Abstract

Inspired by a general theorem on non-radiating sources demonstrated by Devaney and Wolf, a unified theory for invisible and cloaking structures is here proposed. By solving Devaney-Wolf theorem in the quasi-static limit, a weak solution is obtained, demonstrating the existence of Anapole modes, Mantle Cloaking and Plasmonic Cloaking. Beyond the quasi-static regime, a strong solution of Devaney-Wolf theorem can be formulated, predicting general non-scattering devices based on directional invisibility, Transformation Optics, neutral inclusions and refractive index continuity. Both weak and strong solutions are analytically demonstrated to depend on the concept of contrast, mathematically defined as a normalized difference between constitutive parameters (or wave-impedance property) of a material and its surrounding background.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Ideal and nonideal invisibility cloaks

Nina A. Zharova, Ilya V. Shadrivov, Alexander A. Zharov, and Yuri S. Kivshar
Opt. Express 16(26) 21369-21374 (2008)

Closed-form harmonic contrast control with surface impedance coatings for conductive objects

Giuseppe Labate, Symon K. Podilchak, and Ladislau Matekovits
Appl. Opt. 56(36) 10055-10059 (2017)

Fourier optics theory for invisibility cloaks

Kedi Wu, Qiluan Cheng, and Guo Ping Wang
J. Opt. Soc. Am. B 28(6) 1467-1474 (2011)

References

  • View by:
  • |
  • |
  • |

  1. R. W. Wood, “The invisibility of transparent objects,” Phys. Rev. (Series I) 15(2), 123–124 (1902).
    [Crossref]
  2. G. Gbur, “Invisibility physics: past, present, and future,” Prog. Optics 58, 65–114 (2013).
    [Crossref]
  3. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
    [Crossref]
  4. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72(1), 016623 (2005).
    [Crossref]
  5. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [Crossref] [PubMed]
  6. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [Crossref] [PubMed]
  7. S. Guenneau, C. Amra, and D. Veynante, “Transformation thermodynamics: cloaking and concentrating heat flux,” Opt. Express 20(7), 8207–8218 (2012).
    [Crossref] [PubMed]
  8. A. Alù, “Mantle cloak: invisibility induced by a surface,” Phys. Rev. B 80, 245115 (2009).
    [Crossref]
  9. A. J. Devaney and E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
    [Crossref]
  10. A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice Hall, 1991).
  11. A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Comm. 6, 8069 (2015).
    [Crossref]
  12. P. -Y. Chen and A. Alù, “Mantle cloaking using thin patterned metasurfaces,” Phys. Rev. B 84, 205110 (2011).
    [Crossref]
  13. P. -Y. Chen, C. Argyropoulos, and A. Alù, “Broadening the cloaking bandwidth with non-Foster metasurfaces,” Phys. Rev. Lett. 111, 233001 (2013).
    [Crossref]
  14. R. M. Foster, “A reactance theorem,” Bell System Techn. Journ. 3, 259–267 (1924).
    [Crossref]
  15. C. Della Giovampaola and N. Engheta, “Plasmonics without negative dielectrics,” Phys. Rev. B 93, 195152 (2016).
    [Crossref]
  16. O. M. Bucci and T. Isernia, “Electromagnetic inverse scattering: retrievable information and measurement strategies,” Radio Science 32(6), 2123–2137 (1997).
    [Crossref]
  17. R. G. Quarfoth and D. F. Sievenpiper, “Nonscattering waveguides based on tensor impedance surfaces,” IEEE Trans. Antennas Propag. 63(4), 1746–1755 (2015).
    [Crossref]
  18. F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and mirage effects,” Opt. Lett. 32(9), 1069–1071 (2007).
    [Crossref] [PubMed]
  19. S. A. Schelkunoff, “The impedance concept and its application to problems of reflection, refraction, shielding and power absorption,” Bells Labs Techn. Journ. 17(1), 17–48 (1938).
    [Crossref]
  20. G. W. Milton, The Theory of Composites (Cambridge University, 2004).
  21. Microwave Studio, Computer Simulation Technology, 2016.
  22. The cloak external contour is plotted according to this points sequence: [xclk,yclk] = [(0, 8);(6, 4);(6,−0);(6,−4);(0,−8);(−4,−6);(−6,−4); (−4,−2);(−6,−0);(−4, 2);(−6, 4);(−4, 6);(0, 8)].
  23. The object contour is plotted according to this points sequence: [xobj,yobj] = [(4,0);(2,2);(2,4);(0,6);(−2,4);(−4,4);(−2,0);(−4,−4);(−2,−4);(0,−6);(2, −4);(2, −2); (4,0)].
  24. L. Matekovits and T. S. Bird, “Width-modulated microstrip-line based mantle cloaks− for thin single- and multiple cylinders,” IEEE Trans. Antennas Propag.,  62(5), 2606–2615 (2014).
    [Crossref]

2016 (1)

C. Della Giovampaola and N. Engheta, “Plasmonics without negative dielectrics,” Phys. Rev. B 93, 195152 (2016).
[Crossref]

2015 (2)

R. G. Quarfoth and D. F. Sievenpiper, “Nonscattering waveguides based on tensor impedance surfaces,” IEEE Trans. Antennas Propag. 63(4), 1746–1755 (2015).
[Crossref]

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Comm. 6, 8069 (2015).
[Crossref]

2014 (1)

L. Matekovits and T. S. Bird, “Width-modulated microstrip-line based mantle cloaks− for thin single- and multiple cylinders,” IEEE Trans. Antennas Propag.,  62(5), 2606–2615 (2014).
[Crossref]

2013 (2)

P. -Y. Chen, C. Argyropoulos, and A. Alù, “Broadening the cloaking bandwidth with non-Foster metasurfaces,” Phys. Rev. Lett. 111, 233001 (2013).
[Crossref]

G. Gbur, “Invisibility physics: past, present, and future,” Prog. Optics 58, 65–114 (2013).
[Crossref]

2012 (1)

2011 (1)

P. -Y. Chen and A. Alù, “Mantle cloaking using thin patterned metasurfaces,” Phys. Rev. B 84, 205110 (2011).
[Crossref]

2009 (1)

A. Alù, “Mantle cloak: invisibility induced by a surface,” Phys. Rev. B 80, 245115 (2009).
[Crossref]

2007 (1)

2006 (2)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

2005 (1)

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72(1), 016623 (2005).
[Crossref]

1997 (1)

O. M. Bucci and T. Isernia, “Electromagnetic inverse scattering: retrievable information and measurement strategies,” Radio Science 32(6), 2123–2137 (1997).
[Crossref]

1973 (1)

A. J. Devaney and E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
[Crossref]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
[Crossref]

1938 (1)

S. A. Schelkunoff, “The impedance concept and its application to problems of reflection, refraction, shielding and power absorption,” Bells Labs Techn. Journ. 17(1), 17–48 (1938).
[Crossref]

1924 (1)

R. M. Foster, “A reactance theorem,” Bell System Techn. Journ. 3, 259–267 (1924).
[Crossref]

1902 (1)

R. W. Wood, “The invisibility of transparent objects,” Phys. Rev. (Series I) 15(2), 123–124 (1902).
[Crossref]

Alù, A.

P. -Y. Chen, C. Argyropoulos, and A. Alù, “Broadening the cloaking bandwidth with non-Foster metasurfaces,” Phys. Rev. Lett. 111, 233001 (2013).
[Crossref]

P. -Y. Chen and A. Alù, “Mantle cloaking using thin patterned metasurfaces,” Phys. Rev. B 84, 205110 (2011).
[Crossref]

A. Alù, “Mantle cloak: invisibility induced by a surface,” Phys. Rev. B 80, 245115 (2009).
[Crossref]

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72(1), 016623 (2005).
[Crossref]

Amra, C.

Argyropoulos, C.

P. -Y. Chen, C. Argyropoulos, and A. Alù, “Broadening the cloaking bandwidth with non-Foster metasurfaces,” Phys. Rev. Lett. 111, 233001 (2013).
[Crossref]

Bakker, R. M.

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Comm. 6, 8069 (2015).
[Crossref]

Bird, T. S.

L. Matekovits and T. S. Bird, “Width-modulated microstrip-line based mantle cloaks− for thin single- and multiple cylinders,” IEEE Trans. Antennas Propag.,  62(5), 2606–2615 (2014).
[Crossref]

Bucci, O. M.

O. M. Bucci and T. Isernia, “Electromagnetic inverse scattering: retrievable information and measurement strategies,” Radio Science 32(6), 2123–2137 (1997).
[Crossref]

Chen, P. -Y.

P. -Y. Chen, C. Argyropoulos, and A. Alù, “Broadening the cloaking bandwidth with non-Foster metasurfaces,” Phys. Rev. Lett. 111, 233001 (2013).
[Crossref]

P. -Y. Chen and A. Alù, “Mantle cloaking using thin patterned metasurfaces,” Phys. Rev. B 84, 205110 (2011).
[Crossref]

Chichkov, B. N.

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Comm. 6, 8069 (2015).
[Crossref]

Chipouline, A.

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Comm. 6, 8069 (2015).
[Crossref]

Della Giovampaola, C.

C. Della Giovampaola and N. Engheta, “Plasmonics without negative dielectrics,” Phys. Rev. B 93, 195152 (2016).
[Crossref]

Devaney, A. J.

A. J. Devaney and E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
[Crossref]

Engheta, N.

C. Della Giovampaola and N. Engheta, “Plasmonics without negative dielectrics,” Phys. Rev. B 93, 195152 (2016).
[Crossref]

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72(1), 016623 (2005).
[Crossref]

Evlyukhin, A. B.

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Comm. 6, 8069 (2015).
[Crossref]

Foster, R. M.

R. M. Foster, “A reactance theorem,” Bell System Techn. Journ. 3, 259–267 (1924).
[Crossref]

Gbur, G.

G. Gbur, “Invisibility physics: past, present, and future,” Prog. Optics 58, 65–114 (2013).
[Crossref]

Guenneau, S.

Isernia, T.

O. M. Bucci and T. Isernia, “Electromagnetic inverse scattering: retrievable information and measurement strategies,” Radio Science 32(6), 2123–2137 (1997).
[Crossref]

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice Hall, 1991).

Kivshar, Y. S.

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Comm. 6, 8069 (2015).
[Crossref]

Kuznetsov, A. I.

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Comm. 6, 8069 (2015).
[Crossref]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref] [PubMed]

Luk’yanchuk, B.

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Comm. 6, 8069 (2015).
[Crossref]

Matekovits, L.

L. Matekovits and T. S. Bird, “Width-modulated microstrip-line based mantle cloaks− for thin single- and multiple cylinders,” IEEE Trans. Antennas Propag.,  62(5), 2606–2615 (2014).
[Crossref]

Milton, G. W.

G. W. Milton, The Theory of Composites (Cambridge University, 2004).

Miroshnichenko, A. E.

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Comm. 6, 8069 (2015).
[Crossref]

Nicolet, A.

Pendry, J. B.

Quarfoth, R. G.

R. G. Quarfoth and D. F. Sievenpiper, “Nonscattering waveguides based on tensor impedance surfaces,” IEEE Trans. Antennas Propag. 63(4), 1746–1755 (2015).
[Crossref]

Schelkunoff, S. A.

S. A. Schelkunoff, “The impedance concept and its application to problems of reflection, refraction, shielding and power absorption,” Bells Labs Techn. Journ. 17(1), 17–48 (1938).
[Crossref]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

Sievenpiper, D. F.

R. G. Quarfoth and D. F. Sievenpiper, “Nonscattering waveguides based on tensor impedance surfaces,” IEEE Trans. Antennas Propag. 63(4), 1746–1755 (2015).
[Crossref]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
[Crossref]

Veynante, D.

Wolf, E.

A. J. Devaney and E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
[Crossref]

Wood, R. W.

R. W. Wood, “The invisibility of transparent objects,” Phys. Rev. (Series I) 15(2), 123–124 (1902).
[Crossref]

Yu, Y. F.

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Comm. 6, 8069 (2015).
[Crossref]

Zolla, F.

Bell System Techn. Journ. (1)

R. M. Foster, “A reactance theorem,” Bell System Techn. Journ. 3, 259–267 (1924).
[Crossref]

Bells Labs Techn. Journ. (1)

S. A. Schelkunoff, “The impedance concept and its application to problems of reflection, refraction, shielding and power absorption,” Bells Labs Techn. Journ. 17(1), 17–48 (1938).
[Crossref]

IEEE Trans. Antennas Propag. (2)

R. G. Quarfoth and D. F. Sievenpiper, “Nonscattering waveguides based on tensor impedance surfaces,” IEEE Trans. Antennas Propag. 63(4), 1746–1755 (2015).
[Crossref]

L. Matekovits and T. S. Bird, “Width-modulated microstrip-line based mantle cloaks− for thin single- and multiple cylinders,” IEEE Trans. Antennas Propag.,  62(5), 2606–2615 (2014).
[Crossref]

Nat. Comm. (1)

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Comm. 6, 8069 (2015).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. (Series I) (1)

R. W. Wood, “The invisibility of transparent objects,” Phys. Rev. (Series I) 15(2), 123–124 (1902).
[Crossref]

Phys. Rev. B (3)

P. -Y. Chen and A. Alù, “Mantle cloaking using thin patterned metasurfaces,” Phys. Rev. B 84, 205110 (2011).
[Crossref]

A. Alù, “Mantle cloak: invisibility induced by a surface,” Phys. Rev. B 80, 245115 (2009).
[Crossref]

C. Della Giovampaola and N. Engheta, “Plasmonics without negative dielectrics,” Phys. Rev. B 93, 195152 (2016).
[Crossref]

Phys. Rev. D (1)

A. J. Devaney and E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
[Crossref]

Phys. Rev. E (1)

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72(1), 016623 (2005).
[Crossref]

Phys. Rev. Lett. (1)

P. -Y. Chen, C. Argyropoulos, and A. Alù, “Broadening the cloaking bandwidth with non-Foster metasurfaces,” Phys. Rev. Lett. 111, 233001 (2013).
[Crossref]

Prog. Optics (1)

G. Gbur, “Invisibility physics: past, present, and future,” Prog. Optics 58, 65–114 (2013).
[Crossref]

Radio Science (1)

O. M. Bucci and T. Isernia, “Electromagnetic inverse scattering: retrievable information and measurement strategies,” Radio Science 32(6), 2123–2137 (1997).
[Crossref]

Science (2)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
[Crossref]

Other (5)

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice Hall, 1991).

G. W. Milton, The Theory of Composites (Cambridge University, 2004).

Microwave Studio, Computer Simulation Technology, 2016.

The cloak external contour is plotted according to this points sequence: [xclk,yclk] = [(0, 8);(6, 4);(6,−0);(6,−4);(0,−8);(−4,−6);(−6,−4); (−4,−2);(−6,−0);(−4, 2);(−6, 4);(−4, 6);(0, 8)].

The object contour is plotted according to this points sequence: [xobj,yobj] = [(4,0);(2,2);(2,4);(0,6);(−2,4);(−4,4);(−2,0);(−4,−4);(−2,−4);(0,−6);(2, −4);(2, −2); (4,0)].

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.


Figures (5)

Fig. 1
Fig. 1 Phase/antiphase-type mode as zero scalar sum (left) and Loop-type mode as zero vector sum (right), as a result of weak solution.
Fig. 2
Fig. 2 Plasmonic Cloaking for arbitrary dielectric systems in arbitrary homogeneous background: contrast permittivity χ ε 1 covered by a cloak with χ ε 2 , with necessary condition χ ε 1 χ ε 2 < 0 (left). Mantle Cloaking for arbitrary composite structures in arbitrary homogeneous background: contrast permittivity χ ε 1 (same object as before) coated by a surface impedance cloak with Zs (right).
Fig. 3
Fig. 3 Classification of materials as a function of their contrast value (with respect to the considered homogeneous background): positive or negative nature (real part), active or passive behaviour (imaginary part).
Fig. 4
Fig. 4 Non-scattering waveguide: Cartesian (left) and polar coordinate system (right).
Fig. 5
Fig. 5 Absolute value of the scattered electric field Es: uncloaked (left) and cloaked case (right).

Equations (20)

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

J ω r Ω [ j ( r ) exp ( i k r ) ] p ^ d Ω = 0 with | k | = ω / c
Ω j ( r ) p ^ d Ω = 0 for ω 0
j v ( r ) = i ω ε b [ ε r ( r ) 1 ] E t ( r ) with r Σ
j s ( r ) = n ^ × [ H t + ( r ) H t ( r ) ] with r Γ
χ ε Σ E t ( r ) p ^ d Σ = 0 χ ε E ˜ p Σ = 0
i ω ε b χ ε 1 A 1 E t p ^ d A 1 + C [ n ^ × ( H t + H t ) ] p ^ d C = 0
i ω ε b χ ε 1 E ˜ z A 1 + H ˜ ϕ C = 0 E ˜ z H ˜ ϕ Z s T M = + i C ω ε b χ ε 1 A 1
χ ε 1 A 1 E t ( r ) p ^ d A 1 + χ ε 2 A 2 E t ( r ) p ^ d A 2 = 0
χ ε 1 A 1 + χ ε 2 A 2 = 0 A χ ε ( r ) d A = 0
χ P m , b = P m P b P b
A 1 A = χ 2 χ 2 χ 1 and V 1 V = χ 2 χ 2 χ 1
j ( r ) p ^ = 0 ω
k ^ p ^ = 0
χ _ _ Z ( ω , m ) = 0 _ _
μ _ _ = μ b T _ _ H
ε _ _ = ε b T _ _ E
T _ _ H = T _ _ E
Z F F lim r r * E ( r , ω ) p ^ E H ( r , ω ) p ^ H = ( μ ε ) 1 / 2
k ( r , ω ) = ω N ( r , ω ) c k ^
χ _ N ( r , ω ) = N ( r , ω ) N b ( ω ) N b ( ω ) k ^ = 0 _

Metrics