Abstract

A dipole model is used to simulate incoherent Raman and fluorescent scattering by microspheres. The use of the addition theorem for spherical harmonics circumvents the need to evaluate double sums in the final formulas, thereby drastically reducing computational effort. Special attention is paid to consideration of backscattering geometry, which is important for lidar applications. The formulas derived for backscattering geometry decrease the computation time for size parameter x ∼ 100 by a factor of 200 compared with the time for calculations performed at other angles.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Schweiger, “Raman scattering on single aerosol particles and on flowing aerosols: a review,” J. Aerosol. Sci. 21, 483–509 (1990).
    [CrossRef]
  2. R. Vehring, “Linear Raman spectroscopy on aqueous aerosols: influence of nonlinear effects on detection limits,” J. Aerosol. Sci. 29, 65–79 (1998).
    [CrossRef]
  3. H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
    [CrossRef]
  4. H. Chew, M. Kerker, P. J. McNulty, “Raman and fluorescent scattering by molecules embedded in concentric spheres,” J. Opt. Soc. Am. 66, 440–444 (1976).
    [CrossRef]
  5. H. Chew, M. Sculley, M. Kerker, P. J. McNulty, D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: results for coherent optical processes,” J. Opt. Soc. Am. 68, 1686–1689 (1978).
    [CrossRef]
  6. M. Kerker, P. J. McNulty, M. Sculley, H. Chew, D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: results for incoherent optical processes,” J. Opt. Soc. Am. 68, 1676–1686 (1979).
    [CrossRef]
  7. H. Chew, “Total fluorescent scattering cross section,” Phys. Rev. A 37, 4107–4110 (1988).
    [CrossRef] [PubMed]
  8. M. Kerker, S. D. Druger, “Raman and fluorescent scattering by molecules embedded in spheres with radii up to several multiples of the wavelength,” Appl. Opt. 18, 1172–1179 (1979).
    [CrossRef] [PubMed]
  9. S. D. Druger, P. J. McNulty, “Radiation patterns of fluorescence from molecules embedded in small particles: general case,” Appl. Opt. 22, 75–82 (1983).
    [CrossRef] [PubMed]
  10. S. Lange, G. Schweiger, “Structural resonances in the total Raman- and fluorescence-scattering cross section: concentration-profile dependence,” J. Opt. Soc. Am. B 13, 1864–1872 (1996).
    [CrossRef]
  11. J. D. Pendelton, S. C. Hill, “Collection of emission from an oscillating dipole inside a sphere: analytical integration over a circular aperture,” Appl. Opt. 36, 8729–8737 (1997).
    [CrossRef]
  12. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  13. H. C. van der Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  14. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  15. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems and Formulas for Reference and Review (McGraw-Hill, New York, 1961).
  16. I. Veselovskii, V. Griaznov, A. Kolgotin, D. N. Whiteman, “Angle- and size-dependent characteristics of incoherent Raman and fluorescent scattering by microspheres. 2. Numerical simulation,” Appl. Opt. 41, 5783–5791 (2002).
    [CrossRef] [PubMed]

2002

1998

R. Vehring, “Linear Raman spectroscopy on aqueous aerosols: influence of nonlinear effects on detection limits,” J. Aerosol. Sci. 29, 65–79 (1998).
[CrossRef]

1997

1996

1990

G. Schweiger, “Raman scattering on single aerosol particles and on flowing aerosols: a review,” J. Aerosol. Sci. 21, 483–509 (1990).
[CrossRef]

1988

H. Chew, “Total fluorescent scattering cross section,” Phys. Rev. A 37, 4107–4110 (1988).
[CrossRef] [PubMed]

1983

1979

1978

1976

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

H. Chew, M. Kerker, P. J. McNulty, “Raman and fluorescent scattering by molecules embedded in concentric spheres,” J. Opt. Soc. Am. 66, 440–444 (1976).
[CrossRef]

Bohren, C. F.

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

Chew, H.

Cooke, D. D.

Druger, S. D.

Griaznov, V.

Hill, S. C.

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).

Kerker, M.

Kolgotin, A.

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems and Formulas for Reference and Review (McGraw-Hill, New York, 1961).

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems and Formulas for Reference and Review (McGraw-Hill, New York, 1961).

Lange, S.

McNulty, P. J.

Pendelton, J. D.

Schweiger, G.

Sculley, M.

van der Hulst, H. C.

H. C. van der Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Vehring, R.

R. Vehring, “Linear Raman spectroscopy on aqueous aerosols: influence of nonlinear effects on detection limits,” J. Aerosol. Sci. 29, 65–79 (1998).
[CrossRef]

Veselovskii, I.

Whiteman, D. N.

Appl. Opt.

J. Aerosol. Sci.

G. Schweiger, “Raman scattering on single aerosol particles and on flowing aerosols: a review,” J. Aerosol. Sci. 21, 483–509 (1990).
[CrossRef]

R. Vehring, “Linear Raman spectroscopy on aqueous aerosols: influence of nonlinear effects on detection limits,” J. Aerosol. Sci. 29, 65–79 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Phys. Rev. A

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

H. Chew, “Total fluorescent scattering cross section,” Phys. Rev. A 37, 4107–4110 (1988).
[CrossRef] [PubMed]

Other

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

H. C. van der Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems and Formulas for Reference and Review (McGraw-Hill, New York, 1961).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (56)

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

Eincr=lmicn22ω0 αEl, m×jlk02rXlmθ, φ+αMl, mjlk02rXlmθ, φ,
Bincr=lmαEl, m×jlk02rXlmθ, φ-icω0 αMl, mjlk02rXlmθ, φ.
EincOX=E0ex expik02z, EincOY=E0ey expik02z.
αEOXl, m=il4π2l+11/2δm,l-δm,-12i n2E0, αMOXl, m=il4π2l+11/2δm,1+δm,-12 E0, αEOYl, m=-il4π2l+11/2×δm,1+δm,-12 n2E0, αMOYl, m=il4π2l+11/2δm,1-δm,-12i E0.
Etrar=lmicn12ω0 γEl, m×jlk01rXlmθ, φ+γMl, mjlk01rXlmθ, φ,
γEl, m=gElαEl, m,gEl=iμ1M2μ2MψlMxξlx-μ1ξlxψlMx, γMl, m=gMlαMl, mgMl=iμ1Mμ1ψlMxξlx-μ2MξlxψlMx,
E21=expik2rk2rl,m-il+1cM1l, mXlmθ, φ-1n2 cE1l, mer×Xlmθ, φ.
cE1l, m=dElaE1l, m, dEl=iμ2Mμ2MψlpMxξlpx-μ1ξlpxψlpMx, cM1l, m=dMlaM1l, m, dMl=iμ2μ1ψlpMxξlpx-μ2MξlpxψlpMx.
aE1l, m=4πk2Mn2μ1α1Etra×jlpk01rXlm*θ, φ, aM1l, m=4πik3Mn2μ1α1jlpk01rEtra · Xlm*θ, φ
Xlmθ, φ=ill+11/21sin θYlmθ, φφeθ-Ylmθ, φθeφ,
E2θ1=expik2rk2rl,m-il+1ill+11/2×cM1l, m1sin θYlmφ-cE1l, mn2Ylmθ,
E2φ1=-expik2rk2rlm-il+1ill+11/2×cE1l, mn21sin θYlmφ+cM1l, mYlmθ.
E2θ1=4πk3Mn2μ1α1expik2rk2rl,mi-il+1ll+11/2×idMl1sin θ jlpk01rYlmθ, φφE1r· Xlm*θ, φ-dElkn2Ylmθ, φθE1r· ×jlpk01rXlm*θ, φ,
E2φ1=-4πk3Mn2μ1α1expik2rk2rl,mi-il+1ll+11/2×dElkn21sin θYlmθ, φφE1r· ×jlpk01rXlm*θ, φ+idMljlpk01rYlmθ, φθE1r· Xlm*θ, φ.
dσ1dΩ=|E21|2E02 r2=|E2θ1|2+|E2φ1|2E02 r2.
dσOXdΩ=v N dσ1,OXdΩ d3r=v N |E2θ1,OX|2E02 r2d3r+v N |E2φ1,OX|2E02 r2d3r=dσ2θOXdΩ+dσ2φOXdΩ, dσOYdΩ=v N dσ1,OYdΩd3r=v N |E2θ1,OY|2E02 r2d3r+v N |E2φ1,OY|2E02 r2d3r=dσ2θOYdΩ+dσ2φOYdΩ,
dσ2θOXdΩHHIinc, dσ2φOXdΩVHIinc, dσ2θOYdΩHvIinc,dσ2φOYdΩVVIinc,
EtraOXr=E0erArθ, δcos φ+eθAθθ, δcos φ+eφAφθ, δsin φ,
EtraOYr=E0erArθ, δsin φ+eθAθθ, δsin φ-eφAφθ, δcos φ,
Arθ, δ=l=1 il2l+1iM gElψlMxδMxδ2×ρl1θsin θ,
Aθθ, δ=l=1 il2l+1ll+1×iM gElψlMxδτl1θ-gMlψlMxδρl1θ1Mxδ,
Aφθ, δ=l=1 il2l+1ll+1×gMlψlMxδτl1θ-iM gElψlMxδρl1θ1Mxδ,
ρl1θ=Pl1θsin θ, τl1θ=dPl1θdθ, δ=ra,
E2θ1OX=E0expik2rk2r k3Mn2μ1α1×l=1-il+12l+1ll+1iAθθdMl×ψlpMxδpMxδ Sφφ cos φ-iAφθdMlψlpMxδpMxδ Sθφ sin φ-ll+1MArθdElψlpMxδpMxδ2 Sθ cos φ-MAφθdElψlpMxδpMxδ×Sφθ sin φ-MAθθdEl×ψlpMxδpMxδ Sθθ cos φ,
E2φ1OX=E0expik2rk2r k3Mn2μ1α1l=1×-il+12l+1ll+1-iAθθdMl×ψlpMxδpMxδ Sφθ cos φ+iAφθdMlψlpMxδpMxδ Sθθ sin φ-ll+1MArθdElψlpMxδpMxδ2Sφ cos φ-MAφθdElψlpMxδpMxδSφφ sin φ-MAθθdElψlpMxδpMxδSθφ cos φ,
E2θ1OY=E0expik2rk2r k3Mn2μ1α1×l=1-il+12l+1ll+1iAθθdMl×ψlpMxδpMxδ Sφφ sin φ+iAφθdMlψlpMxδpMxδ Sθφ cos φ-ll+1MArθdElψlpMxδpMxδ2 Sθ sin φ+MAφθdElψlpMxδpMxδ Sφθ cos φ-MAθθdElψlpMxδpMxδ Sθθ sin φ,
E2φ1OY=E0expik2rk2r k3Mn2μ1α1l=1×-il+12l+1ll+1-iAθθdMl×ψlpMxδpMxδ Sφθ sin φ-iAφθdMlψlpMxδpMxδ Sθθ cos φ-ll+1MArθdElψlpMxδpMxδ2 Sφ sin φ+MAφθdElψlpMxδpMxδ Sφφ cos φ-MAθθdElψlpMxδpMxδ Sθφ sin φ.
δOXθ, x=VHθ, xHHθ, x,
δOYθ, x=HVθ, xVVθ, x,
dσdΩaerosoldσdΩbulk,
dσdΩbulk=43 πa3Nα12k4μ1μ2M.
ψlρρ2=1/3l=10l>1, ψlρρ=2/3l=10l>1, ψlρρ=0 any l.
Ar=3M2+2sin θ, Aθ=3M2+2cos θ, Aφ=-3M2+2.
E2θ1OX=9k3μ1α1E0 expik2rM2+22k2rcos θ, E2φ1OX=0, E2θ1OY=0, E2φ1OY=9k3μ1α1E0 expik2rM2+22k2r.
dσθOXdΩ=216π2Nα12p4x3μ12n26M2+24λ0cos2 θ, dσφOYdΩ=216π2Nα12p4x3μ12n26M2+24λ0,
msin θ Plmθ180°=dPlmθdθ180°=0 |m|1,
msin θ Plmθ180°=dPlmθdθ180°=-1l12 m=-1,
msin θ Plmθ180°=-1lll+12, dPlmθdθ180°=--1lll+12 m=1.
02πdφ dσθ1OXdΩ=π4 |k2Mμ1α1|23|B1|2+|B3|2+|B4|2-2 ImB1B3*-2 ImB1B4*+2 ReB3B4*+3|B2|2+|B5|2-2 ImB2B5*+-2 ReB1B2*+2 ImB1B5*+2 ImB2B3*+2 ImB2B4*-2 ReB4B5*-2 ReB3B5*,
02πdφ dσφ1OXdΩ=π4 |k2Mμ1α1|2|B1|2+|B2|2+|B3|2+|B4|2+|B5|2+2 ReB1B2*-2 ImB1B3*-2 ImB1B4*-2 ImB1B5*-2 ImB2B3*-2 ImB2B4*-2 ImB2B5*+2 ReB3B4*+2 ReB3B5*+2 ReB4B5*,
B1δ, θ=Aθδ, θl=1 il2l+1ll+1 dMl×ψlpMxδpMxδ ρl1θ, B2δ, θ=Aφδ, θl=1 il2l+1ll+1 dMl×ψlpMxδpMxδ τl1θ, B3δ, θ=MArδ, θl=1 il2l+1ll+1 ll+1dElψlpMxδpMxδ2 ρl1θsin θ, B4δ, θ=MAθδ, θl=1 il2l+1ll+1 dEl×ψlpMxδpMxδ τl1θ, B5δ, θ=MAφδ, θl=1 il2l+1ll+1 dEl×ψlpMxδpMxδ ρl1θ.
Xlm*θ, φ=-ill+11/2×1sin θYlm*θ, φφeθ-Ylm*θ, φθeφ,
×jlpk01rXlmθ, φ*=-ill+11/2×jlpk01rrYlm*θ, φer+1rdrjlpk01rdrer×Xlm*θ, φ=-ill+11/2jlpk01rr Ylm*θ, φer+1rdrjlpk01rdr-ill+11/2 ×1sin θYlm*θ, φφeφ+1rdrjlpk01rdr×-ill+11/2Ylm*θ, φθeθ.
E2θ1=4πk3Mn2μ1α1expik2rk2rl,m-il+1ill+11/2×dMlll+11/21sin θ sin θ×jlpk01rYlm*θ, φφYlmθ, φφ×E1r · eθ-dMlll+11/21sin θ jlpk01r×Ylm*θ, φθYlmθ, φφE1r · eφ+idElkn2ll+11/2jlpk01rr Ylm*θ, φ×Ylmθ, φφE1r · er+idElkn2ll+11/2×1sin θ1rdr0jlpk01rdrYlm*θ, φφ×Ylmθ, φθE1r · eφ+idElkn2ll+11/21rdrjlpk01rdr×Ylm*θ, φθYlmθ, φθE1r · eθ,
E2φ1=-4πk3Mn2μ1α1expik2rk2r×l,m-il+1ill+11/2dMlll+11/21sin θ×jlpk01rYlm*θ, φφYlmθ, φθ×E1r · eθ-dMlll+11/2 jlpk01rYlm*θ, φθ×Ylmθ, φθE1r · eφ-idElkn2ll+11/21sin θjlpk01rr×Ylm*θ, φYlmθ, φφ×E1r · er-idElkn2ll+11/21sin θ sin θ1r ×drjlpk01rdrYlm*θ, φφYlmθ, φφ×E1r · eφ-idElkn2ll+11/21sin θ1r×drjlpk01rdrYlm*θ, φθ ×Ylmθ, φφE1r · eθ.
Plcos γ=4π2l+1m=-ll Ylm*θ, φYlmθ, φ=4π2l+1m=-ll Ylm*θ, φYlmθ, φ,
Sφφ=1sin θ sin θ2Plcos γφφ =4π2l+11sin θ sin θ×m=-llYlm*θ, φφYlmθ, φφ, Sθφ=1sin θ2Plcos γθφ=4π2l+11sin θm=-llYlm*θ, φφYlmθ, φφ, Sθ=Plcos γθ=4π2l+1m=-ll Ylm*θ, φYlmθ, φθ, Sφθ=1sin θ2Plcos γφθ=4π2l+11sin θm=-llYlm*θφφYlmθ, φθ, Sθθ=2Plcos γθθ=4π2l+1m=-llYlm*θ, φθYlmθ, φθ, Sφ=1sin θPlcos γφ=4π2l+11sin θm=-ll Ylm*θ, φYlmθ, φφ.
E2θ1=k3Mn2μ1α1expik2rk2rl=1-il+12l+1ll+1×idMljlpK1rSφφE1r · eθ-idMljlpK1rSθφE1r · eφ-dElkn2 ll+1jlpK1rr SθE1r · er-dElkn21rdrjlpK1rdr SφθE1r · eφ-dElkn21rdrjlpK1rdr SθθE1r · eθ, E2φ1=-k3Mn2μ1α1expik2rk2rl=1-il+12l+1ll+1×idMljlpK1rSφθE1r · eθ-idMljlpK1rSθθE1r · eφ+dElkn2 ll+1jlpK1rr SφE1r · er+dElkn21rdrjlpK1rdr SφφE1r · eφ+dElkn21rdrjlpK1rdr SθφE1r · eθ.
Sφφ=-d2Pldx2sin θ sin θ sin2 φ+dPldxcos φ0, Sθφ=d2Pldx2sin θ sin φ-sin θ cos θ+cos θ sin θ cos φ+dPldxcos θ sin φ, Sφθ=d2Pldx2sin θ sin φcos θ sin θ-sin θ cos θ cos φ-dPldxcos θ sin φ, Sθθ=d2Plx2sin θ cos θ-cos θ sin θ cos φ×cos θ sin θ-sin θ cos θ cos φ+dPlxsin θ sin θ+cos θ cos θ cos φ, Sθ=dPldx-cos θ sin θ+sin θ cos θ cos φ, Sφ=dPldxsin θ sin φ,
2l+1xPl-l+1Pl+1-lPl-1=0,
dPldx=12l-1Pl-1+2l-1x dPl-1dx-l-1dPl-2dx/l, d2Pldx2=l22l-1dPl-1dx+2l-1x d2Pl-1dx2-l-1d2Pl-2dx2/l.
E2θ=expik2rk2rl=1-il+1-1li2l+11/224π×icMl, 1+cMl, -1+cEl, 1-cEl, -1n2, E2φ=expik2rk2rl=1-il+1-1li2l+11/224π×cMl, 1-cMl, -1-i cEl, 1+cEl, -1n2.
Xl,1*+Xl,-1*=22l+11/2ll+14πeφdPl1θdθsin φ-eθPl1θsin θcos φ, Xl,1*-Xl,-1*=2i2l+11/2ll+14πeφdPl1θdθcos φ+eθPl1θsin θsin φ, ×jlpk01rXl,1*+Xl,-1*=-22l+11/2ll+14π×erll+1jlpk01rr Pl1θsin φ+1rdrjlpk01rdreθdPl1θdθsin φ+eφPl1θsin θcos φ, ×jlpk01rXl,1*-Xl,-1*=-2i2l+11/2ll+14π×erll+1jlpk01rr Pl1θcos φ+1rdrjlpk01rdreθdPl1θdθcos φ-eφPl1θsin θsin φ.
E2θ1OX=-E2φ1OY=E0expik2rr k2Mμ1α1×cos2 φ l=1 il2l+1ll+1×dMlAθjlpk01rPl1θsin θ-sin2 φ l=1 il2l+1ll+1 dMlAφjlpk01r×dPl1θdθ-cos2 φ l=1 il2l+1idElArjlpk01rk2r Pl1θ-cos2 φ l=1 il2l+1ill+1 dElAθ×1k2rdrjlpk01rdrdPl1θdθ+sin2 φ l=1 il2l+1ill+1 dElAφ×1k2rdrjlpk01rdrPl1θsin θ,
E2φ1OX=-E2θ1OY =E0expik2rr k2Mμ1α1-sin φcos φ×l=1 il2l+1ll+1 kdMlAθjlpk01r×Pl1θsin θ-sin φ cos φ l=1 il2l+1ll+1×kdMlAφjlpk01rdPl1θdθ+sin φ cos φ l=1 il2l+1idElArjlpk01rk2r Pl1θ+sin φ cos φ l=1 il2l+1ill+1×dElAθ1k2rdrjlpk01rdrdPl1θdθ+sin φ cos φ l=1 il2l+1ill+1×dElAφ1k2rdrjlpk01rdrPl1θsin θ.
E2θ1OX=-E2φ1OY =E0expik2rr k2Mμ1α1B1r, θcos2 φ-B2r, θsin2 φ-iB3r, θcos2 φ-iB4r, θcos2 φ+iB5r, θsin2 φ, E2φ1OX=-E2θ1OY =E0expik2rr k2Mμ1α1-B1r, θ-B2r, θ+iB3r, θ+iB4r, θ+iB5r, θsin φ cos φ.

Metrics