Abstract

To study the interactions between Mie scattering and Raman emissions of spherical particles, we measured the Raman spectra together with the elastically scattered light of the excitation source of an evaporating aqueous sodium nitrate droplet. Resonance structures were observed in the temporal profiles of the elastically scattered light and Raman nitrate and water emissions. The resonance structures in these three profiles occurred in a concerted mode but sometimes occurred independently of each other. A model of inelastic scattering by microspheres by Kerker et al. [ “ Raman and Fluorescent Scattering by Molecules Embedded in Spheres with Radii up to Several Multiples of the Wavelength,” Appl. Opt.18, 1172– 1179 ( 1979); “ Lorenz-Mie Scattering by Spheres: Some Newly Recognized Phenomena,” Aerosol Sci. Technol. 1, 275– 291 ( 1982); “ Inelastic Light Scattering,” in Aerosol Microphysics I: Particle Interaction, W. H. Marlow, Ed. ( Springer-Verlag, New York, 1980); “ Model for Raman and Fluorescent Scattering by Molecules Embedded in Small Particles,” Phys. Rev. A 13, 396– 404 ( 1976)] and the behavior of low order Mie resonances were used to explain the data. This type of data can be used for the determination of chemical compositions of spherical particles.

© 1991 Optical Society of America

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References

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  1. S. Arnold, “Spectroscopy of Single Levitated Micron Sized Particles,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, Ed. (World Scientific, Singapore, 1988).
  2. G. J. Rosasco, E. S. Etz, W. A. Cassatt, “The Analysis of Discrete Fine Particle by Raman Spectroscopy,” Appl. Spectrosc. 29, 396–404 (1975).
    [CrossRef]
  3. R. E. Preston, T. R. Lettieri, H. G. Semerjian, “Characterization of Single Levitated Droplets by Raman Spectroscopy,” Langmuir 1, 365–367 (1985).
    [CrossRef]
  4. R. Thurn, W. Kiefer, “Observations of Structural Resonances in the Raman Spectra of Optically Levitated Dielectric Microspheres,” J. Raman Spectrosc. 15, 411–413 (1984).
    [CrossRef]
  5. R. Thurn, W. Kiefer, “Raman-Microsampling Technique Applying Optical Levitation by Radiation Pressure,” Appl. Spectrosc. 38, 78–83 (1984).
    [CrossRef]
  6. R. Thurn, W. Kiefer, “Structural Resonances Observed in the Raman Spectra of Optically Levitated Liquid Droplets,” Appl. Opt. 24, 1515–1519 (1985).
    [CrossRef] [PubMed]
  7. K. H. Fung, I. N. Tang, “Raman Scattering from Single Solution Droplets,” Appl. Opt. 27, 206–208 (1988).
    [CrossRef] [PubMed]
  8. K. H. Fung, I. N. Tang, “Raman Spectra of Singly Suspended Supersaturated Ammonium Bisulfate Droplets,” Chem. Phys. Lett. 147, 509–513 (1988).
    [CrossRef]
  9. K. H. Fung, I. N. Tang, “Composition Analysis of Suspended Aerosol Particles by Raman Spectroscopy: Sulfates and Nitrates,” J. Colloid and Interface Sci. 130, 219–224 (1989).
    [CrossRef]
  10. I. N. Tang, K. H. Fung, “Characterization of Inorganic Salt Particles by Raman Spectroscopy,” J. Aerosol Sci. 20, 609–617 (1989).
    [CrossRef]
  11. R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of Structure Resonances in the Fluorescence Spectra from Microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
    [CrossRef]
  12. M. Kerker, 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]
  13. M. Kerker, “Lorenz-Mie Scattering by Spheres: Some Newly Recognized Phenomena,” Aerosol. Sci. Technol. 1, 275–291 (1982).
    [CrossRef]
  14. P. J. McNulty, H. W. Chew, M. Kerker, “Inelastic Light Scattering,” in Aerosol Microphysics I: Particle Interaction, W. H. Marlow, Ed. (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  15. H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and Fluorescent Scattering by Molecules Embedded in Small Particles,” Phys. Rev. A 13, 396–404 (1976).
    [CrossRef]
  16. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).
  17. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U.P., Princeton, NJ, 1957).
  18. G. Sageev, R. C. Flagan, J. H. Seinfeld, S. Arnold, “Condensation Rate of Water on Aqueous Droplets in the Transition Regime,” J. Colloid. Interface Sci. 113, 421–429 (1986).
    [CrossRef]
  19. S.-X. Qian, R. K. Chang, “Multi-Order Stokes Emissions from Micrometer Sized Droplets,” Phys. Rev. Lett. 56, 926 (1986).
    [CrossRef] [PubMed]
  20. S. C. Hill, R. E. Benner, “Morphology-Dependent Resonances Associated with Stimulated Processes in Microspheres,” J. Opt. Soc. Am. B 3, 1509–1514 (1986).
    [CrossRef]
  21. S. C. Hill, R. E. Benner, “Morphology-Dependent Resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, Ed. (World Scientific, Singapore, 1988).

1989

K. H. Fung, I. N. Tang, “Composition Analysis of Suspended Aerosol Particles by Raman Spectroscopy: Sulfates and Nitrates,” J. Colloid and Interface Sci. 130, 219–224 (1989).
[CrossRef]

I. N. Tang, K. H. Fung, “Characterization of Inorganic Salt Particles by Raman Spectroscopy,” J. Aerosol Sci. 20, 609–617 (1989).
[CrossRef]

1988

K. H. Fung, I. N. Tang, “Raman Spectra of Singly Suspended Supersaturated Ammonium Bisulfate Droplets,” Chem. Phys. Lett. 147, 509–513 (1988).
[CrossRef]

K. H. Fung, I. N. Tang, “Raman Scattering from Single Solution Droplets,” Appl. Opt. 27, 206–208 (1988).
[CrossRef] [PubMed]

1986

G. Sageev, R. C. Flagan, J. H. Seinfeld, S. Arnold, “Condensation Rate of Water on Aqueous Droplets in the Transition Regime,” J. Colloid. Interface Sci. 113, 421–429 (1986).
[CrossRef]

S.-X. Qian, R. K. Chang, “Multi-Order Stokes Emissions from Micrometer Sized Droplets,” Phys. Rev. Lett. 56, 926 (1986).
[CrossRef] [PubMed]

S. C. Hill, R. E. Benner, “Morphology-Dependent Resonances Associated with Stimulated Processes in Microspheres,” J. Opt. Soc. Am. B 3, 1509–1514 (1986).
[CrossRef]

1985

R. E. Preston, T. R. Lettieri, H. G. Semerjian, “Characterization of Single Levitated Droplets by Raman Spectroscopy,” Langmuir 1, 365–367 (1985).
[CrossRef]

R. Thurn, W. Kiefer, “Structural Resonances Observed in the Raman Spectra of Optically Levitated Liquid Droplets,” Appl. Opt. 24, 1515–1519 (1985).
[CrossRef] [PubMed]

1984

R. Thurn, W. Kiefer, “Raman-Microsampling Technique Applying Optical Levitation by Radiation Pressure,” Appl. Spectrosc. 38, 78–83 (1984).
[CrossRef]

R. Thurn, W. Kiefer, “Observations of Structural Resonances in the Raman Spectra of Optically Levitated Dielectric Microspheres,” J. Raman Spectrosc. 15, 411–413 (1984).
[CrossRef]

1982

M. Kerker, “Lorenz-Mie Scattering by Spheres: Some Newly Recognized Phenomena,” Aerosol. Sci. Technol. 1, 275–291 (1982).
[CrossRef]

1980

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of Structure Resonances in the Fluorescence Spectra from Microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[CrossRef]

1979

1976

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and Fluorescent Scattering by Molecules Embedded in Small Particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

1975

Arnold, S.

G. Sageev, R. C. Flagan, J. H. Seinfeld, S. Arnold, “Condensation Rate of Water on Aqueous Droplets in the Transition Regime,” J. Colloid. Interface Sci. 113, 421–429 (1986).
[CrossRef]

S. Arnold, “Spectroscopy of Single Levitated Micron Sized Particles,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, Ed. (World Scientific, Singapore, 1988).

Barber, P. W.

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of Structure Resonances in the Fluorescence Spectra from Microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[CrossRef]

Benner, R. E.

S. C. Hill, R. E. Benner, “Morphology-Dependent Resonances Associated with Stimulated Processes in Microspheres,” J. Opt. Soc. Am. B 3, 1509–1514 (1986).
[CrossRef]

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of Structure Resonances in the Fluorescence Spectra from Microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[CrossRef]

S. C. Hill, R. E. Benner, “Morphology-Dependent Resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, Ed. (World Scientific, Singapore, 1988).

Cassatt, W. A.

Chang, R. K.

S.-X. Qian, R. K. Chang, “Multi-Order Stokes Emissions from Micrometer Sized Droplets,” Phys. Rev. Lett. 56, 926 (1986).
[CrossRef] [PubMed]

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of Structure Resonances in the Fluorescence Spectra from Microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[CrossRef]

Chew, H.

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and Fluorescent Scattering by Molecules Embedded in Small Particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

Chew, H. W.

P. J. McNulty, H. W. Chew, M. Kerker, “Inelastic Light Scattering,” in Aerosol Microphysics I: Particle Interaction, W. H. Marlow, Ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

Druger, S. D.

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U.P., Princeton, NJ, 1957).

Etz, E. S.

Flagan, R. C.

G. Sageev, R. C. Flagan, J. H. Seinfeld, S. Arnold, “Condensation Rate of Water on Aqueous Droplets in the Transition Regime,” J. Colloid. Interface Sci. 113, 421–429 (1986).
[CrossRef]

Fung, K. H.

I. N. Tang, K. H. Fung, “Characterization of Inorganic Salt Particles by Raman Spectroscopy,” J. Aerosol Sci. 20, 609–617 (1989).
[CrossRef]

K. H. Fung, I. N. Tang, “Composition Analysis of Suspended Aerosol Particles by Raman Spectroscopy: Sulfates and Nitrates,” J. Colloid and Interface Sci. 130, 219–224 (1989).
[CrossRef]

K. H. Fung, I. N. Tang, “Raman Scattering from Single Solution Droplets,” Appl. Opt. 27, 206–208 (1988).
[CrossRef] [PubMed]

K. H. Fung, I. N. Tang, “Raman Spectra of Singly Suspended Supersaturated Ammonium Bisulfate Droplets,” Chem. Phys. Lett. 147, 509–513 (1988).
[CrossRef]

Hill, S. C.

S. C. Hill, R. E. Benner, “Morphology-Dependent Resonances Associated with Stimulated Processes in Microspheres,” J. Opt. Soc. Am. B 3, 1509–1514 (1986).
[CrossRef]

S. C. Hill, R. E. Benner, “Morphology-Dependent Resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, Ed. (World Scientific, Singapore, 1988).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

Kerker, M.

M. Kerker, “Lorenz-Mie Scattering by Spheres: Some Newly Recognized Phenomena,” Aerosol. Sci. Technol. 1, 275–291 (1982).
[CrossRef]

M. Kerker, 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]

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and Fluorescent Scattering by Molecules Embedded in Small Particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

P. J. McNulty, H. W. Chew, M. Kerker, “Inelastic Light Scattering,” in Aerosol Microphysics I: Particle Interaction, W. H. Marlow, Ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

Kiefer, W.

Lettieri, T. R.

R. E. Preston, T. R. Lettieri, H. G. Semerjian, “Characterization of Single Levitated Droplets by Raman Spectroscopy,” Langmuir 1, 365–367 (1985).
[CrossRef]

McNulty, P. J.

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and Fluorescent Scattering by Molecules Embedded in Small Particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

P. J. McNulty, H. W. Chew, M. Kerker, “Inelastic Light Scattering,” in Aerosol Microphysics I: Particle Interaction, W. H. Marlow, Ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

Owen, J. F.

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of Structure Resonances in the Fluorescence Spectra from Microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[CrossRef]

Preston, R. E.

R. E. Preston, T. R. Lettieri, H. G. Semerjian, “Characterization of Single Levitated Droplets by Raman Spectroscopy,” Langmuir 1, 365–367 (1985).
[CrossRef]

Qian, S.-X.

S.-X. Qian, R. K. Chang, “Multi-Order Stokes Emissions from Micrometer Sized Droplets,” Phys. Rev. Lett. 56, 926 (1986).
[CrossRef] [PubMed]

Rosasco, G. J.

Sageev, G.

G. Sageev, R. C. Flagan, J. H. Seinfeld, S. Arnold, “Condensation Rate of Water on Aqueous Droplets in the Transition Regime,” J. Colloid. Interface Sci. 113, 421–429 (1986).
[CrossRef]

Seinfeld, J. H.

G. Sageev, R. C. Flagan, J. H. Seinfeld, S. Arnold, “Condensation Rate of Water on Aqueous Droplets in the Transition Regime,” J. Colloid. Interface Sci. 113, 421–429 (1986).
[CrossRef]

Semerjian, H. G.

R. E. Preston, T. R. Lettieri, H. G. Semerjian, “Characterization of Single Levitated Droplets by Raman Spectroscopy,” Langmuir 1, 365–367 (1985).
[CrossRef]

Tang, I. N.

I. N. Tang, K. H. Fung, “Characterization of Inorganic Salt Particles by Raman Spectroscopy,” J. Aerosol Sci. 20, 609–617 (1989).
[CrossRef]

K. H. Fung, I. N. Tang, “Composition Analysis of Suspended Aerosol Particles by Raman Spectroscopy: Sulfates and Nitrates,” J. Colloid and Interface Sci. 130, 219–224 (1989).
[CrossRef]

K. H. Fung, I. N. Tang, “Raman Spectra of Singly Suspended Supersaturated Ammonium Bisulfate Droplets,” Chem. Phys. Lett. 147, 509–513 (1988).
[CrossRef]

K. H. Fung, I. N. Tang, “Raman Scattering from Single Solution Droplets,” Appl. Opt. 27, 206–208 (1988).
[CrossRef] [PubMed]

Thurn, R.

Aerosol. Sci. Technol.

M. Kerker, “Lorenz-Mie Scattering by Spheres: Some Newly Recognized Phenomena,” Aerosol. Sci. Technol. 1, 275–291 (1982).
[CrossRef]

Appl. Opt.

Appl. Spectrosc.

Chem. Phys. Lett.

K. H. Fung, I. N. Tang, “Raman Spectra of Singly Suspended Supersaturated Ammonium Bisulfate Droplets,” Chem. Phys. Lett. 147, 509–513 (1988).
[CrossRef]

J. Aerosol Sci.

I. N. Tang, K. H. Fung, “Characterization of Inorganic Salt Particles by Raman Spectroscopy,” J. Aerosol Sci. 20, 609–617 (1989).
[CrossRef]

J. Colloid and Interface Sci.

K. H. Fung, I. N. Tang, “Composition Analysis of Suspended Aerosol Particles by Raman Spectroscopy: Sulfates and Nitrates,” J. Colloid and Interface Sci. 130, 219–224 (1989).
[CrossRef]

J. Colloid. Interface Sci.

G. Sageev, R. C. Flagan, J. H. Seinfeld, S. Arnold, “Condensation Rate of Water on Aqueous Droplets in the Transition Regime,” J. Colloid. Interface Sci. 113, 421–429 (1986).
[CrossRef]

J. Opt. Soc. Am. B

J. Raman Spectrosc.

R. Thurn, W. Kiefer, “Observations of Structural Resonances in the Raman Spectra of Optically Levitated Dielectric Microspheres,” J. Raman Spectrosc. 15, 411–413 (1984).
[CrossRef]

Langmuir

R. E. Preston, T. R. Lettieri, H. G. Semerjian, “Characterization of Single Levitated Droplets by Raman Spectroscopy,” Langmuir 1, 365–367 (1985).
[CrossRef]

Phys. Rev. A

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and Fluorescent Scattering by Molecules Embedded in Small Particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

Phys. Rev. Lett.

S.-X. Qian, R. K. Chang, “Multi-Order Stokes Emissions from Micrometer Sized Droplets,” Phys. Rev. Lett. 56, 926 (1986).
[CrossRef] [PubMed]

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of Structure Resonances in the Fluorescence Spectra from Microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
[CrossRef]

Other

P. J. McNulty, H. W. Chew, M. Kerker, “Inelastic Light Scattering,” in Aerosol Microphysics I: Particle Interaction, W. H. Marlow, Ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

S. Arnold, “Spectroscopy of Single Levitated Micron Sized Particles,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, Ed. (World Scientific, Singapore, 1988).

S. C. Hill, R. E. Benner, “Morphology-Dependent Resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, Ed. (World Scientific, Singapore, 1988).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U.P., Princeton, NJ, 1957).

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Figures (12)

Fig. 1
Fig. 1

Electrodynamic balance.

Fig. 2
Fig. 2

Raman spectroscopy of single particles using an OMA.

Fig. 3
Fig. 3

Example of the Raman spectrum of an aqueous sodium nitrate droplet measured by an OMA.

Fig. 4
Fig. 4

Raman water intensity profiles at 3308, 3349, and 3587 cm−1.

Fig. 5
Fig. 5

Intensity profiles of the elastically scattered Ar line and Raman emissions of the nitrate and water peaks of an evaporating sodium nitrate droplet.

Fig. 6
Fig. 6

Scattering intensity profiles from 0 to 400 s.

Fig. 7
Fig. 7

Scattering intensity profiles of nitrate and water peaks from 400 to 1200 s.

Fig. 8
Fig. 8

Scattering intensity profiles of Ar line and nitrate peaks from 400 to 800 s.

Fig. 9
Fig. 9

Scattering intensity profiles from 1200 to 1650 s.

Fig. 10
Fig. 10

Raman nitrate scattering intensity as a function of scattered Ar line intensity.

Fig. 11
Fig. 11

Raman water scattering intensity as a function of scattered Ar line intensity.

Fig. 12
Fig. 12

Raman nitrate intensity as a function of Raman water scattering intensity.

Equations (14)

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E 1 ( r ) = l , m { ( i c / n 1 2 ω 0 ) γ E ( l , m ) x [ j l ( k 1 r ) Y l l m ( r ˆ ) ] + γ M ( l , m ) j l ( k 1 r ) Y l l m ( r ˆ ) } ,
B 1 ( r ) = l , m { γ E ( l , m ) j l ( k 1 r ) Y l l m ( r ˆ ) ( i c / ω 0 ) γ M ( l , m ) x [ j l ( k 1 r ) Y l l m ( r ˆ ) ] } ,
γ E ( l , m ) = ( i n 1 2 μ 1 / k 2 a ) α E ( l , m ) n 1 2 μ 2 j l ( k 1 a ) [ k 2 a h l ( 1 ) ( k 2 a ) ] n 2 2 μ 1 h l ( 1 ) ( k 2 a ) [ k 1 a j l ( k 1 a ) ] ,
γ M ( l . m ) = ( i μ 1 / k 2 a ) α M ( l , m ) μ 2 h l ( 1 ) ( k 2 a ) [ k 1 a j 1 ( k 1 a ) ] μ 1 j l ( k 1 a ) [ k 2 a h l ( 1 ) ( k 2 a ) ] ,
{ x f ( x ) } d d x { x f ( x ) } ,
E sc ( r ) = l , m { ( i c / n 2 2 ω 0 ) β E ( l , m ) x [ h l ( 1 ) ( k 2 r ) Y l l m ( r ˆ ) ] + β M ( l , m ) h l ( 1 ) ( k 2 r ) Y l l m ( r ˆ ) } ,
B sc ( r ) = l , m { β E ( l , m ) h l ( 1 ) ( k 2 r ) Y l l m ( r ˆ ) ( i c / ω 0 ) β M ( l , m ) x [ h l ( 1 ) ( k 2 r ) Y l l m ( r ˆ ) ] } ,
β E ( l , m ) = { n 2 2 μ 1 j l ( k 2 a ) [ k 1 a j l ( k 1 a ) ] n 1 2 μ 2 j l ( k 1 a ) [ k 2 a j l ( k 2 a ) ] } α E ( l , m ) n 1 2 μ 2 j l ( k 1 a ) [ k 2 a h l ( 1 ) ( k 2 a ) ] n 2 2 μ 1 h l ( 1 ) ( k 2 a ) [ k 1 a j l ( k 1 a ) ] ,
β M ( l , m ) = { μ 1 j l ( k 1 a ) [ k 2 a j l ( k 2 a ) ] μ 2 j l ( k 2 a ) [ k 1 a j l ( k 1 a ) ] } α M ( l , m ) μ 2 h l ( 1 ) ( k 2 a ) [ k 1 a j l ( k 1 a ) ] μ 1 j l ( k 1 a ) [ k 2 a h l ( 1 ) ( k 2 a ) ] ,
E 2 ( r ) = l , m { ( i c / n 2 2 ω ) c E ( l , m ) x [ h l ( 1 ) ( k 2 r ) Y l l m ( r ˆ ) ] + c M ( l , m ) h l ( 1 ) ( k 2 r ) Y l l m ( r ˆ ) } ,
B 2 ( r ) = l , m { c E ( l , m ) h l ( 1 ) ( k 2 r ) Y l l m ( r ˆ ) ( i c / ω ) c M ( l , m ) x [ h l ( 1 ) ( k 2 r ) Y l l m ( r ˆ ) ] }
c E ( l , m ) = ( i n 2 2 μ 2 / k 1 a ) a E ( l , m ) n 1 2 μ 2 j l ( k 1 a ) [ k 2 a h l ( 1 ) ( k 2 a ) ] n 2 2 μ 1 h l ( 1 ) ( k 2 a ) [ k 1 a j l ( k 1 a ) ] ,
c M ( l , m ) = ( i μ 2 / k 1 a ) a M ( l , m ) μ 2 h l ( 1 ) ( k 2 a ) [ k 1 a j l ( k 1 a ) ] μ 1 j l ( k 1 a ) [ k 2 a h l ( 1 ) ( k 2 a ) ] ,
n 1 2 = μ 1 1 , n 2 2 = μ 2 2 ,

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