Abstract

We demonstrate the possibility of self-trapping of optical beams by use of quasi phase matching in a counterpropagating configuration in quadratic media. We also show the predominant stability of these beams and estimate the power level required for their observation.

© 1999 Optical Society of America

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References

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  1. M. Segev and G. I. Stegeman, Phys. Today 51(8), 42 (1998), and references therein.
    [CrossRef]
  2. G. I. Stegeman, D. J. Hagan, and L. Torner, J. Opt. Quantum Electron. 28, 1691 (1996).
    [CrossRef]
  3. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
    [CrossRef]
  4. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992)G. M. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, Opt. Lett. 22, 1834 (1997).
    [CrossRef]
  5. V. V. Steblina, Yu. S. Kivshar, and A. V. Buryak, Opt. Lett. 23, 156 (1998)B. Costantini, C. De Angelis, A. Barthelemy, B. Bourliaguet, and V. Kermene, Opt. Lett. 23, 424 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  12. I. V. Barashenkov, D. E. Pelenovsky, and E. V. Zemlyanaya, Phys. Rev. Lett. 80, 5117 (1998)A. De Rossi, C. Conti, and S. Trillo, Phys. Rev. Lett. 81, 85 (1998).
    [CrossRef]
  13. Yu. S. Kivshar, D. E. Pelenovsky, T. Cretegny, and M. Peyrard, Phys. Rev. Lett. 80, 5032 (1998).
    [CrossRef]
  14. A. V. Buryak, Yu. S. Kivshar, and S. Trillo, J. Opt. Soc. Am. B 14, 3110 (1997).
    [CrossRef]
  15. R. C. Tu, Y. K. Su, and S. T. Chou, J. Appl. Phys. 85, 2398 (1999).
    [CrossRef]

1999 (1)

R. C. Tu, Y. K. Su, and S. T. Chou, J. Appl. Phys. 85, 2398 (1999).
[CrossRef]

1998 (5)

I. V. Barashenkov, D. E. Pelenovsky, and E. V. Zemlyanaya, Phys. Rev. Lett. 80, 5117 (1998)A. De Rossi, C. Conti, and S. Trillo, Phys. Rev. Lett. 81, 85 (1998).
[CrossRef]

Yu. S. Kivshar, D. E. Pelenovsky, T. Cretegny, and M. Peyrard, Phys. Rev. Lett. 80, 5032 (1998).
[CrossRef]

M. Segev and G. I. Stegeman, Phys. Today 51(8), 42 (1998), and references therein.
[CrossRef]

V. V. Steblina, Yu. S. Kivshar, and A. V. Buryak, Opt. Lett. 23, 156 (1998)B. Costantini, C. De Angelis, A. Barthelemy, B. Bourliaguet, and V. Kermene, Opt. Lett. 23, 424 (1998).
[CrossRef]

P. M. Lushnikov, P. Lodahl, and M. Saffman, Opt. Lett. 23, 1650 (1998).
[CrossRef]

1997 (3)

1996 (3)

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

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

Y. J. Ding and J. B. Khurgin, Opt. Lett. 21, 1445 (1996)G. D. Landry and T. A. Maldonado, Appl. Opt. 37, 7809 (1998).
[CrossRef] [PubMed]

1995 (1)

S. J. Hewlett and F. Ladouceur, J. Lightwave Technol. 13, 375 (1995).
[CrossRef]

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992)G. M. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, Opt. Lett. 22, 1834 (1997).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Bang, O.

C. B. Clausen, O. Bang, and Yu. Kivshar, Phys. Rev. Lett. 78, 4749 (1997)M. Cha, Opt. Lett. 23, 250 (1998)C. B. Clausen and L. Torner, Opt. Lett. 24, 7 (1999).
[CrossRef]

Barashenkov, I. V.

I. V. Barashenkov, D. E. Pelenovsky, and E. V. Zemlyanaya, Phys. Rev. Lett. 80, 5117 (1998)A. De Rossi, C. Conti, and S. Trillo, Phys. Rev. Lett. 81, 85 (1998).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Burns, W. K.

Buryak, A. V.

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992)G. M. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, Opt. Lett. 22, 1834 (1997).
[CrossRef]

Chou, S. T.

R. C. Tu, Y. K. Su, and S. T. Chou, J. Appl. Phys. 85, 2398 (1999).
[CrossRef]

Clausen, C. B.

C. B. Clausen, O. Bang, and Yu. Kivshar, Phys. Rev. Lett. 78, 4749 (1997)M. Cha, Opt. Lett. 23, 250 (1998)C. B. Clausen and L. Torner, Opt. Lett. 24, 7 (1999).
[CrossRef]

Cretegny, T.

Yu. S. Kivshar, D. E. Pelenovsky, T. Cretegny, and M. Peyrard, Phys. Rev. Lett. 80, 5032 (1998).
[CrossRef]

Ding, Y. J.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Fejer, M. M.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992)G. M. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, Opt. Lett. 22, 1834 (1997).
[CrossRef]

Hagan, D. J.

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

Hewlett, S. J.

S. J. Hewlett and F. Ladouceur, J. Lightwave Technol. 13, 375 (1995).
[CrossRef]

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992)G. M. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, Opt. Lett. 22, 1834 (1997).
[CrossRef]

Kang, J. B.

Khurgin, J. B.

Kivshar, Y.

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

Kivshar, Yu.

C. B. Clausen, O. Bang, and Yu. Kivshar, Phys. Rev. Lett. 78, 4749 (1997)M. Cha, Opt. Lett. 23, 250 (1998)C. B. Clausen and L. Torner, Opt. Lett. 24, 7 (1999).
[CrossRef]

Kivshar, Yu. S.

Ladouceur, F.

S. J. Hewlett and F. Ladouceur, J. Lightwave Technol. 13, 375 (1995).
[CrossRef]

Lodahl, P.

Lushnikov, P. M.

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992)G. M. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, Opt. Lett. 22, 1834 (1997).
[CrossRef]

Melinger, J. S.

Pelenovsky, D. E.

I. V. Barashenkov, D. E. Pelenovsky, and E. V. Zemlyanaya, Phys. Rev. Lett. 80, 5117 (1998)A. De Rossi, C. Conti, and S. Trillo, Phys. Rev. Lett. 81, 85 (1998).
[CrossRef]

Yu. S. Kivshar, D. E. Pelenovsky, T. Cretegny, and M. Peyrard, Phys. Rev. Lett. 80, 5032 (1998).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Peyrard, M.

Yu. S. Kivshar, D. E. Pelenovsky, T. Cretegny, and M. Peyrard, Phys. Rev. Lett. 80, 5032 (1998).
[CrossRef]

Saffman, M.

Segev, M.

M. Segev and G. I. Stegeman, Phys. Today 51(8), 42 (1998), and references therein.
[CrossRef]

Steblina, V. V.

Stegeman, G. I.

M. Segev and G. I. Stegeman, Phys. Today 51(8), 42 (1998), and references therein.
[CrossRef]

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

Su, Y. K.

R. C. Tu, Y. K. Su, and S. T. Chou, J. Appl. Phys. 85, 2398 (1999).
[CrossRef]

Torner, L.

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

Trillo, S.

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, J. Opt. Soc. Am. B 14, 3110 (1997).
[CrossRef]

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

Tu, R. C.

R. C. Tu, Y. K. Su, and S. T. Chou, J. Appl. Phys. 85, 2398 (1999).
[CrossRef]

Zemlyanaya, E. V.

I. V. Barashenkov, D. E. Pelenovsky, and E. V. Zemlyanaya, Phys. Rev. Lett. 80, 5117 (1998)A. De Rossi, C. Conti, and S. Trillo, Phys. Rev. Lett. 81, 85 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992)G. M. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, Opt. Lett. 22, 1834 (1997).
[CrossRef]

J. Appl. Phys. (1)

R. C. Tu, Y. K. Su, and S. T. Chou, J. Appl. Phys. 85, 2398 (1999).
[CrossRef]

J. Lightwave Technol. (1)

S. J. Hewlett and F. Ladouceur, J. Lightwave Technol. 13, 375 (1995).
[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. (4)

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
[CrossRef]

Phys. Rev. Lett. (4)

C. B. Clausen, O. Bang, and Yu. Kivshar, Phys. Rev. Lett. 78, 4749 (1997)M. Cha, Opt. Lett. 23, 250 (1998)C. B. Clausen and L. Torner, Opt. Lett. 24, 7 (1999).
[CrossRef]

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

I. V. Barashenkov, D. E. Pelenovsky, and E. V. Zemlyanaya, Phys. Rev. Lett. 80, 5117 (1998)A. De Rossi, C. Conti, and S. Trillo, Phys. Rev. Lett. 81, 85 (1998).
[CrossRef]

Yu. S. Kivshar, D. E. Pelenovsky, T. Cretegny, and M. Peyrard, Phys. Rev. Lett. 80, 5032 (1998).
[CrossRef]

Phys. Today (1)

M. Segev and G. I. Stegeman, Phys. Today 51(8), 42 (1998), and references therein.
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Parametric interaction of counterpropagating waves: (a) A nonlinear grating is used to couple forward-propagating + and backward-propagating - waves. (b) Square-wave function dz approximates the periodic modulation of the second-order nonlinear susceptibility.

Fig. 2
Fig. 2

Examples of stationary solitons that are due to counterpropagating QPM configuration. The profiles are calculated for the system of Eqs.  (5). Solid curves, forward-propagating waves; dashed curves, backward-propagating waves. Profiles A and B correspond to points A and B in Fig.  3.

Fig. 3
Fig. 3

(a) Existence (shaded) and instability (hatched) domains for the stationary solitons of the system of Eqs.  (5). The solid curves are defined by stability threshold condition  (3). (b), (c) Enlarged plots of unstable regions I and II. Soliton profiles calculated for points A and B (open circles) are shown in Fig.  2. The filled circle corresponds to the exact solution (see text).

Fig. 4
Fig. 4

Examples of unstable propagation of slightly perturbed stationary soliton A shown in Fig.  2. Two typical kinds of instability correspond to (a) increased and (b) re-duced soliton amplitudes. An unstable soliton with a slightly increased amplitude is clearly evolving to some stable state with an excited internal mode (see, e.g., Ref.  13).

Equations (6)

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iEω+z+2Eω+x2-α+Eω++A2ωEω-*=0,-iEω-z+2Eω-x2-α-Eω-+A2ωEω+*=0,iσE2ω+z+2E2ω+x2-σβ+E2ω++Aω+Eω-=0,-iσE2ω-z+2E2ω-x2-σβ-E2ω-+Aω-Eω+=0,
Qp1,p2=-+2p1Eω+2+2p2Eω-2+σp1-p2E2ω+2-E2ω-2dx,
Q1,Q2α+,α-=Q1α+Q2α--Q1α-Q2α+=0,
Q1α+Q1α+Q2α--Q1α-Q2α+<0,Q2α-Q1α+Q2α--Q1α-Q2α+<0.
iV+Z+2V+X2-V++W++W-V-*=0,-iV-Z+2V-X2-γV-+W++W-V+*=0,iσW+Z+2W+X2-σβW+W++2V+V-=0,-iσW-Z+2W-X2-σβW-W-+2V+V-=0,
Imin=Q˜minLeff4πχbulk2r04dl2,

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