Abstract

The S matrix for scattering a diagonally incident focused Gaussian beam by an infinitely long circular cylinder is calculated, and the diagonal and off-diagonal elements of the matrix are physically interpreted. In addition, the TE and TM morphology-dependent resonances of the cylinder for plane-wave incidence are examined, and it is found that the cylinder size parameter at resonance increases as the angle of incidence of the plane wave increases. The size parameter increase is explained in terms of the heuristic model of morphology-dependent resonances. The presence of both TE and TM resonances in each of the -polarized and μ-polarized partial-wave scattering amplitudes is discussed. This mixing is interpreted in terms of successive polarization-preserving and cross-polarized total internal reflections of the morphology-dependent-resonance wave inside the cylinder. Finally, some novel properties of morphology-dependent resonances produced by Gaussian-beam incidence are predicted.

© 1997 Optical Society of America

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References

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  1. S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.
  2. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
    [CrossRef]
  3. G. Chen, R. K. Chang, S. C. Hill, P. W. Barber, “Frequency splitting of degenerate spherical cavity modes: stimulated Raman scattering spectrum of deformed droplets,” Opt. Lett. 16, 1269–1271 (1991).
    [CrossRef] [PubMed]
  4. G. Chen, M. M. Mazumder, Y. R. Chemla, A. Serpenguzel, R. K. Chang, S. C. Hill, “Wavelength variation of laser emission along the entire rim of slightly deformed microdroplets,” Opt. Lett. 18, 1993–1995 (1993).
    [CrossRef] [PubMed]
  5. J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long circular cylinder,” J. Opt. Soc. Am. A 14, 640–652 (1997).
    [CrossRef]
  6. S. de Benedetti, Nuclear Interactions (Wiley, New York, 1964), pp. 322, 326–327.
  7. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1982), pp. 209–210.
  8. J. A. Lock, C. L. Adler, “Debye series analysis of the first-order rainbow produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A (to be published).
  9. Ref. 6, p. 330.
  10. J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, “Enhancement of fluorescence induced by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078 (1981).
    [CrossRef]
  11. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 98.
  12. J. A. Lock, “Improved Gaussian beam scattering algorithm,” Appl. Opt. 34, 559–570 (1995).
    [CrossRef] [PubMed]
  13. R. G. Pinnick, G. L. Fernandez, J.-G. Xie, T. Ruekgauer, J. Gu, R. L. Armstrong, “Stimulated Raman scattering and lasing in micrometer-sized cylindrical liquid jets: time and spectral dependence,” J. Opt. Soc. Am. B 6, 865–870 (1992).
    [CrossRef]

1997 (1)

1995 (1)

1993 (1)

1992 (1)

1991 (1)

1989 (1)

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

1981 (1)

J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, “Enhancement of fluorescence induced by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078 (1981).
[CrossRef]

Adler, C. L.

J. A. Lock, C. L. Adler, “Debye series analysis of the first-order rainbow produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A (to be published).

Alexander, D. R.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

Armstrong, R. L.

Barber, P. W.

G. Chen, R. K. Chang, S. C. Hill, P. W. Barber, “Frequency splitting of degenerate spherical cavity modes: stimulated Raman scattering spectrum of deformed droplets,” Opt. Lett. 16, 1269–1271 (1991).
[CrossRef] [PubMed]

J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, “Enhancement of fluorescence induced by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078 (1981).
[CrossRef]

Barton, J. P.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

Benner, R. E.

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.

Chang, R. K.

Chemla, Y. R.

Chen, G.

de Benedetti, S.

S. de Benedetti, Nuclear Interactions (Wiley, New York, 1964), pp. 322, 326–327.

Dorain, P. B.

J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, “Enhancement of fluorescence induced by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078 (1981).
[CrossRef]

Fernandez, G. L.

Gu, J.

Hill, S. C.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 98.

Lock, J. A.

J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long circular cylinder,” J. Opt. Soc. Am. A 14, 640–652 (1997).
[CrossRef]

J. A. Lock, “Improved Gaussian beam scattering algorithm,” Appl. Opt. 34, 559–570 (1995).
[CrossRef] [PubMed]

J. A. Lock, C. L. Adler, “Debye series analysis of the first-order rainbow produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A (to be published).

Mazumder, M. M.

Owen, J. F.

J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, “Enhancement of fluorescence induced by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078 (1981).
[CrossRef]

Pinnick, R. G.

Ruekgauer, T.

Schaub, S. A.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

Serpenguzel, A.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1982), pp. 209–210.

Xie, J.-G.

Appl. Opt. (1)

J. Appl. Phys. (1)

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, “Enhancement of fluorescence induced by microstructure resonances of a dielectric fiber,” Phys. Rev. Lett. 47, 1075–1078 (1981).
[CrossRef]

Other (6)

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 98.

S. de Benedetti, Nuclear Interactions (Wiley, New York, 1964), pp. 322, 326–327.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1982), pp. 209–210.

J. A. Lock, C. L. Adler, “Debye series analysis of the first-order rainbow produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A (to be published).

Ref. 6, p. 330.

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.

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Figures (9)

Fig. 1
Fig. 1

Geometry of the cylinder and the incident beam. The dominant propagation direction of the incident beam is in the xz plane and makes an angle ξ with the x axis, and the center of its focal waist is at (x0, y0, z0).

Fig. 2
Fig. 2

Partial-wave scattering amplitudes (a) al(-sin ξ) and (b) bl(-sin ξ) in the complex plane for l=50, ξ=0°, and 45.75 ka cos ξ46.50. The refractive index of the cylinder is n = 1.53.

Fig. 3
Fig. 3

Partial-wave scattering amplitudes (a) al(-sin ξ) and (b) bl(-sin ξ) in the complex plane for l=50, ξ=10°, n=1.53, and 45.50ka cos ξ46.25.

Fig. 4
Fig. 4

Magnitude squared of the partial-wave scattering amplitudes (a) |al(-sin ξ)|2, (b) |bl(-sin ξ)|2, and (c) |ql(-sin ξ)|2 as a function of ka cos ξ for l=50, ξ=10°, and n=1.53. The TE50,4 and TM50,4 resonances occur in each scattering amplitude. The TE50,4 resonance in (b) is sufficiently weak that it is not visually apparent.

Fig. 5
Fig. 5

(a), (b) Two physical processes contributing to al(h). The process in (a) is polarization preserving, and that in (b) mixes polarizations during the scattering process. (c), (d) Two physical processes contributing to bl(h). The process in (c) is polarization preserving, and that in (d) mixes polarizations during the scattering process. (e), (f) Two physical processes contributing to ql(h).

Fig. 6
Fig. 6

(a) Ray with grazing normal incidence on a circular cylinder. The ray enters the cylinder at A and becomes a MDR wave. When it has circumnavigated the cylinder one time, the ray location C centers the cylinder. (b) Two rays with grazing diagonal incidence on a circular cylinder. When the MDR wave has spiraled down the cylinder one time from point A to point B, the location C on the second ray enters the cylinder at point B.

Fig. 7
Fig. 7

A μ-polarized incident plane wave produces (a) resonant scattering if the cylinder size parameter corresponds to TM resonance and (b) nonresonant scattering if the cylinder size parameter is slightly larger than that for a TM resonance.

Fig. 8
Fig. 8

Geometry of the far zone. The coordinates of the point P in the far zone are (x, y, z). The line joining P to the origin makes an angle η with the horizontal plane, and its projection onto the horizontal plane makes an angle θ with the x axis.

Fig. 9
Fig. 9

A µ-polarized incident Gaussian beam produces (a) resonant scattering in the horizontal plane and nonresonant scattering above and below the horizontal plane if the cylinder size parameter corresponds to a plane-wave TM resonance, (b) nonresonant scattering in the horizontal plane and resonant scattering at some angle above and below the horizontal plane if the cylinder size parameter is slightly larger than that for a plane-wave TM resonance, and (c) nonresonant scattering in the horizontal plane and resonant scattering at some angle above and below the horizontal plane if the cylinder size parameter is slightly larger than that for a plane-wave TE resonance.

Tables (2)

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Table 1 Cylinder Size Parameter ka for the TM50,4 Resonance

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Table 2 Cylinder Size Parameter ka for the TE50,4 Resonance

Equations (52)

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Al(h)=cos ξ2sπ(1-h2)F×exp{ik[hz0+(1-h2)1/2x0]}×exp{-[h cos ξ+(1-h2)1/2 sin ξ]2/4s2}×exp{-s2[l(1-h2)-1/2+ky0]2/F},
Bl(h)=0,
s1kw0,
F=(cos ξ)[cos ξ-h(1-h2)-1/2×sin ξ]-2is2kx0(1-h2)-1/2.
αl(h)=al(h)Al(h)+ql(h)Bl(h),
βl(h)=-ql(h)Al(h)+bl(h)Bl(h),
al(h)=U2W1-nU3W3W1W2-nW32,
bl(h)=U1W2-nU3W3W1W2-nW32,
ql(h)=2nlh(y2-x2)πx2y2Jl2(y)W1W2-nW32.
xka1-h21/2,ynka1-h2n21/2,
U1=n2xyJl(x)Jl(y)-Jl(x)Jl(y),
U2=nxyJl(x)Jl(y)-nJl(x)Jl(y),
U3=hl(y2-x2)xy2Jl(x)Jl(y),
W1=n2xyHl(1)(x)Jl(y)-Hl(1)(x)Jl(y),
W2=nxyHl(1)(x)Jl(y)-nHl(1)(x)Jl(y),
W3=hl(y2-x2)xy2Hl(1)(x)Jl(y),
γl(h)=cl(h)Al(h)+pl(h)Bl(h),
δl(h)=-npl(h)Al(h)+dl(h)Bl(h),
cl(h)=-2inxπy2W1W1W2-nW32,
dl(h)=-2inxπy2W2W1W2-nW32,
pl(h)=-2nxπy2W3W1W2-nW32.
Einteriorr, θ, z
=E0cos ξ-dhl=-il+1 expikhz×expilθilkrγlhJly¯+h1-h2n21/2×δlhJly¯uˆr-n1-h2n21/2×γlhJly¯-ihlnkrδlhJly¯uˆθ-in1-h2n2δlhJly¯uˆz,
y¯=nkr1-h2/n21/2.
V1=n2xyNl(x)Jl(y)-Nl(x)Jl(y),
V2=nxyNl(x)Jl(y)-nNl(x)Jl(y),
V3=hl(y2-x2)xy2Nl(x)Jl(y),
R1U1U2-nU32,R2V1V2-nV32,
L1U1V2-nU3V3,L2U2V1-nU3V3,
al(h)=(1/2){1+Ml(h)exp[-iϕla(h)]},
bl(h)=(1/2){1+Ml(h)exp[-iϕlb(h)]},
ql(h)=(1/2){Nl(h)exp[-iϕlq(h)]},
Mlh=R1+R22+L1-L22R1-R22+L1+L221/2,
Nl(h)=[1-Ml2(h)]1/2,
ϕlah=arctan2R1L1+R2L2R12-R22-L12-L22,
ϕlbh=arctan2R1L2+R2L1R12-R22+L12-L22,
ϕlqh=arctanL1+L2R1-R2.
ϕla(h)+ϕlb(h)=2ϕlq(h).
ψlexterior,i=Hl2AlihBlih-jSijHl1AlihBlih,
S=SSμSμSμμ=Mlhexp-iϕlah-Nlhexp-iϕlqhNlhexp-iϕlqhMlhexp-iϕlbh.
al=(1/2)[1+exp(-2iζla)],
bl=(1/2)[1+exp(-2iζlb)],
ql=0,
ζla=arctan(V2/U2),ζlb=arctan(V1/U1).
S=exp-2iζla00exp-2iζlb,
(ka)50,4,TMresonant(ξ)=(ka)50,4,TMresonant(0)+9.32ξ2,
(ka)50,4,TEresonant(ξ)=(ka)50,4,TEresonant(0)+12.02ξ2.
lλ=c2πanc,
ka=ln.
sin ξ=n sin σ
lλ+2πa tan σ sin ξ=c2πanc cos σ,
kaln+ξ2l2n3

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