Abstract

A design scheme for low-reflection asymmetric mode conversion structure in three-mode waveguide system is proposed. By using a dark-mode of three-mode system, which can be interpreted in terms of destructive interference of transition amplitudes, the transmission characteristics for forward and backward directions can be designed separately. After explanation of the proposed design scheme, we demonstrate an example of asymmetric mode converter that consists of two gratings. The proposed scheme may be useful for the design of tunable asymmetric transmission devices due to its design flexibility and efficient design process.

© 2014 Optical Society of America

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2014 (4)

C. Pfeiffer, C. Zhang, V. Ray, L. J. Guo, and A. Grbic, “High performance bianisotropic metasurfaces: asymmetric transmission of light,” Phys. Rev. Lett. 113(2), 023902 (2014).
[Crossref] [PubMed]

T. Xu and H. J. Lezec, “Visible-frequency asymmetric transmission devices incorporating a hyperbolic metamaterial,” Nat. Commun. 5, 4141 (2014).
[Crossref] [PubMed]

I. H. Giden, D. Yilmaz, M. Turduev, H. Kurt, E. Çolak, and E. Ozbay, “Theoretical and experimental investigations of asymmetric light transport in graded index photonic crystal waveguides,” Appl. Phys. Lett. 104(3), 031116 (2014).
[Crossref]

A. E. Serebryannikov, E. Ozbay, and S. Nojima, “Asymmetric transmission of terahertz waves using polar dielectrics,” Opt. Express 22(3), 3075–3088 (2014).
[Crossref] [PubMed]

2013 (7)

2012 (10)

C. Huang, Y. Feng, J. Zhao, Z. Wang, and T. Jiang, “Asymmetric electromagnetic wave transmission of linear polarization via polarization conversion through chiral metamaterial structures,” Phys. Rev. B 85(19), 195131 (2012).
[Crossref]

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Diodelike asymmetric transmission of linearly polarized waves using magnetoelectric coupling and electromagnetic wave tunneling,” Phys. Rev. Lett. 108(21), 213905 (2012).
[Crossref] [PubMed]

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

L. Feng, M. Ayache, J. Huang, and Y.-L. Xu, “Response to comments on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

V. Kuzmiak and A. A. Maradudin, “Asymmetric transmission of surface plasmon polaritons,” Phys. Rev. A 86(4), 043805 (2012).
[Crossref]

C. Wang, X.-L. Zhong, and Z.-Y. Li, “Linear and passive silicon optical isolator,” Sci. Rep. 2, 674 (2012).
[PubMed]

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat Commun 3, 1151 (2012).
[Crossref] [PubMed]

H. Kurt, D. Yilmaz, A. E. Akosman, and E. Ozbay, “Asymmetric light propagation in chirped photonic crystal waveguides,” Opt. Express 20(18), 20635–20646 (2012).
[Crossref] [PubMed]

V. Liu, D. A. B. Miller, and S. Fan, “Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect,” Opt. Express 20(27), 28388–28397 (2012).
[Crossref] [PubMed]

2011 (9)

M. Kang, J. Chen, H.-X. Cui, Y. Li, and H.-T. Wang, “Asymmetric transmission for linearly polarized electromagnetic radiation,” Opt. Express 19(9), 8347–8356 (2011).
[Crossref] [PubMed]

J. Xu, C. Cheng, M. Kang, J. Chen, Z. Zheng, Y.-X. Fan, and H.-T. Wang, “Unidirectional optical transmission in dual-metal gratings in the absence of anisotropic and nonlinear materials,” Opt. Lett. 36(10), 1905–1907 (2011).
[Crossref] [PubMed]

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Asymmetric transmission of linearly polarized waves and polarization angle dependent wave rotation using a chiral metamaterial,” Opt. Express 19(15), 14290–14299 (2011).
[Crossref] [PubMed]

C. Lu, X. Hu, H. Yang, and Q. Gong, “Ultrahigh-contrast and wideband nanoscale photonic crystal all-optical diode,” Opt. Lett. 36(23), 4668–4670 (2011).
[Crossref] [PubMed]

C. Wang, C.-Z. Zhou, and Z.-Y. Li, “On-chip optical diode based on silicon photonic crystal heterojunctions,” Opt. Express 19(27), 26948–26955 (2011).
[Crossref] [PubMed]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

E. Plum, V. A. Fedotov, and N. I. Zheludev, “Asymmetric transmission: a generic property of two-dimensional periodic patterns,” J. Opt. 13(2), 024006 (2011).
[Crossref]

C. Lu, X. Hu, Y. Zhang, Z. Li, X. Xu, H. Yang, and Q. Gong, “Ultralow power all-optical diode in photonic crytal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

S. Cakmakyapan, H. Caglayan, A. E. Serebryannikov, and E. Ozbay, “Experimental validation of strong directional selectivity in nonsymmetric metallic gratings with a subwavelength slit,” Appl. Phys. Lett. 98(5), 051103 (2011).
[Crossref]

2010 (4)

C. Menzel, C. Helgert, C. Rockstuhl, E.-B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Phys. Rev. Lett. 104(25), 253902 (2010).
[Crossref] [PubMed]

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
[Crossref]

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[Crossref] [PubMed]

S. Cakmakyapan, A. E. Serebryannikov, H. Caglayan, and E. Ozbay, “One-way transmission through the subwavelength slit in nonsymmetric metallic gratings,” Opt. Lett. 35(15), 2597–2599 (2010).
[Crossref] [PubMed]

2009 (2)

H.-C. Liu and A. Yariv, “Grating induced transparency (GIT) and the dark mode in optical waveguides,” Opt. Express 17(14), 11710–11718 (2009).
[Crossref] [PubMed]

R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B 80(15), 153104 (2009).
[Crossref]

2008 (2)

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8(9), 2940–2943 (2008).
[Crossref] [PubMed]

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101(25), 253903 (2008).
[Crossref] [PubMed]

2006 (1)

V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97(16), 167401 (2006).
[Crossref] [PubMed]

1991 (1)

H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79(10), 1505–1518 (1991).
[Crossref]

1973 (1)

A. Yariv, “Couple-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973).
[Crossref]

Akosman, A. E.

Alamri, M.

Almeida, V. R.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Ayache, M.

L. Feng, M. Ayache, J. Huang, and Y.-L. Xu, “Response to comments on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Azad, A. K.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat Commun 3, 1151 (2012).
[Crossref] [PubMed]

R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B 80(15), 153104 (2009).
[Crossref]

Baets, R.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

Bai, J.

Battal, E.

E. Battal, T. A. Yogurt, and A. K. Okyay, “Ultrahigh contrast one-way optical transmission through a subwavelength slit,” Plasmonics 8(2), 509–513 (2013).
[Crossref]

Brinkmeyer, E.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

Caglayan, H.

S. Cakmakyapan, H. Caglayan, A. E. Serebryannikov, and E. Ozbay, “Experimental validation of strong directional selectivity in nonsymmetric metallic gratings with a subwavelength slit,” Appl. Phys. Lett. 98(5), 051103 (2011).
[Crossref]

S. Cakmakyapan, A. E. Serebryannikov, H. Caglayan, and E. Ozbay, “One-way transmission through the subwavelength slit in nonsymmetric metallic gratings,” Opt. Lett. 35(15), 2597–2599 (2010).
[Crossref] [PubMed]

Cakmakyapan, S.

S. Cakmakyapan, H. Caglayan, A. E. Serebryannikov, and E. Ozbay, “Experimental validation of strong directional selectivity in nonsymmetric metallic gratings with a subwavelength slit,” Appl. Phys. Lett. 98(5), 051103 (2011).
[Crossref]

S. Cakmakyapan, A. E. Serebryannikov, H. Caglayan, and E. Ozbay, “One-way transmission through the subwavelength slit in nonsymmetric metallic gratings,” Opt. Lett. 35(15), 2597–2599 (2010).
[Crossref] [PubMed]

Cao, Z.

Chen, H.-T.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat Commun 3, 1151 (2012).
[Crossref] [PubMed]

Chen, J.

Chen, Y.

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8(9), 2940–2943 (2008).
[Crossref] [PubMed]

V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97(16), 167401 (2006).
[Crossref] [PubMed]

Chen, Y.-F.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Cheng, C.

Cheville, R. A.

R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B 80(15), 153104 (2009).
[Crossref]

Chong, C. T.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[Crossref] [PubMed]

Cicek, A.

A. Cicek and B. Ulug, “Coupling between opposite-parity modes in parallel photonic crystal waveguides and its application in unidirectional light transmission,” Appl. Phys. B 113(4), 619–626 (2013).
[Crossref]

Çolak, E.

I. H. Giden, D. Yilmaz, M. Turduev, H. Kurt, E. Çolak, and E. Ozbay, “Theoretical and experimental investigations of asymmetric light transport in graded index photonic crystal waveguides,” Appl. Phys. Lett. 104(3), 031116 (2014).
[Crossref]

Cui, H.-X.

Cui, T. J.

J. Shi, X. Liu, S. Yu, T. Lv, Z. Zhu, H. F. Ma, and T. J. Cui, “Dual-band asymmetric transmission of linear polarization in bilayered chiral metamaterial,” Appl. Phys. Lett. 102(19), 191905 (2013).
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Doerr, C. R.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
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Eich, M.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

Fainman, Y.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
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Fan, S.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
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V. Liu, D. A. B. Miller, and S. Fan, “Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect,” Opt. Express 20(27), 28388–28397 (2012).
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Fan, Y.-X.

Fedotov, V. A.

E. Plum, V. A. Fedotov, and N. I. Zheludev, “Asymmetric transmission: a generic property of two-dimensional periodic patterns,” J. Opt. 13(2), 024006 (2011).
[Crossref]

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8(9), 2940–2943 (2008).
[Crossref] [PubMed]

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101(25), 253903 (2008).
[Crossref] [PubMed]

V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97(16), 167401 (2006).
[Crossref] [PubMed]

Fegadolli, W. S.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Feng, L.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
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L. Feng, M. Ayache, J. Huang, and Y.-L. Xu, “Response to comments on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Feng, S.

Feng, Y.

C. Huang, Y. Feng, J. Zhao, Z. Wang, and T. Jiang, “Asymmetric electromagnetic wave transmission of linear polarization via polarization conversion through chiral metamaterial structures,” Phys. Rev. B 85(19), 195131 (2012).
[Crossref]

Flach, S.

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
[Crossref]

Freude, W.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

Giden, I. H.

I. H. Giden, D. Yilmaz, M. Turduev, H. Kurt, E. Çolak, and E. Ozbay, “Theoretical and experimental investigations of asymmetric light transport in graded index photonic crystal waveguides,” Appl. Phys. Lett. 104(3), 031116 (2014).
[Crossref]

Giessen, H.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[Crossref] [PubMed]

Gong, Q.

C. Lu, X. Hu, Y. Zhang, Z. Li, X. Xu, H. Yang, and Q. Gong, “Ultralow power all-optical diode in photonic crytal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

C. Lu, X. Hu, H. Yang, and Q. Gong, “Ultrahigh-contrast and wideband nanoscale photonic crystal all-optical diode,” Opt. Lett. 36(23), 4668–4670 (2011).
[Crossref] [PubMed]

Grbic, A.

C. Pfeiffer, C. Zhang, V. Ray, L. J. Guo, and A. Grbic, “High performance bianisotropic metasurfaces: asymmetric transmission of light,” Phys. Rev. Lett. 113(2), 023902 (2014).
[Crossref] [PubMed]

Gu, J.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat Commun 3, 1151 (2012).
[Crossref] [PubMed]

Guo, L. J.

C. Pfeiffer, C. Zhang, V. Ray, L. J. Guo, and A. Grbic, “High performance bianisotropic metasurfaces: asymmetric transmission of light,” Phys. Rev. Lett. 113(2), 023902 (2014).
[Crossref] [PubMed]

Halas, N. J.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[Crossref] [PubMed]

Han, J.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat Commun 3, 1151 (2012).
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H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79(10), 1505–1518 (1991).
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Helgert, C.

C. Menzel, C. Helgert, C. Rockstuhl, E.-B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Phys. Rev. Lett. 104(25), 253902 (2010).
[Crossref] [PubMed]

Hu, X.

C. Lu, X. Hu, H. Yang, and Q. Gong, “Ultrahigh-contrast and wideband nanoscale photonic crystal all-optical diode,” Opt. Lett. 36(23), 4668–4670 (2011).
[Crossref] [PubMed]

C. Lu, X. Hu, Y. Zhang, Z. Li, X. Xu, H. Yang, and Q. Gong, “Ultralow power all-optical diode in photonic crytal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

Huang, C.

C. Huang, Y. Feng, J. Zhao, Z. Wang, and T. Jiang, “Asymmetric electromagnetic wave transmission of linear polarization via polarization conversion through chiral metamaterial structures,” Phys. Rev. B 85(19), 195131 (2012).
[Crossref]

Huang, J.

L. Feng, M. Ayache, J. Huang, and Y.-L. Xu, “Response to comments on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Huang, W.

H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79(10), 1505–1518 (1991).
[Crossref]

Jalas, D.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

Jiang, T.

C. Huang, Y. Feng, J. Zhao, Z. Wang, and T. Jiang, “Asymmetric electromagnetic wave transmission of linear polarization via polarization conversion through chiral metamaterial structures,” Phys. Rev. B 85(19), 195131 (2012).
[Crossref]

Joannopoulos, J. D.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

Kang, M.

Khardikov, V. V.

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8(9), 2940–2943 (2008).
[Crossref] [PubMed]

Kivshar, Y. S.

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
[Crossref]

Kley, E.-B.

C. Menzel, C. Helgert, C. Rockstuhl, E.-B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Phys. Rev. Lett. 104(25), 253902 (2010).
[Crossref] [PubMed]

Kotynski, R.

Krause, M.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

Kurt, H.

I. H. Giden, D. Yilmaz, M. Turduev, H. Kurt, E. Çolak, and E. Ozbay, “Theoretical and experimental investigations of asymmetric light transport in graded index photonic crystal waveguides,” Appl. Phys. Lett. 104(3), 031116 (2014).
[Crossref]

H. Kurt, D. Yilmaz, A. E. Akosman, and E. Ozbay, “Asymmetric light propagation in chirped photonic crystal waveguides,” Opt. Express 20(18), 20635–20646 (2012).
[Crossref] [PubMed]

Kuzmiak, V.

V. Kuzmiak and A. A. Maradudin, “Asymmetric transmission of surface plasmon polaritons,” Phys. Rev. A 86(4), 043805 (2012).
[Crossref]

Lederer, F.

C. Menzel, C. Helgert, C. Rockstuhl, E.-B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Phys. Rev. Lett. 104(25), 253902 (2010).
[Crossref] [PubMed]

R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B 80(15), 153104 (2009).
[Crossref]

Lezec, H. J.

T. Xu and H. J. Lezec, “Visible-frequency asymmetric transmission devices incorporating a hyperbolic metamaterial,” Nat. Commun. 5, 4141 (2014).
[Crossref] [PubMed]

Li, Y.

Li, Z.

C. Lu, X. Hu, Y. Zhang, Z. Li, X. Xu, H. Yang, and Q. Gong, “Ultralow power all-optical diode in photonic crytal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

Li, Z.-Y.

Liu, H.-C.

Liu, V.

Liu, X.

J. Shi, X. Liu, S. Yu, T. Lv, Z. Zhu, H. F. Ma, and T. J. Cui, “Dual-band asymmetric transmission of linear polarization in bilayered chiral metamaterial,” Appl. Phys. Lett. 102(19), 191905 (2013).
[Crossref]

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat Commun 3, 1151 (2012).
[Crossref] [PubMed]

Lu, C.

C. Lu, X. Hu, H. Yang, and Q. Gong, “Ultrahigh-contrast and wideband nanoscale photonic crystal all-optical diode,” Opt. Lett. 36(23), 4668–4670 (2011).
[Crossref] [PubMed]

C. Lu, X. Hu, Y. Zhang, Z. Li, X. Xu, H. Yang, and Q. Gong, “Ultralow power all-optical diode in photonic crytal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

Lu, M.-H.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Luk’yanchuk, B.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[Crossref] [PubMed]

Lusakowski, J.

Lv, T.

J. Shi, X. Liu, S. Yu, T. Lv, Z. Zhu, H. F. Ma, and T. J. Cui, “Dual-band asymmetric transmission of linear polarization in bilayered chiral metamaterial,” Appl. Phys. Lett. 102(19), 191905 (2013).
[Crossref]

Ma, H. F.

J. Shi, X. Liu, S. Yu, T. Lv, Z. Zhu, H. F. Ma, and T. J. Cui, “Dual-band asymmetric transmission of linear polarization in bilayered chiral metamaterial,” Appl. Phys. Lett. 102(19), 191905 (2013).
[Crossref]

Ma, Y.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat Commun 3, 1151 (2012).
[Crossref] [PubMed]

Maier, S. A.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat Commun 3, 1151 (2012).
[Crossref] [PubMed]

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[Crossref] [PubMed]

Maradudin, A. A.

V. Kuzmiak and A. A. Maradudin, “Asymmetric transmission of surface plasmon polaritons,” Phys. Rev. A 86(4), 043805 (2012).
[Crossref]

Melloni, A.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

Menzel, C.

C. Menzel, C. Helgert, C. Rockstuhl, E.-B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Phys. Rev. Lett. 104(25), 253902 (2010).
[Crossref] [PubMed]

R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B 80(15), 153104 (2009).
[Crossref]

Miller, D. A. B.

Miroshnichenko, A. E.

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
[Crossref]

Mladyonov, P. L.

V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97(16), 167401 (2006).
[Crossref] [PubMed]

Mutlu, M.

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Diodelike asymmetric transmission of linearly polarized waves using magnetoelectric coupling and electromagnetic wave tunneling,” Phys. Rev. Lett. 108(21), 213905 (2012).
[Crossref] [PubMed]

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Asymmetric transmission of linearly polarized waves and polarization angle dependent wave rotation using a chiral metamaterial,” Opt. Express 19(15), 14290–14299 (2011).
[Crossref] [PubMed]

Nojima, S.

Nordlander, P.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[Crossref] [PubMed]

Okyay, A. K.

E. Battal, T. A. Yogurt, and A. K. Okyay, “Ultrahigh contrast one-way optical transmission through a subwavelength slit,” Plasmonics 8(2), 509–513 (2013).
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Oliveira, J. E. B.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Ozbay, E.

A. E. Serebryannikov, E. Ozbay, and S. Nojima, “Asymmetric transmission of terahertz waves using polar dielectrics,” Opt. Express 22(3), 3075–3088 (2014).
[Crossref] [PubMed]

I. H. Giden, D. Yilmaz, M. Turduev, H. Kurt, E. Çolak, and E. Ozbay, “Theoretical and experimental investigations of asymmetric light transport in graded index photonic crystal waveguides,” Appl. Phys. Lett. 104(3), 031116 (2014).
[Crossref]

H. Kurt, D. Yilmaz, A. E. Akosman, and E. Ozbay, “Asymmetric light propagation in chirped photonic crystal waveguides,” Opt. Express 20(18), 20635–20646 (2012).
[Crossref] [PubMed]

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Diodelike asymmetric transmission of linearly polarized waves using magnetoelectric coupling and electromagnetic wave tunneling,” Phys. Rev. Lett. 108(21), 213905 (2012).
[Crossref] [PubMed]

S. Cakmakyapan, H. Caglayan, A. E. Serebryannikov, and E. Ozbay, “Experimental validation of strong directional selectivity in nonsymmetric metallic gratings with a subwavelength slit,” Appl. Phys. Lett. 98(5), 051103 (2011).
[Crossref]

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Asymmetric transmission of linearly polarized waves and polarization angle dependent wave rotation using a chiral metamaterial,” Opt. Express 19(15), 14290–14299 (2011).
[Crossref] [PubMed]

S. Cakmakyapan, A. E. Serebryannikov, H. Caglayan, and E. Ozbay, “One-way transmission through the subwavelength slit in nonsymmetric metallic gratings,” Opt. Lett. 35(15), 2597–2599 (2010).
[Crossref] [PubMed]

Papasimakis, N.

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101(25), 253903 (2008).
[Crossref] [PubMed]

Pertsch, T.

C. Menzel, C. Helgert, C. Rockstuhl, E.-B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Phys. Rev. Lett. 104(25), 253902 (2010).
[Crossref] [PubMed]

Petrov, A.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

Pfeiffer, C.

C. Pfeiffer, C. Zhang, V. Ray, L. J. Guo, and A. Grbic, “High performance bianisotropic metasurfaces: asymmetric transmission of light,” Phys. Rev. Lett. 113(2), 023902 (2014).
[Crossref] [PubMed]

Plum, E.

E. Plum, V. A. Fedotov, and N. I. Zheludev, “Asymmetric transmission: a generic property of two-dimensional periodic patterns,” J. Opt. 13(2), 024006 (2011).
[Crossref]

R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B 80(15), 153104 (2009).
[Crossref]

Popovic, M.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

Prosvirnin, S. L.

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8(9), 2940–2943 (2008).
[Crossref] [PubMed]

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101(25), 253903 (2008).
[Crossref] [PubMed]

V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97(16), 167401 (2006).
[Crossref] [PubMed]

Qi, X.

Ray, V.

C. Pfeiffer, C. Zhang, V. Ray, L. J. Guo, and A. Grbic, “High performance bianisotropic metasurfaces: asymmetric transmission of light,” Phys. Rev. Lett. 113(2), 023902 (2014).
[Crossref] [PubMed]

Renner, H.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

Rockstuhl, C.

C. Menzel, C. Helgert, C. Rockstuhl, E.-B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Phys. Rev. Lett. 104(25), 253902 (2010).
[Crossref] [PubMed]

R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B 80(15), 153104 (2009).
[Crossref]

Rogacheva, A. V.

V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97(16), 167401 (2006).
[Crossref] [PubMed]

Scherer, A.

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Schwanecke, A. S.

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8(9), 2940–2943 (2008).
[Crossref] [PubMed]

Serebryannikov, A. E.

A. E. Serebryannikov, E. Ozbay, and S. Nojima, “Asymmetric transmission of terahertz waves using polar dielectrics,” Opt. Express 22(3), 3075–3088 (2014).
[Crossref] [PubMed]

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Diodelike asymmetric transmission of linearly polarized waves using magnetoelectric coupling and electromagnetic wave tunneling,” Phys. Rev. Lett. 108(21), 213905 (2012).
[Crossref] [PubMed]

S. Cakmakyapan, H. Caglayan, A. E. Serebryannikov, and E. Ozbay, “Experimental validation of strong directional selectivity in nonsymmetric metallic gratings with a subwavelength slit,” Appl. Phys. Lett. 98(5), 051103 (2011).
[Crossref]

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Asymmetric transmission of linearly polarized waves and polarization angle dependent wave rotation using a chiral metamaterial,” Opt. Express 19(15), 14290–14299 (2011).
[Crossref] [PubMed]

S. Cakmakyapan, A. E. Serebryannikov, H. Caglayan, and E. Ozbay, “One-way transmission through the subwavelength slit in nonsymmetric metallic gratings,” Opt. Lett. 35(15), 2597–2599 (2010).
[Crossref] [PubMed]

Shi, J.

J. Shi, X. Liu, S. Yu, T. Lv, Z. Zhu, H. F. Ma, and T. J. Cui, “Dual-band asymmetric transmission of linear polarization in bilayered chiral metamaterial,” Appl. Phys. Lett. 102(19), 191905 (2013).
[Crossref]

Singh, R.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat Commun 3, 1151 (2012).
[Crossref] [PubMed]

R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B 80(15), 153104 (2009).
[Crossref]

Stolarek, M.

Szoplik, T.

Taylor, A. J.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat Commun 3, 1151 (2012).
[Crossref] [PubMed]

Tian, Z.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat Commun 3, 1151 (2012).
[Crossref] [PubMed]

Tünnermann, A.

C. Menzel, C. Helgert, C. Rockstuhl, E.-B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Phys. Rev. Lett. 104(25), 253902 (2010).
[Crossref] [PubMed]

Turduev, M.

I. H. Giden, D. Yilmaz, M. Turduev, H. Kurt, E. Çolak, and E. Ozbay, “Theoretical and experimental investigations of asymmetric light transport in graded index photonic crystal waveguides,” Appl. Phys. Lett. 104(3), 031116 (2014).
[Crossref]

Ulug, B.

A. Cicek and B. Ulug, “Coupling between opposite-parity modes in parallel photonic crystal waveguides and its application in unidirectional light transmission,” Appl. Phys. B 113(4), 619–626 (2013).
[Crossref]

Vanwolleghem, M.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

Wang, C.

Wang, H.-T.

Wang, Y.

Wang, Z.

C. Huang, Y. Feng, J. Zhao, Z. Wang, and T. Jiang, “Asymmetric electromagnetic wave transmission of linear polarization via polarization conversion through chiral metamaterial structures,” Phys. Rev. B 85(19), 195131 (2012).
[Crossref]

Xia, K.

Xu, J.

Xu, T.

T. Xu and H. J. Lezec, “Visible-frequency asymmetric transmission devices incorporating a hyperbolic metamaterial,” Nat. Commun. 5, 4141 (2014).
[Crossref] [PubMed]

Xu, X.

C. Lu, X. Hu, Y. Zhang, Z. Li, X. Xu, H. Yang, and Q. Gong, “Ultralow power all-optical diode in photonic crytal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

Xu, Y.-L.

L. Feng, M. Ayache, J. Huang, and Y.-L. Xu, “Response to comments on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Yang, H.

C. Lu, X. Hu, Y. Zhang, Z. Li, X. Xu, H. Yang, and Q. Gong, “Ultralow power all-optical diode in photonic crytal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

C. Lu, X. Hu, H. Yang, and Q. Gong, “Ultrahigh-contrast and wideband nanoscale photonic crystal all-optical diode,” Opt. Lett. 36(23), 4668–4670 (2011).
[Crossref] [PubMed]

Yariv, A.

Yavorskiy, D.

Yilmaz, D.

I. H. Giden, D. Yilmaz, M. Turduev, H. Kurt, E. Çolak, and E. Ozbay, “Theoretical and experimental investigations of asymmetric light transport in graded index photonic crystal waveguides,” Appl. Phys. Lett. 104(3), 031116 (2014).
[Crossref]

H. Kurt, D. Yilmaz, A. E. Akosman, and E. Ozbay, “Asymmetric light propagation in chirped photonic crystal waveguides,” Opt. Express 20(18), 20635–20646 (2012).
[Crossref] [PubMed]

Yogurt, T. A.

E. Battal, T. A. Yogurt, and A. K. Okyay, “Ultrahigh contrast one-way optical transmission through a subwavelength slit,” Plasmonics 8(2), 509–513 (2013).
[Crossref]

Yu, S.

J. Shi, X. Liu, S. Yu, T. Lv, Z. Zhu, H. F. Ma, and T. J. Cui, “Dual-band asymmetric transmission of linear polarization in bilayered chiral metamaterial,” Appl. Phys. Lett. 102(19), 191905 (2013).
[Crossref]

Yu, Z.

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

Zapata Rodríguez, C. J.

Zhang, C.

C. Pfeiffer, C. Zhang, V. Ray, L. J. Guo, and A. Grbic, “High performance bianisotropic metasurfaces: asymmetric transmission of light,” Phys. Rev. Lett. 113(2), 023902 (2014).
[Crossref] [PubMed]

Zhang, G.

Zhang, S.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat Commun 3, 1151 (2012).
[Crossref] [PubMed]

Zhang, W.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat Commun 3, 1151 (2012).
[Crossref] [PubMed]

R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B 80(15), 153104 (2009).
[Crossref]

Zhang, X.

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat Commun 3, 1151 (2012).
[Crossref] [PubMed]

Zhang, Y.

C. Lu, X. Hu, Y. Zhang, Z. Li, X. Xu, H. Yang, and Q. Gong, “Ultralow power all-optical diode in photonic crytal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

Zhao, J.

C. Huang, Y. Feng, J. Zhao, Z. Wang, and T. Jiang, “Asymmetric electromagnetic wave transmission of linear polarization via polarization conversion through chiral metamaterial structures,” Phys. Rev. B 85(19), 195131 (2012).
[Crossref]

Zheludev, N. I.

E. Plum, V. A. Fedotov, and N. I. Zheludev, “Asymmetric transmission: a generic property of two-dimensional periodic patterns,” J. Opt. 13(2), 024006 (2011).
[Crossref]

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[Crossref] [PubMed]

R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B 80(15), 153104 (2009).
[Crossref]

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8(9), 2940–2943 (2008).
[Crossref] [PubMed]

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101(25), 253903 (2008).
[Crossref] [PubMed]

V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97(16), 167401 (2006).
[Crossref] [PubMed]

Zheng, Z.

Zhong, X.-L.

C. Wang, X.-L. Zhong, and Z.-Y. Li, “Linear and passive silicon optical isolator,” Sci. Rep. 2, 674 (2012).
[PubMed]

Zhou, C.-Z.

Zhu, Z.

J. Shi, X. Liu, S. Yu, T. Lv, Z. Zhu, H. F. Ma, and T. J. Cui, “Dual-band asymmetric transmission of linear polarization in bilayered chiral metamaterial,” Appl. Phys. Lett. 102(19), 191905 (2013).
[Crossref]

Zubairy, M. S.

Appl. Phys. B (1)

A. Cicek and B. Ulug, “Coupling between opposite-parity modes in parallel photonic crystal waveguides and its application in unidirectional light transmission,” Appl. Phys. B 113(4), 619–626 (2013).
[Crossref]

Appl. Phys. Lett. (4)

C. Lu, X. Hu, Y. Zhang, Z. Li, X. Xu, H. Yang, and Q. Gong, “Ultralow power all-optical diode in photonic crytal heterostructures with broken spatial inversion symmetry,” Appl. Phys. Lett. 99(5), 051107 (2011).
[Crossref]

I. H. Giden, D. Yilmaz, M. Turduev, H. Kurt, E. Çolak, and E. Ozbay, “Theoretical and experimental investigations of asymmetric light transport in graded index photonic crystal waveguides,” Appl. Phys. Lett. 104(3), 031116 (2014).
[Crossref]

J. Shi, X. Liu, S. Yu, T. Lv, Z. Zhu, H. F. Ma, and T. J. Cui, “Dual-band asymmetric transmission of linear polarization in bilayered chiral metamaterial,” Appl. Phys. Lett. 102(19), 191905 (2013).
[Crossref]

S. Cakmakyapan, H. Caglayan, A. E. Serebryannikov, and E. Ozbay, “Experimental validation of strong directional selectivity in nonsymmetric metallic gratings with a subwavelength slit,” Appl. Phys. Lett. 98(5), 051103 (2011).
[Crossref]

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J. Opt. (1)

E. Plum, V. A. Fedotov, and N. I. Zheludev, “Asymmetric transmission: a generic property of two-dimensional periodic patterns,” J. Opt. 13(2), 024006 (2011).
[Crossref]

Nano Lett. (1)

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8(9), 2940–2943 (2008).
[Crossref] [PubMed]

Nat Commun (1)

J. Gu, R. Singh, X. Liu, X. Zhang, Y. Ma, S. Zhang, S. A. Maier, Z. Tian, A. K. Azad, H.-T. Chen, A. J. Taylor, J. Han, and W. Zhang, “Active control of electromagnetically induced transparency analogue in terahertz metamaterials,” Nat Commun 3, 1151 (2012).
[Crossref] [PubMed]

Nat. Commun. (1)

T. Xu and H. J. Lezec, “Visible-frequency asymmetric transmission devices incorporating a hyperbolic metamaterial,” Nat. Commun. 5, 4141 (2014).
[Crossref] [PubMed]

Nat. Mater. (2)

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[Crossref] [PubMed]

L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. B. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12(2), 108–113 (2012).
[Crossref] [PubMed]

Opt. Express (9)

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Asymmetric transmission of linearly polarized waves and polarization angle dependent wave rotation using a chiral metamaterial,” Opt. Express 19(15), 14290–14299 (2011).
[Crossref] [PubMed]

M. Kang, J. Chen, H.-X. Cui, Y. Li, and H.-T. Wang, “Asymmetric transmission for linearly polarized electromagnetic radiation,” Opt. Express 19(9), 8347–8356 (2011).
[Crossref] [PubMed]

C. Wang, C.-Z. Zhou, and Z.-Y. Li, “On-chip optical diode based on silicon photonic crystal heterojunctions,” Opt. Express 19(27), 26948–26955 (2011).
[Crossref] [PubMed]

S. Feng and Y. Wang, “Unidirectional reciprocal wavelength filters based on the square-lattice photonic crystal structures with the rectangular defects,” Opt. Express 21(1), 220–228 (2013).
[Crossref] [PubMed]

A. E. Serebryannikov, E. Ozbay, and S. Nojima, “Asymmetric transmission of terahertz waves using polar dielectrics,” Opt. Express 22(3), 3075–3088 (2014).
[Crossref] [PubMed]

H. Kurt, D. Yilmaz, A. E. Akosman, and E. Ozbay, “Asymmetric light propagation in chirped photonic crystal waveguides,” Opt. Express 20(18), 20635–20646 (2012).
[Crossref] [PubMed]

V. Liu, D. A. B. Miller, and S. Fan, “Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect,” Opt. Express 20(27), 28388–28397 (2012).
[Crossref] [PubMed]

H.-C. Liu and A. Yariv, “Grating induced transparency (GIT) and the dark mode in optical waveguides,” Opt. Express 17(14), 11710–11718 (2009).
[Crossref] [PubMed]

K. Xia, M. Alamri, and M. S. Zubairy, “Ultrabroadband nonreciprocal transverse energy flow of light in linear passive photonic circuits,” Opt. Express 21(22), 25619–25631 (2013).
[Crossref] [PubMed]

Opt. Lett. (5)

Phys. Rev. A (1)

V. Kuzmiak and A. A. Maradudin, “Asymmetric transmission of surface plasmon polaritons,” Phys. Rev. A 86(4), 043805 (2012).
[Crossref]

Phys. Rev. B (2)

C. Huang, Y. Feng, J. Zhao, Z. Wang, and T. Jiang, “Asymmetric electromagnetic wave transmission of linear polarization via polarization conversion through chiral metamaterial structures,” Phys. Rev. B 85(19), 195131 (2012).
[Crossref]

R. Singh, E. Plum, C. Menzel, C. Rockstuhl, A. K. Azad, R. A. Cheville, F. Lederer, W. Zhang, and N. I. Zheludev, “Terahertz metamaterial with asymmetric transmission,” Phys. Rev. B 80(15), 153104 (2009).
[Crossref]

Phys. Rev. Lett. (5)

C. Menzel, C. Helgert, C. Rockstuhl, E.-B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Phys. Rev. Lett. 104(25), 253902 (2010).
[Crossref] [PubMed]

V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97(16), 167401 (2006).
[Crossref] [PubMed]

M. Mutlu, A. E. Akosman, A. E. Serebryannikov, and E. Ozbay, “Diodelike asymmetric transmission of linearly polarized waves using magnetoelectric coupling and electromagnetic wave tunneling,” Phys. Rev. Lett. 108(21), 213905 (2012).
[Crossref] [PubMed]

C. Pfeiffer, C. Zhang, V. Ray, L. J. Guo, and A. Grbic, “High performance bianisotropic metasurfaces: asymmetric transmission of light,” Phys. Rev. Lett. 113(2), 023902 (2014).
[Crossref] [PubMed]

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101(25), 253903 (2008).
[Crossref] [PubMed]

Plasmonics (1)

E. Battal, T. A. Yogurt, and A. K. Okyay, “Ultrahigh contrast one-way optical transmission through a subwavelength slit,” Plasmonics 8(2), 509–513 (2013).
[Crossref]

Proc. IEEE (1)

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[Crossref]

Rev. Mod. Phys. (1)

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010).
[Crossref]

Sci. Rep. (1)

C. Wang, X.-L. Zhong, and Z.-Y. Li, “Linear and passive silicon optical isolator,” Sci. Rep. 2, 674 (2012).
[PubMed]

Science (3)

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

L. Feng, M. Ayache, J. Huang, and Y.-L. Xu, “Response to comments on Nonreciprocal light propagation in a silicon photonic circuit,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

Other (2)

H. Kim, J. Park, and B. Lee, Fourier Modal Method and Its Applications in Computational Nanophotonics (CRC Press, 2012).

S. C. Rand, Lectures on Light: Nonlinear and Quantum Optics Using the Density Matrix (Oxford University Press, 2010).

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Figures (6)

Fig. 1
Fig. 1 (a) A schematic of the proposed device. A silicon waveguide (εSi = 12.15) of width w1 = 400 nm and a SiO2 waveguide ( ε SiO 2 = 2.25) of width w2 = 400 nm are aligned in parallel with an air gap of width g = 100 nm. The half-infinite areas above the waveguide II and beneath the waveguide I are set to air. (b) When mode 1 is incident from the left, the grating α converts mode 1 to the dark-mode at the grating β and the dark-mode appears at the output because the dark-mode is stationary in the grating β. (c) When mode 1 is incident from the right, mode transitions in the grating β is not prohibited. As a result, various linear combinations of modes 1, 2 and 3 can appear at the output.
Fig. 2
Fig. 2 (a) The dispersion relations of three modes considered in our configuration. The widths of two waveguide are adjusted so that Δ k 23 Δ k 12 for simultaneous coupling of modes. Normalized power flow inside the grating β when (b) Λ β = 1530 nm and (c) Λ β = 1550 nm . Electric field (Ey) distribution inside the grating β when (d) Λ β = 1530 nm and (e) Λ β = 1550 nm . Mode 2 is launched from the left and the waveguide centered at x = 0 is the waveguide II.
Fig. 3
Fig. 3 (a) Normalized power flow in the grating β when the dark-mode is launched. (b) Ey field distribution of the dark-mode in the grating β. Normalized power flow of mode 3 in grating β when there is (c) a variation of normalized input power flow of mode 3 (P3,in) and (d) a variation of relative phase of mode 3 (Δφ) with respect to the exact dark-mode condition at the entrance of grating β.
Fig. 4
Fig. 4 (a) Normalized power flow inside the grating α tuned for the transition from modes 1 to 3. (b) Normalized power flow through two cascaded gratings when a wrong value of D is chosen (D = 5.6 μm). (c) Normalized power transmission after the two cascaded gratings depending on D. The length of the grating β is 40 periods. (d) Normalized power flow inside two cascaded gratings when D = 5.25 μm.
Fig. 5
Fig. 5 (a) Normalized power flows inside the cascaded grating when the length of the grating β is set to 40 periods for the right-to-left propagation case. (b) Normalized power flows inside the grating β for the case of mode 1 incidence. Normalized power flows inside the cascaded gratings when the length of the grating β is set to 11 periods (c) for the left-to-right and (d) right-to-left propagation cases. (e) and (f) are those for the left-to-right and right-to-left propagation when the length of the grating β is set to 16 periods.
Fig. 6
Fig. 6 Ey field distributions on two cascaded gratings for mode 1 incidence when the length of the grating β is set to 11 periods (a) for left-to-right and (b) right-to-left propagation cases. (c) and (d) are those when the length of the grating is set to 16 periods for left-to-right and right-to-left propagation cases, respectively. The field distributions of input and output are shown at the upper parts of each figure.

Equations (10)

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i k 1 d a 1 d z = μ 12 a 2 e i Δ 12 z + μ 13 a 3 e i Δ 13 z , i k 2 d a 2 d z = μ 12 * a 1 e i Δ 12 z + μ 23 a 3 e i Δ 23 z , i k 3 d a 3 d z = μ 13 * a 1 e i Δ 13 z + μ 23 * a 2 e i Δ 23 z ,
i d d z [ a 1 a 2 a 3 ] = [ 0 μ 12 e i Δ z / k 1 0 μ 12 * e i Δ z / k 2 0 μ 23 e i Δ z / k 2 0 μ 23 * e i Δ z / k 3 0 ] [ a 1 a 2 a 3 ] .
i d d z [ a 1 a 2 a 3 ] = [ 0 0 μ 13 ( α ) / k 1 0 0 0 μ 13 ( α ) * / k 3 0 0 ] [ a 1 a 2 a 3 ] ,
a 1 ( z ) = a 1 ( 0 ) cos Ω 13 ( α ) z i k 3 μ 13 ( α ) k 1 μ 13 ( α ) * a 3 ( 0 ) sin Ω 13 ( α ) z , a 2 ( z ) = a 2 ( 0 ) , a 3 ( z ) = a 3 ( 0 ) cos Ω 13 ( α ) z i k 1 μ 13 ( α ) * k 3 μ 13 ( α ) a 1 ( 0 ) sin Ω 13 ( α ) z ,
[ cos Ω 13 ( α ) L α 0 i μ 13 ( α ) * k 1 / μ 13 ( α ) k 3 sin Ω 13 ( α ) L α ] = N [ μ 23 ( β ) e i 2 π ( L α + D ) / Λ β 0 μ 12 ( β ) * e i 2 π ( L α + D ) / Λ β ] ,
[ a 1 a 2 e i Δ z a 3 ] = C 1 [ μ 23 ( β ) 0 μ 12 ( β ) * ] + C 2 [ μ 12 ( β ) / k 1 Ω μ 23 ( β ) * / k 3 ] exp ( i Ω z ) + C 3 [ μ 12 ( β ) / k 1 Ω + μ 23 ( β ) * / k 3 ] exp ( i Ω + z ) , C 1 = μ 23 ( β ) * | μ 23 ( β ) | 2 + k 3 | μ 12 ( β ) | 2 / k 1 a 1 ( 0 ) μ 12 ( β ) | μ 12 ( β ) | 2 + k 1 | μ 23 ( β ) | 2 / k 3 a 3 ( 0 ) , C 2 = k 3 μ 12 ( β ) * | μ 23 ( β ) | 2 + k 3 | μ 12 ( β ) | 2 / k 1 Ω + 2 Ω a 1 ( 0 ) + 1 2 Ω a 2 ( 0 ) + k 1 μ 23 ( β ) | μ 12 ( β ) | 2 + k 1 | μ 23 ( β ) | 2 / k 3 Ω + 2 Ω a 3 ( 0 ) , C 3 = k 3 μ 12 ( β ) * | μ 23 ( β ) | 2 + k 3 | μ 12 ( β ) | 2 / k 1 Ω 2 Ω a 1 ( 0 ) 1 2 Ω a 2 ( 0 ) + k 1 μ 23 ( β ) | μ 12 ( β ) | 2 + k 1 | μ 23 ( β ) | 2 / k 3 Ω 2 Ω a 3 ( 0 ) ,
| a 1 | = cos Ω 13 ( α ) L α , | a 2 | = k 3 | μ 12 ( β ) | | μ 23 ( β ) | 2 + k 3 | μ 12 ( β ) | 2 / k 1 ( Ω 12 ( β ) 2 + Ω 23 ( β ) 2 Ω sin Ω L β ) .
[ a 1 ( z ) a 2 ( z ) a 3 ( z ) ] = [ ( i μ 12 ( β ) / k 1 Ω ) e i Δ z / 2 cos Ω z e i Δ z / 2 cos Ω z + i ( Δ / 2 Ω ) e i Δ z / 2 sin Ω z ( i μ 23 ( β ) * / k 3 Ω ) e i Δ z / 2 cos Ω z ] .
d a 1 d z = i Ω 2 a 2 e i Δ z , d a 2 d z = i Ω 2 a 1 e i Δ z ,
| a 1 ( z ) | 2 = ( 1 + ( κ Ω Δ ) 2 2 ( κ 2 1 ) Ω 2 2 Δ κ Ω ) 2 ( Ω 2 + Δ 2 ) cos ( Ω 2 + Δ 2 z ) ) | a 1 ( 0 ) | 2 .

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