Abstract

Magnetoplasmonic sensors are attractive candidates for ultrasensitive chemical and biomedical sensor applications. A variety of ferromagnetic metal thin films have been used for magnetoplasmonic device applications, yet the dependence of sensor performance on the optical and magneto-optical properties of ferromagnetic metal materials has been rarely studied. In this work, we report the study of enhanced magneto-optical Kerr effect (MOKE) and sensing performance in Au/FexCo1x bilayer magneto-optical surface plasmon resonance (MOSPR) transducers. The optical constants of FexCo1x (x=0, 0.29, 0.47, 0.65, and 1) in a sputter-deposited Au/FexCo1x device are characterized by the attenuated total internal reflection (ATR) method. FexCo1x thin films show different MOKEs as a function of the chemical concentration, with the highest transverse MOKE signal observed in Fe0.7Co0.3. Index sensing performance is closely related to the material’s optical and magneto-optical constants. By studying the sensing performance in the parameter space of the Au/FexCo1x bilayer thicknesses, the highest sensitivity is found to be 0.385 (theoretical) and 0.306  RIU1 (experimental) in the Au/Fe0.7Co0.3 MOSPR devices. Our research highlights the influence of the optical properties of ferromagnetic material to device sensitivity in MOSPR transducers. The high sensitivity in Au/FexCo1x MOSPR devices make these structures attractive candidates for chemical and biomedical sensing applications.

© 2017 Chinese Laser Press

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References

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  1. G. G. Nenninger, P. Tobiška, J. Homola, and S. S. Yee, “Long-range surface plasmons for high-resolution surface plasmon resonance sensors,” Sens. Actuators B 74, 145–151 (2001).
    [Crossref]
  2. G. G. Nenninger, M. Piliarik, and J. Homola, “Data analysis for optical sensors based on spectroscopy of surface plasmons,” Meas. Sci. Technol. 13, 2038–2046 (2002).
    [Crossref]
  3. 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, 1085–1087 (2006).
    [Crossref]
  4. J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. Clavero, A. Cebollada, and R. Clarke, “Surface magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76, 153402 (2007).
    [Crossref]
  5. C. Clavero, K. Yang, J. R. Skuza, and R. A. Lukaszew, “Magnetic field modulation of intense surface plasmon polaritons,” Opt. Express 18, 7743–7752 (2010).
    [Crossref]
  6. 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, 8336–8346 (2011).
    [Crossref]
  7. K. Kämpf, S. Kübler, F. W. Herberg, and A. Ehresmann, “Magneto-optic surface plasmon resonance optimum layers: simulations for biological relevant refractive index changes,” J. Appl. Phys. 112, 034505 (2012).
    [Crossref]
  8. E. Ferreiro-Vila, J. B. González-Díaz, R. Fermento, M. U. González, A. García-Martín, J. M. García-Martín, and E. M. Sandoval, “Intertwined magneto-optical and plasmonic effects in Ag/Co/Ag layered structures,” Phys. Rev. B 80, 125132 (2009).
    [Crossref]
  9. T. J. Wang, K. H. Lee, and T. T. Chen, “Sensitivity enhancement of magneto-optic surface plasmon resonance sensors with noble/ferromagnetic metal heterostructure,” Laser Phys. 24, 036001 (2014).
    [Crossref]
  10. J. Vlček, M. Lesňák, P. Otipka, and J. Sobota, “Magneto-plasmonic properties of Au/Fe/Au planar nanostructures: theory and experiments,” Proc. Mater. Sci. 12, 136–141 (2016).
  11. M. Moradi, S. M. Mohseni, S. Mahmoodi, D. Rezvani, N. Ansari, S. Chung, and J. Akerman, “Au/NiFe magnetoplasmonics: large enhancement of magneto-optical Kerr effect for magnetic field sensors and memories,” Electron. Mater. Lett. 11, 440–446 (2015).
    [Crossref]
  12. H. T. Huang, T. R. Ger, Y. R. Xu, and J. Y. Lai, “Surface plasmon induced enhancement with magneto-optical layer,” J. Appl. Phys. 115, 17E313 (2014).
    [Crossref]
  13. S. Kübler, N. Müglich, and A. Ehresmann, “Magneto-optic surface plasmon resonance of Au/IrMn/Co/Au exchange biased layer systems,” J. Appl. Phys. 116, 064502 (2014).
    [Crossref]
  14. D. Regatos, D. Fariña, A. Calle, A. Cebollada, B. Sepúlveda, G. Armelles, and L. M. Lechuga, “Au/Fe/Au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
    [Crossref]
  15. T. Miihge, T. Zeidler, Q. Wang, C. Morawe, N. Metoki, and H. Zabel, “Structural and magnetic studies of FexCo1-x(001) alloy films on MgO (001) substrates,” J. Appl. Phys. 77, 1055–1060 (1995).
    [Crossref]
  16. S. Višňovský, R. Lopušník, M. Bauer, J. Bok, J. Fassbender, and B. Hillebrands, “Magnetooptic ellipsometry in multilayers at arbitrary magnetization,” Opt. Express 9, 121–135 (2001).
    [Crossref]
  17. J. P. Jay, E. Jêdryka, M. Wójcik, J. Dekoster, G. langouche, and P. Panissod, “On the stability of BCC Co in Co/Fe superlattices an NMR and XRD study,” Z. Phys. B 101, 329–337 (1997).
    [Crossref]
  18. D. N. Tai, A. Chaiken, and T. W. Barbee, “Structural transition in Cu/Fe multilayered thin films,” MRS Online Proc. Library Arch. 441, 353–358 (2011).
    [Crossref]
  19. R. S. Sundar and S. C. Deevi, “Soft magnetic FeCo alloys: alloy development, processing, and properties,” Int. Mater. Rev. 50, 157–192 (2005).
    [Crossref]
  20. D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Benderky, S. E. Lofland, M. Wutting, and I. Takeuchi, “Giant magnetostriction in annealed Co1-xFex thin-films,” Nat. Commun. 2, 518 (2013).
    [Crossref]
  21. Y. B. Xu, Q. Y. Jin, Y. Zhai, M. Lu, Y. Z. Miao, Q. S. Bie, and H. R. Zhai, “Complex magneto‐optical Kerr rotation of Fe, FeCo/Cu multilayer films,” J. Appl. Phys. 74, 3470–3474 (1993).
    [Crossref]
  22. K. J. Kim, S. J. Lee, and D. W. Lynch, “Study of optical properties and electronic structure of ferromagnetic FeCo,” Solid State Commun. 114, 457–460 (2000).
    [Crossref]
  23. A. K. Zvezdin and V. A. Kotov, Modern Magnetooptic and Magnetooptical Materials (CRC Press, 1997).

2016 (1)

J. Vlček, M. Lesňák, P. Otipka, and J. Sobota, “Magneto-plasmonic properties of Au/Fe/Au planar nanostructures: theory and experiments,” Proc. Mater. Sci. 12, 136–141 (2016).

2015 (1)

M. Moradi, S. M. Mohseni, S. Mahmoodi, D. Rezvani, N. Ansari, S. Chung, and J. Akerman, “Au/NiFe magnetoplasmonics: large enhancement of magneto-optical Kerr effect for magnetic field sensors and memories,” Electron. Mater. Lett. 11, 440–446 (2015).
[Crossref]

2014 (3)

H. T. Huang, T. R. Ger, Y. R. Xu, and J. Y. Lai, “Surface plasmon induced enhancement with magneto-optical layer,” J. Appl. Phys. 115, 17E313 (2014).
[Crossref]

S. Kübler, N. Müglich, and A. Ehresmann, “Magneto-optic surface plasmon resonance of Au/IrMn/Co/Au exchange biased layer systems,” J. Appl. Phys. 116, 064502 (2014).
[Crossref]

T. J. Wang, K. H. Lee, and T. T. Chen, “Sensitivity enhancement of magneto-optic surface plasmon resonance sensors with noble/ferromagnetic metal heterostructure,” Laser Phys. 24, 036001 (2014).
[Crossref]

2013 (1)

D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Benderky, S. E. Lofland, M. Wutting, and I. Takeuchi, “Giant magnetostriction in annealed Co1-xFex thin-films,” Nat. Commun. 2, 518 (2013).
[Crossref]

2012 (1)

K. Kämpf, S. Kübler, F. W. Herberg, and A. Ehresmann, “Magneto-optic surface plasmon resonance optimum layers: simulations for biological relevant refractive index changes,” J. Appl. Phys. 112, 034505 (2012).
[Crossref]

2011 (2)

2010 (2)

D. Regatos, D. Fariña, A. Calle, A. Cebollada, B. Sepúlveda, G. Armelles, and L. M. Lechuga, “Au/Fe/Au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
[Crossref]

C. Clavero, K. Yang, J. R. Skuza, and R. A. Lukaszew, “Magnetic field modulation of intense surface plasmon polaritons,” Opt. Express 18, 7743–7752 (2010).
[Crossref]

2009 (1)

E. Ferreiro-Vila, J. B. González-Díaz, R. Fermento, M. U. González, A. García-Martín, J. M. García-Martín, and E. M. Sandoval, “Intertwined magneto-optical and plasmonic effects in Ag/Co/Ag layered structures,” Phys. Rev. B 80, 125132 (2009).
[Crossref]

2007 (1)

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

2006 (1)

2005 (1)

R. S. Sundar and S. C. Deevi, “Soft magnetic FeCo alloys: alloy development, processing, and properties,” Int. Mater. Rev. 50, 157–192 (2005).
[Crossref]

2002 (1)

G. G. Nenninger, M. Piliarik, and J. Homola, “Data analysis for optical sensors based on spectroscopy of surface plasmons,” Meas. Sci. Technol. 13, 2038–2046 (2002).
[Crossref]

2001 (2)

G. G. Nenninger, P. Tobiška, J. Homola, and S. S. Yee, “Long-range surface plasmons for high-resolution surface plasmon resonance sensors,” Sens. Actuators B 74, 145–151 (2001).
[Crossref]

S. Višňovský, R. Lopušník, M. Bauer, J. Bok, J. Fassbender, and B. Hillebrands, “Magnetooptic ellipsometry in multilayers at arbitrary magnetization,” Opt. Express 9, 121–135 (2001).
[Crossref]

2000 (1)

K. J. Kim, S. J. Lee, and D. W. Lynch, “Study of optical properties and electronic structure of ferromagnetic FeCo,” Solid State Commun. 114, 457–460 (2000).
[Crossref]

1997 (1)

J. P. Jay, E. Jêdryka, M. Wójcik, J. Dekoster, G. langouche, and P. Panissod, “On the stability of BCC Co in Co/Fe superlattices an NMR and XRD study,” Z. Phys. B 101, 329–337 (1997).
[Crossref]

1995 (1)

T. Miihge, T. Zeidler, Q. Wang, C. Morawe, N. Metoki, and H. Zabel, “Structural and magnetic studies of FexCo1-x(001) alloy films on MgO (001) substrates,” J. Appl. Phys. 77, 1055–1060 (1995).
[Crossref]

1993 (1)

Y. B. Xu, Q. Y. Jin, Y. Zhai, M. Lu, Y. Z. Miao, Q. S. Bie, and H. R. Zhai, “Complex magneto‐optical Kerr rotation of Fe, FeCo/Cu multilayer films,” J. Appl. Phys. 74, 3470–3474 (1993).
[Crossref]

Akerman, J.

M. Moradi, S. M. Mohseni, S. Mahmoodi, D. Rezvani, N. Ansari, S. Chung, and J. Akerman, “Au/NiFe magnetoplasmonics: large enhancement of magneto-optical Kerr effect for magnetic field sensors and memories,” Electron. Mater. Lett. 11, 440–446 (2015).
[Crossref]

Ansari, N.

M. Moradi, S. M. Mohseni, S. Mahmoodi, D. Rezvani, N. Ansari, S. Chung, and J. Akerman, “Au/NiFe magnetoplasmonics: large enhancement of magneto-optical Kerr effect for magnetic field sensors and memories,” Electron. Mater. Lett. 11, 440–446 (2015).
[Crossref]

Armelles, G.

D. Regatos, D. Fariña, A. Calle, A. Cebollada, B. Sepúlveda, G. Armelles, and L. M. Lechuga, “Au/Fe/Au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
[Crossref]

J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. Clavero, A. Cebollada, and R. Clarke, “Surface magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76, 153402 (2007).
[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, 1085–1087 (2006).
[Crossref]

Barbee, T. W.

D. N. Tai, A. Chaiken, and T. W. Barbee, “Structural transition in Cu/Fe multilayered thin films,” MRS Online Proc. Library Arch. 441, 353–358 (2011).
[Crossref]

Bauer, M.

Benderky, L. A.

D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Benderky, S. E. Lofland, M. Wutting, and I. Takeuchi, “Giant magnetostriction in annealed Co1-xFex thin-films,” Nat. Commun. 2, 518 (2013).
[Crossref]

Bie, Q. S.

Y. B. Xu, Q. Y. Jin, Y. Zhai, M. Lu, Y. Z. Miao, Q. S. Bie, and H. R. Zhai, “Complex magneto‐optical Kerr rotation of Fe, FeCo/Cu multilayer films,” J. Appl. Phys. 74, 3470–3474 (1993).
[Crossref]

Bok, J.

Calle, A.

D. Regatos, D. Fariña, A. Calle, A. Cebollada, B. Sepúlveda, G. Armelles, and L. M. Lechuga, “Au/Fe/Au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
[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, 1085–1087 (2006).
[Crossref]

Carrascosa, L. G.

Cebollada, A.

D. Regatos, D. Fariña, A. Calle, A. Cebollada, B. Sepúlveda, G. Armelles, and L. M. Lechuga, “Au/Fe/Au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
[Crossref]

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

Chaiken, A.

D. N. Tai, A. Chaiken, and T. W. Barbee, “Structural transition in Cu/Fe multilayered thin films,” MRS Online Proc. Library Arch. 441, 353–358 (2011).
[Crossref]

Chen, T. T.

T. J. Wang, K. H. Lee, and T. T. Chen, “Sensitivity enhancement of magneto-optic surface plasmon resonance sensors with noble/ferromagnetic metal heterostructure,” Laser Phys. 24, 036001 (2014).
[Crossref]

Chung, S.

M. Moradi, S. M. Mohseni, S. Mahmoodi, D. Rezvani, N. Ansari, S. Chung, and J. Akerman, “Au/NiFe magnetoplasmonics: large enhancement of magneto-optical Kerr effect for magnetic field sensors and memories,” Electron. Mater. Lett. 11, 440–446 (2015).
[Crossref]

Clarke, R.

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

Clavero, C.

Clavero, J. M.

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

Deevi, S. C.

R. S. Sundar and S. C. Deevi, “Soft magnetic FeCo alloys: alloy development, processing, and properties,” Int. Mater. Rev. 50, 157–192 (2005).
[Crossref]

Dekoster, J.

J. P. Jay, E. Jêdryka, M. Wójcik, J. Dekoster, G. langouche, and P. Panissod, “On the stability of BCC Co in Co/Fe superlattices an NMR and XRD study,” Z. Phys. B 101, 329–337 (1997).
[Crossref]

Ehresmann, A.

S. Kübler, N. Müglich, and A. Ehresmann, “Magneto-optic surface plasmon resonance of Au/IrMn/Co/Au exchange biased layer systems,” J. Appl. Phys. 116, 064502 (2014).
[Crossref]

K. Kämpf, S. Kübler, F. W. Herberg, and A. Ehresmann, “Magneto-optic surface plasmon resonance optimum layers: simulations for biological relevant refractive index changes,” J. Appl. Phys. 112, 034505 (2012).
[Crossref]

Fariña, D.

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, 8336–8346 (2011).
[Crossref]

D. Regatos, D. Fariña, A. Calle, A. Cebollada, B. Sepúlveda, G. Armelles, and L. M. Lechuga, “Au/Fe/Au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
[Crossref]

Fassbender, J.

Fermento, R.

E. Ferreiro-Vila, J. B. González-Díaz, R. Fermento, M. U. González, A. García-Martín, J. M. García-Martín, and E. M. Sandoval, “Intertwined magneto-optical and plasmonic effects in Ag/Co/Ag layered structures,” Phys. Rev. B 80, 125132 (2009).
[Crossref]

Ferreiro-Vila, E.

E. Ferreiro-Vila, J. B. González-Díaz, R. Fermento, M. U. González, A. García-Martín, J. M. García-Martín, and E. M. Sandoval, “Intertwined magneto-optical and plasmonic effects in Ag/Co/Ag layered structures,” Phys. Rev. B 80, 125132 (2009).
[Crossref]

García-Martín, A.

E. Ferreiro-Vila, J. B. González-Díaz, R. Fermento, M. U. González, A. García-Martín, J. M. García-Martín, and E. M. Sandoval, “Intertwined magneto-optical and plasmonic effects in Ag/Co/Ag layered structures,” Phys. Rev. B 80, 125132 (2009).
[Crossref]

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

García-Martín, J. M.

E. Ferreiro-Vila, J. B. González-Díaz, R. Fermento, M. U. González, A. García-Martín, J. M. García-Martín, and E. M. Sandoval, “Intertwined magneto-optical and plasmonic effects in Ag/Co/Ag layered structures,” Phys. Rev. B 80, 125132 (2009).
[Crossref]

Ger, T. R.

H. T. Huang, T. R. Ger, Y. R. Xu, and J. Y. Lai, “Surface plasmon induced enhancement with magneto-optical layer,” J. Appl. Phys. 115, 17E313 (2014).
[Crossref]

González, M. U.

E. Ferreiro-Vila, J. B. González-Díaz, R. Fermento, M. U. González, A. García-Martín, J. M. García-Martín, and E. M. Sandoval, “Intertwined magneto-optical and plasmonic effects in Ag/Co/Ag layered structures,” Phys. Rev. B 80, 125132 (2009).
[Crossref]

González-Díaz, J. B.

E. Ferreiro-Vila, J. B. González-Díaz, R. Fermento, M. U. González, A. García-Martín, J. M. García-Martín, and E. M. Sandoval, “Intertwined magneto-optical and plasmonic effects in Ag/Co/Ag layered structures,” Phys. Rev. B 80, 125132 (2009).
[Crossref]

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

Hattrick-Simpers, J.

D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Benderky, S. E. Lofland, M. Wutting, and I. Takeuchi, “Giant magnetostriction in annealed Co1-xFex thin-films,” Nat. Commun. 2, 518 (2013).
[Crossref]

Herberg, F. W.

K. Kämpf, S. Kübler, F. W. Herberg, and A. Ehresmann, “Magneto-optic surface plasmon resonance optimum layers: simulations for biological relevant refractive index changes,” J. Appl. Phys. 112, 034505 (2012).
[Crossref]

Hillebrands, B.

Homola, J.

G. G. Nenninger, M. Piliarik, and J. Homola, “Data analysis for optical sensors based on spectroscopy of surface plasmons,” Meas. Sci. Technol. 13, 2038–2046 (2002).
[Crossref]

G. G. Nenninger, P. Tobiška, J. Homola, and S. S. Yee, “Long-range surface plasmons for high-resolution surface plasmon resonance sensors,” Sens. Actuators B 74, 145–151 (2001).
[Crossref]

Huang, H. T.

H. T. Huang, T. R. Ger, Y. R. Xu, and J. Y. Lai, “Surface plasmon induced enhancement with magneto-optical layer,” J. Appl. Phys. 115, 17E313 (2014).
[Crossref]

Hunter, D.

D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Benderky, S. E. Lofland, M. Wutting, and I. Takeuchi, “Giant magnetostriction in annealed Co1-xFex thin-films,” Nat. Commun. 2, 518 (2013).
[Crossref]

Jay, J. P.

J. P. Jay, E. Jêdryka, M. Wójcik, J. Dekoster, G. langouche, and P. Panissod, “On the stability of BCC Co in Co/Fe superlattices an NMR and XRD study,” Z. Phys. B 101, 329–337 (1997).
[Crossref]

Jêdryka, E.

J. P. Jay, E. Jêdryka, M. Wójcik, J. Dekoster, G. langouche, and P. Panissod, “On the stability of BCC Co in Co/Fe superlattices an NMR and XRD study,” Z. Phys. B 101, 329–337 (1997).
[Crossref]

Jin, Q. Y.

Y. B. Xu, Q. Y. Jin, Y. Zhai, M. Lu, Y. Z. Miao, Q. S. Bie, and H. R. Zhai, “Complex magneto‐optical Kerr rotation of Fe, FeCo/Cu multilayer films,” J. Appl. Phys. 74, 3470–3474 (1993).
[Crossref]

Kämpf, K.

K. Kämpf, S. Kübler, F. W. Herberg, and A. Ehresmann, “Magneto-optic surface plasmon resonance optimum layers: simulations for biological relevant refractive index changes,” J. Appl. Phys. 112, 034505 (2012).
[Crossref]

Kazantseva, N.

D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Benderky, S. E. Lofland, M. Wutting, and I. Takeuchi, “Giant magnetostriction in annealed Co1-xFex thin-films,” Nat. Commun. 2, 518 (2013).
[Crossref]

Kim, K. J.

K. J. Kim, S. J. Lee, and D. W. Lynch, “Study of optical properties and electronic structure of ferromagnetic FeCo,” Solid State Commun. 114, 457–460 (2000).
[Crossref]

Kotov, V. A.

A. K. Zvezdin and V. A. Kotov, Modern Magnetooptic and Magnetooptical Materials (CRC Press, 1997).

Kübler, S.

S. Kübler, N. Müglich, and A. Ehresmann, “Magneto-optic surface plasmon resonance of Au/IrMn/Co/Au exchange biased layer systems,” J. Appl. Phys. 116, 064502 (2014).
[Crossref]

K. Kämpf, S. Kübler, F. W. Herberg, and A. Ehresmann, “Magneto-optic surface plasmon resonance optimum layers: simulations for biological relevant refractive index changes,” J. Appl. Phys. 112, 034505 (2012).
[Crossref]

Lai, J. Y.

H. T. Huang, T. R. Ger, Y. R. Xu, and J. Y. Lai, “Surface plasmon induced enhancement with magneto-optical layer,” J. Appl. Phys. 115, 17E313 (2014).
[Crossref]

langouche, G.

J. P. Jay, E. Jêdryka, M. Wójcik, J. Dekoster, G. langouche, and P. Panissod, “On the stability of BCC Co in Co/Fe superlattices an NMR and XRD study,” Z. Phys. B 101, 329–337 (1997).
[Crossref]

Lechuga, L. M.

Lee, K. H.

T. J. Wang, K. H. Lee, and T. T. Chen, “Sensitivity enhancement of magneto-optic surface plasmon resonance sensors with noble/ferromagnetic metal heterostructure,” Laser Phys. 24, 036001 (2014).
[Crossref]

Lee, S. J.

K. J. Kim, S. J. Lee, and D. W. Lynch, “Study of optical properties and electronic structure of ferromagnetic FeCo,” Solid State Commun. 114, 457–460 (2000).
[Crossref]

Lesnák, M.

J. Vlček, M. Lesňák, P. Otipka, and J. Sobota, “Magneto-plasmonic properties of Au/Fe/Au planar nanostructures: theory and experiments,” Proc. Mater. Sci. 12, 136–141 (2016).

Lofland, S. E.

D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Benderky, S. E. Lofland, M. Wutting, and I. Takeuchi, “Giant magnetostriction in annealed Co1-xFex thin-films,” Nat. Commun. 2, 518 (2013).
[Crossref]

Lopušník, R.

Lu, M.

Y. B. Xu, Q. Y. Jin, Y. Zhai, M. Lu, Y. Z. Miao, Q. S. Bie, and H. R. Zhai, “Complex magneto‐optical Kerr rotation of Fe, FeCo/Cu multilayer films,” J. Appl. Phys. 74, 3470–3474 (1993).
[Crossref]

Lukaszew, R. A.

Lynch, D. W.

K. J. Kim, S. J. Lee, and D. W. Lynch, “Study of optical properties and electronic structure of ferromagnetic FeCo,” Solid State Commun. 114, 457–460 (2000).
[Crossref]

Mahmoodi, S.

M. Moradi, S. M. Mohseni, S. Mahmoodi, D. Rezvani, N. Ansari, S. Chung, and J. Akerman, “Au/NiFe magnetoplasmonics: large enhancement of magneto-optical Kerr effect for magnetic field sensors and memories,” Electron. Mater. Lett. 11, 440–446 (2015).
[Crossref]

Mehta, A.

D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Benderky, S. E. Lofland, M. Wutting, and I. Takeuchi, “Giant magnetostriction in annealed Co1-xFex thin-films,” Nat. Commun. 2, 518 (2013).
[Crossref]

Metoki, N.

T. Miihge, T. Zeidler, Q. Wang, C. Morawe, N. Metoki, and H. Zabel, “Structural and magnetic studies of FexCo1-x(001) alloy films on MgO (001) substrates,” J. Appl. Phys. 77, 1055–1060 (1995).
[Crossref]

Miao, Y. Z.

Y. B. Xu, Q. Y. Jin, Y. Zhai, M. Lu, Y. Z. Miao, Q. S. Bie, and H. R. Zhai, “Complex magneto‐optical Kerr rotation of Fe, FeCo/Cu multilayer films,” J. Appl. Phys. 74, 3470–3474 (1993).
[Crossref]

Miihge, T.

T. Miihge, T. Zeidler, Q. Wang, C. Morawe, N. Metoki, and H. Zabel, “Structural and magnetic studies of FexCo1-x(001) alloy films on MgO (001) substrates,” J. Appl. Phys. 77, 1055–1060 (1995).
[Crossref]

Mohseni, S. M.

M. Moradi, S. M. Mohseni, S. Mahmoodi, D. Rezvani, N. Ansari, S. Chung, and J. Akerman, “Au/NiFe magnetoplasmonics: large enhancement of magneto-optical Kerr effect for magnetic field sensors and memories,” Electron. Mater. Lett. 11, 440–446 (2015).
[Crossref]

Moradi, M.

M. Moradi, S. M. Mohseni, S. Mahmoodi, D. Rezvani, N. Ansari, S. Chung, and J. Akerman, “Au/NiFe magnetoplasmonics: large enhancement of magneto-optical Kerr effect for magnetic field sensors and memories,” Electron. Mater. Lett. 11, 440–446 (2015).
[Crossref]

Morawe, C.

T. Miihge, T. Zeidler, Q. Wang, C. Morawe, N. Metoki, and H. Zabel, “Structural and magnetic studies of FexCo1-x(001) alloy films on MgO (001) substrates,” J. Appl. Phys. 77, 1055–1060 (1995).
[Crossref]

Müglich, N.

S. Kübler, N. Müglich, and A. Ehresmann, “Magneto-optic surface plasmon resonance of Au/IrMn/Co/Au exchange biased layer systems,” J. Appl. Phys. 116, 064502 (2014).
[Crossref]

Nenninger, G. G.

G. G. Nenninger, M. Piliarik, and J. Homola, “Data analysis for optical sensors based on spectroscopy of surface plasmons,” Meas. Sci. Technol. 13, 2038–2046 (2002).
[Crossref]

G. G. Nenninger, P. Tobiška, J. Homola, and S. S. Yee, “Long-range surface plasmons for high-resolution surface plasmon resonance sensors,” Sens. Actuators B 74, 145–151 (2001).
[Crossref]

Osborn, W.

D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Benderky, S. E. Lofland, M. Wutting, and I. Takeuchi, “Giant magnetostriction in annealed Co1-xFex thin-films,” Nat. Commun. 2, 518 (2013).
[Crossref]

Otipka, P.

J. Vlček, M. Lesňák, P. Otipka, and J. Sobota, “Magneto-plasmonic properties of Au/Fe/Au planar nanostructures: theory and experiments,” Proc. Mater. Sci. 12, 136–141 (2016).

Panissod, P.

J. P. Jay, E. Jêdryka, M. Wójcik, J. Dekoster, G. langouche, and P. Panissod, “On the stability of BCC Co in Co/Fe superlattices an NMR and XRD study,” Z. Phys. B 101, 329–337 (1997).
[Crossref]

Piliarik, M.

G. G. Nenninger, M. Piliarik, and J. Homola, “Data analysis for optical sensors based on spectroscopy of surface plasmons,” Meas. Sci. Technol. 13, 2038–2046 (2002).
[Crossref]

Regatos, D.

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, 8336–8346 (2011).
[Crossref]

D. Regatos, D. Fariña, A. Calle, A. Cebollada, B. Sepúlveda, G. Armelles, and L. M. Lechuga, “Au/Fe/Au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
[Crossref]

Rezvani, D.

M. Moradi, S. M. Mohseni, S. Mahmoodi, D. Rezvani, N. Ansari, S. Chung, and J. Akerman, “Au/NiFe magnetoplasmonics: large enhancement of magneto-optical Kerr effect for magnetic field sensors and memories,” Electron. Mater. Lett. 11, 440–446 (2015).
[Crossref]

Sandoval, E. M.

E. Ferreiro-Vila, J. B. González-Díaz, R. Fermento, M. U. González, A. García-Martín, J. M. García-Martín, and E. M. Sandoval, “Intertwined magneto-optical and plasmonic effects in Ag/Co/Ag layered structures,” Phys. Rev. B 80, 125132 (2009).
[Crossref]

Sepúlveda, B.

Skuza, J. R.

Sobota, J.

J. Vlček, M. Lesňák, P. Otipka, and J. Sobota, “Magneto-plasmonic properties of Au/Fe/Au planar nanostructures: theory and experiments,” Proc. Mater. Sci. 12, 136–141 (2016).

Suchoski, R.

D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Benderky, S. E. Lofland, M. Wutting, and I. Takeuchi, “Giant magnetostriction in annealed Co1-xFex thin-films,” Nat. Commun. 2, 518 (2013).
[Crossref]

Sundar, R. S.

R. S. Sundar and S. C. Deevi, “Soft magnetic FeCo alloys: alloy development, processing, and properties,” Int. Mater. Rev. 50, 157–192 (2005).
[Crossref]

Tai, D. N.

D. N. Tai, A. Chaiken, and T. W. Barbee, “Structural transition in Cu/Fe multilayered thin films,” MRS Online Proc. Library Arch. 441, 353–358 (2011).
[Crossref]

Takahashi, R.

D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Benderky, S. E. Lofland, M. Wutting, and I. Takeuchi, “Giant magnetostriction in annealed Co1-xFex thin-films,” Nat. Commun. 2, 518 (2013).
[Crossref]

Takeuchi, I.

D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Benderky, S. E. Lofland, M. Wutting, and I. Takeuchi, “Giant magnetostriction in annealed Co1-xFex thin-films,” Nat. Commun. 2, 518 (2013).
[Crossref]

Tobiška, P.

G. G. Nenninger, P. Tobiška, J. Homola, and S. S. Yee, “Long-range surface plasmons for high-resolution surface plasmon resonance sensors,” Sens. Actuators B 74, 145–151 (2001).
[Crossref]

Višnovský, S.

Vlcek, J.

J. Vlček, M. Lesňák, P. Otipka, and J. Sobota, “Magneto-plasmonic properties of Au/Fe/Au planar nanostructures: theory and experiments,” Proc. Mater. Sci. 12, 136–141 (2016).

Wang, K.

D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Benderky, S. E. Lofland, M. Wutting, and I. Takeuchi, “Giant magnetostriction in annealed Co1-xFex thin-films,” Nat. Commun. 2, 518 (2013).
[Crossref]

Wang, Q.

T. Miihge, T. Zeidler, Q. Wang, C. Morawe, N. Metoki, and H. Zabel, “Structural and magnetic studies of FexCo1-x(001) alloy films on MgO (001) substrates,” J. Appl. Phys. 77, 1055–1060 (1995).
[Crossref]

Wang, T. J.

T. J. Wang, K. H. Lee, and T. T. Chen, “Sensitivity enhancement of magneto-optic surface plasmon resonance sensors with noble/ferromagnetic metal heterostructure,” Laser Phys. 24, 036001 (2014).
[Crossref]

Wójcik, M.

J. P. Jay, E. Jêdryka, M. Wójcik, J. Dekoster, G. langouche, and P. Panissod, “On the stability of BCC Co in Co/Fe superlattices an NMR and XRD study,” Z. Phys. B 101, 329–337 (1997).
[Crossref]

Wutting, M.

D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Benderky, S. E. Lofland, M. Wutting, and I. Takeuchi, “Giant magnetostriction in annealed Co1-xFex thin-films,” Nat. Commun. 2, 518 (2013).
[Crossref]

Xu, Y. B.

Y. B. Xu, Q. Y. Jin, Y. Zhai, M. Lu, Y. Z. Miao, Q. S. Bie, and H. R. Zhai, “Complex magneto‐optical Kerr rotation of Fe, FeCo/Cu multilayer films,” J. Appl. Phys. 74, 3470–3474 (1993).
[Crossref]

Xu, Y. R.

H. T. Huang, T. R. Ger, Y. R. Xu, and J. Y. Lai, “Surface plasmon induced enhancement with magneto-optical layer,” J. Appl. Phys. 115, 17E313 (2014).
[Crossref]

Yang, K.

Yee, S. S.

G. G. Nenninger, P. Tobiška, J. Homola, and S. S. Yee, “Long-range surface plasmons for high-resolution surface plasmon resonance sensors,” Sens. Actuators B 74, 145–151 (2001).
[Crossref]

Young, M. L.

D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Benderky, S. E. Lofland, M. Wutting, and I. Takeuchi, “Giant magnetostriction in annealed Co1-xFex thin-films,” Nat. Commun. 2, 518 (2013).
[Crossref]

Zabel, H.

T. Miihge, T. Zeidler, Q. Wang, C. Morawe, N. Metoki, and H. Zabel, “Structural and magnetic studies of FexCo1-x(001) alloy films on MgO (001) substrates,” J. Appl. Phys. 77, 1055–1060 (1995).
[Crossref]

Zeidler, T.

T. Miihge, T. Zeidler, Q. Wang, C. Morawe, N. Metoki, and H. Zabel, “Structural and magnetic studies of FexCo1-x(001) alloy films on MgO (001) substrates,” J. Appl. Phys. 77, 1055–1060 (1995).
[Crossref]

Zhai, H. R.

Y. B. Xu, Q. Y. Jin, Y. Zhai, M. Lu, Y. Z. Miao, Q. S. Bie, and H. R. Zhai, “Complex magneto‐optical Kerr rotation of Fe, FeCo/Cu multilayer films,” J. Appl. Phys. 74, 3470–3474 (1993).
[Crossref]

Zhai, Y.

Y. B. Xu, Q. Y. Jin, Y. Zhai, M. Lu, Y. Z. Miao, Q. S. Bie, and H. R. Zhai, “Complex magneto‐optical Kerr rotation of Fe, FeCo/Cu multilayer films,” J. Appl. Phys. 74, 3470–3474 (1993).
[Crossref]

Zvezdin, A. K.

A. K. Zvezdin and V. A. Kotov, Modern Magnetooptic and Magnetooptical Materials (CRC Press, 1997).

Electron. Mater. Lett. (1)

M. Moradi, S. M. Mohseni, S. Mahmoodi, D. Rezvani, N. Ansari, S. Chung, and J. Akerman, “Au/NiFe magnetoplasmonics: large enhancement of magneto-optical Kerr effect for magnetic field sensors and memories,” Electron. Mater. Lett. 11, 440–446 (2015).
[Crossref]

Int. Mater. Rev. (1)

R. S. Sundar and S. C. Deevi, “Soft magnetic FeCo alloys: alloy development, processing, and properties,” Int. Mater. Rev. 50, 157–192 (2005).
[Crossref]

J. Appl. Phys. (6)

H. T. Huang, T. R. Ger, Y. R. Xu, and J. Y. Lai, “Surface plasmon induced enhancement with magneto-optical layer,” J. Appl. Phys. 115, 17E313 (2014).
[Crossref]

S. Kübler, N. Müglich, and A. Ehresmann, “Magneto-optic surface plasmon resonance of Au/IrMn/Co/Au exchange biased layer systems,” J. Appl. Phys. 116, 064502 (2014).
[Crossref]

D. Regatos, D. Fariña, A. Calle, A. Cebollada, B. Sepúlveda, G. Armelles, and L. M. Lechuga, “Au/Fe/Au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
[Crossref]

T. Miihge, T. Zeidler, Q. Wang, C. Morawe, N. Metoki, and H. Zabel, “Structural and magnetic studies of FexCo1-x(001) alloy films on MgO (001) substrates,” J. Appl. Phys. 77, 1055–1060 (1995).
[Crossref]

K. Kämpf, S. Kübler, F. W. Herberg, and A. Ehresmann, “Magneto-optic surface plasmon resonance optimum layers: simulations for biological relevant refractive index changes,” J. Appl. Phys. 112, 034505 (2012).
[Crossref]

Y. B. Xu, Q. Y. Jin, Y. Zhai, M. Lu, Y. Z. Miao, Q. S. Bie, and H. R. Zhai, “Complex magneto‐optical Kerr rotation of Fe, FeCo/Cu multilayer films,” J. Appl. Phys. 74, 3470–3474 (1993).
[Crossref]

Laser Phys. (1)

T. J. Wang, K. H. Lee, and T. T. Chen, “Sensitivity enhancement of magneto-optic surface plasmon resonance sensors with noble/ferromagnetic metal heterostructure,” Laser Phys. 24, 036001 (2014).
[Crossref]

Meas. Sci. Technol. (1)

G. G. Nenninger, M. Piliarik, and J. Homola, “Data analysis for optical sensors based on spectroscopy of surface plasmons,” Meas. Sci. Technol. 13, 2038–2046 (2002).
[Crossref]

MRS Online Proc. Library Arch. (1)

D. N. Tai, A. Chaiken, and T. W. Barbee, “Structural transition in Cu/Fe multilayered thin films,” MRS Online Proc. Library Arch. 441, 353–358 (2011).
[Crossref]

Nat. Commun. (1)

D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick-Simpers, R. Suchoski, R. Takahashi, M. L. Young, A. Mehta, L. A. Benderky, S. E. Lofland, M. Wutting, and I. Takeuchi, “Giant magnetostriction in annealed Co1-xFex thin-films,” Nat. Commun. 2, 518 (2013).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. B (2)

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

E. Ferreiro-Vila, J. B. González-Díaz, R. Fermento, M. U. González, A. García-Martín, J. M. García-Martín, and E. M. Sandoval, “Intertwined magneto-optical and plasmonic effects in Ag/Co/Ag layered structures,” Phys. Rev. B 80, 125132 (2009).
[Crossref]

Proc. Mater. Sci. (1)

J. Vlček, M. Lesňák, P. Otipka, and J. Sobota, “Magneto-plasmonic properties of Au/Fe/Au planar nanostructures: theory and experiments,” Proc. Mater. Sci. 12, 136–141 (2016).

Sens. Actuators B (1)

G. G. Nenninger, P. Tobiška, J. Homola, and S. S. Yee, “Long-range surface plasmons for high-resolution surface plasmon resonance sensors,” Sens. Actuators B 74, 145–151 (2001).
[Crossref]

Solid State Commun. (1)

K. J. Kim, S. J. Lee, and D. W. Lynch, “Study of optical properties and electronic structure of ferromagnetic FeCo,” Solid State Commun. 114, 457–460 (2000).
[Crossref]

Z. Phys. B (1)

J. P. Jay, E. Jêdryka, M. Wójcik, J. Dekoster, G. langouche, and P. Panissod, “On the stability of BCC Co in Co/Fe superlattices an NMR and XRD study,” Z. Phys. B 101, 329–337 (1997).
[Crossref]

Other (1)

A. K. Zvezdin and V. A. Kotov, Modern Magnetooptic and Magnetooptical Materials (CRC Press, 1997).

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Figures (5)

Fig. 1.
Fig. 1. Schematic view of the Au / Fe x Co 1 x MOSPR sensor in a Kretschmann prism coupling system. The external magnetic field is applied along the y axis and perpendicular to the incident plane.
Fig. 2.
Fig. 2. (a) XRD figures of different component Fe x Co 1 x samples with about 18 nm thickness. (b) Reflectivity versus incidence angle for the Au ( 18    nm ) / Fe / Co / Fe 0.7 Co 0.3 ( 18    nm ) device; the inset shows the simulated result (dashed line) compared to the experimental curve (solid line) of Au ( 18    nm ) / Fe 0.7 Co 0.3 ( 18    nm ) . The reflectivity spectra are measured in air.
Fig. 3.
Fig. 3. (a) Room temperature LMOKE hysteresis of Fe x Co 1 x thin films measured under an incident laser wavelength of 650 nm. (b) The LMOKE Kerr rotation and Kerr ellipticity of Fe x Co 1 x thin films with different composition x .
Fig. 4.
Fig. 4. (a) Simulated device sensitivity as a function of Au and Co layer thicknesses in the Au/Co transducer. (b) The optimum sensitivity of Au / Fe x Co 1 x MOSPR transducers with different composition x ; the error bar of sensitivity is determined by the error bar of ϵ x z and ϵ x z as shown in Table 1.
Fig. 5.
Fig. 5. (a) Experimental and simulated angular spectrum of R for p -polarized incident light in three Au / Fe x Co 1 x MOSPR transducers measured in water ( n = 1.332 ) (b) Experimental and simulated angular spectrum of R ( + H ) R ( H ) for p -polarized incident light in three Au / Fe 0.7 Co 0.3 MOSPR transducers. (c) Experimental angular spectrum of R ( + H ) R ( H ) as a function of the index of the sensing medium for the Au / Fe 0.7 Co 0.3 transducer. (d) Comparison of the index sensitivity for Au/Fe, Au / Fe 0.7 Co 0.3 and Au/Co devices as a function of the index of the sensing medium.

Tables (1)

Tables Icon

Table 1. Dielectric Tensor Elements of Fe, Fe x Co 1 x , and Co Thin Films Deposited on Glass Substrates a

Equations (4)

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

S = ( R ( + H ) R ( H ) ) θ × θ n = Δ R n ,
Θ K = Re ( ± r s p ) r s s ( r p p ) = Im sin φ η 2 [ sin φ tan φ ± ( η 2 sin 2 φ ) 1 / 2 ] ( η 2 1 ) ( η 2 tan 2 φ ) ( η 2 sin 2 φ ) 1 / 2 Q ,
ψ K = Im ( ± r s p ) r s s ( r p p ) = Re sin φ η 2 [ sin φ tan φ ± ( η 2 sin 2 φ ) 1 / 2 ] ( η 2 1 ) ( η 2 tan 2 φ ) ( η 2 sin 2 φ ) 1 / 2 Q ,
ϵ x z = i * ϵ x x * Q .

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