Abstract

We investigate nonlinear focusing behavior of light beams propagating in beta-barium-borate crystals under mismatched second-harmonic generation. We clearly identify experimentally multiple self-focusing and defocusing regions against the orientation angle and the condition where competing quadratic and cubic nonlinearities perfectly compensate each other (zero-focusing point).

© 2014 Optical Society of America

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  1. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962).
    [CrossRef]
  2. J. M. R. Thomas and J. P. E. Taran, Opt. Commun. 4, 329 (1972).
    [CrossRef]
  3. N. R. Belashenkov, S. V. Gagarskii, and M. V. Inochkin, Opt. Spectrosc. 66, 806 (1989).
  4. H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
    [CrossRef]
  5. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, Opt. Lett. 17, 28 (1992).
    [CrossRef]
  6. D. C. Hutchings, J. S. Aitchison, and C. N. Ironside, Opt. Lett. 18, 793 (1993).
    [CrossRef]
  7. G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
    [CrossRef]
  8. K. Gallo, P. Baldi, M. De Micheli, D. B. Ostrowsky, and G. Assanto, Opt. Lett. 25, 966 (2000).
    [CrossRef]
  9. P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, Phys. Rev. Lett. 87, 183902 (2001).
    [CrossRef]
  10. K. Beckwitt, F. Wise, L. Qian, L. A. Walker, and E. Canto-Said, Opt. Lett. 26, 1696 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. F. Baronio, C. De Angelis, M. Marangoni, C. Manzoni, R. Ramponi, and G. Cerullo, Opt. Express 14, 4774 (2006).
    [CrossRef]
  14. M. Bache, J. Moses, and F. W. Wise, J. Opt. Soc. Am. B 24, 2752 (2007).
    [CrossRef]
  15. M. Marangoni, G. Sanna, D. Brida, M. Conforti, G. Cirmi, C. Manzoni, F. Baronio, P. Bassi, C. De Angelis, and G. Cerullo, Appl. Phys. Lett. 93, 021107 (2008).
    [CrossRef]
  16. X. Zeng, S. Ashihara, Z. Wang, T. Wang, Y. Chen, and M. Cha, Opt. Express 17, 16877 (2009).
    [CrossRef]
  17. F. W. Wise and J. Moses, in Self-Focusing: Past and Present, R. W. Boyd, S. G. Lukishova, and Y. R. Shen, eds. (Springer, 2009), pp. 481–506.
  18. J. Moses, B. A. Malomed, and F. W. Wise, Phys. Rev. A 76, 021802 (2007).
    [CrossRef]
  19. M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, Phys. Rev. A 82, 063806 (2010).
    [CrossRef]
  20. B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, Phys. Rev. Lett. 109, 04392 (2012).
  21. M. Conforti, F. Baronio, and S. Trillo, Opt. Lett. 37, 1082 (2012).
    [CrossRef]
  22. M. Conforti, N. Westerberg, F. Baronio, S. Trillo, and D. Faccio, Phys. Rev. A 88, 013829 (2013).
    [CrossRef]
  23. S. Trillo, A. V. Buryak, and Y. S. Kivshar, Opt. Commun. 122, 200 (1996).
    [CrossRef]
  24. A. V. Buryak, Y. S. Kivshar, and S. Trillo, Opt. Lett. 20, 1961 (1995).
    [CrossRef]
  25. M. Conforti, F. Baronio, and C. De Angelis, J. Opt. Soc. Am. B 28, 1231 (2011).
    [CrossRef]
  26. M. Bache, H. Guo, B. Zhou, and X. Zeng, Opt. Mater. Express 3, 357 (2013).
    [CrossRef]
  27. M. Conforti and F. Baronio, J. Opt. Soc. Am. B 30, 1041 (2013).
    [CrossRef]
  28. C. R. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994).
    [CrossRef]

2013 (3)

2012 (2)

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, Phys. Rev. Lett. 109, 04392 (2012).

M. Conforti, F. Baronio, and S. Trillo, Opt. Lett. 37, 1082 (2012).
[CrossRef]

2011 (1)

2010 (1)

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, Phys. Rev. A 82, 063806 (2010).
[CrossRef]

2009 (1)

2008 (1)

M. Marangoni, G. Sanna, D. Brida, M. Conforti, G. Cirmi, C. Manzoni, F. Baronio, P. Bassi, C. De Angelis, and G. Cerullo, Appl. Phys. Lett. 93, 021107 (2008).
[CrossRef]

2007 (2)

J. Moses, B. A. Malomed, and F. W. Wise, Phys. Rev. A 76, 021802 (2007).
[CrossRef]

M. Bache, J. Moses, and F. W. Wise, J. Opt. Soc. Am. B 24, 2752 (2007).
[CrossRef]

2006 (1)

2005 (1)

2003 (1)

2001 (2)

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

K. Beckwitt, F. Wise, L. Qian, L. A. Walker, and E. Canto-Said, Opt. Lett. 26, 1696 (2001).
[CrossRef]

2000 (1)

1996 (2)

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

S. Trillo, A. V. Buryak, and Y. S. Kivshar, Opt. Commun. 122, 200 (1996).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

1992 (1)

1990 (1)

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef]

1989 (1)

N. R. Belashenkov, S. V. Gagarskii, and M. V. Inochkin, Opt. Spectrosc. 66, 806 (1989).

1972 (1)

J. M. R. Thomas and J. P. E. Taran, Opt. Commun. 4, 329 (1972).
[CrossRef]

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]

Ashihara, S.

Assanto, G.

Bache, M.

M. Bache, H. Guo, B. Zhou, and X. Zeng, Opt. Mater. Express 3, 357 (2013).
[CrossRef]

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, Phys. Rev. Lett. 109, 04392 (2012).

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, Phys. Rev. A 82, 063806 (2010).
[CrossRef]

M. Bache, J. Moses, and F. W. Wise, J. Opt. Soc. Am. B 24, 2752 (2007).
[CrossRef]

Bakker, H. J.

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef]

Baldi, P.

Bang, O.

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, Phys. Rev. A 82, 063806 (2010).
[CrossRef]

Baronio, F.

Barthélémy, A.

Bassi, P.

M. Marangoni, G. Sanna, D. Brida, M. Conforti, G. Cirmi, C. Manzoni, F. Baronio, P. Bassi, C. De Angelis, and G. Cerullo, Appl. Phys. Lett. 93, 021107 (2008).
[CrossRef]

Beckwitt, K.

Belashenkov, N. R.

N. R. Belashenkov, S. V. Gagarskii, and M. V. Inochkin, Opt. Spectrosc. 66, 806 (1989).

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, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Brida, D.

M. Marangoni, G. Sanna, D. Brida, M. Conforti, G. Cirmi, C. Manzoni, F. Baronio, P. Bassi, C. De Angelis, and G. Cerullo, Appl. Phys. Lett. 93, 021107 (2008).
[CrossRef]

Broderick, N.

Buryak, A. V.

S. Trillo, A. V. Buryak, and Y. S. Kivshar, Opt. Commun. 122, 200 (1996).
[CrossRef]

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

Canto-Said, E.

Cerullo, G.

M. Marangoni, G. Sanna, D. Brida, M. Conforti, G. Cirmi, C. Manzoni, F. Baronio, P. Bassi, C. De Angelis, and G. Cerullo, Appl. Phys. Lett. 93, 021107 (2008).
[CrossRef]

F. Baronio, C. De Angelis, M. Marangoni, C. Manzoni, R. Ramponi, and G. Cerullo, Opt. Express 14, 4774 (2006).
[CrossRef]

Cha, M.

Chen, Y.

Chinaglia, W.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Chong, A.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, Phys. Rev. Lett. 109, 04392 (2012).

Cirmi, G.

M. Marangoni, G. Sanna, D. Brida, M. Conforti, G. Cirmi, C. Manzoni, F. Baronio, P. Bassi, C. De Angelis, and G. Cerullo, Appl. Phys. Lett. 93, 021107 (2008).
[CrossRef]

Conforti, M.

M. Conforti, N. Westerberg, F. Baronio, S. Trillo, and D. Faccio, Phys. Rev. A 88, 013829 (2013).
[CrossRef]

M. Conforti and F. Baronio, J. Opt. Soc. Am. B 30, 1041 (2013).
[CrossRef]

M. Conforti, F. Baronio, and S. Trillo, Opt. Lett. 37, 1082 (2012).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, J. Opt. Soc. Am. B 28, 1231 (2011).
[CrossRef]

M. Marangoni, G. Sanna, D. Brida, M. Conforti, G. Cirmi, C. Manzoni, F. Baronio, P. Bassi, C. De Angelis, and G. Cerullo, Appl. Phys. Lett. 93, 021107 (2008).
[CrossRef]

Conti, C.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Couderc, V.

De Angelis, C.

De Micheli, M.

DeSalvo, R.

Di Trapani, P.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Ducuing, J.

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

Faccio, D.

M. Conforti, N. Westerberg, F. Baronio, S. Trillo, and D. Faccio, Phys. Rev. A 88, 013829 (2013).
[CrossRef]

Gagarskii, S. V.

N. R. Belashenkov, S. V. Gagarskii, and M. V. Inochkin, Opt. Spectrosc. 66, 806 (1989).

Gallo, K.

Guo, H.

Hagan, D. J.

Hutchings, D. C.

Inochkin, M. V.

N. R. Belashenkov, S. V. Gagarskii, and M. V. Inochkin, Opt. Spectrosc. 66, 806 (1989).

Ironside, C. N.

Kilius, J.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Kivshar, Y. S.

S. Trillo, A. V. Buryak, and Y. S. Kivshar, Opt. Commun. 122, 200 (1996).
[CrossRef]

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

Kuipers, L.

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef]

Lagendijk, A.

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef]

Malomed, B. A.

J. Moses, B. A. Malomed, and F. W. Wise, Phys. Rev. A 76, 021802 (2007).
[CrossRef]

Manzoni, C.

M. Marangoni, G. Sanna, D. Brida, M. Conforti, G. Cirmi, C. Manzoni, F. Baronio, P. Bassi, C. De Angelis, and G. Cerullo, Appl. Phys. Lett. 93, 021107 (2008).
[CrossRef]

F. Baronio, C. De Angelis, M. Marangoni, C. Manzoni, R. Ramponi, and G. Cerullo, Opt. Express 14, 4774 (2006).
[CrossRef]

Marangoni, M.

M. Marangoni, G. Sanna, D. Brida, M. Conforti, G. Cirmi, C. Manzoni, F. Baronio, P. Bassi, C. De Angelis, and G. Cerullo, Appl. Phys. Lett. 93, 021107 (2008).
[CrossRef]

F. Baronio, C. De Angelis, M. Marangoni, C. Manzoni, R. Ramponi, and G. Cerullo, Opt. Express 14, 4774 (2006).
[CrossRef]

Menyuk, C. R.

Min, Y.

Minardi, S.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Moses, J.

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, Phys. Rev. A 82, 063806 (2010).
[CrossRef]

J. Moses, B. A. Malomed, and F. W. Wise, Phys. Rev. A 76, 021802 (2007).
[CrossRef]

M. Bache, J. Moses, and F. W. Wise, J. Opt. Soc. Am. B 24, 2752 (2007).
[CrossRef]

F. W. Wise and J. Moses, in Self-Focusing: Past and Present, R. W. Boyd, S. G. Lukishova, and Y. R. Shen, eds. (Springer, 2009), pp. 481–506.

Ostrowsky, D. B.

Pershan, P. S.

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

Pioger, P.

Planken, P. C. M.

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef]

Prawiharjo, J.

Qian, L.

Quiring, V.

Ramponi, R.

Richardson, D. J.

Sanna, G.

M. Marangoni, G. Sanna, D. Brida, M. Conforti, G. Cirmi, C. Manzoni, F. Baronio, P. Bassi, C. De Angelis, and G. Cerullo, Appl. Phys. Lett. 93, 021107 (2008).
[CrossRef]

Schiek, R.

Sheik-Bahae, M.

Sohler, W.

Stegeman, G.

Stegeman, G. I.

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

Taran, J. P. E.

J. M. R. Thomas and J. P. E. Taran, Opt. Commun. 4, 329 (1972).
[CrossRef]

Thomas, J. M. R.

J. M. R. Thomas and J. P. E. Taran, Opt. Commun. 4, 329 (1972).
[CrossRef]

Torner, L.

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

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

Trillo, S.

M. Conforti, N. Westerberg, F. Baronio, S. Trillo, and D. Faccio, Phys. Rev. A 88, 013829 (2013).
[CrossRef]

M. Conforti, F. Baronio, and S. Trillo, Opt. Lett. 37, 1082 (2012).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

S. Trillo, A. V. Buryak, and Y. S. Kivshar, Opt. Commun. 122, 200 (1996).
[CrossRef]

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

Valiulis, G.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Van Stryland, E. W.

Vanherzeele, H.

Walker, L. A.

Wang, T.

Wang, Z.

Westerberg, N.

M. Conforti, N. Westerberg, F. Baronio, S. Trillo, and D. Faccio, Phys. Rev. A 88, 013829 (2013).
[CrossRef]

Wise, F.

Wise, F. W.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, Phys. Rev. Lett. 109, 04392 (2012).

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, Phys. Rev. A 82, 063806 (2010).
[CrossRef]

J. Moses, B. A. Malomed, and F. W. Wise, Phys. Rev. A 76, 021802 (2007).
[CrossRef]

M. Bache, J. Moses, and F. W. Wise, J. Opt. Soc. Am. B 24, 2752 (2007).
[CrossRef]

F. W. Wise and J. Moses, in Self-Focusing: Past and Present, R. W. Boyd, S. G. Lukishova, and Y. R. Shen, eds. (Springer, 2009), pp. 481–506.

Zeng, X.

Zhou, B.

Zhou, B. B.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, Phys. Rev. Lett. 109, 04392 (2012).

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, Phys. Rev. A 82, 063806 (2010).
[CrossRef]

Appl. Phys. Lett. (1)

M. Marangoni, G. Sanna, D. Brida, M. Conforti, G. Cirmi, C. Manzoni, F. Baronio, P. Bassi, C. De Angelis, and G. Cerullo, Appl. Phys. Lett. 93, 021107 (2008).
[CrossRef]

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

Opt. Commun. (2)

S. Trillo, A. V. Buryak, and Y. S. Kivshar, Opt. Commun. 122, 200 (1996).
[CrossRef]

J. M. R. Thomas and J. P. E. Taran, Opt. Commun. 4, 329 (1972).
[CrossRef]

Opt. Express (2)

Opt. Lett. (7)

Opt. Mater. Express (1)

Opt. Quantum Electron. (1)

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

Opt. Spectrosc. (1)

N. R. Belashenkov, S. V. Gagarskii, and M. V. Inochkin, Opt. Spectrosc. 66, 806 (1989).

Phys. Rev. (1)

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

Phys. Rev. A (4)

J. Moses, B. A. Malomed, and F. W. Wise, Phys. Rev. A 76, 021802 (2007).
[CrossRef]

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, Phys. Rev. A 82, 063806 (2010).
[CrossRef]

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, Phys. Rev. A 42, 4085 (1990).
[CrossRef]

M. Conforti, N. Westerberg, F. Baronio, S. Trillo, and D. Faccio, Phys. Rev. A 88, 013829 (2013).
[CrossRef]

Phys. Rev. Lett. (2)

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, Phys. Rev. Lett. 109, 04392 (2012).

Other (1)

F. W. Wise and J. Moses, in Self-Focusing: Past and Present, R. W. Boyd, S. G. Lukishova, and Y. R. Shen, eds. (Springer, 2009), pp. 481–506.

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Figures (4)

Fig. 1.
Fig. 1.

Total effective cubic nonlinearity γ, resulting from both quadratic γ2 and cubic γ3 contributions, plotted versus the angle θ. The shaded domain identifies the region where cascading breaks down and SHG enters the regime of efficient conversion.

Fig. 2.
Fig. 2.

Experimental setup. Nd:YAG laser pump delivering 30 ps pulses, at 1064 nm, 20 Hz repetition rate. L1 and L2, lenses; λ/2, half-wave plates; P, polarizer; BBO, quadratic uniaxial BBO crystal; and a CCD camera.

Fig. 3.
Fig. 3.

Beam diameter (FWHM of intensity) at FF measured at BBO crystal output versus crystal rotation angle θ, for different pulse energy: linear regime (dotted line), 0.08 mJ (blue squares), 0.34 mJ (gold circles), 0.54 mJ (magenta circles), and 1.23 mJ (green diamonds).

Fig. 4.
Fig. 4.

(a)–(c) Measured 2D output beam shapes and (d)–(f) corresponding beam profiles versus y (x=0 section), compared with the input. (a) and (d) θ=22°, (b) and (e) θ=θc=18.6°, (c) and (f) θ=14.7°. Here, the input energy is 0.54 mJ.

Tables (2)

Tables Icon

Table 1. Effective Quadratic Nonlinear Coefficientsa

Tables Icon

Table 2. Effective Cubic Nonlinear Coefficientsa

Equations (3)

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

iE1z+12k12E1+χ2E1*E2eiΔkz+(χ31|E1|2+χ32|E2|2)E1=0,
iE2z+12k22E2+χ2E12eiΔkz+(χ33|E1|2+χ34|E2|2)E2=0,
iE1z+12k12E1+(γ2+γ3)|E1|2E1=0,

Metrics