Abstract

We present an application of the group-theory-based method for solving the problem of Mie scattering of evanescent waves by a dielectric sphere. Comparison with the conventional multipole expansion method is also discussed. The developed method is relevant to the near-field optical scanning microscope theoretical modeling problem.

© 1998 Optical Society of America

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References

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  1. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  2. See, for example, D. W. Pohl, D. Courjon, eds., Near-Field Optics (Kluwer, Dordrecht, The Netherlands, 1993); M. Ohtsu, “Progress of high-resolution photon scanning tunneling microscopy due to a nanometric fiber probe,” J. Lightwave Technol. 13, 1200–1221 (1995).
    [CrossRef]
  3. H. Chew, D. S. Wang, M. Kerker, “Elastic scattering of evanescent electromagnetic waves,” Appl. Opt. 18, 2679–2687 (1979).
    [CrossRef] [PubMed]
  4. C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
    [CrossRef]
  5. J. D. Pendleton, “Mie scattering into solid angles,” J. Opt. Soc. Am. 71, 1029–1033 (1982).
    [CrossRef]
  6. G. Goertzel, “Angular correction of gamma-rays,” Phys. Rev. 70, 897–909 (1946).
    [CrossRef]
  7. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  8. A. V. Zvyagin, “Study of near-field optical microscope image formation,” Ph.D. dissertation (Tokyo Institute of Technology, Tokyo, Japan, 1997).
  9. M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957), Chap. 4.
  10. G. Goertzel, “Angular correction of gamma-rays,” Phys. Rev. 70, 897–909 (1946).
    [CrossRef]
  11. D. Barchiesi, D. Van Labeke, “Application of Mie scattering of evanescent waves to scanning tunneling optical microscopy theory,” J. Mod. Opt. 40, 1239–1254 (1993).
    [CrossRef]
  12. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1965).

1995 (1)

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

1993 (1)

D. Barchiesi, D. Van Labeke, “Application of Mie scattering of evanescent waves to scanning tunneling optical microscopy theory,” J. Mod. Opt. 40, 1239–1254 (1993).
[CrossRef]

1982 (1)

1979 (1)

1946 (2)

G. Goertzel, “Angular correction of gamma-rays,” Phys. Rev. 70, 897–909 (1946).
[CrossRef]

G. Goertzel, “Angular correction of gamma-rays,” Phys. Rev. 70, 897–909 (1946).
[CrossRef]

Barchiesi, D.

D. Barchiesi, D. Van Labeke, “Application of Mie scattering of evanescent waves to scanning tunneling optical microscopy theory,” J. Mod. Opt. 40, 1239–1254 (1993).
[CrossRef]

Bohren, C. F.

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

Chew, H.

Goertzel, G.

G. Goertzel, “Angular correction of gamma-rays,” Phys. Rev. 70, 897–909 (1946).
[CrossRef]

G. Goertzel, “Angular correction of gamma-rays,” Phys. Rev. 70, 897–909 (1946).
[CrossRef]

Huffman, D. R.

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

Jackson, J. D.

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

Kaiser, T.

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

Kerker, M.

Lange, S.

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

Liu, C.

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

Pendleton, J. D.

Rose, M. E.

M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957), Chap. 4.

Schweiger, G.

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

Van Labeke, D.

D. Barchiesi, D. Van Labeke, “Application of Mie scattering of evanescent waves to scanning tunneling optical microscopy theory,” J. Mod. Opt. 40, 1239–1254 (1993).
[CrossRef]

Wang, D. S.

Zvyagin, A. V.

A. V. Zvyagin, “Study of near-field optical microscope image formation,” Ph.D. dissertation (Tokyo Institute of Technology, Tokyo, Japan, 1997).

Appl. Opt. (1)

J. Mod. Opt. (1)

D. Barchiesi, D. Van Labeke, “Application of Mie scattering of evanescent waves to scanning tunneling optical microscopy theory,” J. Mod. Opt. 40, 1239–1254 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

Phys. Rev. (2)

G. Goertzel, “Angular correction of gamma-rays,” Phys. Rev. 70, 897–909 (1946).
[CrossRef]

G. Goertzel, “Angular correction of gamma-rays,” Phys. Rev. 70, 897–909 (1946).
[CrossRef]

Other (6)

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

A. V. Zvyagin, “Study of near-field optical microscope image formation,” Ph.D. dissertation (Tokyo Institute of Technology, Tokyo, Japan, 1997).

M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957), Chap. 4.

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1965).

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

See, for example, D. W. Pohl, D. Courjon, eds., Near-Field Optics (Kluwer, Dordrecht, The Netherlands, 1993); M. Ohtsu, “Progress of high-resolution photon scanning tunneling microscopy due to a nanometric fiber probe,” J. Lightwave Technol. 13, 1200–1221 (1995).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Schematic diagram of the scattering-problem setup. The xz plane is the plane of incidence and lies in the plane of the page. If the angle of incidence I is less than the critical total internal reflection angle, the angle between the wave vector and the z axis, θk is real; otherwise it is pure imaginary.

Equations (43)

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E=E0 exp(ikr),
E(r)=l=1m=-ll{αM(l, m)jl(kr)Xlm(θ,ϕ)+ikαE(l, m)×[jl(kr)Xlm(θ,ϕ)]},
αMs(l, m)jl(kr)=EsXlm*(θ, ϕ)dΩ=E0sl(l+1) [LyYlm(θ, ϕ)]*×exp(ikr)dΩ,
exp(ikr)=4πl=0iljl(kr)m=-ll(-1)m×Ylm(θ, ϕ)Ylm*(θk,ϕk).
αMs(l, m)=(-1)m+1il+1π(2l+1)(l-m)!l(l+1)(l+m)!1/2×[Plm+1(θk)-(l+m)×(l-m+1)Plm-1(θk)]*
Plm+1(θk)-(l+m)(l-m+1)Plm-1(θk)=2 dP(θk)dθk,
αEs(l, m)=(-1)milπ(2l+1)(l-m)!l(l+1)(l+m)!1/2[Pl-1m+1(θk)+(l+m)(l+m-1)Pl-1m-1(θk)]*,
Pl-1m+1(θk)+(l+m)(l+m-1)Pl-1m-1(θk)=-2m Plm(θk)sin θk.
αMp(l, m)=αEs(l, m),αEp(l, m)=-αMs(l, m).
Esc(r)=l,mβM(l, m)hl(1)(kr)Xl,m(θ, ϕ)+ikβE(l, m)×[hl(1)(kr)Xl,m(θ, ϕ)],
βMs(l, m)=Bl(kspa)exp(-γd)αMs(l, m)
βEs(l, m)=Cl(kspa)exp(-γd)αEs(l, m),
Bl(kspa)=jl(ka)[kspajl(kspa)]-nsp2jl(kspa)[kajl(ka)]nsp2jl(kspa)[kahl(1)(ka)]-hl(1)(ka)[kspajl(kspa)],
Cl(kspa)=jl(ka)[kspajl(kspa)]-jl(kspa)[kajl(ka)]jl(kspa)[kahl(1)(ka)]-hl(1)(ka)[kspajl(kspa)],
βMp(l, m)=Bl(kspa)exp(-γd)αMp(l, m)
βEp(l, m)=Cl(kspa)exp(-γd)αEp(l, m).
αM0(l, m)=ilπ(2l+1)(δm,1+δm,-1),
αE0(l, m)=il-1π(2l+1)(δm,1-δm,-1),
Xlm(θ0, ϕ0)=μ=-llDmμl*(α, β, γ)Xlμ(θ,ϕ),
Dmμl(α, β, γ)=exp(-imα)dmμl(β)exp(-iμγ).
E(r)=l=1m=-llμ=-llilπ(2l+1)Dmμl*×(δm,1+δm,-1)jl(kr)Xlμ(θ,ϕ)+1k(δm,1-δm,-1)×[jl(kr)Xlμ(θ,ϕ)].
αM(l, μ)=il-1π(2l+1)[D1,μl*(α, β, γ)-D-1,μl*(α, β, γ)],
αE(l, μ)=ilπ(2l+1)[D1,μl*(α, β, γ)+D-1,μl*(α, β, γ)].
Esc(r)=l=1μ=-llβμ(l, μ)hl(1)(kr)Xlμ(θ, ϕ)+ikβE(l, μ)×[hl(1)(kr)Xlμ(θ,ϕ)].
d±1,μl(β)=(-1)μ+1(l-μ)!l(l+1)(l+μ)!1/2×cos β12Plμ+1(β)-μ sin βPlμ(β)+cos β±12(l-μ+1)(l+μ)Plμ-1(β).
Plμ(-β)=(-1)μPlμ(β).
Ms(α=-π/2, β=θk=i(n2 sin2 I-1)1/2, γ=0),
Mp(α=π, β=θk=i(n2 sin2 I-1)1/2, γ=0).
Ms[α=-π/2, β=-θk=-(1-n2 sin2 I)1/2, γ=0],
Mp[α=π, β=-θk=-(1-n2 sin2 I)1/2, γ=0].
Ms(-π/2, β=-θk, 0),
Mp(π, β=-θk, 0),
exp(ikr)=exp(ikr cos ξ)=l=0il(2l+1)jl(kr)Pl(cos ξ),
Pl(cos ξ)=m=-llBm(θk, ϕk)Plm(cos θ)cos[m(ϕ-ϕk)],
Blm(θk, ϕk)=02πdϕ0πPl(cos ξ)Plm(cos θ)×cos[m(ϕ-ϕk)]sin θdθ.
Pl(cos ξ)=Pl(cos θ)Pl(cos θk)+2m=1l (l-m)!(l+m)!×Plm(cos θ)Plm(cos θk)cos[m(ϕ-ϕk)].
Yl,m(θ, ϕ)=(2l+1)(l-m)!4π(l+m)!1/2Plm(θ)exp(imϕ).
[Plm(θ)]*=(-1)mPlm(θ),
Yl,m*(θ, ϕ)=Yl,m(θ, ϕ),
Plm(θ)=(-1)m2ll! sinm θdd(cos θ)l+m(cos2 θ-1)l.
Pl-m(θ)=(-1)m (l-m)!(l+m)!Plm(θ).
(l+1)Pl+1(θ)=(2l+1)cos θPl-lPl-1.
Pl+1m(θ)=Pl-1m(θ)-(2l+1)sin θPlm-1(θ)

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