Abstract

Evanescent waves on a surface form due to the collective motion of charges within the medium. They do not carry any energy away from the surface and decay exponentially as a function of the distance. However, if there is any object within the evanescent field, electromagnetic energy within the medium is tunneled away and either absorbed or scattered. In this case, the absorption is localized, and potentially it can be used for selective diagnosis or nanopatterning applications. On the other hand, scattering of evanescent waves can be employed for characterization of nanoscale structures and particles on the surface. In this paper we present a numerical methodology to study the physics of such absorption and scattering mechanisms. We developed a MATLAB implementation of discrete dipole approximation with surface interaction (DDA-SI) in combination with evanescent wave illumination to investigate the near-field coupling between particles on the surface and a probe. This method can be used to explore the effects of a number of physical, geometrical, and material properties for problems involving nanostructures on or in the proximity of a substrate under arbitrary illumination.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. A. Hawes, J. T. Hastings, C. Crofcheck, and M. P. Mengüç, “Spatially selective melting and evaporation of nanosized gold particles,” Opt. Lett. 33, 1383–1385 (2008).
    [CrossRef] [PubMed]
  2. E. Purcell and C. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  3. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [CrossRef]
  4. M. A. Taubenblatt and T. K. Tran, “Calculation of light scattering from particles and structures on a surface by the coupled-dipole method,” J. Opt. Soc. Am. A 10, 912–919 (1993).
    [CrossRef]
  5. R. Schmehl, B. M. Nebeker, and E. D. Hirleman, “Discrete-dipole approximation for scattering by features on surfaces by means of a two dimensional fast Fourier transform technique,” J. Opt. Soc. Am. A 14, 3026–3036 (1997).
    [CrossRef]
  6. G. H. Yuan, P. Wang, Y. H. Lu, Y. Cao, D. G. Zhang, H. Ming, and W. D. Xu, “A large-area photolithography technique based on surface plasmons leakage modes,” Opt. Commun. 281, 2680–2684 (2008).
    [CrossRef]
  7. R. Fikri, D. Barchiesi, F. H’Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2d finite element method,” Opt. Commun. 221, 13–22 (2003).
    [CrossRef]
  8. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).
  9. B. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  10. H. Kimura, “Light-scattering properties of fractal aggregates: numerical calculations by a superposition technique and the discrete-dipole approximation,” J. Quant. Spectrosc. Radiat. Transf. 70, 581–594 (2001).
    [CrossRef]
  11. A. Sommerfeld, “Über die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 28, 665–737 (1909).
  12. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).
  13. A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, 1966).
  14. E. K. Burke and G. J. Miller, “Modeling antennas near to and penetrating a lossy interface,” IEEE Trans. Antennas Propag. AP-32, 1040–1049 (1984).
    [CrossRef]
  15. G. J. Burke and A. J. Poggio, Numerical Electromagnetics Code (NEC-4)—Method of Moments, Part I: Program Description—Theory, Tech. Rep. UCID-18834 (Lawrence Livermore Laboratory, 1981).
  16. R. J. Lytle and D. L. Lager, Numerical Evaluation of Sommerfeld Integrals, Tech. Rep. UCRL-51688 (Lawrence Livermore Laboratory, 1974).
  17. D. L. Lager and R. J. Lytle, Fotran Subroutines for the Numerical Evaluation of Sommerfeld Integrals Unter Anderem, Tech. Rep. UCRL-51821 (Lawrence Livermore Laboratory, 1975).
  18. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  19. S. Tojo and M. Hasuo, “Oscillator-strength enhancement of electric-dipole-forbidden transitions in evanescent light at total reflection,” Phys. Rev. A 71, 012508 (2005).
    [CrossRef]
  20. P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
    [CrossRef]
  21. M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, 2000).
  22. D. W. Mackowski, “Discrete dipole moment method for calculation of the T-matrix for nonspherical particles,” J. Opt. Soc. Am. A 19, 881–893 (2002).
    [CrossRef]
  23. V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete-dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transf. 110, 1460–1471 (2009).
    [CrossRef]
  24. D. A. Schultz, “Plasmon resonant particles for biological detection,” Curr. Opin. Biotechnol. 14, 13–22 (2003).
    [CrossRef] [PubMed]
  25. V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Modelling of high numerical aperture imaging of complex scatterers using T-matrix method,” in Proceedings of ELS XII Helsinki (2010), pp. 138–141.
  26. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  27. E.Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).
  28. L. N. Aksyutov, “Temperature dependence of the optical constants of tungsten and gold,” J. Appl. Spectrosc. 26, 656–660 (1977).
    [CrossRef]
  29. B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
    [CrossRef]
  30. H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008).
    [CrossRef] [PubMed]
  31. F. N. Dönmezer, M. P. Mengüç, and T. Okutucu, “Dependent absorption and scattering by interacting nanoparticles,” in Proceedings of the Sixth International Symposium on Radiative Transfer (2010).
  32. H. Zhang and E. D. Hirleman, “Prediction of light scattering from particles on a filmed surface using discrete-dipole approximation,” Proc. SPIE 4692, 38–45 (2002).
    [CrossRef]
  33. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
    [CrossRef]
  34. A. G. Hoekstra, M. Frijlink, L. B. F. M. Waters, and P. M. A. Sloot, “Radiation forces in the discrete-dipole approximation,” J. Opt. Soc. Am. A 18, 1944–1953 (2001).
    [CrossRef]
  35. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9, S196–S203 (2007).
    [CrossRef]

2010 (2)

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Modelling of high numerical aperture imaging of complex scatterers using T-matrix method,” in Proceedings of ELS XII Helsinki (2010), pp. 138–141.

F. N. Dönmezer, M. P. Mengüç, and T. Okutucu, “Dependent absorption and scattering by interacting nanoparticles,” in Proceedings of the Sixth International Symposium on Radiative Transfer (2010).

2009 (1)

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete-dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transf. 110, 1460–1471 (2009).
[CrossRef]

2008 (3)

G. H. Yuan, P. Wang, Y. H. Lu, Y. Cao, D. G. Zhang, H. Ming, and W. D. Xu, “A large-area photolithography technique based on surface plasmons leakage modes,” Opt. Commun. 281, 2680–2684 (2008).
[CrossRef]

H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008).
[CrossRef] [PubMed]

E. A. Hawes, J. T. Hastings, C. Crofcheck, and M. P. Mengüç, “Spatially selective melting and evaporation of nanosized gold particles,” Opt. Lett. 33, 1383–1385 (2008).
[CrossRef] [PubMed]

2007 (1)

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9, S196–S203 (2007).
[CrossRef]

2006 (1)

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[CrossRef]

2005 (1)

S. Tojo and M. Hasuo, “Oscillator-strength enhancement of electric-dipole-forbidden transitions in evanescent light at total reflection,” Phys. Rev. A 71, 012508 (2005).
[CrossRef]

2003 (2)

D. A. Schultz, “Plasmon resonant particles for biological detection,” Curr. Opin. Biotechnol. 14, 13–22 (2003).
[CrossRef] [PubMed]

R. Fikri, D. Barchiesi, F. H’Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2d finite element method,” Opt. Commun. 221, 13–22 (2003).
[CrossRef]

2002 (2)

H. Zhang and E. D. Hirleman, “Prediction of light scattering from particles on a filmed surface using discrete-dipole approximation,” Proc. SPIE 4692, 38–45 (2002).
[CrossRef]

D. W. Mackowski, “Discrete dipole moment method for calculation of the T-matrix for nonspherical particles,” J. Opt. Soc. Am. A 19, 881–893 (2002).
[CrossRef]

2001 (2)

A. G. Hoekstra, M. Frijlink, L. B. F. M. Waters, and P. M. A. Sloot, “Radiation forces in the discrete-dipole approximation,” J. Opt. Soc. Am. A 18, 1944–1953 (2001).
[CrossRef]

H. Kimura, “Light-scattering properties of fractal aggregates: numerical calculations by a superposition technique and the discrete-dipole approximation,” J. Quant. Spectrosc. Radiat. Transf. 70, 581–594 (2001).
[CrossRef]

2000 (1)

M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, 2000).

1999 (1)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

1998 (1)

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).

1997 (1)

1996 (1)

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[CrossRef]

1994 (1)

1993 (1)

1990 (1)

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).

1988 (1)

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

1985 (1)

E.Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

1984 (1)

E. K. Burke and G. J. Miller, “Modeling antennas near to and penetrating a lossy interface,” IEEE Trans. Antennas Propag. AP-32, 1040–1049 (1984).
[CrossRef]

1981 (1)

G. J. Burke and A. J. Poggio, Numerical Electromagnetics Code (NEC-4)—Method of Moments, Part I: Program Description—Theory, Tech. Rep. UCID-18834 (Lawrence Livermore Laboratory, 1981).

1977 (1)

L. N. Aksyutov, “Temperature dependence of the optical constants of tungsten and gold,” J. Appl. Spectrosc. 26, 656–660 (1977).
[CrossRef]

1975 (1)

D. L. Lager and R. J. Lytle, Fotran Subroutines for the Numerical Evaluation of Sommerfeld Integrals Unter Anderem, Tech. Rep. UCRL-51821 (Lawrence Livermore Laboratory, 1975).

1974 (1)

R. J. Lytle and D. L. Lager, Numerical Evaluation of Sommerfeld Integrals, Tech. Rep. UCRL-51688 (Lawrence Livermore Laboratory, 1974).

1973 (1)

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

1972 (1)

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

1966 (1)

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

1965 (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

1909 (1)

A. Sommerfeld, “Über die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 28, 665–737 (1909).

Aksyutov, L. N.

L. N. Aksyutov, “Temperature dependence of the optical constants of tungsten and gold,” J. Appl. Spectrosc. 26, 656–660 (1977).
[CrossRef]

Asakura, T.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[CrossRef]

Bachelot, R.

R. Fikri, D. Barchiesi, F. H’Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2d finite element method,” Opt. Commun. 221, 13–22 (2003).
[CrossRef]

Baños, A.

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

Barchiesi, D.

R. Fikri, D. Barchiesi, F. H’Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2d finite element method,” Opt. Commun. 221, 13–22 (2003).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Branczyk, A. M.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9, S196–S203 (2007).
[CrossRef]

Burke, E. K.

E. K. Burke and G. J. Miller, “Modeling antennas near to and penetrating a lossy interface,” IEEE Trans. Antennas Propag. AP-32, 1040–1049 (1984).
[CrossRef]

Burke, G. J.

G. J. Burke and A. J. Poggio, Numerical Electromagnetics Code (NEC-4)—Method of Moments, Part I: Program Description—Theory, Tech. Rep. UCID-18834 (Lawrence Livermore Laboratory, 1981).

Cao, Y.

G. H. Yuan, P. Wang, Y. H. Lu, Y. Cao, D. G. Zhang, H. Ming, and W. D. Xu, “A large-area photolithography technique based on surface plasmons leakage modes,” Opt. Commun. 281, 2680–2684 (2008).
[CrossRef]

Chen, H.

H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008).
[CrossRef] [PubMed]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).

Christy, R. W.

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

Crofcheck, C.

Dönmezer, F. N.

F. N. Dönmezer, M. P. Mengüç, and T. Okutucu, “Dependent absorption and scattering by interacting nanoparticles,” in Proceedings of the Sixth International Symposium on Radiative Transfer (2010).

Draine, B.

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

Draine, B. T.

Fikri, R.

R. Fikri, D. Barchiesi, F. H’Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2d finite element method,” Opt. Commun. 221, 13–22 (2003).
[CrossRef]

Flatau, P. J.

Frey, B. J.

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[CrossRef]

Frijlink, M.

H’Dhili, F.

R. Fikri, D. Barchiesi, F. H’Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2d finite element method,” Opt. Commun. 221, 13–22 (2003).
[CrossRef]

Harada, Y.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[CrossRef]

Hastings, J. T.

Hasuo, M.

S. Tojo and M. Hasuo, “Oscillator-strength enhancement of electric-dipole-forbidden transitions in evanescent light at total reflection,” Phys. Rev. A 71, 012508 (2005).
[CrossRef]

Hawes, E. A.

Heckenberg, N. R.

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Modelling of high numerical aperture imaging of complex scatterers using T-matrix method,” in Proceedings of ELS XII Helsinki (2010), pp. 138–141.

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete-dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transf. 110, 1460–1471 (2009).
[CrossRef]

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9, S196–S203 (2007).
[CrossRef]

Hirleman, E. D.

H. Zhang and E. D. Hirleman, “Prediction of light scattering from particles on a filmed surface using discrete-dipole approximation,” Proc. SPIE 4692, 38–45 (2002).
[CrossRef]

R. Schmehl, B. M. Nebeker, and E. D. Hirleman, “Discrete-dipole approximation for scattering by features on surfaces by means of a two dimensional fast Fourier transform technique,” J. Opt. Soc. Am. A 14, 3026–3036 (1997).
[CrossRef]

Hoekstra, A. G.

Hovenier, J. W.

M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, 2000).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).

Johnson, P. B.

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

Kimura, H.

H. Kimura, “Light-scattering properties of fractal aggregates: numerical calculations by a superposition technique and the discrete-dipole approximation,” J. Quant. Spectrosc. Radiat. Transf. 70, 581–594 (2001).
[CrossRef]

Knöner, G.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9, S196–S203 (2007).
[CrossRef]

Kou, X.

H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008).
[CrossRef] [PubMed]

Lager, D. L.

D. L. Lager and R. J. Lytle, Fotran Subroutines for the Numerical Evaluation of Sommerfeld Integrals Unter Anderem, Tech. Rep. UCRL-51821 (Lawrence Livermore Laboratory, 1975).

R. J. Lytle and D. L. Lager, Numerical Evaluation of Sommerfeld Integrals, Tech. Rep. UCRL-51688 (Lawrence Livermore Laboratory, 1974).

Leviton, D. B.

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[CrossRef]

Loke, V. L. Y.

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Modelling of high numerical aperture imaging of complex scatterers using T-matrix method,” in Proceedings of ELS XII Helsinki (2010), pp. 138–141.

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete-dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transf. 110, 1460–1471 (2009).
[CrossRef]

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9, S196–S203 (2007).
[CrossRef]

Lu, Y. H.

G. H. Yuan, P. Wang, Y. H. Lu, Y. Cao, D. G. Zhang, H. Ming, and W. D. Xu, “A large-area photolithography technique based on surface plasmons leakage modes,” Opt. Commun. 281, 2680–2684 (2008).
[CrossRef]

Lytle, R. J.

D. L. Lager and R. J. Lytle, Fotran Subroutines for the Numerical Evaluation of Sommerfeld Integrals Unter Anderem, Tech. Rep. UCRL-51821 (Lawrence Livermore Laboratory, 1975).

R. J. Lytle and D. L. Lager, Numerical Evaluation of Sommerfeld Integrals, Tech. Rep. UCRL-51688 (Lawrence Livermore Laboratory, 1974).

Mackowski, D. W.

Madison, T. J.

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[CrossRef]

Mengüç, M. P.

F. N. Dönmezer, M. P. Mengüç, and T. Okutucu, “Dependent absorption and scattering by interacting nanoparticles,” in Proceedings of the Sixth International Symposium on Radiative Transfer (2010).

E. A. Hawes, J. T. Hastings, C. Crofcheck, and M. P. Mengüç, “Spatially selective melting and evaporation of nanosized gold particles,” Opt. Lett. 33, 1383–1385 (2008).
[CrossRef] [PubMed]

Miller, G. J.

E. K. Burke and G. J. Miller, “Modeling antennas near to and penetrating a lossy interface,” IEEE Trans. Antennas Propag. AP-32, 1040–1049 (1984).
[CrossRef]

Ming, H.

G. H. Yuan, P. Wang, Y. H. Lu, Y. Cao, D. G. Zhang, H. Ming, and W. D. Xu, “A large-area photolithography technique based on surface plasmons leakage modes,” Opt. Commun. 281, 2680–2684 (2008).
[CrossRef]

Mischenko, M. I.

M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, 2000).

Nebeker, B. M.

Ni, W.

H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008).
[CrossRef] [PubMed]

Nieminen, T. A.

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Modelling of high numerical aperture imaging of complex scatterers using T-matrix method,” in Proceedings of ELS XII Helsinki (2010), pp. 138–141.

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete-dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transf. 110, 1460–1471 (2009).
[CrossRef]

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9, S196–S203 (2007).
[CrossRef]

Okutucu, T.

F. N. Dönmezer, M. P. Mengüç, and T. Okutucu, “Dependent absorption and scattering by interacting nanoparticles,” in Proceedings of the Sixth International Symposium on Radiative Transfer (2010).

Pennypacker, C.

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

Poggio, A. J.

G. J. Burke and A. J. Poggio, Numerical Electromagnetics Code (NEC-4)—Method of Moments, Part I: Program Description—Theory, Tech. Rep. UCID-18834 (Lawrence Livermore Laboratory, 1981).

Purcell, E.

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

Royer, P.

R. Fikri, D. Barchiesi, F. H’Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2d finite element method,” Opt. Commun. 221, 13–22 (2003).
[CrossRef]

Rubinsztein-Dunlop, H.

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Modelling of high numerical aperture imaging of complex scatterers using T-matrix method,” in Proceedings of ELS XII Helsinki (2010), pp. 138–141.

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete-dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transf. 110, 1460–1471 (2009).
[CrossRef]

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9, S196–S203 (2007).
[CrossRef]

Schmehl, R.

Schultz, D. A.

D. A. Schultz, “Plasmon resonant particles for biological detection,” Curr. Opin. Biotechnol. 14, 13–22 (2003).
[CrossRef] [PubMed]

Sloot, P. M. A.

Sommerfeld, A.

A. Sommerfeld, “Über die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 28, 665–737 (1909).

Stilgoe, A. B.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9, S196–S203 (2007).
[CrossRef]

Taubenblatt, M. A.

Tojo, S.

S. Tojo and M. Hasuo, “Oscillator-strength enhancement of electric-dipole-forbidden transitions in evanescent light at total reflection,” Phys. Rev. A 71, 012508 (2005).
[CrossRef]

Tran, T. K.

Travis, L. D.

M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, 2000).

Vial, A.

R. Fikri, D. Barchiesi, F. H’Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2d finite element method,” Opt. Commun. 221, 13–22 (2003).
[CrossRef]

Wang, J.

H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008).
[CrossRef] [PubMed]

Wang, P.

G. H. Yuan, P. Wang, Y. H. Lu, Y. Cao, D. G. Zhang, H. Ming, and W. D. Xu, “A large-area photolithography technique based on surface plasmons leakage modes,” Opt. Commun. 281, 2680–2684 (2008).
[CrossRef]

Waterman, P. C.

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

Waters, L. B. F. M.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Xu, W. D.

G. H. Yuan, P. Wang, Y. H. Lu, Y. Cao, D. G. Zhang, H. Ming, and W. D. Xu, “A large-area photolithography technique based on surface plasmons leakage modes,” Opt. Commun. 281, 2680–2684 (2008).
[CrossRef]

Yang, Z.

H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008).
[CrossRef] [PubMed]

Yuan, G. H.

G. H. Yuan, P. Wang, Y. H. Lu, Y. Cao, D. G. Zhang, H. Ming, and W. D. Xu, “A large-area photolithography technique based on surface plasmons leakage modes,” Opt. Commun. 281, 2680–2684 (2008).
[CrossRef]

Zhang, D. G.

G. H. Yuan, P. Wang, Y. H. Lu, Y. Cao, D. G. Zhang, H. Ming, and W. D. Xu, “A large-area photolithography technique based on surface plasmons leakage modes,” Opt. Commun. 281, 2680–2684 (2008).
[CrossRef]

Zhang, H.

H. Zhang and E. D. Hirleman, “Prediction of light scattering from particles on a filmed surface using discrete-dipole approximation,” Proc. SPIE 4692, 38–45 (2002).
[CrossRef]

Ann. Phys. (Leipzig) (1)

A. Sommerfeld, “Über die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 28, 665–737 (1909).

Astrophys. J. (2)

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

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

Curr. Opin. Biotechnol. (1)

D. A. Schultz, “Plasmon resonant particles for biological detection,” Curr. Opin. Biotechnol. 14, 13–22 (2003).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propag. (1)

E. K. Burke and G. J. Miller, “Modeling antennas near to and penetrating a lossy interface,” IEEE Trans. Antennas Propag. AP-32, 1040–1049 (1984).
[CrossRef]

J. Appl. Spectrosc. (1)

L. N. Aksyutov, “Temperature dependence of the optical constants of tungsten and gold,” J. Appl. Spectrosc. 26, 656–660 (1977).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9, S196–S203 (2007).
[CrossRef]

J. Opt. Soc. Am. A (5)

J. Quant. Spectrosc. Radiat. Transf. (2)

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete-dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transf. 110, 1460–1471 (2009).
[CrossRef]

H. Kimura, “Light-scattering properties of fractal aggregates: numerical calculations by a superposition technique and the discrete-dipole approximation,” J. Quant. Spectrosc. Radiat. Transf. 70, 581–594 (2001).
[CrossRef]

Langmuir (1)

H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008).
[CrossRef] [PubMed]

Opt. Commun. (3)

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[CrossRef]

G. H. Yuan, P. Wang, Y. H. Lu, Y. Cao, D. G. Zhang, H. Ming, and W. D. Xu, “A large-area photolithography technique based on surface plasmons leakage modes,” Opt. Commun. 281, 2680–2684 (2008).
[CrossRef]

R. Fikri, D. Barchiesi, F. H’Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2d finite element method,” Opt. Commun. 221, 13–22 (2003).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

S. Tojo and M. Hasuo, “Oscillator-strength enhancement of electric-dipole-forbidden transitions in evanescent light at total reflection,” Phys. Rev. A 71, 012508 (2005).
[CrossRef]

Phys. Rev. B (1)

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

Proc. IEEE (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

Proc. SPIE (2)

H. Zhang and E. D. Hirleman, “Prediction of light scattering from particles on a filmed surface using discrete-dipole approximation,” Proc. SPIE 4692, 38–45 (2002).
[CrossRef]

B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” Proc. SPIE 6273, 62732J (2006).
[CrossRef]

Other (11)

F. N. Dönmezer, M. P. Mengüç, and T. Okutucu, “Dependent absorption and scattering by interacting nanoparticles,” in Proceedings of the Sixth International Symposium on Radiative Transfer (2010).

M. I. Mischenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, 2000).

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).

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

E.Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Modelling of high numerical aperture imaging of complex scatterers using T-matrix method,” in Proceedings of ELS XII Helsinki (2010), pp. 138–141.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).

G. J. Burke and A. J. Poggio, Numerical Electromagnetics Code (NEC-4)—Method of Moments, Part I: Program Description—Theory, Tech. Rep. UCID-18834 (Lawrence Livermore Laboratory, 1981).

R. J. Lytle and D. L. Lager, Numerical Evaluation of Sommerfeld Integrals, Tech. Rep. UCRL-51688 (Lawrence Livermore Laboratory, 1974).

D. L. Lager and R. J. Lytle, Fotran Subroutines for the Numerical Evaluation of Sommerfeld Integrals Unter Anderem, Tech. Rep. UCRL-51821 (Lawrence Livermore Laboratory, 1975).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (17)

Fig. 1
Fig. 1

Dipole model of an AFM probe, a particle, and an infinite surface. The evanescent wave above the surface results from the internally reflected plane wave from below the surface.

Fig. 2
Fig. 2

System of equations comprises the interaction matrix ( A ) , the dipole moments ( P ) as unknowns, and the incident field ( E inc ) . A is a square matrix containing N × N of A j k 3 × 3 tensors, where N is the number of dipoles.

Fig. 3
Fig. 3

Radiating dipole over a surface, its image, and the receiving dipole. On the surface, z = 0 .

Fig. 4
Fig. 4

Spherical wave decomposed into cylindrical and planar components.

Fig. 5
Fig. 5

Complex integration contour used in the Bessel function formulation [16]. ψ = ρ or ψ = | z j | + | z k | , whichever is larger.

Fig. 6
Fig. 6

Complex integration contour used in the Hankel function formulation [16]; θ = tan 1 [ ρ / ( | z j | + | z k | ) ] .

Fig. 7
Fig. 7

Total internal reflection of incident (a) TM and (b) TE plane waves from below the substrate surface. Evanescent waves exist above the surface.

Fig. 8
Fig. 8

Scattering frame of a given dipole for calculating the scattered field.

Fig. 9
Fig. 9

Scattering geometry for a particle on a surface. The direction of the incident light is defined by the angle γ from the surface normal, and the incident plane coincides with x = 0 . The scattered far field is calculated for a range of zenith angles θ along the plane at the azimuthal angle ϕ.

Fig. 10
Fig. 10

Normalized intensity versus scattering angle for a 540 nm polystyrene sphere on a flat Si surface. The incident light was s-polarized, the wavelength was λ = 632.8   nm , and the incident angle was γ = 0 ° .

Fig. 11
Fig. 11

Normalized intensity versus scattering angle for a 300 nm polystyrene sphere on a flat Si surface. The incident light was s-polarized, the wavelength was λ = 632.8   nm , and the incident angle was γ = 65 ° .

Fig. 12
Fig. 12

Field intensity of the dipoles under (a) TM and (b) TE evanescent field illumination. The point dipoles are represented here with gray-scaled spheres where the intensity is proportional to the darkness of the shade.

Fig. 13
Fig. 13

Field intensity of the 32 dipoles in the nano-sphere versus the shaft length L, under TM evanescent field illumination ( λ = 632   nm ) .

Fig. 14
Fig. 14

Field intensity of the 32 dipoles in the nano-sphere versus the shaft length L, under TE evanescent field illumination ( λ = 632   nm ) .

Fig. 15
Fig. 15

Field intensity of the 32 dipoles in the nano-sphere versus the AFM tip-particle separation d, with the shaft length constant at L = 60   nm , under TM evanescent field illumination ( λ = 632   nm ) .

Fig. 16
Fig. 16

Field intensity spectra of the 32 dipoles in the nano-sphere, with AFM tip-particle separations of (a) d = 1   nm and (b) d = 14   nm , with the shaft length of L = 60   nm , and (c) in the absence of the AFM tip, illuminated by a TM evanescent wave.

Fig. 17
Fig. 17

Complex refractive index of Si versus wavelength.

Equations (75)

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

P j = α j E j ,
E j = E inc , j k j A j k P j ,
A j k = exp ( i k r j k ) r j k [ k 2 ( r ̂ j k r ̂ j k 1 3 ) + i k r j k 1 r j k 2 ( 3 r ̂ j k r ̂ j k I 3 ) ] ,
j k ,
E inc , j = A j j P j + k j A j k P j ,
k = 1 N A j k P j = E inc , j .
α j CM = 3 d 3 4 π ( m j 2 1 m j 2 + 2 ) = 3 d 3 4 π ( ϵ j 1 ϵ j + 2 ) ,
α j LDR = α j CM 1 + α j CM d 3 [ ( b 1 + m 2 b 2 + m 2 b 3 S ) ( k d ) 2 2 3 i ( k d ) 3 ] ,
b 1 = 1.891 531 ,     b 2 = 0.164 846 9 ,
b 3 = 1.770 000 4 ,     S j = 1 3 ( a ̂ j e ̂ j ) 2 ,
N d 3 = 4 3 π r 3 .
r = ( 3 N 4 π ) 1 / 3 d .
d 1 k | m | .
e i k r 4 π r = i 4 π 0 k ρ k z J 0 ( k ρ ρ ) e i k z | z | d k ρ ,
e i k r 4 π r = i 8 π k ρ k z H 0 ( 1 ) ( k ρ ρ ) e i k z | z | d k ρ ,
R TE = μ r k 2 z k 1 z μ r k 2 z + k 1 z e i k 2 z z k ,
R TM = ϵ r k 2 z k 1 z ϵ r k 2 z + k 1 z e i k 2 z z k ,
( e i k r 4 π r ) TE I = i 4 π 0 k 2 ρ k 2 z J 0 ( k 2 ρ ρ ) μ r k 2 z k 1 z μ r k 2 z + k 1 z e i k 2 z ( z j + z k ) d k ρ ,
( e i k r 4 π r ) TM I = i 4 π 0 k 2 ρ k 2 z J 0 ( k 2 ρ ρ ) ϵ r k 2 z k 1 z ϵ r k 2 z + k 1 z e i k 2 z ( z j + z k ) d k ρ .
( e i k r 4 π r ) TM I = 1 4 π 0 [ 1 γ 2 J 0 ( λ ρ ) e γ 2 ( z j + z k ) + 2 k 1 2 k 1 2 γ 2 + k 2 2 γ 1 J 0 ( λ ρ ) e γ 2 ( z j + z k ) ] λ d λ .
( e i k r 4 π r ) TM I = e i k r 4 π r I + k 1 2 2 π 0 [ 1 k 1 2 γ 2 + k 2 2 γ 1 J 0 ( λ ρ ) e γ 2 ( z j + z k ) ] λ d λ ,
E j I = k 2 2 ϵ 2 G j , k I P k ,
G j , k I = ( I + ϵ 2 k 2 2 ) ( e i k 2 r I , j k 4 π r I , j k ) = { I + ( 1 ϵ 2 k 2 2 ) [ 2 ρ 2 2 ρ ρ ϕ 2 ρ z 2 ρ ρ ϕ 2 ρ 2 ϕ 2 2 ρ ϕ z 2 ρ z 2 ρ ϕ z 2 z 2 ] } ( e i k 2 r I , j k 4 π r I , j k ) .
E ρ V = P ρ 4 π ϵ 2 ( 2 ρ z k 1 2 V 22 + k 1 2 k 2 2 k 1 2 + k 2 2 2 ρ z e i k 2 r I r I ) ,
E z V = P z 4 π ϵ 2 [ ( 2 z 2 + k 2 2 ) k 1 2 V 22 + k 1 2 k 2 2 k 1 2 + k 2 2 ( 2 z 2 + k 2 2 ) e i k 2 r I r I ] ,
V 22 = V 22 2 k 1 2 + k 2 2 e i k 2 r I r I ,
V 22 = 2 0 e γ 2 ( z j + z k ) k 1 2 γ 2 + k 2 2 γ 1 J 0 ( λ ρ ) λ d λ .
E ρ H = P ρ 4 π ϵ 2 cos   ϕ [ 2 ρ 2 k 1 2 V 22 + k 2 2 U 22 + k 1 2 k 2 2 k 1 2 + k 2 2 ( 2 ρ 2 + k 2 2 ) e i k 2 r I r I ] ,
E ϕ H = P ρ 4 π ϵ 2 sin   ϕ [ 2 ρ 2 k 1 2 V 22 + k 2 2 U 22 + k 1 2 k 2 2 k 1 2 + k 2 2 ( 2 ρ 2 + k 2 2 ) e i k 2 r I r I ] ,
E z H = cos   ϕ E 2 , ρ V ,
U 22 = U 22 2 k 2 2 k 1 2 + k 2 2 e i k 2 r I r I ,
U 22 = 2 0 e γ 2 ( z j + z k ) γ 2 + γ 1 J 0 ( λ ρ ) λ d λ .
V 22 = 0 e γ 2 ( z j + z k ) k 1 2 γ 2 + k 2 2 γ 1 H 0 2 ( λ ρ ) λ d λ ,
U 22 = 0 e γ 2 ( z j + z k ) γ 2 + γ 1 H 0 2 ( λ ρ ) λ d λ .
I ρ V = 2 ρ z k 1 2 V 22 ,
I z V = ( 2 z 2 + k 2 2 ) k 1 2 V 22 ,
I ρ H = ( 2 ρ 2 k 2 2 V 22 + k 2 2 U 22 ) ,
I ϕ H = ( 1 ρ ρ k 2 2 V 22 + k 2 2 U 22 ) ,
[ E x E y E z ] = [ x 2 ρ 2 E ρ H y 2 ρ 2 E ϕ H x y ρ 2 ( E ρ H + E ϕ H ) x ρ E ρ V x y ρ 2 ( E ρ H + E ϕ H ) y 2 ρ 2 E ρ H x 2 ρ 2 E ϕ H y ρ E ρ V x ρ E z H y ρ E z H E z V ] .
E x ( x ) = 1 4 π ϵ 2 { [ ( x ρ ) 2 I ρ H ( y ρ ) 2 I ϕ H ] k 1 2 k 2 2 k 1 2 + k 2 2 e i k 2 r r ( 2 x 2 + k 2 2 ) } P x ,
E y ( x ) = 1 4 π ϵ 2 { x y ρ 2 [ I ρ H + I ϕ H ] k 1 2 k 2 2 k 1 2 + k 2 2 e i k 2 r r 2 x y } P x ,
E z ( x ) = 1 4 π ϵ 2 { x ρ I ρ V + k 1 2 k 2 2 k 1 2 + k 2 2 e i k 2 r r 2 x z } P x ,
E x ( y ) = 1 4 π ϵ 2 { x y ρ 2 [ I ρ H + I ϕ H ] k 1 2 k 2 2 k 1 2 + k 2 2 e i k 2 r r 2 x y } P y ,
E y ( y ) = 1 4 π ϵ 2 { [ ( y ρ ) 2 I ρ H ( x ρ ) 2 I ϕ H ] k 1 2 k 2 2 k 1 2 + k 2 2 e i k 2 r r ( 2 x 2 + k 2 2 ) } P y ,
E z ( y ) = 1 4 π ϵ 2 { y ρ I ρ V + k 1 2 k 2 2 k 1 2 + k 2 2 e i k 2 r r 2 y z } P y ,
E z ( z ) = 1 4 π ϵ 2 { x ρ I ρ V + k 1 2 k 2 2 k 1 2 + k 2 2 e i k 2 r r 2 x z } P z ,
E y ( z ) = 1 4 π ϵ 2 { y ρ I ρ V + k 1 2 k 2 2 k 1 2 + k 2 2 e i k 2 r r 2 y z } P z ,
E z ( z ) = 1 4 π ϵ 2 { I z V + k 1 2 k 2 2 k 1 2 + k 2 2 e i k 2 r r ( 2 z 2 + k 2 2 ) } P z .
k = 1 N A j k SI P k = k = 1 N ( B j + k 0 2 G j k + R j k ) P k = E inc , j ,
G j k = exp ( i k 0 r j k ) r j k [ β j k + γ j k r ̂ j k , x 2 γ j k r ̂ j k , x r ̂ j k , y γ j k r ̂ j k , x r ̂ j k , z γ j k r ̂ j k , y r ̂ j k , x β j k + γ j k r ̂ j k , y 2 γ j k r ̂ j k , y r ̂ j k , z γ j k r ̂ j k , z r ̂ j k , x γ j k r ̂ j k , z r ̂ j k , y β j k + γ j k r ̂ j k , z 2 ] ,
r j k = [ ( x j x k ) 2 + ( y j y k ) 2 + ( z j z k ) 2 ] 1 / 2 ,
r ̂ j k , x = r j k , x r j k ,     r ̂ j k , y = r j k , y r j k ,     r ̂ j k , z = r j k , z r j k ,
β j k = [ 1 ( k 0 r j k ) 2 + i ( k 0 r j k ) 1 ] ,
γ j k = [ 1 3 ( k 0 r j k ) 2 + 3 i ( k 0 r j k ) 1 ] .
R j k = [ r ̂ j k x I 2 I ρ H r ̂ j k y I 2 I ϕ H r ̂ j k x I 2 r ̂ j k y I 2 ( I ρ H + I ϕ H ) r ̂ j k x I 2 I ρ V r ̂ j k x I 2 r ̂ j k y I 2 ( I ρ H + I ϕ H ) r ̂ j k x I 2 I ρ H r ̂ j k y I 2 I ϕ H r ̂ j k y I 2 I ρ V r ̂ j k x I 2 I ρ V r ̂ j k y I 2 I ρ V I z V ] k 1 2 k 2 2 k 1 2 + k 2 2 exp ( i k 0 r I , j k ) r I , j k [ ( β j k I + γ j k I r ̂ j k x I 2 ) γ j k I r ̂ j k x I 2 r ̂ j k y I 2 γ j k I r ̂ j k x I 2 r ̂ j k z I 2 γ j k I r ̂ j k y I 2 r ̂ j k x I 2 ( β j k I + γ j k I r ̂ j k y I 2 ) γ j k I r ̂ j k y I 2 r ̂ j k z I 2 γ j k I r ̂ j k z I 2 r ̂ j k x I 2 γ j k I r ̂ j k z I 2 r ̂ j k y I 2 β j k I + γ j k I r ̂ j k z I 2 ] ,
r j k I = [ ( x j x k ) 2 + ( y j y k ) 2 + ( z j + z k ) 2 ] 1 / 2 ,
r ̂ j k x I = r j k x I r j k I ,     r ̂ j k y I = r j k y I r j k I ,     r ̂ j k z I = r j k z I r j k I ,
β j k I = [ 1 ( k 0 r j k I ) 2 + i ( k 0 r j k I ) 1 ] ,
γ j k I = [ 1 3 ( k 0 r j k I ) 2 + 3 i ( k 0 r j k I ) 1 ] .
n 1   sin   θ 1 = n 2   sin   θ 2 .
k = ( 0 , n 2 k 0   sin   θ 2 , n 2 k 0   cos   θ 2 ) ,
sin   θ 2 = ( n 2 / n 1 ) sin   θ 1 ,
cos   θ 2 = i ( n 2 / n 1 ) 2 sin 2 θ 1 1 .
e p = ( 0 , i ( n 2 / n 1 ) 2 sin 2 θ 1 1 , ( n 2 / n 1 ) sin   θ 1 ) ,
e s = ( 1 , 0 , 0 ) ,
T 2 , p = 2 n 1   cos   θ 1 n 1   cos   θ 1 + ( n 1 / n 2 ) n 2 2 n 1 2 sin 2 θ 1 | E 1 , p | ,
T 2 , s = 2 n 1   cos   θ 1 n 1   cos   θ 1 + n 2 2 n 1 2 sin 2 θ 1 | E 1 , s | ,
E 2 , p = e p | T 2 , p | ,
E 2 , s = e s | T 2 , s | .
E sca ( r ) = k 0 2 e i k 0 r 4 π r j = 1 N { e i k sca r j [ ( p j e ̂ 1 ) e ̂ 1 + ( p j e ̂ 2 ) e ̂ 2 ] + e i k sca r I , j [ R TM ( p j e ̂ 1 ) e ̂ 1 + R TE ( p j e ̂ 2 ) e ̂ 2 ] } ,
[ E p , sca E s , sca ] = e i k ( r z ) i k r [ S 2 S 3 S 4 S 1 ] [ E p , inc E s , inc ] ,
S 1 = i k 0 3 4 π ϵ 0 j = 1 N { e i k sca r j + R TE e i k sca r I , j } p j ( 2 ) e ̂ 2 ,
S 2 = i k 0 3 4 π ϵ 0 j = 1 N { e i k sca r j + R TM e i k sca r I , j } p j ( 1 ) e ̂ 1 ,
S 3 = i k 0 3 4 π ϵ 0 j = 1 N { e i k sca r j + R TM e i k sca r I , j } p j ( 2 ) e ̂ 1 ,
S 4 = i k 0 3 4 π ϵ 0 j = 1 N { e i k sca r j + R TE e i k sca r I , j } p j ( 1 ) e ̂ 2 .

Metrics