Abstract

The problem of a vertical electric dipole radiating above a periodic multilayer whose unit cell comprises a layer of a topological insulator (TI) and a columnar thin film (CTF) was solved in order to investigate the left/right asymmetry of the total electric field in the far zone in the half-space containing the dipole. Occurring in a wide range of the polar observation angle, the left/right asymmetry of Eϕ is due to both the CTFs and the TI layers. Occurring in a narrow range of the polar observation angle, the left/right asymmetry of Eθ is entirely due to the TI layers. For presently available values of the magnitude of the surface admittance γTI of TIs, significant left/right asymmetry occurs if the number of unit cells in the periodic TI/CTF multilayer is high enough.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Corrections

15 July 2020: Typographical corrections were made to the body text.

References

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  1. M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
    [Crossref]
  2. Y. G. Semenov, X. Duan, X.-L. Li, and K. W. Kim, “Conductance nonreciprocity on the surface of a topological insulator with magnetic electrodes,” J. Phys. Chem. Solids 128, 196–201 (2019).
    [Crossref]
  3. A. Lakhtakia and T. G. Mackay, “Classical electromagnetic model of surface states in topological insulators,” J. Nanophotonics 10(3), 033004 (2016).
    [Crossref]
  4. A. Lakhtakia and T. G. Mackay, “Left/right asymmetry in reflection and transmission by a planar anisotropic dielectric slab with topologically insulating surface states,” J. Nanophotonics 10(2), 020501 (2016).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2019 (2)

F. Chiadini, V. Fiumara, A. Lakhtakia, and A. Scaglione, “Enhanced left/right asymmetry in reflection and transmission due to a periodic multilayer of a topological insulator and an anisotropic dielectric material,” Appl. Opt. 58(7), 1724–1732 (2019).
[Crossref]

Y. G. Semenov, X. Duan, X.-L. Li, and K. W. Kim, “Conductance nonreciprocity on the surface of a topological insulator with magnetic electrodes,” J. Phys. Chem. Solids 128, 196–201 (2019).
[Crossref]

2016 (2)

A. Lakhtakia and T. G. Mackay, “Classical electromagnetic model of surface states in topological insulators,” J. Nanophotonics 10(3), 033004 (2016).
[Crossref]

A. Lakhtakia and T. G. Mackay, “Left/right asymmetry in reflection and transmission by a planar anisotropic dielectric slab with topologically insulating surface states,” J. Nanophotonics 10(2), 020501 (2016).
[Crossref]

2014 (1)

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

2010 (2)

M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (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(16), 166803 (2010).
[Crossref]

1999 (1)

T. Setälä, M. Kaivola, and A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E 59(1), 1200–1206 (1999).
[Crossref]

Auban-Senzier, P.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Boschetto, D.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Cava, R. J.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Chen, Y. P.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Chiadini, F.

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).

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(16), 166803 (2010).
[Crossref]

Duan, X.

Y. G. Semenov, X. Duan, X.-L. Li, and K. W. Kim, “Conductance nonreciprocity on the surface of a topological insulator with magnetic electrodes,” J. Phys. Chem. Solids 128, 196–201 (2019).
[Crossref]

Durakiewicz, T.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Fiumara, V.

Friberg, A. T.

T. Setälä, M. Kaivola, and A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E 59(1), 1200–1206 (1999).
[Crossref]

Hajlaoui, M.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Hasan, M. Z.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Hodgkinson, I. J.

I. J. Hodgkinson and Q. h. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, 1997).

Ji, H. W.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Jia, S.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Jiang, Z.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Kaivola, M.

T. Setälä, M. Kaivola, and A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E 59(1), 1200–1206 (1999).
[Crossref]

Kane, C. L.

M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

Kim, K. W.

Y. G. Semenov, X. Duan, X.-L. Li, and K. W. Kim, “Conductance nonreciprocity on the surface of a topological insulator with magnetic electrodes,” J. Phys. Chem. Solids 128, 196–201 (2019).
[Crossref]

Lakhtakia, A.

F. Chiadini, V. Fiumara, A. Lakhtakia, and A. Scaglione, “Enhanced left/right asymmetry in reflection and transmission due to a periodic multilayer of a topological insulator and an anisotropic dielectric material,” Appl. Opt. 58(7), 1724–1732 (2019).
[Crossref]

A. Lakhtakia and T. G. Mackay, “Classical electromagnetic model of surface states in topological insulators,” J. Nanophotonics 10(3), 033004 (2016).
[Crossref]

A. Lakhtakia and T. G. Mackay, “Left/right asymmetry in reflection and transmission by a planar anisotropic dielectric slab with topologically insulating surface states,” J. Nanophotonics 10(2), 020501 (2016).
[Crossref]

Li, X.-L.

Y. G. Semenov, X. Duan, X.-L. Li, and K. W. Kim, “Conductance nonreciprocity on the surface of a topological insulator with magnetic electrodes,” J. Phys. Chem. Solids 128, 196–201 (2019).
[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(16), 166803 (2010).
[Crossref]

Mackay, T. G.

A. Lakhtakia and T. G. Mackay, “Left/right asymmetry in reflection and transmission by a planar anisotropic dielectric slab with topologically insulating surface states,” J. Nanophotonics 10(2), 020501 (2016).
[Crossref]

A. Lakhtakia and T. G. Mackay, “Classical electromagnetic model of surface states in topological insulators,” J. Nanophotonics 10(3), 033004 (2016).
[Crossref]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics Publishing, 2001).

Marsi, M.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Mattox, D. M.

D. M. Mattox, The Foundations of Vacuum Coating Technology (Noyes Publications, 2003).

Mauchain, J.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Miotkowski, I.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Moisan, N.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Monteverde, M.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Navarin, F.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Neupane, M.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Papalazarou, E.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Pasquier, C. R.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Perfetti, L.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Qi, X.-L.

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(16), 166803 (2010).
[Crossref]

Scaglione, A.

Semenov, Y. G.

Y. G. Semenov, X. Duan, X.-L. Li, and K. W. Kim, “Conductance nonreciprocity on the surface of a topological insulator with magnetic electrodes,” J. Phys. Chem. Solids 128, 196–201 (2019).
[Crossref]

Setälä, T.

T. Setälä, M. Kaivola, and A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E 59(1), 1200–1206 (1999).
[Crossref]

Taleb-Ibrahimi, A.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Wu, Q. h.

I. J. Hodgkinson and Q. h. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, 1997).

Xu, Y.

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Zhang, S.-C.

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(16), 166803 (2010).
[Crossref]

Appl. Opt. (1)

J. Nanophotonics (2)

A. Lakhtakia and T. G. Mackay, “Classical electromagnetic model of surface states in topological insulators,” J. Nanophotonics 10(3), 033004 (2016).
[Crossref]

A. Lakhtakia and T. G. Mackay, “Left/right asymmetry in reflection and transmission by a planar anisotropic dielectric slab with topologically insulating surface states,” J. Nanophotonics 10(2), 020501 (2016).
[Crossref]

J. Phys. Chem. Solids (1)

Y. G. Semenov, X. Duan, X.-L. Li, and K. W. Kim, “Conductance nonreciprocity on the surface of a topological insulator with magnetic electrodes,” J. Phys. Chem. Solids 128, 196–201 (2019).
[Crossref]

Nat. Commun. (1)

M. Hajlaoui, E. Papalazarou, J. Mauchain, L. Perfetti, A. Taleb-Ibrahimi, F. Navarin, M. Monteverde, P. Auban-Senzier, C. R. Pasquier, N. Moisan, D. Boschetto, M. Neupane, M. Z. Hasan, T. Durakiewicz, Z. Jiang, Y. Xu, I. Miotkowski, Y. P. Chen, S. Jia, H. W. Ji, R. J. Cava, and M. Marsi, “Tuning a Schottky barrier in a photoexcited topological insulator with transient Dirac cone electron-hole asymmetry,” Nat. Commun. 5(1), 3003 (2014).
[Crossref]

Phys. Rev. E (1)

T. Setälä, M. Kaivola, and A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E 59(1), 1200–1206 (1999).
[Crossref]

Phys. Rev. Lett. (1)

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(16), 166803 (2010).
[Crossref]

Rev. Mod. Phys. (1)

M. Z. Hasan and C. L. Kane, “Topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

Other (5)

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

I. J. Hodgkinson and Q. h. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, 1997).

D. M. Mattox, The Foundations of Vacuum Coating Technology (Noyes Publications, 2003).

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics Publishing, 2001).

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).

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Figures (7)

Fig. 1.
Fig. 1. Schematic of the analyzed problem.
Fig. 2.
Fig. 2. Asymmetry functions for $\theta \in \left [0^{\circ },75^{\circ } \right ]$ and $\phi \in \left [0^{\circ },180^{\circ } \right ]$ , when $N=30$ and $\overline {\gamma }_\textrm {TI}=1$ .
Fig. 3.
Fig. 3. Same as Fig. 2 but for $\overline {\gamma }_\textrm {TI}=2$ .
Fig. 4.
Fig. 4. Same as Fig. 2 but for $\overline {\gamma }_\textrm {TI}=3$ .
Fig. 5.
Fig. 5. Maximum values of left/right asymmetry functions in relation to the number $N$ of unit cells and the normalized surface admittance $\overline {\gamma }_\textrm {TI}$ . Black diamonds: $\overline {\gamma }_\textrm {TI}=0$ ; red stars: $\overline {\gamma }_\textrm {TI}=1$ ; blue circles: $\overline {\gamma }_\textrm {TI}=2$ ; magenta triangles: $\overline {\gamma }_\textrm {TI}=3$ .
Fig. 6.
Fig. 6. Reflectances $| r_\textrm {sp}|^{2}$ and $| r_\textrm {pp}|^{2}$ as functions of $\theta$ and $\phi$ for $\bar{\gamma}_{\mathrm{TI}}=2$ .
Fig. 7.
Fig. 7. Reflectances as functions of $\theta$ and $\phi$ for $\bar{\gamma}_{\mathrm{TI}}=2$ . Black curves: $\phi =120^{\circ }$ ; red curves: $\phi =300^{\circ }$ .

Tables (1)

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Table 1. Maximum values of the left/right asymmetry functions | Δ E θ | and | Δ E ϕ | and the direction ( θ , ϕ ) at which they occur.

Equations (19)

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ε _ _ CTF = [ ε b + ( ε a ε b ) sin 2 χ 0 1 2 ( ε a ε b ) sin 2 χ 0 ε c 0 1 2 ( ε a ε b ) sin 2 χ 0 ε a ( ε a ε b ) sin 2 χ ] ,
E prim ( x , y , z ) = i ω 2 μ 0 k 0 2 8 π 2 ( M _ _ ( ξ , η ) ( u ^ z p z ) × exp { i k 0 [ ξ ( x x 0 ) + η ( y y 0 ) + γ | z z 0 | ] } ) d ξ d η ,
M _ _ ( ξ , η ) = 1 k 0 γ [ 1 ( ξ ) 2 ξ η ξ γ ξ η 1 ( η ) 2 η γ ξ γ η γ 1 ( γ ) 2 ]
ξ = sin θ cos ϕ , η = sin θ sin ϕ , γ = cos θ .
E prim ( x , y , z ) = k 0 2 4 π ε 0 e i k 0 r r e i k 0 z 0 cos θ p z [ sin θ cos ϕ cos θ sin θ sin ϕ cos θ sin 2 θ ] .
[ E prim , θ E prim , ϕ ] = k 0 2 4 π ε 0 e i k 0 r r e i k 0 z 0 cos θ p z [ sin θ 0 ] .
V inc ( ξ , η , p ¯ ) = [ sin ϕ cos ϕ cos θ cos ϕ sin ϕ cos θ 0 sin θ ] [ a s ( θ , ϕ , p ¯ ) a p ( θ , ϕ , p ¯ ) ]
a s ( θ , ϕ , p ¯ ) = p y cos ϕ p x sin ϕ k 0 cos θ
a p ( θ , ϕ , p ¯ ) = [ p x cos ϕ k 0 + p y sin ϕ k 0 + p z sin θ k 0 cos θ ]
a p ( θ , ϕ , p z u ^ z ) = p z k 0 tan θ .
V refl ( ξ , η , p z u ^ z ) = [ sin ϕ cos ϕ cos θ cos ϕ sin ϕ cos θ 0 sin θ ] [ r s ( θ , ϕ , p z u ^ z ) r p ( θ , ϕ , p z u ^ z ) ]
[ r s ( θ , ϕ , p z u ^ z ) r p ( θ , ϕ , p z u ^ z ) ] = [ r ss ( θ , ϕ ) r sp ( θ , ϕ ) r ps ( θ , ϕ ) r pp ( θ , ϕ ) ] [ a s ( θ , ϕ , p z u ^ z ) a p ( θ , ϕ , p z u ^ z ) ] .
E refl ( x , y , z ) = i ω 2 μ 0 k 0 2 8 π 2 V refl ( ξ , η , p z u ^ z ) exp { i k 0 [ ξ x + η y + γ ( z + z 0 ) ] } d ξ d η
E refl = k 0 2 4 π ε 0 e i k 0 r r e i k 0 z 0 cos θ p z [ r sp ( θ , ϕ ) sin ϕ sin θ r pp ( θ , ϕ ) cos ϕ sin θ cos θ r sp ( θ , ϕ ) cos ϕ sin θ r pp ( θ , ϕ ) sin ϕ sin θ cos θ 0 r pp ( θ , ϕ ) sin 2 θ ] .
[ E refl , θ E refl , ϕ ] = k 0 2 4 π ε 0 e i k 0 r r e i k 0 z 0 cos θ p z [ r pp ( θ , ϕ ) sin θ r sp ( θ , ϕ ) sin θ ] .
E ( x , y , z ) = E prim ( x , y , z ) + E refl ( x , y , z )
[ E θ E ϕ ] = k 0 2 4 π ε 0 e i k 0 r r p z sin θ [ e i k 0 z 0 cos θ + r pp ( θ , ϕ ) e i k 0 z 0 cos θ r sp ( θ , ϕ ) e i k 0 z 0 cos θ ] .
Δ E θ = | E θ ( r , θ , ϕ ) | | E θ ( r , θ , ϕ + π ) | | E o ( r , θ , ϕ ) |
Δ E ϕ = | E ϕ ( r , θ , ϕ ) | | E ϕ ( r , θ , ϕ + π ) | | E o ( r , θ , ϕ ) | ,

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