Abstract

Coherent anti-Stokes Raman scattering signals may be strongly enhanced when the active molecules are located near the surface of a small silver particle. The theoretical analysis is similar to the electrodynamic mechanism for surface-enhanced Raman scattering, except that there are four instead of two electric fields that stimulate collective electron oscillations within the particle. The general analysis is presented for a sphere of arbitrary size, for arbitrary angle between pump and probe beams, and for arbitrary polarization between pump and probe beams. This is then specialized to the small-particle limit for numerical computation. The peak enhancement for a monolayer of benzene on a silver particle (excitation wavelength 404 nm, Raman shift 992 cm−1) is 1012 when both incident beams are polarized perpendicular to the incident plane and 1021 when these beams are cross polarized. These values are averaged over scattering angle. While the coherent anti-Stokes Raman spectroscopy amplitudes depend on scattering angle, only the enhancement factor for one of the cross-polarized components depends on scattering angle. Enhanced signals from a silver organosol (silver dispersed in neat benzene) should be measurable.

© 1984 Optical Society of America

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  1. H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
    [Crossref]
  2. M. Kerker, P. J. McNulty, M. Sculley, H. Chew, and D. D. Cooke, “Raman fluorescent scattering by molecules embedded in small particles: numerical results for incoherent processes,” J. Opt. Soc. Am. 68, 1676–1685 (1978).
    [Crossref]
  3. H. Chew, M. Sculley, M. Kerker, P. J. McNulty, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: results for coherent optical processes,” J. Opt. Soc. Am. 68, 1686–1689 (1978).
    [Crossref]
  4. M. Kerker and S. D. Druger, “Raman and fluorescent scattering by molecules embedded in spheres with radii up to several multiples of the wavelength,” AppI. Opt. 18, 1172–1179 (1979).
    [Crossref]
  5. H. Chew, M. Kerker, and P. J. McNulty, “Raman and fluorescent scattering by molecules embedded in concentric spheres,” J. Opt. Soc. Am. 66, 440–444 (1976).
    [Crossref]
  6. H. Chew, D. D. Cooke, and M. Kerker, “Raman and fluorescent scattering by molecules embedded in dielectric cylinders,” Appl. Opt. 19, 44–52 (1980).
    [Crossref] [PubMed]
  7. D.-S. Wang, M. Kerker, and H. Chew, “Raman and fluorescent scattering by molecules embedded in dielectric spheroids,” Appl. Opt. 19, 2315–2328 (1980).
    [Crossref] [PubMed]
  8. J. P. Kratohvil, M.-P. Lee, and M. Kerker, “Angular distribution of fluorescence from small particles,” Appl. Opt. 17, 1978–1980 (1978).
    [Crossref] [PubMed]
  9. E.-H. Lee, R. E. Benner, J. B. Fenn, and R. K. Chang, “Angular distribution of fluorescence from monodispersed particles,” Appl. Opt. 17, 1980–1982 (1978).
    [Crossref]
  10. P. J. McNulty, S. D. Druger, M. Kerker, and H. Chew, “Fluorescent scattering by anisotropic molecules embedded in small particles,” Appl. Opt. 18, 1484–1486 (1979).
    [Crossref] [PubMed]
  11. R. E. Benner, R. Dornhaus, M. B. Long, and R. K. Chang, “Inelastic light scattering from a distribution of microparticles,” in Microbeam Analysis, D. E. Newbury, ed. (San Francisco Press, San Francisco, Calif., 1979), pp. 191–195.
  12. R. E. Benner, J. F. Owen, and R. K. Chang, “Radiation patterns of inelastic reemission from microparticles in homogeneous surroundings and near dielectric or metal interfaces,” J. Phys. Chem. 84, 1602–1606 (1980).
    [Crossref]
  13. M. Kerker, M. van Dilla, A. Brunsting, J. P. Kratohvil, P. Hsu, D.-S. Wang, J. Gray, and R. Langlois, “Is the central dogma of flow cytometry true: that fluorescence intensity is proportional to cellular dye content?” Cytometry 3, 71–78 (1982).
    [Crossref] [PubMed]
  14. D.-S. Wang, H. Chew, and M. Kerker, “Enhanced Raman scattering at the surface (SERS) of a spherical particle,” Appl. Opt. 19, 2256–2257 (1980).
    [Crossref] [PubMed]
  15. M. Kerker, D.-S. Wang, and H. Chew, “Surface enhanced Raman scattering (SERS) by molecules adsorbed at spherical particles,” Appl. Opt. 19, 4159–4174 (1980). [This paper is a corrected version of the paper appearing on p. 3373 of that volume, which contains a large number of printer’s errors.]
    [Crossref] [PubMed]
  16. J. A. Creighton, C. G. Blatchford, and M. G. Albrecht, “Plasma resonance enhancement of Raman scattering by pyridine adsorbed on silver or gold sol particles of size comparable to the excitation wavelength,” Faraday Discuss. Chem. Soc. 275, 790 (1979).
  17. M. Kerker, O. Siiman, L. A. Bumm, and D.-S. Wang, “Surface enhanced Raman scattering (SERS) of citrate ion adsorbed on colloidal silver,” Appl. Opt. 19, 3253–3255 (1980).
    [Crossref] [PubMed]
  18. H. Wetzel and H. Gerischer, “Surface enhanced Raman scattering from pyridine and halide ions adsorbed on silver and gold sol particles,” Chem. Phys. Lett. 76, 460–464 (1980).
    [Crossref]
  19. H. Abe, K. Manzel, W. Schulze, M. Moscovits, and D. P. DiLello, “Surface enhanced Raman spectroscopy of CO adsorbed on colloidal silver particles,” J. Chem. Phys. 74, 792–797 (1981).
    [Crossref]
  20. M. E. Lippitsch, “Surface enhanced Raman spectra of biliverdine and pyrromethenone adsorbed to silver colloids,” Chem. Phys. Lett. 79, 224–226 (1981).
    [Crossref]
  21. J. R. Lombardi, E. A. Shields Knight, and R. L. Birke, “Evidence for a Ag adatom-molecule complex in surface enhanced Raman scattering,” Chem. Phys. Lett. 79, 214–218 (1981).
    [Crossref]
  22. D. Akins, “Surface enhanced Raman scattering of 2,2′-cyanine on colloidal silver,” J. Colloid Interface Sci. 90, 373–379 (1982).
    [Crossref]
  23. O. Siiman, L. A. Bumm, R. Callaghan, C. G. Blatchford, and M. Kerker, “Surface-enhanced Raman scattering by citrate on colloidal silver,” J. Phys. Chem. 87, 1014–1023 (1983).
    [Crossref]
  24. J. Gersten and A. Nitzan, “Electromagnetic theory of enhanced Raman scattering by molecules adsorbed on rough surfaces,” J. Chem. Phys. 73, 3023–3037 (1980).
    [Crossref]
  25. F. J. Adrian, “Surface enhanced Raman scattering by surface plasmon enhancement of electromagnetic fields near spheroidal particles on a roughened metal surface,” Chem. Phys. Lett. 78, 45–49 (1981).
    [Crossref]
  26. D.-S. Wang and M. Kerker, “Enhanced Raman scattering by molecules adsorbed at the surface of colloidal spheroids,” Phys. Rev. B 24, 1777–1790 (1981).
    [Crossref]
  27. D.-S. Wang and M. Kerker, “Absorption and luminescence of dye coated silver and gold particles,” Phys. Rev. B 25, 2433–2449 (1982).
    [Crossref]
  28. M. Kerker and C. G. Blatchford, “Elastic scattering, adsorption and surface enhanced Raman scattering by concentric spheres comprised of a metallic and a dielectric region,” Phys. Rev. B 26, 4052–4063 (1982).
    [Crossref]
  29. M. Faraday, “Experimental relations of gold (and other metals) to light,” Phil. Trans. R. Soc. London 147, 145–181 (1857).
    [Crossref]
  30. G. Mie, “Beitrage zur Optik trüber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 378–445 (1908).
  31. J. Cooney and A. Gross, “Coherent anti-Stokes Raman scattering by droplets in the Mie size range,” Opt. Lett. 7, 218–226 (1982).
    [Crossref] [PubMed]
  32. A. Gross, “Coherent anti-Stokes Raman scattering by droplets of the Mie size range,” Ph.D. thesis (Drexel University, Philadelphia, Pa., June1982).
  33. For a review, see M. Kerker, Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  34. H. Chew, M. Kerker, and D. D. Cooke, “Electromagnetic scattering by a dielectric sphere and in diverging radiation field,” Phys. Rev. A 16, 320–323 (1977).
    [Crossref]
  35. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [Crossref]

1983 (1)

O. Siiman, L. A. Bumm, R. Callaghan, C. G. Blatchford, and M. Kerker, “Surface-enhanced Raman scattering by citrate on colloidal silver,” J. Phys. Chem. 87, 1014–1023 (1983).
[Crossref]

1982 (5)

D. Akins, “Surface enhanced Raman scattering of 2,2′-cyanine on colloidal silver,” J. Colloid Interface Sci. 90, 373–379 (1982).
[Crossref]

D.-S. Wang and M. Kerker, “Absorption and luminescence of dye coated silver and gold particles,” Phys. Rev. B 25, 2433–2449 (1982).
[Crossref]

M. Kerker and C. G. Blatchford, “Elastic scattering, adsorption and surface enhanced Raman scattering by concentric spheres comprised of a metallic and a dielectric region,” Phys. Rev. B 26, 4052–4063 (1982).
[Crossref]

J. Cooney and A. Gross, “Coherent anti-Stokes Raman scattering by droplets in the Mie size range,” Opt. Lett. 7, 218–226 (1982).
[Crossref] [PubMed]

M. Kerker, M. van Dilla, A. Brunsting, J. P. Kratohvil, P. Hsu, D.-S. Wang, J. Gray, and R. Langlois, “Is the central dogma of flow cytometry true: that fluorescence intensity is proportional to cellular dye content?” Cytometry 3, 71–78 (1982).
[Crossref] [PubMed]

1981 (5)

H. Abe, K. Manzel, W. Schulze, M. Moscovits, and D. P. DiLello, “Surface enhanced Raman spectroscopy of CO adsorbed on colloidal silver particles,” J. Chem. Phys. 74, 792–797 (1981).
[Crossref]

M. E. Lippitsch, “Surface enhanced Raman spectra of biliverdine and pyrromethenone adsorbed to silver colloids,” Chem. Phys. Lett. 79, 224–226 (1981).
[Crossref]

J. R. Lombardi, E. A. Shields Knight, and R. L. Birke, “Evidence for a Ag adatom-molecule complex in surface enhanced Raman scattering,” Chem. Phys. Lett. 79, 214–218 (1981).
[Crossref]

F. J. Adrian, “Surface enhanced Raman scattering by surface plasmon enhancement of electromagnetic fields near spheroidal particles on a roughened metal surface,” Chem. Phys. Lett. 78, 45–49 (1981).
[Crossref]

D.-S. Wang and M. Kerker, “Enhanced Raman scattering by molecules adsorbed at the surface of colloidal spheroids,” Phys. Rev. B 24, 1777–1790 (1981).
[Crossref]

1980 (8)

J. Gersten and A. Nitzan, “Electromagnetic theory of enhanced Raman scattering by molecules adsorbed on rough surfaces,” J. Chem. Phys. 73, 3023–3037 (1980).
[Crossref]

R. E. Benner, J. F. Owen, and R. K. Chang, “Radiation patterns of inelastic reemission from microparticles in homogeneous surroundings and near dielectric or metal interfaces,” J. Phys. Chem. 84, 1602–1606 (1980).
[Crossref]

D.-S. Wang, H. Chew, and M. Kerker, “Enhanced Raman scattering at the surface (SERS) of a spherical particle,” Appl. Opt. 19, 2256–2257 (1980).
[Crossref] [PubMed]

M. Kerker, D.-S. Wang, and H. Chew, “Surface enhanced Raman scattering (SERS) by molecules adsorbed at spherical particles,” Appl. Opt. 19, 4159–4174 (1980). [This paper is a corrected version of the paper appearing on p. 3373 of that volume, which contains a large number of printer’s errors.]
[Crossref] [PubMed]

M. Kerker, O. Siiman, L. A. Bumm, and D.-S. Wang, “Surface enhanced Raman scattering (SERS) of citrate ion adsorbed on colloidal silver,” Appl. Opt. 19, 3253–3255 (1980).
[Crossref] [PubMed]

H. Wetzel and H. Gerischer, “Surface enhanced Raman scattering from pyridine and halide ions adsorbed on silver and gold sol particles,” Chem. Phys. Lett. 76, 460–464 (1980).
[Crossref]

H. Chew, D. D. Cooke, and M. Kerker, “Raman and fluorescent scattering by molecules embedded in dielectric cylinders,” Appl. Opt. 19, 44–52 (1980).
[Crossref] [PubMed]

D.-S. Wang, M. Kerker, and H. Chew, “Raman and fluorescent scattering by molecules embedded in dielectric spheroids,” Appl. Opt. 19, 2315–2328 (1980).
[Crossref] [PubMed]

1979 (3)

M. Kerker and S. D. Druger, “Raman and fluorescent scattering by molecules embedded in spheres with radii up to several multiples of the wavelength,” AppI. Opt. 18, 1172–1179 (1979).
[Crossref]

P. J. McNulty, S. D. Druger, M. Kerker, and H. Chew, “Fluorescent scattering by anisotropic molecules embedded in small particles,” Appl. Opt. 18, 1484–1486 (1979).
[Crossref] [PubMed]

J. A. Creighton, C. G. Blatchford, and M. G. Albrecht, “Plasma resonance enhancement of Raman scattering by pyridine adsorbed on silver or gold sol particles of size comparable to the excitation wavelength,” Faraday Discuss. Chem. Soc. 275, 790 (1979).

1978 (4)

1977 (1)

H. Chew, M. Kerker, and D. D. Cooke, “Electromagnetic scattering by a dielectric sphere and in diverging radiation field,” Phys. Rev. A 16, 320–323 (1977).
[Crossref]

1976 (2)

H. Chew, M. Kerker, and P. J. McNulty, “Raman and fluorescent scattering by molecules embedded in concentric spheres,” J. Opt. Soc. Am. 66, 440–444 (1976).
[Crossref]

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[Crossref]

1972 (1)

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

1908 (1)

G. Mie, “Beitrage zur Optik trüber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 378–445 (1908).

1857 (1)

M. Faraday, “Experimental relations of gold (and other metals) to light,” Phil. Trans. R. Soc. London 147, 145–181 (1857).
[Crossref]

Abe, H.

H. Abe, K. Manzel, W. Schulze, M. Moscovits, and D. P. DiLello, “Surface enhanced Raman spectroscopy of CO adsorbed on colloidal silver particles,” J. Chem. Phys. 74, 792–797 (1981).
[Crossref]

Adrian, F. J.

F. J. Adrian, “Surface enhanced Raman scattering by surface plasmon enhancement of electromagnetic fields near spheroidal particles on a roughened metal surface,” Chem. Phys. Lett. 78, 45–49 (1981).
[Crossref]

Akins, D.

D. Akins, “Surface enhanced Raman scattering of 2,2′-cyanine on colloidal silver,” J. Colloid Interface Sci. 90, 373–379 (1982).
[Crossref]

Albrecht, M. G.

J. A. Creighton, C. G. Blatchford, and M. G. Albrecht, “Plasma resonance enhancement of Raman scattering by pyridine adsorbed on silver or gold sol particles of size comparable to the excitation wavelength,” Faraday Discuss. Chem. Soc. 275, 790 (1979).

Benner, R. E.

R. E. Benner, J. F. Owen, and R. K. Chang, “Radiation patterns of inelastic reemission from microparticles in homogeneous surroundings and near dielectric or metal interfaces,” J. Phys. Chem. 84, 1602–1606 (1980).
[Crossref]

E.-H. Lee, R. E. Benner, J. B. Fenn, and R. K. Chang, “Angular distribution of fluorescence from monodispersed particles,” Appl. Opt. 17, 1980–1982 (1978).
[Crossref]

R. E. Benner, R. Dornhaus, M. B. Long, and R. K. Chang, “Inelastic light scattering from a distribution of microparticles,” in Microbeam Analysis, D. E. Newbury, ed. (San Francisco Press, San Francisco, Calif., 1979), pp. 191–195.

Birke, R. L.

J. R. Lombardi, E. A. Shields Knight, and R. L. Birke, “Evidence for a Ag adatom-molecule complex in surface enhanced Raman scattering,” Chem. Phys. Lett. 79, 214–218 (1981).
[Crossref]

Blatchford, C. G.

O. Siiman, L. A. Bumm, R. Callaghan, C. G. Blatchford, and M. Kerker, “Surface-enhanced Raman scattering by citrate on colloidal silver,” J. Phys. Chem. 87, 1014–1023 (1983).
[Crossref]

M. Kerker and C. G. Blatchford, “Elastic scattering, adsorption and surface enhanced Raman scattering by concentric spheres comprised of a metallic and a dielectric region,” Phys. Rev. B 26, 4052–4063 (1982).
[Crossref]

J. A. Creighton, C. G. Blatchford, and M. G. Albrecht, “Plasma resonance enhancement of Raman scattering by pyridine adsorbed on silver or gold sol particles of size comparable to the excitation wavelength,” Faraday Discuss. Chem. Soc. 275, 790 (1979).

Brunsting, A.

M. Kerker, M. van Dilla, A. Brunsting, J. P. Kratohvil, P. Hsu, D.-S. Wang, J. Gray, and R. Langlois, “Is the central dogma of flow cytometry true: that fluorescence intensity is proportional to cellular dye content?” Cytometry 3, 71–78 (1982).
[Crossref] [PubMed]

Bumm, L. A.

O. Siiman, L. A. Bumm, R. Callaghan, C. G. Blatchford, and M. Kerker, “Surface-enhanced Raman scattering by citrate on colloidal silver,” J. Phys. Chem. 87, 1014–1023 (1983).
[Crossref]

M. Kerker, O. Siiman, L. A. Bumm, and D.-S. Wang, “Surface enhanced Raman scattering (SERS) of citrate ion adsorbed on colloidal silver,” Appl. Opt. 19, 3253–3255 (1980).
[Crossref] [PubMed]

Callaghan, R.

O. Siiman, L. A. Bumm, R. Callaghan, C. G. Blatchford, and M. Kerker, “Surface-enhanced Raman scattering by citrate on colloidal silver,” J. Phys. Chem. 87, 1014–1023 (1983).
[Crossref]

Chang, R. K.

R. E. Benner, J. F. Owen, and R. K. Chang, “Radiation patterns of inelastic reemission from microparticles in homogeneous surroundings and near dielectric or metal interfaces,” J. Phys. Chem. 84, 1602–1606 (1980).
[Crossref]

E.-H. Lee, R. E. Benner, J. B. Fenn, and R. K. Chang, “Angular distribution of fluorescence from monodispersed particles,” Appl. Opt. 17, 1980–1982 (1978).
[Crossref]

R. E. Benner, R. Dornhaus, M. B. Long, and R. K. Chang, “Inelastic light scattering from a distribution of microparticles,” in Microbeam Analysis, D. E. Newbury, ed. (San Francisco Press, San Francisco, Calif., 1979), pp. 191–195.

Chew, H.

D.-S. Wang, M. Kerker, and H. Chew, “Raman and fluorescent scattering by molecules embedded in dielectric spheroids,” Appl. Opt. 19, 2315–2328 (1980).
[Crossref] [PubMed]

H. Chew, D. D. Cooke, and M. Kerker, “Raman and fluorescent scattering by molecules embedded in dielectric cylinders,” Appl. Opt. 19, 44–52 (1980).
[Crossref] [PubMed]

M. Kerker, D.-S. Wang, and H. Chew, “Surface enhanced Raman scattering (SERS) by molecules adsorbed at spherical particles,” Appl. Opt. 19, 4159–4174 (1980). [This paper is a corrected version of the paper appearing on p. 3373 of that volume, which contains a large number of printer’s errors.]
[Crossref] [PubMed]

D.-S. Wang, H. Chew, and M. Kerker, “Enhanced Raman scattering at the surface (SERS) of a spherical particle,” Appl. Opt. 19, 2256–2257 (1980).
[Crossref] [PubMed]

P. J. McNulty, S. D. Druger, M. Kerker, and H. Chew, “Fluorescent scattering by anisotropic molecules embedded in small particles,” Appl. Opt. 18, 1484–1486 (1979).
[Crossref] [PubMed]

H. Chew, M. Sculley, M. Kerker, P. J. McNulty, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: results for coherent optical processes,” J. Opt. Soc. Am. 68, 1686–1689 (1978).
[Crossref]

M. Kerker, P. J. McNulty, M. Sculley, H. Chew, and D. D. Cooke, “Raman fluorescent scattering by molecules embedded in small particles: numerical results for incoherent processes,” J. Opt. Soc. Am. 68, 1676–1685 (1978).
[Crossref]

H. Chew, M. Kerker, and D. D. Cooke, “Electromagnetic scattering by a dielectric sphere and in diverging radiation field,” Phys. Rev. A 16, 320–323 (1977).
[Crossref]

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[Crossref]

H. Chew, M. Kerker, and P. J. McNulty, “Raman and fluorescent scattering by molecules embedded in concentric spheres,” J. Opt. Soc. Am. 66, 440–444 (1976).
[Crossref]

Christy, R. W.

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

Cooke, D. D.

Cooney, J.

Creighton, J. A.

J. A. Creighton, C. G. Blatchford, and M. G. Albrecht, “Plasma resonance enhancement of Raman scattering by pyridine adsorbed on silver or gold sol particles of size comparable to the excitation wavelength,” Faraday Discuss. Chem. Soc. 275, 790 (1979).

DiLello, D. P.

H. Abe, K. Manzel, W. Schulze, M. Moscovits, and D. P. DiLello, “Surface enhanced Raman spectroscopy of CO adsorbed on colloidal silver particles,” J. Chem. Phys. 74, 792–797 (1981).
[Crossref]

Dornhaus, R.

R. E. Benner, R. Dornhaus, M. B. Long, and R. K. Chang, “Inelastic light scattering from a distribution of microparticles,” in Microbeam Analysis, D. E. Newbury, ed. (San Francisco Press, San Francisco, Calif., 1979), pp. 191–195.

Druger, S. D.

M. Kerker and S. D. Druger, “Raman and fluorescent scattering by molecules embedded in spheres with radii up to several multiples of the wavelength,” AppI. Opt. 18, 1172–1179 (1979).
[Crossref]

P. J. McNulty, S. D. Druger, M. Kerker, and H. Chew, “Fluorescent scattering by anisotropic molecules embedded in small particles,” Appl. Opt. 18, 1484–1486 (1979).
[Crossref] [PubMed]

Faraday, M.

M. Faraday, “Experimental relations of gold (and other metals) to light,” Phil. Trans. R. Soc. London 147, 145–181 (1857).
[Crossref]

Fenn, J. B.

Gerischer, H.

H. Wetzel and H. Gerischer, “Surface enhanced Raman scattering from pyridine and halide ions adsorbed on silver and gold sol particles,” Chem. Phys. Lett. 76, 460–464 (1980).
[Crossref]

Gersten, J.

J. Gersten and A. Nitzan, “Electromagnetic theory of enhanced Raman scattering by molecules adsorbed on rough surfaces,” J. Chem. Phys. 73, 3023–3037 (1980).
[Crossref]

Gray, J.

M. Kerker, M. van Dilla, A. Brunsting, J. P. Kratohvil, P. Hsu, D.-S. Wang, J. Gray, and R. Langlois, “Is the central dogma of flow cytometry true: that fluorescence intensity is proportional to cellular dye content?” Cytometry 3, 71–78 (1982).
[Crossref] [PubMed]

Gross, A.

J. Cooney and A. Gross, “Coherent anti-Stokes Raman scattering by droplets in the Mie size range,” Opt. Lett. 7, 218–226 (1982).
[Crossref] [PubMed]

A. Gross, “Coherent anti-Stokes Raman scattering by droplets of the Mie size range,” Ph.D. thesis (Drexel University, Philadelphia, Pa., June1982).

Hsu, P.

M. Kerker, M. van Dilla, A. Brunsting, J. P. Kratohvil, P. Hsu, D.-S. Wang, J. Gray, and R. Langlois, “Is the central dogma of flow cytometry true: that fluorescence intensity is proportional to cellular dye content?” Cytometry 3, 71–78 (1982).
[Crossref] [PubMed]

Johnson, P. B.

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

Kerker, M.

O. Siiman, L. A. Bumm, R. Callaghan, C. G. Blatchford, and M. Kerker, “Surface-enhanced Raman scattering by citrate on colloidal silver,” J. Phys. Chem. 87, 1014–1023 (1983).
[Crossref]

M. Kerker, M. van Dilla, A. Brunsting, J. P. Kratohvil, P. Hsu, D.-S. Wang, J. Gray, and R. Langlois, “Is the central dogma of flow cytometry true: that fluorescence intensity is proportional to cellular dye content?” Cytometry 3, 71–78 (1982).
[Crossref] [PubMed]

D.-S. Wang and M. Kerker, “Absorption and luminescence of dye coated silver and gold particles,” Phys. Rev. B 25, 2433–2449 (1982).
[Crossref]

M. Kerker and C. G. Blatchford, “Elastic scattering, adsorption and surface enhanced Raman scattering by concentric spheres comprised of a metallic and a dielectric region,” Phys. Rev. B 26, 4052–4063 (1982).
[Crossref]

D.-S. Wang and M. Kerker, “Enhanced Raman scattering by molecules adsorbed at the surface of colloidal spheroids,” Phys. Rev. B 24, 1777–1790 (1981).
[Crossref]

H. Chew, D. D. Cooke, and M. Kerker, “Raman and fluorescent scattering by molecules embedded in dielectric cylinders,” Appl. Opt. 19, 44–52 (1980).
[Crossref] [PubMed]

M. Kerker, O. Siiman, L. A. Bumm, and D.-S. Wang, “Surface enhanced Raman scattering (SERS) of citrate ion adsorbed on colloidal silver,” Appl. Opt. 19, 3253–3255 (1980).
[Crossref] [PubMed]

D.-S. Wang, M. Kerker, and H. Chew, “Raman and fluorescent scattering by molecules embedded in dielectric spheroids,” Appl. Opt. 19, 2315–2328 (1980).
[Crossref] [PubMed]

M. Kerker, D.-S. Wang, and H. Chew, “Surface enhanced Raman scattering (SERS) by molecules adsorbed at spherical particles,” Appl. Opt. 19, 4159–4174 (1980). [This paper is a corrected version of the paper appearing on p. 3373 of that volume, which contains a large number of printer’s errors.]
[Crossref] [PubMed]

D.-S. Wang, H. Chew, and M. Kerker, “Enhanced Raman scattering at the surface (SERS) of a spherical particle,” Appl. Opt. 19, 2256–2257 (1980).
[Crossref] [PubMed]

P. J. McNulty, S. D. Druger, M. Kerker, and H. Chew, “Fluorescent scattering by anisotropic molecules embedded in small particles,” Appl. Opt. 18, 1484–1486 (1979).
[Crossref] [PubMed]

M. Kerker and S. D. Druger, “Raman and fluorescent scattering by molecules embedded in spheres with radii up to several multiples of the wavelength,” AppI. Opt. 18, 1172–1179 (1979).
[Crossref]

J. P. Kratohvil, M.-P. Lee, and M. Kerker, “Angular distribution of fluorescence from small particles,” Appl. Opt. 17, 1978–1980 (1978).
[Crossref] [PubMed]

H. Chew, M. Sculley, M. Kerker, P. J. McNulty, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: results for coherent optical processes,” J. Opt. Soc. Am. 68, 1686–1689 (1978).
[Crossref]

M. Kerker, P. J. McNulty, M. Sculley, H. Chew, and D. D. Cooke, “Raman fluorescent scattering by molecules embedded in small particles: numerical results for incoherent processes,” J. Opt. Soc. Am. 68, 1676–1685 (1978).
[Crossref]

H. Chew, M. Kerker, and D. D. Cooke, “Electromagnetic scattering by a dielectric sphere and in diverging radiation field,” Phys. Rev. A 16, 320–323 (1977).
[Crossref]

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[Crossref]

H. Chew, M. Kerker, and P. J. McNulty, “Raman and fluorescent scattering by molecules embedded in concentric spheres,” J. Opt. Soc. Am. 66, 440–444 (1976).
[Crossref]

For a review, see M. Kerker, Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Kratohvil, J. P.

M. Kerker, M. van Dilla, A. Brunsting, J. P. Kratohvil, P. Hsu, D.-S. Wang, J. Gray, and R. Langlois, “Is the central dogma of flow cytometry true: that fluorescence intensity is proportional to cellular dye content?” Cytometry 3, 71–78 (1982).
[Crossref] [PubMed]

J. P. Kratohvil, M.-P. Lee, and M. Kerker, “Angular distribution of fluorescence from small particles,” Appl. Opt. 17, 1978–1980 (1978).
[Crossref] [PubMed]

Langlois, R.

M. Kerker, M. van Dilla, A. Brunsting, J. P. Kratohvil, P. Hsu, D.-S. Wang, J. Gray, and R. Langlois, “Is the central dogma of flow cytometry true: that fluorescence intensity is proportional to cellular dye content?” Cytometry 3, 71–78 (1982).
[Crossref] [PubMed]

Lee, E.-H.

Lee, M.-P.

Lippitsch, M. E.

M. E. Lippitsch, “Surface enhanced Raman spectra of biliverdine and pyrromethenone adsorbed to silver colloids,” Chem. Phys. Lett. 79, 224–226 (1981).
[Crossref]

Lombardi, J. R.

J. R. Lombardi, E. A. Shields Knight, and R. L. Birke, “Evidence for a Ag adatom-molecule complex in surface enhanced Raman scattering,” Chem. Phys. Lett. 79, 214–218 (1981).
[Crossref]

Long, M. B.

R. E. Benner, R. Dornhaus, M. B. Long, and R. K. Chang, “Inelastic light scattering from a distribution of microparticles,” in Microbeam Analysis, D. E. Newbury, ed. (San Francisco Press, San Francisco, Calif., 1979), pp. 191–195.

Manzel, K.

H. Abe, K. Manzel, W. Schulze, M. Moscovits, and D. P. DiLello, “Surface enhanced Raman spectroscopy of CO adsorbed on colloidal silver particles,” J. Chem. Phys. 74, 792–797 (1981).
[Crossref]

McNulty, P. J.

Mie, G.

G. Mie, “Beitrage zur Optik trüber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 378–445 (1908).

Moscovits, M.

H. Abe, K. Manzel, W. Schulze, M. Moscovits, and D. P. DiLello, “Surface enhanced Raman spectroscopy of CO adsorbed on colloidal silver particles,” J. Chem. Phys. 74, 792–797 (1981).
[Crossref]

Nitzan, A.

J. Gersten and A. Nitzan, “Electromagnetic theory of enhanced Raman scattering by molecules adsorbed on rough surfaces,” J. Chem. Phys. 73, 3023–3037 (1980).
[Crossref]

Owen, J. F.

R. E. Benner, J. F. Owen, and R. K. Chang, “Radiation patterns of inelastic reemission from microparticles in homogeneous surroundings and near dielectric or metal interfaces,” J. Phys. Chem. 84, 1602–1606 (1980).
[Crossref]

Schulze, W.

H. Abe, K. Manzel, W. Schulze, M. Moscovits, and D. P. DiLello, “Surface enhanced Raman spectroscopy of CO adsorbed on colloidal silver particles,” J. Chem. Phys. 74, 792–797 (1981).
[Crossref]

Sculley, M.

Shields Knight, E. A.

J. R. Lombardi, E. A. Shields Knight, and R. L. Birke, “Evidence for a Ag adatom-molecule complex in surface enhanced Raman scattering,” Chem. Phys. Lett. 79, 214–218 (1981).
[Crossref]

Siiman, O.

O. Siiman, L. A. Bumm, R. Callaghan, C. G. Blatchford, and M. Kerker, “Surface-enhanced Raman scattering by citrate on colloidal silver,” J. Phys. Chem. 87, 1014–1023 (1983).
[Crossref]

M. Kerker, O. Siiman, L. A. Bumm, and D.-S. Wang, “Surface enhanced Raman scattering (SERS) of citrate ion adsorbed on colloidal silver,” Appl. Opt. 19, 3253–3255 (1980).
[Crossref] [PubMed]

van Dilla, M.

M. Kerker, M. van Dilla, A. Brunsting, J. P. Kratohvil, P. Hsu, D.-S. Wang, J. Gray, and R. Langlois, “Is the central dogma of flow cytometry true: that fluorescence intensity is proportional to cellular dye content?” Cytometry 3, 71–78 (1982).
[Crossref] [PubMed]

Wang, D.-S.

Wetzel, H.

H. Wetzel and H. Gerischer, “Surface enhanced Raman scattering from pyridine and halide ions adsorbed on silver and gold sol particles,” Chem. Phys. Lett. 76, 460–464 (1980).
[Crossref]

Ann. Phys. (Leipzig) (1)

G. Mie, “Beitrage zur Optik trüber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 378–445 (1908).

AppI. Opt. (1)

M. Kerker and S. D. Druger, “Raman and fluorescent scattering by molecules embedded in spheres with radii up to several multiples of the wavelength,” AppI. Opt. 18, 1172–1179 (1979).
[Crossref]

Appl. Opt. (8)

H. Chew, D. D. Cooke, and M. Kerker, “Raman and fluorescent scattering by molecules embedded in dielectric cylinders,” Appl. Opt. 19, 44–52 (1980).
[Crossref] [PubMed]

D.-S. Wang, M. Kerker, and H. Chew, “Raman and fluorescent scattering by molecules embedded in dielectric spheroids,” Appl. Opt. 19, 2315–2328 (1980).
[Crossref] [PubMed]

J. P. Kratohvil, M.-P. Lee, and M. Kerker, “Angular distribution of fluorescence from small particles,” Appl. Opt. 17, 1978–1980 (1978).
[Crossref] [PubMed]

E.-H. Lee, R. E. Benner, J. B. Fenn, and R. K. Chang, “Angular distribution of fluorescence from monodispersed particles,” Appl. Opt. 17, 1980–1982 (1978).
[Crossref]

P. J. McNulty, S. D. Druger, M. Kerker, and H. Chew, “Fluorescent scattering by anisotropic molecules embedded in small particles,” Appl. Opt. 18, 1484–1486 (1979).
[Crossref] [PubMed]

D.-S. Wang, H. Chew, and M. Kerker, “Enhanced Raman scattering at the surface (SERS) of a spherical particle,” Appl. Opt. 19, 2256–2257 (1980).
[Crossref] [PubMed]

M. Kerker, D.-S. Wang, and H. Chew, “Surface enhanced Raman scattering (SERS) by molecules adsorbed at spherical particles,” Appl. Opt. 19, 4159–4174 (1980). [This paper is a corrected version of the paper appearing on p. 3373 of that volume, which contains a large number of printer’s errors.]
[Crossref] [PubMed]

M. Kerker, O. Siiman, L. A. Bumm, and D.-S. Wang, “Surface enhanced Raman scattering (SERS) of citrate ion adsorbed on colloidal silver,” Appl. Opt. 19, 3253–3255 (1980).
[Crossref] [PubMed]

Chem. Phys. Lett. (4)

H. Wetzel and H. Gerischer, “Surface enhanced Raman scattering from pyridine and halide ions adsorbed on silver and gold sol particles,” Chem. Phys. Lett. 76, 460–464 (1980).
[Crossref]

F. J. Adrian, “Surface enhanced Raman scattering by surface plasmon enhancement of electromagnetic fields near spheroidal particles on a roughened metal surface,” Chem. Phys. Lett. 78, 45–49 (1981).
[Crossref]

M. E. Lippitsch, “Surface enhanced Raman spectra of biliverdine and pyrromethenone adsorbed to silver colloids,” Chem. Phys. Lett. 79, 224–226 (1981).
[Crossref]

J. R. Lombardi, E. A. Shields Knight, and R. L. Birke, “Evidence for a Ag adatom-molecule complex in surface enhanced Raman scattering,” Chem. Phys. Lett. 79, 214–218 (1981).
[Crossref]

Cytometry (1)

M. Kerker, M. van Dilla, A. Brunsting, J. P. Kratohvil, P. Hsu, D.-S. Wang, J. Gray, and R. Langlois, “Is the central dogma of flow cytometry true: that fluorescence intensity is proportional to cellular dye content?” Cytometry 3, 71–78 (1982).
[Crossref] [PubMed]

Faraday Discuss. Chem. Soc. (1)

J. A. Creighton, C. G. Blatchford, and M. G. Albrecht, “Plasma resonance enhancement of Raman scattering by pyridine adsorbed on silver or gold sol particles of size comparable to the excitation wavelength,” Faraday Discuss. Chem. Soc. 275, 790 (1979).

J. Chem. Phys. (2)

J. Gersten and A. Nitzan, “Electromagnetic theory of enhanced Raman scattering by molecules adsorbed on rough surfaces,” J. Chem. Phys. 73, 3023–3037 (1980).
[Crossref]

H. Abe, K. Manzel, W. Schulze, M. Moscovits, and D. P. DiLello, “Surface enhanced Raman spectroscopy of CO adsorbed on colloidal silver particles,” J. Chem. Phys. 74, 792–797 (1981).
[Crossref]

J. Colloid Interface Sci. (1)

D. Akins, “Surface enhanced Raman scattering of 2,2′-cyanine on colloidal silver,” J. Colloid Interface Sci. 90, 373–379 (1982).
[Crossref]

J. Opt. Soc. Am. (3)

J. Phys. Chem. (2)

O. Siiman, L. A. Bumm, R. Callaghan, C. G. Blatchford, and M. Kerker, “Surface-enhanced Raman scattering by citrate on colloidal silver,” J. Phys. Chem. 87, 1014–1023 (1983).
[Crossref]

R. E. Benner, J. F. Owen, and R. K. Chang, “Radiation patterns of inelastic reemission from microparticles in homogeneous surroundings and near dielectric or metal interfaces,” J. Phys. Chem. 84, 1602–1606 (1980).
[Crossref]

Opt. Lett. (1)

Phil. Trans. R. Soc. London (1)

M. Faraday, “Experimental relations of gold (and other metals) to light,” Phil. Trans. R. Soc. London 147, 145–181 (1857).
[Crossref]

Phys. Rev. A (2)

H. Chew, M. Kerker, and D. D. Cooke, “Electromagnetic scattering by a dielectric sphere and in diverging radiation field,” Phys. Rev. A 16, 320–323 (1977).
[Crossref]

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[Crossref]

Phys. Rev. B (4)

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

D.-S. Wang and M. Kerker, “Enhanced Raman scattering by molecules adsorbed at the surface of colloidal spheroids,” Phys. Rev. B 24, 1777–1790 (1981).
[Crossref]

D.-S. Wang and M. Kerker, “Absorption and luminescence of dye coated silver and gold particles,” Phys. Rev. B 25, 2433–2449 (1982).
[Crossref]

M. Kerker and C. G. Blatchford, “Elastic scattering, adsorption and surface enhanced Raman scattering by concentric spheres comprised of a metallic and a dielectric region,” Phys. Rev. B 26, 4052–4063 (1982).
[Crossref]

Other (3)

A. Gross, “Coherent anti-Stokes Raman scattering by droplets of the Mie size range,” Ph.D. thesis (Drexel University, Philadelphia, Pa., June1982).

For a review, see M. Kerker, Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

R. E. Benner, R. Dornhaus, M. B. Long, and R. K. Chang, “Inelastic light scattering from a distribution of microparticles,” in Microbeam Analysis, D. E. Newbury, ed. (San Francisco Press, San Francisco, Calif., 1979), pp. 191–195.

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Figures (9)

Fig. 1
Fig. 1

(a) Geometry of incident beams and particle. (b) Directions of the incident beams in the xz plane.

Fig. 2
Fig. 2

Enhancement factor |τ1|2/225 × |χ + χ′|2 × 4π versus incident wavelength λ1 = 2πc/ω1.

Fig. 3
Fig. 3

Enhancement factor |τ2|2/225 × |χ′|2 × 4π versus incident wavelength λ1 = 2πc/ω1.

Fig. 4
Fig. 4

Enhancement factor |τ1|2/225|χ + χ′|2 × 4π versus distance r′/a for λ = 404 nm.

Fig. 5
Fig. 5

Enhancement factor |τ2|2/225|χ′|2 × 4π versus distance r′/a for λ = 404 nm.

Fig. 6
Fig. 6

Enhancement factor |124 τ ¯ 1/15(χ + χ′)|2 (which is the value averaged over the spherical shell from r′ = a to 5a) versus wavelength λ1 = 2πc/ω1.

Fig. 7
Fig. 7

Enhancement factor |124 τ ¯ 2/15χ′|2 (which is the value averaged over the spherical shell from r′ = a to 5a) versus wavelength λ1 = 2πc/ω1.

Fig. 8
Fig. 8

Enhancement factor |[(d/a)3 − 1] τ ¯ 1 /15(χ + χ′)|2 versus distance d/a for λ = 404 nm. This is averaged over the spherical shell from r′ = a to d.

Fig. 9
Fig. 9

Enhancement factor |[(d/a)3 − 1] τ ¯ 2 /15χ′|2 versus distance d/a for λ = 404 nm. This is averaged over the spherical shell from ra to d.

Tables (4)

Tables Icon

Table 1 Polarization Components and Enhancement Factorsa

Tables Icon

Table 2 Field Amplitudes for Perpendicular Incident Beams and Scattering Angle θ

Tables Icon

Table 3 Averaged Field Amplitudes

Tables Icon

Table 4 Enhancement of CARS in a Benzene Organosol

Equations (62)

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p c = p c ( r ) = 6 χ 1122 ( ω 1 , ω 1 , - ω 2 ) E 1 ( ω 1 , r ) × E 2 * ( ω 2 , r ) E 1 ( ω 1 , r ) + 3 χ 1221 ( ω 1 , ω 1 , - ω 2 ) E 1 ( ω 1 , r ) × E 1 ( ω 1 , r ) E 2 * ( ω 2 , r ) χ E 1 · E 2 * E 1 + χ E 1 · E 1 E 2 * ,
E d ( ω 3 , r , r ) = , m { i c n 2 2 ( ω 3 ) ω 3 a E d ( l , m ) × [ h ( 1 ) ( k 3 r ) Y l l m ( r ^ ) ] + a M d ( , m ) h ( 1 ) ( k 3 r ) Y m ( r ^ ) } ,
a E d ( , m ) = 4 π k 3 2 ( μ 2 ( ω 3 ) 2 ( ω 3 ) ) 1 / 2 × p c ( r ) · { × [ h ( 1 ) ( k 3 r ) Y m * ( r ^ ) ] } ,
a M d ( l , m ) = 4 π i ( k 3 2 ω 3 c ) ( μ 2 ( ω 3 ) 2 ( ω 3 ) ) 1 / 2 × h ( 1 ) ( k 3 r ) p c ( r ) · Y m * ( r ^ ) .
E s c ( ω 3 , r , r ) = , m { i c n 2 2 ( ω 3 ) ω 3 b ( ω 3 ) a E d ( l , m ) × [ h l ( 1 ) ( k 3 r ) Y l l m ( r ^ ) ] + a l ( ω 3 ) a M d ( l , m ) h ( 1 ) ( k 3 r ) Y m ( r ^ ) } ,
E tot ( ω 3 , r , r ) = E d ( ω 3 , r , r ) + E sc ( ω 3 , r , r ) ,
E tot ( ω 3 , r , r ) large r e i k 3 r r [ e d ( r , r ) + e sc ( r , r ) ] = e i k 3 r r E tot ( r , r ) ,
E d ( r , r ) = , m i - l + 1 [ c n 2 2 ( ω 3 ) ω 3 a E d ( l , m ) r ^ × Y m ( r ^ ) - a M d ( l , m ) Y l l m ( r ^ ) ]
E sc ( r , r ) = , m i - l + 1 [ c b l ( ω 3 ) n 2 2 ( ω 3 ) ω 3 a E d ( l , m ) r ^ × Y l l m ( r ^ ) - a l ( ω 3 ) a M d ( l , m ) Y l l m ( r ^ ) ] .
E α i n c ( ω α , r ) = l , m { i c n 2 2 ( ω α ) ω α a E α ( l , m ) × [ j l ( k 2 α r ) Y m ( r ^ ) ] + a M α ( l , m ) j l ( k 2 α r ) Y l l m ( r ^ ) } ,
E α r ( ω α , r ) = l , m { i c n 2 2 ( ω α ) ω α b l ( ω α ) a E α ( l , m ) × [ h l ( 1 ) ( k 2 α r ) Y l l m ( r ^ ) ] + a l ( ω α ) a M α ( l , m ) h l ( 1 ) ( k 2 α r ) Y l l m ( r ^ ) } ,
E α ( ω α , r ) = E α inc ( ω α , r ) + E α r ( ω α , r ) ,             α = 1 , 2.
2 i [ 1 ( ω ) - 2 ( ω ) ] ( k a ) 3 3 [ 1 ( ω ) + 2 2 ( ω ) ]
E 1 inc ( ω 1 , r ) = ( 1 , 0 , 0 ) exp ( i k 21 z ) ,             k 21 = n 2 ( ω 1 ) ω 1 / c ,
E 2 inc ( ω 2 , r ) = ( 0 , 1 , 0 ) exp [ i k 22 ( x sin θ + z cos θ ) ] ,             k 22 = n 2 ( ω 2 ) ω 2 / c .
E 1 ( ω 1 , r ) = ( 1 + 2 δ 1 ) sin θ cos ϕ r ^ + ( 1 - δ 1 ) × ( cos θ cos ϕ θ ^ - sin ϕ ϕ ^ ) ,
E 2 * ( ω 2 , r ) = ( 1 + 2 δ 2 * ) sin θ sin ϕ r ^ + ( 1 - δ 2 * ) × ( cos θ sin ϕ θ ^ + cos ϕ ϕ ^ ) ,
δ α = ( a r ) 3 1 ( ω α ) - 2 ( ω α ) 1 ( ω α ) + 2 2 ( ω α ) ,             α = 1 , 2 , 3.
E 1 inc ( ω 1 , r ) = ( 0 , 1 , 0 ) exp ( i k 21 z ) ,
E 2 inc ( ω 2 , r ) = ( 0 , 1 , 0 ) exp [ i k 22 ( x sin Θ + z cos Θ ) ] ,
E 1 ( ω 1 , r ) = ( 1 + 2 δ 1 ) sin θ sin ϕ r ^ + ( 1 - δ 1 ) × ( cos θ sin ϕ θ ^ + cos ϕ ϕ ^ ) ,
E 2 * ( ω 2 , r ) = ( 1 + 2 δ 2 * ) sin θ sin ϕ r ^ + ( 1 - δ 2 * ) × ( cos θ sin ϕ θ ^ + cos ϕ ϕ ^ ) .
E 1 inc ( ω 1 , r ) = ( 1 , 0 , 0 ) exp ( i k 21 z ) ,
E 2 inc ( ω 2 , r ) = ( cos Θ , 0 , - sin Θ ) × exp [ i k 22 ( x sin Θ + z cos Θ ) ] ,
E 1 ( ω 1 , r ) = ( 1 + 2 δ 1 ) sin θ cos ϕ r ^ + ( 1 - δ 1 ) × ( cos θ cos ϕ θ ^ - sin ϕ ϕ ^ ) ,
E 2 * ( ω 2 , r ) = ( 1 + 2 δ 2 * ) ( cos Θ sin θ × cos ϕ - sin Θ cos θ ) r ^ + ( 1 - δ 2 * ) [ ( cos Θ cos θ cos ϕ + sin Θ sin θ ) θ ^ - cos Θ sin θ ϕ ^ ] .
E sc ( r , r ) = 8 π i k 3 ( k 3 r ) 3 δ 3 3 2 ( ω 3 ) m = - 1 1 p c ( r ) · [ × h 1 ( 1 ) ( k 3 r ) Y m ( r ^ ) ] r ^ × Y m ( r ^ ) ,
E d ( r , r ) = k 3 2 2 ( ω 3 ) [ r ^ × p c ( r ) ] × r ^ ,
E tot ( r , r ) = E sc ( r , r ) + E d ( r , r ) .
E sc ( r ) ¯ = E sc ( r , r ) d Ω ,
E d ( r ) ¯ = E d ( r , r ) d Ω ,
E tot ( r ) ¯ = E sc ( r ) ¯ + E d ( r ) ¯ ,
A = ( 1 + 2 δ 1 ) ( 1 + 2 δ 2 * ) , B = ( 1 - δ 1 ) ( 1 - δ 2 * ) , α = ( 1 + 2 δ 1 ) 2 , β = ( 1 - δ 1 ) 2 , C 1 = χ [ ( 1 + 2 δ 1 ) ( 3 A + 2 B ) + 2 ( 1 - δ 1 ) ( A + 4 B ) ] + χ [ ( 1 + 2 δ 2 * ) ( 3 α + 2 β ) + 2 ( 1 - δ 2 * ) ( α + 4 β ) ] , C 2 = 3 χ ( A - B ) δ 1 + χ [ ( 1 + 2 δ 2 * ) × ( α + 4 β ) + 2 ( 1 - δ 2 * ) ( 2 α + 3 β ) ] , τ 1 = C 1 + 2 δ 3 { χ [ ( 1 + 2 δ 1 ) ( 3 A + 2 B ) - ( 1 - δ 1 ) ( A + 4 B ) ] + χ [ ( 1 + 2 δ 2 * ) ( 3 α + 2 β ) - ( 1 - δ 2 * ) ( α + 4 β ) ] } , τ 2 = C 2 + δ 3 { 3 χ ( A - B ) ( 1 + δ 1 ) + 2 χ [ ( 1 + 2 δ 2 * ) × ( α + 4 β ) - ( 1 - δ 2 * ) ( 2 α + 3 β ) ] } .
E sc ( r ) ¯ = - 4 π k 3 2 δ 3 15 2 ( ω 3 ) { 3 χ ( B - A ) ( 1 + δ 1 ) + 2 χ [ ( 1 - δ 2 * ) ( 2 α + 3 β ) - ( 1 + 2 δ 2 * ) × ( α + 4 β ) ] } ( cos θ sin ϕ θ ^ + cos ϕ ϕ ^ ) ,
E d ( r ) ¯ = 4 π k 3 2 15 2 ( ω 3 ) C 2 ( cos θ sin ϕ θ ^ + cos ϕ ϕ ^ ) ,
E tot ( r ) ¯ = 4 π k 3 2 τ 2 15 2 ( ω 3 ) ( cos θ sin ϕ θ ^ + cos ϕ ϕ ^ ) .
E sc ( r ) ¯ = - 8 π k 3 2 δ 3 15 2 ( ω 3 ) { χ [ ( 1 - δ 1 ) ( A + 4 B ) - ( 1 + 2 δ 1 ) ( 3 A + 2 B ) + χ [ ( 1 - δ 2 * ) × ( α + 4 β ) - ( 1 + 2 δ 2 * ) ( 3 α + 2 β ) ] } × ( cos θ sin ϕ θ ^ + cos ϕ θ ) ,
E d ( r ) ¯ = 4 π k 3 2 C 1 15 2 ( ω 3 ) C 2 ( cos θ sin ϕ θ ^ + cos ϕ ϕ ^ ) ,
E tot ( r ) ¯ = 4 π k 3 2 τ 1 15 2 ( ω 3 ) ( cos θ sin ϕ θ ^ + cos ϕ ϕ ^ ) .
E sc ( r ) ¯ = - 8 π k 3 2 δ 3 15 2 ( ω 3 ) ( χ [ ( 1 + 2 δ 1 ) ( 3 A + 2 B ) - ( 1 - δ 1 ) ( A + 4 B ) ] cos Θ × ( - cos θ cos ϕ θ ^ + sin ϕ ϕ ^ ) + 3 2 ( 1 + δ 1 ) × ( B - A ) ( sin Θ sin θ θ ^ ) + χ { [ 1 + 2 δ 2 * ) ( 3 α + 2 β ) - ( 1 - δ 2 * ) ( α + 4 β ) ] cos Θ × ( - cos θ cos ϕ θ ^ + sin ϕ ϕ ^ ) - [ ( 1 + 2 δ 2 * ) × ( α + 4 β ) - ( 1 - δ 2 * ) ( 2 α + 3 β ) ] × sin Θ sin θ θ ^ } )
E d ( r ) ¯ = 4 π k 3 2 15 2 ( ω 3 ) [ C 1 cos Θ ( cos θ cos ϕ θ ^ - sin ϕ ϕ ^ ) + C 2 sin Θ sin θ θ ^ ]
E tot ( r ) ¯ = 4 π k 3 2 15 2 ( ω 3 ) [ τ 1 cos Θ ( cos θ cos ϕ θ ^ - sin ϕ ϕ ^ ) + τ 2 sin Θ sin θ θ ^ ] .
F E tot ( r ) ¯ / d Ω / E tot ( r ) 2 d Ω = τ 2 2 225 χ 2 for polarization ( 21 ) , = τ 1 2 225 ( χ + χ ) 2 for polarization ( 11 ) , = τ 1 2 cos 2 Θ + τ 2 2 sin 2 Θ 225 [ ( χ χ ) cos 2 Θ + χ 2 sin 2 Θ ] for polarization ( 22 ) ,
E d = 1 r 3 ( 3 n ^ n ^ - 1 ) · p c .
p = δ 3 [ 3 n ^ n ^ - 1 ] · p c ,
p tot = p + p c = [ 1 + δ 3 ( 3 n ^ n ^ - 1 ) ] · p c .
E sc = k 3 2 exp ( i k 3 r ) r ( 1 - n ^ n ^ ) · p tot .
F = ( n ^ n ^ - 1 ) · p tot ,
F θ = ( 1 - δ 3 ) θ ^ · p c + 3 δ 3 ( n ^ · p c ) ( θ ^ × n ^ ) , F ϕ = ( 1 - δ 3 ) ϕ ^ · p c + 3 δ 3 ( n ^ · p c ) ( ϕ ^ × n ^ ) .
G θ i j F θ i j / f θ i j 2 ,             G θ i j F ϕ i j / f ϕ i j 2 .
F θ 12 = F θ 22 = k 3 2 τ 2 / 15 , f θ 12 = f θ 22 = - χ ,
G θ 12 = G θ 22 = 1 225 τ 2 2 χ 2 .
τ ¯ 1 4 π a d τ 1 r 2 d r / [ 4 π ( d 3 - a 3 ) / 3 ] = 3 a 3 { χ [ 5 + 2 f ( 7 Δ 1 Δ 2 * + 7 Δ 1 Δ 3 + 2 Δ 1 2 + 2 Δ 2 * Δ 3 ) + 2 f ( f + 1 ) Δ 1 ( Δ 1 Δ 2 * + 2 Δ 2 * Δ 3 + Δ 1 Δ 3 ) + 8 f ( f 2 + f + 1 ) Δ 1 2 Δ 2 * Δ 3 ] + χ [ 5 + 2 f ( 4 Δ 1 Δ 2 * + 4 Δ 1 Δ 3 + 5 Δ 1 2 + 5 Δ 2 * Δ 3 ) + 2 f ( f + 1 ) Δ 1 ( Δ 1 Δ 2 * + Δ 1 Δ 3 + 2 Δ 2 * Δ 3 ) + 8 f ( f 2 + f + 1 ) Δ I 2 Δ 2 * Δ 3 ] } ,
τ ¯ 2 4 π a d τ 2 r 2 d r / [ 4 π ( d 3 - a 3 ) / 3 ] = 3 a 3 { χ [ 3 f ( Δ 1 Δ 3 + Δ 1 2 + Δ 1 Δ 2 * + Δ 2 * Δ 3 ) + 3 2 f ( f + 1 ) Δ 1 ( Δ 1 Δ 2 * + Δ 1 Δ 3 + 2 Δ 2 * Δ 3 ) + f ( f 2 + f + 1 ) Δ 1 2 Δ 2 * Δ 3 ] + χ [ 5 + 2 f ( 5 Δ 1 2 + 5 Δ 2 * Δ 3 - 2 Δ 1 Δ 2 * - 2 Δ 1 Δ 3 ) - f ( f + 1 ) Δ 1 ( Δ 1 Δ 2 * + Δ 1 Δ 3 + 2 Δ 2 * Δ 3 ) + 6 f ( f 2 + f + 1 ) Δ 1 2 Δ 2 * Δ 3 ] } ,
Δ α = 1 ( ω α ) - 2 ( ω α ) 1 ( ω α ) + 2 2 ( ω α ) ,             α = 1 , 2 , 3.
θ , ϕ i j = | N V [ ( d a ) 3 - 1 ] F ¯ θ , ϕ i j + [ V 0 - N V ( d a ) 3 ] f θ , ϕ i j | 2 .
S ¯ θ , ϕ i j = V 0 f θ , ϕ i j 2 .
G ¯ θ , ϕ i j Σ ¯ θ , ϕ i j / S ¯ θ , ϕ i j = n V [ ( d a ) 3 - 1 ] F ¯ θ , ϕ i j / f θ , ϕ i j + 1 - n V ( d / a ) 3 2 ,
G 1 = 1 225 τ 1 2 χ + χ 2
G 2 = 1 225 τ 2 2 χ 2 .
G ¯ ϕ 11 = | n V [ ( d a ) 3 - 1 ] τ ¯ 1 15 ( χ + χ ) - n V ( d a ) 3 + 1 | 2
G ¯ θ 12 = | n V [ ( d a ) 3 - 1 ] τ ¯ 2 15 χ - n V ( d a ) 3 + 1 | 2 .

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