Enhancement of magneto-optical Kerr effect by surface plasmons in trilayer structure consisting of double-layer dielectrics and ferromagnetic metal

Terunori Kaihara; Takeaki Ando; Hiromasa Shimizu; Vadym Zayets; Hidekazu Saito; Koji Ando; Shinji Yuasa

doi:10.1364/OE.23.011537

Enhancement of magneto-optical Kerr effect by surface plasmons in trilayer structure consisting of double-layer dielectrics and ferromagnetic metal

Terunori Kaihara, Takeaki Ando, Hiromasa Shimizu, Vadym Zayets, Hidekazu Saito, Koji Ando, and Shinji Yuasa

Optics Express
Vol. 23,
Issue 9,
pp. 11537-11555
(2015)
•https://doi.org/10.1364/OE.23.011537

Get Citation
Copy Citation Text
Terunori Kaihara, Takeaki Ando, Hiromasa Shimizu, Vadym Zayets, Hidekazu Saito, Koji Ando, and Shinji Yuasa, "Enhancement of magneto-optical Kerr effect by surface plasmons in trilayer structure consisting of double-layer dielectrics and ferromagnetic metal," Opt. Express 23, 11537-11555 (2015)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Get PDF (3349 KB)
Set citation alerts
Save article

Related Topics
Table of Contents Category
- Plasmonics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Magneto-optical materials (160.3820)
- Magneto-optical materials (210.3820)
- Magneto-optic systems (230.3810)
- Surface plasmons (240.6680)
- Plasmonics (250.5403)

About this Article
History
- Original Manuscript: February 24, 2015
- Revised Manuscript: April 2, 2015
- Manuscript Accepted: April 20, 2015
- Published: April 23, 2015

Accessible

Open Access

Abstract

Giant enhancement of the magneto-optical Kerr effect (MOKE) by surface plasmon polaritons (SPPs) is theoretically shown in a trilayer structure consisting of double-layer dielectrics and a ferromagnetic metal (Al₂O₃/SiO₂/Fe). We calculated the resonant enhancement of the transverse MOKE (TMOKE) and polar MOKE (PMOKE) using the attenuated total reflection (ATR) configuration with the transfer matrix method using a 4 × 4 scattering matrix. At a specific film thickness of the low-index SiO₂ layer, where confinement of the SPPs on the Fe surface becomes close to the cutoff condition, the incident light from the Al₂O₃ couples with the SPPs at the SiO₂/Fe boundary most efficiently, resulting in resonant enhancement of the MOKE at an incident angle corresponding to the wave vector of the SPPs. The calculated PMOKE showed orthogonal transformation (90°-rotation) and almost full-orbed deformation (44°-ellipticity) of the polarization, and the TMOKE showed a change in reflectance of about 34 dB upon magnetization reversal.

Full Article | PDF Article

OSA Recommended Articles

Spin Hall effect of reflected light in dielectric magneto-optical thin film with a double-negative metamaterial substrate

Jie Li, Tingting Tang, Li Luo, Nengxi Li, and Pengyu Zhang
Opt. Express 25(16) 19117-19128 (2017)

Application of strong transverse magneto-optical Kerr effect on high sensitive surface plasmon grating sensors

Kuei-Hsu Chou, En-Ping Lin, Te-Chang Chen, Chih-Huang Lai, Liang-Wei Wang, Ko-Wei Chang, Gwo-Bin Lee, and Ming-Chang M. Lee
Opt. Express 22(16) 19794-19802 (2014)

Localized surface plasmon resonance effects on the magneto-optical activity of continuous Au/Co/Au trilayers

Gaspar Armelles, Juan Bautista González-Díaz, Antonio García-Martín, José Miguel García-Martín, Alfonso Cebollada, María Ujué González, Srdjan Acimovic, Jean Cesario, Romain Quidant, and Gonçal Badenes
Opt. Express 16(20) 16104-16112 (2008)

References

View by:
Article Order
|
Year
|
Author
|
Publication

H. J. Williams, R. C. Sherwood, F. G. Foster, and E. M. Kelley, “Magnetic Writing on Thin Films of MnBi,” J. Appl. Phys. 28(10), 1181–1184 (1957).
[Crossref]
R. J. Gambino and T. Suzuki, Magneto-Optical Recording Materials (IEEE, 2000).
D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is — and what is not — an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]
T. Shintaku and T. Uno, “Optical waveguide isolator based on nonreciprocal radiation,” J. Appl. Phys. 76(12), 8155–8159 (1994).
[Crossref]
T. Shintaku, “Integrated optical isolator based on efficient nonreciprocal radiation mode conversion,” Appl. Phys. Lett. 73(14), 1946–1948 (1998).
[Crossref]
Y. Shoji, Y. Shirato, and T. Mizumoto, “Silicon Mach–Zehnder interferometer optical isolator having 8 nm bandwidth for over 20 dB isolation,” Jpn. J. Appl. Phys. 53(2), 022202 (2014).
[Crossref]
Y. Sobu, Y. Shoji, K. Sakurai, and T. Mizumoto, “GaInAsP/InP MZI waveguide optical isolator integrated with spot size converter,” Opt. Express 21(13), 15373–15381 (2013).
[Crossref] [PubMed]
H. Shimizu and Y. Nakano, “First Demonstration of TE Mode Nonreciprocal Propagation in an InGaAsP/InP Active Waveguide for an Integratable Optical Isolator,” Jpn. J. Appl. Phys. 43(12A12A), L1561–L1563 (2004).
[Crossref]
H. Shimizu and Y. Nakano, “Fabrication and characterization of an InGaAsP/InP active waveguide optical isolator with 14.7 dB/mm TE mode nonreciprocal attenuation,” J. Lightwave Technol. 24(1), 38–43 (2006).
[Crossref]
C. Nylander, B. Liedberg, and T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators 3, 79–88 (1982).
[Crossref]
J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008).
[Crossref] [PubMed]
K. Hering, D. Cialla, K. Ackermann, T. Dörfer, R. Möller, H. Schneidewind, R. Mattheis, W. Fritzsche, P. Rösch, and J. Popp, “SERS: a versatile tool in chemical and biochemical diagnostics,” Anal. Bioanal. Chem. 390(1), 113–124 (2008).
[Crossref] [PubMed]
M. Fleischmann, P. J. Hendra, and A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett. 26(2), 163–166 (1974).
[Crossref]
G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: combining magnetic and plasmonic functionalities,” Adv. Opt. Mater. 1(1), 10–35 (2013).
[Crossref]
P. M. Hui and D. Stroud, “Theory of Faraday rotation by dilute suspensions of small particles,” Appl. Phys. Lett. 50(15), 950–952 (1987).
[Crossref]
J. B. González-Díaz, A. García-Martín, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. B. Wehrspohn, and U. Gösele, “Enhanced Magneto-Optics and Size Effects in Ferromagnetic Nanowire Arrays,” Adv. Mater. 19(18), 2643–2647 (2007).
[Crossref]
J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, G. Armelles, B. Sepúlveda, Y. Alaverdyan, and M. Käll, “Plasmonic Au/Co/Au Nanosandwiches with Enhanced Magneto-Optical Activity,” Small 4(2), 202–205 (2008).
[Crossref] [PubMed]
D. Meneses-Rodríguez, E. Ferreiro-Vila, P. Prieto, J. Anguita, M. U. González, J. M. García-Martín, A. Cebollada, A. García-Martín, and G. Armelles, “Probing the Electromagnetic Field Distribution within a Metallic Nanodisk,” Small 7(23), 3317–3323 (2011).
[Crossref] [PubMed]
G. Armelles, A. Cebollada, A. García-Martín, J. M. Montero-Moreno, M. Waleczek, and K. Nielsch, “Magneto-optical Properties of Core-Shell Magneto-plasmonic Au-CoxFe3 - xO4 Nanowires,” Langmuir 28(24), 9127–9130 (2012).
[Crossref] [PubMed]
L. Wang, C. Clavero, Z. Huba, K. J. Carroll, E. E. Carpenter, D. Gu, and R. A. Lukaszew, “Plasmonics and Enhanced Magneto-Optics in Core-Shell Co-Ag Nanoparticles,” Nano Lett. 11(3), 1237–1240 (2011).
[Crossref] [PubMed]
H. Uchida, Y. Masuda, R. Fujikawa, A. V. Baryshev, and M. Inoue, “Large enhancement of Faraday rotation by localized surface plasmon resonance in Au nanoparticles embedded in Bi:YIG film,” J. Magn. Magn. Mater. 321(7), 843–845 (2009).
[Crossref]
H. Uchida, Y. Mizutani, Y. Nakai, A. A. Fedyanin, and M. Inoue, “Garnet composite films with Au particles fabricated by repetitive formation for enhancement of Faraday effect,” J. Phys. D Appl. Phys. 44(6), 064014 (2011).
[Crossref]
B. Sepúlveda, A. Calle, L. M. Lechuga, and G. Armelles, “Highly sensitive detection of biomolecules with the magneto-optic surface-plasmon-resonance sensor,” Opt. Lett. 31(8), 1085–1087 (2006).
[Crossref] [PubMed]
D. Regatos, B. Sepúlveda, D. Fariña, L. G. Carrascosa, and L. M. Lechuga, “Suitable combination of noble/ferromagnetic metal multilayers for enhanced magneto-plasmonic biosensing,” Opt. Express 19(9), 8336–8346 (2011).
[Crossref] [PubMed]
V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
[Crossref]
V. Zayets, H. Saito, K. Ando, and S. Yuasa, “Optical isolator utilizing surface plasmons,” Materials (Basel) 5(5), 857–871 (2012).
[Crossref]
F. Gervais, “Aluminum Oxide (Al2O3),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, 1991).
H. R. Philipp, “Silicon Dioxide (SiO2) (Glass),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, 1991).
D. W. Lynch and W. R. Hunter, “An introduction to the Data for Several Metals,” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, 1991).
M. B. Stearns, “Optical constants, magneto-optic Kerr or Faraday effect,” in Landolt-Börnstein - Group III Condensed Matter, H. P. J. Wijn, ed. (Springer, 1986).
P. Bertrand, C. Hermann, G. Lampel, J. Peretti, and V. I. Safarov, “General analytical treatment of optics in layered structures: Application to magneto-optics,” Phys. Rev. B 64(23), 235421 (2001).
[Crossref]
V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig relations in optical materials research (Springer, 2005), Chap. 4.
D. Y. Smith, “Dispersion relations and sum rules for magnetoreflectivity,” J. Opt. Soc. Am. 66(6), 547–554 (1976).
[Crossref]
J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. García-Martín, C. Clavero, A. Cebollada, R. A. Lukaszew, J. R. Skuza, D. P. Kumah, and R. Clarke, “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76(15), 153402 (2007).
[Crossref]
V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, “Giant transversal Kerr effect in magneto-plasmonic heterostructures: The scattering-matrix method,” Sov. Phys. JETP 110(5), 816–824 (2010).
[Crossref]
B. Caballero, A. García-Martín, and J. C. Cuevas, “Generalized scattering-matrix approach for magneto-optics in periodically patterned multilayer systems,” Phys. Rev. B 85(24), 245103 (2012).
[Crossref]
S. Zhang, S. Tang, J. Gao, X. Luo, W. Xia, and Y. Du, “Theoretical calculation of magneto-optical properties in cobalt nanotube array with hexagonal symmetry,” Solid State Commun. 170, 19–23 (2013).
[Crossref]
L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, S. Neutzner, D. Dregely, T. Wehlus, I. A. Akimov, M. Bayer, B. Stritzker, and H. Giessen, “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X 3, 041019 (2013).
S. M. Hamidi, M. A. Oskuei, S. Sadeghi, and M. M. Tehranchi, “Enhanced polar magneto-optical Kerr rotation in cobalt thin film incorporating Ag nanoparticles,” J. Supercond. Novel Magn. 27(3), 867–870 (2014).
[Crossref]
H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]
EM Lab - University of Texas at El Paso, “News 2012 | EM Lab,” http://emlab.utep.edu/ee5390cem.htm
R. C. Rumpf, “Improved formulation of scattering matrices for semi-analytical methods that is consistent with convention,” Prog. Electromagn. Res. B 35, 241–261 (2011).
[Crossref]

2014 (3)

Y. Shoji, Y. Shirato, and T. Mizumoto, “Silicon Mach–Zehnder interferometer optical isolator having 8 nm bandwidth for over 20 dB isolation,” Jpn. J. Appl. Phys. 53(2), 022202 (2014).
[Crossref]

S. M. Hamidi, M. A. Oskuei, S. Sadeghi, and M. M. Tehranchi, “Enhanced polar magneto-optical Kerr rotation in cobalt thin film incorporating Ag nanoparticles,” J. Supercond. Novel Magn. 27(3), 867–870 (2014).
[Crossref]

H.-T. Huang, P.-J. Chen, T.-R. Ger, Y.-J. Chi, C.-W. Huang, K.-T. Liao, J.-Y. Lai, J.-Y. Chen, W.-Y. Peng, Q. Zhang, T.-F. Hsieh, W.-J. Sheu, and Z.-H. Wei, “Magneto-optical Kerr effect enhanced by surface plasmon resonance and its application on biological detection,” IEEE Trans. Magn. 50(1), 1001604 (2014).
[Crossref]

2013 (5)

S. Zhang, S. Tang, J. Gao, X. Luo, W. Xia, and Y. Du, “Theoretical calculation of magneto-optical properties in cobalt nanotube array with hexagonal symmetry,” Solid State Commun. 170, 19–23 (2013).
[Crossref]

L. E. Kreilkamp, V. I. Belotelov, J. Y. Chin, S. Neutzner, D. Dregely, T. Wehlus, I. A. Akimov, M. Bayer, B. Stritzker, and H. Giessen, “Waveguide-plasmon polaritons enhance transverse magneto-optical Kerr effect,” Phys. Rev. X 3, 041019 (2013).

Y. Sobu, Y. Shoji, K. Sakurai, and T. Mizumoto, “GaInAsP/InP MZI waveguide optical isolator integrated with spot size converter,” Opt. Express 21(13), 15373–15381 (2013).
[Crossref] [PubMed]

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is — and what is not — an optical isolator,” Nat. Photonics 7(8), 579–582 (2013).
[Crossref]

G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: combining magnetic and plasmonic functionalities,” Adv. Opt. Mater. 1(1), 10–35 (2013).
[Crossref]

2012 (4)

G. Armelles, A. Cebollada, A. García-Martín, J. M. Montero-Moreno, M. Waleczek, and K. Nielsch, “Magneto-optical Properties of Core-Shell Magneto-plasmonic Au-CoxFe3 - xO4 Nanowires,” Langmuir 28(24), 9127–9130 (2012).
[Crossref] [PubMed]

B. Caballero, A. García-Martín, and J. C. Cuevas, “Generalized scattering-matrix approach for magneto-optics in periodically patterned multilayer systems,” Phys. Rev. B 85(24), 245103 (2012).
[Crossref]

V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
[Crossref]

V. Zayets, H. Saito, K. Ando, and S. Yuasa, “Optical isolator utilizing surface plasmons,” Materials (Basel) 5(5), 857–871 (2012).
[Crossref]

2011 (5)

H. Uchida, Y. Mizutani, Y. Nakai, A. A. Fedyanin, and M. Inoue, “Garnet composite films with Au particles fabricated by repetitive formation for enhancement of Faraday effect,” J. Phys. D Appl. Phys. 44(6), 064014 (2011).
[Crossref]

D. Meneses-Rodríguez, E. Ferreiro-Vila, P. Prieto, J. Anguita, M. U. González, J. M. García-Martín, A. Cebollada, A. García-Martín, and G. Armelles, “Probing the Electromagnetic Field Distribution within a Metallic Nanodisk,” Small 7(23), 3317–3323 (2011).
[Crossref] [PubMed]

D. Regatos, B. Sepúlveda, D. Fariña, L. G. Carrascosa, and L. M. Lechuga, “Suitable combination of noble/ferromagnetic metal multilayers for enhanced magneto-plasmonic biosensing,” Opt. Express 19(9), 8336–8346 (2011).
[Crossref] [PubMed]

R. C. Rumpf, “Improved formulation of scattering matrices for semi-analytical methods that is consistent with convention,” Prog. Electromagn. Res. B 35, 241–261 (2011).
[Crossref]

L. Wang, C. Clavero, Z. Huba, K. J. Carroll, E. E. Carpenter, D. Gu, and R. A. Lukaszew, “Plasmonics and Enhanced Magneto-Optics in Core-Shell Co-Ag Nanoparticles,” Nano Lett. 11(3), 1237–1240 (2011).
[Crossref] [PubMed]

2010 (1)

V. I. Belotelov, D. A. Bykov, L. L. Doskolovich, A. N. Kalish, and A. K. Zvezdin, “Giant transversal Kerr effect in magneto-plasmonic heterostructures: The scattering-matrix method,” Sov. Phys. JETP 110(5), 816–824 (2010).
[Crossref]

2009 (1)

H. Uchida, Y. Masuda, R. Fujikawa, A. V. Baryshev, and M. Inoue, “Large enhancement of Faraday rotation by localized surface plasmon resonance in Au nanoparticles embedded in Bi:YIG film,” J. Magn. Magn. Mater. 321(7), 843–845 (2009).
[Crossref]

2008 (3)

J. B. González-Díaz, A. García-Martín, J. M. García-Martín, A. Cebollada, G. Armelles, B. Sepúlveda, Y. Alaverdyan, and M. Käll, “Plasmonic Au/Co/Au Nanosandwiches with Enhanced Magneto-Optical Activity,” Small 4(2), 202–205 (2008).
[Crossref] [PubMed]

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008).
[Crossref] [PubMed]

K. Hering, D. Cialla, K. Ackermann, T. Dörfer, R. Möller, H. Schneidewind, R. Mattheis, W. Fritzsche, P. Rösch, and J. Popp, “SERS: a versatile tool in chemical and biochemical diagnostics,” Anal. Bioanal. Chem. 390(1), 113–124 (2008).
[Crossref] [PubMed]

2007 (2)

J. B. González-Díaz, A. García-Martín, G. Armelles, D. Navas, M. Vázquez, K. Nielsch, R. B. Wehrspohn, and U. Gösele, “Enhanced Magneto-Optics and Size Effects in Ferromagnetic Nanowire Arrays,” Adv. Mater. 19(18), 2643–2647 (2007).
[Crossref]

J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. García-Martín, C. Clavero, A. Cebollada, R. A. Lukaszew, J. R. Skuza, D. P. Kumah, and R. Clarke, “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76(15), 153402 (2007).
[Crossref]

2006 (2)

B. Sepúlveda, A. Calle, L. M. Lechuga, and G. Armelles, “Highly sensitive detection of biomolecules with the magneto-optic surface-plasmon-resonance sensor,” Opt. Lett. 31(8), 1085–1087 (2006).
[Crossref] [PubMed]

H. Shimizu and Y. Nakano, “Fabrication and characterization of an InGaAsP/InP active waveguide optical isolator with 14.7 dB/mm TE mode nonreciprocal attenuation,” J. Lightwave Technol. 24(1), 38–43 (2006).
[Crossref]

2004 (1)

H. Shimizu and Y. Nakano, “First Demonstration of TE Mode Nonreciprocal Propagation in an InGaAsP/InP Active Waveguide for an Integratable Optical Isolator,” Jpn. J. Appl. Phys. 43(12A12A), L1561–L1563 (2004).
[Crossref]

2001 (1)

P. Bertrand, C. Hermann, G. Lampel, J. Peretti, and V. I. Safarov, “General analytical treatment of optics in layered structures: Application to magneto-optics,” Phys. Rev. B 64(23), 235421 (2001).
[Crossref]

1998 (1)

T. Shintaku, “Integrated optical isolator based on efficient nonreciprocal radiation mode conversion,” Appl. Phys. Lett. 73(14), 1946–1948 (1998).
[Crossref]

1994 (1)

T. Shintaku and T. Uno, “Optical waveguide isolator based on nonreciprocal radiation,” J. Appl. Phys. 76(12), 8155–8159 (1994).
[Crossref]

1987 (1)

P. M. Hui and D. Stroud, “Theory of Faraday rotation by dilute suspensions of small particles,” Appl. Phys. Lett. 50(15), 950–952 (1987).
[Crossref]

1982 (1)

C. Nylander, B. Liedberg, and T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators 3, 79–88 (1982).
[Crossref]

1976 (1)

D. Y. Smith, “Dispersion relations and sum rules for magnetoreflectivity,” J. Opt. Soc. Am. 66(6), 547–554 (1976).
[Crossref]

1974 (1)

M. Fleischmann, P. J. Hendra, and A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett. 26(2), 163–166 (1974).
[Crossref]

1957 (1)

H. J. Williams, R. C. Sherwood, F. G. Foster, and E. M. Kelley, “Magnetic Writing on Thin Films of MnBi,” J. Appl. Phys. 28(10), 1181–1184 (1957).
[Crossref]

Ackermann, K.

Akimov, I. A.

Alaverdyan, Y.

Ando, K.

V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
[Crossref]

V. Zayets, H. Saito, K. Ando, and S. Yuasa, “Optical isolator utilizing surface plasmons,” Materials (Basel) 5(5), 857–871 (2012).
[Crossref]

Anguita, J.

Armelles, G.

G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: combining magnetic and plasmonic functionalities,” Adv. Opt. Mater. 1(1), 10–35 (2013).
[Crossref]

Baets, R.

Baryshev, A. V.

Bayer, M.

Belotelov, V. I.

Bertrand, P.

Bykov, D. A.

Caballero, B.

Calle, A.

Carpenter, E. E.

Carrascosa, L. G.

Carroll, K. J.

Cebollada, A.

G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: combining magnetic and plasmonic functionalities,” Adv. Opt. Mater. 1(1), 10–35 (2013).
[Crossref]

Chen, J.-Y.

Chen, P.-J.

Chi, Y.-J.

Chin, J. Y.

Cialla, D.

Clarke, R.

Clavero, C.

Cuevas, J. C.

Doerr, C. R.

Dörfer, T.

Doskolovich, L. L.

Dregely, D.

Du, Y.

Eich, M.

Fan, S.

Fariña, D.

Fedyanin, A. A.

Ferreiro-Vila, E.

Fleischmann, M.

M. Fleischmann, P. J. Hendra, and A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett. 26(2), 163–166 (1974).
[Crossref]

Foster, F. G.

H. J. Williams, R. C. Sherwood, F. G. Foster, and E. M. Kelley, “Magnetic Writing on Thin Films of MnBi,” J. Appl. Phys. 28(10), 1181–1184 (1957).
[Crossref]

Freude, W.

Fritzsche, W.

Fujikawa, R.

Gao, J.

García-Martín, A.

G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: combining magnetic and plasmonic functionalities,” Adv. Opt. Mater. 1(1), 10–35 (2013).
[Crossref]

García-Martín, J. M.

Ger, T.-R.

Giessen, H.

González, M. U.

G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: combining magnetic and plasmonic functionalities,” Adv. Opt. Mater. 1(1), 10–35 (2013).
[Crossref]

González-Díaz, J. B.

Gösele, U.

Gu, D.

Hamidi, S. M.

Hendra, P. J.

M. Fleischmann, P. J. Hendra, and A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett. 26(2), 163–166 (1974).
[Crossref]

Hering, K.

Hermann, C.

Homola, J.

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008).
[Crossref] [PubMed]

Hsieh, T.-F.

Huang, C.-W.

Huang, H.-T.

Huba, Z.

Hui, P. M.

P. M. Hui and D. Stroud, “Theory of Faraday rotation by dilute suspensions of small particles,” Appl. Phys. Lett. 50(15), 950–952 (1987).
[Crossref]

Inoue, M.

Jalas, D.

Joannopoulos, J. D.

Kalish, A. N.

Käll, M.

Kelley, E. M.

H. J. Williams, R. C. Sherwood, F. G. Foster, and E. M. Kelley, “Magnetic Writing on Thin Films of MnBi,” J. Appl. Phys. 28(10), 1181–1184 (1957).
[Crossref]

Kreilkamp, L. E.

Kumah, D. P.

Lai, J.-Y.

Lampel, G.

Lechuga, L. M.

Liao, K.-T.

Liedberg, B.

C. Nylander, B. Liedberg, and T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators 3, 79–88 (1982).
[Crossref]

Lind, T.

C. Nylander, B. Liedberg, and T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators 3, 79–88 (1982).
[Crossref]

Lukaszew, R. A.

Luo, X.

Masuda, Y.

Mattheis, R.

McQuillan, A. J.

M. Fleischmann, P. J. Hendra, and A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett. 26(2), 163–166 (1974).
[Crossref]

Melloni, A.

Meneses-Rodríguez, D.

Mizumoto, T.

Y. Sobu, Y. Shoji, K. Sakurai, and T. Mizumoto, “GaInAsP/InP MZI waveguide optical isolator integrated with spot size converter,” Opt. Express 21(13), 15373–15381 (2013).
[Crossref] [PubMed]

Mizutani, Y.

Möller, R.

Montero-Moreno, J. M.

Nakai, Y.

Nakano, Y.

Navas, D.

Neutzner, S.

Nielsch, K.

Nylander, C.

C. Nylander, B. Liedberg, and T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators 3, 79–88 (1982).
[Crossref]

Oskuei, M. A.

Peng, W.-Y.

Peretti, J.

Petrov, A.

Popovic, M.

Popp, J.

Prieto, P.

Regatos, D.

Renner, H.

Rösch, P.

Rumpf, R. C.

R. C. Rumpf, “Improved formulation of scattering matrices for semi-analytical methods that is consistent with convention,” Prog. Electromagn. Res. B 35, 241–261 (2011).
[Crossref]

Sadeghi, S.

Safarov, V. I.

Saito, H.

V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
[Crossref]

V. Zayets, H. Saito, K. Ando, and S. Yuasa, “Optical isolator utilizing surface plasmons,” Materials (Basel) 5(5), 857–871 (2012).
[Crossref]

Sakurai, K.

Y. Sobu, Y. Shoji, K. Sakurai, and T. Mizumoto, “GaInAsP/InP MZI waveguide optical isolator integrated with spot size converter,” Opt. Express 21(13), 15373–15381 (2013).
[Crossref] [PubMed]

Schneidewind, H.

Sepúlveda, B.

Sherwood, R. C.

H. J. Williams, R. C. Sherwood, F. G. Foster, and E. M. Kelley, “Magnetic Writing on Thin Films of MnBi,” J. Appl. Phys. 28(10), 1181–1184 (1957).
[Crossref]

Sheu, W.-J.

Shimizu, H.

Shintaku, T.

T. Shintaku, “Integrated optical isolator based on efficient nonreciprocal radiation mode conversion,” Appl. Phys. Lett. 73(14), 1946–1948 (1998).
[Crossref]

T. Shintaku and T. Uno, “Optical waveguide isolator based on nonreciprocal radiation,” J. Appl. Phys. 76(12), 8155–8159 (1994).
[Crossref]

Shirato, Y.

Shoji, Y.

Y. Sobu, Y. Shoji, K. Sakurai, and T. Mizumoto, “GaInAsP/InP MZI waveguide optical isolator integrated with spot size converter,” Opt. Express 21(13), 15373–15381 (2013).
[Crossref] [PubMed]

Skuza, J. R.

Smith, D. Y.

D. Y. Smith, “Dispersion relations and sum rules for magnetoreflectivity,” J. Opt. Soc. Am. 66(6), 547–554 (1976).
[Crossref]

Sobu, Y.

Y. Sobu, Y. Shoji, K. Sakurai, and T. Mizumoto, “GaInAsP/InP MZI waveguide optical isolator integrated with spot size converter,” Opt. Express 21(13), 15373–15381 (2013).
[Crossref] [PubMed]

Stritzker, B.

Stroud, D.

P. M. Hui and D. Stroud, “Theory of Faraday rotation by dilute suspensions of small particles,” Appl. Phys. Lett. 50(15), 950–952 (1987).
[Crossref]

Tang, S.

Tehranchi, M. M.

Uchida, H.

Uno, T.

T. Shintaku and T. Uno, “Optical waveguide isolator based on nonreciprocal radiation,” J. Appl. Phys. 76(12), 8155–8159 (1994).
[Crossref]

Vanwolleghem, M.

Vázquez, M.

Waleczek, M.

Wang, L.

Wehlus, T.

Wehrspohn, R. B.

Wei, Z.-H.

Williams, H. J.

H. J. Williams, R. C. Sherwood, F. G. Foster, and E. M. Kelley, “Magnetic Writing on Thin Films of MnBi,” J. Appl. Phys. 28(10), 1181–1184 (1957).
[Crossref]

Xia, W.

Yu, Z.

Yuasa, S.

V. Zayets, H. Saito, K. Ando, and S. Yuasa, “Optical isolator utilizing surface plasmons,” Materials (Basel) 5(5), 857–871 (2012).
[Crossref]

V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
[Crossref]

Zayets, V.

V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
[Crossref]

V. Zayets, H. Saito, K. Ando, and S. Yuasa, “Optical isolator utilizing surface plasmons,” Materials (Basel) 5(5), 857–871 (2012).
[Crossref]

Zhang, Q.

Zhang, S.

Zvezdin, A. K.

Adv. Mater. (1)

Adv. Opt. Mater. (1)

G. Armelles, A. Cebollada, A. García-Martín, and M. U. González, “Magnetoplasmonics: combining magnetic and plasmonic functionalities,” Adv. Opt. Mater. 1(1), 10–35 (2013).
[Crossref]

Anal. Bioanal. Chem. (1)

Appl. Phys. Lett. (2)

P. M. Hui and D. Stroud, “Theory of Faraday rotation by dilute suspensions of small particles,” Appl. Phys. Lett. 50(15), 950–952 (1987).
[Crossref]

T. Shintaku, “Integrated optical isolator based on efficient nonreciprocal radiation mode conversion,” Appl. Phys. Lett. 73(14), 1946–1948 (1998).
[Crossref]

Chem. Phys. Lett. (1)

M. Fleischmann, P. J. Hendra, and A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett. 26(2), 163–166 (1974).
[Crossref]

Chem. Rev. (1)

J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. 108(2), 462–493 (2008).
[Crossref] [PubMed]

IEEE Trans. Magn. (1)

J. Appl. Phys. (3)

H. J. Williams, R. C. Sherwood, F. G. Foster, and E. M. Kelley, “Magnetic Writing on Thin Films of MnBi,” J. Appl. Phys. 28(10), 1181–1184 (1957).
[Crossref]

T. Shintaku and T. Uno, “Optical waveguide isolator based on nonreciprocal radiation,” J. Appl. Phys. 76(12), 8155–8159 (1994).
[Crossref]

V. Zayets, H. Saito, S. Yuasa, and K. Ando, “Enhancement of the transverse non-reciprocal magneto-optical effect,” J. Appl. Phys. 111(2), 023103 (2012).
[Crossref]

J. Lightwave Technol. (1)

J. Magn. Magn. Mater. (1)

J. Opt. Soc. Am. (1)

D. Y. Smith, “Dispersion relations and sum rules for magnetoreflectivity,” J. Opt. Soc. Am. 66(6), 547–554 (1976).
[Crossref]

J. Phys. D Appl. Phys. (1)

J. Supercond. Novel Magn. (1)

Jpn. J. Appl. Phys. (2)

Langmuir (1)

Materials (Basel) (1)

V. Zayets, H. Saito, K. Ando, and S. Yuasa, “Optical isolator utilizing surface plasmons,” Materials (Basel) 5(5), 857–871 (2012).
[Crossref]

Nano Lett. (1)

Nat. Photonics (1)

Opt. Express (2)

Y. Sobu, Y. Shoji, K. Sakurai, and T. Mizumoto, “GaInAsP/InP MZI waveguide optical isolator integrated with spot size converter,” Opt. Express 21(13), 15373–15381 (2013).
[Crossref] [PubMed]

Opt. Lett. (1)

Phys. Rev. B (3)

Phys. Rev. X (1)

Prog. Electromagn. Res. B (1)

R. C. Rumpf, “Improved formulation of scattering matrices for semi-analytical methods that is consistent with convention,” Prog. Electromagn. Res. B 35, 241–261 (2011).
[Crossref]

Sens. Actuators (1)

C. Nylander, B. Liedberg, and T. Lind, “Gas detection by means of surface plasmon resonance,” Sens. Actuators 3, 79–88 (1982).
[Crossref]

Small (2)

Solid State Commun. (1)

Sov. Phys. JETP (1)

Other (7)

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig relations in optical materials research (Springer, 2005), Chap. 4.

F. Gervais, “Aluminum Oxide (Al2O3),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, 1991).

H. R. Philipp, “Silicon Dioxide (SiO2) (Glass),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, 1991).

D. W. Lynch and W. R. Hunter, “An introduction to the Data for Several Metals,” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, 1991).

M. B. Stearns, “Optical constants, magneto-optic Kerr or Faraday effect,” in Landolt-Börnstein - Group III Condensed Matter, H. P. J. Wijn, ed. (Springer, 1986).

R. J. Gambino and T. Suzuki, Magneto-Optical Recording Materials (IEEE, 2000).

EM Lab - University of Texas at El Paso, “News 2012 | EM Lab,” http://emlab.utep.edu/ee5390cem.htm

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

Click here to see a list of articles that cite this paper

Fig. 1 A trilayer structure consisting of double-layer dielectrics and a ferromagnetic metal for nonreciprocal plasmonic propagation. The yellow curve represents the distribution of the SPPs.

Download Full Size | PPT Slide | PDF

Fig. 2 A schematic diagram of the Al₂O₃/SiO₂/Fe trilayer deposited on a SiO₂ substrate. The incident light, defined by orthogonal unit vectors Â_TM and Â_TE, comes from the Al₂O₃ side with p-polarization (ϕ = 0, C_TE = 0) and an arbitrary incident angle, θ. Arrows M indicate the positive magnetization directions of the PMOKE and TMOKE.

Download Full Size | PPT Slide | PDF

Fig. 3 Reflectance, R, and change in reflectance, ΔR, of the TMOKE as a function of incident angle, θ, at SiO₂ thicknesses, t, of (a) 645 nm and (b) 661 nm. Enlarged views depict the reflectance in decibel units. Red, blue, and black curves denote positive, negative, and no magnetization, respectively. The inset in (a) shows the case of s-polarization.

Download Full Size | PPT Slide | PDF

Fig. 4 SiO₂ thickness dependence of (a) maximum ΔR (black) and ΔR/R (green) and (b) minimum reflectance, R, ( + M in red, −M in blue) of the TMOKE. The values of (a) and (b) are obtained at incident angles shown in (c) and (d), respectively.

Download Full Size | PPT Slide | PDF

Fig. 5 (a) Real part and (b) imaginary part of the effective refractive index, N_eff, of the SPPs with the TMOKE as a function of the SiO₂ thickness. Red, blue, and black curves denote positive, negative, and no magnetization, respectively.

Download Full Size | PPT Slide | PDF

Fig. 6 Optical field distribution of TM mode in the Al₂O₃/SiO₂/Fe trilayer waveguide with t = ∞, 1645, 1145, and 645 nm. The curves are normalized at 0 nm.

Download Full Size | PPT Slide | PDF

Fig. 7 Reflectance, R, and change in reflectance, ΔR, of the PMOKE as a function of incident angle, θ, at t = 652 nm. The enlarged view depicts the reflectance in decibel units. The curve for positive magnetization (red) is perfectly overlapped with that for negative magnetization (blue). The black curve denotes the no magnetization case.

Download Full Size | PPT Slide | PDF

Fig. 8 The Kerr rotation (red), ϕ_K, and ellipticity (blue), η_K, at t = 652 nm as a function of the incident angle with positive magnetization, + M. Insets show the polarization normalized by the maximum norm at each angle indicated by arrows.

Download Full Size | PPT Slide | PDF

Fig. 9 SiO₂ thickness dependence of (a) the maximum Kerr rotation, |ϕ_K| (red), ellipticity, |η_K| (blue), and (b) the minimum reflectance, R (black). The values of (a) and (b) are obtained at incident angles shown in (c) and (d), respectively.

Download Full Size | PPT Slide | PDF

Fig. 10 Incident angle dependence at t = 645, 652 and 660 nm: (a) The absolute amplitude of the p-polarization, E_p (red), and induced s-polarization, E_s (blue). (b) The normalized phase shift of each argument, arg(E_p) (red) and arg(E_s)(blue), and phase difference, Δφ (black). (c) The Kerr rotation (red) and ellipticity (blue).

Download Full Size | PPT Slide | PDF

Fig. 11 Reflectance, R (positive (red), negative (blue) and no (black) magnetizations), and change in reflectance, ΔR (broken curve), of The TMOKE as a function of the incident angle. The thickness of the interface layer is assumed to be (a) t_int = 0 nm, (b) 1 nm and (c) 3 nm with the SiO₂ layer of t = 661 − t_int/2 nm and the Fe layer of 300 − t_int/2 nm.

Download Full Size | PPT Slide | PDF

Fig. 12 The Kerr (a) rotation and (b) ellipticity as a function of the incident angle. The thickness of the interface layer is assumed to be t_int = 0, 1, 3 and 5 nm as written beside the curves, and the SiO₂ layer is set to t = 652 − t_int/2 nm and the Fe layer of 300 − t_int/2 nm.

Download Full Size | PPT Slide | PDF

Fig. 13 Reflectance, R (positive (red), negative (blue) and no (black) magnetizations), and change in reflectance, ΔR (broken curve), of The TMOKE as a function of the incident angle. The deviation angle from p-polarization is assumed to be (a) t_int = 0°, (b) 1° and (c) 3° with the SiO₂ layer of t = 661 nm.

Download Full Size | PPT Slide | PDF

Fig. 14 The Kerr (a) rotation and (b) ellipticity as a function of the incident angle. The deviation angle from p-polarization is assumed to be 0° up to 10°.as written beside the curves, and the SiO₂ layer is set to t = 652 nm.

Download Full Size | PPT Slide | PDF

Equations (39)

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

ε_{r} = (\begin{matrix} n^{2} & 0 & γ \\ 0 & n^{2} & 0 \\ - γ & 0 & n^{2} \end{matrix}),

ε_{r} = (\begin{matrix} n^{2} & γ & 0 \\ - γ & n^{2} & 0 \\ 0 & 0 & n^{2} \end{matrix}) .

Δ R = R (- M) - R (+ M) [dB],

\frac{Δ R}{R} = \frac{R (- M) - R (+ M) [%]}{(R (- M) + R (+ M) [%]) / 2},

{\tilde{ϕ}}_{K} = \frac{r_{s p}}{r_{p p}}

| r_{s p} | \propto \int^{​} E_{s} (z) Δ ε_{M} E_{p} (z) d z,

\nabla' \times E = μ_{r} \tilde{H}

\nabla' \times \tilde{H} = ε_{r} E

\frac{\partial}{\partial z'} ψ = Ω ψ

Ω = (\begin{matrix} i (\frac{n_{x} ε_{31}}{ε_{33}} + \frac{n_{x} μ_{23}}{μ_{33}}) & i n_{x} (\frac{ε_{32}}{ε_{33}} - \frac{μ_{23}}{μ_{33}}) & \frac{n_{x} n_{y}}{ε_{33}} + μ_{21} - \frac{μ_{23} μ_{31}}{μ_{33}} & - \frac{n_{x}^{2}}{ε_{33}} + μ_{22} - \frac{μ_{23} μ_{32}}{μ_{33}} \\ i n_{y}^{} (\frac{ε_{31}}{ε_{33}} - \frac{μ_{13}}{μ_{33}}) & i (\frac{n_{y} ε_{32}}{ε_{33}} + \frac{n_{x} μ_{13}}{μ_{33}}) & \frac{n_{y}^{2}}{ε_{33}} - μ_{11} + \frac{μ_{13} μ_{31}}{μ_{33}} & \frac{n_{x} n_{y}}{ε_{33}} - μ_{21} + \frac{μ_{23} μ_{32}}{μ_{33}} \\ ε_{21} - \frac{ε_{23} ε_{31}}{ε_{33}} + \frac{n_{x} n_{y}}{μ_{33}} & ε_{22} - \frac{ε_{23} ε_{32}}{ε_{33}} - \frac{n_{x}^{2}}{μ_{33}} & i (\frac{n_{y} ε_{23}}{ε_{33}} + \frac{n_{x} μ_{31}}{μ_{33}}) & - i n_{x} (\frac{ε_{23}}{ε_{33}} - \frac{μ_{32}}{μ_{33}}) \\ - ε_{11} + \frac{ε_{13} ε_{31}}{ε_{33}} + \frac{n_{y}^{2}}{μ_{33}} & - ε_{12} + \frac{ε_{13} ε_{32}}{ε_{33}} - \frac{n_{x} n_{y}}{μ_{33}} & - i n_{y} (\frac{ε_{13}}{ε_{33}} - \frac{μ_{31}}{μ_{33}}) & i (\frac{n_{x} ε_{13}}{ε_{33}} + \frac{n_{y} μ_{32}}{μ_{33}}) \end{matrix})

ψ = {[E_{x}^{} E_{y}^{} {\tilde{H}}_{x}^{} {\tilde{H}}_{y}^{}]}^{T}

ψ (z^{'}) = W \cdot e^{λ z^{'}} \cdot W^{- 1} \cdot ψ (0)

ψ (z') = W_{i}^{} \cdot e^{λ_{i}^{} z'} \cdot c_{i}^{} = (\begin{matrix} W_{E i}^{+} & W_{E i}^{-} \\ W_{H i}^{+} & W_{H i}^{-} \end{matrix}) (\begin{matrix} e^{- λ_{i}^{} z'} & 0 \\ 0 & e^{λ_{i}^{} z'} \end{matrix}) (\begin{matrix} c_{i}^{+} \\ c_{i}^{-} \end{matrix}) .

\begin{array}{l} {(\begin{matrix} W_{E i}^{+} & W_{E i}^{-} \\ W_{H i}^{+} & W_{H i}^{-} \end{matrix})}^{- 1} (\begin{matrix} W_{E i - 1}^{+} & W_{E i - 1}^{-} \\ W_{H i - 1}^{+} & W_{H i - 1}^{-} \end{matrix}) (\begin{matrix} c_{i - 1}^{+} \\ c_{i - 1}^{-} \end{matrix}) \\ = (\begin{matrix} e^{λ_{i} k_{0} L_{i}^{}} & 0 \\ 0 & e^{- λ_{i} k_{0} L_{i}^{}} \end{matrix}) {(\begin{matrix} W_{E i}^{+} & W_{E i}^{-} \\ W_{H i}^{+} & W_{H i}^{-} \end{matrix})}^{- 1} (\begin{matrix} W_{E i + 1}^{+} & W_{E i + 1}^{-} \\ W_{H i + 1}^{+} & W_{H i + 1}^{-} \end{matrix}) (\begin{matrix} c_{i + 1}^{+} \\ c_{i + 1}^{-} \end{matrix}) \end{array}

(\begin{matrix} W_{E i}^{+}' & W_{E i}^{-}' \\ W_{H i}^{+}' & W_{H i}^{-}' \end{matrix}) (\begin{matrix} c_{i - 1}^{+} \\ c_{i - 1}^{-} \end{matrix}) = (\begin{matrix} e^{λ_{i} k_{0} L_{i}^{}} & 0 \\ 0 & e^{- λ_{i} k_{0} L_{i}^{}} \end{matrix}) (\begin{matrix} W_{E i}^{+}' & W_{E i}^{-}' \\ W_{H i}^{+}' & W_{H i}^{-}' \end{matrix}) (\begin{matrix} c_{i + 1}^{+} \\ c_{i + 1}^{-} \end{matrix})

(\begin{matrix} W_{E i}^{+}' & W_{E i}^{-}' \\ W_{H i}^{+}' & W_{H i}^{-}' \end{matrix}) = {(\begin{matrix} W_{E i}^{+} & W_{E i}^{-} \\ W_{H i}^{+} & W_{H i}^{-} \end{matrix})}^{- 1} (\begin{matrix} W_{E i \pm 1}^{+} & W_{E i \pm 1}^{-} \\ W_{H i \pm 1}^{+} & W_{H i \pm 1}^{-} \end{matrix}) .

(\begin{matrix} c_{i - 1}^{-} \\ c_{i + 1}^{+} \end{matrix}) = (\begin{matrix} S_{11}^{i} & S_{12}^{i} \\ S_{21}^{i} & S_{22}^{i} \end{matrix}) (\begin{matrix} c_{i - 1}^{+} \\ c_{i + 1}^{-} \end{matrix})

S_{11}^{i} = {(A_{1}^{i})}^{- 1} ({(W_{H i}^{-}')}^{- 1} λ_{}^{i} W_{H i}^{+}' {(W_{E i}^{+}')}^{- 1} λ_{}^{i} W_{E i}^{+}' - {(W_{H i}^{-}')}^{- 1} W_{H i}^{+}')

S_{12}^{i} = {(A_{1}^{i})}^{- 1} ({(W_{H i}^{-}')}^{- 1} λ_{}^{i} W_{H i}^{-}' - {(W_{H i}^{-}')}^{- 1} λ_{}^{i} W_{H i}^{-}' {(W_{E i}^{+}')}^{- 1} W_{E i}^{-}')

S_{21}^{i} = {(A_{2}^{i})}^{- 1} (- {(W_{E i}^{+}')}^{- 1} λ_{}^{i} W_{E i}^{-}' {(W_{H i}^{-}')}^{- 1} W_{H i}^{+}' + {(W_{E i}^{+}')}^{- 1} λ_{}^{i} W_{E i}^{+}')

S_{22}^{i} = {(A_{2}^{i})}^{- 1} (- {(W_{E i}^{+}')}^{- 1} W_{E i}^{-}' + {(W_{E i}^{+}')}^{- 1} λ_{}^{i} W_{E i}^{-}' {(W_{H i}^{-}')}^{- 1} λ_{}^{i} W_{H i}^{-}')

A_{1}^{i} = I - {(W_{H i}^{-}')}^{- 1} λ_{}^{i} W_{H i}^{+}' {(W_{E i}^{+}')}^{- 1} λ_{}^{i} W_{E i}^{-}'

A_{2}^{i} = I - {(W_{E i}^{+}')}^{- 1} λ_{}^{i} W_{E i}^{-}' {(W_{H i}^{-}')}^{- 1} λ_{}^{i} W_{H i}^{+}'

S_{A} \otimes S_{B} = (\begin{matrix} S_{11}^{a} + S_{12}^{a} {(I - S_{11}^{b} S_{22}^{a})}^{- 1} S_{11}^{b} S_{21}^{a} & S_{12}^{a} {(I - S_{11}^{b} S_{22}^{a})}^{- 1} S_{12}^{b} \\ S_{21}^{b} {(I - S_{22}^{a} S_{11}^{b})}^{- 1} S_{21}^{a} & S_{21}^{b} {(I - S_{22}^{a} S_{11}^{b})}^{- 1} S_{22}^{a} S_{12}^{b} + S_{22}^{b} \end{matrix})

S_{A / B} = (\begin{matrix} S_{11}^{a / b} & S_{12}^{a / b} \\ S_{21}^{a / b} & S_{22}^{a / b} \end{matrix}) .

(\begin{matrix} c_{ref}^{-} \\ c_{1}^{+} \end{matrix}) = (\begin{matrix} S_{11}^{r e f} & S_{12}^{r e f} \\ S_{21}^{r e f} & S_{22}^{r e f} \end{matrix}) (\begin{matrix} c_{inc}^{+} \\ c_{1}^{-} \end{matrix}),

(\begin{matrix} c_{ref}^{-} \\ c_{trn}^{+} \end{matrix}) = (\begin{matrix} S_{11}^{global} & S_{12}^{global} \\ S_{21}^{global} & S_{22}^{global} \end{matrix}) (\begin{matrix} c_{inc}^{+} \\ c_{trn}^{-} = 0 \end{matrix}) .

\begin{array}{l} ψ (0) = (\begin{matrix} W_{E ref}^{+} & W_{E ref}^{-} \\ W_{H ref}^{+} & W_{H ref}^{-} \end{matrix}) (\begin{matrix} c_{inc}^{+} \\ c_{ref}^{-} \end{matrix}) \\ (\begin{matrix} E_{x}^{ref} (0) \\ E_{y}^{ref} (0) \\ {\tilde{H}}_{x}^{ref} (0) \\ {\tilde{H}}_{y}^{ref} (0) \end{matrix}) = (\begin{matrix} W_{E ref}^{-} \cdot c_{ref}^{-} \\ W_{H ref}^{-} \cdot c_{ref}^{-} \end{matrix}) = (\begin{matrix} W_{E ref}^{-} \cdot S_{11}^{global} \cdot c_{inc}^{+} \\ W_{H ref}^{-} \cdot S_{11}^{global} \cdot c_{inc}^{+} \end{matrix}) \end{array}

\begin{array}{l} ψ (L_{total}) = (\begin{matrix} W_{E trn}^{+} & W_{E trn}^{-} \\ W_{H trn}^{+} & W_{H trn}^{-} \end{matrix}) (\begin{matrix} c_{trn}^{+} \\ c_{trn}^{-} = 0 \end{matrix}) \\ (\begin{matrix} E_{x}^{trn} (L_{total}) \\ E_{y}^{trn} (L_{total}) \\ {\tilde{H}}_{x}^{trn} (L_{total}) \\ {\tilde{H}}_{y}^{trn} (L_{total}) \end{matrix}) = (\begin{matrix} W_{E trn}^{+} \cdot c_{trn}^{+} \\ W_{H trn}^{+} \cdot c_{trn}^{+} \end{matrix}) = (\begin{matrix} W_{E trn}^{+} \cdot S_{21}^{global} \cdot c_{inc}^{+} \\ W_{H trn}^{+} \cdot S_{21}^{global} \cdot c_{inc}^{+} \end{matrix}) \end{array}

\begin{array}{l} E_{z}^{ref/trn} = \frac{i (- {\tilde{H}}_{y}^{ref/trn} n_{x} + {\tilde{H}}_{x}^{ref/trn} n_{y}) - E_{x}^{ref/trn} ε_{31}^{ref/trn} - E_{y}^{ref/trn} ε_{32}^{ref/trn}}{ε_{33}^{ref/trn}} \\ {\tilde{H}}_{z}^{ref/trn} = \frac{i (- E_{y}^{ref/trn} n_{x} + E_{x}^{ref/trn} n_{y}) - {\tilde{H}}_{x}^{ref/trn} μ_{31}^{ref/trn} - {\tilde{H}}_{y}^{ref/trn} μ_{32}^{ref/trn}}{μ_{33}^{ref/trn}} \end{array}

c_{inc}^{+} = {(W_{E ref}^{+})}^{- 1} (\begin{matrix} E_{x}^{inc} (0) \\ E_{y}^{inc} (0) \end{matrix}) .

R = {(\frac{‖ E_{}^{ref} ‖}{‖ E_{}^{inc} ‖})}^{2} .

\begin{matrix} {[x y]}^{T} = {[(E_{x r}^{} + i E_{x i}^{}) e_{}^{i ω t} ​ (E_{y r}^{} + i E_{y i}^{}) e_{}^{i ω t}]}^{T} \\ = {[E_{x r}^{} \cos (ω t) - E_{x i}^{} \sin (ω t) ​ E_{y r}^{} \cos (ω t) - E_{y i}^{} \sin (ω t)]}^{T} \end{matrix}

\begin{array}{l} \cos^{2} (ω t) + \sin^{2} (ω t) \\ = \frac{(E_{y i}^{2} + E_{y r}^{2}) x_{}^{2}}{{(E_{x r}^{} E_{y i}^{} - E_{x i}^{} E_{y r}^{})}^{2}} - 2 \frac{(E_{x i}^{} E_{y i}^{} + E_{x r}^{} E_{y r}^{}) x y}{{(E_{x r}^{} E_{y i}^{} - E_{x i}^{} E_{y r}^{})}^{2}} + \frac{(E_{x i}^{2} + E_{x r}^{2}) y_{}^{2}}{{(E_{x r}^{} E_{y i}^{} - E_{x i}^{} E_{y r}^{})}^{2}} \\ = 1 \end{array}

\begin{array}{l} X_{}^{T} Q X = 1 \\ Q = (\begin{matrix} \frac{E_{y i}^{2} + E_{y r}^{2}}{{(E_{x r}^{} E_{y i}^{} - E_{x i}^{} E_{y r}^{})}^{2}} & - \frac{E_{x i}^{} E_{y i}^{} + E_{x r}^{} E_{y r}^{}}{{(E_{x r}^{} E_{y i}^{} - E_{x i}^{} E_{y r}^{})}^{2}} \\ - \frac{E_{x i}^{} E_{y i}^{} + E_{x r}^{} E_{y r}^{}}{{(E_{x r}^{} E_{y i}^{} - E_{x i}^{} E_{y r}^{})}^{2}} & \frac{E_{x i}^{2} + E_{x r}^{2}}{{(E_{x r}^{} E_{y i}^{} - E_{x i}^{} E_{y r}^{})}^{2}} \end{matrix}) \\ X = {[x y]}^{T} \end{array}

\begin{array}{l} X'_{}^{T} B X' = 1 \Leftrightarrow (X'_{}^{T} P_{}^{- 1}) Q (P X') = 1 \Leftrightarrow {(P X')}^{T} Q (P X') = 1 \\ ∴ X = P X' \end{array}

P = (\begin{matrix} \cos (φ_{K}) & \sin (φ_{K}) \\ - \sin (φ_{K}) & \cos (φ_{K}) \end{matrix})

φ_{K} = arc \tan [\frac{2 (E'_{x i}^{} E_{y i}^{} + E'_{x r}^{} E_{y r}^{})}{E'_{x i}^{2} + E'_{x r}^{2} - E_{y i}^{2} - E_{y r}^{2} + \sqrt{- 4 {(E'_{x r}^{} E_{y i}^{} - E'_{x i}^{} E_{y r}^{})}^{2} + {(E_{y i}^{2} + E_{y r}^{2} + E'_{x i}^{2} + E'_{x r}^{2})}^{2}}}]

η_{K} = arc \tan [\sqrt{\frac{E_{y i}^{2} + E_{y r}^{2} + E'_{x i}^{2} + E'_{x r}^{2} - \sqrt{- 4 {(E'_{x r}^{} E_{y i}^{} - E'_{x i}^{} E_{y r}^{})}^{2} + {(E_{y i}^{2} + E_{y r}^{2} + E'_{x i}^{2} + E'_{x r}^{2})}^{2}}}{E_{y i}^{2} + E_{y r}^{2} + E'_{x i}^{2} + E'_{x r}^{2} + \sqrt{- 4 {(E'_{x r}^{} E_{y i}^{} - E'_{x i}^{} E_{y r}^{})}^{2} + {(E_{y i}^{2} + E_{y r}^{2} + E'_{x i}^{2} + E'_{x r}^{2})}^{2}}}}]