Abstract

We consider the eigen-fields of a two-dimensional negative-refraction photonic crystal and obtain negative effective permittivity and negative effective permeability. Effective permittivity, permeability, and surface impedance are calculated by averaging the eigen-fields. The value of the surface impedance is shown to be location-dependent and is validated by finite-difference time-domain simulations. The unique power propagation mechanism in the photonic crystal is demonstrated through time-evolution of eigen-fields.

© 2007 Optical Society of America

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References

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  1. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773(1996).
    [Crossref] [PubMed]
  2. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theor Tech 47, 2075 (1999).
    [Crossref]
  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 (2000).
    [Crossref] [PubMed]
  4. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77 (2001).
    [Crossref] [PubMed]
  5. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
    [Crossref] [PubMed]
  6. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509 (1968).
    [Crossref]
  7. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic wave: negative refraction by photonic crystals,” Nature 423, 604 (2003).
    [Crossref] [PubMed]
  8. 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]
  9. Z. Lu, C. Chen, C. A. Schuetz, S. Shi, J. A. Murakowski, G. J. Schneider, and D. W. Prather, “Sub-wavelength imaging by a flat cylindrical lens using optimized negative refraction,” Appl. Phys. Lett. 87, 091907 (2005).
    [Crossref]
  10. Z. Lu, J. A. Murakowski, C. A. Schuetz, S. Shi, G. J. Schneider, and D. W. Prather, “Three-dimensional subwavelength imaging by a photonic-crystal flat lens using negative refraction at microwave frequencies,” Phys. Rev. Lett. 95, 153901 (2005).
    [Crossref] [PubMed]
  11. T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic Crystal Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability,” Phys. Rev. Lett. 97, 073905 (2006).
    [Crossref] [PubMed]
  12. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-dormain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173 (2001).
    [Crossref] [PubMed]

2006 (1)

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic Crystal Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[Crossref] [PubMed]

2005 (2)

Z. Lu, C. Chen, C. A. Schuetz, S. Shi, J. A. Murakowski, G. J. Schneider, and D. W. Prather, “Sub-wavelength imaging by a flat cylindrical lens using optimized negative refraction,” Appl. Phys. Lett. 87, 091907 (2005).
[Crossref]

Z. Lu, J. A. Murakowski, C. A. Schuetz, S. Shi, G. J. Schneider, and D. W. Prather, “Three-dimensional subwavelength imaging by a photonic-crystal flat lens using negative refraction at microwave frequencies,” Phys. Rev. Lett. 95, 153901 (2005).
[Crossref] [PubMed]

2004 (1)

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 (1)

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic wave: negative refraction by photonic crystals,” Nature 423, 604 (2003).
[Crossref] [PubMed]

2001 (2)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77 (2001).
[Crossref] [PubMed]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-dormain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173 (2001).
[Crossref] [PubMed]

2000 (2)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
[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 (2000).
[Crossref] [PubMed]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theor Tech 47, 2075 (1999).
[Crossref]

1996 (1)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773(1996).
[Crossref] [PubMed]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509 (1968).
[Crossref]

Aydin, K.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic wave: negative refraction by photonic crystals,” Nature 423, 604 (2003).
[Crossref] [PubMed]

Chen, C.

Z. Lu, C. Chen, C. A. Schuetz, S. Shi, J. A. Murakowski, G. J. Schneider, and D. W. Prather, “Sub-wavelength imaging by a flat cylindrical lens using optimized negative refraction,” Appl. Phys. Lett. 87, 091907 (2005).
[Crossref]

Cubukcu, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic wave: negative refraction by photonic crystals,” Nature 423, 604 (2003).
[Crossref] [PubMed]

Decoopman, T.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic Crystal Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[Crossref] [PubMed]

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]

Enoch, S.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic Crystal Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[Crossref] [PubMed]

Foteinopoulou, S.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic wave: negative refraction by photonic crystals,” Nature 423, 604 (2003).
[Crossref] [PubMed]

Gralak, B.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic Crystal Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[Crossref] [PubMed]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theor Tech 47, 2075 (1999).
[Crossref]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773(1996).
[Crossref] [PubMed]

Joannopoulos, J. D.

Johnson, S. G.

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]

Lu, Z.

Z. Lu, C. Chen, C. A. Schuetz, S. Shi, J. A. Murakowski, G. J. Schneider, and D. W. Prather, “Sub-wavelength imaging by a flat cylindrical lens using optimized negative refraction,” Appl. Phys. Lett. 87, 091907 (2005).
[Crossref]

Z. Lu, J. A. Murakowski, C. A. Schuetz, S. Shi, G. J. Schneider, and D. W. Prather, “Three-dimensional subwavelength imaging by a photonic-crystal flat lens using negative refraction at microwave frequencies,” Phys. Rev. Lett. 95, 153901 (2005).
[Crossref] [PubMed]

Maystre, D.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic Crystal Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[Crossref] [PubMed]

Murakowski, J. A.

Z. Lu, J. A. Murakowski, C. A. Schuetz, S. Shi, G. J. Schneider, and D. W. Prather, “Three-dimensional subwavelength imaging by a photonic-crystal flat lens using negative refraction at microwave frequencies,” Phys. Rev. Lett. 95, 153901 (2005).
[Crossref] [PubMed]

Z. Lu, C. Chen, C. A. Schuetz, S. Shi, J. A. Murakowski, G. J. Schneider, and D. W. Prather, “Sub-wavelength imaging by a flat cylindrical lens using optimized negative refraction,” Appl. Phys. Lett. 87, 091907 (2005).
[Crossref]

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 (2000).
[Crossref] [PubMed]

Ozbay, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic wave: negative refraction by photonic crystals,” Nature 423, 604 (2003).
[Crossref] [PubMed]

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 (2000).
[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]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
[Crossref] [PubMed]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theor Tech 47, 2075 (1999).
[Crossref]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773(1996).
[Crossref] [PubMed]

Prather, D. W.

Z. Lu, C. Chen, C. A. Schuetz, S. Shi, J. A. Murakowski, G. J. Schneider, and D. W. Prather, “Sub-wavelength imaging by a flat cylindrical lens using optimized negative refraction,” Appl. Phys. Lett. 87, 091907 (2005).
[Crossref]

Z. Lu, J. A. Murakowski, C. A. Schuetz, S. Shi, G. J. Schneider, and D. W. Prather, “Three-dimensional subwavelength imaging by a photonic-crystal flat lens using negative refraction at microwave frequencies,” Phys. Rev. Lett. 95, 153901 (2005).
[Crossref] [PubMed]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theor Tech 47, 2075 (1999).
[Crossref]

Schneider, G. J.

Z. Lu, C. Chen, C. A. Schuetz, S. Shi, J. A. Murakowski, G. J. Schneider, and D. W. Prather, “Sub-wavelength imaging by a flat cylindrical lens using optimized negative refraction,” Appl. Phys. Lett. 87, 091907 (2005).
[Crossref]

Z. Lu, J. A. Murakowski, C. A. Schuetz, S. Shi, G. J. Schneider, and D. W. Prather, “Three-dimensional subwavelength imaging by a photonic-crystal flat lens using negative refraction at microwave frequencies,” Phys. Rev. Lett. 95, 153901 (2005).
[Crossref] [PubMed]

Schuetz, C. A.

Z. Lu, J. A. Murakowski, C. A. Schuetz, S. Shi, G. J. Schneider, and D. W. Prather, “Three-dimensional subwavelength imaging by a photonic-crystal flat lens using negative refraction at microwave frequencies,” Phys. Rev. Lett. 95, 153901 (2005).
[Crossref] [PubMed]

Z. Lu, C. Chen, C. A. Schuetz, S. Shi, J. A. Murakowski, G. J. Schneider, and D. W. Prather, “Sub-wavelength imaging by a flat cylindrical lens using optimized negative refraction,” Appl. Phys. Lett. 87, 091907 (2005).
[Crossref]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77 (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 (2000).
[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 (2001).
[Crossref] [PubMed]

Shi, S.

Z. Lu, C. Chen, C. A. Schuetz, S. Shi, J. A. Murakowski, G. J. Schneider, and D. W. Prather, “Sub-wavelength imaging by a flat cylindrical lens using optimized negative refraction,” Appl. Phys. Lett. 87, 091907 (2005).
[Crossref]

Z. Lu, J. A. Murakowski, C. A. Schuetz, S. Shi, G. J. Schneider, and D. W. Prather, “Three-dimensional subwavelength imaging by a photonic-crystal flat lens using negative refraction at microwave frequencies,” Phys. Rev. Lett. 95, 153901 (2005).
[Crossref] [PubMed]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77 (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 (2000).
[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]

Soukoulis, C. M.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic wave: negative refraction by photonic crystals,” Nature 423, 604 (2003).
[Crossref] [PubMed]

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]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theor Tech 47, 2075 (1999).
[Crossref]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773(1996).
[Crossref] [PubMed]

Tayeb, G.

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic Crystal Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[Crossref] [PubMed]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509 (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 (2000).
[Crossref] [PubMed]

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]

Youngs, I.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773(1996).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

Z. Lu, C. Chen, C. A. Schuetz, S. Shi, J. A. Murakowski, G. J. Schneider, and D. W. Prather, “Sub-wavelength imaging by a flat cylindrical lens using optimized negative refraction,” Appl. Phys. Lett. 87, 091907 (2005).
[Crossref]

IEEE Trans. Microwave Theor Tech (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theor Tech 47, 2075 (1999).
[Crossref]

Nature (1)

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic wave: negative refraction by photonic crystals,” Nature 423, 604 (2003).
[Crossref] [PubMed]

Opt. Express (1)

Phys. Rev. Lett. (6)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773(1996).
[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]

Z. Lu, J. A. Murakowski, C. A. Schuetz, S. Shi, G. J. Schneider, and D. W. Prather, “Three-dimensional subwavelength imaging by a photonic-crystal flat lens using negative refraction at microwave frequencies,” Phys. Rev. Lett. 95, 153901 (2005).
[Crossref] [PubMed]

T. Decoopman, G. Tayeb, S. Enoch, D. Maystre, and B. Gralak, “Photonic Crystal Lens: From Negative Refraction and Negative Index to Negative Permittivity and Permeability,” Phys. Rev. Lett. 97, 073905 (2006).
[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 (2000).
[Crossref] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
[Crossref] [PubMed]

Science (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77 (2001).
[Crossref] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509 (1968).
[Crossref]

Supplementary Material (2)

» Media 1: MPG (1510 KB)     
» Media 2: MPG (1531 KB)     

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Figures (4)

Fig. 1.
Fig. 1.

(a) The illustration of the PhC and coordinate system for the study. (b) The photonic dispersion diagram for the PhC and the light in this frequency range. (c) The equi-frequency dispersion contours for the second band and the light cone.

Fig. 2.
Fig. 2.

(a) The amplitude distribution of the eigen-field, E (k) z (r). (b) The phase distribution of the eigen-field,E (k) z (r). [Media 1] (c) The phase distribution of the plane wave, exp(j kr - jωt). [Media 2] (d) The impedance with respect to the location of PhC surface.

Fig. 3.
Fig. 3.

The simulations (amplitudes) of propagation of light from (a, c) the matched medium and (b, d) the air into the PhC. In (a, b), the PhC surface is located at x = 0; in (c, d), the PhC surface is located at x = √3a/4.

Fig. 4.
Fig. 4.

(a) The amplitude distribution of the eigen-field, E (k) z (r)(ωn = 0.236 and θ = 45° ). (b) The phase distribution of the eigen-field, E (k) z (r)(ωn = 0.236 and θ = 45° ). (c) The phase distribution of the plane wave, exp(j kr - jωt). (d) The impedance with respect to the location of PhC surface (ωn = 0.236 and θ = 45°).

Equations (6)

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

1 ε r ( 2 E z x 2 + 2 E z y 2 ) + ( ω c ) 2 E z = 0 .
E z ( k , r , t ) = exp ( j k r jωt ) E z ( k ) ( r ) ,
{ H x ( k , r , t ) = 1 jωμ E z ( k , r , t ) y H y ( k , r , t ) = 1 jωμ E z ( k , r , t ) x
{ E z ( 2 Dav ) = Unit Cell E z ( k ) ( r ) dxdy Area H y ( 2 Dav ) = Unit Cell H y ( k ) ( r ) dxdy Area ,
{ μ eff = 1 H y E z x = k ω E z ( 2 Dav ) H y ( 2 Dav ) ε eff = 1 E z H y x = k ω H y ( 2 Dav ) E z ( 2 Dav ) .
{ E z ( 1 Dav ) = 1 L y Unit Cell E z ( k ) ( r ) dy H y ( 1 Dav ) = 1 L y Unit Cell H y ( k ) ( r ) dy .

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