Role of photonic angular momentum states in

Role of photonic angular momentum states in nonreciprocal diffraction from magnetooptical cylinder arrays Tian-Jing Guo, Li-Ting Wu, Mu Yang, Rui-Peng Guo, Hai-Xu Cui, and Jing Chen Citation: AIP Advances 4, 077133 (2014); doi: 10.1063/1.4891859 View online: http://dx.doi.org/10.1063/1.4891859 View Table of Contents: http://scitation.aip.org/content/aip/journal/adva/4/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Contributed Review: The feasibility of a fully miniaturized magneto-optical trap for portable ultracold quantum technology Rev. Sci. Instrum. 85, 121501 (2014); 10.1063/1.4904066 Study on photonic angular momentum states in coaxial magneto-optical waveguides J. Appl. Phys. 116, 153104 (2014); 10.1063/1.4898317 Enhancement of the transverse non-reciprocal magneto-optical effect J. Appl. Phys. 111, 023103 (2012); 10.1063/1.3677942 Nonlinear magneto-optical diffraction from periodic domain structures in magnetic films Appl. Phys. Lett. 74, 1880 (1999); 10.1063/1.123700 APL Photonics

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AIP ADVANCES 4, 077133 (2014)

Role of photonic angular momentum states in nonreciprocal diffraction from magneto-optical cylinder arrays Tian-Jing Guo, Li-Ting Wu, Mu Yang, Rui-Peng Guo, Hai-Xu Cui, and Jing Chena MOE Key Laboratory of Weak-Light Nonlinear Photonics, School of Physics, Nankai University, Tianjin 300071, China (Received 29 April 2014; accepted 21 July 2014; published online 29 July 2014)

Optical eigenstates in a concentrically symmetric resonator are photonic angular momentum states (PAMSs) with quantized optical orbital angular momentums (OAMs). Nonreciprocal optical phenomena can be obtained if we lift the degeneracy of PAMSs. In this article, we provide a comprehensive study of nonreciprocal optical diffraction of various orders from a magneto-optical cylinder array. We show that nonreciprocal diffraction can be obtained only for these nonzero orders. Role of PAMSs, the excitation of which is sensitive to the directions of incidence, applied magnetic field, and arrangement of the cylinders, are studied. Some interesting phenomena such as a dispersionless quasi-omnidirectional nonreciprocal diffraction and spikes C 2014 Author(s). All associated with high-OAM PAMSs are present and discussed. article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4891859]

I. INTRODUCTION

In recent years, there has been an increased interest to magneto-optical (MO) effects in artificial optical nanostructures due to their exotic physical behaviors and potential applications.1–15 The existence of nonzero off-diagonal tensor elements in either the magnetic permeability or the dielectric permittivity9–15 breaks the time-reversal symmetry of an optical wave propagating inside a MO medium,11 which is a necessary condition to design one-way waveguides, optical isolators and other nonreciprocal optical devices.7, 9–12, 15 It is also shown that MO effects can manipulate the characteristics of extraordinary optical transmission through metamaterials, not only change the frequencies of resonant transmission peaks,1, 3–5, 13 but also enhance the MO Faraday and Kerr effects.2, 6 It is well known that MO effects render different propagation constants or refractive indexes to the optical waves with opposite circular polarizations, an phenomenon that can be termed the circular birefringence.16 Nonreciprocal optical effects can be obtained by using circular birefringence to achieve different reflection or transmission coefficients of the left- and right-handed circular polarizations.16 By noting that optical polarization carries the spin angular momentum (SAM) of light,16, 17 it is then natural to question whether a similar effect can be realized on the other degree of freedoms in photons, the orbital angular momentum (OAM)17 associated with a helical phase wavefront. Artificially optical nanostructures provide a platform to realize such a possibility, where the eigenfrequencies of photonic angular momentum states (PAMSs)14, 15, 18 featured by helical phases of exp (−jmφ), are shown to be sensitive to the off-diagonal tensor elements. Here φ is the azimuthal angle in the cylindrical coordinate frames of (r, φ, z), and m is the quantized topological charge. Wang et. al. showed that a classic analogue of the Zeeman effect on PAMSs

a Electronic address: [email protected]

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can be obtained in a MO cylinder,18 that the frequency degeneracy of ±m PAMSs is broken and the shift in frequency is determined by the topological charge m. Some useful mode-decoupling properties for photonics application18 have been demonstrated. Nonreciprocal optical diffraction effects, especially a nonreciprocal negative directional transmission associated with m = ±1 PAMSs in subwavelength MO cylinder arrays,15 have been demonstrated and briefly discussed. In this article, we provide a comprehensive study on the influence of PAMSs, which are subjected to the classic analogue of the Zeeman effect, to the optical diffraction from MO cylinder arrays. We pay attention to the situation that high-m PAMSs are excited, and that the cylinder array is no long subwavelength and supports high diffraction orders even in normal incidence. We show that transmission of the 0th diffraction order is reciprocal, and nonreciprocal diffraction effects can be obtained only for these nonzero orders. Some exotic phenomena, such as a dispersionless quasi-omnidirectional nonreciprocal diffraction and transmission spikes associated with PAMS of a large m, are present and discussed. Role of PAMSs, the excitation of which are sensitive to the directions of incidence, applied magnetic field, and orientation of the cylinder arrays, are studied. This investigation provides deeper insights into the importance and various applications of PAMSs in different disciplines, which would contribute to the future demands in designing compact optical components for on-chip applications. II. MECHANISM OF THE NON-RECIPROCALITY

Before presenting the numerical simulation and analysis results, we would like to propose a theory to qualitatively explain why the diffraction from a MO cylinder array should be nonreciprocal. Figure 1 shows the required definitions of coordinate frames and geometric parameters of the MO cylinder arrays. Consider the two situations shown in Fig. 1. When the angle of incidence θ is a positive one of +θ 0 , see Fig. 1(a), the phase difference between two adjacent cylinders reads = k0 dsin θ 0 , where k0 = 2π /λ is the wavevector and λ is the free-space wavelength. To compensate for this phase difference, a proper rotating energy flux S such as the clockwise one shown in Fig. 1(a) should be excited in each cylinder. For the reciprocal case, where the angle of incidence is a negative one of −θ 0 [see Fig. 1(b)], the phase difference becomes to = −k0 dsin θ 0 , and consequently a reversed rotating energy flux S is required. Transverse rotating energy fluxes are associated with PAMSs of nonzero topological charges m, and the direction of rotation is correlated to the sign of m.14, 15, 17 If PAMSs are degenerated, that for opposite m values the field distributions m (r), resonant frequencies ωm /2π and widthes m are identical, the two cases shown in Fig. 1 are reciprocal. To see it, let us assume that the excited field in each single MO cylinder of the array for an angle of incidence θ is θ E ext (r, φ) =

+∞

Cmθ m (r ) exp(− jmφ),

(1)

−∞

where m (r) is the eigenfunction of the m PAMS representing its r-dependent distribution of field, and Cmθ is an expansion coefficient to be determined. When the material of the cylinder is optically isotropic, the ±m PAMSs are degenerated so that −m (r) = m (r). Additional, with the degenerated resonant conditions of ±m PAMSs in their eigenfrequencies ω±m /2π and widthes ±m , it is readily +θ0 −θ0 to get that C+m = C−m from Fig. 1. Consequently, the field distributions in these two reciprocal cases are just opposite with each other with respect to the normal axis along x direction, that +θ0 −θ0 (r, +φ) = E ext (r, −φ). The diffraction coefficients of each orders, which are governed by the E ext distribution of field in the array, are identical, i.e. the diffraction is reciprocal. When the cylinder is formed by a MO medium, this reciprocal effect is broken. When a magnetic field H0 is applied in the in-axial z direction, the time-reversal symmetry of an optical wave propagating inside the cylinder is broken, especially to PAMSs with transverse rotating energy fluxes in the x − y plane.18 From Figs. 1(a) and 1(b) we can see PAMSs with opposite m values should be excited in these cases of opposite incidence. However, according to the classic analogue of the Zeeman effect, the ±m PAMSs no longer possess degenerated resonant frequencies ωm /2π


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FIG. 1. Schematic of a MO cylinder array for the situations with (a) a positive angle of incidence +θ 0 , and (b) a negative angle of incidence −θ 0 , respectively. In these two situations different sets of PAMSs are required, as represent by the rotating directions of the transverse energy fluxes S.

+θ0 −θ0 and field distributions m (r).18 Different sets of PAMSs with C+m = C−m are excited in these two cases, leading to nonreciprocal diffraction effects with unequal transmission efficiencies or/and wavelengthes. The classic analogue of the Zeeman effect on PAMSs is the main mechanism of the possible observable optical non-reciprocality.

III. FULL-FIELD SIMULATION

By using full-field three-dimensional finite element optical simulations (COMSOL Multiphysics 4.3a), we study the diffraction of an optical wave from a MO cylinder array by calculating the transmission coefficients of various diffraction orders. A schematic of the MO cylinder array under investigation is shown in Fig. 1. Radius of each MO cylinder is a = 1 cm, and period of the array is d = 2.9 cm. The array is aligned along the y direction, and the incident optical wave forms an angle of incidence θ with respect to the normal of the array in the −x direction. The incident wave is transverse electric (TE) polarized, i.e. with an electric field Ez along the z direction. This polarization can excite the degeneracy-broken PAMSs within our interest in this article. In the coordinate frames of (x, y, z), the magnetic permeability tensor of the MO medium can be expressed as15, 18 ⎛

μr

¯ =⎜ μ ⎝ iμk 0

−iμk

0

⎞

μr

⎟ 0⎠.

0

1

(2)

where the values of tensor elements μr and μk are functions of the applied external magnetic field H0 .15, 18 Usually the tensor elements μr and μk are complex with proper dispersions.15, 18 Here, in order to emphasize the physical mechanism behind the nonreciprocal diffraction we neglect the dispersions in μr and μk , and assume that μr = 0.9 and μk = −0.1 throughout the frequency regime studied in this article. This assumption permits us to reveal the role of the geometric dispersion arisen from the periodically arranged MO cylinders ambiguity. The permittivity of the cylinder is 15.26.15


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FIG. 2. Transmission spectra T0 of the 0th diffraction order versus the angle of incidence θ from −60◦ to +60◦ . The transmission spectra is reciprocal and symmetric with respect to θ = 0, i.e. T0 (−θ 0 ) = T0 (+θ 0 ).

With above parameters we can directly calculate the eigenfrequencies ω/2π of different PAMSs (m, p) in a single MO cylinder,18 where p is the radial index. Within the frequency regime from 14 GHz to 14.4 GHz, the available PAMSs are (m, p) = (−8, 1), (−5, 2), (5, 2), and (8, 1) with eigenfrequencies of 14.044 GHz, 14.042 GHz, 14.189 GHz and 14.299 GHz, respectively.18 High-m PAMSs with m = ±5 and m = ±8 are involved in the possible nonreciprocal diffraction effect discussed in this article.15 Note that according to the conservation of parallel wavevector of k(n) = k0 sin θ + n2π/d 15 for the nth diffraction order, this MO array is subwavelength only for frequency smaller than 10.34 GHz. Unlike that in,15 here the structure is no longer subwavelength from 14 GHz to 14.4 GHz. Below we will show that various nonreciprocal optical transmission effects can be still obtained, and that the simultaneous existence of different diffraction orders, even under a normal incidence, helps us to get a deeper insight into the role of PAMSs in the scattering of an optical wave from the MO cylinder array. A. Transmission spectra of various orders

Figure 2 shows the transmission spectra of the 0th diffraction order T0 versus the angle of incidence θ . It is interesting to note that the transmission of this 0th order is reciprocal, i.e. T0 (+θ 0 ) = T0 (−θ 0 ). In sharp contrast with that of the 0th order, the transmission spectra of other nonzero diffraction orders are nonreciprocal. Examples of the −1st and −2nd orders are shown in Fig. 3, where for opposite angles of incident ±θ 0 the values of transmission coefficient Tn are different. From the transmission spectra shown in Figs. 2 and 3 we can observe two unique features. One feature is that the transmission coefficients Tn varies strongly with the angle of incidence θ around two characteristic frequencies of f1 = 14.090 GHz and f2 = 14.250 GHz. The other feature is that a dispersionless quasi-omnidirectional nonreciprocal optical transmission effect can be observed for the −2st diffraction order, as shown in Fig. 3(b). To be more explicitly, when the angle of incidence θ is negative, a high transmission of the −2st diffraction order takes place around f2 . When θ is positive, a similar high transmission of T−2 can be obtained, albeit at f1 . The positions of the resonant transmission peaks depend very weakly on the magnitude of the incident angle θ . Note that


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FIG. 3. Transmission spectra of (a) the −1st, and (b) −2nd diffraction orders, respectively, versus the angel of incidence θ ◦ ◦ from −60 to +60 . A dispersionless quasi-omnidirectional optical transmission can be observed for the −2nd diffraction order shown in (b), at frequencies f1 =14.090 GHz when θ > 0 and f2 =14.250 GHz when θ < 0, respectively. The transmission spectra are nonreciprocal and asymmetric with respect to θ = 0.

a transmission of the −2nd diffraction order exists only when the magnitude of incident angle θ is greater than a critical value of θ c given by k(−2) = k0 . At frequency f1 the critical value of θ c is given by 27.93◦ . At f2 , θ c equals 26.87◦ . To present above discussed features more clearly, in Fig. 4 we show the transmission spectra ◦ ◦ of the 0th, −1st and −2nd diffraction orders at θ = ±50 and ±30 , respectively. We can see the nonzero (0th) diffraction orders are indeed nonreciprocal (reciprocal). The transmission of the −2st diffraction order is very strong at either f1 or f2 , although its width changes with the angle of incidence θ . The transmission curves near f1 and f2 are of Fano line-shape, implying that there exists an interference between a localized resonance and a collective oscillation. Below, let us investigate what a kind of role the PAMSs play to this nonreciprocal diffraction effect by analyzing the distributions of field and energy flux.


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FIG. 4. Transmission coefficients of the 0th, −1st and −2nd orders versus the frequency f for an angle of incidence θ of (a) −50◦ , (b) +50◦ , (c) −30◦ , and (d) +30◦ , respectively.

B. Field, energy flux, and mode analysis

Paying attention to the two characteristic frequencies of f1 and f2 , where various diffraction orders show noticeable nonreciprocal or reciprocal transmission features, we simulate and analyze the corresponding distributions of field and energy flux. Figure 5 shows the distributions of field and transverse energy flux at f1 and f2 for the angles of incidence at ±50◦ . We can see in all cases a homogenously clockwise or counter-clockwise rotating energy flux is excited in each MO cylinder of the array. For example, at f1 under all angles of incidence a clockwise rotating energy flux is excited, see Figs. 5(a) and 5(c). Consequently, the topological charge m of the dominant PAMS is negative. Furthermore, from the distribution of field Re{Ez } we can see the resonance at f1 is dominated by the (m = −5, p = 2) PAMS, because Re{Ez } varies 5 times when it circles the center, and two rings of high field distributions can be observed along r direction. One ring is localized just at the boundary, while the other one, which is also the strongest one, is inside the cylinder. Similarly, for the frequency at f2 a homogenous counter-clockwise energy flux is excited, which is dominated by the (m = +5, p = 2) PAMS. These results are in consistent with our former analytical calculation, that within the frequency regime from 14 GHz to 14.4 GHz the (m = ±5, p = 2) are available in a single MO cylinder. From above simulation we can see the m = ±5 PAMSs dominate the excited field at both f1 and f2 . The associated rotating energy flux can qualitatively explain some nonreciprocal diffraction effects shown in Fig. 4, especially the dispersionless quasi-omnidirectional optical transmission effect of −2nd orders at f1 and f2 . For example, at f1 the energy flux is clockwise, so the diffraction should favor the clockwise one at the emitting side of the array. Consequently, at the incidence of +50◦ the −2nd diffraction order is greatly favored. The parallel direction of incident wavevector k0 sinθ at the front side of the array also fits well with that of the clockwise rotating energy flux of the MO cylinder, which further enhances the diffraction effect. When the angle of incidence is −50◦ , from the same consideration the diffraction should favor the positive +2nd diffraction order. However, this order is evanescent because k(+2) > k0 and its diffraction is forbidden. For the same


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FIG. 5. Distributions of electric field Re{Ez } and transverse energy flux (by arrows) at transmission peaks of f1 = 14.090 GHz and f2 = 14.250 GHz for the angle of incidence at θ = −50◦ and θ = +50◦ , respectively.

reason, at frequency f2 the energy flux in each MO cylinder is counter-clockwise, so the diffraction should favors the counter-clockwise one at the emitting side, and at the incidence of −50◦ the −2nd diffraction order is enhanced. To get a deeper insight into the mechanism behind this non-reciprocity, especially the role of collective inter-cylinder interaction, we analyze the electric field component Ez by decomposing it into a linear combination of exp (−jmφ). We pick up the excited field inside a MO cylinder within a ring of 0.5 cm < r