Abstract

We analyze (1+1)- and (2+1)-dimensional self-guided beams (spatial solitons) due to three-wave parametric mixing (or type II second-harmonic generation) in diffractive optical materials with χ(2) nonlinearity. We also discuss the optimal conditions for observing self-trapping in different experimental geometries.

© 1997 Optical Society of America

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  1. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 6, 1918 (1962).
    [CrossRef]
  2. D. J. Kaup, A. Rieman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
    [CrossRef]
  3. V. E. Zakharov, S. L. Musher, and A. M. Rubenshik, Phys. Rep. 129, 286 (1985).
    [CrossRef]
  4. A. D. D. Craik, Wave Interactions and Fluid Flows (Cambridge U. Press, Cambridge, 1985).
  5. E. Ibragimov and A. Struthers, Opt. Lett. 21, 1582 (1996).
    [CrossRef] [PubMed]
  6. D. J. Kaup, Stud. Appl. Math. 59, 25 (1978).
  7. S. Trillo, Opt. Lett. 21, 1111 (1996).
    [CrossRef] [PubMed]
  8. Yu. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974); Sov. Phys. JETP 41, 414 (1976).
  9. A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981).
    [CrossRef]
  10. A. V. Buryak and Yu. S. Kivshar, Opt. Lett. 19, 1612 (1994); Phys. Lett. A 197, 407 (1995); L. Torner, C. M. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994).
    [CrossRef] [PubMed]
  11. D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
    [CrossRef] [PubMed]
  12. R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996).
    [CrossRef]
  13. L. Torner, C. M. Menyuk, W. E. Torruellas, and G. I. Stegeman, Opt. Lett. 20, 13 (1995); A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
    [CrossRef] [PubMed]
  14. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
    [CrossRef] [PubMed]
  15. H. T. Tran, Opt. Commun. 118, 581 (1995).
    [CrossRef]
  16. A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996); A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
    [CrossRef] [PubMed]
  17. Our analysis is for the case of the so-called fundamental solitons for which the functions V, U, and W do not have zeros and change monotonically with R. There are also possible multihump solitons; however, they are usually unstable [see, e.g., M. Haelterman, S. Trillo, and P. Ferro, Opt. Lett. 22, 84 (1996)].
    [CrossRef]
  18. D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53, 1940 (1996).
    [CrossRef]
  19. The power threshold is known to exist for the (2+1)-dimensional case [see L. Torner and E. Wright, J. Opt. Soc. Am. B 13, 864 (1996)].
    [CrossRef]
  20. H. He, P. D. Drummond, A. V. Buryak, and Yu. S. Kivshar, “Nonlinear dynamics of Simultons in parametric waveguides,” in Digest of the International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 156.
  21. A. V. Buryak and N. N. Akhmediev, Phys. Rev. E 50, 3126 (1994).
    [CrossRef]
  22. W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, Appl. Phys. Lett. 68, 1449 (1996).
    [CrossRef]

1996 (8)

1995 (4)

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

L. Torner, C. M. Menyuk, W. E. Torruellas, and G. I. Stegeman, Opt. Lett. 20, 13 (1995); A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
[CrossRef] [PubMed]

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

H. T. Tran, Opt. Commun. 118, 581 (1995).
[CrossRef]

1994 (2)

1985 (1)

V. E. Zakharov, S. L. Musher, and A. M. Rubenshik, Phys. Rep. 129, 286 (1985).
[CrossRef]

1981 (1)

A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981).
[CrossRef]

1979 (1)

D. J. Kaup, A. Rieman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
[CrossRef]

1978 (1)

D. J. Kaup, Stud. Appl. Math. 59, 25 (1978).

1974 (1)

Yu. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974); Sov. Phys. JETP 41, 414 (1976).

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Afanasjev, V. V.

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53, 1940 (1996).
[CrossRef]

Akhmediev, N. N.

A. V. Buryak and N. N. Akhmediev, Phys. Rev. E 50, 3126 (1994).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Assanto, G.

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, Appl. Phys. Lett. 68, 1449 (1996).
[CrossRef]

Baek, Y.

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996).
[CrossRef]

Bers, A.

D. J. Kaup, A. Rieman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Buryak, A. V.

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996); A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
[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, Opt. Lett. 19, 1612 (1994); Phys. Lett. A 197, 407 (1995); L. Torner, C. M. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994).
[CrossRef] [PubMed]

A. V. Buryak and N. N. Akhmediev, Phys. Rev. E 50, 3126 (1994).
[CrossRef]

H. He, P. D. Drummond, A. V. Buryak, and Yu. S. Kivshar, “Nonlinear dynamics of Simultons in parametric waveguides,” in Digest of the International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 156.

Craik, A. D. D.

A. D. D. Craik, Wave Interactions and Fluid Flows (Cambridge U. Press, Cambridge, 1985).

Drummond, P. D.

H. He, P. D. Drummond, A. V. Buryak, and Yu. S. Kivshar, “Nonlinear dynamics of Simultons in parametric waveguides,” in Digest of the International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 156.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Ferro, P.

Fuerst, R. A.

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, Appl. Phys. Lett. 68, 1449 (1996).
[CrossRef]

Haelterman, M.

Hagan, D. J.

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

He, H.

H. He, P. D. Drummond, A. V. Buryak, and Yu. S. Kivshar, “Nonlinear dynamics of Simultons in parametric waveguides,” in Digest of the International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 156.

Ibragimov, E.

Kanashov, A. A.

A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981).
[CrossRef]

Karamzin, Yu. N.

Yu. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974); Sov. Phys. JETP 41, 414 (1976).

Kaup, D. J.

D. J. Kaup, A. Rieman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
[CrossRef]

D. J. Kaup, Stud. Appl. Math. 59, 25 (1978).

Kivshar, Yu. S.

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996); A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
[CrossRef] [PubMed]

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53, 1940 (1996).
[CrossRef]

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, Opt. Lett. 19, 1612 (1994); Phys. Lett. A 197, 407 (1995); L. Torner, C. M. Menyuk, and G. I. Stegeman, Opt. Lett. 19, 1615 (1994).
[CrossRef] [PubMed]

H. He, P. D. Drummond, A. V. Buryak, and Yu. S. Kivshar, “Nonlinear dynamics of Simultons in parametric waveguides,” in Digest of the International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 156.

Lawrence, B. L.

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, Appl. Phys. Lett. 68, 1449 (1996).
[CrossRef]

Menyuk, C. M.

Menyuk, C. R.

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

Musher, S. L.

V. E. Zakharov, S. L. Musher, and A. M. Rubenshik, Phys. Rep. 129, 286 (1985).
[CrossRef]

Pelinovsky, D. E.

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53, 1940 (1996).
[CrossRef]

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

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Rieman, A.

D. J. Kaup, A. Rieman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
[CrossRef]

Rubenchik, A. M.

A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981).
[CrossRef]

Rubenshik, A. M.

V. E. Zakharov, S. L. Musher, and A. M. Rubenshik, Phys. Rep. 129, 286 (1985).
[CrossRef]

Schiek, R.

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996).
[CrossRef]

Stegeman, G. I.

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996).
[CrossRef]

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, Appl. Phys. Lett. 68, 1449 (1996).
[CrossRef]

L. Torner, C. M. Menyuk, W. E. Torruellas, and G. I. Stegeman, Opt. Lett. 20, 13 (1995); A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
[CrossRef] [PubMed]

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

Struthers, A.

Sukhorukov, A. P.

Yu. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974); Sov. Phys. JETP 41, 414 (1976).

Torner, L.

Torruellas, W. E.

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, Appl. Phys. Lett. 68, 1449 (1996).
[CrossRef]

L. Torner, C. M. Menyuk, W. E. Torruellas, and G. I. Stegeman, Opt. Lett. 20, 13 (1995); A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
[CrossRef] [PubMed]

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

Tran, H. T.

H. T. Tran, Opt. Commun. 118, 581 (1995).
[CrossRef]

Trillo, S.

Van Stryland, E. W.

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

Wang, Z.

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

Wright, E.

Zakharov, V. E.

V. E. Zakharov, S. L. Musher, and A. M. Rubenshik, Phys. Rep. 129, 286 (1985).
[CrossRef]

Appl. Phys. Lett. (1)

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, Appl. Phys. Lett. 68, 1449 (1996).
[CrossRef]

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

JETP Lett. (1)

Yu. N. Karamzin and A. P. Sukhorukov, JETP Lett. 20, 339 (1974); Sov. Phys. JETP 41, 414 (1976).

Opt. Commun. (1)

H. T. Tran, Opt. Commun. 118, 581 (1995).
[CrossRef]

Opt. Lett. (5)

Phys. Rep. (1)

V. E. Zakharov, S. L. Musher, and A. M. Rubenshik, Phys. Rep. 129, 286 (1985).
[CrossRef]

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Phys. Rev. E (3)

R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996).
[CrossRef]

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53, 1940 (1996).
[CrossRef]

A. V. Buryak and N. N. Akhmediev, Phys. Rev. E 50, 3126 (1994).
[CrossRef]

Phys. Rev. Lett. (3)

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

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

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996); A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
[CrossRef] [PubMed]

Physica D (1)

A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981).
[CrossRef]

Rev. Mod. Phys. (1)

D. J. Kaup, A. Rieman, and A. Bers, Rev. Mod. Phys. 51, 275 (1979).
[CrossRef]

Stud. Appl. Math. (1)

D. J. Kaup, Stud. Appl. Math. 59, 25 (1978).

Other (2)

A. D. D. Craik, Wave Interactions and Fluid Flows (Cambridge U. Press, Cambridge, 1985).

H. He, P. D. Drummond, A. V. Buryak, and Yu. S. Kivshar, “Nonlinear dynamics of Simultons in parametric waveguides,” in Digest of the International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 156.

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Figures (9)

Fig. 1
Fig. 1

Ratios of the soliton energy invariants in the plane of parameters (α, γ): (a) fraction of SH soliton energy, 4QW/Qtot; (b) difference of energy fractions at fundamental harmonics, Qunb/Qtot.

Fig. 2
Fig. 2

Ratios of the soliton energy invariants in the plane of parameters (βu, βv) for σ=1 and Δ=1: (a) fraction of SH soliton energy, 4QW/Qtot; (b) difference of energy fractions at fundamental harmonics, Qunb/Qtot.

Fig. 3
Fig. 3

Domains of the soliton existence and stability in the plane (βu, βv) for σ=1 and Δ=-1 for (a) the (1+1)-dimensional case, (b) the (2+1)-dimensional case.

Fig. 4
Fig. 4

Same as in Fig. 2, but for Δ=-1.

Fig. 5
Fig. 5

Existence–stability diagrams for three-wave χ(2) solitons in the (Qunb, Qtot) plane (σ=1.0, Δ=-1.0): (a) (1+1)-dimensional solitons; (b) (2+1)-dimensional solitons. The thick curves show the instability threshold, and the dashed domains correspond to the existence of unstable solitons. The qualitative dependencies H=H(Qtot) along the vertical (Qunb =constant) lines 1, 2, and 3 in (a) and (b) are presented at the bottom.

Fig. 6
Fig. 6

Evolution of several solitonlike initial beams on the plane (Qtot, H) for (a) Δ<0; (b) Δ>0. For all the cases σ =1.0, and there is no power unbalancing. The thick solid curves represent stable soliton branches; a thick dashed curve represents the unstable soliton branch; open circles indicate various initial conditions, and thin solid curves show their evolution. The dashed areas show the domains of the initial beam parameters that always lead to soliton generation. Point A in (a) corresponds to point A in Fig. 5(a).

Fig. 7
Fig. 7

Generation of solitons in upconversion from highly unbalanced fundamental beams: (a) evolution of the peak amplitudes, (b) evolution of the equivalent polarization angle at the fundamental frequency.

Fig. 8
Fig. 8

Same as in Fig. 7, but for an input with higher unbalancing.

Fig. 9
Fig. 9

Generation of a three-wave soliton via downconversion. Evolution of the peak amplitudes in the (1+1)-dimensional case.

Tables (1)

Tables Icon

Table 1 Parameters for the Optimal Generation of χ(2) Solitons

Equations (19)

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

2ik1 E1z+2E1+χ1E3E2* exp(-iΔkz)=0,
2ik2 E2z+2E2+χ2E3E1* exp(-iΔkz)=0,
2ik3 E3z+2E3+χ3E1E2 exp(iΔkz)=0,
Ly,eff=-+f12dy-+f22dy-+f32dy-+f1f2f3dy2
iσ vz+2v-σβvv+wu*=0,
i(2-σ) uz+2u-(2-σ)βuu+wv*=0,
2i wz+2w-2(βv+βu+Δ)w+uv=0,
H=-+[|w|2+|u|2+|v|2+2Δ|w|2-(uvw*+u*v*w)]dxdy
Qtot=-+[4|w|2+σ|v|2+(2-σ)|u|2]dxdy,
Qunb=-+[σ|v|2-(2-σ)|u|2]dxdy.
V+sRV-V+WU=0,
U+sRU-γU+WV=0,
W+sRW-αW+UV=0,
α2σβu+Δβv+1,γ(2-σ)σβuβv.
(Qtot, Qunb)(βv, βu)=QtotβvQunbβu-QtotβuQunbβv=0.
Q˜tot=Rs(4-n)[4QW+(2-σ)QU+σQV],
Pmin-+(|E3|2+|E1|2+|E2|2)dxdy,
Pmin-+(|E3|2+|E1|2+|E2|2)dx,
IminPminπr02=Q˜tot4πχbulk2r04.

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