Abstract

Using a direct variational technique involving super-Gaussian trial functions, we find approximate solutions for the fundamental radial mode of an optical wave propagating in a nonlinear parabolic-index fiber. A detailed examination is made of the amplitude, width, radial profile, and longitudinal phase shift of the stationary solutions that represent a two-dimensional balance among diffraction, nonlinear self-focusing, and linear wave guiding. For pulse powers below the self-focusing power, the stationary solutions are stabilized by the parabolic index variation.

© 1992 Optical Society of America

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References

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  1. S. A. Akhmanov, R. V. Khoklov, A. P. Sukhorov, “Self-focusing, self-defocusing and self-modulation of Laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972);O. Svelto, “Self-focusing, self-trapping and self-phase modulation,” in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974).
    [CrossRef]
  2. R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
    [CrossRef]
  3. P. L. Kelley, Phys. Rev. Lett. 15, 1005, (1965).
    [CrossRef]
  4. H. A. Haus, Appl. Phys. Lett. 8, 128 (1966).
    [CrossRef]
  5. D. Pohl, Opt. Commun. 2, 305 (1970).
    [CrossRef]
  6. K. J. Blow, N. J. Doran, Proc. Inst. Electr. Eng. Part J 134, 138 (1987).
  7. K. J. Blow, N. J. Doran, B. P. Nelson, D. Wood, IEEE J. Quantum Electron. QE-23, 1108 (1987).
    [CrossRef]
  8. J. T. Manassah, P. L. Baldeck, R. R. Alfano, Opt. Lett. 13, 589 (1988).
    [CrossRef]
  9. R. A. Sammut, C. Pask, J. Opt. Soc. Am. B 8, 395 (1991).
    [CrossRef]
  10. L. Gagnon, C. Paré, J. Opt. Soc. Am. A 8, 601 (1991).
    [CrossRef]
  11. J. Herrmann, J. Opt. Soc. Am. B 8, 1507 (1991).
    [CrossRef]
  12. M. Karlsson, D. Anderson, M. Desaix, “Dynamics of self-focusing and self-phase modulation in a parabolic index optical fiber,” Opt. Lett. 17, 22 (1992).
    [CrossRef] [PubMed]
  13. A. W. Snyder, Y. Chen, L. Poladian, D. J. Mitchell, Electron. Lett. 26, 643 (1990).
    [CrossRef]
  14. A. Ankiewicz, G.-D. Peng, Opt. Commun. 84, 71 (1991).
    [CrossRef]
  15. A. J. Glass, IEEE J. Quantum Electron. QE-10, 705 (1974).
    [CrossRef]
  16. D. Anderson, M. Bonnedal, Phys. Fluids 22, 105 (1979).
    [CrossRef]
  17. D. Anderson, Phys. Rev. A 27, 3135 (1983).
    [CrossRef]
  18. Y. Silberberg, Opt. Lett. 15, 1282 (1990).
    [CrossRef] [PubMed]
  19. M. Desaix, D. Anderson, M. Lisak, J. Opt. Soc. Am. B 8, 2082 (1991).
    [CrossRef]

1992 (1)

1991 (5)

1990 (2)

A. W. Snyder, Y. Chen, L. Poladian, D. J. Mitchell, Electron. Lett. 26, 643 (1990).
[CrossRef]

Y. Silberberg, Opt. Lett. 15, 1282 (1990).
[CrossRef] [PubMed]

1988 (1)

1987 (2)

K. J. Blow, N. J. Doran, Proc. Inst. Electr. Eng. Part J 134, 138 (1987).

K. J. Blow, N. J. Doran, B. P. Nelson, D. Wood, IEEE J. Quantum Electron. QE-23, 1108 (1987).
[CrossRef]

1983 (1)

D. Anderson, Phys. Rev. A 27, 3135 (1983).
[CrossRef]

1979 (1)

D. Anderson, M. Bonnedal, Phys. Fluids 22, 105 (1979).
[CrossRef]

1974 (1)

A. J. Glass, IEEE J. Quantum Electron. QE-10, 705 (1974).
[CrossRef]

1970 (1)

D. Pohl, Opt. Commun. 2, 305 (1970).
[CrossRef]

1966 (1)

H. A. Haus, Appl. Phys. Lett. 8, 128 (1966).
[CrossRef]

1965 (1)

P. L. Kelley, Phys. Rev. Lett. 15, 1005, (1965).
[CrossRef]

1964 (1)

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, R. V. Khoklov, A. P. Sukhorov, “Self-focusing, self-defocusing and self-modulation of Laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972);O. Svelto, “Self-focusing, self-trapping and self-phase modulation,” in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974).
[CrossRef]

Alfano, R. R.

Anderson, D.

Ankiewicz, A.

A. Ankiewicz, G.-D. Peng, Opt. Commun. 84, 71 (1991).
[CrossRef]

Baldeck, P. L.

Blow, K. J.

K. J. Blow, N. J. Doran, Proc. Inst. Electr. Eng. Part J 134, 138 (1987).

K. J. Blow, N. J. Doran, B. P. Nelson, D. Wood, IEEE J. Quantum Electron. QE-23, 1108 (1987).
[CrossRef]

Bonnedal, M.

D. Anderson, M. Bonnedal, Phys. Fluids 22, 105 (1979).
[CrossRef]

Chen, Y.

A. W. Snyder, Y. Chen, L. Poladian, D. J. Mitchell, Electron. Lett. 26, 643 (1990).
[CrossRef]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Desaix, M.

Doran, N. J.

K. J. Blow, N. J. Doran, Proc. Inst. Electr. Eng. Part J 134, 138 (1987).

K. J. Blow, N. J. Doran, B. P. Nelson, D. Wood, IEEE J. Quantum Electron. QE-23, 1108 (1987).
[CrossRef]

Gagnon, L.

Garmire, E.

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Glass, A. J.

A. J. Glass, IEEE J. Quantum Electron. QE-10, 705 (1974).
[CrossRef]

Haus, H. A.

H. A. Haus, Appl. Phys. Lett. 8, 128 (1966).
[CrossRef]

Herrmann, J.

Karlsson, M.

Kelley, P. L.

P. L. Kelley, Phys. Rev. Lett. 15, 1005, (1965).
[CrossRef]

Khoklov, R. V.

S. A. Akhmanov, R. V. Khoklov, A. P. Sukhorov, “Self-focusing, self-defocusing and self-modulation of Laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972);O. Svelto, “Self-focusing, self-trapping and self-phase modulation,” in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974).
[CrossRef]

Lisak, M.

Manassah, J. T.

Mitchell, D. J.

A. W. Snyder, Y. Chen, L. Poladian, D. J. Mitchell, Electron. Lett. 26, 643 (1990).
[CrossRef]

Nelson, B. P.

K. J. Blow, N. J. Doran, B. P. Nelson, D. Wood, IEEE J. Quantum Electron. QE-23, 1108 (1987).
[CrossRef]

Paré, C.

Pask, C.

Peng, G.-D.

A. Ankiewicz, G.-D. Peng, Opt. Commun. 84, 71 (1991).
[CrossRef]

Pohl, D.

D. Pohl, Opt. Commun. 2, 305 (1970).
[CrossRef]

Poladian, L.

A. W. Snyder, Y. Chen, L. Poladian, D. J. Mitchell, Electron. Lett. 26, 643 (1990).
[CrossRef]

Sammut, R. A.

Silberberg, Y.

Snyder, A. W.

A. W. Snyder, Y. Chen, L. Poladian, D. J. Mitchell, Electron. Lett. 26, 643 (1990).
[CrossRef]

Sukhorov, A. P.

S. A. Akhmanov, R. V. Khoklov, A. P. Sukhorov, “Self-focusing, self-defocusing and self-modulation of Laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972);O. Svelto, “Self-focusing, self-trapping and self-phase modulation,” in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974).
[CrossRef]

Townes, C. H.

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Wood, D.

K. J. Blow, N. J. Doran, B. P. Nelson, D. Wood, IEEE J. Quantum Electron. QE-23, 1108 (1987).
[CrossRef]

Appl. Phys. Lett. (1)

H. A. Haus, Appl. Phys. Lett. 8, 128 (1966).
[CrossRef]

Electron. Lett. (1)

A. W. Snyder, Y. Chen, L. Poladian, D. J. Mitchell, Electron. Lett. 26, 643 (1990).
[CrossRef]

IEEE J. Quantum Electron. (2)

K. J. Blow, N. J. Doran, B. P. Nelson, D. Wood, IEEE J. Quantum Electron. QE-23, 1108 (1987).
[CrossRef]

A. J. Glass, IEEE J. Quantum Electron. QE-10, 705 (1974).
[CrossRef]

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

J. Opt. Soc. Am. B (3)

Opt. Commun. (2)

A. Ankiewicz, G.-D. Peng, Opt. Commun. 84, 71 (1991).
[CrossRef]

D. Pohl, Opt. Commun. 2, 305 (1970).
[CrossRef]

Opt. Lett. (3)

Phys. Fluids (1)

D. Anderson, M. Bonnedal, Phys. Fluids 22, 105 (1979).
[CrossRef]

Phys. Rev. A (1)

D. Anderson, Phys. Rev. A 27, 3135 (1983).
[CrossRef]

Phys. Rev. Lett. (2)

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

P. L. Kelley, Phys. Rev. Lett. 15, 1005, (1965).
[CrossRef]

Proc. Inst. Electr. Eng. Part J (1)

K. J. Blow, N. J. Doran, Proc. Inst. Electr. Eng. Part J 134, 138 (1987).

Other (1)

S. A. Akhmanov, R. V. Khoklov, A. P. Sukhorov, “Self-focusing, self-defocusing and self-modulation of Laser beams,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972);O. Svelto, “Self-focusing, self-trapping and self-phase modulation,” in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Super-Gaussian shape for several values of m, ranging from m = 0 to m = 2 with step 0.25.

Fig. 2
Fig. 2

Pm relationship given by Eq. (2.10). The curve is dashed for values of m < 1/2 because the solutions in this domain have ∂E(0)/∂r ≠ 0.

Fig. 3
Fig. 3

Normalized beam radius ω versus Ψ02.

Fig. 4
Fig. 4

Longitudinal phase shift δ versus Ψ02

Fig. 5
Fig. 5

Super-Gaussian coefficient m versus Ψ02.

Fig. 6
Fig. 6

Normalized stationary mode profiles for values of m ranging from m = 0.7 to m = 1 with step 0.05.

Fig. 7
Fig. 7

The stationary fundamental modes for Ψ0 = 1 and 3. Dashed lines are variational approximations, and solid lines are numerical calculations.

Fig. 8
Fig. 8

The stationary fundamental modes for Ψ0 = 5 and 10. Dashed lines are variational approximations, and solid lines are numerical calculations.

Tables (2)

Tables Icon

Table 1 Values of the Normalized Beam Parameters for the Fundamental Mode in the Linear and Nonlinear Limits

Tables Icon

Table 2 Comparison between the Variational Super-Gaussian Approximations and Numerical Calculations for the Eigenvalue δ and the Integrated Power I for Different Values of the Normalized Amplitude Ψ0

Equations (24)

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n = n 0 + n 2 | E | 2 g r 2 2 ,
1 r r ( r E r ) 2 i k E z a 2 r 2 E + k | E | 2 E = 0 ,
L = r | E r | 2 i k r ( E E * z E * E z ) + α 2 r 3 | E | 2 r k 2 | E | 4 .
L d r d z = 0 ,
E ( r , z ) = A ( z ) exp { ½ [ r a ( z ) ] 2 m + i r 2 b ( z ) + i ϕ ( z ) } .
L = 0 L d r = A 2 [ m 2 + a 4 ( b 2 k s b z + α 2 4 ) I 2 ] k ϕ z a 2 A 2 I 1 κ a 2 A 4 2 I 3 ,
I 1 ( m ) = Γ ( 1 + 1 m ) , I 2 ( m ) = Γ ( 1 + 2 m ) , I 3 ( m ) = Γ ( 1 + 1 m ) 2 ( 1 + 1 m ) ,
δ L δ = L z L ( z ) ( = A , a , b , ϕ , m ) .
z ( A 2 a 2 ) = 0 ,
a z = a b k ,
2 a z 2 = 2 m a 3 k 2 I 2 ( 1 ρ ) a g ,
ϕ z = m a 2 k I 1 ( 1 3 2 p ) ,
1 + m { m [ ln ( I 2 I 1 3 ) ] p m [ ln ( I 2 I 3 I 1 3 ) ] } = 0 ,
p = m + 2 [ Ψ ( 1 + 1 / m ) Ψ ( 1 + 2 / m ) ] ln ( 2 ) + 2 [ Ψ ( 1 + 1 / m ) Ψ ( 1 + 2 / m ) ] ,
a b k = 0 ,
α 2 a 4 = m I 2 ( 1 p ) ,
β a 2 = 2 m I 1 ( 3 2 p 1 ) ,
Ψ 0 2 κ α A 2 = m p I 3 [ I 2 2 m ( 1 p ) ] 1 / 2 ,
ω 2 α a 2 = [ 2 m ( 1 p ) I 2 ] 1 / 2 ,
δ β α = 2 m I 1 ( 3 2 p 1 ) [ I 2 2 m ( 1 p ) ] 1 / 2 ,
1 ρ d d ρ ( ρ d Ψ d ρ ) 2 i Ψ ζ ρ 2 Ψ + Ψ 3 = 0 ,
1 ρ d d ρ ( r d Ψ d ρ ) ρ 2 Ψ + Ψ 3 = δ Ψ .
I = 0 Ψ ( ρ ) 2 2 π ρ d ρ .
I = π m p ( m ) 2 ( 1 + 1 / m ) .

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