Abstract

The interaction between two optical soliton beams during type I second-harmonic generation in quadratically nonlinear materials has been investigated numerically. The product of the interaction is found to be highly sensitive to the relative phase between the launched beams. Threshold phenomena were found that persisted even when the input beams were only approximately solitary waves. Cases of both small and large collision angles were investigated, and the effect of walk-off was also taken into account.

© 1997 Optical Society of America

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  1. Y. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974).
  2. Y. N. Karamzin and A. P. Sukhorukov, Zh. Eksp. Teor. Phys. 68, 834 (1975) [Sov. Phys. JETP 41, 414 (1976)].
  3. G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
    [CrossRef]
  4. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, and G. I. Stegeman, Phys. Rev. Lett. 74, 5036 (1995).
    [CrossRef] [PubMed]
  5. R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. A 53, 1138 (1996).
  6. D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995); L. Torner, D. Mihalache, D. Mazilu, and N. N. Akhmediev, Opt. Lett. 20, 2183 (1995); L. Torner, C. R. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994); L. Torner and E. M. Wright, J. Opt. Soc. Am. B 13, 864 (1996).
    [CrossRef] [PubMed]
  7. K. Hayata and M. Koshiba, Phys. Rev. A 50, 675 (1994); D. Mihalache, F. Lederer, D. Mazilu, and L. C. Crasovan, Opt. Eng. 35, 1616 (1996); Y. S. Kivshar, Phys. Rev. E 52, 1156 (1995).
    [CrossRef] [PubMed]
  8. L. Torner, Opt. Commun. 114, 136 (1995); L. Torner, C. R. Menyuk, and G. I. Stegeman, J. Opt. Soc. Am. B 12, 889 (1995).
    [CrossRef]
  9. L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996); D. Mihalache, D. Mazilu, L. C. Crasovan, and L. Torner, Opt. Commun. 137, 113 (1997).
    [CrossRef] [PubMed]
  10. Q. Guo, Quantum Opt. 5, 133 (1993).
    [CrossRef]
  11. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, and G. I. Stegeman, Opt. Lett. 18, 28 (1992).
    [CrossRef]
  12. S. Gatz and J. Hermann, IEEE J. Quantum Electron. 28, 1732 (1992).
    [CrossRef]
  13. D.-M. Baboiu, G. I. Stegeman, and L. Torner, Opt. Lett. 20, 2282 (1995).
    [CrossRef]
  14. C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys. Rev. A 52, R3444 (1995).
    [CrossRef]
  15. J. P. Gordon, Opt. Lett. 8, 596 (1983).
    [CrossRef] [PubMed]

1996 (3)

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. A 53, 1138 (1996).

L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996); D. Mihalache, D. Mazilu, L. C. Crasovan, and L. Torner, Opt. Commun. 137, 113 (1997).
[CrossRef] [PubMed]

1995 (5)

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995); L. Torner, D. Mihalache, D. Mazilu, and N. N. Akhmediev, Opt. Lett. 20, 2183 (1995); L. Torner, C. R. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994); L. Torner and E. M. Wright, J. Opt. Soc. Am. B 13, 864 (1996).
[CrossRef] [PubMed]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, and G. I. Stegeman, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys. Rev. A 52, R3444 (1995).
[CrossRef]

L. Torner, Opt. Commun. 114, 136 (1995); L. Torner, C. R. Menyuk, and G. I. Stegeman, J. Opt. Soc. Am. B 12, 889 (1995).
[CrossRef]

D.-M. Baboiu, G. I. Stegeman, and L. Torner, Opt. Lett. 20, 2282 (1995).
[CrossRef]

1994 (1)

K. Hayata and M. Koshiba, Phys. Rev. A 50, 675 (1994); D. Mihalache, F. Lederer, D. Mazilu, and L. C. Crasovan, Opt. Eng. 35, 1616 (1996); Y. S. Kivshar, Phys. Rev. E 52, 1156 (1995).
[CrossRef] [PubMed]

1993 (1)

Q. Guo, Quantum Opt. 5, 133 (1993).
[CrossRef]

1992 (2)

1983 (1)

1975 (1)

Y. N. Karamzin and A. P. Sukhorukov, Zh. Eksp. Teor. Phys. 68, 834 (1975) [Sov. Phys. JETP 41, 414 (1976)].

1974 (1)

Y. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974).

Baboiu, D.-M.

Baek, Y.

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. A 53, 1138 (1996).

Buryak, A. V.

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995); L. Torner, D. Mihalache, D. Mazilu, and N. N. Akhmediev, Opt. Lett. 20, 2183 (1995); L. Torner, C. R. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994); L. Torner and E. M. Wright, J. Opt. Soc. Am. B 13, 864 (1996).
[CrossRef] [PubMed]

DeSalvo, R.

Etrich, C.

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys. Rev. A 52, R3444 (1995).
[CrossRef]

Gatz, S.

S. Gatz and J. Hermann, IEEE J. Quantum Electron. 28, 1732 (1992).
[CrossRef]

Gordon, J. P.

Guo, Q.

Q. Guo, Quantum Opt. 5, 133 (1993).
[CrossRef]

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, and G. I. Stegeman, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, and G. I. Stegeman, Opt. Lett. 18, 28 (1992).
[CrossRef]

Hayata, K.

K. Hayata and M. Koshiba, Phys. Rev. A 50, 675 (1994); D. Mihalache, F. Lederer, D. Mazilu, and L. C. Crasovan, Opt. Eng. 35, 1616 (1996); Y. S. Kivshar, Phys. Rev. E 52, 1156 (1995).
[CrossRef] [PubMed]

Hermann, J.

S. Gatz and J. Hermann, IEEE J. Quantum Electron. 28, 1732 (1992).
[CrossRef]

Karamzin, Y. N.

Y. N. Karamzin and A. P. Sukhorukov, Zh. Eksp. Teor. Phys. 68, 834 (1975) [Sov. Phys. JETP 41, 414 (1976)].

Y. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974).

Kivshar, Y. S.

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995); L. Torner, D. Mihalache, D. Mazilu, and N. N. Akhmediev, Opt. Lett. 20, 2183 (1995); L. Torner, C. R. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994); L. Torner and E. M. Wright, J. Opt. Soc. Am. B 13, 864 (1996).
[CrossRef] [PubMed]

Koshiba, M.

K. Hayata and M. Koshiba, Phys. Rev. A 50, 675 (1994); D. Mihalache, F. Lederer, D. Mazilu, and L. C. Crasovan, Opt. Eng. 35, 1616 (1996); Y. S. Kivshar, Phys. Rev. E 52, 1156 (1995).
[CrossRef] [PubMed]

Lederer, F.

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys. Rev. A 52, R3444 (1995).
[CrossRef]

Malomed, B.

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys. Rev. A 52, R3444 (1995).
[CrossRef]

Mazilu, D.

L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996); D. Mihalache, D. Mazilu, L. C. Crasovan, and L. Torner, Opt. Commun. 137, 113 (1997).
[CrossRef] [PubMed]

Mihalache, D.

L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996); D. Mihalache, D. Mazilu, L. C. Crasovan, and L. Torner, Opt. Commun. 137, 113 (1997).
[CrossRef] [PubMed]

Pelinovsky, D. E.

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995); L. Torner, D. Mihalache, D. Mazilu, and N. N. Akhmediev, Opt. Lett. 20, 2183 (1995); L. Torner, C. R. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994); L. Torner and E. M. Wright, J. Opt. Soc. Am. B 13, 864 (1996).
[CrossRef] [PubMed]

Peschel, U.

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys. Rev. A 52, R3444 (1995).
[CrossRef]

Schiek, R.

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. A 53, 1138 (1996).

Sheik-Bahae, M.

Stegeman, G. I.

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. A 53, 1138 (1996).

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, and G. I. Stegeman, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

D.-M. Baboiu, G. I. Stegeman, and L. Torner, Opt. Lett. 20, 2282 (1995).
[CrossRef]

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, and G. I. Stegeman, Opt. Lett. 18, 28 (1992).
[CrossRef]

Sukhorukov, A. P.

Y. N. Karamzin and A. P. Sukhorukov, Zh. Eksp. Teor. Phys. 68, 834 (1975) [Sov. Phys. JETP 41, 414 (1976)].

Y. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974).

Torner, L.

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996); D. Mihalache, D. Mazilu, L. C. Crasovan, and L. Torner, Opt. Commun. 137, 113 (1997).
[CrossRef] [PubMed]

L. Torner, Opt. Commun. 114, 136 (1995); L. Torner, C. R. Menyuk, and G. I. Stegeman, J. Opt. Soc. Am. B 12, 889 (1995).
[CrossRef]

D.-M. Baboiu, G. I. Stegeman, and L. Torner, Opt. Lett. 20, 2282 (1995).
[CrossRef]

Torruellas, W. E.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, and G. I. Stegeman, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Van Stryland, E. W.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, and G. I. Stegeman, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Wang, Z.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, and G. I. Stegeman, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

S. Gatz and J. Hermann, IEEE J. Quantum Electron. 28, 1732 (1992).
[CrossRef]

JETP Lett. (1)

Y. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974).

Opt. Commun. (1)

L. Torner, Opt. Commun. 114, 136 (1995); L. Torner, C. R. Menyuk, and G. I. Stegeman, J. Opt. Soc. Am. B 12, 889 (1995).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

Phys. Rev. A (3)

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. A 53, 1138 (1996).

K. Hayata and M. Koshiba, Phys. Rev. A 50, 675 (1994); D. Mihalache, F. Lederer, D. Mazilu, and L. C. Crasovan, Opt. Eng. 35, 1616 (1996); Y. S. Kivshar, Phys. Rev. E 52, 1156 (1995).
[CrossRef] [PubMed]

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys. Rev. A 52, R3444 (1995).
[CrossRef]

Phys. Rev. Lett. (3)

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995); L. Torner, D. Mihalache, D. Mazilu, and N. N. Akhmediev, Opt. Lett. 20, 2183 (1995); L. Torner, C. R. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994); L. Torner and E. M. Wright, J. Opt. Soc. Am. B 13, 864 (1996).
[CrossRef] [PubMed]

L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996); D. Mihalache, D. Mazilu, L. C. Crasovan, and L. Torner, Opt. Commun. 137, 113 (1997).
[CrossRef] [PubMed]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, and G. I. Stegeman, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Quantum Opt. (1)

Q. Guo, Quantum Opt. 5, 133 (1993).
[CrossRef]

Zh. Eksp. Teor. Phys. (1)

Y. N. Karamzin and A. P. Sukhorukov, Zh. Eksp. Teor. Phys. 68, 834 (1975) [Sov. Phys. JETP 41, 414 (1976)].

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Figures (8)

Fig. 1
Fig. 1

Collision of two quadratic solitonlike (Gaussian) beams with amplitude and phase parameters approximately those of quadratic solitons. The normalized amplitudes are A0=2.2 and B0=3 for a phase mismatch β=-3 and collision parameter μ=±0.5. Contour curves are also shown on a logarithmic scale to emphasize the emission of radiation during the interaction and the subsequent evolution.

Fig. 2
Fig. 2

Output field distributions after a propagation distance z=20 for various phase differences between the fundamental waves at the input. The input beams collide with μ=±0.5. Each curve of a plot corresponds to a different numerical experiment characterized by a specific value of the relative phase between the inputs. The plots are shown for phase mismatches of (a) β=-3, (b) β=0, (c) β=+3.

Fig. 3
Fig. 3

Same as in Fig. 2, but for a large collision angle (μ=±1.0).

Fig. 4
Fig. 4

Typical evolution of quadratic solitons at a large collision angle (μ=±1.0) for three different phase differences between the input FW beams, (a) Δϕ=0, (b) π/6, (c) π/2.

Fig. 5
Fig. 5

Interaction of two initially parallel, in-phase solitons for walk-off parameters of (a) δ=0.3, (b) δ=0.9.

Fig. 6
Fig. 6

Output field distribution for initially parallel quadratic solitons at β=-3. Each curve corresponds to a different numerical experiment, characterized by a different value of the walk-off.

Fig. 7
Fig. 7

Repulsive behavior of two in-phase solitons colliding at a moderate collision angle (μ=±0.5) and a relatively large walk-off (δ=0.8).

Fig. 8
Fig. 8

Output field distribution for different initial phase differences between FW input beams. The input beams collide with μ=±0.5 in the presence of walk-off (δ=0.3) at a negative phase mismatch (β=-3).]

Equations (4)

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i a1z-r22a1x2+a1*a2 exp(-iβz)=0,
i a2z-α22a2x2-iδ a2x+a12 exp(iβz)=0,
a1=A0 sech2(x-νz)exp(iκ1z-iμ1x),
a2=B0 sech2(x-νz)exp(iκ2z-iμ2x),

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