Abstract

The transfer matrix method is developed to probe bianisotropic biomolecules via a Kretschmann configuration surface plasmon resonance (SPR) sensor. This method employs wave vectors and 4 × 4 transfer matrices derived by using anisotropic and magnetoelectric coupling constitutive relations. The transfer matrices relate four eigenstates and trace four transverse field components through the multilayer to account for cross-polarization coupling due to the chirality of the biomolecule layer. The validity of the method is confirmed by means of numerical results. It is shown that cross-polarized reflection waves are enhanced around the SPR angle, as the water solution and bianisotropic biomolecules to be detected are placed in contact with the graphene layer of the sensor. The effects of optical activity and bianisotropy on the SPR sensor are investigated. This work enriches the transfer matrix theory for SPR sensors to detect the chirality parameter of bianisotropic chiral material, and may lead to a better design of SPR sensors against the chirality parameter variation.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

2017 (6)

W. Luo, S. Chen, L. Chen, H. Li, P. Miao, H. Gao, Z. Hu, and M. Li, “Dual-angle technique for simultaneous measurement of refractive index and temperature based on a surface plasmon resonance sensor,” Opt. Express 25(11), 12733–12742 (2017).
[Crossref] [PubMed]

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6(9), e17039 (2017).
[Crossref] [PubMed]

L. Li, Y. Liang, J. Guang, W. Cui, X. Zhang, J. F. Masson, and W. Peng, “Dual Kretschmann and Otto configuration fiber surface plasmon resonance biosensor,” Opt. Express 25(22), 26950–26957 (2017).
[Crossref] [PubMed]

L. Luo, X. Qiu, L. Xie, X. Liu, Z. Li, Z. Zhang, and J. Du, “Precision improvement of surface plasmon resonance sensors based on weak-value amplification,” Opt. Express 25(18), 21107–21114 (2017).
[Crossref] [PubMed]

Y. Zhao, A. N. Askarpour, L. Sun, J. Shi, X. Li, and A. Alù, “Chirality detection of enantiomers using twisted optical metamaterials,” Nat. Commun. 8, 14180 (2017).
[Crossref] [PubMed]

F. Wang and B. Wei, “Propagation matrix of plane wave stratified lossy chiral incident obliquely on medium,” Wuli Xuebao 66(6), 064101 (2017).

2016 (5)

S. G. Zhou, H. L. Li, L. Y. Fu, and M. Y. Wang, “Preliminary study on active modulation of polar mesosphere summer echoes with the radio propagation in layered space dusty plasma,” Plasma Sci. Technol. 18(6), 607–610 (2016).
[Crossref]

M. Y. Wang, H. L. Li, Y. L. Dong, G. P. Li, B. J. Jiang, Q. Zhao, and J. Xu, “Propagation matrix method study on THz waves propagation in a dusty plasma sheath,” IEEE Trans. Antenn. Propag. 64(1), 286–290 (2016).
[Crossref]

K. S. Zheng, Z. M. Mu, H. Luo, and G. Wei, “Electromagnetic properties from moving dielectric in high speed with Lorentz-FDTD,” IEEE Antennas Wirel. Propag. Lett. 15, 934–937 (2016).
[Crossref]

L. M. Wu, Z. T. Ling, L. Y. Jiang, J. Guo, X. Y. Dai, Y. J. Xiang, and D. Y. Fan, “Long-range surface plasmon with graphene for enhancing the sensitivity and detection accuracy of biosensor,” IEEE Photonics J. 8(2), 4801409 (2016).
[Crossref]

Q. H. Phan, Y. L. Lo, and C. L. Huang, “Surface plasmon resonance prism coupler for enhanced circular dichroism sensing,” Opt. Express 24(12), 12812–12824 (2016).
[Crossref] [PubMed]

2015 (4)

M. Wang, H. Li, D. Gao, L. Gao, J. Xu, and C. W. Qiu, “Radiation pressure of active dispersive chiral slabs,” Opt. Express 23(13), 16546–16553 (2015).
[Crossref] [PubMed]

C. W. Qiu, W. Q. Ding, M. R. C. Mahdy, D. L. Gao, T. H. Zhang, F. C. Cheong, A. Dogariu, Z. Wang, and C. T. Lim, “Photon momentum transfer in inhomogeneous dielectric mixtures and induced tractor beams,” Light Sci. Appl. 4(4), e278 (2015).
[Crossref]

X. M. Sun, S. Xiao, H. H. Wang, and S. J. Long, “Transportation of gaussian light beam in two-layer clouds by monte carlo simulation,” Wuli Xuebao 64(18), 184204 (2015).

L. Cao and B. Wei, “Near-field coupling characteristics analysis of radome over half-space under HPEM,” IEEE T. Electromagn. C. 57(6), 1637–1644 (2015).
[Crossref]

2014 (3)

2013 (3)

J. M. Auñón and M. Nieto-Vesperinas, “Partially coherent fluctuating sources that produce the same optical force as a laser beam,” Opt. Lett. 38(15), 2869–2872 (2013).
[Crossref] [PubMed]

B. M. Maoz, Y. Chaikin, A. B. Tesler, O. Bar Elli, Z. Fan, A. O. Govorov, and G. Markovich, “Amplification of chiroptical activity of chiral biomolecules by surface plasmons,” Nano Lett. 13(3), 1203–1209 (2013).
[Crossref] [PubMed]

D. Zarifi, M. Soleimani, and A. Abdolali, “Electromagnetic characterization of uniaxial chiral composites using state transition matrix method,” IEEE Trans. Antenn. Propag. 61(11), 5658–5665 (2013).
[Crossref]

2012 (1)

P. K. Maharana and R. Jha, “Chalcogenide prism and graphene multilayer based surface plasmon resonance affinity biosensor for high performance,” Sens. Actuators B Chem. 169, 161–166 (2012).
[Crossref]

2011 (1)

R. Verma, B. D. Gupta, and R. Jha, “Sensitivity enhancement of a surface plasmon resonance based biomolecules sensor using graphene and silicon layers,” Sens. Actuators B Chem. 160(1), 623–631 (2011).
[Crossref]

2010 (1)

C. X. Lei and Z. S. Wu, “A study of radiative properties of randomly distributed soot aggregates,” Wuli Xuebao 59(8), 5692–5699 (2010).

2009 (2)

2008 (2)

H. Chen, H. Cheng, J. Lee, J. H. Kim, M. H. Hyun, and K. Koh, “Surface plasmon resonance spectroscopic chiral discrimination using self-assembled leucine derivative monolayer,” Talanta 76(1), 49–53 (2008).
[Crossref] [PubMed]

J. Ning and E. L. Tan, “Hybrid matrix method for stable analysis of electromagnetic waves in stratified bianisotropic media,” IEEE Microw. Wirel. Co. 18(10), 653–655 (2008).
[Crossref]

2007 (2)

2003 (1)

S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Wave 17(5), 695–706 (2003).
[Crossref]

1998 (2)

D. Y. Khaliullin and S. A. Tretyakov, ““Reflection and transmission coefficients for thin bianisotropic layers,” IEE P. Microw. Anten. P. 145(2), 163–168 (1998).

W. Y. Yin, G. H. Nan, and I. Wolff, “The combined effects of chiral operation in multilayered bianisotropic substrates,” Prog. Electromagnetics Res. 20, 153–178 (1998).
[Crossref]

1996 (2)

S. L. He, “A uniform approximation to the scattering and propagation problem for a stratified Bi-anisotropic slab,” Int. J. of Infrared Milli. 17(2), 415–431 (1996).
[Crossref]

J. Lekner, “Optical properties of isotropic chiral media,” Pure Appl. Opt. 5(4), 417–443 (1996).
[Crossref]

1994 (1)

T. Kusumi, H. Takahashi, P. Xu, T. Fukushima, Y. Asakawa, T. Hashimoto, Y. Kan, and Y. Inouye, “New chiral anisotropic reagents, NMR tools to elucidate the absolute configurations of long-chain organic compounds,” Tetrahedron Lett. 35(25), 4397–4400 (1994).
[Crossref]

1992 (1)

S. A. Tretyakov and M. I. Oksanen, “Electromagnetic waves in layered general biisotropic structures,” J. Electromagnet. Wave. 6(10), 1393–1411 (1992).

1983 (1)

Abdolali, A.

D. Zarifi, M. Soleimani, and A. Abdolali, “Electromagnetic characterization of uniaxial chiral composites using state transition matrix method,” IEEE Trans. Antenn. Propag. 61(11), 5658–5665 (2013).
[Crossref]

Alexander, R. W.

Alù, A.

Y. Zhao, A. N. Askarpour, L. Sun, J. Shi, X. Li, and A. Alù, “Chirality detection of enantiomers using twisted optical metamaterials,” Nat. Commun. 8, 14180 (2017).
[Crossref] [PubMed]

Asakawa, Y.

T. Kusumi, H. Takahashi, P. Xu, T. Fukushima, Y. Asakawa, T. Hashimoto, Y. Kan, and Y. Inouye, “New chiral anisotropic reagents, NMR tools to elucidate the absolute configurations of long-chain organic compounds,” Tetrahedron Lett. 35(25), 4397–4400 (1994).
[Crossref]

Askarpour, A. N.

Y. Zhao, A. N. Askarpour, L. Sun, J. Shi, X. Li, and A. Alù, “Chirality detection of enantiomers using twisted optical metamaterials,” Nat. Commun. 8, 14180 (2017).
[Crossref] [PubMed]

Auñón, J. M.

Bai, B.

Bar Elli, O.

B. M. Maoz, Y. Chaikin, A. B. Tesler, O. Bar Elli, Z. Fan, A. O. Govorov, and G. Markovich, “Amplification of chiroptical activity of chiral biomolecules by surface plasmons,” Nano Lett. 13(3), 1203–1209 (2013).
[Crossref] [PubMed]

Bell, R. J.

Bell, R. R.

Bell, S. E.

Borini, S.

M. Bruna and S. Borini, “Optical constants of graphene layers in the visible range,” Appl. Phys. Lett. 94(3), 031901 (2009).
[Crossref]

Bruna, M.

M. Bruna and S. Borini, “Optical constants of graphene layers in the visible range,” Appl. Phys. Lett. 94(3), 031901 (2009).
[Crossref]

Cao, L.

L. Cao and B. Wei, “Near-field coupling characteristics analysis of radome over half-space under HPEM,” IEEE T. Electromagn. C. 57(6), 1637–1644 (2015).
[Crossref]

Chaikin, Y.

B. M. Maoz, Y. Chaikin, A. B. Tesler, O. Bar Elli, Z. Fan, A. O. Govorov, and G. Markovich, “Amplification of chiroptical activity of chiral biomolecules by surface plasmons,” Nano Lett. 13(3), 1203–1209 (2013).
[Crossref] [PubMed]

Chen, H.

H. Chen, H. Cheng, J. Lee, J. H. Kim, M. H. Hyun, and K. Koh, “Surface plasmon resonance spectroscopic chiral discrimination using self-assembled leucine derivative monolayer,” Talanta 76(1), 49–53 (2008).
[Crossref] [PubMed]

Chen, L.

Chen, S.

Cheng, H.

H. Chen, H. Cheng, J. Lee, J. H. Kim, M. H. Hyun, and K. Koh, “Surface plasmon resonance spectroscopic chiral discrimination using self-assembled leucine derivative monolayer,” Talanta 76(1), 49–53 (2008).
[Crossref] [PubMed]

Cheong, F. C.

C. W. Qiu, W. Q. Ding, M. R. C. Mahdy, D. L. Gao, T. H. Zhang, F. C. Cheong, A. Dogariu, Z. Wang, and C. T. Lim, “Photon momentum transfer in inhomogeneous dielectric mixtures and induced tractor beams,” Light Sci. Appl. 4(4), e278 (2015).
[Crossref]

Chiou, A.

Cui, W.

Dai, X. Y.

L. M. Wu, Z. T. Ling, L. Y. Jiang, J. Guo, X. Y. Dai, Y. J. Xiang, and D. Y. Fan, “Long-range surface plasmon with graphene for enhancing the sensitivity and detection accuracy of biosensor,” IEEE Photonics J. 8(2), 4801409 (2016).
[Crossref]

Ding, W.

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6(9), e17039 (2017).
[Crossref] [PubMed]

Ding, W. Q.

C. W. Qiu, W. Q. Ding, M. R. C. Mahdy, D. L. Gao, T. H. Zhang, F. C. Cheong, A. Dogariu, Z. Wang, and C. T. Lim, “Photon momentum transfer in inhomogeneous dielectric mixtures and induced tractor beams,” Light Sci. Appl. 4(4), e278 (2015).
[Crossref]

Ding, X.

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6(9), e17039 (2017).
[Crossref] [PubMed]

Dogariu, A.

C. W. Qiu, W. Q. Ding, M. R. C. Mahdy, D. L. Gao, T. H. Zhang, F. C. Cheong, A. Dogariu, Z. Wang, and C. T. Lim, “Photon momentum transfer in inhomogeneous dielectric mixtures and induced tractor beams,” Light Sci. Appl. 4(4), e278 (2015).
[Crossref]

Dong, Y. L.

M. Y. Wang, H. L. Li, Y. L. Dong, G. P. Li, B. J. Jiang, Q. Zhao, and J. Xu, “Propagation matrix method study on THz waves propagation in a dusty plasma sheath,” IEEE Trans. Antenn. Propag. 64(1), 286–290 (2016).
[Crossref]

Du, J.

Fan, D. Y.

L. M. Wu, Z. T. Ling, L. Y. Jiang, J. Guo, X. Y. Dai, Y. J. Xiang, and D. Y. Fan, “Long-range surface plasmon with graphene for enhancing the sensitivity and detection accuracy of biosensor,” IEEE Photonics J. 8(2), 4801409 (2016).
[Crossref]

Fan, Z.

B. M. Maoz, Y. Chaikin, A. B. Tesler, O. Bar Elli, Z. Fan, A. O. Govorov, and G. Markovich, “Amplification of chiroptical activity of chiral biomolecules by surface plasmons,” Nano Lett. 13(3), 1203–1209 (2013).
[Crossref] [PubMed]

Fu, L. Y.

S. G. Zhou, H. L. Li, L. Y. Fu, and M. Y. Wang, “Preliminary study on active modulation of polar mesosphere summer echoes with the radio propagation in layered space dusty plasma,” Plasma Sci. Technol. 18(6), 607–610 (2016).
[Crossref]

Fukushima, T.

T. Kusumi, H. Takahashi, P. Xu, T. Fukushima, Y. Asakawa, T. Hashimoto, Y. Kan, and Y. Inouye, “New chiral anisotropic reagents, NMR tools to elucidate the absolute configurations of long-chain organic compounds,” Tetrahedron Lett. 35(25), 4397–4400 (1994).
[Crossref]

Gao, D.

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6(9), e17039 (2017).
[Crossref] [PubMed]

M. Wang, H. Li, D. Gao, L. Gao, J. Xu, and C. W. Qiu, “Radiation pressure of active dispersive chiral slabs,” Opt. Express 23(13), 16546–16553 (2015).
[Crossref] [PubMed]

Gao, D. L.

C. W. Qiu, W. Q. Ding, M. R. C. Mahdy, D. L. Gao, T. H. Zhang, F. C. Cheong, A. Dogariu, Z. Wang, and C. T. Lim, “Photon momentum transfer in inhomogeneous dielectric mixtures and induced tractor beams,” Light Sci. Appl. 4(4), e278 (2015).
[Crossref]

Gao, H.

Gao, L.

Govorov, A. O.

B. M. Maoz, Y. Chaikin, A. B. Tesler, O. Bar Elli, Z. Fan, A. O. Govorov, and G. Markovich, “Amplification of chiroptical activity of chiral biomolecules by surface plasmons,” Nano Lett. 13(3), 1203–1209 (2013).
[Crossref] [PubMed]

Guang, J.

Guo, J.

H. Zhou, J. Guo, K. Xu, Z. Li, J. Tang, and S. Man, “Bistable enhanced total reflection in Kretschmann configuration containing a saturable gain medium,” Opt. Express 26(5), 5253–5264 (2018).
[Crossref] [PubMed]

L. M. Wu, Z. T. Ling, L. Y. Jiang, J. Guo, X. Y. Dai, Y. J. Xiang, and D. Y. Fan, “Long-range surface plasmon with graphene for enhancing the sensitivity and detection accuracy of biosensor,” IEEE Photonics J. 8(2), 4801409 (2016).
[Crossref]

Gupta, B. D.

R. Verma, B. D. Gupta, and R. Jha, “Sensitivity enhancement of a surface plasmon resonance based biomolecules sensor using graphene and silicon layers,” Sens. Actuators B Chem. 160(1), 623–631 (2011).
[Crossref]

Hashimoto, T.

T. Kusumi, H. Takahashi, P. Xu, T. Fukushima, Y. Asakawa, T. Hashimoto, Y. Kan, and Y. Inouye, “New chiral anisotropic reagents, NMR tools to elucidate the absolute configurations of long-chain organic compounds,” Tetrahedron Lett. 35(25), 4397–4400 (1994).
[Crossref]

He, S. L.

S. L. He, “A uniform approximation to the scattering and propagation problem for a stratified Bi-anisotropic slab,” Int. J. of Infrared Milli. 17(2), 415–431 (1996).
[Crossref]

Hu, X.

Hu, Z.

Huang, C. L.

Hyun, M. H.

H. Chen, H. Cheng, J. Lee, J. H. Kim, M. H. Hyun, and K. Koh, “Surface plasmon resonance spectroscopic chiral discrimination using self-assembled leucine derivative monolayer,” Talanta 76(1), 49–53 (2008).
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T. Kusumi, H. Takahashi, P. Xu, T. Fukushima, Y. Asakawa, T. Hashimoto, Y. Kan, and Y. Inouye, “New chiral anisotropic reagents, NMR tools to elucidate the absolute configurations of long-chain organic compounds,” Tetrahedron Lett. 35(25), 4397–4400 (1994).
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P. K. Maharana and R. Jha, “Chalcogenide prism and graphene multilayer based surface plasmon resonance affinity biosensor for high performance,” Sens. Actuators B Chem. 169, 161–166 (2012).
[Crossref]

R. Verma, B. D. Gupta, and R. Jha, “Sensitivity enhancement of a surface plasmon resonance based biomolecules sensor using graphene and silicon layers,” Sens. Actuators B Chem. 160(1), 623–631 (2011).
[Crossref]

Jiang, B. J.

M. Y. Wang, H. L. Li, Y. L. Dong, G. P. Li, B. J. Jiang, Q. Zhao, and J. Xu, “Propagation matrix method study on THz waves propagation in a dusty plasma sheath,” IEEE Trans. Antenn. Propag. 64(1), 286–290 (2016).
[Crossref]

Jiang, L. Y.

L. M. Wu, Z. T. Ling, L. Y. Jiang, J. Guo, X. Y. Dai, Y. J. Xiang, and D. Y. Fan, “Long-range surface plasmon with graphene for enhancing the sensitivity and detection accuracy of biosensor,” IEEE Photonics J. 8(2), 4801409 (2016).
[Crossref]

Kan, Y.

T. Kusumi, H. Takahashi, P. Xu, T. Fukushima, Y. Asakawa, T. Hashimoto, Y. Kan, and Y. Inouye, “New chiral anisotropic reagents, NMR tools to elucidate the absolute configurations of long-chain organic compounds,” Tetrahedron Lett. 35(25), 4397–4400 (1994).
[Crossref]

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D. Y. Khaliullin and S. A. Tretyakov, ““Reflection and transmission coefficients for thin bianisotropic layers,” IEE P. Microw. Anten. P. 145(2), 163–168 (1998).

Kim, J. H.

H. Chen, H. Cheng, J. Lee, J. H. Kim, M. H. Hyun, and K. Koh, “Surface plasmon resonance spectroscopic chiral discrimination using self-assembled leucine derivative monolayer,” Talanta 76(1), 49–53 (2008).
[Crossref] [PubMed]

Koh, K.

H. Chen, H. Cheng, J. Lee, J. H. Kim, M. H. Hyun, and K. Koh, “Surface plasmon resonance spectroscopic chiral discrimination using self-assembled leucine derivative monolayer,” Talanta 76(1), 49–53 (2008).
[Crossref] [PubMed]

Konishi, K.

Kusumi, T.

T. Kusumi, H. Takahashi, P. Xu, T. Fukushima, Y. Asakawa, T. Hashimoto, Y. Kan, and Y. Inouye, “New chiral anisotropic reagents, NMR tools to elucidate the absolute configurations of long-chain organic compounds,” Tetrahedron Lett. 35(25), 4397–4400 (1994).
[Crossref]

Kuwata-Gonokami, M.

Lee, J.

H. Chen, H. Cheng, J. Lee, J. H. Kim, M. H. Hyun, and K. Koh, “Surface plasmon resonance spectroscopic chiral discrimination using self-assembled leucine derivative monolayer,” Talanta 76(1), 49–53 (2008).
[Crossref] [PubMed]

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C. X. Lei and Z. S. Wu, “A study of radiative properties of randomly distributed soot aggregates,” Wuli Xuebao 59(8), 5692–5699 (2010).

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J. Lekner, “Optical properties of isotropic chiral media,” Pure Appl. Opt. 5(4), 417–443 (1996).
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M. Y. Wang, H. L. Li, Y. L. Dong, G. P. Li, B. J. Jiang, Q. Zhao, and J. Xu, “Propagation matrix method study on THz waves propagation in a dusty plasma sheath,” IEEE Trans. Antenn. Propag. 64(1), 286–290 (2016).
[Crossref]

Li, H.

Li, H. L.

M. Y. Wang, H. L. Li, Y. L. Dong, G. P. Li, B. J. Jiang, Q. Zhao, and J. Xu, “Propagation matrix method study on THz waves propagation in a dusty plasma sheath,” IEEE Trans. Antenn. Propag. 64(1), 286–290 (2016).
[Crossref]

S. G. Zhou, H. L. Li, L. Y. Fu, and M. Y. Wang, “Preliminary study on active modulation of polar mesosphere summer echoes with the radio propagation in layered space dusty plasma,” Plasma Sci. Technol. 18(6), 607–610 (2016).
[Crossref]

Li, L.

Li, M.

Li, X.

Y. Zhao, A. N. Askarpour, L. Sun, J. Shi, X. Li, and A. Alù, “Chirality detection of enantiomers using twisted optical metamaterials,” Nat. Commun. 8, 14180 (2017).
[Crossref] [PubMed]

Li, Z.

Liang, Y.

Lim, C.

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6(9), e17039 (2017).
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Lim, C. T.

C. W. Qiu, W. Q. Ding, M. R. C. Mahdy, D. L. Gao, T. H. Zhang, F. C. Cheong, A. Dogariu, Z. Wang, and C. T. Lim, “Photon momentum transfer in inhomogeneous dielectric mixtures and induced tractor beams,” Light Sci. Appl. 4(4), e278 (2015).
[Crossref]

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L. M. Wu, Z. T. Ling, L. Y. Jiang, J. Guo, X. Y. Dai, Y. J. Xiang, and D. Y. Fan, “Long-range surface plasmon with graphene for enhancing the sensitivity and detection accuracy of biosensor,” IEEE Photonics J. 8(2), 4801409 (2016).
[Crossref]

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Liu, Q.

Liu, X.

Lo, Y. L.

Long, L. L.

Long, S. J.

X. M. Sun, S. Xiao, H. H. Wang, and S. J. Long, “Transportation of gaussian light beam in two-layer clouds by monte carlo simulation,” Wuli Xuebao 64(18), 184204 (2015).

Luo, H.

K. S. Zheng, Z. M. Mu, H. Luo, and G. Wei, “Electromagnetic properties from moving dielectric in high speed with Lorentz-FDTD,” IEEE Antennas Wirel. Propag. Lett. 15, 934–937 (2016).
[Crossref]

Luo, L.

Luo, W.

Maharana, P. K.

P. K. Maharana and R. Jha, “Chalcogenide prism and graphene multilayer based surface plasmon resonance affinity biosensor for high performance,” Sens. Actuators B Chem. 169, 161–166 (2012).
[Crossref]

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C. W. Qiu, W. Q. Ding, M. R. C. Mahdy, D. L. Gao, T. H. Zhang, F. C. Cheong, A. Dogariu, Z. Wang, and C. T. Lim, “Photon momentum transfer in inhomogeneous dielectric mixtures and induced tractor beams,” Light Sci. Appl. 4(4), e278 (2015).
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Maoz, B. M.

B. M. Maoz, Y. Chaikin, A. B. Tesler, O. Bar Elli, Z. Fan, A. O. Govorov, and G. Markovich, “Amplification of chiroptical activity of chiral biomolecules by surface plasmons,” Nano Lett. 13(3), 1203–1209 (2013).
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B. M. Maoz, Y. Chaikin, A. B. Tesler, O. Bar Elli, Z. Fan, A. O. Govorov, and G. Markovich, “Amplification of chiroptical activity of chiral biomolecules by surface plasmons,” Nano Lett. 13(3), 1203–1209 (2013).
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S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Wave 17(5), 695–706 (2003).
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Mi, G.

Miao, P.

Mu, Z. M.

K. S. Zheng, Z. M. Mu, H. Luo, and G. Wei, “Electromagnetic properties from moving dielectric in high speed with Lorentz-FDTD,” IEEE Antennas Wirel. Propag. Lett. 15, 934–937 (2016).
[Crossref]

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W. Y. Yin, G. H. Nan, and I. Wolff, “The combined effects of chiral operation in multilayered bianisotropic substrates,” Prog. Electromagnetics Res. 20, 153–178 (1998).
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Nee, T. W.

Nefedov, I.

S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Wave 17(5), 695–706 (2003).
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D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6(9), e17039 (2017).
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J. Ning and E. L. Tan, “Hybrid matrix method for stable analysis of electromagnetic waves in stratified bianisotropic media,” IEEE Microw. Wirel. Co. 18(10), 653–655 (2008).
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A. Novitsky and C. W. Qiu, “Pulling extremely anisotropic lossy particles using light without intensity gradient,” Phys. Rev. A 90(5), 053815 (2014).
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S. A. Tretyakov and M. I. Oksanen, “Electromagnetic waves in layered general biisotropic structures,” J. Electromagnet. Wave. 6(10), 1393–1411 (1992).

Ordal, M. A.

Peng, W.

Phan, Q. H.

Qiu, C. W.

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6(9), e17039 (2017).
[Crossref] [PubMed]

M. Wang, H. Li, D. Gao, L. Gao, J. Xu, and C. W. Qiu, “Radiation pressure of active dispersive chiral slabs,” Opt. Express 23(13), 16546–16553 (2015).
[Crossref] [PubMed]

C. W. Qiu, W. Q. Ding, M. R. C. Mahdy, D. L. Gao, T. H. Zhang, F. C. Cheong, A. Dogariu, Z. Wang, and C. T. Lim, “Photon momentum transfer in inhomogeneous dielectric mixtures and induced tractor beams,” Light Sci. Appl. 4(4), e278 (2015).
[Crossref]

A. Novitsky and C. W. Qiu, “Pulling extremely anisotropic lossy particles using light without intensity gradient,” Phys. Rev. A 90(5), 053815 (2014).
[Crossref]

Qiu, X.

Rahman, M.

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6(9), e17039 (2017).
[Crossref] [PubMed]

Shi, J.

Y. Zhao, A. N. Askarpour, L. Sun, J. Shi, X. Li, and A. Alù, “Chirality detection of enantiomers using twisted optical metamaterials,” Nat. Commun. 8, 14180 (2017).
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A. Sihvola, “Metamaterials in electromagnetics,” Metamaterials (Amst.) 1(1), 2–11 (2007).
[Crossref]

S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Wave 17(5), 695–706 (2003).
[Crossref]

Simovski, C.

S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Wave 17(5), 695–706 (2003).
[Crossref]

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D. Zarifi, M. Soleimani, and A. Abdolali, “Electromagnetic characterization of uniaxial chiral composites using state transition matrix method,” IEEE Trans. Antenn. Propag. 61(11), 5658–5665 (2013).
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Sun, L.

Y. Zhao, A. N. Askarpour, L. Sun, J. Shi, X. Li, and A. Alù, “Chirality detection of enantiomers using twisted optical metamaterials,” Nat. Commun. 8, 14180 (2017).
[Crossref] [PubMed]

Sun, X. M.

X. M. Sun, S. Xiao, H. H. Wang, and S. J. Long, “Transportation of gaussian light beam in two-layer clouds by monte carlo simulation,” Wuli Xuebao 64(18), 184204 (2015).

Svirko, Y.

Takahashi, H.

T. Kusumi, H. Takahashi, P. Xu, T. Fukushima, Y. Asakawa, T. Hashimoto, Y. Kan, and Y. Inouye, “New chiral anisotropic reagents, NMR tools to elucidate the absolute configurations of long-chain organic compounds,” Tetrahedron Lett. 35(25), 4397–4400 (1994).
[Crossref]

Tan, E. L.

J. Ning and E. L. Tan, “Hybrid matrix method for stable analysis of electromagnetic waves in stratified bianisotropic media,” IEEE Microw. Wirel. Co. 18(10), 653–655 (2008).
[Crossref]

Tang, J.

Tesler, A. B.

B. M. Maoz, Y. Chaikin, A. B. Tesler, O. Bar Elli, Z. Fan, A. O. Govorov, and G. Markovich, “Amplification of chiroptical activity of chiral biomolecules by surface plasmons,” Nano Lett. 13(3), 1203–1209 (2013).
[Crossref] [PubMed]

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S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Wave 17(5), 695–706 (2003).
[Crossref]

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D. Y. Khaliullin and S. A. Tretyakov, ““Reflection and transmission coefficients for thin bianisotropic layers,” IEE P. Microw. Anten. P. 145(2), 163–168 (1998).

S. A. Tretyakov and M. I. Oksanen, “Electromagnetic waves in layered general biisotropic structures,” J. Electromagnet. Wave. 6(10), 1393–1411 (1992).

Van, V.

Verma, R.

R. Verma, B. D. Gupta, and R. Jha, “Sensitivity enhancement of a surface plasmon resonance based biomolecules sensor using graphene and silicon layers,” Sens. Actuators B Chem. 160(1), 623–631 (2011).
[Crossref]

Wang, F.

F. Wang and B. Wei, “Propagation matrix of plane wave stratified lossy chiral incident obliquely on medium,” Wuli Xuebao 66(6), 064101 (2017).

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X. M. Sun, S. Xiao, H. H. Wang, and S. J. Long, “Transportation of gaussian light beam in two-layer clouds by monte carlo simulation,” Wuli Xuebao 64(18), 184204 (2015).

Wang, M.

Wang, M. Y.

S. G. Zhou, H. L. Li, L. Y. Fu, and M. Y. Wang, “Preliminary study on active modulation of polar mesosphere summer echoes with the radio propagation in layered space dusty plasma,” Plasma Sci. Technol. 18(6), 607–610 (2016).
[Crossref]

M. Y. Wang, H. L. Li, Y. L. Dong, G. P. Li, B. J. Jiang, Q. Zhao, and J. Xu, “Propagation matrix method study on THz waves propagation in a dusty plasma sheath,” IEEE Trans. Antenn. Propag. 64(1), 286–290 (2016).
[Crossref]

Wang, Z.

C. W. Qiu, W. Q. Ding, M. R. C. Mahdy, D. L. Gao, T. H. Zhang, F. C. Cheong, A. Dogariu, Z. Wang, and C. T. Lim, “Photon momentum transfer in inhomogeneous dielectric mixtures and induced tractor beams,” Light Sci. Appl. 4(4), e278 (2015).
[Crossref]

Ward, C. A.

Wei, B.

F. Wang and B. Wei, “Propagation matrix of plane wave stratified lossy chiral incident obliquely on medium,” Wuli Xuebao 66(6), 064101 (2017).

L. Cao and B. Wei, “Near-field coupling characteristics analysis of radome over half-space under HPEM,” IEEE T. Electromagn. C. 57(6), 1637–1644 (2015).
[Crossref]

Wei, G.

K. S. Zheng, Z. M. Mu, H. Luo, and G. Wei, “Electromagnetic properties from moving dielectric in high speed with Lorentz-FDTD,” IEEE Antennas Wirel. Propag. Lett. 15, 934–937 (2016).
[Crossref]

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W. Y. Yin, G. H. Nan, and I. Wolff, “The combined effects of chiral operation in multilayered bianisotropic substrates,” Prog. Electromagnetics Res. 20, 153–178 (1998).
[Crossref]

Wu, L. M.

L. M. Wu, Z. T. Ling, L. Y. Jiang, J. Guo, X. Y. Dai, Y. J. Xiang, and D. Y. Fan, “Long-range surface plasmon with graphene for enhancing the sensitivity and detection accuracy of biosensor,” IEEE Photonics J. 8(2), 4801409 (2016).
[Crossref]

Wu, Z. S.

C. X. Lei and Z. S. Wu, “A study of radiative properties of randomly distributed soot aggregates,” Wuli Xuebao 59(8), 5692–5699 (2010).

Xiang, Y. J.

L. M. Wu, Z. T. Ling, L. Y. Jiang, J. Guo, X. Y. Dai, Y. J. Xiang, and D. Y. Fan, “Long-range surface plasmon with graphene for enhancing the sensitivity and detection accuracy of biosensor,” IEEE Photonics J. 8(2), 4801409 (2016).
[Crossref]

Xiao, S.

X. M. Sun, S. Xiao, H. H. Wang, and S. J. Long, “Transportation of gaussian light beam in two-layer clouds by monte carlo simulation,” Wuli Xuebao 64(18), 184204 (2015).

Xie, L.

Xu, J.

M. Y. Wang, H. L. Li, Y. L. Dong, G. P. Li, B. J. Jiang, Q. Zhao, and J. Xu, “Propagation matrix method study on THz waves propagation in a dusty plasma sheath,” IEEE Trans. Antenn. Propag. 64(1), 286–290 (2016).
[Crossref]

M. Wang, H. Li, D. Gao, L. Gao, J. Xu, and C. W. Qiu, “Radiation pressure of active dispersive chiral slabs,” Opt. Express 23(13), 16546–16553 (2015).
[Crossref] [PubMed]

Xu, K.

Xu, P.

T. Kusumi, H. Takahashi, P. Xu, T. Fukushima, Y. Asakawa, T. Hashimoto, Y. Kan, and Y. Inouye, “New chiral anisotropic reagents, NMR tools to elucidate the absolute configurations of long-chain organic compounds,” Tetrahedron Lett. 35(25), 4397–4400 (1994).
[Crossref]

Yang, D. M.

Yin, W. Y.

W. Y. Yin, G. H. Nan, and I. Wolff, “The combined effects of chiral operation in multilayered bianisotropic substrates,” Prog. Electromagnetics Res. 20, 153–178 (1998).
[Crossref]

Zarifi, D.

D. Zarifi, M. Soleimani, and A. Abdolali, “Electromagnetic characterization of uniaxial chiral composites using state transition matrix method,” IEEE Trans. Antenn. Propag. 61(11), 5658–5665 (2013).
[Crossref]

Zhang, T.

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6(9), e17039 (2017).
[Crossref] [PubMed]

Zhang, T. H.

C. W. Qiu, W. Q. Ding, M. R. C. Mahdy, D. L. Gao, T. H. Zhang, F. C. Cheong, A. Dogariu, Z. Wang, and C. T. Lim, “Photon momentum transfer in inhomogeneous dielectric mixtures and induced tractor beams,” Light Sci. Appl. 4(4), e278 (2015).
[Crossref]

Zhang, X.

Zhang, Z.

Zhao, Q.

M. Y. Wang, H. L. Li, Y. L. Dong, G. P. Li, B. J. Jiang, Q. Zhao, and J. Xu, “Propagation matrix method study on THz waves propagation in a dusty plasma sheath,” IEEE Trans. Antenn. Propag. 64(1), 286–290 (2016).
[Crossref]

Zhao, Y.

Y. Zhao, A. N. Askarpour, L. Sun, J. Shi, X. Li, and A. Alù, “Chirality detection of enantiomers using twisted optical metamaterials,” Nat. Commun. 8, 14180 (2017).
[Crossref] [PubMed]

Zheng, K. S.

K. S. Zheng, Z. M. Mu, H. Luo, and G. Wei, “Electromagnetic properties from moving dielectric in high speed with Lorentz-FDTD,” IEEE Antennas Wirel. Propag. Lett. 15, 934–937 (2016).
[Crossref]

Zhou, H.

Zhou, S. G.

S. G. Zhou, H. L. Li, L. Y. Fu, and M. Y. Wang, “Preliminary study on active modulation of polar mesosphere summer echoes with the radio propagation in layered space dusty plasma,” Plasma Sci. Technol. 18(6), 607–610 (2016).
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M. Bruna and S. Borini, “Optical constants of graphene layers in the visible range,” Appl. Phys. Lett. 94(3), 031901 (2009).
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D. Y. Khaliullin and S. A. Tretyakov, ““Reflection and transmission coefficients for thin bianisotropic layers,” IEE P. Microw. Anten. P. 145(2), 163–168 (1998).

IEEE Antennas Wirel. Propag. Lett. (1)

K. S. Zheng, Z. M. Mu, H. Luo, and G. Wei, “Electromagnetic properties from moving dielectric in high speed with Lorentz-FDTD,” IEEE Antennas Wirel. Propag. Lett. 15, 934–937 (2016).
[Crossref]

IEEE Microw. Wirel. Co. (1)

J. Ning and E. L. Tan, “Hybrid matrix method for stable analysis of electromagnetic waves in stratified bianisotropic media,” IEEE Microw. Wirel. Co. 18(10), 653–655 (2008).
[Crossref]

IEEE Photonics J. (1)

L. M. Wu, Z. T. Ling, L. Y. Jiang, J. Guo, X. Y. Dai, Y. J. Xiang, and D. Y. Fan, “Long-range surface plasmon with graphene for enhancing the sensitivity and detection accuracy of biosensor,” IEEE Photonics J. 8(2), 4801409 (2016).
[Crossref]

IEEE T. Electromagn. C. (1)

L. Cao and B. Wei, “Near-field coupling characteristics analysis of radome over half-space under HPEM,” IEEE T. Electromagn. C. 57(6), 1637–1644 (2015).
[Crossref]

IEEE Trans. Antenn. Propag. (2)

M. Y. Wang, H. L. Li, Y. L. Dong, G. P. Li, B. J. Jiang, Q. Zhao, and J. Xu, “Propagation matrix method study on THz waves propagation in a dusty plasma sheath,” IEEE Trans. Antenn. Propag. 64(1), 286–290 (2016).
[Crossref]

D. Zarifi, M. Soleimani, and A. Abdolali, “Electromagnetic characterization of uniaxial chiral composites using state transition matrix method,” IEEE Trans. Antenn. Propag. 61(11), 5658–5665 (2013).
[Crossref]

Int. J. of Infrared Milli. (1)

S. L. He, “A uniform approximation to the scattering and propagation problem for a stratified Bi-anisotropic slab,” Int. J. of Infrared Milli. 17(2), 415–431 (1996).
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J. Electromagn. Wave (1)

S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Wave 17(5), 695–706 (2003).
[Crossref]

J. Electromagnet. Wave. (1)

S. A. Tretyakov and M. I. Oksanen, “Electromagnetic waves in layered general biisotropic structures,” J. Electromagnet. Wave. 6(10), 1393–1411 (1992).

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

Light Sci. Appl. (2)

D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light Sci. Appl. 6(9), e17039 (2017).
[Crossref] [PubMed]

C. W. Qiu, W. Q. Ding, M. R. C. Mahdy, D. L. Gao, T. H. Zhang, F. C. Cheong, A. Dogariu, Z. Wang, and C. T. Lim, “Photon momentum transfer in inhomogeneous dielectric mixtures and induced tractor beams,” Light Sci. Appl. 4(4), e278 (2015).
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Metamaterials (Amst.) (1)

A. Sihvola, “Metamaterials in electromagnetics,” Metamaterials (Amst.) 1(1), 2–11 (2007).
[Crossref]

Nano Lett. (1)

B. M. Maoz, Y. Chaikin, A. B. Tesler, O. Bar Elli, Z. Fan, A. O. Govorov, and G. Markovich, “Amplification of chiroptical activity of chiral biomolecules by surface plasmons,” Nano Lett. 13(3), 1203–1209 (2013).
[Crossref] [PubMed]

Nat. Commun. (1)

Y. Zhao, A. N. Askarpour, L. Sun, J. Shi, X. Li, and A. Alù, “Chirality detection of enantiomers using twisted optical metamaterials,” Nat. Commun. 8, 14180 (2017).
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Opt. Express (8)

K. Konishi, T. Sugimoto, B. Bai, Y. Svirko, and M. Kuwata-Gonokami, “Effect of surface plasmon resonance on the optical activity of chiral metal nanogratings,” Opt. Express 15(15), 9575–9583 (2007).
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Figures (8)

Fig. 1
Fig. 1 Schematic diagram of a Kretschmann configuration surface plasmon resonance sensor.
Fig. 2
Fig. 2 Structure of a planar stratified bianisotropic media.
Fig. 3
Fig. 3 Cross-polarized transmission coefficients of a chiral slab versus the incident angle.
Fig. 4
Fig. 4 Total electric field distributions of the SPR based biosensor along the z direction perpendicular to the 2S2G prism base for achiral and bianisotropic biomolecules.
Fig. 5
Fig. 5 Co- and cross-polarized reflectivity of the surface plasmon resonance based biosensor versus the angle of incidence for various biomolecules with different chirality parameters. (a) co-polarized, (b) cross-polarized.
Fig. 6
Fig. 6 Reflectivity as a function of the angle for the surface plasmon resonance based biosensor illuminated by different polarized incident waves.
Fig. 7
Fig. 7 Resonance angles of the SPR sensor versus real and imaginary parts of chirality parameter for various refractive indices. (a) versus Re(γy4), (b) versus Im(γy4).
Fig. 8
Fig. 8 Resonance angle and minimum reflectivity of the SPR sensor versus the thickness of bianisotropic biomolecules for different wavelengths. (a) resonance angle, (b) minimum reflectivity.

Equations (24)

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n 2S2G ( λ ) = 2.24047 + 2.693 × 10 2 / λ 2 + 8.08 × 10 3 / λ 4 ,
n G = 3.0 + i C 1 λ / 3 ,
n m = 1 λ 2 λ c / [ λ p 2 ( λ c + i λ ) ] ,
D = ε ¯ ¯ E + i γ ¯ ¯ H , B = μ ¯ ¯ H i γ ¯ ¯ E .
ε ¯ ¯ = ε 0 [ ε x 0 0 0 ε y 0 0 0 ε z ] , μ ¯ ¯ = μ 0 [ μ x 0 0 0 μ y 0 0 0 μ z ] , γ ¯ ¯ = ε 0 μ 0 [ γ x 0 0 0 γ y 0 0 0 γ z ] .
× E = i ω B , × H = i ω D .
B = ( 1 / i ω ) × E , H = ( B + i γ ¯ ¯ E ) / μ ¯ ¯ ,
D = ( i / ω ) × H = ε ¯ ¯ E + i γ ¯ ¯ H .
B x = ( E z / y E y / z ) / i ω , H x = ( B x + i γ x ε 0 μ 0 E x ) / ( μ x μ 0 ) , B y = ( E x / z E z / x ) / i ω , H y = ( B y + i γ y ε 0 μ 0 E y ) / ( μ y μ 0 ) , B z = ( E y / x E x / y ) / i ω , H z = ( B z + i γ z ε 0 μ 0 E z ) / ( μ z μ 0 ) .
/ y = 0 , / x = i k x , / z = i q ,
[ ( ε x μ y γ x 2 μ y / μ x ) ω 2 / c 2 q 2 ] E x i q ( γ x μ y / μ x + γ y ) E y ω / c + q k x E z = 0 , i q ( γ y μ x μ y + γ x ) ω c E x i k x ( γ y μ x μ y + γ z μ x μ z ) ω c E z + [ ( ε y μ x γ y 2 μ x μ y ) ω 2 c 2 μ x μ z k x 2 q 2 ] E y = 0 , q k x E x + i k x ( γ z μ y / μ z + γ y ) E y ω / c + [ ( ε z μ y γ z 2 μ y / μ z ) ω 2 / c 2 k x 2 ] E z = 0.
| ( ε x μ y γ x 2 μ y μ x ) ω 2 c 2 q 2 i q ω c ( γ x μ y μ x + γ y ) q k x i q ω c ( γ y μ x μ y + γ x ) ( ε y μ x γ y 2 μ x μ y ) ω 2 c 2 k x 2 μ x μ z q 2 i k x ( γ y μ x / μ y + γ z μ x / μ z ) ω / c q k x i k x ω c ( γ z μ y μ z + γ y ) ( ε z μ y γ z 2 μ y / μ z ) ω 2 / c 2 k x 2 | = 0 ,
E x = E x y E y , E x y = E x / E y , H x = ε 0 / μ 0 [ i γ x E x y / μ x c q / ( ω μ x ) ] E y , E z y = E z / E y , H y = ε 0 / μ 0 [ i γ y / μ y + c ( q E x y k x E z y ) / ( ω μ y ) ] E y .
E 0 y = ( A 0 e i q 0 z + B 0 e + i q 0 z ) e i k x x , H 0 x = q 0 ( A 0 e i q 0 z B 0 e +i q 0 z ) e i k x x / ( ω μ s μ 0 ) , E 0 x = c 0 ( C 0 e i q 0 z + D 0 e + i q 0 z ) e i k x x , H 0 y = ε s ε 0 / ( μ s μ 0 ) ( C 0 e i q 0 z + D 0 e +i q 0 z ) e i k x x , E t y = ( A t e i q t z + B t e + i q t z ) e i k x x , H t x = q t ( A t e i q t z B t e + i q t z ) e i k x x / ( ω μ t μ 0 ) , E t x = c t ( C t e i q t z + D t e + i q t z ) e i k x x , H t y = ε t ε 0 / ( μ t μ 0 ) ( C t e i q t z + D t e + i q t z ) e i k x x .
n 0 = ε s μ s , c 0 = cos θ 0 , n t = ε t μ t , c t = cos θ t = 1 ( n 0 sin θ 0 / n t ) 2 , q 0 2 + k x 2 = ε s μ s ω 2 / c 2 , k x = ε s μ s sin θ 0 ω / c , q t = ε t μ t cos θ t ω / c ,
E l y = i f l + i f l + i b l + i b l , E l x = E x y 1 l i f l + E x y 2 l i f l + E x y 3 l i b l + E x y 4 l i b l , H l y = ε 0 μ 0 { i c ω μ y l × [ ( q 1 l E x y 1 l k x E z y 1 l ) f l + + ( k x E z y 2 l q 2 l E x y 2 l ) f l + ( q 3 l E x y 3 l k x E z y 3 l ) b l + + ( k x E z y 4 l q 4 l E x y 4 l ) b l ] + γ y l μ y l ( f l f l + + b l b l + ) } , H l x = ε 0 μ 0 [ γ x l μ x l ( E x y 2 l f l E x y 1 l f l+ + E x y 4 l b l E x y 3 l b l + ) + i c ω μ x l ( q 2 l f l q 1 l f l+ + q 4 l b l q 3 l i b l + ) ] , f l + = f l + e [i( k x x + q 1 l z )] , f l = f l e [i( k x x + q 2 l z )] , b l + = b l + e [i( k x x + q 3 l z )] , b l = b l e [i( k x x + q 4 l z )] .
[ A 0 B 0 C 0 D 0 ] = V 10 [ f 1 + e i q 1 1 d 0 f 1 e i q 2 1 d 0 b 1 + e i q 3 1 d 0 b 1 e i q 4 1 d 0 ] , V 10 = i [ 1 1 0 0 0 0 c 0 c 0 c 0 Y s c 0 Y s 0 0 0 0 Y s Y s ] 1 [ u 11 1 u 12 1 u 13 1 u 14 1 u 21 1 u 22 1 u 23 1 u 24 1 u 31 1 u 32 1 u 33 1 u 34 1 u 41 1 u 42 1 u 43 1 u 44 1 ] , Y s = ε s μ s , u 1 j l = ( 1 ) j + 1 , u 2 j l = ( 1 ) j + 1 E x y j l , u 3 j l = ( 1 ) j + 1 [ i γ x l E x y j l / μ x l c q j l / ( ω μ x l ) ] , u 4 j l = ( 1 ) j + 1 [ ( c q j l E x y j l c k x E z y j l ) / ( ω μ y l ) + i γ y l / μ y l ] , ( j = 1 , 2 , 3 , 4 ) .
[ f l + e i q 1 l d l f l e i q 2 l d l b l + e i q 3 l d l b l e i q 4 l d l ] = V ( l + 1 ) l [ f ( l + 1 ) + e i q 1 ( l + 1 ) d l f ( l + 1 ) e i q 2 ( l + 1 ) d l b ( l + 1 ) + e i q 3 ( l + 1 ) d l b ( l + 1 ) e i q 4 ( l + 1 ) d l ] , V ( l + 1 ) l = [ u 11 l u 12 l u 13 l u 14 l u 21 l u 22 l u 23 l u 24 l u 31 l u 32 l u 33 l u 34 l u 41 l u 42 l u 43 l u 44 l ] 1 [ u 11 l + 1 u 12 l + 1 u 13 l + 1 u 14 l + 1 u 21 l + 1 u 22 l + 1 u 23 l + 1 u 24 l + 1 u 31 l + 1 u 32 l + 1 u 33 l + 1 u 34 l + 1 u 41 l + 1 u 42 l + 1 u 43 l + 1 u 44 l + 1 ] ,
[ f n + e i q 1 n d n f n e i q 2 n d n b n + e i q 3 n d n b n e i q 4 n d n ] = V t n [ A t B t C t D t ] , V t n = i [ u 11 n u 12 n u 13 n u 14 n u 21 n u 22 n u 23 n u 24 n u 31 n u 32 n u 33 n u 34 n u 41 n u 42 n u 43 n u 44 n ] 1 [ Q P 0 0 0 0 c t Q c t P c t Y t Q c t Y t P 0 0 0 0 Y t Q Y t P ] , Y t = ε t / μ t , P = e i q t d n , Q = e i q t d n .
[ A 0 B 0 C 0 D 0 ] = V t 0 [ A t B t C t D t ] = [ v 11 v 12 v 13 v 14 v 21 v 22 v 23 v 24 v 31 v 32 v 33 v 34 v 41 v 42 v 43 v 44 ] [ A t B t C t D t ] = V 10 V 21 V ( l + 1 ) l V n ( n 1 ) V t n [ A t B t C t D t ] ,
r s s = v 12 v 44 v 14 v 42 v 22 v 44 v 24 v 42 , r s p = v 32 v 44 v 34 v 42 v 22 v 44 v 24 v 42 , t s s = v 44 v 22 v 44 v 24 v 42 , t s p = v 42 v 22 v 44 v 24 v 42 .
r p p = v 22 v 34 v 32 v 24 v 22 v 44 v 24 v 42 , r p s = v 14 v 22 v 12 v 24 v 22 v 44 v 24 v 42 , t p p = v 22 v 22 v 44 v 24 v 42 , t p s = v 24 v 22 v 44 v 24 v 42 .
r + ± = ± ( r s s r p p ) /2 i ( r s p ± r p s ) /2, t + ± = ( t p p ± t s s ) /2 + i ( t s p t p s ) /2, r = ± ( r s s r p p ) /2 + i ( r s p ± r p s ) /2, t = ( t p p ± t s s ) /2 i ( t s p t p s ) /2 .
R p p = | r p p | 2 , R s s = | r s s | 2 , R + + = | r + + | 2 , R = | r | 2 , R p s = | r p s | 2 .

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