Abstract

The effect of phase-matched third-harmonic generation on the structure and stability of spatial solitary waves is investigated. A power threshold for the existence of two-frequency spatial solitons is found, and the multistability of solitary waves in a Kerr medium owing to a higher-order nonlinear phase shift caused by cascaded third-order processes is revealed.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. I. V. Tomov and M. C. Richardson, IEEE J. Quantum Electron. 12, 521 (1976).
    [CrossRef]
  2. G. I. Stegeman, D. J. Hagan, and L. Torner, J. Opt. Quantum Electron. 28, 1691 (1996).
    [CrossRef]
  3. A. V. Buryak and Yu. S. Kivshar, Phys. Lett. A 197, 407 (1995).
    [CrossRef]
  4. D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
    [CrossRef] [PubMed]
  5. A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
    [CrossRef] [PubMed]
  6. S. Saltiel, S. Tanev, and A. D. Boardman, Opt. Lett. 22, 148 (1997).
    [CrossRef] [PubMed]
  7. N. A. Ansari, R. A. Sammut, and H. T. Tran, J. Opt. Soc. Am. B 13, 1419 (1996).
    [CrossRef]
  8. After rescaling, this exact solution becomes the same as that found by Y. Chen, Phy. Rev. A 50, 5145 (1994).
    [CrossRef]
  9. To analyze soliton stability, we substitute u=u0x+ξ and w=w0x+η into Eqs.  (3) and (4) and, after linearization in ξ and η, investigate numerically the corresponding linear eigenvalue problem.
  10. A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985); IEEE J. Quantum Electron. QE-21, 1538 (1985).
    [CrossRef] [PubMed]

1997 (1)

1996 (3)

N. A. Ansari, R. A. Sammut, and H. T. Tran, J. Opt. Soc. Am. B 13, 1419 (1996).
[CrossRef]

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

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

1995 (2)

A. V. Buryak and Yu. S. Kivshar, Phys. Lett. A 197, 407 (1995).
[CrossRef]

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

1994 (1)

After rescaling, this exact solution becomes the same as that found by Y. Chen, Phy. Rev. A 50, 5145 (1994).
[CrossRef]

1985 (1)

A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985); IEEE J. Quantum Electron. QE-21, 1538 (1985).
[CrossRef] [PubMed]

1976 (1)

I. V. Tomov and M. C. Richardson, IEEE J. Quantum Electron. 12, 521 (1976).
[CrossRef]

Ansari, N. A.

Boardman, A. D.

Buryak, A. V.

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

A. V. Buryak and Yu. S. Kivshar, Phys. Lett. A 197, 407 (1995).
[CrossRef]

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

Chen, Y.

After rescaling, this exact solution becomes the same as that found by Y. Chen, Phy. Rev. A 50, 5145 (1994).
[CrossRef]

Hagan, D. J.

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

Kaplan, A. E.

A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985); IEEE J. Quantum Electron. QE-21, 1538 (1985).
[CrossRef] [PubMed]

Kivshar, Yu. S.

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

A. V. Buryak and Yu. S. Kivshar, Phys. Lett. A 197, 407 (1995).
[CrossRef]

Pelinovsky, D. E.

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

Richardson, M. C.

I. V. Tomov and M. C. Richardson, IEEE J. Quantum Electron. 12, 521 (1976).
[CrossRef]

Saltiel, S.

Sammut, R. A.

Stegeman, G. I.

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

Tanev, S.

Tomov, I. V.

I. V. Tomov and M. C. Richardson, IEEE J. Quantum Electron. 12, 521 (1976).
[CrossRef]

Torner, L.

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

Tran, H. T.

Trillo, S.

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

I. V. Tomov and M. C. Richardson, IEEE J. Quantum Electron. 12, 521 (1976).
[CrossRef]

J. Opt. Quantum Electron. (1)

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

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

Opt. Lett. (1)

Phy. Rev. A (1)

After rescaling, this exact solution becomes the same as that found by Y. Chen, Phy. Rev. A 50, 5145 (1994).
[CrossRef]

Phys. Lett. A (1)

A. V. Buryak and Yu. S. Kivshar, Phys. Lett. A 197, 407 (1995).
[CrossRef]

Phys. Rev. Lett. (3)

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985); IEEE J. Quantum Electron. QE-21, 1538 (1985).
[CrossRef] [PubMed]

Other (1)

To analyze soliton stability, we substitute u=u0x+ξ and w=w0x+η into Eqs.  (3) and (4) and, after linearization in ξ and η, investigate numerically the corresponding linear eigenvalue problem.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Variation of the normalized total power, Ptot, versus the mismatch α for the three distinct families of solitary-wave solutions of Eqs.  (3) and (4). The dashed curve corresponds to the asymptotic expansion of the cascading limit. Lower curves merge at the bifurcation point O (α=9). Open circles A–D indicate the particular examples presented in Fig.  2. The filled circle on curve B the analytical solution.

Fig. 2
Fig. 2

Examples of the fundamental (thin curves) and the third-harmonic (thick curves) profiles for several solitary-wave solutions that belong to different families. (a), (b), (c), and (d) correspond to points A, B, C, and D, respectively, in Fig.  1.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

2ik1E1z+2E1x2+χE12+2E32E1+E1*2E3 exp-iΔkz=0,
2ik3E3z+2E3x2+9χE32+2E12E3+13E13 expiΔkz=0,
iUZ+2UX2+19U2+2W2U+13U*2W=0,
iσWZ+2WX2-ΔσW+9W2+2U2W+19U3=0,
iuz+2ux2-u+19u2+2w2u+13u*2w=0,
iσwz+2wx2-αw+9w2+2u2w+19u3=0,
Ptot=-u2+3σw2dx,

Metrics