Abstract

We analyze propagation of electromagnetic waves in a photonic crystal at frequencies at which it behaves as an effective medium with a negative index in terms of refraction at its interface with free space. We show that the phase evolution along the propagation direction is positive, despite the fact that the photonic crystal displays negative refraction following Snell’s law, and explain it in terms of the Fourier components of the Bloch wave. Two distinct behaviors are found at frequencies far and close to the band edge of the negative-index photonic band. These findings contrast with the negative phase evolution that occurs in left-handed materials, so care has to be taken when applying the term left-handed to photonic crystals.

© 2006 Optical Society of America

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References

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  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  2. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  3. C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell’s law," Phys. Rev. Lett. 90, 107401 (2003);A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell’s law," Phys. Rev. Lett. 90, 137401 (2003).
    [CrossRef] [PubMed]
  4. A. Grbic and G. V. Eleftheriades, "Overcoming the Diffraction Limit with a Planar Left-Handed Transmission-Line Lens," Phys. Rev. Lett. 92, 117403 (2004).
    [CrossRef] [PubMed]
  5. M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
    [CrossRef]
  6. S. Foteinopoulou and C. M. Soukoulis, "Negative refraction and left-handed behavior in two-dimensional photonic crystals," Phys. Rev. B 67, 235107 (2003).
    [CrossRef]
  7. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative Refraction and Left-Handed Electromagnetism in Microwave Photonic Crystals," Phys. Rev. Lett. 92, 127401 (2004).
    [CrossRef] [PubMed]
  8. A. Martínez and J. Martí, "Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination," Phys. Rev. B 71, 235115 (2005).
    [CrossRef]
  9. A. Martínez, H. Míguez, J. Sánchez-Dehesa, and J. Martí, "Analysis of wave propagation in a two-dimensional photonic crystal with negative index of refraction: plane wave decomposition of the Bloch modes," Opt. Express 13, 4160-4174, (2005).
    [CrossRef] [PubMed]
  10. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, "Fourier analysis of Bloch wave propagation in photonic crystals," J. Opt. Soc. Am. B 22, 1179-1190 (2005).
    [CrossRef]
  11. I. De Leon and F. S. Roux, "Fourier analysis of reflection and refraction in two-dimensional photonic crystals," Phys. Rev. B 71, 235105 (2005).
    [CrossRef]
  12. R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, "Refraction and rightness in photonic crystals," Opt. Express 13, 8596-8605 (2005).
    [CrossRef] [PubMed]
  13. S. Foteinopoulou and C. M. Soukoulis, "Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects," Phys. Rev. B 72, 165112 (2005).
    [CrossRef]
  14. In this article, calculations of photonic bands have been done with BandSolveTM, which implements the plane wave expansion method, and FDTD simulations with FullwaveTM, both by RSoft Design Group.
  15. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature (London) 425, 944-947 (2003).
    [CrossRef]
  16. P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
    [CrossRef]

2005 (6)

A. Martínez and J. Martí, "Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination," Phys. Rev. B 71, 235115 (2005).
[CrossRef]

A. Martínez, H. Míguez, J. Sánchez-Dehesa, and J. Martí, "Analysis of wave propagation in a two-dimensional photonic crystal with negative index of refraction: plane wave decomposition of the Bloch modes," Opt. Express 13, 4160-4174, (2005).
[CrossRef] [PubMed]

B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, "Fourier analysis of Bloch wave propagation in photonic crystals," J. Opt. Soc. Am. B 22, 1179-1190 (2005).
[CrossRef]

I. De Leon and F. S. Roux, "Fourier analysis of reflection and refraction in two-dimensional photonic crystals," Phys. Rev. B 71, 235105 (2005).
[CrossRef]

R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, "Refraction and rightness in photonic crystals," Opt. Express 13, 8596-8605 (2005).
[CrossRef] [PubMed]

S. Foteinopoulou and C. M. Soukoulis, "Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects," Phys. Rev. B 72, 165112 (2005).
[CrossRef]

2004 (2)

A. Grbic and G. V. Eleftheriades, "Overcoming the Diffraction Limit with a Planar Left-Handed Transmission-Line Lens," Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative Refraction and Left-Handed Electromagnetism in Microwave Photonic Crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

2003 (4)

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell’s law," Phys. Rev. Lett. 90, 107401 (2003);A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell’s law," Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell’s law," Phys. Rev. Lett. 90, 107401 (2003);A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell’s law," Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

S. Foteinopoulou and C. M. Soukoulis, "Negative refraction and left-handed behavior in two-dimensional photonic crystals," Phys. Rev. B 67, 235107 (2003).
[CrossRef]

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature (London) 425, 944-947 (2003).
[CrossRef]

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
[CrossRef]

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)

M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

1968 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Akahane, Y.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature (London) 425, 944-947 (2003).
[CrossRef]

Asano, T.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature (London) 425, 944-947 (2003).
[CrossRef]

Brock, J. B.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell’s law," Phys. Rev. Lett. 90, 107401 (2003);A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell’s law," Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Chuang, I. L.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell’s law," Phys. Rev. Lett. 90, 107401 (2003);A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell’s law," Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

De Leon, I.

I. De Leon and F. S. Roux, "Fourier analysis of reflection and refraction in two-dimensional photonic crystals," Phys. Rev. B 71, 235105 (2005).
[CrossRef]

Derov, J. S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative Refraction and Left-Handed Electromagnetism in Microwave Photonic Crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Dunbar, L. A.

Eleftheriades, G. V.

A. Grbic and G. V. Eleftheriades, "Overcoming the Diffraction Limit with a Planar Left-Handed Transmission-Line Lens," Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

Ferrini, R.

Forester, D. W.

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
[CrossRef]

Foteinopoulou, S.

S. Foteinopoulou and C. M. Soukoulis, "Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects," Phys. Rev. B 72, 165112 (2005).
[CrossRef]

S. Foteinopoulou and C. M. Soukoulis, "Negative refraction and left-handed behavior in two-dimensional photonic crystals," Phys. Rev. B 67, 235107 (2003).
[CrossRef]

Gajic, R.

Grbic, A.

A. Grbic and G. V. Eleftheriades, "Overcoming the Diffraction Limit with a Planar Left-Handed Transmission-Line Lens," Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

Greegor, R. B.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell’s law," Phys. Rev. Lett. 90, 107401 (2003);A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell’s law," Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Hingerl, K.

Houck, A. A.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell’s law," Phys. Rev. Lett. 90, 107401 (2003);A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell’s law," Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Houdré, R.

Koltenbah, B. E. C.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell’s law," Phys. Rev. Lett. 90, 107401 (2003);A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell’s law," Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Kuchar, F.

Li, K.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell’s law," Phys. Rev. Lett. 90, 107401 (2003);A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell’s law," Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Lombardet, B.

Loschialpo, P. F.

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
[CrossRef]

Lu, W. T.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative Refraction and Left-Handed Electromagnetism in Microwave Photonic Crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Martí, J.

Martínez, A.

Meisels, R.

Míguez, H.

Noda, S.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature (London) 425, 944-947 (2003).
[CrossRef]

Notomi, M.

M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

Parazzoli, C. G.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell’s law," Phys. Rev. Lett. 90, 107401 (2003);A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell’s law," Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Parimi, P. V.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative Refraction and Left-Handed Electromagnetism in Microwave Photonic Crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Rachford, F. J.

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
[CrossRef]

Roux, F. S.

I. De Leon and F. S. Roux, "Fourier analysis of reflection and refraction in two-dimensional photonic crystals," Phys. Rev. B 71, 235105 (2005).
[CrossRef]

Sánchez-Dehesa, J.

Schelleng, J.

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
[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]

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. L.

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
[CrossRef]

Smith, D. R.

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

Sokoloff, J.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative Refraction and Left-Handed Electromagnetism in Microwave Photonic Crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Song, B.-S.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature (London) 425, 944-947 (2003).
[CrossRef]

Soukoulis, C. M.

S. Foteinopoulou and C. M. Soukoulis, "Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects," Phys. Rev. B 72, 165112 (2005).
[CrossRef]

S. Foteinopoulou and C. M. Soukoulis, "Negative refraction and left-handed behavior in two-dimensional photonic crystals," Phys. Rev. B 67, 235107 (2003).
[CrossRef]

Sridhar, S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative Refraction and Left-Handed Electromagnetism in Microwave Photonic Crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Tanielian, M.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell’s law," Phys. Rev. Lett. 90, 107401 (2003);A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell’s law," Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Vodo, P.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative Refraction and Left-Handed Electromagnetism in Microwave Photonic Crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (1)

Nature (London) (1)

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature (London) 425, 944-947 (2003).
[CrossRef]

Opt. Express (2)

Phys. Rev. B (5)

I. De Leon and F. S. Roux, "Fourier analysis of reflection and refraction in two-dimensional photonic crystals," Phys. Rev. B 71, 235105 (2005).
[CrossRef]

M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

S. Foteinopoulou and C. M. Soukoulis, "Negative refraction and left-handed behavior in two-dimensional photonic crystals," Phys. Rev. B 67, 235107 (2003).
[CrossRef]

A. Martínez and J. Martí, "Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination," Phys. Rev. B 71, 235115 (2005).
[CrossRef]

S. Foteinopoulou and C. M. Soukoulis, "Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects," Phys. Rev. B 72, 165112 (2005).
[CrossRef]

Phys. Rev. E (1)

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, "Electromagnetic waves focused by a negative-index planar lens," Phys. Rev. E 67, 025602 (2003).
[CrossRef]

Phys. Rev. Lett. (3)

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative Refraction and Left-Handed Electromagnetism in Microwave Photonic Crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell’s law," Phys. Rev. Lett. 90, 107401 (2003);A. A. Houck, J. B. Brock, and I. L. Chuang, "Experimental observations of a left-handed material that obeys Snell’s law," Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

A. Grbic and G. V. Eleftheriades, "Overcoming the Diffraction Limit with a Planar Left-Handed Transmission-Line Lens," Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

Science (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]

Sov. Phys. Usp. (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Other (1)

In this article, calculations of photonic bands have been done with BandSolveTM, which implements the plane wave expansion method, and FDTD simulations with FullwaveTM, both by RSoft Design Group.

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Figures (6)

Fig. 1.
Fig. 1.

(a) Schematic EFCs of the PhC under study for frequencies between 0.29 and 0.34. The direction of the input external wave, as well as the direction of k 0,0 and v g inside the PhC, are depicted; (b) vg (normalized to c) vs. k 0,0 (normalized to Gy ) for the frequency range under study: results obtained by derivation of the photonic band (solid line) and by weighted summation of the group velocity of the different Fourier components (open circles). Inset: detailed view of the group velocity near the band edge.

Fig. 2.
Fig. 2.

(a), (b) Amplitudes of the coefficients e m,n and Δm,n obtained from the 2D Fourier transformation as a function of frequency. The inset in (b) shows in detail the region close to the band edge.

Fig. 3.
Fig. 3.

(a)–(d) Distribution of the real part of E z in the computational domain at a normalized frequency of 0.3 at different time steps (see the figure) of the FDTD simulations. (e) Left-hand side: detail of the computation domain at the free space-PhC interface; Right-hand side: detail of E z inside the dashed rectangle in (d).

Fig. 4.
Fig. 4.

Phase evolution of the Bloch wave with normalized frequency of 0.3 along a period in the y direction for different values of the x coordinate in a period. (b) is a the projection of (a) on a plane perpendicular to the x-axis. Values in (b) stand for the values of the x coordinate for which the phase evolution is obtained. The phase shifts that would occur in an ideal LH medium with negative wave vector k 0,0 are also shown as a black line in (b).

Fig. 5.
Fig. 5.

(a)–(d) As in Fig. 3, but for a frequency of 0.346. (e) Detail of Ez inside the dashed rectangle in (d).

Fig. 6.
Fig. 6.

As Fig. 4, but for a normalized frequency of 0.346. In (b) the values indicate the interval of the x axis for which the phase path are obtained.

Equations (3)

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

E z ( x , y ) = E 0 m , n e m , n e i [ m G x x ] e i [ ( k 0,0 + n G y ) y ]
E z ( x , y ) = E 0 e ik 0,0 y ( ( e 0,0 + 2 e 2,0 cos ( 2 G x x ) + ) + ( Δ 0,2 e i 2 G y y + 2 Δ 1,1 cos ( G x x ) e i G y y + ) + ( 2 e 0 , 2 cos ( 2 G y y ) + 4 e 1 , 1 cos ( G x x ) cos ( G y y ) + ) )
v g = c 2 ω m , n ( k 0,0 + m G x + n G y ) e m , n 2

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