Abstract

We reformulate the problem of a steerable left-handed antenna reported by Matsuzawa et al. [IEICE Trans. Electron. E89-C, 1337 (2006)] from the view point of structural electromagnetic resonance of the unit structure. We show that there are two such resonances with different spatial symmetries in the relevant frequency range, which result in the formation of two electromagnetic bands with opposite signs of curvature at the Γ point of the Brillouin zone. We derive an expression of dispersion curves based on the tight-binding picture and show that the dispersion of the two bands is linear in the vicinity of the Γ point in the case of accidental degeneracy only if the symmetry of the two resonance states satisfies certain conditions. We also show that the refraction angle can be designed by changing the lattice constant of the arrayed unit structures, since the band width is modified due to the change in the electromagnetic transfer integral.

© 2010 Optical Society of America

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References

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  1. V. G. Veselago, "Electrodynamics of substances with simultaneously negative values of sigma and mu," Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
    [CrossRef] [PubMed]
  3. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
    [CrossRef] [PubMed]
  4. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  5. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
    [CrossRef] [PubMed]
  6. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
    [CrossRef]
  7. S. A. Ramakrishna, and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (SPIE Press, 2008).
    [CrossRef]
  8. S. Matsuzawa, K. Sato, Y. Inoue, and T. Nomura, ""W-band steerable composite right/left-handed leaky wave antenna for automotive applications," IEICE Trans. Electron. E 89-C, 1337-1344 (2006).
    [CrossRef]
  9. A. Grbic, and G. V. Eleftheriades, "Experimental verification of backward-wave radiation from a negative refractive index metamaterial," J. Appl. Phys. 92, 5930-5935 (2002).
    [CrossRef]
  10. C. Caloz, and T. Ito, "Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line," IEEE-AP-S Int. Symp. Dig. 2, 412-415 (2002).
  11. S. Tokoro, K. Kuroda, A. Kawakubo, K. Fujita, and H. Fujinami, "Electronically scanned millimeter-wave radar for pre-crush safety and adaptive cruise control system," Proc. IEEE Intelligent Vehicles Symp., 304-309 (2003).
  12. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
  13. K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed. (Springer-Verlag, Berlin, 2004).
  14. T. Ito, and K. Sakoda, "Photonic bands of metallic systems. II. Features of surface plasmon polaritons," Phys. Rev. B 64, 045117 (2001).
    [CrossRef]
  15. A. Taflove, Computational Electrodynamics (Artech House, Boston, 1995).
  16. D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE Press, Piscataway, 2000).
    [CrossRef]
  17. N. Peyghambarian, S. W. Koch, and A. Mysyrowicz, Introduction to Semiconductor Optics (Prentice Hall, Englewood Cliffs, 1993) Sec. 2.5.
  18. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, Berlin, 1990).
  19. P. Yeh, "Electromagnetic propagation in birefringent layered media," J. Opt. Soc. Am. 69, 742-756 (1979).
    [CrossRef]
  20. C. Caloz, A. Lai, and T. Itoh, "The challenge of homogenization in metamaterials," N. J. Phys. 7, 167 (2005).
    [CrossRef]
  21. A. Lai, T. Itoh, and C. Caloz, "Composite right/left-handed transmission line metamaterials," IEEE Microwave Magazine, September issue, 34-50 (2004).
    [CrossRef]

2006

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

S. Matsuzawa, K. Sato, Y. Inoue, and T. Nomura, ""W-band steerable composite right/left-handed leaky wave antenna for automotive applications," IEICE Trans. Electron. E 89-C, 1337-1344 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

2005

C. Caloz, A. Lai, and T. Itoh, "The challenge of homogenization in metamaterials," N. J. Phys. 7, 167 (2005).
[CrossRef]

2002

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
[CrossRef]

A. Grbic, and G. V. Eleftheriades, "Experimental verification of backward-wave radiation from a negative refractive index metamaterial," J. Appl. Phys. 92, 5930-5935 (2002).
[CrossRef]

C. Caloz, and T. Ito, "Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line," IEEE-AP-S Int. Symp. Dig. 2, 412-415 (2002).

2001

T. Ito, and K. Sakoda, "Photonic bands of metallic systems. II. Features of surface plasmon polaritons," Phys. Rev. B 64, 045117 (2001).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

2000

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

1979

1968

V. G. Veselago, "Electrodynamics of substances with simultaneously negative values of sigma and mu," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Caloz, C.

C. Caloz, A. Lai, and T. Itoh, "The challenge of homogenization in metamaterials," N. J. Phys. 7, 167 (2005).
[CrossRef]

C. Caloz, and T. Ito, "Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line," IEEE-AP-S Int. Symp. Dig. 2, 412-415 (2002).

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Eleftheriades, G. V.

A. Grbic, and G. V. Eleftheriades, "Experimental verification of backward-wave radiation from a negative refractive index metamaterial," J. Appl. Phys. 92, 5930-5935 (2002).
[CrossRef]

Grbic, A.

A. Grbic, and G. V. Eleftheriades, "Experimental verification of backward-wave radiation from a negative refractive index metamaterial," J. Appl. Phys. 92, 5930-5935 (2002).
[CrossRef]

Inoue, Y.

S. Matsuzawa, K. Sato, Y. Inoue, and T. Nomura, ""W-band steerable composite right/left-handed leaky wave antenna for automotive applications," IEICE Trans. Electron. E 89-C, 1337-1344 (2006).
[CrossRef]

Ito, T.

C. Caloz, and T. Ito, "Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line," IEEE-AP-S Int. Symp. Dig. 2, 412-415 (2002).

T. Ito, and K. Sakoda, "Photonic bands of metallic systems. II. Features of surface plasmon polaritons," Phys. Rev. B 64, 045117 (2001).
[CrossRef]

Itoh, T.

C. Caloz, A. Lai, and T. Itoh, "The challenge of homogenization in metamaterials," N. J. Phys. 7, 167 (2005).
[CrossRef]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Lai, A.

C. Caloz, A. Lai, and T. Itoh, "The challenge of homogenization in metamaterials," N. J. Phys. 7, 167 (2005).
[CrossRef]

Markos, P.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Matsuzawa, S.

S. Matsuzawa, K. Sato, Y. Inoue, and T. Nomura, ""W-band steerable composite right/left-handed leaky wave antenna for automotive applications," IEICE Trans. Electron. E 89-C, 1337-1344 (2006).
[CrossRef]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Nomura, T.

S. Matsuzawa, K. Sato, Y. Inoue, and T. Nomura, ""W-band steerable composite right/left-handed leaky wave antenna for automotive applications," IEICE Trans. Electron. E 89-C, 1337-1344 (2006).
[CrossRef]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Sakoda, K.

T. Ito, and K. Sakoda, "Photonic bands of metallic systems. II. Features of surface plasmon polaritons," Phys. Rev. B 64, 045117 (2001).
[CrossRef]

Sato, K.

S. Matsuzawa, K. Sato, Y. Inoue, and T. Nomura, ""W-band steerable composite right/left-handed leaky wave antenna for automotive applications," IEICE Trans. Electron. E 89-C, 1337-1344 (2006).
[CrossRef]

Schultz, S.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Soukoulis, C. M.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Veselago, V. G.

V. G. Veselago, "Electrodynamics of substances with simultaneously negative values of sigma and mu," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Yeh, P.

IEEE-AP-S Int. Symp. Dig.

C. Caloz, and T. Ito, "Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH line," IEEE-AP-S Int. Symp. Dig. 2, 412-415 (2002).

IEICE Trans. Electron. E

S. Matsuzawa, K. Sato, Y. Inoue, and T. Nomura, ""W-band steerable composite right/left-handed leaky wave antenna for automotive applications," IEICE Trans. Electron. E 89-C, 1337-1344 (2006).
[CrossRef]

J. Appl. Phys.

A. Grbic, and G. V. Eleftheriades, "Experimental verification of backward-wave radiation from a negative refractive index metamaterial," J. Appl. Phys. 92, 5930-5935 (2002).
[CrossRef]

J. Opt. Soc. Am.

N. J. Phys.

C. Caloz, A. Lai, and T. Itoh, "The challenge of homogenization in metamaterials," N. J. Phys. 7, 167 (2005).
[CrossRef]

Phys. Rev. B

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
[CrossRef]

T. Ito, and K. Sakoda, "Photonic bands of metallic systems. II. Features of surface plasmon polaritons," Phys. Rev. B 64, 045117 (2001).
[CrossRef]

Phys. Rev. Lett.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Science

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (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.

V. G. Veselago, "Electrodynamics of substances with simultaneously negative values of sigma and mu," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Other

S. A. Ramakrishna, and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (SPIE Press, 2008).
[CrossRef]

A. Taflove, Computational Electrodynamics (Artech House, Boston, 1995).

D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE Press, Piscataway, 2000).
[CrossRef]

N. Peyghambarian, S. W. Koch, and A. Mysyrowicz, Introduction to Semiconductor Optics (Prentice Hall, Englewood Cliffs, 1993) Sec. 2.5.

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, Berlin, 1990).

S. Tokoro, K. Kuroda, A. Kawakubo, K. Fujita, and H. Fujinami, "Electronically scanned millimeter-wave radar for pre-crush safety and adaptive cruise control system," Proc. IEEE Intelligent Vehicles Symp., 304-309 (2003).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed. (Springer-Verlag, Berlin, 2004).

A. Lai, T. Itoh, and C. Caloz, "Composite right/left-handed transmission line metamaterials," IEEE Microwave Magazine, September issue, 34-50 (2004).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

(a) Structure of the steerable antenna analyzed in Ref. [8] and the present study, where the period of the structure, or the lattice constant, is denoted by a. The specimen consists of a one-dimensional regular array of metallic unit structures fabricated on the surface of a dielectric slab (0.127 mm thick, dielectric constant = 2.2) with a ground electrode on the back surface. (b) Magnified illustration of the metallic unit structure.

Fig. 2
Fig. 2

Schematic dispersion relation of the steerable metamaterial antenna. The vertical and horizontal axes denote the normalized angular frequency and the parallel component of the normalized wave vector, respectively, where a is the lattice constant and c is the light velocity in free space. fl and fu are lower and upper cosine bands generated by two resonant states of the unit structure. The two straight lines denoted by ω = ±ck are the light lines above which propagating waves are leaky so that they escape from the antenna. The upper and lower bounds of frequencies of the leaky waves are labelled by ωu and ωl, respectively. ωi is the angular frequency of the incident wave propagated in the positive x direction in Fig. 1(a) and ki is its crystalline momentum. Only waves with positive group velocities can be excited by such incident waves. See text for details.

Fig. 3
Fig. 3

Diffraction of the leaky mode above the light line excited by an incident wave with angular frequency ωi. Note that the parallel component of the wave vector (crystalline momentum) is conserved during the diffraction process.

Fig. 4
Fig. 4

Frequency dependence of the diffraction angle of the steerable antenna. For Al and νl in Eqs. (26) and (27), three cases are examined: (a) Al = −400 (GHz)2, νl = 67 GHz, (b) Al = −800 (GHz)2, νl = 68 GHz, and (c) Al = −1600 (GHz)2, νl = 69 GHz. Au is 1600 (GHz)2.

Fig. 5
Fig. 5

The z component of the electric field (Ez) of the two resonant states of a single unit structure found at 82.8 GHz ((a) and (c)) and 96.6 GHz ((b) and (d)). In (a) and (b), Ez just above the surface of the dielectric slab is shown. In (c) and (d), Ez on the xz plane (y = 0) is shown. In each figure, Ez is normalized by its maximum value in the plot area. Note that the lower (higher) frequency mode has even (odd) parity with respect to the x coordinate.

Fig. 6
Fig. 6

Dispersion curves of the steerable antenna calculated by the FDTD method. Three modes and the light line are plotted.

Fig. 7
Fig. 7

Dispersion curves calculated by the tight-binding approximation. Following parameters are assumed: ω1 = 82.8 GHz, ω2 = 96.9 GHz, c 2 L 0 ( 22 ) = 400 ( GHz ) 2, c 2 L 1 ( 11 ) = 256 ( GHz ) 2, c 2 L 1 ( 22 ) = 1024 ( GHz ) 2, c 2 L 1 ( 12 ) = 256 ( GHz ) 2, c 2 L 1 ( 21 ) = 256 ( GHz ) 2, and c 2 L 0 ( 11 ) = ( a ) 25, (b) 361, (c) 625, and (d) 1225 (GHz)2.

Fig. 8
Fig. 8

Dependence of the mode frequencies at the Γ point on the lattice constant. The A1 and B1 modes are denoted by black and red colors, respectively. The lattice constant was varied from the lowest value of 0.5 mm, for which the nearest neighbor unit structures are touching each other, to a sufficiently large value of 2.4 mm, for which the two frequencies are well converged to values of an isolated unit structure, 82.8 and 96.9 GHz. When the lattice constant comes close to 0.5 mm, the frequency of the B1 mode decreases rapidly. It becomes smaller than the frequency of the A1 mode when the lattice constant is smaller than 0.576 mm.

Tables (1)

Tables Icon

Table 1 Character table of the C2v point group

Equations (55)

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

B = 4 d r ϕ * ( r a ) U ( r ) ϕ ( r ) ,
E = E 0 B 2 cos ka ,
d r ϕ * ( r ) U ( r ) ϕ ( r ) .
f l 2 ( k ) = ν l 2 A l 2 cos ka , ( A l < 0 )
f u 2 ( k ) = ν u 2 A u 2 cos ka , ( A u > 0 )
ν u 2 = ν l 2 + A u A l 2 ,
ɛ = 1 ω p 2 ω 2
× [ 1 ɛ ( r ) × H ( r , t ) ] = 1 c 2 2 t 2 H ( r , t ) ,
× [ 1 ɛ s ( r ) × H 0 ( r ) ] = ω 0 2 c 2 H 0 ( r ) ,
V d r | H 0 ( r ) | 2 = V ,
H k ( r ) = 1 N n = N / 2 N / 2 1 e ikna H 0 ( r n a ) ,
ω k 2 c 2 = ω 0 2 c 2 + L 0 + 2 Re ( L 1 e ika ) ,
L 0 = V d r H 0 * ( r ) × [ δ χ ( r ) × H 0 ( r ) ] ,
L 1 = V d r H 0 * ( r ) × [ δ χ ( r ) × H 0 ( r a ) ] .
δ χ ( r ) = 1 ɛ ( r ) 1 ɛ s ( r ) ,
L 1 = S d S { H 0 * ( r ) × [ δ χ ( r ) × H 0 ( r a ) ] } n + V d r [ × H 0 * ( r ) ] [ δ χ ( r ) × H 0 ( r a ) ] = ɛ 0 2 ω 0 2 V d r ɛ s 2 ( r ) δ χ ( r ) E 0 * ( r ) E 0 ( r a ) ,
E 0 = i ɛ 0 ɛ s ( r ) ω 0 × H 0 ( r ) .
L 1 = ɛ 0 2 ω 0 2 V d r δ χ ( r ) E 0 ( r ) E 0 ( r a ) .
σ x ϕ ( r ) ϕ ( σ x 1 r ) = ± ϕ ( r ) ,
σ x E ( r ) ( σ x E ) ( σ x 1 r ) = ± E ( r ) .
E x ( x , y , z ) = E x ( x , y , z ) ,
E y ( x , y , z ) = + E y ( x , y , z ) ,
E z ( x , y , z ) = + E z ( x , y , z ) .
L 1 ɛ 0 2 ω 0 2 V d r δ χ ( r ) E 0 z ( r ) E 0 z ( r a ) .
ω i = c k i 2 + k z 2 > c | k i | ,
θ = cos 1 [ c a ω i cos 1 2 ( ν l 2 ω i 2 ) A l ]
θ = cos 1 [ c a ω i cos 1 2 ( ν u 2 ω i 2 ) A u ]
ɛ = 1 ω p 2 ( ω + i δ ) ( ω + i γ ) 1 + i σ ɛ 0 ( ω + i δ ) ,
× H = ɛ 0 E t + σ E ,
ɛ 0 E t + σ E ɛ 0 E ( n + 1 ) Δ t E n Δ t Δ t + σ E ( n + 1 ) Δ t + E n Δ t Δ t
E ( x + a , y , z ) = e ikx E ( x , y , z ) ,
H ( x + a , y , z ) = e ikx H ( x , y , z ) ,
ω ( k ) = ω ( k ) .
ɛ d κ cos k z d k z sin k z d = 0 ,
k z = ɛ d ω 2 c 2 k 2 ,
κ = k 2 ω 2 c 2 .
H 0 ( 1 ) : ( σ x , σ y ) = ( + 1 , + 1 ) ,
H 0 ( 2 ) : ( σ x , σ y ) = ( 1 , + 1 ) .
× [ 1 ɛ s ( r ) × H 0 ( 1 , 2 ) ( r ) ] = ω 1 , 2 2 c 2 H 0 ( 1 , 2 ) ( r ) .
V d r H 0 ( i ) * ( r ) H 0 ( j ) ( r ) = V δ i j ,
H k ( r ) = 1 N n = N / 2 N / 2 1 e ikna { A H 0 ( 1 ) ( r n a ) + B H 0 ( 2 ) ( r n a ) } .
× [ { 1 ɛ s ( r ) + δ χ ( r ) } × H k ( r ) ] = ω k 2 c 2 H k ( r ) .
V d r H 0 ( 1 ) * ( r ) × [ 1 ɛ s ( r ) × H k ( r ) ] A V ω 1 2 c 2 N ,
V d r H 0 ( 1 ) * ( r ) × [ δ χ ( r ) × H k ( r ) ] = A V N n e ikna L n ( i j ) ,
L n ( i j ) = 1 V V d r H 0 ( i ) * ( r ) × [ δ χ ( r ) × H 0 ( j ) ( r n a ) ]
L 0 ( 12 ) = L 0 ( 21 ) = 0 ,
L 1 ( 12 ) = L 1 ( 12 ) ,
L 1 ( 21 ) = L 1 ( 21 ) ,
( ω 1 2 c 2 ω k 2 c 2 + L 0 ( 11 ) + 2 L 1 ( 11 ) cos k a ) A + 2 i B L 1 ( 12 ) sin k a = 0 .
2 i A L 1 ( 21 ) sin ka + ( ω 2 2 c 2 ω k 2 c 2 + L 0 ( 22 ) + 2 L 1 ( 22 ) cos ka ) B = 0 .
| ω 1 2 ω k 2 c 2 + L 0 ( 11 ) + 2 L 1 ( 11 ) cos ka , 2 i L 1 ( 12 ) sin ka 2 i L 1 ( 21 ) sin k a , ω 2 2 ω k 2 c 2 + L 0 ( 22 ) + 2 L 1 ( 22 ) cos k a | = 0 .
ω 1 2 c 2 + L 0 ( 11 ) + 2 L 1 ( 11 ) = ω 2 2 c 2 + L 0 ( 22 ) + 2 L 1 ( 2 ) ω Γ 2 c 2 ,
ω k 2 c 2 = ω Γ 2 c 2 + { L 1 ( 11 ) + L 1 ( 22 ) } ( cos ka 1 ) ± { L 1 ( 11 ) L 1 ( 22 ) } 2 ( cos k a 1 ) 2 4 L 1 ( 12 ) L 1 ( 21 ) sin 2 ka .
ω k ω Γ ± c 2 k a ω Γ | L 1 ( 12 ) L 1 ( 21 ) | .
ω k 2 c 2 = 1 2 [ ω 1 2 + ω 2 2 c 2 + L 0 ( 11 ) + L 0 ( 22 ) + 2 ( L 1 ( 11 ) + L 1 ( 22 ) ) cos ka ] ± 1 2 { [ ω 1 2 ω 2 2 c 2 + L 0 ( 11 ) L 0 ( 22 ) + 2 ( L 1 ( 11 ) L 1 ( 22 ) ) cos k a ] 2 16 L 1 ( 12 ) L 1 ( 21 ) sin 2 ka } 1 / 2 .

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