Abstract

We present a simplified analytic formula that may be used to design gratings intended to couple long-wave infrared radiation to surface plasmons. It is based on the theory of Hessel and Oliner (1965). The recipe is semiempirical, in that it requires knowledge of a surface-impedance modulation amplitude, which is found here as a function of the grating groove depth and the wavelength for silver lamellar gratings at CO2 laser wavelengths. The optimum groove depth for photon-to-surface-plasmon energy conversion was found by experiment and calculation to be 10%15% of the wavelength. This value is about twice what has been reported previously in the visible spectral range for sinusoidal grating profiles.

© 2010 Optical Society of America

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References

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  1. T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61, 44–50 (2008).
    [CrossRef]
  2. J. W. Cleary, R. E. Peale, D. Shelton, G. Boreman, and W. R. Buchwald, “Silicides for infrared surface plasmon resonance biosensors,” Proc. Mater. Res. Soc. 1133, 1133-AA10-03 (2008).
    [CrossRef]
  3. R. Soref, R. E. Peale, and W. R. Buchwald, “Longwave plasmonics on doped silicon and silicides,” Opt. Express 16, 6507–6514 (2008).
    [CrossRef]
  4. D. Heitmann, “Radiative decay of surface plasmons excited by fast electrons on periodically modulated silver surfaces,” J. Phys. C: Solid State Phys. 10, 397–405 (1977).
    [CrossRef]
  5. I. Pockrand, “Resonance anomalies in the light intensity reflected at silver gratings with dielectric coatings,” J. Phys. D: Appl. Phys. 9, 2423–2432 (1976).
    [CrossRef]
  6. M. C. Hutley and V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” J. Mod. Opt. 20, 771–782 (1973).
    [CrossRef]
  7. R. W. Day, S. S. Wang, and R. Magnusson, “Filter-response line shapes of resonant waveguide gratings,” J. Lightwave Technol. 14, 1815–1824 (1996).
    [CrossRef]
  8. F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B. 66, 155412 (2002).
    [CrossRef]
  9. A. V. Kats and A. Y. Nikitan, “Nonzeroth-order anomalous optical transparency in modulated metal films owing to excitation of surface plasmon polaritons: an analytic approach,” JETP Lett. 79, 625–631 (2004).
    [CrossRef]
  10. D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
    [CrossRef]
  11. D. Shin, Z. S. Liu, and R. Magnusson, “Resonant Brewster filters with absentee layers,” Opt. Lett. 27, 1288–1290 (2002).
    [CrossRef]
  12. A. Hessel and A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275–1297 (1965).
    [CrossRef]
  13. M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
    [CrossRef]
  14. R. C. McPhedran and D. Maystre, “Detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” J. Mod. Opt. 21, 413–421 (1974).
    [CrossRef]
  15. D. Maystre and M. Neviere, “Quantitative theoretical study on the plasmon anomalies of diffraction gratings,” J. Opt. 8, 165–174 (1977).
    [CrossRef]
  16. D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface Modes, A.D.Boardman, ed. (Wiley, 1982), Chap. 17.
  17. J. W. Cleary, R. E. Peale, D. J. Shelton, G. D. Boreman, C. W. Smith, M. Ishigami, R. Soref, A. Drehman, and W. R. Buchwald, “IR permittivities for silicides and doped silicon,” J. Opt. Soc. Am. B 27, 730–734 (2010).
    [CrossRef]
  18. Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
  19. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
    [CrossRef]
  20. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1986).
  21. S. Yang, “Effect of surface texture and geometry on spoof surface plasmon dispersion,” Opt. Eng. 47, 029001 (2008).
    [CrossRef]
  22. M. Neviere, P. Vincent, and R. Petit, “Some studies on behavior of surface impedance in vicinity of gratings,” Opt. Commun. 21, 369–373 (1977).
    [CrossRef]
  23. A. Wirgin and A. A. Maradudin, “Resonant response of a bare metallic grating to S-polarized light,” Prog. Surf. Sci. 22, 1–99(1986).
    [CrossRef]
  24. R. A. Depine and V. L. Brudny, “A simple-model for a microrough diffraction grating that predicts diffuse light bands,” J. Mod. Opt. 36, 1257–1271 (1989).
    [CrossRef]

2010 (1)

2008 (4)

R. Soref, R. E. Peale, and W. R. Buchwald, “Longwave plasmonics on doped silicon and silicides,” Opt. Express 16, 6507–6514 (2008).
[CrossRef]

S. Yang, “Effect of surface texture and geometry on spoof surface plasmon dispersion,” Opt. Eng. 47, 029001 (2008).
[CrossRef]

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61, 44–50 (2008).
[CrossRef]

J. W. Cleary, R. E. Peale, D. Shelton, G. Boreman, and W. R. Buchwald, “Silicides for infrared surface plasmon resonance biosensors,” Proc. Mater. Res. Soc. 1133, 1133-AA10-03 (2008).
[CrossRef]

2004 (1)

A. V. Kats and A. Y. Nikitan, “Nonzeroth-order anomalous optical transparency in modulated metal films owing to excitation of surface plasmon polaritons: an analytic approach,” JETP Lett. 79, 625–631 (2004).
[CrossRef]

2002 (2)

F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B. 66, 155412 (2002).
[CrossRef]

D. Shin, Z. S. Liu, and R. Magnusson, “Resonant Brewster filters with absentee layers,” Opt. Lett. 27, 1288–1290 (2002).
[CrossRef]

1997 (1)

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

1996 (1)

R. W. Day, S. S. Wang, and R. Magnusson, “Filter-response line shapes of resonant waveguide gratings,” J. Lightwave Technol. 14, 1815–1824 (1996).
[CrossRef]

1989 (1)

R. A. Depine and V. L. Brudny, “A simple-model for a microrough diffraction grating that predicts diffuse light bands,” J. Mod. Opt. 36, 1257–1271 (1989).
[CrossRef]

1986 (1)

A. Wirgin and A. A. Maradudin, “Resonant response of a bare metallic grating to S-polarized light,” Prog. Surf. Sci. 22, 1–99(1986).
[CrossRef]

1977 (3)

M. Neviere, P. Vincent, and R. Petit, “Some studies on behavior of surface impedance in vicinity of gratings,” Opt. Commun. 21, 369–373 (1977).
[CrossRef]

D. Maystre and M. Neviere, “Quantitative theoretical study on the plasmon anomalies of diffraction gratings,” J. Opt. 8, 165–174 (1977).
[CrossRef]

D. Heitmann, “Radiative decay of surface plasmons excited by fast electrons on periodically modulated silver surfaces,” J. Phys. C: Solid State Phys. 10, 397–405 (1977).
[CrossRef]

1976 (2)

I. Pockrand, “Resonance anomalies in the light intensity reflected at silver gratings with dielectric coatings,” J. Phys. D: Appl. Phys. 9, 2423–2432 (1976).
[CrossRef]

M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

1974 (1)

R. C. McPhedran and D. Maystre, “Detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” J. Mod. Opt. 21, 413–421 (1974).
[CrossRef]

1973 (1)

M. C. Hutley and V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” J. Mod. Opt. 20, 771–782 (1973).
[CrossRef]

1965 (1)

1907 (1)

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Bird, V. M.

M. C. Hutley and V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” J. Mod. Opt. 20, 771–782 (1973).
[CrossRef]

Boreman, G.

J. W. Cleary, R. E. Peale, D. Shelton, G. Boreman, and W. R. Buchwald, “Silicides for infrared surface plasmon resonance biosensors,” Proc. Mater. Res. Soc. 1133, 1133-AA10-03 (2008).
[CrossRef]

Boreman, G. D.

Bozhevolnyi, S. I.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61, 44–50 (2008).
[CrossRef]

Brudny, V. L.

R. A. Depine and V. L. Brudny, “A simple-model for a microrough diffraction grating that predicts diffuse light bands,” J. Mod. Opt. 36, 1257–1271 (1989).
[CrossRef]

Buchwald, W. R.

Cleary, J. W.

J. W. Cleary, R. E. Peale, D. J. Shelton, G. D. Boreman, C. W. Smith, M. Ishigami, R. Soref, A. Drehman, and W. R. Buchwald, “IR permittivities for silicides and doped silicon,” J. Opt. Soc. Am. B 27, 730–734 (2010).
[CrossRef]

J. W. Cleary, R. E. Peale, D. Shelton, G. Boreman, and W. R. Buchwald, “Silicides for infrared surface plasmon resonance biosensors,” Proc. Mater. Res. Soc. 1133, 1133-AA10-03 (2008).
[CrossRef]

Day, R. W.

R. W. Day, S. S. Wang, and R. Magnusson, “Filter-response line shapes of resonant waveguide gratings,” J. Lightwave Technol. 14, 1815–1824 (1996).
[CrossRef]

Depine, R. A.

R. A. Depine and V. L. Brudny, “A simple-model for a microrough diffraction grating that predicts diffuse light bands,” J. Mod. Opt. 36, 1257–1271 (1989).
[CrossRef]

Drehman, A.

Ebbesen, T. W.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61, 44–50 (2008).
[CrossRef]

Friesem, A. A.

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

Garcia-Vidal, F. J.

F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B. 66, 155412 (2002).
[CrossRef]

Genet, C.

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61, 44–50 (2008).
[CrossRef]

Heitmann, D.

D. Heitmann, “Radiative decay of surface plasmons excited by fast electrons on periodically modulated silver surfaces,” J. Phys. C: Solid State Phys. 10, 397–405 (1977).
[CrossRef]

Hessel, A.

Hutley, M. C.

M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

M. C. Hutley and V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” J. Mod. Opt. 20, 771–782 (1973).
[CrossRef]

Ishigami, M.

Kats, A. V.

A. V. Kats and A. Y. Nikitan, “Nonzeroth-order anomalous optical transparency in modulated metal films owing to excitation of surface plasmon polaritons: an analytic approach,” JETP Lett. 79, 625–631 (2004).
[CrossRef]

Liu, Z. S.

Magnusson, R.

D. Shin, Z. S. Liu, and R. Magnusson, “Resonant Brewster filters with absentee layers,” Opt. Lett. 27, 1288–1290 (2002).
[CrossRef]

R. W. Day, S. S. Wang, and R. Magnusson, “Filter-response line shapes of resonant waveguide gratings,” J. Lightwave Technol. 14, 1815–1824 (1996).
[CrossRef]

Maradudin, A. A.

A. Wirgin and A. A. Maradudin, “Resonant response of a bare metallic grating to S-polarized light,” Prog. Surf. Sci. 22, 1–99(1986).
[CrossRef]

Martin-Moreno, L.

F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B. 66, 155412 (2002).
[CrossRef]

Maystre, D.

D. Maystre and M. Neviere, “Quantitative theoretical study on the plasmon anomalies of diffraction gratings,” J. Opt. 8, 165–174 (1977).
[CrossRef]

M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

R. C. McPhedran and D. Maystre, “Detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” J. Mod. Opt. 21, 413–421 (1974).
[CrossRef]

D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface Modes, A.D.Boardman, ed. (Wiley, 1982), Chap. 17.

McPhedran, R. C.

R. C. McPhedran and D. Maystre, “Detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” J. Mod. Opt. 21, 413–421 (1974).
[CrossRef]

Neviere, M.

D. Maystre and M. Neviere, “Quantitative theoretical study on the plasmon anomalies of diffraction gratings,” J. Opt. 8, 165–174 (1977).
[CrossRef]

M. Neviere, P. Vincent, and R. Petit, “Some studies on behavior of surface impedance in vicinity of gratings,” Opt. Commun. 21, 369–373 (1977).
[CrossRef]

Nikitan, A. Y.

A. V. Kats and A. Y. Nikitan, “Nonzeroth-order anomalous optical transparency in modulated metal films owing to excitation of surface plasmon polaritons: an analytic approach,” JETP Lett. 79, 625–631 (2004).
[CrossRef]

Oliner, A. A.

Peale, R. E.

Petit, R.

M. Neviere, P. Vincent, and R. Petit, “Some studies on behavior of surface impedance in vicinity of gratings,” Opt. Commun. 21, 369–373 (1977).
[CrossRef]

Pockrand, I.

I. Pockrand, “Resonance anomalies in the light intensity reflected at silver gratings with dielectric coatings,” J. Phys. D: Appl. Phys. 9, 2423–2432 (1976).
[CrossRef]

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1986).

Rayleigh, Lord

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).

Rosenblatt, D.

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

Sharon, A.

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

Shelton, D.

J. W. Cleary, R. E. Peale, D. Shelton, G. Boreman, and W. R. Buchwald, “Silicides for infrared surface plasmon resonance biosensors,” Proc. Mater. Res. Soc. 1133, 1133-AA10-03 (2008).
[CrossRef]

Shelton, D. J.

Shin, D.

Smith, C. W.

Soref, R.

Vincent, P.

M. Neviere, P. Vincent, and R. Petit, “Some studies on behavior of surface impedance in vicinity of gratings,” Opt. Commun. 21, 369–373 (1977).
[CrossRef]

Wang, S. S.

R. W. Day, S. S. Wang, and R. Magnusson, “Filter-response line shapes of resonant waveguide gratings,” J. Lightwave Technol. 14, 1815–1824 (1996).
[CrossRef]

Wirgin, A.

A. Wirgin and A. A. Maradudin, “Resonant response of a bare metallic grating to S-polarized light,” Prog. Surf. Sci. 22, 1–99(1986).
[CrossRef]

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Yang, S.

S. Yang, “Effect of surface texture and geometry on spoof surface plasmon dispersion,” Opt. Eng. 47, 029001 (2008).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

J. Lightwave Technol. (1)

R. W. Day, S. S. Wang, and R. Magnusson, “Filter-response line shapes of resonant waveguide gratings,” J. Lightwave Technol. 14, 1815–1824 (1996).
[CrossRef]

J. Mod. Opt. (3)

M. C. Hutley and V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” J. Mod. Opt. 20, 771–782 (1973).
[CrossRef]

R. C. McPhedran and D. Maystre, “Detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” J. Mod. Opt. 21, 413–421 (1974).
[CrossRef]

R. A. Depine and V. L. Brudny, “A simple-model for a microrough diffraction grating that predicts diffuse light bands,” J. Mod. Opt. 36, 1257–1271 (1989).
[CrossRef]

J. Opt. (1)

D. Maystre and M. Neviere, “Quantitative theoretical study on the plasmon anomalies of diffraction gratings,” J. Opt. 8, 165–174 (1977).
[CrossRef]

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

J. Phys. C: Solid State Phys. (1)

D. Heitmann, “Radiative decay of surface plasmons excited by fast electrons on periodically modulated silver surfaces,” J. Phys. C: Solid State Phys. 10, 397–405 (1977).
[CrossRef]

J. Phys. D: Appl. Phys. (1)

I. Pockrand, “Resonance anomalies in the light intensity reflected at silver gratings with dielectric coatings,” J. Phys. D: Appl. Phys. 9, 2423–2432 (1976).
[CrossRef]

JETP Lett. (1)

A. V. Kats and A. Y. Nikitan, “Nonzeroth-order anomalous optical transparency in modulated metal films owing to excitation of surface plasmon polaritons: an analytic approach,” JETP Lett. 79, 625–631 (2004).
[CrossRef]

Opt. Commun. (2)

M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

M. Neviere, P. Vincent, and R. Petit, “Some studies on behavior of surface impedance in vicinity of gratings,” Opt. Commun. 21, 369–373 (1977).
[CrossRef]

Opt. Eng. (1)

S. Yang, “Effect of surface texture and geometry on spoof surface plasmon dispersion,” Opt. Eng. 47, 029001 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Philos. Mag. (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Phys. Rev. B. (1)

F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B. 66, 155412 (2002).
[CrossRef]

Phys. Today (1)

T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61, 44–50 (2008).
[CrossRef]

Proc. Mater. Res. Soc. (1)

J. W. Cleary, R. E. Peale, D. Shelton, G. Boreman, and W. R. Buchwald, “Silicides for infrared surface plasmon resonance biosensors,” Proc. Mater. Res. Soc. 1133, 1133-AA10-03 (2008).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).

Prog. Surf. Sci. (1)

A. Wirgin and A. A. Maradudin, “Resonant response of a bare metallic grating to S-polarized light,” Prog. Surf. Sci. 22, 1–99(1986).
[CrossRef]

Other (2)

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1986).

D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface Modes, A.D.Boardman, ed. (Wiley, 1982), Chap. 17.

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Figures (9)

Fig. 1
Fig. 1

Schematic of experiment: p-polarized CO 2 laser radiation is incident on the right grating, and specular reflection is monitored as a function of angle of incidence using power meter P. A second grating 1 cm to the left outcouples the SPP that has traveled to it, and this event is imaged with an infrared camera C. The screen S prevents any rays from the first grating from reaching the camera.

Fig. 2
Fig. 2

Graphical representation of the SPP resonance conditions of this experiment. Solid curves are silver SPP dispersion curves. Light curves (dashed) for different angles of incidence and shifted by different integer multiples of grating momentum are shown. The points of intersection establish the conditions for SPP generation at the wavelengths indicated. Angles of incidence for a, b, a , and b are 32.5 ° , 22.8 ° , 28.1 ° , and 36.1 ° , respectively.

Fig. 3
Fig. 3

Fourier transform of a measured profile (inset) for as- deposited (thin curve) and annealed (thick curve) Ag gratings with amplitude of 1 μm .

Fig. 4
Fig. 4

Angular reflectance spectrum for as-deposited (thin curve) and annealed (thick curve) Ag grating with amplitude of 1 μm at two different CO 2 wavelengths.

Fig. 5
Fig. 5

Measured angular reflectance for two different p- polarized CO 2 laser wavelengths and Ag gratings with different amplitudes.

Fig. 6
Fig. 6

Calculated angular reflectance for Ag gratings of different amplitudes.

Fig. 7
Fig. 7

Surface-impedance modulation parameter M (symbols) determined from fit of theory to observed resonances for grating amplitudes up to 1 μm . Solid curves are quadratic fits. (inset) Measured (thick curve) angular reflectance spectra for Ag grating of 0.5 μm amplitude at 10.591 μm wavelength compared with a best fit calculated spectrum (thin curve).

Fig. 8
Fig. 8

Coefficients α and β plotted as a function of wavelength (symbols). Linear fitting of these values of the form of Eq. (21) are plotted (solid lines).

Fig. 9
Fig. 9

Contour plot of M calculated for Ag gratings as a function of wavelength and grating amplitude.

Tables (2)

Tables Icon

Table 1 Optical Parameters of Ag

Tables Icon

Table 2 Fitting Parameters for M Versus h, according to Eq. (20) a

Equations (22)

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

Z ( x ) = Z 0 ( 1 + M cos [ 2 π x d ] ) ,
Z 0 = μ 0 ( ε + ε ) ε 0 = 377 Ω ε + ε ,
ζ Z 0 377 Ω i | ε | .
ζ = i ( | ε | 2 + ε 2 ) 1 / 4 { cos ( ϕ 2 ) + i sin ( ϕ 2 ) } ,
H s ( x , z ) = n = I n exp [ i ( k s + 2 π n d ) x + i κ n z ] ,
κ n = k 2 ( k s + 2 π n d ) 2 .
R = | I 0 H | 2 = | 2 D 0 4 / M D 0 + A 1 + B 1 1 | 2 ,
D n = 2 M [ 1 + κ n ζ k ] ,
A 1 = { D 1 [ D 2 ( D 3 ... ) 1 ] 1 } 1 ,
B 1 = { D 1 [ D 2 ( D 3 ... ) 1 ] 1 } 1 .
R | 2 D 0 4 / M D 0 1 D 1 1 D 1 1 | 2 ,
Re [ D 1 ] = 2 M [ 1 + ζ κ 1 | ζ | 2 k ] ,
Im [ D 1 ] = 2 | ζ | κ 1 M | ζ | 2 k .
Im [ D 1 ] = 2 ζ | κ 1 | M | ζ | 2 k ,
Re [ D 1 ] = 2 M [ 1 | ζ | | κ 1 | | ζ | 2 k ] .
sin ( θ ) + λ d = 1 + | ζ | 4 | ζ | 2 .
R = ( D 0 + A 1 + B 1 ) 2 + ( 4 cos ( θ ) M | ζ | D 0 A 1 B 1 ) 2 ( D 0 + A 1 + B 1 ) 2 + ( D 0 + A 1 + B 1 ) 2 .
sin ( θ ) + λ d = | ε | 1 | ε | .
sin ( θ ) + n λ d = ± Re [ ε ε + 1 ] = ± c ω Re [ k spp ] ,
M ( h , λ ) = α ( λ ) h + β ( λ ) h 2 .
α ( λ ) = a 0 + a λ 1 β ( λ ) = b 0 + b 1 λ ,
M ( h , λ ) = ( a 0 + a λ 1 ) h + ( b 0 + b 1 λ ) h 2 .

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