Abstract

The scattering by a linear chain of spherical dielectric inclusions, embedded along the axis of an optical fiber, is analyzed using a rigorous integral equation formulation, based on the dyadic Green’s function theory. The coupled electric field integral equations are solved by applying the Galerkin technique with Mie-type expansion of the field inside the spheres in terms of spherical waves. The analysis extends the previously studied case of a single spherical inhomogeneity inside a fiber to the multisphere-scattering case, by utilizing the classic translational addition theorems for spherical waves in order to analytically extract the direct-intersphere-coupling coefficients. Results for the transmitted and reflected power, on incidence of the fundamental HE11 mode, are presented for several cases.

© 2006 Optical Society of America

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References

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  1. G. L. Yip and J. Martucci, "Scattering from a localized inhomogeneity in a cladded fiber optical waveguide. 1: Radiation loss," Appl. Opt. 15, 2131-2136 (1976).
    [CrossRef] [PubMed]
  2. N. Morita and N. Kumagai, "Scattering and mode conversion of guided modes by a spherical object in an optical fiber," IEEE Trans. Microwave Theory Tech. 28, 137-141 (1980).
    [CrossRef]
  3. A. Safaai-Jazi and G. L. Yip, "Scattering from an arbitrarily located off-axis inhomogeneity in a step-index optical fiber," IEEE Trans. Microwave Theory Tech. 28, 24-32 (1980).
    [CrossRef]
  4. A. Safaai-Jazi and G. L. Yip, "Scattering from an off-axis inhomogeneity in step-index optical fibers: radiation loss," J. Opt. Soc. Am. 70, 40-48 (1980).
    [CrossRef]
  5. N. K. Uzunoglu, "Scattering from inhomogeneities inside a fiber waveguide," J. Opt. Soc. Am. 71, 259-273 (1981).
    [CrossRef]
  6. N. K. Uzunoglu, "Resonance of a dielectric cavity inside a fiber," Appl. Opt. 20, 857-861 (1981).
    [CrossRef] [PubMed]
  7. N. K. Uzunoglu and J. G. Fikioris, "Scattering from an inhomogeneity inside a dielectric slab waveguide," J. Opt. Soc. Am. 72, 628-637 (1982).
    [CrossRef]
  8. T. Nobuyoshi, N. Morita, and N. Kumagai, "Scattering and mode conversion of guided modes by an arbitrary cross-sectional cylindrical object in an optical slab waveguide," J. Lightwave Technol. 1, 374-380 (1983).
    [CrossRef]
  9. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).
  10. A. I. Nosich and A. S. Andrenko, "Scattering and mode conversion by a screen-like inhomogeneity inside a dielectric slab waveguide," IEEE Trans. Microwave Theory Tech. 42, 298-307 (1994).
    [CrossRef]
  11. L. Tsang and J. A. Kong, "Effective propagation constants for coherent electromagnetic wave propagation in media embedded with dielectric scatterers," J. Appl. Phys. 53, 7162-7173 (1982).
    [CrossRef]
  12. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  13. K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
    [CrossRef] [PubMed]
  14. I. Psarobas, N. Stefanou, and A. Modinos, "Photonic crystals of chiral spheres," J. Opt. Soc. Am. A 16, 343-347 (1999).
    [CrossRef]
  15. I. Psarobas, N. Stefanou, and A. Modinos, "Scattering of elastic waves by periodic arrays of spherical bodies," Phys. Rev. B 62, 278-291 (2000).
    [CrossRef]
  16. Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, "Elastic wave scattering by periodic structures of spherical objects: theory and experiment," Phys. Rev. B 62, 2446-2457 (2000).
    [CrossRef]
  17. L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, "Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method; Part II. Properties and implementation," J. Opt. Soc. Am. A 17, 2165-2190 (2000).
    [CrossRef]
  18. I. D. Chremmos and N. K. Uzunoglu, "Integral equation analysis of scattering by a spherical microparticle coupled to a subwavelength-diameter wire waveguide," J. Opt. Soc. Am. A 23, 461-467 (2006).
    [CrossRef]
  19. S. Stein, "Addition theorems for spherical wave functions," Q. Appl. Math. 19, 15-24 (1961).
  20. O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1961).
  21. C. Liang and Y. Lo, "Scattering by two spheres," Radio Sci. 2, 1481-1495 (1967).
  22. J. Bruning and Y. Lo, "Multiple scattering of EM waves by spheres. Part I: Multipole expansion and ray-optical solutions; Part II: Numerical and experimental results," IEEE Trans. Antennas Propag. 19, 378-400 (1971).
    [CrossRef]
  23. 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]
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    [CrossRef] [PubMed]
  25. Y. L. Xu, "Calculation of the addition coefficients in electromagnetic multisphere-scattering theory," J. Comput. Phys. 127, 285-298 (1996).
    [CrossRef]
  26. R. J. Pogorzelski and E. Lun, "On the expansion of cylindrical waves in terms of spherical waves," Radio Sci. 11, 753-761 (1976).
    [CrossRef]
  27. R. F. Harrington, Field Computation by Moment Methods (Macmillan, 1983).
  28. C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE Press, 1994).
  29. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, 1960).
  30. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  31. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).

2006 (1)

2000 (3)

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, "Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method; Part II. Properties and implementation," J. Opt. Soc. Am. A 17, 2165-2190 (2000).
[CrossRef]

I. Psarobas, N. Stefanou, and A. Modinos, "Scattering of elastic waves by periodic arrays of spherical bodies," Phys. Rev. B 62, 278-291 (2000).
[CrossRef]

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, "Elastic wave scattering by periodic structures of spherical objects: theory and experiment," Phys. Rev. B 62, 2446-2457 (2000).
[CrossRef]

1999 (1)

1996 (1)

Y. L. Xu, "Calculation of the addition coefficients in electromagnetic multisphere-scattering theory," J. Comput. Phys. 127, 285-298 (1996).
[CrossRef]

1994 (1)

A. I. Nosich and A. S. Andrenko, "Scattering and mode conversion by a screen-like inhomogeneity inside a dielectric slab waveguide," IEEE Trans. Microwave Theory Tech. 42, 298-307 (1994).
[CrossRef]

1990 (1)

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

1988 (2)

1987 (1)

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

1983 (1)

T. Nobuyoshi, N. Morita, and N. Kumagai, "Scattering and mode conversion of guided modes by an arbitrary cross-sectional cylindrical object in an optical slab waveguide," J. Lightwave Technol. 1, 374-380 (1983).
[CrossRef]

1982 (2)

L. Tsang and J. A. Kong, "Effective propagation constants for coherent electromagnetic wave propagation in media embedded with dielectric scatterers," J. Appl. Phys. 53, 7162-7173 (1982).
[CrossRef]

N. K. Uzunoglu and J. G. Fikioris, "Scattering from an inhomogeneity inside a dielectric slab waveguide," J. Opt. Soc. Am. 72, 628-637 (1982).
[CrossRef]

1981 (2)

1980 (3)

A. Safaai-Jazi and G. L. Yip, "Scattering from an off-axis inhomogeneity in step-index optical fibers: radiation loss," J. Opt. Soc. Am. 70, 40-48 (1980).
[CrossRef]

N. Morita and N. Kumagai, "Scattering and mode conversion of guided modes by a spherical object in an optical fiber," IEEE Trans. Microwave Theory Tech. 28, 137-141 (1980).
[CrossRef]

A. Safaai-Jazi and G. L. Yip, "Scattering from an arbitrarily located off-axis inhomogeneity in a step-index optical fiber," IEEE Trans. Microwave Theory Tech. 28, 24-32 (1980).
[CrossRef]

1976 (2)

G. L. Yip and J. Martucci, "Scattering from a localized inhomogeneity in a cladded fiber optical waveguide. 1: Radiation loss," Appl. Opt. 15, 2131-2136 (1976).
[CrossRef] [PubMed]

R. J. Pogorzelski and E. Lun, "On the expansion of cylindrical waves in terms of spherical waves," Radio Sci. 11, 753-761 (1976).
[CrossRef]

1971 (1)

J. Bruning and Y. Lo, "Multiple scattering of EM waves by spheres. Part I: Multipole expansion and ray-optical solutions; Part II: Numerical and experimental results," IEEE Trans. Antennas Propag. 19, 378-400 (1971).
[CrossRef]

1967 (1)

C. Liang and Y. Lo, "Scattering by two spheres," Radio Sci. 2, 1481-1495 (1967).

1961 (2)

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

O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1961).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).

Andrenko, A. S.

A. I. Nosich and A. S. Andrenko, "Scattering and mode conversion by a screen-like inhomogeneity inside a dielectric slab waveguide," IEEE Trans. Microwave Theory Tech. 42, 298-307 (1994).
[CrossRef]

Asatryan, A. A.

Botten, L. C.

Bruning, J.

J. Bruning and Y. Lo, "Multiple scattering of EM waves by spheres. Part I: Multipole expansion and ray-optical solutions; Part II: Numerical and experimental results," IEEE Trans. Antennas Propag. 19, 378-400 (1971).
[CrossRef]

Chan, C. T.

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, "Elastic wave scattering by periodic structures of spherical objects: theory and experiment," Phys. Rev. B 62, 2446-2457 (2000).
[CrossRef]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Chremmos, I. D.

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, 1960).

Cruzan, O. R.

O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1961).

de Sterke, C. M.

Fikioris, J. G.

Fuller, K. A.

Goertzen, A. L.

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, "Elastic wave scattering by periodic structures of spherical objects: theory and experiment," Phys. Rev. B 62, 2446-2457 (2000).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (Macmillan, 1983).

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Kattawar, G. W.

Kong, J. A.

L. Tsang and J. A. Kong, "Effective propagation constants for coherent electromagnetic wave propagation in media embedded with dielectric scatterers," J. Appl. Phys. 53, 7162-7173 (1982).
[CrossRef]

Kumagai, N.

T. Nobuyoshi, N. Morita, and N. Kumagai, "Scattering and mode conversion of guided modes by an arbitrary cross-sectional cylindrical object in an optical slab waveguide," J. Lightwave Technol. 1, 374-380 (1983).
[CrossRef]

N. Morita and N. Kumagai, "Scattering and mode conversion of guided modes by a spherical object in an optical fiber," IEEE Trans. Microwave Theory Tech. 28, 137-141 (1980).
[CrossRef]

Liang, C.

C. Liang and Y. Lo, "Scattering by two spheres," Radio Sci. 2, 1481-1495 (1967).

Liu, Z.

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, "Elastic wave scattering by periodic structures of spherical objects: theory and experiment," Phys. Rev. B 62, 2446-2457 (2000).
[CrossRef]

Lo, Y.

J. Bruning and Y. Lo, "Multiple scattering of EM waves by spheres. Part I: Multipole expansion and ray-optical solutions; Part II: Numerical and experimental results," IEEE Trans. Antennas Propag. 19, 378-400 (1971).
[CrossRef]

C. Liang and Y. Lo, "Scattering by two spheres," Radio Sci. 2, 1481-1495 (1967).

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Lun, E.

R. J. Pogorzelski and E. Lun, "On the expansion of cylindrical waves in terms of spherical waves," Radio Sci. 11, 753-761 (1976).
[CrossRef]

Martucci, J.

McPhedran, R. C.

Modinos, A.

I. Psarobas, N. Stefanou, and A. Modinos, "Scattering of elastic waves by periodic arrays of spherical bodies," Phys. Rev. B 62, 278-291 (2000).
[CrossRef]

I. Psarobas, N. Stefanou, and A. Modinos, "Photonic crystals of chiral spheres," J. Opt. Soc. Am. A 16, 343-347 (1999).
[CrossRef]

Morita, N.

T. Nobuyoshi, N. Morita, and N. Kumagai, "Scattering and mode conversion of guided modes by an arbitrary cross-sectional cylindrical object in an optical slab waveguide," J. Lightwave Technol. 1, 374-380 (1983).
[CrossRef]

N. Morita and N. Kumagai, "Scattering and mode conversion of guided modes by a spherical object in an optical fiber," IEEE Trans. Microwave Theory Tech. 28, 137-141 (1980).
[CrossRef]

Nicorovici, N. A.

Nobuyoshi, T.

T. Nobuyoshi, N. Morita, and N. Kumagai, "Scattering and mode conversion of guided modes by an arbitrary cross-sectional cylindrical object in an optical slab waveguide," J. Lightwave Technol. 1, 374-380 (1983).
[CrossRef]

Nosich, A. I.

A. I. Nosich and A. S. Andrenko, "Scattering and mode conversion by a screen-like inhomogeneity inside a dielectric slab waveguide," IEEE Trans. Microwave Theory Tech. 42, 298-307 (1994).
[CrossRef]

Page, J. H.

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, "Elastic wave scattering by periodic structures of spherical objects: theory and experiment," Phys. Rev. B 62, 2446-2457 (2000).
[CrossRef]

Pogorzelski, R. J.

R. J. Pogorzelski and E. Lun, "On the expansion of cylindrical waves in terms of spherical waves," Radio Sci. 11, 753-761 (1976).
[CrossRef]

Psarobas, I.

I. Psarobas, N. Stefanou, and A. Modinos, "Scattering of elastic waves by periodic arrays of spherical bodies," Phys. Rev. B 62, 278-291 (2000).
[CrossRef]

I. Psarobas, N. Stefanou, and A. Modinos, "Photonic crystals of chiral spheres," J. Opt. Soc. Am. A 16, 343-347 (1999).
[CrossRef]

Robinson, P. A.

Safaai-Jazi, A.

A. Safaai-Jazi and G. L. Yip, "Scattering from an arbitrarily located off-axis inhomogeneity in a step-index optical fiber," IEEE Trans. Microwave Theory Tech. 28, 24-32 (1980).
[CrossRef]

A. Safaai-Jazi and G. L. Yip, "Scattering from an off-axis inhomogeneity in step-index optical fibers: radiation loss," J. Opt. Soc. Am. 70, 40-48 (1980).
[CrossRef]

Sheng, P.

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, "Elastic wave scattering by periodic structures of spherical objects: theory and experiment," Phys. Rev. B 62, 2446-2457 (2000).
[CrossRef]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Stefanou, N.

I. Psarobas, N. Stefanou, and A. Modinos, "Scattering of elastic waves by periodic arrays of spherical bodies," Phys. Rev. B 62, 278-291 (2000).
[CrossRef]

I. Psarobas, N. Stefanou, and A. Modinos, "Photonic crystals of chiral spheres," J. Opt. Soc. Am. A 16, 343-347 (1999).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Stein, S.

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

Tai, C. T.

C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE Press, 1994).

Tsang, L.

L. Tsang and J. A. Kong, "Effective propagation constants for coherent electromagnetic wave propagation in media embedded with dielectric scatterers," J. Appl. Phys. 53, 7162-7173 (1982).
[CrossRef]

Uzunoglu, N. K.

Xu, Y. L.

Y. L. Xu, "Calculation of the addition coefficients in electromagnetic multisphere-scattering theory," J. Comput. Phys. 127, 285-298 (1996).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Yip, G. L.

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

J. Bruning and Y. Lo, "Multiple scattering of EM waves by spheres. Part I: Multipole expansion and ray-optical solutions; Part II: Numerical and experimental results," IEEE Trans. Antennas Propag. 19, 378-400 (1971).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

N. Morita and N. Kumagai, "Scattering and mode conversion of guided modes by a spherical object in an optical fiber," IEEE Trans. Microwave Theory Tech. 28, 137-141 (1980).
[CrossRef]

A. Safaai-Jazi and G. L. Yip, "Scattering from an arbitrarily located off-axis inhomogeneity in a step-index optical fiber," IEEE Trans. Microwave Theory Tech. 28, 24-32 (1980).
[CrossRef]

A. I. Nosich and A. S. Andrenko, "Scattering and mode conversion by a screen-like inhomogeneity inside a dielectric slab waveguide," IEEE Trans. Microwave Theory Tech. 42, 298-307 (1994).
[CrossRef]

J. Appl. Phys. (1)

L. Tsang and J. A. Kong, "Effective propagation constants for coherent electromagnetic wave propagation in media embedded with dielectric scatterers," J. Appl. Phys. 53, 7162-7173 (1982).
[CrossRef]

J. Comput. Phys. (1)

Y. L. Xu, "Calculation of the addition coefficients in electromagnetic multisphere-scattering theory," J. Comput. Phys. 127, 285-298 (1996).
[CrossRef]

J. Lightwave Technol. (1)

T. Nobuyoshi, N. Morita, and N. Kumagai, "Scattering and mode conversion of guided modes by an arbitrary cross-sectional cylindrical object in an optical slab waveguide," J. Lightwave Technol. 1, 374-380 (1983).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

Opt. Lett. (2)

Phys. Rev. B (2)

I. Psarobas, N. Stefanou, and A. Modinos, "Scattering of elastic waves by periodic arrays of spherical bodies," Phys. Rev. B 62, 278-291 (2000).
[CrossRef]

Z. Liu, C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, "Elastic wave scattering by periodic structures of spherical objects: theory and experiment," Phys. Rev. B 62, 2446-2457 (2000).
[CrossRef]

Phys. Rev. Lett. (2)

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Q. Appl. Math. (2)

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

O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1961).

Radio Sci. (2)

C. Liang and Y. Lo, "Scattering by two spheres," Radio Sci. 2, 1481-1495 (1967).

R. J. Pogorzelski and E. Lun, "On the expansion of cylindrical waves in terms of spherical waves," Radio Sci. 11, 753-761 (1976).
[CrossRef]

Other (6)

R. F. Harrington, Field Computation by Moment Methods (Macmillan, 1983).

C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE Press, 1994).

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, 1960).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

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Figures (7)

Fig. 1
Fig. 1

Geometry of a linear chain of spheres embedded in an optical fiber.

Fig. 2
Fig. 2

(a) Transmitted and (b) reflected power fractions versus the normalized frequency parameter V α ( k 1 2 k 0 2 ) 1 2 for chains with α s = 0.3 α , n s = 2 , Λ = α , and different numbers of spheres N s .

Fig. 3
Fig. 3

Ratio 2 n eff Λ λ . The Bragg condition is satisfied when the ratio is equal to an integer m = 1 , 2 , .

Fig. 4
Fig. 4

Reflected power fraction for chains with n s = 2 , Λ = α , N s = 8 spheres, and varying sphere radius α s .

Fig. 5
Fig. 5

Reflected power fraction for chains with α s = 0.3 α , Λ = α , N s = 8 spheres, and varying sphere index n s for (a) n s > n 1 and (b) n s < n 1 .

Fig. 6
Fig. 6

Reflected power fraction for chains with α s = 0.5 α , n s = 2 , Λ = 1.5 α , and different numbers of spheres N s .

Fig. 7
Fig. 7

Reflected power fraction for chains with α s = 0.8 α , n s = 2 , Λ = 2 α , and different numbers of spheres N s . The dotted curve shows the case α s = 0.5 α , n s = 2 , Λ = 2 α , and N s = 4 .

Equations (52)

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

E ( r ) = E inc ( r ) + l = 1 N s [ ( k l s ) 2 k 1 2 ] V l G f ( r , r ) E ( r ) d r ,
[ × × k 2 ( r ) ] G ̱ f ( r , r ) = I ̱ δ ( r r ) ,
G ̱ f ( r , r ) = { G ̱ 1 ( r , r ) + G ̱ 11 ( r , r ) , ρ < α G ̱ 01 ( r , r ) , ρ > α , } ,
G ̱ 11 ( r , r ) = j 8 π + d k m = + ( 1 ) m a 1 2 W m , k ( 1 ) ( r , k 1 ) A 11 ( m , k ) W m , k ( 1 ) ( r , k 1 ) T ,
E ( r l ) = n = 1 + m = n + n [ a n m l m n m ( 1 ) ( r l , k l s ) + b n m l n n m ( 1 ) ( r l , k l s ) ] n , m w n , m ( 1 ) ( r l , k l s ) x n m l ,
x n m l = [ a n m l , b n m l ] T ,
G ̱ 1 ( r l , r l ) = r ̂ l r ̂ l k 1 2 δ ( r l r l ) j k 1 4 π n , m ( 1 ) m 2 n + 1 n ( n + 1 ) { w n , m ( 2 ) ( r l , k 1 ) w n , m ( 1 ) ( r l , k 1 ) T , r l > r l w n , m ( 1 ) ( r l , k 1 ) w n , m ( 2 ) ( r l , k 1 ) T , r l < r l } .
[ ( k l s ) 2 k 1 2 ] V l G 1 ( r l , r l ) E ( r l ) d r l = δ i l E ( r i ) + j k 1 n , m { α l w n , m ( 2 ) ( r l , k 1 ) R n l ( 1 ) x n m l , l i α i w n , m ( 1 ) ( r i , k 1 ) R n i ( 2 ) x n m i , l = i } ,
w n , m ( 2 ) ( r l , k 1 ) = ν = m + w ν , m ( 1 ) ( r i , k 1 ) T i l ( ν , m , n ) ,
[ ( k l s ) 2 k 1 2 ] V l G 1 ( r l , r l ) E ( r l ) d r l = j k 1 α l ν , m w ν , m ( 1 ) ( r i , k 1 ) [ n = m + T i l ( ν , m , n ) R n l ( 1 ) x n m l ] .
W m , k ( 1 ) ( r l , k 1 ) = k 1 ν = m + j ν m ( ν m ) ! ( ν + m ) ! w ν , m ( 1 ) ( r l , k 1 ) Ω ( ν , m , k ) ,
W m , k ( 1 ) ( r l , k 1 ) = W m , k ( 1 ) ( r i , k 1 ) exp [ j k ( Z i Z l ) ]
[ ( k l s ) 2 k 1 2 ] V l G ̱ 11 ( r l , r l ) E ( r l ) d r l = j k 1 α l ν , m w ν , m ( 1 ) ( r i , k 1 ) [ n = m + L i l ( ν , m , n ) R n l ( 1 ) x n m l ] ,
E inc ( r ) = { W m 0 , β ( 1 ) ( r , k 1 ) [ 1 δ m 0 ( β ) ] T , ρ < α A m 0 ( β ) W m 0 , β ( 2 ) ( r , k 0 ) [ 1 δ m 0 ( β ) k 0 k 1 ] T , ρ > α } ,
E inc ( r i ) = ν , m w ν , m ( 1 ) ( r i , k 1 ) I ν , m i ( m 0 , β ) ,
I ν , m i ( m 0 , β ) = δ m , m 0 k 1 exp ( j β Z i ) j ν m 0 ( ν m 0 ) ! ( ν + m 0 ) ! × Ω ( ν , m 0 , β ) [ 1 δ m 0 ( β ) ] T .
R ν i ( 2 ) x ν m i + n = m + L i i ( ν , m , n ) R n i ( 1 ) x n m i + l i α l α i { n = m + [ T i l ( ν , m , n ) + L i l ( ν , m , n ) ] R n l ( 1 ) x n m l } = j k 1 α i I ν , m i ( m 0 , β ) ,
E ( r ) = E inc ( r ) + j k 1 l = 1 N s α l [ n , m w n , m ( 2 ) ( r l , k 1 ) R n l ( 1 ) x n m l ] + j k 1 l = 1 N s α l { + d k exp [ i k Z l ] 2 a 1 2 ( k ) m = + W m , k ( 1 ) ( r , k 1 ) A 11 ( m , k ) × n = m + n ( n + 1 ) 2 n + 1 j m n 2 Ω ( n , m , k ) R n l ( 1 ) x n m l } .
E ( r ) E inc ( r ) + { p = 1 M T ( m p , m 0 ) W m p , β p ( 1 ) ( r , k 1 ) [ 1 δ m p ( β p ) ] T , z + p = 1 M R ( m p , m 0 ) W m p , + β p ( 1 ) ( r , k 1 ) [ 1 δ m p ( + β p ) ] T , z } ,
M m , k ( i ) ( r , k 1 ) = × [ z ̂ C m ( i ) ( a 1 ρ ) exp ( j m ϕ + j k z ) ] ,
N m , k ( i ) ( r , k 1 ) = k 1 1 × M m , k ( i ) ( r , k 1 ) ,
W m , k ( i ) ( r , k 1 ) [ M m , k ( i ) ( r , k 1 ) N m , k ( i ) ( r , k 1 ) ] .
α 11 M m , k ( i ) ( r , k 1 ) M m , k ( j ) ( r , k 1 ) + α 12 M m , k ( i ) ( r , k 1 ) N m , k ( j ) ( r , k 1 ) + α 21 N m , k ( i ) ( r , k 1 ) M m , k ( j ) ( r , k 1 ) + α 22 N m , k ( i ) ( r , k 1 ) N m , k ( j ) ( r , k 1 ) = W m , k ( i ) ( r , k 1 ) [ α 11 α 12 α 21 α 22 ] W m , k ( j ) ( r , k 1 ) T ,
m n , m ( i ) ( r , k 1 ) = × [ r c n ( i ) ( k 1 r ) P n m ( cos θ ) exp ( j m ϕ ) ] ,
n n , m ( i ) ( r , k 1 ) = k 1 1 × m n , m ( i ) ( r , k 1 ) ,
w n , m ( i ) ( r , k 1 ) [ m n , m ( i ) ( r , k 1 ) n n , m ( i ) ( r , k 1 ) ] .
A 11 ( m , k ) = [ a 11 b 11 b 11 c 11 ] ,
{ a 11 c 11 } = H m ( 2 ) ( a 1 α ) J m ( a 1 α ) 2 j Δ m ( k ) π [ a 1 α J m ( a 1 α ) ] 2 { Ψ m ( a 0 α ) ( k 1 k 0 ) 2 Λ m ( a 1 α ) ( k 1 k 0 ) 2 [ Ψ m ( a 0 α ) Λ m ( a 1 α ) ] } ,
b 11 = 2 j m k k 1 [ ( k 1 k 0 ) 2 1 ] Δ m ( k ) π ( a 1 α ) 4 [ a 0 J m ( a 1 α ) ] 2 ,
Λ m ( x ) = J m ( x ) [ x J m ( x ) ] 1 , Ψ m ( x ) = H m ( 2 ) ( x ) [ x H m ( 2 ) ( x ) ] 1 ,
Δ m ( k ) = [ Ψ m ( a 0 α ) Λ m ( a 1 α ) ] [ Ψ m ( a 0 α ) ( k 1 k 0 ) 2 Λ m ( a 1 α ) ] m 2 k 2 k 0 2 [ ( k 1 k 0 ) 2 1 ] 2 ( a 0 a 1 α ) 4 ,
a 0 = ( k 0 2 k 2 ) 1 2 , a 1 = ( k 1 2 k 2 ) 1 2 .
R n l ( i ) = [ E n l ( i ) 0 0 H n l ( i ) ] ,
E n l ( i ) = x l s c n ( i ) ( x l 1 ) j n ( x l s ) x l 1 j n ( x l s ) c n ( i ) ( x l 1 ) ,
H n l ( i ) = x l 1 c n ( i ) ( x l 1 ) j n d ( x l s ) x l s x l s j n ( x l s ) c n d ( i ) ( x l 1 ) x l 1 ,
x l 1 = k 1 α l , x l s = k l s α l ,
T i l ( ν , m , n ) = [ A ν , m , n i l B ν , m n i l B ν , m , n i l A ν , m , n i l ] ,
{ A ν , m , n i l B ν , m , n i l } = ( 1 ) m j ν n 2 ν + 1 2 ν ( ν + 1 ) p j ± p g p ( m , n , m , ν ) h p ( 2 ) ( k 1 d i l ) { n ( n + 1 ) + ν ( ν + 1 ) p ( p + 1 ) ± 2 j m k 1 d i l } ,
P ν m ( x ) P n m ( x ) = p g p ( m , n m , ν ) P p ( x ) ,
g p ( m , ν , m , n ) = g p ( m , n , m , ν ) ,
g p ( m , ν , m , n ) = ( n m ) ! ( ν + m ) ! ( n + m ) ! ( ν m ) ! g p ( m , n , m , ν ) .
g p ( 1 , n , 1 , ν ) = ( 2 p + 1 ) [ n ( n + 1 ) + ν ( ν + 1 ) p ( p + 1 ) ] 2 ν ( ν + 1 ) ( n + ν + p + 1 ) × ( n + ν + p ( n + ν + p ) 2 ) ( n ν + p ( n ν + p ) 2 ) ( n + ν p ( n + ν p ) 2 ) ( n + ν + p ( n + ν + p ) 2 ) 1 .
Ω ( ν , m , k ) = [ j q ν m ( k ) 2 ν + 1 ν ( ν + 1 ) p ν m ( k ) 2 ν + 1 ν ( ν + 1 ) p ν m ( k ) j q ν m ( k ) ] ,
p ν m ( k ) = j m P ν m ( k k 1 ) ,
q ν m ( k ) = P ν 1 m ( k k 1 ) ( ν + m ) ν P ν + 1 m ( k k 1 ) ( ν m + 1 ) ( ν + 1 ) ,
L i l ( ν , m , n ) = k 1 2 j ν n + 2 n ( n + 1 ) 2 n + 1 ( ν m ) ! ( ν m ) ! × + d k exp [ j k ( Z i Z l ) ] a 1 2 ( k ) Ω ( ν , m , k ) A ( m , k ) Ω ( n , m , k )
L i l ( ν , m , n ) = ( 1 ) ν + n X L l i ( ν , m , n ) X , X = [ 1 0 0 1 ] ,
δ m 0 ( β ) = k 1 ( a 1 a 0 α ) 2 ( k 1 2 k 0 2 ) m 0 β [ Ψ m 0 ( a 0 α ) Λ m 0 ( a 1 α ) ] ,
A m 0 ( β ) = a 1 2 J m 0 ( a 1 α ) a 0 2 H m 0 ( 2 ) ( a 0 α ) ,
a 1 = ( k 1 2 β 2 ) 1 2 , a 0 = ( k 0 2 β 2 ) 1 2 = j ( β 2 k 0 2 ) 1 2 .
( T ( m p , m 0 ) R ( m p , m 0 ) ) = ± π k 1 a 1 2 ( β p ) Re s ( a 11 ) l = 1 N s α l exp ( ± j β p z l ) [ 1 δ m p ( β p ) ] n = m p + n ( n + 1 ) 2 n + 1 j m p n Ω ( n , m p , β p ) R n l ( 1 ) x n m p l ,
Re s ( a 11 ) = 2 j [ Ψ m ( a 0 α ) ( k 1 k 0 ) 2 Λ m ( a 1 α ) ] π [ a 1 α J m ( a 1 α ) ] 2 Δ m ( β p ) ,

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