Abstract

The scalar theory of the self-focusing of an optical beam is not valid for a very narrow beam, and a correct description of the beam behavior requires a vector analysis in this case. A vector nonparaxial theory is developed from the vector Maxwell equations by application of an order-of-magnitude analysis method. For the same input beam, the numerical results of self-focusing from both scalar and vector theories are compared. It is found by the vector theory that a linearly polarized circular input beam becomes elliptical in the self-focusing process.

© 1995 Optical Society of America

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References

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  1. R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
    [CrossRef]
  2. P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
    [CrossRef]
  3. W. G. Wagner, H. A. Haus, J. H. Marburger, Phys. Rev. 175, 256 (1968).
    [CrossRef]
  4. For a comprehensive review of experimental work on self-focusing seeY. R. Shen, Prog. Quantum Electron. 4, 1 (1975).
    [CrossRef]
  5. For a comprehensive review of theoretical work on self-focusing seeJ. H. Marburger, Prog. Quantum Electron. 4, 35 (1975); see also N. C. Kothari, S. C. Abbi, Prog. Theor. Phys. 83, 414 (1990).
    [CrossRef]
  6. M. D. Feit, J. A. Fleck, J. Opt. Soc. Am. B 5, 633 (1988).
    [CrossRef]
  7. N. Akhmediev, A. Ankiewicz, J. M. Soto-Crespo, Opt. Lett. 18, 411 (1993).
    [CrossRef] [PubMed]
  8. J. M. Soto-Crespo, N. Akhmediev, Opt. Commun. 101, 223 (1993).
    [CrossRef]
  9. D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982), pp. 9–11.
  10. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 1–5, 25–29.
  11. R. W. Boyd, Nonlinear Optics (Academic, New York, 1992), Chaps. 4.2 and 4.3.
  12. G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), Chap. 2.

1993 (2)

1992 (1)

R. W. Boyd, Nonlinear Optics (Academic, New York, 1992), Chaps. 4.2 and 4.3.

1988 (1)

1975 (2)

For a comprehensive review of experimental work on self-focusing seeY. R. Shen, Prog. Quantum Electron. 4, 1 (1975).
[CrossRef]

For a comprehensive review of theoretical work on self-focusing seeJ. H. Marburger, Prog. Quantum Electron. 4, 35 (1975); see also N. C. Kothari, S. C. Abbi, Prog. Theor. Phys. 83, 414 (1990).
[CrossRef]

1968 (1)

W. G. Wagner, H. A. Haus, J. H. Marburger, Phys. Rev. 175, 256 (1968).
[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]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), Chap. 2.

Akhmediev, N.

Ankiewicz, A.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, New York, 1992), Chaps. 4.2 and 4.3.

Chiao, R. Y.

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

Feit, M. D.

Fleck, J. A.

Garmire, E.

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

Haus, H. A.

W. G. Wagner, H. A. Haus, J. H. Marburger, Phys. Rev. 175, 256 (1968).
[CrossRef]

Kelley, P. L.

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

Marburger, J. H.

For a comprehensive review of theoretical work on self-focusing seeJ. H. Marburger, Prog. Quantum Electron. 4, 35 (1975); see also N. C. Kothari, S. C. Abbi, Prog. Theor. Phys. 83, 414 (1990).
[CrossRef]

W. G. Wagner, H. A. Haus, J. H. Marburger, Phys. Rev. 175, 256 (1968).
[CrossRef]

Marcuse, D.

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982), pp. 9–11.

Shen, Y. R.

For a comprehensive review of experimental work on self-focusing seeY. R. Shen, Prog. Quantum Electron. 4, 1 (1975).
[CrossRef]

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 1–5, 25–29.

Soto-Crespo, J. M.

Townes, C. H.

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

Wagner, W. G.

W. G. Wagner, H. A. Haus, J. H. Marburger, Phys. Rev. 175, 256 (1968).
[CrossRef]

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

Nonlinear Optics (1)

R. W. Boyd, Nonlinear Optics (Academic, New York, 1992), Chaps. 4.2 and 4.3.

Opt. Commun. (1)

J. M. Soto-Crespo, N. Akhmediev, Opt. Commun. 101, 223 (1993).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

W. G. Wagner, H. A. Haus, J. H. Marburger, Phys. Rev. 175, 256 (1968).
[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]

Prog. Quantum Electron. (2)

For a comprehensive review of experimental work on self-focusing seeY. R. Shen, Prog. Quantum Electron. 4, 1 (1975).
[CrossRef]

For a comprehensive review of theoretical work on self-focusing seeJ. H. Marburger, Prog. Quantum Electron. 4, 35 (1975); see also N. C. Kothari, S. C. Abbi, Prog. Theor. Phys. 83, 414 (1990).
[CrossRef]

Other (3)

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, New York, 1982), pp. 9–11.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 1–5, 25–29.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), Chap. 2.

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Figures (2)

Fig. 1
Fig. 1

On-axis normalized intensity versus normalized longitudinal distance ζ for the initial condition: curve a, by our vector model; curve b, by Feit and Fleck’s method; curve c, by Akhmediev and co-workers’ method; curve d, by the paraxial wave equation.

Fig. 2
Fig. 2

Transverse normalized intensity distribution at ζ = 0 (dashed–dotted curve), along the x axis at ζ = ζmax (solid curve), and along the y axis at ζ = ζmax (dashed curve). The input intensity is magnified for a comparison of the beam widths.

Equations (17)

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× E ( r ) = i ω μ 0 H ( r ) ,             × H ( r ) = - i ω D ( r ) ,
D ( r ) = 0
2 E + ω 2 n 0 2 c 2 E + 1 n 0 2 ɛ 0 ( P NL ) + ω 2 c 2 ɛ 0 P NL = 0.
T 2 E x + 2 z 2 E x + ω 2 c 2 n 2 E x = 0 ,
i A z + 1 2 k 0 T 2 A + γ A 2 A = 0 ,
T A x A ( or y A ) A w ,             T 2 A A w 2 ,
z A A k 0 w 2 = σ w A ,             n 2 n 0 A 2 1 k 0 2 w 2 = σ 2 ,
z A / k 0 A ~ σ w A / k 0 A = σ 2 .
E ( x , y , z ) = A ( x , y , z ) exp ( i k 0 z ) ,
T 2 A + 2 z 2 A + 2 i k 0 z A + [ 1 n 0 2 ɛ 0 ( P NL ) + ω 2 c 2 ɛ 0 P NL ] exp ( - i k 0 z ) = 0.
( T A T + i k 0 A z + z A z ) exp ( i k 0 z ) + 1 n 0 2 ɛ 0 P NL = 0.
A z = i k 0 T A T [ 1 + O ( σ 2 ) ] ,
( P NL ) x = 2 ɛ 0 n 0 n 2 exp ( i k 0 z ) ( A x 2 A x + 2 3 A z 2 A x + 1 3 A z 2 A x * ) [ 1 + O ( σ 4 ) ] .
i z A x + 1 2 k 0 t 2 A x + γ A x 2 A x = - 1 2 k 0 2 z 2 A x - γ k 0 2 [ 2 x 2 ( A x 2 A x ) + 2 3 | x A x | 2 A x - 1 3 ( x A x ) 2 A x * ] .
A x ( x , y , z ) = σ n 0 / n 2 u ( ξ , η , ζ ) , ( x , y , z ) = ( w ξ , w η , 1 ζ ) ,
i ζ u + 1 2 ξ , η 2 u + u 2 u = - σ 2 { 1 2 2 ζ 2 u + 2 ξ 2 × [ u 2 u + 2 3 | ξ u | 2 u - 1 3 ( ξ u ) 2 u * ] } ,
ζ u = i 1 2 ξ , η 2 u - i σ 2 8 ξ , η 4 u + i u 2 u - i σ 2 N h u + O ( σ 4 ) ,

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