined by SϕEr×Hz*, 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.

2.4. Phase delay and group index

The most-promising application of the sharp dispersion around the induced transmission dips is slow light [12

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

, 28

28. J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon. 2, 287–318 (2010). [CrossRef]

]. 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 ng versus the magnitude of B.

By investigating the transmission phase delay δ through the structure, we obtain the group index ng versus frequency [12

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

]. 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 ng varies abruptly around the transmission dips, where a high group index ng can be achieved [12

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

]. It is interesting to emphasize that with the introduction of imaginary parts in ε and γ, the evaluated group index ng can be negative. This phenomenon is absence in [12

12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

] 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 ng 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

28. J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon. 2, 287–318 (2010). [CrossRef]

]. 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 ng and (c) absorption coefficient A versus frequency when ωB = 0.19ωp (red lines) and 0.20ωp (blue lines), respectively.

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 ng can reach a great value. When ωB = 0.19ωp, at the frequency range from 149.987 GHz to 149.993 GHz, where the transmission coefficients T are higher than 0.7, group index ng from 3360 to 4400 can be achieved. When ωB = 0.20ωp, 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, ng equals to 488, 554 and 117, respectively. The variation of group index ng versus ωB proves that the slow-light effect is controllable by manipulating the magnitude of B.

3. Conclusion

Acknowledgments

The authors acknowledge the support from the National Natural Science Foundation of China (NSFC) under grants 11174157 and 11074131, and the Specialized Research Fund for the Doctoral Program (SRFDP) under grant 20110031110005.

References and links

1.

M. I. Tribelsky and B. S. Luk’yanchuk, “Anomalous light scattering by small particles,” Phys. Rev. Lett. 97, 263902 (2006). [CrossRef]

2.

L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett. 98, 157403 (2007). [CrossRef] [PubMed]

3.

L. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mat. 8, 643–647 (2009). [CrossRef]

4.

Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett. 105, 013901 (2010). [CrossRef] [PubMed]

5.

J. Du, Z. Lin, S. T. Chui, W. Lu, H. Li, A. Wu, Z. Sheng, J. Zi, X. Wang, S. Zou, and F. Gan, “Optical beam steering based on the symmetry of resonant modes of nanoparticles,” Phys. Rev. Lett. 106, 203903 (2011). [CrossRef] [PubMed]

6.

J. Du, Z. Lin, S. T. Chui, G. Dong, and W. Zhang, “Nearly total omnidirectional reflection by a single layer of nanorods,” Phys. Rev. Lett. 110, 163902 (2013). [CrossRef] [PubMed]

7.

F. I. Baida and D. Van Labeke, “Light transmission by subwavelength annular aperture arrays in metallic films,” Opt. Commun. 209, 17–22 (2002). [CrossRef]

8.

F. I. Baida, D. Van Labeke, G. Granet, A. Moreau, and A. Belkhir, “Origin of the super-enhanced light transmission through a 2-D metallic annular aperture array: a study of photonic bands,” Appl. Phys. B 79, 1–8 (2004). [CrossRef]

9.

M. J. Lockyear, A. P. Hibbins, J. R. Sambles, and C. R. Lawrence, “Microwave transmission through a single subwavelength annular aperture in a metal plate,” Phys. Rev. Lett. 94,193902 (2005). [CrossRef] [PubMed]

10.

P. Banzer, J. Kindler, S. Quabis, U. Peschel, and G. Leuchs, “Extraordinary transmission through a single coaxial aperture in a thin metal film,” Opt. Express 18, 10896–10904 (2010). [CrossRef] [PubMed]

11.

S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mat. 9, 407–412 (2010). [CrossRef]

12.

Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett. 38, 250–252 (2013). [CrossRef] [PubMed]

13.

Q. H. Guo, M. Yang, T. F. Li, T. J. Guo, H. X. Cui, M. Kang, and J. Chen, “Circular polarizer via selective excitation of photonic angular momentum states in metamaterials,” Appl. Phys. Lett. 102, 211906 (2013). [CrossRef]

14.

J. Wang, K. H. Fung, H. Y. Dong, and N. X. Fang, “Zeeman splitting of photonic angular momentum states in a gyromagnetic cylinder,” Phys. Rev. B 84, 235122 (2011). [CrossRef]

15.

T. F. Li, T. J. Guo, H. X. Cui, M. Yang, M. Kang, Q. H. Guo, and J. Chen, “Guided modes in magneto-optical waveguides and the role in resonant transmission,” Opt. Express 21, 9563–9572 (2013). [CrossRef] [PubMed]

16.

J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett. 89, 251115 (2006). [CrossRef]

17.

Y. M. Bahk, J. W. Choi, J. Kyoung, H. R. Park, K. J. Ahn, and D. S. Kim, “Selective enhanced resonances of two asymmetric terahertz nano resonators,” Opt. Express 20, 25644–25653 (2012). [CrossRef] [PubMed]

18.

Y. M. Bahk, H. R. Park, K. J. Ahn, H. S. Kim, Y. H. Ahn, D. S. Kim, J. Bravo-Abad, L. Martin-Moreno, and F. J. Garcia-Vidal, “Anomalous band formation in arrays of terahertz nanoresonators,” Phys. Rev. Lett. 106, 013902 (2011). [CrossRef] [PubMed]

19.

H. R. Park, Y. M. Bahk, K. J. Ahn, Q. H. Park, D. S. Kim, L. Martin-Moreno, F. J. Garcia-Vidal, and J. Bravo-Abad, “Controlling terahertz radiation with nanoscale metal barriers embedded in nano slot antennas,” ACS Nano 5, 8340–8345 (2011). [CrossRef] [PubMed]

20.

H. R. Park, Y. M. Bahk, J. H. Choe, S. Han, S. S. Choi, K. J. Ahn, N. Park, Q. H. Park, and D. S. Kim, “Terahertz pinch harmonics enabled by single nano rods,” Opt. Express 19, 24775–24781 (2011). [CrossRef] [PubMed]

21.

K. Kozuki, T. Nagashima, and M. Hangyo, “Measurement of electron paramagnetic resonance using terahertz time-domain spectroscopy,” Opt. Express 19, 24950–24956 (2011). [CrossRef]

22.

J. Nishitani, T. Nagashima, and M. Hangyo, “Terahertz radiation from antiferromagnetic MnO excited by optical laser pulses,” Appl. Phys. Lett. 103, 081907 (2013). [CrossRef]

23.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008). [CrossRef] [PubMed]

24.

H. Yin and P. M. Hui, “Controlling enhanced transmission through semiconductor gratings with subwavelength slits by a magnetic field: Numerical and analytical results,” Appl. Phys. Lett. 95, 011115 (2009). [CrossRef]

25.

A. K. Geim, “Graphene: status and prospects,” Science 324, 1530–1534 (2009). [CrossRef] [PubMed]

26.

A. Bostwick, T. Ohta, T. Seyller, K. Horn, and E. Rotenberg, “Quasiparticle dynamics in graphene,” Nat. Phy. 3, 36–40 (2007). [CrossRef]

27.

P. B. Catryssea and S. H. Fan, “Understanding the dispersion of coaxial plasmonic structures through a connection with the planar metal-insulator-metal geometry,” Appl. Phys. Lett. 94, 231111 (2009). [CrossRef]

28.

J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon. 2, 287–318 (2010). [CrossRef]

OCIS Codes
(020.7490) Atomic and molecular physics : Zeeman effect
(160.3820) Materials : Magneto-optical materials
(270.1670) Quantum optics : Coherent optical effects
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: July 22, 2013
Revised Manuscript: September 24, 2013
Manuscript Accepted: September 24, 2013
Published: October 14, 2013

Citation
Mu Yang, Teng-Fei Li, Qi-Wen Sheng, Tian-Jing Guo, Qing-Hua Guo, Hai-Xu Cui, and Jing Chen, "Manipulation of dark photonic angular momentum states via magneto-optical effect for tunable slow-light performance," Opt. Express 21, 25035-25044 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-25035


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References

  1. M. I. Tribelsky and B. S. Luk’yanchuk, “Anomalous light scattering by small particles,” Phys. Rev. Lett.97, 263902 (2006). [CrossRef]
  2. L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett.98, 157403 (2007). [CrossRef] [PubMed]
  3. L. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mat.8, 643–647 (2009). [CrossRef]
  4. Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures,” Phys. Rev. Lett.105, 013901 (2010). [CrossRef] [PubMed]
  5. J. Du, Z. Lin, S. T. Chui, W. Lu, H. Li, A. Wu, Z. Sheng, J. Zi, X. Wang, S. Zou, and F. Gan, “Optical beam steering based on the symmetry of resonant modes of nanoparticles,” Phys. Rev. Lett.106, 203903 (2011). [CrossRef] [PubMed]
  6. J. Du, Z. Lin, S. T. Chui, G. Dong, and W. Zhang, “Nearly total omnidirectional reflection by a single layer of nanorods,” Phys. Rev. Lett.110, 163902 (2013). [CrossRef] [PubMed]
  7. F. I. Baida and D. Van Labeke, “Light transmission by subwavelength annular aperture arrays in metallic films,” Opt. Commun.209, 17–22 (2002). [CrossRef]
  8. F. I. Baida, D. Van Labeke, G. Granet, A. Moreau, and A. Belkhir, “Origin of the super-enhanced light transmission through a 2-D metallic annular aperture array: a study of photonic bands,” Appl. Phys. B79, 1–8 (2004). [CrossRef]
  9. M. J. Lockyear, A. P. Hibbins, J. R. Sambles, and C. R. Lawrence, “Microwave transmission through a single subwavelength annular aperture in a metal plate,” Phys. Rev. Lett.94,193902 (2005). [CrossRef] [PubMed]
  10. P. Banzer, J. Kindler, S. Quabis, U. Peschel, and G. Leuchs, “Extraordinary transmission through a single coaxial aperture in a thin metal film,” Opt. Express18, 10896–10904 (2010). [CrossRef] [PubMed]
  11. S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwater, “A single-layer wide-angle negative-index metamaterial at visible frequencies,” Nat. Mat.9, 407–412 (2010). [CrossRef]
  12. Q. H. Guo, M. Kang, T. F. Li, H. X. Cui, and J. Chen, “Slow light from sharp dispersion by exciting dark photonic angular momentum states,” Opt. Lett.38, 250–252 (2013). [CrossRef] [PubMed]
  13. Q. H. Guo, M. Yang, T. F. Li, T. J. Guo, H. X. Cui, M. Kang, and J. Chen, “Circular polarizer via selective excitation of photonic angular momentum states in metamaterials,” Appl. Phys. Lett.102, 211906 (2013). [CrossRef]
  14. J. Wang, K. H. Fung, H. Y. Dong, and N. X. Fang, “Zeeman splitting of photonic angular momentum states in a gyromagnetic cylinder,” Phys. Rev. B84, 235122 (2011). [CrossRef]
  15. T. F. Li, T. J. Guo, H. X. Cui, M. Yang, M. Kang, Q. H. Guo, and J. Chen, “Guided modes in magneto-optical waveguides and the role in resonant transmission,” Opt. Express21, 9563–9572 (2013). [CrossRef] [PubMed]
  16. J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett.89, 251115 (2006). [CrossRef]
  17. Y. M. Bahk, J. W. Choi, J. Kyoung, H. R. Park, K. J. Ahn, and D. S. Kim, “Selective enhanced resonances of two asymmetric terahertz nano resonators,” Opt. Express20, 25644–25653 (2012). [CrossRef] [PubMed]
  18. Y. M. Bahk, H. R. Park, K. J. Ahn, H. S. Kim, Y. H. Ahn, D. S. Kim, J. Bravo-Abad, L. Martin-Moreno, and F. J. Garcia-Vidal, “Anomalous band formation in arrays of terahertz nanoresonators,” Phys. Rev. Lett.106, 013902 (2011). [CrossRef] [PubMed]
  19. H. R. Park, Y. M. Bahk, K. J. Ahn, Q. H. Park, D. S. Kim, L. Martin-Moreno, F. J. Garcia-Vidal, and J. Bravo-Abad, “Controlling terahertz radiation with nanoscale metal barriers embedded in nano slot antennas,” ACS Nano5, 8340–8345 (2011). [CrossRef] [PubMed]
  20. H. R. Park, Y. M. Bahk, J. H. Choe, S. Han, S. S. Choi, K. J. Ahn, N. Park, Q. H. Park, and D. S. Kim, “Terahertz pinch harmonics enabled by single nano rods,” Opt. Express19, 24775–24781 (2011). [CrossRef] [PubMed]
  21. K. Kozuki, T. Nagashima, and M. Hangyo, “Measurement of electron paramagnetic resonance using terahertz time-domain spectroscopy,” Opt. Express19, 24950–24956 (2011). [CrossRef]
  22. J. Nishitani, T. Nagashima, and M. Hangyo, “Terahertz radiation from antiferromagnetic MnO excited by optical laser pulses,” Appl. Phys. Lett.103, 081907 (2013). [CrossRef]
  23. Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett.100, 023902 (2008). [CrossRef] [PubMed]
  24. H. Yin and P. M. Hui, “Controlling enhanced transmission through semiconductor gratings with subwavelength slits by a magnetic field: Numerical and analytical results,” Appl. Phys. Lett.95, 011115 (2009). [CrossRef]
  25. A. K. Geim, “Graphene: status and prospects,” Science324, 1530–1534 (2009). [CrossRef] [PubMed]
  26. A. Bostwick, T. Ohta, T. Seyller, K. Horn, and E. Rotenberg, “Quasiparticle dynamics in graphene,” Nat. Phy.3, 36–40 (2007). [CrossRef]
  27. P. B. Catryssea and S. H. Fan, “Understanding the dispersion of coaxial plasmonic structures through a connection with the planar metal-insulator-metal geometry,” Appl. Phys. Lett.94, 231111 (2009). [CrossRef]
  28. J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon.2, 287–318 (2010). [CrossRef]

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