Abstract

Beam propagation in multimode graded-index parabolic optical fibers in the presence of group-velocity dispersion and Kerr nonlinearity is theoretically investigated. It is shown that a modulational instability arising from the periodic spatial focusing of the beam takes place regardless of the sign of fiber dispersion, leading to a highly nonlinear space–time dynamics and the generation of ultrashort optical pulses.

© 2003 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Optics, 2nd ed. (Academic, New York, 1995).
  2. G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
    [CrossRef] [PubMed]
  3. S. Wabnitz, Phys. Rev. A 38, 2018 (1988).
    [CrossRef] [PubMed]
  4. M. Haelterman and M. Badolo, Opt. Lett. 20, 2285 (1995).
    [CrossRef]
  5. F. Matera, A. Mecozzi, M. Romagnoli, and M. Settembre, Opt. Lett. 18, 1499 (1993).
    [CrossRef]
  6. F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Lett. A 220, 213 (1996).
    [CrossRef]
  7. S. B. Cavalcanti and M. L. Lyra, Phys. Lett. A 211, 276 (1996).
    [CrossRef]
  8. L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, Phys. Rev. A 46, 4202 (1992).
    [CrossRef] [PubMed]
  9. J. E. Rothenberg, Opt. Lett. 17, 1340 (1992).
    [CrossRef]
  10. J. T. Manassah, P. L. Baldeck, and R. R. Alfano, Opt. Lett. 13, 589 (1988).
    [CrossRef]
  11. M. Karlsson, D. Anderson, and M. Desaix, Opt. Lett. 17, 22 (1992).
    [CrossRef] [PubMed]
  12. J. T. Manassah, Opt. Lett. 18, 1259 (1992).
    [CrossRef]
  13. S. S. Yu, C.-H. Chien, Y. Lai, and J. Wang, Opt. Commun. 119, 167 (1995).
    [CrossRef]
  14. It is remarkable that there exist exact nonlinear periodic solutions to Eq. 1. Indeed, if E(x, y,ξ)= F(x,y)exp(-isξ) is a nonlinear guided mode of the fiber, where the mode profile F(x,y) and corresponding propagation constant s are found as the nonlinear eigenmode and the eigenvalue of the equation (1/2k0)∇2F- (k0g/2)r2F+(n2k0/n0)|F|2F=sF, one can show that an exact periodic solution to Eq. 1 is given by Es(x,y,ξ)=1―a(ξ)F(x―a y―a) ×exp[-is∫ξ0ʹa2(x)-ik0 2ada dxr2, where the scaling function a=a(x) is periodic, with period p/g, given by a(x)= a02 cos 2(gx)+sin 2(gx)1/2, and a0=a(0) is an arbitrary scaling constant. For simplicity, an approximate Gaussian solution given by Eq. 2 is considered here.

1996 (2)

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Lett. A 220, 213 (1996).
[CrossRef]

S. B. Cavalcanti and M. L. Lyra, Phys. Lett. A 211, 276 (1996).
[CrossRef]

1995 (2)

S. S. Yu, C.-H. Chien, Y. Lai, and J. Wang, Opt. Commun. 119, 167 (1995).
[CrossRef]

M. Haelterman and M. Badolo, Opt. Lett. 20, 2285 (1995).
[CrossRef]

1993 (1)

1992 (4)

1988 (2)

1987 (1)

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

Abdullaev, F. Kh.

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Lett. A 220, 213 (1996).
[CrossRef]

Agrawal, G. P.

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, Phys. Rev. A 46, 4202 (1992).
[CrossRef] [PubMed]

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

G. P. Agrawal, Nonlinear Optics, 2nd ed. (Academic, New York, 1995).

Alfano, R. R.

Anderson, D.

Badolo, M.

Baldeck, P. L.

Cao, X. D.

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, Phys. Rev. A 46, 4202 (1992).
[CrossRef] [PubMed]

Cavalcanti, S. B.

S. B. Cavalcanti and M. L. Lyra, Phys. Lett. A 211, 276 (1996).
[CrossRef]

Chien, C.-H.

S. S. Yu, C.-H. Chien, Y. Lai, and J. Wang, Opt. Commun. 119, 167 (1995).
[CrossRef]

Darmanyan, S. A.

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Lett. A 220, 213 (1996).
[CrossRef]

Desaix, M.

Haelterman, M.

Karlsson, M.

Kobyakov, A.

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Lett. A 220, 213 (1996).
[CrossRef]

Lai, Y.

S. S. Yu, C.-H. Chien, Y. Lai, and J. Wang, Opt. Commun. 119, 167 (1995).
[CrossRef]

Lederer, F.

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Lett. A 220, 213 (1996).
[CrossRef]

Liou, L. W.

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, Phys. Rev. A 46, 4202 (1992).
[CrossRef] [PubMed]

Lyra, M. L.

S. B. Cavalcanti and M. L. Lyra, Phys. Lett. A 211, 276 (1996).
[CrossRef]

Manassah, J. T.

Matera, F.

McKinstrie, C. J.

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, Phys. Rev. A 46, 4202 (1992).
[CrossRef] [PubMed]

Mecozzi, A.

Romagnoli, M.

Rothenberg, J. E.

Settembre, M.

Wabnitz, S.

S. Wabnitz, Phys. Rev. A 38, 2018 (1988).
[CrossRef] [PubMed]

Wang, J.

S. S. Yu, C.-H. Chien, Y. Lai, and J. Wang, Opt. Commun. 119, 167 (1995).
[CrossRef]

Yu, S. S.

S. S. Yu, C.-H. Chien, Y. Lai, and J. Wang, Opt. Commun. 119, 167 (1995).
[CrossRef]

Opt. Commun. (1)

S. S. Yu, C.-H. Chien, Y. Lai, and J. Wang, Opt. Commun. 119, 167 (1995).
[CrossRef]

Opt. Lett. (6)

Phys. Lett. A (2)

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, Phys. Lett. A 220, 213 (1996).
[CrossRef]

S. B. Cavalcanti and M. L. Lyra, Phys. Lett. A 211, 276 (1996).
[CrossRef]

Phys. Rev. A (2)

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, Phys. Rev. A 46, 4202 (1992).
[CrossRef] [PubMed]

S. Wabnitz, Phys. Rev. A 38, 2018 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

Other (2)

G. P. Agrawal, Nonlinear Optics, 2nd ed. (Academic, New York, 1995).

It is remarkable that there exist exact nonlinear periodic solutions to Eq. 1. Indeed, if E(x, y,ξ)= F(x,y)exp(-isξ) is a nonlinear guided mode of the fiber, where the mode profile F(x,y) and corresponding propagation constant s are found as the nonlinear eigenmode and the eigenvalue of the equation (1/2k0)∇2F- (k0g/2)r2F+(n2k0/n0)|F|2F=sF, one can show that an exact periodic solution to Eq. 1 is given by Es(x,y,ξ)=1―a(ξ)F(x―a y―a) ×exp[-is∫ξ0ʹa2(x)-ik0 2ada dxr2, where the scaling function a=a(x) is periodic, with period p/g, given by a(x)= a02 cos 2(gx)+sin 2(gx)1/2, and a0=a(0) is an arbitrary scaling constant. For simplicity, an approximate Gaussian solution given by Eq. 2 is considered here.

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Figures (3)

Fig. 1
Fig. 1

(a) Behavior of maximum MI gain versus mode spot size of incident Gaussian beam for the first three low-order MI branches in case of a parabolic-index silica fiber. Parameter values are given in the text. (b) Behavior of the MI gain versus frequency for the first MI branch n=1 and for a Gaussian beam spot size a0=10 µm.

Fig. 2
Fig. 2

(a) Temporal behavior of beam intensity (in arbitrary units) on the fiber axis r=0 versus the propagation distance in a 9.83-cm-long silica fiber, as obtained by numerical integration of Eq. (1). Parameter values are the same as in Fig. 1(b), except for P40 kW p=0.1. A small temporal modulation (1% modulation depth) at frequency Ω/2π=8.13 THz has been added to the incident continuous-wave beam to seed the MI. (b) Field intensity (in arbitrary units) versus time on the fiber axis at the exit of the fiber.

Fig. 3
Fig. 3

Space–time plot of the field intensity (in arbitrary units) at the exit of the fiber for the simulation shown in Fig. 2.

Equations (11)

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iEξ=12k02E-k022Eη2-k0g2r2E+n2k0n0E2E,
Esx,y,ξ=pAca0aξexp-12raξ2-ik02adadξr2+ik01-3n2k02Ac2a02p4n00ξdξa2ξ,
aξ=a0cos2gξ+C sin2gξ1/2,
if1ξ=L+2n2k0n0Es2f1+n2k0n0Es2f2,
if2ξ=-n2k0n0Es*2f1-L+2n2k0n0Es2f2,
idF1dξ=2n2k0n0c1F1+n2k0n0c2F2,
idF2dξ=-2n2k0n0c1F2-n2k0n0c2*F1,
μ=±n2k0pAc2θn2n0×1-2θ0θn+n0k0Ω-Ωn2n2k0pAc2θn21/2.
Ωmax=Ωn2-2n2k0pAc2θ0/n0k01/2,
Esx,y,ξ=1aξFxa,ya×exp-is0ξdξa2ξ-ik02adadξr2,
aξ=a02 cos2gξ+sin2 gξ1/2,

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