Abstract

It is well known that the classical optical properties of a bare or metal-film-coated dielectric surface significantly the emission pattern of a fluorophore in close proximity to it. Most previous classical calculations of this perturb model the fluorophore as a continuous fixed-amplitude dipole acting as a simple radiator. However, for effect modeling steady-state excitation, a fixed-power dipole is more appropriate. This modification corresponds to normalizing fixed-amplitude dipole intensities by the total dissipated power, which is itself dependent on fluorophore orientation and proximity to the surface. The results for the fixed-power model differ nontrivially from the fixed-amplitude model. Using the fixed-power dipole model, we calculate the observation-angle-dependent intensity as a function of the fluorophore’s orientation and distance from the surface. The surface can have an intermediate layer of arbitrary thickness on it, which is used to model a metal-film-coated dielectric. In addition, general expressions are derived for the emission power as observed through a circular-aperture collection system (such as a microscope objective) located on either side of the interface. These expressions are applied to several common cases of fluorophore spatial and orientational distributions at bare glass–water and metal-film-coated glass-water interfaces. The results suggest practical experimental approaches for measuring the spatial and orientational distribution of fluorophores adsorbed at a surface, utilizing the distance-dependent fluorescence near a metalized surface and optimizing the collection efficiency from a well-defined volume near a quenching surface.

© 1987 Optical Society of America

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  8. N. L. Thompson, H. M. McConnell, and T. P. Burghardt, “Order in supported phospholipid monolayers detected by dichroism of fluorescence excited with polarized evanescent illumination,” Biophys. J. 46, 739–747 (1984).
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  11. V. Hlady, R. A. Van Wagenen, and J. D. Andrade, “Total internal reflection intrinsic fluorescence (TIRIF) spectroscopy applied to protein adsorption,” in Protein Adsorption, J. D. Andrade, ed., Vol. 2 of Interfacial Aspects of Biomedical Polymers (Plenum, New York, 1985), Chap. 2, p. 80.
  12. V. Hlady, D. R. Reinecke, and J. D. Andrade, “Fluorescence of adsorbed protein layers: I. Quantitation of total internal reflection fluorescence,” J. Colloid Interface Sci. 111, 555–569 (1986).
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  13. S. A. Rockhold, R. D. Quinn, R. A. Van Wagenen, J. D. Andrade, and M. Reichert, “Total internal reflection fluorescence (TIRF) as a quantitative probe of protein adsorption,” J. Electroanal. Chem. 150, 261–275 (1983).
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  16. K. H. Drexhage, “Interaction of light with monomolecular dye layers,” Prog. Opt. 12, 163–232 (1974).
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  17. G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
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  18. K. H. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 12, 693–701 (1970).
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  22. M. R. Philpott, “Effect of surface plasmons on transitions in molecules,” J. Chem. Phys. 62, 1812–1817 (1975).
    [Crossref]
  23. T. Tamir, J. J. Burke, and G. I. Stegeman, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
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  24. B. N. J. Persson, “Theory of the damping of excited molecules located above a metal surface,” J. Phys. C 11, 4251–4269 (1978).
    [Crossref]
  25. W. H. Weber and G. W. Ford, “Enhanced Raman scattering by adsorbates including the nonlocal response of the metal and the excitation of nonradiative modes,” Phys. Rev. Lett. 44, 1774–1777 (1980).
    [Crossref]
  26. P. Ye and Y. R. Shen, “Local-field effect on linear and nonlinear optical properties of adsorbed molecules,” Phys. Rev. B 28, 4288–4294 (1983).
    [Crossref]
  27. I. Pockrand and A. Brillante, “Nonradiative decay of excited molecules near a metal surface,” Chem. Phys. Lett. 69, 449–504 (1980).
    [Crossref]
  28. S. Garoff, A. Weitz, M. S. Alverez, and J. I. Gersten, “Electrodynamics at rough metal surfaces: photochemistry and luminescence of adsorbates near metal-island films,” J. Chem. Phys. 81, 5189–5200 (1984).
    [Crossref]
  29. D. A. Weitz, S. Garoff, J. I. Gersten, and A. Nitzan, “The enhancement of Raman scattering, resonance Raman scattering, and fluorescence from molecules adsorbed on a rough silver surface,” J. Chem. Phys. 78, 5324–5338 (1983).
    [Crossref]
  30. A. Campion, A. R. Gallo, C. B. Harris, H. J. Robota, and P. M. Whitmore, “Electronic energy transfer to metal surfaces: a test of classical image dipole theory at short distances,” Chem. Phys. Lett. 73, 447–450 (1980).
    [Crossref]
  31. E. H. Hellen, R. M. Fulbright, and D. Axelrod, “Total internal reflection fluorescence: theory and applications at biosurfaces,” in Spectroscopic Membrane Probes, L. Loew, ed. (CRC, Boca Raton, Fla., to be published).
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  34. P. Grivet, “Time reversibility in an optical proof of the reciprocity theorem of electromagnetism,” in Modern Optics, J. Fox, ed., Vol. 17 of Microwave Research Institute Symposia (Wiley-Interscience, New York, 1967), pp. 467–479.
  35. D. Axelrod, T. P. Burghardt, and N. L. Thompson, “Total internal reflection fluorescence,” Annu. Rev. Biophys. Bioeng. 13, 247–268 (1984).
    [Crossref] [PubMed]
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    [Crossref]
  38. D. Axelrod, R. M. Fulbright, and E. H. Hellen, “Adsorption kinetics on biological membranes: measurement by total internal reflection fluorescence,” in Applications of Fluorescence in the Biomedical Sciences, D. L. Taylor, A. J. Waggoner, R. F. Murphy, F. Lanni, and R. R. Birge, eds. (Liss, New York, 1986), pp. 461–476.
  39. M. Nakache, A. B. Schrieber, H. Gaub, and H. M. McConnell, “Heterogeneity of membrane phospholipid mobility in endothelial cells depends on cell substrate,” Nature 317, 75–77 (1985).
    [Crossref] [PubMed]
  40. C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, New York, 1976), pp. 154 and 293.

1986 (2)

V. Hlady, D. R. Reinecke, and J. D. Andrade, “Fluorescence of adsorbed protein layers: I. Quantitation of total internal reflection fluorescence,” J. Colloid Interface Sci. 111, 555–569 (1986).
[Crossref]

T. Tamir, J. J. Burke, and G. I. Stegeman, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

1985 (2)

M. Nakache, A. B. Schrieber, H. Gaub, and H. M. McConnell, “Heterogeneity of membrane phospholipid mobility in endothelial cells depends on cell substrate,” Nature 317, 75–77 (1985).
[Crossref] [PubMed]

D. Ausserre, H. Hervet, and F. Rondelez, “Concentration profile of polymer solutions near a solid wall,” Phys. Rev. Lett. 54, 1948–1951 (1985).
[Crossref] [PubMed]

1984 (5)

T. P. Burghardt and N. L. Thompson, “Effect of planar dielectric interfaces on fluorescence emission and detection: evanescent excitation with high aperture collection,” Biophys. J. 46, 729–737 (1984).

N. L. Thompson, H. M. McConnell, and T. P. Burghardt, “Order in supported phospholipid monolayers detected by dichroism of fluorescence excited with polarized evanescent illumination,” Biophys. J. 46, 739–747 (1984).

G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
[Crossref]

D. Axelrod, T. P. Burghardt, and N. L. Thompson, “Total internal reflection fluorescence,” Annu. Rev. Biophys. Bioeng. 13, 247–268 (1984).
[Crossref] [PubMed]

S. Garoff, A. Weitz, M. S. Alverez, and J. I. Gersten, “Electrodynamics at rough metal surfaces: photochemistry and luminescence of adsorbates near metal-island films,” J. Chem. Phys. 81, 5189–5200 (1984).
[Crossref]

1983 (3)

D. A. Weitz, S. Garoff, J. I. Gersten, and A. Nitzan, “The enhancement of Raman scattering, resonance Raman scattering, and fluorescence from molecules adsorbed on a rough silver surface,” J. Chem. Phys. 78, 5324–5338 (1983).
[Crossref]

P. Ye and Y. R. Shen, “Local-field effect on linear and nonlinear optical properties of adsorbed molecules,” Phys. Rev. B 28, 4288–4294 (1983).
[Crossref]

S. A. Rockhold, R. D. Quinn, R. A. Van Wagenen, J. D. Andrade, and M. Reichert, “Total internal reflection fluorescence (TIRF) as a quantitative probe of protein adsorption,” J. Electroanal. Chem. 150, 261–275 (1983).
[Crossref]

1982 (1)

C. Allain, D. Ausserre, and F. Rondelez, “Direct optical observation of interfacial depletion layers in polymer solutions,” Phys. Rev. Lett. 49, 1694–1697 (1982).
[Crossref]

1980 (4)

C. F. Eagen, W. H. Weber, S. L. McCarthy, and R. W. Terhune, “Time dependent decay of surface-plasmon-coupled molecular fluorescence,” Chem. Phys. Lett. 75, 274–277 (1980).
[Crossref]

I. Pockrand and A. Brillante, “Nonradiative decay of excited molecules near a metal surface,” Chem. Phys. Lett. 69, 449–504 (1980).
[Crossref]

A. Campion, A. R. Gallo, C. B. Harris, H. J. Robota, and P. M. Whitmore, “Electronic energy transfer to metal surfaces: a test of classical image dipole theory at short distances,” Chem. Phys. Lett. 73, 447–450 (1980).
[Crossref]

W. H. Weber and G. W. Ford, “Enhanced Raman scattering by adsorbates including the nonlocal response of the metal and the excitation of nonradiative modes,” Phys. Rev. Lett. 44, 1774–1777 (1980).
[Crossref]

1979 (2)

1978 (2)

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[Crossref]

B. N. J. Persson, “Theory of the damping of excited molecules located above a metal surface,” J. Phys. C 11, 4251–4269 (1978).
[Crossref]

1977 (3)

1975 (2)

R. R. Chance, A. Prock, and R. Silbey, “Comments on the classical theory of energy transfer,” J. Chem. Phys. 62, 2245–2253 (1975).
[Crossref]

M. R. Philpott, “Effect of surface plasmons on transitions in molecules,” J. Chem. Phys. 62, 1812–1817 (1975).
[Crossref]

1974 (1)

K. H. Drexhage, “Interaction of light with monomolecular dye layers,” Prog. Opt. 12, 163–232 (1974).
[Crossref]

1972 (2)

C. K. Carniglia, L. Mandel, and K. H. Drexhage, “Adsorption and emission of evanescent photons,” J. Opt. Soc. Am. 62, 479–486 (1972).
[Crossref]

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

1970 (2)

H. Kuhn, “Classical aspects of energy transfer in molecular systems,” J. Chem. Phys. 53, 101–108 (1970).
[Crossref]

K. H. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 12, 693–701 (1970).
[Crossref]

1961 (1)

Allain, C.

C. Allain, D. Ausserre, and F. Rondelez, “Direct optical observation of interfacial depletion layers in polymer solutions,” Phys. Rev. Lett. 49, 1694–1697 (1982).
[Crossref]

Alverez, M. S.

S. Garoff, A. Weitz, M. S. Alverez, and J. I. Gersten, “Electrodynamics at rough metal surfaces: photochemistry and luminescence of adsorbates near metal-island films,” J. Chem. Phys. 81, 5189–5200 (1984).
[Crossref]

Andrade, J. D.

V. Hlady, D. R. Reinecke, and J. D. Andrade, “Fluorescence of adsorbed protein layers: I. Quantitation of total internal reflection fluorescence,” J. Colloid Interface Sci. 111, 555–569 (1986).
[Crossref]

S. A. Rockhold, R. D. Quinn, R. A. Van Wagenen, J. D. Andrade, and M. Reichert, “Total internal reflection fluorescence (TIRF) as a quantitative probe of protein adsorption,” J. Electroanal. Chem. 150, 261–275 (1983).
[Crossref]

V. Hlady, R. A. Van Wagenen, and J. D. Andrade, “Total internal reflection intrinsic fluorescence (TIRIF) spectroscopy applied to protein adsorption,” in Protein Adsorption, J. D. Andrade, ed., Vol. 2 of Interfacial Aspects of Biomedical Polymers (Plenum, New York, 1985), Chap. 2, p. 80.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, New York, 1970), pp. 373–376.

Ausserre, D.

D. Ausserre, H. Hervet, and F. Rondelez, “Concentration profile of polymer solutions near a solid wall,” Phys. Rev. Lett. 54, 1948–1951 (1985).
[Crossref] [PubMed]

C. Allain, D. Ausserre, and F. Rondelez, “Direct optical observation of interfacial depletion layers in polymer solutions,” Phys. Rev. Lett. 49, 1694–1697 (1982).
[Crossref]

Axelrod, D.

D. Axelrod, T. P. Burghardt, and N. L. Thompson, “Total internal reflection fluorescence,” Annu. Rev. Biophys. Bioeng. 13, 247–268 (1984).
[Crossref] [PubMed]

E. H. Hellen, R. M. Fulbright, and D. Axelrod, “Total internal reflection fluorescence: theory and applications at biosurfaces,” in Spectroscopic Membrane Probes, L. Loew, ed. (CRC, Boca Raton, Fla., to be published).

D. Axelrod, R. M. Fulbright, and E. H. Hellen, “Adsorption kinetics on biological membranes: measurement by total internal reflection fluorescence,” in Applications of Fluorescence in the Biomedical Sciences, D. L. Taylor, A. J. Waggoner, R. F. Murphy, F. Lanni, and R. R. Birge, eds. (Liss, New York, 1986), pp. 461–476.

Benner, R. E.

Brillante, A.

I. Pockrand and A. Brillante, “Nonradiative decay of excited molecules near a metal surface,” Chem. Phys. Lett. 69, 449–504 (1980).
[Crossref]

Burghardt, T. P.

N. L. Thompson, H. M. McConnell, and T. P. Burghardt, “Order in supported phospholipid monolayers detected by dichroism of fluorescence excited with polarized evanescent illumination,” Biophys. J. 46, 739–747 (1984).

T. P. Burghardt and N. L. Thompson, “Effect of planar dielectric interfaces on fluorescence emission and detection: evanescent excitation with high aperture collection,” Biophys. J. 46, 729–737 (1984).

D. Axelrod, T. P. Burghardt, and N. L. Thompson, “Total internal reflection fluorescence,” Annu. Rev. Biophys. Bioeng. 13, 247–268 (1984).
[Crossref] [PubMed]

N. L. Thompson and T. P. Burghardt, “Total internal reflection fluorescence: measurement of spatial and orientation distributions of fluorophores near planar dielectric interfaces,” Biophys. Chem. (to be published).

Burke, J. J.

T. Tamir, J. J. Burke, and G. I. Stegeman, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

Campion, A.

A. Campion, A. R. Gallo, C. B. Harris, H. J. Robota, and P. M. Whitmore, “Electronic energy transfer to metal surfaces: a test of classical image dipole theory at short distances,” Chem. Phys. Lett. 73, 447–450 (1980).
[Crossref]

Carniglia, C. K.

Chance, R. R.

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[Crossref]

R. R. Chance, A. Prock, and R. Silbey, “Comments on the classical theory of energy transfer,” J. Chem. Phys. 62, 2245–2253 (1975).
[Crossref]

Chang, R. K.

Christy, R. W.

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

Drexhage, K. H.

K. H. Drexhage, “Interaction of light with monomolecular dye layers,” Prog. Opt. 12, 163–232 (1974).
[Crossref]

C. K. Carniglia, L. Mandel, and K. H. Drexhage, “Adsorption and emission of evanescent photons,” J. Opt. Soc. Am. 62, 479–486 (1972).
[Crossref]

K. H. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 12, 693–701 (1970).
[Crossref]

Eagen, C. F.

C. F. Eagen, W. H. Weber, S. L. McCarthy, and R. W. Terhune, “Time dependent decay of surface-plasmon-coupled molecular fluorescence,” Chem. Phys. Lett. 75, 274–277 (1980).
[Crossref]

W. H. Weber and C. F. Eagen, “Energy transfer from an excited dye molecule to the surface plasmons of an adjacent metal,” Opt. Lett. 4, 236–238 (1979).
[Crossref] [PubMed]

Fen, J. B.

Ford, G. W.

G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
[Crossref]

W. H. Weber and G. W. Ford, “Enhanced Raman scattering by adsorbates including the nonlocal response of the metal and the excitation of nonradiative modes,” Phys. Rev. Lett. 44, 1774–1777 (1980).
[Crossref]

Fulbright, R. M.

D. Axelrod, R. M. Fulbright, and E. H. Hellen, “Adsorption kinetics on biological membranes: measurement by total internal reflection fluorescence,” in Applications of Fluorescence in the Biomedical Sciences, D. L. Taylor, A. J. Waggoner, R. F. Murphy, F. Lanni, and R. R. Birge, eds. (Liss, New York, 1986), pp. 461–476.

E. H. Hellen, R. M. Fulbright, and D. Axelrod, “Total internal reflection fluorescence: theory and applications at biosurfaces,” in Spectroscopic Membrane Probes, L. Loew, ed. (CRC, Boca Raton, Fla., to be published).

Gallo, A. R.

A. Campion, A. R. Gallo, C. B. Harris, H. J. Robota, and P. M. Whitmore, “Electronic energy transfer to metal surfaces: a test of classical image dipole theory at short distances,” Chem. Phys. Lett. 73, 447–450 (1980).
[Crossref]

Garoff, S.

S. Garoff, A. Weitz, M. S. Alverez, and J. I. Gersten, “Electrodynamics at rough metal surfaces: photochemistry and luminescence of adsorbates near metal-island films,” J. Chem. Phys. 81, 5189–5200 (1984).
[Crossref]

D. A. Weitz, S. Garoff, J. I. Gersten, and A. Nitzan, “The enhancement of Raman scattering, resonance Raman scattering, and fluorescence from molecules adsorbed on a rough silver surface,” J. Chem. Phys. 78, 5324–5338 (1983).
[Crossref]

Gaub, H.

M. Nakache, A. B. Schrieber, H. Gaub, and H. M. McConnell, “Heterogeneity of membrane phospholipid mobility in endothelial cells depends on cell substrate,” Nature 317, 75–77 (1985).
[Crossref] [PubMed]

Gersten, J. I.

S. Garoff, A. Weitz, M. S. Alverez, and J. I. Gersten, “Electrodynamics at rough metal surfaces: photochemistry and luminescence of adsorbates near metal-island films,” J. Chem. Phys. 81, 5189–5200 (1984).
[Crossref]

D. A. Weitz, S. Garoff, J. I. Gersten, and A. Nitzan, “The enhancement of Raman scattering, resonance Raman scattering, and fluorescence from molecules adsorbed on a rough silver surface,” J. Chem. Phys. 78, 5324–5338 (1983).
[Crossref]

Grivet, P.

P. Grivet, “Time reversibility in an optical proof of the reciprocity theorem of electromagnetism,” in Modern Optics, J. Fox, ed., Vol. 17 of Microwave Research Institute Symposia (Wiley-Interscience, New York, 1967), pp. 467–479.

Harris, C. B.

A. Campion, A. R. Gallo, C. B. Harris, H. J. Robota, and P. M. Whitmore, “Electronic energy transfer to metal surfaces: a test of classical image dipole theory at short distances,” Chem. Phys. Lett. 73, 447–450 (1980).
[Crossref]

Hass, G.

Hellen, E. H.

D. Axelrod, R. M. Fulbright, and E. H. Hellen, “Adsorption kinetics on biological membranes: measurement by total internal reflection fluorescence,” in Applications of Fluorescence in the Biomedical Sciences, D. L. Taylor, A. J. Waggoner, R. F. Murphy, F. Lanni, and R. R. Birge, eds. (Liss, New York, 1986), pp. 461–476.

E. H. Hellen, R. M. Fulbright, and D. Axelrod, “Total internal reflection fluorescence: theory and applications at biosurfaces,” in Spectroscopic Membrane Probes, L. Loew, ed. (CRC, Boca Raton, Fla., to be published).

Hervet, H.

D. Ausserre, H. Hervet, and F. Rondelez, “Concentration profile of polymer solutions near a solid wall,” Phys. Rev. Lett. 54, 1948–1951 (1985).
[Crossref] [PubMed]

Hlady, V.

V. Hlady, D. R. Reinecke, and J. D. Andrade, “Fluorescence of adsorbed protein layers: I. Quantitation of total internal reflection fluorescence,” J. Colloid Interface Sci. 111, 555–569 (1986).
[Crossref]

V. Hlady, R. A. Van Wagenen, and J. D. Andrade, “Total internal reflection intrinsic fluorescence (TIRIF) spectroscopy applied to protein adsorption,” in Protein Adsorption, J. D. Andrade, ed., Vol. 2 of Interfacial Aspects of Biomedical Polymers (Plenum, New York, 1985), Chap. 2, p. 80.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 395.

Johnson, P. B.

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

Kittel, C.

C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, New York, 1976), pp. 154 and 293.

Kuhn, H.

H. Kuhn, “Classical aspects of energy transfer in molecular systems,” J. Chem. Phys. 53, 101–108 (1970).
[Crossref]

Kunz, R. E.

Lee, E.-H.

Lukosz, W.

Mandel, L.

McCarthy, S. L.

C. F. Eagen, W. H. Weber, S. L. McCarthy, and R. W. Terhune, “Time dependent decay of surface-plasmon-coupled molecular fluorescence,” Chem. Phys. Lett. 75, 274–277 (1980).
[Crossref]

McConnell, H. M.

M. Nakache, A. B. Schrieber, H. Gaub, and H. M. McConnell, “Heterogeneity of membrane phospholipid mobility in endothelial cells depends on cell substrate,” Nature 317, 75–77 (1985).
[Crossref] [PubMed]

N. L. Thompson, H. M. McConnell, and T. P. Burghardt, “Order in supported phospholipid monolayers detected by dichroism of fluorescence excited with polarized evanescent illumination,” Biophys. J. 46, 739–747 (1984).

Nakache, M.

M. Nakache, A. B. Schrieber, H. Gaub, and H. M. McConnell, “Heterogeneity of membrane phospholipid mobility in endothelial cells depends on cell substrate,” Nature 317, 75–77 (1985).
[Crossref] [PubMed]

Nitzan, A.

D. A. Weitz, S. Garoff, J. I. Gersten, and A. Nitzan, “The enhancement of Raman scattering, resonance Raman scattering, and fluorescence from molecules adsorbed on a rough silver surface,” J. Chem. Phys. 78, 5324–5338 (1983).
[Crossref]

Persson, B. N. J.

B. N. J. Persson, “Theory of the damping of excited molecules located above a metal surface,” J. Phys. C 11, 4251–4269 (1978).
[Crossref]

Philpott, M. R.

M. R. Philpott, “Effect of surface plasmons on transitions in molecules,” J. Chem. Phys. 62, 1812–1817 (1975).
[Crossref]

Pockrand, I.

I. Pockrand and A. Brillante, “Nonradiative decay of excited molecules near a metal surface,” Chem. Phys. Lett. 69, 449–504 (1980).
[Crossref]

Prock, A.

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[Crossref]

R. R. Chance, A. Prock, and R. Silbey, “Comments on the classical theory of energy transfer,” J. Chem. Phys. 62, 2245–2253 (1975).
[Crossref]

Quinn, R. D.

S. A. Rockhold, R. D. Quinn, R. A. Van Wagenen, J. D. Andrade, and M. Reichert, “Total internal reflection fluorescence (TIRF) as a quantitative probe of protein adsorption,” J. Electroanal. Chem. 150, 261–275 (1983).
[Crossref]

Reichert, M.

S. A. Rockhold, R. D. Quinn, R. A. Van Wagenen, J. D. Andrade, and M. Reichert, “Total internal reflection fluorescence (TIRF) as a quantitative probe of protein adsorption,” J. Electroanal. Chem. 150, 261–275 (1983).
[Crossref]

Reinecke, D. R.

V. Hlady, D. R. Reinecke, and J. D. Andrade, “Fluorescence of adsorbed protein layers: I. Quantitation of total internal reflection fluorescence,” J. Colloid Interface Sci. 111, 555–569 (1986).
[Crossref]

Robota, H. J.

A. Campion, A. R. Gallo, C. B. Harris, H. J. Robota, and P. M. Whitmore, “Electronic energy transfer to metal surfaces: a test of classical image dipole theory at short distances,” Chem. Phys. Lett. 73, 447–450 (1980).
[Crossref]

Rockhold, S. A.

S. A. Rockhold, R. D. Quinn, R. A. Van Wagenen, J. D. Andrade, and M. Reichert, “Total internal reflection fluorescence (TIRF) as a quantitative probe of protein adsorption,” J. Electroanal. Chem. 150, 261–275 (1983).
[Crossref]

Rondelez, F.

D. Ausserre, H. Hervet, and F. Rondelez, “Concentration profile of polymer solutions near a solid wall,” Phys. Rev. Lett. 54, 1948–1951 (1985).
[Crossref] [PubMed]

C. Allain, D. Ausserre, and F. Rondelez, “Direct optical observation of interfacial depletion layers in polymer solutions,” Phys. Rev. Lett. 49, 1694–1697 (1982).
[Crossref]

Schrieber, A. B.

M. Nakache, A. B. Schrieber, H. Gaub, and H. M. McConnell, “Heterogeneity of membrane phospholipid mobility in endothelial cells depends on cell substrate,” Nature 317, 75–77 (1985).
[Crossref] [PubMed]

Shen, Y. R.

P. Ye and Y. R. Shen, “Local-field effect on linear and nonlinear optical properties of adsorbed molecules,” Phys. Rev. B 28, 4288–4294 (1983).
[Crossref]

Silbey, R.

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[Crossref]

R. R. Chance, A. Prock, and R. Silbey, “Comments on the classical theory of energy transfer,” J. Chem. Phys. 62, 2245–2253 (1975).
[Crossref]

Stegeman, G. I.

T. Tamir, J. J. Burke, and G. I. Stegeman, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

Tamir, T.

T. Tamir, J. J. Burke, and G. I. Stegeman, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

Terhune, R. W.

C. F. Eagen, W. H. Weber, S. L. McCarthy, and R. W. Terhune, “Time dependent decay of surface-plasmon-coupled molecular fluorescence,” Chem. Phys. Lett. 75, 274–277 (1980).
[Crossref]

Thompson, N. L.

T. P. Burghardt and N. L. Thompson, “Effect of planar dielectric interfaces on fluorescence emission and detection: evanescent excitation with high aperture collection,” Biophys. J. 46, 729–737 (1984).

N. L. Thompson, H. M. McConnell, and T. P. Burghardt, “Order in supported phospholipid monolayers detected by dichroism of fluorescence excited with polarized evanescent illumination,” Biophys. J. 46, 739–747 (1984).

D. Axelrod, T. P. Burghardt, and N. L. Thompson, “Total internal reflection fluorescence,” Annu. Rev. Biophys. Bioeng. 13, 247–268 (1984).
[Crossref] [PubMed]

N. L. Thompson and T. P. Burghardt, “Total internal reflection fluorescence: measurement of spatial and orientation distributions of fluorophores near planar dielectric interfaces,” Biophys. Chem. (to be published).

Van Wagenen, R. A.

S. A. Rockhold, R. D. Quinn, R. A. Van Wagenen, J. D. Andrade, and M. Reichert, “Total internal reflection fluorescence (TIRF) as a quantitative probe of protein adsorption,” J. Electroanal. Chem. 150, 261–275 (1983).
[Crossref]

V. Hlady, R. A. Van Wagenen, and J. D. Andrade, “Total internal reflection intrinsic fluorescence (TIRIF) spectroscopy applied to protein adsorption,” in Protein Adsorption, J. D. Andrade, ed., Vol. 2 of Interfacial Aspects of Biomedical Polymers (Plenum, New York, 1985), Chap. 2, p. 80.

Waylonis, J. E.

Weber, W. H.

G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
[Crossref]

C. F. Eagen, W. H. Weber, S. L. McCarthy, and R. W. Terhune, “Time dependent decay of surface-plasmon-coupled molecular fluorescence,” Chem. Phys. Lett. 75, 274–277 (1980).
[Crossref]

W. H. Weber and G. W. Ford, “Enhanced Raman scattering by adsorbates including the nonlocal response of the metal and the excitation of nonradiative modes,” Phys. Rev. Lett. 44, 1774–1777 (1980).
[Crossref]

W. H. Weber and C. F. Eagen, “Energy transfer from an excited dye molecule to the surface plasmons of an adjacent metal,” Opt. Lett. 4, 236–238 (1979).
[Crossref] [PubMed]

Weitz, A.

S. Garoff, A. Weitz, M. S. Alverez, and J. I. Gersten, “Electrodynamics at rough metal surfaces: photochemistry and luminescence of adsorbates near metal-island films,” J. Chem. Phys. 81, 5189–5200 (1984).
[Crossref]

Weitz, D. A.

D. A. Weitz, S. Garoff, J. I. Gersten, and A. Nitzan, “The enhancement of Raman scattering, resonance Raman scattering, and fluorescence from molecules adsorbed on a rough silver surface,” J. Chem. Phys. 78, 5324–5338 (1983).
[Crossref]

Whitmore, P. M.

A. Campion, A. R. Gallo, C. B. Harris, H. J. Robota, and P. M. Whitmore, “Electronic energy transfer to metal surfaces: a test of classical image dipole theory at short distances,” Chem. Phys. Lett. 73, 447–450 (1980).
[Crossref]

Ye, P.

P. Ye and Y. R. Shen, “Local-field effect on linear and nonlinear optical properties of adsorbed molecules,” Phys. Rev. B 28, 4288–4294 (1983).
[Crossref]

Adv. Chem. Phys. (1)

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[Crossref]

Annu. Rev. Biophys. Bioeng. (1)

D. Axelrod, T. P. Burghardt, and N. L. Thompson, “Total internal reflection fluorescence,” Annu. Rev. Biophys. Bioeng. 13, 247–268 (1984).
[Crossref] [PubMed]

Appl. Opt. (1)

Biophys. J. (2)

T. P. Burghardt and N. L. Thompson, “Effect of planar dielectric interfaces on fluorescence emission and detection: evanescent excitation with high aperture collection,” Biophys. J. 46, 729–737 (1984).

N. L. Thompson, H. M. McConnell, and T. P. Burghardt, “Order in supported phospholipid monolayers detected by dichroism of fluorescence excited with polarized evanescent illumination,” Biophys. J. 46, 739–747 (1984).

Chem. Phys. Lett. (3)

C. F. Eagen, W. H. Weber, S. L. McCarthy, and R. W. Terhune, “Time dependent decay of surface-plasmon-coupled molecular fluorescence,” Chem. Phys. Lett. 75, 274–277 (1980).
[Crossref]

A. Campion, A. R. Gallo, C. B. Harris, H. J. Robota, and P. M. Whitmore, “Electronic energy transfer to metal surfaces: a test of classical image dipole theory at short distances,” Chem. Phys. Lett. 73, 447–450 (1980).
[Crossref]

I. Pockrand and A. Brillante, “Nonradiative decay of excited molecules near a metal surface,” Chem. Phys. Lett. 69, 449–504 (1980).
[Crossref]

J. Chem. Phys. (5)

S. Garoff, A. Weitz, M. S. Alverez, and J. I. Gersten, “Electrodynamics at rough metal surfaces: photochemistry and luminescence of adsorbates near metal-island films,” J. Chem. Phys. 81, 5189–5200 (1984).
[Crossref]

D. A. Weitz, S. Garoff, J. I. Gersten, and A. Nitzan, “The enhancement of Raman scattering, resonance Raman scattering, and fluorescence from molecules adsorbed on a rough silver surface,” J. Chem. Phys. 78, 5324–5338 (1983).
[Crossref]

H. Kuhn, “Classical aspects of energy transfer in molecular systems,” J. Chem. Phys. 53, 101–108 (1970).
[Crossref]

M. R. Philpott, “Effect of surface plasmons on transitions in molecules,” J. Chem. Phys. 62, 1812–1817 (1975).
[Crossref]

R. R. Chance, A. Prock, and R. Silbey, “Comments on the classical theory of energy transfer,” J. Chem. Phys. 62, 2245–2253 (1975).
[Crossref]

J. Colloid Interface Sci. (1)

V. Hlady, D. R. Reinecke, and J. D. Andrade, “Fluorescence of adsorbed protein layers: I. Quantitation of total internal reflection fluorescence,” J. Colloid Interface Sci. 111, 555–569 (1986).
[Crossref]

J. Electroanal. Chem. (1)

S. A. Rockhold, R. D. Quinn, R. A. Van Wagenen, J. D. Andrade, and M. Reichert, “Total internal reflection fluorescence (TIRF) as a quantitative probe of protein adsorption,” J. Electroanal. Chem. 150, 261–275 (1983).
[Crossref]

J. Lumin. (1)

K. H. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 12, 693–701 (1970).
[Crossref]

J. Opt. Soc. Am. (4)

J. Phys. C (1)

B. N. J. Persson, “Theory of the damping of excited molecules located above a metal surface,” J. Phys. C 11, 4251–4269 (1978).
[Crossref]

Nature (1)

M. Nakache, A. B. Schrieber, H. Gaub, and H. M. McConnell, “Heterogeneity of membrane phospholipid mobility in endothelial cells depends on cell substrate,” Nature 317, 75–77 (1985).
[Crossref] [PubMed]

Opt. Commun. (1)

W. Lukosz and R. E. Kunz, “Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface,” Opt. Commun. 20, 195–199 (1977).
[Crossref]

Opt. Lett. (1)

Phys. Rep. (1)

G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
[Crossref]

Phys. Rev. B (3)

T. Tamir, J. J. Burke, and G. I. Stegeman, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

P. Ye and Y. R. Shen, “Local-field effect on linear and nonlinear optical properties of adsorbed molecules,” Phys. Rev. B 28, 4288–4294 (1983).
[Crossref]

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

Phys. Rev. Lett. (3)

W. H. Weber and G. W. Ford, “Enhanced Raman scattering by adsorbates including the nonlocal response of the metal and the excitation of nonradiative modes,” Phys. Rev. Lett. 44, 1774–1777 (1980).
[Crossref]

C. Allain, D. Ausserre, and F. Rondelez, “Direct optical observation of interfacial depletion layers in polymer solutions,” Phys. Rev. Lett. 49, 1694–1697 (1982).
[Crossref]

D. Ausserre, H. Hervet, and F. Rondelez, “Concentration profile of polymer solutions near a solid wall,” Phys. Rev. Lett. 54, 1948–1951 (1985).
[Crossref] [PubMed]

Prog. Opt. (1)

K. H. Drexhage, “Interaction of light with monomolecular dye layers,” Prog. Opt. 12, 163–232 (1974).
[Crossref]

Other (8)

V. Hlady, R. A. Van Wagenen, and J. D. Andrade, “Total internal reflection intrinsic fluorescence (TIRIF) spectroscopy applied to protein adsorption,” in Protein Adsorption, J. D. Andrade, ed., Vol. 2 of Interfacial Aspects of Biomedical Polymers (Plenum, New York, 1985), Chap. 2, p. 80.

N. L. Thompson and T. P. Burghardt, “Total internal reflection fluorescence: measurement of spatial and orientation distributions of fluorophores near planar dielectric interfaces,” Biophys. Chem. (to be published).

D. Axelrod, R. M. Fulbright, and E. H. Hellen, “Adsorption kinetics on biological membranes: measurement by total internal reflection fluorescence,” in Applications of Fluorescence in the Biomedical Sciences, D. L. Taylor, A. J. Waggoner, R. F. Murphy, F. Lanni, and R. R. Birge, eds. (Liss, New York, 1986), pp. 461–476.

E. H. Hellen, R. M. Fulbright, and D. Axelrod, “Total internal reflection fluorescence: theory and applications at biosurfaces,” in Spectroscopic Membrane Probes, L. Loew, ed. (CRC, Boca Raton, Fla., to be published).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 395.

G. Arfken, Mathematical Methods for Physicists, 2nd ed. (Academic, New York, 1970), pp. 373–376.

P. Grivet, “Time reversibility in an optical proof of the reciprocity theorem of electromagnetism,” in Modern Optics, J. Fox, ed., Vol. 17 of Microwave Research Institute Symposia (Wiley-Interscience, New York, 1967), pp. 467–479.

C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, New York, 1976), pp. 154 and 293.

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Figures (5)

Fig. 1
Fig. 1

Total power dissipated by fixed-amplitude dipoles oriented parallel and perpendicularly to the interface (dashed lines and arbitrary units) and the ratio of these powers, (solid line) versus dipole distance z measured into the water. a, Bare glas. b Aluminum film (220 Å) on glass. The emission wavelength is λ = 5 200 Å, and the dielectric constants are as follows: for water, ɛ1 = 1.77; for aluminum, ɛ2 = −32.5 + 8.4i; for glass, ɛ3 = 2.13.

Fig. 2
Fig. 2

Intensity emitted at supercritical angle θ = 70° into the glass versus dipole distance z for dipoles oriented perpendicularly to the interface. The solid lines show the normalized intensity Ŝ for fixed-power dipoles with and without a 220-Å aluminum film on the glass. The dashed line shows the exponentially decaying intensity obtained by omitting proper normalization, as explained in the text. All these intensities are independent of ϕ for perpendicular dipoles. Dielectric constants and λ0 are as in Fig. 1.

Fig. 3
Fig. 3

ϕ-averaged normalized intensities S ^ ϕ , versus polar observation angle θ for a layer of dipoles at z = 800 Å from the interface. Two orientations are shown: a, perpendicular to the interface and b, parallel to the interface, both with and without a 220-Å aluminum film on the glass. At each angle, the radial distance from the center circle (surrounding the dipoles’ location) is proportional to the radiated power. The dielectric constants and λ0 are as in Fig. 1.

Fig. 4
Fig. 4

Collection efficiencies Q for an objective with a numerical aperture of 1.4 versus the dipole distance z. Two positions of the objective are indicated: collection of light emitted into the water (dashed lines) and into the glass (solid lines). Two dipole orientations are indicated: perpendicular and parallel to the interface. Dielectric constants and λ0 are same as in Fig. 1. a, Bare glass; b, 220-Å aluminum film on glass; c, expanded scale of (b).

Fig. 5
Fig. 5

The weightings w x , y , z , versus η, used in Eqs. (52) to construct weightings for perpendicular and parallel dipoles in order to predict intensities radiated from randomly oriented distributions of dipoles.

Equations (111)

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

S = c ɛ 1 / 2 8 π E 2
S ^ ( r , r ) S ( r , r ) / P T ( r ) .
I ( r , r ) = μ ab · E ex 2 S ^ ( r , r ) .
J ( r , θ , ϕ ) = k d Ω d z C ( θ , ϕ , z ) I ( r , r ) ,
F = r 2 d Ω J ( r ) .
j ( r , t ) = - i ω μ exp ( - i ω t ) δ ( r - r d ) .
ρ t + · j = 0
ρ ( r , t ) = - exp ( - i ω t ) μ · [ δ ( r - r d ) ] .
E ( r ) = d k exp ( i k · r ) E ( k ) ,             B ( ( r ) = d k exp ( i k · r ) B ( k ) .
E ( r ) = - 1 π ɛ 1 d k r exp ( i k r · ρ ) [ δ ( z 0 - z ) z ^ ( μ · z ^ ) + i 2 q 1 exp ( i q 1 z 0 - z ) k 1 × ( k 1 × μ ) ] ,
k 1 = k r + z 0 - z z 0 - z q 1 z ^ ,
E 1 ( r ) = i 2 π ɛ 1 0 2 π d γ 0 d k r k r ( ξ direct + ξ reflected ) ,
E 3 ( r ) = i 2 π ɛ 1 0 2 π d γ 0 d k r k r ξ transmitted ,
r p , s = r 12 p , s + r 23 p , s exp [ i 2 k 1 t ( ɛ 2 / ɛ 1 - v 2 ) 1 / 2 ] 1 + r 12 p , s r 23 p , s exp [ i 2 k 1 t ( ɛ 2 / ɛ 1 - v 2 ) 1 / 2 ] ,
t p , s = t 12 p , s t 23 p , s exp [ i k 1 t ( ɛ 2 / ɛ 1 - v 2 ) 1 / 2 ] 1 + r 12 p , s r 23 p , s exp [ i 2 k 1 t ( ɛ 2 / ɛ 1 - v 2 ) 1 / 2 ] ,
ξ inc = - 1 q 1 exp ( i k r · ρ ) exp ( i q 1 z 0 - z ) k 1 × ( k 1 × μ ) .
a × ( b × c ) = b ( a · c ) - c ( a · b )
μ = μ · ( z ^ × k ^ r ) ( z ^ × k ^ r ) + ( μ · k ^ r ) k ^ r + ( μ · z ^ ) z ^ .
- k 1 × ( k 1 × μ ) = ( k r z ^ + q 1 k ^ r ) · μ ( k r z ^ + q 1 k ^ r ) + k 1 2 ( z ^ × k ^ r ) · μ ( z ^ × k ^ r ) .
ξ inc = A [ cos γ ( k r z ^ + q 1 k ^ r ) - sin γ k 1 2 q 1 ( z ^ × k ^ r ) ] × exp ( i q 1 z ) exp ( - i q 1 z 0 ) ,
ξ refl = A [ r p cos γ ( k r z ^ - q 1 k ^ r ) - r s sin γ k 1 2 q 1 ( z ^ × k ^ r ) ] × exp ( i q 1 z ) exp ( i q 1 z 0 ) ,
ξ tran = A [ t p ( ɛ 1 ɛ 3 ) 1 / 2 cos γ ( k r z ^ + q 3 k ^ r ) - t 3 sin γ k 1 2 q 1 ( z ^ × k ^ r ) ] × exp ( i q 1 z ) exp [ - i q 3 ( z 0 + t ) ] ;
ξ inc = A [ sin γ ( k r z ^ + q 1 k ^ r ) + cos γ k 1 2 q 1 ( z ^ × k ^ r ) ] × exp ( i q 1 z ) exp ( - i q 1 z 0 ) ,
ξ refl = A [ r p sin γ ( k r z ^ - q 1 k ^ r ) + r s cos γ k 1 2 q 1 ( z ^ × k ^ r ) ] × exp ( i q 1 z ) exp ( i q 1 z 0 ) ,
ξ tran = A [ t p ( ɛ 1 ɛ 3 ) 1 / 2 sin γ ( k r z ^ + q 3 k ^ r ) + t s cos γ k 1 2 q 1 ( z ^ × k ^ r ) ] × exp ( i q 1 z ) exp [ - i q 3 ( z 0 + t ) ] ;
ξ inc = A k r q 1 ( q 1 k ^ r + k r z ^ ) exp ( i q 1 z ) exp ( - i q 1 z 0 ) ,
ξ refl = A r p k r q 1 ( q 1 k ^ r + k r z ^ ) exp ( i q 1 z ) exp ( i q 1 z 0 ) ,
ξ tran = A t p ( ɛ 1 ɛ 3 ) 1 / 2 k r q 1 ( k r z ^ + q 3 k ^ r ) exp ( i q 1 z ) exp [ - i q 3 ( z 0 + t ) ] ,
k r = ( k r cos γ ) p ^ + ( k r sin γ ) s ^ , z ^ × k ^ r = ( - sin γ ) p ^ + ( cos γ ) s ^ .
0 2 π d γ exp ( i k r ρ cos γ ) cos 2 γ = π [ J 0 ( k r ρ ) - J 2 ( k r ρ ) ] ( 2 π k r ρ ) 1 / 2 2 cos ( k r ρ - π 4 ) ,
0 2 π d γ exp ( i k r ρ cos γ ) sin 2 γ = π [ J 0 ( k r ρ ) + J 2 ( k r ρ ) ] 0 ,
0 2 π d γ exp ( i k r ρ cos γ ) sin γ cos γ = 0 ,
0 2 π d γ exp ( i k r ρ cos γ ) cos γ = i 2 π J 1 ( k r ρ ) i ( 2 π k r ρ ) 1 / 2 2 cos ( k r ρ - 3 π 4 ) .
cos ( k r ρ + l ) = ½ { exp [ i ( k r ρ + l ) ] + exp [ - i ( k r ρ + l ) ] } ,
0 d k r exp ( i g ( k r ) ρ ) f ( k r ) .
E dir ( r ) = k 1 2 ɛ 1 ( n ^ × μ ) × n ^ exp ( i k 1 r ) r exp ( - i k 1 z cos θ ) .
E 1 μ p ( r ) = μ k 1 2 exp ( i k 1 r ) ɛ 1 r { cos 2 θ [ exp ( - i k 1 z cos θ ) - r p exp ( i k 1 z cos θ ) ] p ^ + cos θ sin θ [ - exp ( - i k 1 z cos θ ) + r p exp ( i k 1 z cos θ ) ] z ^ } ,
E 3 μ p ( r ) = μ k 3 2 exp ( i k 3 r ) ( ɛ 1 ɛ 3 ) 1 / 2 r t p exp ( i k 3 t cos θ ) exp ( i k 1 α z ) × ( cos 2 θ p ^ + sin θ cos θ ^ z ) .
E 1 μ s ( r ) = μ k 1 2 exp ( i k 1 r ) ɛ 1 r [ exp ( - i k 1 z cos θ ) + r s exp ( i k 1 z cos θ ) ] s ^ ,
E 3 μ s ( r ) = μ k 3 2 ( - cos θ ) exp ( i k 3 r ) ( ɛ 1 ɛ 3 ) 1 / 2 α r t s exp ( i k 3 t cos θ ) exp ( i k 1 α z ) s ^ .
E 1 μ z ( r ) = μ k 1 2 exp ( i k 1 r ) ɛ 1 r sin θ { - cos θ [ exp ( - i k 1 z cos θ ) + r p exp ( i k 1 z cos θ ) ] p ^ + sin θ [ exp ( - i k 1 z cos θ ) + r p exp ( i k 1 z cos θ ) ] z ^ } ,
E 3 μ z ( r ) = μ k 3 2 ( - cos θ ) exp ( i k 3 r ) ɛ 1 α r t p exp ( i k 3 t cos θ ) exp ( i k 1 α z ) × ( - cos θ sin θ p ^ + sin 2 θ z ^ ) ,
μ = μ p p ^ + μ s s ^ + μ z z ^ ,
μ p = μ sin θ cos ( ϕ - ϕ ) ,
μ s = μ sin θ sin ( ϕ - ϕ ) ,
μ z = μ cos θ .
E ( r , z , θ , ϕ ) E ( r , r ) = sin θ cos ( ϕ - ϕ ) E μ p + sin θ sin ( ϕ - ϕ ) E μ s + cos θ E μ z ,
E ( r , r ) = E p p ^ + E s s ^ + E z z ^ ,
E p = sin θ cos ( ϕ - ϕ ) E p μ p + cos θ E p μ z ,
E s = sin θ sin ( ϕ - ϕ ) E s μ s ,
E z = sin θ cos ( ϕ - ϕ ) E z μ p + cos θ E z μ z .
E 2 = cos 2 θ E μ z 2 + sin 2 θ [ cos 2 ( ϕ - ϕ ) E μ p 2 + sin 2 ( ϕ - ϕ ) E μ s 2 ] + sin θ cos θ cos ( ϕ - ϕ ) [ E p μ p ( E p μ z ) * + E z μ z ( E z μ p ) * + c . c . ] .
( region 1 )             E μ p 2 = μ 2 k 1 4 ɛ 1 2 r 2 cos 2 θ exp ( - i 2 k 1 z cos θ ) - r p 2 ,
( region 3 )             E μ p 2 = μ 2 k 3 4 cos 2 θ ɛ 1 ɛ 3 r 2 t p exp ( i k 1 α z ) 2 ;
( region 1 )             E μ s 2 = μ 2 k 1 4 ɛ 1 2 r 2 exp ( - i 2 k 1 z cos θ ) + r s 2 ,
( region 3 )             E μ s 2 = μ 2 k 3 4 cos 2 θ ɛ 1 ɛ 3 r 2 α 2 t s exp ( i k 1 α z ) 2 ;
( region 1 )             E μ z 2 = μ 2 k 1 4 ɛ 1 2 r 2 sin 2 θ exp ( - i 2 k 1 z cos θ ) + r p 2 ,
( region 3 )             E μ z 2 = μ 2 k 3 4 cos 2 θ sin 2 θ ɛ 1 2 r 2 α 2 t p exp ( i k 1 α z ) 2 ,
P T ( z , θ ) = c k 1 4 2 ɛ 1 3 / 2 Re [ 0 d v v ( 1 - v 2 ) 1 / 2 × ( μ 2 v 2 { 1 + r p exp [ i 2 k 1 z ( 1 - v 2 ) 1 / 2 ] } + μ 2 2 { 1 + r s exp [ i 2 k 1 z ( 1 - v 2 ) 1 / 2 ] } + μ 2 2 ( 1 - v 2 ) { 1 - r p exp [ i 2 k 1 z ( 1 - v 2 ) 1 / 2 ] } ) ] ,
P T = ( z , θ ) = cos 2 θ [ P T ( z ) ] + sin 2 θ [ P T ( z ) ] ,
S ^ ( r , r ) = c ɛ i 1 / 2 8 π { cos 2 θ E μ z 2 + sin 2 θ [ cos 2 ( ϕ - ϕ ) E μ p 2 + sin 2 ( ϕ - ϕ ) E μ s 2 ] cos 2 θ P T ( z ) + sin 2 θ P T ( z ) + sin θ cos θ cos ( ϕ - ϕ ) [ E p μ p ( E p μ z ) * + E z μ z ( E z μ p ) * + c . c . ] cos 2 θ P T ( z ) + sin 2 θ P T ( z ) } .
S ^ ϕ ( z , θ , θ ) 1 2 π d ϕ S ^ .
S ^ ϕ ( z , θ , θ ) = S ^ ϕ ( z , θ ) 1 + η ( z ) tan 2 θ + S ^ ϕ ( z , θ ) 1 + ( η ( z ) tan 2 θ ) - 1 ,
S ^ ϕ ( z , θ ) = c ɛ i 1 / 2 8 π E μ z 2 P T ( z )
S ^ ϕ ( z , θ ) = ½ [ S ^ ϕ p ( z , θ ) + S ^ ϕ s ( z , θ ) ] = 1 2 ( c ɛ i 1 / 2 8 π E μ p 2 P T ( z ) + c ɛ i 1 / 2 8 π E μ s 2 P T ( z ) )
cos 2 θ S ^ ϕ + sin 2 θ S ^ ϕ
Q ( z , θ ) = collected power total dissipated power = r 2 d θ sin θ d ϕ S P T ( z , θ )
= r 2 d θ sin θ 2 π S ^ ϕ ( z , θ )
= ( Q ( z ) 1 + η ( z ) tan 2 θ + Q ( z ) 1 + [ η ( z ) tan 2 θ ] - 1 ) ,
Q , ( z ) = 2 π r 2 d θ sin θ S ^ ϕ , ( z , θ ) .
F = k d z d Ω C ( z , ϕ , θ ) μ ab · E ex 2 Q ( z , θ ) ,
F = k ( μ ab ) 2 d z C ( z ) E ex 2 [ w ( z ) Q ( z ) + w ( z ) Q ( z ) ] ,
w ( z ) = α x 2 w x ( z ) + α y 2 w y ( z ) + α z 2 w z ( z ) ,
w ( z ) = α x 2 w x ( z ) + α y 2 w y ( z ) + α z 2 w z ( z ) ,
E x ex = a x E ex ,             E y ex = α y E ex ,             E z ex = α z E ex ,
E ex = E 0 ex exp ( - z k 1 s / 2 ) ( α x x ^ + α z z ^ ) ,
s 2 = ( ɛ 3 ɛ 1 sin 2 θ inc - 1 ) 1 / 2
α x = - i s / 2 ,             α z = ( ɛ 3 ɛ 1 ) 1 / 2 sin θ inc .
w ( z ) = ( ɛ 3 ɛ 1 sin 2 θ inc - 1 ) w x ( z ) + ɛ 3 ɛ 1 sin 2 θ inc w z ( z ) ,
w ( z ) = ( ɛ 3 ɛ 1 sin 2 θ inc - 1 ) w x ( z ) + ɛ 3 ɛ 1 sin 2 θ inc w z ( z ) .
F = 0 d z exp ( - s k 1 z ) g ( z ) ,
g ( z ) = k ( μ ab ) 2 C ( z ) E 0 ex 2 [ w ( z ) Q ( z ) + w ( z ) Q ( z ) ] .
J ( r ) = k d Ω C ( θ , ϕ ) μ ab · E ex ( z = z d ) 2 S ^ ( r , z = z d , ϕ , θ ) ,
F = k C 0 ( μ ab ) 2 E ex ( z = z d ) 2 [ w ( z = z d ) Q ( z = z d ) + w ( z = z d ) Q ( z = z d ) ] ,
F = k 2 π C 0 ( μ ab ) 2 E z ex ( z = z d ) 2 Q ( z = z d ) .
F = k π C 0 ( μ ab ) 2 ( E x ex ( z = z d ) 2 + E y ex ( z = z d ) 2 Q ( z = z d ) .
p P T ( z ) P T ( z = ) ,
f d Ω S ( Ω , Ω , z ) d Ω S ( Ω , Ω , z = ) ,
1 q = ( 1 q 0 + p - 1 ) / f .
τ = q τ 0 / f .
N . A . > n 1 = ɛ 1 1 / 2 ,
N . A . > Re [ ( ɛ 1 ɛ 2 ɛ 1 + ɛ 2 ) 1 / 2 ] ,
ɛ ( ω ) = ɛ b - ω p 2 ω 2 + ν 2 + i ν ω ω p 2 ω 2 + ν 2
μ ab = μ ab ( sin θ cos ϕ x ^ + sin θ sin ϕ y ^ + cos θ z ^ ) .
E ex = E x ex x ^ + E y ex y ^ + E z ex z ^ .
μ ab · E ex 2 = ( μ ab ) 2 { E x ex 2 sin 2 θ cos 2 ϕ + E y ex 2 sin 2 θ sin 2 ϕ + E z ex 2 cos 2 θ + [ E x ex ( E y ex ) * + c . c . ] sin 2 θ cos ϕ sin ϕ + [ E x ex ( E z ex ) * + c . c . ] sin θ cos θ cos ϕ + [ E y ex ( E z ex ) * + c . c . ] sin θ cos θ sin ϕ } .
w x ( z ) = w y ( z ) = π 0 π d θ sin 3 θ cos 2 θ cos 2 θ + η ( z ) sin 2 θ ,
w x ( z ) = w y ( z ) = π 0 π d θ sin 5 θ sin 2 θ + [ η ( z ) ] - 1 cos 2 θ ,
w z ( z ) = 2 π 0 π d θ sin θ cos 4 θ cos 2 θ + η ( z ) sin 2 θ ,
w z ( z ) = 2 π 0 π d θ sin 3 θ cos 2 θ sin 2 θ + [ η ( z ) ] - 1 cos 2 θ .
w x ( z ) = 2 π 1 - η ( a + 2 3 - a ( a + 1 ) × { 1 ( - a ) 1 / 2 tanh - 1 [ - 1 ( - a ) 1 / 2 ] 1 a 1 / 2 tan - 1 ( 1 a 1 / 2 ) } )             a < 0 , η > 1 a > 0 , η < 1 ,
w x ( z ) = 2 π a ( - a - 5 3 + ( a + 1 ) 2 × { 1 ( - a ) 1 / 2 tanh - 1 [ - 1 ( - a ) 1 / 2 ] 1 a 1 / 2 tan - 1 ( 1 a 1 / 2 ) } )             a < 0 , η > 1 a > 0 , η < 1 ,
w y ( z ) = w x ( z ) ,
w y ( z ) = w x ( z ) ,
w z ( z ) = 4 π 1 - η ( 1 3 - a + a 2 { 1 ( - a ) 1 / 2 tanh - 1 [ - 1 ( - a ) 1 / 2 ] 1 a 1 / 2 tan - 1 ( 1 a 1 / 2 ) } )             a < 0 , η > 1 a > 0 , η < 1 ,
w z ( z ) = 4 π a ( 2 3 + a - a ( a + 1 ) × { 1 ( - a ) 1 / 2 tanh - 1 [ - 1 ( - a ) 1 / 2 ] 1 a 1 / 2 tan - 1 ( 1 a 1 / 2 ) } )             a < 0 , η > 1 a > 0 , η < 1 ,
S ( r , r ) = c ɛ i 1 / 2 8 π [ a + b cos 2 ( ϕ - ϕ ) + c sin 2 ( ϕ - ϕ ) + d cos ( ϕ - ϕ ) ] ,
a = cos 2 θ ( E p μ z 2 + E z μ z 2 ) ,             b = sin 2 θ ( E p μ p 2 + E z μ p 2 ) , c = sin 2 θ E s μ s 2 , d = sin θ cos θ [ E p μ p ( E p μ z ) * + E z μ z ( E z μ p ) * + c . c . ] .
J ( r ) = k d z d Ω C ( z ) μ ab · E ex 2 ( c ɛ i 1 / 2 8 π ) × [ a + b cos 2 ( ϕ - ϕ ) + c sin 2 ( ϕ - ϕ ) + d cos ( ϕ - ϕ ) P T ( z ) cos 2 θ + P T ( z ) sin 2 θ ] .
J ( r ) = k ( μ ab ) 2 d z C ( z ) ( E x ex 2 { w x ( z ) S ^ ϕ ( z , θ ) + w x ( z ) × [ ( ¾ cos 2 ϕ + ¼ sin 2 ϕ ) S ^ ϕ p ( z , θ ) + ( ¼ cos 2 ϕ + ¾ sin 2 ϕ ) S ^ ϕ s ( z , θ ) ] } + E y ex 2 { w y ( z ) S ^ ϕ ( z , θ ) + w y ( z ) × [ ( ¼ cos 2 ϕ + ¾ sin 2 ϕ ) S ^ ϕ p ( z , θ ) + ( ¾ cos 2 ϕ + ¼ sin 2 ϕ ) S ^ ϕ s ( z , θ ) } + E z ex 2 [ w z ( z ) S ^ ϕ ( z , θ ) + w z ( z ) S ^ ϕ ( z , θ ) ] + c ɛ i 1 / 2 w x ( z ) 8 π P T [ E p μ p ( E p μ z ) * + E z μ z ( E z μ p ) * + c . c . ] × { [ E x ex ( E z ex ) * + c . c . ] cos ϕ + [ E y ex ( E z ex ) * + c . c . ] sin ϕ } ) ,
J ( r ) = k ( μ ab ) 2 d z C ( z ) E ex 2 [ w z ( z ) S ^ ϕ ( z , θ ) + w z ( z ) S ^ ϕ ( z , θ ) ] .

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