Abstract

Numerical computations were made of scattering of an incident electromagnetic pulse by a coated sphere that is large compared to the dominant wavelength of the incident light. The scattered intensity was plotted as a function of the scattering angle and delay time of the scattered pulse. For fixed core and coating radii, the Debye series terms that most strongly contribute to the scattered intensity in different regions of scattering angle-delay time space were identified and analyzed. For a fixed overall radius and an increasing core radius, the first-order rainbow was observed to evolve into three separate components. The original component faded away, while the two new components eventually merged together. The behavior of surface waves generated by grazing incidence at the core/coating and coating/exterior interfaces was also examined and discussed.

© 2012 Optical Society of America

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References

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2012

2011

2010

P. Laven, “Separating diffraction from scattering: the million dollar challenge,” J. Nanophoton. 4, 041593 (2010).
[CrossRef]

2007

1994

1993

1991

1983

1979

1976

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

1969

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

1951

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

Adam, J. A.

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]

Fraser, A. B.

Fuller, K. A.

Jamison, J. M.

Kerker, M.

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

Khare, V.

V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. dissertation (University of Rochester, 1975).

Langley, D. S.

Laven, P.

Lin, C.-Y.

Lock, J. A.

Morrell, M. J.

Nussenzveig, H. M.

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, “Amplitude functions,” in Light Scattering by Small Particles (Dover, 1981), pp. 124–126.

H. C. van de Hulst, “The localization principle,” in Light Scattering by Small Particles (Dover, 1981), pp. 208–209.

Walker, J. D.

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Am. J. Phys.

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

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]

J. Math. Phys.

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[CrossRef]

J. Nanophoton.

P. Laven, “Separating diffraction from scattering: the million dollar challenge,” J. Nanophoton. 4, 041593 (2010).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Other

H. C. van de Hulst, “The localization principle,” in Light Scattering by Small Particles (Dover, 1981), pp. 208–209.

H. C. van de Hulst, “Amplitude functions,” in Light Scattering by Small Particles (Dover, 1981), pp. 124–126.

V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. dissertation (University of Rochester, 1975).

Supplementary Material (1)

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Figures (9)

Fig. 1.
Fig. 1.

Scattered intensity as a function of the scattering angle θ and delay time t of a 5 fs unpolarized Gaussian pulse by a coated sphere of core radius a12=7.5μm and refractive index m1=1.5, coating refractive index m2=1.3333, and overall radius a23=10μm. The dominant Debye series contributions (N,A,B) in different regions of θt space are indicated.

Fig. 2.
Fig. 2.

(a) Scattered intensity for the pulse and coated sphere parameters of Fig. 1 for 130°θ180° and 150fst230fs in the region of the first-order rainbow. Various ray trajectories as a function of the incident ray impact parameter b are superimposed on the figure; (b) Scattered intensity for the pulse parameters of Fig. 1 but with a12=30μm and a23=40μm, which resolve additional structure of the scattered intensity in the vicinity of the first-order rainbow.

Fig. 3.
Fig. 3.

Scattered intensity for the pulse and coated sphere parameters of Fig. 1 for 0°θ180° and 200fst400fs showing the behavior of the (2, 2, 3), (2, 4, 3), and (2, 6, 3) Debye terms and their αα, αβ, and ββ second-order rainbows.

Fig. 4.
Fig. 4.

(a) Scattered intensity of the (1, 2, 2) Debye term for the pulse and coated sphere parameters of Fig. 1 showing the α first-order rainbow; (b) Scattered intensity of the (1, 4, 2) Debye term showing the β first-order rainbow; (c) Scattered intensity of the (3, 4, 2) Debye term, which has the same time domain trajectory as the (1, 4, 2) term.

Fig. 5.
Fig. 5.

Scattered intensity of the (2, 2, 3) Debye term for the pulse and coated sphere parameters of Fig. 1 showing the glory at θ180° and t190fs caused by the complex ray of the αα second-order rainbow.

Fig. 6.
Fig. 6.

Scattered intensity of the (3, 4, 4) Debye term for the pulse and coated sphere parameters of Fig. 1 showing the ααβ third-order rainbow at b=0.93 and θ20°. As geometric rays with impact parameters on the coated sphere of b=0.7506 and b=0.9968 produce forward scattering (θ=0°), the (3, 4, 4) term also causes two forward glories (at t305fs and t308fs).

Fig. 7.
Fig. 7.

Scattered intensity as a function of θ and t for the incident pulse and coated sphere parameters of Fig. 1 except the core radius a12 is (a) 0, (b) 2, (c) 4, (d) 6, (e) 8, and (f) 10 μm. Figure 7(a) describes an uncoated sphere with m=1.3333 and Fig. 7(f) describes an uncoated sphere with m=1.5. See Media 1.

Fig. 8.
Fig. 8.

Scattered intensity as in Fig. 7 but for 120°θ180° and 130fst220fs in the vicinity of the first-order rainbow showing its evolution for core radius a12 of (a) 0; (b) 2; (c) 4; (d) 6; (e) 8; and (f) 10 μm. The features marked “A” are caused by core/coating surface waves, whereas the features marked “B” are caused by coating/exterior surface waves.

Fig. 9.
Fig. 9.

Scattered intensity of the (3, 4, 0) Debye term as a function of θ and t for the incident pulse and coated sphere parameters of Fig. 1, except that the core radius a12 is (a) 2.5, (b) 5 and (c) 7.5 μm, showing the evolution of the core/coating surface waves (marked as “A”) and coating/exterior surface waves (marked as “B”) of this Debye term. In (c) the two types of surface waves coalesce. The (3, 4, 0) ray trajectory as a function of the incident ray impact parameter b is superimposed on the figures.

Equations (2)

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S1(θ)=n=1{(2n+1)/[n(n+1)]}[anπn(θ)+bnτn(θ)]
S2(θ)=n=1{(2n+1)/[n(n+1)]}[anτn(θ)+bnπn(θ)],

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