Abstract

The prediction is experimentally confirmed that bright spatial solitons can be formed in spontaneously self- defocusing azo-doped polymers through a photoisomerization process, namely, angular hole burning. The experiments are actually the realization of a scheme of light-controlling light. Both linearly polarized and circularly polarized light can propagate as solitons in such media. The solitons with the same intensity but with different polarizations have different widths. The experimental results are further theoretically explained.

© 2010 Optical Society of America

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J. Liang, Z. Cai, Y. Sun, L. Yi, and H. C. Wang, Eur. Phys. J. D 55, 685 (2009).
[CrossRef]

J. Liang, Z. Cai, Y. Sun, S. Xu, and L. Yi, J. Opt. Soc. Am. B 26, 36 (2009).
[CrossRef]

2007 (2)

G. Assanto, A. Fratalocchi, and M. Peccianti, Opt. Express 15, 5248 (2007).
[CrossRef] [PubMed]

J. Liang, H. Zhao, X. Zhou, and H. Wang, J. Appl. Phys. 101, 013106 (2007).
[CrossRef]

2006 (1)

S. V. Serak, N. V. Tabiryan, M. Peccianti, and G. Assanto, IEEE Photonic Technol. Lett. 18, 1287 (2006).
[CrossRef]

2005 (3)

2004 (2)

2001 (1)

T. Buffeteau, F. L. Labarthet, M. Pezolet, and C. Sourisseau, Macromolecules 34, 7514 (2001).
[CrossRef]

1999 (1)

G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

1995 (2)

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

1994 (1)

M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

1993 (1)

G. C. Duree Jr., J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

1992 (2)

Z. Sekkat, D. Morichere, M. Dumont, R. Loucif-Saibi, and J. A. Delaire, J. Appl. Phys. 71, 1543 (1992).
[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992).
[CrossRef] [PubMed]

1991 (1)

1988 (1)

L. Nikolova, P. Markovsky, N. Tomova, V. Dragostinova, and N. Mateva, J. Mod. Opt. 35, 1789 (1988).
[CrossRef]

1985 (1)

A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985).
[CrossRef]

1974 (1)

J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
[CrossRef]

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

Aguilar, P. A.

M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Ashkin, A.

J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
[CrossRef]

Assanto, G.

G. Assanto, A. Fratalocchi, and M. Peccianti, Opt. Express 15, 5248 (2007).
[CrossRef] [PubMed]

S. V. Serak, N. V. Tabiryan, M. Peccianti, and G. Assanto, IEEE Photonic Technol. Lett. 18, 1287 (2006).
[CrossRef]

Barthelemy, A.

A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985).
[CrossRef]

Bashaw, M. C.

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

Bian, S.

S. Bian and M. G. Kuzyk, Appl. Phys. Lett. 85, 1104 (2004).
[CrossRef]

Bjorkholm, J. E.

J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
[CrossRef]

Buffeteau, T.

T. Buffeteau, F. L. Labarthet, M. Pezolet, and C. Sourisseau, Macromolecules 34, 7514 (2001).
[CrossRef]

Cai, Z.

J. Liang, Z. Cai, Y. Sun, L. Yi, and H. C. Wang, Eur. Phys. J. D 55, 685 (2009).
[CrossRef]

J. Liang, Z. Cai, Y. Sun, S. Xu, and L. Yi, J. Opt. Soc. Am. B 26, 36 (2009).
[CrossRef]

Castillo, M. D.

M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Crosignani, B.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

G. C. Duree Jr., J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992).
[CrossRef] [PubMed]

Delaire, J. A.

Z. Sekkat, D. Morichere, M. Dumont, R. Loucif-Saibi, and J. A. Delaire, J. Appl. Phys. 71, 1543 (1992).
[CrossRef]

Di Porto, P.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

G. C. Duree Jr., J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Dragostinova, V.

L. Nikolova, P. Markovsky, N. Tomova, V. Dragostinova, and N. Mateva, J. Mod. Opt. 35, 1789 (1988).
[CrossRef]

Dumont, M.

Z. Sekkat, D. Morichere, M. Dumont, R. Loucif-Saibi, and J. A. Delaire, J. Appl. Phys. 71, 1543 (1992).
[CrossRef]

Duree, G.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

Duree, G. C.

G. C. Duree Jr., J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Fejer, M. M.

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

Fischer, B.

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992).
[CrossRef] [PubMed]

Fratalocchi, A.

Froehly, C.

A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985).
[CrossRef]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Gatz, S.

Herrmann, J.

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

Kuzyk, M. G.

S. Bian and M. G. Kuzyk, Appl. Phys. Lett. 85, 1104 (2004).
[CrossRef]

Labarthet, F. L.

T. Buffeteau, F. L. Labarthet, M. Pezolet, and C. Sourisseau, Macromolecules 34, 7514 (2001).
[CrossRef]

Lee, W. K.

Liang, J.

J. Liang, Z. Cai, Y. Sun, L. Yi, and H. C. Wang, Eur. Phys. J. D 55, 685 (2009).
[CrossRef]

J. Liang, Z. Cai, Y. Sun, S. Xu, and L. Yi, J. Opt. Soc. Am. B 26, 36 (2009).
[CrossRef]

J. Liang, H. Zhao, X. Zhou, and H. Wang, J. Appl. Phys. 101, 013106 (2007).
[CrossRef]

J. Liang and X. Zhou, J. Opt. Soc. Am. B 22, 2468 (2005).
[CrossRef]

Loucif-Saibi, R.

Z. Sekkat, D. Morichere, M. Dumont, R. Loucif-Saibi, and J. A. Delaire, J. Appl. Phys. 71, 1543 (1992).
[CrossRef]

Maneuf, S.

A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985).
[CrossRef]

Markovsky, P.

L. Nikolova, P. Markovsky, N. Tomova, V. Dragostinova, and N. Mateva, J. Mod. Opt. 35, 1789 (1988).
[CrossRef]

Mateva, N.

L. Nikolova, P. Markovsky, N. Tomova, V. Dragostinova, and N. Mateva, J. Mod. Opt. 35, 1789 (1988).
[CrossRef]

Morichere, D.

Z. Sekkat, D. Morichere, M. Dumont, R. Loucif-Saibi, and J. A. Delaire, J. Appl. Phys. 71, 1543 (1992).
[CrossRef]

Morin, M.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

Neurgaonkar, R. R.

G. C. Duree Jr., J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Nikolova, L.

L. Nikolova, P. Markovsky, N. Tomova, V. Dragostinova, and N. Mateva, J. Mod. Opt. 35, 1789 (1988).
[CrossRef]

Peccianti, M.

G. Assanto, A. Fratalocchi, and M. Peccianti, Opt. Express 15, 5248 (2007).
[CrossRef] [PubMed]

S. V. Serak, N. V. Tabiryan, M. Peccianti, and G. Assanto, IEEE Photonic Technol. Lett. 18, 1287 (2006).
[CrossRef]

Pezolet, M.

T. Buffeteau, F. L. Labarthet, M. Pezolet, and C. Sourisseau, Macromolecules 34, 7514 (2001).
[CrossRef]

Salamo, G.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

Salamo, G. J.

G. C. Duree Jr., J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Sanchez-Mondragon, J. J.

M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Segev, M.

G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

G. C. Duree Jr., J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992).
[CrossRef] [PubMed]

Sekkat, Z.

Z. Sekkat, D. Morichere, M. Dumont, R. Loucif-Saibi, and J. A. Delaire, J. Appl. Phys. 71, 1543 (1992).
[CrossRef]

Serak, S. V.

S. V. Serak, N. V. Tabiryan, M. Peccianti, and G. Assanto, IEEE Photonic Technol. Lett. 18, 1287 (2006).
[CrossRef]

Sharp, E.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

Sharp, E. J.

G. C. Duree Jr., J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

She, W. L.

Shultz, J. L.

G. C. Duree Jr., J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Sourisseau, C.

T. Buffeteau, F. L. Labarthet, M. Pezolet, and C. Sourisseau, Macromolecules 34, 7514 (2001).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

Stepanov, S.

M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Sun, Y.

J. Liang, Z. Cai, Y. Sun, S. Xu, and L. Yi, J. Opt. Soc. Am. B 26, 36 (2009).
[CrossRef]

J. Liang, Z. Cai, Y. Sun, L. Yi, and H. C. Wang, Eur. Phys. J. D 55, 685 (2009).
[CrossRef]

Tabiryan, N. V.

S. V. Serak, N. V. Tabiryan, M. Peccianti, and G. Assanto, IEEE Photonic Technol. Lett. 18, 1287 (2006).
[CrossRef]

Taya, M.

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

Tomova, N.

L. Nikolova, P. Markovsky, N. Tomova, V. Dragostinova, and N. Mateva, J. Mod. Opt. 35, 1789 (1988).
[CrossRef]

Torruellas, W. E.

S. Trillo and W. E. Torruellas, Spatial Solitons (Springer-Verlag, 2001).

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Trillo, S.

S. Trillo and W. E. Torruellas, Spatial Solitons (Springer-Verlag, 2001).

Valley, G. C.

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

Vysloukh, V.

M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Wang, H.

J. Liang, H. Zhao, X. Zhou, and H. Wang, J. Appl. Phys. 101, 013106 (2007).
[CrossRef]

Wang, H. C.

J. Liang, Z. Cai, Y. Sun, L. Yi, and H. C. Wang, Eur. Phys. J. D 55, 685 (2009).
[CrossRef]

Wang, X. S.

Wu, S. Z.

Xu, S.

Yariv, A.

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

G. C. Duree Jr., J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992).
[CrossRef] [PubMed]

Yi, L.

J. Liang, Z. Cai, Y. Sun, L. Yi, and H. C. Wang, Eur. Phys. J. D 55, 685 (2009).
[CrossRef]

J. Liang, Z. Cai, Y. Sun, S. Xu, and L. Yi, J. Opt. Soc. Am. B 26, 36 (2009).
[CrossRef]

Zeng, F.

Zhao, H.

J. Liang, H. Zhao, X. Zhou, and H. Wang, J. Appl. Phys. 101, 013106 (2007).
[CrossRef]

Zhou, X.

J. Liang, H. Zhao, X. Zhou, and H. Wang, J. Appl. Phys. 101, 013106 (2007).
[CrossRef]

J. Liang and X. Zhou, J. Opt. Soc. Am. B 22, 2468 (2005).
[CrossRef]

Appl. Phys. Lett. (2)

S. Bian and M. G. Kuzyk, Appl. Phys. Lett. 85, 1104 (2004).
[CrossRef]

M. D. Iturbe Castillo, P. A. Marquez Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and V. Vysloukh, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Eur. Phys. J. D (1)

J. Liang, Z. Cai, Y. Sun, L. Yi, and H. C. Wang, Eur. Phys. J. D 55, 685 (2009).
[CrossRef]

IEEE Photonic Technol. Lett. (1)

S. V. Serak, N. V. Tabiryan, M. Peccianti, and G. Assanto, IEEE Photonic Technol. Lett. 18, 1287 (2006).
[CrossRef]

J. Appl. Phys. (2)

J. Liang, H. Zhao, X. Zhou, and H. Wang, J. Appl. Phys. 101, 013106 (2007).
[CrossRef]

Z. Sekkat, D. Morichere, M. Dumont, R. Loucif-Saibi, and J. A. Delaire, J. Appl. Phys. 71, 1543 (1992).
[CrossRef]

J. Mod. Opt. (1)

L. Nikolova, P. Markovsky, N. Tomova, V. Dragostinova, and N. Mateva, J. Mod. Opt. 35, 1789 (1988).
[CrossRef]

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

Macromolecules (1)

T. Buffeteau, F. L. Labarthet, M. Pezolet, and C. Sourisseau, Macromolecules 34, 7514 (2001).
[CrossRef]

Opt. Commun. (1)

A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (1)

X. S. Wang and W. L. She, Phys. Rev. E 71, 026601 (2005).
[CrossRef]

Phys. Rev. Lett. (5)

G. C. Duree Jr., J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

G. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, and A. Yariv, Phys. Rev. Lett. 74, 1978 (1995).
[CrossRef] [PubMed]

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
[CrossRef]

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[CrossRef] [PubMed]

Science (1)

G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

Other (2)

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

S. Trillo and W. E. Torruellas, Spatial Solitons (Springer-Verlag, 2001).

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Figures (3)

Fig. 1
Fig. 1

Experimental setup for observing CP bright solitons. P: polarizer; λ 4 : quarter-wave plate, which is absent when observing LP bright solitons; C 1 : cylindrical lens, which is used to produce a one-dimensional bright beam; C 2 : cylindrical imaging lens; F: filter; M 1 : reflector; M 1 : selective reflector, which is transparent to 633 nm light and opaque to 514 nm light; DR1/PMMA: sample.

Fig. 2
Fig. 2

Photographs of the bright beams at the input and output faces. Shown in both sides are corresponding intensity profiles. 35 μm input CP beam (1a) goes to self-trapped (1b) with the help of a background light, linearly diffracts to 90 μm (1c), and self-defocuses to 141 μm (1d). 49 μm input LP beam (2a) goes to self-trapped (2b), linearly diffracts to 119 μm (2c), and self-defocuses to 196 μm (2d).

Fig. 3
Fig. 3

(a) Numerical solutions of bright solitons and output solution passing through the 4 mm sample when the loss is considered. The center of the output solitons has been moved rightward for visual effect. (b) Existence curves of bright solitons.

Equations (7)

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d N T d t = N T q ( m σ 1 I 1 cos 2 θ + 1 3 σ 2 I 2 ) + N C q ( σ 1 I 1 + σ 2 I 2 ) + χ N C ,
N = N T ( θ ) + N C ( θ ) ,
Δ n l = 4 π μ N 45 5 γ 9 α β β γ | E l | 2 + β 2 | E l | 2 ,
Δ n c = 8 π μ N 45 5 γ 6 α β 2 β γ | E c | 2 + β 2 | E c | 2 ,
i E ζ + 1 2 k 2 E ξ 2 + Δ n k 0 E + i δ E = 0 ,
i E l ( x , z ) z + 1 2 2 E l ( x , z ) x 2 + 7 | E l | 2 | E l | 2 + 4 E l ( x , z ) + i σ E l ( x , z ) = 0 ,
i E c ( x , z ) z + 1 2 2 E c ( x , z ) x 2 + 13 | E c | 2 | E c | 2 + 2 E c ( x , z ) + i σ E c ( x , z ) = 0 .

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