Abstract

Expressions for the total power radiated by magnetic and electric dipoles of arbitrary orientation located in a medium 1 at distance <i>z</i><sub>0</sub> from the interface to a homogeneous or planar stratified medium 2 are derived. A relation between the normalized powers radiated by magnetic and electric dipoles is established. For a homogeneous loss-free medium 2, curves of the normalized powers <i>L</i>(<i>z</i><sub>0</sub>)/<i>L</i> ∞ radiated by magnetic and electric dipoles versus the normalized distance <i>z</i><sub>0</sub>/λ<sub>1</sub>, are presented for different values of the relative refractive index <i>n</i> = <i>n</i><sub>2</sub>/<i>n</i><sub>1</sub>, as the only parameter. The computer calculations are compared with analytical expressions derived for small and large distances. For <i>n</i> > 1, the contribution of the evanescent waves to the radiated power is calculated separately. We show that the classical results for the normalized radiated power yield the correct normalized spontaneous emission rates from an excited atomic state for electric and magnetic dipole transitions, respectively. We point out that the results for the electric dipole also give the change of the total power scattered by a small dielectric scattering particle when it is placed close to an interface.

© 1978 Optical Society of America

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  1. K. H. Drexhage, M. Fleck, H. Kuhn, F. P. Schäfer, and W. Sperling, Ber. Bunsenges. Phys. Chem. 70, 1179 (1966); K. H. Drexhage, H. Kuhn, and F. P. Schäfer, ibid. 72, 329 (1968).
  2. K. H. Drexhage, Habilitationsschrift (Universität Marburg, 1966).
  3. K. H. Drexhage, "Influence of a dielectric interface on fluorescence decay time," J. Luminescence 1, 2, 693–701 (1970); "Spontaneous emission rate in the presence of a mirror," in Proceedings of the 3rd Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, 1973), p. 187.
  4. K. H. Drexhage, "Interaction of light with monomolecular dye layers, " Progress in Optics, Vol. XII (North-Holland, Amsterdam, 1974), p. 165–232.
  5. K. H. Tews, "Zur Variation von Luminiszenz-Lebensdauern," Ann. Phys. (Leip.) 29, 97–120 (1973).
  6. K. H. Tews, O. Inacker, and H. Kuhn, "Variation of the luminescence lifetime of a molecule near an interface between differently polarizable dielectrics, " Nature (Lond.) 228, 276, erratum 791 (1970).
  7. H. Kuhn, "Classical aspects of energy transfer in molecular systems, " J. Chem. Phys. 53, 101–108 (1970).
  8. K. H. Tews, "On the variation of luminescence lifetimes," J. Lumines. 9, 223–239 (1974).
  9. R. R. Chance, A. Prock, and R. Silbey, "Lifetime of an emitting molecule near a partially reflecting surface, " J. Chem. Phys. 60, 2744-2748 (1974); "Comments on the classical theory of energy transfer," J. Chem. Phys. 62, 2245–2253 (1975).
  10. R. R. Chance, A. Prock, and R. Silbey, "Decay of an emitting dipole between two parallel mirrors, " J. Chem. Phys. 62, 771-772 (1975); R. R. Chance, A. H. Miller, A. Prock, and R. Silbey, "Fluorescence and energy transfer near interfaces: the complete and quantitative description of the Eu+3 / mirror systems," J. Chem. Phys. 63, 1589–1595 (1975).
  11. H. Morawitz, "Self-coupling of a two-level system by a mirror," Phys. Rev. 187, 1792–1796 (1969).
  12. S. R. Barone, "Resonances in the spectral energy density of thermal fields and spontaneous emission, " in Proceedings of the Symposium on Submillimeter Waves (Polytechnic Inst. of Brooklyn, New York, 1970), p. 649–665.
  13. M. R. Philpott, "Fluorescence from molecules between mirrors," Chem. Phys. Lett. 19, 435–439 (1973).
  14. P. W. Milonni and P. L. Knight, "Spontaneous emission between mirrors," Opt. Commun. 9, 119–122 (1973).
  15. G. S. Agarwal, "Quantum electrodynamics in the presence of dielectrics and conductors. III, " Phys. Rev. A 11, 253-264 (1975).
  16. G. S. Agarwal, "Coherence in spontaneous emission in the presence of a dielectric, " Phys. Rev. Lett. 32, 703-706 (1974); "Quantum electrodynamics in the presence of dielectrics and conductors. IV, " Phys. Rev. A 12, 1475-1497 (1975).
  17. A. Sommerfeld, Partielle Differentialgleichungen der Physik (Akademische Verlagsges., Leipzig, 1958), Chap. 36.
  18. A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966).
  19. W. Lukosz and R. E. Kunz, "Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface," Opt. Commun. 20, 195-199 (1977).
  20. See, for instance, Ref. 18, p. 18.
  21. This result is a consequence of our choice of the definition of the reflection coefficients. The reflection coefficient r1,2(s) is the ratio of the magnetic field strengths at the interface of, respectively, the reflected and the incident wave. The reflection coefficient r1,2s is defined analogously in the electric field strengths. In part of the literature the definition of the reflection coefficient for p-polarized light differs from ours by a factor (-1).
  22. R. R. Chance, A. Prock, and R. Silbey, "Comments on the classical theory of energy transfer. H. Extension to higher multipoles and anisotropic media, " J. Chem. Phys. 65, 2527-2531 (1976); 66, 1765 (1977), erratum.
  23. C. K. Carniglia, L. Mandel, and K. H. Drexhage, "Absorption and emission of evanescent photons," J. Opt. Soc. Am. 62, 479-486 (1972).
  24. W. Heitler, The Quantum Theory of Radiation 3rd ed., (Clarendon, Oxford, 1957), Chap. 17.
  25. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 4, Part 1, Relativistic Quantum Theory (Pergamon, New York, 1971), Chap. 45.
  26. Using a different approach Agarwal developed a quantum mechanical theory for electric dipole transitions in Ref. 16. His final expressions for a normalized damping coefficient for atoms in vacuum in front of a homogeneous loss-free dielectric half-space are, for the perpendicular but not for the parallel orientation of the electric dipole moment, equivalent to our results when we insert the Fresnel coefficients into Eqs. (3. 15)-(3. 17) for the electric dipole.
  27. M. Kerker, The Scattering of Light and other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3.2.
  28. For example, we plan to apply Eqs. (3. 15)-(3. 17) to calculate the power radiated by dipoles located close to a planar dielectric waveguide into the modes of the guide.

1977 (1)

W. Lukosz and R. E. Kunz, "Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface," Opt. Commun. 20, 195-199 (1977).

1975 (2)

R. R. Chance, A. Prock, and R. Silbey, "Decay of an emitting dipole between two parallel mirrors, " J. Chem. Phys. 62, 771-772 (1975); R. R. Chance, A. H. Miller, A. Prock, and R. Silbey, "Fluorescence and energy transfer near interfaces: the complete and quantitative description of the Eu+3 / mirror systems," J. Chem. Phys. 63, 1589–1595 (1975).

G. S. Agarwal, "Quantum electrodynamics in the presence of dielectrics and conductors. III, " Phys. Rev. A 11, 253-264 (1975).

1974 (1)

K. H. Tews, "On the variation of luminescence lifetimes," J. Lumines. 9, 223–239 (1974).

1973 (3)

M. R. Philpott, "Fluorescence from molecules between mirrors," Chem. Phys. Lett. 19, 435–439 (1973).

P. W. Milonni and P. L. Knight, "Spontaneous emission between mirrors," Opt. Commun. 9, 119–122 (1973).

K. H. Tews, "Zur Variation von Luminiszenz-Lebensdauern," Ann. Phys. (Leip.) 29, 97–120 (1973).

1972 (1)

1970 (1)

H. Kuhn, "Classical aspects of energy transfer in molecular systems, " J. Chem. Phys. 53, 101–108 (1970).

1969 (1)

H. Morawitz, "Self-coupling of a two-level system by a mirror," Phys. Rev. 187, 1792–1796 (1969).

Agarwal, G. S.

G. S. Agarwal, "Quantum electrodynamics in the presence of dielectrics and conductors. III, " Phys. Rev. A 11, 253-264 (1975).

G. S. Agarwal, "Coherence in spontaneous emission in the presence of a dielectric, " Phys. Rev. Lett. 32, 703-706 (1974); "Quantum electrodynamics in the presence of dielectrics and conductors. IV, " Phys. Rev. A 12, 1475-1497 (1975).

Baños, A.

A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966).

Barone, S. R.

S. R. Barone, "Resonances in the spectral energy density of thermal fields and spontaneous emission, " in Proceedings of the Symposium on Submillimeter Waves (Polytechnic Inst. of Brooklyn, New York, 1970), p. 649–665.

Carniglia, C. K.

Chance, R. R.

R. R. Chance, A. Prock, and R. Silbey, "Decay of an emitting dipole between two parallel mirrors, " J. Chem. Phys. 62, 771-772 (1975); R. R. Chance, A. H. Miller, A. Prock, and R. Silbey, "Fluorescence and energy transfer near interfaces: the complete and quantitative description of the Eu+3 / mirror systems," J. Chem. Phys. 63, 1589–1595 (1975).

R. R. Chance, A. Prock, and R. Silbey, "Lifetime of an emitting molecule near a partially reflecting surface, " J. Chem. Phys. 60, 2744-2748 (1974); "Comments on the classical theory of energy transfer," J. Chem. Phys. 62, 2245–2253 (1975).

R. R. Chance, A. Prock, and R. Silbey, "Comments on the classical theory of energy transfer. H. Extension to higher multipoles and anisotropic media, " J. Chem. Phys. 65, 2527-2531 (1976); 66, 1765 (1977), erratum.

Drexhage, K. H.

C. K. Carniglia, L. Mandel, and K. H. Drexhage, "Absorption and emission of evanescent photons," J. Opt. Soc. Am. 62, 479-486 (1972).

K. H. Drexhage, Habilitationsschrift (Universität Marburg, 1966).

K. H. Drexhage, "Interaction of light with monomolecular dye layers, " Progress in Optics, Vol. XII (North-Holland, Amsterdam, 1974), p. 165–232.

K. H. Drexhage, M. Fleck, H. Kuhn, F. P. Schäfer, and W. Sperling, Ber. Bunsenges. Phys. Chem. 70, 1179 (1966); K. H. Drexhage, H. Kuhn, and F. P. Schäfer, ibid. 72, 329 (1968).

K. H. Drexhage, "Influence of a dielectric interface on fluorescence decay time," J. Luminescence 1, 2, 693–701 (1970); "Spontaneous emission rate in the presence of a mirror," in Proceedings of the 3rd Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, 1973), p. 187.

Fleck, M.

K. H. Drexhage, M. Fleck, H. Kuhn, F. P. Schäfer, and W. Sperling, Ber. Bunsenges. Phys. Chem. 70, 1179 (1966); K. H. Drexhage, H. Kuhn, and F. P. Schäfer, ibid. 72, 329 (1968).

Heitler, W.

W. Heitler, The Quantum Theory of Radiation 3rd ed., (Clarendon, Oxford, 1957), Chap. 17.

Inacker, O.

K. H. Tews, O. Inacker, and H. Kuhn, "Variation of the luminescence lifetime of a molecule near an interface between differently polarizable dielectrics, " Nature (Lond.) 228, 276, erratum 791 (1970).

Kerker, M.

M. Kerker, The Scattering of Light and other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3.2.

Knight, P. L.

P. W. Milonni and P. L. Knight, "Spontaneous emission between mirrors," Opt. Commun. 9, 119–122 (1973).

Kuhn, H.

H. Kuhn, "Classical aspects of energy transfer in molecular systems, " J. Chem. Phys. 53, 101–108 (1970).

K. H. Drexhage, M. Fleck, H. Kuhn, F. P. Schäfer, and W. Sperling, Ber. Bunsenges. Phys. Chem. 70, 1179 (1966); K. H. Drexhage, H. Kuhn, and F. P. Schäfer, ibid. 72, 329 (1968).

K. H. Tews, O. Inacker, and H. Kuhn, "Variation of the luminescence lifetime of a molecule near an interface between differently polarizable dielectrics, " Nature (Lond.) 228, 276, erratum 791 (1970).

Kunz, R. E.

W. Lukosz and R. E. Kunz, "Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface," Opt. Commun. 20, 195-199 (1977).

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 4, Part 1, Relativistic Quantum Theory (Pergamon, New York, 1971), Chap. 45.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 4, Part 1, Relativistic Quantum Theory (Pergamon, New York, 1971), Chap. 45.

Lukosz, W.

W. Lukosz and R. E. Kunz, "Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface," Opt. Commun. 20, 195-199 (1977).

Mandel, L.

Milonni, P. W.

P. W. Milonni and P. L. Knight, "Spontaneous emission between mirrors," Opt. Commun. 9, 119–122 (1973).

Morawitz, H.

H. Morawitz, "Self-coupling of a two-level system by a mirror," Phys. Rev. 187, 1792–1796 (1969).

Philpott, M. R.

M. R. Philpott, "Fluorescence from molecules between mirrors," Chem. Phys. Lett. 19, 435–439 (1973).

Prock, A.

R. R. Chance, A. Prock, and R. Silbey, "Decay of an emitting dipole between two parallel mirrors, " J. Chem. Phys. 62, 771-772 (1975); R. R. Chance, A. H. Miller, A. Prock, and R. Silbey, "Fluorescence and energy transfer near interfaces: the complete and quantitative description of the Eu+3 / mirror systems," J. Chem. Phys. 63, 1589–1595 (1975).

R. R. Chance, A. Prock, and R. Silbey, "Lifetime of an emitting molecule near a partially reflecting surface, " J. Chem. Phys. 60, 2744-2748 (1974); "Comments on the classical theory of energy transfer," J. Chem. Phys. 62, 2245–2253 (1975).

R. R. Chance, A. Prock, and R. Silbey, "Comments on the classical theory of energy transfer. H. Extension to higher multipoles and anisotropic media, " J. Chem. Phys. 65, 2527-2531 (1976); 66, 1765 (1977), erratum.

Schäfer, F. P.

K. H. Drexhage, M. Fleck, H. Kuhn, F. P. Schäfer, and W. Sperling, Ber. Bunsenges. Phys. Chem. 70, 1179 (1966); K. H. Drexhage, H. Kuhn, and F. P. Schäfer, ibid. 72, 329 (1968).

Silbey, R.

R. R. Chance, A. Prock, and R. Silbey, "Decay of an emitting dipole between two parallel mirrors, " J. Chem. Phys. 62, 771-772 (1975); R. R. Chance, A. H. Miller, A. Prock, and R. Silbey, "Fluorescence and energy transfer near interfaces: the complete and quantitative description of the Eu+3 / mirror systems," J. Chem. Phys. 63, 1589–1595 (1975).

R. R. Chance, A. Prock, and R. Silbey, "Lifetime of an emitting molecule near a partially reflecting surface, " J. Chem. Phys. 60, 2744-2748 (1974); "Comments on the classical theory of energy transfer," J. Chem. Phys. 62, 2245–2253 (1975).

R. R. Chance, A. Prock, and R. Silbey, "Comments on the classical theory of energy transfer. H. Extension to higher multipoles and anisotropic media, " J. Chem. Phys. 65, 2527-2531 (1976); 66, 1765 (1977), erratum.

Sommerfeld, A.

A. Sommerfeld, Partielle Differentialgleichungen der Physik (Akademische Verlagsges., Leipzig, 1958), Chap. 36.

Sperling, W.

K. H. Drexhage, M. Fleck, H. Kuhn, F. P. Schäfer, and W. Sperling, Ber. Bunsenges. Phys. Chem. 70, 1179 (1966); K. H. Drexhage, H. Kuhn, and F. P. Schäfer, ibid. 72, 329 (1968).

Tews, K. H.

K. H. Tews, "On the variation of luminescence lifetimes," J. Lumines. 9, 223–239 (1974).

K. H. Tews, "Zur Variation von Luminiszenz-Lebensdauern," Ann. Phys. (Leip.) 29, 97–120 (1973).

K. H. Tews, O. Inacker, and H. Kuhn, "Variation of the luminescence lifetime of a molecule near an interface between differently polarizable dielectrics, " Nature (Lond.) 228, 276, erratum 791 (1970).

Ann. Phys. Leip. (1)

K. H. Tews, "Zur Variation von Luminiszenz-Lebensdauern," Ann. Phys. (Leip.) 29, 97–120 (1973).

Chem. Phys. Lett. (1)

M. R. Philpott, "Fluorescence from molecules between mirrors," Chem. Phys. Lett. 19, 435–439 (1973).

J. Chem. Phys. (2)

R. R. Chance, A. Prock, and R. Silbey, "Decay of an emitting dipole between two parallel mirrors, " J. Chem. Phys. 62, 771-772 (1975); R. R. Chance, A. H. Miller, A. Prock, and R. Silbey, "Fluorescence and energy transfer near interfaces: the complete and quantitative description of the Eu+3 / mirror systems," J. Chem. Phys. 63, 1589–1595 (1975).

H. Kuhn, "Classical aspects of energy transfer in molecular systems, " J. Chem. Phys. 53, 101–108 (1970).

J. Lumines. (1)

K. H. Tews, "On the variation of luminescence lifetimes," J. Lumines. 9, 223–239 (1974).

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

P. W. Milonni and P. L. Knight, "Spontaneous emission between mirrors," Opt. Commun. 9, 119–122 (1973).

W. Lukosz and R. E. Kunz, "Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface," Opt. Commun. 20, 195-199 (1977).

Phys. Rev. (2)

G. S. Agarwal, "Quantum electrodynamics in the presence of dielectrics and conductors. III, " Phys. Rev. A 11, 253-264 (1975).

H. Morawitz, "Self-coupling of a two-level system by a mirror," Phys. Rev. 187, 1792–1796 (1969).

Other (18)

S. R. Barone, "Resonances in the spectral energy density of thermal fields and spontaneous emission, " in Proceedings of the Symposium on Submillimeter Waves (Polytechnic Inst. of Brooklyn, New York, 1970), p. 649–665.

G. S. Agarwal, "Coherence in spontaneous emission in the presence of a dielectric, " Phys. Rev. Lett. 32, 703-706 (1974); "Quantum electrodynamics in the presence of dielectrics and conductors. IV, " Phys. Rev. A 12, 1475-1497 (1975).

A. Sommerfeld, Partielle Differentialgleichungen der Physik (Akademische Verlagsges., Leipzig, 1958), Chap. 36.

A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966).

See, for instance, Ref. 18, p. 18.

This result is a consequence of our choice of the definition of the reflection coefficients. The reflection coefficient r1,2(s) is the ratio of the magnetic field strengths at the interface of, respectively, the reflected and the incident wave. The reflection coefficient r1,2s is defined analogously in the electric field strengths. In part of the literature the definition of the reflection coefficient for p-polarized light differs from ours by a factor (-1).

R. R. Chance, A. Prock, and R. Silbey, "Comments on the classical theory of energy transfer. H. Extension to higher multipoles and anisotropic media, " J. Chem. Phys. 65, 2527-2531 (1976); 66, 1765 (1977), erratum.

W. Heitler, The Quantum Theory of Radiation 3rd ed., (Clarendon, Oxford, 1957), Chap. 17.

L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 4, Part 1, Relativistic Quantum Theory (Pergamon, New York, 1971), Chap. 45.

Using a different approach Agarwal developed a quantum mechanical theory for electric dipole transitions in Ref. 16. His final expressions for a normalized damping coefficient for atoms in vacuum in front of a homogeneous loss-free dielectric half-space are, for the perpendicular but not for the parallel orientation of the electric dipole moment, equivalent to our results when we insert the Fresnel coefficients into Eqs. (3. 15)-(3. 17) for the electric dipole.

M. Kerker, The Scattering of Light and other Electromagnetic Radiation (Academic, New York, 1969), Chap. 3.2.

For example, we plan to apply Eqs. (3. 15)-(3. 17) to calculate the power radiated by dipoles located close to a planar dielectric waveguide into the modes of the guide.

R. R. Chance, A. Prock, and R. Silbey, "Lifetime of an emitting molecule near a partially reflecting surface, " J. Chem. Phys. 60, 2744-2748 (1974); "Comments on the classical theory of energy transfer," J. Chem. Phys. 62, 2245–2253 (1975).

K. H. Tews, O. Inacker, and H. Kuhn, "Variation of the luminescence lifetime of a molecule near an interface between differently polarizable dielectrics, " Nature (Lond.) 228, 276, erratum 791 (1970).

K. H. Drexhage, M. Fleck, H. Kuhn, F. P. Schäfer, and W. Sperling, Ber. Bunsenges. Phys. Chem. 70, 1179 (1966); K. H. Drexhage, H. Kuhn, and F. P. Schäfer, ibid. 72, 329 (1968).

K. H. Drexhage, Habilitationsschrift (Universität Marburg, 1966).

K. H. Drexhage, "Influence of a dielectric interface on fluorescence decay time," J. Luminescence 1, 2, 693–701 (1970); "Spontaneous emission rate in the presence of a mirror," in Proceedings of the 3rd Rochester Conference on Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (Plenum, New York, 1973), p. 187.

K. H. Drexhage, "Interaction of light with monomolecular dye layers, " Progress in Optics, Vol. XII (North-Holland, Amsterdam, 1974), p. 165–232.

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