Abstract

The calculated scattering matrix elements and interior electric fields for a dielectric sphere based on the discrete dipole approximation (DDA) are compared with the exact Mie solution for homogeneous and composite spheres. For homogeneous spheres the macroscopic average field produced at each DDA dipole site by the incident field combined with the field from all DDA sites is found to be approximated by the factor (n12+2)/3 multiplied by the Mie macroscopic field, where n1 is the refractive index. This holds to surprising accuracy, considering the finite wavelength and the small number of dipoles used in the DDA approximation. The approximate relation is most accurate near the center of the sphere and least accurate at the interface. The relation also holds for electric fields within composite spheres, with poorer agreement near each interface, where the refractive index changes. The dependence of this relation on parameters of the model is examined.

© 1999 Optical Society of America

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References

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  1. D. L. Dexter, “A theory of sensitized luminescence in solids,” J. Chem. Phys. 21, 836–850 (1953).
    [CrossRef]
  2. 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]
  3. L. M. Folan, S. Arnold, and S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
    [CrossRef]
  4. S. D. Druger, S. Arnold, and L. M. Folan, “Theory of enhanced energy transfer between molecules in spherical dielectric droplets,” J. Chem. Phys. 87, 2649–2659 (1987).
    [CrossRef]
  5. A. Hoekstra, J. Rahola, and P. Sloot, “Accuracy of internal fields in volume integral equation simulations of light scattering,” Appl. Opt. 37, 8482–8497 (1998), for example, calculate internal fields by using the VIEF method.
    [CrossRef]
  6. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  7. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  8. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [CrossRef]
  9. B. V. Bronk, S. D. Druger, J. Czégé, and W. P. Van De Merwe, “Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering,” Biophys. J. 69, 1170–1177 (1995).
    [CrossRef] [PubMed]
  10. W.-H. Yang, G. C. Schatz, and R. P. Van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shape,” J. Chem. Phys. 103, 869–875 (1995).
    [CrossRef]
  11. T. Jensen, L. Kelly, A. Lazarides, and G. C. Schatz, “Electrodynamics of noble metal nanoparticles and nanoparticle clusters,” J. Cluster Sci. 10, 295–317 (1999).
    [CrossRef]
  12. W. P. Van De Merwe, Z. Z. Li, B. V. Bronk, and J. Czégé, “Polarized light scattering for rapid observation of bacterial size changes,” Biophys. J. 73, 500–506 (1997).
    [CrossRef] [PubMed]
  13. S. D. Druger, M. Kerker, D. S. Wang, and D. D. Cooke, “Light scattering by inhomogeneous particles,” Appl. Opt. 18, 3888–3889 (1979).
    [CrossRef] [PubMed]
  14. A. Hoekstra, M. Grimmick, and P. M. A. Sloot, “Large scale simulations of elastic light scattering by a fast discrete dipole approximation,” Int. J. Mod. Phys. C 9, 87–103 (1998).
    [CrossRef]
  15. B. T. Draine and J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
    [CrossRef]
  16. A. Lakhatia, “Strong and weak forms of the method of moments and the coupled dipole method for scattering from time harmonic electromagnetic fields,” Int. J. Mod. Phys. C 3, 583–603 (1992).
    [CrossRef]
  17. C. R. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  18. Ref. 17, pp. 181–183.
  19. The result follows directly from the development given by J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 152–154.
  20. M. S. Kurdoglyan, “Quantum theory of intermolecular resonance energy transfer in dielectric microsized particles,” Opt. Spectrosc. 75, 488–490 (1993).
  21. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

1999

T. Jensen, L. Kelly, A. Lazarides, and G. C. Schatz, “Electrodynamics of noble metal nanoparticles and nanoparticle clusters,” J. Cluster Sci. 10, 295–317 (1999).
[CrossRef]

1998

A. Hoekstra, M. Grimmick, and P. M. A. Sloot, “Large scale simulations of elastic light scattering by a fast discrete dipole approximation,” Int. J. Mod. Phys. C 9, 87–103 (1998).
[CrossRef]

A. Hoekstra, J. Rahola, and P. Sloot, “Accuracy of internal fields in volume integral equation simulations of light scattering,” Appl. Opt. 37, 8482–8497 (1998), for example, calculate internal fields by using the VIEF method.
[CrossRef]

1997

W. P. Van De Merwe, Z. Z. Li, B. V. Bronk, and J. Czégé, “Polarized light scattering for rapid observation of bacterial size changes,” Biophys. J. 73, 500–506 (1997).
[CrossRef] [PubMed]

1995

B. V. Bronk, S. D. Druger, J. Czégé, and W. P. Van De Merwe, “Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering,” Biophys. J. 69, 1170–1177 (1995).
[CrossRef] [PubMed]

W.-H. Yang, G. C. Schatz, and R. P. Van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shape,” J. Chem. Phys. 103, 869–875 (1995).
[CrossRef]

1994

1993

B. T. Draine and J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

M. S. Kurdoglyan, “Quantum theory of intermolecular resonance energy transfer in dielectric microsized particles,” Opt. Spectrosc. 75, 488–490 (1993).

1992

A. Lakhatia, “Strong and weak forms of the method of moments and the coupled dipole method for scattering from time harmonic electromagnetic fields,” Int. J. Mod. Phys. C 3, 583–603 (1992).
[CrossRef]

1988

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

1987

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

1985

L. M. Folan, S. Arnold, and S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

1979

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]

1973

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

1953

D. L. Dexter, “A theory of sensitized luminescence in solids,” J. Chem. Phys. 21, 836–850 (1953).
[CrossRef]

Arnold, S.

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

L. M. Folan, S. Arnold, and S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

Bronk, B. V.

W. P. Van De Merwe, Z. Z. Li, B. V. Bronk, and J. Czégé, “Polarized light scattering for rapid observation of bacterial size changes,” Biophys. J. 73, 500–506 (1997).
[CrossRef] [PubMed]

B. V. Bronk, S. D. Druger, J. Czégé, and W. P. Van De Merwe, “Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering,” Biophys. J. 69, 1170–1177 (1995).
[CrossRef] [PubMed]

Chew, H.

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]

Cooke, D. D.

Czégé, J.

W. P. Van De Merwe, Z. Z. Li, B. V. Bronk, and J. Czégé, “Polarized light scattering for rapid observation of bacterial size changes,” Biophys. J. 73, 500–506 (1997).
[CrossRef] [PubMed]

B. V. Bronk, S. D. Druger, J. Czégé, and W. P. Van De Merwe, “Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering,” Biophys. J. 69, 1170–1177 (1995).
[CrossRef] [PubMed]

Dexter, D. L.

D. L. Dexter, “A theory of sensitized luminescence in solids,” J. Chem. Phys. 21, 836–850 (1953).
[CrossRef]

Draine, B. T.

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[CrossRef]

B. T. Draine and J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Druger, S. D.

B. V. Bronk, S. D. Druger, J. Czégé, and W. P. Van De Merwe, “Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering,” Biophys. J. 69, 1170–1177 (1995).
[CrossRef] [PubMed]

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

L. M. Folan, S. Arnold, and S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

S. D. Druger, M. Kerker, D. S. Wang, and D. D. Cooke, “Light scattering by inhomogeneous particles,” Appl. Opt. 18, 3888–3889 (1979).
[CrossRef] [PubMed]

Flatau, P. J.

Folan, L. M.

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

L. M. Folan, S. Arnold, and S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

Goodman, J.

B. T. Draine and J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

Grimmick, M.

A. Hoekstra, M. Grimmick, and P. M. A. Sloot, “Large scale simulations of elastic light scattering by a fast discrete dipole approximation,” Int. J. Mod. Phys. C 9, 87–103 (1998).
[CrossRef]

Hoekstra, A.

A. Hoekstra, M. Grimmick, and P. M. A. Sloot, “Large scale simulations of elastic light scattering by a fast discrete dipole approximation,” Int. J. Mod. Phys. C 9, 87–103 (1998).
[CrossRef]

A. Hoekstra, J. Rahola, and P. Sloot, “Accuracy of internal fields in volume integral equation simulations of light scattering,” Appl. Opt. 37, 8482–8497 (1998), for example, calculate internal fields by using the VIEF method.
[CrossRef]

Jensen, T.

T. Jensen, L. Kelly, A. Lazarides, and G. C. Schatz, “Electrodynamics of noble metal nanoparticles and nanoparticle clusters,” J. Cluster Sci. 10, 295–317 (1999).
[CrossRef]

Kelly, L.

T. Jensen, L. Kelly, A. Lazarides, and G. C. Schatz, “Electrodynamics of noble metal nanoparticles and nanoparticle clusters,” J. Cluster Sci. 10, 295–317 (1999).
[CrossRef]

Kerker, M.

S. D. Druger, M. Kerker, D. S. Wang, and D. D. Cooke, “Light scattering by inhomogeneous particles,” Appl. Opt. 18, 3888–3889 (1979).
[CrossRef] [PubMed]

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]

Kurdoglyan, M. S.

M. S. Kurdoglyan, “Quantum theory of intermolecular resonance energy transfer in dielectric microsized particles,” Opt. Spectrosc. 75, 488–490 (1993).

Lakhatia, A.

A. Lakhatia, “Strong and weak forms of the method of moments and the coupled dipole method for scattering from time harmonic electromagnetic fields,” Int. J. Mod. Phys. C 3, 583–603 (1992).
[CrossRef]

Lazarides, A.

T. Jensen, L. Kelly, A. Lazarides, and G. C. Schatz, “Electrodynamics of noble metal nanoparticles and nanoparticle clusters,” J. Cluster Sci. 10, 295–317 (1999).
[CrossRef]

Li, Z. Z.

W. P. Van De Merwe, Z. Z. Li, B. V. Bronk, and J. Czégé, “Polarized light scattering for rapid observation of bacterial size changes,” Biophys. J. 73, 500–506 (1997).
[CrossRef] [PubMed]

McNulty, P. J.

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]

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Rahola, J.

Schatz, G. C.

T. Jensen, L. Kelly, A. Lazarides, and G. C. Schatz, “Electrodynamics of noble metal nanoparticles and nanoparticle clusters,” J. Cluster Sci. 10, 295–317 (1999).
[CrossRef]

W.-H. Yang, G. C. Schatz, and R. P. Van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shape,” J. Chem. Phys. 103, 869–875 (1995).
[CrossRef]

Sloot, P.

Sloot, P. M. A.

A. Hoekstra, M. Grimmick, and P. M. A. Sloot, “Large scale simulations of elastic light scattering by a fast discrete dipole approximation,” Int. J. Mod. Phys. C 9, 87–103 (1998).
[CrossRef]

Van De Merwe, W. P.

W. P. Van De Merwe, Z. Z. Li, B. V. Bronk, and J. Czégé, “Polarized light scattering for rapid observation of bacterial size changes,” Biophys. J. 73, 500–506 (1997).
[CrossRef] [PubMed]

B. V. Bronk, S. D. Druger, J. Czégé, and W. P. Van De Merwe, “Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering,” Biophys. J. 69, 1170–1177 (1995).
[CrossRef] [PubMed]

Van Duyne, R. P.

W.-H. Yang, G. C. Schatz, and R. P. Van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shape,” J. Chem. Phys. 103, 869–875 (1995).
[CrossRef]

Wang, D. S.

Yang, W.-H.

W.-H. Yang, G. C. Schatz, and R. P. Van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shape,” J. Chem. Phys. 103, 869–875 (1995).
[CrossRef]

Appl. Opt.

Astrophys. J.

B. T. Draine and J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Biophys. J.

B. V. Bronk, S. D. Druger, J. Czégé, and W. P. Van De Merwe, “Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering,” Biophys. J. 69, 1170–1177 (1995).
[CrossRef] [PubMed]

W. P. Van De Merwe, Z. Z. Li, B. V. Bronk, and J. Czégé, “Polarized light scattering for rapid observation of bacterial size changes,” Biophys. J. 73, 500–506 (1997).
[CrossRef] [PubMed]

Chem. Phys. Lett.

L. M. Folan, S. Arnold, and S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

Int. J. Mod. Phys. C

A. Hoekstra, M. Grimmick, and P. M. A. Sloot, “Large scale simulations of elastic light scattering by a fast discrete dipole approximation,” Int. J. Mod. Phys. C 9, 87–103 (1998).
[CrossRef]

A. Lakhatia, “Strong and weak forms of the method of moments and the coupled dipole method for scattering from time harmonic electromagnetic fields,” Int. J. Mod. Phys. C 3, 583–603 (1992).
[CrossRef]

J. Chem. Phys.

W.-H. Yang, G. C. Schatz, and R. P. Van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shape,” J. Chem. Phys. 103, 869–875 (1995).
[CrossRef]

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

D. L. Dexter, “A theory of sensitized luminescence in solids,” J. Chem. Phys. 21, 836–850 (1953).
[CrossRef]

J. Cluster Sci.

T. Jensen, L. Kelly, A. Lazarides, and G. C. Schatz, “Electrodynamics of noble metal nanoparticles and nanoparticle clusters,” J. Cluster Sci. 10, 295–317 (1999).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Spectrosc.

M. S. Kurdoglyan, “Quantum theory of intermolecular resonance energy transfer in dielectric microsized particles,” Opt. Spectrosc. 75, 488–490 (1993).

Phys. Rev. A

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]

Other

C. R. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Ref. 17, pp. 181–183.

The result follows directly from the development given by J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 152–154.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

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Figures (4)

Fig. 1
Fig. 1

Comparison of scattering coefficients calculated with the Mie and the DDA methods for a layered sphere of inner size parameter ka1=0.75 with refractive index 1.30 and outer size parameter ka2=1.50 with refractive index n2=1.10. Solid curves, |S1|2 and |S2|2 Mie results; triangles, |S1|2 and, circles, |S2|2 DDA results.

Fig. 2
Fig. 2

Comparison of scattering coefficients calculated by use of Mie theory and by use of the DDA method for a layered sphere with the same parameters as in Ref. 11 except for a larger number of dipoles. The parameters used were ka1=0.75 with n1=1.33 and ka2=1.10 with n2=1.50. Solid curves, |S1|2 and |S2|2 Mie results; triangles, |S1|2 and, circles, |S2|2 DDA results.

Fig. 3
Fig. 3

Comparison of the local fields in the DDA approximation with the Lorentz correction factor times the Mie macroscopic field for homogeneous and layered spheres. The rms fractional difference χ is defined by Eq. (2). The outer sphere size parameter is ka2=1.50 in all cases, and the inner sphere for the layered case is ka1=0.75.

Fig. 4
Fig. 4

Dependence on DDA dipole number of the rms fractional difference χ between DDA local fields and Lorentz-factor-corrected Mie fields for a homogeneous sphere with ka=1.50 and n=1.30.

Tables (5)

Tables Icon

Table 1 Values of the Rms Fractional Difference χs Used in Comparing S1,S2,S3, and S4 Values Based on DDA versus Mie Solutions for a Sphere at Various Orientation Anglesa

Tables Icon

Table 2 Electric Field Amplitudes at DDA Dipole Sites Compared with (n12+2)/3 Multiplied By the Mie Solution for the Electric Field at the Corresponding Point in a Homogeneous Sphere: 47,937 Dipolesa

Tables Icon

Table 3 Electric Field Amplitudes at DDA Dipole Sites Compared with (n12+2)/3 Multiplied by the Mie Solution for the Electric Field at the Corresponding Point in a Homogeneous Sphere: 15,967 Dipolesa

Tables Icon

Table 4 Accuracy of the Relation between DDA Local Fields with Mie Macroscopic Fields as a Function of Dipole Number for Homogeneous and Inhomogeneous Spheresa

Tables Icon

Table 5 Accuracy of the Relation between DDA Local Fields with Mie Macroscopic Fields as a Function of Particle Size at Constant Dipole Numbera

Equations (4)

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

E,sE,s=exp[ik(r-z)]-ikrS2S3S4S1E,iE,i.
χSi=Θ|Si,Mie(Θ)-Si,DDA(Θ)|212Θ|S1,Mie(Θ)|2+Θ|S2,Mie(Θ)|21/2
EDDA,in12+23EMie,i,
χ=i(n12+2)3Ei,Mie-Ei,DDA212i(n12+2)3Ei,Mie2+i|Ei,DDA|21/2,

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