Abstract

Single and double structural resonances in the total Raman- or fluorescence-scattering cross section of spherical dielectric particles are theoretically investigated. We found appreciable differences in the resonance heights and in the distributions of the sources of the Raman-scattered light inside the particle for the emission of dipoles induced parallel to the exciting field (p = αE), designated as directed emission, and the emission of orientation-averaged dipoles. The influence of radial symmetric concentration profiles on Raman or fluorescent scattering in the case of the excitation of double structural resonances is investigated. We show that the resonance heights and the radial positions associated with the spherical layers probed by the resonances depend on the concentration profiles, the mode type of the resonances, and the emission characteristics (directed or orientation-averaged emission) of the probed molecules.

© 1996 Optical Society of America.

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References

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  1. A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354(1977).
    [CrossRef]
  2. P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233(1978).
    [CrossRef]
  3. A. Ashkin and J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803–1814(1981).
    [CrossRef] [PubMed]
  4. A. Ashkin, J. M. Dziedzic, and R. H. Stolen, “Outer diameter measurement of low birefringence optical fibers by a new resonant backscatter technique,” Appl. Opt. 20, 2299–2303(1981).
    [CrossRef] [PubMed]
  5. J. F. Owen, R. K. Chang, and P. W. Barber, “Determination of optical fiber diameter from resonances in the elastic scattering spectrum,” Opt. Lett. 6, 272–274(1981).
    [CrossRef] [PubMed]
  6. R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478(1980).
    [CrossRef]
  7. J. F. Owen, P. W. Barber, P. B. Dorain, and R. K. Chang, “Enhancement of fluorescence by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078(1981).
    [CrossRef]
  8. S. C. Hill, R. E. Benner, C. K. Rushforth, and P. R. Conwell, “Structural resonances observed in the fluorescent emission from small spheres on substrates,” Appl. Opt. 23, 1680–1683(1984).
    [CrossRef]
  9. R. Thurn and W. Kiefer, “Raman-microsampling technique applying optical levitation by radiation pressure,” Appl. Spectrosc. 38, 78–83(1984).
    [CrossRef]
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    [CrossRef] [PubMed]
  11. T. R. Lettieri and R. L. Preston, “Observation of sharp resonances in the spontaneous Raman spectrum of a single optically levitated microdroplet,” Opt. Commun. 54, 349–352(1985).
    [CrossRef]
  12. G. Schweiger, “Observation of morphology dependent resonances caused by the input field in the Raman spectrum of microparticles,” J. Raman Spectrosc. 21, 165–168(1990).
    [CrossRef]
  13. J. F. Owen, R. K. Chang, and P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,”Aerosol Sci. Technol. 1, 293–302(1982).
    [CrossRef]
  14. G. Schweiger, “Observation of input and output structural resonances in the Raman spectrum of a single spheroidal dielectric microparticle,” Opt. Lett. 15, 156–158(1990).
    [CrossRef] [PubMed]
  15. H.-B. Lin, J. D. Eversole, C. D. Merritt, and A. J. Campillo, “Cavity-modified spontaneous-emission rates in liquid microdroplets,” Phys. Rev. A 45, 6756–6760(1992).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  22. P. J. McNulty, S. D. Druger, M. Kerker, and H. W. Chew, “Fluorescent scattering by anisotropic molecules embedded in small particles,” Appl. Opt. 18, 1484–1486(1979).
    [CrossRef] [PubMed]
  23. 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]
  24. J. Zhang and D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: unpolarized emissions,” Appl. Opt. 33, 7132–7139(1992).
    [CrossRef]
  25. J. Zhang and D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: polarized emissions,” Appl. Opt. 33, 7140–7146(1992).
    [CrossRef]
  26. 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]
  27. 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]
  28. S. Lange and G. Schweiger, “Thermal radiation from spherical microparticles: a new model,” J. Opt. Soc. Am. B 11, 2444–2451(1994).
    [CrossRef]
  29. S. D. Druger, S. Arnold, and L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659(1987).
    [CrossRef]
  30. H. Chew, “Total fluorescent scattering cross sections,” Phys. Rev. A 37, 4107–4110(1988).
    [CrossRef] [PubMed]
  31. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon, Oxford, 1946).

1995

1994

1992

H.-B. Lin, J. D. Eversole, C. D. Merritt, and A. J. Campillo, “Cavity-modified spontaneous-emission rates in liquid microdroplets,” Phys. Rev. A 45, 6756–6760(1992).
[CrossRef] [PubMed]

J. Zhang and D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: polarized emissions,” Appl. Opt. 33, 7140–7146(1992).
[CrossRef]

J. Zhang and D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: unpolarized emissions,” Appl. Opt. 33, 7132–7139(1992).
[CrossRef]

1990

G. Schweiger, “Observation of morphology dependent resonances caused by the input field in the Raman spectrum of microparticles,” J. Raman Spectrosc. 21, 165–168(1990).
[CrossRef]

G. Schweiger, “Observation of input and output structural resonances in the Raman spectrum of a single spheroidal dielectric microparticle,” Opt. Lett. 15, 156–158(1990).
[CrossRef] [PubMed]

1988

H. Chew, “Total fluorescent scattering cross sections,” Phys. Rev. A 37, 4107–4110(1988).
[CrossRef] [PubMed]

1987

1985

T. R. Lettieri and R. L. Preston, “Observation of sharp resonances in the spontaneous Raman spectrum of a single optically levitated microdroplet,” Opt. Commun. 54, 349–352(1985).
[CrossRef]

R. Thurn and W. Kiefer, “Structural resonances observed in the Raman spectra of optically levitated liquid droplets,” Appl. Opt. 24, 1515–1519(1985).
[CrossRef] [PubMed]

1984

1982

J. F. Owen, R. K. Chang, and P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,”Aerosol Sci. Technol. 1, 293–302(1982).
[CrossRef]

1981

1980

R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478(1980).
[CrossRef]

1979

1978

1977

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354(1977).
[CrossRef]

1976

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]

Alexander, D. R.

J. Zhang and D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: polarized emissions,” Appl. Opt. 33, 7140–7146(1992).
[CrossRef]

J. Zhang and D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: unpolarized emissions,” Appl. Opt. 33, 7132–7139(1992).
[CrossRef]

Arnold, S.

S. D. Druger, S. Arnold, and L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659(1987).
[CrossRef]

Ashkin, A.

Barber, P. W.

J. F. Owen, R. K. Chang, and P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,”Aerosol Sci. Technol. 1, 293–302(1982).
[CrossRef]

J. F. Owen, R. K. Chang, and P. W. Barber, “Determination of optical fiber diameter from resonances in the elastic scattering spectrum,” Opt. Lett. 6, 272–274(1981).
[CrossRef] [PubMed]

J. F. Owen, P. W. Barber, P. B. Dorain, and R. K. Chang, “Enhancement of fluorescence by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078(1981).
[CrossRef]

R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478(1980).
[CrossRef]

Benner, R. E.

Campillo, A. J.

H.-B. Lin and A. J. Campillo, “Radial profiling of microdroplets using cavity-enhanced Raman spectroscopy,” Opt. Lett. 20, 1589–1591(1995).
[CrossRef] [PubMed]

H.-B. Lin, J. D. Eversole, C. D. Merritt, and A. J. Campillo, “Cavity-modified spontaneous-emission rates in liquid microdroplets,” Phys. Rev. A 45, 6756–6760(1992).
[CrossRef] [PubMed]

Chang, R. K.

J. F. Owen, R. K. Chang, and P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,”Aerosol Sci. Technol. 1, 293–302(1982).
[CrossRef]

J. F. Owen, R. K. Chang, and P. W. Barber, “Determination of optical fiber diameter from resonances in the elastic scattering spectrum,” Opt. Lett. 6, 272–274(1981).
[CrossRef] [PubMed]

J. F. Owen, P. W. Barber, P. B. Dorain, and R. K. Chang, “Enhancement of fluorescence by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078(1981).
[CrossRef]

R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478(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]

Chew, H.

Chew, H. W.

Ching, S. C.

Chylek, P.

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233(1978).
[CrossRef]

Conwell, P. R.

Cooke, D. D.

Dorain, P. B.

J. F. Owen, P. W. Barber, P. B. Dorain, and R. K. Chang, “Enhancement of fluorescence by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078(1981).
[CrossRef]

Druger, S. D.

Dziedzic, J. M.

Eversole, J. D.

H.-B. Lin, J. D. Eversole, C. D. Merritt, and A. J. Campillo, “Cavity-modified spontaneous-emission rates in liquid microdroplets,” Phys. Rev. A 45, 6756–6760(1992).
[CrossRef] [PubMed]

Fenn, J. B.

Folan, L. M.

S. D. Druger, S. Arnold, and L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659(1987).
[CrossRef]

Hill, S. C.

Kerker, M.

Kiefer, W.

Kiehl, J. T.

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233(1978).
[CrossRef]

Ko, M. K. W.

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233(1978).
[CrossRef]

Kratohvil, J. P.

Lai, H. M.

Lange, S.

Lee, E.-H.

Lee, M.-P.

Lettieri, T. R.

T. R. Lettieri and R. L. Preston, “Observation of sharp resonances in the spontaneous Raman spectrum of a single optically levitated microdroplet,” Opt. Commun. 54, 349–352(1985).
[CrossRef]

Lin, H.-B.

H.-B. Lin and A. J. Campillo, “Radial profiling of microdroplets using cavity-enhanced Raman spectroscopy,” Opt. Lett. 20, 1589–1591(1995).
[CrossRef] [PubMed]

H.-B. Lin, J. D. Eversole, C. D. Merritt, and A. J. Campillo, “Cavity-modified spontaneous-emission rates in liquid microdroplets,” Phys. Rev. A 45, 6756–6760(1992).
[CrossRef] [PubMed]

McNulty, P. J.

Merritt, C. D.

H.-B. Lin, J. D. Eversole, C. D. Merritt, and A. J. Campillo, “Cavity-modified spontaneous-emission rates in liquid microdroplets,” Phys. Rev. A 45, 6756–6760(1992).
[CrossRef] [PubMed]

Owen, J. F.

J. F. Owen, R. K. Chang, and P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,”Aerosol Sci. Technol. 1, 293–302(1982).
[CrossRef]

J. F. Owen, R. K. Chang, and P. W. Barber, “Determination of optical fiber diameter from resonances in the elastic scattering spectrum,” Opt. Lett. 6, 272–274(1981).
[CrossRef] [PubMed]

J. F. Owen, P. W. Barber, P. B. Dorain, and R. K. Chang, “Enhancement of fluorescence by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078(1981).
[CrossRef]

R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478(1980).
[CrossRef]

Preston, R. L.

T. R. Lettieri and R. L. Preston, “Observation of sharp resonances in the spontaneous Raman spectrum of a single optically levitated microdroplet,” Opt. Commun. 54, 349–352(1985).
[CrossRef]

Rushforth, C. K.

Schweiger, G.

Sculley, M.

Stolen, R. H.

Thurn, R.

Young, K.

Zhang, J.

J. Zhang and D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: polarized emissions,” Appl. Opt. 33, 7140–7146(1992).
[CrossRef]

J. Zhang and D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: unpolarized emissions,” Appl. Opt. 33, 7132–7139(1992).
[CrossRef]

Aerosol Sci. Technol.

J. F. Owen, R. K. Chang, and P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,”Aerosol Sci. Technol. 1, 293–302(1982).
[CrossRef]

Appl. Opt.

J. Zhang and D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: polarized emissions,” Appl. Opt. 33, 7140–7146(1992).
[CrossRef]

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

A. Ashkin and J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803–1814(1981).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, and R. H. Stolen, “Outer diameter measurement of low birefringence optical fibers by a new resonant backscatter technique,” Appl. Opt. 20, 2299–2303(1981).
[CrossRef] [PubMed]

S. C. Hill, R. E. Benner, C. K. Rushforth, and P. R. Conwell, “Structural resonances observed in the fluorescent emission from small spheres on substrates,” Appl. Opt. 23, 1680–1683(1984).
[CrossRef]

R. Thurn and W. Kiefer, “Structural resonances observed in the Raman spectra of optically levitated liquid droplets,” Appl. Opt. 24, 1515–1519(1985).
[CrossRef] [PubMed]

J. Zhang and D. R. Alexander, “Hybrid inelastic-scattering models for particle thermometry: unpolarized emissions,” Appl. Opt. 33, 7132–7139(1992).
[CrossRef]

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

Appl. Spectrosc.

J. Chem. Phys.

S. D. Druger, S. Arnold, and L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659(1987).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

J. Raman Spectrosc.

G. Schweiger, “Observation of morphology dependent resonances caused by the input field in the Raman spectrum of microparticles,” J. Raman Spectrosc. 21, 165–168(1990).
[CrossRef]

Opt. Commun.

T. R. Lettieri and R. L. Preston, “Observation of sharp resonances in the spontaneous Raman spectrum of a single optically levitated microdroplet,” Opt. Commun. 54, 349–352(1985).
[CrossRef]

Opt. Lett.

Phys. Rev. A

H. Chew, “Total fluorescent scattering cross sections,” Phys. Rev. A 37, 4107–4110(1988).
[CrossRef] [PubMed]

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial wave resonances,” Phys. Rev. A 18, 2229–2233(1978).
[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.-B. Lin, J. D. Eversole, C. D. Merritt, and A. J. Campillo, “Cavity-modified spontaneous-emission rates in liquid microdroplets,” Phys. Rev. A 45, 6756–6760(1992).
[CrossRef] [PubMed]

Phys. Rev. Lett.

R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478(1980).
[CrossRef]

J. F. Owen, P. W. Barber, P. B. Dorain, and R. K. Chang, “Enhancement of fluorescence by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078(1981).
[CrossRef]

A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phys. Rev. Lett. 38, 1351–1354(1977).
[CrossRef]

Other

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon, Oxford, 1946).

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Figures (7)

Fig. 1
Fig. 1

(a) Normalized volume-averaged source function and Raman-scattering efficiency for (b) directed and (c) isotropic dipole emission versus the input size parameter x. Wave-number shift Δν̃ = 3000 cm−1; particle radius r = 10 µm; refractive index n1 = 1.5, n2 = 1.0; size parameter step width Δx = 5 × 10−4.

Fig. 2
Fig. 2

Raman-scattering efficiency for directed (lower curve) and isotropic (upper curve) dipole emission versus the output size parameter xs. The input size parameter is x = 58.719324211 at the resonance TM742. Particle radius r = 10 µm; refractive index n1 = 1.5, n2 = 1.0; size parameter step width Δxs = 10−4.

Fig. 3
Fig. 3

Radial contribution to the Raman-scattering efficiency for two different double resonances. The solid curves correspond to directed dipole emission. The dashed curves correspond to isotropic dipole emission. TE801, x = 58.216740773402; TM801, x = 58.698298220322; TE531, xs = 39.539874096; particle radius r = 10 µm; refractive index n1 = 1.5, n2 = 1.0.

Fig. 4
Fig. 4

Diffusion-controlled concentration profiles of Raman-active molecules inside a spherical particle for consecutive equidistant times 0 ⩽ t ⩽ 10 ms, Δt = 0.5 ms. Particle radius r = 10 µm, diffusivity D = 2 × 10−9 m2 s−1.

Fig. 5
Fig. 5

Normalized inelastically scattered radiant flux as a function of time for the tabulated structural double resonances of ascending order. Refractive index n1 = 1.5, n2 = 1.0; particle radius r = 10 µm; diffusivity D = 2 × 10−9 m2 s−1.

Fig. 6
Fig. 6

Normalized concentration given by the normalized resonance heights versus the radial positions associated with the volume probed by the tabulated structural resonances for varying time. The resonance combinations marked by a, b, c, d, and e are the same as in Fig. 5. The dashed vertical lines are explained in the text. Refractive index n1 = 1.5, n2 = 1.0; 25 µs ⩽ t ⩽ 2.5 ms; particle radius r = 10 µm; diffusivity D = 2 × 10−9 m2 s−1.

Fig. 7
Fig. 7

Normalized inelastically scattered radiant flux and radial positions versus the input-resonance mode number for first-order double resonances of different mode types at the time t = 0.5 ms. Refractive index n1 = 1.5, n2 = 1.0; particle radius r = 10 µm; diffusivity D = 2 × 10−9 m2 s−1.

Tables (1)

Tables Icon

Table 1 Electric Input and Output Resonances Combined to Double Resonances of the Raman-Scattering Cross Section; Refractive Index n1 = 1.5

Equations (47)

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

Einc=E0ex exp(ik02z),
Binc=-eyn2E0 exp(ik02z),
Einc(r)=l,micn12ω0αE(l, m)×[jl(k02r)Xl,m(Ω)]+αM(l, m)jl(k02r)Xl,m(Ω),
Binc(r)=l,mαE(l, m)jl(k02r)Xl,m(Ω)-icω0αM(l, m)[jl(k02r)Xl,m(Ω)],
αE(l, m)=n2il+1E0π(2l+1)(δm,-1-δm,+1),
αM(l, m)=ilE0π(2l+1)(δm,-1+δm,+1).
Etra(r)=l,micn12ω0γE(l, m)×[jl(k01r)Xl,m(Ω)]+γM(l, m)jl(k01r)Xl,m(Ω),
Btra(r)=l,mγE(l, m)jl(k01r)Xl,m(Ω)-icω0γM(l, m)[jl(k01r)Xl,m(Ω)],
γE(l, m)=γ˜E(l)αE(l, m),
γM(l, m)=γ˜M(l)αM(l, m),
γ˜E(l)=iμ1n12/x2μ2n12jl(x1)[x2hl(1)(x2)]-μ1n22hl(1)(x2)[x1jl(x1)],
γ˜M(l)=iμ1/x2μ1jl(x1)[x2hl(1)(x2)]-μ2hl(1)(x2)[x1jl(x1)],
p(r)=αEtra(r)
E1=Edip+Esca,
B1=Bdip+Bsca,
Edip(r)=l,m icn12ωaE(l, m)×[hl(1)(k1r)Xlm(r)]+aM(l, m)hl(1)(k1r)Xlm(r),
Bdip(r)=l,maE(l, m)hl(1)(k1r)Xlm(r)-icωaM(l, m)[hl(1)(k1r)Xlm(r)],
Esca(r)=l,m icn12ωbE(l, m)×[jl(k1r)Xlm(r)]+bM(l, m)jl(k1r)Xlm(r),
Bsca(r)=l,mbE(l, m)jl(k1r)Xlm(r)-icωbM(l, m)[jl(k1r)Xlm(r)],
 E2(r)=l,m icn22ωcE(l, m)×[hl(1)(k2r)Xlm(r)]+cM(l, m)hl(1)(k2r)Xlm(r),
B2(r)=l,mcE(l, m)hl(1)(k2r)Xlm(r)-icωcM(l, m)[hl(1)(k2r)Xlm(r)].
aE(l, m)=p·VE(l, m),
aM(l, m)=p·VM(l, m),
VE(l, m)=4πk12 μ1n1×[jl(k1r)Xlm*(r)],
VM(l, m)=4πik13 μ1n12jl(k1r)Xlm*(r).
cE(l, m)=fE(l)aE(l, m),
cM(l, m)=fM(l)aM(l, m)
fE(l)=in22/(μ1k1a)n12μ1jl(k1a)[k2ahl(k2a)]-n22μ2hl(k2a)[k1ajl(k1a)],
fM(l)=iμ2/(k1a)μ1jl(k1a)[k2ahl(k2a)]-μ2hl(k2a)[k1ajl(k1a)].
dPdipdΩ(r)=c38πμ2n23ω2l,m(-i)l+1[cE(l, m)Xlm(r)+n2cM(l, m)er×Xlm(r)]2.
Pdip(r)=c38πμ2n23ω2l,m[|cE(l, m)|2+n22|cM(l, m)|2],
Pdip(r)=14μ12|n1|4μ2n23ω4c3|p|2l2l(l+1)×(2l+1) |jl(k1r)|2|k1r|2(cos2 δ)+12l+1×|(l+1)jl-1(k1r)-ljl+1(k1r)|2 sin2 δ|fE(l)|2+n22|n1|2×(2l+1)|jl(k1r)|2(sin2 δ)|fM(l)|2.
PR=0a4πPdip(r)ρ(r)r2dΩdr,
PR=14μ12|n1|4μ2n23ω4c3α20al2l(l+1)(2l+1)×|jl(k1r)|2|k1r|2Er2(r)+12l+1|(l+1)jl-1(k1r)-ljl+1(k1r)|2Et2(r)|fE(l)|2+n22|n1|2(2l+1)×|jl(k1r)|2Et2(r)|fM(l)|2ρ(r)r2dr,
Er2(r)=2π E02 n22|n1|2ll(l+1)(2l+1)×|jl(k01r)|2|k01r|2|γ˜E(l)|2,
Et2(r)=2πE02ln22|n1|212l+1|(l+1)jl-1(k01r)-ljl+1(k01r)|2|γ˜E(l)|2+(2l+1)×|jl(k01r)|2|γ˜M(l)|2.
P=2π3ω4c3μ12|n1|4μ2n23|p|2l[(l+1)|jl-1(k1r)|2+l|jl+1(k1r)|2]|fE(l)|2+n22|n1|2×(2l+1)|jl(k1r)|2|fM(l)|2,
f(p)=4πf(p)dΩp.
Sˆ(r, k1)=4πSˆ(r, k1)dΩ
=2πl n22|n1|2[(l+1)×|jl-1(k1r)|2+l|jl+1(k1r)|2]|γ˜E(l)|2+(2l+1)|jl(k1r)|2|γ˜M(l)|2.
CR,iso=23α2μ22 ω4c40aSˆ(r, k01)×Sˆ(r, k1)ρ(r)r2dr.
QR=PR/P0,
P0=4π3μ2n2 ω4c3α2E020aρ(r)r2dr
QR=0aqR(r)ρ(r)r2dr.
|Etra|2=1E02VV|Etra(r)|2d3r
c(r, t)c0=1-arn=1erf (2n+1)a+r2Dt-erf (2n+1)a-r2Dt.
r=0aSˆ(r, k01)Sˆ(r, k1)r3dr0aSˆ(r, k01)Sˆ(r, k1)r2dr.

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