Abstract

A new boundary condition is introduced to calculate the effective impedance matrix of semi-infinite periodic structures such as photonic crystals and metamaterials, which leads to a reduction of the solution space. The obtained effective impedance matrix allows one to relate a matrix to a PC, which includes all of its properties in terms of reflection from its interface. For one-dimensional photonic crystals or multilayer films, it is shown that a closed-form equation can be found for the effective impedance. For two-dimensional photonic crystals the impedance is obtained using the scattering matrices by solving a unilateral quadratic matrix equation. Several examples are outlined to validate the developed scheme. In the examples, the goal is mainly the computation of the reflection from a semi-infinite periodic structure when a plane wave illuminates its boundary.

© 2009 Optical Society of America

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A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Analysis of multilayer frequency selective surfaces on periodic and anisotropic substrates,” Metamaterials 3, 63-74 (2009).
[CrossRef]

2007

2005

W. Jiang, R. T. Chen, and X. Lu, “Theory of light refraction at the surface of a PC,” Phys. Rev. B 71, 245115 (2005).
[CrossRef]

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with PC waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

2004

Y.-C. Hsue and T.-J. Yang, “Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite PC,” Phys. Rev. E 70, 016706 (2004).
[CrossRef]

2003

Z.-Y. Li and K.-M. Ho, “Light propagation in semi-infinite photonic crystals and related waveguide structures,” Phys. Rev. B 68, 155101 (2003).
[CrossRef]

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

2002

K. B. Chung and S. W. Hong, “Wavelength demultiplexers based on the superprism phenomena in photonic crystals,” Appl. Phys. Lett. 81, 1549-1551 (2002).
[CrossRef]

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[CrossRef]

2001

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of PC microcavities for cavity qed,” Phys. Rev. B 65, 016608 (2001).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

2000

E. Popov and M. Nevière, “Grating theory: new equations in fourier space leading to fast converging results for tm polarization,” J. Opt. Soc. Am. A 17, 1773-1784 (2000).
[CrossRef]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212-8222 (2000).
[CrossRef]

1999

F.-R. Yang, K.-P. Ma, Y. Qian, and T. Itoh, “A uniplanar compact photonic-bandgap (UC-PBG) structure and its applications for microwave circuit,” IEEE Trans. Microwave Theory Tech. 47, 1509-1514 (1999).
[CrossRef]

1998

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

P. Dansas and N. Paraire, “Fast modeling of photonic bandgap structures by use of a diffraction-grating approach,” J. Opt. Soc. Am. A 15, 1586-1598 (1998).
[CrossRef]

1996

1995

1994

1993

1987

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dieletric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

1979

D. L. Jaggard, A. R. Mickelson, and C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. A 18, 211-216 (1979).

Adibi, A.

Aghaie, K. Z.

A. Fallahi, K. Z. Aghaie, A. Enayati, and M. Shahabadi, “Diffraction analysis of periodic structures using a transmission-line formulation: principles and applications,” J. Comput. Theor. Nanosci. 4, 649-666 (2007).

Asatryan, A. A.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Aydin, K.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

Badieirostami, M.

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Bijamov, A. Y.

D. D. Karkashadze, F. G. Bogdanov, R. S. Zaridze, A. Y. Bijamov, C. Hafner, and D. Erni, “Simulation of Finite Photonic Crystals Made of Biisotropic or Chiral Material,” in Advances in Electromagnetics of Complex Media and Metamaterials NATO Science Series. II. Mathematics, Physics and Chemistry (2003), Vol. 89, pp. 175-193.

Bogdanov, F. G.

D. D. Karkashadze, F. G. Bogdanov, R. S. Zaridze, A. Y. Bijamov, C. Hafner, and D. Erni, “Simulation of Finite Photonic Crystals Made of Biisotropic or Chiral Material,” in Advances in Electromagnetics of Complex Media and Metamaterials NATO Science Series. II. Mathematics, Physics and Chemistry (2003), Vol. 89, pp. 175-193.

Botten, L. C.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Caloz, C.

C. Caloz and T. Itoh, Electromagnetic Metamaterials Transmission Line Theory and Microwave Applications (Wiley, 2006).

Chen, R. T.

W. Jiang, R. T. Chen, and X. Lu, “Theory of light refraction at the surface of a PC,” Phys. Rev. B 71, 245115 (2005).
[CrossRef]

Chung, K. B.

K. B. Chung and S. W. Hong, “Wavelength demultiplexers based on the superprism phenomena in photonic crystals,” Appl. Phys. Lett. 81, 1549-1551 (2002).
[CrossRef]

Cubukcu, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

Dansas, P.

de Sterke, C. M.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

Eftekhar, A. A.

Eleftheriades, G. V.

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

Enayati, A.

A. Fallahi, K. Z. Aghaie, A. Enayati, and M. Shahabadi, “Diffraction analysis of periodic structures using a transmission-line formulation: principles and applications,” J. Comput. Theor. Nanosci. 4, 649-666 (2007).

Engheta, N.

N. Engheta and R. W. Ziolkowski, Metamaterials Physics and Engineering Explorations (Wiley2006).

Erni, D.

D. D. Karkashadze, F. G. Bogdanov, R. S. Zaridze, A. Y. Bijamov, C. Hafner, and D. Erni, “Simulation of Finite Photonic Crystals Made of Biisotropic or Chiral Material,” in Advances in Electromagnetics of Complex Media and Metamaterials NATO Science Series. II. Mathematics, Physics and Chemistry (2003), Vol. 89, pp. 175-193.

Fallahi, A.

A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Analysis of multilayer frequency selective surfaces on periodic and anisotropic substrates,” Metamaterials 3, 63-74 (2009).
[CrossRef]

A. Fallahi, K. Z. Aghaie, A. Enayati, and M. Shahabadi, “Diffraction analysis of periodic structures using a transmission-line formulation: principles and applications,” J. Comput. Theor. Nanosci. 4, 649-666 (2007).

Fan, S.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212-8222 (2000).
[CrossRef]

Foteinopoulou, S.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

Ghaemi, H. F.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

Grupp, D. E.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

Hafner, C.

A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Analysis of multilayer frequency selective surfaces on periodic and anisotropic substrates,” Metamaterials 3, 63-74 (2009).
[CrossRef]

C. Hafner, “Mmp computation of periodic structures,” J. Opt. Soc. Am. A 12, 1057-1067 (1995).
[CrossRef]

D. D. Karkashadze, F. G. Bogdanov, R. S. Zaridze, A. Y. Bijamov, C. Hafner, and D. Erni, “Simulation of Finite Photonic Crystals Made of Biisotropic or Chiral Material,” in Advances in Electromagnetics of Complex Media and Metamaterials NATO Science Series. II. Mathematics, Physics and Chemistry (2003), Vol. 89, pp. 175-193.

Hamann, H. F.

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with PC waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Ho, K.-M.

Z.-Y. Li and K.-M. Ho, “Light propagation in semi-infinite photonic crystals and related waveguide structures,” Phys. Rev. B 68, 155101 (2003).
[CrossRef]

Hong, S. W.

K. B. Chung and S. W. Hong, “Wavelength demultiplexers based on the superprism phenomena in photonic crystals,” Appl. Phys. Lett. 81, 1549-1551 (2002).
[CrossRef]

Hsue, Y.-C.

Y.-C. Hsue and T.-J. Yang, “Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite PC,” Phys. Rev. E 70, 016706 (2004).
[CrossRef]

Itoh, T.

F.-R. Yang, K.-P. Ma, Y. Qian, and T. Itoh, “A uniplanar compact photonic-bandgap (UC-PBG) structure and its applications for microwave circuit,” IEEE Trans. Microwave Theory Tech. 47, 1509-1514 (1999).
[CrossRef]

C. Caloz and T. Itoh, Electromagnetic Metamaterials Transmission Line Theory and Microwave Applications (Wiley, 2006).

Iyer, A. K.

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

Jaggard, D. L.

D. L. Jaggard, A. R. Mickelson, and C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. A 18, 211-216 (1979).

Jiang, W.

W. Jiang, R. T. Chen, and X. Lu, “Theory of light refraction at the surface of a PC,” Phys. Rev. B 71, 245115 (2005).
[CrossRef]

Joannopoulos, J. D.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[CrossRef]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212-8222 (2000).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

John, S.

S. John, “Strong localization of photons in certain disordered dieletric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Johnson, S. G.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[CrossRef]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212-8222 (2000).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

Karkashadze, D. D.

D. D. Karkashadze, F. G. Bogdanov, R. S. Zaridze, A. Y. Bijamov, C. Hafner, and D. Erni, “Simulation of Finite Photonic Crystals Made of Biisotropic or Chiral Material,” in Advances in Electromagnetics of Complex Media and Metamaterials NATO Science Series. II. Mathematics, Physics and Chemistry (2003), Vol. 89, pp. 175-193.

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Kremer, P. C.

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

Lezec, H. J.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

Li, L.

Li, Z.-Y.

Z.-Y. Li and K.-M. Ho, “Light propagation in semi-infinite photonic crystals and related waveguide structures,” Phys. Rev. B 68, 155101 (2003).
[CrossRef]

Loncar, M.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of PC microcavities for cavity qed,” Phys. Rev. B 65, 016608 (2001).
[CrossRef]

Lu, X.

W. Jiang, R. T. Chen, and X. Lu, “Theory of light refraction at the surface of a PC,” Phys. Rev. B 71, 245115 (2005).
[CrossRef]

Luo, C.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Ma, K.-P.

F.-R. Yang, K.-P. Ma, Y. Qian, and T. Itoh, “A uniplanar compact photonic-bandgap (UC-PBG) structure and its applications for microwave circuit,” IEEE Trans. Microwave Theory Tech. 47, 1509-1514 (1999).
[CrossRef]

Mabuchi, H.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of PC microcavities for cavity qed,” Phys. Rev. B 65, 016608 (2001).
[CrossRef]

McNab, S. J.

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with PC waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

McPhedran, R. C.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

Mickelson, A. R.

D. L. Jaggard, A. R. Mickelson, and C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. A 18, 211-216 (1979).

Mishrikey, M.

A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Analysis of multilayer frequency selective surfaces on periodic and anisotropic substrates,” Metamaterials 3, 63-74 (2009).
[CrossRef]

Momeni, B.

Montiel, F.

Munk, B. A.

B. A. Munk, Frequency Selective Surfaces Theory and Design (Wiley, 2000).
[CrossRef]

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Nicorovici, N. A.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

O'Boyle, M.

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with PC waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Ozbay, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

Papas, C. H.

D. L. Jaggard, A. R. Mickelson, and C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. A 18, 211-216 (1979).

Paraire, N.

Pendry, J. B.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Popov, E.

Qian, Y.

F.-R. Yang, K.-P. Ma, Y. Qian, and T. Itoh, “A uniplanar compact photonic-bandgap (UC-PBG) structure and its applications for microwave circuit,” IEEE Trans. Microwave Theory Tech. 47, 1509-1514 (1999).
[CrossRef]

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Scherer, A.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of PC microcavities for cavity qed,” Phys. Rev. B 65, 016608 (2001).
[CrossRef]

Shahabadi, M.

A. Fallahi, K. Z. Aghaie, A. Enayati, and M. Shahabadi, “Diffraction analysis of periodic structures using a transmission-line formulation: principles and applications,” J. Comput. Theor. Nanosci. 4, 649-666 (2007).

Soukoulis, C. M.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

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H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Thio, T.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Vahldieck, R.

A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Analysis of multilayer frequency selective surfaces on periodic and anisotropic substrates,” Metamaterials 3, 63-74 (2009).
[CrossRef]

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S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212-8222 (2000).
[CrossRef]

Vlasov, Y. A.

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with PC waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

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J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of PC microcavities for cavity qed,” Phys. Rev. B 65, 016608 (2001).
[CrossRef]

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J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

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E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

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F.-R. Yang, K.-P. Ma, Y. Qian, and T. Itoh, “A uniplanar compact photonic-bandgap (UC-PBG) structure and its applications for microwave circuit,” IEEE Trans. Microwave Theory Tech. 47, 1509-1514 (1999).
[CrossRef]

Yang, T.-J.

Y.-C. Hsue and T.-J. Yang, “Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite PC,” Phys. Rev. E 70, 016706 (2004).
[CrossRef]

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D. D. Karkashadze, F. G. Bogdanov, R. S. Zaridze, A. Y. Bijamov, C. Hafner, and D. Erni, “Simulation of Finite Photonic Crystals Made of Biisotropic or Chiral Material,” in Advances in Electromagnetics of Complex Media and Metamaterials NATO Science Series. II. Mathematics, Physics and Chemistry (2003), Vol. 89, pp. 175-193.

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N. Engheta and R. W. Ziolkowski, Metamaterials Physics and Engineering Explorations (Wiley2006).

Appl. Phys. A

D. L. Jaggard, A. R. Mickelson, and C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. A 18, 211-216 (1979).

Appl. Phys. Lett.

K. B. Chung and S. W. Hong, “Wavelength demultiplexers based on the superprism phenomena in photonic crystals,” Appl. Phys. Lett. 81, 1549-1551 (2002).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

F.-R. Yang, K.-P. Ma, Y. Qian, and T. Itoh, “A uniplanar compact photonic-bandgap (UC-PBG) structure and its applications for microwave circuit,” IEEE Trans. Microwave Theory Tech. 47, 1509-1514 (1999).
[CrossRef]

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

J. Comput. Theor. Nanosci.

A. Fallahi, K. Z. Aghaie, A. Enayati, and M. Shahabadi, “Diffraction analysis of periodic structures using a transmission-line formulation: principles and applications,” J. Comput. Theor. Nanosci. 4, 649-666 (2007).

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Metamaterials

A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Analysis of multilayer frequency selective surfaces on periodic and anisotropic substrates,” Metamaterials 3, 63-74 (2009).
[CrossRef]

Nature

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Y. A. Vlasov, M. O'Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with PC waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. B

Z.-Y. Li and K.-M. Ho, “Light propagation in semi-infinite photonic crystals and related waveguide structures,” Phys. Rev. B 68, 155101 (2003).
[CrossRef]

W. Jiang, R. T. Chen, and X. Lu, “Theory of light refraction at the surface of a PC,” Phys. Rev. B 71, 245115 (2005).
[CrossRef]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[CrossRef]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212-8222 (2000).
[CrossRef]

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of PC microcavities for cavity qed,” Phys. Rev. B 65, 016608 (2001).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Phys. Rev. E

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Y.-C. Hsue and T.-J. Yang, “Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite PC,” Phys. Rev. E 70, 016706 (2004).
[CrossRef]

Phys. Rev. Lett.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dieletric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
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Other

C. Caloz and T. Itoh, Electromagnetic Metamaterials Transmission Line Theory and Microwave Applications (Wiley, 2006).

N. Engheta and R. W. Ziolkowski, Metamaterials Physics and Engineering Explorations (Wiley2006).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

K.Busch, S.Lölkes, R.B.Wehrspohn, and H.Föll, eds., Photonic Crystals Advances in Design Fabrication and Characterization (Wiley-VCH, 2004).
[CrossRef]

T.K.Wu, ed., Frequency Selective Surface and Grid Array (Wiley, 1995).

B. A. Munk, Frequency Selective Surfaces Theory and Design (Wiley, 2000).
[CrossRef]

D. D. Karkashadze, F. G. Bogdanov, R. S. Zaridze, A. Y. Bijamov, C. Hafner, and D. Erni, “Simulation of Finite Photonic Crystals Made of Biisotropic or Chiral Material,” in Advances in Electromagnetics of Complex Media and Metamaterials NATO Science Series. II. Mathematics, Physics and Chemistry (2003), Vol. 89, pp. 175-193.

K.Yasumoto, ed., Electromagnetic Theory and Applications for Photonic Crystals (Taylor & Francis, Fukuoka, Japan, 2006).

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Figures (12)

Fig. 1
Fig. 1

Example of a semi-infinite structure, periodic in thez direction. A plane wave is incident on the interface between air and the bulk metamaterial, and the Maxwell equations should be solved to obtain the reflection coefficient. The structure may be either periodic or nonperiodic in the horizontal plane.

Fig. 2
Fig. 2

Geometry of the first considered problem, which is the incidence of a plane wave to a semi-infinite periodic multilayer structure.

Fig. 3
Fig. 3

Truncated multilayer structure stacked on a substrate.

Fig. 4
Fig. 4

Power reflection coefficient versus number of unit cell repetition for the lossless truncated multilayer structure stacked on a substrate. The curve is drawn for four different substrates.

Fig. 5
Fig. 5

Power reflection coefficient versus number of unit cell repetition for the lossy truncated multilayer structure stacked on a substrate. The curve is drawn for four different substrates.

Fig. 6
Fig. 6

Band diagram of the one-dimensional PC considered in the third example. Because of the existing bandgaps in the diagram, an electromagnetic wave with frequencies in these intervals is attenuated within the layers.

Fig. 7
Fig. 7

Power reflection coefficient in terms of frequency for the semi-infinite multilayer film in the third example. The solid line shows the result for reflection from a semi-infinite structure. The dashed and dotted lines are reflection from films with 3 × 2 and 10 × 2 layers, respectively. The underlying substrate is assumed to be air.

Fig. 8
Fig. 8

Lines along which the two roots are traced. The frequency interval is L/?? [0.14, 0.16].

Fig. 9
Fig. 9

Geometry of the second considered problem, which is the incidence of a plane wave on a two-dimensional PC.

Fig. 10
Fig. 10

Power reflection coefficient is depicted in terms of the number of layers for the incidence of TE and TM polarized plane waves on lossy and lossless truncated PCs. The solid lines are for a layered structure stacked on a substrate with the impedance matrix Z, and the dashed line is for reflection from a material with air as the substrate.

Fig. 11
Fig. 11

Geometry of the PC considered in the fifth example. It consists of circular rods embedded in air.

Fig. 12
Fig. 12

Power diffraction efficiency versus normalized frequency for the semi-infinite PC shown in Fig. 12. The results are compared with those for the truncated PC with 10 and 50 layers.

Tables (1)

Tables Icon

Table 1 Computed Diffraction Efficiencies for Reflection from a Semi-Infinite Two-Dimensional Photonic Crystal and Different Truncation Orders a

Equations (24)

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

E ̃ A ( k x , k y ) = Z ̃ A ( k x , k y ) H ̃ A ( k x , k y ) .
Z 0 i = Z 0 ε r i and β i = k 0 ε r i ,
A Z 2 + B Z + C = 0 ,
A = Z 01 tan ( β 2 h 2 ) + Z 02 tan ( β 1 h 1 ) ,
B = j tan ( β 1 h 1 ) tan ( β 2 h 2 ) ( Z 02 2 Z 01 2 ) ,
C = Z 01 Z 02 ( Z 02 tan ( β 2 h 2 ) + Z 01 tan ( β 1 h 1 ) ) .
( E TE E TM ) = ( sin γ cos γ cos γ sin γ ) ( E x E y ) ,
β i = { ε r i k 0 2 k x 2 k y 2 if k 0 ε r i > k x 2 + k y 2 j k x 2 + k y 2 ε r i k 0 2 if k 0 ε r i < k x 2 + k y 2 } ,
Z 0 i TE = ω μ 0 β i ,
Z 0 i TM = β i ( ω ε r i ε 0 ) ,
( E x E y ) = m = + ( E ̃ x m E ̃ y m ) b m t ( x , y ) b m l ( z ) ,
( H x H y ) = m = + ( H ̃ x m H ̃ y m ) b m t ( x , y ) b m l ( z ) ,
( E ̃ x E ̃ y ) = Z ( H ̃ y H ̃ x ) ,
b m t ( x , y ) = e j ( k x + 2 π m d ) x e j k y y .
( V ( h 1 + h 2 ) V ( 0 ) ) = R ( I ( h 1 + h 2 ) I ( 0 ) ) ,
R = ( R 11 R 12 R 21 R 22 ) .
V ( 0 ) = Z I ( 0 ) ,
V ( h 1 + h 2 ) = Z I ( h 1 + h 2 ) .
( Z R 11 ) R 21 1 ( Z R 22 ) I ( 0 ) = R 12 I ( 0 ) .
( R 21 Z 2 + ( R 22 R 11 ) Z R 12 ) I ( 0 ) = 0.
( λ 2 + A λ + B ) q = 0 ,
( V ( 0 ) I ( 0 ) ) = T ( V ( h 1 + h 2 ) I ( h 1 + h 2 ) ) = ( T 11 T 12 T 21 T 22 ) ( V ( h 1 + h 2 ) I ( h 1 + h 2 ) ) .
M = ( 0 I B A ) ,
P = R { q T ( R 21 λ + R 22 ) q } ,

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