Abstract

We propose a general and complete classification of all possible new and old kinds of surface plasmon polaritons (SPPs) that can propagate at boundaries of arbitrary linear, local bi-anisotropic media, including the quartic metamaterials. In particular, we introduce a new kind of generic bi-exponential SPPs with arbitrary frequency, wavelength, propagation direction, and penetration depths and fields. We show that any such an SPP is supported by a broad 72-parametric invariance class of pairs of interfacing media. A member of each invariance class is a pair of anisotropic materials without magnetoelectric couplings.

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

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References

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  1. T. Mulkey, J. Dillies, and M. Durach, “Inverse problem of quartic photonics,” Opt. Lett. 43(6), 1226–1229 (2018).
    [Crossref] [PubMed]
  2. A. Sommerfeld, “Über die Fortpflanzung elektrodynamischer Wellen längs eines Drahtes,” Ann. Phys. 303(2), 233–290 (1899).
    [Crossref]
  3. J. Zenneck, “Über die Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie,” Ann. Phys. 328(10), 846–866 (1907).
    [Crossref]
  4. U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),” J. Opt. Soc. Am. 31(3), 213–222 (1941).
    [Crossref]
  5. R. H. Ritchie, “Plasma losses by fast electrons in thin films,” Phys. Rev. 106(5), 874–881 (1957).
    [Crossref]
  6. Y. Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19(9), 511–514 (1967).
    [Crossref]
  7. J. Polo, T. Mackay, and A. Lakhtakia, Electromagnetic Surface Waves: A Modern Perspective (Newnes, 2013).
  8. D. L. Mills and V. M. Agranovich, eds., Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces (North-Holland, 1982).
  9. S. I. Bozhevolnyi, ed., Plasmonic Nanoguides and Circuits (Pan Stanford, 2009).
  10. M. I. D’yakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP 67, 714 (1988).
  11. D. Artigas and L. Torner, “Dyakonov surface waves in photonic metamaterials,” Phys. Rev. Lett. 94(1), 013901 (2005).
    [Crossref] [PubMed]
  12. Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93(22), 221109 (2008).
    [Crossref]
  13. R. H. Tarkhanyan and D. G. Niarchos, “Nonradiative surface electromagnetic waves at the interface of uniaxially bianisotropic enantiomeric media,” Phys. Status Solidi, B Basic Res. 248(6), 1499–1504 (2011).
    [Crossref]
  14. M. A. Noginov, “Steering Dyakonov-like waves,” Nat. Nanotechnol. 9(6), 414–415 (2014).
    [Crossref] [PubMed]
  15. M. A. Noginov and V. A. Podolskiy, eds., Tutorials in Metamaterials (CRC Press, 2011).
  16. M. A. Noginov, G. Dewar, M. W. McCall, and N. I. Zheludev, Tutorials in Complex Photonic Media (SPIE, 2009).
  17. A. Sihvola, S. Tretyakov, and A. De Baas, “Metamaterials with extreme material parameters,” J. Commun. Technol. Electron. 52(9), 986–990 (2007).
    [Crossref]
  18. W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
    [Crossref] [PubMed]
  19. A. Favaro and F. W. Hehl, “Light propagation in local and linear media: Fresnel-Kummer wave surfaces with 16 singular points,” Phys. Rev. A 93(1), 013844 (2016).
    [Crossref]
  20. A. Sihvola, “Metamaterials in electromagnetics,” Metamaterials (Amst.) 1(1), 2–11 (2007).
    [Crossref]
  21. P. Baekler, A. Favaro, Y. Itin, and F. W. Hehl, “The Kummer tensor density in electrodynamics and in gravity,” Ann. Phys. 349, 297–324 (2014).
    [Crossref]
  22. F. W. Hehl, Yu. N. Obukhov, J.-P. Rivera, and H. Schmid, “Relativistic nature of a magnetoelectric modulus of Cr2O3 crystals: a four-dimensional pseudoscalar and its measurement,” Phys. Rev. A 77(2), 022106 (2008).
    [Crossref]
  23. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).
  24. A. Serdyukov, I. Semchenko, S. Tretyakov, and A. Sihvola, Electromagnetics of Bi-Anisotropic Materials: Theory and Applications (Gordon and Breach, 2001).
  25. F. W. Hehl and Yu. N. Obukhov, Foundations of Classical Electrodynamics: Charge, Flux, and Metric (Birkhauser, 2003)
  26. A. G. Kurosh, Course of Higher Algebra (Nauka, 1968)

2018 (1)

2016 (1)

A. Favaro and F. W. Hehl, “Light propagation in local and linear media: Fresnel-Kummer wave surfaces with 16 singular points,” Phys. Rev. A 93(1), 013844 (2016).
[Crossref]

2015 (1)

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

2014 (2)

P. Baekler, A. Favaro, Y. Itin, and F. W. Hehl, “The Kummer tensor density in electrodynamics and in gravity,” Ann. Phys. 349, 297–324 (2014).
[Crossref]

M. A. Noginov, “Steering Dyakonov-like waves,” Nat. Nanotechnol. 9(6), 414–415 (2014).
[Crossref] [PubMed]

2011 (1)

R. H. Tarkhanyan and D. G. Niarchos, “Nonradiative surface electromagnetic waves at the interface of uniaxially bianisotropic enantiomeric media,” Phys. Status Solidi, B Basic Res. 248(6), 1499–1504 (2011).
[Crossref]

2008 (2)

Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93(22), 221109 (2008).
[Crossref]

F. W. Hehl, Yu. N. Obukhov, J.-P. Rivera, and H. Schmid, “Relativistic nature of a magnetoelectric modulus of Cr2O3 crystals: a four-dimensional pseudoscalar and its measurement,” Phys. Rev. A 77(2), 022106 (2008).
[Crossref]

2007 (2)

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

A. Sihvola, S. Tretyakov, and A. De Baas, “Metamaterials with extreme material parameters,” J. Commun. Technol. Electron. 52(9), 986–990 (2007).
[Crossref]

2005 (1)

D. Artigas and L. Torner, “Dyakonov surface waves in photonic metamaterials,” Phys. Rev. Lett. 94(1), 013901 (2005).
[Crossref] [PubMed]

1988 (1)

M. I. D’yakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP 67, 714 (1988).

1967 (1)

Y. Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19(9), 511–514 (1967).
[Crossref]

1957 (1)

R. H. Ritchie, “Plasma losses by fast electrons in thin films,” Phys. Rev. 106(5), 874–881 (1957).
[Crossref]

1941 (1)

1907 (1)

J. Zenneck, “Über die Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie,” Ann. Phys. 328(10), 846–866 (1907).
[Crossref]

1899 (1)

A. Sommerfeld, “Über die Fortpflanzung elektrodynamischer Wellen längs eines Drahtes,” Ann. Phys. 303(2), 233–290 (1899).
[Crossref]

Artigas, D.

D. Artigas and L. Torner, “Dyakonov surface waves in photonic metamaterials,” Phys. Rev. Lett. 94(1), 013901 (2005).
[Crossref] [PubMed]

Baekler, P.

P. Baekler, A. Favaro, Y. Itin, and F. W. Hehl, “The Kummer tensor density in electrodynamics and in gravity,” Ann. Phys. 349, 297–324 (2014).
[Crossref]

Béri, B.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

D’yakonov, M. I.

M. I. D’yakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP 67, 714 (1988).

De Baas, A.

A. Sihvola, S. Tretyakov, and A. De Baas, “Metamaterials with extreme material parameters,” J. Commun. Technol. Electron. 52(9), 986–990 (2007).
[Crossref]

Dillies, J.

Durach, M.

Fang, F.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

Fano, U.

Favaro, A.

A. Favaro and F. W. Hehl, “Light propagation in local and linear media: Fresnel-Kummer wave surfaces with 16 singular points,” Phys. Rev. A 93(1), 013844 (2016).
[Crossref]

P. Baekler, A. Favaro, Y. Itin, and F. W. Hehl, “The Kummer tensor density in electrodynamics and in gravity,” Ann. Phys. 349, 297–324 (2014).
[Crossref]

Gao, W.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

Hehl, F. W.

A. Favaro and F. W. Hehl, “Light propagation in local and linear media: Fresnel-Kummer wave surfaces with 16 singular points,” Phys. Rev. A 93(1), 013844 (2016).
[Crossref]

P. Baekler, A. Favaro, Y. Itin, and F. W. Hehl, “The Kummer tensor density in electrodynamics and in gravity,” Ann. Phys. 349, 297–324 (2014).
[Crossref]

F. W. Hehl, Yu. N. Obukhov, J.-P. Rivera, and H. Schmid, “Relativistic nature of a magnetoelectric modulus of Cr2O3 crystals: a four-dimensional pseudoscalar and its measurement,” Phys. Rev. A 77(2), 022106 (2008).
[Crossref]

Itin, Y.

P. Baekler, A. Favaro, Y. Itin, and F. W. Hehl, “The Kummer tensor density in electrodynamics and in gravity,” Ann. Phys. 349, 297–324 (2014).
[Crossref]

Jacob, Z.

Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93(22), 221109 (2008).
[Crossref]

Lawrence, M.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

Li, J.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

Liu, F.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

Mulkey, T.

Narimanov, E. E.

Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93(22), 221109 (2008).
[Crossref]

Niarchos, D. G.

R. H. Tarkhanyan and D. G. Niarchos, “Nonradiative surface electromagnetic waves at the interface of uniaxially bianisotropic enantiomeric media,” Phys. Status Solidi, B Basic Res. 248(6), 1499–1504 (2011).
[Crossref]

Noginov, M. A.

M. A. Noginov, “Steering Dyakonov-like waves,” Nat. Nanotechnol. 9(6), 414–415 (2014).
[Crossref] [PubMed]

Obukhov, Yu. N.

F. W. Hehl, Yu. N. Obukhov, J.-P. Rivera, and H. Schmid, “Relativistic nature of a magnetoelectric modulus of Cr2O3 crystals: a four-dimensional pseudoscalar and its measurement,” Phys. Rev. A 77(2), 022106 (2008).
[Crossref]

Ritchie, R. H.

R. H. Ritchie, “Plasma losses by fast electrons in thin films,” Phys. Rev. 106(5), 874–881 (1957).
[Crossref]

Rivera, J.-P.

F. W. Hehl, Yu. N. Obukhov, J.-P. Rivera, and H. Schmid, “Relativistic nature of a magnetoelectric modulus of Cr2O3 crystals: a four-dimensional pseudoscalar and its measurement,” Phys. Rev. A 77(2), 022106 (2008).
[Crossref]

Schmid, H.

F. W. Hehl, Yu. N. Obukhov, J.-P. Rivera, and H. Schmid, “Relativistic nature of a magnetoelectric modulus of Cr2O3 crystals: a four-dimensional pseudoscalar and its measurement,” Phys. Rev. A 77(2), 022106 (2008).
[Crossref]

Sihvola, A.

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

A. Sihvola, S. Tretyakov, and A. De Baas, “Metamaterials with extreme material parameters,” J. Commun. Technol. Electron. 52(9), 986–990 (2007).
[Crossref]

Sommerfeld, A.

A. Sommerfeld, “Über die Fortpflanzung elektrodynamischer Wellen längs eines Drahtes,” Ann. Phys. 303(2), 233–290 (1899).
[Crossref]

Stern, E. A.

Y. Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19(9), 511–514 (1967).
[Crossref]

Tarkhanyan, R. H.

R. H. Tarkhanyan and D. G. Niarchos, “Nonradiative surface electromagnetic waves at the interface of uniaxially bianisotropic enantiomeric media,” Phys. Status Solidi, B Basic Res. 248(6), 1499–1504 (2011).
[Crossref]

Teng, Y. Y.

Y. Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19(9), 511–514 (1967).
[Crossref]

Torner, L.

D. Artigas and L. Torner, “Dyakonov surface waves in photonic metamaterials,” Phys. Rev. Lett. 94(1), 013901 (2005).
[Crossref] [PubMed]

Tretyakov, S.

A. Sihvola, S. Tretyakov, and A. De Baas, “Metamaterials with extreme material parameters,” J. Commun. Technol. Electron. 52(9), 986–990 (2007).
[Crossref]

Yang, B.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

Zenneck, J.

J. Zenneck, “Über die Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie,” Ann. Phys. 328(10), 846–866 (1907).
[Crossref]

Zhang, S.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

Ann. Phys. (3)

A. Sommerfeld, “Über die Fortpflanzung elektrodynamischer Wellen längs eines Drahtes,” Ann. Phys. 303(2), 233–290 (1899).
[Crossref]

J. Zenneck, “Über die Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie,” Ann. Phys. 328(10), 846–866 (1907).
[Crossref]

P. Baekler, A. Favaro, Y. Itin, and F. W. Hehl, “The Kummer tensor density in electrodynamics and in gravity,” Ann. Phys. 349, 297–324 (2014).
[Crossref]

Appl. Phys. Lett. (1)

Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93(22), 221109 (2008).
[Crossref]

J. Commun. Technol. Electron. (1)

A. Sihvola, S. Tretyakov, and A. De Baas, “Metamaterials with extreme material parameters,” J. Commun. Technol. Electron. 52(9), 986–990 (2007).
[Crossref]

J. Opt. Soc. Am. (1)

Metamaterials (Amst.) (1)

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

Nat. Nanotechnol. (1)

M. A. Noginov, “Steering Dyakonov-like waves,” Nat. Nanotechnol. 9(6), 414–415 (2014).
[Crossref] [PubMed]

Opt. Lett. (1)

Phys. Rev. (1)

R. H. Ritchie, “Plasma losses by fast electrons in thin films,” Phys. Rev. 106(5), 874–881 (1957).
[Crossref]

Phys. Rev. A (2)

A. Favaro and F. W. Hehl, “Light propagation in local and linear media: Fresnel-Kummer wave surfaces with 16 singular points,” Phys. Rev. A 93(1), 013844 (2016).
[Crossref]

F. W. Hehl, Yu. N. Obukhov, J.-P. Rivera, and H. Schmid, “Relativistic nature of a magnetoelectric modulus of Cr2O3 crystals: a four-dimensional pseudoscalar and its measurement,” Phys. Rev. A 77(2), 022106 (2008).
[Crossref]

Phys. Rev. Lett. (3)

D. Artigas and L. Torner, “Dyakonov surface waves in photonic metamaterials,” Phys. Rev. Lett. 94(1), 013901 (2005).
[Crossref] [PubMed]

Y. Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19(9), 511–514 (1967).
[Crossref]

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114(3), 037402 (2015).
[Crossref] [PubMed]

Phys. Status Solidi, B Basic Res. (1)

R. H. Tarkhanyan and D. G. Niarchos, “Nonradiative surface electromagnetic waves at the interface of uniaxially bianisotropic enantiomeric media,” Phys. Status Solidi, B Basic Res. 248(6), 1499–1504 (2011).
[Crossref]

Sov. Phys. JETP (1)

M. I. D’yakonov, “New type of electromagnetic wave propagating at an interface,” Sov. Phys. JETP 67, 714 (1988).

Other (9)

M. A. Noginov and V. A. Podolskiy, eds., Tutorials in Metamaterials (CRC Press, 2011).

M. A. Noginov, G. Dewar, M. W. McCall, and N. I. Zheludev, Tutorials in Complex Photonic Media (SPIE, 2009).

J. Polo, T. Mackay, and A. Lakhtakia, Electromagnetic Surface Waves: A Modern Perspective (Newnes, 2013).

D. L. Mills and V. M. Agranovich, eds., Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces (North-Holland, 1982).

S. I. Bozhevolnyi, ed., Plasmonic Nanoguides and Circuits (Pan Stanford, 2009).

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

A. Serdyukov, I. Semchenko, S. Tretyakov, and A. Sihvola, Electromagnetics of Bi-Anisotropic Materials: Theory and Applications (Gordon and Breach, 2001).

F. W. Hehl and Yu. N. Obukhov, Foundations of Classical Electrodynamics: Charge, Flux, and Metric (Birkhauser, 2003)

A. G. Kurosh, Course of Higher Algebra (Nauka, 1968)

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Figures (3)

Fig. 1
Fig. 1 New kind of surface plasmons propagating between a pair of quartic media M1 and M2. (a-b) The constitutive effective parameter matrices for the media. (c-d) The corresponding k-surfaces with real kx, ky, kz. (e) The longitudinal electric field distribution for the proposed surface plasmon wave propagating between M1 and M2.
Fig. 2
Fig. 2 Examples of the proposed surface plasmons with short and long wavelengths and penetration depths. The panels show the longitudinal electric fields of the SPPs and the insets show the corresponding material pairs. The dashed lines in the panels with fields indicate the boundary and the penetration depths.
Fig. 3
Fig. 3 Interference effects of the evanescent waves composing the proposed surface plasmons. (a) The longitudinal field of a surface plasmon that shows bi-exponentiality. (b) A surface plasmon with the phase shifts between the evanescent waves.

Equations (18)

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( D B )= M ^ ( E H ),  M ^ =( ε ^ X ^ Y ^ μ ^ ),
k×E= k 0 B and k×H=- k 0 D,
Q ^ Γ= M ^ Γ,
Q ^ =( 0 R ^ R ^ 0 ),  R ^ = 1 k 0 ( 0 k z k y k z 0 k x k y k x 0 ).
i+j+l+m=4 [ α ijkl k x i k y j k z l k 0 m ]=0 ,
k i (1) =( k || , k zi (1) +i k zi (1) ), i=12 for z > 0,
k i (2) =( k || , k zi (2) i k zi (2) ), i=12 for z <0.
A Γ ||1 (1) +B Γ ||2 (1) =C Γ ||1 (2) +D Γ ||2 (2) ,
| E x1 (1) E x2 (1) E x1 (2) E x2 (2) E y1 (1) E y2 (1) E y1 (2) E y2 (2) H x1 (1) H x2 (1) H x1 (2) H x2 (2) H y1 (1) H y2 (1) H y1 (2) H y2 (2) |=0.
Γ ||1 (j) =( ± i k zj k 0 ε j ,0,0,1 ) for TM polarization,
Γ ||2 (j) =( 0,1,± i k zj k 0 μ j ,0 ) for TE polarization,
( k z1 ε 1 + k z2 ε 2 )( k z1 μ 1 + k z2 μ 2 )=0,
M ^ = P ^ G ^ 1 .
ε ^ = P ^ H E ^ 1 X ^ H ^ E ^ 1 ,  μ ^ = P ^ E H ^ 1 Y ^ E ^ H ^ 1 .
k 0 P ^ E =( k 1 × E 1 , k 2 × E 2 , k 3 × E 3 ),
k 0 P ^ H =( k 1 × H 1 , k 2 × H 2 , k 3 × H 3 ),
E ^ =( E x1 E x2 E x3 E y1 E y2 E y3 E z1 E z2 E z3 ) and   H ^ =( H x1 H x2 H x3 H y1 H y2 H y3 H z1 H z2 H z3 ).
M ^ =( 3 3 κ 0 iκ 0 0 0 3 i 6 κ 0 iκ 0 3 κ 0 3(1iκ) 0 0 iκ iκ 0 0 1 κ/ 2 κ/ 3 0 iκ 0 κ/ 3 1 0 0 0 iκ 0 iκ/ 6 1 ).

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