Abstract

We present a geometrical optics model for the calculation of the internal fields in microspheres with nonlinear optical properties, which occur, for example, in stimulated Raman scattering. Our approach is based on a ray interpretation of the nonlinear wave equations describing the fields of the Stokes modes in a microdroplet. The Stokes modes are assumed to be whispering gallery modes. For such rays the nonlinear wave equations are used to describe the growth of the field along the path of the ray and to calculate the internal fields belonging to various Stokes modes. The interaction between different-order Stokes fields is considered, by using an iteration over the different orders. Although our approach can be extended to stimulated Brillouin scattering as well, in this paper only stimulated Raman scattering is considered.

© 2006 Optical Society of America

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  1. M. Golombok and D. B. Pye, "Droplet sizing in fuel injections by stimulated Raman scattering," Opt. Lett. 15, 872-874 (1990).
    [CrossRef] [PubMed]
  2. M. Golombok and D. B. Pye, "Droplet evaporation measured by nonlinear Raman method," J. Phys. D 23, 1103-1108 (1990).
    [CrossRef]
  3. W. P. Acker, A. Serpengüzel, R. K. Chang, and S. C. Hill, "Stimulated Raman scattering of fuel droplets: chemical concentration and size determination," Appl. Phys. B 51, 9-16 (1990).
    [CrossRef]
  4. G. Roll and G. Schweiger, "Geometrical optics model of Mie resonances," J. Opt. Soc. Am. A 17, 1301-1311 (2000).
    [CrossRef]
  5. J. Schulte and G. Schweiger, "Resonant inelastic scattering by use of geometrical optics," J. Opt. Soc. Am. A 20, 317-324 (2003).
    [CrossRef]
  6. R. W. Hellwarth, "Theory of stimulated Raman scattering," Phys. Rev. 130, 1850-1852 (1963).
    [CrossRef]
  7. H. J. Zeiger, P. E. Tannenwald, S. Kern, and R. Herendeen, "Two-step Raman scattering in nitrobenzene," Phys. Rev. Lett. 11, 419-422 (1963).
    [CrossRef]
  8. R. W. Hellwarth, "Analysis of stimulated Raman scattering of a giant laser pulse," Appl. Opt. 2, 847-853 (1963).
    [CrossRef]
  9. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1818-1939 (1962).
    [CrossRef]
  10. Y. R. Shen and N. Bloembergen, "Theory of stimulated Brillouin and Raman scattering," Phys. Rev. 137, 1787-1805 (1965).
    [CrossRef]
  11. A. Serpengüzel, G. Chen, and R. K. Chang, "Heuristic model for the growth and coupling of nonlinear processes in droplets," J. Opt. Soc. Am. B 9, 871-883 (1992).
    [CrossRef]
  12. A. Penzkofer, A. Lauberau, and W. Kaiser, "High intensity Raman interactions," Prog. Quantum Electron. 6, 55-140 (1979).
    [CrossRef]
  13. S. C. Hill and R. E. Brenner, "Morphology-dependent resonances," in Optical Effects Associated with Small Particles, P.W.Barber and R.K.Chang, eds. (World Scientific, 1988).

2003 (1)

2000 (1)

1992 (1)

1990 (3)

M. Golombok and D. B. Pye, "Droplet sizing in fuel injections by stimulated Raman scattering," Opt. Lett. 15, 872-874 (1990).
[CrossRef] [PubMed]

M. Golombok and D. B. Pye, "Droplet evaporation measured by nonlinear Raman method," J. Phys. D 23, 1103-1108 (1990).
[CrossRef]

W. P. Acker, A. Serpengüzel, R. K. Chang, and S. C. Hill, "Stimulated Raman scattering of fuel droplets: chemical concentration and size determination," Appl. Phys. B 51, 9-16 (1990).
[CrossRef]

1979 (1)

A. Penzkofer, A. Lauberau, and W. Kaiser, "High intensity Raman interactions," Prog. Quantum Electron. 6, 55-140 (1979).
[CrossRef]

1965 (1)

Y. R. Shen and N. Bloembergen, "Theory of stimulated Brillouin and Raman scattering," Phys. Rev. 137, 1787-1805 (1965).
[CrossRef]

1963 (3)

R. W. Hellwarth, "Analysis of stimulated Raman scattering of a giant laser pulse," Appl. Opt. 2, 847-853 (1963).
[CrossRef]

R. W. Hellwarth, "Theory of stimulated Raman scattering," Phys. Rev. 130, 1850-1852 (1963).
[CrossRef]

H. J. Zeiger, P. E. Tannenwald, S. Kern, and R. Herendeen, "Two-step Raman scattering in nitrobenzene," Phys. Rev. Lett. 11, 419-422 (1963).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1818-1939 (1962).
[CrossRef]

Acker, W. P.

W. P. Acker, A. Serpengüzel, R. K. Chang, and S. C. Hill, "Stimulated Raman scattering of fuel droplets: chemical concentration and size determination," Appl. Phys. B 51, 9-16 (1990).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1818-1939 (1962).
[CrossRef]

Bloembergen, N.

Y. R. Shen and N. Bloembergen, "Theory of stimulated Brillouin and Raman scattering," Phys. Rev. 137, 1787-1805 (1965).
[CrossRef]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1818-1939 (1962).
[CrossRef]

Brenner, R. E.

S. C. Hill and R. E. Brenner, "Morphology-dependent resonances," in Optical Effects Associated with Small Particles, P.W.Barber and R.K.Chang, eds. (World Scientific, 1988).

Chang, R. K.

A. Serpengüzel, G. Chen, and R. K. Chang, "Heuristic model for the growth and coupling of nonlinear processes in droplets," J. Opt. Soc. Am. B 9, 871-883 (1992).
[CrossRef]

W. P. Acker, A. Serpengüzel, R. K. Chang, and S. C. Hill, "Stimulated Raman scattering of fuel droplets: chemical concentration and size determination," Appl. Phys. B 51, 9-16 (1990).
[CrossRef]

Chen, G.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1818-1939 (1962).
[CrossRef]

Golombok, M.

M. Golombok and D. B. Pye, "Droplet evaporation measured by nonlinear Raman method," J. Phys. D 23, 1103-1108 (1990).
[CrossRef]

M. Golombok and D. B. Pye, "Droplet sizing in fuel injections by stimulated Raman scattering," Opt. Lett. 15, 872-874 (1990).
[CrossRef] [PubMed]

Hellwarth, R. W.

R. W. Hellwarth, "Analysis of stimulated Raman scattering of a giant laser pulse," Appl. Opt. 2, 847-853 (1963).
[CrossRef]

R. W. Hellwarth, "Theory of stimulated Raman scattering," Phys. Rev. 130, 1850-1852 (1963).
[CrossRef]

Herendeen, R.

H. J. Zeiger, P. E. Tannenwald, S. Kern, and R. Herendeen, "Two-step Raman scattering in nitrobenzene," Phys. Rev. Lett. 11, 419-422 (1963).
[CrossRef]

Hill, S. C.

W. P. Acker, A. Serpengüzel, R. K. Chang, and S. C. Hill, "Stimulated Raman scattering of fuel droplets: chemical concentration and size determination," Appl. Phys. B 51, 9-16 (1990).
[CrossRef]

S. C. Hill and R. E. Brenner, "Morphology-dependent resonances," in Optical Effects Associated with Small Particles, P.W.Barber and R.K.Chang, eds. (World Scientific, 1988).

Kaiser, W.

A. Penzkofer, A. Lauberau, and W. Kaiser, "High intensity Raman interactions," Prog. Quantum Electron. 6, 55-140 (1979).
[CrossRef]

Kern, S.

H. J. Zeiger, P. E. Tannenwald, S. Kern, and R. Herendeen, "Two-step Raman scattering in nitrobenzene," Phys. Rev. Lett. 11, 419-422 (1963).
[CrossRef]

Lauberau, A.

A. Penzkofer, A. Lauberau, and W. Kaiser, "High intensity Raman interactions," Prog. Quantum Electron. 6, 55-140 (1979).
[CrossRef]

Penzkofer, A.

A. Penzkofer, A. Lauberau, and W. Kaiser, "High intensity Raman interactions," Prog. Quantum Electron. 6, 55-140 (1979).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1818-1939 (1962).
[CrossRef]

Pye, D. B.

M. Golombok and D. B. Pye, "Droplet evaporation measured by nonlinear Raman method," J. Phys. D 23, 1103-1108 (1990).
[CrossRef]

M. Golombok and D. B. Pye, "Droplet sizing in fuel injections by stimulated Raman scattering," Opt. Lett. 15, 872-874 (1990).
[CrossRef] [PubMed]

Roll, G.

Schulte, J.

Schweiger, G.

Serpengüzel, A.

A. Serpengüzel, G. Chen, and R. K. Chang, "Heuristic model for the growth and coupling of nonlinear processes in droplets," J. Opt. Soc. Am. B 9, 871-883 (1992).
[CrossRef]

W. P. Acker, A. Serpengüzel, R. K. Chang, and S. C. Hill, "Stimulated Raman scattering of fuel droplets: chemical concentration and size determination," Appl. Phys. B 51, 9-16 (1990).
[CrossRef]

Shen, Y. R.

Y. R. Shen and N. Bloembergen, "Theory of stimulated Brillouin and Raman scattering," Phys. Rev. 137, 1787-1805 (1965).
[CrossRef]

Tannenwald, P. E.

H. J. Zeiger, P. E. Tannenwald, S. Kern, and R. Herendeen, "Two-step Raman scattering in nitrobenzene," Phys. Rev. Lett. 11, 419-422 (1963).
[CrossRef]

Zeiger, H. J.

H. J. Zeiger, P. E. Tannenwald, S. Kern, and R. Herendeen, "Two-step Raman scattering in nitrobenzene," Phys. Rev. Lett. 11, 419-422 (1963).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

W. P. Acker, A. Serpengüzel, R. K. Chang, and S. C. Hill, "Stimulated Raman scattering of fuel droplets: chemical concentration and size determination," Appl. Phys. B 51, 9-16 (1990).
[CrossRef]

J. Opt. Soc. Am. A (2)

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

J. Phys. D (1)

M. Golombok and D. B. Pye, "Droplet evaporation measured by nonlinear Raman method," J. Phys. D 23, 1103-1108 (1990).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (3)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1818-1939 (1962).
[CrossRef]

Y. R. Shen and N. Bloembergen, "Theory of stimulated Brillouin and Raman scattering," Phys. Rev. 137, 1787-1805 (1965).
[CrossRef]

R. W. Hellwarth, "Theory of stimulated Raman scattering," Phys. Rev. 130, 1850-1852 (1963).
[CrossRef]

Phys. Rev. Lett. (1)

H. J. Zeiger, P. E. Tannenwald, S. Kern, and R. Herendeen, "Two-step Raman scattering in nitrobenzene," Phys. Rev. Lett. 11, 419-422 (1963).
[CrossRef]

Prog. Quantum Electron. (1)

A. Penzkofer, A. Lauberau, and W. Kaiser, "High intensity Raman interactions," Prog. Quantum Electron. 6, 55-140 (1979).
[CrossRef]

Other (1)

S. C. Hill and R. E. Brenner, "Morphology-dependent resonances," in Optical Effects Associated with Small Particles, P.W.Barber and R.K.Chang, eds. (World Scientific, 1988).

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Figures (11)

Fig. 1
Fig. 1

Relative error between linear increasing and geometrical-optics-based resonance positions.

Fig. 2
Fig. 2

Energy-density distribution in a particle with refractive index n = 1.33 and size parameter α = 40.41588225 .

Fig. 3
Fig. 3

E 2 d f in plane x = 0 for a particle with refractive index n = 1.33 in case of TE 1 58 resonance (size parameter α = 40.41588225 ) for first- and second-order Stokes fields.

Fig. 4
Fig. 4

Energy-density distribution of first-order Stokes field in a particle with refractive index n = 1.33 and size parameter α = 40.41588225 .

Fig. 5
Fig. 5

Energy-density distribution of second-order Stokes field in a particle with refractive index n = 1.33 and size parameter α = 40.41588225 .

Fig. 6
Fig. 6

E 2 d f in plane x = 0 for a particle with refractive index n = 1.33 in case of TE 1 58 resonance (size parameter α = 48.27186001 ) for first- and second-order Stokes fields.

Fig. 7
Fig. 7

E 2 d f in plane x = 0 for a particle with refractive index n = 1.33 and in case of TE 1 64 resonance (size parameter α = 52.96514596 ) for first- and second-order Stokes fields.

Fig. 8
Fig. 8

Energy-density distribution of first-order Stokes field in a particle with refractive index n = 1.33 and size parameter α = 48.27186001 .

Fig. 9
Fig. 9

Energy-density distribution of second-order Stokes field in a particle with refractive index n = 1.33 and size parameter α = 48.27186001 .

Fig. 10
Fig. 10

Energy-density distribution of first-order Stokes field in a particle with refractive index n = 1.33 and size parameter α = 52.96514596 .

Fig. 11
Fig. 11

Energy-density distribution of second-order Stokes field in a particle with refractive index n = 1.33 and size parameter α = 52.96514596 .

Equations (9)

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n 2 k 0 [ 2 ( a 2 r ci 2 ) 1 2 r ci 2 arccos ( r ci a ) ] + δ B π 2 = ( ν 1 ) 2 π .
r ci = l + 1 2 n 2 k 0 ;
d E 1 S d z = [ g S 2 ( I in I 2 S ) α 1 S 2 L 1 S 2 ] E 1 S g 2 S 2 E in E 1 S * E 2 S × exp ( i Δ k 2 S z ) j = 3 j final g S 2 E in E ( j 1 ) S * E j S exp ( i Δ k j S z ) ,
d E 2 S d z = [ g S 2 ν 2 S ν 1 S ( I 1 S I 3 S ) α 2 S 2 L 2 S 2 ] E 2 S + g 2 S 2 ν 2 S ν 1 S E in E 1 S * E 1 S × exp ( i Δ k 2 S z ) j = 3 j final g S 2 ν 2 S ν 1 S E in E ( j 2 ) S * E j S exp ( i Δ k j S z ) .
Δ k 2 S = k 2 S + k input 2 k 1 S ,
Δ k j S = k j S + k input k 1 S k ( j 1 ) S ,
Δ k j S = k j S + k input k 2 S k ( j 2 ) S .
E 1 S , k l m = { [ g S 2 ( I in , k l m I 2 S , k l m ) α 1 S 2 L 1 S 2 ] E 1 S , k l m g S 2 E in , k l m E 1 S , k l m * E 2 S , k l m exp ( i Δ k 2 S Δ s ) j = 3 j max g S 2 E in , k l m E ( j 1 ) S , k l m * E j S , k l m exp ( i Δ k j S Δ s ) } Δ s + E 1 S , k l m ,
E 2 S , k l m = { [ g S 2 ν 2 S ν 1 S ( I 1 S , k l I 3 S , k l m ) α 2 S 2 L 2 S 2 ] E 2 S , k l m + g S 2 ν 2 S ν 1 S E in , k l m E 1 S , k l m * E 1 S , k l m exp ( i Δ k 2 S Δ s ) j = 3 j max g S 2 ν 2 S ν 1 S E in , k l m E ( j 2 ) S , k l m * E j S , k l m exp ( i Δ k j S Δ s ) } Δ s + E 2 S , k l m .

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