Abstract

The electromagnetic resonances of multilayer metal–dielectric stacks are investigated. These structures support periodic bandpass regions, whose band edges may be predicted by considering the character of the fields inside the different layers. It is shown that the response of the structure is largely independent of its overall length, and that only the geometry of the unit cell is important. In the metal layers, the fields may have either a cosh or a sinh distribution function and match to standing waves inside the adjacent dielectric cavities at the metal–dielectric interface. It is shown that the different boundary conditions, imposed by the evanescent fields, result in the dielectric layers having a different effective length for the two modes. The sinh fields result in an effective length being very close to that of the physical length, and adjacent cavities oscillating out of phase, while the cosh fields may result in a significantly larger effective dielectric length and adjacent cavities oscillating in phase. A bandpass region is opened, with its high frequency edge always being near the dielectric Fabry–Perot limit, while the low frequency band edge is significantly redshifted.

© 2009 Optical Society of America

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References

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  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
    [CrossRef]
  2. 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]
  3. M. Scalora, M. J. Bloemer, and C. M. Bowden, “Laminated photonic band structures with high conductivity and high transparency: metals under a new light,” Opt. Photonics News 10, 23-27 (1998).
  4. M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377-2383 (1998).
    [CrossRef]
  5. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107-4121 (1996).
    [CrossRef]
  6. M. C. Larciprete, C. Sibilia, S. Paolini, and M. Bertolotti, “Accessing the optical limiting properties of metallo-dielectric photonic band gap structures,” J. Appl. Phys. 93, 5013-5017 (2003).
    [CrossRef]
  7. M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and sub-wavelength focusing in the visible range using transparent metallodielectric stacks,” Opt. Express 15, 508-523 (2007).
    [CrossRef] [PubMed]
  8. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  9. N. Fang, H. Lee, C. Sun, and C. X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
    [CrossRef] [PubMed]
  10. K. J. Webb and M. Yang, “Subwavelength imaging with a multilayer silver film structure,” Opt. Lett. 31, 2130-2132 (2006).
    [CrossRef] [PubMed]
  11. C. Sibilia, I. S. Nefedov, M. Scalora, and M. Bertolotti, “Electromagnetic mode density for finite quasi-periodic structures,” J. Opt. Soc. Am. B 15, 1947-1952 (1998).
    [CrossRef]
  12. A. Bichri, J. Lafait, and H. Welsch, “Visible and infrared optical properties of Ag/SiO2 multilayers: radiative virtual modes and coupling effects,” J. Phys.: Condens. Matter 5, 7361-7374 (1993).
    [CrossRef]
  13. A. Bichri, J. Lafait, H. Welsch, and M. Abd-Lefdil, “Characterization of Berreman modes in metal/dielectric and multilayers,” J. Phys.: Condens. Matter 9, 6523-6532 (1997).
    [CrossRef]
  14. A. P. Hibbins, M. J. Lockyear, and J. R. Sambles, “The resonant electromagnetic fields of an array of metallic slits acting as Fabry-Perot cavities,” J. Appl. Phys. 99, 124903 (2006).
    [CrossRef]

2007 (1)

2006 (2)

A. P. Hibbins, M. J. Lockyear, and J. R. Sambles, “The resonant electromagnetic fields of an array of metallic slits acting as Fabry-Perot cavities,” J. Appl. Phys. 99, 124903 (2006).
[CrossRef]

K. J. Webb and M. Yang, “Subwavelength imaging with a multilayer silver film structure,” Opt. Lett. 31, 2130-2132 (2006).
[CrossRef] [PubMed]

2005 (1)

N. Fang, H. Lee, C. Sun, and C. X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

2003 (1)

M. C. Larciprete, C. Sibilia, S. Paolini, and M. Bertolotti, “Accessing the optical limiting properties of metallo-dielectric photonic band gap structures,” J. Appl. Phys. 93, 5013-5017 (2003).
[CrossRef]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

1998 (5)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[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]

M. Scalora, M. J. Bloemer, and C. M. Bowden, “Laminated photonic band structures with high conductivity and high transparency: metals under a new light,” Opt. Photonics News 10, 23-27 (1998).

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377-2383 (1998).
[CrossRef]

C. Sibilia, I. S. Nefedov, M. Scalora, and M. Bertolotti, “Electromagnetic mode density for finite quasi-periodic structures,” J. Opt. Soc. Am. B 15, 1947-1952 (1998).
[CrossRef]

1997 (1)

A. Bichri, J. Lafait, H. Welsch, and M. Abd-Lefdil, “Characterization of Berreman modes in metal/dielectric and multilayers,” J. Phys.: Condens. Matter 9, 6523-6532 (1997).
[CrossRef]

1996 (1)

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107-4121 (1996).
[CrossRef]

1993 (1)

A. Bichri, J. Lafait, and H. Welsch, “Visible and infrared optical properties of Ag/SiO2 multilayers: radiative virtual modes and coupling effects,” J. Phys.: Condens. Matter 5, 7361-7374 (1993).
[CrossRef]

Abd-Lefdil, M.

A. Bichri, J. Lafait, H. Welsch, and M. Abd-Lefdil, “Characterization of Berreman modes in metal/dielectric and multilayers,” J. Phys.: Condens. Matter 9, 6523-6532 (1997).
[CrossRef]

Akozbek, N.

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107-4121 (1996).
[CrossRef]

Bertolotti, M.

M. C. Larciprete, C. Sibilia, S. Paolini, and M. Bertolotti, “Accessing the optical limiting properties of metallo-dielectric photonic band gap structures,” J. Appl. Phys. 93, 5013-5017 (2003).
[CrossRef]

C. Sibilia, I. S. Nefedov, M. Scalora, and M. Bertolotti, “Electromagnetic mode density for finite quasi-periodic structures,” J. Opt. Soc. Am. B 15, 1947-1952 (1998).
[CrossRef]

Bichri, A.

A. Bichri, J. Lafait, H. Welsch, and M. Abd-Lefdil, “Characterization of Berreman modes in metal/dielectric and multilayers,” J. Phys.: Condens. Matter 9, 6523-6532 (1997).
[CrossRef]

A. Bichri, J. Lafait, and H. Welsch, “Visible and infrared optical properties of Ag/SiO2 multilayers: radiative virtual modes and coupling effects,” J. Phys.: Condens. Matter 5, 7361-7374 (1993).
[CrossRef]

Bloemer, M. J.

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and sub-wavelength focusing in the visible range using transparent metallodielectric stacks,” Opt. Express 15, 508-523 (2007).
[CrossRef] [PubMed]

M. Scalora, M. J. Bloemer, and C. M. Bowden, “Laminated photonic band structures with high conductivity and high transparency: metals under a new light,” Opt. Photonics News 10, 23-27 (1998).

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377-2383 (1998).
[CrossRef]

Bowden, C. M.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377-2383 (1998).
[CrossRef]

M. Scalora, M. J. Bloemer, and C. M. Bowden, “Laminated photonic band structures with high conductivity and high transparency: metals under a new light,” Opt. Photonics News 10, 23-27 (1998).

Cappeddu, M. G.

Centini, M.

D'Aguanno, G.

de Ceglia, D.

Dowling, J. P.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377-2383 (1998).
[CrossRef]

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107-4121 (1996).
[CrossRef]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[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]

Fang, N.

N. Fang, H. Lee, C. Sun, and C. X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Fowler, M.

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[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]

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]

Haus, J. W.

Hibbins, A. P.

A. P. Hibbins, M. J. Lockyear, and J. R. Sambles, “The resonant electromagnetic fields of an array of metallic slits acting as Fabry-Perot cavities,” J. Appl. Phys. 99, 124903 (2006).
[CrossRef]

Lafait, J.

A. Bichri, J. Lafait, H. Welsch, and M. Abd-Lefdil, “Characterization of Berreman modes in metal/dielectric and multilayers,” J. Phys.: Condens. Matter 9, 6523-6532 (1997).
[CrossRef]

A. Bichri, J. Lafait, and H. Welsch, “Visible and infrared optical properties of Ag/SiO2 multilayers: radiative virtual modes and coupling effects,” J. Phys.: Condens. Matter 5, 7361-7374 (1993).
[CrossRef]

Larciprete, M. C.

M. C. Larciprete, C. Sibilia, S. Paolini, and M. Bertolotti, “Accessing the optical limiting properties of metallo-dielectric photonic band gap structures,” J. Appl. Phys. 93, 5013-5017 (2003).
[CrossRef]

Lee, H.

N. Fang, H. Lee, C. Sun, and C. X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[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]

Lockyear, M. J.

A. P. Hibbins, M. J. Lockyear, and J. R. Sambles, “The resonant electromagnetic fields of an array of metallic slits acting as Fabry-Perot cavities,” J. Appl. Phys. 99, 124903 (2006).
[CrossRef]

Mandatori, A.

Manka, A. S.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377-2383 (1998).
[CrossRef]

Mattiucci, N.

Nefedov, I. S.

Paolini, S.

M. C. Larciprete, C. Sibilia, S. Paolini, and M. Bertolotti, “Accessing the optical limiting properties of metallo-dielectric photonic band gap structures,” J. Appl. Phys. 93, 5013-5017 (2003).
[CrossRef]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Pethel, A. S.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377-2383 (1998).
[CrossRef]

Sambles, J. R.

A. P. Hibbins, M. J. Lockyear, and J. R. Sambles, “The resonant electromagnetic fields of an array of metallic slits acting as Fabry-Perot cavities,” J. Appl. Phys. 99, 124903 (2006).
[CrossRef]

Scalora, M.

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and sub-wavelength focusing in the visible range using transparent metallodielectric stacks,” Opt. Express 15, 508-523 (2007).
[CrossRef] [PubMed]

M. Scalora, M. J. Bloemer, and C. M. Bowden, “Laminated photonic band structures with high conductivity and high transparency: metals under a new light,” Opt. Photonics News 10, 23-27 (1998).

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377-2383 (1998).
[CrossRef]

C. Sibilia, I. S. Nefedov, M. Scalora, and M. Bertolotti, “Electromagnetic mode density for finite quasi-periodic structures,” J. Opt. Soc. Am. B 15, 1947-1952 (1998).
[CrossRef]

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107-4121 (1996).
[CrossRef]

Sibilia, C.

Sun, C.

N. Fang, H. Lee, C. Sun, and C. X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[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]

Webb, K. J.

Welsch, H.

A. Bichri, J. Lafait, H. Welsch, and M. Abd-Lefdil, “Characterization of Berreman modes in metal/dielectric and multilayers,” J. Phys.: Condens. Matter 9, 6523-6532 (1997).
[CrossRef]

A. Bichri, J. Lafait, and H. Welsch, “Visible and infrared optical properties of Ag/SiO2 multilayers: radiative virtual modes and coupling effects,” J. Phys.: Condens. Matter 5, 7361-7374 (1993).
[CrossRef]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[CrossRef]

Yang, M.

Zhang, C. X.

N. Fang, H. Lee, C. Sun, and C. X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

J. Appl. Phys. (3)

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377-2383 (1998).
[CrossRef]

M. C. Larciprete, C. Sibilia, S. Paolini, and M. Bertolotti, “Accessing the optical limiting properties of metallo-dielectric photonic band gap structures,” J. Appl. Phys. 93, 5013-5017 (2003).
[CrossRef]

A. P. Hibbins, M. J. Lockyear, and J. R. Sambles, “The resonant electromagnetic fields of an array of metallic slits acting as Fabry-Perot cavities,” J. Appl. Phys. 99, 124903 (2006).
[CrossRef]

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

J. Phys.: Condens. Matter (2)

A. Bichri, J. Lafait, and H. Welsch, “Visible and infrared optical properties of Ag/SiO2 multilayers: radiative virtual modes and coupling effects,” J. Phys.: Condens. Matter 5, 7361-7374 (1993).
[CrossRef]

A. Bichri, J. Lafait, H. Welsch, and M. Abd-Lefdil, “Characterization of Berreman modes in metal/dielectric and multilayers,” J. Phys.: Condens. Matter 9, 6523-6532 (1997).
[CrossRef]

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667-669 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Photonics News (1)

M. Scalora, M. J. Bloemer, and C. M. Bowden, “Laminated photonic band structures with high conductivity and high transparency: metals under a new light,” Opt. Photonics News 10, 23-27 (1998).

Phys. Rev. B (1)

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]

Phys. Rev. E (1)

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107-4121 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Science (1)

N. Fang, H. Lee, C. Sun, and C. X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534-537 (2005).
[CrossRef] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

Reflection efficiency response of the multilayer structure comprising five 6.5 nm silver layers separated by four 166 nm air layers. The incident and transmission materials are also air and the structure is illuminated at normal incidence. The permittivity of the silver layers are approximated by the Drude model with the parameters as defined in the main text. The frequency range is 0 × 10 15 rad s 1 < ω 37.7 × 10 15 rad s 1 . The inset is a schematic representation of one unit cell of such a multilayer stack.

Fig. 2
Fig. 2

Reflection efficiency response of the multilayer structure comprising ten 6.5 nm silver layers separated by nine 166 nm air layers (solid curve) and twenty 6.5 nm silver layers separated by nineteen 166 nm air layers (dashed curve). The incident and transmission materials are also air and the structure is illuminated at normal incidence. The permittivity of the silver layers are approximated by the Drude model with the parameters as defined in the main text. The frequency range is 0 × 10 15 rad s 1 < ω 6.28 × 10 15 rad s 1 .

Fig. 3
Fig. 3

Reflection efficiency response of the multilayer structure comprising ten 6.5 nm silver layers separated by nine 166 nm air layers. The incident and transmission materials are also air and the structure is illuminated at normal incidence. The permittivity of the silver layers are approximated by the Drude model with the parameters as defined in the main text (solid curve) and by the Drude model with the imaginary part of the permittivity removed (dashed curve). The frequency range is 0 × 10 15 rad s 1 < ω 6.28 × 10 15 rad s 1 .

Fig. 4
Fig. 4

E ̱ fields for a multilayer structure, comprising ten 20 nm silver layers separated by nine 150 nm air layers. The incident and transmission materials are also air and the structure is illuminated at normal incidence and the permittivity of the silver layers are approximated by the Drude model with parameters as defined in the main text (including the imaginary part). The fields are plotted at two frequencies corresponding to the highest and lowest frequency reflection minima of the first bandpass region. The dashed lines indicate the location of the metal layers.

Fig. 5
Fig. 5

E ̱ fields for an infinite multilayer structure, comprising 20 nm silver layers separated by 150 nm air layers. The incident and transmission materials are also air and the structure is illuminated at normal incidence. The permittivity of the silver layers are approximated by the Drude model, using the parameters defined in the main text, with the imaginary part removed. The fields are plotted (a) using Eqs. (1, 3) and the solution to Eq. (12), and (b) using Eqs. (1, 3) and the solution to Eq. (11), which correspond to the band edges of Fig. 4. The solid lines are the cos standing waves in the dielectric, the dashed lines are the (a) sinh and (b) cosh waves in the metal, and the dotted lines show where the dielectric standing waves would continue if no metal was present. The bold and narrow curves are π out of phase and the vertical dotted-dashed lines indicate the effective cavity length.

Fig. 6
Fig. 6

Reflection efficiency response of the multilayer structure, comprising ten 6.5 nm silver layers separated by nine 166 nm air layers, as a function of the real part of the dielectric permittivity. The black squares are the limit solutions to Eqs. (11, 12) for the same permittivities. The incident and transmission materials are also air and the structure is illuminated at normal incidence. The frequency range is 0 × 10 15 rad s 1 < ω 6.28 × 10 15 rad s 1 ( > λ 300 nm ) , and the permittivity is defined as 101 ϵ r 1 .

Fig. 7
Fig. 7

E ̱ fields for the high frequency band edge of an infinite multilayer structure, comprising 20 nm silver layers separated by 150 nm air layers. The incident and transmission materials are also air, the structure is illuminated at normal incidence, and the permittivity of the silver layers are fixed at either ϵ r = ( a ) 40 or (b) 11.5 . The fields are plotted using Eqs. (1, 3) and the solutions to Eq. (12). The solid lines are the cos standing waves in the dielectric, the dashed lines are the sinh waves in the metal, and the dotted lines show where the dielectric standing waves would continue if no metal was present. The bold and narrow curves are π out of phase and the vertical dotted-dashed lines indicate the effective cavity length.

Fig. 8
Fig. 8

E ̱ fields for the low frequency band edge of an infinite multilayer structure, comprising 20 nm silver layers separated by 150 nm air layers. The incident and transmission materials are also air and the structure is illuminated at normal incidence and the permittivity of the silver layers are fixed at either ϵ r = ( a ) 40 or (b) 11.5 . The fields are plotted using Eqs. (1, 3) and the solutions to Eqs. (11). The solid lines are the cos standing waves in the dielectric, the dashed lines are the cosh waves in the metal, and the dotted lines show where the dielectric standing waves would continue if no metal was present. The bold and narrow curves are π out of phase and the vertical dotted-dashed lines indicate the effective cavity length.

Fig. 9
Fig. 9

Reflection efficiency response of the multilayer structure, comprising ten 6.5 nm silver layers separated by nine 166 nm air layers, as a function of the real part of the dielectric permittivity. The black squares are the limit solutions to Eqs. (11, 12, 13, 14) (and their harmonic solutions) for the same permittivities. The incident and transmission materials are also air and the structure is illuminated at normal incidence. The frequency range is 0 × 10 15 rad s 1 < ω 18.84 × 10 15 rad s 1 ( > λ 100 nm ) , and the permittivity is defined as 101 ϵ r 1 .

Fig. 10
Fig. 10

Reflection efficiency response of the multilayer structure, comprising ten silver layers of thickness 1 nm b 50 nm separated by nine 166 nm air layers, as a function of the thickness of the silver layers. The black squares are the limit solutions to Eqs. (15, 16, 17, 18) (and their harmonic solutions). The incident and transmission materials are also air and the structure is illuminated at normal incidence. The frequency range is 0 × 10 15 rad s 1 < ω 18.84 × 10 15 rad s 1 ( > λ 100 nm ) , and the permittivity is defined by the Drude model with parameters as defined in the main text.

Equations (18)

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

ψ n 1 = A cos ( n 1 k 0 x ) ,
ψ n 1 = A sin ( n 1 k 0 x ) ,
ψ k 2 = B cosh ( k 2 k 0 [ x ( a 2 + b 2 ) ] ) ,
ψ k 2 = B sinh ( k 2 k 0 [ x ( a 2 + b 2 ) ] ) ,
ψ n 1 x = n 1 k 0 A sin ( n 1 k 0 x ) ,
ψ n 1 x = n 1 k 0 A cos ( n 1 k 0 x ) ,
ψ k 2 x = k 2 k 0 B sinh ( k 2 k 0 [ x ( a 2 + b 2 ) ] ) ,
ψ k 2 x = k 2 k 0 B cosh ( k 2 k 0 [ x ( a 2 + b 2 ) ] ) .
A cos ( n 1 k 0 a 2 ) = B cosh ( k 2 k 0 [ b 2 ] ) ,
n 1 k 0 A sin ( n 1 k 0 a 2 ) = k 2 k 0 B sinh ( k 2 k 0 [ b 2 ] ) .
n 1 tan ( n 1 k 0 a 2 ) = k 2 tanh ( k 2 k 0 b 2 ) .
n 1 tan ( n 1 k 0 a 2 ) = k 2 coth ( k 2 k 0 b 2 ) ,
n 1 cot ( n 1 k 0 a 2 ) = k 2 tanh ( k 2 k 0 b 2 ) ,
n 1 cot ( n 1 k 0 a 2 ) = k 2 coth ( k 2 k 0 b 2 ) .
n 1 tan ( n 1 k 0 a 2 ) = n 2 tan ( n 2 k 0 b 2 ) ,
n 1 tan ( n 1 k 0 a 2 ) = n 2 cot ( n 2 k 0 b 2 ) ,
n 1 cot ( n 1 k 0 a 2 ) = n 2 tan ( n 2 k 0 b 2 ) ,
n 1 cot ( n 1 k 0 a 2 ) = n 2 cot ( n 2 k 0 b 2 ) .

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