Abstract

By considering active molecules or ions as a collection of induced oscillating dipoles, we treat the problem of Raman or fluorescent scattering by the active molecules or ions embedded in a single-mode optical fiber theoretically. The analytical expressions and the numerical results for the scattering coefficients and the even–odd mode conversion coefficients of the guided modes are given, based on the method of dyadic Green's functions and on the expansions of the modal fields in terms of the vector cylindrical wave functions. We expect to incorporate the treatments into the analysis of a rare-earth-doped fiber amplifier or a fiber Raman amplifier.

© 1993 Optical Society of America

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References

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  1. 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]
  2. M. Kerker, P. J. McNulty, M. Sculley, H. Chew, D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: numerical results for incoherent optical process,” J. Opt. Soc. Am. 68, 1676–1685 (1978).
    [CrossRef]
  3. M. Kerker, S. D. Druger, “Raman and fluorescent scattering by molecules embedded in spheres and radii up to several multiples of the wavelength,” Appl. Opt. 18, 1172–1179 (1979).
    [CrossRef] [PubMed]
  4. M. Kerker, D.-S. Wang, H. Chew, “Surface enhanced Raman scattering (SERS) by molecules adsorbed at spherical particles,” Appl. Opt. 19, 3373–3388 (1980).
    [CrossRef] [PubMed]
  5. 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]
  6. H. Chew, D. D. Cooke, M. Kerker, “Raman and fluorescent scattering by molecules embedded in dielectric cylinders,” Appl. Opt. 19, 44–52 (1980).
    [CrossRef] [PubMed]
  7. M. Kerker, “Resonances in electromagnetic scattering by objects with negative absorption,” Appl. Opt. 18, 1180–1189 (1979).
    [CrossRef] [PubMed]
  8. D.-S. Wang, M. Kerker, H. Chew, “Raman and fluorescent scattering by molecules embedded in dielectric spheroids,” Appl. Opt. 19, 2315–2328 (1980).
    [CrossRef] [PubMed]
  9. G. Schweiger, “Raman scattering on microparticles: size dependences,” J. Opt. Soc. Am. B 8, 1770–1778 (1991).
    [CrossRef]
  10. Z. L. Wang, W. G. Lin, “Inelastic scattering of Gaussian beams by active molecules embedded in a sphere,” Microwave Opt. Technol. Lett. 1, 179–182 (1988).
    [CrossRef]
  11. Z. L. Wang, W. G. Lin, “Inelastic multiple scattering by active molecules embedded in randomly distributed spherical scatterers,” Microwave Opt. Technol. Lett. 2, 23–26 (1989).
    [CrossRef]
  12. C. D. Cantrell, “Theory of nonlinear optics in dielectric sphere. II. Coupled-partial-wave theory of resonant, resonantly pumped stimulated Brillouin scattering,” J. Opt. Soc. Am. B 8, 2158–2180 (1991).
    [CrossRef]
  13. C. D. Cantrell, “Theory of nonlinear optics in dielectric spheres. III. Partial-wave-index dependence of the gain for stimulated Brillouin scattering,” J. Opt. Soc. Am. B 8, 2181– 2189 (1991).
    [CrossRef]
  14. R. H. Stolen, “Active fiber,” in New Directions in Guided Wave and Coherent Optics, D. B. Ostrowsky, E. Spitz, eds. (Nijhoff, The Hague, 1984), pp. 1–22.
  15. Fiber Laser Sources and Amplifiers, M. J. Digonnet, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1171 (1990).
  16. Fiber Laser Sources and Amplifiers II, M. J. Digonnet, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1373 (1991).
  17. J. R. Armitage, “Three-level fiber laser amplifier: a theoretical model,” Appl. Opt. 27, 4831–4836 (1988).
    [CrossRef] [PubMed]
  18. B. Foley, M. L. Dakes, R. W. Davies, P. Melman, “Gain saturation in fiber Raman amplifiers due to stimulated Brillouin scattering,” J. Lightwave Technol. 7, 2024–2032 (1989).
    [CrossRef]
  19. M. S. Kao, J. Wu, “Signal light amplification by stimulated Raman scattering in an N channel WDM optical fiber communication system,” J. Lightwave Technol. 7, 1290–1299 (1989).
    [CrossRef]
  20. E. Desurvire, J. L. Zyskind, C. R. Giles, “Design optimization for efficient erbium-doped fiber amplifiers,” J. Lightwave Technol. 8, 1730–1741 (1990).
    [CrossRef]
  21. M. Ohasi, “Design considerations for an Er3+-doped fiber amplifier,” J. Lightwave Technol. 9, 1099–1104 (1991).
    [CrossRef]
  22. B. Pederson, A. Bjarklev, J. H. Povlsen, K. Dybdal, C. C. Larsen, “The design of erbium doped fiber amplifiers,” J. Lightwave Technol. 9, 1105–1112 (1991).
    [CrossRef]
  23. M. Montecchi, A. Mecozzi, M. Settembre, M. Tamburrini, L. DiGaaspare, “Gain and noise in rare-earth-doped optical fiber,” J. Opt. Soc. Am. B 8, 134–141 (1991).
    [CrossRef]
  24. N. K. Uzunoglu, “Scattering from inhomogeneties inside a fiber waveguide,” J. Opt. Soc. Am. 71, 259–273 (1981).
    [CrossRef]
  25. C. T. Tai, Dyadic Green's Functions and Electromagnetic Theory (International Textbook, Scranton, Pa., 1971).

1991 (6)

1990 (1)

E. Desurvire, J. L. Zyskind, C. R. Giles, “Design optimization for efficient erbium-doped fiber amplifiers,” J. Lightwave Technol. 8, 1730–1741 (1990).
[CrossRef]

1989 (3)

Z. L. Wang, W. G. Lin, “Inelastic multiple scattering by active molecules embedded in randomly distributed spherical scatterers,” Microwave Opt. Technol. Lett. 2, 23–26 (1989).
[CrossRef]

B. Foley, M. L. Dakes, R. W. Davies, P. Melman, “Gain saturation in fiber Raman amplifiers due to stimulated Brillouin scattering,” J. Lightwave Technol. 7, 2024–2032 (1989).
[CrossRef]

M. S. Kao, J. Wu, “Signal light amplification by stimulated Raman scattering in an N channel WDM optical fiber communication system,” J. Lightwave Technol. 7, 1290–1299 (1989).
[CrossRef]

1988 (2)

J. R. Armitage, “Three-level fiber laser amplifier: a theoretical model,” Appl. Opt. 27, 4831–4836 (1988).
[CrossRef] [PubMed]

Z. L. Wang, W. G. Lin, “Inelastic scattering of Gaussian beams by active molecules embedded in a sphere,” Microwave Opt. Technol. Lett. 1, 179–182 (1988).
[CrossRef]

1981 (1)

1980 (3)

1979 (2)

1978 (1)

1976 (2)

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]

Armitage, J. R.

Bjarklev, A.

B. Pederson, A. Bjarklev, J. H. Povlsen, K. Dybdal, C. C. Larsen, “The design of erbium doped fiber amplifiers,” J. Lightwave Technol. 9, 1105–1112 (1991).
[CrossRef]

Cantrell, C. D.

Chew, H.

Cooke, D. D.

Dakes, M. L.

B. Foley, M. L. Dakes, R. W. Davies, P. Melman, “Gain saturation in fiber Raman amplifiers due to stimulated Brillouin scattering,” J. Lightwave Technol. 7, 2024–2032 (1989).
[CrossRef]

Davies, R. W.

B. Foley, M. L. Dakes, R. W. Davies, P. Melman, “Gain saturation in fiber Raman amplifiers due to stimulated Brillouin scattering,” J. Lightwave Technol. 7, 2024–2032 (1989).
[CrossRef]

Desurvire, E.

E. Desurvire, J. L. Zyskind, C. R. Giles, “Design optimization for efficient erbium-doped fiber amplifiers,” J. Lightwave Technol. 8, 1730–1741 (1990).
[CrossRef]

DiGaaspare, L.

Druger, S. D.

Dybdal, K.

B. Pederson, A. Bjarklev, J. H. Povlsen, K. Dybdal, C. C. Larsen, “The design of erbium doped fiber amplifiers,” J. Lightwave Technol. 9, 1105–1112 (1991).
[CrossRef]

Foley, B.

B. Foley, M. L. Dakes, R. W. Davies, P. Melman, “Gain saturation in fiber Raman amplifiers due to stimulated Brillouin scattering,” J. Lightwave Technol. 7, 2024–2032 (1989).
[CrossRef]

Giles, C. R.

E. Desurvire, J. L. Zyskind, C. R. Giles, “Design optimization for efficient erbium-doped fiber amplifiers,” J. Lightwave Technol. 8, 1730–1741 (1990).
[CrossRef]

Kao, M. S.

M. S. Kao, J. Wu, “Signal light amplification by stimulated Raman scattering in an N channel WDM optical fiber communication system,” J. Lightwave Technol. 7, 1290–1299 (1989).
[CrossRef]

Kerker, M.

Larsen, C. C.

B. Pederson, A. Bjarklev, J. H. Povlsen, K. Dybdal, C. C. Larsen, “The design of erbium doped fiber amplifiers,” J. Lightwave Technol. 9, 1105–1112 (1991).
[CrossRef]

Lin, W. G.

Z. L. Wang, W. G. Lin, “Inelastic multiple scattering by active molecules embedded in randomly distributed spherical scatterers,” Microwave Opt. Technol. Lett. 2, 23–26 (1989).
[CrossRef]

Z. L. Wang, W. G. Lin, “Inelastic scattering of Gaussian beams by active molecules embedded in a sphere,” Microwave Opt. Technol. Lett. 1, 179–182 (1988).
[CrossRef]

McNulty, P. J.

Mecozzi, A.

Melman, P.

B. Foley, M. L. Dakes, R. W. Davies, P. Melman, “Gain saturation in fiber Raman amplifiers due to stimulated Brillouin scattering,” J. Lightwave Technol. 7, 2024–2032 (1989).
[CrossRef]

Montecchi, M.

Ohasi, M.

M. Ohasi, “Design considerations for an Er3+-doped fiber amplifier,” J. Lightwave Technol. 9, 1099–1104 (1991).
[CrossRef]

Pederson, B.

B. Pederson, A. Bjarklev, J. H. Povlsen, K. Dybdal, C. C. Larsen, “The design of erbium doped fiber amplifiers,” J. Lightwave Technol. 9, 1105–1112 (1991).
[CrossRef]

Povlsen, J. H.

B. Pederson, A. Bjarklev, J. H. Povlsen, K. Dybdal, C. C. Larsen, “The design of erbium doped fiber amplifiers,” J. Lightwave Technol. 9, 1105–1112 (1991).
[CrossRef]

Schweiger, G.

Sculley, M.

Settembre, M.

Stolen, R. H.

R. H. Stolen, “Active fiber,” in New Directions in Guided Wave and Coherent Optics, D. B. Ostrowsky, E. Spitz, eds. (Nijhoff, The Hague, 1984), pp. 1–22.

Tai, C. T.

C. T. Tai, Dyadic Green's Functions and Electromagnetic Theory (International Textbook, Scranton, Pa., 1971).

Tamburrini, M.

Uzunoglu, N. K.

Wang, D.-S.

Wang, Z. L.

Z. L. Wang, W. G. Lin, “Inelastic multiple scattering by active molecules embedded in randomly distributed spherical scatterers,” Microwave Opt. Technol. Lett. 2, 23–26 (1989).
[CrossRef]

Z. L. Wang, W. G. Lin, “Inelastic scattering of Gaussian beams by active molecules embedded in a sphere,” Microwave Opt. Technol. Lett. 1, 179–182 (1988).
[CrossRef]

Wu, J.

M. S. Kao, J. Wu, “Signal light amplification by stimulated Raman scattering in an N channel WDM optical fiber communication system,” J. Lightwave Technol. 7, 1290–1299 (1989).
[CrossRef]

Zyskind, J. L.

E. Desurvire, J. L. Zyskind, C. R. Giles, “Design optimization for efficient erbium-doped fiber amplifiers,” J. Lightwave Technol. 8, 1730–1741 (1990).
[CrossRef]

Appl. Opt. (6)

J. Lightwave Technol. (5)

B. Foley, M. L. Dakes, R. W. Davies, P. Melman, “Gain saturation in fiber Raman amplifiers due to stimulated Brillouin scattering,” J. Lightwave Technol. 7, 2024–2032 (1989).
[CrossRef]

M. S. Kao, J. Wu, “Signal light amplification by stimulated Raman scattering in an N channel WDM optical fiber communication system,” J. Lightwave Technol. 7, 1290–1299 (1989).
[CrossRef]

E. Desurvire, J. L. Zyskind, C. R. Giles, “Design optimization for efficient erbium-doped fiber amplifiers,” J. Lightwave Technol. 8, 1730–1741 (1990).
[CrossRef]

M. Ohasi, “Design considerations for an Er3+-doped fiber amplifier,” J. Lightwave Technol. 9, 1099–1104 (1991).
[CrossRef]

B. Pederson, A. Bjarklev, J. H. Povlsen, K. Dybdal, C. C. Larsen, “The design of erbium doped fiber amplifiers,” J. Lightwave Technol. 9, 1105–1112 (1991).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. B (4)

Microwave Opt. Technol. Lett. (2)

Z. L. Wang, W. G. Lin, “Inelastic scattering of Gaussian beams by active molecules embedded in a sphere,” Microwave Opt. Technol. Lett. 1, 179–182 (1988).
[CrossRef]

Z. L. Wang, W. G. Lin, “Inelastic multiple scattering by active molecules embedded in randomly distributed spherical scatterers,” Microwave Opt. Technol. Lett. 2, 23–26 (1989).
[CrossRef]

Phys. Rev. A (1)

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]

Other (4)

R. H. Stolen, “Active fiber,” in New Directions in Guided Wave and Coherent Optics, D. B. Ostrowsky, E. Spitz, eds. (Nijhoff, The Hague, 1984), pp. 1–22.

Fiber Laser Sources and Amplifiers, M. J. Digonnet, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1171 (1990).

Fiber Laser Sources and Amplifiers II, M. J. Digonnet, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1373 (1991).

C. T. Tai, Dyadic Green's Functions and Electromagnetic Theory (International Textbook, Scranton, Pa., 1971).

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Figures (6)

Fig. 1
Fig. 1

Scattering geometry. The fiber axis passes through the origin and lies along the z axis. The active molecules are located at a point r′ inside the fiber, which has a radius of a.

Fig. 2
Fig. 2

Forward [Ttot and T+(+)] and backward [R+(+)] inelastic-scattering coefficients versus ρ/a for V = 2.25, λ10 = 1.1132; ρ is the radial coordinate of the molecules, Ttot denotes the coefficients as the molecules are uniformly embedded in the whole core of a certain length.

Fig. 3
Fig. 3

Foward [T+(+)] and backward [R+(+)] inelastic-scattering coefficients versus ρ/a for V = 2.25, λ10 = 1.1132.

Fig. 4
Fig. 4

Foward [T] inelastic-scattering coefficients versus ρ/a for V = 2.25, 1, Δ = 0.002; 2, Δ = 0.05.

Fig. 5
Fig. 5

Forward [Ttot and T+(−)] and backward [Rtot and R+(−)] inelastic-scattering coefficients versus V for ρ/a = 0.8, λ10 = 1.1132, Δ = 0.002.

Fig. 6
Fig. 6

Forward [Ttot and T+(−)] and backward [Rtot and R+(−)] inelastic-scattering coefficients versus Δ for V = 2.25, λ10 = 1.1132, ρ/a = 1.0.

Equations (12)

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E ie ( r ) = 4 π k 1 2 G ( r , r ) · P ( r ) ,
E e ± ( r ) = M ± 1 , β ( 1 ) ( r , k 1 ) ± δ ( β ) N ± 1 , β ( 1 ) ( r , k 1 ) .
E ie T + ( ± ) [ M + 1 , β ( 1 ) ( r , k 1 ) + δ ( β ) N + 1 , β ( 1 ) ( r , k 1 ) ] + T ( ± ) [ M 1 , β ( 1 ) ( r , k 1 ) + δ ( β ) N 1 , β ( 1 ) ( r , k 1 ) ] = T + ( ± ) E ie + T ( ± ) E ie ,
T υ ( u ) = ( i k 1 2 α / 2 η 1 2 ) Res [ a ( 1 , β ) ] × [ M υ , β ( 1 ) * ( r , k 1 ) + υ δ ( β ) N υ , β ( 1 ) * ( r , k 1 ) ] × [ M u , β ( 1 ) ( r , k 1 ) + u δ ( β ) N u , β ( 1 ) ( r , k 1 ) ] ,
T ( e e ) = [ Y 1 ( η 1 , η 1 ) + X 1 ( η 1 , η 1 ) cos ( 2 φ ) ] ,
T ( o o ) = [ Y 1 ( η 1 , η 1 ) X 1 ( η 1 , η 1 ) cos ( 2 φ ) ] ,
T ( e o ) = T ( o e ) = X 1 ( η 1 , η 1 ) sin ( 2 φ ) ] ,
X 1 ( η 1 , η 1 ) = ϴ { Q 1 + ( η 1 , η 1 ) + [ β δ ( β ) / k 1 β δ ( β ) / k 1 ] × Q 1 ( η 1 , η 1 ) + δ ( β ) δ ( β ) / k 1 k 1 [ η 1 2 η 1 2 × J 1 ( η 1 ρ ) J 1 ( η 1 ρ ) β β Q 1 + ( η 1 , η 1 ) ] } ,
Y ( η 1 , η 1 ) = ϴ { R 1 + ( η 1 , η 1 ) + [ β δ ( β ) / k 1 + β δ ( β ) / k 1 ] × R 1 ( η 1 , η 1 ) + δ ( β ) δ ( β ) / k 1 k 1 × [ η 1 2 η 1 2 J 1 ( η 1 ρ ) J 1 ( η 1 ρ ) + β β R 1 + ( η 1 , η 1 ) ] } ,
ϴ = ( i k 1 2 η 1 / 4 η 1 ) Res [ a ( 1 , β ) ] exp [ i ( β β ) z ] ,
Q 1 ± ( η 1 , η 1 ) = [ J 0 ( η 1 ρ ) J 2 ( η 1 ρ ) ± J 0 ( η 1 ρ ) J 2 ( η 1 ρ ) ] ,
R 1 ± ( η 1 , η 1 ) = [ J 0 ( η 1 ρ ) J 0 ( η 1 ρ ) ± J 2 ( η 1 ρ ) J 2 ( η 1 ρ ) ] .

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