Abstract

We consider periodic multilayers combining ordinary positive index materials and dispersive metamaterials with negative index in some frequency range. These structures can exhibit photonic band gaps which, in contrast with the usual Bragg gaps, are not based on interference mechanisms. We focus on effects produced by the interaction between non-Bragg gaps of different nature: a) the zero averaged refractive index, b) the zero permeability and c) the zero permittivity gaps. Our analysis highlights the role played by the unavoidable dispersive character of metamaterials. We show that the degree of overlap between these bands can be varied by a proper selection of the constructive parameters, a feature that introduces novel degrees of freedom for the design of photonic band gap structures. The numerical examples illustrate the evolution of the dispersion diagrams of a periodic multilayer with the filling fraction of the ordinary material constituent and show a range of filling fractions where propagation in the multilayer is forbidden for any propagation angle and polarization.

© 2006 Optical Society of America

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  1. Y. Fink, J. Winn, S. Fan, C. Chen, J. Michel, J. Joannopoulos, and E. Thomas, "A Dielectric Omnidirectional Reflector," Science 282,1679-1682 (1998).
    [CrossRef] [PubMed]
  2. P. St. J. Russell, S. Tredwell, and P. J. Roberts, "Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures," Opt. Commun. 160, 66-71 (1999).
    [CrossRef]
  3. D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "Observation of total omnidirectional reflection from one-dimensional dielectric latice," Appl. Phys. A 68, 25-28 (1999).
    [CrossRef]
  4. C. Xudong, C. Hafner, R. Vahldieck, and F. Robin, "Sharp trench waveguide bends in dual mode operation with ultra-small photonic crystals for suppressing radiation," Opt. Express 14, 4351-4356 (2006).
    [CrossRef] [PubMed]
  5. D. Delbeke, R. Bockstaele, P. Bienstman, R. Baets, and H. Benisty, "High-efficiency semiconductor resonantcavity light-emitting diodes: a review," IEEE J. Sel. Top. Quantum Electron. 8, 189-206 (2002).
    [CrossRef]
  6. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
    [CrossRef] [PubMed]
  7. J. Pendry (ed), Focus issue: Negative refraction and metamaterials, Opt. Express 11 (2003), http://www.opticsexpress.org/ViewMedia.cfm?id=71852&seq=0.
    [PubMed]
  8. A. Lakhtakia and M. McCall (eds), Focus issue on negative refraction, New J. Phys. 7 (2005), http://www.iop.org/EJ/abstract/-ff30=1/1367-2630/7/1/E03.
    [CrossRef]
  9. D. Smith, J. Pendry, and M. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
    [CrossRef] [PubMed]
  10. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of permittivity and permeability," Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  11. J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic Band Gap from a Stack of Positive and Negative Index Materials," Phys. Rev. Lett. 90, 083901 (2003).
    [CrossRef] [PubMed]
  12. L. Wu, S. He and L Chen, "On unusual narrow transmission bands for a multi-layered periodic structure containing left-handed materials," Opt. Express 11, 1283-1290 (2003).
    [CrossRef] [PubMed]
  13. H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, "Properties of one-dimensional photonic crystals containing single-negative materials," Phys. Rev. E 69, 066607 (2004).
    [CrossRef]
  14. L.-G. Wang, H. Chen, and S.-Y. Zhu, "Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials," Phys. Rev. B 70, 245102 (2004).
    [CrossRef]
  15. N. Panoiu, R. Osgood, S. Zhang, S. Brueck, "Zero-n bandgap in photonic crystal superlattices," J. Opt. Soc. Am. B 23, 506-513 (2006).
    [CrossRef]
  16. L. I. Deych, D. Livdan, and A. A. Lisyansky, "Resonant tunneling of electromagnetic waves through polariton gaps," Phys. Rev. E 57, 7254-7258 (1998).
    [CrossRef]
  17. R. A. Depine, M. L. Mart’ýnez-Ricci, J. A. Monsoriu, E. Silvestre, and P. Andr’es, "Zero permeability and zero permittivity band gaps in 1D metamaterial photonic crystals," arXiv: physics/0606069 (2006).
  18. I. Shadrivov, A. Sukhorukov, and Y. Kivshar, "Complete Band Gaps in One-Dimensional Left-Handed Periodic Structures," Phys. Rev. Lett. 95, 193903 (2005).
    [CrossRef] [PubMed]
  19. P. Yeh, A. Yariv, and C. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 423-438 (1977).
    [CrossRef]

2006 (2)

2005 (2)

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, "Complete Band Gaps in One-Dimensional Left-Handed Periodic Structures," Phys. Rev. Lett. 95, 193903 (2005).
[CrossRef] [PubMed]

A. Lakhtakia and M. McCall (eds), Focus issue on negative refraction, New J. Phys. 7 (2005), http://www.iop.org/EJ/abstract/-ff30=1/1367-2630/7/1/E03.
[CrossRef]

2004 (3)

D. Smith, J. Pendry, and M. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, "Properties of one-dimensional photonic crystals containing single-negative materials," Phys. Rev. E 69, 066607 (2004).
[CrossRef]

L.-G. Wang, H. Chen, and S.-Y. Zhu, "Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials," Phys. Rev. B 70, 245102 (2004).
[CrossRef]

2003 (3)

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic Band Gap from a Stack of Positive and Negative Index Materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

L. Wu, S. He and L Chen, "On unusual narrow transmission bands for a multi-layered periodic structure containing left-handed materials," Opt. Express 11, 1283-1290 (2003).
[CrossRef] [PubMed]

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

2002 (1)

D. Delbeke, R. Bockstaele, P. Bienstman, R. Baets, and H. Benisty, "High-efficiency semiconductor resonantcavity light-emitting diodes: a review," IEEE J. Sel. Top. Quantum Electron. 8, 189-206 (2002).
[CrossRef]

1999 (2)

P. St. J. Russell, S. Tredwell, and P. J. Roberts, "Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures," Opt. Commun. 160, 66-71 (1999).
[CrossRef]

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "Observation of total omnidirectional reflection from one-dimensional dielectric latice," Appl. Phys. A 68, 25-28 (1999).
[CrossRef]

1998 (2)

Y. Fink, J. Winn, S. Fan, C. Chen, J. Michel, J. Joannopoulos, and E. Thomas, "A Dielectric Omnidirectional Reflector," Science 282,1679-1682 (1998).
[CrossRef] [PubMed]

L. I. Deych, D. Livdan, and A. A. Lisyansky, "Resonant tunneling of electromagnetic waves through polariton gaps," Phys. Rev. E 57, 7254-7258 (1998).
[CrossRef]

1977 (1)

1968 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of permittivity and permeability," 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 425, 944-947 (2003).
[CrossRef] [PubMed]

Asano, T.

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

Baets, R.

D. Delbeke, R. Bockstaele, P. Bienstman, R. Baets, and H. Benisty, "High-efficiency semiconductor resonantcavity light-emitting diodes: a review," IEEE J. Sel. Top. Quantum Electron. 8, 189-206 (2002).
[CrossRef]

Benisty, H.

D. Delbeke, R. Bockstaele, P. Bienstman, R. Baets, and H. Benisty, "High-efficiency semiconductor resonantcavity light-emitting diodes: a review," IEEE J. Sel. Top. Quantum Electron. 8, 189-206 (2002).
[CrossRef]

Bienstman, P.

D. Delbeke, R. Bockstaele, P. Bienstman, R. Baets, and H. Benisty, "High-efficiency semiconductor resonantcavity light-emitting diodes: a review," IEEE J. Sel. Top. Quantum Electron. 8, 189-206 (2002).
[CrossRef]

Bockstaele, R.

D. Delbeke, R. Bockstaele, P. Bienstman, R. Baets, and H. Benisty, "High-efficiency semiconductor resonantcavity light-emitting diodes: a review," IEEE J. Sel. Top. Quantum Electron. 8, 189-206 (2002).
[CrossRef]

Brueck, S.

Chan, C. T.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic Band Gap from a Stack of Positive and Negative Index Materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Chen, C.

Y. Fink, J. Winn, S. Fan, C. Chen, J. Michel, J. Joannopoulos, and E. Thomas, "A Dielectric Omnidirectional Reflector," Science 282,1679-1682 (1998).
[CrossRef] [PubMed]

Chen, H.

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, "Properties of one-dimensional photonic crystals containing single-negative materials," Phys. Rev. E 69, 066607 (2004).
[CrossRef]

L.-G. Wang, H. Chen, and S.-Y. Zhu, "Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials," Phys. Rev. B 70, 245102 (2004).
[CrossRef]

Chen, L

Chigrin, D. N.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "Observation of total omnidirectional reflection from one-dimensional dielectric latice," Appl. Phys. A 68, 25-28 (1999).
[CrossRef]

Delbeke, D.

D. Delbeke, R. Bockstaele, P. Bienstman, R. Baets, and H. Benisty, "High-efficiency semiconductor resonantcavity light-emitting diodes: a review," IEEE J. Sel. Top. Quantum Electron. 8, 189-206 (2002).
[CrossRef]

Deych, L. I.

L. I. Deych, D. Livdan, and A. A. Lisyansky, "Resonant tunneling of electromagnetic waves through polariton gaps," Phys. Rev. E 57, 7254-7258 (1998).
[CrossRef]

Fan, S.

Y. Fink, J. Winn, S. Fan, C. Chen, J. Michel, J. Joannopoulos, and E. Thomas, "A Dielectric Omnidirectional Reflector," Science 282,1679-1682 (1998).
[CrossRef] [PubMed]

Fink, Y.

Y. Fink, J. Winn, S. Fan, C. Chen, J. Michel, J. Joannopoulos, and E. Thomas, "A Dielectric Omnidirectional Reflector," Science 282,1679-1682 (1998).
[CrossRef] [PubMed]

Gaponenko, S. V.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "Observation of total omnidirectional reflection from one-dimensional dielectric latice," Appl. Phys. A 68, 25-28 (1999).
[CrossRef]

Hafner, C.

He, S.

Hong, C.

Jiang, H.

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, "Properties of one-dimensional photonic crystals containing single-negative materials," Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Joannopoulos, J.

Y. Fink, J. Winn, S. Fan, C. Chen, J. Michel, J. Joannopoulos, and E. Thomas, "A Dielectric Omnidirectional Reflector," Science 282,1679-1682 (1998).
[CrossRef] [PubMed]

Kivshar, Y.

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, "Complete Band Gaps in One-Dimensional Left-Handed Periodic Structures," Phys. Rev. Lett. 95, 193903 (2005).
[CrossRef] [PubMed]

Lakhtakia, A.

A. Lakhtakia and M. McCall (eds), Focus issue on negative refraction, New J. Phys. 7 (2005), http://www.iop.org/EJ/abstract/-ff30=1/1367-2630/7/1/E03.
[CrossRef]

Lavrinenko, A. V.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "Observation of total omnidirectional reflection from one-dimensional dielectric latice," Appl. Phys. A 68, 25-28 (1999).
[CrossRef]

Li, H.

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, "Properties of one-dimensional photonic crystals containing single-negative materials," Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Li, J.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic Band Gap from a Stack of Positive and Negative Index Materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Lisyansky, A. A.

L. I. Deych, D. Livdan, and A. A. Lisyansky, "Resonant tunneling of electromagnetic waves through polariton gaps," Phys. Rev. E 57, 7254-7258 (1998).
[CrossRef]

Livdan, D.

L. I. Deych, D. Livdan, and A. A. Lisyansky, "Resonant tunneling of electromagnetic waves through polariton gaps," Phys. Rev. E 57, 7254-7258 (1998).
[CrossRef]

McCall, M.

A. Lakhtakia and M. McCall (eds), Focus issue on negative refraction, New J. Phys. 7 (2005), http://www.iop.org/EJ/abstract/-ff30=1/1367-2630/7/1/E03.
[CrossRef]

Michel, J.

Y. Fink, J. Winn, S. Fan, C. Chen, J. Michel, J. Joannopoulos, and E. Thomas, "A Dielectric Omnidirectional Reflector," Science 282,1679-1682 (1998).
[CrossRef] [PubMed]

Noda, S.

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

Osgood, R.

Panoiu, N.

Pendry, J.

D. Smith, J. Pendry, and M. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Roberts, P. J.

P. St. J. Russell, S. Tredwell, and P. J. Roberts, "Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures," Opt. Commun. 160, 66-71 (1999).
[CrossRef]

Robin, F.

Russell, P. St. J.

P. St. J. Russell, S. Tredwell, and P. J. Roberts, "Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures," Opt. Commun. 160, 66-71 (1999).
[CrossRef]

Shadrivov, I.

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, "Complete Band Gaps in One-Dimensional Left-Handed Periodic Structures," Phys. Rev. Lett. 95, 193903 (2005).
[CrossRef] [PubMed]

Sheng, P.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic Band Gap from a Stack of Positive and Negative Index Materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Smith, D.

D. Smith, J. Pendry, and M. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (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 425, 944-947 (2003).
[CrossRef] [PubMed]

Sukhorukov, A.

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, "Complete Band Gaps in One-Dimensional Left-Handed Periodic Structures," Phys. Rev. Lett. 95, 193903 (2005).
[CrossRef] [PubMed]

Thomas, E.

Y. Fink, J. Winn, S. Fan, C. Chen, J. Michel, J. Joannopoulos, and E. Thomas, "A Dielectric Omnidirectional Reflector," Science 282,1679-1682 (1998).
[CrossRef] [PubMed]

Tredwell, S.

P. St. J. Russell, S. Tredwell, and P. J. Roberts, "Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures," Opt. Commun. 160, 66-71 (1999).
[CrossRef]

Vahldieck, R.

Veselago, V. G.

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

Wang, L.-G.

L.-G. Wang, H. Chen, and S.-Y. Zhu, "Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials," Phys. Rev. B 70, 245102 (2004).
[CrossRef]

Wiltshire, M.

D. Smith, J. Pendry, and M. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Winn, J.

Y. Fink, J. Winn, S. Fan, C. Chen, J. Michel, J. Joannopoulos, and E. Thomas, "A Dielectric Omnidirectional Reflector," Science 282,1679-1682 (1998).
[CrossRef] [PubMed]

Wu, L.

Xudong, C.

Yariv, A.

Yarotsky, D. A.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "Observation of total omnidirectional reflection from one-dimensional dielectric latice," Appl. Phys. A 68, 25-28 (1999).
[CrossRef]

Yeh, P.

Zhang, S.

Zhang, Y.

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, "Properties of one-dimensional photonic crystals containing single-negative materials," Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Zhou, L.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic Band Gap from a Stack of Positive and Negative Index Materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Zhu, S.

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, "Properties of one-dimensional photonic crystals containing single-negative materials," Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Zhu, S.-Y.

L.-G. Wang, H. Chen, and S.-Y. Zhu, "Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials," Phys. Rev. B 70, 245102 (2004).
[CrossRef]

Zi, J.

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, "Properties of one-dimensional photonic crystals containing single-negative materials," Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Appl. Phys. A (1)

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, "Observation of total omnidirectional reflection from one-dimensional dielectric latice," Appl. Phys. A 68, 25-28 (1999).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

D. Delbeke, R. Bockstaele, P. Bienstman, R. Baets, and H. Benisty, "High-efficiency semiconductor resonantcavity light-emitting diodes: a review," IEEE J. Sel. Top. Quantum Electron. 8, 189-206 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Nature (1)

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

New J. Phys. (1)

A. Lakhtakia and M. McCall (eds), Focus issue on negative refraction, New J. Phys. 7 (2005), http://www.iop.org/EJ/abstract/-ff30=1/1367-2630/7/1/E03.
[CrossRef]

Opt. Commun. (1)

P. St. J. Russell, S. Tredwell, and P. J. Roberts, "Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures," Opt. Commun. 160, 66-71 (1999).
[CrossRef]

Opt. Express (2)

Phys. Rev. B (1)

L.-G. Wang, H. Chen, and S.-Y. Zhu, "Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials," Phys. Rev. B 70, 245102 (2004).
[CrossRef]

Phys. Rev. E (2)

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, "Properties of one-dimensional photonic crystals containing single-negative materials," Phys. Rev. E 69, 066607 (2004).
[CrossRef]

L. I. Deych, D. Livdan, and A. A. Lisyansky, "Resonant tunneling of electromagnetic waves through polariton gaps," Phys. Rev. E 57, 7254-7258 (1998).
[CrossRef]

Phys. Rev. Lett. (2)

J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic Band Gap from a Stack of Positive and Negative Index Materials," Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

I. Shadrivov, A. Sukhorukov, and Y. Kivshar, "Complete Band Gaps in One-Dimensional Left-Handed Periodic Structures," Phys. Rev. Lett. 95, 193903 (2005).
[CrossRef] [PubMed]

Science (2)

Y. Fink, J. Winn, S. Fan, C. Chen, J. Michel, J. Joannopoulos, and E. Thomas, "A Dielectric Omnidirectional Reflector," Science 282,1679-1682 (1998).
[CrossRef] [PubMed]

D. Smith, J. Pendry, and M. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004).
[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, 509-514 (1968).
[CrossRef]

Other (2)

J. Pendry (ed), Focus issue: Negative refraction and metamaterials, Opt. Express 11 (2003), http://www.opticsexpress.org/ViewMedia.cfm?id=71852&seq=0.
[PubMed]

R. A. Depine, M. L. Mart’ýnez-Ricci, J. A. Monsoriu, E. Silvestre, and P. Andr’es, "Zero permeability and zero permittivity band gaps in 1D metamaterial photonic crystals," arXiv: physics/0606069 (2006).

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Figures (5)

Fig. 1.
Fig. 1.

Frequency dependence of the effective parameters μ 2 and ε 2, as given by Eqs. (7). Note that μ 2 and ε 2 become zero at different frequency values, indicated by horizontal lines (3.133 and 3.787 GHz, respectively).

Fig. 2.
Fig. 2.

Band structure for TE and TM polarizations, different angles of incidence and different filling fractions (f) corresponding to periodic stacks (period d=15 mm) with air layers (µ 1=ε 1=1, d 1=fd) and MM layers (µ 2 and ε 2 shown in Fig. 1, d 2=(1-f)d). Note that the zero permeability and the zero permittivity gaps are absent for normal incidence (central column).

Fig. 3.
Fig. 3.

Projected band structure for TE and TM polarizations corresponding to the periodic stacks considered in Fig. 2.

Fig. 4.
Fig. 4.

Dependency of the region of omnidirectional reflection on the air filling fraction μ.

Fig. 5.
Fig. 5.

TE and TM reflectivity through 16 unit cells corresponding to the band structures in Fig. 2.

Equations (8)

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

cos ( K d ) = cos ( k 1 y d 1 ) cos ( k 2 y d 2 ) 1 2 [ σ 2 k 1 y σ 1 k 2 y + σ 1 k 2 y σ 2 k 1 y ] sin ( k 1 y d 1 ) sin ( k 2 y d 2 ) ,
k 1 y d 1 + k 2 y d 2 = p π , p = ± 1 , ± 2 , . . . ,
σ 2 k 1 y σ 1 k 2 y
k 1 y d 1 q π , q = 1 , 2 , . . .
σ 2 k 1 y σ 1 k 2 y k 1 y d 1 + k 2 y d 2 = 0 }
cos ( K d ) cos ( k 1 d 1 ) 1 2 ω c ε 1 μ 1 μ 2 d 2 sin ( k 1 d 1 ) .
ε 2 ( v ) = 1 + 5 2 0.9 2 v 2 + 10 2 11.5 2 v 2 ,
μ 2 ( v ) = 1 + 3 2 0.902 2 v 2 ,

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