Abstract

We show that a broadband surface plasmon can be excited in a thin metal film. A train of two plasmons can be excited at conditions near the condition of broadband surface plasmon excitation. Also, a method for independent multichannel checks of biochips by wavelength addressing is proposed.

© 2010 Optical Society of America

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References

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  1. S. Maier, Plasmonics: Fundamental and Applications(Springer, 2007).
  2. J. Homola, “Surface plasmon resonance sensor for detection of chemical and biological species,” Chem. Rev. 108, 462–493(2008).
    [CrossRef] [PubMed]
  3. S. Maier and H. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
    [CrossRef]
  4. S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nature Photon. 3, 388–394(2009).
    [CrossRef]
  5. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Grating, Vol. 111 of Springer Tracts in Modern Physics (Springer–Verlag, 1988).
  6. A. Vial, A. S. Grimault, D. Macias, D. Barchiesi, and M. L. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416(2005).
    [CrossRef]
  7. Y. Gong, L. Wang, X. Hu, X. Li, and X. Liu, “Broad-bandgap and low-sidelobe surface plasmon polariton reflector with Bragg-grating-based MIN waveguide,” Opt. Express 17, 13727–13736 (2009).
    [CrossRef] [PubMed]
  8. Y. Jourlin, S. Tonchev, A. Tishchenko, C. Pedri, C. Veillas, O. Parriaux, A. Last, and Y. Lacroute, “Spatially and polarization resolved plasmon mediated transmission through continous metal films,” Opt. Express 17, 12155–12166 (2009).
    [CrossRef] [PubMed]
  9. E. D. Palik, Handbook of Optical Constants in Solids(Academic, 1985).
  10. U. Kreibig, “Electronic properties of small silver particles: the optical constants and their temperature dependence,” J. Phys. F 4, 999–1014 (1974).
    [CrossRef]
  11. K. R. Roy, S. K. Mandal, and A. K. Pal, “Effect of interfacial alloying on the surface plasmon resonance of nanocrystalline Au-Ag multilayer thin films,” Eur. Phys. J. B 33, 109–114(2003).
    [CrossRef]
  12. T. W. H. Oates and A. Mucklich, “Evolution of plasmon resonances during plasma deposition of silver nanoparticles,” Nanotechnology 16, 2606–2611 (2005).
    [CrossRef]
  13. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).
  14. F. Hao and P. Nordlander, Chem. Phys. Lett. 446, 115 (2007).
    [CrossRef]
  15. P. G. Etchegoin, E. C. L. Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
    [CrossRef] [PubMed]
  16. R. Jha and A. K. Sharma, “Chalcogenide glass prism based SPR sensor with Ag-Au bimetallic nanoparticle alloy in infrared wavelength region,” J. Opt. A: Pure Appl. Opt. 11, 045502 (2009) [more exactly we refer to Eq. 5, Eq. 7, and Table 1].
    [CrossRef]
  17. A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37, 5271–5283 (1998).
    [CrossRef]
  18. P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  19. V. P. Drachev, U. K. Chettiar, A. V. Kildishev, H. K. Yuan, W. Cai, and V. M. Shalaev, “The Ag dielectric function in plasmonic metamaterials,” Opt. Express 16, 1186–1195 (2008).
    [CrossRef] [PubMed]
  20. http:/glassbank.com.
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    [CrossRef]
  22. Y. Binfeng, H. Guohua, and C. Yiping, “Bound modes analysis of symmetric dielectric loaded surface plasmon-polariton waveguides,” Opt. Express 17, 3610–3618 (2009).
    [CrossRef] [PubMed]
  23. Z. Chen, T. Holmgaard, S. Bozhevolnyi, A. Krasavin, A. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with directional-loaded plasmonic waveguides,” Opt. Lett. 34, 310–312 (2009).
    [CrossRef] [PubMed]
  24. G. Yuan, P. Wang, Y. Lu, and H. Ming, “Multimode interference splitter based on dielectric-loaded surface plasmon-polariton waveguides,” Opt. Express 17, 12594–12600 (2009).
    [CrossRef] [PubMed]
  25. J. Dostalek, H. Vaisocherova, and J. Homola, “Multichannel surface plasmon resonance biosensor with wavelength division multiplexing,” Sens. Actuators B 108, 758–764 (2004).
    [CrossRef]
  26. I. J. Jen and Y. H. Liao, “Surface plasmon resonance via polarization conversion in a weak anisotropic thin film,” Appl. Phys. Lett. 94, 011105 (2009).
    [CrossRef]
  27. I. Abdulhalim, “Surface plasmon TE and TM waves at the anisotropic film-metal interface,” J. Opt. A: Pure Appl. Opt. 11, 015002 (2009).
    [CrossRef]
  28. S. Otsuki, K. Tamada, and S. Wakida, “Wavelength scanning surface plasmon resonance imaging,” Appl. Opt. 44, 3468–3472(2005).
    [CrossRef] [PubMed]

2009 (9)

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nature Photon. 3, 388–394(2009).
[CrossRef]

Y. Gong, L. Wang, X. Hu, X. Li, and X. Liu, “Broad-bandgap and low-sidelobe surface plasmon polariton reflector with Bragg-grating-based MIN waveguide,” Opt. Express 17, 13727–13736 (2009).
[CrossRef] [PubMed]

Y. Jourlin, S. Tonchev, A. Tishchenko, C. Pedri, C. Veillas, O. Parriaux, A. Last, and Y. Lacroute, “Spatially and polarization resolved plasmon mediated transmission through continous metal films,” Opt. Express 17, 12155–12166 (2009).
[CrossRef] [PubMed]

R. Jha and A. K. Sharma, “Chalcogenide glass prism based SPR sensor with Ag-Au bimetallic nanoparticle alloy in infrared wavelength region,” J. Opt. A: Pure Appl. Opt. 11, 045502 (2009) [more exactly we refer to Eq. 5, Eq. 7, and Table 1].
[CrossRef]

Y. Binfeng, H. Guohua, and C. Yiping, “Bound modes analysis of symmetric dielectric loaded surface plasmon-polariton waveguides,” Opt. Express 17, 3610–3618 (2009).
[CrossRef] [PubMed]

Z. Chen, T. Holmgaard, S. Bozhevolnyi, A. Krasavin, A. Zayats, L. Markey, and A. Dereux, “Wavelength-selective directional coupling with directional-loaded plasmonic waveguides,” Opt. Lett. 34, 310–312 (2009).
[CrossRef] [PubMed]

G. Yuan, P. Wang, Y. Lu, and H. Ming, “Multimode interference splitter based on dielectric-loaded surface plasmon-polariton waveguides,” Opt. Express 17, 12594–12600 (2009).
[CrossRef] [PubMed]

I. J. Jen and Y. H. Liao, “Surface plasmon resonance via polarization conversion in a weak anisotropic thin film,” Appl. Phys. Lett. 94, 011105 (2009).
[CrossRef]

I. Abdulhalim, “Surface plasmon TE and TM waves at the anisotropic film-metal interface,” J. Opt. A: Pure Appl. Opt. 11, 015002 (2009).
[CrossRef]

2008 (2)

2007 (1)

F. Hao and P. Nordlander, Chem. Phys. Lett. 446, 115 (2007).
[CrossRef]

2006 (1)

P. G. Etchegoin, E. C. L. Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

2005 (4)

S. Maier and H. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

A. Vial, A. S. Grimault, D. Macias, D. Barchiesi, and M. L. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416(2005).
[CrossRef]

T. W. H. Oates and A. Mucklich, “Evolution of plasmon resonances during plasma deposition of silver nanoparticles,” Nanotechnology 16, 2606–2611 (2005).
[CrossRef]

S. Otsuki, K. Tamada, and S. Wakida, “Wavelength scanning surface plasmon resonance imaging,” Appl. Opt. 44, 3468–3472(2005).
[CrossRef] [PubMed]

2004 (1)

J. Dostalek, H. Vaisocherova, and J. Homola, “Multichannel surface plasmon resonance biosensor with wavelength division multiplexing,” Sens. Actuators B 108, 758–764 (2004).
[CrossRef]

2003 (1)

K. R. Roy, S. K. Mandal, and A. K. Pal, “Effect of interfacial alloying on the surface plasmon resonance of nanocrystalline Au-Ag multilayer thin films,” Eur. Phys. J. B 33, 109–114(2003).
[CrossRef]

2000 (1)

P. Berini, “Plasmon-polariton waves guided by thin lossy films of finite width: bound modes of symmetric structure,” Phys. Rev. B 61, 10484–10503 (2000).
[CrossRef]

1998 (1)

1974 (1)

U. Kreibig, “Electronic properties of small silver particles: the optical constants and their temperature dependence,” J. Phys. F 4, 999–1014 (1974).
[CrossRef]

1972 (1)

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Abdulhalim, I.

I. Abdulhalim, “Surface plasmon TE and TM waves at the anisotropic film-metal interface,” J. Opt. A: Pure Appl. Opt. 11, 015002 (2009).
[CrossRef]

Atwater, H.

S. Maier and H. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

Barchiesi, D.

A. Vial, A. S. Grimault, D. Macias, D. Barchiesi, and M. L. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416(2005).
[CrossRef]

Berini, P.

P. Berini, “Plasmon-polariton waves guided by thin lossy films of finite width: bound modes of symmetric structure,” Phys. Rev. B 61, 10484–10503 (2000).
[CrossRef]

Binfeng, Y.

Bozhevolnyi, S.

Cai, W.

Chapelle, M. L.

A. Vial, A. S. Grimault, D. Macias, D. Barchiesi, and M. L. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416(2005).
[CrossRef]

Chen, Z.

Chettiar, U. K.

Christy, R.

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Dereux, A.

Djurisic, A. B.

Dostalek, J.

J. Dostalek, H. Vaisocherova, and J. Homola, “Multichannel surface plasmon resonance biosensor with wavelength division multiplexing,” Sens. Actuators B 108, 758–764 (2004).
[CrossRef]

Drachev, V. P.

Elazar, J. M.

Etchegoin, P. G.

P. G. Etchegoin, E. C. L. Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Gong, Y.

Grimault, A. S.

A. Vial, A. S. Grimault, D. Macias, D. Barchiesi, and M. L. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416(2005).
[CrossRef]

Guohua, H.

Hao, F.

F. Hao and P. Nordlander, Chem. Phys. Lett. 446, 115 (2007).
[CrossRef]

Holmgaard, T.

Homola, J.

J. Homola, “Surface plasmon resonance sensor for detection of chemical and biological species,” Chem. Rev. 108, 462–493(2008).
[CrossRef] [PubMed]

J. Dostalek, H. Vaisocherova, and J. Homola, “Multichannel surface plasmon resonance biosensor with wavelength division multiplexing,” Sens. Actuators B 108, 758–764 (2004).
[CrossRef]

Hu, X.

Inouye, Y.

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nature Photon. 3, 388–394(2009).
[CrossRef]

Jen, I. J.

I. J. Jen and Y. H. Liao, “Surface plasmon resonance via polarization conversion in a weak anisotropic thin film,” Appl. Phys. Lett. 94, 011105 (2009).
[CrossRef]

Jha, R.

R. Jha and A. K. Sharma, “Chalcogenide glass prism based SPR sensor with Ag-Au bimetallic nanoparticle alloy in infrared wavelength region,” J. Opt. A: Pure Appl. Opt. 11, 045502 (2009) [more exactly we refer to Eq. 5, Eq. 7, and Table 1].
[CrossRef]

Johnson, P.

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Jourlin, Y.

Kawata, S.

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nature Photon. 3, 388–394(2009).
[CrossRef]

Kildishev, A. V.

Krasavin, A.

Kreibig, U.

U. Kreibig, “Electronic properties of small silver particles: the optical constants and their temperature dependence,” J. Phys. F 4, 999–1014 (1974).
[CrossRef]

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).

Lacroute, Y.

Last, A.

Li, X.

Liao, Y. H.

I. J. Jen and Y. H. Liao, “Surface plasmon resonance via polarization conversion in a weak anisotropic thin film,” Appl. Phys. Lett. 94, 011105 (2009).
[CrossRef]

Liu, X.

Lu, Y.

Macias, D.

A. Vial, A. S. Grimault, D. Macias, D. Barchiesi, and M. L. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416(2005).
[CrossRef]

Maier, S.

S. Maier and H. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

S. Maier, Plasmonics: Fundamental and Applications(Springer, 2007).

Majewski, M. L.

Mandal, S. K.

K. R. Roy, S. K. Mandal, and A. K. Pal, “Effect of interfacial alloying on the surface plasmon resonance of nanocrystalline Au-Ag multilayer thin films,” Eur. Phys. J. B 33, 109–114(2003).
[CrossRef]

Markey, L.

Meyer, M.

P. G. Etchegoin, E. C. L. Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Ming, H.

Mucklich, A.

T. W. H. Oates and A. Mucklich, “Evolution of plasmon resonances during plasma deposition of silver nanoparticles,” Nanotechnology 16, 2606–2611 (2005).
[CrossRef]

Nordlander, P.

F. Hao and P. Nordlander, Chem. Phys. Lett. 446, 115 (2007).
[CrossRef]

Oates, T. W. H.

T. W. H. Oates and A. Mucklich, “Evolution of plasmon resonances during plasma deposition of silver nanoparticles,” Nanotechnology 16, 2606–2611 (2005).
[CrossRef]

Otsuki, S.

Pal, A. K.

K. R. Roy, S. K. Mandal, and A. K. Pal, “Effect of interfacial alloying on the surface plasmon resonance of nanocrystalline Au-Ag multilayer thin films,” Eur. Phys. J. B 33, 109–114(2003).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants in Solids(Academic, 1985).

Parriaux, O.

Pedri, C.

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Grating, Vol. 111 of Springer Tracts in Modern Physics (Springer–Verlag, 1988).

Rakic, A. D.

Roy, K. R.

K. R. Roy, S. K. Mandal, and A. K. Pal, “Effect of interfacial alloying on the surface plasmon resonance of nanocrystalline Au-Ag multilayer thin films,” Eur. Phys. J. B 33, 109–114(2003).
[CrossRef]

Ru, E. C. L.

P. G. Etchegoin, E. C. L. Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Shalaev, V. M.

Sharma, A. K.

R. Jha and A. K. Sharma, “Chalcogenide glass prism based SPR sensor with Ag-Au bimetallic nanoparticle alloy in infrared wavelength region,” J. Opt. A: Pure Appl. Opt. 11, 045502 (2009) [more exactly we refer to Eq. 5, Eq. 7, and Table 1].
[CrossRef]

Tamada, K.

Tishchenko, A.

Tonchev, S.

Vaisocherova, H.

J. Dostalek, H. Vaisocherova, and J. Homola, “Multichannel surface plasmon resonance biosensor with wavelength division multiplexing,” Sens. Actuators B 108, 758–764 (2004).
[CrossRef]

Veillas, C.

Verma, P.

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nature Photon. 3, 388–394(2009).
[CrossRef]

Vial, A.

A. Vial, A. S. Grimault, D. Macias, D. Barchiesi, and M. L. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416(2005).
[CrossRef]

Vollmer, M.

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).

Wakida, S.

Wang, L.

Wang, P.

Yiping, C.

Yuan, G.

Yuan, H. K.

Zayats, A.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

I. J. Jen and Y. H. Liao, “Surface plasmon resonance via polarization conversion in a weak anisotropic thin film,” Appl. Phys. Lett. 94, 011105 (2009).
[CrossRef]

Chem. Phys. Lett. (1)

F. Hao and P. Nordlander, Chem. Phys. Lett. 446, 115 (2007).
[CrossRef]

Chem. Rev. (1)

J. Homola, “Surface plasmon resonance sensor for detection of chemical and biological species,” Chem. Rev. 108, 462–493(2008).
[CrossRef] [PubMed]

Eur. Phys. J. B (1)

K. R. Roy, S. K. Mandal, and A. K. Pal, “Effect of interfacial alloying on the surface plasmon resonance of nanocrystalline Au-Ag multilayer thin films,” Eur. Phys. J. B 33, 109–114(2003).
[CrossRef]

J. Appl. Phys. (1)

S. Maier and H. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

J. Chem. Phys. (1)

P. G. Etchegoin, E. C. L. Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

J. Opt. A: Pure Appl. Opt. (2)

R. Jha and A. K. Sharma, “Chalcogenide glass prism based SPR sensor with Ag-Au bimetallic nanoparticle alloy in infrared wavelength region,” J. Opt. A: Pure Appl. Opt. 11, 045502 (2009) [more exactly we refer to Eq. 5, Eq. 7, and Table 1].
[CrossRef]

I. Abdulhalim, “Surface plasmon TE and TM waves at the anisotropic film-metal interface,” J. Opt. A: Pure Appl. Opt. 11, 015002 (2009).
[CrossRef]

J. Phys. F (1)

U. Kreibig, “Electronic properties of small silver particles: the optical constants and their temperature dependence,” J. Phys. F 4, 999–1014 (1974).
[CrossRef]

Nanotechnology (1)

T. W. H. Oates and A. Mucklich, “Evolution of plasmon resonances during plasma deposition of silver nanoparticles,” Nanotechnology 16, 2606–2611 (2005).
[CrossRef]

Nature Photon. (1)

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nature Photon. 3, 388–394(2009).
[CrossRef]

Opt. Express (5)

Opt. Lett. (1)

Phys. Rev. B (3)

P. Berini, “Plasmon-polariton waves guided by thin lossy films of finite width: bound modes of symmetric structure,” Phys. Rev. B 61, 10484–10503 (2000).
[CrossRef]

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

A. Vial, A. S. Grimault, D. Macias, D. Barchiesi, and M. L. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416(2005).
[CrossRef]

Sens. Actuators B (1)

J. Dostalek, H. Vaisocherova, and J. Homola, “Multichannel surface plasmon resonance biosensor with wavelength division multiplexing,” Sens. Actuators B 108, 758–764 (2004).
[CrossRef]

Other (5)

S. Maier, Plasmonics: Fundamental and Applications(Springer, 2007).

E. D. Palik, Handbook of Optical Constants in Solids(Academic, 1985).

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Grating, Vol. 111 of Springer Tracts in Modern Physics (Springer–Verlag, 1988).

http:/glassbank.com.

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).

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Figures (7)

Fig. 1
Fig. 1

Reflectivity spectra of color plasmons: A, wavelength spectrum; B, angular spectrum.

Fig. 2
Fig. 2

Color plasmons for a four-layer stack; ε 0 corresponds to the required refractive index.

Fig. 3
Fig. 3

Train of plasmons excited: A, in a three-layer stack and B, in a four-layer stack; compare with Fig. 1A and Fig. 2, respectively; simulations are based on Fresnel’s theory.

Fig. 4
Fig. 4

Influence of finite thickness on the plasmon refractive index in a three-layer stack A, of a gold layer and B, of an aluminum layer (the thickness of the metal layer is indicated on the curves).

Fig. 5
Fig. 5

Tuning possibilities of bimetallic nanoparticle alloy: A, plasmon effective index matching with the glass refractive index; B, color plasmons in a four-layer stack.

Fig. 6
Fig. 6

A, Dependence of the incident angle on wavelength provided that Eq. (3) holds for a wide spectral region; B, curved surface prism–metal gives an opportunity for wavelength addressing of the surface.

Fig. 7
Fig. 7

Intensity profile of surface and color surface plasmons at 570 nm in a three-layer stack: quartz, 50 nm gold layer, and air.

Equations (5)

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

k x = 2 π λ p = ω c ( ε 1 ε 2 ε 1 + ε 2 ) 1 / 2 ,
k x = ε 0 ( λ ) ω c sin θ = Re { ω c ( ε 1 ( λ ) ε 2 ( λ ) ε 1 ( λ ) + ε 2 ( λ ) ) 1 / 2 } Re { ω c P x 0 } ,
Δ P x = [ 2 1 + ε 1 ( | ε 1 | | ε 1 | 1 ) 3 / 2 ] exp ( 2 | P x 0 | d ) Re { r 01 p ( P x 0 ) } ,
r 01 p = ε 1 k 0 ε 0 k 1 ε 1 k 0 + ε 0 k 1 , k 0 / 1 = [ ε p / m ( 2 π λ ) ( 2 π λ ε 0 sin ( θ ) ) 2 ] 1 / 2 .
k x = ε 0 ( λ ) ω c sin θ ( λ ) = Re { ω c ( ε 1 ε 2 ε 1 + ε 2 ) 1 / 2 } ,

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