Abstract

We demonstrate experimentally the existence of competing focusing and defocusing nonlinearities in a double-resonant system with quadratic nonlinear response. We use an array of periodically poled coupled optical waveguides and observe inhibition of the nonlinear beam self-action independent on power. This inhibition is demonstrated in both regimes of normal and anomalous beam diffraction.

© 2009 Optical Society of America

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  1. M. Quiroga-Teixeiro and H. Michinel, J. Opt. Soc. Am. B 14, 2004 (1997).
    [CrossRef]
  2. H. Michinel, J. Campo-Táboas, R. García-Fernández, J. R. Salgueiro, and M. L. Quiroga-Teixeiro, Phys. Rev. E 65, 066604 (2002).
    [CrossRef]
  3. S. Orlov, A. Yariv, and M. Segev, Appl. Phys. Lett. 68, 1610 (1996).
    [CrossRef]
  4. A. H. Piekara, J. S. Moore, and M. S. Feld, Phys. Rev. A 9, 1403 (1974).
    [CrossRef]
  5. A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
    [CrossRef]
  6. O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, Opt. Lett. 24, 1413 (1999).
    [CrossRef]
  7. A. V. Buryak, Y. S. Kivshar, and S. Trillo, Opt. Lett. 20, 1961 (1995).
    [CrossRef] [PubMed]
  8. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
    [CrossRef] [PubMed]
  9. W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, Opt. Photonics News 19(1), 24 (2008).
    [CrossRef]
  10. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, Opt. Lett. 17, 28 (1992).
    [CrossRef] [PubMed]
  11. R. Schiek, J. Opt. Soc. Am. B 10, 1848 (1993).
    [CrossRef]
  12. A. Kobyakov and F. Lederer, Phys. Rev. A 54, 3455 (1996).
    [CrossRef] [PubMed]
  13. T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
    [CrossRef]
  14. R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
    [CrossRef] [PubMed]
  15. C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Appl. Phys. Lett. 67, 170 (1995).
    [CrossRef]
  16. T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
    [CrossRef] [PubMed]
  17. K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, Phys. Rev. Lett. 100, 053901 (2008).
    [CrossRef] [PubMed]

2008 (2)

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, Opt. Photonics News 19(1), 24 (2008).
[CrossRef]

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, Phys. Rev. Lett. 100, 053901 (2008).
[CrossRef] [PubMed]

2004 (1)

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

2002 (3)

H. Michinel, J. Campo-Táboas, R. García-Fernández, J. R. Salgueiro, and M. L. Quiroga-Teixeiro, Phys. Rev. E 65, 066604 (2002).
[CrossRef]

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[CrossRef]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

2000 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

1999 (1)

1998 (1)

T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
[CrossRef]

1997 (1)

1996 (2)

S. Orlov, A. Yariv, and M. Segev, Appl. Phys. Lett. 68, 1610 (1996).
[CrossRef]

A. Kobyakov and F. Lederer, Phys. Rev. A 54, 3455 (1996).
[CrossRef] [PubMed]

1995 (2)

C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Appl. Phys. Lett. 67, 170 (1995).
[CrossRef]

A. V. Buryak, Y. S. Kivshar, and S. Trillo, Opt. Lett. 20, 1961 (1995).
[CrossRef] [PubMed]

1993 (1)

1992 (1)

1974 (1)

A. H. Piekara, J. S. Moore, and M. S. Feld, Phys. Rev. A 9, 1403 (1974).
[CrossRef]

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Assanto, G.

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, Phys. Rev. Lett. 100, 053901 (2008).
[CrossRef] [PubMed]

Baldi, P.

C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Appl. Phys. Lett. 67, 170 (1995).
[CrossRef]

Bang, O.

Bräuer, A.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

Büchter, D.

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, Opt. Photonics News 19(1), 24 (2008).
[CrossRef]

Buryak, A. V.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[CrossRef]

A. V. Buryak, Y. S. Kivshar, and S. Trillo, Opt. Lett. 20, 1961 (1995).
[CrossRef] [PubMed]

Campo-Táboas, J.

H. Michinel, J. Campo-Táboas, R. García-Fernández, J. R. Salgueiro, and M. L. Quiroga-Teixeiro, Phys. Rev. E 65, 066604 (2002).
[CrossRef]

Christiansen, P. L.

Clausen, C. B.

De Micheli, M. P.

C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Appl. Phys. Lett. 67, 170 (1995).
[CrossRef]

Delacourt, D.

C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Appl. Phys. Lett. 67, 170 (1995).
[CrossRef]

DeSalvo, R.

Di Trapani, P.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[CrossRef]

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Feld, M. S.

A. H. Piekara, J. S. Moore, and M. S. Feld, Phys. Rev. A 9, 1403 (1974).
[CrossRef]

Gallo, K.

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, Phys. Rev. Lett. 100, 053901 (2008).
[CrossRef] [PubMed]

García-Fernández, R.

H. Michinel, J. Campo-Táboas, R. García-Fernández, J. R. Salgueiro, and M. L. Quiroga-Teixeiro, Phys. Rev. E 65, 066604 (2002).
[CrossRef]

Grundkötter, W.

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, Opt. Photonics News 19(1), 24 (2008).
[CrossRef]

Hagan, D. J.

Herrmann, H.

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, Opt. Photonics News 19(1), 24 (2008).
[CrossRef]

Hu, H.

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, Opt. Photonics News 19(1), 24 (2008).
[CrossRef]

Iwanow, R.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

Kivshar, Y. S.

Kobyakov, A.

A. Kobyakov and F. Lederer, Phys. Rev. A 54, 3455 (1996).
[CrossRef] [PubMed]

Lederer, F.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
[CrossRef]

A. Kobyakov and F. Lederer, Phys. Rev. A 54, 3455 (1996).
[CrossRef] [PubMed]

Michinel, H.

H. Michinel, J. Campo-Táboas, R. García-Fernández, J. R. Salgueiro, and M. L. Quiroga-Teixeiro, Phys. Rev. E 65, 066604 (2002).
[CrossRef]

M. Quiroga-Teixeiro and H. Michinel, J. Opt. Soc. Am. B 14, 2004 (1997).
[CrossRef]

Min, Y.

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, Opt. Photonics News 19(1), 24 (2008).
[CrossRef]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

Moore, J. S.

A. H. Piekara, J. S. Moore, and M. S. Feld, Phys. Rev. A 9, 1403 (1974).
[CrossRef]

Morandotti, R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Nouh, S.

C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Appl. Phys. Lett. 67, 170 (1995).
[CrossRef]

Nouroozi, R.

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, Opt. Photonics News 19(1), 24 (2008).
[CrossRef]

Orlov, S.

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, Opt. Photonics News 19(1), 24 (2008).
[CrossRef]

S. Orlov, A. Yariv, and M. Segev, Appl. Phys. Lett. 68, 1610 (1996).
[CrossRef]

Ostrowsky, D. B.

C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Appl. Phys. Lett. 67, 170 (1995).
[CrossRef]

Papuchon, M.

C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Appl. Phys. Lett. 67, 170 (1995).
[CrossRef]

Pasquazi, A.

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, Phys. Rev. Lett. 100, 053901 (2008).
[CrossRef] [PubMed]

Pertsch, T.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

Peschel, T.

T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
[CrossRef]

Peschel, U.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
[CrossRef]

Piekara, A. H.

A. H. Piekara, J. S. Moore, and M. S. Feld, Phys. Rev. A 9, 1403 (1974).
[CrossRef]

Quiring, V.

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, Opt. Photonics News 19(1), 24 (2008).
[CrossRef]

Quiroga-Teixeiro, M.

Quiroga-Teixeiro, M. L.

H. Michinel, J. Campo-Táboas, R. García-Fernández, J. R. Salgueiro, and M. L. Quiroga-Teixeiro, Phys. Rev. E 65, 066604 (2002).
[CrossRef]

Reza, S.

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, Opt. Photonics News 19(1), 24 (2008).
[CrossRef]

Ricken, R.

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, Opt. Photonics News 19(1), 24 (2008).
[CrossRef]

Salgueiro, J. R.

H. Michinel, J. Campo-Táboas, R. García-Fernández, J. R. Salgueiro, and M. L. Quiroga-Teixeiro, Phys. Rev. E 65, 066604 (2002).
[CrossRef]

Schiek, R.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

R. Schiek, J. Opt. Soc. Am. B 10, 1848 (1993).
[CrossRef]

Segev, M.

S. Orlov, A. Yariv, and M. Segev, Appl. Phys. Lett. 68, 1610 (1996).
[CrossRef]

Sheik-Bahae, M.

Silberberg, Y.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Skryabin, D. V.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[CrossRef]

Sohler, W.

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, Opt. Photonics News 19(1), 24 (2008).
[CrossRef]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

Stegeman, G.

Stegeman, G. I.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Appl. Phys. Lett. 67, 170 (1995).
[CrossRef]

Stivala, S.

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, Phys. Rev. Lett. 100, 053901 (2008).
[CrossRef] [PubMed]

Suche, H.

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, Opt. Photonics News 19(1), 24 (2008).
[CrossRef]

Torner, L.

Trevino-Palacios, C. G.

C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Appl. Phys. Lett. 67, 170 (1995).
[CrossRef]

Trillo, S.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[CrossRef]

A. V. Buryak, Y. S. Kivshar, and S. Trillo, Opt. Lett. 20, 1961 (1995).
[CrossRef] [PubMed]

Van Stryland, E. W.

Vanherzeele, H.

Vannahme, C.

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, Opt. Photonics News 19(1), 24 (2008).
[CrossRef]

Yariv, A.

S. Orlov, A. Yariv, and M. Segev, Appl. Phys. Lett. 68, 1610 (1996).
[CrossRef]

Zentgraf, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

S. Orlov, A. Yariv, and M. Segev, Appl. Phys. Lett. 68, 1610 (1996).
[CrossRef]

C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, P. Baldi, S. Nouh, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Appl. Phys. Lett. 67, 170 (1995).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Lett. (3)

Opt. Photonics News (1)

W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, Opt. Photonics News 19(1), 24 (2008).
[CrossRef]

Phys. Rep. (1)

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[CrossRef]

Phys. Rev. A (2)

A. H. Piekara, J. S. Moore, and M. S. Feld, Phys. Rev. A 9, 1403 (1974).
[CrossRef]

A. Kobyakov and F. Lederer, Phys. Rev. A 54, 3455 (1996).
[CrossRef] [PubMed]

Phys. Rev. E (2)

T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
[CrossRef]

H. Michinel, J. Campo-Táboas, R. García-Fernández, J. R. Salgueiro, and M. L. Quiroga-Teixeiro, Phys. Rev. E 65, 066604 (2002).
[CrossRef]

Phys. Rev. Lett. (4)

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, Phys. Rev. Lett. 100, 053901 (2008).
[CrossRef] [PubMed]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

(a) Calculated phase mismatch for the FW 00 mode in a sample of L = 71 mm . The circles show the interaction with the SH fundamental mode and the double resonance considered here. (b) Measurement of the SH output power of a single waveguide in dependence on the FW input wavelength and profiles of the (c) SH 02 and (d) SH 10 modes intensity distributions.

Fig. 2
Fig. 2

(a) Calculated FW phase shift corresponding to wavelengths of cancellation ( Φ cancellation ) , the maximum phase shift at the SH 10 resonance ( Φ max ) , and at a wavelength 2 nm above this resonance where the two phase shifts are added ( Φ addition ) . (b) Measured SH output power at wavelengths corresponding to (a), showing strong nonlinear interaction at the cancellation wavelength.

Fig. 3
Fig. 3

(a) Output beam width for normal excitation, normalized to the linear output at 1498 nm . The PM wavelengths are indicated by solid lines. (b) Power-independent output pattern at 1501.6 nm [dotted line in (a)].

Fig. 4
Fig. 4

(a) Output beam width for staggered excitation, normalized to the linear output at 1498 nm . The PM wavelengths are indicated by solid lines. (b) Power-independent output pattern at 1502.3 nm [dotted line in (a)].

Equations (1)

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n 2 eff = χ eff 2 ( ω c ) Δ β [ 1 cos ( Δ β z ) ] [ 1 ( 4 χ eff 2 P Δ β 2 ) sin 2 ( Δ β z 2 ) ] ,

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