Coherent-trapped helical mode in parity-time ... - OSA Publishing

Vol. 25, No. 13 | 26 Jun 2017 | OPTICS EXPRESS 15231

Coherent-trapped helical mode in parity-time symmetric metamaterials L U -Q I WANG , 1,2 R UI -D ONG X UE , 1,2 W EI WANG , 1,2 R UI -X UAN WANG , 1 R UI -P ENG G UO, 1 AND J ING C HEN 1,* 1 MOE Key Laboratory of Weak-Light Nonlinear Photonics, School of Physics, Nankai University, Tianjin 300071, China 2 These authors contributed equally to this work * [email protected]

Abstract: Coaxial optical subwavelength elements support helical modes L m with different topological indexes m. Here we propose to couple the two bright L ±1 modes with the dark one L 0 via a parity-time (PT ) symmetric perturbation. We show that the cascading coupled configuration is similar to a three-level atomic system, and supports a special hybridized mode L c via a classic analog of coherent-population-trapping effect. Resonant frequency of L c is independent of the PT -symmetric perturbation. Populations in L ±1 can be manipulated by tuning the PT -symmetric perturbation, and no population is trapped in L 0 . Since the L ±1 modes are associated with optical waves of opposite circular polarizations, the polarization of transmitted wave is independent of the polarization of incidence but solely determined by the PT -symmetric perturbation. Such an effect can be utilized to manipulate the polarization state of light. Numerical simulation in a well-designed coaxial metamaterial verifies our analysis. c 2017 Optical Society of America OCIS codes: (050.1950) Diffraction gratings; (270.0270) Quantum optics; (160.3918) Metamaterials; (050.4865) Optical vortices.

References and links 1. F. J. García de Abajo, “Colloquium: light scattering by particle and hole arrays," Rev. Mod. Phys. 79(4), 1267–1289 (2007). 2. C. Genet and T. W. Ebbesen, “Light in tiny holes," Nature 445(7123), 39–46 (2007). 3. Y. M. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology," Chem. Soc. Rev. 40(5), 2494– 2507 (2011). 4. N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices," Nat. Mater. 11(11), 917–924 (2012). 5. W. J. Fan, S. Zhang, B. Minhas, K. J. Malloy, and S. R. J. Brueck, “Enhanced infrared transmission through subwavelength coaxial metallic arrays," Phys. Rev. Lett. 94(3), 033902 (2005). 6. 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(19), 193902 (2005). 7. F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, “Subwavelength metallic coaxial waveguides in the optical range: role of the plasmonic modes," Phys. Rev. B 74(20), 205419 (2006). 8. Z. Ruan and S. Fan, “Superscattering of light from subwavelength nanostructures," Phys. Rev. Lett. 105(1), 013901 (2010). 9. S. Zhang, H. Wei, K. Bao, U. Hakanson, N. J. Halas, P. Nordlander, and H. Xu, “Chiral surface plasmon polaritons on metallic nanowires," Phys. Rev. Lett. 107(9), 096801 (2011). 10. M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers," Nature 482(7384), 204–207 (2012). 11. 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(23), 235122 (2011). 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(3), 250–252 (2013). 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(21), 211906 (2013). 14. F. I. Baida, “Enhanced transmission through subwavelength metallic coaxial apertures by excitation of the TEM mode," Appl. Phys. B 89(2), 145–149 (2007). 15. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications," Advances in Optics and Photonics 3(2), 161–204 (2011). 16. F. Cardano and L. Marrucci, “Spin-orbit photonics," Nat. Photonics 9(12), 776–778 (2015).

#292952 Journal © 2017

https://doi.org/10.1364/OE.25.015231 Received 25 Apr 2017; revised 18 Jun 2017; accepted 18 Jun 2017; published 22 Jun 2017


17. K. Y. Bliokh, F. J. Rodriguez-Fortuno, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light," Nat. Photonics 9(12), 796–808 (2015). 18. 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(6), 067402 (2015). 19. J. B. Khurgin, “Slow light in various media: a tutorial," Advances in Optics and Photonics 2(3), 287–318 (2010). 20. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997). 21. S. Longhi, “Quantum-optical analogies using photonic structures," Laser and Photon. Rev. 3(3), 243–261 (2009). 22. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having P T symmetry," Phys. Rev. Lett. 80(24), 5243–5246 (1998). 23. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in P T symmetric optical lattices," Phys. Rev. Lett. 100(10), 103904 (2008). 24. S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in P T-symmetric waveguides," Phys. Rev. Lett. 101(8), 080402 (2008). 25. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of P T-symmetry breaking in complex optical potentials," Phys. Rev. Lett. 103(9), 093902 (2009). 26. S. Longhi, “P T-symmetric laser absorber," Phys. Rev. A 82(3), 031801 (2010). 27. W. Wan, Y. Chong, L. Ge, H. Noh, A. D. Stone, and H. Cao, “Time-reversed lasing and interferometric control of absorption," Science 331(6019), 889–892 (2011). 28. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics," Nat. Phys. 6(3), 192–195 (2010). 29. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices," Nature 488(7410), 167–171 (2012). 30. 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(2), 108–113 (2013). 31. H. Hodaei, M. A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers," Science 346(6212), 975–978 (2014). 32. L. Feng, Z. J. Wong, R. M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking," Science 346(6212), 972–975 (2014). 33. K. H. Kim, M. S. Hwang, H. R. Kim, J. H. Choi, Y. S. No, and H. G. Park, “Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains," Nat. Commun. 7, 13893 (2016). 34. B. He, S. B. Yan, J. Wang, and M. Xiao, “Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides," Phys. Rev. A 91(5), 053832 (2015). 35. M. H. Teimourpour, R. El-Ganainy, A. Eisfeld, A. Szameit, and D. N. Christodoulides, “Light transport in P Tinvariant photonic structures with hidden symmetries," Phys. Rev. A 90(5), 053817 (2014). 36. B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, “Lossinduced suppression and revival of lasing," Science 346(6207), 328–332 (2014). 37. M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schoberl, H. E. Tureci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point," Nat. Commun. 5, 4034 (2014). 38. A. K. Jahromi and A. F. Abouraddy, “Observation of Poynting’s vector reversal in an active photonic cavity," Optica 3(11), 1194–1200 (2016). 39. A. K. Jahromi, S. Shabahang, H. E. Kondakci, S. Orsila, P. Melanen, and A. F. Abouraddy, “Transparent perfect mirror," ACS Photonics 4(5), 1026–1032 (2017). 40. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators," Nat. Photon. 8(7), 524–529 (2014). 41. B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities," Nat. Phys. 10(5), 394–398 (2014). 42. H. Hodaei, M. A. Miri, A. U. Hassan, W. E. Hayenga, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Single mode lasing in transversely multi-moded P T-symmetric microring resonators," Laser Photonics Rev. 10(3), 494–499 (2016). 43. F. J. Shu, C. L. Zou, X. B. Zou, and L. Yang, “Chiral symmetry breaking in a microring optical cavity by engineered dissipation," Phys. Rev. A 94(1), 013848 (2016). 44. F. Yang, Z. Fang, Z. Pan, Q. Ye, H. Cai, and R. Qu, “Orthogonal polarization mode coupling for pure twisted polarization maintaining fiber Bragg gratings," Opt. Express 20(27), 28839–28845 (2012). 45. G. D. Valle, M. Ornigotti, T. T. Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, “Adiabatic light transfer via dressed states in optical waveguide arrays," Appl. Phys. Lett. 92(1), 011106 (2008). 46. J. A. Porto, F. J. García-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83(14), 2845–2848 (1999). 47. M. Yang, L. T. Wu, T. J. Guo, R. P. Guo, H. X. Cui, X. W. Cao, and J. Chen, “Study on photonic angular momentum states in coaxial magneto-optical waveguides," J. Appl. Phys. 116(15), 153104 (2014).


1.

Introduction

There has been recently a great deal of interest in understanding the physical mechanism of optical resonances in various nano-structures, e.g. metamaterials [1–4]. The enthusiasm arises due to the facts that the resonances in metamaterials can be geometrically or dynamically manipulated, that the discrete nature of them can be utilized to simulate many quantum effects, and that the sub-wavelength size of the element enables an effective medium approach for various applications. Among all the structures investigated so far, coaxial and cylinder holes are of great importance [5–14] because they are excellent platforms in supporting optical spin-orbit interaction (SOI) [13–18]. To be more explicit, in a coaxial or cylinder hole the discrete helical optical modes L m are characterized by the topological indexes m in the transverse phases exp(− jmθ), which can be understood as the orbital angular momentum (OAM) component [11–13]. Due to a conservation rule over the polarization (spin angular momentum, SAM) and OAM of photons, transmission of a normally incident optical wave through a coaxial or circular hole relies on the excitation of bright L ±1 modes [5–7,13]. Polarization state of transmitted field can be controlled by manipulating the relatively weights of L ±1 . Helical modes with m ±1 are dark and can be utilized to store light energy for slow-light applications [12, 19, 20] because they have giant Q factors. The L ±1 modes possess many celebrated merits. For example, they are energy degenerated and can be directly excited by a normal incidence. They can be converted into opposite circular polarized output via the SOI effect [9, 12, 13]. Also they can interact with dark helical modes via a proper introduced grating, from a defect inside the element, or via aperture scattering. Consequently, the L ±1 modes are appropriate candidates in simulating and testing quantum effects of coupled atomic levels in the spirit of the quantum-optic analogies [21]. In this article we propose a scheme in simulating the coherent population trapping effect of three-level configuration [19–21] by using helical modes. Coupled three-level configuration in atomic system has shown great impact over the interaction of photons with atoms, for example, in refractive index enhancement, electromagnetic induced transparency (EIT), and lasing without reversion (LWI) [19–21]. Here we consider the scenario when the two bright L ±1 modes couple with the central dark one L 0 . Since the topological index of L 0 is just between those of L ±1 , their mutual interaction forms a cascaded process if the coupling mechanism provides the necessary mismatch m of ±1. A unique feature of our investigation presented here is that we utilize parity-time (PT ) symmetry, an important concept of non-Hermitian optics [22–43], as the coupling mechanism. Non-Hermitian optics including the PT symmetry has been shown to render many novel phenomena such as time-reversed lasing or coherent perfect absorber, unidirectional reflectionless, single-mode lasing, loss-induced lasing, and optical isolation [26, 27, 30–32, 36–43]. Here a PT -symmetric grating is utilized to produce nonreciprocal coupling strengths in the forward and backward transitions among L 0 and L ±1 . This mechanism is different from others, e.g. the non-reciprocity originating from twisting a set of coupled fibers [44]. By developing a nonHermitian matrix approach we show that a special hybridized mode L c can be obtained. The formation of L c is similar to the coherent population trapping in atomic system [21], a mechanism that plays a key role in EIT, LWI, stimulated Raman adiabatic passage technique (STIRAP), and adiabatic light transfer via dressed state [19–21, 45]. Formed by the interference of different interaction pathways among L ±1 and L 0 , there is no population trapped in L 0 . Populations in L ±1 are determined by parameters of the PT -symmetric grating. This effect can be used for a polarization-independent optical spin filtering purpose, i.e. the polarization state of transmitted wave is independent of the polarization of incidence but solely determined by the PT -symmetric grating. Finally, we provide numerical simulation in a well-designed metamaterial. From the variation of transmission spectra and polarization of field versus parameters of the PT -symmetric grating we show that the coherent-trapped mode L c is readily to be observed in


Fig. 1. (a) In a coaxial element the optical eigenmodes are helical modes L m with different topological indexes m. (b) The modes L ±1 and L 0 can be cascadingly coupled with each other by a PT -symmetric grating in the form of Δε = ε A exp(− jθ) + ε B exp(+ jθ), which renders nonreciprocal back and forth transitions among L ±1 and L 0 . (c) The hybridized mode can be detected by measuring the spectra of optical transmission and the associated polarization state of field.

future experiments. Discussion and conclusion are provided in the last two sections. 2.

Theory

Figure 1 schematically shows a coaxial element within our interest. Here we do not consider a cylinder hole because it does not support L 0 [7, 12–14]. The coaxial element supports a serial of helical modes L m and let us pay attention to the three ones with the lowest topological indexes m. The L 0 mode is a transverse electromagnetic mode without cutoff, with a resonant frequency ω0 . The L ±1 modes are degenerated in resonant frequency ω1 with opposite helical phases of exp(∓ jθ). By properly designing geometry of the coaxial element, i.e. its inner radius r 1 , outer radius r 2 , thickness h, and dielectric constant ε d inside, we can manage their cutoff wavelengths and resonant frequencies [6, 7, 14] to make ω0 ∼ ω1 while being far away from other modes. When the coaxial elements are periodically arranged, e.g. in a square lattice with a period of a, existence of these resonant modes can be observed from the extraordinary optical transmission (EOT) spectra [1, 2, 5–7, 12–14]. Examples about the dependence of EOT spectra versus geometric parameters r 1 , r 2 , a and h, can be found in [5,7,12–14] and references therein. Here a weak PT -symmetric modulation ε is introduced into the coaxial element (r1 < r < r 2 and −0.5h < z < 0.5h), as ε(θ) = ε d + Δε = ε d + ε A e − jθ + ε B e+ jθ ,

(1)

where parameters ε A and ε B are real and much smaller than the background dielectric constant ε d . The perturbation ε behaves as a helical grating that couples different modes together, as


demonstrated in the single-mode lasing in a ring resonator [32] and the broken of the chiral symmetry of light in traveling wave resonators [43]. Because the component ε A e − jθ (ε B e+ jθ ) can compensate the topological index mismatch Δm from L −1 to L 0 (from L +1 to L 0 ), and then from L 0 to L +1 (from L 0 to L −1 ), the PT -symmetric modulation ε renders a cascaded interaction among these modes, see Fig. 1(b). To develop a proper theory in describing the cascaded coupling, it is necessary to emphasize that unlike former literatures [43], here L m is propagating along z direction, perpendicular to the circular symmetric cross-section. Resonance of L m inside the coaxial element, especially these of the bright L ±1 ones, can be observed from EOT [1, 2, 5–7, 12–14]. A coupled mode theory in a matrix form which can determine the eigenfrequencies and the eigenfunctions is desired. Here we develop a coupled-mode matrix by using a perturbation approach over the Maxwell’s equations. Within the coaxial element the Maxwell’s equations can be expressed as ε A − jθ ε B + jθ ∂2 E 1 ∇ × × E = −(1 + e + e ) 2. ε 0 μ0 ε d εd εd ∂t

(2)

When the perturbation is zero, Eq. (2) reads (ε0 μ0 ε d ) −1 ∇ × × E = −∂2 E/∂t 2 . We can assume that with boundary conditions, its exact solutions are a serial of orthogonal L m modes of Em = em (r, z) exp(− jmθ + jωm t), where em (r, z) is a normalized three-dimensional vector that arises from the waveguide resonances inside the coaxial resonator [1, 6, 10, 46, 47], h/2 r 2π −h/2 r 2 re∗m .em dr dz = 1. 1 The PT -symmetric grating mixes the initial orthogonal modes L m together. To find the solution of Eq. (2) for nonvanishing ε A and/or ε B , we then approximately express the hybridized eigenmode E P T = [c+1 e+1 (r, z)e − jθ + c0 e0 (r, z) + c−1 e −1 (r, z)e+ jθ ]e jωt ,

(3)

where ω is the eigenfrequency to be determined, and cm represents the complex amplitude, or population in the quantum-optic terminology, in L m . Substituting Eq. (3) into Eq. (2), multi∗ exp(+ jθ), e∗ , and e∗ exp(− jθ), respectively, and integrating over the plying both sides by e+1 0 −1 spatial region occupied by the coaxial element, we can get three equations εA κc0 , εd εA εB ω02 c0 = ω2 c0 + ω2 κc−1 + ω2 κ ∗ c+1 , εd εd 2 2 2 εB ∗ κ c0 , ω1 c−1 = ω c−1 + ω εd where, by considering the facts that e+1 = e∗−1 and e0∗ = e0 [7, 45, 46], we have defined h/2 r 2 ∗ re+1 .e0 dr dz κ = 2π ω12 c+1 = ω2 c+1 + ω2

−h/2

(4) (5) (6)

(7)

r1

to represent the field overlap of L ±1 and L 0 . Note that the helical phases are not included in κ because they are matched by the terms exp(± jθ) in ε. Since the perturbation is very weak, i.e. ε A, B ε d , and the overlap integral Λ is also smaller than 1, we can assume that ω associated with ε A, B in Eqs. (4) to (6) equals to ω = (ω0 + ω1 )/2. Now the eigensolution is given by a non-Hermitian matrix ⎤⎡ ⎤ ⎡ ⎤ ⎡ 2 0 ⎥ ⎢ c+1 ⎥ ⎢ c+1 ⎥ ⎢⎢⎢ ω1 A ⎢⎢⎢ B ω2 A ⎥⎥⎥⎥⎥ ⎢⎢⎢⎢⎢ c ⎥⎥⎥⎥⎥ = ω2 ⎢⎢⎢⎢⎢ c ⎥⎥⎥⎥⎥ , (8) ⎥⎦ ⎢⎣ 0 ⎥⎦ ⎢⎣ 0 ⎥⎦ ⎢⎣ 0 c−1 c−1 0 B ω12


Fig. 2. (a) Spectra of eigenfrequencies versus B, when ω0 = 0.90ω1 and A = 0.08ω12 . At the exceptional point of AB = −(ω12 − ω02 ) 2 /8 the ω+ and ω− modes coalescence. Below this value the phase is broken, and the eigenfrequencies are complex. (b) Variations of the magnitude of c±1 and c0 versus B for the coherent-trapped helical mode (ωc = ω1 ).

where the off-diagonal elements are associated with either ε A or ε B εA 2 ω¯ κ, εd

(9)

εB 2 ∗ ω¯ κ . εd

(10)

A=− B=−

The product of AB is always a real number. Because the structure is mirror symmetric with respect to the central plane z = 0, the waveguide resonances em (r, z) is either symmetric or anti-symmetric [1, 6, 7, 46, 47] in the z direction. Parameters κ, A, and B are thus generally real. The cascaded interaction described by Eq. (8) gives three solutions of ω. An example for ω0 = 0.90ω1 is shown in Fig. 2. The first two solutions are

2 + ω2 ω2 − ω02 ω 0 ± ( 1 ) 2 + 2AB. ω±2 = 1 (11) 2 2 Their characteristics are determined by the magnitude of AB. If AB is positive, we get two split branches. When AB is zero, ω+ = ω1 and ω− = ω0 . When AB becomes negative, the two branches approach each other, and merge into [(ω12 + ω02 )/2]0.5 at the exceptional point (EP) of AB = −(ω12 − ω02 ) 2 /8. Below this value the phase is broken and the eigenfrequencies are complex. All these features are general properties of a PT -symmetric two-level system [22–26, 28–32, 40–42], and would not be discussed furthermore in this article. Our attention is paid to the third eigenmode L c [see the blue line in Fig. 2(a)], whose eigenfrequency is given by (12) ωc = ω1 . This solution is independent of A and B, and always coincides with that of the L ±1 modes. As for the eigenfunction, from Eq. (8) we can find c+1 =

εA ε 2A + ε 2B

c0 = 0,

,

(13)

(14)


Fig. 3. COMSOL simulation results on the variation of EOT spectra versus ε B in the PT symmetric metamaterial shown in Fig. 1(c). Parameters are r 1 = 0.5mm, r 2 = 0.7mm, h = 0.6mm, ε d = 12, a = 2.5mm, and ε A = 1. The branch ωc is the coherent trapped mode L c within our interest.

c−1 = −

εB ε 2A + ε 2B

.

(15)

The variations of magnitudes of cm versus B are shown in Fig. 2(b). We can see there is no population trapped in L 0 due to a destructive interference between the two transition pathways of L +1 ↔ L 0 and L 0 ↔ L −1 . Such a phenomenon is a close analogue of the coherent trapping effect that takes place in a three-level atom [19,20], or the population transfer effect via STIRAP [21, 45]. Modes L ±1 are equivalent to the two metastable levels, while L 0 is the third dark one [19–21, 45]. Parameters A and B are equivalent to the Rabi frequencies that stimulate the transitions of L ±1 ↔ L 0 [20, 21, 45]. Note that unlike atomic systems, here the transitions are nonreciprocal, that the strengths A and B in the forward and backward transitions of L ±1 ↔ L 0 are different from each other [see Fig. 1(b)]. Magnitudes of A and B can be artificially tuned by changing the values of ε A and ε B in the PT -symmetric grating [see Eqs. (9) and (10)]. The existence of the coherent-trapped helical mode L c can be observed from the EOT spectra of metamaterials [1, 2, 5–7, 12–14]. Characteristics on the populations c±1 can be detected from the polarization state of transmitted wave. Since SOI would convert the bright L ±1 modes into far-field optical waves with opposite circular polarizations [12–14], that L +1 (L −1 ) is associated with left (right)-handed circular polarization with S3 = +1 (S3 = −1), where S3 is the normalized Stokes parameter, it is readily to show from Eqs. (13) to (15) that the polarization of transmitted optical wave is ε 2 − ε 2B . (16) S3 = 2A ε A + ε 2B In other words, the polarization state of EOT peak is solely determined by ε A, B other than the polarization of incidence. Such an effect can be utilized to manipulate the polarization state of optical field, as demonstrated below.


Fig. 4. (a) EOT spectra when ε A = ε B = 1, and the distribution of field intensity at (b) ω− = 66.8 GHz, (c) ωc = 76.6 GHz, and (d) ω+ = 78.4 GHz, respectively. Plots (c) and (d) share the same colormap.

3.

Simulation

We provide a metamaterial design and simulate the EOT spectra to demonstrate the existence of coherent-trapped helical mode L c . The simulation is based on COMSOL Multiphysics software. The metamaterial is made of a square lattice of coaxial elements in a perfect electrical conductor (PEC) plate, with r 1 = 0.5mm, r 2 = 0.7mm, h = 0.6mm, ε d = 12, and period a = 2.5mm. When the PT -symmetric grating is absent, i.e. ε A = ε B = 0, a single EOT peak at ω1 = 77.04 GHz is obtained. By checking the variation of field pattern with the polarization of incidence we confirm that it is associated with the excitation of L ±1 modes, in agreement with [5, 7, 12–14]. Then, we introduce a PT -symmetric grating in the form of Eq. (1) into the coaxial element. Figure 3 shows the results when ε B varies√from −2 to 2 for ε A = 1. The incident wave is right-handed circularly polarized ( xˆ + j yˆ )/ 2. We can see three EOT branches are obtained. One branch is relatively flat and briefly around ωc = ω1 . This branch is the coherent-trapped mode L c within our interest. As for the other two branches ω+ and ω− , they merge into a single EP-like point at ε B ∼ −1.3. When ε B equals zero, the ω+ branch degenerates with ωc . All these features are in consistence with these shown in Fig. 2(a). In order to prove that the excitation of L ±1 and L 0 modes dominates these EOT peaks, we analyze the distribution of field inside the coaxial elements. Figure 4 shows the results when ε A = ε B = 1. Three EOT peaks are present, at ω− = 66.8 GHz, ωc = 76.6 GHz, and ω+ = 78.4 GHz, respectively. From the field distributions shown in Figs. 4(b) to 4(c) we can see at most two regions of high field are present. It implies that only helical modes with |m| ≤ 1 are excited, i.e. the modes could only be L ±1 and L 0 . Furthermore, the field intensity at ωc [Fig. 4(c)] possesses


Fig. 5. Variation of the polarization state S3 of transmitted field versus ε B on the ωc branch. Red dots are these from COMSOL simulation and the blue solid line is given by Eq. (16).

a better symmetry than those of ω± , and the two regions of high field are almost equal with each other. Such a feature hints that at ωc the L 0 mode is very weakly excited and approaches zero, consistent with Eq. (14). Reasons for the nonzero population of c0 and the weakly dispersion of L c branch are discussed in the next section. Paying attention to the L c branch, we analyze the associated polarization state S3 of transmitted field. The result is shown in Fig. 5. When ε B = 0, the transmitted field is left-handed circularly polarized despite the fact that the incidence has a right-handed circular polarization. When the magnitude of ε B increases, the polarization becomes elliptical and finally turns to linear (S3 = 0) when ε B = ±ε A . If the magnitude of ε B is larger than ε A , the stoke’s parameter S3 is negative. We fit the variation of S3 versus ε B by using Eq. (16). From the solid line shown in Fig. 5 we can see our theory agrees well with the results from COMSOL simulation. 4.

Discussion

Above we have shown that a classic analog of coherent population trapping effect can be realized in PT -symmetric coaxial metamaterials. The energy degeneracy of L ±1 modes and the utilization of L 0 are two keys of this scheme. If L ±1 have different resonant frequencies, we could not get the simple curves shown in Fig. 2(a). An avoid anti-crossing would appear around the otherwise degenerated point of ω+ and ωc branches at ε B = 0, with a strong dispersion on the population cm . As for L 0 , it plays a determinant role because it is the interference between the two transition pathways between L 0 and L ±1 that renders the zero dispersion of ωc = ω1 and the null population in L 0 . In case that the grating reads Δε = ε A exp(− j2θ) + ε B exp(+ j2θ) other than that of Eq. (1), a directly transition pathway between L ±1 is open. The interaction now can be modeled by a 2 × 2 matrix, which is just a standard PT -symmetric two-level system [22–26, 28–32, 40–42] discussed by various authors. As for the discrepancy of COMSOL simulation (Section 3) with the non-Hermitian matrix approach (Section 2), for example, the nonzero dispersion of L c branch versus ε B and the nonzero population c0 , they might arise from below aspects. Firstly, Eq. (8) has neglected the coupling of L ±1 with other dark modes especially L ±2 . Such an assumption is a good one if ω1 is away from other ωm (m ±1). However, resonances in metamaterials, not only inside the coaxial elements but also at the two PEC-air interfaces, are very complex. The lattice considered


in this article is a cubic one, which has a lower symmetry than that of the coaxial cross-section. There might be some other interaction pathways that could enhance the coupling of L ±1 modes with other helical modes or surface optical waves, and consequently violate the assumptions made in our theory. Secondly, the PT -symmetric perturbation is utilized to stimulate the transitions among L ±1 and L 0 . However, energy flux of helical modes are mainly propagating along the z direction. The introduced PT -symmetric perturbation also changes the effective k z component and modifies the waveguiding resonant conditions of various L m modes [5–7]. The resonant frequencies ωm might slightly shift with ε A and ε B . This effect will be considered in our further investigation. Thirdly, the matrix proposed in Eq. (8) is a simplified model. It only considers a static situation that the population in each mode does not change with time. Also it does not include the broadening effect. Consequently, some features on the transmission spectra, for example, the disappearance of transmission peaks when ε B decreases further from the EP-like point, and the appearance of transmission coefficient greater than unity when |ε B | > 1.5, are not predicted. Nevertheless, such a model is a successful one because it predicts most of the features on the EOT spectra. The proposed coherent-trapping effect can be utilized for some obviously applications, for example, to manipulate the polarization of optical wave. In fact, besides the case shown in Section 3 we have also checked the scenarios of other polarized incidences. Similar results with Figs. (3) to (5) are obtained. It proves that the scheme proposed here is of a polarization insensitive nature, and promises a potential utilization of polarization-insensitive optical spin filters. To be more specific, with properly introduced ε A and ε B values, we can always get the desired S3 state on the transmitted wave at the desired frequency of ωc , no matter what the polarization of incidence is. An experimental realization of the proposed coherent-trapped mode needs our further effort. By referring to literatures about optical PT symmetry [25, 28–30, 32, 43], we believe the most feasible scheme is to utilize external fields in modulating the refractive index, e.g. the indiffusion and nonlinear mixing technique in Fe-doped LiNbO3 [28]. For further improvement, we can try to tune the geometric parameters so as to enhance the Q factors of L ±1 and L 0 modes. It might overcome the limitation on the available ε A and ε B values, enhance the transitions among L ±1 and L 0 even for a moderate PT -symmetric modulation, and improve the spin-filtering performance by reducing the line-width of the EOT spectra. Advanced theory in considering the influence of the PT -symmetric modulation over the resonance in z direction, the broadening effect in each helical mode, and the interaction with other high-m helical modes, is also applauded. It might describe some features of our COMSOL simulations that cannot be explained by the matrix proposed in Section 2, and predict other potential applications. 5.

Conclusion

In summary, here we show that by coupling the two bright L ±1 modes inside a coaxial element with the dark L 0 mode via a PT symmetric perturbation, we can get a special hybridized mode L c via a classic analog of coherent-population-trapping effect. Resonant frequency of L c is independent of the PT -symmetric perturbation, and the populations in L ±1 can be artificially manipulated. No population is trapped in the L 0 component. Numerical COMSOL simulation in a well-designed coaxial metamaterial verifies our analysis. This effect can be utilized to realize a polarization-insensitive optical spin filter. Discussion on potential investigations and improvements is provided. Funding Natural National Science Foundation of China (NSFC) (11574162).