Abstract

The Mie formulation for homogeneous spheres is generalized to handle core–shell systems and multiple concentric layers in a manner that exploits an analogy with stratified planar systems, thereby allowing concentric multilayered structures to be treated as photonic bandgap materials. Representative results from a Mie code employing this analogy demonstrate that photonic bands are present for periodic concentric spheres, though not readily apparent in extinction spectra. Rather, the periodicity simply alters the scattering profile, which enhances the ratio of backscattering to forward scattering inside the bandgap, whereas modification of the interference structure is evident in extinction spectra in accordance with the optical theorem.

© 2002 Optical Society of America

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  1. A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  2. R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960–1967 (1985).
    [CrossRef] [PubMed]
  3. D. W. Mackowski, R. A. Altenkirch, and M. P. Menguc, “Internal absorption cross sections in a stratified sphere,” Appl. Opt. 29, 1551–1559 (1990).
    [CrossRef] [PubMed]
  4. Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
    [CrossRef]
  5. J. Sinzig, U. Radtke, M. Quinten, and U. Kreibig, “Binary clusters: homogeneous alloys and nucleus-shell structures,” Z. Phys. D 26, 242–245 (1993).
    [CrossRef]
  6. B. R. Johnson, “Light scattering by a multilayer sphere,” Appl. Opt. 35, 3286–3296 (1996).
    [CrossRef] [PubMed]
  7. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Grehan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36, 5188–5198 (1997).
    [CrossRef] [PubMed]
  8. G. Pan, R. Kesavamoorthy, and S. A. Asher, “Optically nonlinear Bragg diffracting nanosecond optical switches,” Phys. Rev. Lett. 78, 3860–3863 (1997).
    [CrossRef]
  9. A. van Blaaderen and A. Vrij, “Synthesis and characterization of colloidal dispersions of fluorescent, monodisperse silica spheres,” Langmuir 8, 2921–2931 (1992).
    [CrossRef]
  10. S. Westcott, S. Oldenburg, T. R. Lee, and N. J. Halas, “Formation and adsorption of gold nanoparticle clusters on functionalized silica nanoparticle surfaces,” Langmuir 14, 5396–5401 (1998).
    [CrossRef]
  11. D. B. Wolfe, S. J. Oldenburg, S. L. Westcott, J. B. Jackson, M. S. Paley, and N. J. Halas, “Photodeposition of molecular layers on nanoparticle substrates,” Langmuir 15, 2745–2748 (1999).
    [CrossRef]
  12. M. Born and E. Wolf, Principles of Optics (Permagon, Oxford, 1970).
  13. K. A. Fuller, “Scattering of light by coated spheres,” Opt. Lett. 18, 257–259 (1993).
    [CrossRef] [PubMed]
  14. J. E. McDonald, “Large-sphere limit of the radar backscattering coefficient,” Q. J. R. Meteorol. Soc. 88, 183–186 (1962).
    [CrossRef]
  15. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  16. D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Phys. Rev. A 53, 3565–3572 (1996).
    [CrossRef] [PubMed]
  17. K. A. Fuller and D. W. Mackowski, “Electromagnetic scattering by compounded spherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, New York, 2000), pp. 225–272.

1999

D. B. Wolfe, S. J. Oldenburg, S. L. Westcott, J. B. Jackson, M. S. Paley, and N. J. Halas, “Photodeposition of molecular layers on nanoparticle substrates,” Langmuir 15, 2745–2748 (1999).
[CrossRef]

1998

S. Westcott, S. Oldenburg, T. R. Lee, and N. J. Halas, “Formation and adsorption of gold nanoparticle clusters on functionalized silica nanoparticle surfaces,” Langmuir 14, 5396–5401 (1998).
[CrossRef]

1997

1996

D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Phys. Rev. A 53, 3565–3572 (1996).
[CrossRef] [PubMed]

B. R. Johnson, “Light scattering by a multilayer sphere,” Appl. Opt. 35, 3286–3296 (1996).
[CrossRef] [PubMed]

1993

K. A. Fuller, “Scattering of light by coated spheres,” Opt. Lett. 18, 257–259 (1993).
[CrossRef] [PubMed]

J. Sinzig, U. Radtke, M. Quinten, and U. Kreibig, “Binary clusters: homogeneous alloys and nucleus-shell structures,” Z. Phys. D 26, 242–245 (1993).
[CrossRef]

1992

A. van Blaaderen and A. Vrij, “Synthesis and characterization of colloidal dispersions of fluorescent, monodisperse silica spheres,” Langmuir 8, 2921–2931 (1992).
[CrossRef]

1991

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

1990

1985

1962

J. E. McDonald, “Large-sphere limit of the radar backscattering coefficient,” Q. J. R. Meteorol. Soc. 88, 183–186 (1962).
[CrossRef]

1951

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Aden, A. L.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Altenkirch, R. A.

Asher, S. A.

G. Pan, R. Kesavamoorthy, and S. A. Asher, “Optically nonlinear Bragg diffracting nanosecond optical switches,” Phys. Rev. Lett. 78, 3860–3863 (1997).
[CrossRef]

Bhandari, R.

Braunstein, D.

D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Phys. Rev. A 53, 3565–3572 (1996).
[CrossRef] [PubMed]

Fuller, K. A.

Gouesbet, G.

Grehan, G.

Guo, L. X.

Halas, N. J.

D. B. Wolfe, S. J. Oldenburg, S. L. Westcott, J. B. Jackson, M. S. Paley, and N. J. Halas, “Photodeposition of molecular layers on nanoparticle substrates,” Langmuir 15, 2745–2748 (1999).
[CrossRef]

S. Westcott, S. Oldenburg, T. R. Lee, and N. J. Halas, “Formation and adsorption of gold nanoparticle clusters on functionalized silica nanoparticle surfaces,” Langmuir 14, 5396–5401 (1998).
[CrossRef]

Jackson, J. B.

D. B. Wolfe, S. J. Oldenburg, S. L. Westcott, J. B. Jackson, M. S. Paley, and N. J. Halas, “Photodeposition of molecular layers on nanoparticle substrates,” Langmuir 15, 2745–2748 (1999).
[CrossRef]

Johnson, B. R.

Kerker, M.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Kesavamoorthy, R.

G. Pan, R. Kesavamoorthy, and S. A. Asher, “Optically nonlinear Bragg diffracting nanosecond optical switches,” Phys. Rev. Lett. 78, 3860–3863 (1997).
[CrossRef]

Khazanov, A. M.

D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Phys. Rev. A 53, 3565–3572 (1996).
[CrossRef] [PubMed]

Koganov, G. A.

D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Phys. Rev. A 53, 3565–3572 (1996).
[CrossRef] [PubMed]

Kreibig, U.

J. Sinzig, U. Radtke, M. Quinten, and U. Kreibig, “Binary clusters: homogeneous alloys and nucleus-shell structures,” Z. Phys. D 26, 242–245 (1993).
[CrossRef]

Lee, T. R.

S. Westcott, S. Oldenburg, T. R. Lee, and N. J. Halas, “Formation and adsorption of gold nanoparticle clusters on functionalized silica nanoparticle surfaces,” Langmuir 14, 5396–5401 (1998).
[CrossRef]

Mackowski, D. W.

McDonald, J. E.

J. E. McDonald, “Large-sphere limit of the radar backscattering coefficient,” Q. J. R. Meteorol. Soc. 88, 183–186 (1962).
[CrossRef]

Menguc, M. P.

Oldenburg, S.

S. Westcott, S. Oldenburg, T. R. Lee, and N. J. Halas, “Formation and adsorption of gold nanoparticle clusters on functionalized silica nanoparticle surfaces,” Langmuir 14, 5396–5401 (1998).
[CrossRef]

Oldenburg, S. J.

D. B. Wolfe, S. J. Oldenburg, S. L. Westcott, J. B. Jackson, M. S. Paley, and N. J. Halas, “Photodeposition of molecular layers on nanoparticle substrates,” Langmuir 15, 2745–2748 (1999).
[CrossRef]

Paley, M. S.

D. B. Wolfe, S. J. Oldenburg, S. L. Westcott, J. B. Jackson, M. S. Paley, and N. J. Halas, “Photodeposition of molecular layers on nanoparticle substrates,” Langmuir 15, 2745–2748 (1999).
[CrossRef]

Pan, G.

G. Pan, R. Kesavamoorthy, and S. A. Asher, “Optically nonlinear Bragg diffracting nanosecond optical switches,” Phys. Rev. Lett. 78, 3860–3863 (1997).
[CrossRef]

Quinten, M.

J. Sinzig, U. Radtke, M. Quinten, and U. Kreibig, “Binary clusters: homogeneous alloys and nucleus-shell structures,” Z. Phys. D 26, 242–245 (1993).
[CrossRef]

Radtke, U.

J. Sinzig, U. Radtke, M. Quinten, and U. Kreibig, “Binary clusters: homogeneous alloys and nucleus-shell structures,” Z. Phys. D 26, 242–245 (1993).
[CrossRef]

Ren, K. F.

Shuker, R.

D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Phys. Rev. A 53, 3565–3572 (1996).
[CrossRef] [PubMed]

Sinzig, J.

J. Sinzig, U. Radtke, M. Quinten, and U. Kreibig, “Binary clusters: homogeneous alloys and nucleus-shell structures,” Z. Phys. D 26, 242–245 (1993).
[CrossRef]

van Blaaderen, A.

A. van Blaaderen and A. Vrij, “Synthesis and characterization of colloidal dispersions of fluorescent, monodisperse silica spheres,” Langmuir 8, 2921–2931 (1992).
[CrossRef]

Vrij, A.

A. van Blaaderen and A. Vrij, “Synthesis and characterization of colloidal dispersions of fluorescent, monodisperse silica spheres,” Langmuir 8, 2921–2931 (1992).
[CrossRef]

Wang, Y. P.

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Westcott, S.

S. Westcott, S. Oldenburg, T. R. Lee, and N. J. Halas, “Formation and adsorption of gold nanoparticle clusters on functionalized silica nanoparticle surfaces,” Langmuir 14, 5396–5401 (1998).
[CrossRef]

Westcott, S. L.

D. B. Wolfe, S. J. Oldenburg, S. L. Westcott, J. B. Jackson, M. S. Paley, and N. J. Halas, “Photodeposition of molecular layers on nanoparticle substrates,” Langmuir 15, 2745–2748 (1999).
[CrossRef]

Wolfe, D. B.

D. B. Wolfe, S. J. Oldenburg, S. L. Westcott, J. B. Jackson, M. S. Paley, and N. J. Halas, “Photodeposition of molecular layers on nanoparticle substrates,” Langmuir 15, 2745–2748 (1999).
[CrossRef]

Wu, Z. S.

Appl. Opt.

J. Appl. Phys.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Langmuir

A. van Blaaderen and A. Vrij, “Synthesis and characterization of colloidal dispersions of fluorescent, monodisperse silica spheres,” Langmuir 8, 2921–2931 (1992).
[CrossRef]

S. Westcott, S. Oldenburg, T. R. Lee, and N. J. Halas, “Formation and adsorption of gold nanoparticle clusters on functionalized silica nanoparticle surfaces,” Langmuir 14, 5396–5401 (1998).
[CrossRef]

D. B. Wolfe, S. J. Oldenburg, S. L. Westcott, J. B. Jackson, M. S. Paley, and N. J. Halas, “Photodeposition of molecular layers on nanoparticle substrates,” Langmuir 15, 2745–2748 (1999).
[CrossRef]

Opt. Lett.

Phys. Rev. A

D. Braunstein, A. M. Khazanov, G. A. Koganov, and R. Shuker, “Lowering of threshold conditions for nonlinear effects in a microsphere,” Phys. Rev. A 53, 3565–3572 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett.

G. Pan, R. Kesavamoorthy, and S. A. Asher, “Optically nonlinear Bragg diffracting nanosecond optical switches,” Phys. Rev. Lett. 78, 3860–3863 (1997).
[CrossRef]

Q. J. R. Meteorol. Soc.

J. E. McDonald, “Large-sphere limit of the radar backscattering coefficient,” Q. J. R. Meteorol. Soc. 88, 183–186 (1962).
[CrossRef]

Radio Sci.

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Z. Phys. D

J. Sinzig, U. Radtke, M. Quinten, and U. Kreibig, “Binary clusters: homogeneous alloys and nucleus-shell structures,” Z. Phys. D 26, 242–245 (1993).
[CrossRef]

Other

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

M. Born and E. Wolf, Principles of Optics (Permagon, Oxford, 1970).

K. A. Fuller and D. W. Mackowski, “Electromagnetic scattering by compounded spherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, New York, 2000), pp. 225–272.

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Figures (8)

Fig. 1
Fig. 1

Dephasing of n=26, p=1 partial wave for a homogeneous sphere of radius a=1.9 µm and relative index m=2.25 in vacuum. The function is periodic at high frequencies, displaying non-TIR spherical modes, but breaks into MDRs at lower frequencies.

Fig. 2
Fig. 2

Multiple concentric spheres as a sequence of concave–convex spherical resonators. In a manner analogous with the planar Fabry–Perot resonator, multiple reflections are summed to obtain the resultant scattered and internal fields.

Fig. 3
Fig. 3

(a) Extinction efficiency and first partial wave for a core–shell particle (N=3) having refractive indices n1=1.5, n2=1.0, n3=1.5 and radii a2=10.5, a3=10.0. Extinction maxima occur when the incident wave and scattered wave at zero degrees interfere destructively. The indicated spacing of 2.0 µm-1 is predicted by Eq. (30). (b) Extinction efficiency and first partial wave for a multilayered particle (N=5) having refractive indices n1=1.5, n2=1.0, n3=1.5, n4=1.0, n5=1.5 and radii a2=11.5, a3=11.0, a4=10.5, a5=10.0 µm. Comparison of (a) and (b) reveals that the interference structure spacing is N-dependent, in agreement with Eq. (30), but inconsistent with a PBG effect.

Fig. 4
Fig. 4

Photonic bands in the backscattering efficiency. Qback is plotted for (a) a=14-µm homogeneous sphere, (b) N=10 concentric multilayered sphere having a core radius of a10=10 µm, and (c) N=6 concentric multilayered sphere having a core radius of a6=5 µm.

Fig. 5
Fig. 5

Photonic bands in forward and backscattering for incident light polarized parallel to the scattering plane. (a) Scattering angle=45°, (b) scattering angle=150°. The large intensities in (a) and small intensities in (b) at 0.5 µm-1, 1.0 µm-1, 1.5 µm-1, etc., correspond to the passband (transmission band). The regular subsidiary peaks in (b) are the result of interference between the incident wave and light scattered from the various layers.

Fig. 6
Fig. 6

Asymmetry parameter for (a) an N=4 concentric multilayered particle having refractive indices n1=1.0, n2=2.0, n3=1.0, n4=2.0 and radii a2=11.0, a3=10.5, a4=10.0 µm compared with (b) an a=10-µm homogeneous sphere with npart=2.0 and nmed=1.0. Within the PBG stopband, backscattering is enhanced, while within the passband, forward scattering is enhanced.

Fig. 7
Fig. 7

(a) Scattering profile for a multilayered particle with no absorption for incident light polarized parallel to the scattering plane at frequencies (curve a) inside (0.8 µm-1) and (curve b) outside (1.0 µm-1) the gap. Except near 180°, backscattering (θ>90°) is generally enhanced inside the gap at the expense of forward scattering. (b) Scattering profile for a multilayered particle having an absorbing core for light polarized parallel to the scattering plane for frequencies (curve a) inside (0.8 µm-1) and (curve b) outside (1.0 µm-1) the gap. An absorbing core eliminates the gap reversal at 180°.

Fig. 8
Fig. 8

Single-scattering albedo for an N=10 concentric multilayered sphere having refractive indices n1=1.0, n2=2.0, n3=1.0, n4=2.0, etc., and radii a10=10.0 µm, a9=11.0 µm, a8=11.5 µm, etc. Strong scattering bands appear at Δ(1/λ)=0.25 µm-1, 0.75 µm-1, etc., just as predicted for a planar λ/4 stack.

Equations (30)

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Einc=eˆxE0 exp(ikz)=p=12n=1qnpNnp(1),
Esca=p=12n=1qnpanpNnp(3),
Eint=p=12n=1qnpcnpNnp(1),
anp=-m2-pψn(mx)ψn(x)-mp-1ψn(x)ψn(mx)m2-pψn(mx)ξn(x)-mp-1ξn(x)ψn(mx),
cnp=mψn(x)ξn(x)-mξn(x)ψn(x)m2-pψn(mx)ξn(x)-mp-1ξn(x)ψn(mx),
xka=2πnmedλa,
mnpartnmed,
Qsca=2x2p=12n=1(2n+1)|anp|2,
Qext=2x2p=12n=1(2n+1)Re{-anp},
Qback=1x2p=12n=1(2n+1)(-1)n+panp2,
g=cos θ=4x2Qscan=1p=12n(n+2)n+1Re{anpan+1,p*}+2n+1n(n+1)Re{an1an2*},
Δ(1/λ)=12a(npart-nmed),
rl-1:l=pl-1-plpl-1+pl,
tl-1:l=2pl-1pl-1+pl.
rl:l-1=-rl-1:l,
tl:l-1=2plpl-1+pl.
r¯l-1:lt¯l-1:l=rl-1:ltl-1:lexp(iβl-1),
r¯l:l-1t¯l:l-1=rl:l-1tl:l-1exp(iβl),
rl:1=rl:l-1+tl:l-1r¯l-1:1t¯l-1:l1-r¯l-1:lr¯l-1:1,
tl:1=tl:l-1t¯l-1:11-r¯l-1:lr¯l-1:1;
t1:l=t1:l-1t¯l-1:l1-r¯l-1:lr¯l-1:1,
r1:l=r1:l-1+t1:l-1r¯l-1:lt¯l-1:11-r¯l-1:lr¯l-1:1.
anp=-m2-pξn(mx)ξn(x)-mp-1ξn(x)ξn(mx)m2-pψn(mx)ξn(x)-mp-1ξn(x)ψn(mx),
cnp=ξn(mx)ψn(mx)-ψn(mx)ξn(mx)m2-pψn(mx)ξn(x)-mp-1ξn(x)ψn(mx).
anp,l:l+1=-ml2-pψn,l(mlxl)ψn,l(xl)-mlp-1ψn,l(xl)ψn,l(mlxl)ml2-pψn,l(mlxl)ξn,l(xl)-mlp-1ξn,l(xl)ψn,l(mlxl),
a1:l|np=a1:l-1+c1:l-1al-1:lcl-1:11-al-1:lal-1:1np,
c1:l|np=c1:l-1cl-1:l1-al-1:lal-1:1np.
Δ(1/λ)=12[a3(n3-n2)+a2(n2-n1)].
Δ(1/λ)=12l=1Nal+1(nl+1-nl),
Δ(1/λ)=1(N-1)Δnd,

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