Abstract

We investigate the formation and evolution of spatial solitons with light beams propagating in quadratic nonlinear media under conditions for second-harmonic generation in quasi-phase-matched samples. We study the properties of the solitons as a function of the periodicity of the domain-reversal process involved in the quasi-phase-matching techniques, and we show the effects introduced by random shifts of the nominal domain length.

© 1997 Optical Society of America

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  1. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
    [CrossRef]
  2. M. Houe and P. D. Townsend, J. Phys. D 28, 1747 (1995).
    [CrossRef]
  3. L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, J. Opt. Soc. Am. B 12, 2102 (1995).
    [CrossRef]
  4. Yu. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976).
  5. R. Schiek, J. Opt. Soc. Am. B 10, 1848 (1993).
    [CrossRef]
  6. M. J. Werner and P. D. Drummond, J. Opt. Soc. Am. B 10, 2390 (1993).
    [CrossRef]
  7. K. Hayata and M. Koshiba, Phys. Rev. Lett. 71, 3275 (1993).
    [CrossRef] [PubMed]
  8. C. R. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994).
    [CrossRef]
  9. L. Torner, C. R. Menyuk, and G. I. Stegeman, J. Opt. Soc. Am. B 12, 889 (1995); L. Torner and E. M. Wright, J. Opt. Soc. Am. B 13, 864 (1996).
    [CrossRef]
  10. A. V. Buryak and Y. S. Kivshar, Opt. Lett. 19, 1612 (1994); A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
    [CrossRef] [PubMed]
  11. 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]
  12. R. Schiek, Y. Baek, and G. I. Stegeman, Phys. Rev. E 53, 1138 (1996).
    [CrossRef]
  13. G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
    [CrossRef]
  14. C. Bosshard, G. Knöpfle, P. Prêtre, S. Follonier, C. Serbutoviez, and P. Günter, Opt. Eng. 34, 1951 (1995).
    [CrossRef]
  15. L. Torner, J. P. Torres, D. V. Petrov, and G. I. Stegeman, in Nonlinear Guided Waves and Their Applications, Vol. 15 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 65.
  16. N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1990).
  17. M. S. Green, J. Chem. Phys. 20, 1281 (1952); R. Kubo, Can. J. Phys. 34, 1274 (1956).
    [CrossRef]
  18. Notice that phase fluctuations with transverse variations, e.g., random chirps, do affect the amplitude and phase of the nonlinear Schrödinger equation solitons.
  19. F. G. Bass, Yu. S. Kivshar, V. V. Konotop, and Yu. A. Sinitsyn, Phys. Rep. 157, 63 (1988).
    [CrossRef]
  20. F. Abdullaev, S. Darmanyan, and P. Khabibullaev, Optical Solitons (Springer-Verlag, Berlin, 1993), Chap. 4.
  21. We used the RAN1 and GASDEV routines presented in W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1992).
  22. L. Torner, D. Mihalache, D. Mazilu, and N. N. Akhmediev, Opt. Lett. 20, 2183 (1995).
    [CrossRef]
  23. B. A. Malomed, D. Anderson, A. Berntson, M. Florjancyzk, and M. Lisak, Pure Appl. Opt. 5, 941 (1996).
    [CrossRef]
  24. S. Helmfrid and G. Arvidsson, J. Opt. Soc. Am. B 8, 797 (1991).
    [CrossRef]
  25. S. K. Wong, IEEE J. Quantum Electron. 32, 1560 (1996).
    [CrossRef]

1996 (4)

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

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

B. A. Malomed, D. Anderson, A. Berntson, M. Florjancyzk, and M. Lisak, Pure Appl. Opt. 5, 941 (1996).
[CrossRef]

S. K. Wong, IEEE J. Quantum Electron. 32, 1560 (1996).
[CrossRef]

1995 (5)

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]

L. Torner, D. Mihalache, D. Mazilu, and N. N. Akhmediev, Opt. Lett. 20, 2183 (1995).
[CrossRef]

C. Bosshard, G. Knöpfle, P. Prêtre, S. Follonier, C. Serbutoviez, and P. Günter, Opt. Eng. 34, 1951 (1995).
[CrossRef]

M. Houe and P. D. Townsend, J. Phys. D 28, 1747 (1995).
[CrossRef]

L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, J. Opt. Soc. Am. B 12, 2102 (1995).
[CrossRef]

1994 (1)

1993 (3)

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

1991 (1)

1988 (1)

F. G. Bass, Yu. S. Kivshar, V. V. Konotop, and Yu. A. Sinitsyn, Phys. Rep. 157, 63 (1988).
[CrossRef]

1976 (1)

Yu. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976).

Akhmediev, N. N.

Anderson, D.

B. A. Malomed, D. Anderson, A. Berntson, M. Florjancyzk, and M. Lisak, Pure Appl. Opt. 5, 941 (1996).
[CrossRef]

Arvidsson, G.

Baek, Y.

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

Bass, F. G.

F. G. Bass, Yu. S. Kivshar, V. V. Konotop, and Yu. A. Sinitsyn, Phys. Rep. 157, 63 (1988).
[CrossRef]

Berntson, A.

B. A. Malomed, D. Anderson, A. Berntson, M. Florjancyzk, and M. Lisak, Pure Appl. Opt. 5, 941 (1996).
[CrossRef]

Bosenberg, W. R.

Bosshard, C.

C. Bosshard, G. Knöpfle, P. Prêtre, S. Follonier, C. Serbutoviez, and P. Günter, Opt. Eng. 34, 1951 (1995).
[CrossRef]

Byer, R. L.

L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, J. Opt. Soc. Am. B 12, 2102 (1995).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Drummond, P. D.

Eckardt, R. C.

Fejer, M. M.

L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, J. Opt. Soc. Am. B 12, 2102 (1995).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Florjancyzk, M.

B. A. Malomed, D. Anderson, A. Berntson, M. Florjancyzk, and M. Lisak, Pure Appl. Opt. 5, 941 (1996).
[CrossRef]

Follonier, S.

C. Bosshard, G. Knöpfle, P. Prêtre, S. Follonier, C. Serbutoviez, and P. Günter, Opt. Eng. 34, 1951 (1995).
[CrossRef]

Günter, P.

C. Bosshard, G. Knöpfle, P. Prêtre, S. Follonier, C. Serbutoviez, and P. Günter, Opt. Eng. 34, 1951 (1995).
[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, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Hayata, K.

K. Hayata and M. Koshiba, Phys. Rev. Lett. 71, 3275 (1993).
[CrossRef] [PubMed]

Helmfrid, S.

Houe, M.

M. Houe and P. D. Townsend, J. Phys. D 28, 1747 (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).
[CrossRef]

Karamzin, Yu. N.

Yu. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976).

Kivshar, Yu. S.

F. G. Bass, Yu. S. Kivshar, V. V. Konotop, and Yu. A. Sinitsyn, Phys. Rep. 157, 63 (1988).
[CrossRef]

Knöpfle, G.

C. Bosshard, G. Knöpfle, P. Prêtre, S. Follonier, C. Serbutoviez, and P. Günter, Opt. Eng. 34, 1951 (1995).
[CrossRef]

Konotop, V. V.

F. G. Bass, Yu. S. Kivshar, V. V. Konotop, and Yu. A. Sinitsyn, Phys. Rep. 157, 63 (1988).
[CrossRef]

Koshiba, M.

K. Hayata and M. Koshiba, Phys. Rev. Lett. 71, 3275 (1993).
[CrossRef] [PubMed]

Lisak, M.

B. A. Malomed, D. Anderson, A. Berntson, M. Florjancyzk, and M. Lisak, Pure Appl. Opt. 5, 941 (1996).
[CrossRef]

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Malomed, B. A.

B. A. Malomed, D. Anderson, A. Berntson, M. Florjancyzk, and M. Lisak, Pure Appl. Opt. 5, 941 (1996).
[CrossRef]

Mazilu, D.

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]

C. R. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994).
[CrossRef]

Mihalache, D.

Myers, L. E.

Pierce, J. W.

Prêtre, P.

C. Bosshard, G. Knöpfle, P. Prêtre, S. Follonier, C. Serbutoviez, and P. Günter, Opt. Eng. 34, 1951 (1995).
[CrossRef]

Schiek, R.

Serbutoviez, C.

C. Bosshard, G. Knöpfle, P. Prêtre, S. Follonier, C. Serbutoviez, and P. Günter, Opt. Eng. 34, 1951 (1995).
[CrossRef]

Sinitsyn, Yu. A.

F. G. Bass, Yu. S. Kivshar, V. V. Konotop, and Yu. A. Sinitsyn, Phys. Rep. 157, 63 (1988).
[CrossRef]

Stegeman, G. I.

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

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, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Sukhorukov, A. P.

Yu. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976).

Torner, L.

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, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

L. Torner, D. Mihalache, D. Mazilu, and N. N. Akhmediev, Opt. Lett. 20, 2183 (1995).
[CrossRef]

C. R. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994).
[CrossRef]

Torruellas, W. E.

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]

Townsend, P. D.

M. Houe and P. D. Townsend, J. Phys. D 28, 1747 (1995).
[CrossRef]

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]

Werner, M. J.

Wong, S. K.

S. K. Wong, IEEE J. Quantum Electron. 32, 1560 (1996).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

S. K. Wong, IEEE J. Quantum Electron. 32, 1560 (1996).
[CrossRef]

J. Opt. Soc. Am. B (5)

J. Phys. D (1)

M. Houe and P. D. Townsend, J. Phys. D 28, 1747 (1995).
[CrossRef]

Opt. Eng. (1)

C. Bosshard, G. Knöpfle, P. Prêtre, S. Follonier, C. Serbutoviez, and P. Günter, Opt. Eng. 34, 1951 (1995).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

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

Phys. Rep. (1)

F. G. Bass, Yu. S. Kivshar, V. V. Konotop, and Yu. A. Sinitsyn, Phys. Rep. 157, 63 (1988).
[CrossRef]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (2)

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]

K. Hayata and M. Koshiba, Phys. Rev. Lett. 71, 3275 (1993).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

B. A. Malomed, D. Anderson, A. Berntson, M. Florjancyzk, and M. Lisak, Pure Appl. Opt. 5, 941 (1996).
[CrossRef]

Sov. Phys. JETP (1)

Yu. N. Karamzin and A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976).

Other (8)

L. Torner, C. R. Menyuk, and G. I. Stegeman, J. Opt. Soc. Am. B 12, 889 (1995); L. Torner and E. M. Wright, J. Opt. Soc. Am. B 13, 864 (1996).
[CrossRef]

A. V. Buryak and Y. S. Kivshar, Opt. Lett. 19, 1612 (1994); A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
[CrossRef] [PubMed]

F. Abdullaev, S. Darmanyan, and P. Khabibullaev, Optical Solitons (Springer-Verlag, Berlin, 1993), Chap. 4.

We used the RAN1 and GASDEV routines presented in W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1992).

L. Torner, J. P. Torres, D. V. Petrov, and G. I. Stegeman, in Nonlinear Guided Waves and Their Applications, Vol. 15 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 65.

N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1990).

M. S. Green, J. Chem. Phys. 20, 1281 (1952); R. Kubo, Can. J. Phys. 34, 1274 (1956).
[CrossRef]

Notice that phase fluctuations with transverse variations, e.g., random chirps, do affect the amplitude and phase of the nonlinear Schrödinger equation solitons.

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Figures (7)

Fig. 1
Fig. 1

Periodic domain reversal in 1st-order QPM. The function d(ξ) is scaled to the relevant nonlinear coefficient of the material, so ideally it reverses sign from +1 to -1 periodically.

Fig. 2
Fig. 2

Typical evolution of the amplitude of the fundamental and the second-harmonic beams, normalized to the corresponding input values, as a function of the propagation coordinate in a quasi-phase-matched sample. Solid curves, evolution from the full governing equations; dashed curves, evolution from truncated relations (6). Conditions: Δτ=0, A=4, B=3, a=2, b=2. In (a), ϕ0=π/2; in (b), ϕ0=π/2 and ϕ0=0. In (b) only the evolution of the amplitude of the fundamental beam is shown.

Fig. 3
Fig. 3

Output beams at ξ=10 for the conditions shown in Fig. 2(b).

Fig. 4
Fig. 4

Typical output fundamental beams at ξ=10, propagated in a 1st-order quasi-phase-matched sample with random duty-cycle domain-length shifts. Dashed curves, output for an ideal structure; solid curves, outputs for different realizations of the random shifts. In (a), σ2=0.1; in (b), σ2=0.2. Conditions: Δτ/τ=0.02, A=4, a=1, and B=0.

Fig. 5
Fig. 5

Same as in Fig. 4, but for A=6.

Fig. 6
Fig. 6

Typical evolution of the fundamental beam under the conditions of Fig. 4(b).

Fig. 7
Fig. 7

Same as in Fig. 4, but for random-walk domain-length shifts. In (a), σ2=0.02; in (b), σ2=0.04.

Equations (17)

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

i a1ξ-r22a1+d(ξ)a1*a2 exp(-iβξ)=0,
i a2ξ-α22a2-iδδˆ·a2+d(ξ)a12 exp(iβξ)=0,
dm(ξ)=n=-Cn exp(inqmξ)
C2n=0,C±(2n+1)=±2iπ(2n+1).
=2πΔττm(τm+Δτ).
i a1ξ-r22a1+C1a1*a2 exp(-iξ)
=-C2a1*a2 exp[i(β-2)ξ]+,
i a2ξ-α22a2+C-1a12 exp(iξ)
=-C-2a12 exp[-i(β-2)ξ]+.
i a1ξ-r22a1-i 2πa1*a2 exp(-iξ)0,
i a2ξ-α22a2+i 2πa12 exp(iξ)0.
a1(s, ξ=0)=A secha(s),
a2(s, ξ=0)=B sechb(s)exp(iϕ0),
ξ˜n=nξ˜0+Rn,
ξ˜n=ξ˜n-1+Rn.
i a1ξ-r22a1+γ1a1*a2exp(-iξ)0,
i a2ξ-α22a2+γ2a12 exp(iξ)0,

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