Abstract

We consider the propagation of broad optical beams through slab waveguides with a purely quadratic nonlinearity and containing linear and nonlinear long-period quasi-phase-matching gratings. An exact Floquet analysis of the periodic, plane-wave solution shows that the periodicity can drastically alter the growth rate of the modulational instability but that it never completely removes the instability. The results are confirmed by direct numerical simulation as well as through a simpler, approximate theory for the averaged fields that accurately predicts the low-frequency part of the gain spectrum.

© 2002 Optical Society of America

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  23. P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  36. H. He, A. Arraf, C. M. de Sterke, P. D. Drummond, and B. A. Malomed, “Theory of modulational instability in Bragg gratings with quadratic nonlinearity,” Phys. Rev. E 59, 6064–6078 (1996).
    [CrossRef]
  37. T. Cretegny, T. Dauxois, S. Ruffo, and A. Torcini, “Localization and equipartition of energy in the β-FPU chain: chaotic breathers,” Physica D 121, 109–126 (1998).
    [CrossRef]
  38. 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, “Quasi phase matching in GaAs-AlAs superlattice waveguides through bandgap tuning by use of quantum-well intermixing,” Opt. Lett. 25, 1370–1372 (2000).
    [CrossRef]

2001

J. F. Corney and O. Bang, “Solitons in quadratic nonlinear photonic crystals,” Phys. Rev. E 64, 047601 (2001).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

J. F. Corney and O. Bang, “Modulational instability in periodic quadratic nonlinear materials,” Phys. Rev. Lett. 87, 133901 (2001).
[CrossRef] [PubMed]

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[CrossRef] [PubMed]

V. Couderc, E. Lopez Lago, C. Simos, and A. Barthélémy, “Experiments in quadratic spatial soliton generation and steering in a noncollinear geometry,” Opt. Lett. 26, 905–907 (2001).
[CrossRef]

O. Bang, T. W. Graversen, and J. F. Corney, “Accurate switching intensities and length scales in quasi-phase-matched materials,” Opt. Lett. 26, 1007–1009 (2001).
[CrossRef]

2000

1999

1998

1997

O. Bang, “Dynamical equations for wave packets in material with both quadratic and cubic response,” J. Opt. Soc. Am. B 14, 51–61 (1997).
[CrossRef]

O. Bang, Y. S. Kivshar, and A. Buryak, “Bright spatial solitons in defocusing Kerr media supported by cascaded nonlinearities,” Opt. Lett. 22, 1680–1682 (1997).
[CrossRef]

B. A. Malomed, P. D. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
[CrossRef]

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[CrossRef]

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

1996

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138–1141 (1996).
[CrossRef]

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, “All-optical switching by spatial walkoff compensation and solitary-wave locking,” Appl. Phys. Lett. 68, 1449–1451 (1996).
[CrossRef]

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity,” Opt. Commun. 123, 394–402 (1996).
[CrossRef]

H. He, A. Arraf, C. M. de Sterke, P. D. Drummond, and B. A. Malomed, “Theory of modulational instability in Bragg gratings with quadratic nonlinearity,” Phys. Rev. E 59, 6064–6078 (1996).
[CrossRef]

D. V. Petrov, “Nonlinear phase shift by cascaded quasi-phase-matched second harmonic generation,” Opt. Commun. 131, 102–106 (1996).
[CrossRef]

1995

L. Torner, C. R. Menyuk, W. E. Torruellas, and G. I. Stegeman, “Two-dimensional solitons with second-order nonlinearities,” Opt. Lett. 20, 13–15 (1995).
[CrossRef] [PubMed]

S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation,” Opt. Lett. 20, 438–440 (1995).
[CrossRef] [PubMed]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

L. Bergé, V. K. Mezentsev, J. J. Rasmussen, and J. Wyller, “Formation of stable solitons in quadratic nonlinear media,” Phys. Rev. A 52, R28–R31 (1995).
[CrossRef] [PubMed]

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, “Instability of solitons governed by quadratic nonlinearities,” Phys. Rev. Lett. 75, 591–595 (1995).
[CrossRef] [PubMed]

1994

1993

K. Hayata and M. Koshiba, “Multidimensional solitons in quadratic nonlinear media,” Phys. Rev. Lett. 71, 3275–3278 (1993).
[CrossRef] [PubMed]

1992

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

1990

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
[CrossRef]

1986

B. Jaskorzynska, G. Arvidsson, and F. Laurell, “Periodic structures for phase-matching in second harmonic generation in titanium lithium niobate waveguides,” in Integrated Optical Circuit Engineering III, R. Th. Kersten, ed., Proc. SPIE 651, 221–228 (1986).
[CrossRef]

1973

C. L. Tang and P. P. Bey, “Phase matching in second-harmonic generation using artificial periodic structure,” IEEE J. Quantum Electron. 9, 9–17 (1973).
[CrossRef]

1962

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Aitchison, J. S.

Alexander, T. J.

Anderson, D.

B. A. Malomed, P. D. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Arnold, J. M.

Arraf, A.

H. He, A. Arraf, C. M. de Sterke, P. D. Drummond, and B. A. Malomed, “Theory of modulational instability in Bragg gratings with quadratic nonlinearity,” Phys. Rev. E 59, 6064–6078 (1996).
[CrossRef]

Arvidsson, G.

B. Jaskorzynska, G. Arvidsson, and F. Laurell, “Periodic structures for phase-matching in second harmonic generation in titanium lithium niobate waveguides,” in Integrated Optical Circuit Engineering III, R. Th. Kersten, ed., Proc. SPIE 651, 221–228 (1986).
[CrossRef]

Assanto, G.

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, “All-optical switching by spatial walkoff compensation and solitary-wave locking,” Appl. Phys. Lett. 68, 1449–1451 (1996).
[CrossRef]

Baboiu, D.-M.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Baek, Y.

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138–1141 (1996).
[CrossRef]

Bang, O.

Barthélémy, A.

Beckwitt, K.

X. Liu, K. Beckwitt, and F. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E 62, 1328–1340 (2000).
[CrossRef]

Bergé, L.

L. Bergé, V. K. Mezentsev, J. J. Rasmussen, and J. Wyller, “Formation of stable solitons in quadratic nonlinear media,” Phys. Rev. A 52, R28–R31 (1995).
[CrossRef] [PubMed]

Berntson, A.

B. A. Malomed, P. D. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
[CrossRef]

Bey, P. P.

C. L. Tang and P. P. Bey, “Phase matching in second-harmonic generation using artificial periodic structure,” IEEE J. Quantum Electron. 9, 9–17 (1973).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bramati, A.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Brown, C. T. A.

Bryce, A. C.

Buryak, A.

Buryak, A. V.

T. J. Alexander, A. V. Buryak, and Y. S. Kivshar, “Stabilization of dark and vortex parametric spatial solitons,” Opt. Lett. 23, 670–672 (1998).
[CrossRef]

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, “Instability of solitons governed by quadratic nonlinearities,” Phys. Rev. Lett. 75, 591–595 (1995).
[CrossRef] [PubMed]

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Chinaglia, W.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[CrossRef] [PubMed]

Christiansen, P. L.

Clausen, C. B.

O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, “Engineering competing nonlinearities,” Opt. Lett. 24, 1413–1415 (1999).
[CrossRef]

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[CrossRef]

Conti, C.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Corney, J. F.

J. F. Corney and O. Bang, “Solitons in quadratic nonlinear photonic crystals,” Phys. Rev. E 64, 047601 (2001).
[CrossRef]

O. Bang, T. W. Graversen, and J. F. Corney, “Accurate switching intensities and length scales in quasi-phase-matched materials,” Opt. Lett. 26, 1007–1009 (2001).
[CrossRef]

J. F. Corney and O. Bang, “Modulational instability in periodic quadratic nonlinear materials,” Phys. Rev. Lett. 87, 133901 (2001).
[CrossRef] [PubMed]

Couderc, V.

Cretegny, T.

T. Cretegny, T. Dauxois, S. Ruffo, and A. Torcini, “Localization and equipartition of energy in the β-FPU chain: chaotic breathers,” Physica D 121, 109–126 (1998).
[CrossRef]

Dauxois, T.

T. Cretegny, T. Dauxois, S. Ruffo, and A. Torcini, “Localization and equipartition of energy in the β-FPU chain: chaotic breathers,” Physica D 121, 109–126 (1998).
[CrossRef]

de Sterke, C. M.

H. He, A. Arraf, C. M. de Sterke, P. D. Drummond, and B. A. Malomed, “Theory of modulational instability in Bragg gratings with quadratic nonlinearity,” Phys. Rev. E 59, 6064–6078 (1996).
[CrossRef]

Di Trapani, P.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[CrossRef] [PubMed]

Drummond, P. D.

B. A. Malomed, P. D. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
[CrossRef]

H. He, A. Arraf, C. M. de Sterke, P. D. Drummond, and B. A. Malomed, “Theory of modulational instability in Bragg gratings with quadratic nonlinearity,” Phys. Rev. E 59, 6064–6078 (1996).
[CrossRef]

H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity,” Opt. Commun. 123, 394–402 (1996).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Ebrahimzadeh, M.

Fang, H.

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[CrossRef] [PubMed]

H. Fang, R. Malendevich, R. Schiek, and G. I. Stegeman, “Spatial modulational instability in one-dimensional lithium niobate slab waveguides,” Opt. Lett. 25, 1786–1788 (2000).
[CrossRef]

Fejer, M. M.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Ferro, P.

Fuerst, R. A.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, “All-optical switching by spatial walkoff compensation and solitary-wave locking,” Appl. Phys. Lett. 68, 1449–1451 (1996).
[CrossRef]

Graversen, T. W.

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Hayata, K.

K. Hayata and M. Koshiba, “Multidimensional solitons in quadratic nonlinear media,” Phys. Rev. Lett. 71, 3275–3278 (1993).
[CrossRef] [PubMed]

He, H.

B. A. Malomed, P. D. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
[CrossRef]

H. He, A. Arraf, C. M. de Sterke, P. D. Drummond, and B. A. Malomed, “Theory of modulational instability in Bragg gratings with quadratic nonlinearity,” Phys. Rev. E 59, 6064–6078 (1996).
[CrossRef]

H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity,” Opt. Commun. 123, 394–402 (1996).
[CrossRef]

Helmy, A. Saher

Hutchings, D. C.

Jaskorzynska, B.

B. Jaskorzynska, G. Arvidsson, and F. Laurell, “Periodic structures for phase-matching in second harmonic generation in titanium lithium niobate waveguides,” in Integrated Optical Circuit Engineering III, R. Th. Kersten, ed., Proc. SPIE 651, 221–228 (1986).
[CrossRef]

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Kilius, J.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Kivshar, Y. S.

Kleckner, T. C.

Kobyakov, A.

Koshiba, M.

K. Hayata and M. Koshiba, “Multidimensional solitons in quadratic nonlinear media,” Phys. Rev. Lett. 71, 3275–3278 (1993).
[CrossRef] [PubMed]

Laurell, F.

B. Jaskorzynska, G. Arvidsson, and F. Laurell, “Periodic structures for phase-matching in second harmonic generation in titanium lithium niobate waveguides,” in Integrated Optical Circuit Engineering III, R. Th. Kersten, ed., Proc. SPIE 651, 221–228 (1986).
[CrossRef]

Lawrence, B.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Lawrence, B. L.

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, “All-optical switching by spatial walkoff compensation and solitary-wave locking,” Appl. Phys. Lett. 68, 1449–1451 (1996).
[CrossRef]

Lederer, F.

Lisak, M.

B. A. Malomed, P. D. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
[CrossRef]

Liu, X.

X. Liu, K. Beckwitt, and F. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E 62, 1328–1340 (2000).
[CrossRef]

Lopez Lago, E.

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Malendevich, R.

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[CrossRef] [PubMed]

H. Fang, R. Malendevich, R. Schiek, and G. I. Stegeman, “Spatial modulational instability in one-dimensional lithium niobate slab waveguides,” Opt. Lett. 25, 1786–1788 (2000).
[CrossRef]

Malomed, B. A.

B. A. Malomed, P. D. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
[CrossRef]

H. He, A. Arraf, C. M. de Sterke, P. D. Drummond, and B. A. Malomed, “Theory of modulational instability in Bragg gratings with quadratic nonlinearity,” Phys. Rev. E 59, 6064–6078 (1996).
[CrossRef]

H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity,” Opt. Commun. 123, 394–402 (1996).
[CrossRef]

Marsh, J. H.

Menyuk, C. R.

Mezentsev, V. K.

L. Bergé, V. K. Mezentsev, J. J. Rasmussen, and J. Wyller, “Formation of stable solitons in quadratic nonlinear media,” Phys. Rev. A 52, R28–R31 (1995).
[CrossRef] [PubMed]

Minardi, S.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[CrossRef] [PubMed]

Moutzouris, K.

Nishihara, H.

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
[CrossRef]

Pelinovsky, D. E.

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, “Instability of solitons governed by quadratic nonlinearities,” Phys. Rev. Lett. 75, 591–595 (1995).
[CrossRef] [PubMed]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Petrov, D. V.

D. V. Petrov, “Nonlinear phase shift by cascaded quasi-phase-matched second harmonic generation,” Opt. Commun. 131, 102–106 (1996).
[CrossRef]

Piskarskas, A.

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[CrossRef] [PubMed]

Rasmussen, J. J.

L. Bergé, V. K. Mezentsev, J. J. Rasmussen, and J. Wyller, “Formation of stable solitons in quadratic nonlinear media,” Phys. Rev. A 52, R28–R31 (1995).
[CrossRef] [PubMed]

Ruffo, S.

T. Cretegny, T. Dauxois, S. Ruffo, and A. Torcini, “Localization and equipartition of energy in the β-FPU chain: chaotic breathers,” Physica D 121, 109–126 (1998).
[CrossRef]

Schiek, R.

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[CrossRef] [PubMed]

H. Fang, R. Malendevich, R. Schiek, and G. I. Stegeman, “Spatial modulational instability in one-dimensional lithium niobate slab waveguides,” Opt. Lett. 25, 1786–1788 (2000).
[CrossRef]

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138–1141 (1996).
[CrossRef]

C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2)(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1994).
[CrossRef]

Simos, C.

Stanley, C. R.

Stegeman, G. I.

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[CrossRef] [PubMed]

H. Fang, R. Malendevich, R. Schiek, and G. I. Stegeman, “Spatial modulational instability in one-dimensional lithium niobate slab waveguides,” Opt. Lett. 25, 1786–1788 (2000).
[CrossRef]

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138–1141 (1996).
[CrossRef]

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, “All-optical switching by spatial walkoff compensation and solitary-wave locking,” Appl. Phys. Lett. 68, 1449–1451 (1996).
[CrossRef]

L. Torner, C. R. Menyuk, W. E. Torruellas, and G. I. Stegeman, “Two-dimensional solitons with second-order nonlinearities,” Opt. Lett. 20, 13–15 (1995).
[CrossRef] [PubMed]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Suhara, T.

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
[CrossRef]

Tang, C. L.

C. L. Tang and P. P. Bey, “Phase matching in second-harmonic generation using artificial periodic structure,” IEEE J. Quantum Electron. 9, 9–17 (1973).
[CrossRef]

Torcini, A.

T. Cretegny, T. Dauxois, S. Ruffo, and A. Torcini, “Localization and equipartition of energy in the β-FPU chain: chaotic breathers,” Physica D 121, 109–126 (1998).
[CrossRef]

Torner, L.

O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, “Engineering competing nonlinearities,” Opt. Lett. 24, 1413–1415 (1999).
[CrossRef]

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

L. Torner, C. R. Menyuk, W. E. Torruellas, and G. I. Stegeman, “Two-dimensional solitons with second-order nonlinearities,” Opt. Lett. 20, 13–15 (1995).
[CrossRef] [PubMed]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2)(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1994).
[CrossRef]

Torruellas, W. E.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, “All-optical switching by spatial walkoff compensation and solitary-wave locking,” Appl. Phys. Lett. 68, 1449–1451 (1996).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

L. Torner, C. R. Menyuk, W. E. Torruellas, and G. I. Stegeman, “Two-dimensional solitons with second-order nonlinearities,” Opt. Lett. 20, 13–15 (1995).
[CrossRef] [PubMed]

Trillo, S.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation,” Opt. Lett. 20, 438–440 (1995).
[CrossRef] [PubMed]

Valiulis, G.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[CrossRef] [PubMed]

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, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Wabnitz, S.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Wang, Z.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Wise, F.

X. Liu, K. Beckwitt, and F. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E 62, 1328–1340 (2000).
[CrossRef]

Wyller, J.

L. Bergé, V. K. Mezentsev, J. J. Rasmussen, and J. Wyller, “Formation of stable solitons in quadratic nonlinear media,” Phys. Rev. A 52, R28–R31 (1995).
[CrossRef] [PubMed]

Appl. Phys. Lett.

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, “All-optical switching by spatial walkoff compensation and solitary-wave locking,” Appl. Phys. Lett. 68, 1449–1451 (1996).
[CrossRef]

IEEE J. Quantum Electron.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
[CrossRef]

C. L. Tang and P. P. Bey, “Phase matching in second-harmonic generation using artificial periodic structure,” IEEE J. Quantum Electron. 9, 9–17 (1973).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity,” Opt. Commun. 123, 394–402 (1996).
[CrossRef]

D. V. Petrov, “Nonlinear phase shift by cascaded quasi-phase-matched second harmonic generation,” Opt. Commun. 131, 102–106 (1996).
[CrossRef]

Opt. Lett.

L. Torner, C. R. Menyuk, W. E. Torruellas, and G. I. Stegeman, “Two-dimensional solitons with second-order nonlinearities,” Opt. Lett. 20, 13–15 (1995).
[CrossRef] [PubMed]

S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation,” Opt. Lett. 20, 438–440 (1995).
[CrossRef] [PubMed]

O. Bang, Y. S. Kivshar, and A. Buryak, “Bright spatial solitons in defocusing Kerr media supported by cascaded nonlinearities,” Opt. Lett. 22, 1680–1682 (1997).
[CrossRef]

A. Kobyakov, F. Lederer, O. Bang, and Y. S. Kivshar, “Nonlinear phase shift and all-optical switching in quasi-phase-matched quadratic media,” Opt. Lett. 23, 506–508 (1998).
[CrossRef]

T. J. Alexander, A. V. Buryak, and Y. S. Kivshar, “Stabilization of dark and vortex parametric spatial solitons,” Opt. Lett. 23, 670–672 (1998).
[CrossRef]

O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, “Engineering competing nonlinearities,” Opt. Lett. 24, 1413–1415 (1999).
[CrossRef]

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, “Quasi phase matching in GaAs-AlAs superlattice waveguides through bandgap tuning by use of quantum-well intermixing,” Opt. Lett. 25, 1370–1372 (2000).
[CrossRef]

H. Fang, R. Malendevich, R. Schiek, and G. I. Stegeman, “Spatial modulational instability in one-dimensional lithium niobate slab waveguides,” Opt. Lett. 25, 1786–1788 (2000).
[CrossRef]

V. Couderc, E. Lopez Lago, C. Simos, and A. Barthélémy, “Experiments in quadratic spatial soliton generation and steering in a noncollinear geometry,” Opt. Lett. 26, 905–907 (2001).
[CrossRef]

O. Bang, T. W. Graversen, and J. F. Corney, “Accurate switching intensities and length scales in quasi-phase-matched materials,” Opt. Lett. 26, 1007–1009 (2001).
[CrossRef]

Opt. Quantum Electron.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Phys. Rev.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Phys. Rev. A

L. Bergé, V. K. Mezentsev, J. J. Rasmussen, and J. Wyller, “Formation of stable solitons in quadratic nonlinear media,” Phys. Rev. A 52, R28–R31 (1995).
[CrossRef] [PubMed]

Phys. Rev. E

B. A. Malomed, P. D. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997).
[CrossRef]

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138–1141 (1996).
[CrossRef]

X. Liu, K. Beckwitt, and F. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E 62, 1328–1340 (2000).
[CrossRef]

J. F. Corney and O. Bang, “Solitons in quadratic nonlinear photonic crystals,” Phys. Rev. E 64, 047601 (2001).
[CrossRef]

H. He, A. Arraf, C. M. de Sterke, P. D. Drummond, and B. A. Malomed, “Theory of modulational instability in Bragg gratings with quadratic nonlinearity,” Phys. Rev. E 59, 6064–6078 (1996).
[CrossRef]

Phys. Rev. Lett.

J. F. Corney and O. Bang, “Modulational instability in periodic quadratic nonlinear materials,” Phys. Rev. Lett. 87, 133901 (2001).
[CrossRef] [PubMed]

C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000).
[CrossRef] [PubMed]

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001).
[CrossRef] [PubMed]

K. Hayata and M. Koshiba, “Multidimensional solitons in quadratic nonlinear media,” Phys. Rev. Lett. 71, 3275–3278 (1993).
[CrossRef] [PubMed]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, “Instability of solitons governed by quadratic nonlinearities,” Phys. Rev. Lett. 75, 591–595 (1995).
[CrossRef] [PubMed]

Physica D

T. Cretegny, T. Dauxois, S. Ruffo, and A. Torcini, “Localization and equipartition of energy in the β-FPU chain: chaotic breathers,” Physica D 121, 109–126 (1998).
[CrossRef]

Proc. SPIE

B. Jaskorzynska, G. Arvidsson, and F. Laurell, “Periodic structures for phase-matching in second harmonic generation in titanium lithium niobate waveguides,” in Integrated Optical Circuit Engineering III, R. Th. Kersten, ed., Proc. SPIE 651, 221–228 (1986).
[CrossRef]

Other

E. Lopez Lago, V. Couderc, C. de Angelis, F. Gringoli, and A. Barthélémy, “Experimental demonstration of the self-trapping of a weak probe induced by a quadratic spatial soliton,” in Nonlinear Guided Waves and Their Applications, Vol. 55 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 379–381.

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Figures (13)

Fig. 1
Fig. 1

Periodic linear and quadratically nonlinear gratings, αj(Z) and χ(Z), with period 2L0=2π/|kg|.

Fig. 2
Fig. 2

Ratio of intensities L=log(v¯s2/w¯s2) for plane-wave solutions, as a function of γ˜ and σ, for r=-1.

Fig. 3
Fig. 3

Parameter rη versus averaged intensity I¯ for β˜=-10 (solid curves), β˜=0 (dashed curves), and β˜=10 (dotted curves), with d0=0, d=1, and kg=100.

Fig. 4
Fig. 4

Regions of stability and instability for the r=-1 solutions.

Fig. 5
Fig. 5

Maximum gain of unstable branches (a) G and (b) U1 and U2 of the r=-1 solutions (black is the largest gain). Hatching indicates the absence of a solution.

Fig. 6
Fig. 6

Gain profiles of the r=-1 solutions for γ˜=0.

Fig. 7
Fig. 7

Evolution of the fundamental intensity w, corresponding to the solutions (a) Eq. (11) with γ=0.08 and (b) Eq. (12) with γ=-0.08. The gratings parameters are d0/d=2.8 and a/κ=0.36. r=-1.

Fig. 8
Fig. 8

(a) Ratio R=|v0|2/|w0|2 versus mismatch parameter σ calculated exactly (asterisks) and by the averaged-field theory (solid curves). (b) Fourier components of the solution for κ=700, σ=1 calculated exactly (solid curves) and by the averaged-field theory (asterisks for wn and circles for vn). r=-1.

Fig. 9
Fig. 9

Gain curves for the dual plane-wave solution when (a) ρ=0 and (b) |ρ|=2/(15π), for γ<0 and σ=-0.5 (β˜=-3/2). Dashed curves, averaged-field results; solid curves, exact Floquet results.

Fig. 10
Fig. 10

Maximum gain for solutions with (a) 5κ30 and (b) 1<κ<10. The regions of zero gain predicted from the averaged-field theory are shaded. d0/d=5/3 and a/κ=0.316.

Fig. 11
Fig. 11

Floquet gain spectra for (a) the LiNbO3 and (b) the GaAs/AlAs gratings, for σ=-2. The diamonds show the averaged-field results in (c) and the equivalent non-phase-matched homogeneous results in (d).

Fig. 12
Fig. 12

Gain curves for (a) κ=2 and σ=4 and (b) κ=6 and σ=1, calculated from Floquet theory (solid curves) and simulations (dotted curves). The averaged theory predicts a stable solution in (a) and a gain marked by the dashed curve in (b). d0/d=5/3 and a/κ=0.316.

Fig. 13
Fig. 13

(a) Gain profile calculated from propagative simulations for the same parameters as in Fig. 11(a). Comparisons with the Floquet theory (solid curves) are given in (b) and (c).

Equations (26)

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

iZE1+½X2E1+α1E1+χE1*E2 exp(iβZ)=0,
iZE2+¼X2E2+2α2E2+χE12 exp(-iβZ)=0,
 izw+½x2w-rw+χw*v exp(iκz)=0,
izv+¼x2v-σv+αv+χw2 exp(-iκz)=0,
α=a n gn exp(inκz),
χ=d0+d n gn exp(inκz),
w=n wn(x, z)exp(inκz),
v=n vn(x, z)exp(inκz).
wn0=(dgn-1+d0δn,1)w0*v0nκ,
vn0=agnv0+(dgn+1+d0δn,-1)w02nκ,
izw¯+½x2w¯-rw¯+ρw¯*v¯+γ(|v¯|2-|w¯|2)w¯=0,
izv¯+¼x2v¯-σv¯+ρ*w¯2+2γ|w¯|2v¯=0.
0=4γ˜3y3+γ˜[1+4γ˜(r-σ)] y2+σ(γ˜σ-4rγ˜-1)y+rσ2.
σ(γ˜)=[5+16γ˜+3(f3-+f3+)]/(8γ˜),
w¯(z)=w¯s+δ¯1(z)exp(-iνx)+δ¯2*(z)exp(iνx),
v¯(z)=v¯s+δ¯3(z)exp(-iνx)+δ¯4*(z)exp(iνx),
Δ¯=δ¯1δ¯2δ¯3δ¯4,M¯=iabcd-b-a-d-c2c2dg0-2d-2c0-g,
w¯=tanh(x)/γ,v¯=0(γ>0),
w¯=tanh(x)/|4γ|,v¯=±5/|4γ|tanh(x)(γ<0),
-(nκ+r)wn+l,p Dn+l-p-1wl*vp=0,
-(nκ+σ)vn+a l gn-lvl
+l,p Dn-l-p+1wlwp=0,
{J}l,q=FlwqFlvqFlwq*Flvq*Fl*wqFl*vqFl*wq*Fl*vq*.
w(x, z)=ws(z)+δ1(z)exp(-iνx)+δ2*(z)exp(iνx),
v(x, z)=vs(z)+δ3(z)exp(-iνx)+δ4*(z)exp(iνx),
P=δ1δ2δ3δ4,M=iabc*0-b*-a0-c2c0d00-2c*0-d,

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