Abstract

Impedance matching refers to the suppression of reflected radiation from an interface and is a concept that applies right across the electromagnetic spectrum. In particular it has come to prominence in relation to the propagation of light in metallic structures and associated meta-materials. Whilst established for microwaves and electrical circuits, this concept has only very recently been observed in the optical domain, yet is not well defined or understood. We present a framework to elucidate the concept of optical impedance. We describe using a scattering matrix approach the characteristic, iterative, image and wave impedances of an optical system. With a numerical model, we explore each form of impedance matching in metal-dielectric structures. Thin gold layers may extend the concept of Brewster’s angle to normal incidence and s polarization. Optical impedance for recently realized metallic gold nano-pillars which has shown negative permeability is also explored and we show that current measurements are inconclusive to robustly state its characteristic impedance is matched to the vacuum.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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]
  2. 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]
  3. V. G. Veselago, "Electrodynamics of substances with simultaneously negative values of sigma and mu," Sov. Phys. Usp. 10, 509 (1968).
    [CrossRef]
  4. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  5. N. Fang, H. Lee, C. Sun and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
    [CrossRef] [PubMed]
  6. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
    [CrossRef] [PubMed]
  7. .P. Lorrain and D. R. Corson, Electromagnetic fields and waves, (W. H. Freeman, 1970) Chap. 13.
  8. R. Yorke, Electric circuit theory (Pergamon Press, 1986) Chap. 8.
  9. S. A. Ramakrishna, "Physics of negative refractive index materials," Rep. Prog. Phys. 68, 449-521 (2005).
    [CrossRef]
  10. R. Biswas, Z. Y. Li and K. M. Ho, "Impedance of photonic crystals and photonic crystal waveguides," Appl. Phys. Lett. 84, 1254-1256 (2004).
    [CrossRef]
  11. D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
    [CrossRef]
  12. X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004).
    [CrossRef]
  13. U. Leonhardt, "Optical Conformal Mapping," Science 312, 1777 (2006).
    [CrossRef] [PubMed]
  14. J. B. Pendry, D. Schuring and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
    [CrossRef] [PubMed]
  15. R. J. Potton, "Reciprocity in optics," Rep. Prog. Phys. 67, 717-754 (2004).
    [CrossRef]
  16. J. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114,185-200 (1994).
    [CrossRef]
  17. P. B. Johnson, "Optical constants of noble metals," Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]
  18. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander and C. A. Ward, "Optical-properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared," Appl. Opt. 22, 1099-1119 (1983).
    [CrossRef] [PubMed]
  19. I. H. Malitson, "Interspecimen comparison of the refractive index of fused silica," J. Opt. Soc. Am. 55, 1205-1209 (1965).
    [CrossRef]
  20. M. Mazilu, V. Donchev and A. Miller, "A modular method for the calculation of transmission and reflection in multilayered structures," Appl. Opt. 40, 6670-6676 (2001).
    [CrossRef]
  21. R. Biswas, Z. Y. Li, and K. M. Ho, "Impedance of photonic crystals and photonic crystal waveguides," Appl. Phys. Lett. 84, 1254-1256 (2004).
    [CrossRef]

2006 (2)

U. Leonhardt, "Optical Conformal Mapping," Science 312, 1777 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schuring and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

2005 (3)

N. Fang, H. Lee, C. Sun and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

S. A. Ramakrishna, "Physics of negative refractive index materials," Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

2004 (4)

R. Biswas, Z. Y. Li and K. M. Ho, "Impedance of photonic crystals and photonic crystal waveguides," Appl. Phys. Lett. 84, 1254-1256 (2004).
[CrossRef]

X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004).
[CrossRef]

R. J. Potton, "Reciprocity in optics," Rep. Prog. Phys. 67, 717-754 (2004).
[CrossRef]

R. Biswas, Z. Y. Li, and K. M. Ho, "Impedance of photonic crystals and photonic crystal waveguides," Appl. Phys. Lett. 84, 1254-1256 (2004).
[CrossRef]

2002 (1)

D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
[CrossRef]

2001 (1)

2000 (2)

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

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

1998 (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]

1994 (1)

J. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114,185-200 (1994).
[CrossRef]

1983 (1)

1972 (1)

P. B. Johnson, "Optical constants of noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

1968 (1)

V. G. Veselago, "Electrodynamics of substances with simultaneously negative values of sigma and mu," Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

1965 (1)

Alexander, R. W.

Bell, R. J.

Bell, R. R.

Bell, S. E.

Berenger, J.

J. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114,185-200 (1994).
[CrossRef]

Biswas, R.

R. Biswas, Z. Y. Li, and K. M. Ho, "Impedance of photonic crystals and photonic crystal waveguides," Appl. Phys. Lett. 84, 1254-1256 (2004).
[CrossRef]

R. Biswas, Z. Y. Li and K. M. Ho, "Impedance of photonic crystals and photonic crystal waveguides," Appl. Phys. Lett. 84, 1254-1256 (2004).
[CrossRef]

Chen, X.

X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Donchev, V.

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]

Fang, N.

N. Fang, H. Lee, C. Sun and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Firsov, A. A.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Geim, A. K.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

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]

Gleeson, H. F.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Grigorenko, A. N.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Grzegorczyk, T. M.

X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Ho, K. M.

R. Biswas, Z. Y. Li and K. M. Ho, "Impedance of photonic crystals and photonic crystal waveguides," Appl. Phys. Lett. 84, 1254-1256 (2004).
[CrossRef]

R. Biswas, Z. Y. Li, and K. M. Ho, "Impedance of photonic crystals and photonic crystal waveguides," Appl. Phys. Lett. 84, 1254-1256 (2004).
[CrossRef]

Johnson, P. B.

P. B. Johnson, "Optical constants of noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Khrushchev, I. Y.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Kong, J. A.

X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Lee, H.

N. Fang, H. Lee, C. Sun and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt, "Optical Conformal Mapping," Science 312, 1777 (2006).
[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]

Li, Z. Y.

R. Biswas, Z. Y. Li and K. M. Ho, "Impedance of photonic crystals and photonic crystal waveguides," Appl. Phys. Lett. 84, 1254-1256 (2004).
[CrossRef]

R. Biswas, Z. Y. Li, and K. M. Ho, "Impedance of photonic crystals and photonic crystal waveguides," Appl. Phys. Lett. 84, 1254-1256 (2004).
[CrossRef]

Long, L. L.

Malitson, I. H.

Markos, P.

D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Mazilu, M.

Miller, A.

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]

Ordal, M. A.

Pacheco, J.

X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004).
[CrossRef]

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]

Pendry, J. B.

J. B. Pendry, D. Schuring and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

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

Petrovic, J.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Potton, R. J.

R. J. Potton, "Reciprocity in optics," Rep. Prog. Phys. 67, 717-754 (2004).
[CrossRef]

Ramakrishna, S. A.

S. A. Ramakrishna, "Physics of negative refractive index materials," Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

Schultz, S.

D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
[CrossRef]

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]

Schuring, D.

J. B. Pendry, D. Schuring and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

Smith, D. R.

J. B. Pendry, D. Schuring and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
[CrossRef]

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.

D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Sun, C.

N. Fang, H. Lee, C. Sun and 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]

Veselago, V. G.

V. G. Veselago, "Electrodynamics of substances with simultaneously negative values of sigma and mu," Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Vier, D. C.

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

Ward, C. A.

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]

Wu, B.

X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Zhang, X.

N. Fang, H. Lee, C. Sun and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Zhang, Y.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

R. Biswas, Z. Y. Li, and K. M. Ho, "Impedance of photonic crystals and photonic crystal waveguides," Appl. Phys. Lett. 84, 1254-1256 (2004).
[CrossRef]

R. Biswas, Z. Y. Li and K. M. Ho, "Impedance of photonic crystals and photonic crystal waveguides," Appl. Phys. Lett. 84, 1254-1256 (2004).
[CrossRef]

J. Comput. Phys. (1)

J. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114,185-200 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

Nature (2)

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]

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Phys. Rev. B (2)

P. B. Johnson, "Optical constants of noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Phys. Rev. E (1)

X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Phys. Rev. Lett. (2)

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]

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

Rep. Prog. Phys. (2)

S. A. Ramakrishna, "Physics of negative refractive index materials," Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

R. J. Potton, "Reciprocity in optics," Rep. Prog. Phys. 67, 717-754 (2004).
[CrossRef]

Science (3)

N. Fang, H. Lee, C. Sun and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

U. Leonhardt, "Optical Conformal Mapping," Science 312, 1777 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schuring and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780 (2006).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, "Electrodynamics of substances with simultaneously negative values of sigma and mu," Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Other (2)

.P. Lorrain and D. R. Corson, Electromagnetic fields and waves, (W. H. Freeman, 1970) Chap. 13.

R. Yorke, Electric circuit theory (Pergamon Press, 1986) Chap. 8.

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 (5)

Fig. 1.
Fig. 1.

Matching the iterative reflectivity (a) and the image reflectivity (b).

Fig. 2.
Fig. 2.

(a). Real (red) and imaginary part (blue) of the iterative impedance for the black gold structure. (b) Intensity reflectance (red) and transmittance (blue) as a function of wavelength for 100 periods of the black gold structure.

Fig. 3.
Fig. 3.

Iterative reflectivity as a function of wavelength and angle for the black gold structure. For both polarizations, the black region corresponds to a generalized Brewster angle matching where the black gold structure does not reflect (|riter|2<0.01) and all light enters the structures to be absorbed giving reflection invisibility.

Fig. 4.
Fig. 4.

(a). Amplitude reflectance of the coupled gold nano-pillars structure on top of the substrate (|r| and |r|) compared to the amplitude reflectance |r0| of the substrate alone. |r| correspond to the electric field being parallel to the displacement between the nano-pillars and |r| to it being orthogonal. (b) Same as on the left but in the opposite direction.

Fig. 5.
Fig. 5.

Real (left) and imaginary (right) parts of the two image impedances (Zimage;1 and Zimage;2) of the coupled gold nano-pillars.

Tables (2)

Tables Icon

Table 1. Comparison between the electrical impedance and optical characteristic reflectivity

Tables Icon

Table 2. Characteristic and wave impedances matching at a wavelength of 500nm. The structure is defined by three thicknesses corresponding to three successive layers of gold, glass and gold in nm.

Equations (47)

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

S t = S 12 = S 21
S r = S 11 = S 22
r in = r out = r w = S r + S t r w S t 1 r w S r
r w = ( 1 + S r 2 S t 2 q ( 1 + S r 2 S t 2 ) 2 4 S r 2 )
Z w = 1 r w 1 + r w = ( 1 S r ) 2 S t 2 ( 1 + S r ) 2 S t 2
r w = k x 0 k x ; eff ε eff k x 0 + k x ; eff ε eff
p w = exp ( i k x ; eff h )
k x ; eff i Log ( p w ) h ( mod 2 π h )
ε eff = k x ; eff k x 0 ( 1 + r w ) ( 1 r w ) μ eff = 1 ε eff ( k k ; eff 2 + k y 2 k x 0 0 + k y 2 )
n eff = Sign ( Re ( k x ; eff ) ) ε eff μ eff Z c = n eff ε eff
r iter ; 1 = S 11 + S 12 r iter ; 1 S 21 1 r iter ; 1 S 22
r iter ; 2 = S 22 + S 12 r iter ; 2 S 12 1 r iter ; 2 S 11
r iter ; 1 = 1 2 S 22 ( 1 + S 11 S 22 S 12 S 21 q ( 1 + S 11 S 22 S 12 S 21 ) 2 4 S 11 S 22 )
r iter ; 2 = S 22 S 11 r iter ; 1
Z iter ; 1 = 1 r iter ; 1 1 + r iter ; 1
Z iter ; 2 = 1 r iter ; 2 1 + r iter ; 2
r image ; 1 = S 11 + S 12 r image ; 2 S 21 1 r image ; 2 S 22
r image ; 2 = S 22 + S 12 r image ; 1 S 21 1 r image ; 1 S 11
Z image ; 1 = ( 1 S 11 + S 22 + S 12 S 21 S 11 S 22 ) ( 1 + S 11 + S 22 + S 12 S 21 + S 11 S 22 ) ( 1 + S 11 + S 22 S 12 S 21 + S 11 S 22 ) ( 1 S 11 + S 22 S 12 S 21 + S 11 S 22 )
Z image ; 2 = ( 1 + S 11 S 22 + S 12 S 21 S 11 S 22 ) ( 1 + S 11 + S 22 + S 12 S 21 S 11 S 22 ) ( 1 + S 11 + S 22 S 12 S 21 + S 11 S 22 ) ( 1 + S 11 S 22 S 12 S 21 + S 11 S 22 )
b 2 c = b 1 c
( b 1 c p 1 c b ̅ 1 c ) ( b 2 c p 2 c b ̅ 2 c ) = b 1 c p 1 + 2 c b ̅ 2 c
b ̅ 1 c b 2 c = I
p 1 c p 2 c = p 1 + 2 c
( b iter 1 p iter 1 b ̅ iter 1 ) ( b iter 1 p iter 1 b ̅ iter 1 ) = b iter 1 p iter 1 + 1 b ̅ iter 1
( b 1 image ; 1 p 1 image b ̅ 1 image ; 2 ) ( b 1 image ; 2 p ̂ 1 image b ̅ 1 image ; 1 ) = b 1 image ; 1 ( p 1 image p ̂ 1 image ) b ̅ 1 image ; 1
S r + S t 2 r 0 1 S r r 0 = r 0
S t = ± ( 1 S r r 0 ) ( r 0 S r ) r 0
( S 11 S 12 S 21 S 22 ) = ( S ' 11 S ' 12 S ' 21 S ' 22 ) ( S " 11 S " 12 S " 21 S " 22 )
S 11 = S ' 11 + S ' 11 S " 11 S ' 21 1 S " 11 S ' 22 ; S 12 = S ' 12 S " 12 1 S " 11 S ' 22 S 21 = S ' 21 S " 21 1 S " 11 S ' 22 ; S 22 = S " 22 + S " 21 S ' 22 S " 22 1 S " 11 S ' 22
I = ( 0 1 1 0 )
s ̅ = ( S ̅ 11 S ̅ 12 S ̅ 21 S ̅ 22 ) = ( S 11 S 11 S 22 S 12 S 21 S 21 S 11 S 22 S 12 S 21 S 12 S 11 S 22 S 12 S 21 S 22 S 11 S 22 S 12 S 21 )
s ̂ = ( S 22 S 21 S 12 S 11 ) .
s = b w p w b ̅ w
p w = b ̅ w s b w .
b w = ( r w 1 + r w 1 r w r w )
p w = ( 0 p w p w 0 )
p w = 1 2 S t ( 1 S r 2 + S t 2 q ( 1 + S r 2 S t 2 ) 2 4 S r 2 ) .
s = b iter p iter b ̅ iter
b iter = ( r iter ; 1 1 + r iter ; 1 r iter ; 2 1 r iter ; 1 r iter ; 2 r iter ; 2 )
p iter = ( 0 p iter ; 12 p iter ; 21 0 )
p iter ; 12 = 1 2 S 21 ( 1 S 11 S 22 + S 12 S 21 q ( 1 + S 11 S 22 S 12 S 21 ) 2 4 S 11 S 22 )
p iter ; 21 = S 21 S 12 p iter ; 12
s = b image ; 1 p image b ̅ image ; 2
b image ; 1 = ( r image ; 1 1 + r image ; 1 1 r image ; 1 r image ; 1 )
b ̅ image ; 2 = ( r image ; 2 1 r image ; 2 1 + r image ; 2 r image ; 2 )
p image ; 1 = b ̅ image ; 1 s b image ; 2

Metrics