Abstract

An electrodynamic approach is developed to solve the problem of stimulated Raman scattering (SRS) in a transparent dielectric spherical particle. In this approach it is proposed that optical fields for time-dependent amplitudes of coupled waves at Stokes and pump frequencies in a spherical particle be represented as an expansion in terms of eigenfunctions of the stationary scattering problem in which the expansion coefficients determine the temporal behavior of the field and comply with inhomogeneous differential equations. Solutions of these equations for initial phase SRS and under steady-state conditions are analyzed. The SRS threshold is determined, and the threshold for steady-state SRS at a given intensity is found for when there is double resonance between the fields. It is shown that, to excite SRS, one should compensate for the loss of the Stokes wave that is due to absorption and emission through the particle surface. To provide steady-state SRS generation it is necessary to compensate additionally for the energy loss that is due to depletion of pump intensity.

© 2003 Optical Society of America

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References

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  1. G. Schweiger, “Raman scattering on single aerosol particles and on flowing aerosols: a review,” J. Aerosol Sci. 21, 483–509 (1990).
    [CrossRef]
  2. Yu. E. Geints, A. A. Zemlyanov, V. E. Zuev, A. M. Kabanov, and V. A. Pogodaev, Nonlinear Optics of Atmospheric Aerosol (SB RAS, Novosibirsk, Russia, 1999).
  3. Y. R. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, New York, 1984).
  4. A. A. Zemlyanov and Yu. E. Geints, “Nonstationary elastic linear light scattering by spherical microparticles,” Atmos. Ocean. Opt. 15, 619–627 (2002).
  5. J. Schwarz, P. Rambo, L. Giuggioli, and J. C. Daniels, in Nonlinear Guided Waves and Their Applications, Vol. 55 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 285–288.
  6. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  7. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  8. C. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the position, widths, and strengths of reso-nances in the Mie scattering,” J. Opt. Soc. Am. B 9, 1585–1592 (1992).
    [CrossRef]
  9. H.-B. Lin, J. D. Eversole, and A. J. Campillo, “Continuous-wave stimulated Raman scattering in microdroplets,” Opt. Lett. 17, 828–830 (1992).
    [CrossRef] [PubMed]
  10. R. Vehring and G. Schweiger, “Threshold of stimulated Raman scattering in microdroplets,” J. Aerosol Sci. 26, Suppl. 1, 235–236 (1995).
    [CrossRef]
  11. R. G. Pinnick, A. Biswas, J. Pendleton, and R. L. Armstrong, “Aerosol-induced laser breakdown thresholds: effect of resonant particles,” Appl. Opt. 31, 311–317 (1992).
    [CrossRef] [PubMed]
  12. Yu. E. Geints, A. A. Zemlyanov, and E. K. Chistyakova, “Energetic threshold of SRS generation in transparent droplets,” Atmos. Ocean. Opt. 8, 803–807 (1995).
  13. P. Chŷlek, M. A. Jarzembski, V. Srivastava, R. G. Pinnick, J. D. Pendleton, and J. P. Cruncleton, “Effect of spherical particles on laser-induced breakdown of gases,” Appl. Opt. 26, 760–762 (1987).
    [CrossRef]
  14. A. Biswas, H. Latifi, R. L. Armstrong, and R. G. Pinnik, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. B 40, 7413–7416 (1989).
    [CrossRef]

2002 (1)

A. A. Zemlyanov and Yu. E. Geints, “Nonstationary elastic linear light scattering by spherical microparticles,” Atmos. Ocean. Opt. 15, 619–627 (2002).

1995 (2)

R. Vehring and G. Schweiger, “Threshold of stimulated Raman scattering in microdroplets,” J. Aerosol Sci. 26, Suppl. 1, 235–236 (1995).
[CrossRef]

Yu. E. Geints, A. A. Zemlyanov, and E. K. Chistyakova, “Energetic threshold of SRS generation in transparent droplets,” Atmos. Ocean. Opt. 8, 803–807 (1995).

1992 (3)

1990 (1)

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

1989 (1)

A. Biswas, H. Latifi, R. L. Armstrong, and R. G. Pinnik, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. B 40, 7413–7416 (1989).
[CrossRef]

1987 (1)

Armstrong, R. L.

R. G. Pinnick, A. Biswas, J. Pendleton, and R. L. Armstrong, “Aerosol-induced laser breakdown thresholds: effect of resonant particles,” Appl. Opt. 31, 311–317 (1992).
[CrossRef] [PubMed]

A. Biswas, H. Latifi, R. L. Armstrong, and R. G. Pinnik, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. B 40, 7413–7416 (1989).
[CrossRef]

Biswas, A.

R. G. Pinnick, A. Biswas, J. Pendleton, and R. L. Armstrong, “Aerosol-induced laser breakdown thresholds: effect of resonant particles,” Appl. Opt. 31, 311–317 (1992).
[CrossRef] [PubMed]

A. Biswas, H. Latifi, R. L. Armstrong, and R. G. Pinnik, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. B 40, 7413–7416 (1989).
[CrossRef]

Campillo, A. J.

Chistyakova, E. K.

Yu. E. Geints, A. A. Zemlyanov, and E. K. Chistyakova, “Energetic threshold of SRS generation in transparent droplets,” Atmos. Ocean. Opt. 8, 803–807 (1995).

Chylek, P.

Cruncleton, J. P.

Eversole, J. D.

Geints, Yu. E.

A. A. Zemlyanov and Yu. E. Geints, “Nonstationary elastic linear light scattering by spherical microparticles,” Atmos. Ocean. Opt. 15, 619–627 (2002).

Yu. E. Geints, A. A. Zemlyanov, and E. K. Chistyakova, “Energetic threshold of SRS generation in transparent droplets,” Atmos. Ocean. Opt. 8, 803–807 (1995).

Jarzembski, M. A.

Lam, C. C.

Latifi, H.

A. Biswas, H. Latifi, R. L. Armstrong, and R. G. Pinnik, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. B 40, 7413–7416 (1989).
[CrossRef]

Leung, P. T.

Lin, H.-B.

Pendleton, J.

Pendleton, J. D.

Pinnick, R. G.

Pinnik, R. G.

A. Biswas, H. Latifi, R. L. Armstrong, and R. G. Pinnik, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. B 40, 7413–7416 (1989).
[CrossRef]

Schweiger, G.

R. Vehring and G. Schweiger, “Threshold of stimulated Raman scattering in microdroplets,” J. Aerosol Sci. 26, Suppl. 1, 235–236 (1995).
[CrossRef]

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

Srivastava, V.

Vehring, R.

R. Vehring and G. Schweiger, “Threshold of stimulated Raman scattering in microdroplets,” J. Aerosol Sci. 26, Suppl. 1, 235–236 (1995).
[CrossRef]

Young, K.

Zemlyanov, A. A.

A. A. Zemlyanov and Yu. E. Geints, “Nonstationary elastic linear light scattering by spherical microparticles,” Atmos. Ocean. Opt. 15, 619–627 (2002).

Yu. E. Geints, A. A. Zemlyanov, and E. K. Chistyakova, “Energetic threshold of SRS generation in transparent droplets,” Atmos. Ocean. Opt. 8, 803–807 (1995).

Appl. Opt. (2)

Atmos. Ocean. Opt. (2)

Yu. E. Geints, A. A. Zemlyanov, and E. K. Chistyakova, “Energetic threshold of SRS generation in transparent droplets,” Atmos. Ocean. Opt. 8, 803–807 (1995).

A. A. Zemlyanov and Yu. E. Geints, “Nonstationary elastic linear light scattering by spherical microparticles,” Atmos. Ocean. Opt. 15, 619–627 (2002).

J. Aerosol Sci. (2)

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

R. Vehring and G. Schweiger, “Threshold of stimulated Raman scattering in microdroplets,” J. Aerosol Sci. 26, Suppl. 1, 235–236 (1995).
[CrossRef]

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

Opt. Lett. (1)

Phys. Rev. B (1)

A. Biswas, H. Latifi, R. L. Armstrong, and R. G. Pinnik, “Double-resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. B 40, 7413–7416 (1989).
[CrossRef]

Other (5)

Yu. E. Geints, A. A. Zemlyanov, V. E. Zuev, A. M. Kabanov, and V. A. Pogodaev, Nonlinear Optics of Atmospheric Aerosol (SB RAS, Novosibirsk, Russia, 1999).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, New York, 1984).

J. Schwarz, P. Rambo, L. Giuggioli, and J. C. Daniels, in Nonlinear Guided Waves and Their Applications, Vol. 55 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 285–288.

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

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

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Figures (6)

Fig. 1
Fig. 1

Dependence of the ratio ge/gS, on the Q factor of Stokes field resonance modes QS for SRS excitation in water droplets at single- (1) and double- (2) field resonance. Solid curves were plotted according to a least-squares approximation.

Fig. 2
Fig. 2

SRS threshold intensities in water droplets with various size parameters. The values were plotted accordingly to Eq. (27) at η=0(1), η=2(2), and η=9(3). The pump radiation wavelength is assumed to be λL=0.532μm; na=1.33 and index of absorption κa=10-8. The experimental data were taken from Refs. 13 (curve 5), 14 (curve 6), and 9 (curve 7). The threshold intensity of optical breakdown of water droplets14 (curve 4) is shown also.

Fig. 3
Fig. 3

Overlapping coefficient Bc as a function of the relative half-width of resonance modes of a spherical particle Γ¯S at SRS (curve 1), and double-resonance SRS (curve 2). The upper scale shows the Q factor of the Stokes field resonance modes.

Fig. 4
Fig. 4

Radial distribution of field inhomogeneity factor BL=[EL(r)EL*(r)]/E02 (averaged over spherical angles) inside a water particle. Curves 1 represent nonresonant optical field profile at λL=0.532 μm; curves 2 corespond to resonance field excitation in (a) the TE801 mode and (b) the TE421 modes, respectively, at wavelength λL=0.65 μm.

Fig. 5
Fig. 5

Spatial distribution of function BLS (on the principal cross section of the particle) for (a) nonresonance and (b)–(d) resonance excitation of the Stokes wave in a spherical water droplet with radius a0=5μm and the following MDR combinations: nonresonant field, (a) -TE601; resonant field (b) TE741-TE601, (c) TE702-TE601, and (d) TE702-TE602. The laser radiation is incident from right to left.

Fig. 6
Fig. 6

Dependence of normalized coefficient of overlap fields B¯c on input resonance order p at various values of output resonance order p.

Equations (77)

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××ES(r, t)+ac22ES(r, t)t2+4πσc2ES(r, t)t
=-4πc22PNS(r, t)t2,
××EL(r, t)+ac22EL(r, t)t2+4πσc2EL(r, t)t
=-4πc22PNL(r, t)t2,
(EL)θ,φ=(ELsc)θ,φ+(ELi)θ,φ,
(HL)θ,φ=(HLsc)θ,φ+(HLi)θ,φ,
(ES)θ,φ=(ESsc)θ,φ,(HS)θ,φ=(HSsc)θ,φ,
2E(r, t)=E˜(r, t)exp(jωt)+E˜*(r, t)exp(-jωt),
2PN(r, t)=P˜N(r, t)exp(jωt)+P˜N*(r, t)exp(-jωt),
EL,S(r, t)=M[AML,S(t)EMTE(r)-jBML,S(t)EMTM(r)],
HL,S(r, t)=aM[jAML,S(t)HMTE(r)+BML,S(t)HMTM(r)],
d2dt2 AML,S(t)+2ΓML,Sddt AML,S(t)+ωM2AML,S(t)
=JML,S(t),
JML(t)=FMi(t)-4πaVaEM*2PNLt2dr,
JMS(t)=-4πaVaEM*2PNSt2dr.
FMi(t)
=-jcaSaωM(Ei×HM*)-j t (Hi×EM*)nrds
=E0f(t)KMi.
KMi=jc2Rn21/2akLzMVaψn(kLa0)ψn*(nakMa0)-1naωLωM ψn(kLa0)ψn*(nakMa0),
zM-2=02πdφ0πsin θdθ 0a0r2dr|Mn1(1)|2;
PNL(r, t)=χR(3)(ωL)[ES(r, t)ES*(r, t)]EL(r, t),
PNS(r, t)=χR(3)(ωS)[EL(r, t) EL*(r, t)]×ES(r, t)+PNsp(r, t),
χR(3)(ωS)=-j N0T216MΩRαqk2,
gs=-32π2ωSac2Im[χR(3)],
2qkt2+2Γkqkt+ΩR2qk=fE(r, t)+fsp(r, t),
qk(t)=exp(-Γkt)ΩR0tsinΩoR(t-t)×exp(Γkt)[fE(t)+fsp(t)]dt=qkE(t)+qksp(t),
PNsp(r, t)=N0αqk (qksp)*EL(r, t)+c.c.-N0ΩRαqkE˜L(r, t)exp(jωst-Γkt)×0tsinΩoR(t-t)×exp(Γkt)fsp*(r, t)dt+c.c.,
JMS(t)=-4πχR(3)(ωS)aMd2dt2|AML|2M AMS(t)×Va(EMEM*)(EMEM*)dr+FMsp(t),
FMsp(t)=-4πaVaEM*2PNspt2dr
JMS(t)=-j c2gS8πωSMd2dt2 [|AML(t)|2AMS(t)]×SMM+FMsp(t),
SMM=Va[(EMEM*)(EMEM*)]dr,
2jωS+2ΓMS+2jωSdGMS(t)dt+jωsGMS(t)×dA˜MS(t)dt+ωS2ΔMS2+2jΓMSωS-2ωS2dGMS(t)dt
+iωS GMS(t)A˜MS(t)=F˜Msp(t),
GMS(t)=c2gS8πM|AML(t)|2SMM.
AMS(t)=A˜MS0(t)exp[DMS(t)]exp[jωoMSt+jφMS(t)].
I¯S(t)=1VaVaIS(r, t)dr=cna8πVaM|AMS(t)|2,
I¯S(t)=M I¯Msp(t) exp[2DMS(t)].
I¯Msp(t)=28πN0F0ΓknaΩRαqk20texp[-2DMS(t)]×M|AML(t)|2SMMdt
AML(t)=A˜ML0(t)exp[DML(t)]exp[ωoML(t)+jσML(t)],
ωoML=ωL1-ΔML22,ΔML=ωM-ωLωL,
φML(t)=-1/(2ωL)0t[GML(t)+ΓML]2dt,
DML(t)=½(1-ΔML2)0t[GML(t)+2ΓML]dt,
A˜ML0(t)=1/(2jωL)0texp[DML(t)]F˜Mi(t)dt,
GML(t)=c2gSωL8πωsM|AMS(t)|2SMM.
GMS(t)=(cgS/na)B¯cMI¯L(t),
I¯L(t)=1VaVa IL(r, t)dr=cna8πVaM|AML(t)|2.
B¯cM(ωL, ωM)=cna8πI¯LM|AML(t)|2SMM
B¯cM(ωL, ωS)=VaVa(ELEL*)dr-1Va(EMEM*)×M|AML|2(EMEM*)dr.
B¯c(ωL, ωS)=VaVa(ELEL*)drVa(ESES*)dr-1×Va(ELEL*)(ESES*)dr.
I¯LI0=B¯L=1VaE02VaEL(r)EL*(r)dr.
w0>w0th=naωst/(cgeQSB¯L).
I0>I0th=naωScgeQSB¯L=na2VaωLωSc2geσex(a0, ωL)QLQS,
I¯Lst=2naΓS/(cge).
I¯L(t)=I¯L0(t)exp[-2DL(t)],
I¯Lst=4π|KMin|2I0/[(GstL+2ΓL)2cnaωL2].
I0th=na2Va(1+η)2ωLωSc2geσex(a0, ωL)QLQS.
ηcgeωL2ΓLnaωS I¯sst=QLQSI¯SstI¯Lst.
×ES(r; t)=-1ctHS(r; t),
×HS(r; t)=-actES(r; t)+4πσcES(r; t)+4πctPNS(r; t).
×EM(r)=-j ωMcHM(r),
×HM(r)=j aωMcEM(r),
Va(×ESHM*)dr
=Va(ES×HM*)dr+Sa(ES×HM*)nrds
=-j aωMc AM+Sa(ES×HM*)nrds=-acddt AM,
Va(×HSEM*)dr
=Va(HS×EM*)dr+Sa(HS×EM*)nrds 
=-aωMc AM+Sa(HS×EM*)nrds=acddt AM+4πσc AM+4πcVaEM*tPNSdr.
d2dt2 AM(t)+4πσaddt AM(t)+ωM2AM(t)
=ΠM(t)-4πaVaEM*tPNSdr,
ΠM(t)=-jcaSaωM(ES×HM*)-j t (HS×EM*)ds.
ΠM(t)=2jωM2AM(t)1+j2QMR.
d2dr2 AM(t)+ωMQMaddt AM(t)+ωM1+j2QMR2AM(t)
=-4πaVaEM*2PNSt2dr,
d2dt2 AM(t)+2ΓMddt AM(t)+ωM2AM(t)=JM(t),
JM(t)=-4πaVaEM*2PNSt2dr,
1QM=1QMa+1QMR,
ΠM(t)=2jωM2AM(t)1+j2QMR+FMi(t),

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