Abstract

The generalized multiparticle Mie-solution (GMM), a Lorenz–Mie-type rigorous theory for the scattering of a monochromatic plane wave by an arbitrary configuration of nonintersecting scattering bodies, has lately been revisited and further developed. A recent progress is the initiation of a special version applied to one- and two-dimensional (1D and 2D) periodic arrays (PAs) of identical particles [J. Opt. Soc. Am. A 30, 1053 (2013)]. As a continuous advance, the present work extends the initiative PA-type solution from 1D and 2D to the more involved three-dimensional (3D) regular arrays. Analytical formulations applicable to the 3D PAs are derived, including the special PA-type explicit expressions for cross sections of extinction, scattering, backscattering, and radiation pressure. The specific PA-version is a complement to the general formulation and solution process of the standard GMM. In either 1D and 2D or 3D cases, the newly devised PA-approach is capable of providing expeditiously theoretical predictions of radiative scattering characteristics for periodic structures consisting of a huge number of identical unit cells, which the general approach of the GMM is unable to handle in practical calculations, owing to excessive computing time and/or computer memory requirements. To illustrate practical applications, sample numerical solutions obtained via the PA-approach are shown for 3D PAs of finite lengths that have 5×107 component particles, including structures having a rectangular opening. Also discussed is potential future work on the theory and its tests.

© 2014 Optical Society of America

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References

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  1. L. V. Lorenz, “Sur la Lumière réfléchie et réfractée par une sphère transparente,” in Oeuvres Scientifiques, L. de Lorenz and H. Valentiner, revues et annotées (Librairie Lehman et Stage, 1898), pp. 405–529.
  2. G. Mie, “Beiträge zur Optik trüber Medien speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
    [CrossRef]
  3. Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
    [CrossRef]
  4. Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres: far field,” Appl. Opt. 36, 9496–9508 (1997).
    [CrossRef]
  5. Y.-L. Xu and R. T. Wang, “Electromagnetic scattering by an aggregate of spheres: theoretical and experimental study of the amplitude scattering matrix,” Phys. Rev. E 58, 3931–3948 (1998).
    [CrossRef]
  6. Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres: asymmetry parameter,” Phys. Lett. A 249, 30–36 (1998).
    [CrossRef]
  7. Y.-L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365 (1999).
    [CrossRef]
  8. Y.-L. Xu and B. Å. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
    [CrossRef]
  9. Y.-L. Xu and N. G. Khlebtsov, “Orientation-averaged radiative properties of an arbitrary configuration of scatterers,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1121–1137 (2003).
    [CrossRef]
  10. Y.-L. Xu, “Radiative scattering properties of an ensemble of variously shaped small particles,” Phys. Rev. E 67, 046620 (2003).
    [CrossRef]
  11. Y.-L. Xu, “Scattering Mueller matrix of an ensemble of variously shaped small particles,” J. Opt. Soc. Am. A 20, 2093–2105 (2003).
    [CrossRef]
  12. Y.-L. Xu and B. Å. S. Gustafson, “Light scattering by an ensemble of small particles,” in Recent Research Developments in Optics (Research Signpost, 2003), pp. 599–648.
  13. P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
    [CrossRef]
  14. P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969).
    [CrossRef]
  15. P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [CrossRef]
  16. P. C. Waterman, “Numerical solution of electromagnetic scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed. (Pergamon, 1973), pp. 97–157.
  17. P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
    [CrossRef]
  18. G. Kristensson and P. C. Waterman, “The T-matrix for acoustic and electromagnetic scattering by circular disks,” J. Acoust. Soc. Am. 72, 1612–1625 (1982).
    [CrossRef]
  19. P. C. Waterman, “Surface fields and the T-matrix,” J. Opt. Soc. Am. A 16, 2968–2977 (1999).
    [CrossRef]
  20. P. C. Waterman, “The T-matrix revisited,” J. Opt. Soc. Am. A 24, 2257–2267 (2007).
    [CrossRef]
  21. GMM public-domain FORTRAN codes are currently available at http://code.google.com/p/scatterlib .
  22. Y.-L. Xu, “Scattering of electromagnetic waves by periodic particle arrays,” J. Opt. Soc. Am. A 30, 1053–1068 (2013).
    [CrossRef]
  23. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University, 1996).
  24. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
  25. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  26. M. Kerker, The Scattering of Light (Academic, 1969).
  27. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  28. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  29. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

2013 (1)

2007 (1)

2003 (3)

Y.-L. Xu and N. G. Khlebtsov, “Orientation-averaged radiative properties of an arbitrary configuration of scatterers,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1121–1137 (2003).
[CrossRef]

Y.-L. Xu, “Radiative scattering properties of an ensemble of variously shaped small particles,” Phys. Rev. E 67, 046620 (2003).
[CrossRef]

Y.-L. Xu, “Scattering Mueller matrix of an ensemble of variously shaped small particles,” J. Opt. Soc. Am. A 20, 2093–2105 (2003).
[CrossRef]

2001 (1)

Y.-L. Xu and B. Å. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
[CrossRef]

1999 (2)

Y.-L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365 (1999).
[CrossRef]

P. C. Waterman, “Surface fields and the T-matrix,” J. Opt. Soc. Am. A 16, 2968–2977 (1999).
[CrossRef]

1998 (2)

Y.-L. Xu and R. T. Wang, “Electromagnetic scattering by an aggregate of spheres: theoretical and experimental study of the amplitude scattering matrix,” Phys. Rev. E 58, 3931–3948 (1998).
[CrossRef]

Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres: asymmetry parameter,” Phys. Lett. A 249, 30–36 (1998).
[CrossRef]

1997 (1)

1995 (1)

1982 (1)

G. Kristensson and P. C. Waterman, “The T-matrix for acoustic and electromagnetic scattering by circular disks,” J. Acoust. Soc. Am. 72, 1612–1625 (1982).
[CrossRef]

1979 (1)

P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
[CrossRef]

1971 (1)

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

1969 (1)

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969).
[CrossRef]

1965 (1)

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

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Blum, J.

Y.-L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365 (1999).
[CrossRef]

Bohren, C. F.

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

Born, M.

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

de Lorenz, L.

L. V. Lorenz, “Sur la Lumière réfléchie et réfractée par une sphère transparente,” in Oeuvres Scientifiques, L. de Lorenz and H. Valentiner, revues et annotées (Librairie Lehman et Stage, 1898), pp. 405–529.

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University, 1996).

Giovane, F.

Y.-L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365 (1999).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gustafson, B. Å. S.

Y.-L. Xu and B. Å. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
[CrossRef]

Y.-L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365 (1999).
[CrossRef]

Y.-L. Xu and B. Å. S. Gustafson, “Light scattering by an ensemble of small particles,” in Recent Research Developments in Optics (Research Signpost, 2003), pp. 599–648.

Huffman, D. R.

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

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

Kerker, M.

M. Kerker, The Scattering of Light (Academic, 1969).

Khlebtsov, N. G.

Y.-L. Xu and N. G. Khlebtsov, “Orientation-averaged radiative properties of an arbitrary configuration of scatterers,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1121–1137 (2003).
[CrossRef]

Kristensson, G.

G. Kristensson and P. C. Waterman, “The T-matrix for acoustic and electromagnetic scattering by circular disks,” J. Acoust. Soc. Am. 72, 1612–1625 (1982).
[CrossRef]

Lorenz, L. V.

L. V. Lorenz, “Sur la Lumière réfléchie et réfractée par une sphère transparente,” in Oeuvres Scientifiques, L. de Lorenz and H. Valentiner, revues et annotées (Librairie Lehman et Stage, 1898), pp. 405–529.

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Tehranian, S.

Y.-L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365 (1999).
[CrossRef]

Valentiner, H.

L. V. Lorenz, “Sur la Lumière réfléchie et réfractée par une sphère transparente,” in Oeuvres Scientifiques, L. de Lorenz and H. Valentiner, revues et annotées (Librairie Lehman et Stage, 1898), pp. 405–529.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

Wang, R. T.

Y.-L. Xu and R. T. Wang, “Electromagnetic scattering by an aggregate of spheres: theoretical and experimental study of the amplitude scattering matrix,” Phys. Rev. E 58, 3931–3948 (1998).
[CrossRef]

Waterman, P. C.

P. C. Waterman, “The T-matrix revisited,” J. Opt. Soc. Am. A 24, 2257–2267 (2007).
[CrossRef]

P. C. Waterman, “Surface fields and the T-matrix,” J. Opt. Soc. Am. A 16, 2968–2977 (1999).
[CrossRef]

G. Kristensson and P. C. Waterman, “The T-matrix for acoustic and electromagnetic scattering by circular disks,” J. Acoust. Soc. Am. 72, 1612–1625 (1982).
[CrossRef]

P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
[CrossRef]

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

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969).
[CrossRef]

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

P. C. Waterman, “Numerical solution of electromagnetic scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed. (Pergamon, 1973), pp. 97–157.

Wolf, E.

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

Xu, Y.-L.

Y.-L. Xu, “Scattering of electromagnetic waves by periodic particle arrays,” J. Opt. Soc. Am. A 30, 1053–1068 (2013).
[CrossRef]

Y.-L. Xu, “Radiative scattering properties of an ensemble of variously shaped small particles,” Phys. Rev. E 67, 046620 (2003).
[CrossRef]

Y.-L. Xu, “Scattering Mueller matrix of an ensemble of variously shaped small particles,” J. Opt. Soc. Am. A 20, 2093–2105 (2003).
[CrossRef]

Y.-L. Xu and N. G. Khlebtsov, “Orientation-averaged radiative properties of an arbitrary configuration of scatterers,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1121–1137 (2003).
[CrossRef]

Y.-L. Xu and B. Å. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
[CrossRef]

Y.-L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365 (1999).
[CrossRef]

Y.-L. Xu and R. T. Wang, “Electromagnetic scattering by an aggregate of spheres: theoretical and experimental study of the amplitude scattering matrix,” Phys. Rev. E 58, 3931–3948 (1998).
[CrossRef]

Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres: asymmetry parameter,” Phys. Lett. A 249, 30–36 (1998).
[CrossRef]

Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres: far field,” Appl. Opt. 36, 9496–9508 (1997).
[CrossRef]

Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
[CrossRef]

Y.-L. Xu and B. Å. S. Gustafson, “Light scattering by an ensemble of small particles,” in Recent Research Developments in Optics (Research Signpost, 2003), pp. 599–648.

Ann. Phys. (1)

G. Mie, “Beiträge zur Optik trüber Medien speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Appl. Opt. (2)

J. Acoust. Soc. Am. (2)

G. Kristensson and P. C. Waterman, “The T-matrix for acoustic and electromagnetic scattering by circular disks,” J. Acoust. Soc. Am. 72, 1612–1625 (1982).
[CrossRef]

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969).
[CrossRef]

J. Appl. Phys. (1)

P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
[CrossRef]

J. Opt. Soc. Am. A (4)

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

Y.-L. Xu and B. Å. S. Gustafson, “A generalized multiparticle Mie-solution: further experimental verification,” J. Quant. Spectrosc. Radiat. Transfer 70, 395–419 (2001).
[CrossRef]

Y.-L. Xu and N. G. Khlebtsov, “Orientation-averaged radiative properties of an arbitrary configuration of scatterers,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1121–1137 (2003).
[CrossRef]

Phys. Lett. A (1)

Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres: asymmetry parameter,” Phys. Lett. A 249, 30–36 (1998).
[CrossRef]

Phys. Rev. D (1)

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

Phys. Rev. E (3)

Y.-L. Xu, B. Å. S. Gustafson, F. Giovane, J. Blum, and S. Tehranian, “Calculation of the heat-source function in photophoresis of aggregated spheres,” Phys. Rev. E 60, 2347–2365 (1999).
[CrossRef]

Y.-L. Xu, “Radiative scattering properties of an ensemble of variously shaped small particles,” Phys. Rev. E 67, 046620 (2003).
[CrossRef]

Y.-L. Xu and R. T. Wang, “Electromagnetic scattering by an aggregate of spheres: theoretical and experimental study of the amplitude scattering matrix,” Phys. Rev. E 58, 3931–3948 (1998).
[CrossRef]

Proc. IEEE (1)

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

Other (11)

L. V. Lorenz, “Sur la Lumière réfléchie et réfractée par une sphère transparente,” in Oeuvres Scientifiques, L. de Lorenz and H. Valentiner, revues et annotées (Librairie Lehman et Stage, 1898), pp. 405–529.

Y.-L. Xu and B. Å. S. Gustafson, “Light scattering by an ensemble of small particles,” in Recent Research Developments in Optics (Research Signpost, 2003), pp. 599–648.

GMM public-domain FORTRAN codes are currently available at http://code.google.com/p/scatterlib .

P. C. Waterman, “Numerical solution of electromagnetic scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed. (Pergamon, 1973), pp. 97–157.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University, 1996).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

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

M. Kerker, The Scattering of Light (Academic, 1969).

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

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

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Figures (10)

Fig. 1.
Fig. 1.

Comparison between PA- and GA-type GMM scattering calculations for a linear chain of identical nonabsorbing spheres, illuminated by a monochromatic plane wave of wavelength 31.416 mm. Its spatial orientation is specified by the Euler-angle triad (0°,45°,20°), referring to the initial orientation of x alignment, i.e., all sphere centers are initially aligned along the x axis. The component spheres are of 1 mm diameter and of refractive index 1.6, equally spaced 1 mm apart. The left panel shows the relative PA-to-GA deviations (%) of the calculated cross sections of extinction, backscattering, and radiation pressure when the total number of component spheres in the 1D array gradually increases. The right panel shows the CPU time elapsed in both the GA- and PA-type calculations.

Fig. 2.
Fig. 2.

Similar to Fig. 1 but for square arrays of densely packed identical spheres of 1 mm diameter and of complex refractive index (1.6, 0.1). The orientation of the 2D arrays is specified by the Euler-angle triad (0°,20°,30°). Their initial orientation is such that all component sphere centers are in the initial xy plane with rows parallel to the x axis and columns parallel to the y axis.

Fig. 3.
Fig. 3.

Dependence of the Mueller matrix element S11 on scattering angle θ and azimuth angle ϕ (shown as variation with x=sinθcosϕ and y=sinθsinϕ) for a cubic array of 365×365×365 densely packed identical spheres of 1 mm diameter, illuminated by a monochromatic plane wave of wavelength 31.416 mm. The cubic array is in its initial orientation and the dielectric component spheres are nonabsorbing, having a refractive index of 1.6. This is Case 1 as listed in Table 1. The lower two panels are equivalent to the upper two, simply presenting the same data in an alternative way.

Fig. 4.
Fig. 4.

Dependence of the Stokes parameter I on scattering angle θ and azimuth angle ϕ (shown as variation with x=sinθcosϕ and y=sinθsinϕ) for the same cubic array and for the same Case 1 as shown in Fig. 3. The incident plane wave is unpolarized. Note that the tip of the central peak is removed in both the forward and backward directions, as shown in the figures. Otherwise, all the figures would look exactly the same as those in Fig. 3.

Fig. 5.
Fig. 5.

Dependence of the Stokes parameters I and Q on scattering angle θ and azimuth angle ϕ (shown as variation with x=sinθcosϕ and y=sinθsinϕ) for the same cubic array in the same Case 1 as shown in Figs. 3 and 4. The incident plane wave is linearly x-polarized. The parameter I is shown in dB.

Fig. 6.
Fig. 6.

Dependence of the Stokes parameters I and Q on scattering angle θ and azimuth angle ϕ (shown as variation with x=sinθcosϕ and y=sinθsinϕ) for the same cubic array shown in Figs. 35 but for Case 2 (see Table 1), i.e., the component spheres are metallic, electrically conducting. The incident plane wave is linearly y polarized.

Fig. 7.
Fig. 7.

Dependence of the Mueller matrix element S11 and S12/S11 on scattering angle θ and azimuth angle ϕ (shown as variation with x=sinθcosϕ and y=sinθsinϕ) for the same cubic array shown in Figs. 36 but for Case 3 (see Table 1), i.e., the Euler-angle triad specifying the orientation of the 3D array is (0°,20°,30°). The Mueller matrix element S11 is shown in decibels.

Fig. 8.
Fig. 8.

Dependence of the Stokes parameters I and Q on scattering angle θ and azimuth angle ϕ (shown as variation with x=sinθcosϕ and y=sinθsinϕ) for the same cubic array in the same Case 3 as shown in Fig. 7. The incident plane wave is right-circularly polarized.

Fig. 9.
Fig. 9.

Dependence of the Stokes parameter I on scattering angle θ and azimuth angle ϕ (shown as variation with x=sinθcosϕ and y=sinθsinϕ) for a square array consisting of 301×301 identical dielectric spheres of 1 mm diameter, in which the inside 201×201 spheres are removed. The refractive index of the spheres is (1.6, 0.1). This 2D structure is illuminated by a monochromatic plane wave of 31.416 mm wavelength. It is in its initial orientation, i.e., all component sphere centers are in the xy plane and the “sides” of the array and the window are parallel to the x or y axis, respectively. Along either the x or y directions, adjacent spheres are in contact. The incident plane wave is left-circularly polarized.

Fig. 10.
Fig. 10.

Same as Fig. 9 but for a cubic array having a square window opening, which is the simple extension in the z direction of the 2D structure shown in Fig. 9, i.e., every component sphere in the 2D structure becomes the same linear chain of 301 contacting spheres. The incident plane wave is right-circularly polarized.

Tables (1)

Tables Icon

Table 1. Three Cases Calculated Using the FORTRAN Program “gmm01_PA.f” for a Regular Cubic Array of 365×365×365 Identical Spheres

Equations (23)

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

R(αβγ)=Rz(γ)Ry(β)Rz(α).
Rz(α)=[cosαsinα0sinαcosα0001],Ry(β)=[cosβ0sinβ010sinβ0cosβ].
(XlYlZl)=R(αβγ)(X0lY0lZ0l),
Einc=E0exp(ik^·riωt),
k^=e^xsinϑinccosφinc+e^ysinϑincsinφinc+e^zcosϑinc.
amnp=amnpl·Φ(θ,ϕ),S=Sl·Φ(θ,ϕ),Φ(θ,ϕ)=j=1Lexp[idj·(k^r^)].
Φ(θ,ϕ)=Φx·Φy·Φz,Φx=sin(NxΔx/2·ux)sin(Δx/2·ux),Φy=sin(NyΔy/2·uy)sin(Δy/2·uy),Φz=sin(NzΔz/2·uz)sin(Δz/2·uz),
ux=t1cosαs1sinα,uy=t1sinα+s1cosα,uz=cosη(β,πγ;ϑinc,φinc)cosη(β,πγ;θ,ϕ),
t1=cosη(βπ2,πγ;ϑinc,φinc)cosη(βπ2,πγ;θ,ϕ),s1=sinϑinc·sin(γ+φinc)sinθ·sin(γ+ϕ),
Φ(θ,ϕ)=sin(NxΔx/2·sinθcosϕ)sin(Δx/2·sinθcosϕ)×sin(NyΔy/2·sinθsinϕ)sin(Δy/2·sinθsinϕ)sin[NzΔz/2·(1cosθ)]sin[Δz/2·(1cosθ)].
S(0°)=L·Sl(0°),Cext=L·Cextl,Cextl=4πk2ReSl(0°),
Sl(0°)=n=1Nmaxp=122n+12×[a1npl(βp)exp(iβp)+(1)pa1npl(βp)exp(iβp)],
ux=2sinβcosα,uy=2sinβsinα,uz=2cosβ,
Φ(180°)=sin(NxΔx·sinβcosα)sin(Δx·sinβcosα)×sin(NyΔy·sinβsinα)sin(Δy·sinβsinα)sin(NzΔz·cosβ)sin(Δz·cosβ).
Cbak=4πk2|S(180°)|2,S(180°)=Sl(180°)Φ(180°),
Sl(180°)=n=1Nmaxp=12(1)n+p2n+12×[a1npl(βp)exp(iβp)+(1)pa1npl(βp)exp(iβp)].
Φ(180°)=Nx·Ny·sin(Nz·Δz)sinΔz.
Cscal=4πk2Ren=1Nmaxlm=nnp=12amnpl*a˜mnp(l),
a˜mnp(l)=n=1Nmaxlm=nnp=12C˜mnpmnplamnpl,C˜mnpmnpl=j=1Lexp(ik^·dlj)A˜mnpmnplj,
A˜mnpmnplj=14π02π0πsinθdθdϕ×exp(ir^·djl)exp[i(mm)ϕ]×[τ˜mnp(θ)τ˜mnp(θ)+τ˜mn,3p(θ)τ˜mn,3p(θ)],
cosθ¯l=4πk2CscaRen=1Nmaxlm=nnp=12amnpl*s˜mnp(l),
s˜mnp(l)=f1a˜mn,3p(l)+f2a˜m,n+1,p(l)+f3a˜m,n1,p(l),
f1=mn(n+1),f2=1n+1[n(n+2)(nm+1)(n+m+1)(2n+1)(2n+3)]1/2,f3=1n[(n1)(n+1)(nm)(n+m)(2n1)(2n+1)]1/2.

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