Abstract

We derive an analytic expression for the local field intensity in an aggregate of particles illuminated by diffuse light (T matrix formalism). To be precise, the diffuse light average is obtained by averaging the electromagnetic response from plane waves over all possible incident field direction polarizations. We applied this new averaging formula to analyze variations in the electromagnetic couplings between two isotropic spheres as a function of their separation distance. The numerical calculations were performed with the recursive centered T matrix algorithm (RCTMA), one of the known analytical solutions of the multiple scattering equation of light. Illustrative calculations clearly demonstrate that diffuse averaging has a large smoothing effect on the strong angular and local variations in the field intensities that are omnipresent in orientation fixed calculations. We believe that this formalism can be a valuable tool in the analysis of electromagnetic couplings in dense random heterogeneous media, in which light propagation is dominated by a scalar diffusion behavior.

© 2008 Optical Society of America

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References

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  1. P. G. Etchegoin, R. C. Maher, and L. F. Cohen, “Amplification of locals in disordered metallic structures,” New J. Phys. 6, 142 (2004).
    [CrossRef]
  2. A. J. Ward and J. B. Pendry, “Calculating photonic Green's function using a non-orthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252 (1998).
    [CrossRef]
  3. J. R. Nestor and E. R. Lippincott, “The effect of internal field on Raman scattering cross sections,” J. Raman Spectrosc. 1, 305-318 (2005).
    [CrossRef]
  4. R. H. Harding, B. Golding, and R. A. Morgen, “Optics of light-scattering films. Study of effects of pigment size and concentration,” J. Opt. Soc. Am. 50, 446-455 (1960).
    [CrossRef]
  5. D. F. Tunstall and M. J. Hird, “Effect of particle crowding on scattering power of TiO2 pigments,” J. Paint Technol. 46, 33-40 (1974).
  6. E. S. Thiele and R. H. French, “Light-scattering properties of representative, morphological rutile titanium particles using a finite-element method,” J. Am. Ceram. Soc. 81, 469-479 (1998).
    [CrossRef]
  7. L. E. McNeil, A. R. Hanuska, and R. H. French, “Orientation dependence in near-field scattering from TiO2 particles,” Appl. Opt. 40, 3726-3736 (2001).
    [CrossRef]
  8. K. S. Kunz and R. J. Luebbers, Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).
  9. R. Garg, R. K. Prud'homme, and I. A. Aksay, “Optical transmission in highly concentrated dispersion,” J. Opt. Soc. Am. A 15, 932-935 (1998).
    [CrossRef]
  10. D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851-2861 (1994).
    [CrossRef]
  11. B. Stout, J. C. Auger, and J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105-2128 (2001).
  12. B. Stout, J. C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129-2152 (2002).
    [CrossRef]
  13. D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266-2278 (1996).
    [CrossRef]
  14. S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19(1), 15-24 (1961).
  15. P. C. Waterman, “Symmetry, unitary and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825-839 (1971).
    [CrossRef]
  16. A.-K. Hamid, “Electromagnetic scattering by an arbitrary configuration of dielectric spheres,” Can. J. Phys. 68, 1419-1428 (1990).
    [CrossRef]
  17. K. A. Fuller and G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. I: Linear chains,” Opt. Lett. 13, 90-92 (1988).
    [CrossRef] [PubMed]
  18. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).
  19. T. Wriedt, “Using the T-matrix method for light scattering computations by nonaxisymmetric particles: superellipsoids and realistically shaped particles,” Part. Part. Syst. Charact. 19, 256-268 (2002).
    [CrossRef]
  20. D. Petrov, Y. Shkuratov, and G. Videen, “Analytical light-scattering solution for Chebyshev particles,” J. Opt. Soc. Am. A 24, 1103-1119 (2007).
    [CrossRef]
  21. F. M. Schultz, K. Stamnes, and J. J. Stamnes, “Scattering of electromagnetic waves by spheroid particles: a novel approach exploiting the T-matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875-7896 (1998).
    [CrossRef]
  22. L. Cruz, L. F. Fonseca, and M. Gómez, “T-matrix approach for the calculation of local fields in the neighborhood of small clusters in the electrodynamic regime,” Phys. Rev. B 40, 7491-7500 (1989).
    [CrossRef]
  23. W. Vargas, L. Cruz, L. F. Fonseca, and M. Gómez, “T-matrix approach for calculation local fields around clusters of rotated spheroids,” Appl. Opt. 32, 2164-2170 (1993).
    [CrossRef] [PubMed]
  24. J. C. Auger, V. Martinez, and B. Stout, “Absorption and scattering properties of dense ensemble of nonspherical particles,” J. Opt. Soc. Am. A 24, 3508-3516 (2007).
    [CrossRef]
  25. S. Fitzwater and J. W. Hook III, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coat. Technol. 57, 39-47 (1985).
  26. W. E. Vargas, “Optical properties of pigmented coating taking into account the particle interactions,” J. Quant. Spectrosc. Radiat. Transfer 78, 187-195 (2003).
    [CrossRef]
  27. J. C. Auger, V. Martinez, and B. Stout, “Scattering efficiency of aggregated clusters of spheres: dependence on configuration and composition,” J. Opt. Soc. Am. A 22, 2700-2708 (2005).
    [CrossRef]

2007 (2)

2005 (2)

J. C. Auger, V. Martinez, and B. Stout, “Scattering efficiency of aggregated clusters of spheres: dependence on configuration and composition,” J. Opt. Soc. Am. A 22, 2700-2708 (2005).
[CrossRef]

J. R. Nestor and E. R. Lippincott, “The effect of internal field on Raman scattering cross sections,” J. Raman Spectrosc. 1, 305-318 (2005).
[CrossRef]

2004 (1)

P. G. Etchegoin, R. C. Maher, and L. F. Cohen, “Amplification of locals in disordered metallic structures,” New J. Phys. 6, 142 (2004).
[CrossRef]

2003 (1)

W. E. Vargas, “Optical properties of pigmented coating taking into account the particle interactions,” J. Quant. Spectrosc. Radiat. Transfer 78, 187-195 (2003).
[CrossRef]

2002 (2)

B. Stout, J. C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129-2152 (2002).
[CrossRef]

T. Wriedt, “Using the T-matrix method for light scattering computations by nonaxisymmetric particles: superellipsoids and realistically shaped particles,” Part. Part. Syst. Charact. 19, 256-268 (2002).
[CrossRef]

2001 (2)

L. E. McNeil, A. R. Hanuska, and R. H. French, “Orientation dependence in near-field scattering from TiO2 particles,” Appl. Opt. 40, 3726-3736 (2001).
[CrossRef]

B. Stout, J. C. Auger, and J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105-2128 (2001).

1998 (4)

F. M. Schultz, K. Stamnes, and J. J. Stamnes, “Scattering of electromagnetic waves by spheroid particles: a novel approach exploiting the T-matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875-7896 (1998).
[CrossRef]

R. Garg, R. K. Prud'homme, and I. A. Aksay, “Optical transmission in highly concentrated dispersion,” J. Opt. Soc. Am. A 15, 932-935 (1998).
[CrossRef]

A. J. Ward and J. B. Pendry, “Calculating photonic Green's function using a non-orthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252 (1998).
[CrossRef]

E. S. Thiele and R. H. French, “Light-scattering properties of representative, morphological rutile titanium particles using a finite-element method,” J. Am. Ceram. Soc. 81, 469-479 (1998).
[CrossRef]

1996 (1)

1994 (1)

1993 (1)

1990 (1)

A.-K. Hamid, “Electromagnetic scattering by an arbitrary configuration of dielectric spheres,” Can. J. Phys. 68, 1419-1428 (1990).
[CrossRef]

1989 (1)

L. Cruz, L. F. Fonseca, and M. Gómez, “T-matrix approach for the calculation of local fields in the neighborhood of small clusters in the electrodynamic regime,” Phys. Rev. B 40, 7491-7500 (1989).
[CrossRef]

1988 (1)

1985 (1)

S. Fitzwater and J. W. Hook III, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coat. Technol. 57, 39-47 (1985).

1974 (1)

D. F. Tunstall and M. J. Hird, “Effect of particle crowding on scattering power of TiO2 pigments,” J. Paint Technol. 46, 33-40 (1974).

1971 (1)

P. C. Waterman, “Symmetry, unitary and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825-839 (1971).
[CrossRef]

1961 (1)

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19(1), 15-24 (1961).

1960 (1)

Aksay, I. A.

Auger, J. C.

J. C. Auger, V. Martinez, and B. Stout, “Absorption and scattering properties of dense ensemble of nonspherical particles,” J. Opt. Soc. Am. A 24, 3508-3516 (2007).
[CrossRef]

J. C. Auger, V. Martinez, and B. Stout, “Scattering efficiency of aggregated clusters of spheres: dependence on configuration and composition,” J. Opt. Soc. Am. A 22, 2700-2708 (2005).
[CrossRef]

B. Stout, J. C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129-2152 (2002).
[CrossRef]

B. Stout, J. C. Auger, and J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105-2128 (2001).

Cohen, L. F.

P. G. Etchegoin, R. C. Maher, and L. F. Cohen, “Amplification of locals in disordered metallic structures,” New J. Phys. 6, 142 (2004).
[CrossRef]

Cruz, L.

W. Vargas, L. Cruz, L. F. Fonseca, and M. Gómez, “T-matrix approach for calculation local fields around clusters of rotated spheroids,” Appl. Opt. 32, 2164-2170 (1993).
[CrossRef] [PubMed]

L. Cruz, L. F. Fonseca, and M. Gómez, “T-matrix approach for the calculation of local fields in the neighborhood of small clusters in the electrodynamic regime,” Phys. Rev. B 40, 7491-7500 (1989).
[CrossRef]

Etchegoin, P. G.

P. G. Etchegoin, R. C. Maher, and L. F. Cohen, “Amplification of locals in disordered metallic structures,” New J. Phys. 6, 142 (2004).
[CrossRef]

Fitzwater, S.

S. Fitzwater and J. W. Hook III, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coat. Technol. 57, 39-47 (1985).

Fonseca, L. F.

W. Vargas, L. Cruz, L. F. Fonseca, and M. Gómez, “T-matrix approach for calculation local fields around clusters of rotated spheroids,” Appl. Opt. 32, 2164-2170 (1993).
[CrossRef] [PubMed]

L. Cruz, L. F. Fonseca, and M. Gómez, “T-matrix approach for the calculation of local fields in the neighborhood of small clusters in the electrodynamic regime,” Phys. Rev. B 40, 7491-7500 (1989).
[CrossRef]

French, R. H.

L. E. McNeil, A. R. Hanuska, and R. H. French, “Orientation dependence in near-field scattering from TiO2 particles,” Appl. Opt. 40, 3726-3736 (2001).
[CrossRef]

E. S. Thiele and R. H. French, “Light-scattering properties of representative, morphological rutile titanium particles using a finite-element method,” J. Am. Ceram. Soc. 81, 469-479 (1998).
[CrossRef]

Fuller, K. A.

Garg, R.

Golding, B.

Gómez, M.

W. Vargas, L. Cruz, L. F. Fonseca, and M. Gómez, “T-matrix approach for calculation local fields around clusters of rotated spheroids,” Appl. Opt. 32, 2164-2170 (1993).
[CrossRef] [PubMed]

L. Cruz, L. F. Fonseca, and M. Gómez, “T-matrix approach for the calculation of local fields in the neighborhood of small clusters in the electrodynamic regime,” Phys. Rev. B 40, 7491-7500 (1989).
[CrossRef]

Hamid, A.-K.

A.-K. Hamid, “Electromagnetic scattering by an arbitrary configuration of dielectric spheres,” Can. J. Phys. 68, 1419-1428 (1990).
[CrossRef]

Hanuska, A. R.

Harding, R. H.

Hird, M. J.

D. F. Tunstall and M. J. Hird, “Effect of particle crowding on scattering power of TiO2 pigments,” J. Paint Technol. 46, 33-40 (1974).

Hook, J. W.

S. Fitzwater and J. W. Hook III, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coat. Technol. 57, 39-47 (1985).

Kattawar, G. W.

Kunz, K. S.

K. S. Kunz and R. J. Luebbers, Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Lafait, J.

B. Stout, J. C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129-2152 (2002).
[CrossRef]

B. Stout, J. C. Auger, and J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105-2128 (2001).

Lippincott, E. R.

J. R. Nestor and E. R. Lippincott, “The effect of internal field on Raman scattering cross sections,” J. Raman Spectrosc. 1, 305-318 (2005).
[CrossRef]

Luebbers, R. J.

K. S. Kunz and R. J. Luebbers, Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

Mackowski, D. W.

Maher, R. C.

P. G. Etchegoin, R. C. Maher, and L. F. Cohen, “Amplification of locals in disordered metallic structures,” New J. Phys. 6, 142 (2004).
[CrossRef]

Martinez, V.

McNeil, L. E.

Mishchenko, M. I.

D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266-2278 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Morgen, R. A.

Nestor, J. R.

J. R. Nestor and E. R. Lippincott, “The effect of internal field on Raman scattering cross sections,” J. Raman Spectrosc. 1, 305-318 (2005).
[CrossRef]

Pendry, J. B.

A. J. Ward and J. B. Pendry, “Calculating photonic Green's function using a non-orthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252 (1998).
[CrossRef]

Petrov, D.

Prud'homme, R. K.

Schultz, F. M.

Shkuratov, Y.

Stamnes, J. J.

Stamnes, K.

Stein, S.

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19(1), 15-24 (1961).

Stout, B.

J. C. Auger, V. Martinez, and B. Stout, “Absorption and scattering properties of dense ensemble of nonspherical particles,” J. Opt. Soc. Am. A 24, 3508-3516 (2007).
[CrossRef]

J. C. Auger, V. Martinez, and B. Stout, “Scattering efficiency of aggregated clusters of spheres: dependence on configuration and composition,” J. Opt. Soc. Am. A 22, 2700-2708 (2005).
[CrossRef]

B. Stout, J. C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129-2152 (2002).
[CrossRef]

B. Stout, J. C. Auger, and J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105-2128 (2001).

Thiele, E. S.

E. S. Thiele and R. H. French, “Light-scattering properties of representative, morphological rutile titanium particles using a finite-element method,” J. Am. Ceram. Soc. 81, 469-479 (1998).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Tunstall, D. F.

D. F. Tunstall and M. J. Hird, “Effect of particle crowding on scattering power of TiO2 pigments,” J. Paint Technol. 46, 33-40 (1974).

Vargas, W.

Vargas, W. E.

W. E. Vargas, “Optical properties of pigmented coating taking into account the particle interactions,” J. Quant. Spectrosc. Radiat. Transfer 78, 187-195 (2003).
[CrossRef]

Videen, G.

Ward, A. J.

A. J. Ward and J. B. Pendry, “Calculating photonic Green's function using a non-orthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252 (1998).
[CrossRef]

Waterman, P. C.

P. C. Waterman, “Symmetry, unitary and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825-839 (1971).
[CrossRef]

Wriedt, T.

T. Wriedt, “Using the T-matrix method for light scattering computations by nonaxisymmetric particles: superellipsoids and realistically shaped particles,” Part. Part. Syst. Charact. 19, 256-268 (2002).
[CrossRef]

Appl. Opt. (3)

Can. J. Phys. (1)

A.-K. Hamid, “Electromagnetic scattering by an arbitrary configuration of dielectric spheres,” Can. J. Phys. 68, 1419-1428 (1990).
[CrossRef]

J. Am. Ceram. Soc. (1)

E. S. Thiele and R. H. French, “Light-scattering properties of representative, morphological rutile titanium particles using a finite-element method,” J. Am. Ceram. Soc. 81, 469-479 (1998).
[CrossRef]

J. Coat. Technol. (1)

S. Fitzwater and J. W. Hook III, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coat. Technol. 57, 39-47 (1985).

J. Mod. Opt. (2)

B. Stout, J. C. Auger, and J. Lafait, “Individual and aggregate scattering matrices and cross sections: conservation laws and reciprocity,” J. Mod. Opt. 48, 2105-2128 (2001).

B. Stout, J. C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129-2152 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

J. Paint Technol. (1)

D. F. Tunstall and M. J. Hird, “Effect of particle crowding on scattering power of TiO2 pigments,” J. Paint Technol. 46, 33-40 (1974).

J. Quant. Spectrosc. Radiat. Transfer (1)

W. E. Vargas, “Optical properties of pigmented coating taking into account the particle interactions,” J. Quant. Spectrosc. Radiat. Transfer 78, 187-195 (2003).
[CrossRef]

J. Raman Spectrosc. (1)

J. R. Nestor and E. R. Lippincott, “The effect of internal field on Raman scattering cross sections,” J. Raman Spectrosc. 1, 305-318 (2005).
[CrossRef]

New J. Phys. (1)

P. G. Etchegoin, R. C. Maher, and L. F. Cohen, “Amplification of locals in disordered metallic structures,” New J. Phys. 6, 142 (2004).
[CrossRef]

Opt. Lett. (1)

Part. Part. Syst. Charact. (1)

T. Wriedt, “Using the T-matrix method for light scattering computations by nonaxisymmetric particles: superellipsoids and realistically shaped particles,” Part. Part. Syst. Charact. 19, 256-268 (2002).
[CrossRef]

Phys. Rev. B (2)

L. Cruz, L. F. Fonseca, and M. Gómez, “T-matrix approach for the calculation of local fields in the neighborhood of small clusters in the electrodynamic regime,” Phys. Rev. B 40, 7491-7500 (1989).
[CrossRef]

A. J. Ward and J. B. Pendry, “Calculating photonic Green's function using a non-orthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252 (1998).
[CrossRef]

Phys. Rev. D (1)

P. C. Waterman, “Symmetry, unitary and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825-839 (1971).
[CrossRef]

Q. Appl. Math. (1)

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19(1), 15-24 (1961).

Other (2)

K. S. Kunz and R. J. Luebbers, Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

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Figures (3)

Fig. 1
Fig. 1

(a) Mapping of the scattered electric field intensity (outside the particles) of a system composed of an isolated dielectric sphere with index of refraction 2.8 and radius 0.12 μ m embedded into an infinite nonabsorbing medium with refractive index 1.5. The incident field with wavelength of 0.545 μ m propagates in the positive direction of the O z axis. (b) Same as Fig. 1a but instead the system is composed of two identical dielectric spheres separated by | z 2 z 1 | = 17 r s . (c) Same as Fig. 1b with separations | z 2 z 1 | = 12 r s . (d) Same as Fig. 1b with separations | z 2 z 1 | = 8 r s . (e) Same as Fig. 1b with separations | z 2 z 1 | = 4 r s . (f) Same as Fig. 1b with separations | z 2 z 1 | = 2 r s .

Fig. 2
Fig. 2

(a) Mapping of the scattered electric field intensity (outside the particles) of a system composed of an isolated dielectric sphere with index of refraction 2.8 and radius 0.12 μ m embedded into an infinite nonabsorbing medium with refractive index 1.5. The incident field with wavelength of 0.545 μ m propagates in the negative direction of the O y axis. (b) Same as Fig. 2a but instead the system is composed of two identical dielectric spheres separated by | z 2 z 1 | = 17 r s . (c) Same as Fig. 2a with separations | z 2 z 1 | = 12 r s . (d) Same as Fig. 2a with separations | z 2 z 1 | = 8 r s . (e) Same as Fig. 2a with separations | z 2 z 1 | = 4 r s . (f) Same as Fig. 2a with separations | z 2 z 1 | = 2 r s .

Fig. 3
Fig. 3

(a) Mapping of the orientation average scattered electric field intensity (outside the particles) of a system composed of an isolated dielectric sphere with index of refraction 2.8 and radius 0.12 μ m embedded into an infinite nonabsorbing medium with refractive index 1.5. The wavelength of the incident field is 0.545 μ m . (b) Same as Fig. 3a but instead the system is composed of two identical dielectric spheres separated by | z 2 z 1 | = 17 r s . (c) Same as Fig. 3a with separations | z 2 z 1 | = 12 r s . (d) Same as Fig. 3a with separations | z 2 z 1 | = 8 r s . (e) Same as Fig. 3a with separations | z 2 z 1 | = 4 r s . (f) Same as Fig. 3a with separations | z 2 z 1 | = 2 r s .

Equations (24)

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

E i ( k r ) = E 0 Rg { Ψ t ( k r ) } a γ ,
E s ( k r j ) = E 0 Ψ t ( k r j ) f N ( j ) ,
T N ( j ) = T 1 ( j ) [ I + l = 1 l j N H ( j , l ) T N ( l ) J ( l , j ) ] , j = 1 , , N ,
e N ( j ) = J ( j , 0 ) a γ + l = 1 , l j N H ( j , l ) f N ( l ) .
E tot ( k r ) = E 0 ( e ˆ x , e ˆ y , e ˆ z ) [ ( E i x E i y E i z ) t + j = 1 N C ( r j ) Ψ t ( k r j ) f N ( j ) ] ,
C ( r j ) = [ sin θ j cos ϕ j cos θ j cos ϕ j sin ϕ j sin θ j sin ϕ j cos θ j sin ϕ j cos ϕ j cos θ j sin θ j 0 ] .
E tot ( k r ) 2 = E 0 2 [ A + B + C ] ,
A = 1 ,     B = 2 Re [ ( E i x E i y E i z ) * j = 1 N C ( r j ) Ψ t ( k r j ) f N ( j ) ] , C = j = 1 N l = 1 N [ f N ( j ) ] Ψ * ( k r j ) C t ( r j ) C ( r l ) Ψ t ( k r l ) f N ( l ) ,
a γ M a γ 0 = ϕ = 0 2 π θ = 0 π γ = 0 2 π a γ M a γ d γ sin θ d θ d ϕ d γ ϕ = 0 2 π θ = 0 π γ = 0 2 π d γ sin θ d θ d ϕ d γ ,
a γ M a γ 0 = 1 2 1 4 π ϕ = 0 2 π θ = 0 π [ a p M a p + a q M a q ] sin θ d θ d ϕ .
a γ M a γ 0 = 2 π Tr ( M ) ,
T N ( j ) = l = 1 N τ N ( j , l ) J ( l , j ) , j = 1 , , N .
E tot ( k r ) 2 = E 0 2 ( e ˆ x , e ˆ y , e ˆ z ) [ C ( r ) Rg { Ψ t ( k r ) } a γ + j = 1 N l = 1 N C ( r j ) Ψ t ( k r j ) τ N ( j , l ) J ( l , 0 ) a γ ] 2 ,
E tot ( k r ) 2 = E 0 2 [ a γ Rg { Ψ * ( k r ) } R g { Ψ t ( k r ) } a γ + a γ R g { Ψ * ( k r ) } j = 1 N l = 1 N P ( r ˆ , r ˆ j ) Ψ t ( k r j ) τ N ( j , l ) J ( l , 0 ) a γ + j = 1 N l = 1 N a γ J ( 0 , l ) [ τ N ( j , l ) ] Ψ * ( k r j ) P ( r ˆ j , r ˆ ) Rg { Ψ t ( k r ) } a γ + j = 1 N l = 1 N i = 1 N k = 1 N a γ J ( 0 , k ) [ τ N ( i , k ) ] Ψ * ( k r i ) P ( r ˆ i , r ˆ j ) Ψ t ( k r j ) τ N ( j , l ) J ( l , 0 ) a γ ] ,
E i ( k r ) = E 0 C ( r l ) Rg { Ψ t ( k r l ) } J ( l , 0 ) a γ .
E tot ( k r ) 2 = E 0 2 [ a γ Rg { Ψ * ( k r l ) } Rg { Ψ t ( k r j ) } a γ + 2 Re ( j = 1 N l = 1 N a γ J ( 0 , l ) Rg { Ψ * ( k r l ) } P ( r ˆ l , r ˆ j ) Ψ t ( k r j ) τ N ( j , l ) J ( l , 0 ) a γ ) + j = 1 N l = 1 N i = 1 N k = 1 N a γ J ( 0 , k ) [ τ N ( i , k ) ] Ψ * ( k r i ) P ( r ˆ i , r ˆ j ) Ψ t ( k r j ) τ N ( j , l ) J ( l , 0 ) a γ ] .
E tot ( k r ) 2 o = 2 π E 0 2 [ A o + B o + C o ] ,
A o = 1 / 2 π , B o = 2 Re Tr ( j = 1 N l = 1 N Rg { Ψ * ( k r l ) } P ( r ˆ l , r ˆ j ) Ψ t ( k r j ) τ N ( j , l ) ) , C o = Tr ( j = 1 N l = 1 N i = 1 N k = 1 N J ( l , k ) [ τ N ( i , k ) ] Ψ * ( k r i ) P ( r ˆ i , r ˆ j ) Ψ t ( k r j ) τ N ( j , l ) ) ,
B o = 2 Re Tr ( j = 1 N Rg { Ψ * ( k r j ) } Ψ t ( k r j ) τ N ( j , j ) ) + 2 Re Tr ( l = 1 N j > l N Rg { Ψ * ( k r l ) } P ( r ˆ l , r ˆ j ) Ψ t ( k r j ) τ N ( j , l ) ) ,
C o = Tr ( l = 1 N i = 1 N k = 1 N J ( l , k ) [ τ N ( i , k ) ] Ψ * ( k r i ) Ψ t ( k r i ) τ N ( i , l ) ) + Tr ( i = 1 N l = 1 N j > i N k = 1 N J ( l , k ) [ τ N ( i , k ) ] Ψ * ( k r i ) P ( r ˆ i , r ˆ j ) Ψ t ( k r j ) τ N ( j , l ) ) .
E i ( r ) = E 0 exp ( i k i · r ) e ˆ i , γ .
E i ( r ) = [ e ˆ k e ˆ θ k e ˆ ϕ k ] [ 0 cos γ sin γ ] E 0 exp ( i k i · r ) .
E i ( k r ) = [ e ˆ x e ˆ y e ˆ z ] [ cos θ k cos ϕ k cos γ sin ϕ k sin γ cos θ k sin ϕ i cos γ + cos ϕ k sin γ sin θ k sin γ ] E 0 exp ( i k i · r ) ,
k i · r = k r ( sin θ r sin θ k cos ( ϕ r ϕ k ) + cos θ r cos θ k ) .

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