Abstract

We present an analysis for self-focusing of an elliptic Gaussian laser beam in a saturable nonlinear medium. It is shown that stationary self-trapped propagation is forbidden in a saturable medium. Though self-trapped propagation does not occur, a virtual threshold power for self-focusing can be defined. Above this threshold power value, but not far from it, the beam focuses. Below this threshold the beam defocuses. It is also shown that the effective beam radius never reaches zero, which is a property of a Gaussian laser beam.

© 1994 Optical Society of America

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References

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  1. S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
    [CrossRef]
  2. M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, Self Focusing of Laser Beams in Dielectrics, Plasmas and Semiconductors (Tata McGraw-Hill, New Delhi, 1974).
  3. M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. XIII, pp. 169–265.
    [CrossRef]
  4. V. I. Kruglov and R. A. Vlasov, Phys. Lett. 111A, 401 (1985).
  5. F. Cornolti, M. Lucchesi, and B. Zambon, Opt. Commun. 75, 129 (1990).
    [CrossRef]
  6. H. Hora, Z. Phys. 226, 156 (1969).
    [CrossRef]
  7. P. K. Kaw, G. Schmidt, and P. Wilcox, Phys. Fluids 16, 1522 (1973).
    [CrossRef]
  8. E. L. Kane and H. Hora, Laser Interaction and Related Plasma Phenomena, H. J. Schwartz and H. Hora, eds. (Plenum, New York, 1977), p. 913.
    [CrossRef]
  9. G. Schmidt and W. Horton, Comments Plasma Phys. Controlled Fusion 9, 85 (1985).
  10. R. R. Jones, W. C. Mead, S. V. Coggeshal, C. H. Aldrich, L. Norton, G. D. Pollak, and J. M. Wallace, Phys. Fluids 31, 1249 (1988).
    [CrossRef]
  11. D. P. Garuchava, N. L. Tsintsadze, and D. D. T. Tshakaya, Proceedings of the 1989 International Conference in Plasma Physics, A. Sen and P. K. Kaw, eds. (Indian Academy of Sciences, Bangalore, 1991), pp. 329–348.
  12. M. S. Sodha, S. Konar, and K. P. Maheshwari, J. Plasma Phys. 48, 107 (1992).
    [CrossRef]
  13. Y. Chen, Opt. Lett. 16, 4 (1991).
    [CrossRef] [PubMed]

1992 (1)

M. S. Sodha, S. Konar, and K. P. Maheshwari, J. Plasma Phys. 48, 107 (1992).
[CrossRef]

1991 (1)

1990 (1)

F. Cornolti, M. Lucchesi, and B. Zambon, Opt. Commun. 75, 129 (1990).
[CrossRef]

1988 (1)

R. R. Jones, W. C. Mead, S. V. Coggeshal, C. H. Aldrich, L. Norton, G. D. Pollak, and J. M. Wallace, Phys. Fluids 31, 1249 (1988).
[CrossRef]

1985 (2)

V. I. Kruglov and R. A. Vlasov, Phys. Lett. 111A, 401 (1985).

G. Schmidt and W. Horton, Comments Plasma Phys. Controlled Fusion 9, 85 (1985).

1973 (1)

P. K. Kaw, G. Schmidt, and P. Wilcox, Phys. Fluids 16, 1522 (1973).
[CrossRef]

1969 (1)

H. Hora, Z. Phys. 226, 156 (1969).
[CrossRef]

1968 (1)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Aldrich, C. H.

R. R. Jones, W. C. Mead, S. V. Coggeshal, C. H. Aldrich, L. Norton, G. D. Pollak, and J. M. Wallace, Phys. Fluids 31, 1249 (1988).
[CrossRef]

Chen, Y.

Coggeshal, S. V.

R. R. Jones, W. C. Mead, S. V. Coggeshal, C. H. Aldrich, L. Norton, G. D. Pollak, and J. M. Wallace, Phys. Fluids 31, 1249 (1988).
[CrossRef]

Cornolti, F.

F. Cornolti, M. Lucchesi, and B. Zambon, Opt. Commun. 75, 129 (1990).
[CrossRef]

Garuchava, D. P.

D. P. Garuchava, N. L. Tsintsadze, and D. D. T. Tshakaya, Proceedings of the 1989 International Conference in Plasma Physics, A. Sen and P. K. Kaw, eds. (Indian Academy of Sciences, Bangalore, 1991), pp. 329–348.

Ghatak, A. K.

M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, Self Focusing of Laser Beams in Dielectrics, Plasmas and Semiconductors (Tata McGraw-Hill, New Delhi, 1974).

M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. XIII, pp. 169–265.
[CrossRef]

Hora, H.

H. Hora, Z. Phys. 226, 156 (1969).
[CrossRef]

E. L. Kane and H. Hora, Laser Interaction and Related Plasma Phenomena, H. J. Schwartz and H. Hora, eds. (Plenum, New York, 1977), p. 913.
[CrossRef]

Horton, W.

G. Schmidt and W. Horton, Comments Plasma Phys. Controlled Fusion 9, 85 (1985).

Jones, R. R.

R. R. Jones, W. C. Mead, S. V. Coggeshal, C. H. Aldrich, L. Norton, G. D. Pollak, and J. M. Wallace, Phys. Fluids 31, 1249 (1988).
[CrossRef]

Kane, E. L.

E. L. Kane and H. Hora, Laser Interaction and Related Plasma Phenomena, H. J. Schwartz and H. Hora, eds. (Plenum, New York, 1977), p. 913.
[CrossRef]

Kaw, P. K.

P. K. Kaw, G. Schmidt, and P. Wilcox, Phys. Fluids 16, 1522 (1973).
[CrossRef]

Khokhlov, R. V.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Konar, S.

M. S. Sodha, S. Konar, and K. P. Maheshwari, J. Plasma Phys. 48, 107 (1992).
[CrossRef]

Kruglov, V. I.

V. I. Kruglov and R. A. Vlasov, Phys. Lett. 111A, 401 (1985).

Lucchesi, M.

F. Cornolti, M. Lucchesi, and B. Zambon, Opt. Commun. 75, 129 (1990).
[CrossRef]

Maheshwari, K. P.

M. S. Sodha, S. Konar, and K. P. Maheshwari, J. Plasma Phys. 48, 107 (1992).
[CrossRef]

Mead, W. C.

R. R. Jones, W. C. Mead, S. V. Coggeshal, C. H. Aldrich, L. Norton, G. D. Pollak, and J. M. Wallace, Phys. Fluids 31, 1249 (1988).
[CrossRef]

Norton, L.

R. R. Jones, W. C. Mead, S. V. Coggeshal, C. H. Aldrich, L. Norton, G. D. Pollak, and J. M. Wallace, Phys. Fluids 31, 1249 (1988).
[CrossRef]

Pollak, G. D.

R. R. Jones, W. C. Mead, S. V. Coggeshal, C. H. Aldrich, L. Norton, G. D. Pollak, and J. M. Wallace, Phys. Fluids 31, 1249 (1988).
[CrossRef]

Schmidt, G.

G. Schmidt and W. Horton, Comments Plasma Phys. Controlled Fusion 9, 85 (1985).

P. K. Kaw, G. Schmidt, and P. Wilcox, Phys. Fluids 16, 1522 (1973).
[CrossRef]

Sodha, M. S.

M. S. Sodha, S. Konar, and K. P. Maheshwari, J. Plasma Phys. 48, 107 (1992).
[CrossRef]

M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, Self Focusing of Laser Beams in Dielectrics, Plasmas and Semiconductors (Tata McGraw-Hill, New Delhi, 1974).

M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. XIII, pp. 169–265.
[CrossRef]

Sukhorukov, A. P.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Tripathi, V. K.

M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. XIII, pp. 169–265.
[CrossRef]

M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, Self Focusing of Laser Beams in Dielectrics, Plasmas and Semiconductors (Tata McGraw-Hill, New Delhi, 1974).

Tshakaya, D. D. T.

D. P. Garuchava, N. L. Tsintsadze, and D. D. T. Tshakaya, Proceedings of the 1989 International Conference in Plasma Physics, A. Sen and P. K. Kaw, eds. (Indian Academy of Sciences, Bangalore, 1991), pp. 329–348.

Tsintsadze, N. L.

D. P. Garuchava, N. L. Tsintsadze, and D. D. T. Tshakaya, Proceedings of the 1989 International Conference in Plasma Physics, A. Sen and P. K. Kaw, eds. (Indian Academy of Sciences, Bangalore, 1991), pp. 329–348.

Vlasov, R. A.

V. I. Kruglov and R. A. Vlasov, Phys. Lett. 111A, 401 (1985).

Wallace, J. M.

R. R. Jones, W. C. Mead, S. V. Coggeshal, C. H. Aldrich, L. Norton, G. D. Pollak, and J. M. Wallace, Phys. Fluids 31, 1249 (1988).
[CrossRef]

Wilcox, P.

P. K. Kaw, G. Schmidt, and P. Wilcox, Phys. Fluids 16, 1522 (1973).
[CrossRef]

Zambon, B.

F. Cornolti, M. Lucchesi, and B. Zambon, Opt. Commun. 75, 129 (1990).
[CrossRef]

Comments Plasma Phys. Controlled Fusion (1)

G. Schmidt and W. Horton, Comments Plasma Phys. Controlled Fusion 9, 85 (1985).

J. Plasma Phys. (1)

M. S. Sodha, S. Konar, and K. P. Maheshwari, J. Plasma Phys. 48, 107 (1992).
[CrossRef]

Opt. Commun. (1)

F. Cornolti, M. Lucchesi, and B. Zambon, Opt. Commun. 75, 129 (1990).
[CrossRef]

Opt. Lett. (1)

Phys. Fluids (2)

R. R. Jones, W. C. Mead, S. V. Coggeshal, C. H. Aldrich, L. Norton, G. D. Pollak, and J. M. Wallace, Phys. Fluids 31, 1249 (1988).
[CrossRef]

P. K. Kaw, G. Schmidt, and P. Wilcox, Phys. Fluids 16, 1522 (1973).
[CrossRef]

Phys. Lett. (1)

V. I. Kruglov and R. A. Vlasov, Phys. Lett. 111A, 401 (1985).

Sov. Phys. Usp. (1)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Z. Phys. (1)

H. Hora, Z. Phys. 226, 156 (1969).
[CrossRef]

Other (4)

M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, Self Focusing of Laser Beams in Dielectrics, Plasmas and Semiconductors (Tata McGraw-Hill, New Delhi, 1974).

M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. XIII, pp. 169–265.
[CrossRef]

E. L. Kane and H. Hora, Laser Interaction and Related Plasma Phenomena, H. J. Schwartz and H. Hora, eds. (Plenum, New York, 1977), p. 913.
[CrossRef]

D. P. Garuchava, N. L. Tsintsadze, and D. D. T. Tshakaya, Proceedings of the 1989 International Conference in Plasma Physics, A. Sen and P. K. Kaw, eds. (Indian Academy of Sciences, Bangalore, 1991), pp. 329–348.

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Figures (2)

Fig. 1
Fig. 1

Variation of normalized beam widths a and b and effective beam radius ρ with the normalized distance of propagation ξ for p = 0.0445 (p < pcr1 = 0.047), g = 0.1, M1/2 = 500, a(o) = 2.0, and b(o) = 0.5.

Fig. 2
Fig. 2

Behavior of the beam at the virtual threshold. Variation of normalized beam widths a and b and effective radius ρ with normalized distance of propagation ξ for p = pcr1 = 0.047; other parameters are same as in Fig. 1.

Equations (21)

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E 2 = E 0 2 a b exp { - [ 2 ( x / r a ) 2 + 2 ( y / r b ) 2 ] } ,
= 0 + Φ ( E 2 ) ,             Φ ( E 2 ) = s α E 2 1 + α E 2 ,
c 2 ω 2 | 1 2 ln | 1 ,
2 E + ω 2 c 2 E = 0.
- 2 i k A z + 2 A + k 2 Φ ( A A * ) 0 A = 0 ,
k = ω c 0 ,             2 = 2 x 2 + 2 y 2 .
2 S z + ( S x ) 2 + ( S y ) 2 = 1 k 2 A 0 2 A 0 + Φ ( A 0 A 0 * ) 0 ,
A 0 2 z + S x A 0 2 x + S y A 0 2 y + A 0 2 2 S = 0.
A 0 2 = E 0 2 ( a b ) exp ( - { 2 [ x ( r a ) ] 2 + 2 [ y ( r b ) ] 2 } ) ,
S = x 2 2 β 1 + y 2 2 β 2 + η ( z ) ,
β 1 = 1 a d a d z ,             β 2 = 1 b d b d z .
d 2 a d z 2 = 4 k 2 r 4 a 3 - 2 E 0 2 Φ ( E 0 2 / a b ) r 2 0 a 2 b ,
d 2 b d z 2 = 4 k 2 r 4 b 3 - 2 E 0 2 Φ ( E 0 2 / a b ) r 2 0 b 2 a ,
M d 2 a d ξ 2 = 1 a 3 - g p M 1 / 2 a 2 b ( 1 + p / a b ) 2 ,
M d 2 b d ξ 2 = 1 b 3 - g p M 1 / 2 b 2 a ( 1 + p / a b ) 2 ,
M 4 d 2 ρ 2 d ξ 2 = M 2 ( a ˙ 2 + b ˙ 2 ) + 1 2 [ 1 a 2 + 1 b 2 - 2 g p M 1 / 2 a b ( 1 + p / a b ) 2 ] ,
d ρ 2 d ξ | ξ = 0 = 0.
d 2 ρ 2 d ξ 2 | ξ = 0 = 0 ,
1 2 [ 1 a 2 ( o ) + 1 b 2 ( o ) ] = g p cr M 1 / 2 a ( o ) b ( o ) [ 1 + p cr / a ( o ) b ( o ) ] 2 ,
d 2 ρ 2 d ξ 2 | ξ = 0 = 0.
d ρ 2 d ξ | ξ = 0 = 0 ,             d 2 ρ 2 d ξ 2 | ξ = 0 = 0 ,

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