Abstract

Dispersion curves of metamaterial steerable antennas composed of two-dimensional arrays of metallic unit structures with the C 4v and C 6v symmetries are calculated both qualitatively by the tight-binding approximation and quantitatively by the finite-difference time-domain method. Special attention is given to the case of eigenmodes of the E symmetry of the C 4v point group and those of the E 1 and E 2 symmetries of the C 6v point group, since they are doubly degenerate on the Γ point of the Brillouin zone so that they naturally satisfy the steerability condition. We show that their dispersion curves have quadratic dependence on the wave vector in the vicinity of the Γ point. To get a linear dispersion, which is advantageous for steerable antennas, we propose a method of controlled symmetry reduction. The present theory is an extension of our previous one [Opt. Express 18, 27371 (2010)] to two-dimensional systems, for which we can achieve the deterministic degeneracy due to symmetry and the controlled symmetry reduction becomes available. This design of metamaterial steerable antennas is advantageous in the optical frequency.

©2011 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).
  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. E89-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. K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express 18, 27371–27386 (2010).
    [Crossref]
  13. K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed. (Springer-Verlag, 2004).
  14. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and its Applications in Physics (Springer, 1990).
    [Crossref]
  15. A. Taflove, Computational Electrodynamics (Artech House, 1995).
  16. D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method (IEEE Press, 2000).
    [Crossref]

2010 (1)

2006 (3)

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]

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. E89-C, 1337–1344 (2006).
[Crossref]

2003 (1)

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).

2002 (2)

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 (1)

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 (1)

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]

1968 (1)

V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

1951 (1)

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).

Caloz, C.

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]

Fujinami, H.

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).

Fujita, K.

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).

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]

Grzegorczyk, T. M.

S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (SPIE Press, 2008).
[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. E89-C, 1337–1344 (2006).
[Crossref]

Inui, T.

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

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]

Kawakubo, A.

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).

Kuroda, K.

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).

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).

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. E89-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. E89-C, 1337–1344 (2006).
[Crossref]

Onodera, Y.

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and its Applications in Physics (Springer, 1990).
[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]

Ramakrishna, S. A.

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

Sakoda, K.

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. E89-C, 1337–1344 (2006).
[Crossref]

Schultz, S.

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]

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).

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]

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]

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).

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).

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]

Sullivan, D. M.

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

Taflove, A.

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

Tanabe, Y.

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and its Applications in Physics (Springer, 1990).
[Crossref]

Tokoro, S.

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).

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]

Zhou, H.-F.

IEEE-AP-S Int. Symp. Dig. (1)

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. (1)

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. E89-C, 1337–1344 (2006).
[Crossref]

J. Appl. Phys. (1)

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]

Opt. Express (1)

Phys. Rev. B (1)

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).

Phys. Rev. Lett. (1)

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]

Proc. IEEE Intelligent Vehicles Symp. (1)

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).

Science (3)

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. (1)

V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

Other (5)

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

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

T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and its Applications in Physics (Springer, 1990).
[Crossref]

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

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

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Figures (7)

Fig. 1
Fig. 1 Conceptual dispersion curves of a metamaterial steerable antenna. The vertical axis is the angular frequency ω of the electromagnetic field normalized by the light velocity in free space c and the lattice constant of the regular array of unit structures a. The horizontal axis is the wave vector in the first Brillouin zone. The dispersion consists of two curves: The upper one denoted by fu is concave-up and the lower one denoted by fl is concave-down. They touch each other on the Γ point (k = 0), which can be realized by accidental degeneracy of eigen frequencies due to appropriate combination of device parameters or by deterministic degeneracy caused by spatial symmetry of device structure. When an incident wave with angular frequency ωi is propagated in the positive k direction, it excites an eigenmode (denoted by wave vector ki ) with a positive group velocity, which is given by ∂ω/∂k. So, only eigenmodes on dispersion branches with positive slopes denoted by the blue color are excited. If the angular frequency of the excited mode is located above the light lines, which are given by ω = ±ck, the incident wave is leaky and diffracted in the direction determined by ωi and ki . Thus, the upper and lower limits of working frequency are given by ωu and ωl . Reproduced with permission from Opt. Express 18, 27371 (2010). Copyright 2010 The Optical Society (OSA).

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Fig. 2
Fig. 2 Illustration of dispersion curves in the case of accidental degeneracy, which can be realized by A 1 and B 1 modes, or by A 2 and B 2 modes on the Γ point of one-dimensional metamaterial steerable antennas composed of regular arrays of unit structures of the C 2v symmetry.

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Fig. 3
Fig. 3 Symmetry operations of the C 4v group. There are two sets of two equivalent mirror reflections, which are denoted by (σx , σy ) and (σ′d , σ″d ). Lattice points are denoted by solid circles.

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Fig. 4
Fig. 4 Symmetry operations of the C 6v group. There are two sets of three equivalent mirror reflections that are denoted by (σx , σ′x , σ″x ) and (σy , σ′y , σ″y ). Seven lattice points (the origin and its nearest neighbors) are denoted by integers from 0 to 6.

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Fig. 5
Fig. 5 Illustration of metallic unit structures of (a) square and (b) hexagonal symmetries. These unit structures were assumed to be fabricated on a dielectric slab with a ground electrode on its back surface. Based on the device design of Ref. [8], the dielectric constant and thickness of the slab were assumed to be 2.2 and 0.127 mm, respectively. For calculating the dispersion relation, (c) a periodic square array of the unit structure shown in (a) was assumed, whose lattice constant a was 0.6 mm.

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Fig. 6
Fig. 6 Distribution of Hz of resonance states of a unit structure. (a), (b) E mode at 158 GHz, (c), (d) E 1 mode at 177 GHz, and (e), (f) E 2 mode at 295 GHz. Hz on the horizontal plane in the middle of the dielectric slab is shown.

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Fig. 7
Fig. 7 Dispersion curves of the regular square array of unit structures illustrated in Fig. 5(c). In the analyzed frequency range from 100 to 250 GHz, there are four electromagnetic modes: (1) two modes originating from a degenerate E resonance state shown in Fig. 6(a) and 6(b), (2) one mode originating from a non-degenerate B 2 resonance state, and (3) one mode that has the character of the lowest TM waveguide mode of the dielectric slab whose original dispersion is very close to the light line given by ω = ck. The parity of the electric field with respect to the y coordinate is denoted by py , which the reader should note is opposite to that of the magnetic field. Because modes with the same parity mix with each other when their dispersion curves come close, they show apparent anti-crossing behaviors.

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Tables (4)

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Table 1 Parity of Eigenmodes on the Γ Point for a Regular Array of Metallic Unit Structure of the C 2v Symmetry

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Table 2 Character Table of the C 4v Point Group

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Table 3 Character Table of the C 6v Point Group

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Table 4 Character Table of the Cs Point Group

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Equations (122)

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θ   =   cos − 1   c k i ω i ,
ω − k   =   ω k ,
∇   ×   [ 1 ɛ ( r ) ∇   ×   H ( r ,   t ) ]   =   − 1 c 2 ∂ 2 ∂ t 2 H ( r ,   t ) ,
∇   ×   [ 1 ɛ s ( r ) ∇   ×   H 0 ( 1 , 2 ) ( r ) ]   =   ω 0 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 , m e i k ⋅ r n m   [ A H 0 ( 1 )   ( r   −   r n m )   +   B H 0 ( 2 )   ( r   −   r n m ) ] ,
r n m   =   n a 1   +   m a 2 .
a 1   =   ( a 0 ) ,   a 2   =   ( 0 a ) ,
H k ( r + a i ) = e i k ⋅ a i H k ( r ) .
ℒ H k ( r ) = ω k 2 c 2 H k ( r ) ,
ℒ H ( r ) = ∇ × [ 1 ɛ ( r ) ∇ × H ( r ) ] .
∫ V d r   H 0 ( 1 ) * ( r ) ⋅ ℒ H k ( r ) = V N ∑ n , m e i k ⋅ r n m [ A L n m ( 11 ) + B L n m ( 12 ) ] ,
L n m ( i j ) = 1 V ∫ V d r   H 0 ( i ) * ( r ) ⋅ [ ℒ H 0 ( j ) ( r − r n m ) ] .
L 0 , 0 ( 11 ) = L 0 , 0 ( 22 ) ≡ ω 0 2 c 2 + M 1 ,
L 0 , 0 ( 12 ) = L 0 , 0 ( 21 ) = 0 ,
L ± 1 , 0 ( 11 ) = L 0 , ± 1 ( 22 ) ≡ M 2 ,
L 0 , ± 1 ( 11 ) = L ± 1 , 0 ( 22 ) ≡ M 3 ,
L ± 1 , 0 ( 12 ) = L 0 , ± 1 ( 12 ) = L ± 1 , 0 ( 21 ) = L 0 , ± 1 ( 21 ) = 0 ,
L ± 1 , ± 1 ( 11 ) = L ± 1 , ± 1 ( 22 ) ≡ M 4 ,
L 1 , 1 ( 12 ) = L − 1 , − 1 ( 12 ) = − L 1 , − 1 ( 12 ) = − L − 1 , 1 ( 12 ) = L 1 , 1 ( 21 ) = L − 1 , − 1 ( 21 ) = − L 1 , − 1 ( 21 ) = − L − 1 , 1 ( 21 ) ≡ M 5
A [ ω 0 2 − ω k 2 + c 2 ( M 1 + 2 M 2 cos k x a + 2 M 3 cos k y a + 4 M 4 cos k x a cos k y a ) ] = 4 B c 2 M 5 sin k x a sin k y a .
4 A c 2 M 5 sin k x a sin k y a = B [ ω 0 2 − ω k 2 + c 2 ( M 1 + 2 M 3 cos k x a + 2 M 2 cos k y a + 4 M 4 cos k x a cos k y a . ) ]
ω k 2 c 2 = ω 0 2 c 2 + M 1 + ( M 2 + M 3 ) ( cos k x a + cos k y a ) + 4 M 4 cos k x a cos k y a ± [ ( M 2 − M 3 ) 2 ( cos k x a − cos k y a ) 2 + 16 M 5 2 sin 2 k x a sin 2 k y a ] 1 / 2 .
ω Γ = ω 0 2 + c 2 ( M 1 + 2 M 2 + 2 M 3 + 4 M 4 ) .
ω − k = ω k ,
ω k = ω Γ − a 2 c 2 k 2 4 ω Γ [ M 2 + M 3 + 4 M 4 ± 2 F ( ϕ ) ] ,
k = k x 2 + k y 2 ,
ϕ = tan − 1 k y k x ,
F ( ϕ ) = ( M 2 − M 3 2 ) 2 cos 2 2 ϕ + 4 M 5 2 sin 2 2 ϕ .
a 1 = ( a 0 ) , a 2 = ( a / 2 3 a / 2 ) .
L 0 ( 11 ) = L 0 ( 22 ) ≡ ω 0 2 c 2 + M 1 ,
L 0 ( 12 ) = L 0 ( 21 ) = 0 ,
L 1 ( 11 ) = L 4 ( 11 ) ≡ M 2 ,
L 1 ( 22 ) = L 4 ( 22 ) ≡ M 3 ,
L 1 ( 12 ) = L 4 ( 12 ) = L 1 ( 21 ) = L 4 ( 21 ) = 0 ,
L 2 ( 11 ) = L 3 ( 11 ) = L 5 ( 11 ) = L 6 ( 11 ) ≡ M 4 ,
L 2 ( 22 ) = L 3 ( 22 ) = L 5 ( 22 ) = L 6 ( 2 ) ≡ M 5 ,
L 2 ( 12 ) = − L 3 ( 12 ) = L 5 ( 12 ) = − L 6 ( 12 ) = L 2 ( 21 ) = − L 3 ( 21 ) = L 5 ( 21 ) = − L 6 ( 21 ) ≡ M 6 .
M 4 = M 2 + 3 M 3 4 ,
M 5 = 3 M 2 + M 3 4 ,
M 6 = 3 ( M 2 − M 3 ) 4 .
ω k 2 c 2 = ω 0 2 c 2 + M 1 + ( M 2 + M 3 ) ( cos k x a + 2 cos k x a 2 cos 3 k y a 2 ) ± ( M 2 − M 3 ) [ ( cos k x a − cos k x a 2 cos 3 k y a 2 ) 2 + 3 sin 2 k x a 2 sin 2 3 k y a 2 ] 1 / 2
ω Γ = ω 0 2 + c 2 ( M 1 + 3 M 2 + 3 M 3 ) .
ω k = ω Γ − 3 a 2 c 2 k 2 M 16 ω Γ ,
M = { M 2 + 3 M 3 , 3 M 2 + M 3 .
χ ( H ) ( R ) = det R χ ( E ) ( R ) ,
∇ × [ 1 ɛ s ( r ) ∇ × H 1 , 2 ( r ) ] = ω 1 , 2 2 c 2 H 1 , 2 ( r ) .
H k ( r ) = 1 N ∑ n , m e i k ⋅ r n m [ A H 1 ( r − r n m ) + B H 2 ( r − r n m ) ] .
L n m ( i j ) = 1 V ∫ V d r   H i * ( r ) ⋅ [ ℒ H j ( r − r n m ) ] .
L 0 , 0 ( 11 ) ≡ ω 1 2 c 2 + M ′ 1 ,               L 0 , 0 ( 22 ) ≡ ω 2 2 c 2 + M ″ 1 ,
L 0 , 0 ( 12 ) = L 0 , 0 ( 21 ) = 0 ,
L 1 , 0 ( 11 ) = L − 1 , 0 ( 11 ) ≡ M 2 ,
L 1 , 0 ( 22 ) = L − 1 , 0 ( 22 ) ≡ M 3 ,
L 1 , 0 ( 12 ) = − L − 1 , 0 ( 12 ) = L − 1 , 0 ( 21 ) * = − L 1 , 0 ( 21 ) * ≡ L 1 ,
L 0 , 1 ( 11 ) = L 0 , − 1 ( 11 ) * ≡ M ′ 3 ,
L 0 , 1 ( 22 ) = L 0 , − 1 ( 22 ) * ≡ M ′ 2 ,
L 0 , 1 ( 12 ) = L 0 , − 1 ( 12 ) = L 0 , 1 ( 21 ) = L 0 , − 1 ( 21 ) = 0.
ω k 2 c 2 = 1 2 ( ω 1 2 + ω 2 2 c 2 + M ′ 1 + M ″ 1 ) + ( M 2 + M 3 ) cos k x a + ( M ′ 2 + M ′ 3 ) cos k y a ± 1 2 { [ ω 1 2 − ω 2 2 c 2 + M ′ 1 − M ″ 1 + 2 ( M 2 − M 3 ) cos k x a + 2 ( M ′ 3 − M ″ 2 ) cos k y a ] 2 + 16 L 1 2 sin 2 k x a } 1 / 2 .
ω Γ ≡ ω 1 2 + c 2 ( M ′ 1 + 2 M 2 + 2 M ′ 3 ) = ω 2 2 + c 2 ( M ″ 1 + 2 M 3 + 2 M ′ 2 ) .
ω k 2 c 2 = ω Γ 2 c 2 + ( M 2 + M 3 ) ( cos k x a − 1 ) + ( M ′ 2 + M ′ 3 ) ( cos k y a − 1 ) ± { [ ( M 2 − M 3 ) ( cos k x a − 1 ) + ( M ′ 3 − M ′ 2 ) ( cos k y a − 1 ) ] 2 + 4 L 1 2 sin 2 k x a } 1 / 2 .
ω k = ω Γ ± a c 2 k x L 1 ω Γ ,
f 1 ( r ) = x       and       f 2 ( r ) = y .
R f 1 , 2 ( r ) = f 1 , 2 ( R − 1 r ) .
σ x f 1 ( r ) = − x = − f 1 ( r ) ,
σ x f 2 ( r ) = y = f 2 ( r ) .
σ x ( f 1 f 2 ) = ( − 1 , 0 0 , 1 ) ( f 1 f 2 ) .
σ x : ( − 1 , 0 0 , 1 ) , σ y : ( 1 , 0 0 , − 1 ) , σ ′ d : ( 0 , 1 1 , 0 ) , σ ″ d : ( 0 , − 1 − 1 , 0 ) , C 4 : ( 0 , 1 − 1 , 0 ) , C 4 − 1 : ( 0 , − 1 1 , 0 ) , C 2 : ( − 1 , 0 0 , − 1 ) , E : ( 1 , 0 0 , 1 ) .
[ σ x H 0 ( i ) ] ( r ) ≡ σ x H 0 ( i ) ( σ x − 1 r ) .
ℒ H ( r ) = ∇ × [ 1 ɛ ( r ) ∇ × H ( r ) ] .
L n m ( i j ) = 1 V ∫ V d r   H 0 ( i ) * ( r ) ⋅ [ ℒ H 0 ( j ) ( r − r n m ) ] .
V L 00 ( 12 ) = ∫ V d r ′ H 0 ( 1 ) * ( σ x − 1 r ′ ) ⋅ [ ℒ H 0 ( 2 ) ( σ x − 1 r ′ ) ] = ∫ V d r ′ [ σ x − 1 σ x H 0 ( 1 ) * ( σ x − 1 r ′ ) ] ⋅ [ σ x − 1 σ x ℒ σ x − 1 σ x H 0 ( 2 ) ( σ x − 1 r ′ ) ] ,
V L 00 ( 12 ) = ∫ V d r ′ [ σ x H 0 ( 1 ) * ( σ x − 1 r ′ ) ] ⋅ [ ℒ ′ σ x H 0 ( 2 ) ( σ x − 1 r ′ ) ] ,
ℒ ′ = σ x ℒ σ x − 1 .
V L 00 ( 12 ) = ∫ V d r ′ [ − H 0 ( 1 ) * ( r ′ ) ] ⋅ [ ℒ ′ H 0 ( 2 ) ( r ′ ) ] = − V L 00 ( 12 ) .
L 00 ( 12 ) = 0.
L 00 ( 21 ) = 0.
L 00 ( 11 ) = ω 0 2 c 2 + M 1 ,
V L 00 ( 11 ) = ∫ V d r ′ [ C 4 H 0 ( 1 ) * ( C 4 − 1 r ′ ) ] ⋅ [ ℒ ′ C 4 H 0 ( 1 ) ( C 4 − 1 r ′ ) ] = ∫ V d r ′ H 0 ( 2 ) * ( r ′ ) ⋅ [ ℒ ′ H 0 ( 2 ) ( r ′ ) ] = V L 00 ( 22 ) .
L 00 ( 11 ) = L 00 ( 22 ) ≡ ω 0 2 c 2 + M 1 .
V L 10 ( 12 ) = ∫ V d r ′ [ σ y H 0 ( 1 ) * ( σ y − 1 r ′ ) ]                   ⋅ [ ℒ ′ σ y H 0 ( 2 ) ( σ y − 1 ( r ′ − r 10 ) ) ] = ∫ V d r ′ [ H 0 ( 1 ) * ( r ′ ) ] ⋅ [ − ℒ ′ H 0 ( 2 ) ( r ′ − r 10 ) ] = − V L 10 ( 12 ) .
L 10 ( 12 ) = 0.
L ± 1 , 0 ( 12 ) = L 0 , ± 1 ( 12 ) = L ± 1 , 0 ( 21 ) = L 0 , ± 1 ( 21 ) = 0 .
V L 10 ( 11 ) = ∫ V d r ′ [ C 4 H 0 ( 1 ) * ( C 4 − 1 r ′ ) ] ⋅ [ ℒ ′ C 4 H 0 ( 1 ) ( C 4 − 1 r ′ − r 10 ) ] = ∫ V d r ′ H 0 ( 2 ) * ( r ′ ) ⋅ [ ℒ ′ H 0 ( 2 ) ( r ′ − r 01 ) ] = V L 01 ( 22 ) ,
L 10 ( 11 ) = L − 1 , 0 ( 11 ) = L 0 , − 1 ( 22 ) .
L ± 1 , 0 ( 11 ) = L 0 , ± 1 ( 22 ) ≡ M 2 .
L 0 , ± 1 ( 11 ) = L ± 1 , 0 ( 22 ) ≡ M 3 .
L ± 1 , ± 1 ( 11 ) = L ± 1 , ± 1 ( 22 ) ≡ M 4 .
L 1 , 1 ( 12 ) = L − 1 , − 1 ( 12 ) = − L 1 , − 1 ( 12 ) = − L − 1 , 1 ( 12 ) = L 1 , 1 ( 21 ) = L − 1 , − 1 ( 21 ) = − L 1 , − 1 ( 21 ) = − L − 1 , 1 ( 21 ) ≡ M 5 .
σ x : ( − 1 , 0 0 , 1 ) , σ y : ( 1 , 0 0 , − 1 ) , C 6 : ( 1 / 2 , 3 / 2 − 3 / 2 , 1 / 2 ) , C 6 − 1 : ( 1 / 2 , − 3 / 2 3 / 2 , 1 / 2 ) , C 3 : ( − 1 / 2 , 3 / 2 − 3 / 2 , − 1 / 2 ) , C 3 − 1 : ( − 1 / 2 , − 3 / 2 3 / 2 , − 1 / 2 ) , C 2 : ( − 1 , 0 0 , − 1 ) , E : ( 1 , 0 0 , 1 ) .
V L 0 ( 11 ) = ∫ V d r ′ H 0 ( 1 ) * ( C 6 − 1 r ′ ) ⋅ [ ℒ H 0 ( 1 ) ( C 6 − 1 r ′ ) ] = ∫ V d r ′ [ C 6 H 0 ( 1 ) * ( C 6 − 1 r ′ ) ] ⋅ [ ℒ ′ C 6 H 0 ( 1 ) ( C 6 − 1 r ′ ) ] = V ( L 0 ( 11 ) + 3 L 0 ( 21 ) + 3 L 0 ( 12 ) + 3 L 0 ( 22 ) ) 4 .
L 0 ( 11 ) = L 0 ( 11 ) − 3 L 0 ( 21 ) − 3 L 0 ( 12 ) + 3 L 0 ( 22 ) 4 .
L 0 ( 12 ) + L 0 ( 21 ) = 0 ,
L 0 ( 11 ) = L 0 ( 22 ) ≡ ω 0 2 c 2 + M 1 .
V L 0 ( 12 ) = ∫ V d r ′ [ − H 0 ( 1 ) * ( r ′ ) ] ⋅ [ ℒ ′ H 0 ( 2 ) ( r ′ ) ] = − V L 0 ( 12 ) .
L 0 ( 12 ) = L 0 ( 21 ) = 0.
V L 1 ( 11 ) = ∫ V d r ′ [ − H 0 ( 1 ) * ( r ′ ) ] ⋅ [ − ℒ ′ H 0 ( 1 ) ( r ′ − r 4 ) ] = V L 4 ( 11 ) ,
V L 1 ( 22 ) = ∫ V d r ′ [ − H 0 ( 2 ) * ( r ′ ) ] ⋅ [ − ℒ ′ H 0 ( 2 ) ( r ′ − r 4 ) ] = V L 4 ( 22 ) .
L 1 ( 11 ) = L 4 ( 11 ) ≡ M 2 ,
L 1 ( 22 ) = L 4 ( 22 ) ≡ M 3 .
L 1 ( 12 ) = L 4 ( 12 ) ,   L 1 ( 21 ) = L 4 ( 21 )
L 1 ( 12 ) = − L 4 ( 12 ) ,   L 1 ( 21 ) = − L 4 ( 21 )
L 1 ( 12 ) = L 4 ( 12 ) ,   L 1 ( 21 ) = L 4 ( 21 ) = 0.
L 2 ( 11 ) = L 3 ( 11 ) = L 5 ( 11 ) = L 6 ( 11 ) ≡ M 4 ,
L 2 ( 22 ) = L 3 ( 22 ) = L 5 ( 22 ) = L 6 ( 22 ) ≡ M 5 .
L 2 ( 12 ) = − L 3 ( 12 ) = L 5 ( 12 ) = − L 6 ( 12 ) = L 2 ( 21 ) = − L 3 ( 21 ) = L 5 ( 21 ) = − L 6 ( 21 ) ≡ M 6 .
M 4 = L 2 ( 11 ) = 1 V ∫ V d r ′ H 0 ( 1 ) ( r ′ ) − 3 H 0 ( 2 ) * ( r ′ ) 2 ⋅ [ ℒ ′ H 0 ( 1 ) ( r ′ − r 1 ) − 3 H 0 ( 2 ) ( r ′ − r 1 ) 2 ] = M 2 + 3 M 3 4 ,
M 5   =   3 M 2   +   M 3 4 ,
M 6   =   3 ( M 2   −   M 3 ) 4 .
σ x : ( − 1 , 0 0 , 1 ) , σ y : ( − 1 , 0 0 , 1 ) , C 6 : ( − 1 / 2 , − 3 / 2 3 / 2 , − 1 / 2 ) , C 6 − 1 : ( − 1 / 2 , 3 / 2 − 3 / 2 , − 1 / 2 ) , C 3 : ( − 1 / 2 , 3 / 2 − 3 / 2 , − 1 / 2 ) , C 3 − 1 : ( − 1 / 2 , − 3 / 2 3 / 2 , − 1 / 2 ) , C 2 : ( 1 , 0 0 , 1 ) , E : ( 1 , 0 0 , 1 ) .
L 00 ( 11 )   ≡   ω 1 2 c 2   +   M 1 ′ ,   L 00 ( 22 )   ≡   ω 2 2 c 2   +   M 1 ″ ,
L 00 ( 12 )   =   L 00 ( 21 )   =   0.
L 10 ( 11 )   =   L − 1 , 0 ( 11 )   ≡   M 2 ,
L 10 ( 22 )   =   L − 1 , 0 ( 22 )   ≡   M 3 .
T nm ℒ T nm − 1 = ( T nm ∇   ×   T nm − 1 )   ( T nm   1 ɛ ( r ) T nm − 1 )   ( T nm ∇   ×   T nm − 1 ) = ℒ .
∫ V d r Q 1 *   ( r )   ⋅   [ ℒ Q 2   ( r ) ]     =   ∫ V d r [ ℒ Q 1   ( r ) ] *   ⋅   Q 2   ( r ) ,  
L 10 ( 11 ) = 1 V   ∫ V d r ′ H 1 *   ( r ′   +   r 10 )   ⋅   [ ( T − 1 , 0 ℒ T − 1 , 0 − 1 ) ] H 1   ( r ′ )   = 1 V   ∫ V d r ′ [ ℒ H 1   ( r ′   −   r − 1 , 0 ) ] *   ⋅ H 1   ( r ′ )     = L − 1 , 0 ( 11 ) * .
L 10 ( 22 )   =   L − 1 , 0 ( 22 ) * .
L 10 ( 12 )   =   − L − 1 , 0 ( 12 )   =   L − 1 , 0 ( 21 ) * .
L 10 ( 21 )   =   − L − 1 , 0 ( 21 )   =   L − 1 , 0 ( 12 ) * .
L 10 ( 12 )   =   − L − 1 , 0 ( 12 )   =   L − 1,0 ( 21 ) *   =   − L 10 ( 21 ) *   ≡   L 1 .
L 01 ( 11 )   =   L 0 , − 1 ( 11 ) *   ≡   M 3 ′ ,   L 01 ( 22 )   =   L 0 , − 1 ( 22 ) *   ≡   M 2 ′ .
L 01 ( 12 )   =   L 0 , − 1 ( 12 )   =   L 01 ( 21 )   =   L 0 , − 1 ( 21 )   =   0.

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