Abstract

We show numerically that, contrary to what has been believed so far, multidimensional optical solitons can propagate in cubic nonlinear media. To avoid a collapse along the propagation axis, we take advantage of the cross-phase modulations between fundamental- and harmonic-field components. A compact analytical expression for new solitonlike fields with an arbitrary transverse dimension is derived through a self-consistent-field approximation.

© 1994 Optical Society of America

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References

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  1. E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos (Cambridge U. Press, Cambridge, 1990), Chap. 9, pp. 298–328.
  2. V. G. Makhankov, Soliton Phenomenology (Muwer, Dordrecht, The Netherlands, 1990), part VI, pp. 391–448.
  3. S. V. Manakov, V. E. Zakharov, L. A. Bordag, A. R. Its, V. B. Matveev, Phys. Lett. A 63, 205 (1977).
    [CrossRef]
  4. A. Hasegawa, K. Mima, Phys. Fluids 21,87 (1978).
    [CrossRef]
  5. J. Satsuma, M. J. Ablowitz, J. Math. Phys. 20, 1496 (1979).
    [CrossRef]
  6. V. I. Petviashvili, JETP Lett. 32, 619 (1980).
  7. M. Makino, T. Kamimura, T. Taniuti, J. Phys. Soc. Jpn. 50, 980 (1981).
    [CrossRef]
  8. M. Boiti, J. J.-P. Leon, F. Pempinelli, Phys. Lett. A 141, 101 (1989).
    [CrossRef]
  9. H. Iwasaki, S. Toh, T. Kawahara, Physica D 43, 293 (1990).
    [CrossRef]
  10. K. Hayata, M. Koshiba, Phys. Rev. Lett. 71,3275 (1993); erratum K. Hayata, M. Koshiba, Phys. Rev. Lett. 72, 178 (1994).
    [CrossRef] [PubMed]
  11. D. Suter, T. Blasberg, Phys. Rev. A 48, 4583 (1993).
    [CrossRef] [PubMed]
  12. J. J. Rasmussen, K. Rypdal, Phys. Scr. 33, 481 (1986).
    [CrossRef]
  13. G. H. Derrick, J. Math. Phys. 5, 1252 (1964).
    [CrossRef]
  14. K. Hayata, Y. Uehira, M. Koshiba, in Technical Research Reports (Institute of Electrical Engineers of Japan/Institute of Electronics, Information and Communication Engineers, Tokyo, 1993), paper EMT-93–117; in Extended Abstracts of 41st Spring Meeting (Japan Society of Applied Physics, Tokyo, 1994), abstract 28p-E-6.
  15. K. Hayata, M. Koshiba, Electron. Lett. 26, 1566 (1990).
    [CrossRef]
  16. K. Hayata, A. Misawa, M. Koshiba, Trans. Inst. Electron. Inform. Commun. Eng. J73-C-I, 151 (1990) [Electron. Commun. Jpn. (Part II) 74, 13 (1991)]; J. Opt. Soc. Am. B 7, 1772 (1990).
  17. K. Hayata, A. Misawa, M. Koshiba, Trans. Inst. Electron. Inform. Commun. Eng. E73, 855 (1990); J. Opt. Soc. Am. B 7, 1268 (1990).
  18. V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).
  19. W. L. Smith, in CRC Handbook of Laser Science and Technology, M. J. Weber, ed. (CRC, Boca Raton, Fla., 1986), Vol. 3.

1993

K. Hayata, M. Koshiba, Phys. Rev. Lett. 71,3275 (1993); erratum K. Hayata, M. Koshiba, Phys. Rev. Lett. 72, 178 (1994).
[CrossRef] [PubMed]

D. Suter, T. Blasberg, Phys. Rev. A 48, 4583 (1993).
[CrossRef] [PubMed]

1990

K. Hayata, M. Koshiba, Electron. Lett. 26, 1566 (1990).
[CrossRef]

K. Hayata, A. Misawa, M. Koshiba, Trans. Inst. Electron. Inform. Commun. Eng. J73-C-I, 151 (1990) [Electron. Commun. Jpn. (Part II) 74, 13 (1991)]; J. Opt. Soc. Am. B 7, 1772 (1990).

K. Hayata, A. Misawa, M. Koshiba, Trans. Inst. Electron. Inform. Commun. Eng. E73, 855 (1990); J. Opt. Soc. Am. B 7, 1268 (1990).

V. G. Makhankov, Soliton Phenomenology (Muwer, Dordrecht, The Netherlands, 1990), part VI, pp. 391–448.

H. Iwasaki, S. Toh, T. Kawahara, Physica D 43, 293 (1990).
[CrossRef]

1989

M. Boiti, J. J.-P. Leon, F. Pempinelli, Phys. Lett. A 141, 101 (1989).
[CrossRef]

1986

J. J. Rasmussen, K. Rypdal, Phys. Scr. 33, 481 (1986).
[CrossRef]

1981

M. Makino, T. Kamimura, T. Taniuti, J. Phys. Soc. Jpn. 50, 980 (1981).
[CrossRef]

1980

V. I. Petviashvili, JETP Lett. 32, 619 (1980).

1979

J. Satsuma, M. J. Ablowitz, J. Math. Phys. 20, 1496 (1979).
[CrossRef]

1978

A. Hasegawa, K. Mima, Phys. Fluids 21,87 (1978).
[CrossRef]

1977

S. V. Manakov, V. E. Zakharov, L. A. Bordag, A. R. Its, V. B. Matveev, Phys. Lett. A 63, 205 (1977).
[CrossRef]

1964

G. H. Derrick, J. Math. Phys. 5, 1252 (1964).
[CrossRef]

Ablowitz, M. J.

J. Satsuma, M. J. Ablowitz, J. Math. Phys. 20, 1496 (1979).
[CrossRef]

Blasberg, T.

D. Suter, T. Blasberg, Phys. Rev. A 48, 4583 (1993).
[CrossRef] [PubMed]

Boiti, M.

M. Boiti, J. J.-P. Leon, F. Pempinelli, Phys. Lett. A 141, 101 (1989).
[CrossRef]

Bordag, L. A.

S. V. Manakov, V. E. Zakharov, L. A. Bordag, A. R. Its, V. B. Matveev, Phys. Lett. A 63, 205 (1977).
[CrossRef]

Derrick, G. H.

G. H. Derrick, J. Math. Phys. 5, 1252 (1964).
[CrossRef]

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

Gurzadyan, G. G.

V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

Hasegawa, A.

A. Hasegawa, K. Mima, Phys. Fluids 21,87 (1978).
[CrossRef]

Hayata, K.

K. Hayata, M. Koshiba, Phys. Rev. Lett. 71,3275 (1993); erratum K. Hayata, M. Koshiba, Phys. Rev. Lett. 72, 178 (1994).
[CrossRef] [PubMed]

K. Hayata, A. Misawa, M. Koshiba, Trans. Inst. Electron. Inform. Commun. Eng. J73-C-I, 151 (1990) [Electron. Commun. Jpn. (Part II) 74, 13 (1991)]; J. Opt. Soc. Am. B 7, 1772 (1990).

K. Hayata, A. Misawa, M. Koshiba, Trans. Inst. Electron. Inform. Commun. Eng. E73, 855 (1990); J. Opt. Soc. Am. B 7, 1268 (1990).

K. Hayata, M. Koshiba, Electron. Lett. 26, 1566 (1990).
[CrossRef]

K. Hayata, Y. Uehira, M. Koshiba, in Technical Research Reports (Institute of Electrical Engineers of Japan/Institute of Electronics, Information and Communication Engineers, Tokyo, 1993), paper EMT-93–117; in Extended Abstracts of 41st Spring Meeting (Japan Society of Applied Physics, Tokyo, 1994), abstract 28p-E-6.

Infeld, E.

E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos (Cambridge U. Press, Cambridge, 1990), Chap. 9, pp. 298–328.

Its, A. R.

S. V. Manakov, V. E. Zakharov, L. A. Bordag, A. R. Its, V. B. Matveev, Phys. Lett. A 63, 205 (1977).
[CrossRef]

Iwasaki, H.

H. Iwasaki, S. Toh, T. Kawahara, Physica D 43, 293 (1990).
[CrossRef]

Kamimura, T.

M. Makino, T. Kamimura, T. Taniuti, J. Phys. Soc. Jpn. 50, 980 (1981).
[CrossRef]

Kawahara, T.

H. Iwasaki, S. Toh, T. Kawahara, Physica D 43, 293 (1990).
[CrossRef]

Koshiba, M.

K. Hayata, M. Koshiba, Phys. Rev. Lett. 71,3275 (1993); erratum K. Hayata, M. Koshiba, Phys. Rev. Lett. 72, 178 (1994).
[CrossRef] [PubMed]

K. Hayata, A. Misawa, M. Koshiba, Trans. Inst. Electron. Inform. Commun. Eng. J73-C-I, 151 (1990) [Electron. Commun. Jpn. (Part II) 74, 13 (1991)]; J. Opt. Soc. Am. B 7, 1772 (1990).

K. Hayata, A. Misawa, M. Koshiba, Trans. Inst. Electron. Inform. Commun. Eng. E73, 855 (1990); J. Opt. Soc. Am. B 7, 1268 (1990).

K. Hayata, M. Koshiba, Electron. Lett. 26, 1566 (1990).
[CrossRef]

K. Hayata, Y. Uehira, M. Koshiba, in Technical Research Reports (Institute of Electrical Engineers of Japan/Institute of Electronics, Information and Communication Engineers, Tokyo, 1993), paper EMT-93–117; in Extended Abstracts of 41st Spring Meeting (Japan Society of Applied Physics, Tokyo, 1994), abstract 28p-E-6.

Leon, J. J.-P.

M. Boiti, J. J.-P. Leon, F. Pempinelli, Phys. Lett. A 141, 101 (1989).
[CrossRef]

Makhankov, V. G.

V. G. Makhankov, Soliton Phenomenology (Muwer, Dordrecht, The Netherlands, 1990), part VI, pp. 391–448.

Makino, M.

M. Makino, T. Kamimura, T. Taniuti, J. Phys. Soc. Jpn. 50, 980 (1981).
[CrossRef]

Manakov, S. V.

S. V. Manakov, V. E. Zakharov, L. A. Bordag, A. R. Its, V. B. Matveev, Phys. Lett. A 63, 205 (1977).
[CrossRef]

Matveev, V. B.

S. V. Manakov, V. E. Zakharov, L. A. Bordag, A. R. Its, V. B. Matveev, Phys. Lett. A 63, 205 (1977).
[CrossRef]

Mima, K.

A. Hasegawa, K. Mima, Phys. Fluids 21,87 (1978).
[CrossRef]

Misawa, A.

K. Hayata, A. Misawa, M. Koshiba, Trans. Inst. Electron. Inform. Commun. Eng. E73, 855 (1990); J. Opt. Soc. Am. B 7, 1268 (1990).

K. Hayata, A. Misawa, M. Koshiba, Trans. Inst. Electron. Inform. Commun. Eng. J73-C-I, 151 (1990) [Electron. Commun. Jpn. (Part II) 74, 13 (1991)]; J. Opt. Soc. Am. B 7, 1772 (1990).

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

Pempinelli, F.

M. Boiti, J. J.-P. Leon, F. Pempinelli, Phys. Lett. A 141, 101 (1989).
[CrossRef]

Petviashvili, V. I.

V. I. Petviashvili, JETP Lett. 32, 619 (1980).

Rasmussen, J. J.

J. J. Rasmussen, K. Rypdal, Phys. Scr. 33, 481 (1986).
[CrossRef]

Rowlands, G.

E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos (Cambridge U. Press, Cambridge, 1990), Chap. 9, pp. 298–328.

Rypdal, K.

J. J. Rasmussen, K. Rypdal, Phys. Scr. 33, 481 (1986).
[CrossRef]

Satsuma, J.

J. Satsuma, M. J. Ablowitz, J. Math. Phys. 20, 1496 (1979).
[CrossRef]

Smith, W. L.

W. L. Smith, in CRC Handbook of Laser Science and Technology, M. J. Weber, ed. (CRC, Boca Raton, Fla., 1986), Vol. 3.

Suter, D.

D. Suter, T. Blasberg, Phys. Rev. A 48, 4583 (1993).
[CrossRef] [PubMed]

Taniuti, T.

M. Makino, T. Kamimura, T. Taniuti, J. Phys. Soc. Jpn. 50, 980 (1981).
[CrossRef]

Toh, S.

H. Iwasaki, S. Toh, T. Kawahara, Physica D 43, 293 (1990).
[CrossRef]

Uehira, Y.

K. Hayata, Y. Uehira, M. Koshiba, in Technical Research Reports (Institute of Electrical Engineers of Japan/Institute of Electronics, Information and Communication Engineers, Tokyo, 1993), paper EMT-93–117; in Extended Abstracts of 41st Spring Meeting (Japan Society of Applied Physics, Tokyo, 1994), abstract 28p-E-6.

Zakharov, V. E.

S. V. Manakov, V. E. Zakharov, L. A. Bordag, A. R. Its, V. B. Matveev, Phys. Lett. A 63, 205 (1977).
[CrossRef]

Electron. Lett.

K. Hayata, M. Koshiba, Electron. Lett. 26, 1566 (1990).
[CrossRef]

J. Math. Phys.

G. H. Derrick, J. Math. Phys. 5, 1252 (1964).
[CrossRef]

J. Satsuma, M. J. Ablowitz, J. Math. Phys. 20, 1496 (1979).
[CrossRef]

J. Phys. Soc. Jpn.

M. Makino, T. Kamimura, T. Taniuti, J. Phys. Soc. Jpn. 50, 980 (1981).
[CrossRef]

JETP Lett.

V. I. Petviashvili, JETP Lett. 32, 619 (1980).

Phys. Fluids

A. Hasegawa, K. Mima, Phys. Fluids 21,87 (1978).
[CrossRef]

Phys. Lett. A

S. V. Manakov, V. E. Zakharov, L. A. Bordag, A. R. Its, V. B. Matveev, Phys. Lett. A 63, 205 (1977).
[CrossRef]

M. Boiti, J. J.-P. Leon, F. Pempinelli, Phys. Lett. A 141, 101 (1989).
[CrossRef]

Phys. Rev. A

D. Suter, T. Blasberg, Phys. Rev. A 48, 4583 (1993).
[CrossRef] [PubMed]

Phys. Rev. Lett.

K. Hayata, M. Koshiba, Phys. Rev. Lett. 71,3275 (1993); erratum K. Hayata, M. Koshiba, Phys. Rev. Lett. 72, 178 (1994).
[CrossRef] [PubMed]

Phys. Scr.

J. J. Rasmussen, K. Rypdal, Phys. Scr. 33, 481 (1986).
[CrossRef]

Physica D

H. Iwasaki, S. Toh, T. Kawahara, Physica D 43, 293 (1990).
[CrossRef]

Soliton Phenomenology

V. G. Makhankov, Soliton Phenomenology (Muwer, Dordrecht, The Netherlands, 1990), part VI, pp. 391–448.

Trans. Inst. Electron. Inform. Commun. Eng.

K. Hayata, A. Misawa, M. Koshiba, Trans. Inst. Electron. Inform. Commun. Eng. J73-C-I, 151 (1990) [Electron. Commun. Jpn. (Part II) 74, 13 (1991)]; J. Opt. Soc. Am. B 7, 1772 (1990).

K. Hayata, A. Misawa, M. Koshiba, Trans. Inst. Electron. Inform. Commun. Eng. E73, 855 (1990); J. Opt. Soc. Am. B 7, 1268 (1990).

Other

V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

W. L. Smith, in CRC Handbook of Laser Science and Technology, M. J. Weber, ed. (CRC, Boca Raton, Fla., 1986), Vol. 3.

K. Hayata, Y. Uehira, M. Koshiba, in Technical Research Reports (Institute of Electrical Engineers of Japan/Institute of Electronics, Information and Communication Engineers, Tokyo, 1993), paper EMT-93–117; in Extended Abstracts of 41st Spring Meeting (Japan Society of Applied Physics, Tokyo, 1994), abstract 28p-E-6.

E. Infeld, G. Rowlands, Nonlinear Waves, Solitons and Chaos (Cambridge U. Press, Cambridge, 1990), Chap. 9, pp. 298–328.

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Figures (3)

Fig. 1
Fig. 1

Evolution of a monodimensional solitonlike wave (ω component) along the propagation (ζ) axis (D = 1, m = 2, K = 0.375, and η = 2.0). The total propagation distance is 5 × 104(λ0). The third-order nonlinear-optical interaction in LiNbO3 is implied; however, to isolate the effect that is due to the Kerr-type nonlinearity, we have disregarded the contribution of quadratic nonlinearities. With respect to photonic polarization, the fundamental (ω) wave (λ0 = 1.064 μm) is assumed to be ordinary, whereas the second-harmonic (2ω) wave (λ0′ = 0.532 μm) is assumed to be extraordinary. The material data are extracted from Refs. 18 and 19. To check the solitonic feature, we assumed that the medium is lossless. The amplitude on the ordinate is in units of volts per meter. Similar results were obtained for other K values as well.

Fig. 2
Fig. 2

Same as Fig. 1 but with a two-dimensional quasi-soliton (D = 2). Similar results were obtained for other K values as well.

Fig. 3
Fig. 3

Same as Fig. 1 but with a three-dimensional localization (D = 3). Although the growth rate tends to decrease with increasing K, the results for other K values were found to be qualitatively similar to the present result.

Equations (13)

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- i 2 B ζ e = ρ 1 - D ρ ( ρ D - 1 ρ e ) + ( ɛ - B 2 ) e + κ ( e 2 + 2 e 2 ) e ,
- i 2 m B ζ e = ρ 1 - D ρ ( ρ D - 1 ρ e ) + m 2 ( ɛ - B 2 ) e + m 2 κ ( e 2 + 2 e 2 ) e ,
e ( ρ ) = e 0 sech ( α ρ ) ,             e ( ρ ) = e 0 sech ( α ρ ) ,
B = [ ɛ + Δ ɛ / ( m 2 - 1 ) ] 1 / 2 ,
α = m [ Δ ɛ / ( m 2 - 1 ) ] 1 / 2 ,
e 0 = ± m [ ( m 2 - 1 ) ( 2 η 2 + 1 ) ] - 1 / 2 ( 2 Δ ɛ / κ ) 1 / 2 ,
e 0 = η e 0 ,
η = ± [ ( 1 - 2 m 2 K ) / ( m 2 K - 2 ) ] 1 / 2 ,
1 / ( 2 m 2 ) < K < 2 / m 2 .
e ( x 1 , x 2 , , x D ) = e 0 f ( x 1 ) f ( x 2 ) f ( x D ) ,
e ( x 1 , x 2 , , x D ) = e 0 f ( x 1 ) f ( x 2 ) f ( x D ) ,
α = m [ ( m 2 - 1 ) ( 4 - D ) ] - 1 / 2 ( 3 Δ ɛ ) 1 / 2 ,
e 0 = ± ( 3 / 2 ) ( D - 1 ) / 2 m [ ( m 2 - 1 ) ( 4 - D ) ( 2 η 2 + 1 ) ] - 1 / 2 × ( 6 Δ ɛ / κ ) 1 / 2 .

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