Abstract

We consider unseeded type  I second-harmonic generation in quasi-phase-matched quadratic nonlinear materials and derive an accurate analytical expression for the evolution of the average intensity. The intensity-dependent nonlinear phase mismatch that is due to the cubic nonlinearity induced by quasi phase matching is found. The equivalent formula for the intensity of maximum conversion, the crossing of which changes the one-period nonlinear phase shift of the fundamental abruptly by π, corrects earlier estimates [Opt.  Lett.    23, 506 (1998)] by a factor of 5.3. We find the crystal lengths that are necessary to obtain an optimal flat phase versus intensity response on either side of this separatrix intensity.

© 2001 Optical Society of America

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References

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  1. For a review, see G. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
    [CrossRef]
  2. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
    [CrossRef]
  3. M. M. Fejer, in Beam Shaping and Control with Nonlinear Optics, F. Kajzar and R. Reinisch, eds. (Plenum, New York, 1998), pp. 375–406.
  4. A. Saher Helmy, D. C. Hutchings, T. C. Kleckner, J. H. Marsh, A. C. Bryce, J. M. Arnold, C. R. Stanley, J. S. Aitchison, C. T. A. Brown, K. Moutzouris, and M. Ebrahimzadeh, Opt. Lett. 25, 1370 (2000).
    [CrossRef]
  5. C. Balslev Clausen, O. Bang, and Y. S. Kivshar, Phys. Rev. Lett. 78, 4749 (1997).
    [CrossRef]
  6. J. F. Corney and O. Bang, “Solitons in quadratic nonlinear photonic crystals,” March20, 2001, arXiv.org e-Print archive, Preprint nlin.PS/0007042, http://xxx.lanl.gov .
  7. O. Bang, C. Balslev Clausen, P. L. Christiansen, and L. Torner, Opt. Lett. 24, 1413 (1999).
    [CrossRef]
  8. A. Kobyakov, F. Lederer, O. Bang, and Y. S. Kivshar, Opt. Lett. 23, 506 (1998).
    [CrossRef]
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    [CrossRef]
  10. L. S. Telegin and A. S. Chirkin, Kvant. Elektron. 9, 2086 (1982) Sov. J. Quantum Electron. 12, 1354 (1982).
  11. Y. Zhao, G. Town, and M. Sceats, Opt. Commun. 115, 129 (1995).
    [CrossRef]
  12. C. R. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994).
    [CrossRef]
  13. O. Bang, J. Opt. Soc. Am. B 14, 51 (1997).
    [CrossRef]
  14. O. Bang, Y. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, Phys. Rev. E 58, 5057 (1998).
    [CrossRef]
  15. P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Killius, and G. Valiulis, in Digest of Nonlinear Guided Waves and Their Applications (Optical Society of America, Washington, D.C., 2001), pp. 179–181.

2000 (1)

1999 (1)

1998 (2)

A. Kobyakov, F. Lederer, O. Bang, and Y. S. Kivshar, Opt. Lett. 23, 506 (1998).
[CrossRef]

O. Bang, Y. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, Phys. Rev. E 58, 5057 (1998).
[CrossRef]

1997 (2)

O. Bang, J. Opt. Soc. Am. B 14, 51 (1997).
[CrossRef]

C. Balslev Clausen, O. Bang, and Y. S. Kivshar, Phys. Rev. Lett. 78, 4749 (1997).
[CrossRef]

1996 (1)

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

1995 (1)

Y. Zhao, G. Town, and M. Sceats, Opt. Commun. 115, 129 (1995).
[CrossRef]

1994 (1)

1991 (1)

1982 (1)

L. S. Telegin and A. S. Chirkin, Kvant. Elektron. 9, 2086 (1982) Sov. J. Quantum Electron. 12, 1354 (1982).

1962 (1)

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

Aitchison, J. S.

Armstrong, J. A.

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

Arnold, J. M.

Balslev Clausen, C.

O. Bang, C. Balslev Clausen, P. L. Christiansen, and L. Torner, Opt. Lett. 24, 1413 (1999).
[CrossRef]

C. Balslev Clausen, O. Bang, and Y. S. Kivshar, Phys. Rev. Lett. 78, 4749 (1997).
[CrossRef]

Banerjee, P. P.

Bang, O.

O. Bang, C. Balslev Clausen, P. L. Christiansen, and L. Torner, Opt. Lett. 24, 1413 (1999).
[CrossRef]

A. Kobyakov, F. Lederer, O. Bang, and Y. S. Kivshar, Opt. Lett. 23, 506 (1998).
[CrossRef]

O. Bang, Y. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, Phys. Rev. E 58, 5057 (1998).
[CrossRef]

C. Balslev Clausen, O. Bang, and Y. S. Kivshar, Phys. Rev. Lett. 78, 4749 (1997).
[CrossRef]

O. Bang, J. Opt. Soc. Am. B 14, 51 (1997).
[CrossRef]

J. F. Corney and O. Bang, “Solitons in quadratic nonlinear photonic crystals,” March20, 2001, arXiv.org e-Print archive, Preprint nlin.PS/0007042, http://xxx.lanl.gov .

Bloembergen, N.

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

Bramati, A.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Killius, and G. Valiulis, in Digest of Nonlinear Guided Waves and Their Applications (Optical Society of America, Washington, D.C., 2001), pp. 179–181.

Brown, C. T. A.

Bryce, A. C.

Buryak, A. V.

O. Bang, Y. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, Phys. Rev. E 58, 5057 (1998).
[CrossRef]

Caimi, F. C.

Chinaglia, W.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Killius, and G. Valiulis, in Digest of Nonlinear Guided Waves and Their Applications (Optical Society of America, Washington, D.C., 2001), pp. 179–181.

Chirkin, A. S.

L. S. Telegin and A. S. Chirkin, Kvant. Elektron. 9, 2086 (1982) Sov. J. Quantum Electron. 12, 1354 (1982).

Choe, W.

Christiansen, P. L.

Conti, C.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Killius, and G. Valiulis, in Digest of Nonlinear Guided Waves and Their Applications (Optical Society of America, Washington, D.C., 2001), pp. 179–181.

Corney, J. F.

J. F. Corney and O. Bang, “Solitons in quadratic nonlinear photonic crystals,” March20, 2001, arXiv.org e-Print archive, Preprint nlin.PS/0007042, http://xxx.lanl.gov .

De Rossi, A.

O. Bang, Y. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, Phys. Rev. E 58, 5057 (1998).
[CrossRef]

Di Trapani, P.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Killius, and G. Valiulis, in Digest of Nonlinear Guided Waves and Their Applications (Optical Society of America, Washington, D.C., 2001), pp. 179–181.

Ducuing, J.

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

Ebrahimzadeh, M.

Fejer, M. M.

M. M. Fejer, in Beam Shaping and Control with Nonlinear Optics, F. Kajzar and R. Reinisch, eds. (Plenum, New York, 1998), pp. 375–406.

Hagan, D. J.

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

Hutchings, D. C.

Killius, J.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Killius, and G. Valiulis, in Digest of Nonlinear Guided Waves and Their Applications (Optical Society of America, Washington, D.C., 2001), pp. 179–181.

Kivshar, Y. S.

O. Bang, Y. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, Phys. Rev. E 58, 5057 (1998).
[CrossRef]

A. Kobyakov, F. Lederer, O. Bang, and Y. S. Kivshar, Opt. Lett. 23, 506 (1998).
[CrossRef]

C. Balslev Clausen, O. Bang, and Y. S. Kivshar, Phys. Rev. Lett. 78, 4749 (1997).
[CrossRef]

Kleckner, T. C.

Kobyakov, A.

Lederer, F.

Marsh, J. H.

Menyuk, C. R.

Minardi, S.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Killius, and G. Valiulis, in Digest of Nonlinear Guided Waves and Their Applications (Optical Society of America, Washington, D.C., 2001), pp. 179–181.

Moutzouris, K.

Pershan, P. S.

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

Saher Helmy, A.

Sceats, M.

Y. Zhao, G. Town, and M. Sceats, Opt. Commun. 115, 129 (1995).
[CrossRef]

Schiek, R.

Stanley, C. R.

Stegeman, G.

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

Telegin, L. S.

L. S. Telegin and A. S. Chirkin, Kvant. Elektron. 9, 2086 (1982) Sov. J. Quantum Electron. 12, 1354 (1982).

Torner, L.

Town, G.

Y. Zhao, G. Town, and M. Sceats, Opt. Commun. 115, 129 (1995).
[CrossRef]

Trillo, S.

O. Bang, Y. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, Phys. Rev. E 58, 5057 (1998).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Killius, and G. Valiulis, in Digest of Nonlinear Guided Waves and Their Applications (Optical Society of America, Washington, D.C., 2001), pp. 179–181.

Valiulis, G.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Killius, and G. Valiulis, in Digest of Nonlinear Guided Waves and Their Applications (Optical Society of America, Washington, D.C., 2001), pp. 179–181.

Zhao, Y.

Y. Zhao, G. Town, and M. Sceats, Opt. Commun. 115, 129 (1995).
[CrossRef]

J. Opt. Soc. Am. B (3)

Kvant. Elektron. (1)

L. S. Telegin and A. S. Chirkin, Kvant. Elektron. 9, 2086 (1982) Sov. J. Quantum Electron. 12, 1354 (1982).

Opt. Commun. (1)

Y. Zhao, G. Town, and M. Sceats, Opt. Commun. 115, 129 (1995).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

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

Phys. Rev. (1)

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

Phys. Rev. E (1)

O. Bang, Y. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, Phys. Rev. E 58, 5057 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

C. Balslev Clausen, O. Bang, and Y. S. Kivshar, Phys. Rev. Lett. 78, 4749 (1997).
[CrossRef]

Other (3)

J. F. Corney and O. Bang, “Solitons in quadratic nonlinear photonic crystals,” March20, 2001, arXiv.org e-Print archive, Preprint nlin.PS/0007042, http://xxx.lanl.gov .

M. M. Fejer, in Beam Shaping and Control with Nonlinear Optics, F. Kajzar and R. Reinisch, eds. (Plenum, New York, 1998), pp. 375–406.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, S. Trillo, C. Conti, J. Killius, and G. Valiulis, in Digest of Nonlinear Guided Waves and Their Applications (Optical Society of America, Washington, D.C., 2001), pp. 179–181.

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Figures (3)

Fig. 1
Fig. 1

Nonlinear phase shift of the FW versus intensity I for bulk LiNbO3, as predicted by the averaged model [Eqs.  (4) and (5)] (dashed curve) and found numerically from Eqs.  (1) and (2) (solid curve). The QPM domain length is Ld=8.8 μm, and the crystal length is L=351 μm 174 μm for unseeded (seeded) SHG. The vertical dotted lines mark the separatrices Is and Is0.

Fig. 2
Fig. 2

Nonlinear phase shift of the FW versus intensity I for unseeded SHG in bulk LiNbO3. The QPM domain length is Ld=8.8 μm. The crystal lengths, L, given in the figure. The vertical dotted line marks the separatrix Is=44 GW/cm2.

Fig. 3
Fig. 3

Left, period versus intensity I for unseeded SHG in bulk QPM LiNbO3 with the QPM domain length given in micrometers at the curves. Right, optimal crystal length and separatrix intensity Is versus domain length.

Equations (8)

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idE1/dz+Gzχ1E1*E2expiΔkz=0,
idE2/dz+Gzχ2E12exp-iΔkz=0,
E1z=nwnzexpinκz,E2z=σnvnzexpinκ-βz,
idw0/dz+iρ1ω0*v0+γ1v02-γ2w02w0=0,
idv0/dz+βv0-iρ2ω02+2γ2w02v0=0,
nκwn0=χ1gn-1w0*v0,nκwn0=χ2gn+1w02.
uz=[u3 sn2(rz|m)+u0p]/[sn2(rz|m)+p],
Is=-η0n221-8π2βγ1,

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