Abstract

Topological insulators (TIs) show unusual optical responses resulting from a topological magnetoelectric (TME) effect. In this paper, we study theoretically the scattering of electromagnetic waves by circular TI cylinders. In certain configurations, the bulk scattering can be suppressed, leading to strong scattering in the backward direction in both Rayleigh and Mie scattering regimes due to the TME effect. At antiresonances, an interesting filed trapping phenomenon is found which is absent in conventional dielectric cylinders.

© 2014 Optical Society of America

1. Introduction

Recently, topological insulators (TIs) have been received considerable interest as an emerging phase in condensed matter physics [13]. TIs resemble ordinary insulators with a bulk energy gap but possess gapless edge or surface states within the bulk energy gap which are protected by the time-reversal symmetry. This unique feature gives rise to many unusual physical properties. In addition to many exotic electronic and transport properties, a novel quantized topological magneto-electric (TME) effect – an applied electric (magnetic) field can induce magnetic (electric) polarization – has been revealed in TIs [3]. An additional term in the Maxwell Lagrangian, Δ = (θα/4π2)E · B, is introduced to describe the TME effect [3, 4], where α = e2/h̄c is the fine structure constant, θ = (2p + 1)π is the quantized axion angle [4] with p being an integer, and E and B are the electric field and magnetic induction, respectively. The TME effect plays a key role in the electromagnetic (EM) responses of TIs and many unconventional optical phenomena due to the TME effect have been found [511].

The scattering of EM waves by small particles is an interesting optical problem and has been treated systematically in the literatures [12, 13] for conventional scatterers. The EM scattering depends strongly on the optical properties of the constituent materials of the scatterers. Owing to the TME effect, EM scattering by TI scatterers should be different from that by conventional ones. Indeed, a parity violating and a strong perturbation of dipole radiation are found in axionic scatterers [14]. In the Rayleigh limit, the quantization of the TME effect can be simply determined by the scattered electric fields in the far field [15]. In this paper, we study theoretically the EM scattering by circular TI cylinders in both the Rayleigh and Mie scattering regimes. Interestingly, strong scattering in the backward direction due to the TME is revealed, which is absent for bi-isotropic [16] (chiral or Tellegen) scatterers since the TME effect is basically a surface effect rather than a bulk one. Moreover, antiresonances for TI cylinders are found in the Mie scattering regime showing an interesting field trapping phenomenon.

2. Scattering coefficients

The propagation of EM waves in a conventional homogeneous medium can be described by the Helmholtz equation ∇2F + k2F = 0, where F is either the vector field E or B, k is the corresponding wavevector, ω is the angular frequency, and c is the light speed in vacuum. In TIs, however, the constitutive relations should be modified as [3]

D=εEα¯B,
H=B/μ+α¯E,
where D and H are respectively the electric displacement and magnetic field, = θα/π is an odd-numered multiple of α, and ε and μ are respectively the permittivity and permeability. It can be easily verified that, if the modified constitutive relations are applied in the Maxwell equations, the corresponding master equation remains the same form as the one for the topologically trivial case with = 0. Interestingly, the dispersion relation k=εμω/c is independent of the factor . This implies that the TME effect is basically a surface effect, different from chiral media in which the chirality parameter κ enters the corresponding dispersion relation [17]. Therefore, the unusual optical properties of TIs should be contributed from the TI boundaries.

The EM scattering by a TI cylinder could be solved by the standard multiple expansion method [15]. The system under study is schematically shown in Fig. 1. In the present work, we consider an EM wave Einc = E0(eiêxeiêz)eikby is incident perpendicular to the cylinder, where E0 is the amplitude, kb=εbμbω/c is the wavevector in the background medium. In this configuration, there are two independent polarizations of the incident EM wave: TE (with the electric field perpendicular to the cylinder) and TM (with electric field parallel to the cylinder). Here we use ei and ei represent the polarization-vector components of the incident wave with ei = 1 and ei = 0 standing for the TE polarization, and ei = 0 and ei = 1 for the TM polarization. Similar to that in conventional circular cylinders, the EM wave functions can be expanded by the vector cylindrical harmonics Mn(I)(kρ,φ) and Nn(I)(kρ,ϕ) [12], in which n is an integer, ρ is the radial distance, φ is the azimuth angle, and I stands for which kind of the Bessel (Hankel) function we use. The scattered electric field and the internal field inside the TI cylinder can be expanded respectively in terms of these vector cylindrical harmonics

Esca=n=En[ianMn(3)+bnNn(3)],
Eint=n=En[cnMn(1)+dnNn(1)],
where an and bn are the scattering coefficients, En = E0/kb. The superscripts 1 and 3 stand for the Bessel function of the first kind and the Hankel function of the first kind, respectively. Note that the scattering coefficients an and bn are associated with the electric and magnetic multipoles of order n, respectively.

 figure: Fig. 1

Fig. 1 Schematic view of a circular TI cylinder placed along the z axis. The incident EM wave with TE or TM polarization is propagating along y axis.

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It can be shown that by imposing the standard boundary conditions the scattering coefficients for circular TI cylinders under normal incidence can be found as

an=eiAnDn+α˜fnAnBn+α˜2tn,
bn=eiBnCn+α˜gnAnBn+α˜2tn,
where α˜=α˜/εb (α̃ = as the background is vacuum), An, Bn, Cn, and Dn are the same functions as defined in circular dielectric cylinders [12]
An(x)=mJn(mx)Hn(1)(x)Jn(mx)Hn(1)(x),Bn(x)=mJn(mx)Hn(1)(x)Jn(mx)Hn(1)(x),Cn(x)=mJn(mx)Jn(x)Jn(mx)Jn(x),Dn(x)=mJn(mx)Jn(x)Jn(mx)Jn(x).
Here, Jn and Hn(1) are respectively the Bessel and Hankel functions of the first kind, x = kbr is the size parameter with r being the cylinder radius, and m=εμ/εbμb is the relative refractive index. In this study we assume all materials are nonmagnetic, i.e., μ = μb = 1. The coefficients related to the internal fields inside the cylinders are
cn=iJn(x)ei+iHn(1)(x)anmJn(mx),
dn=Jn(x)ei+Hn(1)(x)bnmJn(mx).
The auxiliary functions fn, gn, and tn with the prefactor α̃ (or α̃2) are unique in TI cylinders due to the TME effect, given by
fn(x)=Jn(mx)Jn(mx)[ei2iπx+α˜eiJn(x)Hn(1)(x)],gn(x)=Jn(mx)Jn(mx)[ei2iπx+α˜eiJn(x)Hn(1)(x)],tn(x)=Jn(mx)Jn(mx)Hn(1)(x)Hn(1)(x).
It can be verified that an = an and bn = bn, similar to those in conventional cylinders [12]. Note that an and bn are polarization dependent and there hence exist two independent sets, {an,TE, bn,TE} and {an,TM, bn,TM} for TE and TM polarizations, respectively. For TI cylinders, we have another identity an,TM = bn,TE which can be easily verified. For topological-trivial insulators, the corresponding axion angle θ = 0 and the scattering coefficients are reduced to the conventional ones given in [12]. Compared with ordinary dielectric cylinders, extra contributions resulting from the TME effect appear in both {an} and {bn} [18, 19], leading to many unusual scattering properties. For example, for an ordinary dielectric cylinder a TE incident wave cannot excite the magnetic multipoles because bn,TE = 0 (not valid for bn,TM generally). However, for a TI cylinder, bn,TE does not vanish in general, implying that the magnetic multi-poles can be excited. The scattering efficiency defined by Q(θ)=2xn={|an|2+|bn|2} is now a function of θ. To estimate the weight of the scattering efficiency contributed from the TME effect, we define a relative scattering efficiency, Qr = [Q(θ ≠ 0) − Q(θ = 0)]/Q(θ = 0), where Q(θ ≠ 0) and Q(θ = 0) are the scattering efficiency with an axion angle θ ≠ 0 and θ = 0, respectively.

3. Unusual scattering pattern

To obtain the fields of scattered waves in the far field, an amplitude scattering matrix T is usually introduced which relates the electric field of the scattered wave to that of incident wave [12]

[EsEs]=ei3π/42πkρeikρ[T1T4T3T2][EiEi],
where E and E are the electric-field components parallel and perpendicular to the cylinder, respectively; the subscripts i and s stand for the incident and scattered wave, respectively; and Ti (i = 1, 2, 3, 4) are the elements of amplitude scattering matrix defined by
T1=n=einϕbn,TM,
T2=n=einϕan,TE,
T3=n=einϕan,TM,
T4=n=einϕbn,TE.
Here, ϕ = π/2 − φ is the scattering angle [15]. From the electric field of the scattered wave, we can obtain easily its intensity. The intensity of the scattered wave I can be decomposed as two components according to the polarizations, one parallel and the other perpendicular to the cylinder, given by
I=|E|2=2πkρ|T1ei+T4ei|2,
I=|E|2=2πkρ|T3ei+T2ei|2,
where we assume that the incident plane wave has a unit intensity.

We first discuss the scattering in the Rayleigh scattering limit (x, mx ≪ 1). With the known scattering coefficients [15], the intensity for the TE incident wave can be obtained as

I,TE=2kρπx4|α˜4α˜cosϕ2m2+2+α˜2|2,
I,TE=2kρπx4|(2m22+α˜2)cosϕ2(2m2+2+α˜2)|2.
Note that I‖,TE should vanish for conventional cylinders. For TI cylinders, however, the contributions from the magnetic monopolar and dipolar terms induced by the TME effect lead to a non-zero I‖,TE. Interestingly, the scattered fields from the magnetic monopole and dipole interfere destructively in forward direction (ϕ = 0°), while constructively in the backward direction (ϕ = 180°). In contrast, I⊥,TE(ϕ), which is contributed from the electric dipolar term induced by the bulk, has the same magnitude in both the forward and backward directions. In general cases, the intensity of the scattered wave is dominated by I⊥,TE since α̃ is very small. In the case of m ∼ 1 (the refractive index of the TI cylinder is very close to that of the background), however, the scattering from the bulk is negligibly small such that I‖,TEI⊥,TE. The scattering by TI cylinders is now dominated by I‖,TE, leading to strong scattering in the backward direction. In fact, the ratio of the backward-to-forward scattering intensity takes a large value (∼ 1/α̃2) as m ∼ 1.

For the TM incident wave, the intensity components of the scattered wave are given by

I,TM=2kρπx4|m21+α˜24α˜2cosϕ2(2m2+2+α˜2)|2,
I,TM=2kρπx4|α˜4α˜cosϕ2m2+2+α˜2|2.
Obviously, I‖,TM is mainly from the monopolar term of the scattered wave by the bulk. However, I⊥,TM has the similar property as I‖,TE, showing the interesting destructive interference in the forward direction and constructive interference in the backward direction. As m ∼ 1, we have I‖,TMI⊥,TM, manifesting strong scattering in the backward direction. The ratio of the backward-to-forward scattered intensity also takes a large value in order of 1/α̃2 as m ∼ 1. Note that Rayleigh scattering in conventional dielectric cylinders displays a symmetric scattering pattern in forward and backward directions. Thus, the strong backward scattering in TI cylinders stem from the TME effect.

From the above discussions, the key to realizing strong backward scattering is to suppress the contribution of bulk scattering. The choice of m ∼ 1 can indeed eliminate the bulk scattering to attain strong backward scattering. But it is rather difficult to realize experimentally. Practically, we can adopt instead a hollow TI cylinder to implement strong backward scattering as shown in Fig. 2. The axion angles for the hollow TI cylinder should be properly chosen since there are two interfaces. We assume that the axion angle of the hollow TI cylinder is (2p + 1)π, whereas the axion angle of the vacuum outside and inside the hollow cylinder is respectively 0 and (2p + 1)2π, as in [7] for TI thin films. Although the axion angles 0 and (2p + 1)2π are equivalent in the topological sense, this proper choice guarantees that the surface currents at the two interfaces do not cancel each other.

 figure: Fig. 2

Fig. 2 Scattering of TE incident waves by a hollow TI cylinder with ε = 30. The background is vacuum with εb = 1. The cross-sectional profile of the hollow cylinder is shown in the inset of (b). (a) |Qr| as a function of the thickness parameter d = d0/r0. Here, x = kbr0 is the size parameter. (b) Ratio of the scattered radiant intensities I between the backward and forward directions. In (a) and (b), the red, blue and black curves correspond to the hollow TI cylinder with the axion angle θ = π, 3π and 5π, respectively. (c)–(f) Field distributions |E|2 of the scattered waves for d = 0.001. In (c) and (e), the axion angle of the cylinder is 0, standing for a conventional dielectric hollow cylinder; whereas in (d) and (f) the axion angle is 5π, standing for a TI hollow cylinder. In (c) and (d), x = 0.05; and in (e) and (f), x = 1.

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In Fig. 2(a), we calculate the relative scattering efficiency |Qr| for the hollow cylinder. The outer radius and the wall thickness of the hollow cylinder are denoted by r0 and d0, respectively. It can be observed from Fig. 2(a) that |Qr| decreases with increasing d(= d0/r0), indicating that the bulk scattering increases with d. However, for small d (< 0.001) the scattering efficiency from the TME effect is manifest in both the Rayleigh (e.g., x = 0.05) and Mie (e.g., x = 1) scattering regimes. For small d, the intensity of the backward scattering can surpass the forward scattering in both the Rayleigh and Mie scattering regimes, as can be seen from Fig. 2(b), especially for large axion angles. Figures 2(c)–2(f) show the field distributions of the scattered EM wave. Obviously, the conventional hollow cylinder (θ = 0) displays an almost symmetric scattering pattern in the backward and forward directions in the Rayleigh scattering regime. In the Mie scattering regime, the scattering pattern shows strong forward scattering. The symmetric and forward scattering in respectively the Rayleigh and Mie scattering regimes is the known features for conventional dielectric scatterers. For a TI hollow cylinder with a small wall thickness, however, both the Rayleigh and Mie scattering show a pattern of strong backward scattering. For TM incident waves, similar results are also observed (data not shown).

The underlying physics for the strong backward scattering in TI hollow cylinders can be understood by the interference between electric and magnetic multipoles. Note that such backward scattering has also been found in other kinds of scatterers such as magnetic [20, 21], metallic [22], or semiconducting [2325] particles. To achieve strong backward scattering in magnetic spheres, the optical constants should satisfy the condition ε = (4 − μ)/(2μ + 1) [20]. Under this condition, the radiation fields from the induced electric and magnetic dipoles have the same strength but are out of phase in the forward direction and in phase in the backward direction. For metallic spheres [22] or semiconducting spheres [2325], the strong backward scattering is frequency-dependent. The magnetic dipole moment arising from the closed loops of the induced displacement currents are usually caused by resonances. As a result, the observed strong backward scattering exists only in a narrow frequency range. In hollow TI cylinders with a small wall thickness, the interference between the electric and magnetic multipoles is constructive in the backward direction and destructive in the forward direction, eventually leading to strong backward scattering. Note that the frequency range for the strong backward scattering is wide since the TME effect is a non-resonant effect. Moreover, such strong backward scattering can occur not only in Rayleigh scattering but also in Mie scattering.

4. Antiresonances

In Fig. 3, the scattering coefficients of the four lowest order modes for a TI cylinder in vacuum as a function of the size parameter x for TE incident waves are shown. Obviously, there exist resonances and antiresonances in both an,TE and bn,TE. Physically, an and bn are associated with the n-th order electric and magnetic multiplolar modes, respectively. The resonances at x = 0.425 in |a0,TE| and x = 0.670 in |a1,TE| correspond to the excitation of the electric monopolar and dipolar modes, respectively. In contrast, at the antiresonances (x = 0.946 in |a0,TE| and x = 0.727 in |a1,TE|) no such electric monopolar and dipolar modes can be excited since at these particular x the corresponding scattering coefficients are almost zero. The asymmetric lineshapes in Fig. 3 can be explained by the interference between the resonant and direct scattering similar to the cases considered in [26].

 figure: Fig. 3

Fig. 3 Scattering coefficients an and bn as functions of the size parameter x with n = 0, 1 for the TE incident wave. The dielectric constant of the TI cylinder is ε = 30 and its axion angle is π. The background medium is vacuum.

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The scattering coefficients an,TE of a TI cylinder should be similar to those of a dielectric cylinder with the same refractive index [27] since the fine structure constant is much smaller than one. As a result, the TME effect could not be manifest in an,TE. However, it emerges in bn,TE since bn,TE should vanish in conventional dielectric cylinders. Interestingly, there exist also resonances and antiresonances in bn,TE. Right at these antiresonances, the corresponding scattering coefficients vanish. In order to enable bn,TE = 0, the function gn in Eq. (3) should be zero, or in other words, the condition Jn(mx)J′n(mx) = 0 should be satisfied. Therefore, the antiresonances correspond exactly to the roots of Jn(mx) = 0 or J′n(mx) = 0. At the antiresonances, all the introduced auxiliary functions that are related to the TME effect should be zero, namely, fn = gn = tn = 0. The scattering now is reduced to the one for conventional dielectric cylinders, implying the absence of the TME effect. For TM polarization, similar conclusions can be drawn since an,TM = bn,TE.

To verify our analysis, we show in Fig. 4 the field distributions for two antiresonances at x = 0.439 and 0.336, corresponding to J0(mx) = 0 and J′1(mx) = 0, or b0,TE = 0 and b1,TE = 0, respectively. For TE incident waves, from the definition of the vector cylindrical harmonics Mn and Nn, the electric filed component Eφ and the magnetic induction component Bz are proportional to J′n(mx) and Jn(mx), respectively. For the magnetic monopole mode at x = 0.439, it is obvious that Bz = 0 at the boundary since J0(mx) = 0. Inside the TI cylinder, Eφ is nonzero which can induce a nonzero Bφ inside the cylinder via the TME effect. However, the components of the scattered fields Hφ and Bρ should be zero since b0,TE = 0. The magnetic field component Bφ is not continuous at the boundary. Interestingly, Bφ is completely trapped inside the cylinder. For the magnetic dipole mode at x = 0.336, Eφ = 0 at the boundary since it corresponds to J′1(mx) = 0. As a result, Hφ cannot be induced inside the cylinder by the TME effect. However, Dz can be induced by the TME effect since Bz inside the cylinder may not be zero. Similar to Bφ for the monopole mode, Dz for the dipole mode is trapped completely inside the cylinder.

 figure: Fig. 4

Fig. 4 Field distributions at two antiresonances x = 0.439 (upper panels) and 0.336 (lower panels). (a) and (d) are for |Eφ|, (b) and (e) for |Bz|, (c) for |Bφ|, and (f) for |Dz|. The boundary of the TI cylinder is indicated by dashed lines.

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The vanishment of the field component Eφ or Bz at antiresonances for TE incident waves is an exotic phenomenon in TI cylinders and is also a strong evidence that the TME effect is intrinsically a surface effect. For TM incident waves, similar antiresonances also exist. The corresponding vanishing field component at the boundary is now Bφ or Ez instead. The field trapping of Bz or Dφ inside the cylinder can also be observed. For conventional dielectric cylinders, the scattering coefficients an and bn also possess antiresonances at particular size parameters. However, the cylinder is perfectly transparent to the n–th harmonic of the incident wave at any antiresonance [27]. There is no such a field trapping phenomenon in conventional dielectric cylinders.

A few points should be mentioned. First, one should introduce a thin magnetic layer coating on the TI surface to open up a surface energy gap Eg [3, 28]. With this magnetic layer, the time reversal symmetry is broken at the surface and the value of θ an be specified definitely. For a typical value Eg ∼ 10 meV, the corresponding EM wavelength λg is about 0.12 mm, the incident wavelength λ should be much larger than λg, i.e. λgλ, which can be attained with microwaves. Second, if the TI is doped, the doping level we considered should in the gap [29]. Therefore the axion angle is quantized and the “axion electrodynamics” can be applied to describe the EM responses of the system. Third, the role that the magnetic layer plays in the EM scattering should be rigorously restricted in the total EM scattering. Since the magnetic layer can be as thin as several nanometers [30], the optical distance of this layer is extremely small in the microwave regime.

5. Conclusion

In summary, we study systematically the EM scattering properties of circular TI cylinders. Owing to the TME effect, scattering in TI cylinders shows unusual feathers. For TI cylinders with a refractive index close to that of the background or for hollow TI cylinders with a small wall thickness with respect to the wavelength, strong backward scattering are found in both the Rayleigh and Mie scattering regimes, which is completely different from conventional dielectric cylinders. The underlying physics for the strong backward scattering can be understood by the interference between electric and magnetic multipoles, which are destructive in the forward direction and constructive for the backward scattering. For conventional media, the spectral responses of magnetic dipoles are usually narrow-band since they are caused by resonances in general. However, the enhanced backscattering for TIs is a broadband effect for the non-resonant nature of TME effect. Interesting antiresonances are found showing a field-trapping phenomenon, which does not exist in conventional dielectric cylinders. At anti-resonances, the EM fields generated by the surface Hall currents are completely trapped inside the cylinder.

Acknowledgments

This work is supported by the 973Program (Grant Nos. 2013CB632701 and 2011CB922004). The research of H.D.Z., X.H.L. and J.Z. is further supported by the National Natural Science Foundation of China (Grant No. 11304038 and 11234010). The work of H.D.Z is also supported by the Fundamental Research Funds for the Central Universities (Grant no. CQDXWL-2014-Z005).

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References

  • View by:

  1. X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
    [Crossref]
  2. M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
    [Crossref]
  3. X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B. 78, 195424 (2008).
    [Crossref]
  4. F. Wilczek, “Two applications of axion electrodynamics,” Phys. Rev. Lett. 58, 1799–1802 (1987).
    [Crossref] [PubMed]
  5. X. L. Qi, R. D. Li, J. D. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science 323, 1184–1187 (2009).
    [Crossref] [PubMed]
  6. W. K. Tse and A. H. MacDonald, “Giant magneto-optical kerr effect and universal faraday effect in thin-film topological insulators,” Phys. Rev. Lett. 105, 057401 (2010).
    [Crossref]
  7. W. K. Tse and A. H. MacDonald, “Magneto-optical faraday and kerr effects in topological insulator films and in other layered quantized Hall systems,” Phys. Rev. B 84, 205327 (2011).
    [Crossref]
  8. J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
    [Crossref]
  9. M. C. Chang and M. F. Yang, “Optical signature of topological insulators,” Phys. Rev. B 80, 113304 (2009).
    [Crossref]
  10. J. Inoue, “An optical test for identifying topological insulator thin films,” Opt. Express 21, 8564 (2013).
    [Crossref] [PubMed]
  11. X. Xiao, S. Li, K. T. Law, B. Hou, C. T. Chan, and W. J. Wen, “Thermal coherence properties of topological insulator slabs in time-reversal symmetry breaking fields,” Phys. Rev. B 87, 205424 (2013).
    [Crossref]
  12. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, 1983).
  13. P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopoic Phenomena (Springer-Verlag, 2006).
  14. T. Ochiai, “Theory of light scattering in axion electrodynamics,” J. Phys. Soc. Japan 81, 094401 (2012),
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  15. L. X. Ge, T. R. Zhan, D. Z. Han, X. H. Liu, and J. Zi, “Determination of the quantized topological magneto-electric effect in topological insulators from Rayleigh scattering”, arXiv:1404.2384 (2014).
  16. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).
  17. B. N. Wang, J. F. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A: Pure Appl. Opt. 11, 114003 (2009).
    [Crossref]
  18. L. W. Zeng, R. X. Song, and X. L. Jian, “Scattering of electromagnetic radiation by a time reversal perturbation topological insulator circular cylinder,” Mod. Phys. Lett. B 27, 1350098 (2013).
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  19. M. Akhtar, N. A. Naz, M.A. Fiaz, and Q. A. Naqvi, “Scattering from topological insulator circular cylinder buried in a semi-infinite medium,” J. Mod. Opt. 61, 697–702 (2014).
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  20. M. Kerker, D.-S. Wang, and C. L. Giles, “Electromagnetic scattering by magnetic spheres,” J. Opt. Soc. Am. 73, 765–767 (1983).
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  21. R. V. Mehta, R. Patel, R. Desai, R. V. Upadhyay, and K. Parekh, “Experimental evidence of zero forward scattering by magnetic spheres,” Phys. Rev. Lett. 96, 127402 (2006).
    [Crossref] [PubMed]
  22. B. S. Lukyanchuk, M. I. Tribelsky, Z. B. Wang, Y. Zhou, M. H. Hong, L. P. Shi, and T. C. Chong, “Extraordinary scattering diagram for nanoparticles near plasmon resonance frequencies,” Appl. Phys. A 89, 259–264 (2007).
    [Crossref]
  23. Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nat. Commun. 4, 1527 (2013).
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  24. J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171–1178 (2012).
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  25. B. Rolly, B. Stout, and N. Bonod, “Boosting the directivity of optical antennas with magnetic and electric dipolar resonant particles,” Opt. Express 20, 20376–20386 (2012).
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  26. W. D. Zhou, D. Y. Zhao, Y. C. Shuai, H. J. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Q. Ma, and S. H. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quant. Electron. 38, 1–74 (2014).
    [Crossref]
  27. J. A. Schuller and M. L. Brongersma, “General properties of dielectric optical antennas,” Opt. Express 17, 24084–24095 (2009).
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  28. W. D. Luo and X. L. Qi, “Massive Dirac surface states in topological insulator/magnetic insulator heterostructures,” Phys. Rev. B 87, 085431 (2013).
    [Crossref]
  29. A. G. Grushin and F. Juan, “Finite-frequency magnetoelectric response of three-dimensional topological insulators,” Phys. Rev. B 86, 075126 (2012).
    [Crossref]
  30. P. Wei, F. Katmis, B. A. Assaf, H. Steinberg, P. Jarillo-Herrero, D. Heiman, and J. S. Moodera, “Exchange-coupling-induced symmetry breaking in topological insulators,” Phys. Rev. Lett. 110, 186807 (2013).
    [Crossref] [PubMed]

2014 (2)

M. Akhtar, N. A. Naz, M.A. Fiaz, and Q. A. Naqvi, “Scattering from topological insulator circular cylinder buried in a semi-infinite medium,” J. Mod. Opt. 61, 697–702 (2014).
[Crossref]

W. D. Zhou, D. Y. Zhao, Y. C. Shuai, H. J. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Q. Ma, and S. H. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quant. Electron. 38, 1–74 (2014).
[Crossref]

2013 (6)

Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nat. Commun. 4, 1527 (2013).
[Crossref] [PubMed]

W. D. Luo and X. L. Qi, “Massive Dirac surface states in topological insulator/magnetic insulator heterostructures,” Phys. Rev. B 87, 085431 (2013).
[Crossref]

P. Wei, F. Katmis, B. A. Assaf, H. Steinberg, P. Jarillo-Herrero, D. Heiman, and J. S. Moodera, “Exchange-coupling-induced symmetry breaking in topological insulators,” Phys. Rev. Lett. 110, 186807 (2013).
[Crossref] [PubMed]

L. W. Zeng, R. X. Song, and X. L. Jian, “Scattering of electromagnetic radiation by a time reversal perturbation topological insulator circular cylinder,” Mod. Phys. Lett. B 27, 1350098 (2013).
[Crossref]

J. Inoue, “An optical test for identifying topological insulator thin films,” Opt. Express 21, 8564 (2013).
[Crossref] [PubMed]

X. Xiao, S. Li, K. T. Law, B. Hou, C. T. Chan, and W. J. Wen, “Thermal coherence properties of topological insulator slabs in time-reversal symmetry breaking fields,” Phys. Rev. B 87, 205424 (2013).
[Crossref]

2012 (4)

T. Ochiai, “Theory of light scattering in axion electrodynamics,” J. Phys. Soc. Japan 81, 094401 (2012),
[Crossref]

A. G. Grushin and F. Juan, “Finite-frequency magnetoelectric response of three-dimensional topological insulators,” Phys. Rev. B 86, 075126 (2012).
[Crossref]

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171–1178 (2012).
[Crossref] [PubMed]

B. Rolly, B. Stout, and N. Bonod, “Boosting the directivity of optical antennas with magnetic and electric dipolar resonant particles,” Opt. Express 20, 20376–20386 (2012).
[Crossref] [PubMed]

2011 (2)

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
[Crossref]

W. K. Tse and A. H. MacDonald, “Magneto-optical faraday and kerr effects in topological insulator films and in other layered quantized Hall systems,” Phys. Rev. B 84, 205327 (2011).
[Crossref]

2010 (3)

J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[Crossref]

W. K. Tse and A. H. MacDonald, “Giant magneto-optical kerr effect and universal faraday effect in thin-film topological insulators,” Phys. Rev. Lett. 105, 057401 (2010).
[Crossref]

M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[Crossref]

2009 (4)

X. L. Qi, R. D. Li, J. D. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science 323, 1184–1187 (2009).
[Crossref] [PubMed]

M. C. Chang and M. F. Yang, “Optical signature of topological insulators,” Phys. Rev. B 80, 113304 (2009).
[Crossref]

B. N. Wang, J. F. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A: Pure Appl. Opt. 11, 114003 (2009).
[Crossref]

J. A. Schuller and M. L. Brongersma, “General properties of dielectric optical antennas,” Opt. Express 17, 24084–24095 (2009).
[Crossref]

2008 (1)

X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B. 78, 195424 (2008).
[Crossref]

2007 (1)

B. S. Lukyanchuk, M. I. Tribelsky, Z. B. Wang, Y. Zhou, M. H. Hong, L. P. Shi, and T. C. Chong, “Extraordinary scattering diagram for nanoparticles near plasmon resonance frequencies,” Appl. Phys. A 89, 259–264 (2007).
[Crossref]

2006 (1)

R. V. Mehta, R. Patel, R. Desai, R. V. Upadhyay, and K. Parekh, “Experimental evidence of zero forward scattering by magnetic spheres,” Phys. Rev. Lett. 96, 127402 (2006).
[Crossref] [PubMed]

1987 (1)

F. Wilczek, “Two applications of axion electrodynamics,” Phys. Rev. Lett. 58, 1799–1802 (1987).
[Crossref] [PubMed]

1983 (1)

Akhtar, M.

M. Akhtar, N. A. Naz, M.A. Fiaz, and Q. A. Naqvi, “Scattering from topological insulator circular cylinder buried in a semi-infinite medium,” J. Mod. Opt. 61, 697–702 (2014).
[Crossref]

Albella, P.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171–1178 (2012).
[Crossref] [PubMed]

Assaf, B. A.

P. Wei, F. Katmis, B. A. Assaf, H. Steinberg, P. Jarillo-Herrero, D. Heiman, and J. S. Moodera, “Exchange-coupling-induced symmetry breaking in topological insulators,” Phys. Rev. Lett. 110, 186807 (2013).
[Crossref] [PubMed]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, 1983).

Bonod, N.

Brongersma, M. L.

Chadha, A.

W. D. Zhou, D. Y. Zhao, Y. C. Shuai, H. J. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Q. Ma, and S. H. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quant. Electron. 38, 1–74 (2014).
[Crossref]

Chan, C. T.

X. Xiao, S. Li, K. T. Law, B. Hou, C. T. Chan, and W. J. Wen, “Thermal coherence properties of topological insulator slabs in time-reversal symmetry breaking fields,” Phys. Rev. B 87, 205424 (2013).
[Crossref]

Chang, M. C.

M. C. Chang and M. F. Yang, “Optical signature of topological insulators,” Phys. Rev. B 80, 113304 (2009).
[Crossref]

Chong, T. C.

B. S. Lukyanchuk, M. I. Tribelsky, Z. B. Wang, Y. Zhou, M. H. Hong, L. P. Shi, and T. C. Chong, “Extraordinary scattering diagram for nanoparticles near plasmon resonance frequencies,” Appl. Phys. A 89, 259–264 (2007).
[Crossref]

Chuwongin, S.

W. D. Zhou, D. Y. Zhao, Y. C. Shuai, H. J. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Q. Ma, and S. H. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quant. Electron. 38, 1–74 (2014).
[Crossref]

Desai, R.

R. V. Mehta, R. Patel, R. Desai, R. V. Upadhyay, and K. Parekh, “Experimental evidence of zero forward scattering by magnetic spheres,” Phys. Rev. Lett. 96, 127402 (2006).
[Crossref] [PubMed]

Drew, H. D.

J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[Crossref]

Eyraud, C.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171–1178 (2012).
[Crossref] [PubMed]

Fan, S. H.

W. D. Zhou, D. Y. Zhao, Y. C. Shuai, H. J. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Q. Ma, and S. H. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quant. Electron. 38, 1–74 (2014).
[Crossref]

Fiaz, M.A.

M. Akhtar, N. A. Naz, M.A. Fiaz, and Q. A. Naqvi, “Scattering from topological insulator circular cylinder buried in a semi-infinite medium,” J. Mod. Opt. 61, 697–702 (2014).
[Crossref]

Froufe-Pérez, L. S.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171–1178 (2012).
[Crossref] [PubMed]

Fu, Y. H.

Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nat. Commun. 4, 1527 (2013).
[Crossref] [PubMed]

García-Cámara, B.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171–1178 (2012).
[Crossref] [PubMed]

Ge, L. X.

L. X. Ge, T. R. Zhan, D. Z. Han, X. H. Liu, and J. Zi, “Determination of the quantized topological magneto-electric effect in topological insulators from Rayleigh scattering”, arXiv:1404.2384 (2014).

Geffrin, J. M.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171–1178 (2012).
[Crossref] [PubMed]

Giles, C. L.

Gómez-Medina, R.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171–1178 (2012).
[Crossref] [PubMed]

González, F.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171–1178 (2012).
[Crossref] [PubMed]

Grushin, A. G.

A. G. Grushin and F. Juan, “Finite-frequency magnetoelectric response of three-dimensional topological insulators,” Phys. Rev. B 86, 075126 (2012).
[Crossref]

Han, D. Z.

L. X. Ge, T. R. Zhan, D. Z. Han, X. H. Liu, and J. Zi, “Determination of the quantized topological magneto-electric effect in topological insulators from Rayleigh scattering”, arXiv:1404.2384 (2014).

Hasan, M. Z.

M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[Crossref]

Heiman, D.

P. Wei, F. Katmis, B. A. Assaf, H. Steinberg, P. Jarillo-Herrero, D. Heiman, and J. S. Moodera, “Exchange-coupling-induced symmetry breaking in topological insulators,” Phys. Rev. Lett. 110, 186807 (2013).
[Crossref] [PubMed]

Hong, M. H.

B. S. Lukyanchuk, M. I. Tribelsky, Z. B. Wang, Y. Zhou, M. H. Hong, L. P. Shi, and T. C. Chong, “Extraordinary scattering diagram for nanoparticles near plasmon resonance frequencies,” Appl. Phys. A 89, 259–264 (2007).
[Crossref]

Hou, B.

X. Xiao, S. Li, K. T. Law, B. Hou, C. T. Chan, and W. J. Wen, “Thermal coherence properties of topological insulator slabs in time-reversal symmetry breaking fields,” Phys. Rev. B 87, 205424 (2013).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, 1983).

Hughes, T. L.

X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B. 78, 195424 (2008).
[Crossref]

Inoue, J.

Jarillo-Herrero, P.

P. Wei, F. Katmis, B. A. Assaf, H. Steinberg, P. Jarillo-Herrero, D. Heiman, and J. S. Moodera, “Exchange-coupling-induced symmetry breaking in topological insulators,” Phys. Rev. Lett. 110, 186807 (2013).
[Crossref] [PubMed]

Jian, X. L.

L. W. Zeng, R. X. Song, and X. L. Jian, “Scattering of electromagnetic radiation by a time reversal perturbation topological insulator circular cylinder,” Mod. Phys. Lett. B 27, 1350098 (2013).
[Crossref]

Juan, F.

A. G. Grushin and F. Juan, “Finite-frequency magnetoelectric response of three-dimensional topological insulators,” Phys. Rev. B 86, 075126 (2012).
[Crossref]

Kafesaki, M.

B. N. Wang, J. F. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A: Pure Appl. Opt. 11, 114003 (2009).
[Crossref]

Kane, C. L.

M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[Crossref]

Katmis, F.

P. Wei, F. Katmis, B. A. Assaf, H. Steinberg, P. Jarillo-Herrero, D. Heiman, and J. S. Moodera, “Exchange-coupling-induced symmetry breaking in topological insulators,” Phys. Rev. Lett. 110, 186807 (2013).
[Crossref] [PubMed]

Kerker, M.

Koschny, T.

B. N. Wang, J. F. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A: Pure Appl. Opt. 11, 114003 (2009).
[Crossref]

Kuznetsov, A. I.

Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nat. Commun. 4, 1527 (2013).
[Crossref] [PubMed]

Law, K. T.

X. Xiao, S. Li, K. T. Law, B. Hou, C. T. Chan, and W. J. Wen, “Thermal coherence properties of topological insulator slabs in time-reversal symmetry breaking fields,” Phys. Rev. B 87, 205424 (2013).
[Crossref]

Li, R. D.

X. L. Qi, R. D. Li, J. D. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science 323, 1184–1187 (2009).
[Crossref] [PubMed]

Li, S.

X. Xiao, S. Li, K. T. Law, B. Hou, C. T. Chan, and W. J. Wen, “Thermal coherence properties of topological insulator slabs in time-reversal symmetry breaking fields,” Phys. Rev. B 87, 205424 (2013).
[Crossref]

Lindell, I. V.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Litman, A.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171–1178 (2012).
[Crossref] [PubMed]

Liu, V.

W. D. Zhou, D. Y. Zhao, Y. C. Shuai, H. J. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Q. Ma, and S. H. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quant. Electron. 38, 1–74 (2014).
[Crossref]

Liu, X. H.

L. X. Ge, T. R. Zhan, D. Z. Han, X. H. Liu, and J. Zi, “Determination of the quantized topological magneto-electric effect in topological insulators from Rayleigh scattering”, arXiv:1404.2384 (2014).

Luk’yanchuk, B.

Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nat. Commun. 4, 1527 (2013).
[Crossref] [PubMed]

Lukyanchuk, B. S.

B. S. Lukyanchuk, M. I. Tribelsky, Z. B. Wang, Y. Zhou, M. H. Hong, L. P. Shi, and T. C. Chong, “Extraordinary scattering diagram for nanoparticles near plasmon resonance frequencies,” Appl. Phys. A 89, 259–264 (2007).
[Crossref]

Luo, W. D.

W. D. Luo and X. L. Qi, “Massive Dirac surface states in topological insulator/magnetic insulator heterostructures,” Phys. Rev. B 87, 085431 (2013).
[Crossref]

Ma, Z. Q.

W. D. Zhou, D. Y. Zhao, Y. C. Shuai, H. J. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Q. Ma, and S. H. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quant. Electron. 38, 1–74 (2014).
[Crossref]

MacDonald, A. H.

W. K. Tse and A. H. MacDonald, “Magneto-optical faraday and kerr effects in topological insulator films and in other layered quantized Hall systems,” Phys. Rev. B 84, 205327 (2011).
[Crossref]

W. K. Tse and A. H. MacDonald, “Giant magneto-optical kerr effect and universal faraday effect in thin-film topological insulators,” Phys. Rev. Lett. 105, 057401 (2010).
[Crossref]

Maciejko, J.

J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[Crossref]

Mehta, R. V.

R. V. Mehta, R. Patel, R. Desai, R. V. Upadhyay, and K. Parekh, “Experimental evidence of zero forward scattering by magnetic spheres,” Phys. Rev. Lett. 96, 127402 (2006).
[Crossref] [PubMed]

Miroshnichenko, A. E.

Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nat. Commun. 4, 1527 (2013).
[Crossref] [PubMed]

Moodera, J. S.

P. Wei, F. Katmis, B. A. Assaf, H. Steinberg, P. Jarillo-Herrero, D. Heiman, and J. S. Moodera, “Exchange-coupling-induced symmetry breaking in topological insulators,” Phys. Rev. Lett. 110, 186807 (2013).
[Crossref] [PubMed]

Moreno, F.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171–1178 (2012).
[Crossref] [PubMed]

Naqvi, Q. A.

M. Akhtar, N. A. Naz, M.A. Fiaz, and Q. A. Naqvi, “Scattering from topological insulator circular cylinder buried in a semi-infinite medium,” J. Mod. Opt. 61, 697–702 (2014).
[Crossref]

Naz, N. A.

M. Akhtar, N. A. Naz, M.A. Fiaz, and Q. A. Naqvi, “Scattering from topological insulator circular cylinder buried in a semi-infinite medium,” J. Mod. Opt. 61, 697–702 (2014).
[Crossref]

Nieto-Vesperinas, M.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171–1178 (2012).
[Crossref] [PubMed]

Ochiai, T.

T. Ochiai, “Theory of light scattering in axion electrodynamics,” J. Phys. Soc. Japan 81, 094401 (2012),
[Crossref]

Parekh, K.

R. V. Mehta, R. Patel, R. Desai, R. V. Upadhyay, and K. Parekh, “Experimental evidence of zero forward scattering by magnetic spheres,” Phys. Rev. Lett. 96, 127402 (2006).
[Crossref] [PubMed]

Patel, R.

R. V. Mehta, R. Patel, R. Desai, R. V. Upadhyay, and K. Parekh, “Experimental evidence of zero forward scattering by magnetic spheres,” Phys. Rev. Lett. 96, 127402 (2006).
[Crossref] [PubMed]

Qi, X. L.

W. D. Luo and X. L. Qi, “Massive Dirac surface states in topological insulator/magnetic insulator heterostructures,” Phys. Rev. B 87, 085431 (2013).
[Crossref]

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
[Crossref]

J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[Crossref]

X. L. Qi, R. D. Li, J. D. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science 323, 1184–1187 (2009).
[Crossref] [PubMed]

X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B. 78, 195424 (2008).
[Crossref]

Rolly, B.

Sáenz, J. J.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171–1178 (2012).
[Crossref] [PubMed]

Schuller, J. A.

Seo, J. H.

W. D. Zhou, D. Y. Zhao, Y. C. Shuai, H. J. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Q. Ma, and S. H. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quant. Electron. 38, 1–74 (2014).
[Crossref]

Sheng, P.

P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopoic Phenomena (Springer-Verlag, 2006).

Shi, L. P.

B. S. Lukyanchuk, M. I. Tribelsky, Z. B. Wang, Y. Zhou, M. H. Hong, L. P. Shi, and T. C. Chong, “Extraordinary scattering diagram for nanoparticles near plasmon resonance frequencies,” Appl. Phys. A 89, 259–264 (2007).
[Crossref]

Shuai, Y. C.

W. D. Zhou, D. Y. Zhao, Y. C. Shuai, H. J. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Q. Ma, and S. H. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quant. Electron. 38, 1–74 (2014).
[Crossref]

Sihvola, A. H.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Song, R. X.

L. W. Zeng, R. X. Song, and X. L. Jian, “Scattering of electromagnetic radiation by a time reversal perturbation topological insulator circular cylinder,” Mod. Phys. Lett. B 27, 1350098 (2013).
[Crossref]

Soukoulis, C. M.

B. N. Wang, J. F. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A: Pure Appl. Opt. 11, 114003 (2009).
[Crossref]

Steinberg, H.

P. Wei, F. Katmis, B. A. Assaf, H. Steinberg, P. Jarillo-Herrero, D. Heiman, and J. S. Moodera, “Exchange-coupling-induced symmetry breaking in topological insulators,” Phys. Rev. Lett. 110, 186807 (2013).
[Crossref] [PubMed]

Stout, B.

Tretyakov, S. A.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Tribelsky, M. I.

B. S. Lukyanchuk, M. I. Tribelsky, Z. B. Wang, Y. Zhou, M. H. Hong, L. P. Shi, and T. C. Chong, “Extraordinary scattering diagram for nanoparticles near plasmon resonance frequencies,” Appl. Phys. A 89, 259–264 (2007).
[Crossref]

Tse, W. K.

W. K. Tse and A. H. MacDonald, “Magneto-optical faraday and kerr effects in topological insulator films and in other layered quantized Hall systems,” Phys. Rev. B 84, 205327 (2011).
[Crossref]

W. K. Tse and A. H. MacDonald, “Giant magneto-optical kerr effect and universal faraday effect in thin-film topological insulators,” Phys. Rev. Lett. 105, 057401 (2010).
[Crossref]

Upadhyay, R. V.

R. V. Mehta, R. Patel, R. Desai, R. V. Upadhyay, and K. Parekh, “Experimental evidence of zero forward scattering by magnetic spheres,” Phys. Rev. Lett. 96, 127402 (2006).
[Crossref] [PubMed]

Vaillon, R.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171–1178 (2012).
[Crossref] [PubMed]

Viitanen, A. J.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Wang, B. N.

B. N. Wang, J. F. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A: Pure Appl. Opt. 11, 114003 (2009).
[Crossref]

Wang, D.-S.

Wang, K. X.

W. D. Zhou, D. Y. Zhao, Y. C. Shuai, H. J. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Q. Ma, and S. H. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quant. Electron. 38, 1–74 (2014).
[Crossref]

Wang, Z. B.

B. S. Lukyanchuk, M. I. Tribelsky, Z. B. Wang, Y. Zhou, M. H. Hong, L. P. Shi, and T. C. Chong, “Extraordinary scattering diagram for nanoparticles near plasmon resonance frequencies,” Appl. Phys. A 89, 259–264 (2007).
[Crossref]

Wei, P.

P. Wei, F. Katmis, B. A. Assaf, H. Steinberg, P. Jarillo-Herrero, D. Heiman, and J. S. Moodera, “Exchange-coupling-induced symmetry breaking in topological insulators,” Phys. Rev. Lett. 110, 186807 (2013).
[Crossref] [PubMed]

Wen, W. J.

X. Xiao, S. Li, K. T. Law, B. Hou, C. T. Chan, and W. J. Wen, “Thermal coherence properties of topological insulator slabs in time-reversal symmetry breaking fields,” Phys. Rev. B 87, 205424 (2013).
[Crossref]

Wilczek, F.

F. Wilczek, “Two applications of axion electrodynamics,” Phys. Rev. Lett. 58, 1799–1802 (1987).
[Crossref] [PubMed]

Xiao, X.

X. Xiao, S. Li, K. T. Law, B. Hou, C. T. Chan, and W. J. Wen, “Thermal coherence properties of topological insulator slabs in time-reversal symmetry breaking fields,” Phys. Rev. B 87, 205424 (2013).
[Crossref]

Yang, H. J.

W. D. Zhou, D. Y. Zhao, Y. C. Shuai, H. J. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Q. Ma, and S. H. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quant. Electron. 38, 1–74 (2014).
[Crossref]

Yang, M. F.

M. C. Chang and M. F. Yang, “Optical signature of topological insulators,” Phys. Rev. B 80, 113304 (2009).
[Crossref]

Yu, Y. F.

Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nat. Commun. 4, 1527 (2013).
[Crossref] [PubMed]

Zang, J. D.

X. L. Qi, R. D. Li, J. D. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science 323, 1184–1187 (2009).
[Crossref] [PubMed]

Zeng, L. W.

L. W. Zeng, R. X. Song, and X. L. Jian, “Scattering of electromagnetic radiation by a time reversal perturbation topological insulator circular cylinder,” Mod. Phys. Lett. B 27, 1350098 (2013).
[Crossref]

Zhan, T. R.

L. X. Ge, T. R. Zhan, D. Z. Han, X. H. Liu, and J. Zi, “Determination of the quantized topological magneto-electric effect in topological insulators from Rayleigh scattering”, arXiv:1404.2384 (2014).

Zhang, S. C.

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
[Crossref]

J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[Crossref]

X. L. Qi, R. D. Li, J. D. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science 323, 1184–1187 (2009).
[Crossref] [PubMed]

X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B. 78, 195424 (2008).
[Crossref]

Zhao, D. Y.

W. D. Zhou, D. Y. Zhao, Y. C. Shuai, H. J. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Q. Ma, and S. H. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quant. Electron. 38, 1–74 (2014).
[Crossref]

Zhou, J. F.

B. N. Wang, J. F. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A: Pure Appl. Opt. 11, 114003 (2009).
[Crossref]

Zhou, W. D.

W. D. Zhou, D. Y. Zhao, Y. C. Shuai, H. J. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Q. Ma, and S. H. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quant. Electron. 38, 1–74 (2014).
[Crossref]

Zhou, Y.

B. S. Lukyanchuk, M. I. Tribelsky, Z. B. Wang, Y. Zhou, M. H. Hong, L. P. Shi, and T. C. Chong, “Extraordinary scattering diagram for nanoparticles near plasmon resonance frequencies,” Appl. Phys. A 89, 259–264 (2007).
[Crossref]

Zi, J.

L. X. Ge, T. R. Zhan, D. Z. Han, X. H. Liu, and J. Zi, “Determination of the quantized topological magneto-electric effect in topological insulators from Rayleigh scattering”, arXiv:1404.2384 (2014).

Appl. Phys. A (1)

B. S. Lukyanchuk, M. I. Tribelsky, Z. B. Wang, Y. Zhou, M. H. Hong, L. P. Shi, and T. C. Chong, “Extraordinary scattering diagram for nanoparticles near plasmon resonance frequencies,” Appl. Phys. A 89, 259–264 (2007).
[Crossref]

J. Mod. Opt. (1)

M. Akhtar, N. A. Naz, M.A. Fiaz, and Q. A. Naqvi, “Scattering from topological insulator circular cylinder buried in a semi-infinite medium,” J. Mod. Opt. 61, 697–702 (2014).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

B. N. Wang, J. F. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A: Pure Appl. Opt. 11, 114003 (2009).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. Soc. Japan (1)

T. Ochiai, “Theory of light scattering in axion electrodynamics,” J. Phys. Soc. Japan 81, 094401 (2012),
[Crossref]

Mod. Phys. Lett. B (1)

L. W. Zeng, R. X. Song, and X. L. Jian, “Scattering of electromagnetic radiation by a time reversal perturbation topological insulator circular cylinder,” Mod. Phys. Lett. B 27, 1350098 (2013).
[Crossref]

Nat. Commun. (2)

Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nat. Commun. 4, 1527 (2013).
[Crossref] [PubMed]

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171–1178 (2012).
[Crossref] [PubMed]

Opt. Express (3)

Phys. Rev. B (5)

X. Xiao, S. Li, K. T. Law, B. Hou, C. T. Chan, and W. J. Wen, “Thermal coherence properties of topological insulator slabs in time-reversal symmetry breaking fields,” Phys. Rev. B 87, 205424 (2013).
[Crossref]

W. K. Tse and A. H. MacDonald, “Magneto-optical faraday and kerr effects in topological insulator films and in other layered quantized Hall systems,” Phys. Rev. B 84, 205327 (2011).
[Crossref]

M. C. Chang and M. F. Yang, “Optical signature of topological insulators,” Phys. Rev. B 80, 113304 (2009).
[Crossref]

W. D. Luo and X. L. Qi, “Massive Dirac surface states in topological insulator/magnetic insulator heterostructures,” Phys. Rev. B 87, 085431 (2013).
[Crossref]

A. G. Grushin and F. Juan, “Finite-frequency magnetoelectric response of three-dimensional topological insulators,” Phys. Rev. B 86, 075126 (2012).
[Crossref]

Phys. Rev. B. (1)

X. L. Qi, T. L. Hughes, and S. C. Zhang, “Topological field theory of time-reversal invariant insulators,” Phys. Rev. B. 78, 195424 (2008).
[Crossref]

Phys. Rev. Lett. (5)

F. Wilczek, “Two applications of axion electrodynamics,” Phys. Rev. Lett. 58, 1799–1802 (1987).
[Crossref] [PubMed]

W. K. Tse and A. H. MacDonald, “Giant magneto-optical kerr effect and universal faraday effect in thin-film topological insulators,” Phys. Rev. Lett. 105, 057401 (2010).
[Crossref]

J. Maciejko, X. L. Qi, H. D. Drew, and S. C. Zhang, “Topological quantization in units of the fine structure constant,” Phys. Rev. Lett. 105, 166803 (2010).
[Crossref]

P. Wei, F. Katmis, B. A. Assaf, H. Steinberg, P. Jarillo-Herrero, D. Heiman, and J. S. Moodera, “Exchange-coupling-induced symmetry breaking in topological insulators,” Phys. Rev. Lett. 110, 186807 (2013).
[Crossref] [PubMed]

R. V. Mehta, R. Patel, R. Desai, R. V. Upadhyay, and K. Parekh, “Experimental evidence of zero forward scattering by magnetic spheres,” Phys. Rev. Lett. 96, 127402 (2006).
[Crossref] [PubMed]

Prog. Quant. Electron. (1)

W. D. Zhou, D. Y. Zhao, Y. C. Shuai, H. J. Yang, S. Chuwongin, A. Chadha, J. H. Seo, K. X. Wang, V. Liu, Z. Q. Ma, and S. H. Fan, “Progress in 2D photonic crystal Fano resonance photonics,” Prog. Quant. Electron. 38, 1–74 (2014).
[Crossref]

Rev. Mod. Phys. (2)

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
[Crossref]

M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[Crossref]

Science (1)

X. L. Qi, R. D. Li, J. D. Zang, and S. C. Zhang, “Inducing a magnetic monopole with topological surface states,” Science 323, 1184–1187 (2009).
[Crossref] [PubMed]

Other (4)

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, 1983).

P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopoic Phenomena (Springer-Verlag, 2006).

L. X. Ge, T. R. Zhan, D. Z. Han, X. H. Liu, and J. Zi, “Determination of the quantized topological magneto-electric effect in topological insulators from Rayleigh scattering”, arXiv:1404.2384 (2014).

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

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Figures (4)

Fig. 1
Fig. 1 Schematic view of a circular TI cylinder placed along the z axis. The incident EM wave with TE or TM polarization is propagating along y axis.
Fig. 2
Fig. 2 Scattering of TE incident waves by a hollow TI cylinder with ε = 30. The background is vacuum with εb = 1. The cross-sectional profile of the hollow cylinder is shown in the inset of (b). (a) |Qr| as a function of the thickness parameter d = d0/r0. Here, x = kbr0 is the size parameter. (b) Ratio of the scattered radiant intensities I between the backward and forward directions. In (a) and (b), the red, blue and black curves correspond to the hollow TI cylinder with the axion angle θ = π, 3π and 5π, respectively. (c)–(f) Field distributions |E|2 of the scattered waves for d = 0.001. In (c) and (e), the axion angle of the cylinder is 0, standing for a conventional dielectric hollow cylinder; whereas in (d) and (f) the axion angle is 5π, standing for a TI hollow cylinder. In (c) and (d), x = 0.05; and in (e) and (f), x = 1.
Fig. 3
Fig. 3 Scattering coefficients an and bn as functions of the size parameter x with n = 0, 1 for the TE incident wave. The dielectric constant of the TI cylinder is ε = 30 and its axion angle is π. The background medium is vacuum.
Fig. 4
Fig. 4 Field distributions at two antiresonances x = 0.439 (upper panels) and 0.336 (lower panels). (a) and (d) are for |Eφ|, (b) and (e) for |Bz|, (c) for |Bφ|, and (f) for |Dz|. The boundary of the TI cylinder is indicated by dashed lines.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

D = ε E α ¯ B ,
H = B / μ + α ¯ E ,
E sca = n = E n [ i a n M n ( 3 ) + b n N n ( 3 ) ] ,
E int = n = E n [ c n M n ( 1 ) + d n N n ( 1 ) ] ,
a n = e i A n D n + α ˜ f n A n B n + α ˜ 2 t n ,
b n = e i B n C n + α ˜ g n A n B n + α ˜ 2 t n ,
A n ( x ) = m J n ( m x ) H n ( 1 ) ( x ) J n ( m x ) H n ( 1 ) ( x ) , B n ( x ) = m J n ( m x ) H n ( 1 ) ( x ) J n ( m x ) H n ( 1 ) ( x ) , C n ( x ) = m J n ( m x ) J n ( x ) J n ( m x ) J n ( x ) , D n ( x ) = m J n ( m x ) J n ( x ) J n ( m x ) J n ( x ) .
c n = i J n ( x ) e i + i H n ( 1 ) ( x ) a n m J n ( m x ) ,
d n = J n ( x ) e i + H n ( 1 ) ( x ) b n m J n ( m x ) .
f n ( x ) = J n ( m x ) J n ( m x ) [ e i 2 i π x + α ˜ e i J n ( x ) H n ( 1 ) ( x ) ] , g n ( x ) = J n ( m x ) J n ( m x ) [ e i 2 i π x + α ˜ e i J n ( x ) H n ( 1 ) ( x ) ] , t n ( x ) = J n ( m x ) J n ( m x ) H n ( 1 ) ( x ) H n ( 1 ) ( x ) .
[ E s E s ] = e i 3 π / 4 2 π k ρ e i k ρ [ T 1 T 4 T 3 T 2 ] [ E i E i ] ,
T 1 = n = e i n ϕ b n , TM ,
T 2 = n = e i n ϕ a n , TE ,
T 3 = n = e i n ϕ a n , TM ,
T 4 = n = e i n ϕ b n , TE .
I = | E | 2 = 2 π k ρ | T 1 e i + T 4 e i | 2 ,
I = | E | 2 = 2 π k ρ | T 3 e i + T 2 e i | 2 ,
I , TE = 2 k ρ π x 4 | α ˜ 4 α ˜ cos ϕ 2 m 2 + 2 + α ˜ 2 | 2 ,
I , TE = 2 k ρ π x 4 | ( 2 m 2 2 + α ˜ 2 ) cos ϕ 2 ( 2 m 2 + 2 + α ˜ 2 ) | 2 .
I , TM = 2 k ρ π x 4 | m 2 1 + α ˜ 2 4 α ˜ 2 cos ϕ 2 ( 2 m 2 + 2 + α ˜ 2 ) | 2 ,
I , TM = 2 k ρ π x 4 | α ˜ 4 α ˜ cos ϕ 2 m 2 + 2 + α ˜ 2 | 2 .

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