Abstract

We consider periodic multilayers combining ordinary positive index materials and dispersive metamaterials with negative index in some frequency ranges. These structures can exhibit photonic bandgaps which, in contrast with the usual Bragg gaps, are not based on interference mechanisms. Changing the dispersion models for the constituent metamaterial, we investigate its role in the production of zero-average-index bandgaps. In particular, we show the effect of each constitutive parameter on both bandgap edges. Finally, we give some approximated analytical expressions in terms of average parameters for the determination of the upper and lower limits of the zero-average refractive-index bandgap.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B 10, 283-295 (1993).
    [CrossRef]
  2. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  3. 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]
  4. D. Smith, J. Pendry, and M. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788-792 (2004).
    [CrossRef] [PubMed]
  5. N.Engheta and R.W.Ziolkowski (eds.), Metamaterials: Physics and Engineering Explorations (Wiley-IEEE, 2006).
  6. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  7. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
    [CrossRef]
  8. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
    [CrossRef] [PubMed]
  9. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53-55 (2007).
    [CrossRef]
  10. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41-48 (2007).
    [CrossRef]
  11. J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic bandgap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
    [CrossRef] [PubMed]
  12. N. Panoiu, R. Osgood, S. Zhang, and S. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506-513 (2006).
    [CrossRef]
  13. M. de Dios-Leyva and O. E. González-Vásquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 77, 125102 (2008).
    [CrossRef]
  14. R. A. Depine, M. L. Martínez-Ricci, J. A. Monsoriu, E. Silvestre, and P. Andrés, “Zero permeability and zero permittivity bandgaps in 1D metamaterial photonic crystals,” Phys. Lett. A 364, 352-355 (2007).
    [CrossRef]
  15. J. A. Monsoriu, R. A. Depine, M. L. Martínez-Ricci, and E. Silvestre, “Interaction between non-Bragg bandgaps in 1D metamaterial photonic crystals,” Opt. Express 14, 12958-12967 (2006).
    [CrossRef] [PubMed]
  16. 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]
  17. L. Wu, S. He, and L. Chen, “On unusual narrow transmission bands for a multilayered periodic structure containing left-handed materials,” Opt. Express 11, 1283-1290 (2003).
    [CrossRef] [PubMed]
  18. J. A. Monsoriu, R. A. Depine, and E. Silvestre, “Non-Bragg bandgaps in 1D metamaterial aperiodic multilayers,” J. Eur. Opt. Soc. Rapid Publ. 2, 07002 (2007).
    [CrossRef]
  19. M. de Dios-Leyva, J. Sabin, and J. López-Gondar, “Band and branch inversion phenomena in GaAs-AlxGa1−xAs superlattices,” Phys. Status Solidi B 134, 615-620 (1986).
    [CrossRef]

2008 (1)

M. de Dios-Leyva and O. E. González-Vásquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 77, 125102 (2008).
[CrossRef]

2007 (4)

R. A. Depine, M. L. Martínez-Ricci, J. A. Monsoriu, E. Silvestre, and P. Andrés, “Zero permeability and zero permittivity bandgaps in 1D metamaterial photonic crystals,” Phys. Lett. A 364, 352-355 (2007).
[CrossRef]

J. A. Monsoriu, R. A. Depine, and E. Silvestre, “Non-Bragg bandgaps in 1D metamaterial aperiodic multilayers,” J. Eur. Opt. Soc. Rapid Publ. 2, 07002 (2007).
[CrossRef]

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41-48 (2007).
[CrossRef]

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53-55 (2007).
[CrossRef]

2006 (3)

2004 (1)

D. Smith, J. Pendry, and M. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788-792 (2004).
[CrossRef] [PubMed]

2003 (2)

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic bandgap 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 multilayered periodic structure containing left-handed materials,” Opt. Express 11, 1283-1290 (2003).
[CrossRef] [PubMed]

2000 (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

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 Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

1998 (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]

1995 (1)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

1993 (1)

1986 (1)

M. de Dios-Leyva, J. Sabin, and J. López-Gondar, “Band and branch inversion phenomena in GaAs-AlxGa1−xAs superlattices,” Phys. Status Solidi B 134, 615-620 (1986).
[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]

Andrés, P.

R. A. Depine, M. L. Martínez-Ricci, J. A. Monsoriu, E. Silvestre, and P. Andrés, “Zero permeability and zero permittivity bandgaps in 1D metamaterial photonic crystals,” Phys. Lett. A 364, 352-355 (2007).
[CrossRef]

Brueck, S.

Chan, C. T.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic bandgap 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, L.

de Dios-Leyva, M.

M. de Dios-Leyva and O. E. González-Vásquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 77, 125102 (2008).
[CrossRef]

M. de Dios-Leyva, J. Sabin, and J. López-Gondar, “Band and branch inversion phenomena in GaAs-AlxGa1−xAs superlattices,” Phys. Status Solidi B 134, 615-620 (1986).
[CrossRef]

Depine, R. A.

J. A. Monsoriu, R. A. Depine, and E. Silvestre, “Non-Bragg bandgaps in 1D metamaterial aperiodic multilayers,” J. Eur. Opt. Soc. Rapid Publ. 2, 07002 (2007).
[CrossRef]

R. A. Depine, M. L. Martínez-Ricci, J. A. Monsoriu, E. Silvestre, and P. Andrés, “Zero permeability and zero permittivity bandgaps in 1D metamaterial photonic crystals,” Phys. Lett. A 364, 352-355 (2007).
[CrossRef]

J. A. Monsoriu, R. A. Depine, M. L. Martínez-Ricci, and E. Silvestre, “Interaction between non-Bragg bandgaps in 1D metamaterial photonic crystals,” Opt. Express 14, 12958-12967 (2006).
[CrossRef] [PubMed]

Dolling, G.

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]

González-Vásquez, O. E.

M. de Dios-Leyva and O. E. González-Vásquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 77, 125102 (2008).
[CrossRef]

He, S.

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 Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

Hong, C.

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]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Li, J.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic bandgap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Linden, S.

López-Gondar, J.

M. de Dios-Leyva, J. Sabin, and J. López-Gondar, “Band and branch inversion phenomena in GaAs-AlxGa1−xAs superlattices,” Phys. Status Solidi B 134, 615-620 (1986).
[CrossRef]

Martínez-Ricci, M. L.

R. A. Depine, M. L. Martínez-Ricci, J. A. Monsoriu, E. Silvestre, and P. Andrés, “Zero permeability and zero permittivity bandgaps in 1D metamaterial photonic crystals,” Phys. Lett. A 364, 352-355 (2007).
[CrossRef]

J. A. Monsoriu, R. A. Depine, M. L. Martínez-Ricci, and E. Silvestre, “Interaction between non-Bragg bandgaps in 1D metamaterial photonic crystals,” Opt. Express 14, 12958-12967 (2006).
[CrossRef] [PubMed]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

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]

Monsoriu, J. A.

J. A. Monsoriu, R. A. Depine, and E. Silvestre, “Non-Bragg bandgaps in 1D metamaterial aperiodic multilayers,” J. Eur. Opt. Soc. Rapid Publ. 2, 07002 (2007).
[CrossRef]

R. A. Depine, M. L. Martínez-Ricci, J. A. Monsoriu, E. Silvestre, and P. Andrés, “Zero permeability and zero permittivity bandgaps in 1D metamaterial photonic crystals,” Phys. Lett. A 364, 352-355 (2007).
[CrossRef]

J. A. Monsoriu, R. A. Depine, M. L. Martínez-Ricci, and E. Silvestre, “Interaction between non-Bragg bandgaps in 1D metamaterial photonic crystals,” Opt. Express 14, 12958-12967 (2006).
[CrossRef] [PubMed]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Osgood, R.

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Panoiu, N.

Pendry, J.

D. Smith, J. Pendry, and M. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

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 Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

Sabin, J.

M. de Dios-Leyva, J. Sabin, and J. López-Gondar, “Band and branch inversion phenomena in GaAs-AlxGa1−xAs superlattices,” Phys. Status Solidi B 134, 615-620 (1986).
[CrossRef]

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Shalaev, V. M.

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41-48 (2007).
[CrossRef]

Sheng, P.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic bandgap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

Silvestre, E.

R. A. Depine, M. L. Martínez-Ricci, J. A. Monsoriu, E. Silvestre, and P. Andrés, “Zero permeability and zero permittivity bandgaps in 1D metamaterial photonic crystals,” Phys. Lett. A 364, 352-355 (2007).
[CrossRef]

J. A. Monsoriu, R. A. Depine, and E. Silvestre, “Non-Bragg bandgaps in 1D metamaterial aperiodic multilayers,” J. Eur. Opt. Soc. Rapid Publ. 2, 07002 (2007).
[CrossRef]

J. A. Monsoriu, R. A. Depine, M. L. Martínez-Ricci, and E. Silvestre, “Interaction between non-Bragg bandgaps in 1D metamaterial photonic crystals,” Opt. Express 14, 12958-12967 (2006).
[CrossRef] [PubMed]

Smith, D.

D. Smith, J. Pendry, and M. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Smith, D. R.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Soukoulis, C. M.

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 Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

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]

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]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Wegener, M.

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]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Wu, L.

Yablonovitch, E.

Yariv, A.

Yeh, P.

Zhang, S.

Zhou, L.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic bandgap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

IEEE Trans. Microwave Theory 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 Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ. (1)

J. A. Monsoriu, R. A. Depine, and E. Silvestre, “Non-Bragg bandgaps in 1D metamaterial aperiodic multilayers,” J. Eur. Opt. Soc. Rapid Publ. 2, 07002 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Nat. Photonics (1)

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41-48 (2007).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Lett. A (1)

R. A. Depine, M. L. Martínez-Ricci, J. A. Monsoriu, E. Silvestre, and P. Andrés, “Zero permeability and zero permittivity bandgaps in 1D metamaterial photonic crystals,” Phys. Lett. A 364, 352-355 (2007).
[CrossRef]

Phys. Rev. B (1)

M. de Dios-Leyva and O. E. González-Vásquez, “Band structure and associated electromagnetic fields in one-dimensional photonic crystals with left-handed materials,” Phys. Rev. B 77, 125102 (2008).
[CrossRef]

Phys. Rev. Lett. (2)

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic bandgap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Phys. Status Solidi B (1)

M. de Dios-Leyva, J. Sabin, and J. López-Gondar, “Band and branch inversion phenomena in GaAs-AlxGa1−xAs superlattices,” Phys. Status Solidi B 134, 615-620 (1986).
[CrossRef]

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. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

N.Engheta and R.W.Ziolkowski (eds.), Metamaterials: Physics and Engineering Explorations (Wiley-IEEE, 2006).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

(a) Frequency dependence of the effective parameters μ 2 and ϵ 2 , as given by Eqs. (5). Note that ϵ 2 and μ 2 become zero at different frequency values [ ν ϵ = 1.878 GHz and ν μ = 3.133 GHz (out of the plot range)]. (b) Projected band structure for TE polarization and different angles of incidence corresponding to periodic stacks with air layers ( ϵ 1 = μ 1 = 1 ) and MM layers [ ϵ 2 and μ 2 shown in (a)], both of the same width ( d 1 = d 2 = 6 mm ) . Dots represent the low-frequency limit of the zero- n ¯ bandgap edges given by Eqs. (7).

Fig. 2
Fig. 2

Half of the trace of the unit cell translation matrix, ξ, as given by Eq. (1), for the structure considered in Fig. 1, at normal incidence, when either ϵ 2 (a), or μ 2 (c), or both (b) are considered dispersive.

Fig. 3
Fig. 3

Evolution of the zero- n ¯ bandgap when the constitutive parameters, μ 2 ( ν ) and ϵ 2 ( ν ) , change their slope around ν n ¯ [see Eqs. (6)] for TE polarization and two different angles of incidence (solid curves, 0°; dashed curves, 45°; dots, band edge low-frequency limits). (a) α ϵ = 1 and α μ ranging from 1 to 0. (b) α μ = 1 and α ϵ ranging from 1 to 0. The gray- and white-striped zone is forbidden for one angle of incidence and allowed for the other one. This insets show the dispersion relations for two cases close to the limits of the considered ranges.

Fig. 4
Fig. 4

Evolution of the zero- n ¯ gap as a function of Δ μ [see Eq. (9)] for TE polarization and two different angles of incidence (solid lines, 0°; dashed lines, 45°; dots, band edge low-frequency limits). At Δ μ = 2.28 , when the transverse impedance matching between both materials is achieved, the zero- n ¯ gap vanishes. Gray- and white-striped zones are forbidden for one angle of incidence and allowed for the other one. The inset shows the dispersion relations for the two cases in the limits of the considered range.

Fig. 5
Fig. 5

Reflectance spectra of two stacks of 30 periods of the multilayer analyzed in Fig. 1b when a lossy MM is considered: (a) γ = 0.01 GHz and (b) γ = 0.1 GHz . The polarization is TE and the angle of incidence ranges from 0 to 90°.

Fig. 6
Fig. 6

Projected band structure for TE polarization and different angles of incidence corresponding to two periodic stacks with layers made of air ( ϵ 1 = μ 1 = 1 ) , a dispersive MM [ ϵ 2 ( ν ) = ϵ ̂ ( ν ) , and μ 2 ( ν ) = μ ̂ ( ν ) ], and two different high-index dielectrics with (a) ϵ 3 = 4 and μ 3 = 3 (solid lines) and (b) ϵ 3 = 4 and μ 3 = 3 (dashed lines), the three of them of the same width ( d 1 = d 2 = d 3 = 6 mm ) . In both cases, dots represent the low-frequency limit of the zero- n ¯ bandgap edges given by Eqs. (7). Gray- and white-striped zones are forbidden for one case and allowed for the other one.

Equations (20)

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

ξ 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 ,
k 1 y d 1 + k 2 y d 2 = 0 ,
σ 2 k 1 y σ 1 k 2 y ,
q ( ν ) = sin d 1 k 1 y 2 cos d 2 k 2 y 2 + σ 2 k 1 y σ 1 k 2 y cos d 1 k 1 y 2 sin d 2 k 2 y 2 ,
r ( ν ) = sin d 1 k 1 y 2 cos d 2 k 2 y 2 + σ 1 k 2 y σ 2 k 1 y cos d 1 k 1 y 2 sin d 2 k 2 y 2 ,
ϵ ̂ ( ν ) = 1 1.62 2 ν 2 0.95 2 ,
μ ̂ ( ν ) = 1 3 2 ν 2 0.902 2 ,
ϵ 2 ( ν ; α ϵ ) = ϵ ̂ ( ν n ¯ ) + α ϵ [ ϵ ̂ ( ν ) ϵ ̂ ( ν n ¯ ) ] ,
μ 2 ( ν ; α μ ) = μ ̂ ( ν n ¯ ) + α μ [ μ ̂ ( ν ) μ ̂ ( ν n ¯ ) ] ,
μ ¯ = 0 ,
( ω c ) 2 ϵ ¯ + k x 2 μ 1 ¯ = 0 ,
ϵ ¯ = 0 ,
( ω c ) 2 μ ¯ + k x 2 ϵ 1 ¯ = 0 ,
ϵ 2 ( ν ; Δ ϵ ) = ϵ ̂ ( ν ) + Δ ϵ ,
μ 2 ( ν ; Δ μ ) = μ ̂ ( ν ) + Δ μ ,
ϵ 2 ( ν ; γ ) = 1 1.62 2 ν 2 0.95 2 + i γ ν ,
μ 2 ( ν ; γ ) = 1 3 2 ν 2 0.902 2 + i γ ν .

Metrics