Abstract

We introduce the concept of multipole spatial optical vector solitons associated with higher-order guided modes trapped by a soliton-induced waveguide in a bulk medium. Such stationary localized waves include previously predicted vortex- and dipole-mode vector solitons and also describe new higher-order vector solitons and necklace-type beams. We present the theoretical and experimental results of the structure, formation, and instability development of the quadrupole vector solitons.

© 2001 Optical Society of America

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  1. G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
    [CrossRef] [PubMed]
  2. D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 53 (1988).
    [CrossRef] [PubMed]
  3. E. A. Ostrovskaya, Yu. S. Kivshar, D. V. Skryabin, and W. J. Firth, Phys. Rev. Lett. 83, 296 (1999).
    [CrossRef]
  4. See, e.g., M. Haelterman and A. P. Sheppard, Phys. Rev. E 49, 3376 (1994).
    [CrossRef]
  5. Z. H. Musslimani, M. Segev, D. N. Christodoulides, and M. Soljačić, Phys. Rev. Lett. 84, 1164 (2000).
    [CrossRef] [PubMed]
  6. J. N. Malmberg, A. H. Carlsson, D. Anderson, M. Lisak, E. A. Ostrovskaya, and Yu. S. Kivshar, Opt. Lett. 25, 643 (2000).
    [CrossRef]
  7. J. J. García-Ripoll, V. Perez-García, E. A. Ostrovskaya, and Yu. S. Kivshar, Phys. Rev. Lett. 85, 82 (2000).
    [CrossRef]
  8. W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Yu. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
    [CrossRef] [PubMed]
  9. T. Carmon, C. Anastassiou, S. Lan, D. Kip, Z. H. Musslimani, M. Segev, and D. N. Christodoulides, Opt. Lett. 25, 1113 (2000).
    [CrossRef]
  10. A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, Cambridge, 1998), Chap.  8.
    [CrossRef]
  11. D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, Appl. Phys. Lett. 68, 1763 (1996).
    [CrossRef]
  12. A. Barthelemy, C. Froehly, and M. Shalaby, Proc. SPIE 2041, 104 (1994).
    [CrossRef]
  13. M. Soljačić, S. Sears, and M. Segev, Phys. Rev. Lett. 81, 4851 (1998).
    [CrossRef]

2000 (5)

Z. H. Musslimani, M. Segev, D. N. Christodoulides, and M. Soljačić, Phys. Rev. Lett. 84, 1164 (2000).
[CrossRef] [PubMed]

J. J. García-Ripoll, V. Perez-García, E. A. Ostrovskaya, and Yu. S. Kivshar, Phys. Rev. Lett. 85, 82 (2000).
[CrossRef]

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Yu. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

J. N. Malmberg, A. H. Carlsson, D. Anderson, M. Lisak, E. A. Ostrovskaya, and Yu. S. Kivshar, Opt. Lett. 25, 643 (2000).
[CrossRef]

T. Carmon, C. Anastassiou, S. Lan, D. Kip, Z. H. Musslimani, M. Segev, and D. N. Christodoulides, Opt. Lett. 25, 1113 (2000).
[CrossRef]

1999 (2)

G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

E. A. Ostrovskaya, Yu. S. Kivshar, D. V. Skryabin, and W. J. Firth, Phys. Rev. Lett. 83, 296 (1999).
[CrossRef]

1998 (1)

M. Soljačić, S. Sears, and M. Segev, Phys. Rev. Lett. 81, 4851 (1998).
[CrossRef]

1996 (1)

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, Appl. Phys. Lett. 68, 1763 (1996).
[CrossRef]

1994 (2)

A. Barthelemy, C. Froehly, and M. Shalaby, Proc. SPIE 2041, 104 (1994).
[CrossRef]

See, e.g., M. Haelterman and A. P. Sheppard, Phys. Rev. E 49, 3376 (1994).
[CrossRef]

1988 (1)

Anastassiou, C.

Anderson, D.

Barthelemy, A.

A. Barthelemy, C. Froehly, and M. Shalaby, Proc. SPIE 2041, 104 (1994).
[CrossRef]

Carlsson, A. H.

Carmon, T.

Carvalho, M. I.

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, Appl. Phys. Lett. 68, 1763 (1996).
[CrossRef]

Christodoulides, D. N.

T. Carmon, C. Anastassiou, S. Lan, D. Kip, Z. H. Musslimani, M. Segev, and D. N. Christodoulides, Opt. Lett. 25, 1113 (2000).
[CrossRef]

Z. H. Musslimani, M. Segev, D. N. Christodoulides, and M. Soljačić, Phys. Rev. Lett. 84, 1164 (2000).
[CrossRef] [PubMed]

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, Appl. Phys. Lett. 68, 1763 (1996).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 53 (1988).
[CrossRef] [PubMed]

Denz, C.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Yu. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Firth, W. J.

E. A. Ostrovskaya, Yu. S. Kivshar, D. V. Skryabin, and W. J. Firth, Phys. Rev. Lett. 83, 296 (1999).
[CrossRef]

Froehly, C.

A. Barthelemy, C. Froehly, and M. Shalaby, Proc. SPIE 2041, 104 (1994).
[CrossRef]

García-Ripoll, J. J.

J. J. García-Ripoll, V. Perez-García, E. A. Ostrovskaya, and Yu. S. Kivshar, Phys. Rev. Lett. 85, 82 (2000).
[CrossRef]

Geisser, M.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Yu. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Ghatak, A.

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, Cambridge, 1998), Chap.  8.
[CrossRef]

Haelterman, M.

See, e.g., M. Haelterman and A. P. Sheppard, Phys. Rev. E 49, 3376 (1994).
[CrossRef]

Joseph, R. I.

Kip, D.

Kivshar, Yu. S.

J. J. García-Ripoll, V. Perez-García, E. A. Ostrovskaya, and Yu. S. Kivshar, Phys. Rev. Lett. 85, 82 (2000).
[CrossRef]

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Yu. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

J. N. Malmberg, A. H. Carlsson, D. Anderson, M. Lisak, E. A. Ostrovskaya, and Yu. S. Kivshar, Opt. Lett. 25, 643 (2000).
[CrossRef]

E. A. Ostrovskaya, Yu. S. Kivshar, D. V. Skryabin, and W. J. Firth, Phys. Rev. Lett. 83, 296 (1999).
[CrossRef]

Krolikowski, W.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Yu. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Lan, S.

Lisak, M.

Luther-Davies, B.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Yu. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Malmberg, J. N.

McCarthy, G.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Yu. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Musslimani, Z. H.

Z. H. Musslimani, M. Segev, D. N. Christodoulides, and M. Soljačić, Phys. Rev. Lett. 84, 1164 (2000).
[CrossRef] [PubMed]

T. Carmon, C. Anastassiou, S. Lan, D. Kip, Z. H. Musslimani, M. Segev, and D. N. Christodoulides, Opt. Lett. 25, 1113 (2000).
[CrossRef]

Ostrovskaya, E. A.

J. J. García-Ripoll, V. Perez-García, E. A. Ostrovskaya, and Yu. S. Kivshar, Phys. Rev. Lett. 85, 82 (2000).
[CrossRef]

J. N. Malmberg, A. H. Carlsson, D. Anderson, M. Lisak, E. A. Ostrovskaya, and Yu. S. Kivshar, Opt. Lett. 25, 643 (2000).
[CrossRef]

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Yu. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

E. A. Ostrovskaya, Yu. S. Kivshar, D. V. Skryabin, and W. J. Firth, Phys. Rev. Lett. 83, 296 (1999).
[CrossRef]

Perez-García, V.

J. J. García-Ripoll, V. Perez-García, E. A. Ostrovskaya, and Yu. S. Kivshar, Phys. Rev. Lett. 85, 82 (2000).
[CrossRef]

Sears, S.

M. Soljačić, S. Sears, and M. Segev, Phys. Rev. Lett. 81, 4851 (1998).
[CrossRef]

Segev, M.

T. Carmon, C. Anastassiou, S. Lan, D. Kip, Z. H. Musslimani, M. Segev, and D. N. Christodoulides, Opt. Lett. 25, 1113 (2000).
[CrossRef]

Z. H. Musslimani, M. Segev, D. N. Christodoulides, and M. Soljačić, Phys. Rev. Lett. 84, 1164 (2000).
[CrossRef] [PubMed]

G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

M. Soljačić, S. Sears, and M. Segev, Phys. Rev. Lett. 81, 4851 (1998).
[CrossRef]

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, Appl. Phys. Lett. 68, 1763 (1996).
[CrossRef]

Shalaby, M.

A. Barthelemy, C. Froehly, and M. Shalaby, Proc. SPIE 2041, 104 (1994).
[CrossRef]

Sheppard, A. P.

See, e.g., M. Haelterman and A. P. Sheppard, Phys. Rev. E 49, 3376 (1994).
[CrossRef]

Singh, S. R.

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, Appl. Phys. Lett. 68, 1763 (1996).
[CrossRef]

Skryabin, D. V.

E. A. Ostrovskaya, Yu. S. Kivshar, D. V. Skryabin, and W. J. Firth, Phys. Rev. Lett. 83, 296 (1999).
[CrossRef]

Soljacic, M.

Z. H. Musslimani, M. Segev, D. N. Christodoulides, and M. Soljačić, Phys. Rev. Lett. 84, 1164 (2000).
[CrossRef] [PubMed]

M. Soljačić, S. Sears, and M. Segev, Phys. Rev. Lett. 81, 4851 (1998).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

Thyagarajan, K.

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, Cambridge, 1998), Chap.  8.
[CrossRef]

Weilnau, C.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Yu. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, Appl. Phys. Lett. 68, 1763 (1996).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. E (1)

See, e.g., M. Haelterman and A. P. Sheppard, Phys. Rev. E 49, 3376 (1994).
[CrossRef]

Phys. Rev. Lett. (5)

Z. H. Musslimani, M. Segev, D. N. Christodoulides, and M. Soljačić, Phys. Rev. Lett. 84, 1164 (2000).
[CrossRef] [PubMed]

J. J. García-Ripoll, V. Perez-García, E. A. Ostrovskaya, and Yu. S. Kivshar, Phys. Rev. Lett. 85, 82 (2000).
[CrossRef]

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Yu. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

E. A. Ostrovskaya, Yu. S. Kivshar, D. V. Skryabin, and W. J. Firth, Phys. Rev. Lett. 83, 296 (1999).
[CrossRef]

M. Soljačić, S. Sears, and M. Segev, Phys. Rev. Lett. 81, 4851 (1998).
[CrossRef]

Proc. SPIE (1)

A. Barthelemy, C. Froehly, and M. Shalaby, Proc. SPIE 2041, 104 (1994).
[CrossRef]

Science (1)

G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

Other (1)

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, Cambridge, 1998), Chap.  8.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Dipole-mode soliton components obtained numerically for s=0.5. (b) Profiles of the dipole-mode vector soliton components at s=0.5: solid curves, numerical solutions; dotted curves, variational solutions for λ=0.5. (c) Power diagram of the dipole-mode soliton family shown for the partial (Pu and Pw) and total Ptot powers. Solid curves, results of the variational approach; points, numerical results.

Fig. 2
Fig. 2

Examples of two-component soliton solutions of Eqs.  (2) at s=0.5 for (a), (b) a quadrupole m=2 at λ=0.46 and (c), (d) a dodecagon multipole m=6 at λ=0.6.

Fig. 3
Fig. 3

Experimental observation of the quadrupole solitons: (a) input beams; (b) individual propagation; (c) simultaneous propagation, the formation of a quadrupole structure; (d) effect of a misalignment of the components with the initial upward shift of the fundamental beam. Input powers are 3.9 and 3.3 μW.

Equations (3)

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

iE1,2z+ΔE1,2-E1,21+E12+E22=0,
Δu-u+FIu=0,Δw-λw+FIw=0,
Δru+s-11-s-f1u,wu=0,Δrw+s-11-sλ-m2s/r2-f2u,ww=0,

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