Abstract

We show that the propagating modes in a strongly-guided chiral one-way photonic crystal are not backscattering-immune even though they are indeed insensitive to many kinds of scatters. Since these modes are not protected by the nonreciprocity, the backscattering does occur under certain circumstances. We use a perturbative method to derive criteria for the prominent backscattering in such chiral structures. From both our theory and numerical examinations, we find that the amount of backscattering critically depends on the symmetry of scatters. Additionally, for these chiral photonic modes, disturbances at the most intense parts of field profiles do not necessarily lead to the most effective backscattering.

© 2015 Optical Society of America

Full Article  |  PDF Article
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    [Crossref]
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    [Crossref] [PubMed]
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2014 (1)

2013 (4)

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nature Mater. 12, 233–239 (2013).
[Crossref]

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nature Photon 7, 579–582 (2013).
[Crossref]

K. Fang and S. Fan, “Controlling the flow of light using the inhomogeneous effective gauge field that emerges from dynamic modulation,” Phys. Rev. Lett. 111, 203901 (2013).
[Crossref] [PubMed]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

2012 (2)

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon 6, 782–787 (2012).
[Crossref]

L. Zhang, W. He, K. Ronald, A. Phelps, C. Whyte, C. Robertson, A. Young, C. Donaldson, and A. Cross, “Multimode coupling wave theory for helically corrugated waveguide,” IEEE Trans. Microw. Theory Techn. 60, 1–7 (2012).
[Crossref]

2011 (3)

W.-J. Chen, Z. H. Hang, J.-W. Dong, X. Xiao, H.-Z. Wang, and C. T. Chan, “Observation of backscattering-immune chiral electromagnetic modes without time reversal breaking,” Phys. Rev. Lett. 107, 023901 (2011).
[Crossref] [PubMed]

Y. Poo, R.-x. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” Phys. Rev. Lett. 106, 093903 (2011).
[Crossref] [PubMed]

K. Fang, Z. Yu, and S. Fan, “Microscopic theory of photonic one-way edge mode,” Phys. Rev. B 84, 075477 (2011).
[Crossref]

2010 (3)

J.-X. Fu, R.-J. Liu, and Z.-Y. Li, “Robust one-way modes in gyromagnetic photonic crystal waveguides with different interfaces,” Appl. Phys. Lett. 97, 041112 (2010).
[Crossref]

W.-M. Ye, X.-D. Yuan, C.-C. Guo, and C. Zen, “Unidirectional transmission in non-symmetric gratings made of isotropic material,” Opt. Express 18, 7590–7595 (2010).
[Crossref] [PubMed]

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

2009 (5)

S. Gennady, T. Simeon, I. K. Victor, N. Daniel, and Z. G. Azriel, “Polarization properties of chiral fiber gratings,” J. Opt. A: Pure Appl. Opt. 11, 074007 (2009).
[Crossref]

A. E. Serebryannikov and E. Ozbay, “Isolation and one-way effects in diffraction on dielectric gratings with plasmonic inserts,” Opt. Express 17, 278–292 (2009).
[Crossref] [PubMed]

A. E. Serebryannikov and E. Ozbay, “Unidirectional transmission in non-symmetric gratings containing metallic layers,” Opt. Express 17, 13335–13345 (2009).
[Crossref] [PubMed]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref] [PubMed]

X. Ao, Z. Lin, and C. T. Chan, “One-way edge mode in a magneto-optical honeycomb photonic crystal,” Phys. Rev. B 80, 033105 (2009).
[Crossref]

2008 (3)

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref] [PubMed]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[Crossref]

2007 (3)

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater 19, 207–210 (2007).
[Crossref]

V. I. Kopp, V. M. Churikov, G. Zhang, J. Singer, C. W. Draper, N. Chao, D. Neugroschl, and A. Z. Genack, “Single- and double-helix chiral fiber sensors,” J. Opt. Soc. Am. B 24, A48–A52 (2007).
[Crossref]

L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76, 045302 (2007).
[Crossref]

2006 (2)

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757–1761 (2006).
[Crossref] [PubMed]

G. Shvets, “Optical polarizer/isolator based on a rectangular waveguide with helical grooves,” Appl. Phys. Lett. 89, 141127 (2006).
[Crossref]

2005 (1)

C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95, 146802 (2005).
[Crossref]

1995 (1)

H. Cory, “Chiral devices-an overview of canonical problems,” J. Electromagnet Wave 9, 805–829 (1995).
[Crossref]

1991 (1)

X. G. Wen, “Gapless boundary excitations in the quantum Hall states and in the chiral spin states,” Phys. Rev. B 43, 11025–11036 (1991).
[Crossref]

1987 (2)

O. Pankratov, S. Pakhomov, and B. Volkov, “Supersymmetry in heterojunctions: band-inverting contact on the basis of Pb1−xSnxTe and Hg1−xCdxTe,” Solid State Commun. 61, 93–96 (1987).
[Crossref]

S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5, 5–15 (1987).
[Crossref]

1982 (1)

B. I. Halperin, “Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential,” Phys. Rev. B 25, 2185–2190 (1982).
[Crossref]

1980 (1)

K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45, 494–497 (1980).
[Crossref]

Ao, X.

X. Ao, Z. Lin, and C. T. Chan, “One-way edge mode in a magneto-optical honeycomb photonic crystal,” Phys. Rev. B 80, 033105 (2009).
[Crossref]

Azriel, Z. G.

S. Gennady, T. Simeon, I. K. Victor, N. Daniel, and Z. G. Azriel, “Polarization properties of chiral fiber gratings,” J. Opt. A: Pure Appl. Opt. 11, 074007 (2009).
[Crossref]

Baets, R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nature Photon 7, 579–582 (2013).
[Crossref]

Bernevig, B. A.

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757–1761 (2006).
[Crossref] [PubMed]

Chan, C. T.

W.-J. Chen, Z. H. Hang, J.-W. Dong, X. Xiao, H.-Z. Wang, and C. T. Chan, “Observation of backscattering-immune chiral electromagnetic modes without time reversal breaking,” Phys. Rev. Lett. 107, 023901 (2011).
[Crossref] [PubMed]

Y. Poo, R.-x. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” Phys. Rev. Lett. 106, 093903 (2011).
[Crossref] [PubMed]

X. Ao, Z. Lin, and C. T. Chan, “One-way edge mode in a magneto-optical honeycomb photonic crystal,” Phys. Rev. B 80, 033105 (2009).
[Crossref]

Chang, S. W.

Chao, N.

Chen, W.-J.

W.-J. Chen, Z. H. Hang, J.-W. Dong, X. Xiao, H.-Z. Wang, and C. T. Chan, “Observation of backscattering-immune chiral electromagnetic modes without time reversal breaking,” Phys. Rev. Lett. 107, 023901 (2011).
[Crossref] [PubMed]

Chong, Y.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref] [PubMed]

Chong, Y. D.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref] [PubMed]

Chuang, S. L.

S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5, 5–15 (1987).
[Crossref]

Churikov, V. M.

Cory, H.

H. Cory, “Chiral devices-an overview of canonical problems,” J. Electromagnet Wave 9, 805–829 (1995).
[Crossref]

Cross, A.

L. Zhang, W. He, K. Ronald, A. Phelps, C. Whyte, C. Robertson, A. Young, C. Donaldson, and A. Cross, “Multimode coupling wave theory for helically corrugated waveguide,” IEEE Trans. Microw. Theory Techn. 60, 1–7 (2012).
[Crossref]

Daniel, N.

S. Gennady, T. Simeon, I. K. Victor, N. Daniel, and Z. G. Azriel, “Polarization properties of chiral fiber gratings,” J. Opt. A: Pure Appl. Opt. 11, 074007 (2009).
[Crossref]

Decker, M.

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater 19, 207–210 (2007).
[Crossref]

Deubel, M.

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater 19, 207–210 (2007).
[Crossref]

Doerr, C. R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nature Photon 7, 579–582 (2013).
[Crossref]

Donaldson, C.

L. Zhang, W. He, K. Ronald, A. Phelps, C. Whyte, C. Robertson, A. Young, C. Donaldson, and A. Cross, “Multimode coupling wave theory for helically corrugated waveguide,” IEEE Trans. Microw. Theory Techn. 60, 1–7 (2012).
[Crossref]

Dong, J.-W.

W.-J. Chen, Z. H. Hang, J.-W. Dong, X. Xiao, H.-Z. Wang, and C. T. Chan, “Observation of backscattering-immune chiral electromagnetic modes without time reversal breaking,” Phys. Rev. Lett. 107, 023901 (2011).
[Crossref] [PubMed]

Dorda, G.

K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45, 494–497 (1980).
[Crossref]

Draper, C. W.

Dreisow, F.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Eich, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nature Photon 7, 579–582 (2013).
[Crossref]

Fan, S.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nature Photon 7, 579–582 (2013).
[Crossref]

K. Fang and S. Fan, “Controlling the flow of light using the inhomogeneous effective gauge field that emerges from dynamic modulation,” Phys. Rev. Lett. 111, 203901 (2013).
[Crossref] [PubMed]

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon 6, 782–787 (2012).
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Microscopic theory of photonic one-way edge mode,” Phys. Rev. B 84, 075477 (2011).
[Crossref]

Fang, K.

K. Fang and S. Fan, “Controlling the flow of light using the inhomogeneous effective gauge field that emerges from dynamic modulation,” Phys. Rev. Lett. 111, 203901 (2013).
[Crossref] [PubMed]

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon 6, 782–787 (2012).
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Microscopic theory of photonic one-way edge mode,” Phys. Rev. B 84, 075477 (2011).
[Crossref]

Freude, W.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nature Photon 7, 579–582 (2013).
[Crossref]

Fu, J.-X.

J.-X. Fu, R.-J. Liu, and Z.-Y. Li, “Robust one-way modes in gyromagnetic photonic crystal waveguides with different interfaces,” Appl. Phys. Lett. 97, 041112 (2010).
[Crossref]

Fu, L.

L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76, 045302 (2007).
[Crossref]

Genack, A. Z.

Gennady, S.

S. Gennady, T. Simeon, I. K. Victor, N. Daniel, and Z. G. Azriel, “Polarization properties of chiral fiber gratings,” J. Opt. A: Pure Appl. Opt. 11, 074007 (2009).
[Crossref]

Guo, C.-C.

Haldane, F. D. M.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[Crossref]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

Halperin, B. I.

B. I. Halperin, “Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential,” Phys. Rev. B 25, 2185–2190 (1982).
[Crossref]

Hang, Z. H.

W.-J. Chen, Z. H. Hang, J.-W. Dong, X. Xiao, H.-Z. Wang, and C. T. Chan, “Observation of backscattering-immune chiral electromagnetic modes without time reversal breaking,” Phys. Rev. Lett. 107, 023901 (2011).
[Crossref] [PubMed]

Hasan, M. Z.

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

He, W.

L. Zhang, W. He, K. Ronald, A. Phelps, C. Whyte, C. Robertson, A. Young, C. Donaldson, and A. Cross, “Multimode coupling wave theory for helically corrugated waveguide,” IEEE Trans. Microw. Theory Techn. 60, 1–7 (2012).
[Crossref]

Hughes, T. L.

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757–1761 (2006).
[Crossref] [PubMed]

Jalas, D.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nature Photon 7, 579–582 (2013).
[Crossref]

Joannopoulos, J. D.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nature Photon 7, 579–582 (2013).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref] [PubMed]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref] [PubMed]

Kane, C. L.

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

L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76, 045302 (2007).
[Crossref]

C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95, 146802 (2005).
[Crossref]

Kargarian, M.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nature Mater. 12, 233–239 (2013).
[Crossref]

Khanikaev, A. B.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nature Mater. 12, 233–239 (2013).
[Crossref]

Klitzing, K. v.

K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45, 494–497 (1980).
[Crossref]

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory. (EMW Publishing, 2008).

Kopp, V. I.

Li, Z.-Y.

J.-X. Fu, R.-J. Liu, and Z.-Y. Li, “Robust one-way modes in gyromagnetic photonic crystal waveguides with different interfaces,” Appl. Phys. Lett. 97, 041112 (2010).
[Crossref]

Lin, Z.

Y. Poo, R.-x. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” Phys. Rev. Lett. 106, 093903 (2011).
[Crossref] [PubMed]

X. Ao, Z. Lin, and C. T. Chan, “One-way edge mode in a magneto-optical honeycomb photonic crystal,” Phys. Rev. B 80, 033105 (2009).
[Crossref]

Linden, S.

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater 19, 207–210 (2007).
[Crossref]

Liu, R.-J.

J.-X. Fu, R.-J. Liu, and Z.-Y. Li, “Robust one-way modes in gyromagnetic photonic crystal waveguides with different interfaces,” Appl. Phys. Lett. 97, 041112 (2010).
[Crossref]

Lumer, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

MacDonald, A. H.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nature Mater. 12, 233–239 (2013).
[Crossref]

Mele, E. J.

C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95, 146802 (2005).
[Crossref]

Melloni, A.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nature Photon 7, 579–582 (2013).
[Crossref]

Mousavi, S. H.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nature Mater. 12, 233–239 (2013).
[Crossref]

Neugroschl, D.

Nolte, S.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Ozbay, E.

Pakhomov, S.

O. Pankratov, S. Pakhomov, and B. Volkov, “Supersymmetry in heterojunctions: band-inverting contact on the basis of Pb1−xSnxTe and Hg1−xCdxTe,” Solid State Commun. 61, 93–96 (1987).
[Crossref]

Pankratov, O.

O. Pankratov, S. Pakhomov, and B. Volkov, “Supersymmetry in heterojunctions: band-inverting contact on the basis of Pb1−xSnxTe and Hg1−xCdxTe,” Solid State Commun. 61, 93–96 (1987).
[Crossref]

Pepper, M.

K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45, 494–497 (1980).
[Crossref]

Petrov, A.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nature Photon 7, 579–582 (2013).
[Crossref]

Phelps, A.

L. Zhang, W. He, K. Ronald, A. Phelps, C. Whyte, C. Robertson, A. Young, C. Donaldson, and A. Cross, “Multimode coupling wave theory for helically corrugated waveguide,” IEEE Trans. Microw. Theory Techn. 60, 1–7 (2012).
[Crossref]

Plotnik, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Podolsky, D.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Poo, Y.

Y. Poo, R.-x. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” Phys. Rev. Lett. 106, 093903 (2011).
[Crossref] [PubMed]

Popovic, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nature Photon 7, 579–582 (2013).
[Crossref]

Raghu, S.

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[Crossref]

Rechtsman, M. C.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Renner, H.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nature Photon 7, 579–582 (2013).
[Crossref]

Robertson, C.

L. Zhang, W. He, K. Ronald, A. Phelps, C. Whyte, C. Robertson, A. Young, C. Donaldson, and A. Cross, “Multimode coupling wave theory for helically corrugated waveguide,” IEEE Trans. Microw. Theory Techn. 60, 1–7 (2012).
[Crossref]

Ronald, K.

L. Zhang, W. He, K. Ronald, A. Phelps, C. Whyte, C. Robertson, A. Young, C. Donaldson, and A. Cross, “Multimode coupling wave theory for helically corrugated waveguide,” IEEE Trans. Microw. Theory Techn. 60, 1–7 (2012).
[Crossref]

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics, revised ed. (Addison Wesley, 1994).

Segev, M.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Serebryannikov, A. E.

Shvets, G.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nature Mater. 12, 233–239 (2013).
[Crossref]

G. Shvets, “Optical polarizer/isolator based on a rectangular waveguide with helical grooves,” Appl. Phys. Lett. 89, 141127 (2006).
[Crossref]

Simeon, T.

S. Gennady, T. Simeon, I. K. Victor, N. Daniel, and Z. G. Azriel, “Polarization properties of chiral fiber gratings,” J. Opt. A: Pure Appl. Opt. 11, 074007 (2009).
[Crossref]

Singer, J.

Soljacic, M.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref] [PubMed]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref] [PubMed]

Szameit, A.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Thiel, M.

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater 19, 207–210 (2007).
[Crossref]

Tse, W.-K.

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nature Mater. 12, 233–239 (2013).
[Crossref]

Vanwolleghem, M.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nature Photon 7, 579–582 (2013).
[Crossref]

Victor, I. K.

S. Gennady, T. Simeon, I. K. Victor, N. Daniel, and Z. G. Azriel, “Polarization properties of chiral fiber gratings,” J. Opt. A: Pure Appl. Opt. 11, 074007 (2009).
[Crossref]

Volkov, B.

O. Pankratov, S. Pakhomov, and B. Volkov, “Supersymmetry in heterojunctions: band-inverting contact on the basis of Pb1−xSnxTe and Hg1−xCdxTe,” Solid State Commun. 61, 93–96 (1987).
[Crossref]

von Freymann, G.

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater 19, 207–210 (2007).
[Crossref]

Wang, H.-Z.

W.-J. Chen, Z. H. Hang, J.-W. Dong, X. Xiao, H.-Z. Wang, and C. T. Chan, “Observation of backscattering-immune chiral electromagnetic modes without time reversal breaking,” Phys. Rev. Lett. 107, 023901 (2011).
[Crossref] [PubMed]

Wang, Z.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref] [PubMed]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref] [PubMed]

Wegener, M.

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater 19, 207–210 (2007).
[Crossref]

Wen, X. G.

X. G. Wen, “Gapless boundary excitations in the quantum Hall states and in the chiral spin states,” Phys. Rev. B 43, 11025–11036 (1991).
[Crossref]

Whyte, C.

L. Zhang, W. He, K. Ronald, A. Phelps, C. Whyte, C. Robertson, A. Young, C. Donaldson, and A. Cross, “Multimode coupling wave theory for helically corrugated waveguide,” IEEE Trans. Microw. Theory Techn. 60, 1–7 (2012).
[Crossref]

Wu, R.-x.

Y. Poo, R.-x. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” Phys. Rev. Lett. 106, 093903 (2011).
[Crossref] [PubMed]

Xiao, X.

W.-J. Chen, Z. H. Hang, J.-W. Dong, X. Xiao, H.-Z. Wang, and C. T. Chan, “Observation of backscattering-immune chiral electromagnetic modes without time reversal breaking,” Phys. Rev. Lett. 107, 023901 (2011).
[Crossref] [PubMed]

Yang, Y.

Y. Poo, R.-x. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” Phys. Rev. Lett. 106, 093903 (2011).
[Crossref] [PubMed]

Ye, W.-M.

Young, A.

L. Zhang, W. He, K. Ronald, A. Phelps, C. Whyte, C. Robertson, A. Young, C. Donaldson, and A. Cross, “Multimode coupling wave theory for helically corrugated waveguide,” IEEE Trans. Microw. Theory Techn. 60, 1–7 (2012).
[Crossref]

Yu, Z.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nature Photon 7, 579–582 (2013).
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon 6, 782–787 (2012).
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Microscopic theory of photonic one-way edge mode,” Phys. Rev. B 84, 075477 (2011).
[Crossref]

Yuan, X.-D.

Zen, C.

Zeuner, J. M.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Zhang, G.

Zhang, L.

L. Zhang, W. He, K. Ronald, A. Phelps, C. Whyte, C. Robertson, A. Young, C. Donaldson, and A. Cross, “Multimode coupling wave theory for helically corrugated waveguide,” IEEE Trans. Microw. Theory Techn. 60, 1–7 (2012).
[Crossref]

Zhang, S.-C.

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757–1761 (2006).
[Crossref] [PubMed]

Adv. Mater (1)

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater 19, 207–210 (2007).
[Crossref]

Appl. Phys. Lett. (2)

G. Shvets, “Optical polarizer/isolator based on a rectangular waveguide with helical grooves,” Appl. Phys. Lett. 89, 141127 (2006).
[Crossref]

J.-X. Fu, R.-J. Liu, and Z.-Y. Li, “Robust one-way modes in gyromagnetic photonic crystal waveguides with different interfaces,” Appl. Phys. Lett. 97, 041112 (2010).
[Crossref]

IEEE Trans. Microw. Theory Techn. (1)

L. Zhang, W. He, K. Ronald, A. Phelps, C. Whyte, C. Robertson, A. Young, C. Donaldson, and A. Cross, “Multimode coupling wave theory for helically corrugated waveguide,” IEEE Trans. Microw. Theory Techn. 60, 1–7 (2012).
[Crossref]

J. Electromagnet Wave (1)

H. Cory, “Chiral devices-an overview of canonical problems,” J. Electromagnet Wave 9, 805–829 (1995).
[Crossref]

J. Lightwave Technol. (2)

S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5, 5–15 (1987).
[Crossref]

S. W. Chang, “Bidirectionality in bianistropic but reciprocal photonic crystals and its usage in active photonics,” J. Lightwave Technol. 32, 10–19 (2014).
[Crossref]

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

S. Gennady, T. Simeon, I. K. Victor, N. Daniel, and Z. G. Azriel, “Polarization properties of chiral fiber gratings,” J. Opt. A: Pure Appl. Opt. 11, 074007 (2009).
[Crossref]

J. Opt. Soc. Am. B (1)

Nature (2)

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref] [PubMed]

Nature Mater. (1)

A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nature Mater. 12, 233–239 (2013).
[Crossref]

Nature Photon (2)

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nature Photon 6, 782–787 (2012).
[Crossref]

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is–and what is not–an optical isolator,” Nature Photon 7, 579–582 (2013).
[Crossref]

Opt. Express (3)

Phys. Rev. A (1)

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).
[Crossref]

Phys. Rev. B (5)

K. Fang, Z. Yu, and S. Fan, “Microscopic theory of photonic one-way edge mode,” Phys. Rev. B 84, 075477 (2011).
[Crossref]

X. Ao, Z. Lin, and C. T. Chan, “One-way edge mode in a magneto-optical honeycomb photonic crystal,” Phys. Rev. B 80, 033105 (2009).
[Crossref]

B. I. Halperin, “Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential,” Phys. Rev. B 25, 2185–2190 (1982).
[Crossref]

X. G. Wen, “Gapless boundary excitations in the quantum Hall states and in the chiral spin states,” Phys. Rev. B 43, 11025–11036 (1991).
[Crossref]

L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76, 045302 (2007).
[Crossref]

Phys. Rev. Lett. (7)

C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95, 146802 (2005).
[Crossref]

W.-J. Chen, Z. H. Hang, J.-W. Dong, X. Xiao, H.-Z. Wang, and C. T. Chan, “Observation of backscattering-immune chiral electromagnetic modes without time reversal breaking,” Phys. Rev. Lett. 107, 023901 (2011).
[Crossref] [PubMed]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref] [PubMed]

K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45, 494–497 (1980).
[Crossref]

Y. Poo, R.-x. Wu, Z. Lin, Y. Yang, and C. T. Chan, “Experimental realization of self-guiding unidirectional electromagnetic edge states,” Phys. Rev. Lett. 106, 093903 (2011).
[Crossref] [PubMed]

K. Fang and S. Fan, “Controlling the flow of light using the inhomogeneous effective gauge field that emerges from dynamic modulation,” Phys. Rev. Lett. 111, 203901 (2013).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

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

Science (1)

B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in HgTe quantum wells,” Science 314, 1757–1761 (2006).
[Crossref] [PubMed]

Solid State Commun. (1)

O. Pankratov, S. Pakhomov, and B. Volkov, “Supersymmetry in heterojunctions: band-inverting contact on the basis of Pb1−xSnxTe and Hg1−xCdxTe,” Solid State Commun. 61, 93–96 (1987).
[Crossref]

Other (2)

J. A. Kong, Electromagnetic Wave Theory. (EMW Publishing, 2008).

J. J. Sakurai, Modern Quantum Mechanics, revised ed. (Addison Wesley, 1994).

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Figures (7)

Fig. 1
Fig. 1 The schematic diagram of the one-way chiral PhC covered by PECs. The top inset shows a generic cross section inside the PhC. The bottom inset indicates a chiral scatter to be inserted into the the wave-guiding structure.
Fig. 2
Fig. 2 (a) Bandstructures of the 1D chiral PhC. The dispersion curves of unperturbed TE±1,1 modes are split into those of the LHCP-like and RHCP-like modes. The RHCP-like mode has a chiral bandgap at the BZ center. (b) The incidence of a forward-propagating TE+1,1 mode from the circular WG into chiral PhC. The cross-sectional distributions of square field magnitudes in the WG and PhC regions are shown at the bottom. Except for areas near the bump, the two field patterns look similar.
Fig. 3
Fig. 3 The two functions f(ρs) versus ρs. The function f(ρs) vanishes at ρs = 0 but peaks at ρs ≈ 0.81R, while f+(ρs) is the largest and smallest at ρs = 0 and R, respectively.
Fig. 4
Fig. 4 (a) Cross sections of 2-fold LH scatters as Rs is varied. The cross-sectional areas are fixed at 0.3πR2. (b) The reflectivities of the 2-fold LH scatters as a function of Rs at various Δεr,s. The maximal backscattering occurs at the largest Rs in these calculations. (c) The reflectivities of the 2-fold RH scatters as a function of Rs at various Δεr,s. They are much smaller than their counterparts in (b). (d) The field distributions near the 2-fold LH (top) and RH (bottom) scatters with Rs = 0.7R and Δεr,s = 0.75 on the yz plane, respectively. The incident field is hardly backscattered by the RH scatter.
Fig. 5
Fig. 5 (a) The reflectivities of the 3-fold LH scatters versus Rs at various Δεr,s, and (b) the counterparts of the 4-fold LH scatters. Both scatters do not backscatter significantly. (c) The field distributions near the 3-fold (left) and 4-fold (right) LH scatters with Rs = 0.3R and Δεr,s = 0.75 on the yz plane, respectively. Except for the prominent local fields, the magnitudes of the propagating waves change little in these two cases.
Fig. 6
Fig. 6 (a) The reflectivities of the rotationally asymmetric LH scatters versus Rs at various Δεr,s. While the scatters have no azimuthal Fourier components at l = −2, the backscattering is prominent. (b) The reflectivities of the rotationally asymmetric LH scatters with Rs = 0.27R and 0.7R versus Δεr,s, respectively. The first-order Born approximation breaks down as Δεr,s becomes too large.
Fig. 7
Fig. 7 The reflectivity of a small copper block with a size of 0.3R × 0.5R × 0.3R versus the position ρs. The fitting curve shows a decent agreement with the numerical data.

Equations (26)

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× × E ( r ) ( ω c ) 2 ε r ( r , ω ) E ( r ) = 0 , 0 < ρ < R , ε r ( r , ω ) = Δ ε r , s ( r , ω ) + ε r , c ( r , ω ) , ε r , c ( r , ω ) = Δ ε r , c ( r , ω ) + ε r , b ( ω ) ,
× × E i ( r ) ( ω c ) 2 ε r , c ( r , ω ) E i ( r ) = 0 .
× × E s ( r ) ( ω c ) 2 ε r , c ( r , ω ) E s ( r ) i ω μ 0 J s ( r ) , J s ( r ) i ω ε 0 Δ ε r , s ( r , ω ) E i ( r ) ,
z A d ρ z ^ [ E 1 ( r ) × H 2 ( r ) E 2 ( r ) × H 1 ( r ) ] = i ω ε 0 A d ρ [ ε r , 1 ( r , ω ) ε r , 2 ( r , ω ) ] E 1 ( r ) E 2 ( r ) A d ρ [ E 1 ( r ) J 2 ( r ) E 2 ( r ) J 1 ( r ) ] .
[ E m n s σ ( r ) , H m n s σ ( r ) ] = [ m n s σ ( ρ ) , m n s σ ( ρ ) ] e i m ϕ 2 π e i σ β m n s z ,
[ E 1 ( r ) , H 1 ( r ) ] = [ E m n s σ ( r ) , H m n s σ ( r ) ] , [ E 2 ( r ) , H 2 ( r ) ] = [ E m n s σ ( r ) , H m n s σ ( r ) ] ,
i ( σ β m n s + σ β m n s ) A d ρ z ^ [ m n s σ ( ρ ) × m n s σ ( ρ ) m n s σ ( ρ ) × m n s σ ( ρ ) ] e i ( m + m ) ϕ 2 π = 0 ,
A d ρ z ^ [ m n s σ ( ρ ) × m n s σ ( ρ ) m n s σ ( ρ ) × m n s σ ( ρ ) ] e i ( m + m ) ϕ 2 π = σ 𝒩 m n s ( ω ) δ m + m , 0 δ n n δ s s δ σ + σ , 0 ,
z A d ρ z ^ [ E m , n s σ ( r ) × H s ( r ) E s ( r ) × H m , n s σ ( r ) ] = i ω ε 0 A d ρ Δ ε r , c ( r , ω ) E m , n s σ ( r ) E s ( r ) A d ρ E m , n s σ ( r ) J s ( r ) ,
Δ ε r , c ( r , ω ) = Δ ε r , c ( ρ , ϕ q z , ω ) = l = Δ ε r , c ( l ) ( ρ , ω ) e i l ( ϕ q z ) ,
[ E s ( r ) , H s ( r ) ] m n s σ A m n s σ ( z ) [ m n s σ ( ρ ) , m n s σ ( ρ ) ] e i m ϕ 2 π ,
A m n s σ ( z ) z i σ β m , n s A m n s σ ( z ) = i σ ω ε 0 m n s σ e i ( m m ) q z κ ( m n s ) , ( m n s ) σ , σ ( ω ) A m n s σ ( z ) + σ J m n s σ ( z ) ,
κ ( m n s ) , ( m n s ) σ , σ ( ω ) 0 R d ρ ρ Δ ε r , c ( m m ) ( ρ , ω ) m , n s σ ( ρ ) m n s σ ( ρ ) 𝒩 m , n s ( ω ) ,
J m n s σ ( z ) A d ρ e i m ϕ 2 π m , n s σ ( ρ ) J s ( r ) 𝒩 m , n s ( ω ) ,
[ A m σ ( z ) , J m σ ( z ) ] = e i σ q z [ A ˜ m σ ( z ) , σ J ˜ m σ ( z ) ] .
z A ˜ ( z ) = i M A ˜ ( z ) + J ˜ ( z ) , A ˜ ( z ) ( A ˜ + f ( z ) A ˜ f ( z ) A ˜ + b ( z ) A ˜ b ( z ) ) , J ˜ ( z ) ( J ˜ + f ( z ) J ˜ f ( z ) J ˜ + b ( z ) J ˜ b ( z ) ) , M = ( Δ κ ( ω ) 0 0 0 0 Δ κ ( ω ) κ ( 2 ) ( ω ) 0 0 κ ( 2 ) ( ω ) Δ κ ( ω ) 0 0 0 0 Δ κ ( ω ) ) , Δ κ ( ω ) = β ( ω ) κ ( 0 ) ( ω ) q , κ ( 0 ) ( ω ) = κ m m σ σ ( ω ) , κ ( 2 ) ( ω ) = κ + bf ( ω ) , κ ( 2 ) ( ω ) = κ + fb ( ω ) ,
λ 1 ( ω ) = λ 4 ( ω ) = Δ κ ( ω ) , λ 2 ( ω ) = λ 3 ( ω ) = Δ κ 2 ( ω ) κ ( 2 ) ( ω ) κ ( 2 ) ( ω ) .
E i ( r ) = A + , i f ( z ) + f ( ρ ) e i ϕ 2 π , A + , i f ( z ) = 𝒜 + , i f e i [ β ( ω ) κ ( 0 ) ( ω ) ] ( z z L ) ,
J b ( z ) = i ω ε 0 𝒩 + ( ω ) 𝒜 + , i f e i [ β ( ω ) κ ( 0 ) ( ω ) ] ( z z L ) A d ρ e 2 i ϕ 2 π [ Δ ε r , s ( r , ω ) + f ( ρ ) + f ( ρ ) ] .
z A b ( z ) + i [ β ( ω ) κ ( 0 ) ( ω ) ] A b ( z ) = J b ( z ) .
A b ( z ) = z z R d z e i [ β ( ω ) κ ( 0 ) ( ω ) ] ( z z ) J b ( z ) .
r ( ω ) = A b ( z L ) 𝒜 + , i f = i ω ε 0 𝒩 + ( ω ) Ω s d r e 2 i { ϕ + [ β ( ω ) κ ( 0 ) ( ω ) ] z } 2 π [ Δ ε r , s ( r , ω ) + f ( ρ ) + f ( ρ ) ] ,
Δ ε r , s ( r , ω ) = U ( r ) l = Δ ε r , s ( l ) ( ρ , ω ) e i l ( ϕ + q s z ) ,
[ ± , ρ f ( ρ ) , ± , ϕ f ( ρ ) ] ~ [ J 1 ( k t ρ ) k t ρ , ± J 1 ( k t ρ ) ] ,
r ( ω ) = ρ s Δ ρ s / 2 ρ s + Δ ρ s / 2 d ρ ρ Δ ε r , s ( r , ω ) [ | + , ρ f ( ρ ) | 2 | + , ϕ f ( ρ ) | 2 ] , ρ s Δ ρ s f ( ρ s ) Δ ε r , s ( r , ω ) ρ = ρ s , Δ ε r , s ( r , ω ) ρ = ρ s = ρ s Δ ρ s / 2 ρ s + Δ ρ s / 2 d ρ Δ ε r , s ( r , ω ) Δ ρ s , f ( ρ s ) = [ J 1 ( k t ρ s ) ( k t ρ s ) ] 2 [ J 1 ( k t ρ s ) ] 2 ,
r ( ω ) a f ( ρ s ) + b ,

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